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 asi 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 to 100%, or expressed as mole fractions
XA + XB + Xc = 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
Ternary equilibrium
49
60 % A
Fig. 1.40 The Gibbs triangle.
I ich the three binary systems AB, BC and CA are simple eutec-M rgy surfaces exist for three solid phases a, j3 and 7 and the liquid M this temperature the liquid phase is most stable for all alloy At lower temperatures the GL surface moves upwards and
tics. Free' phase, L composition^.
\oiuually intersects the G° surface as shown in Fig. 1.41b. Alloys with nmnositions in the vicinity of the intersection of the two curves consist of l' - L at equilibrium. In order for the chemical potentials to be equal in both
(a)
Fig- 1.41   (a) Free energies of a liquid and three solid phases of a ternary syste
50
Thermodynamics and phase diagrams
Ternary equilibrium
51
lb) Ic)
Fig. 1.41 (Cont.) (b) A tangential plane construction to the free energy surfaces defines equilibrium between s and I in the ternary system, (c) Isothermal section through a ternary phase diagram obtained in this way with a two-phase region (L+S) and various tie-lines. The amounts of / and s at point x are determined by the lever rule. (After P. Haasen, Physical Metallurgy, Cambridge University Press, Cambridge, 1978.)
phases the compositions of the two phases in equilibrium must be given by points connected by a common tangential plane, for example .s and / inj Fig. 1.41b. These points can be marked on an isothermal section of the equilibrium phase diagram as shown in Fig. 1.41c. The lines joining the compositions in equilibrium are known as tie-lines. By rolling the tangential) plane over the two free energy surfaces a whole series of tie-lines will be| generated, such as pr and qt, and the region covered by these tie-lines pqtr is| a two-phase region on the phase diagram. An alloy with composition x in-Fig. 1.41c will therefore minimize its free energy by separating into solid a| with composition s and liquid with composition /. The relative amounts of a and L are simply given by the lever rule. Alloys with compositions within Apq will be a homogeneous a phase at this temperature, whereas alloys within BCrt will be liquid.
On further cooling the free energy surface for the liquid will rise through \ the other free energy surfaces producing the sequence of isothermal sections:! shown in Fig. 1.42. In Fig. 1.42f, for example, the liquid is stable near the-centre of the diagram whereas at the corners the a, p and 7 solid phases are';: stable. In between are several two-phase .regions containing bundles of tie-; lines. In addition there are three-phase regions known as tie-triangles. The^ L + a + p triangle for example arises because the common tangential plane simultaneously touches the Ga, Gp and GL surfaces. Therefore any alloy with; a composition within the L 4- a + (J triangle at this temperature will be in: equilibrium as a three-phase mixture with compositions given by the corners \
(9) T=E
(h) E^r
Fig. 1.42 Isothermal sections through Fig. 1.44. (After A. Prince, Alloy Phase tquihbna, Elsevier, Amsterdam, 1966.)
of the triangle. If the temperature is lowered still further the L region shrinks to a point at which four phases are in equilibrium L + a + 0 + 7. This is known as the ternary eutectic point and the temperature at which it occurs is he ternary eutectic temperature, Fig. 1.42g. Below this temperature the iquid is no longer stable and an isothermal section contains three two-phase 2"s and one three-phase tie triangle a + 0 + 7 as shown in Fig. 1.42h. If _ thermal sections are constructed for ail temperatures they can be combined '"in nHee"difleriSi0nal ternary Phase diagram as shown in Fig. 1.44. eouTb     t0 the COUrse of solidification of a ternary alloy, assuming
q 1 ibnum is maintained at all temperatures, it is useful to plot the liquidus
52
Thermodynamics and phase diagrams A
Fig. 1.43  A projection of the liquidus surfaces of Fig. 1.44 onto the Gibbs triangle. '\
surface contours as shown in Fig. 1.43. During equilibrium freezing of alloy X the liquid composition moves approximately along the line Xe (drawn through A and X) as primary a phase is solidified; then along the eutectic valley eE as both a and p solidify simultaneously. Finally at E, the ternary eutectic point, the liquid transforms simultaneously into a + 3 + 7. This sequence of events is also illustrated in the perspective drawing in Fig. 1.44.
The phases that form during solidification can also be represented on a vertical section through the ternary phase diagram. Figure 1.45 shows such a section taken through X parallel to AB in Fig. 1.44. It can be seen that on cooling from the liquid phase the alloy first passes into the L + a region, then into L + a + (}, and finally all liquid disappears and the ot + p + 7 region is entered, in agreement with the above.
An important limitation of vertical sections is that in general the section will not coincide with the tie-lines in the two-phase regions and so the diagram only shows the phases that exist in equilibrium at different temperatures and not their compositions. Therefore they can not be used like binary phase diagrams, despite the superficial resemblance.
1.8  Additional Thermodynamic Relationships for Binary Solutions
It is often of interest to be able to calculate the change in chemical potential (d|x) that results from a change in alloy composition (dX). Considering Fig. 1.46 and comparing triangles it can be seen that
dp-/
d(u-b - u.A)
Xq     A*a 1 and that the slope of the free energy-composition curve is given by
dG_ = u,B - u.A dXn 1
(1.63)
(1.64)
Additional thermodynamic relationships
53
Fig. 1 44 The equilibrium solification of alloy X. (After A. Prince, Alloy Phase Equilibria, Elsevier, Amsterdam, 1966.)
z 2
A*flnv:il   A^ertical section between points 1, 2 and Xin Fig. 1.44. (After A. Prince, 0y Fhase Equilibria, Elsevier, Amsterdam, 1966.)
54
Thermodynamics and phase diagrams
dG
Fig. 1.46  Evaluation of the change in chemical potential due to a change in composition. (After M. Hillert, in Lectures on the Theory of Phase Transformations, H I Aaronson (Ed.),©The American Society for Metals and The Metallurgical Society of AIME, New York, 1969.)
Substituting this expression into Equation 1.63 and multiplying throughout by ^a-^b leads to the following equalities:
-XAd|xA = XBd\j.B = XAXB-j-^dXB
'dX2
(1.65)
which are the required equations relating d|xA, du,B and dXB. The first ,., equality in this equation is known as the Gibbs-Duhem relationship for a binary solution. Note that the B subscript has been dropped from d2G/dX2 as d2G/dXB = d2G/dXA.  For  a regular solution differentiation of Equation 1.39 gives
For an ideal solution Q,
0 and
dX2
RT XAXB
(1.67)
Equation 1.65 can be written in a slightly different form by making use of activity coefficients. Combining Equations 1.41 and 1.43 gives
[XB = Gb + rt In Yb^b (1.68)
Therefore
dXB    XB[     lB dXB
*B1 dlnXB
(1.69)
The kinetics of phase transformations
55
iiTill (( i0iationship can be derived for d^A/dXB. Equation 1.65 therefore becomes
- v ^ - xBd»B « Rt\i + ^~}<^b - Rt{
d In XA
Cumpaiini: Equations 1.65 and 1.70 gives
d In XB\
\2G        f     dl" 7a
rt\ \ +
d 1" 7b dlnXB
(1.70)
(1.71)
1.9   Tlii' Kinetics of Phase Transformations
The ihfi nu'dynamic functions that have been described in this chapter apply tn sWv in that are in stable or metastable equilibrium. Thermodynamics can theiitir. 'le used to calculate the driving force for a transformation, Equation 1.4, but it cannot say how fast a transformation will proceed. The study of how fast processes occur belongs to the science of kinetics.
- Let us redraw Fig. 1.1 for the free energy of a single atom as it takes part in a phase transformation from an initially metastable state into a state of lower free energy, Fig. 1.47. If Gx and G2 are the free energies of the initial and
2?-^-2f
Initial Activated Final state      state state
hiM^V Transfo™ations from initial to final state through an activated state of tugner free energy.
56
Thermodynamics and phase diagrams
Exercises
57
final states, the driving force for the transformation will be AG = G2 ~ Gx, However, before the free energy of the atom can decrease from Gj to G2 the atom must pass through a so-called transition or activated state with a free energy AGa above Gl. The energies shown in Fig. 1.47 are average energies associated with large numbers of atoms. As a result of the random thermal motion of the atoms the energy of any particular atom will vary with time and occasionally it may be sufficient for the atom to reach the activated state. This process is known as thermal activation.
According to kinetic theory, the probability of an atom reaching the activated state is given by exp (-&Ga/kT) where k is Boltzmann's constant (R/Nz) and AGa is known as the activation free energy barrier. The rate at which a transformation occurs will depend on the frequency with which atoms reach the activated state. Therefore we can write
AGa kT
rate   exp —
Putting AG* = AHa - TA5a and changing from atomic to molar quantities enables this equation to be written as
rate « exp
AH* ' RT
(1.72)
This equation was first derived empirically from the observed temperature dependence of the rate of chemical reactions and is known as the Arrhenius rate equation. It is also found to apply to a wide range of processes and transformations in metals and alloys, the simplest of these is the process of diffusion which is discussed in Chapter 2.
References
1. See D.R. Gaskell, Introduction to Metallurgical Thermodynamics, p. 342, McGraw-Hill, 1973.
2. For a more detailed treatment of the structures of solid solutions, inter-metallic compounds and intermediate phases see, for example, Chapters 4 and 5 in Physical Metallurgy, R.W. Cahn (Ed.), North-Holland, 1974 (second edition).
3. For a treatment of nonspherical precipitates see M. Ferrante and R.D. Doherty, Acta Metallurgica, 27: (1979) 1603.
4. See, for example, M. Hillert, Phase Transformations, American Society for Metals, Ohio, 1970, Chapter 5.
Further Reading
A.H. Cottrell, An Introduction to Metallurgy, Chapter 14 'Alloys' and Chapter 15. 'The phase diagram', Edward Arnold, London, 1967.
Jl Gaskell Introduction to Metallurgical Thermodynamics, McGraw-Hill, New V"»ik, 1973.
p Gordon, Principles of Phase Diagrams in Materials Systems, McGraw-Hill, New Vuk, 1968.
Hillen Calculation of Phase Equilibria', Chapter 5 in Phase Transformations, \merican Society for Metals, Ohio, 1970.
M Hillert, 'The uses of the Gibbs free energy-composition diagrams', Chapter 1 in Lectures on the Theory of Phase Transformations, H.I. Aaronson (Ed.) published by the The Metallurgical Society of AIME, New York, 1975.
A Prince Alloy Phase Equilibria, Elsevier, London, 1966.
G V. Ravnor, 'Phase diagrams and their determination', Chapter 7 in R.W. Cahn (Ed ) Physical Metallurgy, North-Holland, 1970.
F.N. Rhine5, Phase Diagrams in Metallurgy, McGraw-Hill, New York, 1956.
p!c Shewmon, 'Metallurgical thermodynamics', Chapter 6 in R.W. Cahn and V I laasen (Eds.) Physical Metallurgy, North-Holland, 1983.
■\ D l\.ll"n, 'Phase diagrams', Chapter 7 in R.W. Cahn and P. Haasen I HI* i Physical Metallurgy, North-Holland, 1983.
R \ s» iiin; Thermodynamics of Solids, Wiley, New York, second edition, 1972
D R h   W sst'j Ternary Equilibrium Diagrams, Chapman & Hall, 2nd diti.-u, 1.982.
C.H.P.. Lupis, Chemical Thermodynamics of Materials, North Holland, 1983, :
Exercises
1.1 .The specific heat of solid copper above 300 K is given by
Cp = 22.64 + 6.28 x 10~3 T Jmor'KT1
By how much does the entropy of copper increase on heating from 300 to 1358 K?
1.2 With the aid of Equation 1.11 and Fig. 1.5, draw schematic free energy-^. pressure curves for pure Fe at 1600, 800, 500 and 300 °C.
1.3 Estimate the change in the equilibrium melting point of copper caused by a change of pressure of 10 kbar. The molar volume of copper is 8.0 x 10-6 m3 for the liquid, and 7.6 x 10~6 for the solid phase. The
..latent heat of fusion of copper is 13.05 kJ mol-1. The melting point is ,1085°C.
1.4. For a single component system, why do the allotropes stable at high temperatures have higher enthalpies than allotropes stable at low temperatures, e.g. H(y-Fe) > H(ct-Fe)? • Determine, by drawing, the number of distinguishable ways of arranging two black balls and two white balls in a square array. Check your answer with Equation 1.24.
/ nermoaynamics and phase diagrams
1.6 By using Equations 1.30 and 1.31, show that the chemical potentials of A and B can be obtained by extrapolating the tangent to the G~X Jf curve to XA = 0 and XB = 0.
1.7 Derive Equation 1.40 from 1.31 and 1.39.
1.8 15 g of gold and 25 g of silver are mixed to form a single-phase ideal solid solution.
(a) How many moles of solution are there?
(b) What are the mole fractions of gold and silver?
(c) What is the molar entropy of mixing?
(d) What is the total entropy of mixing?
(e) What is the molar free energy change at 500 °C?
(f) What are the chemical potentials of Au and Ag at 500 °C taking the free energies of pure Au and Ag as zero?
(g) By how much will the free energy of the solution change at 500 °C if one Au atom is added? Express your answer in eV/atom.
1.9 In the Fe~C system Fe3C is only a metastable phase, whilst graphite is. the most stable carbon-rich phase. By drawing schematic free energy-composition diagrams show how the Fe-graphite phase diagram compares to the Fe-Fe3C phase diagram from 0 to 2 wt% Fe. Check your answer with the published phase diagram in the Metals Handbook for example.
1.10 Consider a multicomponent system A, B, C . . . containing several phases a, (3, 7 ... at equilibrium. If a small quantity of A (dnAmoi) is taken from the a phase and added to the |3 phase at constant T and P what are the changes in the free energies of the a and (3 phases, dG" and dGp? Since the overall mass and composition of the system is unchanged by the above process the total free energy change dG = dG01 + dGp = 0. Show, therefore, that u.A = |xA. Repeating for other pairs of phases and other components gives the general equilibrium conditions, Equation 1.48.
1.11 For aluminium A//v = 0.8eV atom-1 and ASJR = 2. Calculate the equilibrium vacancy concentration at 660 °C (Tm) and 25 "G.
1.12 The solid solubility of silicon in aluminium is 1.25 atomic % at 550 °C and 0.46 atomic % at 450 °C. What solubility would you expect at 200 °C? Check your answer by reference to the published phase diagram.
1.13 The metals A and B form an ideal liquid solution but are almost immiscible in the solid state. The entropy of fusion of both A and B is S^Jmor'K-1 and the melting temperatures are 1500 and 1300 K respectively. Assuming that the specific heats of the solid and liquid are identical calculate the eutectic composition and temperature in the A-B phase diagram.
1.14 Write down an equation that shows by how much the molar free energy of solid Cu is increased when it is present as a small sphere of radius r in liquid Cu. By how much must liquid Cu be cooled below Tm before a
Exercises
59
solid particle of Cu can grow if the particle diameter is (i) 2 fun, iii) 2 nm (20 A)? (Cu: Tm = 1085 °C = 1358 K. Atomic weight 63.5. Density 8900 kg m-3. Solid/liquid interfacial energy y = 0.144 J m-2. Latent heat of melting L = 13 300 J moP1.) 1 15 Suppose a ternary alloy containing 40 atomic % A, 20 atomic % B, 40 atomic % C solidifies through a ternary eutectic reaction to a mixture of a £ and 7 with the following compositions: 80 atomic % A, 5 ■atomic % B, 15 atomic % C; 70 atomic % B, 10 atomic % A, 20 atomic % C; and 20 atomic % B, 10 atomic % A, 70 atomic % C. What will be the mole fractions of a, $ and -y in the microstructure? 116 Show that a general expression for the chemical potential of a component in solution is given by
(xA = Ga + SA(T0 - T) + RT In yAXA + (P - PQ)Vm where G°A is the free energy of pure A at temperature T0 and pressure P0, SA is the entropy of A, R is the gas constant, -yA the activity coefficient for A, XA the mole fraction in solution, Vm is the molar volume which is assumed to be constant. Under what conditions is the above equation valid?