phase diagrams - progress in solid state chemistry (10) 1975 pelton

Progres~ in Solid State Chemistry Vol. 10, Part 3, pp. 119-155. Pergamon Press. Printed in Great Britain

P H A S E D I A G R A M S

A. D. PELTON

D6pt. de g6nie m6tallurgique, t~cole Polytechnique, Montr6al, Canada

and W. T. THOMPSON

Dept. of Mining and Metallurgical Engineering, McGill University, Montreal, Canada

Notation

Except in the case of pressure, where the at- mosphere (101.325kPa) is employed throughout, units conform to SI recommendations.

Extensive thermodynamic quantities are indicated by upper case letters, molar quantities are indicated by corresponding lower case letters.

aA activity of component A area

A, B . . . . components A~ coefficients in an orthogonal series a~, b,, c~ coefficients in a power series c, constant pressure molar heat capacity C~, Co, CD integration functions e isobaric molar expansivity G Gibbs free energy g molar Gibbs free energy g., partial Gibbs free energy of component

A g~, standard molar Gibbs free energy of

component A Ag integral molar Gibbs free energy of

mixing AgA relative partial Gibbs free energy of

component A gV. integral molar excess Gibbs free

energy H enthalpy h molar enthalpy Ah integral molar enthalpy of mixing hA partial enthalpy of component A h,~ standard molar enthalpy of compo-

nent A K equilibrium constant k general constant nA moles of component A P hydrostatic pressure P~(x) i th Legendre polynomial over range

0~<x<~l PA partial pressure of A p, coefficient in a power series Q~ general entensive function R gas constant S entropy s molar entropy As integral molar entropy of mixing SA partial entropy of component A s~ standard molar entropy of component

A T temperature (K) t ternary composition variable (see eqn.

144) U internal energy

H

V 1)

X A

Y

Z

'~A

ff

tx~

A / ~ A

E P,A

P (3"

E (O

,of

molar internal energy volume molar volume mole fraction of component A ternary composition variable (see eqn.

143) coordination number phases activity coefficient of component A value of ~A at infinite dilution first order interaction parameter mole ratio nx/(nA + n~) in ternary sys-

tem A - B - X of two metals and one non-metal

chemical potential of component A standard chemical potential of compo-

nent A relative chemical potential of compo-

nent A excess chemical potential of compo-

nent A mole ratio nB/(nA + riB) in ternary sys-

tem A - B - X of two metals and one non-metal

second-order interaction parameter surface tension general thermodynamic potential func-

tion general integral molar excess property general partial excess property of com-

ponent A

I. Introduction

An important application of classical thermodynamics in both fundamental research and engineering is the determination of the conditions required for phases to coexist in a state of equilibrium. The purpose of this article is to correlate the graphical representation of these requirements with the analytical thermodynamic calculations. In particular, we shall show how the use of various diagrams can be of assistance in the coordination, interpretation and evaluation of thermodynamic data, calculations and approximations.

A phase diagram may be defined as a geometrical representation of the loci of thermodynamic parameters when equilibrium between different phases under a specified set of conditions is established. Since there are many parameters such as the temperature, pressure, chemical potentials, etc., which may be used to define the axes of a diagram, it can readily be appreciated that many different kinds of phase diagrams can be constructed. For our purpose, we shall consider only a portion of the

I P g.S.C.. Vol. 10. Part 3 - -A 1 19

120 A. D. PELTON and W. T. THOMPSON

total number of possible types; namely those two- dimensional diagrams which of necessity fall into one of three easily recognizable topological forms.m

In what shall be referred to as a type 1 diagram, as exemplified by the pressure-temperature phase diagram for sulfur (Fig. 2a), the conditions for two-phase equilibrium are represented by lines. The lines may terminate at a critical point or join together in three-fold fashion to specify the conditions for co-existence of three phases. Although the one-component pressure-temperature diagram is perhaps the best known of this topological type, similar diagrams involving systems of more than one component may also be constructed. For example, the temperature-chemical potential diagrams for the iron-oxygen system shown in Figs. 15 b and c are type 1, as is the isothermal chemical potential of sulfur versus chemical potential of oxygen phase diagram illustrated for the iron-sulfur-oxygen system at 800 K in Fig. 27.

In a type 2 diagram, two-phase equilibrium is represented by an area and three-phase equilibrium by a straight line. The well-known binary temperature-composition phase diagrams such as that illustrated in Fig. 15a for the iron-oxygen system are of this type. However, topologically similar diagrams exist for one- and three- component systems. This is illustrated by the pressure-molar entropy diagram for sulfur in Fig. 2b and by the isothermal diagram of log Po2 versus the molar ratio nCr/(nCr + nvo) for the iron-chromium-oxygen system at 1573 K shown in Fig. 20a.

In a type 3 diagram, three-phase equilibrium is depicted by a triangular area. The most familiar examples of this type are isothermal diagrams for ternary systems in which the compositions of phases in equilibrium are represented on the "Gibbs' triangle" as in Fig. 20d. A topological equivalent for a one-component system is the molar volume-molar entropy diagram of sulfur shown in Fig. 2d. For a two-component system, an example of a type-3 diagram is the molar entropy-composition diagram for the gold-copper system in Fig. 6m.

An examination of the diagrams referred to above indicates that, in all types, lines do not cross, and intersecting phase boundaries obey certain angular relationships. For example, in the pressure- temperature diagram for sulfur in Fig. 2a, the extension of any phase boundary at a triple-point passes between the other two boundaries. These generalizations, together with the topological requirements for triple-point construction for the three types of diagrams, collectively constitute a set of construction rules which are obeyed when the relationships between the variables used for the axes meet certain conditions now to be discussed.

II. General considerations

The thermodynamic potential functions T (ther- mal potential), - P (mechanical potential), and p-A (chemical potential of component A) may be defined as follows:

T =-(OU/OS)v. ........ - . . , (1)

- P =- (OU/OV)s . . . . . . . . . . . . , (2)

/x A ==- ( ~ U / OnA)v.s ...... . . . (3)

where U is internal energy. The entropy S, volume V, and number of moles of A, nA are the conjugate extensive variables of the potentials T, - P , and tXa. The choice of the mechanical potential and extensive variable as - P and V rather than P and - V is arbitrary. In the present context we are considering the extensive variables as the fundamental properties defining the state of the system and we consider the potential functions as being defined by eqns. (1-3). Thus, it is preferable to let all extensive variables be positive quantities.

Following the notation of Pelton and Schmal- zried, (') we let &i and Q; be general symbols for thermodynamic potential functions and conjugate extensive variables respectively. Then,

~)i ~ ( O U ] OQi )Q,O,o. (4)

Generalized potentials other than those in eqns. (1-3) can be defined in terms of other extensive variables if these variables are important to the particular system being studied. For instance, if Q~ is the surface area, ~/, then tr is the surface tension:

o- =- (oUIos¢)s.,,.,;. (5)

For any single phase containing components A, B , . . . the Gibbs-Duhem equation in general form may be written as:

S d T - V d P + nadlXA + nadp,~ + . . . . 0. (6)

Or, using the general symbolism of eqn. (4):

Y Q,d&, = 0. (7) i

In the present article, we shall consider only those potential functions defined by eqns. (1-3). Thus, for a one-component system there will be three terms in eqn. (6): (T, - P , ~). For a two-component system A-B there are four terms: (T, - P, IXA,/xB), and so on. In certain special cases, one may wish to include the effect of other potential functions such as surface tension, electrical potential, magnetic potential, etc., upon the phase equilibria. This can be done within the framework of the general methodology to be developed simply by adding extra terms to eqn. (7).

2.1. T y p e I d i a g r a m s

At equilibrium, the state of a phase within a heterogeneous system is specified by assigning values to the extensive variables Q~. The potential functions &; may then be defined by eqn. (4), such that each &~ can be expressed as a function of the Q, Alternatively, these equations can be inverted to express each Q, as a function of the &;. Substitution into eqn. (7) thus yields, for each phase, one equation relating all of the ok; so that, for any phase, any particular potential function 4~; can be written as a function solely of the others: &;(&~, &2...). This may be regarded as an equation of state for the phase. For example, for an ideal monatomic gas phase, the three potential functions T, - P and Ix are related by the equation:

Ix = R T I n P - 5 / 2 R T I n T + k ~ T + k 2 (8)

where k~ and k2 are constants.

Phase diagrams

In a type 1 diagram, we plot two potentials which we shall arbitrarily designate 6~ and 6: as ordinate and abscissa. All other potentials, except one, which we shall call 63, are given specified constant values; 63 assumes values to satisfy eqn. (7). For example, in the three-component F e - S - O system, the potential functions 6,, 62 . . . . 65 are/Xs, tzo,/xvo, T and - P . In Fig. 27, log ps, is plotted versus logpo~ keeping T and P constant, with tzF, unspecified. Since, at constant temperature, t~o is proportional to log po. and tzs is proportional to log ps~, Fig. 27 is thus a type 1 diagram. In the two-component F e - O system, the potential functions (h,, 6z, &3, 64 are T, txo, txF~ and - P . Figure 15b is a type 1 diagram of T versus tZo at constant P with /xr~ unspecified. As a final example, for the one-component sulfur system, we have only three potentials 6~, 6"- and 63: - P , T and ~s, and so the P - T diagram of Fig. 2a is a type 1 diagram.

A general type 1 diagram is plotted in Fig. la. When the phases a and /3 are in equilibrium we may write:

67 = 6~, (9)

6~ = ch~, (10)

,:/,';' (6 ;, 6~) = 6~(6f, 6~). (11)

121

(d) ~ . . . . .

. . . . . ,, q:~ 01103 :; I,:

[ I I ! I I ~ --%]-4- L I I I l i [

02103 " (~2

(c)

(a)

FIG. 1. Interrelationship of corresponding type I, type 2, and type 3 phase diagrams. (~b,, &~ . . . . are held constant;

6~ is unspecified.)

There are thus three equations relating the four variables 67, 6 , e, 6L 6~, and so we have one degree of freedom in the sense of the Gibbs phase-rule. The condition for two-phase a- /3 equilibrium is thus represented on Fig. la as a line. Similarly, it can be seen that the condition for three-phase c~-/3-7 equilibrium is represented by a triple- point. The two-phase lines may end at critical points as shown in Fig. la and also in Fig. 2a. They

( d )

.017.011651 I / ' ' l i ! I " i / 7I i ' l l D I / ' . .' ' ' , LIQUID / Q,

° , 5 ,o ,s so'/ ;o i~o

, ~ .3 vAPouR'~/

"'~ J 217 ' ~0 LIQUID

,, __ 15

( c )

104 v~"~

2l/3

LIQUID

L V .... ~ - ~

1"35 10 ,15 5'O'Jl 8'0 100 S/n J K -1 mot -1

10 2

_ ld; E (b)~ 16"

VA P 0 U~.!

L l t l I I _

o(

j / 350

J

I I ! f /8 I I I LIQUID I

VAPOUR 0 ~50 5

T (K)

104 I

• 102 !

1

16 2

~6 6 1500

(a)

FIG. 2. Corresponding phase diagrams for the one-component sulfur system; /Zs is the unspecified potential.

122 A.D. PELTON and W. T. THOMPSON

may also pass through minima or maxima (points where dd),/d4)2 = 0) as in Fig. la or as above the congruently melting compound magnetite in Fig. 15b.

Along the boundary separating the phases a and /3 in Fig. la, one may write from eqn. (7), (with 4'4, 4'5.. • held constant):

QTdchl + Q~dd~2 = - Q~d03, (12)

Q~,d4', + Qfd4'2 = - Q~d4,3. (13)

where Q7 and Q~ refer to values in the two phases a and/3 along the phase boundary. Equating d4,3 in eqns. (12) and (13) yields, for the slope of the boundary,

d4',= \ Q 3 3 ] - \ - ~ 3 ] _ _ : \ Q 3 } ~ (14) d4,2 o, ° a(ol

In Fig. 2a, for instance, Q z / Q 3 = S / n = s and Q~/Q3 = V / n = v, where s and v are molar entropy and volume respectively. Since 4,~ = - P and 4,2 = T, eqn. (14) becomes:

dP As d--T = a--~ (15)

which is the Clapeyron equation. Equation (14) may thus be regarded as the generalized Clapeyron equation. "~

From eqn. (14) we can prove the "extension rule" that at any triple-point in a type 1 diagram, the extension of any two-phase line must pass into the field of the third phase. The proof is given in Appendix I.

We have used the general symbol 4~, to represent the potential functions T, - P , IZA, /ZB . . . . In the general case, we may also let th~ represent certain independent combinations of potential functions. For instance, for the F e - S - O system we let ~b~ =/Xs and 4,2 =/Xo, and in Fig. 27 we plotted log ps~ versus log po , recognizing that at constant T these are proportional to /Xs and /Xo respectively. However, a similar and useful diagram could be obtained were we to let 4,1 = tZso~ = (/Xs + 2tzo) and 4,2 =/xo. That is, we have taken the Gibbs-Duhem equation:

nsdlxs + nodlxo + nv~dlXF~ + S d T - VdP = 0 (16)

and rearranged it to:

nsd0zs + 2tZo) + (no - 2ns)d~zo + nvodtzF~ + SdT - V d V = 0 (17)

which is of the form of the general equation (7). The log pso~- log Po2 diagram in Fig. 28 is thus a type 1 diagram to which the generalized Clapeyron equation still applies. Hence, the above mentioned "extension rule" applies to both Figs. 27 and 28. In this example, of course, we could also treat the problem by redefining the components of the system as Fe, O, and SO2, as long as we were willing to deal with negative mole fractions in some phases.

2.2 Type 2 diagrams

In Fig. lb is shown a general type 2 diagram. In this diagram we again keep ~b4, 4'5 . . . . constant, and

we replace the abscissa, cbz, of Fig. la by the ratio Q2/Q3 of conjugate extensive variables.

It is shown in Appendix II that Q~ must increase monotonically as 4'i increases. Moving along the horizontal arrow in Fig. la, then, 62 increases at constant 4'~, th4, 4,5 . . . . It is also shown in Appen- dix II, on the basis of the relationship in eqn. (7), that q~3 must simultaneously decrease. Thus, as we move along the horizontal arrow in Fig. la, Q: increases and Q~ decreases, and so the ratio Q2/Q3 increases. This increase is discontinuous at the phase boundary, and so the topological relationships between Figs. la and lb are understandable.

Figure lb can be called a "corresponding" type 2 diagram to the type 1 diagram of Fig. la. At a phase boundary, the discontinuity in Q2/Q3 causes a two- phase line of Fig. la to become a two-phase region with horizontal tie-lines in Fig. lb. The triple-points of Fig. la have become horizontal lines in Fig. lb. The triple-point at the upper right in Fig. la which has two lines emanating from it in an upwards direction and one line in a downwards direction has become a "eutecular"~ construction in Fig. lb, and the triple-point at the lower left in Fig. la with two lines emanating from it in a downwards direction and one line in an upwards direction has become a "peri tecular" construction in Fig. lb. The lines which end at critical points in Fig. la have become "miscibili ty-gaps" in Fig. lb.

From Fig. 1 it is apparent that for each type 1 diagram, Fig. la, there are two corresponding type 2 diagrams: Figs. lb and lc. Figure lc is more familiar when viewed from the left side of the page. An example of Fig. 1 for a real system appears in Fig. 2. The P - T diagram of sulfur is shown along with the corresponding P-S In and T - V / n diagrams.

In certain cases it may be more convenient to use the ratio Q2/(Q2 + Q3) as the abscissa in a type 2 diagram rather than Q2/Q3. This substitution can always be made without altering the topology of the diagram, since Q2/(Q2 + Q3) varies monotonically as QdQ3. This substitution is particularly useful when Q2 and Q3 represent moles in a system of more than one component. In the binary system Fe-O, for example, the type 1 T-p. diagrams in Figs. 15b and c correspond to the familiar type 2 T-no(nve + no) phase diagram in Fig. 15a. In addition to the graphical advantage that attends restricting the composition variable to the range 0 to 1, the use of mole fraction as the abscissa permits the "lever principle" to be employed to graphically assess the relative quantities of each phase present in two- phase regions. This principle of course would not apply were the ratio no/nFo used as abscissa. An examination of Fig. 15 will reveal that, just as in the general diagrams of Fig. 1, the triple-points of the

?The adjectives "eutecular" and "peritecular" have been coined here to refer to any three-phase construction on a general type 2 diagram which are "eutectic-like" or peritectic-like". Even for binary temperature-composition diagrams, there has, up to the present, been no collective term for the various topologically identical eutectic-type (eutectic, eutectoid, monotectic, etc.) and topologically identical peritectic-type (peritectic, peritectoid, syntectic, etc.) constructions.

Phase diagrams

type 1 diagrams in Figs. 15 b and c correspond to eutecular and peritecular constructions in the type 2 diagram in Fig. 15a,

It should also be noted that at point Z in Fig. lb, we have an illustration of the "extension rule" that at any such point in any type 2 diagram, the metastable extensions of the phase boundaries must pass into the two-phase regions.

2.3 Type 3 diagrams

Figure Id is a plot of Q,/Q3 versus Q,_[Q3. The correspondence with the type 1 and type 2 diagrams is evident from an examination of the figure. Each triple-point in the (b~ - 4~2 plot of Fig. la has become a triangle in Fig. 1 d. In the triangle on the right, the perpendiculars to two sides, shown by arrows, point upwards into two-phase fields, and the per- pendicular to the third side points downwards into a two-phase field. This corresponds to the eutecular constructions in Figs. lb and c. The other triangle, corresponding to the peritecular constructions, has perpendiculars from two sides of the tie-triangle pointing downwards and one upwards into two- phase fields. Examples of type 3 diagrams include the plot of molar volume versus molar entropy for sulfur in Fig. 2d, and the isothermal diagram for the Ag-Cu-C1 system in Figs. 19c and d.

In Fig. 19c, rather than plot QJQ3 versus QJQ3, we have plotted QI/(Q2 + Q3) versus Q2/(Q2 + Q3). It is shown in Appendix III that the denominators of both these ratios must be the same in order that tie-lines in the type 3 plot be linear. The diagram shown in Fig. 19c is sometimes said to be plotted on J~inecke coordinates. ~2~ This diagram can be "closed up" as in Fig. 19d to the more familiar "Gibbs ' triangle" representation of an isothermal ternary composition diagram. It may be noted that only in the "Gibbs ' triangle" representation can the "centre of gravity principle" be applied to determine the relative quantities of phases in equilibrium in a three-phase field.

In the remainder of this article, we shall examine in order, one-, two-, three- and multicomponent systems. In each case we shall develop examples of type l, type 2 and type 3 diagrams with a view to showing the inter-relationships between these diagrams and the thermodynamic properties of the systems.

I lL One-component systems

In a one-component system there are three potential functions of interest in the present context, namely - P , T, p. Thus, a plot of pressure versus temperature with /z unspecified is a type 1 diagram.

Figure 2a is the pressure-temperature diagram for sulfur established by direct experimental measurement. '3'4~ The /3 or monoclinic crystal field of stability is enclosed by three triple-points joined by essentially linear pressure- temperature boundaries which appear curved on the logarithmic scale. The diagram is particularly suited to understanding the effects of isobaric and isothermal changes in state. For other applications, however, it is more useful to identify the axes with different variables. For exam-

123

pie, in compression studies it may be useful to employ a pressure-molar entropy diagram, since reversible adiabatic changes in volume are isen- tropic.

The pressure-molar entropy diagram for sulfur, with the molecular weight taken as 32.1, is shown in Fig. 2b. This is a type 2 diagram corresponding to the type 1 diagram in Fig. 2a. That is, in Fig. 2a, (h~ = - P , &-~= T, ~b3= Vs- In Fig. 2b, we have replaced qS: by Q2/Q3= S/ns = s, where s is the molar entropy. The calculation of Fig. 2b was performed using the relationship:

ds = cpd In T - evdO (18)

where e is the isobaric volumetric expansivity. Provided the pressure is not too high, it is often justifiable to regard e and v of condensed phases as constants. This was the assumption used in generating the present diagram, and as a result, the positioning of molar entropy values at the a - /3 - liquid triple-point is subject to error in the order of l J K-' mol -~. In the case of the entropy of condensed phases in equilibrium with vapor at low pressure, the second term on the right of eqn. (18) is negligible, and the first term may be evaluated by the indicated integration or read directly from tables of entropy valuesJ 5) The completed P - s diagram shows, for example, that reversible adiabatic compression of monoclinic /3 sulfur from am- bient pressure is not associated with partial melting as a result of the accompanying temperature rise.

It is clear that another corresponding type 2 diagram could be generated by replacing ~ = - P by the ratio Q~/Q3 = V/n = v, where v is the molar volume. The resultant v - T diagram, constructed from density data together with a knowledge of the topological requirements of a type 2 diagram, appears in Fig. 2c.

The corresponding type 3 diagram, not often constructed for a one-component system, could be prepared by plotting the molar volume versus the molar entropy. This diagram appears in Fig. 2d. The tie-triangle construction characteristic of this type of diagram is only apparent for the a-/3-1iquid triple-point since the volumes associated with the vapour phase at the other triple-points are ex- tremely large and consequently cannot be shown on the scale of Fig. 2d.

In the cycle of diagrams shown in Fig. 2, the chemical potential /zs is unspecified. Two other cycles of diagrams could be prepared by selecting pressure or temperature as the unspecified of the three potential functions. Consider the case where P is unspecified and/z is plotted versus T. Figure 3 is t h e / z - T diagram for sulfur. The ordinate scale is taken relative to /z of pure rhombic a sulfur at T = 298 K and P = 1 atm. Generally speaking, this presentation would not be so useful as some shown in Fig. 2, since physically /~ is not a convenient variable to adjust independently. For this reason, the corresponding type 2 and type 3 diagrams in this cycle are not constructed, and isobars are indicated on the phase fields to assist in understanding how the alteration of ~z could be brought about. Diag- rams of this type, emphasizing the kinks in the isobars, have been used extensively to rationalize the construction of P - T diagrams) 6~ (In a one-


'=~" -5

-g 300 350 400 450

Temperature (K) FIG. 3. Type 1 chemical potential-temperature phase

diagram for sulfur.

component system, tx is equivalent to g, the molar free energy.)

It is interesting to note that the slopes of the lines in Fig. 3 are given by the generalized Clapeyron equation (14) as:

dixldT A(SIV) (19) A(n/V)"

For the case of a condensed phase in equilibrium with vapor at relatively low pressure, the molar volume of the vapor is very much larger than that of the condensed phase so that eqn. (19) reduces to

dix / dT = - s (20)

whers s is the molar entropy of the condensed phase. Equation (20) is simply the Gibbs- Helmholtz equation. This implies that pressure has little effect on the chemical potential of a condensed phase.

A number of other diagrams may be constructed for one-component systems that do not fall into the classification scheme of type 1, -2 and -3 diagrams. Two of these that are widely used for engineering purposes are P - v and T-s diagramsJ 7~ Such diagrams involving a potential function and its own conjugate extensive variable are admirably suited, for example, to the design of heat engines, since areas on such diagrams can be interpreted as reversible heat or work transfers to the surroundings. Although such diagrams may often appear to be topologically similar to type 2 diagrams, this is not true in general for reasons outlined previously. A case in point is that of water where the increase in density accompanying the melting of ice results in the situation at the triple-point depicted in Fig. 4. Not only may P - v projections be confusing to interpret, but the possible failure of such diagrams to follow the topological rules of type 2 construction makes them unreliable as guides in the organ- ization of thermodynamic information.

IV. Two-component systems

Unlike the potential functions - P and T, the chemical potential of component A,/zA, can only be

FIG. 4. Pressure-temperature-volume interrelationships for a material such as water that expands on freezing. The

P-v projection is not a type 2 phase diagram.

specified relative to a standard potential. In view of this intrinsic complication which is inherent in all heterogeneous equilibrium calculations, but which is particularly important when more than one component is involved, discussions in this section will emphasize the role of the chemical potential in isobaric, isothermal equilibrium and ways of specifying the standard chemical potential, IXL

4.1. Principles of calculation

Minimum free energy, the condition for equilibrium, is reached in a closed heterogeneous system when the chemical potentials of each component in each phase are equal. Since

OG oU

(21)

it follows that the partial molar Gibbs free energy of component A, gA is identical to the chemical potential IXA. These two functions are thus often used interchangeably. We shall use t~A in the present article, except where attention is to be drawn to this equivalency. From the preceding definition, the integral molar free energy of a phase a in a binary system A-B, is given by

g ° = XAIz ~ + XBIX ~ (22)

where XA = nA/(nA+ nB) is the mole fraction of component A.

By combining eqn. (21) with eqn. (22), the following differential relationship between the chemical potential of a component and the integral molar free energy of the phase can be established:

dg ~ + ° ~ ^ = X B ~ g (23)

The pressure and temperature are understood to be fixed. This equation has the graphical interpretation evident from an examination of Fig. 5a. The intercept of the tangent to the g~ curve on the pure component-A axis is IZA for the composition corresponding to the point of tangency.

ga

Xs

o ct _ j -

Ca)

X B /

/ I

g (b)

[ I

A B

FIG. 5. Relationship between Ag and chemical potential, and between Ag and conjugate phase compositions at constant temperature and pressure in a binary system

A-B.

Equation (23) facilitates the graphical understanding of the conditions for phase equilibrium. In Fig. 5b, the points on the molar free energy isotherms for the a - and /3-phases determined by the common tangent locate the equilibrium compositions of the conjugate a - and/3-phases, since by this construction the chemical potential of each component is made the same in each phase. Al- though it may not always be desirable to determine the compositions of conjugate phases by drawing free energy isotherms for each phase, diagrams of the type shown in Fig. 5b are useful as guides in the coordination of calculations and the discussion thereof.

The shape of the isotherms in Fig. 5 is related to the integral molar Gibbs free energy of mixing which is defined as

Ag ~ = g" - (XA~/~ -~ X B j ~ ~ ) (24)

where ~ is the standard chemical potential of component A.

The equation

A g " = A h ° - T A s °, (25)

where ±h" and As ~ are the integral molar euthalpy and entropy of mixing, is a convenient way to express the temperature dependence of Ag" because of a generalization known as Kopp's rule. '8~ This states that the heat capacity of most binary solutions is closely approximated by

c~ = XA(C,~,)~ + XB(C ~)~. (26)

Accordingly, h" and s ~ vary with respect to temperature in approximately the same manner as the expressions (XAhX + XBh~) and (X,,s?, + XBsg) . As a result, for the formation of one mole of a -phase from its components A and B (each in the a-phase) as represented by

XAA ~ + XBB ~ ~ (A, B) ~ (27)

Phase diagrams 125

the associated enthalpy and entropy changes, Ah ~ and As" are practically independent of temperature. Therefore, a knowledge of the variation of Ah ° and As ~ with respect to composition at one temperature defines the free energy change, Ag ° , for the process as a function of temperature and composition.

Many experimental techniques are available for the measurement of Ah ~, As ~ and Ag e. It is not the purpose here to enumerate these methods. The two basic principles involved, however, merit brief consideration. In some instances, it is physically possible to carry out the process indicated by eqn. (27); this is particularly true for the measurement of 2~h" when fluid phases are involved, and the heat liber- ated during mixing can be directly measured. This, however, is an exceptional situation. In a large proportion of experimental studies it is only possible to measure the thermodynamic properties associated with the general process

A ° + v ( A , B ) ~ ( v + I ) ( A , B ) ", (28) v ~ l .

The free energy change of this process, as estab, lished for instance by determination of the partial pressure or the voltage of a suitably designed electrochemical cell, is

A a ~ ix A /~ 7, - ~ ~, (29)

where A/x~ is the relative chemical potential of component A. When the isothermal (and, strictly, isobaric) variation of A/x~ with respect to composition has been established, the variation of Ag ° with respect to composition may be determined by an integration based upon rearranging eqns. (23) and (24) into the form:

Ag ~ _

\ x ~ / "

An examination of Fig. 6 will indicate how the various solution properties relate to the calculation of a type 2 isobaric temperature-composit ion diagram for a system in which a - and/3-phases extend from pure component A to pure component B. The figure refers to the Au-Cu system.

Consider for the moment only the upper nine panels (Figs. 6a-i). The three panels on the left refer to pure Au; the three on the right refer to pure Cu; the total pressure is 1 atm in both cases. The enthalpy and entropy changes associated with rais- ing the temperature and melting the metals are available in this case in tabulated formY ~ The free energy changes can be evaluated with the aid of the entropies of the pure metals at 298 K using the relationship:

g~-g~ga = (h~-- h~98)- T ( s ~ - s°gs)- s % d T - 298). (31)

The plots of (g}-g~98) versus T may be recog- nized as an isobar on a diagram of the type given in Fig. 3. The central upper three panels Figs. 6 b, e and h refer to the binary alloy under a total pressure of 1 atm at 1250 K. Enthalpies, entropies and free energies of solid and liquid phases in stable and metastable conditions are given with respect to a mechanical mixture of the pure solid components at 298 K of the same overall composition. The most

126

Au

% 50 . ,

E 2 o

T (a)

L

~ oo~° ,~ 5 - 4 0

,~ T (d)

- 5 0 , ,

, - 8 0

"901000 12L00 1400 TIK)

(g)

A. D. PELTON and W. T. THOMPSON

AHoy Cu

. . . . "' , , 50

t , T~-,-/ '~,°~I . ~ , , 12° Xcu (b) T (¢)

. . . . ~L ~ , -S0

- 7 0

80

0 02 0,4 06 08 10 1000 1200 1400 90 Xcu: nCu/(nCu* nAu) TIKI

(h) (J)

. . . . . . 1400

1000 -40 -20

1 4 0 0 ~ t,Qu,d

~_ 1200

1000 -20 -40 (IJ .;j°S) kJmol-1

Au Au T (j) 7-- 120 , , , ,

~.~ ~ , ~ c ° 100B0 Liau,a

~ 60

~. 40 Xcu (m)

FIG. 6. Thermodynamic interrelationships associated with the construction of corresponding type 1, type 2, and type 3 phase diagrams for Au-Cu alloys when P is

constant.

important features in this sequence of plots are the cross-overs in the free energy isotherms in Fig. 6h. By the common tangent rule, this situation results in the formation of two sets of conjugate phases at this temperature. The scale employed in Fig. 6 serves to draw attention to the very small free energy differences involved in phase equilibrium calculations. However, since only the relative positioning of the isotherms is involved in determining the composition of conjugate phases, the locations, for example, of both g]~ and g ~ at any temperature are arbitrary. The relative positioning of g ~ and g~S, for instance, then involves calculating the free energy change for the S--* L transformation. This is most readily accomplished by integrating the entropy change for the transformation from the equilibrium transformation temperature T,,.

In general for an a-/3 transformation at temperature T:

I; g°O - g°~ = - (s °~ - s °~) dT (32) T r

where

. . . . frl c~° - c~° " d r s ° ~ - s = ( s ° ~ - s )r,,+ T t

(33)

where (s o8 _ s °")r,, is the molar entropy of transformation at T~. If the difference in molar heat capacity can be represented by

c°p~-c~ ~ = A a + A b T + A c T -2 (34)

where Aa, Ab and Ac are constants, then:

g°O - g ° " = (s °~ - s°~)r,,(Ttr - T)

+ A a ( T - Ttr+ T l n ~ ) - ~ ( T , r - T):

,5c ( T - Ttr) 2 2 ' T. T~r " (35)

In many cases, particularly when T is near the transformation temperature T,,, the magnitude of the second term on the right in eqn. (33) is small compared with the first, and, all the terms in eqn. (35) containing Aa, Ab or Ac are negligible.

4.2. Geometrical representation

The four lower panels of Figs. 6 j-m relate to various ways of displaying the conditions for phase equilibrium. In a two-component system, the potential functions of interest are T, /ZA, /XB and - P . Accordingly, two type 1 diagrams will result if temperature is plotted versus one chemical potential at constant P with the other chemical potential left unspecified. Fig. 61 is a type 1 phase diagram of temperature versus /xco relative to the chemical potential of pure solid Cu at the same temperature and pressure. The two points indicated on the phase boundary at 1250 K result from a calculation involving the tangent intercepts in Fig. 6h. Another type 1 phase diagram, similar in form, could be prepared by selecting the relative chemical potential of Au as the abscissa and leaving /zc, unspecified. This phase diagram appears in Fig. 6j. The most familiar and widely used phase diagram for a binary alloy system is the type 2 temperature versus mole fraction diagram at a fixed total pressure, usually of 1 atm. This phase diagram for the Au-Cu system, corresponding to the type 1 diagrams which flank it, appears in Fig. 6k. Figure 6m shows the corresponding type 3 presentation. Temperature has been replaced as the ordinate by its conjugate extensive variable, the entropy, di- vided by the total number of moles.

In all of the presentations indicated in Fig. 6, the potential function - P has been kept constant at a value corresponding to a pressure of 1 atm. It is possible to develop a similar series of phase diagrams in which the temperature is fixed, say at a value of 1250K, and the pressure is treated as a variable in the range 0 to 1 atm.

The development of these diagrams is considera- bly simplified by restricting the pressure to the range indicated, since the vapor phase which appears at low pressure approaches ideal gas behavior and the effect of pressure on the enthalpy and entropy of condensed phases is negligible. In the case of the enthalpy, for example, it may be shown that

(O h) = v ( 1 - e T ) (36) " ~ T, nA,n B

where v is the molar volume, and e is the isobaric volumetric expansivity. In view of the usually small magnitude of e and v of condensed phases, a change in P of 1 atm results in only a few J mol -I change in h, which is negligible in view of the typical magnitude of the enthalpy of mixing and the accuracy to which the latter is usually known. A similar conclusion regarding the effect of pressure on the entropy of condensed phases can be reached

Phase diagrams

using eqn. (18) as mentioned previously. The net result of the preceding development is that Figs. 6 b and e are essentially unaltered by changing the pressure; thus the composition of the conjugate condensed phases are, practically speaking, un- changed as the pressure is changed.

As the total pressure is reduced, a vapor phase comes into equilibrium with the condensed phase(s). If the vapor is ideal it may be shown that, in general:

= p~. exp ~A-R----~I~ A (37) pA

where pa is the partial pressure of component A in the vapor phase, and p~, is the saturated partial pressure over pure component A. The vapor pressures of Au and Cu at 1250 K are known ~9~ and these are indicated on Fig. 7. Equation (37) may now be employed in conjunction with Fig. 6 to determine the partial pressures of Au and Cu when solid and liquid phases coexist. The total pressures and compositions of the vapor when solid, liquid and vapor coexist at 1250 K can therefore be determined. This information is displayed in Fig. 7 together with the two-phase boundaries established by further application of eqn. (37).

x

E

30 -

2~-

I

2E ~

l:J

F

5-

0

1250 K

~ ~ Vapour

0'2 o'.~ o16 o'.B Au Cu

Xcu:nCu / ( ~ C u " ~Au ) FIG. 7. Type 2 pressure-composition phase diagram for

Au-Cu alloys when T is constant at 1250 K.

Diagrams of this type have been prepared for metal-oxygen and metal-sulfur systems and have been of use in the understanding of the high temperature destruction of materials by evaporation processes."° ~

4.3. Solu t ion propert ies

In some instances, all of the information indicated in the various upper panels of Fig. 6 is available. Kubaschewski and coworkers have graphically calculated a number of binary phase diagrams for alloy and oxide systems from thermodynamic data using free energy-composit ion

127

diagrams. (See ref. 11 for a discussion of this work.) In by far the larger number of cases, however, incomplete information must be supplemented with estimated values of thermodynamic properties. Ac- cordingly, in the remainder of this section, we will emphasize the effects of various assumptions on the form of calculated phase diagrams, restricting consideration to type 2, temperature-composit ion diagrams. For an extensive list of references to calculations of this sort the reader is referred to ref. 11.

Ideal solut ions. The simplest solution to consider is one in which the activity of component A in the a-phase , defined by

~ - ~ (38) a ] = exp R T '

is equal to the mole fraction of component A in that phase at all compositions and temperatures. From an examination of eqn. (38), it is clear that for the activity to equal the mole fraction as XA approaches 1, the standard p o t e n t i a l / ~ must refer to pure component A in the a-phase.

Suppose that two ideal solutions, a and/3, are in equilibrium. Since g~ = ~] , we may write

(/x ~, - ~ ~,°)- (~ AB- /~ ~ ) = (/X Y,~- /X.~"), (39)

that is, for an ideal solution

R T In X~A -- R T In X~A = (/z~fl -- p.~,~ ). (40)

Since, at the same time, IX g =/~ ~ :

R T l n X ~ 3 - R T l n X ~ B =(/x~ ~ -/z?3°). (41)

These equations may be solved analytically in conjunction with eqn. (35) to yield the compositions of conjugate phases for specified temperatures.

One of the difficulties in the application of eqn. (40) is that pure component A may not be physically known to exist in, for example, the a -phase at any temperature or pressure. Therefore, values for ( /~f l - /x~, °) obtained by extrapolation of actual experimental measurements (as in eqn. (35)) cannot be determined. The classical approach to this situation has been to use carefully chosen standard chemical potentials. For example, when c~- and /3-phases are in equilibrium, eqn. (39) may be written in the form:

( / ~ - p.~fl) - ( / ~ - / z ~ ) = 0. (42)

This implies that the activity of component A is the same in the a - and /3-phases at equilibrium since activities in both phases are now defined with respect to the same standard state. In spite of the conceptual simplication, the fundamental difficulty is obscured and not, of course, overcome, since the activity of component A in the a-phase , given with respect to pure component A in the/3-phase, can no longer be equated to XA particularly as unity is approached.

Recently, progress has been made at estimating free energy changes for metallic transformations that cannot be observed directly. A wide variety of interpolation and extrapolation procedures have been employed for this purpose. Referring to such information as relative "lattice-stabilities", Kauf- man and Berstein ~6~ have eqns. (40) and (41) to develop isobaric temperature-composit ion phase


diagrams. Consider the Re-Pt system. Re melts at 3450 K from the hcp structure (e) and Pt melts at 2040 K from the fcc structure (a) . With the entropy of melting of both elements estimated at 8.4 J K -~ mol -~ (Richard's rule):

(/z oL _ /z O,)R~ = 8.4 (3450- T) (43)

(/~oL _/x O~)pt = 8.4 (2040- T) (44)

where the superscript L refers to the liquid phase. Kaufman and Bernstein ~6~ estimate melting of

unstable a -Re and E-Pt at 2890 and 1670 K respectively with entropy changes of 9 .6JK-~mol -'. Hence:

(~U~ ° L - - ~ ° ~ ) R e = 9.6 (2890- T), (45)

(/.t °L - / z °" )p, = 9.6 (1670 - T). (46)

These equations together with eqns. (40) and (41) permit solidus and liquidus boundaries extending from pure Re and Pt to be calculated. Solidus and' liquidus lines are displayed in Fig. 8a. Since the a - and e-phases must be separated by a two-phase

(sL- 5£)Re= B.? (sL'-sE)R =8,~ J K-'mo[ -I ($ L- 5E)Re= 101

,,~ 3000

o 1 0 02 04 O 08 10 Re Pt Re I~ Re Pt

Xpt Xpt Xpt

(b) (a) (c)

FIG. 8. Calculated (~ Re-Pt type 2 temperature- composition phase diagrams for different values of the entropy of melting of Re (in J K ~ mol-'). Observed

diagram "2~ shown by the lighter lines.

field, it is clear, after establishing these solidus and liquidus lines, that a peritectic will be formed, the temperature being given by the intersection of the two liquidus lines. The boundaries separating the a - and e-phases may be calculated in a manner similar to that described for the liquidus-solidus boundaries, by combining eqns. (43--46) to give the standard chemical potential changes for the a ~ e transformations for Re and Pt. The complete calculated diagram ~6~ is shown in Fig. 8a superimposed on the known diagram for this system "2~ obtained by direct experimentation. It is apparent that the calculated diagram is qualitatively correct.

Since the positioning of the phase boundaries is dependent on the free energy changes for transformations in the pure components, it is instructive to consider the effects that result when these values are changed. Figures 8 b and c illustrate the effect of changing only one parameter, the entropy of fusion of Re, in eqn. (43) by 20%. Although the diagram is qualitatively unaltered, the boundaries are significantly shifted.

D e v i a t i o n s f r o m ideali ty . The behavior of real solutions may conveniently be expressed in terms of the departure from ideal behavior. Since for the ideal solution described above

tZA - Ix ~ = R T In XA (47)

it follows that the partial enthalpy and entropy of mixing for an ideal solution are:

AhA = 0 (48)

ASA = - R In XA (49)

Partial excess properties may then be defined as follows:

t~ ~ = (tXA-- tz ~)-- R T ln XA = R T ln TA (50)

where 3'A is the activity coefficient of component A.

SEA = ASA + R lnXA, (51)

h~ = AhA. (52)

From which it follows that:

ix ~ = A h A - - TS~. (53)

Integral excess properties can also be defined. For example, the integral molar excess free energy is:

ge = XAp~ + XB/xg. (54)

It may be shown that equations similar to eqns. (23) and (30) relate partial and integral excess quantities. Hence, the previously mentioned graphical relationships between integral and partial properties such as in Fig. 5a apply also to excess properties.

Let ~o~ be a partial excess property of component A in a binary system A-B, such as g~, hl , or s~. Then, as first suggested by Margules, (13~ we may expand ~0~ in terms of the mole fraction, XB, as follows:

oJ ~ = X~(ao + a lXB + a2Xg + ' ' "). (55)

The form of the series provides ideal behavior for component A in solutions concentrated in component A (Raoult 's Law). That is:

if XB = 0,

then w ~ = 0 (56)

and when XB ~ 0,

then 0w ~, ~ 0 (57) OX~ "

In other respects, the series is empirical. The corresponding partial excess property of

component B, o~, and the corresponding integral molar excess property, w E, can also be expressed as simple power series:

to ~ = X~(bo + bIXB + b2X~ + . • .) + kB, (58)

w E = X A X B ( p o + p t X B + p 2 X ~ + . . . ) + kBXB. (59)

The coefficients a,, b, and p, of eqns. (55), (58) and (59) are related through the Gibbs-Duhem equation. The relationships are:

(n + a, = b. - k~-~)~n-- l , (60)

b, = (n + 1) p,. (61)

The arrangement of the power series as above is convenient, "'~ since when eqn. (55) applies over a concentration range up to XB = 1, the integration constant kB is zero.

When non-ideal phases are in equilibrium, the calculation of the composition of the conjugate

Phase diagrams

phases is similar to that previously described for the ideal case. Equations (40) and (41) are simply modified as follows:

( R T In XT, + # ~°) - ( R T In XeA + IxA Eel = (IX ~fl - Ix ~,° ) , ( 6 2 )

(RT In X~ + IX~") - (RT In X~ + IX~") = (IX ~ - IX ~ " ) . (63)

Although conceptually simple, the solution of these equations for various temperatures is complex, and numerical techniques are required.

Regular solutions. When the entropy of a solution approaches the ideal behavior indicated by eqn. (49) and when, furthermore, one term is sufficient to describe the excess free energy over a range of temperature, i.e.

g t~ = poXAXB (64)

for 0 ~< XB <~ 1, the solution is termed "regular". "~ The constant po, characterizing the solution behavior, may be interpreted "~' as being related to the difference in energy of A-A, A-B and B-B pair interactions:

Pc, = Z(2UAB -- UAA- UBB) (65)

where Z is the first coordination number and UAB is the molar energy of an A-B pair interaction, etc. Various methods of numerically estimating p0 from electronegativities, atomic sizes, etc., have been p r o p o s e d 9 -~8~

From eqns. (60) and (61) it follows that:

tx ~ = poX~ (66) and

tx ~ = poX~. (67)

In view of the simplicity of a regular solution, the concept has been extensively used to approximately describe non-ideal solution behavior when little experimental information is available.

Since the integral molar free energy of mixing of a regular binary phase is given by:

Ag = R T ( X A In XA -[- XB In XB) + poXAXB (68)

it may be seen that for positive values of Po, Ag isotherms calculated from eqn. (68) have points of inflection provided T is sufficiently small (see Fig. 9). This implies from the common-tangency rule, discussed previously, that at constant temperature, two different compositions, a ' and a", of the a solution will have the same chemical potentials for both components. In this situation, the a solution has a miscibility-gap extending from the composition associated with a ' to that for a". The consolute temperature occurs when the points of inflection come together. From the symmetry of eqn. (68), it is evident that the consolute point occurs when X A = X B = 0 . 5 . The consolute temperature T, is thus given by solving the equation

1 d2Ag RT(X~A+-~B) - 2p° = = (69)

for XA = XB = 0.5. This yields:

T¢ = po/2R. (70)

129

- 500

~ , -1000

o', -1500 <3

-200O

XB B

02 0.4 0.6 0.8 1 0 T

PO : 15 kJmoF ~

J

_ TWO PHASES / ///

1000 K

L I I i

FIG. 9. Development of a miscibility-gap in a binary regular solution.

Equation (70) is useful for providing a rough estimation of po. A case where agreement between po as determined by calorimetric measurements and as calculated by eqn. (70) from the known consolute temperature is particularly good is found in liquid Cd-Ga solutionsJ 9) Compositions of conjugate phases at temperatures below T,. are calculated most simply by setting the first derivative of eqn. (68) with respect to XB equal to zero.

Even when such an elementary mathematical form describes the behaviour of the phases, a wide variety of features can be generated in a type 2 temperature-composition phase diagram as illustrated in Fig. 10. The panel in Fig. 10n shows the typical lenticular diagram when a - and/3-phases of components A and B are ideal and extend over the entire range 0 <~ XB ~< 1. The other panels surround- ing this figure indicate the effects of systematic changes in the regular solution parameters of each phase. For purposes of developing this series of diagrams, the transformation temperatures of pure A and pure B were arbitrarily taken as 800 and 1200 K respectively with entropies of transformation of 10.0 J K -~ mol '. The trends in the diagrams of Fig. 10 which accompany increasing positive or negative deviations of each phase from the ideal case of Fig. 10n are evident from an examination of the figure. Of particular interest is the fact that with only the simple regular solution model one can produce a simple peritectic diagram (Fig. 10i), a syntectic construction (Fig. 10k) and retrograde solubility (Fig. 10d). The retrograde solubility in Fig. 10d can be understood as an "anticipation" of Fig. 10e in which the slope of the solidus abruptly changes sign at the monotectic temperature. This ability of the regular solution model to reproduce a large number of the features of binary temperature-composition diagrams was pointed out long ago by Van LaarJ 19~ The calculation of "y- loops" in the Fe-Cr, Fe-Mo and Fe-W systems using regular solution theory has also recently been performedJ :°~

Using both theoretical and empirical methods, Kaufman and Bernstein ~6~ have estimated regular solution parameters when refractory metals form binary solution phases that are liquid, hcp, fcc or bcc. Using these estimates together with their esti-

130

-20

(a)


-10

(b)

pS(kd moF 1)

o

! '(c)

(f) (g) (h)

*10 .20

(d) (e)

/3

*30

• / 3 ~ 800

1600

1400

1200 / O.

I000

r / ~ 600

600

40O

( i ) ( j ) (k)

( t )

m

(q)

{a

B (m) A XB (n )

cl

(r)

(Z

(0)

1400

1200

a 1000

800

600

400

(S) ( t )

1400

1200 "~

D 1000 ,~

800 ~,~

500

400 ~

(p)

FIO. I0. Topological changes in the type 2 temperature-composition phase diagram of a binary system A-B with regular phases a and 3 brought about by systematic changes in the regular solution

parameters p g and p o ~. (Entropies of transformation of pure A and B taken as 10.0 J K-' mol-~.)

mates for the free energies of metastable phase transformations in the pure components (relative "lattice stabilities"), they have been able to generate a large number of binary temperature- composition phase diagrams of these metals. Fig- ure 11 illustrates the agreement between the calculated (6~ and known m~ diagram for the Hf-Ta system. This is typical of the best agreement which it is possible to achieve when all solutions are assumed to be regular.

Quadratic formalism. Although few solutions approach regular behavior, it has been shown mm'28~ that many binary systems A-B are represented quite well in the limiting concentration region as XA "~ 1 by

tx ~ = R T In 7A = aoX~ (71)

as might be expected in view of the form of eqn. (55).

Substituting this expression into the Gibbs- Duhem relationship and integrating, we obtain for the solute:

sooo

~ 200C

p-

1000 ÷

I I I i i I I I I I I Hf 0.2 0.4 0.6 0.8 T a

X T a

F I G . 1 1. Temperature-composition phase diagram of the Hf-Ta system calculated from regular solution model for all phases (after Kaufman and Bernstein(6~). Observed

diagram (2" shown in light lines.

Ix~ = R T In T, = kB + aoX~ (72)

where the constant of integration, kB, is equal to ( R T In y~ - ao), where y~ is the limiting (Henrian)

Phase

activity coefficient of B at X^ = 1. Thus, the solution is not "regular" with respect to component B (unless by chance kB = 0). However, near XA = 1, the solution could be considered to be regular with respect to component B if the standard state of B were changed to a hypothetical standard state of pure B having the same properties as those of a reference state of an infinitely dilute solution in A. The constant k. is then equal to the difference between the standard free energies of the real and hypothetical standard states, and could thus be approximated in a manner similar to. that used to approximate a "relative lattice stability" as discussed previously. This consideration may be useful for purposes of estimating thermodynamic properties of solutions.

Sub -regular solut ions. One of the principal drawbacks of the regular solution concept is that deviations from ideal behavior are symmetrical with respect to concentration. For example, the consolute point of a miscibility gap in a binary regular solution A-B is always at X , = 0.5. This difficulty may be overcome by including additional terms in eqns. (55, 58, 59). In the case where the two-term equation

g ~ = XA " X , (po + p~X,) (73)

is a satisfactory approximation of the solution behavior over a temperature range for 0 ~< X, ~< 1, the solution A-B is termed "sub-regular"J 24)

The variation of the bracketed term in eqn. (73) with composition could, for instance, be physically interpreted as reflecting the dependence of coordination number on the composition of the solution.

By applying considerations similar to those al- ready mentioned for the regular solution (cf. eqn. (70)), it may be shown ~25) that the composition and temperature of the consolute point of a miscibility gap in the system A-B may be related to p0 and p, in eqn. (73) by solving the following equations:

RT~ X A X . - - 3p~Xa - 2(p~ - po) (74)

RT, 6pt 2 2 - (75)

X A X B 2XB-- 1

where XA and XB refer to the mole fractions of the components at the consolute point and T~ is the consolute temperature. We shall make use of these equations later in Section 5.1.

Alternatively, a knowledge of the compositions of the conjugate phases on either side of the miscibility-gap at some temperature below T¢ may be used to evaluate the constants p0 and p~. This information is, of course, of use in estimating the position of the entire miscibility-gap on a temperature-composi t ion diagram when very little information is available. Wreidt (25) has discussed these approximations in details.

Defini t ion o f ideal solution. In eqn. (50) the excess chemical potential, lL~ was defined in terms of an "ideal" relative chemical potential, AlL~ d~a"= R T In XA. For certain systems, however, we may use our knowledge of the molecular structure of the solution in order to obtain a more suitable definition of ideality. The excess chemical potential, now

diagrams 131

defined as the deviation from this more correctly defined ideal relative chemical potential, will then generally be smaller, and it will be found that far fewer terms of a series expansion will be required for an adequate analytical representation of the excess property.

As an example, A/xAsm has been measured in the quasi-binary AgBr-CsCI system at 1073 K 26. If we define

E lL Agm = A l L A g m - R T I n XAgBr (76)

as in eqn. (50), then the plot of lL ~Agm[X2c+a as shown in the lower curve in Fig. 12 results. Obviously, a

40

, (eq. 78)

~ 2C

1C T= 1073K 2¢

?3 6) -1C

X -2(

,- -3C if3

- 5(

- 60

- 70

- 80 i l k t t 0.2 0.4 0.6 0.8

X C s C t

FIG. 12. Illustration of the effect of improper definition of ideal solution behavior in the AgBr-CsCI system.

very large number of terms will be required in any power series expansion (cf. eqn. (55)) to represent this curve. However, a more appropriate definition of ideality for such a "reciprocal salt system" would be the Temkin definition:

m (ideal) lLAgBr = R T In (XAg+XBr) (77)

where XAg+ and Xm are ionic fractions on the cationic and anionic sub-lattices. In this quasi- binary system, XAg+ = Xm = XA+B,, and so we can define

E -- XAgBr - ~L AgBr = A l L AgBr R T I n 2 (78)

E IX2 A plot of lLAsm/ csc~ as defined by eqn. (78) is shown as the upper curve in Fig. 12. This curve is well represented by a simple three-coefficient series: am

E lL ~B~ = X~,~c~(37614 - 49662Xc~c, - 33262Xc~c3 (79)

(units of J tool-) . As a further example, consider a solid solution of

iron and chromium oxides in a spinel phase. If we assume that Fe304 is a normal spinel with Fe :÷ ions on tetrahedral sites and Fe 3+ ions on octahedral sites, then pure Fe304 can be represented as

2+ 3+ FeA (Fe2),04, and the solution can be represented 2+ 3+ 3+ as: FeA (FeE 3~r3+).04 where the subscripts A and

B refer to tetrahedral and octahedral sites respectively, and where it is assumed that all the chromium is present as Cr 3÷ ions which substitute only on octahedral sites. ~ is the ratio of Cr 3+ ions to total cations in the solution. The ideal activity of

132 A.D. PELTON and

Fe~O, in solution is then equal to X~e~+~) where X~0,+(,) is the cationic fraction of Fe ~+ ions on

* (idea[) octahedral sites. Thus, a/~ro,o, is best defined as follows:

A (ideal) RT in 2 ) 3 - 3+~ /z~o,o, = ~ 3~ = R T In (1 - 3~:/2). (80)

A further example is found for relatively dilute interstitial solutions such as solid Fe-C solutions. Considerations of the structure of such systems lead to the following definition of excess chemical potential:

Xc (81) /xc ~ = Ap, c - R T In X F ~ - - Xc"

4.4. Intermediate phases

When, in a binary system, a phase is formed that does not extend to either pure component at any temperature or pressure, and when the integral free energy of mixing isotherms for that phase can be located with respect to those for the other phases, then phase boundaries can be established by using in effect the (lowest) common tangent method discussed previously. Consider the formation of the intermediate T-phase from components A and B in the a - and /3-phases respectively.

X^A ~ + XBB ~ ~ (A, B) v (82)

With reference to Fig. 13, it is clear that the position of the integral molar free energy of mixing isotherm for the y-phase may be located in more than one way. For example, as eqn. (82) suggests, g ~ could be specified with respect t o / ~ a n d / ~ , i.e. with respect to pure A and B in the a - and /3-phases respectively. This would be a convenient and obvi- ous choice if experimental measurements were to be represented. Another approach would be to locate g~ with respect to /~]~ and /z~ v, i.e. with respect to pure A and B each in the metastable

g

, J

(a)

T

(b)

A x 8

FIG. 13. Isothermal, isobaric free energy relationships in a binary system containing an intermediate phase, and resultant type 2 temperature-composition phase diagram.

W. T. THOMPSON

y-phase. Although conceptually attractive, this procedure would require knowledge of the free energy changes associated with transformation of the pure components into the y-phase. Unfortu- nately, these data are usually impossible to obtain by experimental means. There are, however, some cases where this may be a fruitful approach. An example is the Mo-Rh system. Kaufman and Bernstein, (6) from estimates of free energy changes of fcc(a), bcc(/3), hcp(E) and liquid interconver- sions, together with estimated regular solution parameters for each possible binary phase, have generated the temperature-composition diagram shown in Fig. 14. Comparison with the experimental diagram ~2' on which it is superimposed indicates surprisingly good agreement in this particular case.

3000

2000 // 1000 /

Me 0.2 0.4 XR h

0.6 0.8 Rh

FIG. 14. Calculated temperature-composition phase diagram of the Mo-Rh system (after Kaufman and Berns-

tein~6)). Observed diagram ~2" shown in light lines.

(6) Similar calculations for other binary metallic systems were, however, less successful.

This problem of standard states can perhaps be clarified by the following analytical development. Let Aft~ be the relative chemical potential of A in the y-phase. This quantity can be expanded as follows:

m'

Aft ~ = I~ ~ - I~ = R T In XA + X~ ~ a,X~ + kA m = 0

(83) where kA is a constant. Suppose that the standard state is chosen as pure A in the metastable y-phase. That is: /z,~=tt~ v. In this case, when XA= 1, A/~ = 0, and furthermore, Raoult's Law will be obeyed (Ap.A-->RT InXA as XA"-> 1). We can thus set kA ---- 0, and eqn. (83) is now equivalent to eqn. (55). Suppose, however, that the standard state is chosen as pure A in the a-phase. That is:/x~, =/zY, °. In this case, when X A = I : A~LY, = / ~ - - / ~ = k ^ # 0 . If direct experimental measurements of A l ~ = R T In aA are available, they will probably be with respect to pure A in its stable a-phase. These data should then be fitted to a power series of the form of eqn. (83) with kA as another constant to be determined by the curve-fitting technique. By applying the Gibbs-Duhem equation to eqn. (83) it may be shown that:

n '

AIx~ = lx~ - I ~ = R T l n X B + X 2 ~ b,X~ + kB, nffi0

(84)

Phase diagrams

Ag" = XAA/Xl + XBA/x ~ n'

= R T ( X A I n X A + X B l n X B ) + X A X B ~, p,Xg n=0

+ (XAkA + XBkB). (85)

These equations should be compared to eqns. (58-59). The coefficients a,, b, and p, of eqns. (83-85) are related by eqns. (60-61). If we refer Ag * to pure A and B each in the metastable ,/-phase, then Ag ~ = 0 when XA = 0 and also when XB = 0, and so kA = kB = 0. If pure A in the stable a -phase and pure B in the stable /3-phase are chosen as standard states, however, then the constants kA and kB may be associated with the "relative lattice stabilities". That is: kA=tZ~-- /x~,° ; and kB = ~ g ' - t L ~ °. The geometrical interpretation can be seen by reference to Fig. 13.

From an examination of Fig. 13 it is clear that the necessary condition for nearly stoichiometric intermediate phases, such as the magnetite and hematite phases in the F e - O system (Fig. 15a), is steep free energy curves g~ with sharp changes of curvature near the minimum of the curve. When the composition range over which the intermediate phase is stable is sufficiently narrow as to be negligible, considerable simplification in calculations can be effected by regarding the minimum of the free energy curve as a point. When such an approximation is made, it is often desirable to redefine components. For example, for the /3-phase isotherm

( OG'~ ~ + ( OG'~ ~ = ( OG ~o (86) OnA/ \ O n a / \OnAa/ "

Therefore

d l n a ~ + d l n a g = d l n a ~ B . (87)

The last equation could also have been established by writing the equilibrium constant for the homogeneous equilibrium

A e + B ° ~ A B ~. (88)

133

Equation (87) can be simplified by using the Gibbs-Duhem relationship between aA and aB:

- X B ~ d In a6. (89) d lna A B= 1 XA/

The activity of pure AB in the /3-phase may be determined by integration of eqn. (89) for XB/> 0.5 from XB = 0.5 at which point a~B may be assigned the value unity. From a knowledge of the activity of AB in the/3-phase, the phase boundary (composition of the/3-phase when saturated with y ; see Fig. 13) may be determined by solving the equation:

RT In aAa = - 2 ( g ~ - g ~ ) x . o~. (90)

(The term on the right of eqn. (90) may be recog- nized as the free energy of transformation of the compound AB. Compare with eqns. (62-63).)

If the equilibrium in eqn. (88) is displaced strongly to the right, then, when XB I> 0.5, an ideal activity may be defined as:

a~de~, X ~ - XA (91) XB

This implies that a t approaches ideal (Raoultian solution) behavior with respect to mixing of the components AB and B. Truly ideal behavior is impossible, of course, in the AB-B sub-system, since aB cannot become zero in the pure AB-B phase without the equilibrium constant for the process in eqn. (88) reaching the value of infinity. Nevertheless, in systems where equilibrium of the type in eqn. (88) are physically meaningful, the sudden fall to near zero activity for one primary component with the simultaneous rise in the activity of the other as composition is altered in the vicinity of AB, results in a sharp change in curvature of the integral free energy of mixing isotherms. For this reason, maxima in d2g~/dX~, sometimes called the "stability parameter", are observed. °'8)

Jordad TM has discussed a "regular associated

~LIQUID r LIQUID IRON I P~ = 1aim ~, IRON I~ / + LIQUID OXIDE // u2

,i,q I / -<f . . . . . . / I RON I / IMA'N" E1 r~-i~ON/ / _ _/__11'°°

1500i / MAGNETITJ ] - Y-ITITE I / W U S i j ETIr ] ~ _ _ ~ / MA~//~I[IE t

/ 1 . o

" I / -v~. .... I HEMmlTE/ I,RON/ / / I HEMATITE WUSTITE I / I OXYGEN / I I I / I

" V II [Y 'oo O( IRON ÷MAGNEIITE

/~ 700 70 I I [ I 1 I I O -50 -100 -~50 -200 -250 .50 .S/* .5B .62 -250 -200 -150 -100 -50 0 (~U Fe-P °Fe)-r( k J m o I, -~ ) no / ( no ÷ riFe) ( )J O - )J%)T ( k J me l-~ )

(C ) ( a ) ( b )

FIG. 15. Corresponding type 1 and type 2 phase diagrams for the Fe-O system. (3°)

134

solution" model for binary systems A-B in which an equilibrium as in eqn. (88) is set up between A, B and AB species which then form a regular solution. Calculated liquidi in the Zn-Te and Cd-Te systems are in good agreement with the observed liquidi.

The type 2 isobaric temperature--composition diagram is unquestionably the most popular and often most useful representation of phase equilibrium in a two-component system. However, a corresponding type 1 diagram, in which (/ZA --/Z~)r for one of the components, A, is plotted versus T at constant total P and unspecified p~B is of considerable value when nearly stoichiometric intermediate phases are involved, and/or when A is an element such as oxygen, sulfur, etc., whose chemical potential can be easily fixed or measured via a gaseous phase. Consider the Fe -O system ~3°) in Fig. 15. When pure Fe.,O3 (hematite) is in equilibrium with pure Fe304 (magnetite), the following equilibrium may be written:

2Fe304 + ~O2~-~,3Fe:O3 (92) for which

AG ° = - R T In K = R T lnpg~ (93)

where AG ° is the standard free energy change for the reaction (92). Since

R T In p~2 = (Ix0-/Z~)r (94)

it follows that (/Zo-/Zg) along the two-phase Fe304/Fe203 boundary in Fig. 15b is equivalent to the standard free energy of reaction (92). In general, along any boundary in Fig. 15b, (/~0 - / x ~) is the standard free energy change of an oxidation reaction from one phase to the other based upon ~ mole of O2.

In addition, the slope of a phase boundary in Fig. 15b corresponds to minus the reciprocal of the standard entropy change for the appropriate oxidation reaction. This can be proven through use of the generalized Clapeyron equation (14) as follows. Once again, let us take as example the Fe304/Fe203 boundary of Fig. 15b. For the phase change Fe304~ Fe:O3, we have, from eqn. (14), setting tb~ = T, ~2 = go, 4'~ = gF0, ¢', = - P :

no/nF~ _ 3/2 - 4/3 d T / d t x o = - A - - S/nEe .o 12 S ° r~ F e 2 0 ~ / - - F e 3 O J J

(2s ° 3s ° ~ = Fo~o, -- Fo~O~J (95)

where s~o~ and s o Fc~O, are the standard molar entropies of the compounds. Since

d ~ / d T = - s ~ i o = - ~ s o~ ( 9 6 )

we have, by combination of eqns. (95-96):

= - ( 3 s Fo~O~ - - 2 S F~O, - - ~ S O2) d T / d ( l ~ o - i x n ) ° ° ~ ° -~

(97)

The right-hand side of eqn. (97) is - ( A S ° ) -~ for reaction (92).

It can readily be shown that the standard enthalpy change at any particular temperature for the oxidation associated with reaction (92) is given by the T = 0 intercept of the tangent to the phase boundary in Fig. 15b. In view of the quantity of fundamental information displayed on this type of diagram, it has been extensively used in engineering calculations where families of such diagrams (ox-


ides, sulfides, chlorides, etc.) are referred to as Ellingham diagramsJ ~') In comparing these families of diagrams with Fig. 15b, we realize that boundaries parallel to the ( /Zo-/z~) axis in published Ellingham diagrams have been eliminated for clar- ity. This does not introduce any confusion, since kinks in the remaining boundaries in the Ellingham diagram define the triple-points. However, this has tended to obscure the recognition of Ellingham plots as families of type 1 phase diagrams.

In the F e - O system, another type 1 diagram could be prepared by plotting (IXFo- /~ ~c) versus T at constant total P leaving /x0 unspecified. The diagram is shown in Fig. 15c. The Fe304/Fe203 boundary, for example, relates in the manner discussed above to the standard free energy change for the reaction

Fe + 4Fe20~ ~ 3Fe304. (98)

Since ( d t x ~ e / d T ) changes discontinuously at the transformation temperatures of pure iron, care must be taken in this diagram to distinguish such a kink in a two-phase boundary from a triple-point at which one boundary may have been omitted. In Fig. 15c, the scale is such that this does not become a practical difficulty since the kinks are impercep- tible. In principle, a family of reactions involving Fe could be represented as an Ellingham diagram as is the case for oxides. The diagrams would, however, be much less useful since the chemical potential of iron cannot be, in the physical sense, as easily controlled or altered as that of oxygen.

4.5. O r t h o g o n a l s e r i e s

For relatively simple systems for which a series expansion with only a few terms can give a satisfactory representation of excess solution properties, a simple power series as in eqn. (55) is adequate. For many systems, however, which show large deviations from simple regular solution behavior, a large number of terms in the series expansion will be necessary in order to represent the excess property analytically, and in such cases there are drawbacks to the use of a simple power series. For example, in Fig. 16 are plots of / z ~ and of ~ 2 t x M , / X a i in the Mg-Bi system (~) at 973 K. In order to reproduce the curve in Fig. 16 to within the thickness of the line drawn, it has been shown (14) that a 16-term power series expansion in XB~ is required. The problem now arises that for large values of n, the function X~ becomes very small for small values of Xa~. Therefore, if we expand/z ~ as in eqn. (55), we find that for larger values of n the coefficients ao must become very large in order that the term a,, X~ can make any significant contribution to / z ~ when XB~ is small. The coefficients obtained by a least- squares analysis to a 16-term series are listed in Table 1. Details of the actual computational procedure used in the analysis are discussed in ref. 14. Many of the higher coefficients are of the order of 108. Therefore, although tz~g is never greater than about 50kJmol -~ we see that near XB~= 1 the higher terms a ° X ~ are all of the order 108 and so in order to reproduce the curve to, say, 0.1 kJ tool -~ we must record many of the coefficients to 10

Phase diagrams 135

, , ~ , , , , r T

- 1 0 ).~

-20

t~ ~ - 3 0 ::5..

40 (a)

s e r i e s is r e g r o u p e d a s f o l l o w s :

E 2 jU~Mg/XBi = Ao x ( 1 ) + A, x (2XB~- 1 + ' ' '

A 2 x ( 6 X ~ i - 6 X m + I ) + " • • A3 x ( 2 0 X ~ - 3 0 X ~ + 12XB~- 1) + - • '

(99)

E a c h b r a c k e t e d t e r m is t h e n t h " L e g e n d r e P o l y n o - m i a l " P,(XB~) f o r t h e i n t e r v a l 0 ~< X B ~ < 1. T h a t is:

o

20

- 40

~ : 60

a3 80 X

~ - l O O

tn ~ 120 =L

- 140

-~60

o.'i o.'~ o13 o.', oi~ oI~ o17 o.'8 o.'9 i.o X B i

(b)

FIG. 16. Orthogonal series representat ion of the excess chemical potential of Mg in the liquid Mg-Bi sys tem (9~ at

973 K.

E 2 ~ tx MdX ,i = A,,P,. (Xs j . (100) n =o

T h e f u n c t i o n s P° (Xsj) a re g e n e r a t e d by t h e r e c u r - s ion r e l a t i o n s h i p :

Po(x) = 1.0. (101) (2n - 1)(2x - 1) (n - 1)

P,,(x) = x P,, , ( x ) - - - - n i 1

x P,, 2(x). (102)

T h e L e g e n d r e f u n c t i o n s P, (x ) c o n s t i t u t e a n or- t h o g o n a l s e t o f f u n c t i o n s o v e r t h e i n t e r v a l 0 ~< x ~< 1.

T h a t is:

f [ P , ( x ) X Pm(x) dx = 0 fo r m # n. (103)

s i g n i f i c a n t d ig i t s a s is d o n e in T a b l e 1. A f u r t h e r d r a w b a c k to t h e u s e o f a s i m p l e p o w e r s e r i e s is t h a t t h e c o e f f i c i e n t s a r e h i g h l y c o r r e l a t e d . T h a t is , t h e n u m e r i c a l v a l u e o f a n y coe f f i c i en t d e p e n d s u p o n t h e to ta l n u m b e r o f c o e f f i c i e n t s u s e d . F o r e x a m p l e , if w e ref i t t h e M g - B i s y s t e m u s i n g a to ta l o f 10 c o e f f i c i e n t s , t h e n t h e v a l u e o f a n y p a r t i c u l a r coeff i- c i e n t is c o m p l e t e l y d i f f e r e n t t h a n w h e n 16 coeff i- c i e n t s w e r e u s e d ( see T a b l e 1). I t is t h u s i m p o s s i b l e to c o m p a r e t h e p r o p e r t i e s o f d i f f e r e n t s y s t e m s b y c o m p a r i n g c o e f f i c i e n t s , o r to a t t a c h a n y p h y s i c a l o r m a t h e m a t i c a l s i g n i f i c a n c e to t h e n u m e r i c a l v a l u e s o f t h e coe f f i c i en t s .

T h e s e p r o b l e m s c a n be o v e r c o m e °4~ if t h e p o w e r

T h e i m p o r t a n t r e s u l t o f t h i s f a c t is t h a t t h e coeff i- c i e n t s A . a re c o m p l e t e l y u n c o r r e l a t e d . "4) V a l u e s o f t h e c o e f f i c i e n t s A . in t h e M g - B i s y s t e m c a l c u l a t e d b y a l e a s t - s q u a r e s r e g r e s s i o n a n a l y s i s u s i n g a to ta l o f 16, 10 a n d 4 c o e f f i c i e n t s a r e a l so s h o w n in T a b l e 1. T h e v a l u e o f a n y spec i f i c coe f f i c i en t c a n b e s e e n to b e i n d e p e n d e n t o f t h e to ta l n u m b e r o f coeff i- c i e n t s u s e d . It c a n b e p r o v e n t h a t a n y c u r v e c a n be f i t ted to a n y r e q u i r e d d e g r e e o f a c c u r a c y w i t h a f in i te n u m b e r o f t e r m s u s i n g an e x p a n s i o n in t e r m s o f o r t h o g o n a l f u n c t i o n s a s in eqn . (100); all t h e coe f f i c i en t s will be o f t h e s a m e o r d e r o r s m a l l e r t h a n t h e f i rs t f e w c o e f f i c i e n t s ; a n d t h e A , will t e n d to z e ro as n b e c o m e s la rger .

T A B L E 1. C O E F F I C I E N T S O F S I M P L E S E R I E S A N D O F O R T H O G O N A L L E G E N D R E S E R I E S

E X P A N S I O N S O F ~ 2 • 9 Id, Mg/Xal IN THE Mg-Bl SYSTEM AT 973 K (kJ mol ')

O. Mg/Xu~ = a,,X~ = A,,P,,(Xm) . = o n = )

16 coefficients 10 coefficients 16 coefficients ao = -2 . 13 ao = - 7 . 2 4 Ao = -68 .12 a, = - 387.27 a~ = 647.06 A, = - 24.94 a2 = 11,810.01 a2 = - 18,210.53 A2 = 72.72 a3 = - - 155,124.94 a3 = 170,981.59 A~ = 2.68 a , = 854,405.27 a4 = - 798,535.35 Aa = - 47.03 a5 = - 1,805,467.80 a5 = 2,018,694.23 A5 = 1.34 a6 = - 1,646,526.59 a6 = -2,895,454.40 A6 = 27.66 aT = 14,885,978.79 a7 = 2,352,073.38 AT = --2.55 a8 = 125,405,734.52 as = --1,003,482.69 As = 19.62 a9 = 9,538,731.90 a9 = 173,247.98 A9 = 4.35

am = 20,075,423.32 Am = 12.93 a , = - 24,210,355.34 A , = - 4 . 3 9 a, : = 6,024,236.34 A ,~ = - 9.54 a,3 = 2,920,943.66 A,3 = 4.85 a~4 = -742,160.10 A~4 = 5.65 a, , = -345,818.39 A~5 = - 3 . 6 4

10 coefficients Ao = - 67.99 A, = - 25.06 A2 = 73.35 A3 =2.38 A, = -45 .90 A5 = 0.84 A , = 29.29 A7 = - 3 . 1 8 As = - 17.41 A9 = 3.56

4 coefficients Ao = - 68.66 A1 = - 24.98 Az = 70.00 A3 = 2.59

3.P.S.S.C., VoL 10. Part 3 - -B


Finally, we note that the Legendre series is based upon algebraic (rather than trigonometric or other transcendental) terms, since each function P~(x) can be expressed as a polynomial. Furthermore, the first term, A0 x (I) and the second term A, x (2X - I) in the expansion eqn. (99) correspond to "regular" and "sub-regular" solution terms. In fact, the coefficients A0 and A~, being independent of the total number of coefficients used, can truly be called the "regular" and "sub-regular" coefficients, whereas the coefficients a0 and a~ of the expansion eqn. (55), being correlated with all the rest of the coefficients, cannot be accorded any such significance.

Other series expansions in terms of orthogonal sets of functions are discussed in ref. 14. Pos- sibilities are Fourier series, Chebyshev series, and the "Z-series" proposed by Williams. ~32) For systems in which thermodynamic data are available only over a very narrow composition range (a few mole percent or less), as is the case, for example, when log po~ has been measured over the composition range of stoichiometry of a solid metal oxide, it has been shown °3~ that the most suitable series expansion is an orthogonal Fourier cosine expansion:

"' {XM -- X~ \ logpo ,= A0+E,=, A,. cos n ' r r l ~ ) (104)

where X~ and X~ are the mole fractions of metal M at the upper and lower composition limits. Kellogg ~3~) has also discussed the problem of analytical representation over a very narrow composition range.

4.6. Recent calculations oftemperature- composition diagrams

Recent review articles" L35.36) have provided a very complete source of references to calculations of specific phase diagrams. Discussions have been made of the determination of phase diagrams for I I I -V semiconductor systems. °7'38~ An international project named "CALPHAD" is concerned with providing tabulations and bibliographies of thermodynamic data on binary and ternary alloy systems in order to permit the calculation of phase diagramsJ 39'4°~ Hillert ~27~ has discussed the thermodynamic relationships governing phase equilibria in binary systems, as well as analytical methods of representation.

V. Three-component systems containing only quasi-binary phases

A quasi-binary phase is one in which composition can be expressed by only one nB/(nA + riB) variable. For example, in the Co-Ni-O system at 1600 K, an alloy phase (Co, Ni) is observed as well as an oxide phase (Co, Ni)O. If we ignore the slight solubility of oxygen in the solid alloy and the slight deviation from stoichiometry of the oxide phase, then each phase is a quasi-binary phase which can be represented as a straight line on the Gibbs' composition triangle in Fig. 17d. Calculation of the phase diagrams of ternary systems is simplified if the systems contain only quasi-binary phases, since the calculations will be very similar to, and no more complicated than, those for binary systems as

CoO/_e ~kNiO

-,

~ CoC e f NiO

(c) o

c0, ,i , 'i ,

I . O~s~ -6.6 ii(Co,Ni )

-7.4 '~ . . . . ;~

-7.8 ~ °y(sl

~*o~coo~- 0'2 0'~ o'~ 0'o Co Ni

~=nN i / ( i qCo ' * ~Ni ) (U-LI*)Ni ( Jmot "1 ) (a ) (b)

FIG. 17. Corresponding type 1, type 2 and type 3 phase diagrams for the Co-Ni-O system at 1600 K.

Ni ~ / ~ P*0~ (Ni/Ni0)

(Co,Ni) 0is I

i -20 -10 i 0

discussed in Section IV. Ternary systems with two metallic and one non-metallic component, such as Co-Ni-O or Fe-Ni-S, etc., or systems with one metal and two non-metals, such as Fe-S-O, often contain only phases which closely approach quasi- binary behavior, particularly in the solid state, and therefore, many ternary ceramic systems are in- cluded in this section.

The potential functions of interest in a ternary system A-B-C are T, - P , /XA, /ZB and/xc. In order to obtain a type 1, -2, or -3 diagram, we need to maintain two potential functions constant. We shall first consider diagrams in which T and P are fixed. The development will be with respect to specific examples.

5.1. Isothermal isobaric diagrams Co-Ni-O system at 1600K. In the Co-Ni-O

system, let &l =/Zo, ~b2 = ~t, gNi, ~ 3 = [.gCo, (~4 : T, and ~ = - P . Figure 17b is a type 1 diagram of ~bl versus ~b2 at constant ~b4 and ~5. (As ordinate, we have actually plotted log Po2 = (/zo-/~)/2.303 RT, which, however, varies directly with/~o at constant T). Figure 17a is a corresponding type 2 diagram in which ~2 has been replaced by Qd(Q2+ Q3)= nNJ(nN~ + nco)= ~:. The symbol ~: will henceforth be reserved for the molar ratio nd(nA +nB) of metals in a system A - B - X of two metals and one non- metal. Figure 17b consists of one lenticular two- phase region between the solid quasi-binary oxide and alloy phase fields. The corresponding type 3 diagram is shown (in J~inecke coordinates) in Fig. 17c. Here, d'~ has been replaced by the ratio QI/( Q2 + Q~) = no/(nN~ + nco) to which we shall assign the symbol "0. The correspondence between the tie-lines of the type 2 and type 3 diagrams is illustrated in Fig. 17. Finally, Fig. 17c may be "folded-up" to the more usual isothermal "Gibbs' triangle" representation of Fig. 17d.

The isothermal free energy-composition diagram of the Co-Ni-O system at 1600 K is shown in Fig. 18. This should be compared to the free energy-composition diagram for a binary system in

Ni

FIG. 18. Gibbs free energy surfaces for the Co-Ni-O system at 1600 K associated with the development of

Fig. 17.

Fig. 5. The free energy surfaces for the alloy and oxide phases are virtually vertical knife-edges, since these phases do indeed very closely approach quasi-binary behavior. The absolute values of the standard chemical potentials /X~o and t ~ on the vertical scale in Fig. 18 are arbitrary. Similarly, the value of txg~ (not shown in Fig. 18) is arbitrary.

The position of the point/Xo~co;coo~ is determined from the relationship:

I o /x o~co/coo~-/x~ = ,RT In p o~co;coo~ (105)

where /Xo~co~c,,o~ and p g~:~co;coo) are the chemical potential of O and the partial pressure of O~. in equilibrium with pure Co and CoO. The point for /XO~Nim~o~ is determined similarly. The position of point a, which defines the value ~ ° ~/Zcoo on the vertical scale, is then found by joining the points /X~o and go~co;coo~ by a line lying in the face of the prism. The position of point b is found in a similar manner.

The plane of common tangency is tangent to the two free energy surfaces at points c and d, thus defining the ends of the tie-line e - f in the type 3 diagram shown at the base of the prism in Fig. 18 and also in Fig. 17c and d. The points of intersection of the common tangent plane with the three vertical edges of the prism give the values of/xn~,/Zco, and /Zo when the two phases at points c and d are in equilibrium. From these values, the type 1 and type 2 diagrams in Figs. 17a and 17b can be constructed.

Phase diagrams 137

If we can express the chemical potentials of all three components in both the oxide and alloy phases as functions of composition, then we can calculate the points of common tangency by solving the three equations:

/x °X~de = tx~ "°y (i = Co, Ni, O). (106)

In the general case of a ternary phase, each tx; in each phase will be a function of two composition variables, and the three simultaneous equations (106) would have to be solved to determine the four composition variables which define the ends of the tie-line e-f. However, since in the present case each phase is a quasi-binary phase, only two composition variables suffice to define the tie-line e-f, and the calculations can be simplified.

It is most convenient to start with the equilibrium constants K for the oxidation of each pure metal:

Ni + ~O~ = NiO K = aNio/ONi I/2 o 1.2 (107)

+l Co ~O:=CoO K = a c o o / a c o × p l ~ = (p~c,,.c,,,,) ~.2. (108)

In the alloy phase, aco and aN~ are functions of ~a,oy, where, in this quasi-binary phase, ~a,or is numerically equal to the mole fraction of nickel. Thus:

R T In a n i = RT In s c""°~ + /x Ni(SC~ ~"°~), (109) E allo~ R T In aco = R T In (1 - ~alloy) + t.£ ( , o (~ ). (110)

where the excess chemical potentials in the alloy phase could be expressed, for instance, as power series in s ca"°y in the form of eqn. (55). Similarly, in the quasi-binary oxide phase, acoo and aNio are functions of ~:ox~d~, where ~oxide iS numerically equal to the mole fraction XN~O = nN~o/(nN~o + ncoo) in this phase for this particular case.

E oxide RTlnanio=RTln,~"~ide+lxnio(, ~ ), (111)

R T In acoo = R T In (1 o~ide ~ oxide - ~ )+tXcoo(~ ). (112)

The excess chemical potentials in the oxide phase could be expressed as power series in ~oxid~ in the form of eqn. (55). Taking the logarithms of eqns. (107-108), and substituting from eqns. (109-112), we obtain the following two equations which are applicable when equilibrium is attained between the phases:

E alloy ~L£ Ni()(~ ) RT in ~:aUoy + /X~(~: ) _ RT in ~:o~ide__ E oxide

= ~R T In p g~'~n'm'°', (113) po:

R T In (1 - ~,.oy) +/x ~o(~noy) _ R T In (1 - ~ox~d~)

E oxide P ~)2{CoJCoO) - / x coo(~: ) = ~RT In (114) PO2

Values of p~N,m~o, and p~co/coo~ are obtained from the literature/ '~ At 1600 K, it is a reasonable assumption that both the oxide and alloy phases are close to being ideal solutions, so that all the excess chemical potential terms in eqns. (113-114) can be set equal to zero. Equations (113-114) can then be solved to obtain ~.oy and ~o~d~ when the two phases are in equilibrium for suitable values of po:, thus generating the type 2 diagram of Fig. 17a. When this is done, the type 3 diagram of Fig. 17c (or Fig. 17d) is then immediately obtained from the geometrical


construction illustrated in Fig. 17. The complete analogy of these calculations to the calculation of a binary temperature-composition diagram as discussed in Section IV can be seen from a comparison of eqns. (113-114) with eqns. (62-63). The right-hand side of eqn. (113) is the free energy change for the phase change from pure Ni to pure NiO at Po:

~RT" P~2(NimiO) Ni(~.r~) + ½0~l~o~l = N i O ( ~ r e ) AG = ~ m Po2

(115)

whereas the right-hand side of eqn. (62) is the molar free energy change for the phase change of pure component A from a-phase to/3-phase at temperature T.

Finally, the type 1 diagram in Fig. 17b can now be calculated from the relation:

alloy E alloy (IX - IX°)s~ = R T In a~i = R T In ~ + IXN~(£ ) (116)

Ag-Cu-C1 s y s t em at 1000K . In the Ag-Cu-C1 system at 1000 K in Fig. 19, the alloy phase exhibits

Ct Ct

A g C I / / / ~t~ Cue[

(c)

~" Ag I , u P~t (Ag/AgCt) . . . . 1 ' ' , I,

, ~ J r . . (Ag.Co) c% 'I - " I I I

° '° 86 ] ~ i~ct(Cu/CuCt )

-9.4 (

0.2 0.4 0.6 03 0.9 1D Ag Cu

~' : rlcu / ( h A g • ~qCu) OCu

(a ) (b ) FIG. 19. Corresponding type 1, type 2 and type 3 phase

diagrams for the Ag-Cu-CI system at 1000 K.

a region of immiscibility ~9~ extending from ~ = 0.105 at point a in Fig. 19a, to ~: = 0.965 at point b in Fig. 19a, whereas the liquid chloride phase exhibits complete miscibility at all compositions. The type 2 diagram of Fig. 19a could be called a "simple eutecular diagram".

Let us consider the two-phase region in Fig. 19a between the chloride phase and the Ag-rich a alloy. Equations analogous to eqns. (113-114) may be formulated: R T In ~ : " + ~ " chloride Ixc~(~ ) - R T In ~ -- IX CuCl(~E chloride)

= ½RT In p ~(c./c~c~!, (117) P C12

E a chloride RT In ( 1 - ~ ) + IXAS(~ ) - - R T l n ( 1 - ~ ) c~o.de ½R T In p ~I2(Ag/AgCI) - ~ Agcx(~ ) = - . (118)

P el2 Values of p ~:,/c~c~) and p ~Ag/A~C~) are obtained from the literature, re'd3) Thermodynamic measurements ~"~

show that the molten chloride phase is a regular solution with:

E -840(1 ~ chloride)2 /~c~cl = - Jmo1-1, (119) (~oh,o.do)~

]J~ AgCI ~--- -840 J mol-'. (120)

The assumption is made that the a-phase is a "Henrian" solution with IXA~g=0 and IX~= constant. If we further assume that the Cu-rich/3 alloy is also Henrian, then the value of this constant can be determined by equating the activity of Cu at points a and b in Fig. 19a:

Ix~c~ R T ln~U 0.965 = = R T In ~ = 18.4 kJ mol -~.

(121)

By making these substitutions into eqns. (117-118), the boundaries of the two-phase a-chloride region in Fig. 19a were calculated. The boundaries of the /3-chloride region were calculated in a similar fashion. The type 3 diagrams in Figs. 19 c, d were then determined by simple geometric construction as illustrated, and the type 1 diagram of Fig. 19b was calculated by a relationship analogous to eqn. (116).

Fe-Cr-O at 1573 K and Fe-Ni-O at 1273 K . In Fig. 20 are shown diagrams for the system Fe-Cr-O at 1573 K in which three different approximately quasi-binary oxide phases appear. The two tie- triangles in Fig. 20c or 20d become the two "eutecular" constructions in Fig. 20a, as well as the two triple-points in Fig. 20b. The diagrams shown in Figs. 20a and 20c were measured directly "5) by the method of equilibration under a known partial

0 5esqui°xideo

Spinel W(istite

(d)

Ire C r Fe 0 ' Sesauioxide - - C r 0 z ~ • 2 3

. . . . I ~ / / / / ~ / ~ ~ p l n e l ~ WOstite

~o (c)

~- F e V / / / Z / / j / / A / / /, / / II°cr

gezO3 --4 Spinet, ~ _ ~ " ~ Cr)zO

Sesquioxide i ~ Sesquioxide

. ":--L~°'cr~" I / ~Wdst ite ~ J 1 -10

F e / F e O ~ S m e,~_2.Q~.~_~_~. ~. _ Aistite 7 - - - - - ~ / . . . . . . ~

-12 Fe*Spinet ]

-14 ~'-Attoy ÷ Sesqui0xide ~_ . . . . . . . . .

-16 o~-Attoy ~ - ~'--

0.2 04 0.6 O.g Or 200 -100 0 o('* k Cr/Cr203

=nCr /(nFe+ nCr) (P-P°)Cr (kJ mot "1) (a) (b)

FIG. 20. Corresponding type 1, type 2 and type 3 phase diagrams for the Fe-Cr-O system ~'~) at 1573 K.

Phase diagrams 139

pressure of oxygen fixed via a gaseous mixture followed by quenching and analysis. (Only the al loy-sesquioxide boundary in Fig. 20a was not measured directly, but was calculated from these experimental results by methods to be discussed shortly.)

Similar experimentally determined ~46~ type 2 and -3 diagrams for the F e - N i - O system at 1273 K are shown in Fig. 21. The experimental technique involved the use of a solid calcia-stabilized zirconia electrolyte to measure the equilibrium partial pressures of oxygen.

/Spinet+ Wust ite ÷ Art oy W0stite! / '\ / /SpineL÷NiO* Ni

/' I p 1

C ~ F ~ . . . . . . . . . . . . . . . . . . .

-10~- ' I 'SpineL* ' / " J ~ N i / N i O - (Fe.Ni) 0

CL°-11~ / - Al[0y

° t / / o -12 Spinel * Aktoy

Fe/Fe 0 ~ Fe 0.2 0.4 0.6 0.8 N i

{ ='rlN i /(riFe *rlNi) FIG. 21. Corresponding type 2 and type 3 phase diagrams

for the Fe-Ni-O system ~"~ at 1273 K.

The four preceding examples show the advan- tages of the type 2 log px=-~ representation in A - B - X (metal-metal-non-metal) systems. As long as all phases are truly quasi-binary phases, these diagrams contain all the information to be found on the corresponding type 3 diagrams. For example, the only information on Fig. 20c not also contained in Fig. 20a is the extent of the slight solubility of oxygen in the wiistite and spinel phases. Further- more, the type 2 diagrams tend to be easier to interpret since their topology is the same as that of the very familiar type 2 binary temperature- composition diagrams. Such log px~- ~ plots have appeared infrequently in the literature, ~-5" and the thermodynamics of such diagrams has only recently been formulated/" This type of representation should prove to be of aid in correlating complex experimental phase equilibria data and in examining these data for thermodynamic inconsis- tencies. Some of the areas of engineering in which these diagrams are useful are high temperature corrosion of alloys, preparation of ceramic materials, internal oxidation, reduction of ores, preparation of semiconductor devices, etc.

We shall now examine in detail the F e - M n - O system in order to show how one can correlate incomplete experimental determinations of phase boundaries with free energy data and with thermodynamic approximations in order to calculate a complete type 2 and type 3 phase diagram for ternary systems containing only quasi-binary

phases. The complete analogy to the calculation of temperature-composit ion diagrams in binary systems as discussed in Section 4 will also be illustrated.

F e - M n - O system at 1823 K. The type 2 diagram for the F e - M n - O system as at 1823 K as calculated by Sticher and Schmalzried t~2~ is shown in Fig. 22a.

(b) , i , J,

0 / J ' ~ l F e , M n ) 2 0 3 ~

FSe pinet ~ /

~ L Fe'M n) O ( s ' ° " e

_

Uo (a)

- 0'.4 016 018 Fe Mn

FIG. 22. Corresponding type 2 and type 3 phase diagrams for the Fe-Mn-O system at 1823 K.

At this temperature, pure " F e e " is liquid, but pure MnO is solid. The melting temperature Tm and entropies of melting at Tm are: (43~ 1650K and 18.99 J K -1 mol- ' for pure " F e e " and 2058 K and 26 .43JK- 'mo l - ' for pure MnO. Therefore, at 1823 K the free energies of melting are : t

1823 K

F e e s - , FeO~

AG~s2~ ~ 0.01899(1650 - 1823) = - 3.29 kJ, (122)

1823 K

M n O s - - - - ~ MnO,

AG?~> ~-0.02643(2058- 1823)= + 6.21 kJ. (123)

If it is assumed that both the solid manganow(istite (Mws) and liquid manganow0stite (Mwl) phases are ideal solutions, then the values of ~Mw~ and ~Mw, along the boundaries a - b and c - d in Fig. 22a are given by (cf. eqns. (40-41)):

R T In ~M~_ R T In ~Mw, = -6.21 kJ c o l -~, (124)

R T In (1 - ~:M,,) _ R T In (1 - ~Mw,) = +3.29 kJ moI-', (125)

?Although "Fee" does not melt congruently, the free energy of "melting" can be defined unambiguously by an extrapolation of the appropriate phase boundary on Fig. 15b.

140 A.D. PELTON and

whence: ~Uwl= 0.28 and ~:uw, = 0.42. These values are very close to the directly observed <53> phase boundaries between solid and liquid (Fe, Mn)O solutions at 1823 K, thus showing that the assumption of ideal solution behavior was justified. The phase boundaries a - b and c - d are assumed to be vertical in Fig. 22a since both phases have the same stoichiometry with respect to oxygen (both are monoxides), and it is reasonable to assume that the stoichiometries of FeO and MnO in these phases are little affected by po~.f

To calculate the boundaries g - f and c - f of the two-phase spinel-solid manganowtistite region, we formulate the oxidation reactions for the pure iron and manganese oxides (cf. eqns. (107-108)):

MnOs+102 ) {Mn304 K ,/3 • .. ,t6 = aMn)o4]aMno X PO2 = (p82(M.Os/Mn304)) -'16, (126)

F e O s +102 ) ~Fe304 K = aF,,O,t'/3 /aFeOXpO2-- ,/6 i ( O ~-1/6 - ~ P o : o o ~ / F o ~ O , ) • ( 1 2 7 )

The required value of P82(M,Os/M,30,) at point f in Fig. 22a is obtained from the literature. "3) At point e in Fig. 22 is plotted the literature <43) value of log p?~:~o:,,o,~ = - 6 . 6 2 for the equilibrium between Fe304 and liquid FeO. This value could also be taken from Fig. 15b. To obtain the value of

o P o~(v~o~/v~,o,> required in eqn. (127) for the metastable equilibrium between Fe304 and solid FeO at 1823 K, we utilize the free energy of "melting" of FeO from eqn. (122) to obtain, after some rearrangement, the relation:

~RT In (p3:FeOs/Fe304/p ~)2 FeOljFe304)) = -- 3.29 kJ mol-' , (128)

whence: log p 02(FeOslFe30,)° = -6 .82. Geometrically on Fig. 22a, this is the value of log pS~ at which the lines g - f and c - f would intersect the axis at ~ = 0 to form a lens. The situation here is analogous to the case in binary systems in which, as discussed in Section 4.3, it is often necessary to calculate standard free energies of unstable or metastable phases ("lattice stabilities") of the pure components.

Taking logarithms of eqns. (126-127), we obtain the following equations which are applicable for equilibrium between the spinel and solid manganowiistite phases:

.Mw,+ ~ ,.Mw,, i R r t E ~p R T In ¢ /XM,Ot(; )--g In (~sp)3 --3jl~Mn304(~ )

~RT In P 8~CM.O~Mo~O,~ (129) PoE

R T In (1 - ~Mw~)+ /Z F~eo(~M~,) --~RT In (1 - ~,p)3

_1 E sp p ~2(FeO$/Fe304) --~/XFo,O,(~ ) = ~RT In (130) Po~

We note that the spinel phase is an ionic solution, and so the "ideal" activities of Mn~O4 and Fe30, have been defined as (~")~ and ( 1 - ~ ) ~ respectively. Assuming now that both the solid manganowiistite and spinel phases are ideal solutions, we set all excess chemical potentials in eqns.

*The oxygen content of "FexO" can vary by about 4 atom% depending upon Po2 at 1823 K as can be seen from Fig. 15a. This is negligible in the present calculations.

W. T. THOMPSON

(12%130) equal to zero, and solve to calculate the phase boundaries g - f and c - f in Fig. 22a. All other phase boundaries in Fig. 22a were calculated in a similar manner, each necessitating the calculation of a metastable PS~ value. In these calculations, the liquid alloy phase was taken to be a regular solution. <9)

Since all phases are quasi-binary phases, the corresponding type 3 diagram, Fig. 22b, is obtained from Fig. 22a by simple geometrical construction.

F e - M n - O system at 1173K. A partially experimental, partially calculated type 2 diagram is shown by the heavy lines in Fig. 23a. The P82 values

Hemat i te B i x b y i t e 1.5 ~__..._ ~ - -~ - - - - - -~ .~ - - - . . , .~ H . . . . . . . ire

' / / / Mant: jano~'6st i te / - / ~ " A ~

11

II ' ' 11 8i'xbyite' "

(Fe,Mn)203 . H e m a t i t e - (Fe,Mn)zO 3 ell f ~

4ausmannite

~" m ~ o w u e t l t e o °~ -Infi.~ - (Fe,Mn) 0

ge 0.2 04 0.6 0.8 Mn

~'= nMn/ (nFe, nMn)

FIG. 23. Corresponding type 2 and type 3 phase diagrams for the Fe-Mn-O system at 1173 K. Heavy lines are calculated diagram, and light lines are observed diagram.

(points a, i, p, h, m, n) are obtained from the literature. °'4~'43'54) The phase boundaries between the rhombohedral hematite and orthorhombic bixbyite phases have been determined experimentally t49'55) and lie at ~hcm= 0.10 and ~bix= 0.47. These phase boundaries are assumed to be independent of Po2, and so are represented by vertical lines in Fig. 23a. Using these data, and assuming that both the hematite and bixbyite phases are ideal ionic solutions, we can calculate the molar free energy change for the transformation of pure Mn203~. to hypothetical unstable Mn203h,. having the hematite structure:

Mn203h,~ • Mn203,,~ AG ° = - R T In (0"10~ 2 \0.47]

= 30.20 kJ. (131)

Similarly for Fe203:

FezO3h,~ ) Fe203~ {I - 0.47~ 2 AG ° = - R T In k ~ /

= 10.30 kJ. (132)

Once again, this is a "lattice stability" calculation of the sort discussed previously in Section 4.3.

Phase

The phase boundary c - j on the spinel side of the two-phase region between the (cubic) spinel and (tetragonal) hausmannite phases has been determined experimentally ~49~56~ and lies at ~sp = 0.70. As before, we assume that this boundary is vertical in Fig. 23a. Now, pure Mn304 undergoes a phase change at 1445 K from tetragonal hausmannite to a cubic structure with an entropy of transformation ~43~ of 14.39 J K- ' mol ', and so the free energy of the metastable phase change at 1173 K is given by:

1173K Mn304 ...... ) M n 3 0 % AG~I73 K

~ 0.01439 (1445 - 1173) = 3.914 kJ. (133)

The composition ~hau~ of the hausmannite side of the spinel-hausmannite region can now be calculated, under the assumption that both phases are ideal ionic solutions, from the relation (cf. eqn. (40)):

R T In (~P)~ - R T In (~h~s)3 = --3.914 = R T In (0.70/~h"~) ~ (134)

whence: ~h,os= 0.81. This value is in agreement with the measurements of Ono et al. "9~ but not those of Van Hook and KeithJ 56~ These two different measurements were performed at different oxygen partial pressures, and it is possible that the boundary d - k is not vertical in Fig. 23a. In the present calculation of the diagram, however, it was assumed that line d - k is indeed vertical at ~h,.~= 0.81. Finally, for the metastable phase change of pure Fe~O4, we calculate that:

(1-0.81 V redO4,, ~ Fe304 .... AG ° = - R T In \1 - 0.79]

= 13.36 kJ. (135)

This value cannot be independently verified, since Fe304 does not transform to a tetragonal structure at any temperature.

The various other two-phase regions in Fig. 23a were all calculated by assuming ideal behavior for every phase. To perform these calculations, values of p3: for metastable phase changes are required. For example, to calculate the hematite-spinel boundaries h - e and h - g , we need to know P~,~Mo,m,~/Mo:O,~o,,,, This quantity can be determined by combining the experimental value of p 3,:M°,O,,,.~:M°~O,~I~ at point a in Fig. 23a with the free energy changes for the metastable transformations in eqns. (131) and (133).

Finally, under the assumption that all phases are quasi-binary phases, the type 3 diagram in Fig. 23b was determined by projection from Fig. 23a.

The light lines in Fig. 23a show the type 2 diagram as estimated by Sticher and Schmalzried ~5:) mainly on the basis of the experimental measurements of Ono et al. ~49~ The aforementioned discre- pancy between values of the position of the line d - k as measured by different authors was re- solved ~52~ by assuming that this line is, in fact, not vertical. Considering the experimental difficulties involved, we can see that the calculated diagram lies nearly within the error limits of the experimental.

This agreement is illustrative of the general ob- servation that approximate calculations of phase diagrams based upon simple ideal or regular solution models have a greater chance of success in

diagrams 141

ionic systems than in metallic systems, since it may reasonably be assumed that electronic contribu- tions to the excess entropy are much smaller and the approximation of additive pair-bond energies is much better in the former case. Thus, a general computerized scheme for calculating a large number of phase diagrams based upon simple regular solution behavior might be more successful when applied to the calculation of isothermal log px~-~ diagrams in A - B - X systems than when applied to calculating temperature-composit ion diagrams in alloy systems as discussed in Section 4.3.

Fe -M n-O sys tem at 1573 K. The diagram in Fig. 24 was calculated by assuming ideal behavior of the

(ge,Mn)2 0 3

(Fe,Mn)30 ~

-6

-8 (ge, Mn) 0

-10

#~ -12 - ~ -14

-16

-18 At~ t0y Attoy(S)

- 2 0 o12 o'.~ 0'.6 0.8 M~ Fe

~= nMn/(nFe" r~Mn )

FiG. 24. Type 2 phase diagram for the Fe-Mn-O system at 1573 K.

alloy, the (Fe, Mn)O and the (Fe, Mn)304 phases. The (Fe, Mn)203 phase was assumed to be "sub- regular". The two parameters of eqn. (73) were obtained from the known values ~7~ of the consolute temperature and consolute composition of the solid-solid miscibility-gap in the Fe,.O3-Mn203 quasi-binary system, using eqns. (74--75).t The resultant excess chemical potentials in the (Fe, Mn)203 phase are positive in Fe203-rich solutions and negative in Mn2Orrich solutions, thus giving rise to both a maximum and a minimum in

tThe ideal activity of Fe20~ or Mn203 in the ionic phase was defined as the square of the appropriate mole fraction, thus resulting in a factor 2 on the left of eqns. (74-75) and in the first term on the right of eqn. (74).


the two-phase (Fe, Mn)20~-(Fe, Mn)~O4 region in Fig. 24.

Fe-Si-O system at 1473K. Diagrams for the Fe-Si-O system at 1473 K, based <~) partly upon thermodynamic calculations and partly upon direct experimental measurements, are shown in Fig. 25.

Ag-Sb-S at 673 K. The diagrams shown in Fig. 26 are based upon direct experimental measurements <~s°) of phase equilibria. In this system, the activity of Ag can readily be measured by an electrochemical technique, and so most of the type 1 diagram of Fig. 26b has been obtained by direct measurement.

2.0

1.8

1.6

Magnetite 1.4 Fe~O 4

Fe304+Fe O* Li -

1.0

,-G-_

0.8

0.6

=o 0.4

F 0.2

Fe 4 it ~ K ', a[Fe,S~ AU%L I \, ~,[F~.S 0 FeSi

Tridy mite SiO 2

Fe30 ( + FeiO) + Si 0 l Fe~04+ Li q .S iO 2

Hemat i te /FelOn

", si "FeSi 2

Fe2% + S iO 2 -2

Fe30 ~ + S iO 2 -E

-~0 ~:o U d , ~ \ s,o~

\ (Fe* Fayalit e)

-14 2 [Fe.S i ] ÷ S i02

o .

L~ uoy(S) (b)

+ AIIoytL) A i, i / IIo y(L) ~.Fe Si ¢,, Fe Si z

' / ' ° ,

~ [ , < . , s , ] - ' 1 . . / / , : . ~ _ f l t~l . Fe 0.2 0.4 0.6 0.8 Si

~ = h S i / ( ~ F e ÷ I ' lS i )

FIG. 25. Corresponding type 2 and type 3 phase diagrams ~) for the Fe-Si-O system at 1473 K.

The compound fayalite could be called a "congruently oxidizing compound", while the inter- metallic compounds FeSi and FeS6 are "incongruently oxidizing compounds" by analogy to congruently and incongruently melting compounds. That is, fayalite oxidizes to form a higher oxide of the same Fe/Cr ratio, whereas FeSiz oxidizes to form FeSi + SiO2. In a complicated system such as this, it is apparent how the type 2 presentation of Fig. 25a aids in the interpretation and visualization of the phase relationships.

Application of generalized Clapeyron equation. The generalized Clapeyron equation (14) can be used for the thermodynamic analysis of experimentally-determined type 2 log Px2- ~: diagrams. (" Consider the boundaries a - d and a-e in the Fe-Cr-O system in Fig. 20a which have been measured by Katsura and Muan/45) Equation (14), applied along these boundaries, with ~b, =/~o, 4)2 = p~Fe, and ~b3 = /ZCr, becomes:

dp, o/dtXFe d log po2 (nFdncF) '°~ -- (nFe/ncr) sp 2d log aFe (no~no,) ~°' -- ( n o / n c r f p

(136) where "ses" and "sp" refer to sesquioxide and spinel phases. Now, in the sesquioxide phase:

d log aFo~o~ = 2d log aF, + -~d log Po~. (137)

Substitution into eqn. (136) yields:

{3 (no/ncr) ~°~- ( n o / n c r f P ~ . d log aFo:o~ = "n " "~°~ ~. Fe /nc r ) ~ p j d log Po~.

(138)

The ratios of mole numbers in eqn. (138) are obtained by taking experimental points from the boundaries a - d and a -e in Fig. 20a. In these calculations, deviations from quasi-binary behavior, if known, can be taken into account ex- plicitly. Equation (138) may now be graphically integrated along the two-phase boundary, with point a, where a Fo:o~ = l, being used as end-point in the integration, in order to calculate a Fo~O2 along the line a-e.

Further discussion of this type of calculation may be found in refs. 1 and 46. It is through calculations of this sort that the type 1 diagram shown in Fig. 20b was determined.

Predominance area diagrams. The type 1 isothermal ~A--p~B diagrams of the sort just discussed above are particularly useful in systems of one metal and two non-metals in which the chemical potentials of the two non-metals can readily be fixed or measured via the gaseous phase. In Fig. 27 is shown a plot of log Ps2 versus log po2 at constant T = 800 K and P = 1 atm in the Fe-S-O system. Here we have defined ~b~ =/xs, ~: =/xo, 4h = P~F0, ~b4 = T, and ~b5 = - P . (At constant temperature, log Ps2 and log po~ vary directly as /xs and/Zo respectively.) We note that all phases in Fig. 27 are quasi-unary phases. Such diagrams have been called "predominance area diagrams". The construction of these diagrams and their application in the field of extractive metallurgy have been extensively discussed. <6~-u~ Garrels and Christ (63> have discussed and presented a large number of such diagrams involving carbonates, sulphates, etc., of interest in the fields of mineralogy and geology. Complete log ac-log Po2 diagrams for the Ca-C-O system have recently been presentedJ 65)

Phase diagrams 143

1 S'

< (c)

b , / : l i v / , i , g . . . . •

/ % ,o.,°,. -'11 "°""' I ,:l r / ":° '11 I Ag,S ÷ A_.q

9 g,SbS~ 1 ._~

I 91' ~ [~ ~ I 1 [ N i ' ~ t ' " ' b . ] I I l+ , I , 4 ~ _ j . J t 4 t+ t , 1 : 1 . . . . ~ . ~ ) 0 I 20 r ~ 0 I 6 O ~ --1OO Ag 01 02 0"3 0"4 05' Sb

('U--'U*)Ag(k J mot -1) ~': nSb/ (r~Ag+r~Sb) ( b ) ( a )

FIG. 26. Corresponding type 1, type 2 and type 3 phase diagramd ~-~°) for the Ag-Sb-S system at 673 K.

5

0

-5

-10

-15

-20

Fe -25

- 30

-3b

FeS 2

FeS /

a

Fe 30~,

tog po 2

0~

Fe 2 (S0~,)3

Fe203

-115 -10 -5

FIG. 27. Type 1, predominance area phase diagram for the Fe-S-O system at 800 K.

Figure 27 was calculated completely from thermodynamic data. (43'66) For instance, along the line a-b , the following equilibrium is attained:

2FeS+~O. , . " ]Fe304+S~.AG~0o=-353kJ. (139)

Therefore, line a - b is given by the equation:

log ps: - ]log po~ = 353[2.303RT. (140)

Alternatively, once point a had been calculated from considerations of the Fe/FeS and Fe/Fe304 equilibria, line a - b could immediately have been drawn without further calculation, since it is evident from eqns. (139) and (140) that the slope of this line is ~.

The slope of line a - b for the phase change FeS ~ Fe304 can also be calculated through the use of the generalized Clapeyron equation (14), as

follows:

d log Ps2 _ (no/nre) F°~°'- (no/nF,) ~es dl.~s/dlzo = d log po~- - (ns/n~o) ~°~°'-t s/n~o) i n / xFeS

_ ~ - 0 4 (141) 0 - 1 3"

The formulation using the generalized Clapeyron equation is somewhat trivial in this case, but becomes more useful when the phases are not stoichiometric so that the molar ratios in eqn. (141) are no longer ratios of simple whole numbers and it is no longer readily apparent what the slope is.

By redefining the potentials a s (~1 = /d, SO2, (l)2 = i~J,O,

63 =/~F0, etc., as was discussed in Section 2.1, we arrive at the log pso : log po2 diagram in Fig. 28 which may be more useful in certain applications. Figures 27 and 28 are topologically identical. Thus, for example, once Fig. 27 has been calculated, Fig. 28 can be constructed simply by calculating log pso:

5 0 F e /

Fe30 ~

-30 -25 -2~0 tog po z

10

-5 o" m. o ~ -10

-15

- 20

FI2(S O 4)J

Fe203

-I'5 -I'o -5

FIG. 28. Type 1, predominance area phase diagram for the Fe-S-O system at 800 K equivalent to Fig. 27.


at each triple-point, and then joining up these triple-points in the same way as in Fig. 27.

5.2. Isobaric iso-tx diagrams

Type ! diagrams. For the F e - S - O system shown in Fig. 29, let us set ~b~ = Ixo, 4,2 = T, ~3 = IXFo, 64 = - P , ~b5 = Ixso,. Figure 29 is a diagram of ~bl versus ~b2 with ~b4 and Pso: held constant, where we note that In pso, = (ixso,-IX~o=)/RT and where Ix~o, is a function of T. Thus, rather than simply keeping 4,4 and th5 fixed, we are fixing ~b4 and some function of &~ and 4~2. Nevertheless, Fig. 29 is still a type 1 diagram as can be appreciated from the arguments presented in Section II. Figure 29 was calculated entirely from experimental free energy data. ~43'~)

PS02? 1,0 atm

- 5 0

4 I / I e,o,

= -1001-// /

; 15o 700 900 1100

Temperature (K)

FIG. 29. Type 1, phase diagram for the Fe-S-O system. The partial pressure of SO2 is held constant at I atm.

Figure 29 can be called (~ a free energy- temperature (Ellingham) diagram. For instance, if we consider the reaction:

FeSO, + ½02 + ½SO2 . , ½Fe2(SO33 (142)

then the line a-b is a plot of AG ° of this reaction versus T.

Figures 28 and 29 may be considered to be isothermal and iso-log pso~ sections respectively of a three-dimensional RT log pso~-RT log po,-T diagram. Such a diagram or a topologically equivalent log pso:-log po:-l/T diagram is called a "predominance volume diagram". The uses of such representations have been discussed] 62)

Type 2 diagrams. In Fig. 30 is shown a type 2 diagram (67~ for the Fe -Cr -O system in which T is plotted versus ~ = no[(n~o +ncr) at constant po, = 0.21 atm. Often, in the literature, one sees such a diagram referred to as a " temperature-composi t ion diagram for the quasi-binary system Fe:O3-Cr203 in air" with the abscissa being called the mole fraction of CrzO3. This terminology is incorrect.

Figures 20 and 30 are isothermal and iso-log po, sections respectively of a three-dimensional T-log Po~-~: diagram for the F e - C r - O system.

250,

230,

2100

1900 E

1700

i i i i

Liquid . /1L iqu id / ÷

/ /

, " , , '11 , ' / I , I / / / X 7" Spine[ / /Sesq+uioxide

/ / Sp;net

Sesquioxide

I

Fe 0!2 0,~ 0',6 018 or ~ :ncr / (riFe + nCr)

FIG. 30. Type 2, temperature-~ diagram ~67~ for the Fe-Cr-O system. The partial pressure of O= is held

constant at 0.21 atm.

There are, of course, many other type 1, -2, and -3 diagrams, involving P, v, s, etc., as coordinates, which we could draw for ternary systems containing quasi-binary phases. Only a few of the most important examples have been presented here.

VI. Ternary systems

In this section we shall discuss general ternary systems such as alloy systems in which most phases are not quasi-binary phases. We shall limit our discussion to the familiar isobaric isothermal (type 3) composition diagrams in the Gibbs ' triangle representation. Although the corresponding type 1 and type 2 diagrams of the sort just discussed in Section 5.1 (i.e. log px,-~ and "predominance area" diagrams) can, of course, be drawn for a general ternary system, these representations are most useful in systems in which all phases are approximately quasi-binary phases, and so we shall not discuss these types of diagrams further in this section.

6.1. Analytical representation of thermodynamic properties

Two variables are required to define the composition of a ternary phase. In order to calculate ternary phase diagrams, we must be able to express the partial excess properties of each phase as analytical functions of these two variables. It is desirable to have a general series representation for ternary systems analogous to eqns. (55, 58, 59) for binary systems.

In this section we shall illustrate the use of such a general series expansion by expressing the experimental thermodynamic properties of the liquid Cd-Bi -Sn phase in this way, and then using these expressions to calculate the liquidus surface of the phase diagram.

The choice of composition variables is somewhat arbitrary. In the following calculations we shall use

Phase diagrams

two variables, called y and t, which are particularly convenient in this particular case. Afterwards, we shall discuss the use of other composition variables. The following development has been treated in more detail in refs. 68 and 69.

The components Cd, Bi and Sn are lettered A, B and C respectively as in Fig. 31. It is most convenient to let component "A" be the component for

A(Od) y=0

",?

B(Bi) C(Sn)

Fro. 31. Graphical interpretation of the composition variables y and t on the Gibbs' Triangle.

which experimental partial properties have been measured. The composit ion variables y and t are defined as:

y = (1 - XA), (143)

t = Xc / (XB + Xc ) (144)

where X is mole fraction. The geometrical interpretation of the variables on the Gibbs' triangle is illustrated in Fig. 31.

The excess chemical potential and enthalpy of mixing of Cd,/xce~ and Ahcd, in the liquid phase have been measured by means of an emf technique (7°~ at twenty-five compositions covering the entire ternary system. In addition, values at ten compositions in each of the Cd-Bi and Cd-Sn systems have also been tabulated9 ~ The forty-five isothermal experimental values of p.c~d~773~ at 773 K and of Ahca (assumed to be independent of temperature) were each fitted, using a least-squares technique, to a general power series expansion of the form:

i' k'

o9~ = ~ , ~'~ a~yit k (145) ] - 2 k - 0

where w~ =/xga~77~ or Ahca. It was found by trial and error that the use of fourteen term expansions in each case gave good fits to the experimental points. In the case of jb~Cd1773)E, the root-mean-square deviation was 32Jmol -~ and for Ahca it was 150 J mol '. The expansions are reproduced in eqns. (146-147).

E ~tt c~77~ (J mol 1) = (8652.74 - 1173.08t - 9693.28t ~ + 9028.04t ~- 1537.95t~)y 2 + (-49352.11 + 53060.59t + 1495.64t' - 8594.50ta)y 3 + (68671.25 - 80937.64t + 14449.14t2)y ~ + ( - 27987.29 + 27350.61 t)y ~. (146)

145

Ahcd (J tool -1) -- (3%28.29 - 50634.77t + 73784.25 t 2 - 10126.63t 3 - 31828.63t4)y 2 + ( - 137592.66 + 169980.18t - 168219.71 t ~ + 88330.25t3)y 3 + (168444.78- 153646.35t + 43070.35t2)y 4 + (-67246.85 +42350.87t)y 5 (147)

It may be noted that along a line of constant t (see Fig. 31) the expressions in eqns. (146-147) reduce to power series expansions in y = (1 -XA) of the form of the "Margules" expansion in eqn. (55) discussed previously for binary systems.

In order now to obtain analytical expressions for E the excess properties ~o~ and ~oc of Bi and Sn, we

apply the Gibbs-Duhem integration to eqn. (145). Just as in the case of the binary Gibbs-Duhem integration (cf. eqn. (58)), we obtain "constants" of integration, but now, since we are dealing with a partial differential equation in ternary systems, these "constants" will be functions of one composition variable. It can be shown that this is the variable t as follows.

At constant temperature and pressure the Gibbs-Duhem equation:

XAdto ~ + XBdto ~ + Xcdto~ = 0 (148)

may be rewritten in terms of t and y as:

( l ~ ) d w ~ + ( l - t ) & o ~ , + t d w ~ = O . (149)

Let us now express w~ and tog as power series expansions in the form:

to~ = ~_, bjkyJt k + CB(t), (150) j = 1 k =11

] ' k '

w~, -- ~ ~ , qky~t k + Cc(t) (151) i - I k 0

where CB(t) and Cc(t) are functions of t related by the equation:

(1 - t) dCB( t )+ tdCc( t ) = 0. (152)

By substitution of eqns. (150-152) into eqn. (149), it may be shown (u~ that the following relationships between the coefficients ajk, bik and q~ exist:

j + l - k b~k = ai, - - x a~i~llk, (153)

J k + !

cjk = bjk - j x alj+ll~k+t,. (154)

However, when this substitution is made, all terms containing CB(t) and Cc(t) drop out, and so these functions cannot be determined simply from a knowledge of the coefficients aik of the expansion for to~, but rather, they must be obtained from

E independent measurements ~ of rob or ~oEc. In the present case of the Cd-Bi-Sn system, experimental values ~9'7~ of excess chemical potentials and partial enthalpies of Bi and of Sn in the B-C (Bi-Sn) binary system are available. For instance, in the B-C system, Ah a~B_c~ may be expressed as a power series of the form of eqn. (55) in the mole fraction of tin, Xc, as:

hh BI(B C)= 701.24Xc - 818.81X 3 + 506.26Xc' (J mol -I) (155)

However, in the B-C binary system: y = 1 and


t = Xc. Substitution into eqn. (150) then yields the following expression for CB(t) in the expansion for Ahm:

j' k'

Ca(t) = 701.24t 2 - 818.81t3+ 506.26t 4 - ~'~ ~ bid k. j=l k -0

(156)

The coefficients bsk are determined via eqn. (153) from the known coefficients ajk of eqn. (147), and the resultant expansion for aha~ in the ternary system, obtained by the combination of eqns. (147), (150) and (156) is:

Aha~ (J tool -1) = 701.24t 2 - 818-81t 3 + 506.26t4 + ( - 79256.58 + 50634.77t + 0 - 10126.63t 3 - 63657.26t')(y - 1) + (246017.28 - 220614.95t + 157894.11t 2 - 10126.63t 3 - 31828.63t4)(y 2 - 1) + ( - 362185.70 + 323636.52t - 196933.28t2 + 88330.25t3)(y 3 - 1) + (252503.33 - 195997.22t + 43070.35t2)(y 4 - 1) + ( - 67246.85 + 42350.87t)(y 5 - 1).

(157)

Similar expansions were derived for E I1~ Bi(773)~ ~l/- Sn(773)

and Ahs. in the ternary system. It is not necessary, in general, to use data from the

B-C binary system to obtain values of the functions Cdt) and Cc(t), but known values of to~ or ~oc ~ along phase boundaries, etc., can also be easily used for this purpose J 6s'69)

In complete analogy with the previously discussed case of binary systems (cf. eqns. (62-63)), the calculation of the composition coordinates of the two ends of a tie-line in a two-phase a-/3 region are obtained by simultaneous solution of the three equations:

RTlnX7 + I ~ - R T l n X ~ - ~ =( /x°B- / z °~) ( i = A , B , C ) . (158)

It is known (9~ that there is negligible solid solubility of Bi or Sn in Cd-rich solid solutions. Therefore, the calculation of the ternary liquidus surface when Cd is the primary crystallization product reduces to the solution of the one equation:

RT In X~zd + p. CEd = -- (/~ ~ - p.~s) (159)

where p.cEd is given as a function of y, t and T by combining eqns. (146-147):

A hcd -- P, gd(773) il,~Ed ~-~ Ahcd-- T ( 7-73 ) (160)

and where the standard free energy of melting, ( ~ o~ --/-~Cd), can be expressed as a function of temperature as in eqn. (35) by using literature ~'3~ values of entropy and temperature of melting and heat capacities. Equation (159) is thus an equation in three variables: y, t and T. For chosen values of t and T, we can solve eqn. (159) numerically to obtain y, thus generating the liquidus surface.

Since there is negligible solid solubility of Cd or Sn in Bi, ~9~ the liquidus surface when Bi is the primary crystallization product can be calculated in an entirely analogous manner through use of the calculated expansions for ~Bi(773) and AhB~ as in eqn. (157).

In order to calculate the liquidus surface when Sn is the primary crystallization product, account must be taken of the solubilities of Cd and Bi in the solid Sn a and/3 phases. Since no thermodynamic data are available for this solid phase, an approximation procedure was used based upon the known tg~ solubilities in the Sn-Cd and Sn-Bi binary systems and upon the assumption that aS, = xsS.. Details have been discussed elsewhere tn~ for a similar case. The correction for solid solubility is small, and even if solid solubility is completely ignored, the calculated liquidus surface is changed by no more than 4 or 5 mole% at any temperature.

The calculated phase diagram (projection of the liquidi of several isothermal type 3 diagrams) is shown in Fig. 32 along with the diagram determined by direct observationJ TM

Cd

Bi Sn

FIG. 32. Liquidus surface of the Cd-Bi-Sn system. Solid lines calculated analytically from thermodynamic data. Dashed lines indicate the experimentally observed diag-

ram. m~ (Temperature in K.)

Other series representations. In certain cases, we may wish to use composition variables other than y and t. For example, in phases rich in component A, the mole fractions of the solutes, XB and Xc, are convenient variables. Let us assume that the excess property of component A, (o~ has been expressed as a double power series in these variables, viz.

j' k'

.'~A : Y , X ajkX~X~. (161) (j+k)~2

By use of the Gibbs-Duhem equation, it may be shown (69) that the corresponding expansions for to

E and ~Oc are:

r k' j + l

o+k)~l (162) r v k + l

tog = E E (aik - - ~ " ~ >( aj(k.l ,)X~X~ + C c ( t ) . (j+k)~'l

(163)

The integration functions CB(t) and Cc(t) will once again be functions of the variable t defined in eqn. (114), and eqn. (152) will apply. As can be appreciated from eqn. (149), this will be the case no matter what composition variables we use in the series expansion of WA ~.

Phase

In certain cases, orthogonal Fourier series expansions may be preferable to power series, particularly when the phase in question exists only over a narrow composition range. Such series have been used °3"7"~ to express the properties of the liquid sulfide phase of the Fe-Cu-S system at 1473 K.

The use of various different composition variables and of different functional forms of the series expansions for ternary systems are discussed in detail in ref. 69.

Dilute solutions--Interaction parameters. For a dilute solution of several solutes (components B, C . . . ) in a solvent (component A), Wagner ~ has proposed that the excess chemical potentials of the solutes be e~pressed as Taylor expansions about X~ = I. For component i:

B C D I ~ / R T = l n y~ = l n 3,°+ e~Xa+ •~Xc+ • ~X~+" • - B 2 C 2 D 2

+ p i X ~ + p i X c + p i X o + " "

+ E = ~ , c ~ = , . c (164)

(j,~k)

+ (higher order terms) (i = B , C , D . . . . )

where

el = (o In 3",/oxj)~.~:×,,~,. ( 1 6 5 )

2 p{ = (02 In 3',/OXj)rp x,-,. (166) ik p~ = (02 In 3",]OX~OXDr.z:x~, (167)

where 3'~ is the limiting activity coefficient at XA = 1. The notation of Lupis and Elliotf TM has been used apart from the use of letters rather than numbers for components. The "first-order interaction parameters" e~ can be measured as the limiting slopes of plots of In 3'~ versus Xj at XA = 1. The "self-interaction parameters" e~ can be determined from data in binary systems, while the "cross- interaction parameters" ¢~ require ternary solution data. The "second-order parameters" p~ and pl k are obtained in a similar manner as limiting slopes at XA = I of appropriate plots. Available data is rarely of sufficient accuracy to permit third-order terms to be determined.

It may be shown ~76~ from the Gibbs-Duhem equation that there exist certain thermodynamically essential relations among the parameters e and p. The most important of these is:

e~ = el. (168)

Interaction parameter representations similar to eqn. (164) for partial enthalpies and excess entropies have also been discussedJ rG~

One main advantage of the interaction parameter representation in dilute solutions is that the parameters can be interpreted, at least semi- quantitatively, in terms of interatomic interac- tionsJ TM For example, in liquid Fe at 1873K, metal-oxygen interaction parameters tend to be quite large and negative, indicating attractive interactions. For instance, ~77~ A~ eo = - 4 3 3 , and eov'= -118 . Metal-metal interactions tend to be smaller, and may be either positive or negative: ~77~ e~l = 5.6;

~ = -8.4. Interaction parameters have been measured and tabulated for many liquid alloy systems '7~'~' particularly for liquid Fe as solute. As yet, however, there appears to be little information available

diagrams 147

for dilute liquid solutions other than alloys, or for solid solutions. A detailed treatment of the theory and application of interaction parameters has been presented by Lupis and ElliottJ ~'v~ See also refs. 75 and 78.

The iron-rich corner of phase diagrams of ternary Fe-O-M systems, where M is Si, Mn, C, Al, etc., have been calculated from thermodynamic data with the use of first-order interaction parametersff '~ When e~ is very negative, an interesting result occurs. In Fig. 33 is the Fe-rich corner of the observed ~"' Fe-O-A1 phase diagram at 1873K.

10-3

o ~ Liquid. Metat X10-~ ~ At203 ~

Liquid ~ Metal

0 s 10" 10 -3 10 -~ XAt

FIG. 33. Deoxidation minimum in the Fe-AI-O system at 1873 Kff"

Aluminium "deoxidizer" is commonly added to molten steel to precipitate dissolved oxygen as solid A1203. As the aluminium is added, some dissolves in the steel. The phase boundary in Fig. 33 may be calculated from the known equilibrium constant K = aA~2o,/a~a 3. We may set aAi2O3 = 1, and therefore, if aA~ and ao were to vary directly as XA~ and Xo respectively (Henry's law), Xo would continually decrease as XAJ increased, and the phase boundary shown in Fig. 33 would have a negative slope everywhere. As can be seen, however, this is not the case. The reason for this is that eo A~ is very negative, so that as XA, increases, 3'0 decreases (see eqn. 164), thus increasing the ten- dency of oxygen to enter solution. Eventually, this effect becomes so marked that a further addition of deoxidizer will actually result in an increase in the dissolved oxygen content. Equations relating the position of the "deoxidation minimum" (see Fig. 33) to the equilibrium constant K and first-order interaction parameters have been derivedff 2~ Calcu- lated and observed curves are in good agreement for many systems. '8'~ General considerations regarding the calculation of phase equilibria in dilute solution have been discussed ~3~ and used to calculate the portion of the Fe-C-Mn system relevant to the pro-eutectoid transformation with the aid of first-order interaction parameters. '8~

For a binary system A-B, eqn. (164) reduces to:

ln3 'B=ln3 '~+ B s 2 (169) • sX~ + p BXB.

Application of the Gibbs-Duhem relationship then yields, for the solvent (component A):

B 2 In 3'A = (e~ + 2psS)(Xs + In XA) + psXs. (170)

B 2 When Xa = 1, In XA = -XB, and so In 3'A -- pBX~. It may be noted that the "quadratic formalism" (eqns.


(71-72)) is simply a special case of eqns. (169-170) with ~ = - 2 p ~ .

For ternary systems, we can devise various forma- lisms by truncating the series in eqn. (145) or eqn. (161) in certain ways. For instance, in solutions concentrated in component A, we could postulate a quadratic formalism by using three terms in eqn. (161):

In ~/A = a2oX~ + m~XBXc + ao2X2c. (171)

Then, from eqns. (162-163) we have, for the solutes:

In TB = In 3'~ - 2a20XB - a~Xc + a2oX~ + ao2X~ + a .XBXc, (172)

In 3'c = In 3'8 - a.XB - 2a02Xc + a2oX~ + ao2X~ + m~XBXc. (173)

Comparison of eqns. (172-173) with eqn. (164) then indicates that this "polynomial formalism ''~69'j°2) is identical to a second-order interaction parameter formalism with the restraints that:

2 s E ~ = - 2 p ~ = - pc, (174) ~.c c c = -2p¢ = - 2 0 c, (175)

d - ~ = - p . ~ = 0~ ~ - c - . (176)

Alternatively, we could truncate eqn. (145) as:

In yA = (a20 + a2~t + a~d:)y z. (177)

This formalism is identical to that of eqn. (171). If we further set a22 = 0 in eqn. (177), then we have a quadratic formalism in which In 3'A varies directly with t at constant X~. It can be shown from eqns. (150-154) that this is equivalent to an interaction parameter formalism with the restraints of eqns. (174--176), and with the further condition that:

c = ec B = (~B + Ecc)/2. (178)

It must be remembered that interaction parameters are defined by eqns. (165-167) only at XA = 1. That is, the parameters of eqn. (164) are not obtained by a curve-fitting technique to thermodynamic data over a finite composition range, as in the case of eqns. (145 or 161) for instance, but rather, these parameters are all obtained as limiting slopes at XA = 1 as discussed previously. There- fore, the numerical values of the parameters are independent of the total number of terms used in eqn. (164) as well as of the number of components. If experimental data are available over a finite

Nb

range of composition, then the coefficients of eqn. (164) could be obtained by, say, a least-squares analysis. The coefficients so obtained, however, should not be called "interaction parameters" since, because eqn. (164) is not an orthogonal series, these coefficients will be correlated with one another, and one must exercise great caution in truncating a series obtained in this manner. ~u~

6.2. Methods of calculation and approximation

Recent review articles (1''35'36~ provide a source of references to calculations of specific ternary phase diagrams, mainly for alloy systems.

It is only very rarely that enough thermodynamic solution data are available to permit a complete ternary phase diagram to be calculated without the necessity of some approximation procedure. Most approximations are based upon some truncated form of a series expansion of g E. A "regular" ternary solution may be defined as one in which:

gE = XAXBpAB 4- XBXcPBc 4- XcXApcA (179)

where pAB, pBC, and pCA are the regular solution parameters for the three binary systems. Kauf- mant6.~L3~) has extended his methodology of calculating phase diagrams as discussed in Section 4.3 to ternary systems by assuming regular (or sometimes sub-regular) solution behavior, and by estimating the parameters of eqn. (179) as well as "lattice stabilities", free energies of compound phases, etc., partially on the basis of theoretical considerations and partially by comparison with known binary equilibrium diagrams. One of the more successful calculations"" by this method is that of the Nb-Cr-A1 diagram at 1273 K which is reproduced in Fig. 34.

Rudy and Chang m4~ have discussed the thermodynamic relationships governing phase equilibria in ternary systems containing intermediate phases.

Often, excess solution properties wE for some phase of the A-B, B-C and C-A binary systems have been determined either by direct measurement or by calculationsbased uponknownbi- nary phase diagrams. Several methods have been proposed for using these binary data to estimate the solution properties of the corresponding ternary phase.

In the method proposed by Alcock and Richard- son ~sT) and by Toop, tSs~ the value of o) ~p~ at some point p in the ternary system (see Fig. 31) is given from

N b

Cr AI Cr p AI FIG. 34. Calculated m~ and observed ~8~'s6~ isothermal section of the Nb-Cr-AI system at 1273 K. Reproduced with permission, from "Theoretical Approaches to the D, • crmination of Phase Diagrams" by L. Kaufman and N. Nesor, Ann. Rev. Mat. Sci., Vol. 3. Copyright © 1973 by Annual Reviews Inc. All

rights reserved.

Phase diagrams

values of ~OA ~ at points a and b, and from the value of the integral property to ~ at point c by the equation:

E WA(,~ =- t x o~ ~,,,~ + (1 - t) x o~,~,- y-" x ~o~ (180)

where y and t are the variables defined in eqns. (143-144). If the three binary phases as well as the ternary phase are regular solutions, then eqn. (180) reduces to eqn. (179), and the approximation is exact.

If E ~o A~o, and to ~,~) are expanded as power series in y. and if ¢o~ is represented as a power series in t (where t = Xc in the B-C binary system), then eqn. (180) can be seen to be simply a truncated form of the general expansion of eqn. (145). Accordingly, eqns. (150-154) can be used directly to generate the corresponding series for ~o~ and ~oc.~ The phase diagram for the Cd-Bi -Sn system calculated solely from binary data ~9'7n with the use of eqn. (180) for / ~ and Ahc~ is compared with the observed diagram in Fig. 35.

Cd

52~

"" .~,...

x

/ / /

$ n

FIG. 35. Liquidus surface of the Cd-Bi-Sn system based on calculations employing the Alcock-Richardson-Toop approximation, eqn. (180) (solid lines). Dashed lines indicate the experimentally observed diagram/TM

(Temperature in K.)

In the method proposed by Kohler ~89~ and by E Olson and Toop, ~9°) the value of to~ in Fig. 31 is

related to values of E at points c, d, and e according to:

E w,e, = (1 _ X2A)W~, + (1 , E 2 E - X~)W~d~ + (1 -- Xc)w~,. (181)

Once again, if the binary and ternary phases are regular, the approximation is exact, and eqn. (181) reduces to eqn. (179). A physical interpretation of the approximation in eqn. (181) is as follows. The factor ~o(~) is related to the B-C interatomic pair interactions in the B-C binary system at the same

F thus contains a molar ratio X c / X B as at point p. co~p~ term due to B-C pair interactions which is equal to

E ~o,~ times a factor (1 - XA 2) which accounts for the dilutive effect of the "A" species upon the B-C interactions. The approximation in eqn. (181) has the advantage of being symmetrical with respect to the three components, whereas the approximation of eqn. (180) is dependent upon which component is called component "A".

149

Equations for ~OA E, COS, and w~ corresponding to eqn. (181) have been calculatedJ 9n The phase diagram for the Cd-Bi -Sn system calculated by Ansara and Bonnier m~ solely on the basis of binary data using eqn. (181) is compared with the observed diagram in Fig. 36.

Cd

. : - . . .

\ x 3oe t / x x t i

~' \ J )

\ \ \ \ x ~ ¢ z I

Bi Sn

FIG. 36. Liquidus surface of the Cd-Bi-Sn system based on calculations ~' employing the Kohler-Olson-Toop approximation, eqn. (181) (solid lines). Dashed lines indicate the experimentally observed diagram/TM

(Temperature in K.)

Similar approximation methods based upon simple solution models have been reviewed. (3~m~ Such approximations have been used by Bonnier, An- sara, and co-workers t36'9~-93~ and by Olson and Toop (8s9°'9"~ to calculate a large number of ternary phase diagrams, mainly for alloy systems. This sort of approximation, based upon simple solution models, can only be applied with confidence to phases in which deviation from ideal behavior is not large. Liquid phases in relatively simple alloy systems can thus be reasonably approximated in this way. Two examples have just been given in Figs. 35-36. However, solid phases, or liquid phases in which there are strong interactions, will, in general, not be amenable to such approximation procedures. This explains why the method has found its greatest successes in the calculation of miscibility-gaps in ternary liquid phasesJ 93)

In general, for ternary systems in which limited experimental data are available, no general formula for approximating the missing data and for calculating the phase diagram can exist. Rather, one must make use of what is available and apply a measure of experience and judgment in making approximations. For example, ternary phase equilibrium data in the F e - M o - W system have been analyzed (9~' in terms of a regular solution model, and the resultant equations have then been used to synthesize the phase boundaries. As a further example, we may consider the calculation by Counsell et al . ~9~) of the miscibility-gap in the solid fcc phase of the F e - C u - N i system. Enthalpies of mixing, Ah, have been measured in the single- phase fcc ternary alloys as well as in the Fe -Ni and Ni-Cu binary alloys. These data were fitted ~9~) to a

150

series expansion of the form

Ah = XAXB(aBAXA + aABX,) + X.Xc(acBXB + a~cXc) + XcX~(a~cXc + ac~X~) + a~cX~X~Xc.

(182)

Excess entropies are available for the Fe -Ni and Ni-Cu alloys. For the Fe -Cu alloys, excess entropies were estimated from the known phase diagram of the binary system in combination with values of Ah for this binary system obtained by extrapolation of eqn. (182) to XN~ = 0. Excess entropy values for the three binary systems were then combined to give an equation of the form of eqn. (182). With the "ternary interaction constant" a^ac set equal to zero, this provided an estimate of s ~ in the ternary system. The resultant equation gave miscibility-gap boundaries which agreed well with the observed boundaries only at higher temperatures. With the addition of a small ternary term aABC to the expression for sE, however, the calculated and observed miscibility-gap boundaries were in good agreement as is shown in Fig. 37.


7.1. Analytical representation of thermodynamic properties

In a quaternary system A - B - C - D , any excess property of component A, to~, may be expressed as a power series "°°~ as follows:

j' k' I'

= = a ~ l y ~ y 2 y 3 (184)

where the composition variables are defined as:

y, = (x~ + Xc + X~)l(X~ + X , + X~ + XD) = (1 - x . ) , (185)

Y2 = (Xc + XD)/(XB "Jr- Xc -}- X o ) , (186)

y3 = XoI(Xc + X.). (187)

Equation (184) is an extension to quaternary systems of eqn. (145). Experimental values of/x CEd, A hcd, and S~d in the liquid Cd-B i -Pb -Sn system have been fitted "°" to eqn. (184). The agreement of a series representation with eighteen coefficients to the experimental values of p. Cd at 773 K over the entire composition range of the quaternary system is shown in Fig. 38. Similar power series (analogous to

Ni

Cu 0 .2 0 . 4 0 .6 0 .8 F'e

FIG. 37. Calculated ~ (solid lines) and observed c~ (dashed lines) miscibility-gap in the Fe-Cu-Ni system at

1123 Kand 1323 K.

Krupkowski ~98~ proposes that excess chemical potentials in binary systems be fitted to equations of the form

/ ~ = (a - bT)(1 - XA)" (183)

where a, b and m are adjustable parameters. He then proposes equations for estimating ternary properties using only the parameters a, b and m for each of the three binary systems. This method has been used with success t99~ for simple liquid ternary alloy systems, although the method is, of course, limited by the fact that approximations for the ternary phase can only be made if all three binaries can be expressed by equations of the form of eqn. (183).

VII. M u l t i c o m p o n e n t sys tems

For quaternary and higher order systems, there have been very few attempts to calculate phase diagrams or to search for novel means of presentation. Accordingly, this section will be brief.

Cd(A)

~ S n ( D )

P b ( c ~

200£ 8 i 150(J 1000 5000

2000 Path I Y2 = • 1 5 0 C = ~ 100C

'~ 500

1500 w3.'~ 1000

500 O= ~'ath

1500 3 ~ 1000 500 ~ y2 = 0.800

"""~Path 4 Y3 =0"250 i i i I i i i i I

Cd 0.9 0.0 0.7 0.5 0.5 0.4 0.3 02 01 Xcd=(1-yl)

FIG. 38. Excess chemical potential of Cd at 773 K in the Cd-Bi-Pb--Sn system. "°" Curves shown are from least-

squares analysis using eqn. (184) with 18 coefficients.

eqns. (150-151)) for the proper t ies /zg, /x~, and/x~ of the other components can be written, and simple relationships among the coefficients (analogous to eqns. (153-154)) have been derived. "°°~ The integration functions for these Gibbs-Duhem integrations are now functions of the two composition variables y2 and y3. "°°-~°2) Experimental and estimated values of the properties of Bi, Pb, and Sn in the B-C-D(Bi -Pb-Sn) ternary system have been used "°" as end-point data in order to calculate these integration functions. In this way, explicit analytical series representations for tz~, Ah, and s~ for all

Phase diagrams

four components in the liquid C d - B i - P b - S n phase over the entire range of compositions can be written. With these explicit functions it should then be a relatively simple matter to calculate the liquidus surface of the quaternary system, although this calculation has not yet been performed.

This analytical technique has been extended to systems of any number of components. "°°)

Analytical representation in quaternary phases can also be achieved through an extension of eqn. (161). 169}

z a~k~X~X(,X~. (j+/+ll;:,2

The corresponding expansions for co~, co~ and rOD E a r e :(69)

'°.~=E E Y.(a~,, j+1 ~+k*ll-~, j + k + 1

x a{~+ ~k~ X~XcXD + CB(y2, y3), (189)

k + l ~S+k*l~;~ j + k + I

) × a~+,,, X~X~X~+ Cdy.~,

/ + I ° ~ g : E E E a,k, j + k + l

j k I x as,,+~} XBXcXD+ Co(y2, Y3)

y~), (190)

(191)

where the integration functions CB, Cc and CD are functions of the two variables y2 and y3 defined in eqns. (186-187). The use of eqn. (188) would appear to have the advantage of simplicity over eqn. (184) for representing the properties of quaternary systems. Extension of eqns. (188-191) to systems of any number of components is easily accomplished. <1°2)

Procedures in which quaternary properties may be estimated from known properties of the ternary or binary systems by means of truncated forms of series expansions can be proposed. An example for the liquid C d - B i - P b - S n phase has been presented, "°n and agreement with the measured quaternary properties was excellent. It has also been proposed ~9u°3~ to extend the semi-empirical approximation techniques of eqns. (179-181) to quaternary systems. The estimation of the properties of an n-component system from the properties of the constituent (n - 1)-component systems should become better as n becomes larger, since, in effect, the (n - l)-component data contains a larger fraction of information regarding the interactions in the n-component system as n increases.

Interaction parameters. Wright and Elliott ~°4~ have extended the investigations of ternary iron systems discussed (m~ in Section 6.1 to quaternary systems and have used the formalism of eqn. (164) to calculate the phase diagrams of the Fe-rich corners of the Fe-C-O-A1 and F e - C - O - S i systems. Methods of graphically presenting the phase relationships using various projections are discussed by the authors.

-5

cr~ -10 o

151

I I I I

NiS04 *Cr203

Ni504 *NiCrz04 1

NiCr204 NiO *NiCr20 ~ Or203

--'30

" 20

tD--

k-10

~ N i3 S.._~ 2 +N I Cr204

15 NiS *NtCr20 ~ I I 0

Ni 0.2 0.4 016 01.8 Cr

~ ' : r'lcr / ( nNi + ncr)

FIG. 39. Estimated type 2 phase diagram of the Ni-Cr-S-O system at T=1000K, pso_~=0.1atm, P =

I arm.

7.2. Geometrical representation

In order to obtain a type 1, -2 or -3 phase diagram for a quaternary system, it is necessary to hold three potential functions constant. An example of a type 2 diagram is shown in Fig. 39 which is a plot of log Po2 versus ~ = nCr/(nNi+ncr) at T = 1000K, pso2= 0.1 atm, and P = 1 arm in the N i - C r - S - O system. At the same time, this is also a plot of log ps, versus ~, since log ps~ varies directly as log po: when pso2 and T are constant. Data for the free energies of formation of the pure compounds were taken from the literature) 43''~'°8~ Values of log Po2 less than - 16 are not shown, since at lower po.. the partial pressure of sulfur becomes greater than 1 atm. In the construction of Fig. 39 it was assumed that the other observed °°9~ compounds (Ni, Cr)2S3 and (Ni, Cr)3S4, for which no free energy data are available, do not appear in the po:-range shown. Furthermore, the solubility of Cr in the NiS phase and the various other solubility limits have only been estimated, with the observed phase diagram of the N i -Cr -S system "°9~ being used as a guide. Figure 39 should thus not be taken as quantitatively correct; it is presented here to illustrate the use of such type 2 diagrams in organizing and presenting quaternary phase equilibrium relationships. These diagrams may be of use in the study of the high-temperature corrosion of alloys or of roasting processes. See ref. 1 for a further example.

The general considerations of Section II can be used as a guide in a search for other useful phase diagrams in multicomponent systems. For example, in a system of three metals and one non-metal, A - B - C - X , a Gibbs' triangle plot with A, B and C at the corners at constant T, P and /Xx is a type 3 diagram which would be useful in the study of slags or of the corrosion of ternary alloys.

VIII. Conclusion

There are many parameters such as temperature, pressure, chemical potential, composition, etc.. which influence phase equilibria and which can be used to define the axes of a phase diagram. By properly specifying the relationships between the

J . P . S . S . C . . Vok I0, Par t 3 - - C

152 A.D. PELTON and

variables which define the axes of the diagram and those which are held constant, however, a large number of phase diagrams can be classified as one of three familiar topological types. This simplification aids in the interpretation and visualization of the phase relationships. For example, the isothermal type 2 phase diagram of log po: - ncd(nFo +ncr) in the Fe-Cr-O system in Fig. 20a can immediately be interpreted by anyone familiar with the topological rules of construction of the well-known and topologically similar type 2 temperature- composition diagrams for binary systems. For clearly presenting the phase relationships in a complicated system, the use of a set of "corresponding" type 1, -2 and -3 diagrams, as for example in Figs. 20, 25 or 26, is often very useful.

In a system of components A, B, C . . . . the thermodynamic properties of conjugate phases a and/3 at equilibrium can be calculated by solving the set of equations:

/~ =/z~ (i = A,B,C . . . . ). (192)

For calculating phase diagrams for systems of two or more components, it is essential to be able to express the chemical potentials of the components as analytical functions of composition. The representation of thermodynamic properties of solution by series expansions in the composition variables has thus been discussed in detail. It is now possible to express any isothermal partial or integral thermodynamic solution property as an analytical function of composition to any desired degree of precision in systems of two, three or more components, and to perform the Gibbs-Duhem integration and associated thermodynamic calculations analytically.

Because of the transcendental nature of eqns. (192), their solution requires the use of numerical techniques. Such calculations present no difficulties in principle, at least for binary and ternary systems. With the rapid increase in the speed of computers, considerations of computational time, which were important in the past, would now appear to be less of an issue, and it would seem that the simpler the numerical technique the better. For example, the binary temperature-composition diagrams of Fig. l0 were calculated using the very simple Gauss-Seidel "~°~ technique. For the two phases a and /3 in equilibrium in this case, the problem of solving eqns. (192) reduces to the solution of the two equations (62-63). The right-hand sides of eqns. (62-63) are known functions of temperature T, and the left sides can be expressed as transcendental analytical functions of the mole fraction of A in the two conjugate phases: XT, and X~ (XB = 1 - XA). For a chosen value of T, we start by assuming a value of X~, in eqn. (62) and then we solve eqn. (62) by a simple iteration technique to obtain a value of X~. This value is then substituted into eqn. (63) which is solved by an iteration technique to obtain a new value of XT, which is then substituted into eqn. (62), etc. No more than seven such steps were ever required to generate the diagrams of Fig. 10 within 1 mole%. This technique can easily be extended t~°~ to solve the three simultaneous equations required to calculate the compositions of conjugate phases in ternary systems. A certain

W. T. THOMPSON

amount of judgment must, of course, be exercised in using this technique, since only two-phase regions are calculated, and these various regions must then be coordinated to give the final phase diagram, by using the topological restrictions discussed previously. This usually presents no problems in prac- tice, although computational techniques have been developed tiN) for deciding which pair of all possible pairs of conjugate phases does, in fact, have the minimum free energy (lowest common tangent on a free energy-composition diagram). This permits a graphical computer output in the form of a complete phase diagram. Various other computational techniques for calculating binary and ternary phase diagrams have been presented ~6'38'~'m~ such as the Newton-Raphson technique ~6~ or stepwise in- crementation techniques which make use of previ- ous calculations at neighboring temperatures as discussed by Gaye and LupisJ"" A somewhat different formulation of the problem than that given in eqns. (192) is obtained by writing an expression for the total free energy of the system and then minimizing this expression with respect to the number of moles of each component in each phase, under the constraint of constant total number of moles of each component. This technique has been discussed, for example, by Counsell et al. ~ who employed "hill-climbing" computational techniques, and by Gaye and Lupis. ""~

Attempts to calculate phase diagrams from first principles, for example by using pseudopotential theory to calculate "lattice stabilities ''~35~ and by using a priori estimates of solution properties such as regular solution parameters, free energies of compounds, etc., ~6~ on the basis of atomic volumes, compressibilities, electron densities, etc., have been made. For binary and ternary systems, qualitatively correct results are sometimes ob- tainedJ 6'~ There is much room for future work in this area, which is, of course, the ultimate goal of phase diagram calculations. Other semi-empirical methods of approximation aim at estimating properties of ternary phases from known data for the binary phases. Most such methods are based upon extension of regular solution theory, quasi- chemical-type models, truncations of series expansions, etc. These are reviewed in refs. 36 and 113. Such procedures are usually only successful when applied to phases in which intermolecular interactions are small so that deviations from ideal solution behavior are not great.

Although we are still a long way from the a priori calculation of the conditions for phase equilibria, it is nevertheless presently possible to obtain good estimations for a large number of systems on the basis of limited experimental phase equilibrium and thermodynamic data by the judicious use of interpolation, extrapolation and semi-empirical models. The graphical aid of a phase diagram which obeys well-defined topological rules of construction is indispensible in the coordination and interpretation of these computational efforts.

Acknowledgements

The authors are indebted to Professor H. Schmalzried for his inspiration in this work and to

Phase diagrams

Dr. C. W. Bale for the many ideas which he contributed. Thanks are also due to Mr. J. Des- rochers who made all the drawings.

This work was supported by the National Re- search Council of Canada.

Appendix I Proof of "extension rule" at a triple-point in a

type I diagram

Since (QdQ3) is an extensive property, it follows that, at the triple-point:

A(Qt/Q3) ~-~ = A(Qt/Q3) °-~ + A(QdQ~) ~-~ (A1)

where

A(Q,/Q~) °-~ = (Q,/Q3) ~ - (Q,/Q3) ~ (A2)

etc. An analogous equation can be written for

(Q2/Q3). Let (dda~/dgo,) ~ ~, (dc~,/dc~2) t~ v and (dcb~/dcb2) ~-~

be the slopes of the two-phase boundary lines at the triple-point. Then, from eqn. (14):

(dcht/d4~,~) ~-~ = - A(Q2/Q3) ~ ~/A(Q,/Q3) ~-~, (A3)

(dch,/drh:) ~-" = -A(Q2/Q3) t3 '/A(QI/Q3) ~-v (A4)

and so, from eqn. (AI):

(d&ddfby-~ = - (A(Q2/Q~) o ~ (AS) + A(Q,/Q3)~-~)/(A(QdQ3) °-e + A(Q,/Q3) ~ ~)

For any four numbers a, b, c and d, where a/b < c/d, it can be proved that:

a + ¢ a/b < ~ < c/d. (A6)

Therefore:

(dcht/drh2) ~-~ < (dd~ddch:) ~-~ < (dc~ddrb,.)" ~ (A7)

which proves the "extension rule" that at any triple-point in a type 1 diagram, the extension of any two-phase boundary must pass into the field of the third phase.

Appendixll Proof of topological relationships between corresponding type I,-2 and -3 diagrams

It is evident that the abscissa of a type 2 diagram must be a ratio of extensive variables, since the diagram should be independent of the total mass of the system. We wish to prove, then, that Q2/Q3 increases as ~b.~ is increased when ~ , 4,4, d~5. • • are held constant.

We first note that if there exists a gradient of potential &, there will be a "flow" of the corresponding extensive quantity Q~ down the gradient from high to low qS~. We now imagine a homogeneous isolated system at internal equilibrium, and we suppose that a random fluctuation causes an increase in the potential 4~ by an amount 6& > 0 in some region of the system. There will then be a flow of Q~ out of this region. This flow must serve to decrease ~b~, for if it did otherwise, the fluctuation would tend to grow and the system would not be at stable equilibrium, contrary to our original assump-

153

tion. That is, a decrease in Q, is always accompanied by a decrease in & and vice versa. This then proves that Q2 increases as we increase ~b2. It also proves that Q3 decreases as we decrease q~3.

Finally, the general Gibbs-Duhem equation (7) for the case of constant 4~, ~b4, ~b5 . . . . is as follows:

Q:dcb2 = - Q3dc~3. (A8)

Since we have chosen all the Q, to be positive quantities, it follows that an increase in th,, at constant the, ~4, 65 . . . . is always accompanied by a decrease in 4~3, and thus by a decrease in Q3. Therefore, our original hypothesis that an increase in ~b: with 0,, 4',, ~b5 . . . . held constant is accompanied by an increase in QE/Q3 has been proven.

Similarly, it is evident that the ratio Q2/(Q,-+ Q3) = 1/(1 + Q3/Q,.) will also increase monotonically as tb_, is increased.

AppendixIII Coordinates of type 3 diagrams

Consider a type 3 diagram in which Q d Y is plotted versus Q2/Z where Y and Z are some extensive properties or functions thereof. It will be shown that it is necessary that Y ~ Z in order that tie-lines be straight lines.

Suppose that an a - and a 13-phase with properties QT, Q~, Y° and Z ° and Q{, Q2 °, Y~ and Z ~ are in equilibrium in a two-phase region with the overall extensive properties Q~, Q2, Y and Z. That is

Q, = Q7 + Q~, (A9)

Therefore

Q2 = Q~ + Q~, (AIO)

Y = Y° + Y~, (Al l )

Z = Z ~ + Z ~. (A12)

y y y -~--¢ = a + ( l - a )

(AI3)

~ - = ~ - _ ~ - A ~-~ = b + ( l - b ) .

(AI4)

In order that the tie-line in the two-phase region on the plot of Q d Y versus.Q2/Z be a straight line it is necessary that a =- b. This means that Y =- Z. A plot of QdQ3 versus Q2[Q3 satisfies this criterion, as does a plot of, say, QI/(Q: + Q3) versus Qd(Q2 + Q3) and so forth. However, a plot of Q,[(Q2+ Q3) versus QdQ3, for example, would not be allowed. It therefore follows that in the representation of composition for a ternary system A-B-C, the following plots result in straight tie-lines:

(a) (nAl(nB + nc)) vs. (nc/(nB + nc)) (J~inecke coordinates);

(b) (nA/(nA + nB+ no)) vs. (nc/(nA +nB + nc));

(C) (V3/'---2nA/(na+ no+ nc)) vs. (nc+ n~,/2)/(nA+ nB+ nc).

The condition that the total number of moles be equal to or greater than zero restricts compositional points to a right triangle in case (b) and an equila- teral triangle (Gibbs' triangle) in case (e).


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