High temperature phase equilibria and phase diagrams

1 o 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 syst

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First edition 1990 This edition of High Temperature Phase Equilibria and Phase Diagrams is published by arrangement with Shanghai Scientific and Technical Press, Shanghai, China

L i b r a r y 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 D a t a

Kuo Chu-Kun High temperature phase equilibria and phase diagrams/Kuo Chu-Kun, Lin Zu-Xiang, and Yan Dong-Sheng.—Isted

Preface

HIGH 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 fabrica-tion processes, the phase relationships of different components at high temperature are usually consulted Moreover, equihbrium or non-equihbrium 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 century 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

half-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

÷ 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, experimen­tal 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

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 under­standing 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 signifi­cantly 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 Two 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

4 High temperature phase equilibria and phase diagrams

6 High temperature phase equilibria and phase diagrams

3(0.7) 4(6.1) 30(5.2) 19(5.3) 18(18) 14(13.3)

T Í O 2 69(15.4) 53(80.3) 147(25.7) 43(11.9) 41(41) 28(26.7)

G e O j 3(0.8) 2(3.0) 19(3.3) 24(6.6) 16(16) 16(15.2) ZrO^ 43(10.7) 16(24.2) 56(9.8) 43(11.9) 36(36) 27(25.7)

U O 2 6(1.3) 6(9.1) 17(3.0) 14(3.9) 7(7) 10(9.5)

T h 0 2 10(2.2) 3(4.5) 15(2.6) 4(1.1) 10(10) 23(21.9) Nb^Os 1(0.2) 5(7.6) 19(3.3) 15(4.2) 34(34) 11(10.5)

V2O5 4(0.9) 10(15.1) 22(3.8) 45(12.5) 15(15) 61(58.1)

G a 2 0 3 4(0.9) 1(1.5) 27(4.7) 10(2.8) 4(4) 6(5.7) WO3 18(4.0) 15(22.7) 30(5.2) 33(9.1) 25(25) 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 , TÍO2,

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

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

8 High temperature phase equilibria and phase diagrams

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 success­fully 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

1 o 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 iron-titanium-, 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 temperature (X-T) diagram to describe equilibria in a condensed system

composition-at low pressures, where tempercomposition-ature is considered to be the only external variable However, pressure has to be considered as a variable in gas-containing systems since the gas pressure affects the equilibria appreciably

Hence the pressure-temperature (P-T), the composition-pressure {X-P)

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 tempera­ture 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 micro-probe 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

(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

2 1 2 T h e p h a s e r u l e

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:

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

(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

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^ pidn^ (2.4)

1 = 1 , 2 , , 1 , i = l , 2 , J

At constant pressure and temperature, dP and drare equal to zero, and

the resulting free energy changes of these phases are given by:

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

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

2.2 P H A S E E Q U I L I B R I A A N D P H A S E 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 O N E - C O M P O N E N T S Y S T E M S

2 3 1 T w o - p h a s e e q u i l i b r i a a n d p h a s e d i a g r a m s

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

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

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

18 High temperature phase equilibria and phase diagrams

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 a

dT 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 T h r e e - p h a s e e q u i l i b r i a

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

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= ^^ A' {Ä = constant)

RT

FIG 2.2 Phase diagram for water

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 co­ordinates are: ^=610.4833 pa, Γ=273.16Κ 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

20 High temperature phase equilibria and phase diagrams

2 3 3 P o l y m o r p h i s m

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

liquid

It can be seen from Fig 2.3(b) that in the whole temperature range where a

solid phase exists, the vapour pressure of β is always higher than that of

(x{Pß>PJ, and therefore, only α is the stable phase In addition, the

transition temperature from α to j8 is higher than the melting points of both

α and β, and obviously, the transition can not be observed Hence, when

the liquid is slowly cooled, only the stable phase α is crystallized Although

the metastable phase β can also be obtained if cooling is sufficiently rapid However, once metastable β has transformed into a, the phase β can only

be obtained again by fusion of α followed by rapid cooling of the hquid This is the difference between a reversible and irreversible transition

G = H-TS

and is valid, because the influence of pressure is negligible Therefore, if the Gibbs free energies of all possible phases in a system may be calculated at a given temperature and constant pressure as a function of composition, it is possible to determine the limits of composition over which any phase, or combination of phases, is stable

When a component A is mixed with a component B, one of the following

cases may occur: no solid solution or intermediate compound is formed, formation of solid solution with limited solubility, formation of a continuous sohd solution, or the formation of an intermediate compound The molar free energy of a binary solid solution consisting of component

A and component Β may be represented by

where and x^ are the mole fractions of components A and B, and and

Gß are their partial free energies in respective solid solutions At constant

The Ρ - Γ diagram of an irreversible transition is shown in Fig 2.3(b), and the symbols have the same meaning as in Fig 2.3(a) The equilibria relations between phases for irreversible polymorphic transitions may be expressed as follows:

22 High temperature phase equilibria and phase diagrams

G = RT{x^ In x^ + Xß In x^) H-x^G^ -hXßGß (2.16) Therefore, the free energy terms may be evaluated as a function of

composition at a given temperature and an isothermal plot constructed

Then a phase diagram may be drawn from a number of isothermal free

energy-composition curves of different temperature

When components A and Β are mixed together, if the resultant product

is a mixture, the free energy of the mixture will obey the rule of addition

The dependence of the free energy on composition may be expressed by a

straight line (curve a in Fig 2.4) If the two components form continuous

solid solution, the free energy-composition curve will no longer be Unear,

but concave downward (curve b in Fig 2.4) It means that the free energy

of the solid solution is lower than that of the mixture, hence the solid

solution is stable If, on the other hand, the curve is concave upward as

A Β

FIG 2.4 Free energy curves of a two-component mixture

temperature Γ, the partial molar free energies may be expressed in terms of

the activities of components A and Β (i.e and

G = RT{x^ In α^ + Χβ In αβ)Λ-χ^0''^-^ΧΒ01 (2.15)

x^G^ and XßGl represent the free energies of pure components and are

constant under conditions of constant pressure and temperature

For an ideal solution, the activity of a component in solution is equal to

its mole fraction, i.e

Substitution in equation (2.15) yields

(1) For a system containing continuous solid solution For a system

exhibiting complete miscibility in both solid and liquid states, there are two concave curves at different temperatures One of them represents the free energy of the solid solution and the other the liquid solution The phase diagram may be different depending on the mutual position of the two curves

For the first case, the two curves intersect at one point only as shown in Fig 2.5 At temperatures higher than the liquidus, the free energy-composition curve of liquid solution lies completely below that of solid

solution (e.g at temperature T^) At temperatures lower than the solidus,

the curve of the solid solution lies wholly below that of the liquid solution

as at Γ5 At an intermediate temperature Γ3 the free energy curve of hquid

solution intersects that of solid solution and gives a range of composition,

T4

C, G

FIG 2.5 Free energy curves for binary systems with continuous solid solution.^

curve c is in Fig 2.4, the free energy of the mixture will be lower than that

of the solid solution, and this implies the greater stability of the mixture compared with the solid solution

24 High temperature phase equilibria and phase diagrams

FIG 2.6 Free energy curves for binary eutectic systems.^

from Xß = 0 to x^ = C i , in which the homogeneous sohd solution is stable,

and a second range, from Xb = C2 to Xß= 1.0, in which the homogenous

liquid solution is stable For compositions between Q and C2, the stable

form consists of a phase mixture of solid solution Q and liquid solution

C2 Q and C2 are the common tangent points of the two curves and also

represent the conjugate compositions at equilibrium

For the second case, the free energy curves of solid (GJ and liquid (Gi)

intersect at one and then at two points when the temperature is decreased Further, the two points move gradually closer and merge completely at a temperature This type of phase diagram is shown in Fig 2.6 is the

melting point of the component 5, at which G^ and Gi^ intersect at one

point At temperature Γ2, the free energy curves intersect dXc.C^ and C2

are the conjugate compositions of the liquid and the solid solutions at

equilibrium At temperature Γ3, G^ and intersect at two points, and two

conjugate equilibrium pairs appear for two compositions Gradually the two intersects move towards one another, implying the approach of the

two compositions At last, they merge into one at temperature Γ4, the

minimum melting point of the system Assembhng the results of all the isothermal free energy-composition diagrams, we obtain the phase

FIG 2.7 Free energy curves for binary system with limited solubility.^

diagram for a binary system consisting of a single solid solution with a minimum melting point as shown in Fig 2.6

The third case is contrary to the second With a decrease in temperature, and G^^ intersect each other one point at first, and then intersect at two points, which gradually separate from each other when the temperature decreases further and intersect at one point at last This is the binary system consisting of a single solid solution with a maximum melting point The phase diagram will be the same as in Fig 2.6, if the latter is placed upside down

( 2 ) For a system with limited solubility The free energy-composition

curves and the phase diagram for this type of system are shown in Fig 2.7

The components A and Β form α and β solid solution respectively At the melting point of ^ ( Γ ^ ) , both α phase and liquid A have the same free energy

26 High temperature phase equilibria and phase diagrams

(3) For a system containing an intermediate phase The free

energy-composition curves and the phase diagram are shown in Fig 2.8 At a temperature , the liquid phase has the lowest free energy over the entire composition range, hence, it is the most stable phase at this temperature

At temperature Γ2, the free energy of hquid is still the lowest but

intersects that of the intermediate phase β At still lower temperatures of Γ3 and Γ4, Gi^ is intersected not only by Gß but also by G^ and G^ Thus, two

eutectics are formed between the intermediate phase and each of the terminal phases

Χβ-FiG 2.8 Free energy curves for binary systems containing an intermediate phase.^

and are equilibrium phases A similar condition is observed for β phase at

temperature T2 When the temperature is further lowered to Γ3, the free

energy curves represent the equilibrium of phases α and β with their

respective hquid phases of definite compositions At temperature Γ4, the

lowest points of the three free energy curves are on a common tangent

representing the coexistence of a, β and a liquid phase Hence, Γ4 is the

eutectic temperature Below the eutectic temperature, as Γ5 and Γ^, only

the mixture of α and β phases are the stable configurations

In the phase diagram of a binary system, the horizontal axis represents

the composition of a mixture of A and 5, and the perpendicular axis the

temperature The main types of binary systems may be concluded as follows

(1) A binary system with a eutectic point There is neither an

intermediate compound or a solid solution formed in this type of system

The typical phase diagram is shown in Fig 2.9 The curve T^E

representing a gradual decrease in the melting point with an increase in the

content of 5 in ^4 is called the liquidus, and T^E similarly represents a decrease in the melting from point B Ε is the intersection point of T^E and

ΤβΕ is called the eutectic point, at temperatures below the eutectic a liquid

phase no longer appears The whole phase diagram is divided into four

parts by the curves T^E, T^E and Τ^Τ^ The upper part is located at temperatures higher than the melting points of both components A and 5,

so all compositions are in a liquid state Since the number of phases is equal

to one, the number of degrees of freedom is 2 Hence, both temperature and composition can be changed independently without causing any variation

in the number of phases In both the region of T^ETQ and TßETc ,ei solid

phase coexists with a liquid phase, so the number of degrees of freedom is one In these regions, the composition of the liquid phase at equilibrium with the solid phase is dependent on the temperature For example, at

28 High temperature phase equilibria and phase diagrams

(2) A binary system with a congruently melting compound The phase

diagram of this type of system is presented in Fig 2.10 Differing from the above phase diagram, a maximum point appears on the liquidus, which is

the melting point of the binary compound A^B^ Consequently, there are

two eutectic points in this kind of system, and the whole phase diagram can

be considered to be constructed from two binary phase diagrams with one eutectic point in each system

{2) A binary system with an incongruently melting compound The

model phase diagram is shown in Fig 2.11 and there is no longer a maxima

on the liquidus due to the incongruent melting of A^B^ at temperature ,

however, a break in the liquidus curve is evident at this temperature, due to

the decomposition reaction of A^B^: /1^5,,^^-hliquid The crystallized

temperature T2, the composition of the hquid phase at equiHbrium with A

is F Below the eutectic temperature Τ^Τ^ is the region of coexistence of the two solid phase A and B The number of degrees of freedom in this region is also equal to one At the eutectic point E, three phases {A, Β and the liquid phase with composition H) coexist Since neither the temperature or

composition can be changed without the disappearence of one or more

phases, the point Ε is invariant

The following is the crystallization path of the composition D

( 2 5 % 5 + 7 5 % ^ ) At temperature Γ 4 , it is in a liquid state, as the

temperature is decreased to 7\, it reaches on the liquidus curve, and

begins to crystalHze phase A The composition of the liquid phase at equilibrium with A will be G With further cooling, A will continue to

crystallize from to T2 Meanwhile, the composition of liquid phase will

vary from to fe, i.e G to F\ this will continue until the eutectic point Ε is reached, at which point both A and Β wiU crystaHize out from an equilibrium liquid phase of composition H, Below the eutectic tempera­ ture, the liquid phase will disappear and only A and Β will remain

Therefore, the phase diagram gives not only the crystallization sequence

of the solid phase, but also the variation of the liquid composition with temperature Both the liquid composition after crystallization and the relative quantities of solid to liquid phases can be determined from the phase diagram In order to determine the relative quantities of sohd phase

A to liquid phase L for composition G at temperature Γ2, a parallel line to

the horizontal axis from T2 is drawn It crosses the liquidus T^L'at point ¿,

quantity of solid phase to that of liquid phase is equal to the ratio of a2b to

T2a2 The relative quantities of solid to liquid phases may be calculated out

from the length of the two sections of the line The above relationship is called the Lever rule

solid phase is A when the temperature is higher than Γ^, but A^B^

crystallizes at temperatures below At temperature (point F ) , three

phases {A, A^B„ and liquid) are in equilibrium Therefore, Fis an invariant

point with zero degrees of freedom

When a melt with a composition to the left of A^B„ is cooled, the primary crystallization phase is A, and the composition of the liquid varies along

T^F When the temperature is reached, the crystallized A reacts with

remaining liquid and forms compound A^B„ This kind of reaction is

called a peritectic With further cooling, the reaction proceeds until all the remaining hquid is exhausted, resulting in the coexistence of two solid

phases A and A^B„

30 High temperature phase equilibria and phase diagrams

A+B

A+B

FIG 2.12 Phase diagram for binary systems with a subsoHdus binary compound

When a meU with composition A^B^ is cooled, the primary phase crystallized is also A, and again it reacts with the liquid of composition F when the temperature is reached Since the total composition of A plus Fconforms with that of A^B„, only the compound A^B^ is formed without

A or Β remaining

For a composition between phase A^B„ and F, A crystallizes at first on cooling; however, since the quantity of Β in the original composition is excessive for formation of A^B^ alone, the crystallization process will not

end after the peritectic reaction Further coding of the mixture below temperature will see the crystallization of v4^B„, and the composition of

the remaining liquid will change along FE until at the eutectic point F,

A^B^ and Β co-crystallize

The relative quantities of either solid phase to liquid phase or two sohd phases may be determined by the Lever rule

(4) A binary system with a sub solidus compound In the system

described above, the formation or decomposition of the compound takes place in presence of a liquid phase However, this can occur also at temperatures below the eutectic point, as shown in Fig 2.12 The

crystalline A and Β react with each other according to mA-\-nB^A^B„

when the temperature is decreased to T2 The compound A^B„ is stable over the temperature range between T2 and Γ3 and decomposes into A and

Β again on further cooling

(5) A binary system with polymorphism If the components or

compounds in a binary system have polymorphs, in the phase diagram there will be horizontal lines separating the stable regions of each polymorph The simplest example is that one of the two components of the binary system has two polymorphs Two situations may occur, (a) The transformation temperature is higher than the eutectic point as shown in

[α)

L Aß^L

Aa-^B

Fig 2.13 Phase diagram for binary systems with polymorphic transition

Fig 2.13(a) The region of the high temperature modification Aß is located above the line CD, so it is stable only when a liquid phase is present and

transforms to the low temperature modification before solidification of the

mixture is complete Therefore, below the eutectic point, phase A^ coexist with phase B (b)The transformation temperature is lower than the eutectic point as shown in Fig 2.13(b) In this case A^ and Aß are stable

over different temperature ranges

(6) The binary system with solid solutions The phase diagrams of

several types of binary system with solid solutions have been given in section 2.4.1, and there is no need to repeat here

The main difference between the binary systems with or without solid solutions include the following In the system without solid solution, the crystallized solid phases have definite compositions and are not changed with temperature, although the composition of the hquid phase varies when the temperature is changed However, in the system with solid solutions, the composition of the crystallized phase also changes when the temperature is altered

A solid solution is a homogeneous phase with changeable composition formed by two components in variable proportions A continuous solid solution is formed when the two components are miscible in any ratio Conversely, a discontinuous solid solution will be formed

The crystallization path in the systems with solid solutions is similar to the systems without solid solution in addition to the variation of the composition of the solid phase with temperature

Figure 2.14 shows the phase diagram of a binary system with a peritectic reaction in which solid solutions are formed In this system, there are two

solid solution regions and ^2 based on components A and Β

respectively In the systems without solid solution, the addition of the second component always causes a decrease in the melting point of the first

( b )

I

Tb

32 High temperature phase equilibria and phase diagrams

FIG 2.14 Phase diagram for binary systems having solubiUty of the two

components in solid phase

component This is not true in the systems with soHd solutions It can be seen from Fig 2.14, that the addition of ^ to 5 leads to the decrease of the

melting temperature of Β on the one hand, but the addition oi Β Xo A causes the increase of the melting temperature of A on the other hand

Changes in the system illustrated in Fig 2.14 may be followed by varying either temperature or composition When melt 1 is cooled to

temperature Γ 3 , the solid solution 5^ with composition Ν starts to

crystallize and the liquid phase with composition Μ is in equiUbrium with

it On further cooling, the compositions of the liquid and solid phases vary

gradually along the liquidus DC and solidus FC until the end of

crystallization at C When a melt of composition 3 is cooled to , an

equilibrium between liquid phase Ρ and solid phase a is formed with

further cooling the compositions of the liquid and solid phases change

along PD and QG, and at temperature T2 the peritectic reaction D -h G^=^F

occurs According to the Lever rule, the quantity of liquid phase D should

be less than that of the solid phase Therefore, the reaction results in the

disappearance of the liquid phase Z), with the soUd solutions and 5*2

with compositions F and G remaining On further cooUng, the composi­

tions of solid solutions and ^2 vary along FR and G/respectively When

a melt of composition 2 is cooled to liquidus temperature, the

crystallization mechanism is similar to that of melt 3 until temperature T2,

where the peritectic reaction occurs However, since the quantity of liquid

phase is more than that of solid phase for melt 2 at Γ2, the reaction results

in disappearence of the solid phase G The crystallization path of D and Fis

similar to that of melt 1 on further cooling

Consequently, the system shown in Fig 2.14 includes three cases: (a) the crystallization of a single solid solution (melt 1), (b) the crystalliza­tion of a solid solution with limit solubiUty (melt 3), and (c) the crystallization of a solid solution with peritectic reaction (melt 2)

2 '•

2 4 3 S y s t e m s w i t h l i q u i d i m m i s c i b i l i t y

(1) The thermodynamics of immiscible liquids Liquid immiscibility is

sometimes observed in binary systems In the immiscible region, two

hquids with different compositions are in equilibrium The decomposition

of a homogeneous liquid into two immiscible liquids is called

phase-separation

From the view of thermodynamics, the necessary condition for the

liquid immiscibility is that the sum of the free energies of the two

immiscible hquids produced after phase-separation should be less than

that of the original homogeneous hquid It may be also illustrated by the

free energy-composition curve as in section 2.4.L

In a binary system without liquid immiscibility, the free energy curve of

the liquid phase is situated under the line joining the free energies of the

melts of the two components as shown in Fig 2.15 As seen, there is a

minimum on the free energy curve of the liquid phase On the contrary, if

the immiscible liquids are more stable than the original homogeneous

liquid, there should be a maximum on the free energy curve of the liquid

phase Consequently, the free energy curve of the liquid phase in a binary

system with liquid immiscibility should have both minimum and

maximum as illustrated in Fig 2.16 A maximum is located between two

minima, and the sign of the second derivative for a maximum is negative

Thus the necessary condition for the phase-separation is

34 High temperature phase equilibria and phase diagrams

A Β

FIG 2.16 Free energy curve of two immiscible liquids

The immiscible Hquids in a binary system may be stable or metas table

The phase diagram and free energy-composition curves of the metastable

immiscible liquids are shown in Fig 2.17 Its liquidus curve has a S shape

indicating the tendency toward liquid immiscibility The dashed curve

gives the metastable immiscibility gap with a critical temperature C just

below the inflection point of the liquidus curve Figure 2.18 shows the

stable immiscibihty gap over the liquidus curve

It is seen from Fig 2.17 that at temperature the system is completely

in a liquid state At temperature T2 solid A is in equilibrium with a liquid of

composition obtained by drawing a tangent Une from to the free

energy curve of the liquid At temperature Γ3 the metastable immiscibility

gap has been reached, and on the free energy curve of the liquid there

appears one maximum and two minima The tangent from one minimum

of the liquid curve to solid A is below that portion of the liquid curve

showing the two minima Consequently, phase A and a liquid are the

stable equilibrium phases, whereas the two Uquid phases whose composi­

tions are defined by a tangent are in metastable equilibrium At

temperature the metastable immiscibility gap has widened, and a region

liquid appears in addition At temperature Γ5 the values of G^, Gß and

Gi^ fall on a common tangent through all the three points Γ5 is the eutectic

point where three phases A, Β and L coexist

The phase diagram for the binary system with stable immiscibility gap is

shown in Fig 2.18 shows the imminence of immiscibihty At T2 the

immiscibility gap has widened, and since the double tangency represents

the lowest free energy, the two Hquids of compositions b and e are stable In

addition, the region of ^ - f liquid has appeared At temperature Γ3 two

Gl