32                           Thermodynamics and phase diagrams
two phases can also be expressed using the activity concept defined for homogeneous systems in Fig. 1.16. In heterogeneous systems containing more than one phase the pure components can, at least theoretically, exist in different crystal structures. The most stable state, with the lowest free energy, is usually defined as the state in which the pure component has unit activity. In the present example this would correspond to defining the activity of A in pure a — A as unity, i.e. when XA = 1, aA = 1. Similarly when XB — 1, aB — 1. This definition of activity is shown graphically in Fig. 1.28a; Fig. 1.28b and c show how the activities of B and A vary with the composition of the a and ß phases. Between A and ac, and ßc and B, where single phases are stable, the activities (or chemical potentials) vary and for simplicity ideal solutions have been assumed in which case there is a straight line relationship between a and X. Between ac and ßc the phase compositions in equilibrium do not change and the activities are equal and given by points q and r. In other words, when two phases exist in equilibrium, the activities of the components in the system must be equal in the
two phases, i.e.
(1.47)
a% = al,
„o.  __   „P
aB — aB
-RT[naA
(a)
(b)
(c)
RTioa,
Fig. 1.28   The variation of öa and aB with composition for a binary system containing two ideal solutions, a and ß.
Binary phase diagrams                                       33
1.5   Binary Phase Diagrams
In the previous section it has been shown how the equilibrium state of an alloy can be obtained from the free energy curves at a given temperature. The next step is to see how equilibrium is affected by temperature.
1.5.1    A Simple Phase Diagram
The simplest case to start with is when A and B are completely miscible in both the solid and liquid states and both are ideal solutions. The free energy of pure A and pure B will vary with temperature as shown schematically in Fig. 1.4. The equilibrium melting temperatures of the pure components occur when Gs = GL, i.e. at rm(A) and rm(B). The free energy of both phases decreases as temperature increases. These variations are important for A-B alloys also since they determine the relative positions of GA, GA, GB and G& on the molar free energy diagrams of the alloy at different temperatures, Fig. 1.29.
At a high temperature 7: > T"m(A) > 7"m(B) the liquid will be the stable phase for pure A and pure B, and for the simple case we are considering the liquid also has a lower free energy than the solid at all the intermediate compositions as shown in Fig. 1.29a.
Decreasing the temperature will have two effects: firstly GA and GB will increase more rapidly than G% and GB, secondly the curvature of the G curves will be reduced due to the smaller contribution of ~TASmix to the free energy.
At Tm(A), Fig. 1.29b, GA = GA, and this corresponds to point a on the A-B phase diagram, Fig. 1.29f. At a lower temperature T2 the free energy curves cross, Fig. 1.29c, and the common tangent construction indicates that alloys between A and b are solid at equilibrium, between c and B they are liquid, and between b and c equilibrium consists of a two-phase mixture (S + L) with compositions b and c. These points are plotted on the equilibrium phase diagram at T2.
Between T2 and 7m(B) GL continues to rise faster than Gs so that points b and c in Fig. 1.29c will both move to the right tracing out the solidus and liquidus lines in the phase diagram. Eventually at Tm(B) b and c will meet at a single point, d in Fig. 1.29f. Below rm(B) the free energy of the solid phase is everywhere below that of the liquid and all alloys are stable as a single phase solid.
1.5.2    Systems with a Miscibility Gap
Figure 1.30 shows the free energy curves for a system in which the liquid phase is approximately ideal, but for the solid phase AHmix > 0, i.e. the A and B atoms 'dislike' each other. Therefore at low temperatures (T3) the free energy curve for the solid assumes a negative curvature in the middle,
ô<\
i nermoaynamics ana pnase aiagrams
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Binary phase diagrams                                       35
(C)
Fig. 1.30 energy v
The derivation of a phase diagram where Aíř^ix > A//^ = °-  Free composition curves for (a) 7",, (b) T2, and (c) 7"3.
Fig. 1.30c, and the solid solution is most stable as a mixture of two phases a' and a" with compositions e and f. At higher temperatures, when - TASmix becomes larger, e and f approach each other and eventually disappear as shown in the phase diagram, Fig. 1.30d. The a' + a" region is known as a miscibility gap.
The effect of a positive A/fmix in the solid is already apparent at higher temperatures where it gives rise to a minimum melting point mixture. The reason why all alloys should melt at temperatures below the melting points of both components can be qualitatively understood since the atoms in the alloy 'repel' each other making the disruption of the solid into a liquid phase possible at lower temperatures than in either pure A or pure B.
1.5.3    Ordered Alloys
The opposite type of effect arises when A#mix < 0. In these systems melting will be more difficult in the alloys and a maximum melting point mixture may
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Fig. 1.32 The derivation of a eutectic phase diagram where both solid phases have the same crystal structure. (After A.H. Cottrell, Theoretical Structural Metallurgy, Edward Arnold, London, 1955,©Sir Alan Cottrell.)
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Fig. 1.33 The derivation of a eutectic phase diagram where each solid phase has a different crystal structure. (After A. Prince, Alloy Phase Equilibria, Elsevier, Amsterdam, 1966.)

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(e)                                           (f)
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40
Thermodynamics and phase diagrams
B
A             7|
Stable compositions       '
Stoichiometric
composition    {Am Sp) Minimum G
Fig. 1.35   Free energy diagram to illustrate that the range of compositions over which a phase is stable depends on the free energies of the other phases in equilibrium.
containing several components and phases is that the chemical potential of each component must be identical in every phase, i.e.
M-A = M-a = M-X =
P-B = H-
l£
= M-B - ■ ■ • Ml: = M-2: = • • •
(1.48)
The proof of this relationship is left as an exercise for the reader (see Exercise 1.10). A consequence of this general condition is the Gibbs phase rule. This states that if a system containing C components and P phases is in equilibrium the number of degrees of freedom F is given by
P + F = C+ 2                                                                               (1.49)
A degree of freedom is an intensive variable such as T, P, XA, XB . . . that can be varied independently while still maintaining equilibrium. If pressure is maintained constant one degree of freedom is lost and the phase rule becomes
P + F = C + 1                                                                               (1.50)
At present we are considering binary alloys so that C = 2 therefore
P + F= 3
This means that a binary system containing one phase has two degrees of freedom, i.e. T and XB can be varied independently. In a two-phase region of
Binary phase diagrams                                       41
hase diagram P == 2 and therefore F = 1 which means that if the tempera-1     is chosen independently the compositions of the phases are fixed. When three phases are in equilibrium, such as at a eutectic or peritectic tempera-there are no degrees of freedom and the compositions of the phases and
the temperature of lhe system are a11 fixed'
.5.7   The Effect of Temperature on Solid Solubility
"he equations for free energy and chemical potential can be used to derive he effect of temperature on the limits of solid solubility in a terminal solid olution. Consider for simplicity the phase diagram shown in Fig. 1.36a where í is soluble in A, but A is virtually insoluble in B. The corresponding free nergy curves for temperature T\ are shown schematically in Fig. 1.36b. Since ^ is almost insoluble in B the Gß curve rises rapidly as shown. Therefore the laximum concentration of B soluble in A (XB) is given by the condition
For a regular solid solution Equation 1.40 gives
tä = G% + ft(l - XBf + RT\n XB
But from Fig. 1.36b, GB - u-g = AGB, the difference in free energy between pure B in the stable ß-form and the unstable ot-form. Therefore for XB = XB
-RT\nX% - i)(l - X%f = AGB                                                (1.51)
If the solubility is low X% < I and this gives
*! = exp[-^}                                 (1.52)
Putting
AGB = A//B - TASB gives
x%=AQXv-Řf                                                                í1-53)
where A is a constant equal to exp (ASB/tf) and
0 = AHB + a                                                                                (1.54)
AHB is^the difference in enthalpy between the ß-form of B and the a-form in J mol . ft is the change in energy when 1 mol of B with the a-structure dissolves in A to make a dilute solution. Therefore Q is just the enthalpy change, or heat absorbed, when 1 mol of B with the ß-structure dissolves in A to make a dilute solution.
42
Thermodynamics and phase diagrams
<b)
(a)      A  i
Fig. 1.36   Solubility of B in A.
Binary phase diagrams
43
A c  ,c the difference in entropy between ß-B and ot-B and is approximately
ľ indent of temperature. Therefore the solubility of B in a increases
pntially with temperature at a rate determined by Q. It is interesting to
eTľhat except at absolute zero, X% can never be equal to zero, that is, no
^components are ever completely insoluble in each other.
15 8   Equilibrium Vacancy Concentration
q far it has been assumed that in a metal lattice every atom site is occupied. However let us now consider the possibility that some sites remain without inms that is there are vacancies in the lattice. The removal of atoms from ľheir sites not only increases the internal energy of the metal, due to the broken bonds around the vacancy, but also increases the randomness or confieurational entropy of the system. The free energy of the alloy will depend on the concentration of vacancies and the equilibrium concentration X* will be that which gives the minimum free energy.
K for simplicity, we consider vacancies in a pure metal the problem of calculating X% is almost identical to the calculation of AGmix for A and B atoms when Ai/mix is positive. Because the equilibrium concentration of vacancies is small the problem is simplified because vacancy-vacancy interactions can be ignored and the increase in enthalpy of the solid (AH) is directly proportional to the number of vacancies added, i.e.
AH = AHVXV where Xv is the mole fraction of vacancies and AHV is the increase in enthalpy per mole of vacancies added. (Each vacancy causes an increase of AHv/iVa where Na is Avogadro's number.)
There are two contributions to the entropy change AS on adding vacancies. There is a small change in the thermal entropy of A5V per mole of vacancies added due to changes in the vibrational frequencies of the atoms around a vacancy. The largest contribution, however, is due to the increase in con-figurational entropy given by Equation 1.25. The total entropy change is thus
AS = XVASV - R(XV In Xw + (1 - Xv) In (1 - Xv)) The molar free energy of the crystal containing Xv mol of vacancies is therefore given by
G = GA + AG = GA + &HVXV - TASVXV
+ RT(XV In Xv + (1 - Xv) In (1 - Xv))         (1-55)
This is shown schematically in Fig. 1.37. Given time the number of vacancies will adjust so as to reduce G to a minimum. The equilibrium concentration of vacancies XI is therefore given by the condition
dG
44
Thermodynamics ana phase diagrams
A"VXV
-ThS = RT lnXv
Fig. 1.37   Equilibrium vacancy concentration.
Differentiating Equation 1.55 and making the approximation Xv < 1 gives
AHV - TA5V + RT\nXl = 0 Therefore the expression for X$ is
-AH„
A5V X* = exp — • exp
or, putting AGV = AHV --AGV
AT TA5V gives
X* = exp
P7/
(1.56)
(1.57)
The first term on the right-hand side of Equation 1.56 is a constant ~~3, independent of T, whereas the second term increases rapidly with increasing T. In practice APřv is of the order of 1 eV per atom and X% reaches a value of about 10_4-10~3 at the melting point of the solid.
1.6   The Influence of Interfaces on Equilibrium
The free energy curves that have been drawn so far have been based on the molar free energies of infinitely large amounts of material of a perfect single crystal. Surfaces, grain boundaries and interphase interfaces have been ignored. In real situations these and other crystal defects such as dislocations do exist and raise the free energies of the phases. Therefore the minimum free
^r/re^innuancc uj iíucijuk-ců u n eifuiuur lurrt                              tj
*''         of an alloy, i-e. the equilibrium state, is not reached until virtually
^"nterfaces and dislocations have been annealed out. In practice such a state is unattainable within reasonable periods of time.
Interphase interfaces can become extremely important in the early stages of phase transformations when one phase, ß, say, can be present as very fine oarticles in the other phase, a, as shown in Fig. 1.38a. If the a phase is acted on bv a pressure of 1 atm the ß phase is subjected to an extra pressure AP due to the curvature of the a/ß interface, just as a soap bubble exerts an extra pressure AP on its contents. If 7 is the a/ß interfacial energy and the particles are spherical with a radius r, AP is given approximately by
2-y AP = —
r
(a)
Atmospheric pressure
G
!g- 1.38   The effect of interfacial energy on the solubility of small particles.
46                           Thermodynamics and phase diagrams
By definition, the Gibbs free energy contains a lPV" term and an increase ol pressure P therefore causes an increase in free energy G. From Equation 1.9 at constant temperature
AG = A/> ■ V
Therefore the ß curve on the molar free energy-composition diagram in Fig. 1.38b will be raised by an amount
where Vm is the molar volume of the ß phase. This free energy increase due tc interracial energy is known as a capillarity effect or the Gibbs-Thomson effect.
The concept of a pressure difference is very useful for spherical liquid particles, but it is less convenient in solids. This is because, as will be discussed in Chapter 3, finely dispersed solid phases are often non-spherical For illustration, therefore, consider an alternative derivation of Equation 1.58 which can be more easily modified to deal with non-spherical cases3
Consider a system containing two ß particles one with a spherical interface of radius r and the other with a planar interface (r = <*>) embedded in an o matrix as shown in Fig. 1.39. If the molar free energy difference between the two particles is AG7, the transfer of a small quantity (án mol) of ß from the large to the small particle will increase the free energy of the system by a small amount (dG) given by
dG = AGydn
If the surface area of the large particle remains unchanged the increase in free energy will be due to the increase in the interfacial area of the spherical
Fig. 1.39   Transfer of dn mol of ß from large to a small particle.
The influence of interfaces on equilibrium                        47
HĽle (üVl;- Therefore assuming 7 is constant
dG = 7^ Equating these two expressions gives ÓA
Mi " ^
(1.59)
S'nco n = 4irr3/3Vm and A = 4-rrr2 it can easily be shown that M _ ôA/ár = 2V™ d« ~" dn/dr        r
from which Equation 1.58 can be obtained.
\p important practical consequence of the Gibbs-Thomson effect is that the solubility of ß in a is sensitive to the size of the ß particles. From the innimon tangent construction in Fig. 1.38b it can be seen that the concentration of solute B in a in equilibrium with ß across a curved interface (Xr) is ciciter than X«,, the equilibrium concentration for a planar interface. Assuming for simplicity that the a phase is a regular solution and that the ß phase is almost pure B, i.e. X% ~ 1, Equation 1.52 gives
agr + n
V. - exp           RT
Similarly Xr can be obtained by using (AGB - 2yVm/r) in place of AGB
AGB + íl - 2yVJr]
Thciefore
*'-*-«P7äf                                                      í1-60*
aihl for small values of the exponent
*--4+^)                   (L61)
T.iking the following typical values: 7 = 200 mJ m"2, Vm = 10~5 m3, K   ■ 8.31 J mol"1 K"1, T = 500 K gives
XT     t        1
**            r(nm)
c c for r = 10 nm XT/X*> ~ 1.1. It can be seen therefore that quite large Mobility differences can arise for particles in the range r = 1-100 nm. Hin* ever, for particles visible in the light microscope (r > 1 |xm) capillarity Ľlk\tsare very small.
48                         Thermodynamics and phase diagrams                             '%
1.7   Ternary Equilibrium
Since most commercial alloys are based on at least three components, an understanding of ternary phase diagrams is of great practical importance. The ideas that have been developed for binary systems can be extended to systems with three or more components4.
The composition of a ternary alloy can be indicated on an equilateral triangle (the Gibbs triangle) whose corners represent 100% A, B or C as shown in Fig. 1.40. The triangle is usually divided by equidistant lines parallel to the sides marking 10% intervals in atomic or weight per cent. All points on lines parallel to BC contain the same percentage of A, the lines parallel to AC represent constant B concentration, and lines parallel to AB constant C concentrations. Alloys on PQ for example contain 60% A, on RS 30% B, and TU 10% C. Clearly the total percentage must sum tp 100%, or expressed as mole fractions
*a + *b + *c = 1                                                                 (1-62)
The Gibbs free energy of any phase can now be represented by a vertical distance from the point in the Gibbs triangle. If this is done for all possible compositions the points trace out the free energy surfaces for all the possible phases, as shown in Fig. 1.41a. The chemical potentials of A, B and C in any phase are then given by the points where the tangential plane to the free energy surfaces intersects the A, B and C axes. Figure 1.41a is drawn for a
A     T                             GL                                                           C
Fig. 1.40   The Gibbs triangle.
Ternary equilibrium
49
in which the three binary systems AB, BC and CA are simple eutec-;e energy surfaces exist for three solid phases a, ß and 7 and the liquid
ph.!1«1. L- At this temperature the liquid phase is most stable for all alloy o impositions. At lower temperatures the GL surface moves upwards and owiiNiilly intersects the Ga surface as shown in Fig. 1.41b. Alloys with compositions in the vicinity of the intersection of the two curves consist of n - I at equilibrium. In order for the chemical potentials to be equal in both
(a) Free energies of a liquid and three solid phases of a ternary system.