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 equilaleral 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 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 A T Q C Fig. 1.40 The Gibbs triangle. Ternary equilibrium 49 i the three binary systems AB, BC and CA are simple eutec-ŕ'i ae energy surfaces exist for three solid phases a, ß and 7 and the liquid "h "l At this temperature the liquid phase is most stable for all alloy ' ' 'iti^r,^ At lower temperatures the GL surface moves upwards and comport10115*- ■lirually intersects the C* surface as shown in Fig. 1.41b. Alloys with ompi^itions m the vicinity of the intersection of the two curves consist of L *t equilibrium. In order for the chemical potentials to be equal in both '§■ 1-41 (a) Free energies of a liquid and three solid phases of a ternary system. 50 Thermodynamics and phase diagrams (b) (c) 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 / in 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, ß 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 + ß triangle for example arises because the common tangential plane simultaneously touches the G01, Gß and GL surfaces. Therefore any alloy with a composition within the L + a + ß triangle at this temperature will be in equilibrium as a three-phase mixture with compositions given by the corners Ternary equilibrium 51 0/ c c Cg) T=E (h) E^T Fig. 1.42 Isothermal sections through Fig. 1.44. (After A. Prince, Alloy Phase tqwhbria, 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 + ß + 7. This is Known as the ternary eutectic point and the temperature at which it occurs is liau'dernaryieUteCtÍC temPerature> Fi§- l-42g. Below this temperature the q 1 is no longer stable and an isothermal section contains three two-phase isoth« a 1 °ne three"Phase tie trian§,e a + ß + 7 as shown in Fig. 1.42h. If into a tí Se.Ctl0ns are instructed for all temperatures they can be combined In ordf ninSÍOnal ternary phase diagram as shown in Fig. 1.44. equilibr'Cr the C°UfSe °f solidification of a ternary alloy, assuming mm is maintained at all temperatures, it is useful to plot the liquidus 52 Thermodynamics and phase diagrams 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 ß solidify simultaneously. Finally at E, the ternary eutectic point, the liquid transforms simultaneously into a + ß + 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 + ß, and finally all liquid disappears and the ot + ß + 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 oVa dp,B d(n,B - n,A) "*T = xT = —I— (L63) and that the slope of the free energy-composition curve is given by AG = M-B - txA dZB 1 (1.64) ■í--------^£ľ* Additional thermodynamic relationships 53 Fig. 1 44 The equilibrium solification of alloy X. (After A. Prince, Alloy Phase Equilibria, Elsevier, Amsterdam, 1966.) 1 f X z 2 AnJ,'il A*ertical section between points 1, 2 and Xin Fig. 1.44. (After A. Prince, UOy Phase EwHbria, Elsevier, Amsterdam, 1966.) 54 Thermodynamics and phase diagrams -d(u,B_uA) dG Fig. 1.46 Evaluation of the change in chemical potential due to a change in composi--tion. (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 XAXB leads to the following equalities: d2G -XAd\xA - XBd\^B - XAXB-^dXB (1-65); which are the required equations relating djxA, dp,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 ^ - -51- - 20 (1.66) dX2 XAXB K ' For an ideal solution O. - 0 and d2G RT dX2~ xAxB 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 \iB = GB + RT\nyBXB Therefore du,B dXv RTÍ ZßdTB *b[ Tb^b. RT\ d In 7b XB{ dlnZe (1.68) (1.69) The kinetics of phase transformations 55 A 'mil n icl.ttionship can be derived for dfjt,A/dX"B. Equation 1.65 therefore become*. (1.70) Gompanni: tquations 1.65 and 1.70 gives í d In -vol (1.71) ■v^ä-?-^ i+ d In 7a dlnXA = RT\1 + d In 7B dlnXB 1.9 ľhľ Kinetics of Phase Transformations The ihei nit «dynamic functions that have been described in this chapter apply to «swonv» ill it are in stable or metastable equilibrium. Thermodynamics can then hue N. used to calculate the driving force for a transformation, rqu.tti'in I a but it cannot say how fast a transformation will proceed. The sludv o[ ii-n*. fast processes occur belongs to the science of kinetics. Lei u* icdiawFig. 1.1 for the free energy of a single atom as it takes part in a ph.i*.».' li information from an initially metastable state into a state of lower tree eiK-i-:\. Fig. 1.47. If G: and G2 are the free energies of the initial and ^----------2£------------2f Initial Activated Final state state state hü*.' / Transřormations from initial to final state through an activated state of mgner free energy. 56 Thermodynamics and phase diagrams final states, the driving force for the transformation will be AG = G2 - Gx. However, before the free energy of the atom can decrease from Gi to G2 the atom must pass through a so-called transition or activated state with a free energy AGa above Gx. 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/Na) 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 / AG rate « exp I — kT Putting AGa = A//a - TASa and changing from atomic to molar quantities enables this equation to be written as (a ua\ -—J (1.72) This equation was first derived empirically from the observed temperature dependence of the rate of chemical reactions and is known as the Arrheniusi 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. Exercises 57 Gi^ktll, Introduction to Metallurgical Thermodynamics, McGraw-Hill, New ^oik. Ll>73. fi0rdon.''«'" Wles °f Fhase Diagrams in Materials Systems, McGraw-Hill, Ne« Yoik lu,)8- \a Hillťri t iKi'lationof Phase Equilibria', Chapter 5 in Phase Transformations, Amu k .n Society for Metals, Ohio, 1970. M HilltTE ''I hi. uses of the Gibbs free energy-composition diagrams', Chapter 1 in LnUires on the Theory of Phase Transformations, H.I. Aaronson (Ed.) publice I by the The Metallurgical Society of AIME, New York, 1975. A- Prince, -\Ho\ Phase Equilibria, Elsevier, London, 1966. G V. Ramm Phase diagrams and their determination', Chapter 7 in R.W. Cahn (Ed ) Physical Metallurgy, North-Holland, 1970. F.N. Rhine*. Phase Diagrams in Metallurgy, McGraw-Hill, New York, 1956. P.G SliLwnii'n 'Metallurgical thermodynamics', Chapter 6 in R.W. Cahn and P ILi.i^n (Eds.) Physical Metallurgy, North-Holland, 1983. A D l\.lion Phase diagrams', Chapter 7 in R.W. Cahn and P. Haasen (Eds i ľh *.. .'Metallurgy, North-Holland, 1983. R.A Sv ihn /- t'.rmodynamics of Solids, Wiley, New York, second edition, vri DRI- W^i ulitími l"0 C H P 1 up-1«\3 ľernary Equilibrium Diagrams, Chapman & Hall, 2nd Chemical Thermodynamics of Materials, North Holland, Exercises 1.3 1.1 The bpccinc heat of solid copper above 300 K is given by Cp = 22.64 + 6.28 x KT3 T J mol-1 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. 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 ätiotropes stable at high temperatures have higher enthalpies than ätiotropes stable at low temperatures, e.g. H(y-Fe) > H(a-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. 1.5 i 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~ \ 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 ide ľ 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 taki;1-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 ■: 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 i-the most stable carbon-rich phase. By drawing schematic free energ. composition diagrams show how the Fe~graphite phase diagram compares to the Fe-Fe3C phase diagram from 0 to 2 wt% Fe. Check you answer with the published phase diagram in the Metals Handbook f->i example. 1.10 Consider a multicomponent system A, B, C . . . containing sevet il phases a, ß, y ... at equilibrium. If a small quantity of A (d«Amol) i^ taken from the a phase and added to the ß phase at constant T and /' what are the changes in the free energies of the a and ß phases, dO and dGp? Since the overall mass and composition of the system is unchanged by the above process the total free energy chan_^ áG = dGw + dGß = 0. Show, therefore, that |xA = |xA. Repeating f"1 other pairs of phases and other components gives the general equilibrium conditions, Equation 1.48. 1.11 For aluminium AHV = 0.8~eV atom"1 and ASV/R = 2. Calculate tru-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 '< and 0.46 atomic % at 450 °C. What solubility would you expect ■• 200 °C? Check your answer by reference to the published phase di ■ gram. 1.13 The metals A and B form an ideal liquid solution but are almc-: immiscible in the solid state. The entropy of fusion of both A and B i-8A J morlK~l and the melting temperatures are 1500 and 1300 K respectively. Assuming that the specific heats of the solid and liquid a -identical calculate the eutectic composition and temperature in the A- M phase diagram. 1.14 Write down an equation that shows by how much the molar free enerj * of solid Cu is increased when it is present as a small sphere of radius r 11 liquid Cu. By how much must liquid Cu be cooled below Tm before i Exercises 59 lid particle of Cu can grow if the particle diameter is (i) 2 (xm, (ii) 2 nm (20 Á)? (Cu: 7"^ = 1085 °C = 1358 K. Atomic weight 63.5. Density 8900 kg m"3. Solid/liquid interracial energy 7 = 0.144 J m~2. Latent heat of melting L - 13 300 J mol-1.) 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, ß and 7 w'tn tne 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, ß and 7 in the microstructure? í Show that a general expression for the chemical potential of a component in solution is given by Ha = Gl + SA(T0 -T) + RT In 7a*a + (P - Po)Vm where GA is the free energy of pure A at temperature T0 and pressure P0, SA is the entropy of A, R is the gas constant, 7A 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?