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MAR 01 '05 16=01 FR IBM 



9149453281 TO 84941475 



P. 03/06 



REMARKS 

Attached herewith are Attachments A to Z and AA. The following is a list of 
these attachments: 

Attachment A - "CRC Handbook of Chemistry and Physics" 59th Edition 
1978-1979, page B262 and Periodic Table of the Elements, 

Attachment AA - list of Hg containing solid state compounds from the 
1 989 Powder Diffraction File Index. 

Attachment B - "Introduction to Ceramics," Kingery et al., Second Edition, 
John Wiley & Sons, 1976, in particular pages 5-20, 269-319, 381-447 and 
448-513. 

Attachment C - "Polar Dielectrics and Their Applications," Burfoot et al., 
University of California Press, 1979, in particular pages 13-33. 

Attachment D - "Ceramic Processing Before Firing," Onoda et al., John 
Wiley & Sons, 1978, the entire book. 

Attachment E - "Structure, Properties and Preparation of Perovskite-Type 
Compounds," F. S. Glasso , Pergamon Press, 1969, in particular pages 
159-186. 

Attachment F - Oxygen Defect K2MF4 - Type Oxides: The Compounds 
La 2 . x Sr x Cu04.,</2v, Nguyen et al., Journal of Solid State Chemistry 39, 
120-127(1981). 



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Attachment G - "The Oxygen Defect Perovskite BaLa 4 Cus.0 13 .4, A Metallic 
Conductor," C. Michel et at., Mat. Res. Bull., Vol. 20, pp. 667-671, 1985. 

Attachment H - Oxygen Intercalation in Mixed Valence Copper Oxides 
Related to the Perovskite, C. Michel et al., Revue de Chemie Minerale, p. 
407, 1984. 

Attachment I - Thermal Behaviour of Compositions in the Systems x 
BaTi0 3 + (1-x) Ba(Ln 05 Bos) 0 3 , V.S. Chincholkar et al., Therm. Anal. 6th, 
Vol. 2., p. 251-6, 1980.. 

Attachment J - Phys. Rev. B. Vol. 38, No. 16, p. 6531 (1988). 

Attachment K - Jap. Joun. of Appl. Phys., Vol. 27, No. 2, p. L209-L210 
(1988). 

Attachment L - "Letter to Nature", Vol. 38, No. 2, p. 226 (18 March 1993) . 

Attachment M - Nature, Vol. 336, p. 21 1 (17 November 1988). 

Attachment N - "Landholt-Bbrnstein", Volumn 4, "Magnetic and Other 
Properties of Oxides and Related Compounds Part A", 1970, pages 
148-206. 

Attachment O - The Langenscheidfs German-English, English-German 
Dictionary 1970, page 446. 

Attachment P - "Landholt-Bdrnstein, Volume 3, Ferro- and 
Antiferroelectric Substances," 1969, pages 571-584. 



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Attachment Q - "Structure and Properties of Inorganic Solids," Francis S. 
Galasso, Pergamon Press (1969) at page 190. 

Attachment R - "Crystal Structures" Volume 4, by Ralph W. G. Wyckoff, 
Interscience Publishers. 1960, pages 96-97 and 142-143. 

Attachment S - "Modern Oxide Materials Preparation, Properties and 
Device Applications" edited by Cockayne and Jones, Academic Press 
(1972), page 155. 

Attachment T - "Landholt-B6rnstein, Volume 3, Ferro- and Antiferroelectric 
Substances" (1969), pages 107-1 14. 

Attachment U - "Structure, Properties and Preparation of Perovskite-Type 
Compounds" by F. S. Galasso, published in 1969, pages 191-207. 

Attachment V - "Mixed bismuth oxides with layer lattices", B. Aurivillius, 
Arkiv Kemi 1, pages 463-480, (1950). 

Attachment W - "Mixed bismuth oxides with layered lattices ", B. 
Aurivillius, Arkiv Kemi 1, pages 499-512, (1950). 

Attachment X - "Mixed bismuth oxides with layered lattices ", B. Aurivillius, 
Arkiv Kemi 2, pages 519-527, (1951). 

Attachment Y - "The structure of Bi 2 Nb0 5 F and isomorphous compounds", 
B. Aurivillius, Arkiv Kemi 5, pages 39-47, (1952). 

Attachment Z - "Superconductivity," Charles Poole et al., 1995, Academic 
Press, Chapter 7 on "Perovskite and Cuprate Crystallographic Structures". 



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Please charge any fee necessary to enter this paper and any previous paper to 
deposit account 09-0468. 




(914)945-3217 



IBM CORPORATION 
Intellectual Property Law Dept. 
P.O. Box 218 

Yorktown Heights, New York 10598 



Serial No.: 08/479,810 



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Docket: YO987-074BZ 



** TOTAL PAGE. 06 ** 






OF 







A Ready-Reference Book of Chemical and Physical Date 




EDITOR 

ROBERT C. WEAST.Ph.D. 

f^ormeriyftofesw of Chemistry at One Institute ofTedfmobgy 
ASSOCIATE EDff OR 

MELVIN J. AST LE, Ph.D. 

Formerly Professor of Organic Chemistry at Owe Institute of Technology 

and 

imager of Research at Gtidden-Dwkee Division ofSCM Corporation 

In cotiaboratioa with a large number of professional chemists and physicists 
l^>m^k&*9rtri&to the fet ofgene^ coB^ 
connection with the particular ^ t^fcs or secdcms uivolwd. 




CRC PRESS, Inc. 
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Alphabetical Index (Chemical and Mineral Name) 



INTERNATIONAL CENTRE FOR DIFFRACTION DATA 



BEST AVAILABLE COPY 



Powder Diffraction File 

Alphabetical Index 
Inorganic Phases 
1989 



eration with the Amencan Ceram, '^^^^ f^^n X-Ray 
ciation, American Soc,ety or Te ^ an J Association , T he Clay M,n- 
Analytical Associate, British C 7j^°^X*aft The Institute of Physics, 
erals Society, Deutsche M^^^^MincraloBicd Society of 
The Mineralogical Association^ f££*M and Ireland , Nati onal Asso- 
America, Mineralogical Society ^^^^ansata de M ineralogie et de 
ciation of Corrosion Engineers, and Societe tranc 



Cristallographie. 



Published by the 

INTERNATIONAL CENTRE FOR DIFFRACTION DATA 

,60, PARK LANE • SWARTHMORE. PA 19081-2389 • U.S.A. 



Copyright • JCPDS International Centre for Diffraction Data 1989 

formerly the 

Joint Committee on Powder Diffraction Standards 

All riehts reserved. No part of this publication may be reproduced or transmitted in 
a fo g rm or b^any means, e.ectronic or mechanica., including photocopy, record^ 
or anyTnformation storage and retrieval system, without perm.ss.on m wnUng from 

the publisher. 



Printed in U.S.A. 
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Introduction 
to Ceramics 

Second Edition 



W. D. Kingery 

PROFESSOR OF CERAMICS 

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 

H. K. Bowen 

ASSOCIATE PROFESSOR OF CERAMICS 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 

D. R. Uhlmann 

PROFESSOR OF CERAMICS AND POLYMERS 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 



A Wiley-Interscience Publication 

JOHN WILEY & SONS, New York • Chichester- Brisbane . Toronto 

rfEST AVAILABLE COPY 



Copyright © I960, 1976 by John Wiley & Sons, Inc. 
AH rights reserved. Published simultaneously in Canada. 

Reproduction or translation of any part of this work beyond that 
permitted by Sections 107 or 108 of the 1976 United States 
Copyright Act without the permission of the copyright owner is 
unlawful. Requests for permission or further information should be 
addressed to the Permissions Department, John Wiley & Sons, Inc. 

Library of Congress Cataloging in Publication Data: 

Kingery, W. D. 
Introduction to ceramics. 

(Wiley series on the science and technology of 
materials) 

*'A Wiley-Interscience publication. " 

Includes bibliographical references and index. 

1. Ceramics. I. Bowen, Harvey Kent, joint author. 
II. Uhlmann, Donald Robert, joint author. III. Title. 

TP807.K52 1975 666 \ 75-22248 

ISBN 0-471-47860-1 

Printed in the United States of America 
10 9 8 7 6 5 



During the fiftee 
published, the ap 
ticed. However, t 
developing new c 
modifications in t 
new material. 

In particular, 
noncrystalline so 
j new insight into 

I spinodal decomp 

recognition of tru 
ment of glass-cer 
sintering phenom 
transmission elei 
microstructure, a 
and a myriad o 
magnetic ceramic 
I these advances r 

| competence. 

The necessary 
the recent availab 
turing methods [1 
(1970); F. H. Nor 
Norton, Element 
(1974); F. V. Tc 
Ogden Publ. Co 



CERAMIC PROCESSES AND PRODUCTS 



5 



or prepare fine particles, shape them, and then stick them back together 
by heating. The second basic process is to melt the material to form a 
liquid and then shape it during cooling and solidification; this is most 
widely practiced in forming glasses. For completeness, we should also 
mention forming shapes in a mold or by dipping a form with a slurry 
containing a ceramic binder such as portland cement or ethyl silicate. 

Raw Materials. The types of minerals found in nature are controlled 
mainly by the abundance of the elements and their geochemical charac- 
teristics. Since oxygen, silicon, and aluminum together account for 90% 
of the elements in the earth's crust, as shown in Fig. 1.1, it is not 
surprising that the dominant minerals are silicates and aluminum silicates. 
These, together with other mineral compounds of oxygen, constitute the 
great bulk of naturally occurring ceramic raw materials. 

The mineral raw materials used in the ceramic industry are mainly 



54 




Element 



Fig. 1 . 1 Abundance of common elements in the earth's crust. 



6 



INTRODUCTION TO CERAMICS 



inorganic nonmetallic crystalline solids formed by complex geologic 
processes. Their ceramic properties are largely determined by the crystal 
structure and the chemical composition of their essential constituents and 
the nature and amounts of accessory minerals present. The mineralogic 
characteristics of such materials and therefore their ceramic properties 
are subject to wide variation among different occurrences or even within 
the same occurrence, depending on the geological environment in which 
the mineral deposit was formed as well as the physical and chemical 
modifications that have taken place during subsequent geological history. 

Since silicate and aluminum silicate materials are widely distributed, 
they are also inexpensive and thus provide the backbone of high-tonnage 
products of the ceramic industry and determine to a considerable extent 
its form. Low-grade clays are available almost everywhere; as a result, 
the manufacture of building brick and tile not requiring exceptional 
properties is a localized industry for which extensive beneficiation of the 
raw material is not appropriate. In contrast, for fine ceramics requiring the 
use of better-controlled raw materials, the raw materials are normally 
beneficiated by mechanical concentration, froth floatation, and other 
relatively inexpensive processes. For materials in which the value added 
during manufacture is high, such as magnetic ceramics, nuclear-fuel 
materials, electronic ceramics, and specialized refractories, chemical 
purification and even chemical preparation of raw materials may be 
necessary and appropriate. 

The raw materials of widest application are the clay minerals— fine- 
particle hydrous aluminum silicates which develop plasticity when mixed 
with water. They vary over wide limits in chemical, mineralogical, and 
physical characteristics, but a common characteristic is their crystalline 
layer structure, consisting of electrically neutral aluminosilicate layers, 
which leads to a fine particle size and platelike morphology and allows the 
particles to move readily over one another, giving rise to physical 
properties such as softness, soapy feel, and easy cleavage. Clays perform 
two important functions in ceramic bodies. First, their characteristic 
plasticity is basic to many of the forming processes commonly used; the 
ability of clay-water compositions to be formed and to maintain their 
shape and strength during drying and firing is unique. Second, they fuse 
over a temperature range, depending on composition, in such a way as to 
become dense and strong without losing their shape at temperatures 
which can be economically attained. 

The most common clay minerals and those of primary interest to 
ceramists, since they are the major component of high-grade clays, are 
based on the kaolinite structure, Al 2 (Si 2 0 5 )(OH) 4 . Other compositions 
often encountered are shown in Table 1.1. 



CERAMIC PROCESSES AND PRODUCTS 
Table 1.1. Ideal Chemical Formulas of the Clay Minerals 



7 



Kaolinite 

Halloysite 

Pyrophyllite 



Al 2 (Si 2 0 5 )(OH) 4 

Al 2 (Si 2 0 5 )(OH) 4 -2H 2 0 

Al 2 (Si 2 0 5 ) 2 (OH) 2 



Montmorillonite 



Al 2 K(Sii. 6 Alo. 6 0 5 ) 2 (OH) 2 
Al 2 - I M gx K 1 _ z _ v (Sii. 6 _ w AIo. 5 - h /0 6 ) 2 (OH) 2 




Mica 
Illite 



A related material is talc, a hydrous magnesium silicate with a layer 
structure similar to the clay minerals and having the ideal formula 
Mg 3 (Si 2 0 5 ) 2 (OH) 2 . Talc is an important raw material for the manufacture 
of electrical and electronic components and for making tile. Asbestos 
minerals are a group of hydrous magnesium silicates which have a fibrous 
structure. The principal variety is chrysotile, Mg 3 Si 2 0 5 (OH) 4 . 

In addition to the hydrous silicates already discussed, anhydrous silica 
and silicate materials are basic raw materials for much of the ceramic 
industry. Si0 2 is a major ingredient in glass, glazes, enamels, refractories, 
abrasives, and whiteware compositions. It is widely used because it is 
inexpensive, hard, chemically stable, and relatively infusable and has the 
ability to form glasses. There is a variety of mineral forms in which silica 
occurs, but by far the most important as a raw material is quartz. It is used 
as quartzite rock, as quartz sand, and as finely ground potter's flint. The 
major source of this material is sandstone, which consists of lightly 
bonded quartz grains. A denser quartzite, gannister, is used for refractory 
brick. Quartz is also used in the form of large, nearly perfect crystals, but 
these have been mostly supplanted by synthetic crystals, manufactured 
by a hydrothermal process. 

Together with quartz, which serves as a refractory backbone con- 
stituent, and clay, which provides plasticity, traditional triaxial porcelains 
(originally invented in China) include feldspar, an anhydrous aluminosili- 
cate containing K\ Na\ or Ca 2+ as a flux which aids in the formation of a 
glass phase. The major materials of commercial interest are potash 
feldspar (microcline or orthoclase), K(AlSi 3 )0 8 , soda feldspar (albite), 
Na(AlSi 3 )0 8 , and lime feldspar (anorthite), Ca(Al 2 Si 2 )0 8 . Other related 
materials sometimes used are nepheline syenite, a quartzfree igneous 
rock composed of nephelite, Na 2 (Al 2 Si 2 )0 8 , albite, and microcline; also 
wollastonite, CaSi0 3 . One group of silicate minerals, the sillimanite' 
group, having the composition Al 2 Si0 5 , is used for the manufacture of 
refractories. 



INTRODUCTION TO CERAMICS 



Most of the naturally occurring nonsilicate materials are used primarily 
as refractories^ Aluminum oxide is mostly prepared from the mineral 
bauxite by the Bayer process, which involves the selective leaching of the 
alumma by caustic soda, followed by the precipitation of aluminum 
hydroxide. Some bauxite is. used directly in the electric-furnace produc- 
tion of alumina, but most is first purified. Magnesium oxide is produced 
SSnST "f Ura ' ma * nesite ' WO,, and from magnesium hydroxide, 
Mg(OH) 2 , obtained from seawater or brines. Dolomite, a solid solution of 
calcium and magnesium carbonates with the formula CaMg(CO,) 2 is used 
to make basic brick for use in the steel industry. Another refractory 
widely used for metallurgical purposes is chrome ore, which consists 
primarily of a complex solid solution of spinels, (Mg,Fe)(Al,Cr) 2 0 4 , which 
make up most of the material; the remainder consists of various mag- 
nesium silicates. B 

Other mineral-based materials which are widely used include soda ash 
NaC0 3 , mostly manufactured from sodium chloride; borate materials' 
mcludmg kern.te, Na 2 B,0 7 -4H 2 0, and borax, Na 2 B 4 0 7 - 10H 2 O, used as 
fluxing agents; fluorspar, CaF 2 , used as a powerful flux for some glazes 
S(OH a ,F) e (Po!)" d Ph ° Sphate materials most 'y deriv ^ from apatite, 

Although most traditional ceramic formulations are based on the use of 
natural mineral materials which are inexpensive and readily available an 
increasing fraction of specialized ceramic ware depends on the availabil- 
ity of chemically processed materials which may or may not start directly 
from mined products and in which the particle-size characteristics and 
chemical purity are closely controlled. Silicon carbide for abrasives is 
manufactured by electrically heating mixtures of sand and coke to a 
temperature of about 2200°C where they react to form SiC and carbon 
monoxide. Already mentioned, seawater magnesia, Bayer alumina, and 
soda ash are widely used chemical products. In the manufacture of 
barium titanate capacitors, chemically purified titania and barium carbo- 
nate are used as raw materials. A wide range of magnetic ceramics is 
manufactured from chemically precipitated iron oxide. Nuclear-fuel 
elements are manufactured from chemically prepared UO, Single crys- 

oxL J aPP 3 !l d / Uby alS ° P ° refree Po'ycystalline aluminum 
ca efu.r^? rom aluminum oxide made by precipitating and 
careful y calcining alum in order to maintain good control of both 
chemmry and particle size. Special techniques of material preparation 

^ drOP,e,S ° f SOlUti ° n t0 form h^ogeneous particles 
of small size and high purity are receiving increasing attention as is the 

Znhv eP ° S !T n ° f I thi "- film materia ' S in a careful, y controlled'chemical 
and phys.cal form. In general, raw-material preparation is clearly headed 



towar 
tion a 
partic 

On IT1£ 

For 

firing ] 
conce 
the ra 
which 
prepai 
have 
micro 
be les; 
suspei 
depen 
stable 
propo: 
cohen 
of the 
partic; 
size, i 
Mixin; 
the cc 
achiev 
partic) 
result: 
numb* 
with a 
during 
In a 
intima: 
within 
firing t 
usual f 
ball mi 
impro\ 
tion ol 
either 
which 
mainta 
gates, 
formin 



CERAMIC PROCESSES AND PRODUCTS 



9 



ily toward the increasing use of mechanical, physical, and chemical purifica- 

ral tion and upgrading of raw materials together with special control of 
he — particle size and particle-size distribution and away from the sole reliance ^ 

im on materials in the form found in nature. 

ic- Forming and Firing. The most critical factors affecting forming and 

ed firing processes are the raw materials and their preparation. We have to be 

le, concerned with both the particle size and the particle-size distribution of 

of the raw materials. Typical clay materials have a particle-size distribution 

?d which ranges from 0.1 to 50 microns for the individual particles. For the 

ry preparation of porcelain compositions the flint and feldspar constituents 

.ts have a substantially larger particle size ranging between 10 and 200 

;h microns. The fine-particle constituents, which for special ceramics may 

g- be less than 1 micron, are essential for the forming process, since colloidal 

suspensions, plastic mixes with a liquid-phase binder, and dry pressing all 

h, depend on very small particles flowing over one another or remaining in a 

Is stable suspension. For suspensions, the settling tendency is directly 

as 1 proportional to the density and particle size. For plastic forming the 

es coherence of the mass and its yield point are determined by the capillarity 

e, of the liquid between particles; this force is inversely proportional to the 

particle size. However, if all the material were of a uniformly fine particle 

Df size, it would not be feasible to form a high concentration of solids, 

in Mixing in a coarser material allows the fines to fill the interstices between 

1- the coarse particles such that a maximum particle-packing density is 

iy achieved at a ratio of about 70% coarse and 30% fine material when two 

id particle sizes are used. In addition, during the drying process, shrinkage 

is > results from the removal of water films between particles. Since the 

a number of films increases as the particle size decreases, bodies prepared 

n with a liquid binder and all fine-particle materials have a high shrinkage 

d during drying and the resultant problems of warping and distortion. 

>f In addition to a desired particle size and particle-size distribution, 

>- intimate mixing of material is necessary for uniformity of properties 

s within a body and for the reaction of individual constituents during the 

il firing process. For preparing slurries or a fine-grain plastic mass, it is the 

usual practice to use wet mixing, with the raw materials placed together in 
n ball mills or a blunger. Shearing stresses developed in the mixing process 
i ' improve the properties of a plastic mix and ensure the uniform distribu- 
te tion of the fine-grain constituent. For dewatering the wet-milled mix, 
n either a filter press may be used, or more commonly spray-drying, in 
s which droplets of the slurry are dried with a countercurrent of warm air to 
e maintain their uniform composition during drying. The resulting aggre- 
.1 gates, normally 1 mm or so in size, flow and deform readily in subsequent 
J forming. 



10 



INTRODUCTION TO CERAMICS 



Since the firing process also depends on the capillary forces resulting 
from surface energy to consolidate and densify the material and since 
these forces are inversely proportional to particle size, a substantial 
percentage of fine-particle material is necessary for successful firing. The 
clay minerals are unique in that their fine particle size provides both the 
capability for plastic forming and also sufficiently large capillary forces 
for successful firing. Other raw materials have to be prepared by chemical 
precipitation or by milling into the micron particle range for equivalent 
results to be obtained. 

Perhaps the simplest method of compacting a ceramic shape consists of 
forming a dry or slightly damp powder, usually with an organic binder in 
a metal die at sufficiently high pressures to form a dense, strong piece 
This method is used extensively for refractories, tiles, special electrical 
and magnetic ceramics, spark-plug insulators and other technical 
ceramics, nuclear-fuel pellets, and a variety of products for which large 
numbers of simple shapes are required. It is relatively inexpensive and 
can form shapes to close tolerances. Pressures in the range of 3000 to 
30,000 psi are commonly used, the higher pressures for the harder 
materials such as pure oxides and carbides. Automatic dry pressing at 
high rates of speed has been developed to a high state of effectiveness 
One limitation is that for a shape with a high length-to-diameter ratio the 
fnctional forces of the powder, particularly against the die wall, lead to 
pressure gradients and a resulting variation of density within the piece 
During firing these density variations are eliminated by material flow 
during sintering; it necessarily follows that there is a variation in 
shrinkage and a loss of the original tolerances. One modification of the 
dry-pressing method which leads to a more uniform density is to enclose 
the sample in a rubber mold inserted in a hydrostatic chamber to make 
pieces by hydrostatic molding, in which the pressure is more uniformly 
applied. Variations in sample density and shrinkage are less objection- 
able. This method is widely used for the manufacture of spark-plug 
insulators and for special electrical components in which a high degree of 
uniformity and high level of product quality are required. 

A quite different method of forming is to extrude a stiff plastic mix 
through a die orifice, a method commonly used for brick, sewer pipe 
hollow tile, technical ceramics, electrical insulators, and other materials' 
having an axis normal to a fixed cross section. The most widely practiced 
method is to use a vacuum auger to eliminate air bubbles, thoroughly mix 
the body with 12 to 20% water, and force it through a hardened steel or 
carbide die. Hydraulic piston extruders are also widely used. 

The earliest method of forming clay ware, one still widely used is to 
add enough water so that the ware can readily be formed at low pressures 



CERAMIC PROCESSES AND PRODUCTS 



11 



This may be done under hand pressure such as building ware with coils, 
free-forming ware, or hand throwing on a potter's wheel. The process can 
= be,mechanized,by,soft-plastic=pressing^between=porous=plaster molds and 
also by automatic jiggering, which consists of placing a lump of soft 
plastic clay on the surface of a plaster-of-paris mold and rotating it at 
about 400 rpm while pulling a profile tool down on the surface to spread 
the clay and form the upper surface. 

When a larger amount of water is added, the clay remains sticky plastic 
until a substantial amount has been added. Under a microscope it is seen 
that individual clay particles are gathered in aggregates or floes. However, 
if a small quantity of sodium silicate is added to the system, there is a 
remarkable change, with a substantial increase in fluidity resulting from 
the individual particles being separated or deflocculated. With proper 
controls a fluid suspension can be formed with as little as 20% liquid, and 
a small change in the liquid content markedly affects the fluidity. When a 
suspension such as this is cast into a porous plaster-of-paris mold, the 
mold sucks liquid from the contact area, and a hard layer is built on the 
surface. This process can be continued until the entire interior of the mold 
is filled (solid casting) or the mold can be inverted and the excess liquid 
poured out after a suitable wall thickness is built up (drain casting). 

In each of the processes which require the addition of some water 
content, the drying step in which the liquid is removed must be carefully 
controlled for satisfactory results, more so for the methods using a higher 
liquid content. During drying, the initial drying rate is independent of the 
water content, since in this period there is a continuous film of water at 
the surface. As the liquid evaporates, the particles become pressed more 
closely together and shrinkage occurs until they are in contact in a solid 
structure free from water film. During the shrinkage period, stresses, 
warping, and possibly cracks may develop because of local variations in 
the liquid content; during this period rates must be carefully controlled. 
Once the particles are in contact, drying can be continued at a more rapid 
rate without difficulty. For the dry-pressing or hydrostatic molding 
process, the difficulties associated with drying are avoided, an advantage 
for these methods. 

After drying, ceramic ware is normally fired to temperatures ranging 
from 700 to 1800°C, depending on the composition and properties desired. 
Ware which is to be glazed or decorated may be fired in different ways. 
The most common procedure is to fire the ware without a glaze to a 
sufficiently high temperature to mature the body; then a glaze is applied 
and fired at a low temperature. Another method is to fire the ware initially 
to a low temperature, a bisque fire; then apply the glaze and mature the 
body and glaze together at a higher temperature. A third method is to 




12 



INTRODUCTION TO CERAMICS 



apply the glaze to the unfired ware and heat them together in a one-fire 
process. 

During the firing process, either a viscous liquid or sufficient atomic 
mobility in the solid is developed to permit chemical reactions, grain 
^growth, and^sinteringH^^ 
tension to consolidate the ware and reduce the porosity. The volume 
shrinkage which occurs is just equal to the porosity decrease and varies 
from a few to 30 or 40 vol%, depending on the forming process and the 
ultimate density of the fired ware. For some special applications, com- 
plete density and freedom from all porosity are required, but for other 
applications some residual porosity is desirable. If shrinkage proceeds at 
an uneven rate during firing or if part of the ware is restrained from 
shrinking by friction with the material on which it is set, stresses, warping, 
and cracking can develop. Consequently, care is required in setting the 
ware to avoid friction. The rate of temperature rise and the temperature 
uniformity must be controlled to avoid variations in porosity and shrin- 
kage. The nature of the processes taking place is discussed in detail in 
Chapters 11 and 12. 

Several different types of kilns are used for firing ware. The simplest is 
a skove kiln in which a benchwork of brick is set up inside a surface 
coating with combustion chambers under the material to be fired. 
Chamber kilns of either the up-draft or down-draft type are widely used 
for batch firing in which temperature control and uniformity need not be 
too precise. In order to achieve uniform temperatures and maximum use 
of fuel, chamber kilns in which the air for combustion is preheated by the 
cooling ware in an adjacent chamber, the method used in ancient China, is 
employed. The general availability of more precise temperature controls 
for gas, oil, and electric heating and the demands for ware uniformity 
have led to the increased use of tunnel kilns in which a temperature profile 
is maintained constant and the ware is pushed through the kiln to provide 
a precise firing schedule under conditions such that effective control can 
be obtained. 

Melting and Solidification. For most ceramic materials the high vol- 
ume change occurring during solidification, the low thermal conductivity, 
and the brittle nature of the solid phase have made melting and solidifica- 
tion processes comparable with metal casting and foundry practice 
inappropriate. Recently, techniques have been developed for unidirec- 
tional solidification in which many of these difficulties can be substantially 
avoided. This process has mainly been applied to forming controlled 
structures of metal alloys which are particularly attractive for applica- 
tions such as turbine blades for high-temperature gas turbines. So far as 
we are aware, there is no large scale manufacture of ceramics in this way, 



■i 

i 



CERAMIC PROCESSES AND PRODUCTS 



13 



^-fire 

omic 

>rain 

face 

ume 

aries 

I the 

:om- 

>ther 

is at 

"rom 

>ing, 

; the 

ture 

lrin- 

il in 

st is 
face 
red. 
ised 
t be 

use 

the 
a, is 
rols 
lity 
>file 
'ide 
can 

/ol- 
ity, 
ca- 
ice 
ec- 
tlly 
led 
ca- 
as 

ay, 



but we anticipate that the development of techniques for the unidirec- 
tional solidification of ceramics will be an area of active research during 
the next decade. — ■ ■ 

Another case in which these limitations do not apply is that of 
glass-forming materials in which the viscosity increases over a broad 
temperature range so that there is no sharp volume discontinuity during 
solidification and the forming processes can be adjusted to the fluidity of 
the glass. Glass products are formed in a high-temperature viscous state 
by five general methods: (1) blowing, (2) pressing, (3) drawing, (4) rolling, 
and (5) casting. The ability to use these processes depends to a large 
extent on the viscous flow characteristics of the glass and its dependence 
on temperature. Often surface chilling permits the formation of a stable 
shape while the interior remains sufficiently fluid to avoid the buildup of 
dangerous stresses. Stresses generated during cooling are relieved by 
annealing at temperatures at which the force of gravity is insufficient to 
cause deformation. This is usually done in an annealing oven or lehr 
which, for many silicate glasses, operates at temperatures in the range of 
400 to 500°C. 

The characteristics most impressive about commercial glass-forming 
operations are the rapidity of forming and the wide extent of automation. 
Indeed, this development is typical of the way in which technical progress 
affects an industry. Before the advent of glass-forming machinery, a 
major part of the container industry was based on ceramic stoneware. 
Large numbers of relatively small stoneware potters existed solely for the 
manufacture of containers. The development of automatic glass-forming 
machinery allowing the rapid and effective production of containers on a 
continuous basis has eliminated stoneware containers from common use. 

Special Processes. In addition to the broadly applicable and widely 
used processes discussed thus far, there is a variety of special processes 
which augment, modify, extend, or replace these forming methods. These 
include the application of glazes, enamels, and coatings, hot-pressing 
materials with the combined application of pressure and temperature, 
methods of joining metals to ceramics, glass crystallization, finishing and 
machining operations, preparation of single crystals, and vapor- 
deposition processes. 

Much ceramic ware is coated with a glaze, and porcelain enamels are 
commonly applied on a base of sheet steel or cast iron as well as for 
special jewelry applications. Glazes and enamels are normally prepared in 
a wet process by milling together the ingredients and then applying the 
coating by brushing, spraying, or dipping. For continuous operation, 
spray coating is most frequently used, but for some applications more 
satisfactory coverage can be obtained by dipping or painting. For 



14 



INTRODUCTION TO CERAMICS 



coa et with IT Ir ° n ' Iar8C CaSti " 8S heated in a f »™ ce are 

ove tJ f V e u amd P ° Wder WhiGh mUst be distributed uniform y 
over the surface, where it fuses and sticks. In addition to these wTddv 
used processes, special coatings for technical ware have been applTed by 
- ■^Tte factory dense layer; vacuum-dTp" sit d" 

coatings have been formed by evaporation or cathodic sputtering coat- 
.ngs have been applied by chemical vapor deposition; electrophone 

Jtsj^zk 8 * and other speciaii - ~ 

To obtain a high density together with fine particle size oarticularlv fnr 
materials such as carbides and borides, the combination o^ £st£ ^ w th 
h gh temperature ,s an effective technique mostly used for sma 1 samples 

formeZ t e H COnfi8 r i0n - At ,OWer tem P^«-s, glass-bonded 2 
formed ,n this way for use as an inexpensive insulation One of the main 

^^l*T p ? ssins method is that material P™nl S ™ 

critical than for the sintering processes, which require a high decree of 
material uniformity for successful applications of the h fLst qua lhv 

t^me^o K ffiCUltieS h0t - preSSing techniques ar ppl^ 
the method to large shapes and the time required for heatinp tho m «w 5 
sample, which makes the method slow and expensive 8 

For many applications, joining processes are necessary to form fabri 
™n ""S i" manufacturi «8 teacups, for example, the hLdle is no " 

cue y S Tn^ S « P , arateIy ', diPPCd " 3 SHP ' and stuck on the body of the 
ZLl T 7 T SS ° f C ° mplex desi * n are s ™«arl y built up from 

o essu e Lhr 6 ? PartS - F ° r many deCtr0nic ^Plications requ ring 
pressure-tight seals, it ,s necessary to form a bond between metals and 
ceramics. For glass-metal seals, the main problem is machine S 

ea i" Hal s T h TT "T' deVe '° P in USe; S P ecial ™etal afloyJand 
sealing glasses have been designed for this purpose. For crystalline 
ceramics, the most widely applied method has been to use iZ£b£^ 
manganese layer which, when fired under partially oxidizing condit'ns" 
£T In Zt that rCaCtS thC CCramic to B-e an adheLe bo dTg 

iro r ni::rave%rLd active metai brazes - 



oc 
cr 
be 
fo: 
nu 
au 
pr 

re< 
sh 
m; 
in 
on 

ce 
of 
inl 
wl 
pr 
su 
ga 
pr 
fill 

ah 
be 
m< 
ca 
pn 

<de 
eh 
nu 
is 
ru 
m; 
m; 
sii 
gr 
wi 

• us 



CERAMIC PROCESSES AND PRODUCTS 



15 



ire occurs; subsequent reheating into the growth region develops proper 

lly crystallite sizes for the colloidal ruby color. In the past 10 years there has 

i\y been extensive development of glasses in ^"w^ the volume of crystals " 

by formed is much larger than the volume of the residual glass. By controlled 

ed nucleation and growth, glass-ceramics are made in which the advantage of 

at- automatic glass-forming processes is combined with some of the desirable 

tic properties of a highly crystalline body. 

ad For most forming operations, some degree of finishing or machining is 
required which may range from fettling the mold lines from a slip-cast 

: or shape to diamond-grinding the final contour of a hard ceramic. For hard 

ith materials such as aluminum oxide, as much machining as feasible is done 

ies in the unfired state or the presintered state, with final finishing only done 

is on the hard, dense ceramic where required. 

lin A number of processes have been developed for the formation of 

:ss ceramics directly from the vapor phase. Silica is formed by the oxidation 

of of silicon tetrachloride. Boron and silicon carbide fibers are made by 

; ty introducing a volatile chloride with a reducing agent into a hot zone, 

ng where deposition occurs on a fine tungsten filament. Pyrolytic graphite is 

ad prepared by the high-temperature deposition of graphite layers on a 
substrate surface by the pyrolytic decomposition of a carbon-containing 

ri- gas. Many carbides, nitrides, and oxides have been formed by similar 

>r- processes. For electronic applications, the development of single-crystal 

he films by these techniques appears to have many potential applications. 

>m Thin-wafer substrates are formed by several techniques, mostly from 

ng alumina. A widely used development is the technique in which a fluid 

id body is prepared with an organic binder and uniformly spread on a 

he moving nonporous belt by a doctor blade to form thin, tough films which 

ae can subsequently be cut to shape; holes can be introduced in a high-speed 

id punch press. 

le There is an increasing number of applications in which it is necessary or 

n- I desirable to have single-crystal ceramics because of special optical, 

s, * electrical, magnetic, or strength requirements. The most widespread 

ig method of forming these is the Czochralski process, in which the crystal 

or is slowly pulled from a molten melt, a process used for aluminum oxide, 
ruby, garnet, and other materials. In the Verneuil process a liquid cap is 

:n maintained on a growing boule by the constant-rate addition of powdered 

;d material at the liquid surface. For magnetic and optical applications thin 

id single-crystal films are desirable which have been prepared by epitaxial 

>y growth from the vapor phase. Hydrothermal growth from solution is 

Id widely used for the preparation of quartz crystals, largely replacing the 

is use of natural mineral crystals for device applications. 



16 INTRODUCTION TO CERAMICS 

1.3 Ceramic Products 

The diversity of ceramic products, which range from microscopic 
smgle-crys al wh, s kers, tiny magnets, and substrate chips to m l 
refractory furnace blocks, from single-phase c.osely controlled compos " 
t.ons to mulUphase multicomponent brick, and from porefree transparent 
crystals and glasses to lightweight insulating foams is such that no simpL 
classification is appropriate. From the point of view of historical develop! 
ZXZ f & ? Pr ° dU r d ' U iS convenient *> insider the mineral-raw- 
Z^tZ ' ""^ Si ' iCateS ' fr0m " ewer plicate 

Traditional Ceramics. We can define traditional ceramics as those 
SeSser indUStrieS -P rima "«y clay products, cement, and 
The art of making pottery by forming and burning clay has been 
pranced from the earliest civilizations. Indeed, the examinat on of 
pottery fragments has been one of the best tools of the archeologist 
Burnt clayware has been found dating from about 6500 b.c. and was we 1 
developed as a commercial product by about 4000 b c 

Similarly the manufacture of silicate glasses is an ancient art. Naturally 
occurring glasses (obsidian) were used during the Stone Age and the e 
was a stable industry in Egypt by about 1500 b c 

In contrast, the manufacture of portland cement has only been prac 
feed for about 100 years. The Romans combined burned Hme whh 
volcanic ash to make a natural hydraulic cement; the art seems then to 

lime 6 , w aPPe H red ' bUt i he hydraU ' iC Pr ° perties of burned clayed 

hTrnZf r f ,SCOVered ,n En * ,and -bout 1750, and in the next 100 years 

deveToped " ng Pr ° CeSS ' CSSentia,ly the Same as that <»ed nowf was 
By far the largest segment of the silicate ceramic industry i s the 
manufacture of various glass products. These are manufactured mo ly as 

Slfrr 8,aSSCS - ThB nCXt **** of the ceraml 

industry Is lime and cement products. In this category the lareest B ro,,n«f 
materials is hydraulic cements such as those X 
.on. A much more diverse group of products is included in the class fica" 
tion of whitewares. This group includes pottery, porcelain, and 1^ 
fine-grained porcelainlike compositions which comprise a wide var^v of 
specific products and uses. The next classification of traditional ce amies 

ieZ c T 0 rT t - ? ich are main,y siiicate * ,ass,ike ^s^s 

metals Ano her. distinct group is the structural clay products which 
consist mamly of brick and tile but include a variety of similar products 
such as sewer pipe. A particularly important group of th radklon 



CERAMIC PROCESSES AND PRODUCTS 



17 



ceramics industry is refractories. About 40% of the refractory industry 
consists of fired-clay products, and another 40% consists of heavy 
rionclay refractories such as magnesite, chromite, and similar composi- 
tions. In addition there is a sizable demand for various special refractory 
compositions. The abrasives industry produce mainly silicon carbide and 
aluminum oxide abrasives. Finally, a segment of the ceramic industry 
which does not produce ceramic products as such is concerned with the 
mineral preparation of ceramic and related raw materials. 

Most of these traditional ceramics could be adequately defined as the 
silicate industries, which indeed was the description originally proposed 
for the American Ceramic Society in 1899. The silicate industries still 
compose by far the largest part of the whole ceramic industry, and from 
this point of view they can be considered the backbone of the field. 

New Ceramics. In spite of its antiquity, the ceramic industry is not 
stagnant. Although traditional ceramics, or silicate ceramics, account for 
the large bulk of material produced, both in tonnage and in dollar volume, 
a variety of new ceramics has been developed in the last 20 years. These 
are of particular interest because they have either unique or outstanding 
properties. Either they have been developed in order to fulfill a particular 
need in greater temperature resistance, superior mechanical properties, 
special electrical properties, and greater chemical resistivity, or they have 
been discovered more or less accidentally and have become an important 
part of the industry. In order to indicate the active state of development, it 
may be helpful to describe briefly a few of these new ceramics. 

Pure oxide ceramics have been developed to a high state of uniformity 
and with outstanding properties for use as special electrical and refrac- 
tory components. The oxides most often used are alumina (A1 2 0 3 ), 
zirconia (Zr0 2 ), thoria (Th0 2 ), beryllia (BeO), magnesia (MgO), spinel 
(MgAl 2 0 4 ), and forsterite (Mg 2 Si0 4 ). 

Nuclear fuels based on uranium dioxide (U0 2 ) are widely used. This 
material has the unique ability to maintain its good properties after long 
use as a fuel material in nuclear reactors. 

Electrooptic ceramics such as lithium niobate (LiNb0 3 ) and 
lanthanum-modified lead zirconate titanate (PLZT) provide a medium by 
which electrical information can be transformed to optical information or 
by which optical functions can be performed on command of an electrical 
signal. 

Magnetic ceramics with a variety of compositions and uses have been 
developed. They form the' basis of magnetic memory units in large 
computers. Their unique electrical properties are particularly useful in 
high-frequency microwave electronic applications. 

Single crystals of a variety of materials are now being manufactured, 



18 



INTRODUCTION TO CERAMICS 



either to replace natural crystals which are unavailable or for their own 
unique properties. Ruby and garnet laser crystals and sapphire tubes and 
substrates are grown from a melt; large quartz crystals are grown by a 
hydrothermal process. ~'~ " 

Ceramic nitrides with unusually good properties for special applica- 
tions have been developed. These include aluminum nitride, a laboratory 
refractory for melting aluminum; silicon nitrides and SiAION, commer- 
cially important new refractories and potential gas turbine components; 
and boron nitride, which is useful as a refractory. 

Enamels for aluminum have been developed and have become an 
important part of the architectural industry. 

Metal -ceramic composites have been developed and are now an 
important part of the machine-tool industry and have important uses as 
refractories. The most important members of this group are various 
carbides bonded with metals and mixtures of a chromium alloy with 
aluminum oxide. 

Ceramic carbides with unique properties have been developed. Silicon 
carbide and boron carbide in particular are important as abrasive 
materials. 

Ceramic borides have been developed which have unique properties of 
high-temperature strength and oxidation resistance. 

Ferroelectric ceramics such as barium titanate have been developed 
which have extremely high dielectric constants and are particularly 
important as electronic components. 

Nonsilicate glasses have been developed and are particularly useful for 
infrared transmission, special optical properties, and semiconducting 
devices. 

Molecular sieves which are similar to, but are more controlled than, 
natural zeolite compositions are being made with controlled structures so 
that the lattice spacing, which is quite large in these compounds, can be 
used as a means of separating compounds of different molecular sizes. 

Glass -ceramics are a whole new family of materials based on fabricat- 
ing ceramics by forming as a glass and then nucleating and crystallizing to 
form a highly crystalline ceramic material. Since the original introduction 
of Pyroceram by the Corning Glass Works the concept has been extended 
to dozens of compositions and applications. 

Porefree poly crystalline oxides have been made based on alumina, 
yttria, spinel, magnesia, ferrites, and other compositions. 
. Literally dozens of other new ceramic materials unknown 10 or 20 
years ago are now being manufactured and used. From this point of view 
the ceramic industry is one of our most rapidly changing industries, with 
new products having new and useful properties constantly being de- 



vek 

for 

sen 

mar 

sati: 

N 
bett 
ava: 
pro] 
acci 
thei 

0 
occ 
hys 
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mer 
and 

A 
urai 
thoi 
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Ii 
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coa 



CERAMIC PROCESSES AND PRODUCTS 



19 



veloped. These ceramics are being developed because there is a real need 
for new materials to transform presently available designs into practical, 
serviceable productsrBy^far the majorhindrance'to the development of 
many new technologically feasible structures and systems is the lack of 
satisfactory materials. New ceramics are constantly filling this need. 

New Uses for Ceramics. In the same way that the demand for new and 
better properties has led to the development of new materials, the 
availability of new materials had led to new uses based on their unique 
properties. This cycle of new ceramics-new uses-new ceramics has 
accelerated with the attainment of a better understanding of ceramics and 
their properties. 

One example of the development of new uses for ceramics has 
occurred in the field of magnetic ceramic materials. These materials have 
hysteresis loops which are typical for ferromagnetic materials. Some have 
very nearly the square loop that is most desirable for electronic computer 
memory circuits. This new use for ceramics has led to extensive studies 
and development of materials and processes. 

Another example is the development of nuclear power, which requires 
uranium-containing fuels having large fractions of uranium (or sometimes 
thorium), stability against corrosion, and the ability to withstand the 
fissioning of a large part of the uranium atoms without deterioration. For 
many applications U0 2 is an outstanding material for this fuel. Urania 
ceramics have become an important part of reactor technology. 

In rocketry and missile development two critical parts which must 
withstand extreme temperatures and have good erosion resistance are the 
nose cone and the rocket throat. Ceramic materials are used for both. 

For machining metals at high speeds it has long been known that oxide 
ceramics are superior in many respects as cutting tools. However, their 
relatively low and irregular strength makes their regular use impossible. 
The development of alumina ceramics with high and uniform strength 
levels has made them practicable for machining metals and has opened up 
a new field for ceramics. 

In 1946 it was discovered that barium titanate had a dielectric constant 
100 times larger than that of other insulators. A whole new group of these 
ferroelectric materials has since been discovered. They allow the man- 
ufacture of capacitors which are smaller in size but have a larger capacity 
than other constructions, thus improving electronic circuitry and develop- 
ing a new use for ceramic materials. 

In jet aircraft and other applications metal parts have had to be formed 
from expensive, and in wartime unobtainable, alloys to withstand the 
moderately high temperatures encountered. When a protective ceramic 
coating is applied, the temperature limit is increased, and either higher 



20 



INTRODUCTION TO CERAMICS 



temperatures can be reached or less expensive and less critical alloys can 
be substituted. 

Many further applications of ceramics which did not even exist a few 
years ago can be cited, and we may expect new uses to develop that we 
cannot now anticipate. 



Suggested Reading 

1. F. H. Norton, Elements of Ceramics, 2d ed., Addison Wesley Publishing 
Company, Inc., Reading, Mass., 1974. 

2. F. H. Norton, Fine Ceramics, McGraw-Hill Book Company, New York, 
1970. 

3. F. H. Norton, Refractories, 4th ed., McGraw-Hill Book Company, New 
York, 1968. 

4. Institute of Ceramics Textbook Series: 

(a) W. E. Worrall, Raw Materials, Maclaren & Sons, Ltd., London, 1964. 

(b) F. Moore, Rheology of Ceramic Systems, Maclaren & Sons, Ltd., 
London, 1965. 

(c) R. W. Ford, Drying, Maclaren & Sons, Ltd., London, 1964. 

(d) W. F. Ford, The Effect of Heat on Ceramics, Maclaren & Sons, Ltd., 
London, 1967. 

5. "Fabrication Science," Proc. Brit. Ceram. Soc, No. 3 (September, 1965). 

6. "Fabrication Science: 2," Proc. Brit. Ceram. Soc, No. 12 (March, 1969). 

7. J. E. Burke, Ed., Progress in Ceramic Science, Vols. 1-4, Pergamon Press, 
Inc., New York, 1962-1966. 

8. W. D. Kingery, Ed., Ceramic Fabrication Processes, John Wiley & Sons, 
Inc., New York, 1958. 

9. F. V. Tooley, Ed., Handbook of Glass Manufacture, 2 Vols., Ogden 
Publishing Company, New York, 1961. 

10. A. Davidson, Ed., Fabrication of Non-metals: Handbook of Precision 
Engineering, Vol. 3, McGraw-Hill Book Company, New York, 1971. 



3^ '■'*■, " " ■ 



7 



Ceramic Phase- 
Equilibrium 
Diagrams 

At equilibrium a system is in its lowest free energy state for the 
composition, temperature, pressure, and other imposed conditions. When 
a given set of system parameters is fixed, there is only one mixture of 
phases that can be present, and the composition of each of these phases is 
determined. Phase-equilibrium diagrams provide a clear and concise 
method of graphically representing this equilibrium situation and are an 
invaluable tool for characterizing ceramic systems. They record the 
composition of each phase present, the number of phases present, and the 
amounts of each phase present at equilibrium. 

The time that it takes to reach this equilibrium state from any arbitrary 
starting point is highly variable and depends on factors other than the final 
equilibrium state. Particularly for systems rich in silica the high viscosity 
of the liquid phase leads to slow reaction rates and very long times before 
equilibrium is established; equilibrium is rarely achieved. For these 
systems and for others, metastable equilibrium, in which the system tends 
to a lower but not the lowest free energy state, becomes particularly 
important. 

It is obvious that the phases present and their composition are an 
essential element in analysing, controlling, improving, and developing 
ceramic materials. Phase diagrams are used for determining phase and 
composition change occurring when the partial pressure of oxygen or 
other gases is changed, for evaluating the effects of heat treatments on 
crystallization and precipitation processes, for planning new composi- 
tions, and for many other purposes. We have already seen the importance 
of thermodynamic equilibrium in our discussions of single-phase systems: 
crystalline solid solutions (Chapter 2), crystalline imperfections (Chapter 

4) , structure of glasses (Chapter 3), and surfaces and interfaces (Chapter 

5) . In this chapter we concentrate our attention on equilibria involving 
two or more phases. 



269 



1 >v: : ^ J 





270 



INTRODUCTION TO CERAMICS 



7.1 Gibbs's Phase Rule 

When a system is in equilibrium, it is necessary that the temperature 
and pressure be uniform throughout and that the chemical potentiator 

toTuEE l ^t "TT * *» ™ phase 
nZl7Z ? tendency for heat or material to be transferred from one 
part of the system to some other part. In 1874 J. Willard Gibbs* showed 
that these equihbnum conditions can occur only if the relationship 

P + V = C + 2 (7-1) 

lltf TH t iS iS kn ° Wn 35 thC Pkase mle ' with P bei "g the number of 
phases present at equihbrium, V the variance or number of degrees of 
reed om , and c the ^ of components Thjs relations « es J 

for preparing and using phase-equilibrium diagrams 

hnml ha$e ,S de 5"t d 38 any Part 0f the s y stem whi <* is physically 
homogeneous and bounded by a surface so that it is mechanics Iv 
separable from other parts of the system. It need not be conUnuouT th t 
js, two ice cubes in a drink are one phase. The number of de^ee s o 
freedom or the vanance is the number of intensive variables (pressure 
TZTT* C ° mpOSition) that can be a "ered independently and ibTa 
•ly without bringing about the disappearance of a phase or the appearance 
of a new phase. The number of components is the smallest number o 
independently vanable chemical constituents necessary and suffident to 
express he composition of each phase present. The meaning of the e 

SZff^ Cl6arer - thCy ^ * 

Deduction of the phase rule follows directly from the requirement that 
the chemical potenUal Mi of each constituent / be the same in ever^se 

f^^T rium - The chemicai potentia « is ^ » *» is mot: 



G ' = (f) 



which is the change ,n free energy of a system at constant temperature 
and pressure resultmg from the addition of one mole of constituent / to 
such a large quantity of the system that there is no appreciaol hang n 
the concentration. In a system with C components we have an indepen 

P^ nUals Tor I ^ ^T™ ^ of ch ^ a ' 

potentials. For a system containing P phases, we have 



: Mi 



(7.2) 



'Collected Works, Vol. 1, Longmans, Green & Co., Ltd., London, 1928. 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 271 



erature 
itial or 
lerwise 
)m one 
howed 
lip 

(7.1) 

iber of 
r ees of 
e basis 

sically 
nically 
s; that 
ees of 
assure, 
bitrar- 
arance 
ber of 
lent to 
these 
in the 

nt that 
phase 
molar 



rature 
it i to 
nge in 
lepen- 
jmical 

(7.2) 



fjL 2 a = ^2 b — fx 2 c = •*• = i^i P (7.3) 
etc. 

which constitute C(P - 1) independent equations which serve to fix 
C(P - 1) variables. Since the composition of each phase is defined by 
C - 1 concentration terms, completely defining the composition of P 
phases requires P(C - 1) concentration terms, which together with the 
imposed conditions of temperature and pressure give 

Total number of variables = P(C - 1) + 2 (7.4) 

Variables fixed by equality of chemical potentials = C(P — 1) (7.5) 

Variables remaining to be fixed = P(C - 1) + 2 - C(P - 1) (7.6) 

V = C - P + 2 (7.7) 

which is Gibbs's phase rule (Eq. 7.1). 

The main limitation on the phase rule is that it applies only to 
equilibrium states, requiring homogeneous equilibrium within each phase 
and heterogeneous equilibrium between phases. Although a system in 
equilibrium always obeys the phase rule (and nonconformance proves 
that equilibrium does not exist), the reverse is not always true. That is, 
conformation with the phase rule is not a demonstration of equilibrium. 



7.2 One-Component Phase Diagrams 

In a single-component system the phases that can occur are vapor, 
liquid, and various polymorphic forms of the solid. (The energy of 
different polymorphic forms as related to temperature and crystallo- 
graphic structure has been discussed in Section 2.10, and might well be 
reviewed by the reader, since it is closely related to the present section.) 
The independent variables that cause appearance or disappearance of 
phases are temperature and pressure. For example, when we heat water, 
it boils; if we cool it, it freezes. If we put it in an evacuated chamber, the 
water vapor pressure quickly reaches some equilibrium value. These 
changes can be diagrammatically represented by showing the phases 
present at different temperatures and pressures (Fig. 7.1). 

Since this is a one-component system, even the air phase is eliminated, 
and different phase distributions correspond to Fig. 7.2a to c. In actual 
practice measurements in which the vapor phase is unimportant are 
usually made at constant atmospheric pressure in a way similar to Fig. 
7.2d. Although this is not an ideal closed system, it closely approximates 
one as long as the vapor pressure is low compared with atmospheric 
pressure (so that we can ignore the insignificant vapor phase which would 



272 INTRODUCTION TO CERAMICS 



E 



1U,UUU 

1,000 




1 ) 1 1 1 1 
-3^- Ice III 


I I I I 


II 


1 


100 












10 






Liquid 






1.0 












0.1 
0.01 




Ice I *C 


■ — """"b 


Vapor 




0.001 




-kh i i i 


1 1 1 


1 1 1 1 


1 



Temperature (°C) 

Fig. 7.1. Pressure-temperature diagram for H 2 0. 




M (b) ( C ) (d) 

Fig. 7.2. Experimental conditions for a single-component system with (a) one phase, 
(b) two phases, (c) three phases, and (d) common conditions, with the condensed phase 
exposed to a gas atmosphere. 



not exist at all in a closed system) or is equal to or greater than 
atmospheric pressure (so that the vapor phase has the partial pressure 
predicted by the phase diagram). For many condensed systems of 
interest, the first criterion is satisfied. 

In a one-component system the largest number of phases that can occur 
at equilibrium is given when the variance is zero: P + V = C + 2, P 4- 0 = 
1+2, P = 3. When three phases are present at equilibrium (ice, water, 
vapor), as at point A in Fig. 7.1, any change in pressure or temperature 
causes the disappearance of a phase. The lines on the diagram represent 
conditions for two phases to exist together at equilibrium; for example, 
when liquid and vapor are present, as at point B, P + V = C + 2, 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



273 



2+ V = 1 + 2, V = 1, and the variance is one. This means that either 
pressure or temperature, but not both, can be changed arbitrarily without 
the disappearance of a phase. If we change T, to T 2 , Pi must also change 
to P 2 if both phases are to remain present. If only one phase is present, as 
at C, 

p + y = C+ 2, + 2, V = 2, 

and both pressure and temperature can be arbitrarily changed without the 
appearance of a new phase. 

At 1 atm pressure, as shown in Fig. 7.1, equilibrium between the solid 
and liquid occurs at 0°C, the freezing point. Equilibrium coexistence of 
liquid and vapor occurs at 100°C, the boiling point. The slope of these 
phase-boundary curves is at any point determined by the Clausius- 
Clapeyron equation 



dp = AH 
dT TAV 



(7-8) 



where AH is the molar heat of fusion, vaporization, or transformation, 
A V is the molar volume change, and T is the temperature. Since AH is 
always positive and AV is usually positive on going from a low- 
temperature to a high-temperature form, the slopes of these curves are 
usually positive. Since A V is usually small for condensed-phase transfor- 
mations, lines between solid phases are often almost vertical. 

There are a number of applications of one-component phase diagrams 
in ceramics. Perhaps the most spectacular of these is the development of 
the commercial production of synthetic diamonds from graphite. High 
temperatures and high pressures are necessary, as shown in Fig. 7.3. In 
addition, the presence of a liquid metal catalyst or mineralizer such as 
nickel is required for the reaction to proceed at a useful rate. Another 
system which has been extensively studied at high pressure and tempera- 
ture is Si0 2 . At pressures above 30 to 40 kilobars a new phase, coesite, 
appears which has been found to occur in nature as a result of meteorite 
impacts. At even higher pressures, above 100 kilobars, another new 
phase, stishovite, has been found. 

Of greater interest for ceramic applications are the low-pressure phases 
of silica, still subject to some dispute as to the role of minor impurities, 
but illustrated schematically in Fig. 7.4. There are five condensed phases 
which occur at equilibrium — a -quartz, ($ -quartz, /3 2 -tridymite, 0- 
cristobalite, and liquid silica. At 1 atm pressure the transition tempera- 
tures are as shown. As discussed in Section 2.10, the a -quartz-/3 -quartz 
transition at 573° is rapid and reversible. The other transformations shown 
are sluggish, so that long periods of time are required to reach equilib- 




1000 



£ 600 



200 




0 1000 3000 5000 

T(°K) 

Fig. 7.3. High-pressure, high-temperature phase equilibrium diagram for carbon. From C 
G. Suits, Am. ScL, 52, 395 (1964). 



rium. The 
chemical \ 
diagram c 
may occu: 
lines in th 
phase. H< 
quartz is : 
into a-cri 
and /3 -trie 
the forms 
when coo 
this state 
At any 
into anott 
reverse tr 
however, 
at 1100° 
j3 2 -tridym 



573° 870* 



1470° 1713° 



1 atm 




Temperature 

Fig. 7.4. Equilibrium diagram for Si0 2 . 



274 



Fig. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



275 



rium. The vapor pressure shown in the diagram is a measure of the 
chemical potential of silica in the different phases, and this same kind of 
diagram can be extended to include the metastable forms of silica which 
may occur (Fig. 7.5). The phase with the lowest vapor pressure (the heavy 
lines in the diagram) is the most stable at any temperature, the equilibrium 
phase. However, once formed, the transition between cristobalite and 
quartz is so sluggish that )3 -cristobalite commonly transforms on cooling 
into a -cristobalite. Similarly, j3 2 -tridymite commonly transforms into a- 
and /3-tridymite rather than into the equilibrium quartz forms. These are 
the forms present in the refractory silica brick, for example. Similarly, 
when cooled, the liquid forms silica glass, which can remain indefinitely in 
this state at room temperature. 

At any constant temperature there is always a tendency to transform 
into another phase of lower free energy (lower vapor pressure), and the 
reverse transition is thermodynamically impossible. It is not necessary, 
however, to transform into the lowest energy form shown. For example, 
at 1100° silica glass could transform into p -cristobalite, )3 -quartz, or 
j3 2 -tridymite. Which of these transformations actually takes place is 





^Liquid 




^^^-0-Cristobalite 


















/ ^^a-Quartz 





Temperature 

Fig. 7.5. Diagram including metastable phases occurring in the system Si0 3 . 



276 



INTRODUCTION TO CERAMICS 



determined by the kinetics of these changes. In practice, when silica glass 
is heated for a long time at this temperature, it crystallizes, or devitrifies, 
to form cristobalite, which is not the lowest energy form but is structur- 
ally the most similar to silica glass. On cooling, /3 -cristobalite. transforms 
into a -cristobalite. 

The silica system illustrates that the phase-equilibrium diagram graphi- 
cally represents the conditions for minimum free energy in a system; 
extension to include metastable forms also allows certain deductions 
about possible nonequilibrium behavior. Almost always, however, a 
number of alternative nonequilibrium courses are possible, but there is 
only one equilibrium possibility. 



7.3 Techniques for Determining Phase-Equilibrium Diagrams 

The phase-equilibrium diagrams discussed in the last section and in the 
rest of this chapter are the product of experimental studies of the phases 
present under various conditions of temperature and pressure. In using 
phase-equilibrium diagrams it is important to remember this experimental 
basis. In critical cases, for example, diagrams should not be used without 
referring directly to the original experimenter's description of exactly 
how the diagram was determined and with what detail the measurements 
were made. As additional measurements are carried out, diagrams are 
subject to constant revision. 

There is a large body of literature describing methods of determining 
phase equilibrium. In general, any physical or chemical difference be- 
tween phases or effect occurring on the appearance or disappearance of a 
phase can be used in determining phase equilibrium. Two general 
methods are used: dynamic methods use the change in properties of a 
system when phases appear or disappear, and static methods use a sample 
held under constant conditions until equilibrium is reached, when the 
number and composition of the phases present are determined. 

Dynamic Methods. The most common dynamic method is thermal 
analysis, in which the temperature of a phase change is determined from 
changes in the rate of cooling or heating brought about by the heat of 
reaction. Other properties such as electrical conductivity, thermal expan- 
sion, and viscosity have also been used. Under the experimental condi- 
tions used, the phase change must take place rapidly and reversibly at the 
equilibrium temperature without undercooling, segregation, or other 
nonequilibrium effects. In silicate systems the rate of approach toward 
equilibrium is slow; as a result thermal-analysis methods are less Useful 
for silicates than they are for metals, for example. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



277 



Dynamic methods are suitable for determining the temperature of 
phase changes but give no information about the exact reactions taking 
place. In addition to the measurements of temperature changes then, 
phase identification before and after any phase change is required. This 
analysis is usually carried out by chemical determination of composition, 
determination of optical characteristics, X-ray determination of crystal 
structure, and microscopic examination of phase amounts and phase 
distribution. 

Static Methods. In contrast to dynamic measurements, static measure- 
ments often consist of three steps. Equilibrium conditions are held at 
elevated temperatures or pressures, the sample is quenched to room 
temperature sufficiently rapidly to prevent phase changes during cooling, 
and then the specimen is examined to determine the phases present. By 
carrying out these steps at a number of different temperatures, pressures, 
and compositions, the entire phase diagram can be determined. Some- 
times high-temperature X-ray and high-temperature microscopic exami- 
nations can determine the phases present at high temperatures, making 
quenching unnecessary. 

For silicate systems the major problem encountered in determining 
phase-equilibrium diagrams is the slow approach toward equilibrium and 
the difficulty in ensuring that equilibrium has actually been reached. For 
most systems this means that static measurements are necessary. A 
common technique is to mix together carefully constituents in the correct 
ratio to give the final composition desired. These are held at a constant 
temperature in platinum foil; after rapid cooling, the mixture is reground 
in a mortar and pestle and then heated for a second time and quenched. 
The phases present are examined, the sample mixture remixed, reheated, 
and quenched again. The resulting material is then reexamined to ensure 
that the phase composition has not changed. 

This process requires much time and effort; since several thousand 
individual experiments, such as those just described, may be necessary 
for one ternary diagram, we can understand why only a few systems have 
been completely and exhaustively studied. 

Reliability of Individual Diagrams. In general, the original experi- 
menter investigating a particular phase diagram is usually concerned with 
some limited region of composition, temperature, and pressure. His effort 
is concentrated in that area, and the other parts of the phase diagram are 
determined with much less precision and detail. As reported in summariz- 
ing descriptions (such as those given in this chapter), the diagram is not 
evaluated as to which parts are most reliable. As a result, although the 
general configuration of diagrams given can be relied on, the exact 
temperatures and compositions of individual lines or points on the 



V. 




278 INTRODUCTION TO CERAMICS 

diagram should only be accepted with caution. They represent the results 
of difficult experimental techniques and analysis. 

These cautions are particularly applicable to regions of limited crystal- 
line solution at high temperatures, since for many systems exsolution 
occurs rapidly on cooling and for many systems this was not a feature of 
the experimenters' interest. Similarly, phase separation at moderate and 
low temperatures often results in submicroscopic phases which are not 
recognized without the use of electron microscopy and electron diffrac- 
tion, which have not as yet been widely applied to crystalline solid 
solutions. 



7.4 Two-Component Systems 

In two-component systems one additional variable, the composition is 
introduced so that if only one phase is present, the variance is three: 
P+V-C + 2, 1 + V= 2 + 2, V = 3. In order to represent the pressure 
temperature, and composition region of the stability of a single phase a 
three-dimensional diagram must be used. However, the effect of pressure 
is. small for many condensed-phase systems, and we are most often 
concerned with the systems at or near atmospheric pressure Conse- 
quently, diagrams at constant pressure can be drawn with temperature 
and composition as variables. A diagram of this kind is shown in Fig 7 6 

If one phase is present, both temperature and composition can be 
arbitrarily varied, as illustrated for point A. In the areas in which two 
phases are present at equilibrium, the composition of each phase is 





CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



279 



indicated by lines on the diagram. (In binary r^^ 8 ^ 
will often be shaded, single-phase re gl ons not.) The 'Jtcrsectoon o a 
Istant-temperature "tie line" with the phase boundary g.v s the 
compositions of the .phases in temperature. WUh two 

CZ~ny ibVafy thange in either temperature or composition 
oftTof the phases present requires a corresponding 
variable. The maximum number of phases that can be present where 
pressure is arbitrarily fixed (V = 1) is 

P + v = C+2,P + l=2 + 2,P=3. 

When three phases are present, the ^^^^f^^ 
temperature are fixed, as indicated by the solid horizontal line a C. 

Systems in Which a Gas Phase Is Not Important. Systems containing 
on y stab e o X 7de t which the valence of the cations is fixed compose a 
large fraction of the systems of interest for ceramics and can adequately 
be represented at a constant total pressure of 1 atm. A e « u ^"^ 
chemical potential of each constituent must be equal in each phase 
present As a result the variation of chemical potential with composition 
fs the underlying thermodynamic consideration which determines phase 
stabmty If we consider a simple mechanical mixture of two pure 
components, the free energy of the mixture G is 

G M = X A G A +X B G B ( 7 - 9 ) 

For the simplest case, an ideal solution in which the heat of mixing and 
changes in vibrational entropy terms are zero, random mixing gives r.se to 
a configuration^ entropy of mixing AS m which has been derived ,n Eq. 
4.14; the free energy of the solution is 

G fcus = G M_ TASm (7.10) 

and under all conditions the free energy of the solution is less , thw > that of 
a mechanical mixture; the free energy curves for the solid and I liquid 
soLions and the resulting phase-equilibrium diagram are similar to those 
aheady illustrated in Fig. 4.2. Since very dilute ^ons^c^l 
behavior, Eq. 7.10 requires that there is always at least some m.nute 
solubility on the addition of any solute to any pure 

Most concentrated solutions are not ideal, but many can be well 
represented as regular solutions in which the ^ * "J 

solution is negligible, but the excess enthalpy or heat of mixing AH is 
significant. In this case the free energy of the regular solut.on ,s 



280 



INTRODUCTION TO CERAMICS 



The resulting forms of typical free-energy-composition curves for an 
ideal solution and for regular solutions with positive or negative excess 
enthalpies are shown in Fig. 7.7. In Fig. 7.7c the minimum free energy for 
the system at compositions intermediate between a and p consists of a 
mixture of a and p in which these two solution compositions have the 
same chemical potential for each component and a lower free energy than 
intermediate single-phase compositions; that is, phase separation occurs. 
When differences of crystal structure occur (as discussed in Chapter 2), a 
complete series of solid solutions between two components is not 
possible, and the free energy of the solution increases sharply after an 
initial decrease required by the configurational entropy of mixing. This 




Composition 
(a) 





Composition 
ib) 




Composition Composition 
<c> idi 

Fig. 7.7. Free-energy-composition diagrams for (a) ideal solution, (b) and (c) regular 
solutions, and (d) incomplete solid solution. 



o: 
sr 
oi 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



281 



situation is illustrated in Fig. 7. Id, in which the minimum system free 
energy again consists of a mixture of the two solutions a and 0. 

When, for any temperature and composition, free-energy curves such 
as shown in Fig. 7.7 are known for each phase which may exist; these 
phases actually occur at equilibrium which give the lowest system free 
energy consistent with equal chemical potentials for the components in 
each phase. This has been illustrated for an ideal solution in Fig. 4.2, 
compound formation in Fig. 4.3, and phase separation in Fig. 3.10 and is 
illustrated for a series of temperatures in a eutectic system in Fig. 7.8. 

Systems in Which a Gas Phase Is Important. In adjusting the oxygen 
pressure in an experimental system, it is often convenient to use the 
equilibria 

(7.12) 



co+p 2 = co 2 



H 2 + iQ 2 = H 2 0. 



(7.13) 



gular 



In this case, with no condensed phase present, P + V- C+2, 1 + V 
2 + 2 V = 3 and it is necessary to fix the temperature, system total 
pressure, and the gas composition, that is, C0 2 /CO or H 2 /H 2 0 ratio, in 
order to fix the oxygen partial pressure. If a condensed phase, that is, 
graphite, is in equilibrium with an oxygen-containing vapor phase, P + 
V = C + 2, 2 + V = 2 + 2, V = 2, and fixing any two independent variables 
completely defines the system. 

The most extensive experimental data available for a two-component 
system in which the gas phase is important is the Fe-0 system, in which a 
number of condensed phases may be in equilibrium with the vapor phase. 
A useful diagram is shown in Fig. 7.9, in which the heavy lines are 
boundary curves separating the stability regions of the condensed phases 
and the dash-dot curves are oxygen isobars. In a single condensed-phase 
region (such as wustite) P + V = C + 2, 2 + V = 2 + 2, V = 2, and both the 
temperature and oxygen pressure have to be fixed in order to define the 
composition of the condensed phase. In a region of two condensed phases 
(such as wustite plus magnetite) P+V = C+ 2, 3+V = 2 + 2, V = 1, and 
fixing either the temperature or oxygen pressure fully defines the system. 
For this reason, the oxygen partial-pressure isobars are horizontal, that is, 
isothermal, in these regions, whereas they run diagonally across single 
condensed-phase regions. 

An alternative method of representing the phases present at particular 
oxygen pressures is shown in Fig. 7.9b. In this representation we do not 
show the O/Fe ratio, that is, the composition of the condensed phases, but 
only the pressure-temperature ranges for each stable phase. 




liquid iron - 
+ 

liquid oxide 



■\ magnetite + liquid 
7 



hematite 

+ ) 

liquid/ 



400 



200 



a-iron+ magnetite 



hematite. 

—vf- 

air r 



—■-10-^ 

magnetite — 
+ hematite 

— 10 --i — i 




'KeO 10 16 3CT 40 50 60 FeOFe 2 0 3 80 90 Fe 2 0, 
Weight % 

Fig 7 9. (a) Phase relations in the FeO-Fe.O, system. Dash-dot lines are oxygen isobars. 
Alternate solidification paths for composition A are discussed in text. Frorr, , AJuar, , and 
F. Osborn, Phase Equilibria among Oxides in Steelmaking, Add.son-Wesley Publ.sh.ng 
Company, Inc., Reading, Mass., 1965. 



283 



284 



INTRODUCTION TO CERAMICS 



5.0 



1700 




-8 -6 

log po 2 (atm) 



1500 



1100 



1200 



1300 



1400 



1600 



Fig. 7.9 (continued), (b) Temperature-oxygen pressure diagram for the Fe-Fe 2 0 3 system. 
From J. B. Wagner, Bull. Am. Cer. Soc, 53, 224 (1974). 



7.5 Two-Component Phase Diagrams 

Phase-equilibrium diagrams are graphical representations of experi- 
mental observations. The most extensive collection of diagrams useful in 
ceramics is that published by the American Ceramic Society in two large 
volumes, which are an important working tool of every ceramist.* Phase 
diagrams can be classified into several general types. 

Eutectic Diagrams. When a second component is added to a pure 
material, the freezing point is often lowered. A complete binary system 
consists of lowered liquidus curves for both end members, as illustrated 
in Fig. 7.8. The eutectic temperature is the temperature at which the 
liquidus curves intersect and is the lowest temperature at which liquid 
occurs. The eutectic composition is the composition of the liquid at this 
temperature, the liquid coexisting with two solid phases. At the eutectic 
temperature three phases are present, so the variance is one. Since 
pressure is fixed, the temperature cannot change unless one phase 
disappears. 

In the binary system BeO-Al 2 0 3 (Fig. 7.10) the regions of solid solution 
that are necessarily present have not been determined and are presumed 

*E. M. Levin, C R. Robbins, and H. F. McMurdie, Phase Diagrams for Ceramists, 
American Ceramic Society, Columbus, 1964; Supplement, 1969. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



285 



system. 



<peri- 
:ful in 
large 
Phase 

pure 
/stem 
trated 
h the 
liquid 
it this 
tectic 
Since 
phase 

lution 
umed 

amists, 




BeAl 2 0 4 
+ 

BeAl 6 0 l0 



80 

1:1 1:3 A1 2 0 3 



40 

Weight % A1 2 0 3 
Fig. 7.10. The binary system BeO-Al 2 0 3 . 



to be of limited extent, although this is uncertain, and are not shown in the 
diagram. The system can be divided into three simpler two-component 
systems (BeO-BeAl 2 0 4 , B e AI2O4-B e A1 6 0 1 0 , and BeAUO„r-Al 2 0 3 ) in each 
of which the freezing point of the pure material is lowered by addition of 
the second component. The BeO-BeAl 2 0 4 subsystem contains a com- 
pound, Be 3 Al 2 0 6 , which melts incongruently, as discussed in the next 
section. In the single-phase regions there is only one phase present, its 
composition is obviously that of the entire system, and it comprises 100% 
of the system (point A in Fig. 7.10). In two-phase regions the phases 
present are indicated in the diagram (point B in Fig. 7.10); the composi- 
tion of each phase is represented by the intersection of a constant 
temperature tie line and the phase-boundary lines. The amounts of each 
phase can also be determined from the fact that the sum of the 
composition times the amount of each phase present must equal the 
composition of the entire system. For example, at point C in Fig. 7.10 the 
entire system is composed of 29% A1 2 0 3 and consists of two phases, BeO 
(containing no A1 2 0 3 ) and 3BeOAl 2 0 3 (which contains 58% A1 2 0 3 ). There 



286 



INTRODUCTION TO CERAMICS 



must be 50% of each phase present for a mass balance to give the correct 
overall composition. This can be represented graphically in the diagram 
by the lever principle, in which the distance from one phase boundary to 
the overall system composition, divided by the distance from that 
boundary to the second phase" boundary, is trie fraction of the second 
phase present. That is, in Fig. 7.10, 

OC 

— (100) = Per cent 3BeOAl 2 0 3 

A little consideration indicates that the ratio of phases is given as 

DC = BeO 
OC 3Be0Al 2 0 3 

This same method can be used for determining the amounts of phases 
present at any point in the diagram. 

Consider the changes that occur in the phases present on heating a 
composition such as E, which is a mixture of BeAl 2 0 4 and BeAl 6 0 10 . 
These phases remain the only ones present until a temperature of 1850°C 
is reached; at this eutectic temperature there is a reaction, BeAI 2 0 4 + 
BeAl 6 O 10 = Liquid (85% A1 2 0 3 ), which continues at constant temperature 
to form the eutectic liquid until all the BeAl 6 Oi 0 is consumed. On further 
heating more of the BeAl 2 0 4 dissolves in the liquid, so that the liquid 
composition changes along GF until at about 1875°C all the BeAl 2 0 4 has 
disappeared and the system is entirely liquid. On cooling this liquid, 
exactly the reverse occurs during equilibrium solidification. 

As an exercise students should calculate the fraction of each phase 
present for different temperatures and different system compositions. 

One of the main features of eutectic systems is the lowering of the 
temperature at which liquid is formed. In the BeO-Al 2 0 3 system, for 
example, the pure end members melt at temperatures of 2500°C and 
2040°C, respectively. In contrast, in the two-component system a liquid is 
formed at temperatures as low as 1835°C. This may be an advantage or 
disadvantage for different applications. For maximum temperature use as 
a refractory we want no liquid to be formed. Addition of even a small 
amount of BeO to A1 2 0 3 results in the formation of a substantial amount 
of a fluid liquid at 1890°C and makes it useless as a refractory above this 
temperature. However, if high-temperature applications are not of major 
importance, it may be desirable to form the liquid as an aid to firing at 
lower temperatures, since liquid increases the ease of densification. This 
is true, for example, in the system Ti0 2 -U0 2 , in which addition of 1% 
Ti0 2 forms a eutectic liquid, which is a great aid in obtaining high 
densities at low temperatures. The structure of this system, shown in Fig. 





CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



287 



ct 
m 
to 
at 
id 



es 

. a 

>io. 

°C 
.+ 
ire 
ler 
lid 
as 
id, 

ise 

he 
cor 
nd 
(is 
or 
as 
tall 
jnt 
his 
jor 
at 
his 
1% 
igh 



7.11, consists of large grains of U0 2 surrounded by the eutectic composi- 
te effectiveness of eutectic systems in lowering the melting point is 
made use of in the Na 2 0-Si0 2 system, in which glass compositions can be 
melted at low temperatures (Fig. 7.12). The liquidus is lowered from 
mVC in pure Si0 2 to about 790° for the eutectic composition at 
approximately 75% Si02-25% Na 2 0. 

Formation of low-melting eutectics also leads to some severe limita- 
tions on the use of refractories. In the system CaO-Al 2 0 3 the liquidus is 
strongly lowered by a series of eutectics. In general, strongly basic oxides 
such as CaO form low-melting eutectics with amphoteric or basic oxides 
and these classes of materials cannot be used adjacent to each other, even 
though they are individually highly refractive. 

Incongruent Melting. Sometimes a solid compound does not melt to 
form a ifquid of its own composition but instead 

solid phase and a liquid. This is true of enstatite (MgSiO,) at 1557 C (Fig. 
7 ay this compound forms solid M g2 S:0 4 plus a liquid containing about 
6 1% S O, At this incongruent melting point or peritecUc temperature there 




Fig. 7.1 1 Structure of 99% UO,-l% TiO, ceramic (228X, HNO, etch). UO, is the primary 
phase, bonded by eutectic composition. Courtesy G. Ploetz. 



1800 
1700 
1600 
1500 
1400 
1300 
1200 
1100 
1000 
900 
800 
700 



Cristobalite + liquid 




Na 2 Si0 3 +/Na (i Si 8 0 19 +\ 
"Na 2 Si 2 0 5 \3 Na 2 Si 2 0/' 





Na fi Si fi Qj f) + liquid 
Na 2 Si 2 0s 

liquid 



Tridymite + liquid 
Na G Si s 0 19 + a Na 2 Si 2 0 5 



Quartz + liquid 



HNa 6 Si 8 O iy + quartz/' Metastable 



50 



60 



8 Na 2 Si 2 0 5 + quartz / 
A 



two liquids 



70 80 
Weight % Si0 2 



90 



100 



Fig. 7.12. The binary system Na 2 Si(VSi0 2 . The dashed line shows metastable liquid-liquid 
phase separation. 





CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 

are three phases present (two solids and a liquid), so that the temperature 
remains fixed until the reaction is completed. Potash feldspar (Fig. 7.14) 
also melts in this way. 

Phase Separation. When a liquid or crystalline solution is cooled, it 
separates into two separate phases at the consolute temperature as long as 
the excess enthalpy is positive (see Fig. 7.7). This phenomenon is particu- 
larly important relative to the development of substructure in glasses, as 
discussed in Chapter 3 (Figs. 3.11, 3.12, 3.14 to 3.19). Although it has been 
less fully investigated for crystalline oxide solid solutions, it is probably 
equally important for these systems when they are exposed to moderate 
temperatures for long periods of time. The system CoO-NiO is shown in 
Fig. 7.15. 



00 
[uid 



1800 



1713 + 5° 



1686+5° 




ttflff 



0 20 
Leucite Potash 
K 2 0-Al 2 0 3 -4Si02 feldspar 

K 2 0-Al 2 0 3 -6Si02 

Weight per cent Si0 2 

Fig. 7.14. The binary system K^O AljO, 4Si0 2 (leucite)-Si0 2 . From J. F. SchairerandN. L. 
Bowen, Bull. Soc. Geoi Fin/., 20, 74 (1947). Two-phase regions are shown shaded in this 
diagram. 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 291 

Solid Solutions. As discussed in Chapter 4 and in Section 7.4, a 
complete series of solid solutions occurs for some systems such as 
illustrated in Fig. 4.2 and Fig. 7.15, and some minute or significant limited 
solid solution occurs for all systems, as shown in Figs. 4.3, 7. 1 3, and 7. 15. 

It has only been in the last decade or so that careful experimentation 
has revealed the wide extent of solid solubility, reaching several percent 
at high temperatures in many systems, as shown in Figs. 4.3, 7. 13, and 7.15 
and for the MgO-CaO system in Fig. 7.16 and the MgO-Cr 2 0 3 system in 
Fig. 7.17. For steel-plant refractories directly bonded magnesia-chromite 
brick is formed when these materials are heated together at temperatures 
above 1600°C as a result of the partial solubility of the constituents; 
exsolution occurs on cooling. Almost all open-hearth roofs are formed of 
either direct-bonded, rebonded fine-grain, or fusion-cast magnesia- 
chromite refractories. In the basic oxygen-furnace process for steel 
making MgO-CaO refractories bonded with pitch are widely used, and 
the solid solubility at high temperatures forms a high-temperature bond. 
In magnesia refractories the lower solid solubility of Si0 2 as compared 




292 INTRODUCTION TO CERAMICS 

with CaO in MgO requires that excess CaO be added to prevent the 
formation of low-melting intergranular silicates. 

In the MgO-Al 2 0 3 system (Fig. 4.3) there is extensive solubility of MgO 
and of AI2O3 in spinel. As spinel in this composition range is cooled, the 

-solubility decreasesrand corundum precipitates as a separate solid^phase^ 
(Fig. 7.18). 

This same sort of limited solid solution is observed in the CaO-Zr0 2 
system (Fig. 7.19); in this system there are three different fields of solid 
solution, the tetragonal form, the cubic form, and the monoclinic form. 
Pure ZrQ 2 exhibits a monoclinic tetragonal phase transition at 1000°C, 
which involves a large volume change and makes the use of pure zirconia 
impossible as a ceramic material. Addition of lime to form the cubic solid 
solution, which has no phase transition, is one basis for stabilized 
zirconia, a valuable refractory. 

Complex Diagrams. All the basic parts of binary phase-equilibrium 
diagrams have been illustrated; readers should be able to identify the. 
number of phases, composition of phases, and amounts of phases present 
at any composition and temperature from any of these diagrams with ease 
and confidence. If they cannot, they should consult one of the more 
extensive treatments listed in the references. 




Fig. 7.18. Precipitation of A1 2 0 3 from spinel solid solution on cooling (400x H 2 S0 4 etch). 
Courtesy R. L. Coble. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



293 



the 

go 

the 

ase 

:0 2 
>lid 
cm. 
°C, 
nia 
>lid 
zed 

urn 
the 
ent 
ase 
ore 



m ■ 



Liquid 




Fig. 7.19. The binary system CaO- 
J. Am. Ceram. Soc, 35, 109 (1952) 



Mole per cent CaO 
-ZrOj. From P. Duwez, F. Odell, and F. H. Brown, Jr. 
Two-phase regions are shown shaded in this figure. 



tch). 



Combinations of simple elements in one system sometimes appear 
frightening in their complexity but actually offer no new problems in 
interpretation. In the system Ba 2 TiO<-Ti0 2 (Fig. 7.20), for example, we 
find two eutectics, three incongruently melting compounds, polymorphic 
forms of BaTiO,, and an area of limited solid solution. All of these have 
already been discussed. 

Generally phase diagrams are constructed at a total pressure of 1 atm 
with temperature and composition as independent variables. Since the 
interesting equilibrium conditions for many ceramics involve low oxygen 
partial pressures, phase diagrams at a fixed temperature but with oxygen 



1 



294 



INTRODUCTION TO CERAMICS 



1900 




33.3 
Ba 2 Ti0 4 



60 

Moie per cent TiC>2 



syst€ 

Po* = 
(Fe,( 
chro; 
betw 

7.6 

Th 
comj 
pres* 
fix tr 
gives 
ternE 
tions 
a ve 
two-i 
equil 



c 8 * , wfnc^^ SyStCm Ba2Ti0 - Ti0 - Fr ^ D- E. Rase and R. Roy, J. Am. Ceram. 
boc. 38, III (1955). Two-phase regions are shown shaded in this figure. 



pressure and composition as variables become a useful alternative for 
describing phase equilibria, for example, Fig. 7.9b. Figure 7 21(a-l) 
shows such a diagram for Co-Ni-0 at 1600°K. The lens-shaped two- 
phase region between (Co,Ni)0 and the NiCo alloy is similar to that 
between the liquid oxides and (Co,Ni)0 in a temperature-composition 
plot (Fig. 7.15). Figure 7.21(a-2) shows the oxygen isobar tie lines 
between the metal alloy and the oxide solid solution; for example the 
dotted line represents the equilibrium at P G2 = 1.5 x l(T 7 atm between 
Nio.6 2 Coo.3 8 0 and Nio. 9 Co 0 .,. (A tie line connects phases in equilibrium and 
designates the composition of each phase. For example, a constant 
temperature tie line in Fig. 7.17 at 2600°C specifies the composition of the 
solid solution, 10 w/o Cr 2 0„ in equilibrium with the liquid, which contains 
40 w/o'Cr 2 0 3 .) A plot of the nickel activity as a function of P G2 is shown in 
Fig. 7.21(a-3). In systems which form intermediate compounds, such as 
spinels, the diagrams become more complex. The Fe-Cr-O ternary 



Fig. 7. 
(2) ox; 
nickel 




33p| 



IBS 



tit 



lliflil 



ftfb 



100 
Ti0 2 

Am. Ceram. 



iative for 
7.21(a-l) 
iped two- 
ir to that 
^position 

tie lines 
mple, the 

between 
>rium and 

constant 
ion of the 
a contains 
i shown in 
s, such as 
3 ternary 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



295 



system at 1573°K is shown in Fig. ™ b -"™™^r^g 
Po,= 10- ,0 atm, the stable phases may be FeO, FeO + (Fe,Cr),0«, 
(F: ) Cr)30 < + (Fe ( Cr) J 03 > or (Fe,Cr) 2 C>3, 

chromium. The oxygen isobars shown in Ftg 7.21(b-2) are tie fines 
between the compositions in equilibrium at 1573 K. 

7.6 Three-Component Phase Diagrams 

Three-component systems are fundamentally no different from two- 
component systems, except that there are t™™^^™?^ 
pressure, temperature, and the concentrations of two components (which 
fix the third). If pressure is arbitrarily fixed, the presence of four phases 
givls rise to an invariant system. A complete graphical representation of 
ternary systems is difficult, but if the pressure is held constant, composi- 
tions can be represented on an equilateral triangle and the temperature on 
a vertical ordinate to give a phase diagram such as Fig. 7.22. hor 
two-dimensional representation the temperatures can be projected on an 
equilateral triangle, with the liquidus temperatures represented by 



CoO 



*T = 1600°K 
Po 2 = const 
= 1.5 x 10 



-6.6 



-7.0 



-7.4 




0.2 0.4 0.6 0.8 1.0 



ia) <3) 

Fig 7 21 (a) Co-Ni-0 system. (1) Composition of condensed phases as a function of P^; 
(2) oxygen isobars for equilibrium between the oxide solid solution and the alloy solution. (3) 
nickel activity as a function of Poj. 



296 INTRODUCTION TO CERAMICS 



Fe,0 



(2) 

Sesquioxide 



Wustite 




Cr 2 0 3 



Fig. 7.21 (continued), (b) Fe-Cr-0 system. (1) Composition — diagram and (2) oxygen 
isobars for equilibrium between two phases. From A. Pelton and H. Schmalzried, Met. Trans., 
4, 1395 (1973). 



isotherms. The diagram is divided into areas representing equilibrium 
between the liquid and a solid phase. Boundary curves represent equilib- 
rium between % two solids and the liquid, and intersections of three 
boundary curves represent points of four phases in equilibrium (invariant 
points in the constant-pressure system). Another method of tWO- 



CX 

E 



Fig. 7.22. 



dimensio 
the diagr: 
ture. 

Interpr 
that of bi 
composit 
phase-bo 
phase an 
composr 
Fig. 7.22 
field of : 
when th< 
along Al 
applies, 
100(BA/ 
path rea 
and two 
changes 
are a liq 
composi 
mixture 
(100) = ) 
That is. 
compos 



i 



mm. 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



297 





(b) 

Fig. 7.22. Space diagram of (a) ternary eutectic and (b) complete series of solid solutions. 



dimensional representation is to take a constant-temperature cut through 
the diagram, indicating the phases at equilibrium at some fixed tempera- 
ture. 

Interpretation of ternary diagrams is not fundamentally different from 
that of binary diagrams. The phases in equilibrium at any temperature and 
composition are shown; the composition of each phase is given by the 
phase-boundary surfaces or intersections; the relative amounts of each 
phase are determined by the principle that the sum of the individual phase 
compositions must equal the total composition of the entire system. In 
Fig. 7.22 and Fig. 7.23, for example, the composition A falls in the primary 
field of X. If we cool the liquid A, X begins to crystallize from the melt 
when the temperature reaches T,. The composition of the liquid changes 
along AB because of the loss of X. Along this line the lever principle 
applies, so that at any point the percentage of X present is given by 
100(BA/XB). When the temperature reaches T 2 and the crystallization 
path reaches the boundary representing equilibrium between the liquid 
and two solid phases X and Z, Z begins to crystallize also, and the liquid 
changes in composition along the path CD. At L, the phases in equilibrium 
are a liquid of composition L and the solids X and Z, whereas the overall 
composition of the entire system is A. As shown in Fig. 7.23b, the only 
mixture of L, X, and Z that gives a total corresponding to A is xAlxX 
(100) = Per cent X, z A/zZ (100) = Per cent Z, /A//L (100) = Per cent L. 
That is, the smaller triangle XZL is a ternary system in which the 
composition of A can be represented in terms of its three constituents. 





Composition 

Fig. 7.23. (a) Crystallization path illustrated in Fig. 7.22a and (b) application of center of 
gravity principle to a ternary system. 



Many tei 
ogy. Two c 
system, an 
the MgO- 
K 2 0-A1 2 0: 
compositic 
silica-mull 
discussed 
the basis < 
systems a 
provides a 
Si0 2 syste 
tions fall 
Na 2 0-3Ca 



Si0 2 

Cristobalite 
Tridymite \ 90 /£CI5t7\ 10 



Quartz ^80^^ \ ^20 Potash feldspar 
K 2 0.4Si0 2 O^^^rV^^ K 2 0-Al 2 0 3 -6Si0 2 



K 2 0-2Si0 2 




Mutlite 
3A1 2 0 3 * 2SiO z 



Fig. 7.24. The ternary system K 2 0-Al 2 03-Si0 2 . From J. F. Schairer and N. L. Bowen, Am. J. 
ScL, 245, 199 (1947). 



2Na 

2Na 2 0< 

Na 
To Na 2 



Fig. 7.25. 
Glass Tei 



298 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 299 

Many ternary systems are of interest in ceramic science and technol- 
ogy. Two of these, the K 2 0-Al 2 C>3-Si0 2 system and the Na 2 0-CaO-Si0 2 
system, are illustrated in Figs. 7.24 and 7.25. Another important system, 
the MgO-Al 2 0 3 -Si0 2 system, is discussed in Section 7.8. The 
K 2 0-Al 2 0 3 -Si0 2 system is important as the basis for many porcelain 
compositions. The eutectic in the subsystem potash-feldspar- 
silica-mullite determines the firing behavior in many compositions. As 
discussed in Chapter 10, porcelain compositions are adjusted mainly on 
the basis of (a) ease in forming and (b) firing behavior. Although real 
systems are usually somewhat more complex, this ternary diagram 
provides a good description of the compositions used. The Na 2 0-CaO- 
Si0 2 system forms the basis for much glass technology. Most composi- 
tions fall along the border between the primary phase of devitrite, 
Na 2 03CaO-6Si0 2 , and silica; the liquidus temperature is 900 to 1050°C. 



To CaO 




Weight per cent Na 2 0 

Fig. 7.25. The Na 2 0-0-CaO-Si0 2 system. From G. W. Morey and N. L. Bowen, /. Soc. 
Glass TechnoL, 9, 232 (1925). 



300 



INTRODUCTION TO CERAMICS 



This is a compositional area of low melting temperature, but the glasses 
formed contain sufficient calcium oxide for reasonable resistance to 
chemical attack. When glasses are heated for extended times above the 
transition range, devitrite or cristobalite is the crystalline phase formed as 
the devitrification product. 

Very often constant-temperature diagrams are useful. These are illus- 
trated for subsolidus temperatures in Figs. 7.24 and 7.25 by lines between 
the forms that exist at equilibrium. These lines form composition triangles 
in which three phases are present at equilibrium, sometimes called 
compatibility triangles. Constant-temperature diagrams at higher temper- 
atures are useful, as illustrated in Fig. 7.26, in which the 1200° isothermal 



Si0 2 




Weight per cent 

] Single -phase region 




Two -phase regions 
Three-phase regions 



Fig. 7.26. Isothermal cut in the K : O-Al 2 O 3 -Si0 2 diagram at 1200°C. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



301 



isses 
;e to 
,e the 
ed as 

illus- 
ween 
ngles 
ailed 
nper- 
;rmal 



plane is shown for the K 2 0-Al 2 0 3 -Si0 2 diagram. The liquids formed in 
this system are viscous; in order to obtain vitrification, a substantial 
amount of liquid must be present at the firing temperature. From 
isothermal diagrams the composition of liquid and amount of liquid for 
different compositions can be easily determined at the temperature 
selected Frequently it is sufficient to determine an isothermal plane 
rather than an entire diagram, and obviously it is much easier. 

Although our discussion of three-component diagrams has been brief 
and we do not discuss phase-equilibrium behavior for four or more 
component systems at all, students would be well advised to become 
familiar with these as an extra project. 

7.7 Phase Composition versus Temperature 

One of the useful applications for phase equilibrium diagrams in 
ceramic systems is the determination of the phases present at different 
temperatures. This information is most readily used in the form of plots of 
the amount of phases present versus temperature. 

Consider, for example, the system MgO-Si0 2 (Fig. 7.13). For a compos- 
ition of 50 wt% MgO-50 wt% Si0 2 , the solid phases present at equilibrium 
are forsterite and enstatite. As they are heated, no new phases are formed 
until 1557°C. At this temperature the enstatite disappears and a composi- 
tion of about 40% liquid containing 61% Si0 2 is formed. On further 
heating the amount of liquid present increases until th ^ u i du ^^% h " d 
at some temperature near 1800°C. In contrast, for a 60% MgO-40% Si0 2 
composition the solid phases present are forsterite, MfcSiO, and peric- 
lase MgO. No new phase is found on heating until 1850 C, when the 
composition becomes nearly all liquid, since this temperature is near the 
eutectic composition. The changes in phase occurring for these two 
compositions are illustrated in Fig. 7.27. 

Several things are apparent from this graphical representation. One is 
the large difference in liquid content versus temperature for a relatively 
small change in composition. For compositions containing greater than 
42% silica, the forsterite composition, liquids are formed at relatively low 
temperatures. For compositions with silica contents less than 42% no 
liquid is formed until 1850°C. This fact is used in the treatment of 
chromite refractories. The most common impurity present is serpentine 
3MgO-2Si0 2 -2H 2 0, having a composition of about 50 wt% Si0 2 . It 
sufficient MgO is added to put this in the MgO-forsterite field, it no longer 
has a deleterious effect. Without this addition a liquid is formed at low 

temperatures. . 

Another application of this diagram is in the selection of compositions 



i 




Forsterite 



0 20 40 60 80 100 
Weight per cent of each phase present 
(a) 40% Si0 2 




0 20 40 60 80 100 
Weight per cent of each phase present 
(b) 45% Si0 2 



0 20 40 60 80 100 
Weight per cent of each phase present 
(c) 55% Si0 2 



0 20 40 60 80 100 
Weight per cent of each phase present 
(d) 60% Si0 2 




0 20 40 60 80 100 
Weight per cent of each phase present 
(e) 65% Si0 2 



0 20 40 60 80 100 
Weight per cent of each phase present 
(/) 70% Si0 2 



Fig. 7.27. Phase composition versus temperature for samples in the MgO-Si0 2 system. 



that have desirable firing characteristics. It is necessary to form a 
sufficient amount of liquid for vitrification, but not so much that ware 
slumps or warps during firing. The limits of liquid required vary with the 
properties of the liquid but are in the range of 20 to 50 wt%. To have, a 
sufficient range of firing temperature, it is desirable that the liquid content 
not change much with temperature. Forsterite compositions cannot be 
fired until very high temperatures if the composition is exactly 42% SiOz, 
since no liquid is formed below 1850°C. Compositions in the forsterite- 
enstatite field which are mainly forsterite form a liquid at 1557°C, and 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 303 

since the liquidus curve ,s steep the 

slightly with temperature as shown m F*. 12T • Con*g uen y 

cond nsed Phase, appear and their "-P?*^ » ™ *Vsys m 
the dotted line : wim £ ^ Qxygen pressure , 

mmmmm 

Sled d^stn . of crystallisation paths in f 
references should be consulted. The following summary' can serve as a 

review. 

i When a liouid is cooled, the first phase to appear is the primary 
phase for"h« Pan of the system in which the composition of the melt ts 

T ^'crystallization curve follows to the nearest boundary the 

einTn oSe Uh, line ^££F££Z£3& 
liquid with that of the primary phase of that held, i ne o v m 

the primary phase) with the liquidus surface. 

3 At the boundary line a new phase appears which is i the primary 
phase of fheadjacentfield. The two phases separate together along this 

A^^S^ ™ with a line connecting the compos- 
er E M. Levin, H. F. McMurdie, and F. P. HaU, PHase Digram, for Ceramists, 
American Ceramic Society, Cleveland, Ohio, 1956. 



304 



INTRODUCTION TO CERAMICS 



tions of the two solid phases. Two things can occur. If this tangent line 
runs between the compositions of the two solid phases, the amount of 
each of these phases present increases. If the tangent line intersects an 
extension of the line between solid compositions, the first phase de- 
creases in amount (is resorbed; Reaction A + Liquid = B) as crystalliza- 
tion proceeds. In some systems the crystallization curve leaves the 
boundary curve if the first phase is completely resorbed, leaving only the 
second phase. Systems in which this occurs may be inferred from a study 
of the mean composition of the solid separating between successive 
points on the crystallization path. 

5. The crystallization curve always ends at the invariant point which 
represents equilibrium of liquid with the three solid phases of the three 
components within whose composition triangle the original liquid com- 
position was found. 

6. The mean composition of the solid which is crystallizing at any point 
on a boundary line is shown by the intersection at that point of the tangent 
with a line joining the composition of the two solid phases which are 
crystallizing. 

7. The mean composition of the total solid that has crystallized up to 
any point on the crystallization curve is found by extending the line 
connecting the given point with the original liquid composition to the line 
connecting the compositions of the phases that have been separating. 

8. The mean composition of the solid that has separated between two 
points on a boundary is found at the intersection of a line passing through 
these two points with a line connecting the compositions of the two solid 
phases separating along this boundary. 



7.8 The System AI2O3-S1O2 

As an example of the usefulness of phase diagrams for considering 
high-temperature phenomena in ceramic systems, the Al 2 0 3 -Si0 2 system 
illustrates many of the features and problems encountered. In this system 
(Fig. 7.28), there is one compound present, mullite, which is shown as 
melting incongruently. (The melting behavior of mullite has been con- 
troversial; we show the metastable extensions of the phase boundaries in 
Fig. 7.28. For our purposes this is most important as indicative of the fact 
that experimental techniques are difficult and time consuming; the diag- 
rams included here and in standard references are summaries of experi- 
mental data. They usually include many interpolations and extrapolations 
and have been compiled with greater or lesser care, depending on the 
needs of the original investigator.) The eutectic between mullite and 



2300 
2200 
2100 
* 2000 

o 

« 1900 

ro 
cu 

t 1800 

CL) 
I — 

1700 

1600 

1500 

1400 
Si0 2 

Fig. 7.28. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



ent line 
ount of 
;ects an 
ase de- 
stalliza- 
ves the 
mly the 
a study 
xessive 

t which 
ie three 
id com- 

ny point 
tangent 
nich are 

;d up to 
the line 

the line 
rating, 
een two 

through 
wo solid 



isidering 
2 system 
s system 
hown as 
jen con- 
daries in 
the fact 
:he diag- 
f expec- 
tations 
g on the 
llite and 



0 10 20 30 40 
2300 r i i i i I — 1 — I — \ r 



2200 



— Metastable extension of 

liquidus and solidus lines 



2100 



2000 



1900 



E 1800 



MM 1 ' - 

" Stable equilibrium diagram 



A1 2 0 3 <% by weight) 
50 60 70 

T — i — I — 1 T" 



80 



90 



305 



100 



i — r 



2054° ± 6° 




Alumina + 
mullite (ss) 



70 80 90 A1 2 0 3 



A1 2 0 3 (mole %) 



Fig . 7.28. The binary system A.,0,-SiO„ From Aksay and Pask, Science, 183, 69 (1974). 

cristobalite occurs at 1587°C to form a liquid containing «^95mofc% 
Pftrtnrs affecting the fabrication and use 01 severm iw j *- 

sintered alumina (>90 wt% AM),). d f 

At one end of the composition range are silica bricks _wiaeiy 
furnace roofs and similar structures requiring high atM^ 
temperatures. A major application was «^££J^£"S 
furnaces in which temperatures ,ot ^^J^^ sta y te . In the 

zssz set « " Prb^^rs^ 

aluminum oxide are particularly deleterious to bnck properties because 



306 



INTRODUCTION TO CERAMICS 



the eutectic composition is close to the silica end of the diagram. 
Consequently, even small additions of aluminum oxide mean that sub- 
stantial amounts of liquid phase are present at temperatures above 
1600°C. For this reason supersilica brick, which has a lower alumina 
content through special raw-material selection or treatment, is used in 
structures that will be heated to high temperatures. 

Fire-clay bricks have a composition ranging from 35 to 55% aluminum 
oxide. For compositions without impurities the equilibrium phases pres- 
ent at temperatures below 1587°C are mullite and silica (Fig. 7.29). The 
relative amounts of these phases present change with composition, and 
there are corresponding changes in the properties of the brick. At 
temperatures above 1600°C the amount of liquid phase present is sensitive 
to the alumina-silica ratio, and for these high-temperature applications the 
higher-alumina brick is preferred. 




Fig. 7.29. Mullite crystals in silica matrix formed by heating kaolinite (37,000*). Courtesy 
J. J. Comer. 




igram. 
it sub- 
above 
lurnina 
ised in 

minum 
s pres- 
)). The 
>n, and 
ck. At 
:nsitive 
ons the 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



307 



Refractory properties of brick can be substantially improved if suffi- 
cient alumina is added to increase the fraction of mullite present until at 
greater than 72 wt% alumina the brick is entirely mullite or a mixture of 
mullite plus alumina. Under these conditions no liquid is present until 
temperatures above 1828°C are reached. For some applications fused 
mullite brick is used; it has superior ability to resist corrosion and 
deformation at high temperatures. The highest refractoriness is obtained 
with pure alumina. Sintered A1 2 0 3 is used for laboratory ware, and 
fusion-cast AI2O3 is used as a glass tank refractory. 

7.9 The System MgO-Al 2 0 3 -Si0 2 

A ternary system important in understanding the behavior of a number 
of ceramic compositions is the MgO-Al 2 0 3 -Si0 2 system, illustrated in 
Fig. 7.30. This system is composed of several binary compounds which 



. Courtesy 



MgO-Al 2 0. 
(spinel) 
2135±20 



2030120* 



l 2 0 3 .2Si0 2 
(mullite) 



1810±10* 




2MgO-2Al 2 0 3 -5Si0 2 
(cordierite) 



MgO 
2800* ± 



2MgO*Si0 2 MgO-SiOa i*** 5 ' 
(forstertte) (enstatite) 
1890 + 40* 



171315* 



Fig. 7.30. The ternary system MgO-Al 2 0,-Si0 2 . From M. L. Keith and J. F. Schairer, J. 
GeoU 60, 182 (1952). Regions of solid solution are not shown; see Figs. 4.3 and 7.13. 



308 



INTRODUCTION TO CERAMICS 



have already been described, together with two ternary compounds, 
cordierite, 2Mg02Al 2 03*5Si0 2 , and sapphirine, 4MgO-5 Al 2 0 3 -2Si0 2 , both 
of which melt incongruently. The lowest liquidus temperature is at the 
tridymite-protoenstatite-cordierite eutectic at 1345°C, but the cordierite- 
enstatite-forsterite eutectic at 1360°C is almost as low-melting. 

Ceramic compositions that in large part appear on this diagram include 
magnesite refractories, forsterite ceramics, steatite ceramics, special 
low-loss steatites, and cordierite ceramics. The general composition areas 
of these products on the ternary diagram are illustrated in Fig. 7.31. In all 
but magnesite refractories, the use of clay and talc as raw materials is the 
basis for the compositional developments. These materials are valuable in 
large part because of their ease in forming; they are fine-grained, and 
platey and are consequently plastic, nonabrasive, and easy to form. In 
addition, the fine-grained nature of these materials is essential for the 



AI 2 0 3 




Fig. 7.31. Common compositions in the ternary system MgO-Al 2 O 3 -Si0 2 . See text for 
other additives. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



309 



ands, \ 
both 
t the 
;rite- 

:lude 
>ecial 
areas 
In all 
is the 
ble in 
1 and 
m. In 
»r the 



firing process, which is described in more detail in Chapter 12. On heating, 
clay decomposes at 980°C to form fine-grained mullite in a silica matrix. 
Talc decomposes and gives rise-to a similar mixture of fine-grained 
protoenstatite crystals, MgSi0 3 , in a silica matrix at about 1000°C. Further 
heating of clay gives rise to increased growth of mullite crystals, 
crystallization of the silica matrix as cristobalite, and formation of a 
eutectic liquid at 1595°C. Further heating of pure talc leads to crystal 
growth of the enstatite, and liquid is formed at a temperature of 1547°C. 
At this temperature almost all the composition melts, since talc (66.6% 
Si0 2 , 33.4% MgO) is not far from the eutectic composition in the 
MgO-Si0 2 system (Fig. 7.13). 

The main feature which characterizes the melting behavior of cordier- 
ite, steatite porcelain, and low T loss steatite compositions is the limited 
firing range which results when pure materials are carried to partial 
fusion. In general, for firing to form a vitreous densified ceramic about 20 
to 35% of a viscous silicate liquid is required. For pure talc, however, as 
indicated in Fig. 7.32, no liquid is formed until 1547°C, when the entire 
composition liquifies. This can be substantially improved by using talc- 
clay mixtures. For example, consider the composition A in Fig. 7.31 
which is 90% talc-10% clay, similar to many commercial steatite composi- 
tions. At this composition about 30% liquid is formed abruptly at the 
liquidus temperature, 1345°C; the amount of liquid increases quite rapidly 
with temperature (Fig. 7.32), making close control of firing temperature 
necessary, since the firing range is short for obtaining a dense vitreous 



Si0 9 



text for 




1200 1300 1400 1500 1600 1700 1800 
Temperature ("C) 

Fig. 7.32. Amount of liquid present at different temperatures for compositions illustrated 
in Fig. 7.31. 



3 



310 



INTRODUCTION TO CERAMICS 



body (this composition would be fired at 1350 to 1370°C). In actual fact, 
however, the raw materials used contain Na 2 0, K 2 0, CaO, BaO, Fe 2 0 3 , 
and Ti0 2 as minor impurities which both lower and widen the fusion 
range. Additions of more than 10% clay again so shorten the firing range 
that they are not feasible, and only limited compositions are practicable. 
The addition of feldspar greatly increases the firing range and the ease of 
firing and has been used in the past for compositions intended as 
low-temperature insulators. However, the electrical properties are not 
good. 

For low-loss steatites, additional magnesia is added to combine with the 
free silica to bring the composition nearer the composition triangle for 
forsterite-cordierite-enstatite. This changes the melting behavior so that 
a composition such as B in Fig. 7.31 forms about 50% liquid over a 
temperature range of a few degrees, and control in firing is very difficult 
(Fig. 7.32). In order to fire these compositions in practice to form vitreous 
bodies, added flux is essential. Barium oxide, added as the carbonate, is 
the most widely used. 

Cordierite ceramics are particularly useful, since they have a very low 
coefficient of thermal expansion and consequently good resistance to 
thermal shock. As far as firing behavior is concerned, compositions show 
a short firing range corresponding to a flat liquidus surface which leads to 
the development of large amounts of liquid over a short temperature 
interval. If a mixture consisting of talc and clay, with alumina added to 
bring it closer to the cordierite composition, is heated, an initial liquidus is 
formed at 1345°C, as for composition C in Fig. 7.31. The amount of liquid 
rapidly increases; because of this it is difficult to form vitreous bodies. 
Frequently when these compositions are not intended for electrical 
applications, feldspar (3 to 10%) is added as a fluxing medium to increase 
the firing range. 

Magnesia and forsterite compositions are different in that a eutectic 
liquid is formed of a composition widely different from the major phase 
with a steep liquidus curve so that a broad firing range is easy to obtain. 
This is illustrated for the forsterite composition D in Fig. 7.31 and the 
corresponding curve in Fig. 7.32. The initial liquid is formed at the 1360°C 
eutectic, and the amount of liquid depends mainly on composition and 
does not change markedly with temperature. Consequently, in contrast to 
the steatite and cordierite bodies, forsterite ceramics present few prob- 
lems in firing. 

In all these compositions there is normally present at the firing 
temperature an equilibrium mixture of crystalline and liquid phases. This 
is illustrated for a forsterite composition in Fig. 7.33. Forsterite crystals 
are present in a matrix of liquid silicate corresponding to the liquidus 



compositi 
phase at t 
and the c 
liquid ph£ 
(or a parti 
for fixing 
deduced ii 

7.10 Noi 

The kin 
in the nex 
in Chapte 
lowest en 
systems. ] 
free ener 
illustrated 
phases th 
equilibriu: 
solution o 
liquid anc 
phases in 
the comir 
slowness 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



311 



;tual fact, 
O, Fe 2 0 3) 
he fusion 
ring range 
acticable. 
he ease of 
ended as 
;s are not 

ie with the 
iangle for 
ior so that 
lid over a 
ry difficult 
m vitreous 
rbonate, is 




Fig. 7.33. Crystal-liquid structure of a forsterite composition (150x). 



a very low 
istance to 
tions show 
zh leads to 
mperature 
a added to 
liquidus is 
nt of liquid 
ms bodies. 

electrical 
to increase 

a eutectic 
iajor phase 
/ to obtain. 
31 and the 
the 1360°C 
osition and 
contrast to 
: few prob- 

the firing 
hases. This 
ite crystals 
he liquidus 



composition at the firing temperature. For other systems the crystalline 
phase at the firing temperature is protoenstatite, periclase, or cordiente, 
and the crystal size and morphology are usually different as well. The 
liquid phase frequently does not crystallize on cooling but forms a glass 
(or a partly glass mixture) so that the compatibility triangle cannot be used 
for fixing the phases present at room temperature, but they must be 
deduced instead from the firing conditions and subsequent heat treatment. 

7.10 Nonequilibrium Phases 

The kinetics of phase transitions and solid-state reactions is considered 
in the next two chapters; however, from our discussion of glass structure 
in Chapter 3 and atom mobility in Chapter 6 it is already apparent that the 
lowest energy state of phase equilibria is not achieved in many practical 
systems. For any change to take place in a system it is necessary that the 
free energy be lowered. As a result the sort of free-energy curves 
illustrated in Figs. 3.10, 4.2, 4.3, 7.7, and 7.8 for each of the possible 
phases that might be present remain an important guide to metastable 
equilibrium. In Fig. 7.8, for example, if at temperature T 2 the solid 
solution a were absent for any reason, the common tangent between the 
liquid and solid solution j3 would determine the composition of those 
phases in which the constituents have the same chemical potential. One of 
the common types of nonequilibrium behavior in silicate systems is the 
slowness of crystallization such that the liquid is supercooled. When this 



4 



312 INTRODUCTION TO CERAMICS 

happens, metastable phase separation of the liquid is quite common, 
discussed in Chapter 3. 

Glasses. One of the most common departures from equilibrium be- 
havior in ceramic systems is the ease with which many silicates are cooled 
from the liquid state to form noncrystalline products. This requires that 
the driving force for the liquid-crystal transformation be low and that the 
activation energy for the process be high. Both of these conditions are 
fulfilled for many silicate systems. 

The rate of nucleation for a crystalline phase forming from the liquid is 
proportional to the product of the energy difference between the crystal 
and liquid and the mobility of the constituents that form a crystal, as 
discussed in Chapter 8. In silicate systems, both of these factors change 
so as to favor the formation of glasses as the silica content increases. 
Although data for the diffusion coefficient are not generally available, the 
limiting mobility is that of the large network-forming anions and is 
inversely proportional to the viscosity. Thus, the product of AH//T mp and 
1/tj can be used as one index for the tendency to form glasses on cooling, 
as shown in Table 7.1. 



Table 7.1. Factors Affecting Glass-Forming Ability 


Compo- 




AHf/T mp 




(AH f /T mp ) X 




sition 


T mp (°C) 


(cal/mole/°K) 


(poise -1 ) 


(lA?)mp 


Comments 


B 2 0 3 


450 


7.3 


2 X 10" 5 


1.5 X 10~ 4 


Good glass 












former 


Si0 2 


1713 


1.1 


1 X 10~ 6 


1.1 X 10" 6 


Good glass 












former 


Na 2 Si 2 0 5 


874 


7.4 


5 X 10" 4 


3.7 X lO" 3 


Good glass 












former 


Na 2 Si0 3 


1088 


9.2 


5 X 10" 3 


4.5 X 10~ 2 


Poor glass 












former 


CaSiCh 


1544 


7.4 


lO" 1 


0.74 


Very diffi- 












cult to 












form as 












glass 


NaCl 


800.5 


6.9 


50 


345 


Not a glass 












former 



Metastable Crystalline Phases. Frequently in ceramic systems crystal- 
line phases are present that are not the equilibrium phases for the 
conditions of temperature, pressure, and composition of the system. 
These remain present in a metastable state because the high activation 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



313 



ite common, 

uilibrium be- 
es are cooled 
requires that 
' and that the 
editions are 

1 the liquid is 
n the crystal 
a crystal, as 
ctors change 
nt increases, 
ivailable, the 
lions and is 
4H//T mp and 
s on cooling, 



Comments 

Good glass 
former 

Good glass 
former 

Good glass 
former 

Poor glass 
former 

Very diffi- 
cult to 
form as 
glass 

Not a glass 
former 



;ms crystal- 
ies for the 
he system, 
i activation 



energies required for their conversion into more stable phases cause a low 
rate of transition. The energy relationships among three phases of the 
same composition might be represented as given in Fig. 7.34. Once any 
one of these phases is formed, its rate of transformation into another 
more stable phase is slow. In particular, the rate of transition to the lowest 
energy state is specially slow for this system. 

The kinetics of transformation in systems such as those illustrated in 
Fig. 7.34 are discussed in Chapter 9 in terms of the driving force and 
energy barrier. Structural aspects of transformations of this kind have 
been discussed in Chapter 2. In general, there are two common ways in 
which metastable crystals are formed. First, if a stable crystal is brought 
into a new temperature or pressure range in which it does not transform 
into the more stable form, metastable crystals are formed. Second, a 
precipitate or transformation may form a new metastable phase. For 
example, if phase 1 in Fig. 7.34 is cooled into the region of stability of 
phase 3, it may transform into the intermediate phase 2, which remains 
present as a metastable crystal. 

The most commonly observed metastable crystalline phases not under- 
going transformation are the various forms of silica (Fig. 7.5). When a 
porcelain body containing quartz as an ingredient is fired at a temperature 
of 1200 to 1400°C, tridymite is the stable form but it never is observed; the 
quartz always remains as such. In refractory silica brick, quartz used as a 
raw material must have about 2% calcium oxide added to it in order to be 
transformed into the tridymite and cristobalite forms which are desirable. 
The lime provides a solution-precipitation mechanism which essentially 
eliminates the activation energy barrier, shown in Fig. 7.34, and allows 




Fig. 7.34. Illustration of energy barriers between three different states of a system. 



314 



INTRODUCTION TO CERAMICS 



the stable phase to be formed. This is, in general, the effect of mineraliz- 
ers such as fluorides, water, and alkalies in silicate systems. They provide 
a fluid phase through which reactions can proceed without the activation 
energy barrier present for the solid-state process. 

Frequently, when high-temperature crystalline forms develop during 
firing of a ceramic body, they do not revert to the more stable forms on 
cooling. This is particularly true for tridymite and cristobalite, which 
never revert to the more stable quartz form. Similarly, in steatite bodies 
the main crystalline phase at the firing temperature is the protoenstatite 
form of MgSi0 3 . In fine-grained samples this phase remains as a metasta- 
ble phase dispersed in a glassy matrix after cooling. In large-grain samples 
or on grinding at low temperature, protoenstatite reverts to the equilib- 
rium form, clinoenstatite. 

A common type of nonequilibrium behavior is the formation of a 
metastable phase which has a lower energy than the mother phase but is 
not the lowest-energy equilibrium phase. This corresponds to the situa- 
tion illustrated in Fig. 7.34 in which the transition from the highest-energy 
phase to an intermediate energy state occurs with a much lower activation 
energy than the transition to the most stable state. It is exemplified by the 
devitrification of silica glass, which occurs in the temperature range of 
1200 to 1400°C, to form cristobalite as the crystalline product instead of 
the more stable form, tridymite. The reasons for this are usually found in 
the structural relationships between the starting material and the final 
product. In general, high-temperature forms have a more open structure 
than low-temperature crystalline forms and consequently are more nearly 
like the structure of a glassy starting material. These factors tend to favor 
crystallization of the high-temperature form from a supercooled liquid or 
glass, even in the temperature range of stability of a lower-temperature 
modification. 

This phenomenon has been observed in a number of systems. For 
example, J. B. Ferguson and H. E. Merwin* observed that when calcium- 
silicate glasses are cooled to temperatures below 1125°C, at which 
wollastonite (CaSi0 3 ) is the stable crystalline form, the high-temperature 
modification, pseudowollastonite, is found to crystallize first and then 
slowly transform into the more stable wollastonite. Similarly, on cooling 
compositions corresponding to Na 2 O Al 2 0 3 2Si0 2 , the high-temperature 
crystalline form (carnegieite) is observed to form as the reaction product, 
even in the range in which nephelite is the stable phase; transformation of 
carnegieite into nephelite occurs slowly. 

In order for any new phase to form, it must be lower in free energy than 
the starting material but need not be the lowest of all possible new phases. 

Mm. J. Science, Series 4, 48, 165 (1919). 



This requin 
the phase e 
diagram mi 
other phase 
precipitate. 
Fig. 7.35. 1 
difficulty s 
frequently 
potassium < 
eutectic ter 
which both 
the liquid c 
this system 
cristobalite 
rium quarts 
by the dott 
Extensio 



o 

°u 1( 



Fig. 7.35. 
system. 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



315 



mineraliz- 
iey provide 
; activation 

lop during 
e forms on 
lite, which 
tite bodies 
toenstatite 
a metasta- 
in samples 
he equilib- 

ation of a 
hase but is 
the situa- 
est-energy 
activation 
Red by the 
; range of 
instead of 
y found in 
1 the final 
> structure 
ore nearly 
id to favor 
J liquid or 
nperature 

terns. For 
1 calcium- 
at which 
nperature 
and then 
n cooling 
nperature 
i product, 
oiation of 

ergy than 
* phases. 



This requirement means that when a phase does not form as indicated on 
the phase equilibrium diagram, the liquidus curves of other phases on the 
diagram must be extended to determine the conditions under which some 
other phase becomes more stable than the starting solution and a possible 
precipitate. This is illustrated for the potassium disilicate-silica system in 
Fig. 7.35. Here, the compound K 2 04Si0 2 crystallizes only with great 
difficulty so that the eutectic corresponding to this precipitation is 
frequently not observed. Instead, the liquidus curves for silica and for 
potassium disilicate intersect at a temperature about 200° below the true 
eutectic temperature. This nonequilibrium eutectic is the temperature at 
which both potassium disilicate and silica have a lower free energy than 
the liquid composition corresponding to the false eutectic. Actually, for 
this system the situation is complicated somewhat more by the fact that 
cristobalite commonly crystallizes from the melt in place of the equilib- 
rium quartz phase. This gives additional possible behaviors, as indicated 
by the dotted line in Fig. 7.35. 

Extension of equilibrium curves on phase diagrams, such as has been 



1400 




56 60 
K 2 0-2Si0 2 



100 
Si0 2 



70 80 
K 2 0-4Si0 2 

Weight per cent Si0 2 

Fig- 7.35. Equilibrium and nonequilibrium liquidus curves in the potassium disilicate-silica 
system. 



316 



INTRODUCTION TO CERAMICS 



shown in Fig. 7.35 and also in Fig. 7.5, provides a general method of using 
equilibrium data to determine possible nonequilibrium behavior. It pro- 
vides a highly useful guide to experimental observations. The actual 
behavior jr^any^ysterri^may follow m of several possible ^courses, 
so that an analysis of the kinetics of these processes (or more commonly 
experimental observations) is also required. 

Incomplete Reactions. Probably the most common source of non- 
equilibrium phases in ceramic systems are reactions that are not com- 
pleted in the time available during firing or heat treatment. Reaction rates 
in condensed phases are discussed in Chapter 9. The main kinds of 
incomplete reactions observed are incomplete solution, incomplete solid- 
state reactions, and incomplete resorption or solid-liquid reactions. All of 
these arise from the presence of reaction products which act as barrier 
layers and prevent further reaction. Perhaps the most striking example of 
incomplete reactions is the entire metallurgical industry, since almost all 
metals are thermodynamically unstable in the atmosphere but oxidize and 
corrode only slowly. 

A particular example of incomplete solution is the existence of quartz 
grains which are undissolved in a porcelain body, even after firing at 
temperatures of 1200 to 1400°C. For the highly siliceous liquid in contact 
with the quartz grain, the diffusion coefficient is low, and there is no fluid 
flow to remove the boundary layer mechanically. The situation is similar 
to diffusion into an infinite medium, illustrated in Fig. 6.5. To a first 
approximation, the diffusion coefficient for Si0 2 at the highly siliceous 
be andary may be of the order of 10~ 8 to 1(T 9 cm 2 /sec at 1400°C. With these 
da a it is left as an exercise to estimate the thickness of the diffusion layer 
after 1 hr of firing at this temperature. 

The way in which incomplete solid reactions can lead to residual 
starting material being present as nonequilibrium phases will be clear 
from the discussion in Chapter 9. However, new products that are not the 
final equilibrium composition can also be formed. For example, in heating 
equimolar mixtures of CaC0 3 and Si0 2 to form CaSi0 3 , the first product 
formed and the one that remains the major phase through most of the 
reaction is the orthosilicate, Ca 2 SiC>4. Similarly, when BaC0 3 and Ti0 2 are 
reacted to form BaTi0 3 , substantial amounts of Ba 2 Ti0 4 , BaTi 3 0 7 , and 
BaTU09 are formed during the reaction process, as might be expected 
from the phase-equilibrium diagram (Fig. 7.20). When a series of inter- 
mediate compounds is formed in a solid reaction, the rate at which each 
grows depends on the effective diffusion coefficient through it. Those 
layers for which the diffusion rate is high form most rapidly. For the 
CaO-Si0 2 system this is the orthosilicate. For the BaO-Ti0 2 system the 
most rapidly forming compound is again the orthotitanate, Ba 2 Ti0 4 . 




thod of using 
avior. It pro- 
The actual 
iible courses, 
re commonly 

jrce of non- 
are not com- 
paction rates 
ain kinds of 
-mplete solid- 
ctions. All of 
ict as barrier 
ig example of 
ice almost all 
it oxidize and 

ice of quartz 
ifter firing at 
lid in contact 
^re is no fluid 
ion is similar 
5. To a first 
;hly siliceous 
2. With these 
iffusion layer 

i to residual 
will be clear 
at are not the 
>Ie, in heating 
first product 
most of the 
and Ti0 2 are 
BaTi 3 0 7 , and 
be expected 
ries of inter- 
t which each 
gh it. Those 
idly. For the 
^system the 
Ba 2 Ti0 4 . 



CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



317 




A AB 
Fig. 7.36. Nonequilibrium crystallization path with (1) Liquid - 
Liquid -> AB, (4) Liquid AB + B, (5) Liquid -* AB + B + C. 



-A, (2) A 



B 

- liquid - 



•AB,<3) 



A final example of nonequilibrium conditions important in interpreting 
phase-equilibrium diagrams is the incomplete resorption that may occur 
whenever a reaction, A + Liquid = AB, takes place during crystallization. 
This is the case, for example, when a primary phase reacts with a liquid to 
form a new compound during cooling. A layer tends to build up on the 
surface of the original particle, forming a barrier to further reaction. As 
the temperature is lowered, the final products are not those anticipated 
from the equilibrium diagram. A nonequilibrium crystallization path for 
incomplete resorption is schematically illustrated in Fig. 7.36. 



Suggested Reading 

1. E. M. Levin, C. R. Robbins, and H. F. McMurdie, Phase Diagrams for 
Ceramists, American Ceramic Society, Columbus, Ohio, 1964. 

2. E. M. Levin, C. R. Robbins, H. F. McMurdie, Phase Diagrams for Ceramists, 
1969 Supplement, American Ceramic Society, Columbus, Ohio, 1969. 

3. A. M. Alper, Ed., Phase Diagrams: Materials Science and Technology, Vol. I, 
"Theory, Principles, and Techniques of Phase Diagrams," Academic Press, 
Inc., New York, 1970; Vol. II, "The Use of Phase Diagrams in Metal, 
Refractory, Ceramic, and Cement Technology," Academic Press, Inc., New 



318 



INTRODUCTION TO CERAMICS 



York, 1970; Vol. Ill, "The Use of Phase Diagrams in Electronic Materials and 7.4. Dis 

Glass Technology," Academic Press, Inc., New York, 1970. refr 

4. A. Muan and E. F. Osborn, Phase Equilibria among Oxides in Steelmaking, ont 
Addison-Wesley, Publishing Company, Inc., Reading, Mass., 1965. and 

5. A. Reisman, Phase Equilibria, Academic Press, Inc., New York, 1970. imp 

6. P. Gordon, Principles of Phase Diagrams in Materials Systems, McGraw Hill 7.5. A t 



Book Company, New York, 1968. 

7. A. M. Alper, Ed., High Temperature Oxides, Part I, "Magnesia, Lime and 
Chrome Refractories," Academic Press, Inc., New York, 1970; Part II, 
"Oxides of Rare Earth, Titanium, Zirconium, Hafnium, Niobium/and Tan- 
talum," Academic Press, Inc., New York, 1970; Part III, "Magnesia, Alumina, 
and Beryllia Ceramics: Fabrication, Characterization and Properties," 
Academic Press, Inc., New York; Part IV, "Refractory Glasses, Glass- 
Ceramics, Ceramics," Academic Press, New York, Inc., 1971. 

8. J. E. Ricci, The Phase Rule and Heterogeneous Equilibrium, Dover Books 
New York, 1966. 



Problems 



7.1. 



7.2. 



7.3. 



(b) 
(c) 

W) 



A power failure allowed a furnace used by a graduate student working in the K 2 0- 
CaO-Si0 2 system to cool down over night. For the fun of it, the student analyzed the 
composition he was studying by X-ray diffraction. To his horror, he found 0-CaSiO 3 
2K I OCaO-3Si0 2 , 2K 2 O CaO-6Si0 2 , K 3 O-3Ca0-6SiO 2 , and K 2 02Ca06Si0 2 in his 
sample. 

(a) How could he get more than three phases? 

Can you tell him in which composition triangle his original composition was? 
Can you predict the minimum temperature above which his furnace was 
operating before power failure? 

He thought at first he also had some questionable X-ray diffraction evidence for 
K 2 OCaOSi0 2 , but after thinking it over he decided K 2 O CaO Si0 2 should not 
crystallize out of his sample. Why did he reach this conclusion? 
According to Alper, McNally, Ribbe, and Doman,* the maximum solubility of A1 2 0 3 in 
MgO is 18 wt% at 1995°C and of MgO in MgAl 2 0, is 39% MgO, 51% AI 2 0 3 . Assuming 
the NiO-Al 2 0 3 binary is similar to the MgO-Al 2 0 3 binary, construct a ternary. Make 
isothermal plots of this ternary at 2200°C, 1900°C, and 1700°C. 
You have been assigned to study the electrical properties of calcium metasilicate 
by the director of the laboratory in which you work. If you were to make the 
material synthetically, give a batch composition of materials commonly obtainable in 
high purity. From a production standpoint, 10% liquid would increase the rate of 
sintering and reaction. Adjust your composition accordingly. What would be the 
expected firing temperature? Should the boss ask you to explore the possibility of 
lowering the firing temperature and maintain a white body, suggest the direction to 
procede. What polymorphic transformations would you be conscious of in working 
with the above systems? 



An 

7.6. Tri 
terr 
(mi 
are 
for 
phi 

7.7. Fo! 
in 
for 
tioi 
riui 
stit 

7.8. If j 
to 
tha 

7.9. Th< 
Al 2 
ten 
tha 
for 



*/. Am. Ceram. Soc. 45(6), 263-268 (1962). 




CERAMIC PHASE-EQUILIBRIUM DIAGRAMS 



319 



terials and 
ze\ making, 
1970. 

Graw Hill 

Lime and 
; Part II, 
and Tan- 
Alumina, 
operties," 
s, Glass- 

er Books, 



7.4. 



the K 2 0- 
^alyzed the 
Q-CaSiO,, 
i0 2 in his 



tion was? 
nace was 

idence for 
ihould not 

Jf A1 2 0, in 
Assuming 
ary. Make 

etasilicate 
make the 
ainable in 
ie rate of 
Id be the 
sibility of 
rection to 
i working 



7.5. 



7.6. 



7.7. 



7.8. 



7.9. 



Discuss the importance of liquid-phase formation in the production and utilization of 
refractory bodies. Considering the phase diagram for the MgO-Si0 2 system, comment 
on the relative desirability in use of compositions containing 50MgO-50SiO 2 by weight 
and 60MgO-40SiO 2 by weight. What other characteristics of refractory bodies are 
important in their use? 

A binary silicate of specified composition is melted from powders of the separate 
oxides and cooled in different ways, and the following observations are made: 





Condition 


Observations 


(a) 


Cooled rapidly 


Single phase, no evidence of 






crystallization 


(b) 


Melted for 1 hr, held 


Crystallized from surface with 




80°C below liquidus 


primary phases Si0 2 plus glass 




for 2 hr 




(c) 


Melted for 3 hr, held 


Crystallized from surface with 




80°C below liquidus for 


primary phases compound AO Si0 2 




2hr 


plus glass 


(d) 


Melted for 2 hr, cooled 


No evidence of crystallization 




rapidly to 200°C below 


but resulting glass is cloudy 




liquidus, held for 1 hr, 






and then cooled rapidly 





Are all these observations self consistent? How do you explain them? 
Triaxial porcelains (flint-feldspar-clay) in which the equilibrium phases at the firing 
temperature are mullite and a silicate liquid have a long firing range; steatite porcelains 
(mixtures of talc plus kaolin) in which the equilibrium phases at the firing temperature 
are enstiatite and a silicate liquid have a short firing range. Give plausible explanations 
for this difference in terms of phases present, properties of phases, and changes in 
phase composition and properties with temperature. 

For the composition 40MgO-55SiO2-5Al 2 O 3 , trace the equilibrium crystallization path 
in Fig. 7.30. Also, determine the crystallization path if incomplete resorption of 
forsterite occurs along the forsterite-protoenstatite boundary. How do the composi- 
tions and temperatures of the eutectics compare for the equilibrium and nonequilib- 
rium crystallization paths? What are the compositions and amounts of each con- 
stituent in the final product for the two cases? 

If a homogeneous glass having the composition 13Na 2 0-13CaO-74Si0 2 were heated 
to 1050°C, 1000°C, 900°C, and 800°C, what would be the possible crystalline products 
that might form? Explain. 

The clay mineral kaolinite, AI 2 Si 2 0 3 (OH)„ when heated above 600°C decomposes to 
Al 2 Si 2 0 7 and water vapor. If this composition is heated to 1600°C and left at that 
temperature until equilibrium is established, what phase(s) will be present. If more 
than one is present, what will be their weight percentages. Make the same calculations 
for 1585°C. 




9 



Reactions with 
and between 
Solids 

In heterogeneous reactions there is a reaction interface between the 
reacting phases, such as nucleus and matrix or crystal and melt. In order 
for the reaction to proceed, three steps must take place in series- 
material transport to the interface, reaction at the phase boundary, and 
sometimes transport of reaction products away from the interface. In 
addition, reactions at the phase boundary liberate or absorb heat, chang- 
ing the boundary temperature and limiting the rate of the process. Any of 
these steps may determine the overall rate at which a heterogeneous 
reaction takes place, since the overall reaction rate is determined by the 
slowest of these series steps. 

In this chapter we consider these rate-determining steps as applied to 
changes taking place in ceramic systems. Decomposition of hydrates and 
carbonates, solid-state reactions, oxidation, corrosion, and many other 
phenomena must be considered on the basis of limitations imposed by the 
rates of phase-boundary reactions, material transport, and heat flow. 

9.1 Kinetics of Heterogeneous Reactions 

Reaction Order. Classical chemical-reaction kinetics has been mainly 
concerned with homogeneous reactions and cannot be directly applied to 
many phenomena of particular interest in ceramics, but it provides the 
basis for understanding rate phenomena. Reaction rates are frequently 
classified as to molecularity— the number of molecules or atoms formally 
taking part in the reaction. Overall reactions are also commonly classified 
as to reaction order — the sum of the powers to which concentrations c,, 
c 2> and so on, must be raised to give empirical agreement with a rate 
equation of the form 

^^KcfcfcS- (9.1) 



381 



382 



INTRODUCTION TO CERAMICS 



In a first-order reaction, for example, 




(9.2) 



On integration this gives 



ln± = K(t-t 0 ) 



(9.3) 



where K is a constant and c 0 is the initial concentration at time t 0 . For the 
simplest overall reactions which involve but one elementary step, the 
order and molecularity are the same. For more complex reactions which 
consist of several consecutive elementary steps involving different 
species and for heterogeneous reactions in general, the molecularity and 
order are quite different, and characterization by reaction order is a 
purely formal empirical method. In fact, zero and fractional reaction 
orders are sometimes found. Although the reaction-order concept is 
useful as a means of representing data for heterogeneous reactions, these 
cannot usually be interpreted simply in terms of molecular interactions. 

Activation Energy and Reaction Rate. The effect of temperature on the 
rate of processes taking place is frequently great. The historical basis for 
its understanding is the Arrhenius equation, in which it was found that for 
many processes the specific reaction-rate constant could be related to 
temperature by the relation logK-l/T or, alternatively, K = 
A exp(-Q/£7), where Q is the experimental activation energy. The 
basis of this relationship in a general theory of rate processes has been 
discussed in Chapter 6 in connection with diffusion as an activated 
process. In general, an activation energy is required for each of the steps 
involved in an overall rate process (Fig. 9.1). 

Two general considerations are the basis for interpreting most kinetic 
data. The first of these is that each individual step in a rate process must 
be relatively simple and that the reaction path of each step, such as an 
individual atom jump in diffusion, a molecular decomposition, or a new 
chemical bond being formed, involves an activated complex or transition 
state of maximum energy along the reaction path. Of all possible parallel 
paths of reaction, the one with the lowest energy barrier is the most rapid 
and the major contributor to the overall process. This activated-complex 
theory has provided a general form of equation for rate processes and a 
model that allows semiempirical calculations for simple processes. The 
second general consideration has been that the overall rate of a complex 
process involving a series of consecutive steps is fixed by the rate of the 
slowest individual step. 
If we plot energy along a distance coordinate corresponding to the 




1.1 

1.0 
.9 

~S .8 

% 

Cl 

E .7 
o 
o 

I -6 

£ .5 
*o 

o *^ 

2 3 



— m — i — i — i — i — i i 1 1 












/ \ 

a 7 / 






" Rio s 


^-^^"^ Rib 


L_ 


1 1 1 1 1 i J — 1 — 


1 2 


3 4 5 6 7 8 9 10 1 
Relative time 




Reaction distance *■ 

<M 

Fia 9 1 Schematic representation of (a) multipath process in which each path contains 
T vera step ^process is dominated by the fastest path (path3).(6)Mult,ste P P ath,n wh.ch 

eachstX^ 
step. . 



383 



384 



INTRODUCTION TO CERAMICS 



reaction path of lowest energy between the reactant and product, there is 
an energy maximum, actually a saddle point, corresponding to the 
activated complex or transition state, such as discussed for diffusion in 
Chapter 6. This concept of an activated complex has been generally 
accepted as the basis for reaction-rate studies and, as discussed in 
Chapter 6, leads to a specific reaction-rate constant given by 

where k is the Boltzmann constant, h the Planck constant, and Aif f and 
AS* the enthalpy and entropy of activation, respectively. Individual 
reaction steps in an overall reaction process are usually simple and may 
be designated as monomolecular or bimolecular. Semiempirical treat- 
ments of the unit steps on the basis of activated-complex theory allow a 
rational theoretical approach to reaction processes. 

Complex Processes. Overall processes are frequently complex and 
require a series of individual separate unit steps. In such a sequence the 
rate of any individual step depends on the specific reaction-rate constant 
and the concentration of the reactants for this step. For a series of 
consecutive steps, 

A x = A 2 = •■• Ai = A i+l = A n (9.5) 

We can define a virtual maximum rate for each step as the rate that would 
be found if equilibria were established for all previous and following 
steps. Under these conditions the reaction with the lowest virtual 
maximum rate controls the overall rate if it is much lower than the rates of 
other steps. Under these conditions equilibrium will have been virtually 
established for all previous steps but will not necessarily be established 
for the following steps. As shown schematically in Fig. 9.1, the reaction 
rate for path 1 is determined by step lb of the process; it has the slowest 
rate and the largest activation energy barrier and accounts for 85% of the 
reaction time; steps la and lc occur more rapidly. Reaction step R la will 
be slowed to a rate giving a virtual equilibrium concentration of products; 
reaction R ic will be slowed because R lb is producing few reactants for 
step R lc . 

We have already noted that most condensed-phase processes of 
interest in ceramics involve heterogeneous systems; changes take place at 
a phase boundary. The overall process involves (1) transport of reactants , 
to the phase boundary, (2) reaction at the phase boundary, (3) transport of 
products away from the phase boundary. This series of reaction steps has 
relatively simple kinetics, provided the virtual maximum rate of one step 
is much slower than that of any of the other steps. If we assume this to be 



REACTIONS WITH AND BETWEEN SOLIDS 



385 



the case, we have two general classes of heterogeneous reactions: (1) 
those controlled by transport rate and (2) those controlled by phase- 
boundary reaction rate. In general, both the transport process and the 
phase-boundary process involve a number of individual steps, one of 
which has the lowest virtual maximum rate. In going from reactants to 
products, there may be several possible reaction paths for transport 
processes and for phase-boundary reactions. There are three different 
possible reaction paths shown in Fig. 9- la. 

9.2 React ant Transport through a Planar Boundary Layer 

Slip Casting. As an example of the usefulness of determining the 
rate-limiting step for deriving kinetic equations, we begin with the 
ceramic processing technique of slip casting, in which a slurry containing 
clay particles dispersed in water is poured into a plaster of paris (gypsum) 
mold which contains fine capillaries (see Fig. 11.36) that absorb water 
from the slip. This causes a compact layer of clay particles to form at the 
mold-slip interface (Fig. 9.2). The rate of the process is determined by the 



Slip 

Suspended clay 
particles 



o 



O o 

0 



CD 

0 



O 

Oo 

& -CD 



o 



Cast layer 




Plaster of paris 
mold 



-Capillaries 



Pm 




Fig. 9.2. Schematic representation of the formation of a slip-cast layer formed by the 
extraction of water by capillary action of a plaster of paris mold. 



386 



INTRODUCTION TO CERAMICS 



transport of water out of the slip and into the capillaries; the rate-limiting 
step is the flow of water through the compact clay layer. As the layer 
thickness increases, the overall rate of material transport decreases 
because of the increased permeation distance (similar to gas permeation 
through glasses, discussed in Chapter 6). 
We begin by writing the flux equation for water, 

> = (9-6) 

where we assume a planar deposit (unidirectional flow) and that the water 
flux / H2 o is proportional to the pressure gradient resulting from the 
capillaries of the plaster mold. The permeation coefficient K depends on 
the clay particle size, particle packing, and the viscosity of water and is 
temperature-sensitive. The water pressure in the slip, P s , is 1 atm; in the 
mold, P mt it is determined by capillarity, AP = P s - P m =2y/r (Chapter 
5). The surface tension is a function of the deflocculating agents used. 
Until the capillaries become filled with water, AP is approximately 
constant, and the flux can be related to the change of the layer thickness 
dx\dt y 



\Kp/dt dx x rx 



(9.7) 



where p is the density of the cast layer and k is a factor for converting the 
volume of water removed to the volume of clay particles deposited. 
Integration of Eq. 9.7 gives 



1/2 

>l/2 



or in the general parabolic form 

x = (K'ty n ( 9. 8) 

That is, the wall thickness of a planar casting should increase with the 
square root of time (Fig. 9.3). 

This parabolic rate law is commonly observed for kinetic processes in 
which the limiting step is mass transport through a reaction layer. 

Interdiffusion between Solids. In Section 6.3, we discussed the chemi- 
cal diffusion coefficient and its formulation in terms of the tracer diffusion 
coefficients for the case of interdiffusion. If we measure the rate at which 
two ceramics interdiffuse, this too can be considered the formation of a 
reaction product which is a solid solution rather than a distinct or separate 
phase. Let us consider the interdiffusion between crystals of NiO and 
CoO at a high temperature. The solid solution that forms is nearly ideal; 



REACTIONS WITH AND BETWEEN SOLIDS 



387 



rate-limiting 
Vs the layer 
t decreases 
permeation 



(9.6) 

t the water 
from the 
epends on 
iter and is 
tm; in the 
(Chapter 
*nts used. 
>ximately 
thickness 



(9.7) 

rting the 
^posited. 



(9.8) 
ith the 

sses in 
r. 

:hemi- 
Fusion 
which 
n of a 
karate 
and 
ideal; 




Fig. 9.3. 
mold. 



Time2(min)2 m 



(concentration) by in , = v- ■ +K 1 ln ? c ' w " 
equal to one. Thus Eq. 6.41 becomes 

This is the familiar Darken ^J^SS^SSSS. 
everywhere in .he interdiffusror, zone and ,s no ^ „ f p 

£51= atKSSfJS- spec, moves 

in Fig. 9.4« ^ The curves ; or ,th plot behavior . In 

diffusivities (F,g£4b) 9 ^ U ^. 9 is no t directly applicable 

the case of interdiffusion of NiO-MgO, ttj. *J concent ration. 
because the tracer diffusivities are a t anct on of the mete ^ 
The experimental interdiffus.on coefficenj ^^ mckd and 
dependence on the concentrate of n eke! jnva. ^ 
cation vacancies become associated /sections were 
increase the tran = o dominate the 

made in air, so that there is sumu * { the vac ancies arise 

cation vacancy •—,•"^££3 Ts^ions 4.7 and 6.4, a 
from the presence of Nw M uis^ vacancies are 

significant fraction of the trivalent n,cke ■»« '"^^ ^ nown in Fig . 
coupled by association. Data for some other systems 

"Next ietus consider a reaction in a— 

reaction iayer, <°'™^ 0 ^TJy ^ -tion paths; 



10" 



10" 



E 

u 

'5 



io~ y 



10" 



bill 



I'll 

- Calculated D 
Experimenta 



T3 




1.0 



J I L 



0.5 

c in (Co c Ni !_,.)() 
(a) 



10 



= 1 I 1 I 1 U I i m | r 




0.8 0.6 0.4 0.2 0 
c in(Co ( .Ni,_ c )0 

vl^at ^S^S^-^IST in,e S fUSi0 " ? - experi m e„ t a, 

(r„ m; \n . . V. ( } Tracer dlffus '°n coefficients of "Co and "Ni in 

^ 7 ^ CrySta ' S at 1445 3nd ' 300 ° C in air - Plolted l°8 D versus c From W K rL- 
and N. L. Petersen, J. Pfcy S . C /,em. Sac. 34. 1093 (1973). Che " 



388 




389 



Reaction occurs at AB 2 0 4 -B 2 0 3 interface: 
oxygen gas phase transport with A 2+ ion and 
electron transport through AB 2 0 4 : 

A 2+ + 2e ~ + Iq 2 + B 2 0 3 = AB 2 0 4 



Reaction occurs at AO-AB 2 0 4 interface: 
oxygen gas phase transport with B 3+ ion and 
electron transport through AB 2 0 4 : 

AO + 2B 3+ +6e+|o 2 = AB 2 0 4 



AO I AB 2 Or 



AO 



2c- 



|0 2 



AB,0 4 



I. 2B' 



6e- 



B,0 3 



might I 
the tra 
interfa 
Whe 
throug 
9.8), in 
Figure 
two d: 
planar 
when 
sed in 



Oxygen and cation transport through AB 2 0 4 : 

(1) Both cations diffuse (j B >* = |/ A ^. 

Reactions occur at 

AO-AB 2 0 4 interface 
2B 3+ +4AO = AB 2 0 4 +3A 2+ 

and at 

AB 2 0 4 -B 2 0 3 interface 
3A 2+ +4B 2 0 3 = 3AB 2 0 4 +2B 3+ 

(2) A 2+ and O 2 " diffuse. 
Reaction at 

AB 2 0 4 -B 2 0 3 interface 
A 2+ +0 2 +B 2 0 3 - AB 2 0 4 

(3) B 3+ and O 2 " diffuse. 
Reaction at 

AO-AB 2 0 4 interface 
AO+2B 3+ +30 2 = AB 2 0 4 



AO 



AB,0, 



2B :,+ 



3A' + 



2B :1 



30- 



B,0 :: 



0) 



<2> 



(3» 



Fig. 9.6. Schematic representation of several mechanisms which may control the rate of 
AB 2 0 4 (e.g., spinel) formation. From Ref. 1. 



Fig. 9 

COUpll 

(1966) 

Ti 

defe 

*/ 



390 



REACTIONS WITH AND BETWEEN SOLIDS 391 

might be controlled by the diffusion of A 2+ ions; B 3+ ions or O 2 " ions, by 
the transport of electrons (holes), by the transport of 0 2 gas, or by the 
interface reactions at A0-AB 2 O 4 or AB 2 04-B 2 0 3 . 

When the rate of reaction-product formation is controlled by diffusion 
through the planar product layer, the parabolic rate law is observed (Eq. 
9.8), in which the diffusion coefficient is that for the rate-limiting process. 
Figure 9/7 shows the parabolic time dependence for NiAl 2 0 4 formation at 
two different temperatures, and Fig. 9.8 is a photomicrograph of the 
planar spinel reaction product on A1 2 0 3 . (More complex situations arise 
when several phases are formed as reaction products. These are discus- 
sed in reference 1 and by C. Wagner *) 




Time (Hr) 

Fig 9 7 Thickness of NiAl 2 0« formed in NiO-Al 2 0, couples as a function of time for 
couples heated in argon at 1400 and 1500°C. From F. S. Pettit et aL, I Am. Ceram. Soc.,49, 199 
(1966). 

The Electrochemical Potential in Ionic Solids/ When considering point 
defects (Chapter 4) and atom mobility (Chapter 6), we noted that a 



*Acta Met., 17, 99 (1969). 



392 




INTRODUCTION TO CERAMICS 




SPINEL 



Al 2 0 3 



20 Ji. 



Fig. 9.8. Cross-sectional view of a tVDi'cal NiAl n j - 

after 73 hr a, I400°C. From F. S PeS e aT } kl r, c M U[0 ~ M ^ "Me 

o. reim, et ai., J. ,4m. Ceram. Soc, 49, 199 (1966). 

distinguishing feature of ionic crystals is the effective charge that an 
atom,c specie may have within the crystal lattice. When there is mass 
transport ma ceramic, the transport of one charged spede fs usZ y 
coupled to the transport of an ion or defect of the opposite charge We 
mus therefore consider the electrochemical potentials the ZoLZl 
caZ , V" 1385 t I anSP ° rt rathCr than J' ust the ch ^ical potential or 
is h TlTrtTT The , e,eCtrochemi -I Potential of the £h specie" 
acting onl P ° tent ' al * and the dectrical P° te "^' * 

V: = M. + Z,F<I> (9 , 0) 
where Z, is the effective charge and F is the Faraday constant We have 
already noted ,n Table 6.1 the interrelationship between he mobimies 
expressed ,n terms of electrical and chemical driving forces The flux due 
to an electrochermcal potential gradient is thus given by ' hemJXdUe 

(9,1) 

^^^ n ^Z^^ Ltmr °" Sh ° WS ^ 
oppositely charged species. For example, ^^sVls^Z 



packed oxid 
NiO-CoO ii 
case in whi 
coefficient r 
ions in alur 
motion of P 
(9.11) are n< 
Oxidation 
law in whicl 
reaction pre 
metals. The 
begin with 1 
results exte 
coherent o>: 
Poa* and thf 
determined 
of the oxid; 



The oxygen 
layer (Fig. \ 
the metal-o 
metal ion ii 
the oxygen 
also a net f 
determines 
cations and 
and then \ 
determining 
The flux 
changed to 




:ouple 



it an 
mass 
ually 
. We 
ating 
il or 
;ie r]i 
ial <f> 

9.10) 

have 
lities 
due 



J.ll) 

the 
ition 
y an 
tion. 
/een 
ose- 





REACTIONS WITH AND BETWEEN SOLIDS 



393 



nacked oxides diffuse more rapidly than oxygen, as for the NiO-MgO and 
E>-CoO interdiff usion already discussed. If this begins to happen in the 
case in which there is a net mass flow (not for the case of diffusion 
coefficient measurements using radioactive tracers) for example, Al 
ons in alumina, a net electrical field results and thereby couples the 
motion of Al +3 ions and O" 2 ions. Several solid reactions based on Eq. 
(9 11) are now considered. 

Oxidation of a Metal. The most extensive stud.es of the parabolic rate 
law in which the process is controlled by diffusive transport through the 
eaction product are investigations into the formation of ox.de layers or, 
metals. The analysis techniques were developed by Carl Wagner which 
beein with Eq. (9.1 1). They are described here in some detail because the 
results extend to many ceramic problems. Consider the format.on of a 
coherent oxide layer on a metal where the ambient oxygen pressure is 
Po« and the effective oxygen pressure at the ox.de-metal interface P 0j is 
determined by the temperature and the standard free energy of format.on 
of the oxidation reaction (see Fig. 9.9): 



2Me + 0 2 = 2M eO A G?ormalion 

Po > =e ^-mr 



(9.12) 



The oxygen concentration gradient (chemical potential) across the ox de 
layer (Fig 9.10) provides the driving force for oxygen diffus.on towards 
tne metal-oxide interface. A gradient of the chemical potential of the 
metal ion in the opposite direction produces meta Uon diffusion toward 
the oxygen atmosphere. If one atomic flux .s larger than the other there .s 
also a net flux of electrons or electron holes. The net transport which 
determines the rate of oxide growth, is the sum of fhn > o ^ 
cations and electrons or holes. First we must consider each of these fluxes 
and then we shall look for circumstances when one specie is rate- 
determining and the complex relationships reduce to more simple oms 

The flux of the atomic and electronic species given by Eq. 9 11 can be 
changed to the flux of charged particles by multiplying by the valence: 

/o=-|Zo|CoBo^ 2 =|Zo|/o 



(9.13) 




Fig. 9.9. Standard free energy of formation of oxides as a function of temperature. From F. 
D. Richardson and J. H. E. JefTes, J. iron Steel Inst,, 160, 261 (1948); modified by L. S. Darken 
and R. W. Gurry, Physical Chemistry of Metals, McGraw-Hill Book Company, New York 
1953. 



394 



REACTIONS WITH AND BETWEEN SOLIDS 



395 




For a given oxide layer, either electrons or holes are predominant, so that 
only one of the last two equations is necessary. The constraint on the net 
flux is electrical neutrality. If we assume electrons to be the important 
electronic defect, this constraint requires that 

J. + /.. = /„. ( 914 ) 

The net flux and therefore the rate of oxidation is the sum /<» = |/o| + |/m«|- 
The general result can be expressed in terms of the conductivity a and the 
transference number t, (see also Chapter 17), which represents the 
fraction of the total charge flux carried by a particular specie; 



/o *~|Z M «|F 2t ' 0 'I 9x 
~\Zj[F Ate+tMc) \dx\ 



(9.15) 



Although the composition varies through the layer, average values can be 
assumed for t f and a to simplify the result, which yields a form of the 



396 INTRODUCTION TO CERAMICS 

parabolic rate law; 

dx__ 1 I" 51.. - - 1 x 

dt x [\Z Mt \F 2 ( '° + 'Mc)|A/x Me | J = - 



(9-16) 



Recalling that, 



t i a = ( r i ^eZ i c i „ i =^fl Di 



(9.17) 

we can see that the oxidation rate is governed by the atomic mobilities or 
diffusivities. Let us now consider specific rate limiting cases- 

1. The electrical current is carried primarily by electronic defects 
U-~ 1 or t h = I: 

a. If D 0 >D Mc , then 



' \Z Mc \F 2 



Me 



(9.18) 



which for an oxide for predominant W defects reduces to the approxi- 
mation FF 



(9.19) 

since = 1/2^ = 1/ 2(m 4 + J?T In P 0l ). If we assume that [ V 0 "l <* P -« 
as d.scussed in Chapter 6, and that the diffusion coefficient varies' 
similarly; 



(9.20) 



and the rate constant becomes 



(iVr" 6 } 



(9.21) 

b. If D Me §> Do and we assume singly charged metal vacancies Vl> the 
rate constant is 



(9.22) 



* ~ mv^Dyjap^r - (/» 0j ')'« }Cm . 

note that D v ^{ V^] = D Me . 

2. If the electral current is carried primarily by the ions, (f„+ f„ c ) - 1 

the rate constant from Eq. 9.16 becomes •««/•, 

^"Siz^pjj^^ 1 "^ (9.23) 

where a-., is the conduction due to electrons and holes which have 
mobilities ft. and ti h , respectively (see Table 6.1 /x, = B"). 



If v 

OVf 



An 
tra 
sio 
Th 
foi 

OX 
OX 
CO 

[V 
th- 



If 

de 



REACTIONS WITH AND BETWEEN SOLIDS 



397 



(9.16) 

(9.17) 
lobilities or 
ic defects, 

(9.18) 
e approxi- 

(9.19) 

] « Po;' 5 , 
nt varies 



(9.24) 



If we assume that the defect concentration does not have a large variation 
over the oxide layer 



K 



'&\Z Me \e 2 



[In Po/~ln Po,'] 



(9.25) 



An example of the applicability of this relationship is the diffusive 
transport of oxygen through calcia-stabilized zirconia. The oxygen diffu- 
sion coefficient plotted in Fig. 6.11 is very large and accounts for f 0 ~ 1. 
Thus, the slower-moving specie, the electron hole, becomes rate-limiting 
for oxygen permeation (Eq. 9.25), as shown in Fig. 9.11. 

3. If the metal undergoing oxidation has an impurity with a different 
oxidation state, for example, Li in Ni, the defect concentration in the 
oxide may be determined by the impurity concentration. As an example, 
consider the analogous case to Eq. 9.22 for which D Mc >D 0 but where 
[ Vm c ] = [Fm c ]. The thickness of this extrinsic layer is again determined by 
the parabolic rate law, Eq. 9.16, but with the reaction constant, 



K cx = 2D wJ Vie][In Po 2 * "I" ^>„ e 



(9.26) 



If the impurity concentration and oxygen pressure are such that the 
defect concentrations are in an intermediate range, an intrinsic layer may 



(9.20) 

(9.21) 
VU„ the 

(9.22) - 

*M C )~1, 

(9.23) 
ch have 



Temperature <°C) 
1600 1400 1200 1000 800 



600 




0.8 

l/rnTx lo^) 

Fig. 9.1 1 . Oxygen permeation through calcia-stabilized zirconia as a function of tempera- 
ture. The oxygen transport is controlled by the concentration and mobility of electron holes, 
Eq. 9.25. From K. Kitazawa, Ph.D. thesis, MIT, 1972. 



398 



INTRODUCTION TO CERAMICS 



form on the oxygen-rich side (external) and an extrinsic oxide layer on the 
metal-rich side (at the oxide-metal interface) 

Short-circuit Diffusion Paths. In each of the examples of metal 
ox.dat.on, latt.ce d.ffusion D, was assumed to be the rate-determining 
transport process. In Section 6.6 the importance of other more rapid 
d.ffus.on paths was discussed. The effects of short-circuit paths can be 
incorporated Int0 the parabolic rate equations. For example, an apparent 
d,ffus.v.ty D from Eq. 6.67 can be used in Eq. 9.16 to include the 
contr.but.ons from lattice D, and boundary diffusion D„ ; 



D a = D,(l-f) + fD b 

dx_ K'D a 
dt x 



(9.27) 



where the diffusion coefficient has been extracted from the rate constant 
to . g,ve another constant K'. Low-temperature oxidation and oxide layers 
with .fine gram s.zes are expected to form by boundary diffusion 

Chem.cal Diffusion in Nonstoichiometric Oxides. The chemical diffu- 
sion coefficient for the counter diffusion of cations and anions can also be 
determined from the Wagner analysis. If we assume that electrical 
conduction ,s mainly electronic 1) that is, movement of electrical 
charge ,s not the rate-limiting step for mass transport (ions), the chemical 
d.ffus.on coefficient D can be determined. In terms of diffusion coeffi- 
cients rather than transference numbers Eq. 9.15 becomes 

J~ = ^\Z Mc \D, + \Z 0 \D Mc )±<^ (9 . 28) 
In terms of Fick's first law this can be rewritten 

/o ' = [ (|Z - |D ° + ,Z ^ra(^t)]f^f *»> 
where c represents the excess (or deficit) of the metal or oxygen in the 
nonsto.ch.ometnc compound. The chemical diffusion coefficient is the 
bucketed ^Consider, for example, the transition metal monoxide! 
iff.'", ' k- ' Co, : s °- • ) f or which c«[V M .--j, where « is the 
effect.ve charge on the vacancy and where D Mc >D 0 The chemical 
d.ffus,on coefficient can be written from Eq. 9.29 n the form 



D M ^ d ln , P °L. 



2[V„/"]" M < d"ln[V Me -] 



(9-30) 



SZl ,iCH t ! ,e k SUbstitUtion " 2 *™ 1" Po, has been made. From the 

defect equ.hbr.um reaction, the mass action law gives 



1/2 



(9.31) 





r on the 

- metal 
rmining 
e rapid 
can be 
pparent 
ide the 



(9.27) 



onstant 
; layers 
1. 

il diffu- 
also be 
ectrical 
ectrical 
lemical 
coeffi- 



(9.28) 



(9.29) 

i in the 
: is the 
loxides 
is the 
lemical 



(9.30) 
om the 

(9.31) 



REACTIONS WITH AND BETWEEN SOLIDS 399 

The derivative in Eq. 9.30 can now be determined; 

_dJnPo^ = 2(a + 1) (9.32) 
d\n[V M< T] 

Substituting this into Eq. 9.30 and recalling that c Mc D Mc = c v D v , the 
chemical diffusion coefficient is given by 

£ = (a+l)Dv M - (933) 
Thus for singly charged vacancies, D = 2Dv Me > and for doubly charged 
T^^ssure is changed from one value to another * .new 

controlled changes in the composition. The value of the cnem 



1100°C 



-7.0 




• 1.125 to 1.050 
a 1.125 to 1.075 
■ 1.125 to 1.100 



.72 ts *r7T7 84 

Temperature, lO'VT < K > 

several composition increments. From R. L. Lev.n and J. tl- vv g 
(1965). 



400 



INTRODUCTION TO CERAMICS 



sion coefficient correlates with the tracer value (Eq. 9.30) when the defect 
equilibrium relationships are known. 

Ambipolar Diffusion. The formality used to derive Eqs 9 11 and 9 13 
also allows us to determine the effective diffusion constants when cations' 
and anions are flowing in the same direction. Referred to as ambipolar 
diffusion, a description of the atomistic process must again consider the 
coupling between the oppositely charged species when the transport of 
electrons and holes is slower than ion transport. If the flux of cations 
becomes excessive, a local internal electric field builds up to "drag" along 
the anions. This behavior is important in processes involving reactions 
which cause product formation, in processes which are in response to an 
applied electric field, and in processes which result in a shape change due 
to mechanical or surface tension forces such as sintering and creep. 

As an example, consider a pure oxide for which f eI = 0. Equation 9.11 
can be written for anion and cation transport as in Eq. 9.13. Since the 
transport of each ion is in the same direction, electrical neutrality is 
maintained when 

jT -jz^rm 0-34) 

where J T refers to the total molecular flux. Equating the anion and cation 
charge flux allows for the solution of the internal electric field, Idx, 

|Z 0 |c oBo [^ + Z 0 Ff] = |Z M£ |c MeBMe [fe + z Me F|f] (9 .35) 

in terms of the chemical potential of the oxide, ft (Me^J. The chemical 
potential of the oxide is the sum of the chemical potentials of cations and 
anions, 

dfi. (Me^0 2M .) = Z Mc dft 0 + Z 0 dn Mc (9 36) 

[\Z 0 \c 0 B 0 -\Z Mc \c Mt B»J A f . n 
dx F[Z Mc \Z Mc \c Mc B Mc -Z 0 \Z 0 \c 0 B 0 ]~jf < 9 - 37 > 

tTn h n r n e f W p ha Q V ! 7 aSS , Um ^ d l0c o al equilibrium. |Z Me | dc Mc = |Z„| dc 0 . Substitu- 
tion of Eq. 9.37 into Eqs. 9.34 and 9.13 yields 

Z Mc \Z Mc \c Mc B Mc - Z 0 \Z 0 \c 0 B 0 Jx—^ 1 (9-38) 

This term is the correction due to ambipolar effects to the diffusion 
transport resulting from a chemical potential gradient. Consider as an 



i defect 

nd 9.13 
cations 
ibipolar 
der the 
port of 
cations 
along 
^actions 
se to an 
nge due 
reep. 
on 9.11 
nee the 
alky is 



(9.34) 

1 cation 
dx, 

(9.35) 

hemical 
ons and 

(9.36) 
>ns, and 



(9.37) 
jbstitu- 

(9.38) 

iffusion 
r as an 



REACTIONS WITH AND BETWEEN SOLIDS 



401 



example of the applicability of Eq. 9.38 to sintering of pure MgO for 
which the values of Z M , = |Zj = 2 and c Mg - c„ - c . 



Jr = - 



(9.39) 



[B Mlt + Bo] dx 

Since ^ = ^ + RT In a ~ ^ + RT In c. Eq. (9.39) can be expressed 

as 

c MM B Mx BoRT dlncw = _Bu £ B^Tdc^ (9 4Q) 
Jt = ~~ [B Ms + B 0 ] ' dx [B Mt + B 0 ] dx 

where dw> is the concentration gradient due to curvature (Chapter 10). 
Recalling^ the tracer diffusion coefficient and mobility are related by 
Eq. (6.11), 



D M JD 0 T dc 



(9.41) 



'[D M / + D 0 T ] dx 
Thus the total molecular transport may be governed by ^ 'lowest- 
moving specie if there is a large difference in diffus.vities (e g., 
H Do) or by an intermediate value when they are not too d.s.m.lar (e.g., 

*3n£2on£ ^transport more rapidly in boundaries or 
tions, a relationship for ambipolar diffusion can bedenved when pa h 
other than the lattice are assumed. A simple case has ' *^^f° 
steady-state grain boundary and lattice transport * The effective area ot 
ransport in the lattice A' and boundary A b must be ^triin h 
equation for total mass flow. For the case of 
effective diffusion coefficient is similar in form to Eq. 9.41 and given by 

(A'DJ + A b D M b )(A , Do' +A"ql ±_ (942) 
" (A 'Dm J + A "D M b ) + (A U' + A Do ) 
where D' refers to lattice diffusion and D' ' refers to boundary diffusion 
In many oxides, it has been observed that A D„ > A D 0 and 
A'DJ > A b D M b \ thus Eq. 9.42 reduces to 

A'DjA'Do" 



D 



effcoiv.- A , D J +A » D 



(9.43) 



Diffusive transport in real materials is more complex owing to impurities 
and imperfections, but relationships like these can be der.ved to mclude 
more complex situations.t 

*R. S. Gordon, J. Am. Ceram. Soc. 56, 147 (1973) 
tD. W. Readey, I Am. Ceram. Soc, 49, 366 (1966). 




402 



INTRODUCTION TO CERAMICS 



9.3 Reactant Transport through a Fluid Phase 



As discussed in Section 9.1, heterogeneous reactions at hieh temn~ 
- tures^equ.rernrstrmaterial-transfer to Wreacdof n^rfS tS"^ 

™vZl t phase , boundary ' and in some cases -fflK??pSS; 

away from the reaction site. Any of these steps can have the low2 
virtual reacnon rate and be rate-controHing for the ote a proce 2 

dZ^Z'r CC 3 ,f eaCti ° n iS in, ' tiated ' material-transfer phenomena 
determine the overall rate in the high-temperature systems of importance 
in ceramics. As discussed in the previous serf mn this * • ,P°" ance 
electrons through a stable ^^^^Jf^^^ 

e react,on rate. If, however, the film forms with cracks an^ fis h 
ra e may be determined by gaseous diffusion through these channe s I" 
this section we wish to consider several important example ^of the wav 

sasc^^ssr gases and ,iquids and to " - 

he solid. Section 9.4 contains a discussion of the decompos.S of a solid 
to a gas and another solid; in this section we are primarily concerned with 
reasons ,n which the solid forms only gaseous produc The rat^of 
decomposition is dependent on the thermodynamic dnving forces on the 

he rr? 0 " kme i;' CS ' ° n thC C ° nditi0n 0f the rea ^ion Lice and on 
the ajnbient atmosphere; for example, at high temperatures' oxkles 
volatilize much more rapidly in a vacuum than in air 
The loss of silica from glass and refractories in reducing atmospheres is 

CoZ t *Tn T WhiCH limitS thC USefulness of *«e S produ" 
Colder the following reaction which can cause the vo.ati.izaUon of 



(9.44) 



2Si0 2 (s) = 2SiO(g) + 0 2 (g) 
At 1320 C C, the equilibrium constant is 

SET" 10 (9.45) 

Assuming unit activity for the silica, it is apparent that the ambient 
oxygen partial pressure controls the pressure of SiOfc > and therefore the 

(0.23 torr). atm > the SiO pressure is 3 x 1(T atm 

The rate of evaporation near equilibrium is given by the Knudsen 



equation:* 



. drit . 
where -j- is i 
at 

sample area, c 
weight of i, an 
gas flow rate » 
sample is not 
evaporation n 
be in equilibri 
total pressure 



where A is ir 
When the o 
mixtures Po 2 c 



where Poj is c 
tion reaction 
For the vap 
rate of about 
actual Si0 2 lo: 
in hydrogen. 



The effect of . 
from Fig. 9.] 
decreased by 
Chemical Y 
active transpc 
vapor-phase t 
film electroni 
controlling th 
rate of deposi 
the temperati) 

*M. Knudsen 




REACTIONS WITH AND BETWEEN SOLIDS 



403 



equation: 



dt ~ V2irM ( J?T 



(9.46) 



where ^! is the loss of component i in moles per unit time, A is the 
sample area, a, is the evaporation coefficient (a, * l), M, is the molecular 
weight of i, and P, is the pressure of i above the sample. If there is a high 
gas flow rate over the sample or if the evaporation is into a vacuum the 
sample is not able to maintain its equilibrium vapor pressure P„ and the 
evaporation rate is controlled by the interface reaction rate. For the gas to 
be in equilibrium with the solid, the gas flow rate S (moles/sec) and the 
total pressure P T (atm) must satisfy the inequality 

Aa,P T 



S(M,T) 



m >2.1x 10" 



(9-47) 



where A is in square centimeters and T in degrees Kelvin. 

When the oxygen partial pressure in the gas phase is controlled by gas 
mixtures P 0 "\ the equation (9.46) becomes 



Jo 2 ~~ 



(fo,-Po,">o i 
(2ttMo 2 RT) 



"2 



(9.48) 



where Po, is calculated from the standard free energy of the decomposi- 
tion reaction (e.g. Eq. 9.45). 

For the vaporization of Si0 2 by reaction (9.44), Eq. 9.46 predicts a loss 
rate of about 5x10"' moles SiC/cm'sec at 1320°C. Figure 9.13 shows 
actual SiO, loss rates from various silica-containing refractories annealed 
in hydrogen. The overall decomposition reaction in this case is 

H 2 (g) + SiCMs) = SiO(g) + H 2 0 (9-49) 
The effect of a few mole percent water vapor in the gas stream is evident 
from Fig. 9.14. As predicted from Eq. 9.49, the SiO(g) pressure is 
decreased by an increase in the H 2 0(g) pressure. 

Chemical Vapor Transport. Next let us consider the react.on of an 
active transport gas with a ceramic. The net effect is to increase the 
vapor-phase transport. Some high-temperature ceramics and many thin- 
film electronic devices are prepared by chemical vapor deposition. By 
controlling the chemical potential (concentration) of reaction gases, the 
rate of deposition can be controlled. Generally the rate of deposition and 
the temperature of deposition determine the reaction kinetics and rates at 



nudsen 



*M. Knudsen, Ann. Phys., 47, 697 (1915). 





Fig. 9.14. 
vapor way 



which t 
surface, 
tion oc 
supersa 
a polyc 
porosit) 
materia 
temper? 
single-c 
epitaxia 
orientat 
To ur 
a know 
respecti 
reactani 
nations 
of the s 
system 
Consid* 
held at 
chambe 



REACTIONS WITH AND BETWEEN SOLIDS 



405 




Water vapor added 




26% 



8% 
_j 



20 



40 



60 



80 



140 



160 180 200 



100 120 
Time (Hr) 

Fig. 9.14. Weight loss of brick at 1370°C in 75% H 2 -25% N 2 atmosphere. After 32 hr water 
vapor was added for 150 hr. From M. S.Crowley , Bull. Am. Ceram. Soc, 46,679 (1967). 



which the decomposition products can "crystallize" on the reaction 
surface. If the supersaturation is large, homogeneous gas-phase nuclea- 
tion occurs; that is, a heterogeneous surface is not needed. As the 
supersaturation is reduced, the gases react in the vicinity of a surface, and 
a poly crystalline deposit is formed. The perfection of the deposit, 
porosity, preferred grain orientation, and so on, depend on the particular 
material and the rate of deposition; usually slower deposition and higher 
temperatures result in a more perfect reaction product. Finally, when a 
single-crystal substrate is used as the heterogeneous reaction surface, 
epitaxial deposition occurs. In the latter case, a single crystal with an 
orientation determined by the substrate is formed. 

To understand the kinetics of chemical vapor deposition fully requires 
a knowledge of all of the thermodynamic equilibria involved and the 
respective kinetic processes for the generation of reactants, mixing of 
reactant gases, diffusion through the boundary layers, molecular combi- 
nations at the interface, exsolution of gaseous products, surface diffusion 
of the solid products, and so on. We have chosen, as an example, a simple 
system for which the rate-determining step is diffusion in the gas phase. 
Consider the closed system shown in Fig. 9.15 in which two chambers are 
held at thermal equilibrium. Assume that the chemical reactions in each 
chamber reach thermodynamic equilibrium such that the diffusion flux of 




406 



INTRODUCTION TO CERAMICS 




FeCl 2 + H 2 0 



Distance - 
fA 



Powder 



XX 



■HCl 



tai 



FeO(s) + 2HCI<£) 



FeCl 2 fe) + H 2 0(g) 



D(gas) = 4 x 10"' 



cm 2 /sec 



System 



Fig. 9.15. Schematic diagram of chemical vapor transport of iron oxide in a temperature 
gradient. 



matter is from the hot chamber to the cooler chamber because of the 
concentration gradient (the direction of transport is determined by the 
sign of the enthalpy of the reaction). 

The kinetics of mass transport as determined by the diffusion of the 
rate-limiting specie — for example, diffusion of FeCl 2 (g) — is given by 
Fick's law: 

where n is the number of moles transported, A the cross-sectional area of 
the connecting tube (cm 2 ), D the diffusion coefficient of the rate-limiting 
specie, and c the concentrations in the respective isothermal chambers. 
For an ideal gas 



and the composition difference is 



(c h -c c ) 



(9.51) 



(9.52) 



Thus, the tr; 



The equilibr 
of formatior 



In a close 
B atm res 
formation o 
(9.54) reduc 



which can 
prediction 
In generf 
total pressi 
gas-phase r 
line-of-sigh 
diffusion-li 
pressures t 
convectior, 
boundary I 
Liquid-S 
of the kine 
in liquids, 
by molten 
componen 
ceramic b< 
required f< 
rate of the 
fixed by tfc 
crystal gn 
leads to ai 
away fron 
material t 
the liquid 
natural cc 
For a s 



REACTIONS WITH AND BETWEEN SOLIDS 



407 



mperature 



e of the 
d by the 

>n of the 
jiven by 



(9.50) 

al area of 
e-limiting 
hambers. 



(9.51) 



(9.52) 



Thus, the transport rate is determined by 



dn 
dt 



AD 



(Ph-Pc) 



(9.53) 



The equilibrium pressures can be determined by the standard free energy 
formation at each temperature; for example, at the higher temperature 



AOS — RT.lnfe^ 

r HC1« FcO 



(9.54) 



In a closed system such as a quartz ampoule an initial HC1 P^re of 
B atm results in the adjustment of the formation reaction by the 
formaln of equivalent numbers of moles FeCU and H 2 0. The expression 
(9.54) reduces to 

P FeCI 2 



AGS=-KT h ln^ Z 2p^ 



(9.55) 



which can be solved for each temperature and therefore leads to a 
prediction of the transport rate from Eq. 9.53. 

P "n general, the rate-limiting gas-phase transport step is a funcUon oi Mhe 
total Pressure of the system. At very low pressures (P,o^<H) atm) 
tX-molecular colnsions are infrequent and tfcus 
line-of-sight. At intermediate pressures (10 <*-< 10 
diffusion-limited case discussed above becomes m f^ A ^f£ 
nressures (P. ,..> 10"' atm) convective mass transport is more rapid. It 
S^^ToicS flow becomes rapid, gas-phase diffusion through the 
boundary layer may become the rate-determining process. 

Liquid-SoL Reactions: Refractory Corrosion. An important example 
of the kinetics of liquid-solid reactions is the rate of ^ 
in liquids, particularly important in connect.on with refractory corrosion 
by molten 'slags and glasses, with the rate of co^^teh 
components to glass in the glass-making process, and with the firing of a 
c3ic body in which a liquid phase develops. No nucleate step is 
r?«Z d for the dissolution of a solid. One process that can determine the 
ra?e of the overall reaction is the phase-boundary reaction rate which s 
fixed by the movement of ions across the interface in a way equivalent to 
crystal growth (Section 8.4). However, reaction at the phase boundary 
o 8 increased concentration at the interface. Material mus diffuse 
away from the interface in order for the reaction to continue. The rate of 
mate" alTansfer, the dissolution rate, is controlled by mass ^transport ,„ 
the liquid which may fall into three regimes: (1) molecular d.ffusion, (2) 
natural convection, and (3) forced convection. 

For a stationary specimen in an unstirred liquid or in a liquid with no 



408 



INTRODUCTION TO CERAMICS 



discussed in Chapter 6 on £ **? * those 

wh.ch mass is transported is proposal Tva^f** ° Ver 
change in thickness of the specimen wMrh ' ther efore the 

dissolved, varies with ,« ^7^1^°^ * the mass 
tion due to hydrodynamic instabiaies fronT^ t Underg0 Convec - 
from thermal gradients or t^m^^^V Mmta which 

the initial dissolute ^^If^^^Z 

^^sT^ > considered in 

various possible species mus * £££ , Chem ' Cal effect *°fthe 
dissolution of Al 2 0,i n a sS e s L Z k ^ F ° r CXam P ,e ' ** 
cations or anions in the Aiotl * COntro,,ed b V any of the 
Eq. 9.41). An exam^o VZlZ^ & C ° mbin ^ 

diffusion is shown in Fig 1? f th 7 T™ 1 ' 6 by mo,ecu ^ 

Ca^A. 2 0 3 -Si0 2 mel^onLU 2^% ° f in a 

in the h£id^^ 1 hydrodynamic instabilities 

dissolution kinetics It ha often bee o^T' , S ° Hd - Th '' S Cnhances the 



0.006 - 



0.004 




REACTIONS WITH AND BETWEEN SOLIDS 



409 



SSSSSssasrssss: 

sion for mass transport during convection is 



• _ dnldt _ D(c,-c.) 
J A 8(1 -CiV) 



(9.56) 



s- r r D rf T^rr;.°" 

" and " is o.a, volume. The boundary ,ayer is shewn ,n 

Fig. 9.17 and defined by 

(9.57) 



5 = 



d - c- 
(dcldy) 



1 " Uid ^^^XS^S^ .hLer boundary 

in glasses and silicate sla B s, .he high vrscosr.y and slo w "^J^ 
combine to give relatively thick boundary layers Th ttactae» o he 




Distance from interface (cm) 
Fig. 9.17. Concentration gradient through diffusion layer at a so.ution interface. 



410 



INTRODUCTION TO CERAMICS 



in aqueous solutions, so that there is more of a tendency for the reaction 
process to be controlled by material-transfer phenomena rather than 
interface reactions. 

Values for the boundary-layer thickness have been derived for special 
cases in fluid flow. The boundary-layer thickness for mass transport from 
a vertical slab with natural convection caused by density-difference 
driving forces is 



Igx (p, - p«) J 



(9.58) 



where x is the distance from the leading edge of the plate, v is the 
kinematic viscosity Vp, g is the gravitational constant, p„ is the density 
of the bulk liquid, and p t is the density of the saturated liquid (the liquid at 
the interface). Thus the average dissolution rate for a plate of height h is 
given by 



J = 



^ = „. 726D (^)'» (c ,_ c ., ( , 59) 

The boundary-layer thickness for mass transport from a rotating disc is 

where o> is the angular velocity (rad/sec). The mass transfer for a rotating 
disc is proportional to the square root of the angular velocity: 



dnldt 



= 0.62D 



2/3 -1/6 1/2 (C| — C») 

V 0) — 

V) 



(9.61) 



A (l-c,l 
Figure 9.18 shows the dissolution kinetics of sapphire into CaO-AhOr- 
Si0 2 for the free convection kinetics and in Fig. 9.19 for forced flow. In 
each case the kinetics are time-independent, as predicted by Eqs. 9 59 and 
9.61. 

Comparison of the data for sapphire dissolution at 1550°C for kinetics 
limited by molecular diffusion, free convection, and forced convection 
(126 rad/sec) show the dimensional change AR (cm) to be related to time 



as 



AR (molecular diffusion) = (1.77 x 1(T 4 cm/sec" 2 ):" 2 

AR (free convection) = (3.15 x l(T 6 cm/sec)f 
AR (forced convection) = (9.2 x 10" 5 cm/sec)f 



(9.62) 



The important parameters for connective dissolution are fluid velocity, 
kinematic viscosity, the diffusivity, and the composition gradient. 



0.06 



~ 0.05 

E 

o 

°3 0.04 



0.03 
0.02 
0.01 
0 



Fig. 9.18. Dissolu 
21 wt% A1 2 0 3 ver: 



It is clear f rc 
extremely ten 
limited kinetict 
exponential tei 
dence of the c< 
in Fig. 9.20. 



Fig. 9.19. (a)D 
angular velocity. 




59) 

.60) 
:ing 



REACTIONS WITH AND BETWEEN SOLIDS 



411 



)n 




an 


0.06 - 


ial 


~ 0.05 - 
E 


>m 


o 

* 0.04 - 


ce 






o 




g 0.03 - 


58) 


o 
o 

0.02 - 


the 


0.01 


;ity 




I at 


0 

( 


t is 






10,000 
Time tsec) 



15,000 



20,000 



Fig 9.18. Dissolution at relatively ion* ».»~ - 
21 wt% Al.O, versus time. From Ref . 6. 

. r- o 1* o«H 0 19 that the dissolution rate is 
It is clear from the data in Bgs. 9.1 and 9.19 transport . 

in Fig. 9.20. 



.61) 

Or- 
: In 
and 

;tics 
tion 
time 



).62) 
city, 




10 20 30 _ 

Square root of angular velocity (rad /sec )i 
(a) 

Fig. 9,9. (.) Dependence of rate of disso.ution of face of sapphire disK on square root of 

angular velocity. 





0.06 



£ 0.04 



0.02 



' . i — i — r 

,1550 C ' 
il480°C 



n 1 r 



Rotating disk oj = 126 rad/sec = 1200 rpm 
(21% A1 2 0 3 > 




1340°C 



2,000 4,000 6,000 

Time (sec) 
(b) 



8,000 10,000 



Fig. 9. 19 (continued), (b ) Rate of dissolution of face of sapphire disk rotating at 126 rad/sec 
on CaO-Al 2 0 3 -Si0 2 with 21 wt% A1 2 0 3 . From Ref. 6. 



E 
u 

b 



40 

20 

10 
8 
6 



2 — 



8 1 
t 08 

(3 0.6. 



0.2 



O.t 



— V 


V ^ Mi 


tllite 












t Silic 

\ 

\ 


0 






\ 


\ 
\ 

\ 

\ 

A. 

\ 





5.4 



5.6 



5.8 60 
I0 4 /T(°K) 



6.2 



6.4 



Fig. 9.20. Temperature dependence of forced convection corrosion in the 40CaO~20AI 2 O 3 - 
40SiO 2 slag of alumina, mullite, and fused silica. From Ref. 6. 



Fig. 9.21. Corr 
slag of indicatec 
From Ref. 6. 



Refractor 
ities in the 
ideal surfac 
bodies and 1 
corrosion, s 
ceramics, ct 
from the da 
is about 40 

9.4 Reacti 

Of partic 
tions whicl 
the dehyd 
polymorph 
products ir 
of these re 



412 



REACTIONS WITH AND BETWEEN SOLIDS 



413 



0.250r 




2000 3000 4000 5000 
Time (second) 

Fi* 9 21 Corrosion rate under forced convection conditions in the ^aCWOAUO^OSiO, 
Z "LiS^-en. of sapphire. po.ycrystalHne alumina, mu.lite, and v.treous su.ca. 

From Ref . 6. 



Refractory corrosion is often much more complex. Besides complex- 
ities inTe hydrodynamics of a molten bath, refractories seWorn have 

deal surtes'andare usually not ^.^^-^SSSS 
bodies and brick with extensive porosity provide centers for accelerated 
c onllZ, spalling, and penetration by the Hquid^In d ™ 
ceramics, corrosion may be greatest at grain boundaries. This can be seen 
"om the data in Fig. 9.21, in which the corrosion of noncrystalline AW, 
is about 40% greater than sapphire after 2500 sec. 

9.4 Reactant Transport in Particulate Systems 

Of particular interest to ceramists is the large numb 
tions which occur with granular or powdered raw 
the dehydration of minerals, decarbomzation of carbonates, and 
SyZSTS^onMtion.. In general, the minerals and Reaction 
SctTinvolved are used in large volumes; thus even though the nature 
of ttfe reactions is complex, study of a few examples ,s unportant and 



414 



INTRODUCTION TO CERAMICS 



elucidates the important kinetic parameters and illustrates the concept of 
the rate-limiting step. 

Calcination and Dehydration Reactions. Calcination reactions are 
common for the production of many oxides from carbonates, hydroxides, 
sulfates, nitrates, acetates, oxalates, alkoxides, and so on. In general the 
reactions produce an oxide and a volatile reaction product (e.g., C0 2 , S0 2 , 
H 2 0, . . .). The most extensively studied reactions are the decomposition 
of Mg(OH) 2 , MgC0 3 , and CaC0 3 . Depending on the particular conditions 
of temperature, time, ambient pressure, particle size, and so on, the 
process may be controlled (1) by the reaction rate at the reaction surface, 
(2) by gas diffusion or permeation through the oxide product layer, or (3) 
by heat transfer. The kinetics of each of these rate-limiting steps is 
considered. 

Let us first consider the thermodynamics of decomposition, for exam- 
ple, the calcination of CaC0 3 : 



CaC0 3 (s ) -+ CaO(s ) + C0 2 (g ) 



AHSL, = 44.3kcal/mole 



(9.63) 



The standard heat of reaction is 44.3 kcal/mole, that is, strongly endother- 
mic, which is typical for most decomposible salts of interest. This means 
that heat must be supplied to the decomposing salt. 

The standard free energy for the decomposition of CaC0 3 , MgC0 3 , and 
Mg(OH) 2 is plotted in Fig. 9.22. The equilibrium partial pressure of the gas 
for each of the reactions is also plotted in Fig. 9.22. Note, for example, 
that when AG° becomes zero, P C o 2 above MgC0 3 and CaC0 3 and P H2 o 
above Mg(OH) 2 have become 1 atm. The temperatures at which this 
occurs are 1156°K (CaC0 3 ), 672°K (MgC0 3 ), and 550°K (Mg(OH) 2 ). The 
Pco, normally in the atmosphere and the range of P H2 o (humidity) in air 
are also shown in Fig. 9.22. From these values we can determine the 
temperature at which the salt becomes unstable when fired in air. For 
example, CaC0 3 becomes unstable over 810°K, MgC0 3 above 480°K. 
Depending on the relative humidity, Mg(OH) 2 becomes unstable above 
445 to 465°K. Because acetates, sulfates, oxalates, and nitrates have 
essentially zero partial pressure of product gases in the ambient atmos- 
phere, it is clear that they are unstable at room temperature. That they 
exist as salts to a decomposition temperature of about 450°K indicates 
that their decomposition is governed by atomistic kinetic factors and not 
by thermodynamics. 

The kinetics, as noted above, may be limited by the reaction at the 
surface, the flow of heat from the furnace to the reaction surface, or the 
diffusion (permeation) of the product gas from the reaction surface to the 
ambient furnace atmosphere. This is shown schematically in Fig. 9.23, 
which also includes the appropriate heat and mass flow equations. The 



£ 



30 
26 
22 
18 
14 
10 



I 

II 

h 



2 

> o 
1 -2 

-6 

-10 

-14 

-18 

-22 



Fig. 9.22. S 
the equilibri 



rate-limiti 
decompo: 
atures the 
tures su$ 
atomistic 



REACTIONS WITH AND BETWEEN SOLIDS 



415 



ept of 

s are 
xides, 
al the 
,S0 2 , 
sition 
itions 
i, the 
rface, 
or (3) 
ips is 

ixam- 

(9.63) 

Dther- 
neans 

3 , and 
le gas 
mple, 

Ph 2 o 

i this 
The 
in air 
e the 
. For 
80°K. 
ibove 
have 
tmos- 
they 
cates 
d not 

it the 
>r the 

:o the 
.9.23, 
. The 



30 
26 
22 
18 
14 



o 

I o 10 

£ pf 

g£ 6 

• Sa 0 

U < -2 

o 

-6 



-10 
-14 
-18 
-22 



P H2 o(atm> ,''10 ' 




_L 



_L 



_L 



300 400 500 600 700 800 900 1000 1100 1200 1300 
Temperature (°K) 

Fig. 9.22. Standard free energy of reaction as a function of temperature. The dashed lines are 
the equilibrium gas pressure above the oxide and carbonate (hydroxide). 



rate-limiting step depends on the particular substance which is undergoing 
decomposition and the relative temperature. For example, at low temper- 
atures the existence of unstable salts which decompose at higher tempera- 
tures suggests that the initial decomposition must be controlled by 
atomistic processes because there is no reaction-product interference in 



416 



INTRODUCTION TO CERAMICS 




P t = e -*°' IRT f 

P= density of CaC0 3 

5 = boundary-layer thickness 

M- molecular weight 



h, = heat-transfer coefficient 
k = thermal conductivity of CaO 

Fig. 9.23. Schematic representation of the decomposition of a spherical particle (e.g., 
CaC0 3 ) of a salt which yields a porous oxide product (e.g., CaO) and a gas (C0 2 ). The reaction 
is endothermic, requiring heat transfer. The driving forces for heat and mass transport for 
steady-state decomposition are expressed as temperatures and pressures in the furnace 
(7),P,), at the particle surface (T„P S ), and at the reaction interface {T„P t ). 



the transport of heat to the reaction interface or gaseous product away 
from the interface. 

The reaction shown schematically in Fig. 9.23 is heterogeneous; that is, 
the reaction occurs at a sharply defined reaction interface. Figure 9.24 
shows this interfacial area for MgC0 3 for which the reaction proceeds 
from nucleation sites on the surface of the MgCO a platelets. The 



Fig. 9.24. Transmission electron micrographs of MgO prepared by thermal decompos.t.on 
of basic magnesium carbonate, (a) Pseudomorphed MgO (550«C calcme ; (b) crystallite 
approaching cube form (900*C calcine); (c) two-dimensional mo.re pattern Uon flapped 
crystallites (550»C calcine). From A. F. Moodie, C. E. Warble, and L. S. Williams. I Am. 



Ceram. Soc, 49. 676 (1966). 



417 



418 



INTRODUCTION TO CERAMICS 



decomposition kinetics for cylindrical geometry is 

(1 -«)'"= 1-to/r. (9 64) 

where a i| the fraction decomposed, k is the thermally activated kinetic 
constant, t is the time (assumed constant temperature), and r„ is the initial 
particle radius. The first-order kinetics (Eq. 9.2) /or this reaction al 
several temperatures is shown in Fig. 9.25 for decomposition of Mg(OH), 
The importance of the surface on the decomposition rate is indicated by 
the time to decompose (700°C) a cleaved calcite crystal (CaCO,) 60 hr 
compared with an equivalent mass of the same material in powder form,' 

At low temperatures the crystallite size strongly affects the decomposi- 
tion rate; however, at higher temperatures, as the chemical driving force 
increases and as the thermal energy to motivate diffusional processes and 
reaction kinetics increases, other steps may become rate-controlling for 
example the rate of heat transfer. Figure 9.26 shows the centeH ne 
temperature of a cylindrical sample of pressed CaCO, powder which was 
hrust into a hot furnace. The sample temperature increases to a ma X - 
mum, at which nucleation of CaO finally occurs. The decrease in 
temperature represents the endothermic heat absorbed by the reaction 

The effect of varying the ambient CO, pressure is illustrated in Fig' 




V 374°C 
J I L 



° 20 40 60 80 '«> '20 140 160 l80 200 220 240 260 28 0 

Time (min) 

WD 2 L^Tf ti0 n ° f T H) ' ShOWi " 8 firS, -° rder kinetics - From R- S. Gordon and 
W. U. Kingery, /. Am. Ceram. Soc. t SO, 8 (I%7). 



REACTIONS WITH AND BETWEEN SOLIDS 



419 



.64) 

etic 
itial 
1 at 

3)2. 

Iby 
)hr, 
>rm, 

>osi- 
Drce 
and 
.for 
•line 
was 
lax- 
t in 
;ion. 
Fig. 



1800 



1760 



1720 



1680 



1640 




1600 



Center temperature 



0 



20 



40 



80 



100 120 



60 

Time (min) 

Fig 9 26 Comparison of the furnace temperature to center-line temperature of a cylindrical 
sample of CaCO, thrust into a preheated furnace. From C. N. Satterfield and F. Feales, 
AJ.CH.EJ., 5, 1 (1959). 

9.27. As the Pco, is increased, the driving potential for the reaction 
decreases, and thus the reaction rate is decreased. 

Some of the clay minerals, kaolin in particular, do not decompose in the 
manner shown in Fig. 9.23; that is, they do not have a heterogeneous 



280 
n and 




0.1 0.2 



0.7 0.8 0.9 



0.3 0.4 0.5 0.6 
C0 2 pressure <atm> 

Fie 9 27 Rate of decomposition of CaCO, in CO, atmosphere; = 
n - P IP- M(BP^+\lRo) where Pco, = equilibrium C0 2 pressure, B = constant, and 
S. SS££55££ * a pure neutra. ^sphere. From E. P. Hyatt. I. B. Cufer, and M. 
E. Wadsworth, J. Am. Ceram. Soc, 41, 70 (1958). 



420 



INTRODUCTION TO CERAMICS 



reaction interface or a reaction product which breaks up into small 
crystallites. Above 500°C the water of crystallization is evolved, and a 
pseudomorphic structure remains until 980°C. The pseudomorph is a 
matrix of the original crystal structure containing large concentrations of 
vacant anion sites. Above 980°C the structure collapses irreversibly into 
crystalline mulhte and silica, which releases heat (see Fig 9 28) 
^T 3 ^ 011 ? inetiCS iS controlled b V th e diffusion of hydroxylions in 
I a ^", • thC heter °e enous surface decomposition illustrated in 
♦J 8 ' S" ? kinet,cs IS thus homogeneous and controlled by diffusion in 
the solid, wh,ch gives a parabolic rate law. The dehydration kinetics of 
kaohnite is given (1) in Fig. 9.29 for size fractions. A similar situation is 
observed for the decomposition of AI(OH) 3 . 

Powder Reactions. In most processes of interest in ceramic technol- 
ogy solid-state reactions are carried out by intimately mixing fine 
powders. This changes the geometry from that considered in Fig 9 6 and 
the actual reaction is more like that illustrated in Fig 9 30 

If the reaction is carried out isothermally, the rate of formation of the 
reaction zone depends on the rate of diffusion. For the initial parts of the 
reaction the rate of growth of the interface layer is given to a good 



0 




! - 


1 


i 

i 




i 




HI 






Kc 


olinite 

i 

! 

t 


\ 


j 


j 

i 
1 


J h 

! 

J » 


) 






Di< 


:kite 






y 




i 


0 10 


0 20 


0 30 


* En 

n /in 


dellite 

1 

n «;fi 


V 


n "7 a 









Furnace temperature 

mi?."., ^r^ 1 ,hermi " ana ' ySiS CUrV " ° f ka °' in Clays - The sa "P |e temperature leads 
SiS!?^ fUn,aCe temPera,UrC 31 leVC,S " WhkH hea « iS ^ - absorbed by 




h is a 
ions of 
ly into 

ions in 
ated in 
ision in 
^tics of 
ation is 

echnol- 
ng fine 
?.6, and 

a of the 
s of the 
a good 



rature leads 
bsorbed by 



30 



25 



20 



.2 15 



± 10 



-56b "600 700 800 900 1000 1100 1200 
Time (minutes) 



0 100 200 300 400 




Fig . 9.30. Static representation of reaction-product .ayers forming on surface of 
particles in powder mixture. 

421 




422 



INTRODUCTION TO CERAMICS 



approximation by the parabolic relationship in Eq. 9.8. If V is the volume 
of material still unreacted at time r, then 

V=±< rr (r-yy (9.65) 

The volume of unreacted material is also given by 

4 

V=-nr\l-a) (9.66) 

where a is the fraction of the volume that has already reacted. Combining 
Eqs. 9.65 and 9.66, 

y = r(l -Vl -a) (9.67) 
Combining this with Eq. 9.8 gives for the rate of reaction 

o-vr^f= (™) t (9 . 68) 

Note that this is for spheric al geo metry where Eq. 9.64 is for cylindrical 
geometry. By plotting (1 -^1 -a) 2 against time, a reaction-rate constant 
equivalent to KD/r 2 can be obtained which is characteristic of the 
reaction conditions. The constant K is determined by the chemical- 
potential difference for the species diffusing across the reaction layer and 
by details of the geometry. 

The relationship given in Eq. 9.68 has been found to hold for many 
solid-state reactions, including silicate systems, the formation of ferrites, 
reactions to form titanates, and other processes of interest in ceramics. 
The dependence on different variables is illustrated for the reaction 
between silica and barium carbonate in Fig. 9.31. In Fig. 9.31a it is 
observed that there is a linear dependence of the function (1 - W^) 2 on 
time. The dependence on particle size illustrated in Fig. 9.31b shows that 
the rate of the reaction is directly proportional to II r 2 in agreement with Eq. 
9.68: in 9.31c it is shown that the temperature dependence of the 
reaction-rate constant follows an Arrhenius equation, K' = 
Kiexp (-Q/KT), as expected from its major dependence on diffusion 
coefficient. 

There are two oversimplifications in Eq. 9.68 which limit its applicabil- 
ity and the range over which it adequately predicts reaction rates. First, 
Eq. 9.68 is valid only for a small reaction thickness, Ay ; and second, there 
was no consideration of a change in molar volume between the reactants 
and the product layer. The time dependence of the fraction reacted 



!! 1 



424 



INTRODUCTION TO CERAMICS 



corrected for these two constraints is given as* 

[1 + (Z - Da]* 3 + (Z - 1)(1 - af> = Z + (1 - Z)(^P) t (9.69) 

where Z is the volume of particle formed per unit volume of the spherical 
particle which is consumed, that is, the ratio of equivalent volumes. A 
demonstration that Eq. 9.69 is valid even to 100% reaction is shown in Fig. 
9.32 for the reaction ZnO + AI 2 0 3 = ZnAl 2 0 4 . 



1.55 



B l - 51 



I 1.47 
+ 

5 1.43 
+ 1.39 



1.35 



19 microns 
25 microns 




0.0 
0.2 
0.4 



- 0.60 



0.80 



0.8 



_L 



_L 



0.90 -° 



0.95 



1.0 



3.2 



3.6 



1.6 2.4 
Time <hr) 

Fig. 9.32. Reaction between ZnO and A^O, to form ZnAI 2 0 4 at MOOT in air (two spherical 
particle sizes). See reference 1, p. 102. 

Calculating the reaction rate given in Eqs. 9.68 and 9.69 on an absolute 
basis requires knowledge of the diffusion coefficient for all the ionic 
species together with a knowledge of the system's geometry and the 
chemical potential for each specie as related to their position in the 
reaction-product layer. The diffusing species which control the reaction 

*R. E. Carter, /. Chem. Phys., 34, 2010 35, 1137 (1961). 



RE 

rate are the most 
arriving at a pha 
Section 9.2 must 
Another diffici 
dependence of re 
many cases the i 
coherent with the 
with many defec 
tunity for surfac 
indicated in Eqs. 
through a single- 
lower limit for t 
rate. When new ] 
temperatures, th- 
be some nonequ 
structure with t] 
and growth in 
nonequilibrium ] 
product. For exe 
observed at a p 
This effect is n 
changes at the 
extensively in q 
Also, at the train 
forms tends to c 
which increase 
transfer. At pre: 
a quantitative t 
Coarsening o 
may undergo a 
represents a va 
Generally term 
dispersed in so 
varying size in 
ones dissolve a 
of the interf aci; 
ter 5) relates th 
a relative to tl 



where 7 is the 




.69) 

ical 
. A 
Fig. 



rical 



lute 
onic 
the 
the 
tion 



REACTIONS WITH AND BETWEEN SOLIDS 



dependence of reaction rates on me » h t it ■ not 

structure with the reactant as aisc coefficients for this 

and growth in Sections 8.3 and A D, f US1 ° h£ final equilibrium 
nonequilibrium lattice are normally larger ^ an ^ at£ is f requently 
product. For example, an increase in solid-state react on ^r a m 
observed at a polymorphic transition ^^^^ *™ 
This effect is related to lattice ^^^^fije. occur 
changes at the transition ^^^^Zn* change is large, 
extensively m quartz, for example, ,n ~ m between two polymorphic 

Tra^^ 

a quantitative ^ ipitated , the par ticles 

Coarsening of Particles. Aiie variation in particle size 

may undergo a coarsening ^J^™™ from particle to particle, 
represents a variation «^ ■ ch ^.^jS5» apply to precipitates 
Generally termed ^^^^J^S^ particles of 
dispersed in solids or liquid i. bor ine y solubility> the smaller 

varying size in a medium in which they have s om 
ones dissolve and the larger ^^^^Lk equation (Chap- 

a relative to that for a planar interface c„,.. 



^ c 2y M 
RTln — = 



(9.70) 



where y is the interfacia. energy (ergs/cm*), M the molecular weight, and 



426 



INTRODUCTION TO CERAMICS 




P the density of the precipitate particle. This relation also assumes that 
the activity is given by the concentration. If i, the increased 

solubility is given by ™ ""— ' 



\ RTap J c "' + RTap 



(9.71) 



For simplicity, consider a system of two particle sizes a, and a 2 where 
a, > a 2 . The a 2 particles are more soluble in the matrix and thus tend to 
dissolve because of the concentration driving force: 



Rip \a, a 2 ) 



(9.72) 



From Fick's law we can determine the rate of growth of these particles if 
we assume the rate is controlled by diffusion in the matrix, that is, solvent 
(see F lg . 9.33a). The rate of mass gain by a, is 



dQ 
dt 



= ~ D (f)( c °- c °J 



(9.73) 



where Ajx is a representative area-to-length ratio for diffusion between 
two dissimilar particles. Substitution of Eq. 9.72 into 9.73 yields 



_£> (A\ 2MyCpA. /_L__j\ 
* \x) RTp U, a 2 j 



(9.74a) 



As we have assumed spherical particles and must conserve mass, 

ida. 



dQ_ 2 da 2 



P4™, — 



(9.74b) 



the growth of a, is 



da, 
dt 



\xj RTp \a, aj 



(9.75) 

Equation 9.75 can be integrated under various approximations, however 
the same solution results by considering the following approximate 
solution. If we assume that the small particles contribute solute to the 
matrix faster than the solute is precipitated onto the large particles the 
growth of large particles can be treated as a diffusion-limited-growth 
problem. The rate-limiting step is assumed to be diffusion of matter to the 
large particle from the matrix. Assume a diffusion field of r(r > a ) 



c 



Fig. 9.33. <a)Coi 
in a diffusion field c 



around the gro 
symmetry is 



where 

Recalling that 
J 



REACTIONS WITH AND BETWEEN SOLIDS 



427 




Fig 9 33. (a) Coarsening of particles in a two-size particle system; (b) growth of particle a, 

around the growing particle (Fig. 9.33b); thus Fick's first law of spherical 

symmetry is 

(9 - 76) 

_ 2Myc p .i. 

where Ac Rjpa, 

Recalling that for dispersed particles r > a t , the flux is given by 

AirD2Myc v .x. ( _a 1 r_\ M 4irDc P .,.2M7 = cpnst (9 77) 

J ~ RTpa, \r - a,/ RTp 



428 



INTRODUCTION TO CERAMICS 



The flux is a constant, independent of the growing particle radius, 



/ = constant = p^= p4 W ^ = 4irPc»,.2My 



which after integration becomes 



or 



where 



p kt 

_ 6Dc P ) .My 



(9.78) 



(9.79) 



(9.80) 



More rigorous analyses give essentially the same result for a distribution 
of precipitates.* The variation in the growth rate for varying particle size 
and for increases in the mean radius is illustrated in Fig. 9.33c. The 
diffusion-limited growth of precipitates and of grains during liquid-phase 
sintering have been observed to have this cubic time dependence (Figs 
9.34 and 9.35). 



200 




1.0 1.5 2.0 
. . i i 

Ageing timea (hr) 3 



2.5 



3.0 



3.5 



Fig. 9.34. Coarsening of Mg L2 Fe L8 0 3 . 9 precipitates in MgO. From G. P. Wirtz and M E Fine 
/. Am. Ceram. Soc, 51, 402 (1968). 

The coarsening relationships discussed above assumed spherical parti, 
cles. The following discussion demonstrates that faceted particles and 
even those with different surface energies can be included in the growth 
expressions by properly defining Ac, the concentration difference. 

(1956) WagnCr ' Z * EleCtr0cHem '' 6S * 581-591 (1961): G " W - Greenwood, Acta Met., 4, 243-248 



RE 



5.4 



5.0 - 



icff 4.6 



a? 4.2 



3.8 



3.4 



Fig. 9.35. Isotherr 
microns, t = time in 
Publishing Corpor? 



Consider a si 
to-volume ratio 



The chemical \ 
the planar inte 



REACTIONS WITH AND BETWEEN SOLIDS 



429 



us, 



5.4 



5.0 



ic£ 4.6 
I CD 



I CD 



PP 4.2 



o 85 MgO: 15 Ca 2 Fe 2 0s<wt %) 
o 85 MgO: 15CaMgSi0 4 
_ + 80 MgO: 15 CaMgSiO<: 5 Fe z 0 3 
■ 80 MgO: 15 CaMgSi0 4 : 5 Cr 2 0 3 



3.8 



3.4 



1725°C 




-0.2 



0.0 



1.2 



0.2 0.4 0.6 
log [t - t 0 \ 

Fig. 9.35 . Isothermal grain growth in systems containing MgO and liquid (GS = grain size in 
microns, / = time in hours) (G&at. From J. White in Materials Sci Research, Vol. 6, Plenum 
Publishing Corporation, New York, 1973, p. 81. 



Consider a size class of particles of constant shape but with a surface- 
to-volume ratio S v = SIV. The surface contribution to the free energy is 

G-G p .L = yS = ySvV (9.81) 

The chemical potential difference between the faceted particle (^) and 
the* planar interface (fi P .i) is 



Sv 



(9.82) 




430 



INTRODUCTION TO CERAMICS 



where V is the molar volume for the particle phase. Let x be some linear 
parameter of the particle size such that S = ax> = S£ where a and a are 
characteristic shape constants. Then 



and 



3 jc 



(9.83) 
(9.84) 



Equations 9.81 to 9.84 hold for any system of constant-shape particles 
"respective of whether they are spherical or faceted. If we assume h ' 
activity ,s given by the concentration 6 



Ac _ (M ~ /i.p...)c„,i. -y V Cp , /2 q\ 



(9-85) 

For spheres «=3andx=r,we have the Thompson-Freundlich equation 

Ac - 2 T Vc pi , _ gyAfc^ 
where V is the molar volume, M the molecular weight and p the densitv 

^H^fi?) (9.86) 
where x, is the distance from the ith facet to the particle center. 

9.5 Precipitation in Crystalline Ceramics 

The nucleation and growth of a new phase has been discussed in 
Chapter 8 and applied there to processes occurring in a Hau o 11 
matrix. Polymorphic phase transformations in crySaHine ^ I e 
discussed in Chapter 2. Precipitation processes from a crystall ne matr x 
" ! ch the P rec 'P itate h *s * composition different from th S 
crystal are important ,n affecting the properties of many ceramL sys ems 
and as techniques such as transmission electron microscopy ca P ab Ie of 

th e IT* 3 ^ ent,fying ^ PrCCipitate P artic,es « ^re 7u," appHed 
the widespread occurrence and importance of precioitation i 
more fully recognized. Initiation of'the 

process or by discrete particle nuc.eationVhapter 8) w"^ a driving To' e 



R] 

for phase separa 
mobility (Chapte 
For nucleatioi 
volume between 
free-energy chai 
replaced by 



where the strain 
strain, and b is 2 
can be calculatt 
expression for A 
AG*, which cor 
and /3 structur 
crystalline phas( 
stable nuclei an 
sites. The strain 
when a decomp 
of precipitates r 
minimum increa 
spherical particl 
is proportional t 
the volume cha 
Strain energy : 
energy of the in 
which phase se 
lamellae in pre! 

The energy f 
structure and o 
general kinds c 
planes of atom 
second coordin 
boundary. In c 
planes of atoms 
giving rise to di 
described in Ch 
order of magn 
formation of r 
mother matrix 
commonly the 
mation in whic 



REACTIONS WITH AND BETWEEN SOLIDS 



431 



for phase separation occurs (Chapter 7); growth rates are limited by atom 
mobility (Chapter 6). . , 

For nucieation in solid s7 strain energy resulting from differences m 
volume between precipitate and matrix must be included in evaluating the 
free-energy change on forming a nucleus. In these cases, Eq. 8.19 is 
replaced by 



AG, = 47rr 2 y +^7rr 3 (AG„ + AG.) (9.87) 



3 



where the strain energy per unit volume is given by AG. - be , e is the 
strain, and b is a constant which depends on the shape of the nucleus and 
can be calculated from elasticity theory. The presence of AG. in the 
expression for AG r results in a free energy on forming the critical nucleus, 
AG*, which corresponds to a definite crystallographic relation of the a 
and structures and the boundary between them when both are 
crystalline phases. Inclusion of AG. can affect greatly the morphology of 
stable nuclei and increase the tendency for nucieation at heterogeneous 
sites. The strain energy typically causes the formation of parallel platelets 
when a decomposition (precipitation) reaction occurs. The configuration 
of precipitates as parallel platelets allows growth to take place with the 
minimum increase in strain energy. In general, the formation of thick or 
spherical particles produces large values of strain; since the strain energy 
is proportional to e J , precipitates with a platelike habit are preferred when 
the volume change on precipitation is appreciable, as is often the case. 
Strain energy also effects spinodal decomposition by increasing the 
energy of the inhomogeneous solution and depressing the temperature at 
which phase separation occurs and by causing separation to occur as 
lamellae in preferred crystallographic directions. 

The energy for nucieation of a new phase depends on the interface 
structure and orientation, as discussed in Chapter 5. We can define two 
general kinds of precipitate. In a coherent precipitate, as in Fig. 9.36, 
planes of atoms are continuous across the interface so that only the 
second coordination of individual atoms is changed, similar to a twin 
boundary. In contrast, a noncoherent precipitate is one in which the 
planes of atoms, or some of them, are discontinuous across the interface, 
giving rise to dislocations or a random structure in the boundary layer, as 
described in Chapter 5. The interface energy of a coherent boundary is an 
order of magnitude less than that of an incoherent boundary, so that 
formation of new phases with- definite structural relationships to the 
mother matrix is strongly preferred. In addition, the oxygen ions are 
commonly the more slowly moving in oxide structures, so that a transfor- 
mation in which these ions must migrate to new positions is bound to be 



432 



INTRODUCTION TO CERAMICS 































































































































■3ft* 
























— ( 
— ( 


>-< 

f— ( 


>- ' 






























y:- : ,:" 






: <-; : - 




:p 








>— C 


>— 
















t •* 




w 


















*-;■;: 










w 
















































>— < 













































































































(a) 















i 






























































i 






















































































> — < 






























































i — ( 


( 1 




n 


>— < 




i 


































r — 














































v— < 


( 










) 
































li 











































































































































































































































































(6) 

Fig. 9.36. (o ) Coherent precipitate with continuous planes of atoms across the interface; (b ) 
noncoherent precipitate with discontinuous planes of atoms across the interface. 



relatively slow. Therefore, coherency of the oxygen ion lattice is favor- 
able both for the driving force of nucleation and for the rate of nucleation 
and crystal growth. 

Precipitation Kinetics. The kinetics of precipitation in a crystalline 
solid depend on both the rate of initiation or nucleation of the process and 
the rate of crystal growth, as discussed in Chapter 8. When the precipita- 
tion process consists of a combination of nucleation and growth, the 
sigmoidal curve characteristic of the Johnson-Mehl or Arrami relations 
(Chapter 8) results in an apparent incubation time period, as illustrated in 
Fig. 9.37. For precipitation processes far from an equilibrium phase 
boundary, which is the most usual case, both the nucleation rate and the 
growth rate increase with temperature, as illustrated in Fig. 9.38 such that 
the incubation time is decreased and the transformation time for forma- 
tion of the new phase is decreased at higher temperatures, such as occurs 
for the process illustrated in Fig. 9.37. In many cases, however, the 



REACTIONS WITH AND BETWEEN SOLIDS 



433 



1.00 



0.80 



0.60 - 



0.40 - 



0.20 - 




000— '—^2 T 6~ 8 

Time (hr) 

Fie 9 37 Effect of temperature on the decomposition of solid solutions of 80 mole % 
Z 0,-20 mofe % MgO . So!M solutions prepared at 1520°C for . hr and then decomposed at A, 
fSc; B i07?C; C, . 150°C;D, 1250°C; E, 1350T; F, .375'C. From V. S. Stub.can and D. J. 
Viechnicki, J. Appl. Phys., 37, 2751 (1%6). 



nucleation process occurs rapidly during cooling, such that a large 
number of nuclei are available for growth. This is particular!, , the ca e ,n 
which heterogeneous nucleation sites are avatlable in a no ver P£ 
matrix crystal or in which the interface energy term is low for a coherent 
precipitate. When this occurs the overall precipitation P^ess as meas- 
ured by the fraction of material transformed, corresponds^ growth of 
existing nuclei and no incubation period is observed, as illustrated for 
p'cipLion of MgAl,0, spinel from MgO in Fig. 9.39. For F™*^" 
of magnesium ferrite, MgFe.O,, from MgO, appl.cat.on of super- 
paramagnetic measurements capable of identifying newly formed crystals 
Ling an average diameter of about 15 A has shown no md.cat.on o an 
incubation period; that is, the critical nucleus size « very very small in 
accordance with a low energy for the coherent interface. 

Precipitate Orientation. The influence of stra.n energy and coherent 
internes leads to a high degree of precipitate orientat.on for many ox.de 
precipitation processes. These relationships are particularly strong for the 
many oxide structures based on close-packed arrangements of oxygen 
ions described in Chapter 2. In the case of magnesium alum.nate sp.nel 
contain.ng excess aluminum oxide in solid solution the 
energy and coherency relationships leads to prec.p.tat.on of a metastable 

*G. P. Wirtz and M. E. Fine, J. Am. Ceram. Soc, 51, 402 (1968). 




10* 



10" 



5 io- 



10" 



lo- 



re 





i .1 






\ A Cristobalite 
1 /\ growth 






Kaolinite \ 
dehydration • 


~ 7-a-Al 2 0 3 \ 
transition \ 




Cristobalite 
nucleation 

L. 1 1 | 


1 ' 



0.7 



0.8 



1.0 

1000/7TK) 



1.1 



1.2 



1.3 



Fig. 9.38. Temperature dependence of rate of (a) nucleation (b ) erowth of rr.\rnh a iit. f 
s.lica gel, (c) formation of «-A, 2 0 3 , and W , dehydration of :^^ rfc ™^'"»n 




LOO - 



0.80 



0.60 



0.40 



0.20 



8 12 16 20 
Time <hr) 



24 



28 



RE 

intermediate wit] 
product* which 
product, a-alumi 
metastable prec; 
alumina. After lo 
expense of the r 

Synthetic star 
titaniferous prec 
0.3% Ti0 2 . Whe 
cence causes the 
times for precipii 
1500°C. The lath 
As for precipitat 
equilibrium phas 

Strong orient 
believed to exhi 
Sn02-Ti0 2 syste 
5-min anneal at 
corner of Fig. 9.' 
to the 001 direci 
formed by spim 
Al 2 03-Cr 2 0 3 and 
this manner. A si 
spinel phase fro 
large metal defic 
Chapter 4) is be 
agglomerates m< r 
forming the spir 
the resulting hig 
in this and relat< 
300°C. On coolii 
defect clusters, 

When growth 
change, the rate 
the morphology 
dissipated or ma 
inverse radius o 
dendritic forms 
remaining small 

* H. Jagodzinski 
(1962). 



434 



REACTIONS WITH AND BETWEEN SOLIDS 



435 



te from 



intermediate with a structure similar to spinel as the first precipitation 
product* which is more easily nucleated than the stable equilibrium 
product, a-alumina. In fact, as shown in Fig. 9.40, two different types of 
metastable precipitates initially form, plus a smaller amount of a- 
alumina. After long annealing at 850°C the a-alumina particles grow at the 
expense of the metastable intermediate precipitates. 

Synthetic star sapphires are produced by precipitating an alumina-nch 
titaniferous precipitate from single crystals of sapphire containing 0.1 to 
0 3% Ti0 2 . When viewed in the direction of the c-axis stellate opales- 
cence causes the reflected light to form a well-defined six-ray star. Aging 
times for precipitation range from approximately 72 hr at 1 100°C to 2 hr at 
1500°C. The lath-shaped precipitates formed are illustrated in Fig. 9.41. 
As for precipitation from spinel, the precipitate particle formed is not the 
equilibrium phase (Al 2 Ti0 5 ) but a metastable product. 

Strong orientation effects are also observed in systems which are 
believed to exhibit spinodal decomposition, shown in Fig. 9.42, for the 
SnOz-TiO* system in which a lamellar microstructure is formed after a 
5-min anneal at 1000°C. The electron diffraction pattern at the lower 
corner of Fig. 9.42 shows streaking of the diffraction spots perpendicular 
to the 001 direction, which is to be expected for the periodic structure 
formed by spinodal decomposition. Other crystalline systems such as 
AUOj-CrjOj and CoFe 2 O4-Co,04 are also believed to phase separate in 
this manner. A similar structure, Fig. 9.43, is found for precipitation of the 
spinel phase from an FeO-MnO solid solution at low temperature. The 
large metal deficit in this highly nonstoichiometric system (discussed in 
Chapter 4) is believed to result in defect association on cooling; defect 
agglomerates may serve as nucleation sites for the precipitation reaction 
firming the spinel phase. Because of the high defect concentration and 
the resulting high diffusivity of the cations, precipitation processes occur 
in this and related systems at quite low temperatures, in this case about 
300°C. On cooling a sample, it is not possible to prevent the formation of 
defect clusters, even with the most rapid quench. 

When growth is rapid or occurs at low temperatures with a composition 
change, the rate of flow of heat or material limits the growth rate and fixes 
the morphology. Under these conditions the rate at which heat is 
dissipated or material added to a growing precipitate is proportional to the 
inverse radius of curvature of the growing tip of the crystal. As a result, 
dendritic forms result, with the radius of curvature of the growing tip 
remaining small and side arms developing to form a treelike structure. 



•cimens 
1966). 



* H. Jagodzinski, Z. Krist, 109. 388 (1957), and H. Saalfeld. Ber. Deut. Keram Ges., 39, 52 
(1962). 



I! *! ll 



SI 



n 




in.e rm edia,e precipices. Cour.esy G. hZ?™ * "* ""^ °' ,he 

436 



1& 



Fig. 9.41 
Courtesy 



Depen< 
illustra 
basic i 
tion oi 
along ( 
long p< 
rate-d( 
graphi 
limitec 
period 
sphere 
minim 
Het* 
that pi 
when 
may s 
precip 
grain 
wustit 



REACTIONS WITH AND BETWEEN SOLIDS 



437 




Depending on the conditions of formation, different structures arise as 
illustrated for the precipitation of magnesiofernte "-^^ 
basic refractory brick (Fig. 9.44). Sometimes a crysta Hog rapine onenta 
tion of the precipitate occurs in which the platelets of MgFe,0, form 
along 1W) planes in the parent magnesia phase. On prec.pitat.on during 
SZsof time at a lower temperature at which .diffusion is probably 
rtte 8 determining, dendritic precipitates form wh* j rtdlhave cry^tallo- 
Eraohic orientations with the matrix but in which the rate of growth is 
S£ so that starlike crystals result (Fig. 9.450); 
periods at the higher temperature levels, there .s a tendency for a 
spheroidal precise to develop in which the total surface energy is a 
minimum and the strain energy may be relieved by plas ic flow. 

Heterogeneous Precipitation. It is frequently observed (F.g. 9.45a) 
that preefpitation of a new phase occurs primarily along gram boundaries; 
wh n' more extensive precipitation occurs (F* MSb) [ ■^J^S 
may show precipitates surrounded by an area of material which ,s nearly 
pTcipitaJn free. This can result from het erogeneous nuc eation a the 
grain boundary, although in the case of prec.p.tation of Fe 3 0, from 
SteL microstore observed at low magnificat.ons results pnmar- 




438 



INTRODUCTION TO CERAMICS 




f^C 4 E.eS 0 U nTff r T i0j " Sn ° ! CryS ' aI ho ™S™™<> « '600»C and annealed 5mi „ at 
ie Sy E M%r fndT^r ' 0Wer r, ' 8h,: ° P,iCal diffraC,i °" — » "PP^ 



han f rnm a T ?- ,n ^ ^ adjaCent t0 grain boundaries rather 

than from a nucleat.on process. In this system the grain boundaries act as 
h gh diffusiv.ty paths, discussed in Chapter 6, which allow nuclei at the 
gram boundary to grow initially at a faster rate than nuclei in the bulk 
wh,ch tends to denude the area adjacent to the grain boundary of o^^ te-' 
at later stages ,n the precipitation process (Fig. 9.456) there is an area 
adjacent to the grain boundaries which tends to be predpitaL free ?n ^ 
system, as for many of those previously described the prec P i a te 
parties are coherent with the matrix crystal, and a. have the same 
orientation in each grain of wustite. 

bounl SamP ' eS A n T hiCh SOlUbiHty iS Sma "' direct observation of grain 
boundaries and d.slocations indicates that second-phase precipitation a" 
he Se Slt es is very common indeed. Particularly for Ly Z 
contaming s.hcates as minor impurities, coherency's not to be expected 



Fig. 9.43. Lo 
Courtesy C. A 



440 



INTRODUCTION TO CERAMICS 





rat v>. >%• a 



morphology( 2 32x)Xourte S TTSandK Tr** (975><): (C) SpheroidaI 

B. Tavasci, Radex Rdsch., Ijjfi^ * ™ C ' L - 1 ' a ' 149(!948),and 



and gram boundaries, as illustrated in Fig. 9.46 dislocations 

9.6 Nonisothermal Processes 

We have considered diffusional processes a* th™ . . 

mal conditions- however ma™ ' pr0CesSes as tne y occ "r under isother- 

co^"™^" *• lEmPe ' a,U ' e * <"««. diffusion 

D = D oe - Q,KT 
'See W. D. Kingery, / Am. Ceram. Soc, 57, 1 (| 9 74). 




;k as (a) platelets 
<); (c) spheroidal 
',8, 149(1948), and 



ition enhances 
xt dislocations 



under isother- 
dures include 
>f the effect of 
ecipitation of 
from a high 

, the diffusion 



Fig. 9.45. Precipitate of Fe>0 4 from wustite{Fe, O) containing (a) 52.67 and (b) 53.10 atom 
% oxygen (95 x). Courtesy L. Himmel. 

441 



Precipi,a,e parses a, ^Z:^^^^^ M. Pa U | US . 

442 



REACTIONS WITH AND BETWEEN SOLIDS 



443 



and we assume the temperature to vary from Tt to T 2 at a linear rate of a, 
the time-dependent diffusion coefficient is 



An approximate diffusion length / may be estimated from the integral; 



Let us consider, as an example of the use of Eq. 9.89, A1 2 0 3 impurities in 
MgO. The diffusion of supersaturated aluminum ions from within a grain 
to the grain boundary is essentially that for the defect diffusion (vacancy) 
because of the impurity-vacancy pair which tends to form (see Section 
6.4). From the data of impurity diffusion into MgO a value of 2 to 3 eV (50 
to 75kcal/mole) seems a reasonable activation energy for vacancy 
diffusion. Assume a sample of MgO annealed at high temperature 
contains 100 ppm A1 2 0 3 . If the sample is cooled at 0.1°C/sec, the solubility 
limit at 1300°C produces the onset of grain-boundary precipitation. For an 
assumed diffusivity of 10" 8 cm 2 /sec at 1300°C and Q=2eV, Eq. 9.89 
yields a value of 30 microns for the effective diffusion distance. A similar 
calculation for 100 ppm MgO in A1 2 0 3 (T, = 1530°C, Q-3eV, and 
D =5 x 10" 8 cm 2 /sec) yields a segregation thickness of 60 microns. 

There are many other examples of ceramic processes which occur 
during nonisothermal annealing. As porcelain or refractories are proces- 
sed in production kilns, much of the densification and reaction between 
granular components takes place during the heating cycle. We consider 
finally two examples of nonisothermal kinetic processes which are 
described in detail in Section 9.4 and Section 10.3 for isothermal condi- 
tions. 

First, let us consider the nonisothermal decomposition reaction (Eq. 
9.63) in which CaC0 3 decomposes to CaO and C0 2 . The reaction rate is 
determined by decomposition at the surface and obeys linear kinetics. 
The reaction rate R is equal to the change in weight per unit area of the 
CaC0 3 with time, d(cola)ldL Thus Eq. 9.4 can be rewritten 





(9.89) 




(9.90) 



d{(ola) 
dt 




exp 



/AS f \ / AH r \ , / AH'N 

t^) ex n-^r Aexp rKr) 



(9.91) 



444 



INTRODUCTION TO CERAMICS 



If the temperature of the CaCO, is changed at a constant rate, T = at, the 
weight change as a function of temperature is obtained from 

djtola) A I AH'\ 

~df « exp (--rt) ' (9-92) 

The integration of Eq. 9.92, assuming that A is not a strong function of 
temperature, yields the approximate solution 

A<a ART 2 I AH'\ 

6> 0 ~aAH ,eXp rW/ (9-93) 

The form of the equation is similar to Eq. 9.89. A plot of the non- 
lsothermal decomposition in vacuum of a single crystal of CaCOj is 
given in Fig. 9.47. For this reaction and for several other endothermic 
decomposition reactions the activation energy for decomposition is 
identical with the heat of reaction (Eq. 9.63). 

As a final example of nonisothermal kinetic processes consider the 
sintering of glass spheres (discussed in Chapter 10). The shrinkage rate 
djAL/Lo) .... 

j t , which is a function of the surface tension y, the viscosity 

t, = Be Q,RT , and the particle radius a, can be determined from nonisother- 




REACTIONS WITH AND BETWEEN SOLIDS 



445 



3 x 10 



Temperature (°C) 
680 660 640 620 600 



580 




5 X 10 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 
1000 / T <°K) 

Fig. 9.48. Kinetic data for nonisothermal sintering of 025-fim glass particles (soda-lime- 
silica). From I. B. Cutler, /. Am. Ceram. Soc. y 52, 14 (1969). 



mal sintering from* 

Kinetic data illustrating Eq. 9.94 are given in Fig. 9.48 for the sintering of 
25-micron soda-lime-silica particles in an atmosphere of oxygen and 
water vapor. 

Suggested Reading 

1. H. Schmalzried, Solid State Reactions, Academic Press, New York, 1974. 

2. G. C. Kuczynski, N. A. Hooton, and C. F. Gibbon, Eds., Sintering and Related 
Phenomena, Gordon and Breach, New York, 1967. 

3. G. C. Kuczynski, Ed., "Sintering and Related Phenomena Materials Science 
Research, Vol. 6, Plenum Press, New York, 1973. 

4. T. J. Gray and V. D. Frechette, Eds., "Kinetics of Reaction in Icnic Systems/" 
Materials Science Research, Vol. 4, Plenum Press, New York, 1969. 



*L B. Cutler, J. Am. Ceram. Soc. t 52, 14 (1969). 



446 



INTRODUCTION TO CERAMICS 



5. P. Kofstad, Nonstoichiometry, Diffusion, and Electrical Conductivity in 
Binary Metal Oxides, John Wiley & Sons, New York., 1972. 

6. For a discussion of dissolution kinetics see A. R. Cooper, Jr., B. N. Samaddar 
Y. Oishi, and W. D. Kingery, J. Am.Ceram. Soc.,41, 37 (1964) --47 249 f l964V 
and 48, 88 (1965). ,^y U w), 

G. M. Schwab, Ed., Reactivity of Solids, Elsevier Publishing Company, New 
York, 1965. 

W. D. Kingery, Ed., Kinetics of High Temperature Processes, John Wiley & 
Sons, New York, 1959. 

M. E. Fine, Introduction to Phase Transformations in Condensed Systems, 
McGraw-Hill, New York, 1964; Bull. Am. Ceram. Soc, 51, 510 (1972). 



7. 



9.1. 
9.2. 



9.3. 



9.4. 



9.5. 



Problems 

Rates of solution can be controlled by (a ) diffusion in the liquid, (b) diffusion through 
a reaction layer, or (c) phase-boundary reaction. How would you distinguish these? 
While measuring the rate of decomposition of alumina monohydrate, a student finds 
the weight loss to increase linearly with time up to about 50% reacted during an 
rsothermal experiment. Beyond 50%, the rate of the weight loss is less than linear 
lhe linear isothermal rate increases exponentially with temperature. An increase of 
temperature from 451 to 493X increases the rate tenfold. Compute the activation 
energy Is this a d 1 ffusion-controlled reaction, a first-order reaction, or an interface- 
controlled reaction? 

Consider formation of NiCr 2 0 4 from spherical particles of NiO and Cr 2 0, when the 
rate is controlled by diffusion through the product layer 

(a) Carefully sketch an assumed geometry, and then derive a relation for the rate of 
formation early in the process. 

(b) What governs the particles on which the product layer forms'' 

(C) NicJSJ^yf ° m > D ° N, ' Crj ° 4 - Wh ' Ch C ° n,r0lS thC ra,C ° f forma,ion of 

Polymorphic transformations in solids result in polycrystalline materials of small size 
(fine-gramed) or large size (coarse-grained), depending on the rates of nucleation and 
nude, growth. How do these rates vary to produce fine-grained and coarse-grained 
products? Draw a tune versus size for an individual grain illustrating the growth of a 

trLTo3io„ * " ^ ^ ^ time °" ' ime 3XiS 3t ,he moment of 

IX? £ t ? " y Welght 3t 17 °°° C - 1% at 180O ° C - ,2% at «WC. and 0% at 
1500 C. They observed crystallization of spinel crystals from the solid solution region 

temtratu're 'T"?* "J** lhe >° M « a single phase at room 

temperature. The exsolved spinel appeared uniformly without regard to grain 

IZXT W P l riC ' aSe 8niinS bUt ° n Specific p,anes - < fl > Is 'he nucleation of 

spinel homogeneous or heterogeneous within the periclase grains? (b) Account for 
he appearance of spinel crystals along specific planes of periclase crystals. Predict 

L «T, r . a - e 0f cr y stalliza,ion ve ™ temperature for nucleated periclase 
sohd solution containing 5% Al,0, over the temperature range 0«C to I850°C 




REACTIONS WITH AND BETWEEN SOLIDS 



447 



mductivity in 

N. Samaddar, 
H, 249 (1964); 

3mpany, New 

lohn Wiley & 

zsed Systems, 
0 (1972). 



9.6. In the previous problem, we described a solid solution of A1 2 0 3 in MgO. Assuming a 
manufacturer of basic refractories uses MgO contaminated with 5 to 1% A1 2 0 3 , what 
microstructure differences will exist in slow-cooled refractory compared to fast- 
cooled material? Would you predict sintering by self -diffusion (bulk), grain growth, 
and cation diffusion in this material would be different than in pure MgO? Why? 

9.7. Suppose that the formation of mullite from alumina and silica powder is a diffusion- 
controlled process. How would you prove it? If the activation energy is 50 kcal/mole 
and the reaction proceeds to 10% of completion at 1400°C in 1 hr, how far will it go in 
1 hr at 1500°C? in 4 hr at 1500°C? 

9.8. An amorphous Si0 2 film on SiC builds up, limiting further oxidation. The fraction of 
complete oxidation was determined by weight gain measurements and found to obey 
a parabolic oxidation law. For a particular-particle-sized SiC and pure 0 2 the 
following data were obtained. Determine the apparent activation energy in kcal/mole. 
How can it be shown that this is a diffusion-controlled reaction? 



ffusion through 
iguish these? 
a student finds 
cted during an 
;ss than linear. 
An increase of 
the activation 
ir an interface- 

t 2 0 3 when the 
i for the rate of 

>f formation of 

Is of small size 
nucleation and 
coarse-grained 
he growth of a 
the moment of 

0 3 is soluble in 
0°C, and 0% at 
solution region 
phase at room 
gard to grain 
; nucleation of 
) Account for 
ystals. Predict 
ated periclase 
3.1850°C. 



Temp (°C) 


Fraction Reacted 


Time (hr) 


903 


2.55 x 10" 2 


100 


1135 


1.47 x 10" 2 


10 




4.26 x 10 -2 


100 


1275 


1.965 xlO" 2 


10 




6.22 x 10" 2 


100 


1327 


1.50 x 10~ 2 


5 




4.74 x 10 -2 


50 



9.9. The slow step in the precipitation of BaSO« from aqueous solution is the interface 
addition of the individual Ba** and S0 4 ". Diffusion to the surface is assumed 
sufficiently fast that we may neglect any concentration differences in the solution. 
Assume that the rate of addition is first-order in both Ba ++ and SO/. 

(a) Derive an expression for the approach to equilibrium in terms of the rate 
constants for the forward and back reaction and the surface area. 

(b) What is the effect of an excess of Ba ++ ? 

(c) Why can you assume the surface area to be constant? 

(d) How would you modify your approach to include a correction for diffusion? 

9.10. One-micron spheres of Al 2 O a are surrounded by excess MgO powder in order to 
observe the formation of spinel. Twenty percent of the A1 2 0 3 was reacted to form 
spinel during the first hour of a constant-temperature experiment. How long before all 
the A1 2 0 3 will be reacted? Compute the time for completion on the basis of (a) no 
spherical geometry correction and {b) the Jander equation for correction of spherical 
geometry. 

9. 11. In fired chrome ore refractories, an R 2 0 3 phase precipitates as platelets in the spinel 
phase matrix. Write the chemical equation for this reaction, and explain why it 
occurs. The precipitate is oriented so that the basal plane in the R 2 0 3 phase is parallel 
to the (111) plane in the spinel. Explain why this should occur in terms of crystal 
structure. 



1 




I !i 



1 1 



li 



t r 




Grain Growth, 
Sintering, 
and 

Vitrification 



Object of this chapter. 8 * treatment befo ^ use; these are the 

D «nng the usual processino of o 
Powde rs are co^JZ^T^l °' "-™,Iine 

f n j t Ve °P usefu ' Properties. During the firino temperatUre efficient to 

n fally beca us e of decomposition £ 5^ 

the phases present. On further h J ; ransform ations in some of 

srs %r major changes 

grain s 12e , there ,s a change in pore shaoe »^ ^ * " S an incre ase in 

be U f SU t a " y t0 8IVe 3 d «^SSSt? to " in P ° re size »»<• 

be sohd-state reactions forming new nhl " y Ceramics th ere may 

t-ons, decompositions of crystaflin e l m 7° lym0rphic ""ansforma 
gases, and a variety of btherTh, C ° mpOUnds to form new phases or 
-Portance in particular case^ ^ ^ "« ^quenUyT^ 
events. are not essentia, to the main stream of 

happen, and so many varies that ^ ^ -»ich can 

mere cataloging of phenomena T ° Ccas,0n a«y important that nn 
s t»dy- In general, we shaH h " Pr ° V ' de a sou ^ basis for f„l 
^n-growth PbenomVnt , le b cVnd"rth e fi d t "? 
systems, and finally with more corne l , dens ' fic ation of single-phase 

many — 

448 



10.1 R eery si 

The terms i 
indefinite usa. 
been used to 
and other phe 
are mainly coi 
taliization is 
generation of 
plastically def 
grain size of ; 
ously during h 
Secondary rec 
grain growth, i 
grow at the ex 
Although all t\ 
secondary rec 
Primary Re 
increased enei 
energy stored 
Although this 
(which is a 10 
change to eff< 
If the isotl 
deformed mat 
constant rate 
size is d, 



where U is th 
period. This i 
chloride cryst 
470°C. The i 
nucleation pr< 
mined by the 
The nucleai 
nucleus to be 
which the lov 
surface free 
required for 
nucleus. If an 
increases to 




th, 

nd 
on 



sforma- 
fabrica- 
are the 
are the 

/stalline 
cient to 
y occur 
some of 
porous 
rease in 
size and 
ere may 
isforma- 
tiases or 
of great 
tream of 

lg of the 
tich can 
that no 
further 
tion and 
.le-phase 
here are 



GRAIN 



GROWTH, SINTERING, AND VITRIFICATION 



449 



10 1 Recrystallization and Grain Growth 

indefinite usage in much of *«^^S^cipiLion. exsolution, 
been used to mclude the microstructure. We 

and other phenomena which produce changes 

^^^^^^^l^ Q ? a new 
tallizahon is the process by wn ' c ix which has been 

generation of stramfree grams ° C ™ T *"*™ hy which the average 
p,asti C ally deformed. Gram grovth * ^the proce ss by ^ 
grain size of strainfree or dilution, 
ously during heat treatment or discontinuous 

increased energy of a matrix which has ; been ^^Jf 5 tQ , ^ 
energy stored in the deformed matrix is of th order of ^ 

Although this is small ^^^^ a h ^^ a .uflteient enw 
(which is a 1000 or more times this value) pro 

change to effect grain-boundary movement ]s in a 

size is d, 



d = l/(f -<o) 



(10.1) 



/ „\ t ; c the time and f 0 is the induction 
where U is the growth rate (cm/se ) ^^Jza^ of a sodium 
period. This is illustrated in F,g^ 1& for recrysta ^ ^ 

chloride crystal which had been deformed ired for a 

^ t .^ rhTtL™at; 2 realization , S deter- 

The nucleation processes »^°^^ , |ne cri tical diameter at 
nucleus to be stable, its size must ™ , to the increased 

which the lowered free energy of the new .^ am ^ • hc time 



450 

3 



INTRODUCTION TO CERAMICS 




10 

Time (min) 



F'g- 10.1. Recrystallization of NaCI 
deformed at 40O°C (stress - 
40O0g/ mmJ) d recrysta , lized 

Sf 19 sr Mu " er ' zp ^-- 96 ' 



observed that nuclei ^d™™ ^ 

corners, for example As th* t - ! naea to form first at grain 

nuc.ea.ion increase : xp tel,; y emPe ' a,Ure K """ eased . «* ™. », 



. ... , (10.2) 
where W is the number of nuclei and A n *u 
for nucleatio, Conse^t^" £££ 

creases rap.dly as the temperature level is r.S t '~UWI*). 

from the constant driving Ce^^TfT 8n>Wth ratC resu,ts 

the strained matrix and ^^^f^" V**?*'™* 
to*™^\»iten^« n ^ t ^*^\^ eram size is 
present when they finally impinge on onTanother t£ T ■ ° f ^ 
necessary for grain growth is the jumping of an atorJ f miS, ' C Pr ° CeSS 
boundary to the other and is similar to^^ZSZ^tn'l ^ ° f " 
Co "^-'ythe,emperatured^^ 



. (10.3) 

ne ^rowth-rate-temperature curve for 

*Z P/iyj., 96, 279 (1935). 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 



451 



recrystallization of sodium chloride has a knee similar to that observed 
for diffusion and conductivity data, as discussed in Chapter 6. 
■ Since both the nucleation^ rate and the growth rate are strongly 
temperature-dependent, the overall rate of recrystallization changes 
rapidly with temperature. For a fixed holding time, experiments at 
different temperatures tend to a show either little or nearly complete 
recrystallization. Consequently, it is common to plot data as the amount 
of cold work or the final grain size as a function of the recrystallization 
temperature. Since the final grain size is limited by impingement of the 
grains on one another, it is determined by the relative rates of nucleation 
and growth. As the temperature is raised, the final grain size is larger, 
since the growth rate increases more rapidly than the rate of nucleation. 
However, at higher temperatures recrystallization is completed more 
rapidly, so that the larger grain size observed in constant-time experi- 
ments (Fig. 10.2) may be partly due to the greater time available for grain 
growth following recrystallization. The growth rate increases with in- 
creasing amounts of plastic deformation (increased dnving force), 
whereas the final grain size decreases with increasing deformation. 

In general, it is observed that (1) some minimum deformation is 
required for recrystallization, (2) with a small degree of deformation a 
higher temperature is required for recrystallization to occur, (3) an 
increased annealing time lowers the temperature of recrystallization, and 




500 600 700 800 900 1000 1100 1200 1300 1400 
Temperature CO 

Fig 10.2. Effect of annealing temperature on grain size of CaF, following compression at 
80,000 psi and 10 hr at temperature. From M. J. Buerger. Am. Mineral.. 32, 2% (1947). 




452 



INTRODUCTION TO CERAMICS 



Ml 
!{ 

Hi 

\\l 
if 

ti 



such as sodium chloride Tea cium T ^ d /°7 f e,at,ve, y soft Aerials, 
recrystallization do occur I h « 2» be^' J* 0 ™? 1 ™ and P ri ™ry 

heated at elevated temperatures A th, £C 8ra ' n S,Ze when 

obvious that some gES ^ JT' ^ 5,26 inCreaSCS ' il » 
of looking at grain growth Ts a *J < < d ' Sappear - An equivalent way 
the driving fo^ ?^ h ?p^ s ^ s ^ d t d,MpPe ?™ e ° f 8rai - 
fine-grained material al/tZ T the dlfference «" energy between the 
decrease in H^^^^^^* ^ 

.fVa'^ 

d.fference across a curved grain boundary which is gTven by ^ 



AG 



(10.4) 



are the principal radii of curvature ™ ' te /"° l "™l™«.»nd r, and r, 
and discussed in Chapter 5 Th»,T , ? ° nsh ' P has bee " d «"™1 

* m .a„i„ g ,s .^ssr s : h „r y b 0 ; re rr if 

the two sides of a grain boundarv k th. 1 i * 8y ° f matenal on 
boundary move toward its cen f cur ST ^ make ' the 

boundary moves is proportional to H>wJ^Jt t T T t * 
atoms can jump across the boundary ^ me at wh,ch 

<££J^^ thC abso.ute-react.on- 

a boundary (Fig. ,0 3 rate J Mh? . ^ C ° nS ' der StrUCt " re ° f 
which atoms jum ^ across efn " ^ by ,he rate «' 
atom's posit.on is shown n Fig e " er ^ an 

rig- lU.Jb, and the frequency of atomic jumps 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 



e initial 
ntinued 
of grain 

lich are 
taterials 
primary 
aterials, 
primary 
in mag- 
;apter 4 

;urs, an 
e when 
;es, it is 
;nt way 
s. Then 
een the 
;om the 
y. This 
from a 

/ith the 
-energy 



(10.4) 

curved 
i and r 2 
derived 
ewed if 
jrial on 
.<es the 
/hich a 
; which 

action- 
;ture of 
: rate at 
vith an 
. jumps 





Position 
(b) 



Fig. 10.3. (<i) Structure of boundary and (b) energy change for atom jump. 



in the forward direction is given by 



RT ( &G\ 



(10.5) 



(10.6) 



and the frequency of reverse jumps is given by 

RT I AG f + AG \ 

' BA= NK eXP l KT / 

so that the net growth process, U = A/, where A is the distance of each 
jump is given by 

W .tf-x</--M-^(»)«p(-^)('-»5T) (10J) 

and since l-«pf?-f?. + § ^ 4 °' " 

AH' - T AS 1 , 

which is equivalent in form to Eq. 10.3 given previously^That is the rate 
of growth increases exponentially with temperature. The un.t step m- 
volved is the jump of an atom across the boundary, so that the activation 
energy should correspond approximately to the act.vat.on energy for 
boundary diffusion. 



454 



INTRODUCTION TO CERAMICS 



of 1 20° I bOUnd , aneS ^ CqUal energy ' the y meet t° form angles 
ot 120 . If we cons.der a two-dimensional example for illustrative 
purpo Ses angles of ,20°betwecn grains with straight sides can occu only 
for s.x-s.ded grams. Grains with fewer sides have boundaries that a e 
concave when observed from the center of the grain. Shapes of grains 
having different numbers of.sides are illustrated in Fig. 104" a sample 
with un.form grain size is shown in Fig. 10.5. Since grain bounds 
m.grate toward their center of curvature, grains with less than six side 
tend t 0 grow smaller> and grajns wjth more than s . des x s des 

larger. For any one gram, the radius of curvature of a side is directlv 
proporuonal to the grain diameter, so that the driving force, and ^ herefc e 
the rate of gram growth, is inversely proportional to grain size 



d = 



dt 



and integrating, 



k 
d 



d-d 0 = (2k)'' 



(10.9) 



(10.10) 



where d 0 .s the grain diameter at time zero. Experimentally it is found that 
when log d ,» plotted versus log t, a straight line is obtaLd (Rg ,0 6) 
Frequently the slope of curves plotted in this way is smaller than one-half,' 




Fig. 10.4. Schematic drawing of polycrystalline snerimpn tk 0 ■ 

boundaries changes as ,he number of sfdes a e fZ e" slhl T, ° f « hc 

•he radius of curvature is less, the more ,he number" Z differ fro T ^ ^ md 

•he directions in which boundaries migrate. From J E B J * ' ^ '"^ 




usually fall: 
one being t 
reason is th 
growth. 

A somev^ 
Bi such tha 
force Fi re: 



Fig. 10.6. 
Burke. 



Lo, 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 455 



angles 
Native 
ir only 
at are 
grains 
ample 
daries 
sides 
grow 
rectly 
refore 



(10.9) 




Fig. 10.5. PolycrystallineCaF 2 illustrat- 
ing normal grain growth. Average angle at 
grain junctures is 120°. 



10.10) 

dthat 
10.6). 
;-half, 



usually falling between 0.1 and 0.5. This may occur for several reasons, 
one being that d Q is not a large amount smaller than d \ another common 
reason is that inclusions or solute segregation or sample size inhibits grain 
growth. 

A somewhat different approach is to define a grain-boundary mobility 
Bi such that the boundary velocity v is proportional to the applied driving 
force Fi resulting from boundary curvature: 



(10.11a) 



of the 




idicate Fig. 10.6. Log grain diameter versus log time for grain growth in pure a-brass. From J. E. 

Burke. 



456 



INTRODUCTION TO CERAMICS 



For the atomic-jump mechanism illustrated in Fig. 10.3, the boundary 
mobility is given by the atomic mobility divided by the number of atoms 
involved, n a : . . 



where D b is the grain-boundary diffusion coefficient, fl is the atomic 
volume, S is the boundary area, and w is the boundary width. Since the 
average boundary velocity is equal to v and the driving force is inversely 
proportional to grain size, a grain-growth law of the form of Eqs. 10.9 and 
10.10 results. However, as discussed in Chapter 5, the actual structure of 
a ceramic grain boundary is not quite so simple as pictured in deriving 
Eqs. 10.8 and 10.1 If?. Even for a completely pure material there is a 
space-charge atmosphere of lattice defects associated with the boundary 
and usually solute segregation as well, as shown in Figs. 5.11, 5.12, 5.17, 
and 5.18. The effect of this lattice defect and impurity atmosphere is to 
sharply reduce the grain-boundary velocity at low driving forces, as 
shown in Fig. 10.7 and analysed by J. Cahn* and K. Liicke and H. D. 
Stuwe.t The influence of this atmosphere becomes stronger as the grain 



Force (10 4 dyne/cm 2 ) 

Fig. 10.7. Variation of boundary velocity v with driving force F at 750°C for a 20° tilt 
boundary in NaCl. From R. C. Sun and C. L. Bauer, Acta Met., 18, 639 (1970). 

size increases, the solute segregate concentration increases, and the 
average boundary curvature decreases. Additions of MgO to Al 2 0 3 , CaCI 2 



to KC1 ; 
proved * 
When 
specime 
when th 
to form 
boundai 
occurs, 
movemt 
boundai 
decreasi 
area of 
pull it 
inclusio 
insuffici 
It has b 



where c 
and f di i 
only ap 
creases 
For t 
attache 
Anothe 



/ 



( 





*Acta Met., 10, 789 (1962). 
lActa Met., 19, 1087 (1971). 




I 




boundary 
of atoms 

(10.11b) 

le atomic 
Since the 
inversely 
10.9 and 
ructure of 
i deriving 
there is a 
boundary 
5.12, 5.17, 
here is to 
['orces, as 
ind H. D. 
. the grain 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 457 

to KC1 and of Th0 2 to Y 2 0 3 in amounts below the solubility limit have 
proved effective as grain-growth inhibitors. 

When grains grow to such a size that they are nearly equal to the 
specimen size, grain growth is stopped. In a rod sample, for example, 
when the grain size is equal to the rod diameter, the grain boundaries tend 
to form flat surfaces normal to the axis so that the driving force for 
boundary migration is eliminated and little subsequent grain growth 
occurs. Similarly, inclusions increase the energy necessary for the 
movement of a grain boundary and inhibit grain growth. If we consider a 
boundary such as the one illustrated in Fig. 10.8, the boundary energy is 
decreased when it reaches an inclusion proportional to the cross-sectional 
area of the inclusion. The boundary energy must be increased again to 
pull it away from the. inclusion. Consequently, when a number of 
inclusions are present on a grain boundary, its normal curvature becomes 
insufficient for continued grain growth after some limiting size is reached. 
It has been found that this size is given by 



where d t is the limiting grain size, d, is the particle size of the inclusion, 
and f di is the volume fraction of inclusions. Although this relationship is 
only approximate, it indicates that the effectiveness of inclusions in- 
creases as their particle size is lowered and the volume fraction increases. 

For the process illustrated in Fig. 10.8, the boundary approaches, is 
attached to, and subsequently breaks away from a second-phase particle. 
Another possibility is that the grain boundary drags along the particle 




>, and the 

l 2 0 3 , CaCl 2 



for a 20° tilt 
'70). 




Fig. 10.8. Changing configuration of a boundary while 
passing an inclusion. 



J 




458 



INTRODUCTION TO CERAMICS 



which remains attached to the boundary as it moves. This requires 
material transport across the particle, which may occur by interface or 
surface or volume diffusion, by viscous flow, or by solution (precipitation 
in a liquid or glass inclusion), or by evaporation (condensation in a gas 
inclusion). We can define an inclusion particle mobility B p relating the 
driving force and particle velocity v p = B P F P in the same way as has been 
done for the boundary (Eq. 10.1 lb) and for atomic diffusion in Chapter 6. 
When the inclusion is dragged by the boundary, their velocities are 
identical; in the case in which B p < B b we can neglect the intrinsic 
boundary mobility, and the resulting grain-boundary velocity is controlled 
by the driving force on the boundary together with the mobility and 
number of inclusions per grain boundary, p: 



v b =■ 



B p F b 



(10.13) 



The inclusion particle moves along with the boundary, gradually becom- 
ing concentrated at boundary intersections and agglomerating into larger 
particles as grain growth proceeds. This is illustrated for the special case 
of pore agglomeration in Figs. 10.9 and 10.10. 

Thus, second-phase inclusions can either (1) move along with boun- 
daries, offering little impedance; (2) move along with boundaries, with the 
inclusion mobility controlling the boundary velocity; or (3) be so im- 
mobile that the boundary pulls away from the inclusion, depending on the 
relative values of the boundary driving force (inversely proportional to 
grain size), the boundary mobility (Fig. 10.7), and the inclusion particle 
mobility, which, depending on the assumed mechanism and particle 
shape, may be proportional to r~\ r~\ or r P " 4 .* As grain growth proceeds, 
the driving force diminishes, and any inclusions dragged along by the 
boundary increase in size so that their mobility decreases. As a result, the 
exact way in which second-phase inclusions inhibit grain growth not only 
depends on the properties of the particular system but also can easily 
change during the grain-growth process. Sorting out these effects requires 
a careful evaluation of the microstructure evolution in combination with 
the kinetics of grain growth and a detailed knowledge of system proper- 
ties. Inhibition of grain growth by solid second-phase inclusions has been 
observed for MgO additions to Al 2 0„ for CaO additions to Th0 2 , and in 
other systems. 

A second phase that is always present during ceramic sintering and in 
almost all ceramic products prepared by sintering is residual porosity 

*P. G. Shewmon. Trans. A./.M.E., 230, 1 134 (1964); M. F. Ashby and R. M. A. Ccntamorc 
Acta Met., 16, 1081 (1968). 



rerru 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 459 



s requires 
terface or 
scipitation 
n in a gas 
bating the 
s has been 
Chapter 6. 
>cities are 
; intrinsic 
controlled 
bility and 



(10.13) 

ly becom- 
nto larger 
ecial case 

'ith boun- 
>, with the 
oc so im- 
ing on the 
>rtional to 
n particle 
i particle 
proceeds, 
lg by the 
result, the 
i not only 
:an easily 
s requires 
ition with 
n proper- 
has been 
) 2 , and in 

ng and in 
porosity 

"Centamore, 




Fig. 10.9. {a) Pore shape distorted from spherical by moving boundary and (b) pore 
agglomeration during grain growth. 

remaining from the interparticle space present in the initial powder 
compact. This porosity is apparent both on the grain boundaries 
(intergranular) and within the grains (intragranular) in the sintered CaF 2 
sample shown in Fig. 10.5. It is present almost entirely at the grain corners 
(intergranular) in the sintered U0 2 samples shown in Fig. 10.10. As with 
particulate inclusions, pores on the grain boundaries may be left behind 
by the moving boundary or migrate with the boundary, gradually ag- 
glomerating at grain corners, as illustrated in Figs. 10.9 and 10.10. In the 
early stages of sintering, when the boundary curvature and the driving 
force for boundary migration are high, pores are often left behind, and a 
cluster of small pores in the center of a grain is a commonly observed 
result (see Fig. 10.5). In the later stages of sintering, when the grain size is 
larger and the driving force for boundary migration is lower, it is more 
usual for pores to be dragged along by the boundary, slowing grain 
growth. 



460 



INTRODUCTION TO CERAMICS 




Fig. 10.10. Grain growth and pore growth in sample of UO ; after (a) 2 min, 91.5% dense, 
and (b) 5 hr, 91.9% dense, at 1600°C (400x). From Francois and Kingery. 

Another factor that may restrain grain growth is the presence of a liquid 
phase. If a small amount of a boundary liquid is formed, it tends to slow 
grain growth, since the driving force is reduced and the diffusion path is 
increased. There are now two solid-liquid interfaces, and the driving force 
is the difference between them, that is, (1/r, + l/r.) A - (1/r, 4- l/r 2 ) B , which 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 



461 



>1.5% dense, 



of a liquid 
ds to slow 
on path is 
ving force 
\)b, which 



is smaller than either alone; in addition, if the liquid wets the boundary, 
the interface energy must be lower than the pure-grain-boundary energy 
Also the process of solution, diffusion through a liquid film and 
precipitation is usually slower than the jump across a boundary. How- 
ever this case is more complex in that grain growth may be enhanced by 
the presence of a reactive liquid phase during the densification process, as 
discussed in Section 10.4. In addition, a very small amount of liquid may 
enhance secondary recrystallization, as discussed later, whereas larger 
amounts of liquid phase may give rise to the grain-growth process 
described in Chapter 9. In practice, it is found that addition of a moderate 
amount of silicate liquid phase to aluminum oxide prevents the extensive 
erain growth which frequently occurs with purer materials. 

Secondary Recrystallization. The process of secondary recrystalliza- 
tion, sometimes called discontinuous or exaggerated grain growth, occurs 
when some small fraction of the grains grow to a large size, consuming the 
uniform-grain-size matrix. Once a single grain grows to such a size that it 
na many more sides than the neighboring grains (such as the gram with 
S ty sides illustrated in Fig. 10.4), the curvature of each side increases 
and it grows more rapidly than the smaller grains with fewer sides The 
increased curvature on the edge of a large grain ,s particularly evident in 
Fig. 10.11, which shows a large alumina crystal growing at the expense of 
a uniform-particle-size matrix. 

Secondary crystallization is particularly likely to occur when continu- 
ous grain growth is inhibited by the presence of impurities or pores. 
Under these conditions the only boundaries able to move are those wi h a 
curvature much larger than the average; that is, the «^ ed «™™ 
with highly curved boundaries are able to grow, whereas the matrix 
material remains uniform in grain size. The rate of growth of the large 
Tains is initially dependent on the number of sides. However, after 
growth has reached the point at which the exaggerated grain diameter is 
much larger than the matrix diameter, d g > d the cu^ 's deter- 
mined by the matrix grain size and is proportional to 1/<L- That s, there is 
an induction period corresponding to the increased growth rate and the 
formation of a grain large enough to grow at the expense of the constant- 
grarsize matrix. Therefore, the growth rate is constant as long as the 
Sain size of the matrix remains unchanged. Consequently, the kinet.cs of 
secondary recrystallization is similar to that of primary recrystall.z t.on, 
even though the nature of the nucleation and driving force is different. 

Secondary recrystallization is common for oxide, titanate, and fern te 
ceramics in which grain growth is frequency inhibited by --or amounts 
of second phases or by porosity during the sintering process A typical 
resultant structure is illustrated for barium titanate in Fig. 10.12, and the 



464 



INTRODUCTION TO CERAMICS 



progressive growth of aluminum oxide crystals during secondary recrys- 
tallization is illustrated in Fig. 10.13. 

When polycrystalline bodies are made from tine powder, the extent of 
secondary recrystallization depends on the particle size of the starting 
material. Coarse starting material gives a much smaller relative grain 
growth, as illustrated in Fig. 10.14 for beryllia. This is caused by both the 
rate of nucleation and the rate of growth. There are almost always present 
in the fine-grained matrix a few particles of substantially larger particle 
size than the average; these can act as embryos for secondary recrystalli- 
zation, since already d g > d mj and growth proceeds to a rate proportional 
to 1/cL. In contrast, as the starting particle size increases, the chances of 
grains being present which are much larger in particle size than the 
average are much decreased, and consequently the nucleation of secon- 
dary recrystallization is much more difficult; the growth rate, proportional 
to \ld my is also smaller. In the data shown in Fig. 10.14, material having a 
starting particle size of 2 microns grows to a final particle size of about 50 
microns, whereas material with an initial particle size of 10 microns shows 
a final grain size of only about 25 microns. This result of a much larger 
final grain size for a smaller initial particle size would be very puzzling if 
the process of secondary recrystallization was not known to occur. 

Secondary recrystallization has been observed to occur with the 
boundaries of the large grains apparently perfectly straight (Fig. 10.15). 
Here the previous discussion of the surface tension and curvature of the 
phase boundary does not apply directly. That is, the boundary energy is 




GRAIN GROWTH, SINTERING, AND VITRIFICATION 465 



:ecrys- 

tent of 
tarting 
■ grain 
3th the 
tresent 
article 
ystalli- 
rtional 
ices of 
an the 
secon- 
rtional 
iving a 
)OUt 50 
shows 
larger 
ding if 
ur. 

th the 
10.15). 
of the 
ergy is 



f .h during 
)f BeO 
Duwefc, 
. Ceram. 



not independent of crystal directions, and the growth planes are those of 
low surf ace energy. These structures all seem to occur in systems having 
a small concentration of impurity which gives rise to a small amount of a 
boundary phase. The driving force for secondary recrystallization is the 
lower surface energy of the large grain compared with the high-surface- 
energy faces or small radius of curvature of adjacent grains. Transfer of 
material under these conditions can only occur when there is an inter- 
mediate boundary phase separating the surfaces of the small and large 
grains. The amount of second phase present tends to increase at the 
boundaries of the large crystals compared with that at other boundaries in 
the system, and a large grain continues to grow once it is initiated. If the 
amount of boundary phase is increased, however, normal grain growth 
and this kind of secondary recrystallization are both inhibited, as discus- 
sed previously. 

Secondary recrystallization affects both the sintering of ceramics and 
resultant properties. Excessive grain growth is frequently harmful to 
mechanical properties (see Sections 5.5 and 15.5). For some electrical and 
magnetic properties either a large or a small grain size may contribute to 
improved properties. Occasionally grain growth has been discussed in the 
literature as if it were an integral part of the densification process. That 
this is not true can best be seen from Fig. 10.16. A sample of aluminum 
oxide with an initial fine pore distribution was heated to a high tempera- 
ture so that secondary recrystallization occurred. The recrystallization 
has left almost the same amount of porosity as was present in the initial 



Fig 10 15 (a ) Isomorphic grains in a polycrystalline spinel. The large grain edges appear 
straight, whereas the shape of the small grains is controlled by surface tension (350x). 
Courtesy R. L. Coble. 





n 




466 



INTRODUCTION TO CERAMICS 



*• ''A^ >-a -? 1 .-,» 



''AAA^'i. 



^r-/ ..v>xw-r:f.--- ;r A : ^H . 



;;'\ 



. v. 
•v . 



..'•AA A * : * 

;AW'-.A.,. 



i. j ^a* * ■ 



':aA-, 



Fig. 10.15 (Continued), (b) Idiomorphic grains of a-6H SiC in a (3-S'iC matrix (lOOOx). 

compact. Elimination of porosity is a related but separate subject and is 
considered in following sections. An application in which secondary 
recrystallization has been useful is in the development of preferred 
orientation on firing of the magnetically hard ferrite, BaFe, 2 Oiv-* For this 

*A. L. Stuijts, Trans. Brit. Ceram. Sac, 55, 57 (1956). 



GRAIN i 



Fig. 10.15 (Continued 



magnetic material 
degree of prefern 
powdered materia 
them to a high ma, 
alignment after Y 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 467 




Fig. 10.15 (Continued), (c) Detail of boundary (75,000 x). Courtesy S. Prochazka. 



magnetic material it is desirable to obtain a high density as well as a high 
degree of preferred orientation in the sintered product. Particles of the 
powdered material can be oriented to a considerable extent by subjecting 
them to a high magnetic field while forming. On sintering there was a 57% 
alignment after heating at 1250°C. On further heating at 1340°C the 





468 



INTRODUCTION TO CERAMICS 




Fig. 10.16. A specimen of alumina {a) sintered 1 hrat I800°C and (b) heated I hrat 1900T 
to Secondary recrystallization. Note that the pore spacing has not changed. Courtesy J. 



preferred orientation increased to 93% alignment, corresponding to the 
structural change brought about by secondary recrystallization. It seems 
apparent that the few large grains in the starting material are more 
uniformly aligned than the fine surrounding material. These grains serve 
as nuclei for the secondary recrystallization process and give rise to a 
highly oriented final product. 



GRAIN • 

10.2 Solid-State ; 

Changes that o< 
in grain size and s 
size. In Section 1( 
the following sect 
that is, the change 
porous compact 
compact, before 
separated by bei 
particular materi; 
properties such a: 
desirable to elimir 
applications it ma 
ing the gas perme 
transfer of materi 
of changes that m 
present can chans 
necessarily chang 
shape of the pon 
becoming more sr. 

Driving Force f< 
to densification is 
free energy by t 
takes place with 




Fig. 10.17. ( 



i 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 



469 



10.2 Solid-State Sintering 

Changes that occur during the firing process are related to (1) changes 
in grain size and shape, (2) changes in pore shape, and (3) changes in pore 
size. In Section 10.1 we concentrated on changes in grain size; in this and 
the following section we are mainly concerned with changes in porosity, 
that is, the changes taking place during the transformation of an originally 
porous compact to a strong, dense ceramic. As formed, a powder 
compact, before it has been fired, is composed of individual grains 
separated by between 25 and 60 vol% porosity, depending on the 
particular material used and the processing method. For maximizing 
properties such as strength, translucency, and thermal conductivity, it is 
desirable to eliminate as much of this porosity as possible. For some other 
applications it may be desirable to increase this strength without decreas- 
ing the gas permeability. These results are obtained during firing by the 
transfer of material from one part of the structure to the other. The kind 
of changes that may occur are illustrated in Fig. 10.17. The pores initially 
present can change shape, becoming channels or isolated spheres, without 
necessarily changing in size. More commonly, however, both the size and 
shape of the pores present change during the firing process, the pores 
becoming more spherical in shape and smaller in size as firing continues. 

Driving Force for Densification. The free-energy change that gives rise 
to densification is the decrease in surface area and lowering of the surface 
free energy by the elimination of solid-vapor interfaces. This usually 
takes place with the coincidental formation of new but lower-energy 




Change in shape 



Changes in 
pore shape 



and shrinkage 




— *-i AL 

h l 0 H 

Fig. 10.17. Changes in pore shape do not necessarily require shrinkage. 



470 INTRODUCTION TO CERAMICS 



size, and consequently the radius of curvature is small tL « 

be of a substantial magnitude. As indicSta Ch ap e I PtutlT™* 

large when the radius of curvature is less than a SK^S 

p r otrL^ 

experimental data with si m ple models Since ou m^r Z i £ ZT* 

and plasfc deformation are among those that occur I u o "h se' 
diffusion and viscous flow are important in the lara<t mini * ' 
evaporation-condensation is perLps l"« i ^Z£° f 

Evaporanon-Condensation. During the sintering proce s there is a 
tendency for material transfer because nf th* .?_ p,utess * nere ,s a 

«n«« «d conseque „ tly , he S£Z££££Z, '." 

', 6 T em ' MaKria ' " ans " ! ' ^ou,r, h Twa' s "l 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 



471 



Material 
transfer 




Fig. 10.18. Initial stages of sintering by evaporization-condensation. 

lens between the spheres with the increase in its volume. The vapor 
pressure over the small negative radius of curvature is decreased because 
of the surface energy in accordance with the Thomson-Freundlich 
(Kelvin) equation discussed in Chapter 5: 



p 0 dRT 



(10.14) 



where p x is the vapor pressure over the small radius of curvature, M is the 
molecular weight of the vapor, and d is the density. In this case the neck 
radius is much larger than the radius of curvature at the surface, p, and the 
pressure difference p 0 - p. is small. Consequently, to a good approxima- 
tion, lnp,/p 0 equals Ap/p 0 , and we can write 



Ap = 



dpRT 



(10.15) 



where Ap is the difference between the vapor pressure of the small 
negative radius of curvature and the saturated vapor in equilibrium with 
the nearly flat particle surfaces. The rate of condensation is proportional 
to the difference in equilibrium and atmospheric vapor pressure and is 
given by the Langmuir equation to a good approximation as 

m=aAp(J T )" g/cm 2 /sec (10.16) 

where a is an accommodation coefficient which is nearly unity. Then the 
rate of condensation should be equal to the volume increase. That is, 



mA 

d 



dv 
dt 



cm 3 /sec 



(10.17) 



From the geometry of the two spheres in contact, the radius of curvature 
at the contact points is approximately equal to x 2 l2r for x/r less than 0.3; 



472 



INTRODUCTION TO CERAMICS 



the area of the surface of the lens between spheres is approximately equal 
to 7r 2 jc 3 /r; the volume contained in the lenticular area is approximately 
77X 4 /2r. That is, 

P= Yr - A =-r- v =^~ < 10 - 18 > 

Substituting values for m in Eq. 10.16, A and v in Eq. 10.18 into Eq. 10.17 
and integrating, we obtain a relationship for the rate of growth of the bond 
area between particles: 



3V^yM 3 >\' /3 m , /3 



This equation gives the relationship between the diameter of the contact 
area between particles and the variables influencing its rate of growth. 

The important factor from the point of view of strength and other 
material properties is the bond area in relation to the individual particle 
size, which gives the fraction of the projected particle area which is 
bonded together — the main factor in fixing strength, conductivity, and 
related properties. As seen from Eq. 10.19, the rate at which the area 
between particles forms varies as the two-thirds power of time. Plotted on 
a linear scale, this decreasing rate curve has led to characterizations of 
end point conditions corresponding to a certain sintering time. This 
concept of an end point is useful, since periods of time for sintering an 
not widely changed; however, the same rate law is observed for the entin 
process (Fig. 10.19b). 




Time (min) Time (min) 

(a) (b) 
Fig. 10.19. (a) Linear and (b) log-log plots of neck growth between spherical particles of 
sodium chloride at 725°C. 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 473 



timately equal 
ipproximately 



(10.18) 

into Eq. 10.17 
th of the bond 



(10.19) 

)f the contact 
:e of growth. 
;th and other 
idual particle 
irea which is 
fuctivity, and 
rich the area 
te. Plotted on 
:erizations of 
g time. This 
sintering an 
for the entin 



) 20 30 



cal particles of 



If we consider the changes in structure that take place during a process 
such as this, it is clear that the distance between centers of spherical 
particles (Fig. 10.18) is not affected by the transfer of material from the 
particle surface to the interparticle neck. This means that the total 
shrinkage of a row of particles, or of a compact of particles, is unaffected 
by vapor-phase-material transfer and that only the shape of pores is 
changed. This changing shape of pores can have an appreciable effect on 
properties but does not affect density. 

The principal variables in addition to time that affect the rate of 
pore-shape change through this process are the initial particle radius (rate 
proportional to l/r 2 ' 3 ) and the vapor pressure (rate proportional to p 0 ,/3 ). 
Since the vapor pressure increases exponentially with temperature, the 
process of vapor-phase sintering is strongly temperature-dependent. 
From a processing point of view, the two main variables over which 
control can be exercised for any given material are the initial particle size 
and the temperature (which fixes the vapor pressure). Other variables are 
generally not easy to control, nor are they strongly dependent on 
conditions of use. 

The negligible shrinkage corresponding to vapor-phase-material trans- 
fer is perhaps best illustrated in Fig. 10.20, which shows the shape 
changes that occur on heating a row of initially spherical sodium chloride 
particles. After long heating the interface contact area has increased; the 




Fig. 10.20. Photomicrographs of sintering sodium chloride at 750°C: (a) 1 min; {b)90 min. 



474 



INTRODUCTION TO CERAMICS 



particle diameter has been substantially decreased, but the distance 
between particle centers, that is, the shrinkage, has not been affected. 

Vapor-phase-material transfer requires that materials be heated to a 
temperature sufficiently high for the vapor pressure to be appreciable. For 
micron-range particle sizes this requires vapor pressures in the order of 
1(P to 10~ 5 atm, a pressure higher than those usually encountered during 
sintering of oxide and similar phases. Vapor-phase transfer plays an 
important part in the changes occurring during treatment of halides such 
as sodium chloride and is important for the changes in configuration 
observed in snow and ice technology. 

Solid-State Processes. The difference in free energy or chemical poten- 
tial between the neck area and the surface of the particle provides a 
driving force which causes the transfer of material by the fastest means 
available. If the vapor pressure is low, material transfer may occur more 
readily by solid-state processes, several of which can be imagined. As 
shown in Fig. 10.21 and Table 10.1, in addition to vapor transport (process 
3), matter can move from the particle surface, from the particle bulk, or 
from the grain boundary between particles by surface, lattice, or grain- 
boundary diffusion. Which one or more of these processes actually 
contributes significantly to the sintering process in a particular system 
depends on their relative rates, since each is a parallel method of lowering 
the free energy of the system (parallel reaction paths have been discussed 
in Chapter 9). There is a most significant difference between these paths 
for matter transport: the transfer of material from the surface to the neck 
by surface or lattice diffusion, like vapor transport, does not lead to any 
decrease in the distance between particle centers. That is, these processes 
do not result in shrinkage of the compact and a decrease in porosity. Only 



Fig. 10.21. 
Courtesy M. 



Table 10.1. Alternate Paths for Matter Transport During the Initial Stages of 

Sintering" 



Mechanism 








Number 


Transport Path 


Source of Matter 


Sink of Matter 


1 


Surface diffusion 


Surface 


Neck 


2 


Lattice diffusion 


Surface 


Neck 


3 


Vapor transport 


Surface 


Neck 


4 


Boundary diffusion 


Grain boundary 


Neck 


5 


Lattice diffusion 


Grain boundary 


Neck 


6 


Lattice diffusion 


Dislocations 


Neck 



°See Fig. 10.21. 



transfer o\ 
between p 
Let us o 
to the necl 
is exactly 
vapor-pha 
surface an 
The geom- 



The proct 




; e distance 
i affected, 
eated to a 
ciable. For 
ie order of 
red during 
plays an 
Iides such 
lfiguration 

cal poten- 
•rovides a 
est means 
:cur more 
gined. As 
t (process 
* bulk, or 
or grain- 
actually 
r system 
lowering 
(iscussed 
;se paths 
the neck 
d to any 
rocesses 
ity. Only 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 



475 




Fig. 10.21. Alternate paths for matter transport during the initial stages of sintering. 
Courtesy M. A. Ashby. (See Table 10.1.) 



Stages of 



latter 



transfer of matter from the particle volume or from the gram boundary 
between particles causes shrinkage and pore elimination. 

Let us consider mechanism 5, matter transport from the grain boundary 
to the neck by lattice diffusion. Calculation of the kinetics of this process 
is exactly analogous to determination of the rate of s.ntenng by a 
vapor-phase process. The rate at which matenal is discharged at the 
surface area is equated to the increase in volume of matenal transferred. 
The geometry is slightly different: 

(10.20) 



A = 



tt'x' 
2r 



V = 



■ax 
4r 



The process can be visualized most easily by considering the rate of 




476 



INTRODUCTION TO CERAMICS 



migration of vacancies. In the same way' that there are differences in 
vapor pressure between the surface of high negative curvature and the 
nearly flat surfaces, there is a difference in vacancy.concentration. If c is 
the concentration of vacancies and Ac is the excess concentration over 
the "concentration on a plane surface c 0 , then, equivalent to Eq. 10.15, 



Ac 



ya Co 
kTp 



(10.21) 



where a 3 is the atomic volume. of the diffusing vacancy and k is the 
Boltzmann constant. The flux of vacancies diffusing away from the neck 
area per second per centimeter of circumferential length under this 
concentration gradient can be determined graphically and is given by 

J = 4D v Ac (10.22) 
Where D v is the diffusion coefficient for vacancies, D v equals D*/a 3 c 0 if 
D* is the self-diffusion coefficient. Combining Eqs. 10.22 and 10.21 with 
the continuity equation similar to Eq. 10.17, we obtain the result 



r 



(40ya 3 D* \ 
\ kT ) 



*\ "5 



r- 3 'Y' 5 



(10.23) 



With diffusion, in addition to the increase in contact area between 
particles, there is an approach of particles centers. The rate of this 
approach is given by d(x 2 l2r)ldt. Substituting from Eq. 10.23, we obtain 



AV = 3AL = /20ya 3 D* \ 2fs 
V 0 L 0 \ V2kT I 



r -6/5 j 2/5 



(10.24) 



These results indicate that the growth of bond formation between 
particles increases as a one-fifth power of time (a result which has been 
experimentally observed for a number of metal and ceramic systems) and 
that the shrinkage of a compact densified by this process should be 
proportional to the two-fifths power of time. The decrease in densification 
rate with time gives rise to an apparent end-point density if experiments 
are carried out for similar time periods. However, when plotted on a 
log-log basis, the change in properties is seen to occur as expected from 
Eq. 10.24. Experimental data for sodium fluoride and aluminum oxide are 
shown in Fig. 10.22. 

The relationships derived in Eqs. 10.23 and 10.24 and similar relation- 
ships for the alternate matter transport processes, which we shall not 
derive, are important mainly for the insight that they provide on the 
variables which must be controlled in order to obtain reproducible 
processing and densification. It is seen that the sintering rate steadily 
decreases with time, so that merely sintering for longer periods to obtain 



GR 



L 



GRAIN GROWTH, SINTERING, AND VITRIFICATION 



477 



differences in 
ature and the 
tration. If c is 
;ntration over 
to Eq. 10.15, 

(10.21) 

and k is the 
Tom the neck 
:h under this 
is given by 

(10.22) 

alsD*/a 3 c 0 if 
nd 10.21 with 
i result 

(10.23) 

irea between 
rate of this 
■ , we obtain 

(10.24) 

don between 
lich has been 
systems) and 
ss should be 
densification 
experiments 
plotted on a 
tpected from 
urn oxide are 

lilar relation- 
we shall not 
Dvide on the 
reproducible 
rate steadily 
ods to obtain 




100 200 300 400 500 600 
Time (min) 
(a) 



700 



0.2 
0.1 
0.05 



AL 

T- 0.02 



0.01 
0.005 

0 002 



NaF, 726 




T 



A1 2 0 3 , 1300°C 



J_ 



_L 



1000 



10 100 
Time (min) 

Fig. 10.22. (a) Linear and (b) log-log plots of shrinkage of sodium fluoride and aluminum 
oxide compacts. From J. E. Burke and R. L. Coble. 



improved properties is impracticable. Therefore, time is not a major or 
critical variable for process control. 

Control of particle size is very important, since the sintering rate is 
roughly proportional to the inverse of the particle size. The interface 
diameter achieved after sintering for a period of 100 hr at 1600°C is 
illustrated in Fig. 10.23 as a function of particle size. For large particles 
even these long periods do not cause extensive sintering; as the particle 
size is decreased, the rate of sintering is raised. 



478 



INTRODUCTION TO CERAMICS 




0.20 



0.25 



0.05 0.10 0.15 

-V -V 
r * (micron *) 

Fig. 10.23. Effect of particle size on the contact area growth in A1 2 0 3 heated 100 hr at 
1600°C. From R. L. Coble. 



The other variable appearing in Eqs^ 10.22 and 10.24 that is subject to 
analysis and some control is the diffusion coefficient; it is affected by 
composition and by temperature; the relative effectiveness of surfaces, 
boundaries, and volume as diffusion paths is affected by the microstruc- 
ture. A number of relationships similar to Eqs. 10.23 and 10.24 have been 
derived, and it has been shown that surface diffusion is most important 
during early stages of sintering (these affect the neck diameter between 
particles but not the shrinkage or porosity); grain-boundary diffusion and 
volume diffusion subsequently become more important. In ionic 
ceramics, as discussed in Chapter 9, both the anion and the cation 
diffusion coefficients must be considered. In A1 2 0 3 , the best studied 
material, oxygen diffuses rapidly along the grain boundaries, and the more 
slowly moving aluminum ion at the boundary or in the bulk controls the 
overall sintering rate. As discussed in Chapter 5, the grain-boundary 
structure, composition, and electrostatic charge are influenced strongly 
by temperature and by impurity solutes; as discussed in Chapter 6, the 
exact mechanism of grain-boundary diffusion remains controversial. 
Estimates of the grain-boundary-diffusion width from sintering data range 
from 50 to 600 A. These complications require us to be careful not to 
overanalyze data in terms of specific numerical results, since the time or 
temperature dependence of sintering may be in accordance with several 
plausible models. In genera] the presence of solutes which enhance either 



GF 

boundary or 
sintering. As 
coefficients i 
sintering rati 

In order t< 
solid-state pi 
particle size 
temperature, 

As an ex* 
effect of tita: 
in a region 
processes ar 
ally as Ti +3 



from which 



100 



§ io 



0.1 



i 100 hr at 



lbject to 
jcted by 
mrfaces, 
rrostruc- 
we been 
nportant 
between 
sion and 
n ionic 
; cation 

studied 
he more 
trols the 
oundary 
strongly 
;r 6, the 
wersial. 
ta range 

1 not to 

time or 
... several 

e either 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



479 



■^S^^S/^^tuie^lep^ which means that the 
C ° e ,?Hn e ate 7s strong," dependent on the temperature level. 
Sm r Irlo effectWely control sintering processes which take place by 
S state S^M^talLntUd to maintain close control of the in.tial 
sohd- tate processes^ t dislribution of the material, the s.ntenng 

rr^eratu e tie comptiUon and frequently the sintering atmosphere. 
TsTn example of the influence of solutes, Fig. 10.24 ulustrates the 

l^^^^^ -t Ti enters AW. substitution- 
ally as IT 3 and Tf 4 (Ti A1 and Ti A1 ). At equilibrium 

(10.25) 
(10.26) 



from which 



3Ti A1 + ^0 2 (g) = 3Ti A ,+ V A , + 20o 

K, ~[Ti A1 ] 3 [Po 2 r 




log atom % Ti 



Fig 10 24. Data for the relative sintering process 
diffusion coefficient with Ti additions to A1 2 0 3 . 
Da[Ti] 3 . From R. D. Bagley, I. B. Cutler, and D. L. 
Johnson, J. Am. Ceram. Soc 53, 136 (1970); R. J- 
Brook, J. Am. Ceram. Sac, 55, U4 (1972). 



11 



480 



INTRODUCTION TO CERAMICS 



In the powders used, divalent impurities such as magnesium exceed in 
concentrations the intrinsic defect levels, so that overall charge neutrality 
at moderate titania levels is achieved by 

|Ti A1 ) = [Mg AI ] (10.27) 

and at constant impurity and oxygen pressure levels, combining Eos 
10.26 and 10.27 gives ' 

= K 2 [Ti A ,] 3 (10.28) 
Since the total Ti addition (Ti A1 + T, A ,) is much greater than the impurity 
levels, [Ti] To ,„ = [Ti AI ] and [ VZ] = K 2 [Ti£ Mal . The dependence of lattice 
defect concentrations on titania concentration is shown in Fig 10 25 for 
the proposed model. As discussed in Chapter 6, the diffusion coefficient is 
proportional to the vacancy concentration; as a result the effect of this 
model is to anticipate an increase in the sintering rate proportional to the 
hird power of titania concentration as experimentally observed (Fig 
10.24) At higher concentrations the dependence on titania concentration 
should become less steep, which is suggested by the sintering data 

Thus far our discussion of the variables influencing the sintering 
process has been based on the initial stages of the process, in which 
models are based on solid particles in contact. As the process continues 
an intermediate microstructure forms in which the pores and solid are' 
both contmuous, followed by a later stage in which isolated pores are 
separated from one another. A number of analytical expressions have 







M 




Mg A1 




/ " f 






/ ' 


-7r-t 




Al|" 






Ti A y/ 




l /\ 1 

'/ V 




/ / 




1 1 
. 1 1 





Gl 

been derm 
processes li 
mechanisms 
boundary ai 
spherical p( 



where D v i 
concentrate 
material-soi 
sort of anal; 



log concentration of titanium 

Im°L M R 0d r el R 0r T 'T^r 6 ° f defCCt COncentrali ^ 0" the Ti concentration in 
A1 2 0 5 . hrom R. J. Brook, J. Am. Ceram. Soc. y 55, 114 (1972). 



Fig. 10.26. T 
more of the s 



with a largt 
diffusion d 
elimination 
porosity. T 
volume or 
solute com 
the geomet 
variety of 
With fin 
increase in 
treatment, 
results fro 
sinter rapi 
rapid grain 
the bound; 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



uum exceed in 
arge neutrality 

(10.27) 
>mbining Eqs. 

(10.28) 

t the impurity 
nee of lattice 
Fig. 10.25 for 
t coefficient is 
effect of this 
rtional to the 
>served (Fig. 
oncentration 
ing data, 
he sintering 
ss, in which 
is continues, 
nd solid are 
id pores are 
ssions have 



been derived from specific rnicrostructural models for the transport 
processes listed in Table 10.1. In the later stages of the process only two 
mechanisms are important: boundary diffusion from sources on the 
boundary and lattice diffusion from sources on the boundary. For a nearly 
spherical pore the flux of material to a pore can be approximated as 

/ = 4 ? rDvAc(^ 7 ) (10.29) 

where D v is the volume diffusion coefficient, Ac is the excess vacancy 
concentration (Eq. 10.21), r is the pore radius, and R is the effective- 
material-source radius. The importance of microstructure in applying this 
sort of analysis to specific systems is illustrated in Fig. 10.26. For a sample 




(a) 

Fig. 10.26. The mean diffusion distance for material transport is smaller when there are 
more of the same size of pores in a boundary. 



icentration in 



with a larger number of pores, all the same size, on a boundary the mean 
diffusion distance is smaller when there are more pores, and pore 
elimination is accomplished more quickly for the sample with the higher 
porosity. Thus, although the terms which influence the rate of sintering— 
volume or boundary diffusion coefficient (and therefore temperature and 
solute concentration) surface energy and pore size-are well established, 
the geometrical relationship of grain boundaries to the pores may have a 
variety of forms and is critical in determining what actually occurs. 

With fine-grained materials such as oxides, it is usual to observe an 
increase in both grain size and pore size during the early stages of heat 
treatment, as illustrated for Lucalox alumina in Fig. 10.27. This partially 
results from the presence of agglomerates of the fine particles which 
sinter rapidly, leaving interagglomerate pores, and is partly due to the 
rapid grain growth during which pores are agglomerated by moving with 
the boundaries, as illustrated in Fig. 10.9. In cases in which agglomeration 



484 



INTRODUCTION TO CERAMICS 



GR 




of fine precipitated particles into clumps is severe, ball milling to break up 
the agglomerates leads to a remarkable increase in the sintering rate. Even 
minor variations in the original particle packing are exaggerated during 
the pore growth process; in addition, spaces between agglomerates and 
occasional larger voids resulting from the bridging of particles or agglom- 
erates are present. As a result, during intermediate stages of the sintering 
process there is a range of pore sizes present, and the slower elimination 
of the larger pores leads to variations in pore concentration in the later 
stages of the sintering process, as illustrated in Fig. 10.28c. 

In addition to local agglomerates and packing differences, pore- 
concentration variations in the later stages of sintering can result from 
particle-size variations in the starting material, from green density varia- 




o 

c 

o 



Fig. 10.28. Poi 
die friction, (c 
elimination nea 

tions caused 
elimination < 
during heatir 
in pore cone 
containing pi 
That is, the 
adjacent gra 
orders of m 
sintered oxi< 
Not only 
residual poi 
thermodynai 



GRAIN GROWTH, SINTERING 



AND VITRIFICATION 485 





oooo ooooooo 
jooooooQdOd&o, 

Rooo°oooo 0 o Q 
°oo8o 0 oo 0 °o ° 

°n Wo On °o° 



(d) 

Fig. 10.28. Pore-concentration variations resulting from (a) a variation in grain sizes, (b) 
die friction, (c) local packing and agglomeration differences, and {d) more rapid pore 
elimination near surfaces. 



tions caused by die-wall friction during pressing, and from the more rapid 
elimination of porosity near surfaces caused by temperature gradients 
during heating, as shown in Fig. 10.28. The importance of local variations 
in pore concentration results from the fact that the part of the sample 
containing pores tends to shrink but is restrained by other porefree parts. 
That is, the effective diffusion distance is no longer from the pore to an 
adjacent grain boundary but a pore-pore or pore-surface distance many 
orders of magnitude larger. An example of residual pore clusters in a 
sintered oxide is shown in Fig. 10.29. 

Not only the kinetics of pore elimination can lead to "stable" and 
residual porosity, but it is also possible in some cases to have a 
thermodynamically metastable equilibrium pore configuration. In Fig. 




10.26 we have drawn spherical pores located on a grain boundary, the 
usual model description, but we know from our discussion of interface 
energies in Chapter 5 that there is a dihedral angle <j> at the pore-boundary 
intersection determined by the relative interface energies; 

cos f = (10.30) 

In most cases the dihedral angle for pure oxides is about 150°, and the 
spherical pore approximation is quite good; but for A1 2 0 3 + 0.1% MgO the 



value i 
carbide 
nonsph 
Asd 
and 10. 
on the 
grains, 
polyhe 
of the i 
angle a 
When 




o 

Q. 
O 

IA 
3 
"O 
TO 

o> 
a> 

CL 
iA 
TD 

a> 
.o 

u 
tn 

E 
u 
o 

G 

o 
o 

cr 



Fig. 10 
numbe 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



487 



value is 130°, for UO 2 + 30ppm C the value is 88°, and for impure boron 
carbide the value is about 60°. For these materials the consequences of 
nonspherical pores have to be considered. ............. 

As discussed for discontinuous grain growth and illustrated in Figs. 10.4 
and 10.11, the boundary curvature between grains or phases depends both 
on the value of the dihedral angle and on the number of surrounding 
grains. If we take r as the radius of a circumscribed sphere around a 
polyhedral pore surrounded by grains, the ratio of the radius of curvature 
of the pore surfaces p to the spherical radius depends both on the dihedral 
angle and on the number of surrounding grains, as shown in Fig. 10.30a. 
When rip decreases to zero, the interfaces are flat and have no tendency 




488 



INTRODUCTION TO CERAMICS 




for shrinkage; when r/p is negative, the pore tends to grow. This is 
illustrated in Fig. 10.30b. For a uniform grain size the space-filling form is 
a tetrakaidecahedron with 14 surrounding grains. From an approximate 
relationship between the number of surrounding grains and the pore- 
diameter to grain-diameter ratio we can derive a relationship for pore 
stability as a function of dihedral angle and the ratio of pore size to grain 
size, as shown in Fig. 10.31. From this figure we can see why large pores 
present in poorly compacted powder such as shown in Fig. 10.32 not only 
remain stable but grow. It is also seen that an enormous disparity between 




0 0.5 1.0 1.5 2.0 2.5 

Ratio of pore diameter to grain diameter 



Fig. 10.31. 



Conditions for pore stability. 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



489 



us is 
rm is 
mate 
pore- 
pore 
grain 
>ores 
only 
ween 




Fig .0 32 Large voids formed by bridging of agglomerates in fine AW, powder viewed 
with scanning electron microscope at 2000x. Courtesy C. Greskov.ch. 

grain size and pore size is not necessary for pore stability. That is, the site 
and size of the porosity relative to the grain-boundary network not on* 
aff ects "the necessary distance for diffusion but also the dnv.ng force for the 

^"'interaction of grain boundaries and porosity is, of course a 
two-way street. When many pores are present dur.ng the .mttal , stages of 
sintering, grain growth is inhibited. However, as d.scussed in Section 10.1 , 
once he porosity has decreased to a value such that secondary gram 
growt can occu'r, extensive grain growth may result at high smtermg 
femperatures. When grain growth occurs, many pores become .solated 
r'm gTain boundaries, and the diffusion distance between pores and a 
grain boundary becomes large, and the rate of sintenng decreases Th s .s 
fl ustrated in Fig. 10.16b. in which extensive secondary -recrystall.zat.on 
has occurred, with the isolation of pores in the intenor of grams and a 
reduc ion in he densification rate. Similarly, the sample of alum.num 
oxide hown in Fig. 10.33 has been sintered at a high temperature at wh.ch 
dLcontuous grain growth occurred. Porosity is ™* 
grain boundaries, which act as the vacancy s.nk. The tmportance of 



490 



INTRODUCTION TO CERAMICS 



Fig 10.33 Sintered A1.0, illustrating elimination of porosity adjacent to grain boundaries 
with residual porosity remaining at grain centers. Courtesy J. E. Burke. 



controll.ng grain growth as an integral part of controlling sintering 
phenomena cannot be overestimated. Consequently, the grain-growth 
processes discussed in Section 10.1 must be actively prevented in order to 
obtain complete densification. Usually densification continues by a diffu- 
sion process until about 10% porosity is reached; at this point rapid grain 
growth occurs by secondary recrystallization, and the rate of densifica- 
tion is sharply reduced. In order to obtain densification much beyond this 
level, prevention of secondary recrystallization is essential The most 
satisfactory way of doing this is with additives which prevent or slow 
down boundary migration to a point at which it is possible to obtain pore 
elimination. Additions of MgO to Al 2 0 3 , ThO. to Y 2 0 3 , and CaO to Th0 2 
among others, have been found to slow boundary migration and allow 
complete pore elimination by solid-state sintering in these systems The 
porefree microstructure of a polycrystalline ceramic having optical 
transparancy suitable for use as a laser material is shown in Fig. 10.34. 

10.3 Vitrification 

To vitrify is to make glasslike and the vitrification process— 
densification with the aid of a viscous liquid phase— is the major firing 
process for the great majority of silicate systems. (In some current 
glossaries vitrification is defined as being identical to densification on 
firing, but the more specific usage is preferred.) A viscous liquid silicate is 
formed at the firing temperature and serves as a bond for the body For 



GRAIN GROWTH, SINTERING AND VITRIFICATION 




satisfactory finng the amount and viscosity of the l.qu.d ph e rnusl b 
such that deification occurs in a - asonab, ^ ,m ^^ thOUt ;^^ r t e e 
slumping or warping under the force of gravity The relative and .*so ute 
rates of these two processes (shrinkage and deformation) , determine to a 
Se extent the temperature and compositions suitable for satisfactory 

fir pr 8 ocess Kinetics. If we consider two particles initially in contact (Fig. 
10.21), there is a negative pressure at the small negative radius of 



492 



INTRODUCTION TO CERAMICS 



curvature p compared with the surface of the particles. This causes a 
viscous flow of material into the pore region. By an analysis similar to that 
derived for the diffusion process, the rate of initial neck growth is given = .. 

as* 




(10.31) 



The increase in contact diameter is proportional to r" 2 ; the increase in 
area between particles is directly proportional to time. Factors of most 
importance in determining the rate of this process are the surface tension, 
viscosity, and particle size. The shrinkage which takes place is deter- 
mined by the approach between particle centers and is 

AV = 3AL = ^r t (1032) 
V 0 L 0 4i7r 

That is, the initial rate of shrinkage is directly proportional to the surface 
tension, inversely proportional to the viscosity, and inversely propor- 
tional to the particle size. 

The situation after long periods of time can best be represented as small 
spherical pores in a large body (Fig. 10.35). At the interior of each pore 




Fig. 10.35. Compact with isolated spherical pores near the end of the sintering process. 

there is a negative pressure equal to 2y/r; this is equivalent to an equal 
positive pressure on the exterior of the compact tending to consolidate it. 
j k Mackenzie and R. Shuttlewortht have derived a relation for the rate 
of shrinkage resulting from the presence of isolated equal-size pores in a 
viscous body. The effect of surface tension is equivalent to a pressure of 
-2y/r inside all pores or, for an incompressible material, to the applica- 

*J. Frenkel, J. Phys (USSR), 9, 385 (1945). 
tProc. Phys. Soc. (London), B62. K33 (1949). 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



493 



This causes a 
similar to that 
owth is given 



(10.31) 

e increase in 
:tors of most 
face tension, 
ace is deter- 



(10.32) 

3 the surface 
sely propor- 

nted as small 
jf each pore 



tion of a hydrostatic pressure of +2y/r to the compact. The real problem 
is to deduce the properties of the porous material from the porosity and 
viscosity of the dense material. The method of approximation used gives 
an equation of the form 

f = l(Tr« ,/3 ^'-P')V." 3 (10.33) 

where p' is the relative density (the bulk density divided by the true 
density or the fraction of true density which has been reached) and n is 
the number of pores per unit volume of real material. The number of 
pores depends on the pore size and relative density and is given by 



n r 3 = Pore volume _ 1 - p' 
3 Solid volume p' 



By combining with Eq. 10.33, 



dp' _ 3y 



dt 2r 0 T] 



d-p') 



(10.34) 
(10.35) 

(10.36) 



where r 0 is the initial radius of the particles. 

The general course of the densification process is best represented by a 
plot of relative density versus nondimensional time, illustrated in Fig. 
10.36 following Eq. 10.33. Spherical pores are formed very quickly to 
reach a relative density of about 0.6. From this point until the completion 
of the sintering process about one unit of nondimensional time is 



ring process. 

to an equal 
nsolidate it. 
for the rate 
i pores in a 
pressure of 
he applica- 




0.2 0.4 



1.6 1.8 2.0 



0.6 0.8 1.0 1.2 1.4 
Reduced time, 7/1 **(*■*- t 0 )Iv 
Fig. 10.36. Increase in relative density of compact with reduced time for a viscous 
material. From J. K. Mackenzie and R. Shuttleworth, Proc. Phys. Soc. (London) B62 833 
(1949). 



494 INTRODUCTION TO CERAMICS 

required. For complete densification 



l-5r 0 -rj 

~T~ (10.37) 



Some experimental data for the densification of a viscous body are 
shown ,n Fig. 10.37, in which the strong effect of temperature, tha° is the 

ratTSe ' H r aten ' al 'J S illUStrated by thC rapid chan * e in si «t- ng 
rates. The solid lines in F.g. 10.37 are calculated from Eq. 10.33. The 




0.6 1 1 L 

O24 

Fig. 10.37. Densification of a soda-lime-silica glass. 



T . 6 8 10 12 

Time (hr) 



from a, ^ teS .0 O ^ in TK rin8 in , d,Cated by thC daShed CUrves are calcu '^ed 
from Eq. 10 32. The good agreement of these relationships with the 

■ '' np ° rtant j Variab,es - The particular importance of Eqs. 10 31 to 10 37 
.s the dependence of the rate of densification on three major varilbles- 
he particle size viscosity, and surface tension. For silicate materials he 
urface tens.on ,s not changed much by composition, although there are 

n Chaote? 5 S H° r ^ "IT ^ " ' OW > as "'"^ ted 

caused h T Ver ' T taCe tenSi0 " is " 0t 3 Variable that normally 

causes d.fficulty dunng the design of compositions or the control of 
process, The particle size has a strong effect on the sintering rl and 
cnnt J f° t I COn,r ° I,ed ' f thC dens '"fi^ion process is going to be 
controlled. In changmg from a 10-micr On to .-micron particle the Je o^ 
smtering ,s increased by a factor of .0. Even more important for con ro 



purposes 
typical so 
over an in 
factor ove 
be closely 
discussed 
by changii 
The relatr 
viscosity i 
under the i 
makes it r 
stresses di 
due to grz 
supported 
obtaining 
fine-graine 
ment is o 
systems a: 
naturally i 
vitrificatioi 
Silicate £ 
fact that ro 
ture and th 
the pressui 
much liqui 
7.26, whicl 
system; thi 
ous porcel 
Si0 2 ), 25% 
are in the j 
between m 
75Si0 2 , 12. 
eutectic liq 
only a sma 
and the coi 
as well as ( 
silica whic 
compositio 
high viscos 
relative arr 
very fine-g 
those of a 1 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



495 



(10.37) 

>dy are 
: is, the 
ntering 
3. The 



ated 
the 
ition 

0.37 

iS — 

the 
are 
ited 
ally 

of 
and 

be 

:Of 

Vol 



purposes is the viscosity and its rapid change with temperature. For a 
typical soda-lime-silica glass the viscosity changes by a factor of 1000 
over an interval of 100°C; the rate of densification changes by an equal 
factor over the temperature range. This means that the temperature must 
be closely controlled. Viscosity is also much changed by composition, as 
discussed in Chapter 3. The rate of densification, then, can be increased 
by changing the composition to lower the viscosity of the glassy material. 
The relative values of viscosity and particle size are also important; the 
viscosity must not be so low that appreciable deformation takes place 
under the forces of gravity during the time required for densification. This 
makes it necessary for the particle size to be in such a range that the 
stresses due to surface tension are substantially larger than the stresses 
due to gravitational forces. Materials sintered in a fluid state must be 
supported so that deformation does not occur. The best means of 
obtaining densification without excessive deformation is to use very 
fine-grained materials and uniform distribution of materials. This require- 
ment is one of the reasons why successful compositions in silicate 
systems are composed of substantial parts of talc and clays that are 
naturally fine-grained and provide a sufficient driving force for the 
vitrification process. 

Silicate Systems. The importance of the vitrification process lies in the 
fact that most silicate systems form a viscous glass at the firing tempera- 
ture and that a major part of densification results from viscous flow under 
the pressure caused by fine pores. Questions that naturally arise are how 
much liquid is present and what are its properties. Let us consider Fig. 
7.26, which shows an isothermal cut at 1200°C in the K20-Al 2 0 3 -Si0 2 
system; this is the lower range of firing temperatures used for semi vitre- 
ous porcelain bodies composed of about 50% kaolin (45% A1 2 0 3 , 55% 
Si0 2 ), 25% potash-feldspar, and 25% silica. This and similar compositions 
are in the primary field of mullite, and at 1200°C there is an equilibrium 
between mullite crystals and a liquid having a composition approximately 
75Si0 2 , 12.5K 2 0, 12.5A1 2 0 3 , not much different in composition from the 
eutectic liquid in the feldspar-silica system (Fig. 7.14). In actual practice 
only a small part of the silica present as flint enters into the liquid phase, 
and the composition of the liquid depends on the fineness of the grinding 
as well as on the overall chemical composition. However, the amount of 
silica which dissolves does not have a large effect on the amount and 
composition of the liquid phase present. The liquid is siliceous and has a 
high viscosity; the major effect of compositional changes is to alter the 
relative amounts of mullite and liquid phases present. Since mullite is 
very fine-grained, the fluid flow properties of the body correspond to 
those of a liquid having a viscosity greater than the pure liquid phase. For 



496 



INTRODUCTION TO CERAMICS 



some systems the overall flow process corresponds to plastic flow with a 
yield point rather than to true viscous flow. This changes the kinetics of 
the vitrification process by introducing an additional term in Eqs. 10.33 
and 10.36 but does not change the relative effects of different variables. 

Although phase diagrams are useful, they do not show all the effects of 
small changes in composition. For example, a kaolinite composition 
should show equilibrium between mullite and tridymite at 1400°C with no 
glassy material. However, it is observed experimentally that even after 
24 hr about 60 vol% of the original starting material is amorphous and 
deforms as a liquid. The addition of a small amount of lithium oxide as 
Li 2 C0 3 has been observed to give a larger content of glass than additions 
of the same composition as the fluoride. Similar small amounts of other 
mineralizers can also have a profound effect in the firing properties of 
particular compositions. That fine grinding and intimate mixing reduce the 
vitrification temperature follows from the analysis in Eqs. 10.31 to 10.37. 
S. C. Sane and R. L. Cook* found that ball milling for 100 hr reduced the 
final porosity of a clay-feldspar-flint composition from 17.1 to 0.3% with 
the same firing conditions. This change is caused in part by increased 
tendencies toward fusion equilibrium and uniform mixing of constituents 
and in part by the smaller initial particle and pore size. In contrast to 
triaxial (flint-feldspar-clay) porcelains, which frequently do not reach 
fusion equilibrium, many steatite bodies and similar compositions which 
are prepared with fine-particle, intimately mixed material and form a less 
siliceous liquid reach phase equilibrium early in the firing process. 

The time-temperature relationship and the great dependence of vitrifi- 
cation processes on temperature can perhaps be seen best in the experi- 
mental measurements illustrated in Fig. 10.38. As shown there, the time 
required for a porcelain body to reach an equivalent maturity changes by 
almost an order of magnitude with a 50° temperature change. There are 
changes in both the amount and viscosity of the glassy phase during firing, 
so that it is difficult to elucidate a specific activation energy for the 
process with which to compare the activation energy for viscous flow. 
However, the temperature dependence of the vitrification rate of a 
composition such as this (a mixture of clay, feldspar, and flint) is greater 
than the temperature dependence of viscosity alone. This is to be 
expected from the increased liquid content at the higher firing tempera- 
tures. 

In summary, the factors determining the vitrification rate are the pore 
size, viscosity of the overall composition (which depends on amount of 
liquid phase present and its viscosity), and the surface tension. Equivalent 



*J. Am. Ceram. Soc, 34, 145 (1951). 



'lastic flow with a 
es the kinetics of 
■rm in Eqs. 10.33 
rent variables. 
' all the effects of 
lite composition 
it 1400°C with no 
i that even after 
amorphous and 
lithium oxide as 
;s than additions 
mounts of other 
fig properties of 
uxing reduce the 
?. 10.31 to 10.37. 
Ohr reduced the 
7.1 to 0.3% with 
rt by increased 
of constituents 
. In contrast to 
/ do not reach 
positions which 
and form a less 
g process, 
lence of vitrifi- 
t in the experi- 
there, the time 
ity changes by 
nge. There are 
:e during firing, 
energy for the 
viscous flow, 
ion rate of a 
flint) is greater 
This is to be 
iring tempera- 

e are the pore 
on amount of 
>n. Equivalent 



GRAIN GROWTH, SINTERING AND VITRIFICATION 




0.65 



1000. 



1050 



1100 
Temperature (*C) 



1150 



1200 



1.00 
0.95 
, 0.90 
f 0.85 

a> 

I 0.80 

a> 
ce 

075 
070 



0.65 















~ noo w C/S 




I 1 


1 1 1 



0.1 



10 100 1000 10,000 1 00,000 
Time (hr) 



Fig. 10.38. Effect of time and temperature on the vitrification of a porcelain body. Data 
from F. H. Norton and F. B. Hodgdon, J. Am. Ceram. Soc. y 14, 177 (1931). 



densification results from longer periods of time at the same temperature. 
In controlling the process, the temperature dependence is great because 
of the increase in liquid content and lowered viscosity at higher tempera- 
tures. Changes in processing and changes in composition affect the 
vitrification process as they affect these parameters. 



498 



INTRODUCTION TO CERAMICS 



10.4 Sintering with a Reactive Liquid 

: Appier quite different process which leads to densification is sintering 
m the presence of a reactive liquid. Here we are referring to systems 7n 
w hl ch the sohd phase shows a certain limited so.ubility in the liqu d aTthe 
sintering temperature; the essential part of the sintering process is Je 

ssr j£ !cindT itation of soiids to give --sed^^rr^ 

density. This k nd of process occurs m cermet systems such as bonded 
carbides and also ,n oxide systems when the liquid phase is fluid and 
reactive such as magnesium oxide with a small amount of 1 quid phase 

im h L°- 3 , 9) ' U ° 2 with the addition of 3 sma » ~ o?rfo!^ 

bon ingt I 1 "" 11113 b ° dieS WhiCh ^ a " aIka,i " e silicate ast 




Studies of a large number of systems indicate that for densification to 

£iIZ s r^ ly * ,S eSS t? tial t0 h3Ve (1) 3n appreciab,r mo ^o 
liquid phase (2) an appreciable solubility of the solid in the liquid and G 

Z S3 ?om e t he°c d ^ ^ driV ' ng f ° rCe f ° r ^nSa on ( 

thTfin ?5 ? P ,3ry PreSSUre ° f the Hc > uid P ha «e Seated between 
the fine solid particles, as illustrated in Fig. 10.40 When the liaZ 

wets the solid particles, each interpartic.e 8 space bTcols a Tpl' v n 
which a substantial capillary pressure is developed. For suWron 
particle sizes, capillaries with diameters in the range of 0. U , " on 



wintering 
;tems in 
id at the 
s is the 
ize and 
bonded 
uid and 
d phase 
0 2 (Fig. 
ate as a 



ve-liquid 



ttion to 
>unt of 
and (3) 
ttion is 
etween 
1 phase 
lary in 
micron 
micron 




Fig 10 40 (a) Surface of solid-liquid composite with varying amounts of liquid phase, {b) 
Drop of liquid between two solid spheres exerts pressure to pull them together, (c) Surface 
of forsterite ceramic showing liquid capillary depression between crystals. 

499 




500 INTRODUCTION TO CERAMICS 

develop pressures in the range of 175 to 1750 n<i f 

the range of 975 , 0 9750psi for a ° IJf, ?J* S ' ,,Cate Mds and i n 
d.scuss,on in Chapter 5 and Table 5 2™ "* * HqUid Coba,t («£ 

^S5^T5 o n n?o enS,fiCati0n by d ' ff — Passes 
rearranges of l^to^SSi^ 3 *^ *« £ 
can lead to complete densification 7 th , * Packing ' This Process 
sufficient to fil, in the inte,,^^^; 0 ^ °< Present * 

where there are bridges between ■ ,1 eCOnd ' at cont act points 
Plastic deformation and creeT wMch n § ^ ,0Cal stress « lead 0 
Third, there is during the s^e^g p oces H f ^ — ^men ° 
and growth of larger particles by Tater J T ° f Sma,,er P^icles 
Phase^The kinetics of this solu^on p/ec 1 ' thr ° Ugh the ""^ d 

been d.scussed in Chapter 9. Because th ' " PTOCeSS ^ 
capillary pressure, additional part cle relrraL" " C ° nStant,y ' mpOSed 
grain-growth and grain-shape cLgel and "v ^t" 1 Can ° CCUr du ™g 
d,scussed for vapor transport and sur fac e ' densificatio "- (A 

mg mere solution-precipitation materiart?!/ 10 " ™ "^'e ^inter- 

ca P1 llary pressure would not give S to Jen « OUt the im P° sed 

which liquid penetrates between ZZ? " ,ncat,on )- Fourth , in cases in 

contact points leads to an increased ^ 1 , . mcreased Pressure at the 
transfer away from the contac "eat s^S™* 11 : 1 " ^ is ™tenj 
one another and shrinkage results thelt Centers ^oach 

the contact pressure hasbeen S^^ST ™ S f ™* -suiting from 
•s complete wetting, recrystallizaZ IV* P 5 " Flna,, y> Un 'ess there 
soHd skeleton occur, ^n/t^TcZ^T. ^ SUffident to fo ™ a 
Perhaps even more than fo he oZ Z? 8 ' S S,0Wed and ^PPed. 
Presence of a liquid phase is a com^x D oo Pr ° CeSS ' S,nterin 8 in 
Phenomena occur simultaneously Ea C ha ° h 3 number of 

expenmental systems in which a s^J" ^ * own to occu '> but 
analysed during sintering have nof bJ ■ ^ bee " iso,ated and 
Clearly, the process requires a fine part cl e t^n"? 8 ^ dem °nstrated. 
necessary C a P i„ary pressures whfc ar ? " ^ ^ to deve '°P the 
cap.llary diameter. Clearly the lia, ?h P ro P° r tional to the inverse 
Particle packing must ^^^^ ^ to ^ 
necessary capillary, pressure. Cleariy ff^Z t **" deVe, ° pin 8 ,h * 
develops by particle coalescence Z 'rt ^ 3 SO,id ske 'eton 

A critical and still controversial , " St ° PS - 
required for the process to p'ce Td ? ZT " ^ degree ° f 
^ngsten carbide-c obalt an 5 ZalL ' mp ° rtant «>»««ns such as 
dihedral angle is zero. In other yst em Tsuc^ » ^^^num the 

s,,,cate - - - - - 



'4 




quids and i n 
cobalt (see 

t processes 
se there is a 
his process 

present is 
(tact points 
ies lead to 
rangement. 
^r particles 

the liquid 
ve already 
y imposed 
cur during 
:ation. (As 
ate sinter- 
2 imposed 
in cases in 
ure at the 
s material 

approach 
Iting from 
(less there 

to form a 
stopped, 
ng in the 
umber of 
ccur, but 
!ated and 
mstrated. 
/elop the 
inverse 

the solid 
ping the 

skeleton 

wetting 
such as 
mm the 
agnesia- 
angle is 



GRAIN GROWTH, SINTERING AND VITRIFICATION 

,ow and the solid is wetted by the liquid phase, as required to develop the 
necessary capillary pressure. For grain growth of penclase particles in a 
" cate hquid, the dihedral angle has a large effect on the grain-growth 
voTss as illustrated in Fig. 10.41. Although zero dihedral angle is not 
essential for liquid-phase sintering to occur, the process becomes more 
effective as this ideal is approached. 



26 



a> 

E 

ro 
x> 
c 
ra 
ab 

a> 
to 
ro 



24 



22 



20 



> 
< 



1 


r [ 1 i 1 

\ 1 1 m No R 2 0 3 




\ A • Cr 2 0 3 




\ AFe 2 0 3 




\ oCr 2 0 3 &Fe 2 0 3 








° \ — 


) - 




I - 








6 - 

1 


i i lJLi 




10 



20 

Dihedral angle (deg) 

Fi, 10.4,. Grain srowtn of peruse parses in ^^^^^1 
compositions as a function of dihedral angle. From B. JacKson, 
Trans. Brit. Ceram. Soc, 62, 577 (1963). 



10.5 Pressure Sintering and Hot Pressing 

The sintering processes thus far discussed depend on the capillary 
pressures resulting from surface energy to provide the dnvtng force for 
Location. Another method is to apply an external pressure ^usuaHy a 
elevated temperature, rather than relying ent.rely on capillanty^ Th.s ts 
desirable in that it eliminates the need for very nne-parttcle matc^ and 
also removes large pores caused by nonumform mtxtng. An fddiuonal 
advantage is that in some cases densificat.on can be obtamed at a 
temperature at which extensive grain growth or secondary recrystall.za- 

*R. L. Coble, J. Appl Phys. 41, 4798 (1970). 



i - 

III; 
I '>il 



502 



INTRODUCTION TO CERAMICS 



tion does not occur. Since the mechanical properties of many ceramic 
systems are maximized with high density and small grain size, optimum 
properties can be obtained by hot-pressing techniques. The effect of 
added pressure on the densification of a beryllium oxide body is illus- 
trated in Fig. 10.42. The main disadvantages of hot pressing for oxide 




10 15 20 25 30 35 40 45 
Time (min) 



Fig. 10.42. Densification of beryllia by sintering and by hot pressing at 2000 psi\ 



bodies are the unavailability of inexpensive and long-life dies for high 
temperatures and the difficulty in making the process into an automatic 
one to achieve high-speed production. Both factors make the hot-pressing 
process an expensive one. For oxide materials which have to be pressed 
at temperature above 1200 or 1300°C (often at 1800 to 2000°C) graphite is 
the most satisfactory die material available; the maximum stress is limited 
to a few thousand pounds per square inch, and the life of dies is usually 
limited to seven or eight pieces. The entire die must be heated and cooled 
with the formation of each piece. Techniques for using high temperatures 
in a process in which the die is maintained cool with the material heated 
have shown some promise in laboratory tests but have not been de- 
veloped for production. 

For lower-temperature materials, such as glasses or glass-bonded 
compositions which can be pressed in metal dies at temperatures below 
800 to 900°C, the hot-pressing process can be developed as an automatic 
and inexpensive forming method. This is similar to the normal pressing of 



glass as 
shape r; 

Densi 
which 1 
liquid-p 
when hi 
materia 
densific 
pressur 
eraturef 
optimui 
nearly 
carbide 
It is of ti 
MgO, I 
inducec 

10.6 S 

The j 
connec 
growth 
these c 
occur ( 
chemic 
closed 
pressui 
general 
major 
them ii 
possibi 

Oxid 
which i 
as bine 
oxidize 
organic 
temper 
temper 
this pr< 
carbon 
before 
tempei 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



503 



ceramic 
optimum 
effect of 
/ is illus- 
for oxide 



Opsi. 

s for high 
automatic 
)t-pressing 
be pressed 
graphite is 
s is limited 
i is usually 
and cooled 
nperatures 
rial heated 
t been de- 
ass-bonded 
ures below 
i automatic 
pressing of 



elasS a S a giass-fomting method in which it is used to „bta,n the destred 
Ce LthUan - * — <*; the mechafflsms ■ 

which have been * sc » ssc ° narticu i a rlv during the early stages, 

liqui d-phase sintering. In P^^wtart ^tats, and for «ift 

when high stresses are present at the P^^miaiton is an important 

densification mode Since the gram g v ures and moderate temp- 

pressure, P"»^ nt ™. OTd "?j^^^-^ * am P les With 
er atures allows ^-J^^Jj^tSL^ tow porosity to be 
optimum mechanical properties ma w 

nearly transparent. Covalent ^2?!^"^ comptele density, 
carbide, and silicon nitride can h ^^^° 0 ^ q J d phaS e (i . e ., LiF to 

Luced liquid phase, or liquid-film, sintering to occur. 

10 6 Secondary Phenomena 

The primary processes ^.^J^J^Z^ 
connection with the firing behavior of all eramK ^comp 
growth and densification, as discussed ^5 which 
these changes, there are a large These include 

occur during the firing of ^ m \^ l ^ ^s oi gas trapped in 
chemical reactions, oxidation, phase trans application of 
closed pores, effects of »^^ m ^^oc«es" the most 
pressure during heating. Alt ^ugh they s and the 

general importance, they frequency " use ^ e m ^ cannot discuss 

- ^ with some of the 

rSSSl Many natural clays ^5-^3 
which must be oxidized during firing, ^f^"^^^. m ust be 
as binders, as well as *™*%J^Z^£ normal conditions 
oxidized during firing, or difficulties res utt. U n ^ 

organic materials *-«^|^t^y with low-firing- 
temperatures ranging from 300 to 4W ^. r h rate for 
temperature compos^ 

this process to be completed before shr.nkag b vitrificat ion occurring 
carbonaceous material is ^^J^'j j| act!3 as a reducing agent at higher 
^^ZZ^t: affect the color, g.ving rise to 



ft 



504 



INTRODUCTION TO CERAMICS 



black coring of brick and heavy clay products whose interiors are in , 




In ferrite and titania compositions control of oxidation reactions during 
finng , s part.cularly important. As illustrated for the Ti-TC and Z n 
systems (Chapter 7), the phases present depend on thl ox^n « 2 
In add-on as d.scussed in Chapter 4, the composition of SseTh^' 
covers a substant,al range of stoichiometry and depends I th * 
pressure. It is common practice in the m JZtZ7te rZ t cITol 
the oxygen pressure during firing so that the composition of eac^n^ 
present, and the overall phase composition of the body ,s ma n a'n'ed 0 
give the best magnetic properties. maintained to 

Decomposition Reactions. Many of the constituents used in ceramic 
bodes are ,n the form of carbonates or hydrated compounds Zse 
decompose dunng firing to form the oxide plus a gaseous produci (CO 

IS a T PU " t,eS ^ a,S ° incor P°™ted as carbonates hydral anS 
suites and decompose during firing (see Section 9 4) 
PWrates decompose over a wide temperature range between 100 and 

Particmar composition. For each tem^re \£ %TS£% 




GRAIN GROWTH, SINTERING AND VITRIFICATION 



equilibrium pressure of the gaseous product; if this pressure is exceeded, 
further decomposition does not take place, leading to the major problem 
encountered, the sealing of pores before complete dissociation. As the 
temperature is raised, the decomposition pressure increases and forms 
large pores, blistering, and bloating. (This is, of course, the method used 
to form cellular glass products in which the surface is intentionally sealed 
off before chemical reaction or decomposition takes place to form a gas 
phase that expands and produces a foamed product.) This kind of defect 
is particularly common when high heating rates are used, for then there is 
a temperature gradient between the surface and interior of the ware, and 
the surface layer vitrifies, sealing off the interior. This temperature 
gradient and the time required for oxidation of constituents or impurities 
are the two most important reasons for limiting the rate of heating during 
firing. 

Sulfates create a particular problem in firing because they do not 
decompose until a temperature of 1200 to 1300°C is reached. Therefore 
they remain stable during the firing process used for burning many clay 
bodies. In particular, CaS0 4 is stable but slightly soluble in water, so that 
a high sulfate content leads to a high concentration of soluble salts in the 
burned brick. This causes efflorescence— the transport of slightly soluble 
salts to the surface, forming an undesirable white deposit. Addition of 
barium carbonate prevents the deposit from forming by reacting with 
calcium sulfate to precipitate insoluble barium sulfate. 

Decomposition also occurs in some materials to form new solid phases. 
A particular example used in refractory technology is the decomposition 
of kyanite, AMVSiCfe, to form mullite and silica at a temperature of 1300 
to 1450°C. This reaction proceeds with an increase in volume, since both 
mullite and the silica glass or cristobalite formed have lower densities 
than kyanite. The reaction is useful, since the addition of kyanite to a 
composition can counteract a substantial part of the firing shrinkage if the 
other constituents are carefully selected. Similarly, reaction of MgO with 
A1 2 0 3 to form spinel occurs with a decrease in volume. By incorporating 
magnesia and alumina in a refractory mix, or more commonly in a 
high-temperature ramming mix or cement, the shrinkage taking place on 
heating can be decreased. 

Phase Transformations. Polymorphic transformations may be desira- 
ble or undesirable, depending on the particular composition and the 
anticipated use. If a large volume change accompanies the polymorphic 
transformation, difficulties result; owing to the induced stresses. Refrac- 
tories cannot be made containing pure zirconium oxide, for example, 
since the tetragonal monoclinic transformation at about 1000° involves 
such a large volume change that the ware is disrupted. The source of these 




i 

I 

i 



506 



INTRODUCTION TO CERAMICS 



esses has been discussed in Chapter 5 in connection with boundary 
s resses caused by differential thermal expansion or contraction 3 

lead?," \r nS - e T nSi0 " ° r contracti o" of a crystal in a matr* 
lead to the same sort of stresses that may give rise to actual cracking 
illustrated for quartz grams in a porcelain body in Fig. 10.44. The stresses' 
" "dividual grams can be reduced if the grain size is reduced; proper, e 
of po ce,ams are unproved if fine-grained flint is used rather than coars 




Sometimes desirable phase transformations only occur sluggishly This 
.s what happens, for example, with the firing of refractoryTilica brick 

tobal fr0m H qUa ; tZ ' the Startin 8 materia '- to tridymite and cris^ 

tobal.te, the des.red end constituents, occurs only slowly. In order to 
increase the rate of transformation, calcium oxide is added as a mineral 

h t u m ° XldC f ° rmS a ,iquid in which sili <* is soluble 
Consequently the quartz dissolves and precipitates as tridymite, which is 

he more stable phase (Chapter 7). Some of the quartz transform directly 
to cristobahte during the process as well. In general, minera.izers help in 
achieving equilibrium conditions by providing a mechanism of material 
trans er-solut,on or vaporization-that circumvents energy barriers to 
direct transformations. In silicate systems the addition of fluorides or 
hydroxyl ions is particularly helpful in this regard, since they greatly 
increase the fluidity of the liquid phase present * 

Trapped Gases. In addition to the bloating occasioned by decomposi- 
te reactions, trapping of gases within closed pores imposes a limitation 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



507 



nn the ultimate density that can be reached during firing. Gases such as 
water vapor, hydrogen, and oxygen (to a lesser extent) are able to escape 
from closed pores by solution and diffusion. In contrast, gases such as 
carbon monoxide, carbon dioxide, and particularly nitrogen have a lower 
solubility and do not normally escape from closed pores. If, for example, 
spherical pores are closed at a total porosity of 10% and a partial pressure 
of 0 8atm nitrogen, the pressure has increased to 8atm (about llOpsi) 
when they have shrunk to a total porosity of 1%, and further shrinkage is 
limited At the same time that the gas pressure is increasing, however, the 
negative radius of curvature of the pore becomes small so that the 
negative pressure produced by surface tension is increased proportional 
to 1/r the gas pressure builds up proportional to 1/r 3 . For sintering m air 
this factor usually limits densification; where very high densities are 
required, as for optical materials or dental porcelains requiring high 
translucency, vacuum or hydrogen atmosphere is preferred. 

Nonuniform Mixing. Although not mentioned in most discussions of 
sintering, the most important reason why densification and shrinkage stop 
short of complete elimination of pores is that gross defects caused by 
imperfect mixing and compact consolidation prior to firing are usually 
present. Examination of typical production ceramics shows that they 
commonly contain upward of 10% porosity in the millimeter size range 
(that is, pores much larger than the particle size of the raw materials 
introduced in the composition). These pores are caused by local varia- 
tions induced during forming, and there is no tendency for elimination of 
these pores during firing. Corrective treatment must be taken in the 
forming method. 

Overfiring. Ware is commonly referred to as overfired if for any of a 
variety of reasons a higher firing temperature leads to poorer properties or 
a reduced shrinkage. For solid-state sintering, such as ferrites and 
titanates, a common cause is secondary recrystallization occurring at the 
higher temperature before the elimination of porosity. Consequently, 
there is some maximum temperature at which the greatest density or 
optimum properties are obtained. For vitreous ceramics the most com- 
mon cause of overfiring is the trapping of gases in pores or the evolution 
of gases which cause bloating or blistering. 

10.7 Firing Shrinkage 

As formed, green ware contains between 25 and 50 vol% porosity. The 
amount depends on the particle size, particle-size distribution, and 
forming method (Chapter 1). During the firing process this porosity is 
removed; the volume firing shrinkage is equal to the pore volume 



508 



INTRODUCTION TO CERAMICS 



-^ofnJlXS&S^ S fi UbStantia,,y *«— by 
manufactured with grog (^^5 .SS- 5 ? * COmmon| y 
finng shrinkage. Similarly ^f^^T ^ ^ t0 
porcelain body; it provides a nln!T ?• * funct,ons of the flint in the 
shrinkage during liKl^.^ reduces the 

of fired grog ^V^^^SSS ° f 
the raw material has been prefired and L fi • " b f Cause a large, part of 

If firing is carried to complete denti ♦ " 8 Shnnkage is low - 
originally present is equal T^^^ \ he fractional Porosity 
commonly amounts to as much as 35^ 8 P ' 3Ce durin S firing. This 
hnear shrinkage and causes tmj^n^ ° T U l ° 15% 

However, the main difficulties^ wl T*™*™? 8 C,ose tolerances. 
Afferent amounts of firing 2i2 J77« " Caused by 

Nonuniform shrinking can sometime 'v ^ parts ° f the w ^ 

Warping. A maio cl7J n ? CaUSe cracks to open 

jn the grefn ^^^^^^ ^nsity Liations 
the green ware. The density aft.r « differences in porosity in 

higher shnnkage for the -^Z^^ ^ ther " « 

had a high density i„ the green ware In Zt T* **" fW ^ parts ^ 
•» die (Chapter 1) cause £ 11! f 7*"' Variati ™ 
Parts of a pressed pi ece - usuallv the ,h T com P a ction at different 
the shrinkage at the" ends, an Z houSSf " ""^ is ^ tha " 
cylindrical sample (Fig. 10.45a). Pe feSU,tS from 

an initially 

^sTwt: jj^ d ^ ~ of temperature 

temperature difference to^i^^J?!? 1 fr ° m abov *> *ere is a 
cause greater shrinkage at the top than a, th 1 ^ ° f the Ware that «™y 
warping. I n some cases the grav ^ 

make the ware lie flat c^^^J^ may be suffi cient to 
relationship between tempera ^ure dist^utn 86 " n ° nunif °™- The 
under the stresses developed 7s 1 ? t,0 , n ' Warpa ^ and deformation 
quantitatively. Another soufce o waZ "! n *** d, ' ffiCU,t l ° ana '^ 
t.on of the platey Cay Particles dulgT ormi^ " Pr6ferred 0rienta ' 
the drymg and firing shrinkage to have dlec Z! Pr ° CeSS - Th ' S C3USes 
Vitreous ware is also wamtn u T air ectional properties. 

developed. In , he forming ot vLeou s LI! stresses are 

a close, ^ (Fig . I0 J c) or «~ ware , e upper surface of 

with a greater curvature than is desired !%, A ' 
3e.tl.ne whieh occurs on firing p o dU e S a final h T" 1 " SO ,I|M "«= 



illy decreased by 
orick is commonly 
serve to decrease f| 
of the flint in the 
vhich reduces the 
posed of mixtures 
use a large part of 
ikage is low. 
ractional porosity 
during firing. This 
:age or 12 to 15% 
close tolerances. 
:>rtion caused by 
irts of the ware, 
ks to open, 
density variations 
ces in porosity in 
orm, and there is 
i for the parts that 
ressure variations 
iction at different 
iter is larger than 
; from an initially 

;e of temperature 
above, there is a 
he ware that may 
I a corresponding 
' be sufficient to 
lonuniform. The 
and deformation 
icult to analyze 
referred orienta- 
ess. This causes 
>erties. 

* gravity. This is 
itial stresses are 
upper surface of 
lust be designed 
duct so that the 
t is satisfactory, 
lal force or drag 



GRAIN 



GROWTH, SINTERING AND VITRIFICATION 



509 




Green 



Green 



Fired 



(b) 





(c) 

factional force of setting. 

other sources of porosity variafon. Pressmg "^l^^^.* 



mi 



512 



INTRODUCTION TO CERAMICS 



way that the final shape is satisfactory. In the same way, when plates are 
tired m the horizontal position there is a tendency for the rims to settle- 
thiAcan be compensated for by adjusting the shape of the initial piece " ' 
Correct setting methods are important in eliminating difficulties caused 
by firing-shrinkage variations. These have been most extensively de- 
veloped for porcelain compositions in which complete vitrification is 
desired and high shrinkages result. Some of the standard setting methods 
are illustrated in Fig. 10.46. Cups and bowls are commonly boxed as 
indicated ,n Fig. 10.46a. This keeps the rim circular, since warpage of one 
restricts warpage of the other; in addition, it prevents the thin rims from 
being too rap,dly heated. For larger pieces, unfired setters are necessary 
as a means of controlling shrinkage and maintaining circular rims A 
variety of methods is used for setting different kinds of plate composi- 
tions depending on the amount of shrinkage expected. For ware fired to 
complete vitr.fication individual setting and support are essential. For 
ware fired to partial densification, plates can be stacked with no ill effects 
In general, large tiles and brick do not cause much difficulty 





(a) 



(b) 





SM°H 7 ' S f n8 meth .°? S f ° r SPCCial ShapCS - {a) Large ,iles se < a " a "8le of repose- 
Norton SUPP " C0 " ar: (C> SPCCial ShapC: W) Scul " ,ured » ie « From F H.' 





tes are 

settle; 
:e. 

caused 
ly de- 
tion is 
;thods 
ted as 
of one 
s from 
essary 
ms. A 
mposi- 
ired to 
ll. For 
;ffects. 



•repose; 
n F. H. 



GRAIN GROWTH, SINTERING AND VITRIFICATION 



513 



2. 



3. 



Snecial shapes may require special setting methods to eliminate ad- 
verse effects of firing shrinkage. Large refractory tile can be set at an 
Inde of repose on a flat surface (Fig. 10.47a). This allows the tile to 
shrink without much stress. In the same way rods or tubes may be set in 
an inclined V groove or supported by a collar from the upper end (Fig 
10 47b) Gravitational forces keep the tubes straight up to lengths of 
several feet. Unique shapes can always be supported on special setters 
designed for the particular sample. Some experience is necessary to 
handle unique shapes efficiently. Small pieces of sculptured vitrified ware 
are particularly difficult. The safest setting provides complete support 
from unfired struts (Fig. 10.47d). 

Suggested Reading 

1 G. C. Kuczyuski, N. A. Hooton, and C. F. Gibson, Eds., Sintering and Related 
Phenomena, Gordon and Breach, New York, 1967. 

G. C. Kuczyuski, Ed., "Sintering and Related Phenomena", Materials Science 
Research, Vol. 6, Plenum Press, New York, 1973. 

R. L. Coble and J. E. Burke, Progress in Ceramic Science, Vol. HI, J. E. 
Burke, Ed., Pergamon Press, 1963. 
4 W D Kingery , Ed., Ceramic Fabrication Process, Part IV, Technology Press, 
' Cambridge, Mass., and John Wiley & Sons, New York, 1958. 
5. W. D. Kingery, Ed., Kinetics of High -Temperature Process^ Part IV .Technol- 
ogy Press Cambridge, Mass., and John Wiley & Sons, New York, 1959. 
J. E. Burke and D. Turnbull, "Recrystallization and Grain Growth in Metals, 
Prog. Met. Phys., 3, 220 (1952). 

E. Schramm and F. P. Hall, "The Fluxing Effect of Feldspar in Whiteware 
Bodies," J. Am. Ceram. Soc, 15, 159 (1936). 

For additional papers on sintering see: R. L. Coble J. AppL P hys 41, 4798 
(1970); D. L. Johnson and I. B. Cutler, /. Am., Ceram. Soc. 46, 541 (1963). 



Problems 

10 1 Distinguish between primary recrystallization. grain growth and secondary recrys- 

taxation as to («) source of driving force, (0) magnitude of dnv.ng force, and (c) 

importance in ceramic systems. 
10 2 Explain why the activation energy , for grain-boundary migration corresponds 

approximately with that for boundary diffusion, even though no concentrate 

gradient exists in the former case. 
10.3. Can grain growth during sintering cause compaction of cerarmcs? Explain. Can 

grain growth affect the sintering rate? Explain. 



6. 



7. 



POLAR DIELECTRICS AND 
THEIR APPLICATIONS 

JACK C. BURFOOT and GEORGE W. TAYLOR 



University of California Press 
Berkeley and Los Angeles 



BEST AVAILABLE COPY 




University of California Press 
Berkeley and Los Angeles, California 

ISBN: 0-520-03749-9 

Library of Congress Catalog Card Number: 78-62835 



Copyright © 1979 by Jack C. Burfoot and George W. Taylor 



Printed in Great Britain 



2 



Preparation of Polar 
Materials 



This chapter summarises the techniques used for fabricating polar materials in 
smgle crystal ceramic, thin film and glass forms. It then goes on to discuss the 
various post-fabrication procedures such as annealing, poling, cutting thin- 
ning, polishing and electroding which are usually needed before the polar 
materials can be used for either experiments or applications 

For basic studies, where the polar material should be as near perfect as 
possib e, it ,s desirable to use single crystals. For application* where 
dimens ons, reproduc.b.hty and cost are crucial factors then the polar material 
.s usually fabricated ,n ceramic, thin film or glass form. This division is not 
precise. For example, some materials have not yet been grown as single crystals 

fair Zf.T "* T' ^ " ,he Cefamic form - In other ^e 
larger dielectric, piezoelectric, pyroelectric, electro-optic, etc., coefficients 

obtainable with particular single crystal materials can make Ms fabrication 

lir 1 ; h y ? rtai " a PP ,ications - And again thin films offer an 
important advantage for studying surface layer physics 

The two major problems in fabricating large samples of high quality polar 

of 2a ^ m rf a " C f ° f the stoichiometry and the avoidance 

of strains. Chemical, thermal and optical methods, as well as sophisticated 




14 

fabrication techniques tb£^££^™^ By comparison, thf 
variety of 'recipes' reported for ££££££ Th* h^ * ^ 
changing. For example, it is now customarvfn^T Ths , however may be 
the phase diagram of the maS ° asTo cZ T'™ determina «<>n of 
conditions. These diagrams can be mniV , the ° ptImum fab ri<ation 
2.1 and 2.2, the phase diaSS tet^r ■ Can U Seen ^figures 
in figure 2.2 beiause S^^™*'^™™™***- Notfth* 
best considered a Nb doped PbZrO pSnO 0^'' W the material is 
different phase diagram is needed for £5 t 3 T, ° 3 S ° ,id Solution > <*» * 
the scientific approach l^J^'^tST 1 ^ ™ mpk of 

understand citation currents * * " 




1200- 



80 Ti0 2 
Ti0 2 (mole%) 

Fiflure 2.1 Barium titanate. Phase diagram (Rase and Roy') 

Many measurements can be m^P 
ceramic, thin film or glass m^ria, Lnrn f ""'f the degree to which * 
crystal counterpart. The ^^^^^ PWpCrties of its ™& 
making this dLmnatk^^^^T^ "!! P r erfuI Criterion for 
-on, then the better the orient £ ^Zt^^t 



PREPARATION OF POLAR MATERIALS 



15 




Figure 2.2 Part of the phase diagram of Pb 0 . 99 (ZrSnTi) 098 Nb O 02O 3 at 25° C 
(poled at 25°) F = Ferroelectric, A = antiferroelectric, T = tetragonal, 
R = rhombohedral, LT = low. temperature phase, H.T = high temperature phase 
(Raider and Cook 2 ) 

percentage of non-polar material and voids present. In general, doctor-bladed 
ceramics and epitaxial thin films are the fabrication techniques producing 
materials closest to the single crystal form, whilst the glasses and con- 
ventionally sintered ceramics are the furthest away. 



2.1 Growth of Single Crystals 

The two major methods that have been used to date for fabricating single 
crystals have been the solution (or flux) growth technique and the melt growth 
technique. 

2.1a Solution Grown Crystals 

This method has been successfully used for both water soluble materials and 
those that are soluble in other liquids or fluxes. 

The first stage in growing a water soluble crystal is to prepare an aqueous 
saturated solution of the polar material. Crystals can be grown by either 
keeping the solution at a constant temperature and allowing a gradual 
evaporation of the solvent or by slowly lowering the temperature while keeping 
the solution saturated. Slow growth, taking a thousand hours or more, usually 
produces the best and largest crystals. Other factors which affect the crystal 



17 

PREPARATION OF POLAR MATERIALS 

growth are the purity of the materials, the solubility-temperature characteris- 
tics of the solution, the fineness of the temperature control, the use of stirring to 
prevent temperature and concentration gradients from developing in the 
solution, and the use of seed crystals suspended in the solution to enhance 
growth. 

Table 2. 1 contains a fairly comprehensive listing of the polar materials which 
have been successfully grown frojn aqueous solution. Figure 2.3 shows a large 
solution-grown crystal of triglycine sulphate, TGS. The original seed crystal is 
visible in the centre of the photograph. The organic polar material thiourea can 
be grown from an aqueous solution; however, better results are obtained from 
an alcoholic solution. 




Figure 2.3 Photograph of a TGS crystal (Linz 4 ) 



Polar materials that are not soluble in water or alcohol can often be 
dissolved at high temperature in other materials, usually referred to as fluxes. 
For example, 26 different fluxes have been reported 5 for barium titanate, 
BaTi0 3 . In some cases, an excess of one of the constituents can act as a flux. 
For example, additional Bi 2 0 3 will serve as a flux in growing crystals of 

bismuth titanate, Bi 4 Ti 3 0 12 - 6 

Besides the particular flux used there are many other variables in flux 
growth. They include the purity and particle size of the component materials, 
the time-temperature cycle used for forming the molten flux solution, the 
crucible material and its shape and size, the method of heating (both resistive 
and r.f. induction heating are used), the time-temperature cycle used for 
cooling, the thermal gradients established in the furnace (both vertical and 




18 

BASIC PROPERTIES 

fTnace taI 8fadientS ^ ^ ^ thC atmos P here maintained in the 
Some of the crystals that have been grown by flux techniques are shown in 
table 2.1. Space does not permit detailing the growth conditions used for each 
material. To giye some idea, however, it is worth summarising the successful 
technique developed by Remeika 7 for the flux growth of BaTiO, crystals A 
platinum crucible ^containing 30% BaTiQ 3 powder, 70% KF (the flux) and a 
small trace of Fe 2 0 3 is heated for 8 h at 1 175°C. The Fe 2 0 3 compensates for 
the loss of oxygen at high temperature. The crucible is then cooled slowly to 
875 C at which stage the excess liquid flux is poured off. The crystals thus 
formed are then cooled slowly to room temperature. Any residual flux is 
removed by acid etching. The crystals have a plate-like morphology and some 
typical examples are shown in figure 2.4a. 





1cm 



(a) 



(b) 



Figure 2.4 Photograph of barium titanate crystals, 

(a) Crystals grown by flux method (Epstein 8 ). 

(b) Crystals grown by Czochralski method (Belruss et a/. 1 ') 



2.1b Melt Growth 

If a polar material melts congruently, that is, if stoichiometry is maintained 
then the crystal can be grown directly from the melt. As the crystal grows' 
either by spontaneous nucleation on to a chemically inert platinum or iridium 
wire or onto a seed crystal, it is gradually withdrawn from the molten liquid In 
he Stockbarger method, this is done by withdrawal of the crucible containing 
the melt. In the Czochralski method (figure 2.5), the crystal is gradually 
pulled out of the melt, and it is usual to rotate the crystal while pulling to 
minimise thermal and stress gradients. Also suitable optics are provided 'for 



viewin; 
growei 
design) 
methoi 
and les 
are pos 
and an 
used. F 
at 70 r 
atmosj 
Beln 
times r 
gradua 
grown 
crystal; 
132°C. 
figure 2 



PREPARATION OF POLAR MATERIALS 



19 



Pt/Pl -Rh 
THERMCOUPLE 
IN GROUNDED 
Pt SHIELD 



PULLING MECHANISM 

-CERAMIC INSULATOR 
ROTATING CONTACT, 

\/)0-10mA 



PLATINUM 
WOUND 
AFTERHEATER 
TEMP. 
~1050°C 




§) PLATINUM 
CRUCIBLE 



-MOLTEN LiNb0 3 
I lcm 



Figure 2.5 Czochralski crystal grower for lithium niobate (Nassau 5 



viewing the crystal during growth. Figure 2.5 shows a Czochralski crystal 
grower used for growing lithium niobate, LiNb0 3 . The apparatus is also 
designed to pole the crystal during the growth process. The Czochralski 
method is usually the best for polar materials, in that it produces less strains 
and less twinning in the crystal. A large number of variations in the technique 
are possible including variation of the pulling and rotation rates, the method 
and amount of after-heating used as the crystal emerges, and the atmosphere 
used. Figure 2.6 is a photograph of a Gd 2 (Mo0 4 ) 3 crystal pulled by Kumada 
at 70mm/h using a rotational speed of lOOrev/min in an oxygen rich 



d, atmosphere 10 . Most Czochralski grown crystals have the form of figure 2.6. 

s, Belruss et ai 11 have developed a modified Czochralski technique, some- 

m times referred to as 'top seeding', in which the temperature of the melt is 

In gradually dropped (0.2°C/h) as pulling proceeds. The crystal of figure 2.4b was 

ig grown by this technique, using a pulling rate of 0.7 mm/h. Pulled BaTi0 3 

ly crystals have a polyhedral morphology and a transition temperature, T c of 

to 132°C By comparison, the Remeika 7 flux grown BaTi0 3 crystals, shown in 

or figure 2.4a have a plate-like structure and a T c of 1 20°C. The lower T c is due to 



A 



20 



BASIC PROPERTIES ' 




H3D 




Figure 2.6 Single crystal of gadolinium molybdate (Kumada'°) 



the substitution of K atoms at some Ba sites and Fe at some Ti sites in the 
BaTiQ 3 crystal lattice; the K and Fe impurities originate from the flux used 
The polar material single crystals that have been successfully grown from the 
melt by Czochralski type techniques are listed in table 2. 1 . 

2.1c Other Techniques 

Single crystals of both BaTi0 3 and antimony sulpho-iodide,SbSI, have been 
grown by vapour transport'*. Hydrothermal methods, which involve a 
combination of high pressure and temperature have been used to grow single 
crystals of several types of polar materials "-'^-see also table 2. 1 

2.2 Ceramic Fabrication 

™J* SSi ? te ? hniqUe f ° r f ° rming Ceramics is sinterin 8 at atmospheric 
posing 6 vanat,ons on Ms Process are doctor-blading and hot 

2.2a Sintering 

The constituents (or their oxides) of the polar material are mixed in the correct 
proportions with an organic binder and then pressed at room temperature into 
a structure having the desired shape and dimensions. In most cases this is a 
cylinder, although for some applications, for example sonar, more complex 
shapes are used. The pressed structure is sintered or fired at an appropriate 
temperature in an appropriate atmosphere. This causes the organic binder to 
be burnt out and the pressed materials to react chemically and form the desired 
polar material. Table 2.2 contains a list of some of the polar materials which 
have, been fabricated as sintered ceramics. 



PREPARATION OF POLAR MATERIALS 



21 



Table 2.2 Ceramic polar materials that have been fabricated by various 

techniques 

CERAMICS 



SINTERED 



AgNb0 3 
AgTa0 3 
AgV0 3 

Ba 2 NaNb 5 0 15 

BaTi0 3 

B' 4 Ti 3 0 12 

Cd 2 Nb 2 0 

CdTi0 3 

(K, Li,Sr)Nb l0 0 13 

KNb0 3 

KTa0 3 

K(Nb,Ta)0 3 

UNb0 3 

LiTa0 3 

(Na, K) Nb0 3 

NaNb0 3 

NaTa0 3 

Na(NbTa)0 3 



Pb(Fe,Nb)0 6 
PbHf0 3 

Pb 3 (Mg.Nb 2 )0 9 
PbNb 2 0 6 
PbTi0 3 
PbZr0 3 

PbBi(Zr,Sn,Ti)0 3 

PbNb(Zr,Sn,Ti)0 3 

PbBi(Zr,Ti)0 3 

PbLa(Zr,Ti)0 3 

PbNb(Zr,Ti)0 3 

RbTa0 3 

SrBaNb 2 0 6 

SrTi0 3 

WO, 



DOCTOR B LADED 

I 

BaTi0 3 

PbNb(Zr,Sn,Ti)0 3 
Pb(Zr, Ti)0 3 



HOT PRESSED 

I 

(Na,K)Nb0 3 
Pb(Zr,Ti)0 3 
PbBi(Zr,Ti)0 3 
PbLa(Zr, Ti)0 3 
PbNb(Zr, Ti)0 3 
Pb(Sn,Zr, Ti)0 3 



2.2b Doctor- Blading 

Doctor-Blading 16 is particularly suited for forming large area, thin sheets of 
ceramic. The constituents of the polar materials are mixed in a liquid together 
with a suitable plasticiser and the resultant slurry is poured onto flat glass. A 
stainless steel blade, accurately positioned a small distance, S, above the sub- 
strate is then drawn through the slurry. The resulting sheet is allowed to dry after 
which it can be peeled ofT the glass. At this stage the material is termed 'green' 
because it can be easily cut or punched into any two dimensional shape. The 
sintering is done in a two step process, viz. a lower temperature firing to burn 
out the plasticiser and then a higher temperature firing in a controlled 
atmosphere to form the polar material. 

Whenever a ceramic is sintered there is a large amount of shrinkage. For a 
doctor-bladed material, the shrinkage is particularly evident as a decrease in 
thickness Figure 2.7 shows the relationship between the thickness of 'green' 
and fired materials for a PbNb(Zr, Sn, Ti)0 3 ceramic. A fired and electroded 
doctor-bladed ceramic strip is shown in figure 2.14. 

2.2c Hot Pressing 

In the hot-pressing process the ceramic is sintered under pressure, typically 
developed in a hydraulic press. Hot pressing can result in ceramic densities 



22 



BASIC PROPERTIES 




0 250 500 750 1000 

BLADE SETTING S 



Fig lire 2.7 Tickness of 'green' and fired materials as a function of blade setting, S, 
for a doctor-bladed Pb(ZrSnTiNb) 0 3 ceramic (Wentworth and Taylor 16 ) 

even higher than 99.9% of the theoretical maximum. As a resut, such 
materials have properties approaching those of a single crystal. For example, 
the hot-pressed ceramic can have a high value of remanent polarisation, a low 
value of coercive field and, as shown in figure 2.8, can be transparent. 




Figure 2.8 Photograph of polished transparent Pb(ZrTiLa)0 3 hot-pressed 
ceramic samples. (Haertling and Land 18 ) 



PREPARATION OF POLAR MATERIALS 23 

Haertling 17 ' 18 has extensively studied how the parameters of time, pressure, 
temperature, chemical purity and firing atmosphere affect the properties of 
hot-pressed Nb, Sn, Bi and La doped and Sn, Ba, and La modified f 
Pb(Zr, Ti)0 3 ceramics. Figure 2.9 is typical of such results. In this case the 
effects of time, pressure, and temperature, on grain size of a 
PbNb(Zr, Sn, Ti)0 3 ceramic are shown. As discussed in section 15.2a the grain 
-size is importantMn determining the electro-optic properties of certain ceramic- 
compositions. It also affects many other properties of the ceramic including the 
permittivity. Other ceramics that have been hot pressed are listed in table 2.2. 




HOT PRESSING TIME t (h) 

Figure 2.9 Effect of hot pressing time, temperature and pressure on average grain 
diameter of a hot pressed PbNb(ZrSnTi)0 3 ceramic (Haertling 17 ) 



2.3 Thin Film Fabrication 

There has recently been much general interest in thin films of polar materials 
(i.e. less than about 10 fim thick). In particular, for device applications, thin 
films have important advantages which include (a)^ formation of large 
capacitances, (b) low switching voltages, (c) the possibility of forming the 
film directly on the integrated semiconductor 'driving' circuits. 

The various techniques used for making thin films are described below. The 
materials that have been made by these techniques are summarised in table 2.3. 
With the exception of r.f. sputtering, the fabrication techniques generally 

+ As described in sections 1 5.2 and 1 7.4 T the properties of the Pb(Zr ( Ti)0 3 ceramics vary greatly 
depending on whether the amount of additive is less or greater than about 5 atom per cent. As a 
result it is convenient to use the term doped Pb(ZrTi)0 3 when the additive is less than 5 atom per 
cent and the term modified Pb(ZrTi)0 3 when the additive is greater than 5 atom per cent. 



PREPARATION OF POLAR MATERIALS 25 

f 

produce polycrystalline films, which have properties more similar to ceramics 
than to "single crystals. 

2.3a Solution Deposition - : 

Three types of solution deposition have been used for forming thin films. They 
are casting, hydrolysis and electrophoresis. 

Casting 

Beerman 19 has made thin films of TGS (a water soluble polar material) by 
spraying an aqueous solution of TGS onto a suitable substrate. Chapman 20 
has formed thin films of the complex ferroelectric Pb(BiLaFeNbZr)0 3 by first 
making a colloidal suspension, or slurry, of the basic oxides of the composition. 
The suspension was then centrifuged onto a metallic substrate and sintered at 
900°C to form the ferroelectric thin film. 

Hydrolysis 

Lure et a/. 21 have deposited a mixture of Pb, Zr, Sn and Ti oxides on a 
metallic substrate by hydrolysing a solution of Pb, Zr, Sn and Ti tetrachlorides. 
The oxides were then sintered to form a ferroelectric film of Pb(ZrSnTi)0 3 . 

Electrophoresis 

Lamb et ai 22 placed two noble metal electrodes into a suspension of BaTi0 3 
particles in ether. The application of about 200 V cm" 1 between the electrodes 
caused a film to be formed on the anode. Subsequent sintering in an 
atmosphere of 98 % helium and 2 % oxygen at 1 350°C created a stable BaTi0 3 
film. 

2.3b Melting 

Nolta etal 23 have shown that a thin layer of potassium nitrate, KN0 3 , can be 
easily formed by melting KN0 3 powder onto a metal substrate. However, 
KN0 3 has the disadvantage that it is only ferroelectric at room temperature 
and atmospheric pressure for a short time, before reverting to a non- 
ferroelectric phase 24 . Sodium nitrite, NaN0 2 , and barium titanate, BaTi0 3 
films have also been prepared by melting. 

2.3c Vacuum Deposition 

Evaporation and sputtering techniques have been used for the vacuum 
deposition of thin films. 

Evaporation 

In his early work Feldman 25 evaporated BaTi0 3 from a coated tungsten 



26 



BASIC PROPERTIES 



filament onto a metallic substrate in a vacuum of less than 5 x 10" 5 mm Hg. 
Due to the difference in volatility of the constituent oxides, the resultant film 
consisted of separated layers of BaO and Ti0 2 . To combine the oxides the film 
had to be subsequently heated in air at 1100°C 

Flash evaporation onto a heated substrate is a technique developed by 
Miiller et ai 26 and Burfoot et ai 21 which overcomes the dissociation problem. 
The polar material is evaporated in small-thickness increments, typically 
corresponding to a few crystal lattice spacings, by dropping the source 
material, a grain at a time, onto a filament heated to a temperature of about 
2000°C. One variation which has proved successful in improving the quality of 
the evaporated films has been to leak a small amount of oxygen into the 
vacuum chamber. This improved stoichiometry by overcoming oxygen 
deficiencies. Another successful technique has been the use of multiple 
evaporation sources. For example, in forming BaTi0 3 , a source of BaO and a 
source of Ti0 2 are used. The multiple sources can reduce the amount of 
filament material in the film. 

In addition to BaTi0 3 , thin films of lead titanate, PbTi0 3 , and bismuth 
titanate, Bi 4 Ti 3 0 12 , have been evaporated. Films evaporated onto metal 
substrates such as platinum are polycrystalline in nature. An epitaxial film of 
BaTi0 3 with its c-axis uniformly aligned perpendicular to the substrate can be 
evaporated if the substrate is a freshly cleaved alkali halide crystal such as LiF 
or NaF. Burfoot 27 has developed a technique for removing the film from the 
insulating substrate, so that an electrode can be placed on the thin film for 
electrical measurements. 

Sputtering 

Francombe 28 has suggested that sputtering offers several advantages over 
evaporation, namely (a) better control of stoichiometry, especially for the 
more complex oxide materials, (b) better thickness control and (c) freedom 
from material contamination. Most sputtering has been r.f. sputtering from 
ceramic targets. However, diode, triode and tetrode sputtering has also been 
used. The polar materials which have been sputtered are listed in table 2.3. Two 
of these materials, BaTi0 3 and Bi 4 Ti 3 0 12 , have also been sputtered 
epitaxially. 

It is instructive to summarise the sputtering technique developed for 
monoclinic Bi 4 Ti 3 0 12 , since the problems which occur are rather typical, and 
the results obtained have been the most impressive. Takei et ai 29 used a 
10 cm 2 ceramic target of bismuth titanate mounted on a water-cooled metallic 
base and positioned 4 cm from a heated substrate. Using a 4 mm atmosphere 
of 0 2 and Ar they were able to sputter Bi 4 Ti 3 O l2 films at a rate of about 
1 As" 1 using a power level of 1 W cm" 2 and a self bias of 700 V. 

The problem in vacuum deposition, as with single crystal and ceramic 
fabrication, is to develop the correct stoichiometry so that only the required 
phase is formed. In the case of sputtering Bi 4 Ti 3 0 12 , this is done by using a 



PREPARATION OF POLAR MATERIALS 



27 



EXCESS f 
Ti0 2 | 



Bi 4 Ti 3 0 t2 

EXCESS I 
-Bi 2 0 3 \ 



400 " 600 eoo 1000 

DEPOSITION TEMPERATURE T (°C) 

Figure 2.10 Dependence of composition on substrate temperature for films r.f. 
sputtered from a target of composition 80% Bi 4 Ti 3 0 12 , 20% Bi 12 Ti0 20 . 
(Francombe 28 ) 

ceramic target of 0.8 Bi 4 Ti 3 0 1 2 , 0.2 Bi , 2 TiO 20 and a substrate temperature of 
650°C. The bismuth enriched target compensates for the high volatility of that 
oxide, while as can be seen from figure 2.10, the chosen substrate temperature 
favours Bi 4 Ti 3 0 12 over the other phases. 

The Bi 4 Ti 3 0 12 film can be sputtered epitaxially and aligned along any 
preferred direction by choosing a substrate having a suitable crystal lattice. 
Substrates which have been used are Pt, single crystal Bi 4 Ti 3 0 12 and MgO 
and MgAl 2 0 4 . Except for a higher coercive field, the electrical and optical 
properties of the epitaxial films closely match those of the bulk (single crystal) 
material; see, for example, figure 2.11. The most dramatic evidence of the 
quality of the sputtered films of Bi 4 Ti 3 0 12 is their ability, as shown in figure 
2.12, to duplicate the complex electro-optic light valve switching properties of 
the single crystal material described in section 15.2a. 

2.4 Fabrication of Polar Glasses 

Borrelli, Herczog and colleagues 31, 32 at Corning have succeeded in crystallis- 
ing NaNb0 3 and (NaK)Nb0 3 in glass matrices of Si0 2 , and BaTi0 3 and 
SrTi0 3 in a matrix of BaAl 2 Si 2 0 8 . Also, Isard and his colleagues 33 * 34 at 
Sheffield have made glasses ofPbTi0 3 and(PbBa)Ti0 3 in B 2 0 3 matrices and 
(KTa)Nb0 3 in a Si0 2 matrix. These polar glasses are made by the rapid 
quench cooling of the appropriate molten oxides, followed by suitable 
annealing treatment to crystallise the polar material in the low permittivity, 






Figure 2.11 Birefringence values and extinction angle of bismuth titanate as a 
funcfon of temperature measured at wavelength 5890 A. The s^d lines are from 
single crystal, and thedashed linesfrom film. The subscripts a. b. and ^^5,6 
monochmc ax.s along which light was directed for each measurement (J u e 




in 9 (Tn»- H S ,v " er ^ ^ iSmUth ^ anate film viewed betwee " crossed polarisers 
Im^gS e C £r a ( X ° n C ° nd ' ,,0n ^ ^'^ 



high res.stw.ty glass matrix. The volume per cent of glass matrix present 
should be kept as small as possible: 20% is an achievable amount The 
electncal properties of the glasses are enhanced by a large crystallite grain size 
for the polar material portion, while the optical properties are favoured by a 
small gram s.ze. For electro-optic applications, 0.5 m is a typical compromise 



PREPARATION OF POLAR MATERIALS 



29 



The polar glass materials have the characteristic advantages associated with 
glass, namely an absence of pores and hence a high dielectric breakdown and 
mechanical strength. Also, they can be conveniently fabricated in thin uniform 
sheets. On the other hand, the presence of the glass matrix with its relatively 
low dielectric permittivity as compared to the polar materials means that the 
electrical and optical coefficients are smaller than those of the corresponding — 
ceramic or single crystal material. The properties are, however, still sufficiently 
significant for some electro-optic applications and for some cryogenic 
temperature, capacitor applications. 32 

2.5 Post-Fabrication Procedures 

2.5a Annealing 

Crystals grown from the melt often develop strains which lead to undesirable 
domain structures (twinning) and mechanical cracking. The strains are mainly 
caused by the mechanical constraints imposed during the growth process. The 
easiest way to remove these strains is by thermal annealing. This can be done 
by cooling the crystal through its transition temperature at a prescribed rate in 
a suitable temperature gradient. For example, Hopkins and Miller 35 have 
found that the strains due to domain structures in Bi 4 Ti 3 0 12 can be 
successfully annealed out by placing the crystals on an alumina surface in a 
tube furnace with a temperature gradient of 5°C cm " 1 and cooling at the rate 
of 2°C min~ Annealing is also used to remove strains which can develop in 
thin film materials and ceramics while they are being fabricated. 

2.5b Poling 

In a newly fabricated ceramic, the crystallites, and hence the polar axis, can lie 
in a large number of directions. This will cause the material to have only a 
quasi-isotropic response in its electrical, optical, piezoelectric and other 
characteristics. A lack of orientation can also exist in certain freshly prepared 
single crystal and thin film materials due to there being more than one possible 
orientation for the polar axis. For example in BaTi0 3 there are three possible 
orientations for the polar axis and each orientation has two possible directions. 

Before such unaligned materials can be used for measurements and 
applications, it is necessary to orient the polar axis in a common direction 
usually normal to the major surfaces. (However, in the case of the electro-optic 
PLZT ceramics it is necessary, for some modes of operation (see section 15.2), 
to pole parallel to the surface.) 

Polar axis orientation is done by a poling technique which consists of 
applying a D.C. voltage for a sufficient time to suitable electrodes on the 
material. The amplitude and duration of the voltage required for poling vary 
substantially between materials. Some materials will pole easily and if used in 



30 



BASIC PROPERTIES 




Figure 2.13 Single crystal bismuth titanate poled in centra! region as viewed 
between crossed polarisers (Hopkins and Miller 35 ) :scale has 1 mm divisions. 

large-signal switching applications may not even require a preliminary poling 
cycle. Others can only be poled near their Curie point where the coercive field 
of the material is small. Sometimes the poling is done during the crystal 
growth— see figure 2.5. The maximum poling voltage that can be applied is 
limited by breakdown and arcing in the material. Hence poling is often done in 
an oil bath. Figure 2.13 showsa Bi 4 Ti 3 O l2 crystal which has been poled in its 
central section. The crystal is viewed between crossed polars and hence only 
the poled region is uniformly extinguished. 

2.5c Cutting, Thinning and Polishing 

Depending on the intended usage, it is often necessary to cut and thin the 4 as 
grown' single crystal or the 'as sintered' ceramic slug or disc to smaller 
dimensions. If the material is water soluble then the cutting can be done with a 
water soaked string. Other materials may require cutting with a high speed 
diamond coated wire. The thinning can be done by slicing, mechanical 
grinding or etching. In the latter process, depending on the material, the etch 
solution can vary from water to aqua regia. Some materials will, with a suitably 
applied high impulse pressure, naturally cleave into thin platelets. Desputter- 
mg, where-in the polar material is made the sputtering source, is a technique 
that permits very carefully controlled thinning of a material. 38 

For optical use it is almost mandatory to have a finely polished surface to 
minimise light scattering. Several methods are used for polishing. One of the 
most common is a series of polishes with alumina powdenof progressively 
smaller particle size. 



PREPARATION OF POLAR MATERIALS 



31 



2.5d Electroding 

Many measurements and most devices require the application of electric fields 
to the material. This is done by applying a voltage between electrodes placed 
on the surfaces of the polar material. Because of the high c of polar materials, 
the electrodes must be in intimate contact with the surfaces. Virtually any 
airgap will cause an intolerable reduction in the voltage developed in the polar 
material. 



EL CONNECTIONS 




DIODE 

, 1cm t CONNECTIONS 

Figure 2.14 An evaporated gold electrode pattern on a doctor bladed ceramic 
strip (Taylor 36 ). 

For quick measurements, air drying silver paste is often used as an electrode. 
For applications which require accurately defined electrodes, it is usual to 
evaporate metal through photomechanically formed masks. Figure 2. 14 shows 
such an electrode pattern formed with evaporated gold on a doctor-bladed 
PZT-type ceramic. Silk screening and sputtering are two other techniques 
which have been used for forming electrodes. For electro-optic applications, it 
is often necessary to have transparent electrodes. Very thin (less than 500 A 
thick) evaporated metal electrodes can sometimes be used. However, r.f. 
sputtered indium and/or tin oxide has proved to be a more satisfactory 
transparent electrode. Such electrodes can have a resistance of less than 10 Q 
per square and an optical transmission of greater than 90% in the visible 
spectrum. 

At first sight, electroding would seem to be no more than a technical process. 
However, as discussed in detail in section 3.2c, electrodes can have a significant 
effect on the ageing phenomena in polar materials. For example, if metal 
electrodes are used, single crystal BaTi0 3 ages after repeated polarisation 
reversals. With conducting liquid electrodes (e.g. an aqueous lithium chloride 




32 

solution) no ageing occurs 37 TW , h ,u BAS, ° PR0PERTies 

Physics invoked if °^ W ^ ^ tbesuAce 

field gradients which exfst n<Lr te,^J"^ O0d The ,a '8 e el <**c 
contribute to the effects that tte^E^^™^^™*** 
properties of polar materials. Can have on the switching 

References 

i S e ^ ase and R Roy - J - Am - cer <»»- Soc 38 no no™ 

3. J. R. Carruthers and K Nassa.. / „ , 11 ,\ (1967) 
4- A. Linz, private colunSo"' ^ ^ 3 * 5205 (1968 > 
5. M. L. Sholokhovich and I. N Belvaev 7h,. r nu L . 

218 (1954); ibid., 1118 (1954) ' ' ' 0bscheheL ™i m ., 24, 

7 f P^e? 011 ; F , ^ ^ ^ A - M,1,er > ^"electrics 1 75 (1970) 

7. J P. Remeika, J. Am. chem. Soc, 76, 940 (1954) ' ' ° 970) 

8. D. L Epstein, U. S. Air Force Contract Report AFAL-TR 70 m 

9. K. Nassau, Ferroelectricity (ed E F Welfcri piL ■ a rR - 70 ~ 17 2 
(1967) y K ■ We,ler) ' EIs evier, Amsterdam, p. 266 

10. A. Kumada, Ferroelectrics, 3, 115 (1972) 

"' L^X 8 9909°7>f A ' LinZ ^ R ' F< " We ""' ***** 
14. V G HU1 anri IK r 7 ' ^ C W»-. 13, 483 (1968) 

» n w ^ "• P"'^""", 2. 123 (1971) ' ' 

21 S r.^'ST J - ""<*■ Fh r s - *»• 2381 (1969) 

1» (££ VaSir ™ Md V ' * « -tat « WS«, 24, 

S 3 V p A N^ b r d d N H w ^z:: tt P rr mic «• 781 

R. A. Dork, N. W. ^Imbring and J p N^T"' ?' S 5 (1 *«> 
(1964) 8 J ' K Nol,a - J a PPl- Phys., 35, 1984 

24. £ W.?S 

25. C. Feldman, P hys . R^H'^Z) ^ * ^ ° 968) 
C. Feldman, Rev. sciem. Instrum., 26, 463 (1955) 

TecHnoiolTi; 158 Im^ 1 ™ ^ ™ H < F ™° mb *, Electrochem. * 



26. 



it 



PREPARATION OR POLAR MATERIALS 



27. J. C. Burfoot and J. R. Slack, Proc. European Meeting, Saarbruchen (1969) 
J. C Burfoot and J, R. Slack, J. phys. Soc. Japan, 28, Supplement, 417 
(1970) 

28. M. H. Francombe, Ferroelectrics, 3, 199 (1972) 

29. W. J. Takei, N. P. Formigoni and M. H. Francombe, AppL Phys. Letters, 
15, 256 (1969) 

W. J. Takei, N. P. Formigoni and M H. Francombe, J. Vacuum Sci. Tech., 7, 
- 442 (1970) - - 

30. S. Y. Wu, W. J. Takei and M. H. Francombe, Ferroelectric*, 10, 209 (1976) 

31. N. F. Borrelli, A. Herczog and R. D. Mauer, AppL Phys. Letters, 7, 117 
(1965) 

N. F. Borrelli, J. appi Phys., 38, 4243 (1967) 

N. F. Borrelli and M. M. Layton, IEEE Trans. Electron Devices, ED-16, 
511 (1969) 

31 W. N. Lawless, Ferroelectrics, 3, 287 (1972); ibid., 7, 379 (1974) 

33. D. G. Grossman and J. O. Isard, J. Phys. D. appl. Phys., 3, 1058 (1970) 

34. D. V. Keight and J. O. Isard, paper given at British Ceramic Society 
Meeting (Dec. 1972) 

35. M. M. Hopkins and A. Miller, Ferroelectrics, 1, 37 (1970) 

36. G. W. Taylor, IEEE Trans. Electron Devices, ED16, 565 (1969) 

37. G. W. Taylor, Ph.D. Thesis, University of London (1961) 

38. J. Vossen, private communication 



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