High temperature phase equilibria and phase diagrams

P e r g a m o n T i t l e s of R e l a t e d Interest
ASHBY & JONES
Engineering Materials 1 & 2

Η EAR Ν
M e c h a n i c s of Materials, 2 n d Edition

HULL & BACON
Introduction to Dislocations, 3 r d Edition

KELLY
Concise Encyclopedia of C o m p o s i t e Materials

KHAN & MASATAKA
A d v a n c e s in Plasticity 1 9 8 9

LANSDOWN
Materials t o Resist W e a r

NIKU-LARI
A d v a n c e s in Surface Treatments 4

SCULLY
Fundamentals of Corrosion, 3 r d Edition

TAYA & A R S E N A U L T
Metal Matrix Composites

P e r g a m o n R e l a t e d Journals
A c t a M e c h a n i c a Solida Sinica
International J o u r n a l of Solids a n d Structures

J o u r n a l of Physics a n d Chemistry of Solids
Materials a n d Society
Materials Research Bulletin
Progress in Materials Science
Progress in Solid State Chemistry

{Free Specimen Copy Gladly Sent on Request)


HIGH TEMPERATURE
PHASE EQUILIBRIA
and

PHASE DIAGRAMS
KUO CHU-KUN
The Institute of Ceramics of Academia Si nica,
Shanghai, China

LIN ZU-XIANG
The Institute of Ceramics of Academia Sνnica,
Shanghai, China

YAN DONG-SHENG
Vice President of Academia Sinica,
Beijing 100045, China

PERGAMON PRESS
M e m b e r of M a x w e l l M a c m i l l a n P e r g a m o n Publishing C o r p o r a t i o n

OXFORD · NEW YORK

BEIJING · FRANKFURT

SAO PAULO

TOKYO · TORONTO

SYDNEY


U.K.

Pergamon Press pic, Headington Hill Hall,
Oxford 0 X 3 OBW, England

U.S.A.

Pergamon Press, Inc., Maxwell House, Fairview Park,
Elmsford, New York 10523, U.S.A.

PEOPLE'S REPUBLIC
OF C H I N A

Pergamon Press, Room 4 0 3 7 , Qianmen Hotel, Beijing,
People's Republic of China

FEDERAL REPUBLIC
OF G E R M A N Y

Pergamon Press GmbH, Hammerweg 6,

D - 6 2 4 2 Kronberg, Federal Republic of Germany

BRAZIL

Pergamon Editora Ltda, Rua Ega de Queiros, 346,
CEP 0 4 0 1 1 , Paraνso, Sao Paulo, Brazil

AUSTRALIA

Pergamon Press (Australia) Pty Ltd., PO Box 544,
Potts Point, N S W 2 0 1 1 , Australia

JAPAN

Pergamon Press, 5th Floor, Matsuoka Central Building,
1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan

CANADA

Pergamon Press Canada Ltd., Suite No 2 7 1 ,
253 College Street, Toronto, Ontario, Canada M 5 T 1 R5

Copyright © 1 9 9 0 Pergamon Press pic

All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted in
any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publisher

First edition 1 9 9 0

High Temperature Phase Equilibria and
Phase Diagrams is published by arrangement with

This edition of

Shanghai Scientific and Technical Press, Shanghai, China
Library o f C o n g r e s s C a t a l o g i n g - i n - P u b l i c a t i o n Data
Kuo Chu-Kun
High temperature phase equilibria and phase
diagrams/Kuo Chu-Kun, Lin Zu-Xiang, and Yan
Dong-Sheng.—Isted.
p. cm.
1. Phase rule and equilibrium. 2. High temperature
chemistry.
I. Lin Tsu-Hsiang. II. Yen T. S. (Tung-Sheng), 1 9 1 8 - .
III. Title.
QD503.K86 1 9 9 0
541.3'687—dc20
90-7211
B r i t i s h L i b r a r y C a t a l o g u i n g in P u b l i c a t i o n D a t a
Kuo Chu-Kun
High temperature phase equilibria and phase diagrams
1. Phase diagrams
I. Title II. LinZu-Xiang III. Yan Dong-Sheng
541.363
ISBN 0 - 0 8 - 0 3 5 8 9 2 - 6

PrintoH

in

^rMt

Rritain hv

BPCC Wheatons Ltd., Exeter


Preface
H I G H temperature phase equihbria studies, materials, science and
engineering have had a close relationship through a number of decades.
For the purpose of improving the properties of existing materials,
development of new materials or designing and innovating new fabrication processes, the phase relationships of different components at high
temperature are usually consulted. Moreover, equihbrium or nonequihbrium states can usually be observed between different phases due to
reaction kinetics or some other factor and these will, to a large extent,
govern or strongly influence the microstructure and properties of the final
material produced.
The study of high temperature phase diagrams of nonmetallic systems
dates back to the turn of the century. Silica, and mineral systems
containing silica, where the first to be studied since they relate, by and
large, to the composition of traditional ceramics, refractories, glasses, and
cement. The phase diagrams of these systems were immediately found
useful by ceramists of that period and since then a new field of phase
equihbria studies and phase diagrams of oxide systems has opened up as
an important and pioneering part of ceramic science. Over the past halfcentury or more, the progress of high temperature phase research can be
summarized into two main streams: firstly, the diversity of components in
the systems studied and secondly, the advancement of phase diagram
studies themselves.

The components involved in early studies were essentially silica,
alumina, alkali and alkali earth oxides. However, with the evolution of
technical ceramics in the forties and fifties, such as pure oxide ceramics,
electronic ceramics, and special glasses, etc., new components have been
introduced into the phase diagram systems being studied; for example,
oxides of titanium, niobium, zirconium and tantalum are a few of the
materials currently being used in research and development. The last
couple of decades have also seen an expansion in the components being
researched in high temperature phase studies; the more traditional oxides
being joined by nitrides, oxynitrides, carbides, chalcogenides, etc. These
¡X


÷

Preface

new ceramics are interrelated with various kinds of structural and
functional ceramics, and this area of work has attracted widespread
attention from metallurgists, geologists, mineralogists, solid state chemists
and solid state physicists, as well as material scientists and engineers.
Progress in phase diagram study includes the improvement of
experimental methods, innovation of new techniques and adaption of
those from other fields. As the field of high temperature study has
expanded this has resulted in a concurrent expansion in related fields such
as high temperature generation, temperature measurement and control,
controlled gas pressure, high pressure techniques, phase analyses,
structural and microstructural analyses and so on. In addition, since the
seventies, with the general application of computers, the accumulation of
thermochemical data of various compounds, and the advances in the

theory of high temperature thermodynamics and phase equilibria
relationships, the thermodynamical calculation of phase diagrams and
multi-phase equilibria data have become a reality. These calculations are
usually checked by a few experimental results, thus rendering both
approaches self compatible and complementary.
In the mid-eighties, Shanghai Scientific and Technical Publishers and
the authors of this book. Professors Kuo Chu-kun and Lin Zu-xiang, set
out a plan for "Phase Diagrams of High Temperature Systems". This
would certainly be an invaluable text from any point of view. Through
their unfaiUng effOrts, this arduous task has been successfully completed
and the contents include the fundamentals of phase diagrams, experimen­
tal and computational methods, examples of applications, as well as the
experience and results accumulated by the authors throughout their years
of work on high temperature phase diagram studies. I am sure that its
publication will be welcomed and have the full support of readers from
various disciplines. Any amendments or corrections of any part of this text
will also be greatly appreciated by the authors. I personally look upon this
as an important contribution that is worthy of recommendation.
Y A N D O N G ( T . S. Y A N )

Member, Chinese Academy of Sciences


CHAPTER

1

Introduction
FROM the viewpoint of conventional terminology we would not expect a
universally acceptable definition of the term "high temperature". Plasma

physicists may call some hundred million degrees Kelvin in an ionized
gaseous atmosphere a high temperature. However, on the other hand, high
temperature is often considered to be temperatures as low as minus one
hundred degrees centigrade in the thinking of scientists and engineers who
work with low temperature systems or the physics and chemistry of
superconductivity. In material science the term "high temperature" may be
used to cover the temperature interval from 5 0 0 ° C (for polymer chemists)
to 2 0 0 0 ° C (for some ceramists and metallurgists). At present only a few
solid materials can stand at temperatures above 2 0 0 0 ° C . This temperature
may be considered as an acceptable upper limit for phase diagram studies
in most experimental laboratories. In this book we roughly define high
temperature phase diagrams or systems as those in which the liquidus
temperatures are above 5 0 0 ° C . In accordance with this definition, most, if
not all, systems of interest in the non-metallic materials will be included.
So far the investigation of high temperature equiHbria and phase
diagrams concerns only the systems containing condensed phases, with or
without the participation of a gaseous phase. The heterogeneous equilibria
research of high temperature systems may stem from the beginning of this
century when members of the staff* at the Geophysical Laboratory of The
Carnegie Institution of Washington developed and established the
quenching technique which is extremely useful for the examination of
rock-forming systems and which has been used to examine many siHcate
phase diagrams. These early studies have led to a fundamental under­
standing of the reactions and solidification process occurring in silicate
and aluminosilicate melts, and the results have been directly employed to
interpret the formation of minerals and rocks. The research work in this
field attracted the attention of ceramists since the chemical composition
and fusion behaviour of the systems are so similar to those of cement,
porcelain, glass and refractory materials. In the early thirties, with the
support of the American Ceramic Society, Hall and Insley compiled and

2

High temperature phase equilibria and phase diagrams

published the first collection of phase diagrams which consisted of mainly
the silicate and oxide systems.
Slightly later, German chemists studied fusion diagrams and phase
relationships between high temperature oxides. Although the fusion
diagrams deviated, more or less, from the equilibrium condition due to
insufficient reaction time and vaporization, this work still provided
information and built an important foundation for application and further
investigation of the equilibrium phase relations.
During the past thirty years the development of high technology
advanced ceramics and glasses stimulated the research programme of
phase diagrams. At the same time the interest in high temperature system
studies extended to a series of new components, such as TÍO2, Z r 0 2 ,
B2O3, N b 2 0 5 and, in addition, a new group of phase diagrams of
non-oxide components came into being. Since then equilibria and phase
diagram studies have become not only the basis of mineralogy and
petrology but also a fundamental discipline of material science.
In China the study of high temperature phase diagrams began in the
fifties. At that time a project on high temperature oxide and ffuoride
systems was being carried out at the Ceramic Department of the Institute
of Metallurgy and Ceramics, the predecessor of the Shanghai Institute of
Ceramics. Several years later the phase diagram studies were redirected
towards rare earth sesquioxide-containing systems, with a view to
searching for new materials. More recently, Chinese researchers have
aimed their investigation at the systems relevant to heat engine ceramics

and to crystal growth technology, as well as the exploitation of new
materials. This work is currently being carried out in the Shanghai
Institute of Ceramics and the Institute of Physics.
Table 1.1 lists some seventy high temperature phase diagrams published
in Chinese journals.
The recent progress in high temperature phase diagram research can be
summarized as follows.
(1) Applications of phase diagrams to material science
The applications of and interest in phase equilibria and phase
transformations in various areas of material science has grown signifi­
cantly. On the one hand, the experimental results of phase equilibria and
phase diagrams convey information about development of new materials,
improvement of existing materials and estimation of potential use of
products. On the other hand, the appearance of new materials also
introduces new components for examination. T w o essential points of
change in experimental phase diagram investigation are:
(i) In addition to rock-forming oxides, many new components are
introduced into the high temperature systems. Listed in Table L 2 are the
statistics of frequency of appearance of twenty-one oxides in the phase


Lin Zu Xiang, Yu Hui Jun

Lin Zu Xiang, Yu Hui Jun

Kuo Chu Kun, Yen Tung Sheng

Han Wen Long, Kuo Chu Kun

Gd203-Zr02

Y203-Zr02

La203-BeO

Gd203-BeO

Sm 2 0 3-BeO, H0 20 3-BeO,
Y2 0 3-BeO
RE 2 0 3 (RE = La, Nd, Gd, Ho,
Y}-Ti0 2
SrNb 2 0 6 -NaNb0 3 -LiNb0 3

Kuo Chu Kun, Yen Tung Sheng
Kuo Chu Kun, Yen Tung Sheng
Lin Zu Xiang, Yu Hiu Jun

CaF
CaF2-La203
2-La 20 3
CaF2-Al203
CaF2-A1203
La
La203-Zr02
20 3-Zr0 2

Han Wen Long, Huang Yu Zhen,
Kuo Chu Kun, Yen Tung Sheng
Kuo Chu Kun, Huang Yu Zhen,
Huang Yuan Mou, Yen Tung Sheng

Huang Zhen Kun, Lin Zu Xiang,
Yen Tung Sheng
Tang Oi Sheng, Liang Jin Kui,
Shi Tin Jun, Zhang Yu Lin,
Tian Jin Hua, Li Wen Xiu

Tan Bin Vi, Tan Hao Ran
Kuo Chu Kun, Yen Tung Sheng

Authors

Publication

Wuli Xuebao (Acta Physica Sinica) 28,62 (1979).

Guisuanyan Xuebao (Journal Chinese Silicate Society) 7, 1 (1979).

Kexue Tongbao (Science Bulletin) 26, 411 (1981).

Kexue Tongbao (Science Bulletin) 26,414 (1981).

Xisuanyan (Silicate) 2,150 (1958).
1959 yan Guisuanyan Yanjiu Guongzhou Baogaohui Lunwenji
(Proceedings National Meeting on Silicate Research, 1959), p. 283,
Kexue Chubanshe (Science Press) 1962.
Guisuanyan Xuebao (Journal Chinese Silicate Society) 1, 1 (1962).
Huaxue Xuebao (Acta Chimica Sinica) 30,381 (1964).
Guisuanyan Xuebao (Journal Chinese Silicate Society) 3, 159
(1964).
Guisuanyan Xuebao (Journal Chinese Silicate Society) 3,229

(1964).
Guisuanyan Xuebao (Journal Chinese Silicate Society) 4,22
(1965).
Guisuanyan Xuebao (Journal Chinese Silicate Society) 4, 82
(1965).
Guisuanyan Xuebao (Journal Chinese Silicate Society) 4, 211
(1965).

1.1 Phase diagrams published in Chinese journals

MgO-AI 20 3- Ti0 2
CaF2-CaA12Si20s

System

TABLE

.

Introduction
3


Zr0 2-AI 20 3-Si0 2

Guisuanyan Xuebao (Journal Chinese Silicate Society) 11, 380
(1983).
Guisuanyan Xuebao (Journal Chinese Silicate Society) 10,412
(1982).

Gao Zhen Xin

Sun Wei Rong, Huang Zhen Kun,
Chen Jian Xin

Guisuanyan Xuebao (Journal Chinese Silicate Society) 11, 189
(1983).

Fu Zhen Min, Li Wen Xiu

and phase

Guisuanyan Xuebao (Journal Chinese Silicate Society) 10, 141
(1982).

equilibria

Wang Pei Ling, Liu Jan Chen,
Chao Guo Bing, Wu Jing Oi,
Li Oe Yu

U^WO^-UίiO^^U^GeO^^
Che Guan
Sheng
Wuli Xuebao {Acta
Physica
Sinica)
1061 (1983).
GuanTang
Chan,DiTang

Oi Sheng
CheChan,
Wuli
Xuebao
(Acta32,Physica
Sinica) 32, 1061 (1983).
Li
3 V04-Li4Si04-Li4Ge04
BaB204-Na20,
BaB204Huang
Qing
Zhen,
Liang
Jin
Kui
Wuli
Xuebao
{Acta
Physica
Sinica)
30,
559 (1981).
BaB
0
-Na
0,
BaB
0
Huang
Qing

Zhen,
Liang
Jin
Kui
Wuli
Xuebao
(Acta
Physica
Sinica)
30, 559 (1981).
2 4
2
2 4
Na2C03,
BaB204-Na2B204
Na
2C0 3 , BaB204-Na2B204
BaB204-LÍ20,
BaB204Huang Qing
Qing Zhen,
Zhen, Wang
Wang Guo
Guo Fu,
Fu, Wuli Xuebao
Physica
33,Sinica)
76 (1984),
Huang
BaB
Wuli {Acta

Xuebao
(ActaSinica)
Physica
33, 76 (1984).
20 420 4-Li 20, BaB
LÍ2B2O4
Liang Jin Kui
Liang Jin Kui
Li
2 B2 0 4
BaB204-SrO,
BaB204-SrB204
Wang Guo
Guo Fu,
Fu, Huang
Huang Qing
Qing Zhen,
Zhen,
Huaxue Xuebao
Xuebao (Acta
{Acta Chimica
Chimica Sinica)
Sinica) 42,
42, 503
503 (1984).
(1984).
BaB
Wang
Huaxue
20 4-SrO, BaB

20 4-SrB 20 4
Liang Jin
Jin Kui
Kui
Liang

Guisuanyan Xuebao (Journal Chinese Silicate Society) 9, 253
(1981 ).

Wuli Xuebao (Acta Physica Sinica) 29, 1497 (1980).

Wuli Xuebao (Acta Physica Sinica) 33, 1427 (1984).

Guisuanyan Xuebao (Journal Chinese Silicate Society) 9, 143
(1981).

phase

Li 4Ge0 4-Zn 2Ge0 4

Tang Oi Sheng, Che Guan Chan,
Chen Li Quan
Li Shi Chun, Lin Zu Xiang

Zao Zhong Yuan, Tang Oi Sheng,
Che Guan Chan, Bi Jian Qing,
Chen Li Quan
Fu Zhen Min, Li Wen Xiu,
Xu Pan Xiang, Zan Jin Yu,
Qi Xiao Zhen

Publication

temperature

Li 4Ge0 4-Zn 2Ge0 4

S

Authors

1.1 Phase diagrams published in Chinese journals-(continued)
High

LiNb0 3 -Zn 3 Nb 20

System

TABLE

4
diagrams


Chao

Tang

Tang

LilOj-RblOa

LiI03-Zn(I03)2

Liang

HoCl3-NaCl, ErClj-NaCl

H0CI3-KCI, ErClg-KCl

KCl-SrCl2

LiS04-MgS04, LÍNO3Mg(N03)2
Su
LiCl-KCl-PbS04
Shang

BeS04-Al2(S04)3-Na2S04

BeS04-Na2S04

LaOBr-BiOBr

Mg(I03)2-LiI03-HI03 Liang

LilOj-NalOj

KIO3-CSIO3

RblOj-HIOj, CSIO3-HIO3

Liang

LÍIO3-KIO3

LÍ2SO4-LÍ2B2O4, LÍ4SO4-

Y2O3-AI2O3-SÍ2N2O

Qian Jio Xin, Tan Bo Yun,
Ma Jin Hua, Su Main Zheng

Zhang Qui Yun, Ru Jing Zhi,
Sun Shu Ren
Shang Bao Xu, Chao Yuan Chun
N. P. Luzhlaya, Xu Xiao Bai

P. I. Fedolov, Zhang Qi Yun
P. I. Fedolov, Zhang Qi Yun

Liang Jin Kui, Liu Hong Bing,
Zhang Sun Min, Xu De Zhong
Tang Di Sheng, Li Wen Xiu,
Yu Cui Zhen, He Bao Xiang
Tang Di Sheng, Fu Zhen Min,
Li Wen Xiu
Fu Zhen Min, Li Wen Xiu,
Chen Li Quan
Fu Zhen Min, Li Wen Xiu,
Zhang Yu Ling, Tian Jin Hua

Liang Jin Kui, Che Guan Chan,
Zhang Yu Ling
Liang Jin Kui, Yu Yu De
Su Main Zheng, Wang Van Ji

Chao Guo Zhong, Huang Zhen Kun,
Fu Xi Ren, Yen Tung Sheng
Che Guan Chan, Chen Li Quan

Wuli (Physics) 11, 222 (1982).
Gaodeng Xuexiao Huaxue Xuebao (Chern. Journal Chinese
University) 3, 433 (1982).
Huaxue Xuebao (Acta Chirnica Sinica) 9,23 (1957).
Gaodeng Xuexiao Huaxue Xuebao (Chern. Journal Chinese
University) 4, 159 (1983).
Gaodeng Xuexiao Huaxue Xuebao (Chern. Journal Chinese
University) 4, 163 (1983).
Huaxue tongbao (Chemistry) 11, 665 (1985).
Huaxue Xuebao (Acta Chimica Sinica) 24,356 (1958).
Beijing Daxue Xuebao (Journal Beijing University) No.4, 401
(1963).
Kexue Tongbao (Science Bulletin) 17, 70 (1966).

Guisuanyan Xuebao (Journal Chinese Silicate Society) 9,90
(1981 ).
Wuli Xuebao (Acta Physica Sinica) 31, 621 (1982).

Wuli Xuebao (Acta Physica Sinica) 30, 1383 (1981).

Wuli Xuebao (Acta Physica Sinica) 30, 234 (1981).

Wuli (Physics) 3,395 (1980).

Wuli Xuebao (Acta Physica Sinica) 28, 518 (1979).

Wuli Xuebao (Acta Physica Sinica) 30, 1219 (1981).

Zhongguo Kexue (Scientia Sinica) Ser. A, No.4, 379 (1985).

Introduction
5


Kexue Tongbao (Science Bulletin) 29, 602 (1984).

Liu Su Qi, Rao Hong,
Zhang Qi Yun
Zhang Qi Yun, Li Wen Hua

Kexue Tongbao (Science Bulletin) 4, 370 (1965).
Gaodeng Xuexiao Huaxue Xuebao (Chern. Journal Chinese
University) 5, 755 (1984).

Van Li Chen, Zu Ying Ying,
Chen Nain Yi

and phase

Zhang Gui Cheng

Gaodeng Xuexiao Huaxue Xuebao (Chern. Journal Chinese
University) 4, 443 (1982).

Gaodeng Xuexiao Huaxue Xuebao (Chem. Journal Chinese
University) 5, 765 (1984).

Liu Su Qi, Liu Ying, Zhang Qi Yun

phase equilibria

CaF2-LiCI-NaCI
LiF-KCI-KBr, KF-NaCI-NaBr

Jingshu Xuebao (Journal Chinese Metal Society) 8, 187 (1965).

Li Xi Qiang
Mo Wen Jin, Wu Si Min,
Qiao Zi Yu, Zhu Yuan Kai

KCI-TiCI 2, NaCI-TiCI 2
NaCI-CaCI 2-SrCI 2

Gaodeng Xuexiao Huaxue Xuebao (Chem. Journal Chinese
University) 4, 395 (1983).

Kexue Tongbao (Science Bulletin) 17, 72 (1966).

Publication

Su Main Zheng, Qiu Bin Yi

Authors

1.1 Phase diagrams published in Chinese journals-(continued)

High temperature

YCI 3-LiCl, YCI 3 -NaCl, YCI 3 KCl, YCI 3 -RbCl, YCI 3-CsCL

System

TABLE

6
diagrams


Introduction
TABLE 1.2 Appearance frequency of certain oxides in Phase Diagrams for Ceramists
{the numerals in brackets represent the relative appearance frequency when setting
SÍO2 as 100)
The year of publication
Oxides
SÍO2
LÍ2O

BeO
FeO
ZnO
PbO

P2O3
AI2O3
Fe,03
Y2O3
TÍO2

GeOj
ZrO^
UO2

Th02
Nb^Os
Ta^Os
V2O5
BÍ2O3
Ga203
WO3

1956

1959

1964

1969

1975

1981

448(100)
39(8.7)
31(6.9)
70(15.6)
11(2.46)
40(8.9)
39(8.7)
274(61.2)
62(13.8)
3(0.7)
2(0.4)
69(15.4)
3(0.8)
43(10.7)
6(1.3)
10(2.2)
1(0.2)
0(0)
4(0.9)
3(0.7)
4(0.9)
18(4.0)

66(100)
44(66.7)
3(4.5)
26(39.4)
8(12.1)
24(36.4)
17(25.8)

57(86.4)
35(53.0)
4(6.1)
0(0)
53(80.3)
2(3.0)
16(24.2)
6(9.1)
3(4.5)
5(7.6)
2(3.0)
10(15.1)
0(0)
1(1.5)
15(22.7)

572(100)
94(16.4)
33(5.8)
101(17.7)
17(29.7)
72(12.6)
69(12.1)
396(69.2)
162(28.3)
30(5.2)
16(2.8)
147(25.7)
19(3.3)
56(9.8)

17(3.0)
15(2.6)
19(3.3)
10(1.7)
22(3.8)
43(7.5)
27(4.7)
30(5.2)

361(100)
33(9.1)
3(0.8)
78(21.6)
39(10.8)
54(15.0)
21(5.8)
270(74.8)
107(29.6)
19(5.3)
21(5.8)
43(11.9)
24(6.6)
43(11.9)
14(3.9)
4(1.1)
15(4.2)
12(3.3)
45(12.5)
8(2.2)
10(2.8)

33(9.1)

100(100)
28(28)
8(8)
12(12)
18(18)
17(17)
60(60)
68(68)
21(21)
18(18)
14(14)
41(41)
16(16)
36(36)
7(7)
10(10)
34(34)
36(36)
15(15)
6(6)
4(4)
25(25)

103(100)
41(39.0)
1(1.0)
35(33.3)
24(22.9)

31(29.5)
19(18.1)
67(63.8)
51(48.6)
14(13.3)
21(20)
28(26.7)
16(15.2)
27(25.7)
10(9.5)
23(21.9)
11(10.5)
2(1.9)
61(58.1)
7(6.7)
6(5.7)
44(41.9)

diagrams collected in Phase Diagrams for Ceramists published between
1956 and 1981. The newly-included oxides are P b O , Z r 0 2 , Z n O , T Í O 2 ,
N b 2 0 5 , T a 2 0 5 ,etc.
(ii) Recently phase diagrams and equilibria studies have extended to
measurement of properties in the subsolidus regions, due to their
theoretical and practical significance in material science. A new type of
phase diagram is therefore designated to describe the relationship between
properties and composition or structure patterns. For example, ferroelec­
tric phase diagrams of titanate and niobate systems.
Figure 1.1 consists of a general review describing the connection
between high temperature phase equilibria and phase diagram knowledge
and the various non-metallic products and processes.

( 2 ) Progress in experimental techniques
In addition to optical microscopy and X-ray powder diffraction
methods, the more recent phase characterization techniques, such as
electron microscopy, microprobe analysis and various spectroscopic
methods, are often used to complement the more classical techniques,
though not superseding them due to the development of X-ray powder


_

_

_

_

I

I

g.

0
:J

0..

g

5.

t-1

.b'

S.CD

~

I

~

~

Glass melt

MseatlatlblUartghiCal slag

Electrofused product

Metallurgical slage

Crystalline coating
Phase separation

Coating

Crystallized glaze

;: ,;

Devitrification of glass

Enamel

Glaze

Crystal grO\ith

U

~

ceramic sealing

and ceramic-

crystals
r

Sintering of poly-

material
Cement production

refractory

Corrosion on

.

Hydrothermal reaction

Material production

Compound synthesis

Raw material synthesis

I

Sintering of polycrystals

Phase transition

I

I
J

I

I
I
I
I

- -------!-----,

Metal-ceramic

~

~
0:

.....

High pressure equilibrium Crystal growth
~ Phase transition

L....

relatIonshIp

I

FIG. 1.1 The connection between high temperature phase equiUbrium and phase diagram knowledge and the various non-metallic products and
processes

I

L

I

I
I

§I

U

I

I

~.

~

S.

formation

and phase

~

~I

w

~

~hase

lcomposit.ion-?,roPertY-Phase

....r

New
New material exploitation,
design and prediction

phase equilibria

ffi

synthesis and

'

1

High temperature phase
IL....,~
equilibrium and diagram

I 1

equllibriL'lTI

. ~
arlO
;as-solid

CUI' a'

I.....

I

T

1

Gas-~l'

,--t------- _t --- ~-

§ I Ra\v material
Z
~~e~~~

grO\ith from gas phase

on material
I Corrosion
High temperature vaporization

§ I Crystal

u

__ _

I Nonstoichiometric compound

I CVD reaction

_

------------------~

High temperature

a3

m

..

__

Application of materials

2hase transition
properties change
8
diagrams


Introduction

9

diffraction instrumentation and data processing techniques which greatly
increases both the accuracy and applicability of the X-ray analytical

methods.
An electron microscope has a much higher resolution than an ordinary
optical microscope. The application of electron microscopy has success­
fully identified multiple-phase separation and accurately determined the
stable and metastable phase separation in glass systems. The combination
of a microscope and microprobe analyser offers a new experimental
approach capable of structural determination and compositional analysis
of very small regions of sample, thus aiding identification of solid solutions
and intermediate phases.
A method making use of diffusion couples is most attractive in phase
diagram research since it can be directly applied to constitute phase
diagrams with a few equilibrated or non-equilibrated samples.
With respect to equilibria reactions in the presence of gases, important
progress was made some thirty years ago by the application of high
temperature mass spectrometry. It provided a method by which both
partial pressure and gaseous species can be determined simultaneously. By
using the high temperature spectrometer, thermodynamic functions can
also be measured.
Infrared, Raman spectra and magnetic resonance techniques are usually
applied to particular high temperature systems. Spectra measurements not
only contribute to the phase identification and structural observation in
crystalline materials but also to those of amorphous materials.
(3) Calculation of phase diagrams
Considering 200 components, their combination yields about 20,000
binary systems, 1.3x10^ ternary systems and 8.5x10^^ comprising
between two and six components. Of these, only a small number have been
experimentally studied. It would be expected that more new components
would take part in high temperature systems due to the development of
new materials. However, the experimental work for phase diagram
construction is usually time consuming. It is roughly estimated that two or

three man-years are required for establishing a ternary phase diagram.
Much more time is required when working on multicomponent systems.
So it is hard to think that the requirement of phase diagram construction
can be satisfactorily met by relying on only experimental measurement.
Computer calculation may be the key to speeding up phase diagram
accumulation and in the last decade great progress has been made in phase
diagram calculation. Moreover, the computation of phase diagrams is
based on the thermochemical data of individual species and, therefore, the
success of equilibria computation should be contributed also to the
collection of high temperature thermochemical data and the development
of non-ideal solution thermodynamics at and before that period. T o


1o

High temperature phase equilibria and phase diagrams

present a number of computer programs, facilities have been established
and used to calculate phase diagrams of a variety of systems. However, it
must be stated that the reliability and accuracy of calculated phase
diagrams rely upon the free energies derived for the individual species and
mixtures. An additional advantage of computer calculation is that
difficulties that may be encountered in experiments due to chemical
corrosion and material loss can be circumvented.
(4) Investigation of non-oxide systems
The non-oxide phase diagrams have been studied by ceramists,
metallurgists, and solid state and high temperature chemists. Gradually a
new category of high temperature materials have been discovered, and
experiments have shown that many non-oxides possess extraordinary
properties, such as high melting point, high hardness, chemical inertness,

characteristic electrical, semiconducting and optical properties.
(5) Gas participation and high pressure phase diagrams
There are a number of recent materials and processes that give
considerable attention to systems containing a gaseous phase. This type of
phase equilibria depends upon the partial and total pressures of gases
involved. The following systems have a participating gas phase:
(i) Non-stoichiometric compounds. It is well known that at equilibrium
the chemical composition, structure and properties of oxides or sulphides
of elements having several valence states are dependent on the partial
pressure of oxygen or sulphur. Studies have revealed that a series of
continuous and discontinuous structural forms occur in the irontitanium-, cerium-, and uranium-oxygen systems when the oxygen partial
pressure is varied. Even for some compounds containing only one stable
valence state, the atmospheric environment may still considerably affect
certain properties because of the formation of defects.
(ii) Chemical vapour deposition (CVD). In recent years various C V D
processes have been developed in high technology ceramic preparation.
These processes involve chemical reactions between gases to produce the
desired solid product which may be monocrystalline, polycrystalline or
amorphous and in the form of a powder, thin film, coating or bulk
material. Equilibria studies in which a gaseous phase is participating may
lead to an understanding of reaction mechanisms and prediction of final
products and also the efficiency of the gaseous reactions.
(iii) Solid-gas phase diagrams. Usually we use a compositiontemperature (X-T) diagram to describe equilibria in a condensed system
at low pressures, where temperature is considered to be the only external
variable. However, pressure has to be considered as a variable in gascontaining systems since the gas pressure affects the equilibria appreciably.
Hence the pressure-temperature (P-T), the composition-pressure {X-P)


Introduction

11

and the composition-pressure-temperature {X-P-T)
phase diagrams
must be studied. Additionally, gas-solid phase relationships plots are also
employed to characterize the phase and structure stabilities at specified
temperature and pressure conditions.
In early high pressure phase diagrams, gas was used as a pressure
transmission medium in pressure vessels. This method is favourable for the
observation of equilibria in the presence of gases. However, the upper limit
is restricted by the solidification of the gas transmission medium.
Techniques now available for generating dry static pressures have greatly
increased the limit of pressure, previously supplied by gas pressure
techniques. Moreover, the solid media vessel can be more easily equipped
with phase analysis instruments, thus making in situ observation possible
even at ultra high pressures.
Besides the subjects which have already been introduced, topics such as
metastable phase diagrams, low concentration solid solutions, data
evaluation and storage may also be of interest. It has been found that the
phenomenon of meta-equilibrium is frequently observed in the course of
phase transformations, crystallization and high temperature reactions.
From the viewpoint of thermodynamics, it is possible that more than one
metastable phase assemblage can exist under a given initial condition, thus
resulting in a number of meta-equilibrium phase diagrams.
The experimental data for very dilute solid solutions in high tempera­
ture systems is still very sparse or absent, even though such data may be of
great technological importance to semiconducting and optical properties
of certain materials. In addition, the solubilities of these additives,
although small, may influence considerably the sintering ability of a
polycrystalline body as well as condition formation of metastable phases.

In order to study solid solutions of extremely low concentrations, new or
improved experimental techniques are often required. Electron microprobe analysis, neutron activation analysis, solid electrochemical methods
and the measurement of characteristic properties may be helpful for
particular systems.
Thermodynamic phase data evaluation has been undertaken for
metallic systems in the last thirty years. Recently least squares refinement
procedures have been developed for approximating the phase diagram
data from different sources. However, fewer phase equilibrium and phase
diagram data are available for ceramic systems. Hence the evaluation of
the phase equilibria data can only be accomplished by comparison
between measured and calculated results. Furthermore, the deposit,
withdrawal and resolution of high temperature diagrams appears to be of
special significance in the establishment of the high temperature data bank
and depository base. It may be expected that the incorporation of the
computer-aided equilibrium calculation, phase diagram resolution and
data storage may open a new field of phase diagram science.


CHAPTER

2

The phase rule, phase equilibria
and phase diagrams
2.1

T H E PHASE RULE

2.1.1

Basic c o n c e p t s

In dealing with phase equilibria, terms such as system, phase,
component and degree of freedom are frequently encountered. Therefore,
it is important to have these terms well defined before discussing the
principles of phase equilibria.
(1) System. A system is a portion of materials which can be isolated
completely and arbitrarily from all other materials for consideration of the
changes which may occur within it when the conditions are varied. If the
state of a system is not changed with time, the equilibrium state of the
system is attained.
(2) Phase. A phase is a homogeneous and physically distinct part of a
system which is separated from other parts by a definite bounding surface.
Gases, either pure or mixed, constitute one phase. Liquids, in addition to
immiscible liquids, are considered to be a single phase. Solids with different
chemical compositions constitute separate phases, but homogeneous solid
solutions are considered to be single phase.
(3) Component. The number of components of a system is the smallest
number of independently variable constituents necessary and sufficient to
express the composition of each phase involved in the equilibrium. For
example, in the system CaO-Si02, not only CaO and SÍO2 exist but also
intermediate compounds such as 3CaO.Si02, 2CaO.Si02 and
3Ca0.2Si02. However, all these compounds can be formed by the
reactions between CaO and SÍO2. Therefore, the number of components
of this system is 2 and the system is binary.
12


The phase rule, phase equilibria and phase diagrams

13

(4) Degree of freedom. A degree of freedom is a thermodynamic
variable which can be altered without bringing about a change of the phase
number. The number of degrees of freedom is the number of independent
variables such as temperature, pressure, and concentration of components
that need to be fixed in order that the equilibrium condition of a system
may be completely defined.

2.1.2

T h e phase rule

The phase rule represents the relation between the numbers of
components (i), phases ( ; ) and degrees of freedom ( / ) and can be expressed
by the following formula:

M-j+2
The phase rule is fundamental in studying phase equilibria. Below is the
derivation of the formula.
( 1 ) Chemical potentials. For a small change of composition in a
multicomponent system, if G represents the free energy of a phase, then a
change in the free energy dG of the system can be expressed by the
following complete differential:

Ρ, Τ, »2, ns,

. . , , dni

(2.1)

+
.

«2

DM

where /i^, « 2 , « 3 , . . . are the numbers of moles of the various components
represented by 1, 2, 3, . . .
According to the definition of thermodynamics
dG\
.SP,

^ ^

ídG\
ρ. Πι ,

=
, 12 ,

.

-S

. .

is the partial molar free energy expressed by G i .
Thus, formula (2.1) can be rewritten as:

dG= VdP-SdP^G,dn,^G^DN2

+ · · '^G^dn,

(2.2)

The physical meaning of the partial molar free energy of a component is
the change in the free energy of the system resulting from the addition, at
constant pressure and temperature, of 1 mole of that component to the
system so that there is no appreciable change in the concentration.


14

High temperature phase equilibria and phase diagrams
Since chemical potential

Then equation (2.1) can be written as:
dG= VdP-SdT+

μ^αη^+ P2dn2 + · · · -\-μidn^

(2.3)

( 2 ) Phase equilibria. In a system containing several phases at
equiHbrium, equation (2.3) for each phase may be written as follows:
dG^= V^dP-SUT^Y^
1 = 1,2,... ,1,

pidn^

(2.4)

i = l , 2 , . . .J

At constant pressure and temperature, dP and d r a r e equal to zero, and
the resulting free energy changes of these phases are given by:
dG^-'=ii¡-'dnr'
(2.5)

dG^ = p{dn{
The total free energy change of the system is
dG = dG^-' + dGJ = μ{-'dnj-'

4- μ{dn|

(2.6)

Since —dn{~^=dn{, it follows that
dG = (p{-p{-')dn,

(2.7)

At equiUbrium, the free energy change must be equal to zero, i.e. dG = 0.
Therefore,
μί = ΐ4-'

(2.8)

Thus for any heterogeneous system at equilibrium, the chemical potential

of each component has an identical value in all phases.
(3) Derivation of the phase rule. In a system consisting of i
components distributed between j phases, if the concentrations of i —1
components are given, the composition of each phase is completely
defined. Therefore, in order to define the compositions of j phases, it is
necessary to know
concentration terms. In order words, the total
number of concentration variables is equal to j(/—1). Further, since the
temperature and pressure are the same for all the phases at equilibrium in
the system, there are two variables to be considered in addition to the
concentration terms. The total number of variables are thus equal to
It has been demonstrated in the previous section that in a system


The phase rule, phase equilibria and phase diagrams

15

containing a number of phases at equiHbrium, the chemical potential of
each component is the same throughout the system. Hence for a system
containing / phases and j components, we have

μ\=μΙ=μΙ='"=μ{

μΙ=μί

= μ!='"=μ{

(2.9)

constituting a total of iij—l) equations which consist of j(i—1) + 2
variables. For a group of independent equations, if the number of
unknown terms is larger than the number of equations, then the number of
independent variables will be equal to the total number of variables minus
the number of equations. Therefore, for a system at equilibrium, we have

j(i-l)

+ 2-iU-l)

= i-j + 2

The number of independent variables is called the number of degrees of
freedom ( / ) , and the phase rule is expressed by
/=i-J + 2

(2.10)

If a component is absent in one phase, the number of variables will be
one less (i.e. the concentration term of this component in the phase).
However, similarly, the number of equations for chemical potential will
also be one less. Therefore, the difference between the number of variables
and equations is the same as before, and equation (2.10) is still applicable.
According to equation (2.10), if the difference between the number of
phases and components in a system is two, the number of degrees of
freedom will be zero, and the equilibrium state of this system can exist only
under completely fixed conditions. The alteration of any variable will lead
to a change in the number of phases present in the system. The system with
zero degrees of freedom is called an invariant. If the number of phases in a
system is larger than the number of components by one, the number of

degrees of freedom will be one. In such a system, only one parameter can be
changed independently without bringing about a change in the number of
phases. After the value of the first parameter is given, the values of the other
parameters are consequently fixed. The system with one degree of freedom
is called mono variant. The same argument is true for di variant and
trivariant systems.
In equation (2.10), both temperature and pressure are considered as
variables. If the pressure is constant, then the number of variables will be
decreased by one. The phase rule may be written as
/=/-;·+1

(2.11)


16

High temperature phase equilibria arid phase diagrams

Generally, pressure has little influence on condensed systems and can be
assumed to be constant. Therefore, formula (2.11) is used for studying
phase equilibrium in condensed systems.
However, if the parameters required to determine the state of the system
consist of supplementary terms in addition to temperature, pressure and
concentration, such as electrical ñeld, magnetic held and so on, then
equation (2.10) may be written as

2.2

PHASE EQUILIBRIA A N D PHASE D I A G R A M S

Illustrated above is the general rule of phase equilibria which is valid for
all systems at equiHbrium. In studying a specific system, it is not only
necessary to know the number of phases or components of the system but
also the variation of its physical properties with the parameters defining
the state of the equilibrium. In other words, it is necessary to study the
functional relation of these physical properties to temperature, pressure,
etc.
This type of study can be carried out by different methods. The relation
between the properties and components may be expressed by (1) tables, (2)
mathematical equations, and (3) diagrams. Of the three, the diagrammatic
representation is the most easily interpreted. For a two-component
system, a clear composition-property diagram may be obtained by
expressing the composition on the horizontal axis and properties on
perpendicular axes. Using the composition-property diagram, one can
find not only the variation of the properties with composition but also the
number and chemical nature of the phases and their compositional ranges.
Therefore, phase diagram may be considered as a geometrical method for
studying chemical reactions and various representations emphasizing
particular features, for example diagrams showing X-T, X-P,
X-P-T,
μ-Γ, etc. (Ä'= composition, Γ = temperature, P=pressure, μ = chemical
potential).
2.3

ONE-COMPONENT SYSTEMS

2.3.1

T w o - p h a s e equilibria and phase diagrams

In one-component systems, the number of components ¿ = 1 , so
/ = / - 7 + 2 = 3 - ; , i.e. the number of degrees of freedom depends on the
number of phases. When the latter is equal to one, the former wiU equal to
two. When two or three phases coexist, the number of degrees of freedom
will be equal to one or zero respectively. It is evident from the phase rule
that in a one-component system, the number of phases at equilibrium can
not be greater than 3.


The phase rule, phase equilibria and phase diagrams

17

The mono-variant equiUbrium will arise when two phases coexist in a
one-component system, such as (1) evaporation; where liquid and vapour
phases coexist, (2) sublimation, where solid and vapour phases coexist, (3)
melting, where solid and liquid phases coexist, and (4) polymorphic
transition where two solid phases coexist.
The state of the mono-variant system may be defined by two
parameters: temperature and pressure. When two phases coexist in
equilibrium, it is enough to fix one parameter to determine the other. For
example, in vaporization, the temperature of the system determines its
vapour pressure and vice versa.
Figure 2.1 represents the Ρ - Γ diagram of a one-component system in
which the horizontal axis gives temperature, and the vertical axis,
pressure. The solid fines OA, 0 5 and OC each define the coexistence of two
phases (.S, L, V represent solid, liquid and gas phases respectively). The
number of degrees of freedom at any point on the lines is equal to one.
Where one can change arbitrarily one parameter without altering the state
of coexistence of the two phases.

The Clausius-Clapeyron equation can be applied to the mono-variant
equilibrium:
^= Γ ^ Δ Κ

(2.12)

Here, q is the latent heat of transformation of one phase to another at
equilibrium and Δ Κ is the volume change due to the phase transformation.
If one of the two phases is a gas, A Κ will be the difference between the molar
volume of gas and Hquid or solid. Obviously, the volume of liquid or solid
can be neglected compared to that of the gas. Hence A F = V^^^. For one
mole of gas, V^^^ = RT/P. Substituting in equation (2.12), the second
expression of the Clausius-Clapeyron equation can be written as
din Ρ
dT

(2.13)

T-

T(a)
FIG. 2.1

(h)
Phase diagram for a one-component system.


18

High temperature phase equilibria and phase diagrams

For evaporation and sublimation processes {OC and OA in Fig. 2.\)q can
be considered approximately as constant (the difference between
and
T2 is small). Integrating equation (2.13), we obtain:
\nP=--^^

RT

A'

{Ä = constant)

After simplification
\ogP = A - ^

(2.14)

where ^ = .472.303, B = q/2303R. Thus, log Ρ is approximately a linear
function of l/T. The equilibrium curves for evaporation and sublimation
can be derived from equation (2.14),
According to the Clausius-Clapeyron equation, the curve for sublima­
tion should have a greater slope than the curve of evaporation. It is known
from equation (2.12) that
dP
dT

a
T{^V)

Since the heat of sublimation is larger than that of evaporation, the
increase in the vapour pressure with temperature will be greater for
subHmation than for evaporation.
For melting, the Clausius-Clapeyron equation can be also used to
calculate the variation of melting point with pressure:
dT_T{AV)
dP~
q
Two situations must be considered: (1) the increase of melting point with
the increase of pressure, and (2) the decrease of melting point with the
increase of pressure. The sign of Δ Κ governs which situation is
encountered. If the volume of material decreases on melting, dT/dP will be
negative and the melting point will decrease with increasing pressure. The
equilibrium curve on the phase diagram will decline to the left (Fig. 2.1 (a)).
On the contrary, if the melting point increases with increasing pressure, the
equilibrium curve will decline to the right (Fig. 2.1(b)). Since the influence
of pressure on melting point is small, the declination of the melting curve to
the pressure axis is small.

2.3.2

Three-phase equilibria

In Fig. 2.1, the curves of evaporation, sublimation and melting intersect
each other at point O, and O is called the three-phase point at which the


The phase rule, phase equilibria and phase diagrams
solid, liquid and gas phases are in equilibrium at a definite temperature
and pressure. Changing either one of these two parameters will dispel one

or two of the three phases. This implies that the number of degrees of
freedom is equal to zero at the three-phase point.
For any one-component system, the three-phase point has its own
definite value of temperature and pressure. Figure 2.2 shows the phase
diagram of water. OA represents the relation of the saturated vapour
pressure of ice to temperature, OB the melting point of ice (or freezing
point of water) to pressure, and OC the saturated vapour pressure of water
to temperature. These curves divide the diagram into three parts
representing the various states of water. In the mono-phase areas,
temperature and pressure can be changed in a considerable range without
the appearance of a second or third, therefore the number of degrees of
freedom equals two. The curves OA, OB, OC as described above, represent
the equilibrium between ice-vapour, ice-water and water-vapour respec­
tively, and their number of degrees of freedom is one. Point O expresses the
condition under which the three phases coexist at equiHbrium. Its co­
ordinates are: ^=610.4833 pa, Γ = 2 7 3 . 1 6 Κ . The equilibrium wiU be
destroyed by a small variation of temperature or pressure. For example, at
constant pressure, the decrease or increase of temperature will convert the
system into ice or vapour, and at constant temperature, the change of
pressure will also produce a similar effect. Therefore it is impossible to
maintain the equiHbrium state if the temperature or pressure is changed. In
Fig. 2.2, OD is the extension of curve OC. It expresses the saturated
vapour pressure of supercooled water existing in a metastable state.
Therefore, the vapour pressure is higher than that of ice at the same
temperature.

c

Β

s

L

jL.579
~b

1 ^
1
rc—i—

FIG. 2.2

Phase diagram
10. ()()7Γ) for water.

0

19


20
2.3.3

High temperature phase equilibria and phase diagrams
Polymorphism

Materials can exist in different modifications, and each has its own
region of stability in a phase diagram. The boundary line between two
regions represents the coexistence of two modifications at equiHbrium and

the three-phase point has the same meaning as described above.
There are two kinds of polymorphism: reversible and irreversible.
Figure 2.3(a) shows diagrammaticaHy the relation between a stable phase
and an unstable phase in a reversible transition.
Suppose a, jß, L, represent a low temperature phase, a high temperature
phase and a liquid phase respectively. Solid lines represent a stable state,
and dotted lines a metastable state. As shown in Fig. 2.3(a), the transition
temperature from α to
is Τ^β and the melting point of j3 is Τ β. The
equilibrium can be expressed as follows:
οί:^β^

liquid

If cooling is carried out rapidly, a supercooled liquid phase may appear
which is expressed by the dotted part of curve LL.
is the intersection of
the dotted part of αα and L L , and this is the melting point of the metastable
phase a. These phenomena are observed only under conditions of
supercooling or superheating respectively.
Comparison of αα with β β shows that at temperatures lower than Τ^β the
vapour pressure of β is higher than that of a(P^ > PJ and therefore, α is the
stable phase. On the contrary, at temperatures higher than Γ^^, the
phase β is stable, and both phases become stable at Τ^β when they have
identical vapour pressures.
The essential condition required for the reversible transition to occur is
the transition temperature must be lower than the melting point of the two
polymorphs involved.

FIG. 2.3

Phase diagram for systems with polymorphic transition, (a) reversible;
(b) irreversible.