Phase Transformations and Phase Diagrams W-151. Appendix 5A The GFE and Available Work

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1 Phase Transformations and Phase Diagrams W-151 Appendix 5 Appendix 5A The GFE and Available Work In this appendix we show that the GFE is the energy available to do work at constant temperature, pressure, and number of atoms. If the number of atoms is constant, the system is closed and no atoms are entering or leaving the system. At constant temperature and pressure, the differential of the GFE according to Equation 4.14 written in differential form is equal to Equation 5A.1. dg 5 de 2 T ds 1 P dv From the first law of thermodynamics, energy is conserved. If heat and work are the only forms of energy transfer possible, the change in internal energy de is given by Equation 5A.2, where dq is the heat input to the material and dw is the work input. de 5 dq 1 dw 5A.1 5A.2 Inserting de from Equation 5A.2 into Equation 5A.1 results in Equation 5A.3. dg 5 dq 2 T ds 1 dw 5A.3 If the heat (dq) is transferred reversibly, then from Equation 5.1, dq is equal to T ds, with Equation 5A.4 as a result. dg 5 dw 5A.4 The integral of Equation 5A.4 shows that the available reversible work is equal to the GFE for a process at constant temperature, pressure, and number of atoms. All forms of work are included. Appendix 5B The Thermodynamics and Kinetics of Nucleation and Growth Phase Transformations 5B.1 The Thermodynamics of Phase Transformations The following thermodynamic analysis applies to isothermal phase transformations at a temperature close to the equilibrium temperature, such as the transformation from FCC iron to BCC iron, or the transformation from water to ice. Transformations far from equilibrium conditions and the martensite transformation in iron-carbon alloys are not covered by this analysis, because equilibrium thermodynamics is utilized for the analysis. For a single-component system such as the pure iron at the temperature where two phases are in equilibrium, the GFE (G) for each of the two phases is equal, as shown in Equation 5B B.1

2 W-152 CHAPTER 5 If -phase iron transforms to -phase iron at the equilibrium temperature (T C), the change in GFE (DG ) is 0, as shown in Equation 5B.2. D B.2 From Equation 4.14, at constant temperature and pressure, the change in the GFE is the enthalpy change in the transformation (DH ) minus the temperature (T ) times the entropy change in the transformation (DS ), as shown in Equation 5B.3. D 5DH 2 T DS 5B.3 From Equations 5B.3 and 5B.2, the entropy change in the transformation at the equilibrium temperature is determined from the enthalpy of transformation and the equilibrium transformation temperature, as shown in Equation 5B.4. DS 5 DH T The entropy change in the transformation is related to the change in structure that occurs as a result of the phase transformation. We assume that the entropy change is the same if the transformation occurs at a temperature T 2 DT rather than at T. This should be valid as long as the final structure is the same resulting from the transformation at T and at T 2 DT. However, if the transformation occurs at the temperature T 2 DT, the change in GFE is no longer equal to 0 because the and phases are no longer in equilibrium at the temperature T 2 DT. Solving for the change in GFE from Equations 5B.3 and 5B.4 yields Equation 5B.5. D 5 DH 2 TDS 5 DH 2 T D 5 DH st 2 T d T 5 DH DT T DH T 5B.4 D 5 DS DT 5B.5 The term DT is the super-cooling; it is the equilibrium temperature minus the temperature of the transformation. The change in the GFE is directly proportional to the amount of super-cooling (DT). Example Problem 5B.1 Determine the change in the GFE of iron if 1 mole of -phase iron transforms to -phase iron at 1085 K (812 C) and at 1 atmosphere of pressure during cooling. The equilibrium transformation temperature is 1185 K (912 C) and the latent heat of this transformation upon heating is 900 J per mole of iron atoms. Solution At the equilibrium temperature of 1185 K and at constant pressure of 1 atmosphere, the change in the GFE of transforming pure iron from the phase to the phase is 0. D DH 2 T DS

3 Phase Transformations and Phase Diagrams W-153 Everything in the above equation is known except the entropy of the transformation, so the entropy change can be determined. DH 2900 J /m o le J DS T 1185 K m o le? K If the super-cooling is 100 K, then the change in the GFE is from Equation 5B.5. J J D 5 DS DT s100 K d m o le? K m o le The change in the GFE is negative, as it must be for a spontaneous transformation. The calculation of the change in the GFE (DG ) assumes that all of the atoms in the material transform. However, the material does not transform all at once. As we discussed in Section 5.6, for materials that transform by nucleation and growth at a constant temperature, the transformation requires time, as is demonstrated in Figure B.2 The Thermodynamics of Homogeneous Nucleation We continue the study of phase transformations by using the example of the -to- -phase transformation in iron. The nucleation is homogenous if the -phase nucleus forms in the volume of the phase where no significant defects are present. If we assume that the -phase nucleus is a sphere of radius r, then there must be an interface between the sphere of the phase and the matrix of the phase, as shown in Figure 5B.1. Nuclei can also be in other shapes, such as cubes or tetrahedrons. The interface has an energy in units of energy per unit area. Table 3.2 presents some surface energies between some solids and air; also presented in Table 5B.1 are some interface energies between the liquid and solid phases of some metals. The energy required to form an interface around a spherical nucleus of radius r is 4 r 2 ; this is an increase in energy. Table 5B.1 Experimentally Determined Solid-Liquid Interfacial Energies in mj per m 2 Material G LS Material G LS Material G LS Mercury 24.4 Germanium 181 Cobalt (FCC) 234 Water 32.1 Silver 126 Iron (BCC) 204 Gallium 55.9 Gold 132 Palladium 209 Tin 54.5 Copper 177 Platinum 240 Lead 33.3 Manganese 206 Bismuth 54.4 Aluminum 93 Nickel 255 Antimony 101 Based on data from Turnbull, D., Journal of Applied Physics, 21, (1950) Figure 5B.1 Interface ~ Nucleus of Matrix An -phase nucleus of radius r forming in a matrix of phase, with a 2 phase interface.

4 W-154 CHAPTER 5 In addition to the increase in energy associated with formation of an interface, there is an energy decrease associated with the volume of material in the nucleus that transforms from the to the phase, as we calculated in Example Problem 5B.1 above. Change DG from energy per mole to energy per unit volume. Then the total GFE change (DG T ) when a nucleus of radius r forms equals the sum of the negative volume and positive surface energy terms, as shown in Equation 5B.6. DG T 5 4 r3 D 1 4 r 2 3 5B.6 Since the positive surface energy term is proportional to the radius squared, and the negative volume energy term is proportional to the radius cubed, it is possible at very small values of radius for the surface energy term to be larger than the volume energy. Figure 5B.2 shows the two functions (4 r 3 /3) D and 4 r 2 plotted as a function of radius. The sum of these two plots is the total GFE of the nucleus (DG T ). There is a maximum in the plot of DG T ; the radius corresponding to this maximum is the critical nucleus size (r * ). If the radius is less than the critical nucleus size (r * ), the total GFE is reduced by shrinking in size until the nucleus disappears. If the radius is greater than the critical nucleus size (r * ), the total GFE is reduced if the nucleus grows. The value of the critical nucleus size (r * ) is found by determining the value of the radius for the maximum of the total GFE, by taking the derivative of the total GFE change with respect to radius, and setting the derivative equal to 0, as shown in Equation 5B.7. ddg T dr 5 d 1 4 r3 3 DG 2 dr 1 ds4 r2 d dr 5 0, for r 5 r * 5B.7 4 r 2 +DG r* DG* r DG DG T 4 3 pr3 DG ga Figure 5B.2 radius r. Volume, surface, and total energy changes associated with nucleation of a spherical particle of

5 Phase Transformations and Phase Diagrams W-155 Taking the derivative of Equation 5B.7 and setting r 5 r * results in Equation 5B.8. 4 r *2 D 1 8 r * 5 0 5B.8 Solving for the critical nucleus radius results in Equation 5B.9. r * 2G 5 2 D Since is positive and D is negative, the value of the critical nucleus radius (r * ) is positive. Since the critical nucleus radius (r * ) is a function of D, when D is 0, such as when the temperature is at the equilibrium temperature, then the critical nucleus radius (r * ) goes to infinity, or it requires an infinitely large nucleus to form before the total GFE can start to decrease with an increase in nucleus size. This can be seen by substituting for D from Equation 5B.5 into Equation 5B.9, to make Equation 5B.10. 5B.9 r * T DH DT 5B.10 Snowflakes, which are solids that form from water vapor, demonstrate Equation 5B.10. When the temperature is just below the equilibrium freezing temperature, DT is small and the snowflakes are large. When the temperature is very cold, DT is large and the snowflakes are small. Figure 5.5 shows that the transformation rate goes to 0 as the super-cooling goes to 0. One reason for the transformation rate of zero is that the change in the GFE for the transformation (D ) goes to 0, so there is no chemical force driving the atoms to the new phase. From Equation 4.37, if the gradient of the GFE with respect to position is 0, there is no net flux of atoms. A second reason is that the critical nucleus size becomes infinitely large at the equilibrium temperature, where DT is equal to 0, as demonstrated by Equation 5B.9. At the equilibrium temperature, in an infinitely large nucleus an infinite number of atoms must rearrange themselves into a new structure, with no driving force for this to happen. The probability of this happening approaches 0. The magnitude of the activation GFE (DG * ) is found by substituting the critical nucleus radius (r * ) into the total GFE in Equation 5B.6. The value of DG * is positive, and it is the activation energy that must be provided to a system to make it change to a lower energy state, as shown in Figure 5B.2. Since atoms are in continuous motion, they have some probability of moving to atom positions of the new structure. If only a few atoms are necessary to form a critical-size nucleus, then there is a significant probability that this will occur. Example Problem 5B.2 Calculate the critical nucleus size for the transformation of -phase iron to -phase iron at 1085 K. The density of -phase iron at 1085 K is kg/m 3, the atomic mass is kg/mole, and the interface energy (G ) is 0.55 J/m 2. Solution From Example Problem 5B.1, at 1085 K the change in the GFE is 276 J/mole. Converting 276 J/mole to J/m 3 by dimensional analysis results in the following. D J m o le 1 1 m o le g2 103 g k g k g 2 m J m 3

6 W-156 CHAPTER 5 The critical nucleus size is determined from Equation 5B.9. r * 5 2 2G DG 5 2s0.55 J /m 2 d J /m m n m Since the atomic radius is typically 0.1 nm, the critical nucleus radius at this temperature is a few thousand atoms across. 5B.3 Thermodynamics of Heterogeneous Nucleation Heterogeneous nucleation occurs when a nucleus of the new phase forms on some defect in the crystal, such as at a surface or an interface, as is demonstrated in Figure 5.4. A schematic of heterogeneous nucleation is shown in Figure 5B.3b, where an -phase nucleus of radius r forms at a grain boundary in the matrix of the phase. The -phase nucleus eliminates an area of 2 r 2 from the grain boundary, with an energy per unit area of G. The change in the GFE is shown in Equation 5B.11. gb DG T 5 4 r3 D 1 4 r r 2 G 3 gb DG T 5 4 r3 3 D 1 4 r G gb 2 2 5B.11 Grain boundary energies are approximately 0.2 to 0.4 times the surface energy and depend upon the orientation of the grains relative to each other, as we discussed in Section For heterogeneous nucleation, in Equations 5B.9 and 5B.10 is replaced by ( 2 G /2), and Equation 5B.12 shows the gb critical nucleus size for heterogeneous nucleation G gb r * D G gb 2 2 T DH DT 5B.12 Comparing Equation 5B.12 with 5B.9, the critical-size nucleus for heterogeneous nucleation at a grain boundary must be smaller than for homogeneous nucleation. Also, the GFE to form a critical-size nucleus (DG T ) calculated with Equation 5B.11 is reduced because of the smaller of critical nucleus size. The smaller critical nucleus size makes it more likely that nuclei form at grain boundaries than within a grain. Figure 5B.4 shows precipitate particles that have formed along the grain boundary of a highstrength aircraft-grade aluminum alloy. Precipitates along grain boundaries lead to more-brittle alloys. Processing procedures have been developed to suppress grain boundary nucleation and growth, which Grain 1 Grain 2 (a) Grain boundary Nucleus ~ (b) Figure 5B.3 boundary. (a) A grain boundary in the phase. (b) A nucleus of -phase material forming at the -phase grain

7 Phase Transformations and Phase Diagrams W-157 Figure 5B.4 A scanning electron micrograph of a 6061 aluminum alloy with AlFeSi intermetallic particles that have formed along a grain boundary. The grain boundary particles are the large particles that form a wavy line in the photo. Magnification 142,0003. (Photo courtesy of Claves, S.R., and Misiolek, W.Z., Lehigh University, Department of Materials Science and Engineering, the Institute for Metal Forming.) we will cover in Chapter 7, on strong solids. In other applications, heterogeneous nucleation is promoted to make a transformation occur more rapidly, as in the seeding of clouds to produce rain, and in catalytic converters. Example Problem 5B.3 Assume that the grain boundary energy for -phase iron is 0.85 J/m 2. What is the critical nucleus size for the heterogeneous nucleation of an -phase nucleus in a -phase grain boundary, under the same conditions as in Example Problem 5B.2? Solution From Equation 5B.12, 21 G 2 G 2 gb J /m 0.85 J /m r * m 5 24 n m D J /m 3 The nucleus forming heterogeneously in the grain boundary of the phase is approximately one-fourth the size of the homogeneous nucleus. The smaller nucleus forms more easily. 5B.4 The Kinetics of Nucleation and Growth Transformations The activation GFE for nucleation (DG T ) is calculated by substituting the critical nucleus size (r * ) into Equation 5B.6 for homogeneous nucleation, or into Equation 5B.11 for heterogeneous nucleation, for

8 W-158 CHAPTER 5 a spherically shaped nucleus. Then the probability of forming a critical-size nucleus during one atomvibration cycle at any nucleation site (P n ) is given in Equation 5B.13. P n 5 ex p 2 1 DG T kt 2 In homogeneous nucleation, the number of sites per unit volume is the number of atoms per unit volume (N), and in heterogeneous nucleation the number of sites is the number of atoms at defects per unit volume, such as along a grain boundary. The equilibrium number of critical-size nuclei per unit volume (N * ) is the product of the probability that the nucleus forms (P n ) times the number of formation sites per unit volume (N), as shown in Equation 5B.14. N * 5 N ex p 2 1 DG T kt 2 Growth of a new phase occurs as atoms surrounding the critical-size nuclei diffuse across the interface between the old phase and the new phase. We assume that there are N s atoms that surround a criticalsize nucleus of the new phase, and that the frequency of attempts at diffusing from the old phase to the new nucleus is equal to the vibrational frequency ( ) of the atoms, and the activation GFE for diffusion is DG D. Then the rate (R a ) at which atoms diffuse to the critical-size nucleus is given by Equation 5B.15. R a 5 N s ex p 2 1 DG D kt 2 5B.13 5B.14 5B.15 The growth rate per unit volume of the equilibrium number of critical-size nuclei (R * ) is then given by Equation 5B.16. R * 5 N * R a 5 N ex p 2 1 DG T kt 2 3 N s ex p 2 1 DG D kt 24 5B.16 The analysis of the transformation rate given above is that due to Volmer and Weber. More advanced analyses of phase transformation rates have been developed, and for these analysis see some of the references at the end of this chapter. The shape of the transformation (growth) rate as a function of temperature, shown schematically in Figure 5.5, can now be explained with the theory developed above. At the equilibrium transformation temperature, DG T goes to infinity, because the critical nucleus size (r * ) goes to infinity and because the interface energy term is always positive and finite. Therefore, the first exponential term (nucleation rate) in Equation 5B.16 goes to 0 at the equilibrium temperature (T ). Both of the exponential terms go to 0 as the temperature goes to 0. In between these two limits of 0, the transformation rate reaches a maximum at some temperature. This explains the shape of the growth-rate curve in Figure 5.5. Appendix 5C Continuous-Solid Solution Phase Diagrams and Gibbs Free Energy Possible GFE versus composition plots of the mixture of atoms as a function of chemical composition at different temperatures are used to explain the appearance of phase diagrams. For example, the reason for the different equilibrium chemical compositions of the liquid and solid in the two-phase region of

9 Phase Transformations and Phase Diagrams W-159 the Cu-Ni phase diagram in Figure 5.7a is demonstrated. The GFE can be calculated as a function of composition to predict phase diagrams; however, this calculation is beyond the scope of this textbook. In these appendices the general shape of the GFE versus composition plots is justified based upon our understanding of the strength of the bonding between different atoms and the entropy of atom mixtures, but it is not rigorously calculated. Then, knowing the appearance of the phase diagram, some reverse engineering is used to develop reasonable GFE versus composition plots that would result in this phase diagram. The total GFE of the mixed atoms (G T ) is calculated from the total enthalpy (H T ) of the mixed atoms minus the temperature times the entropy (2TS T ) of the mixed atoms, as shown in Equation 5C.1 G T 5 H T 2 TS T 5C.1 If there is no change in the enthalpy as A and B atoms are mixed to form a solution relative to the enthalpy of A and B separated, then A and B mix ideally. If the A and B atoms mix ideally, the enthalpy (H T ) of mixing N A A-type atoms and N B B-type atoms is shown in Equation 5C.2 H T (ideal) 5 N A H A 1 N B H B 5C.2 where H A is the enthalpy per atom of the A-type material, and H B is the enthalpy per atom of the B-type material. In Figure 5C.1 the enthalpy for ideal mixing (H T ) is plotted on the y axis, and the x axis is the chemical composition of the mixture of atoms in percent (C B 5 [N B /N] 3 100). In Figure 5C.1 material B would have a higher melting temperature than material A does, because B has the more negative enthalpy, indicating stronger bonding. Also plotted in Figure 5C.1 is the negative of temperature (T) times the total entropy (S T ). For an ideal solution, we assume that the entropy of mixing is the primary contribution to the total entropy (S T ) that changes with composition. Other contributions to the entropy, such as that due to the thermal vibration of the atoms, are assumed to be relatively constant with respect to composition variation, and configurational entropy due to displacements resulting from atom substitution is neglected. Therefore, the shape of the plot of the total entropy times temperature (2TS T ) as a function of composition has the same shape as the plot of the mixing entropy times temperature. The mixing entropy is calculated for an ideal mixture of atoms from Equation 4.12, and the shape is presented in Figure 4.9. Figure 5C.1 shows that the total enthalpy (H T ) minus the temperature-entropy term (TS T ) is the total GFE (G T ) of the mixture shown as a function of chemical composition. Figure 5C.2 shows hypothetical GFE curves from Equations 5C.1 and 5C.2 for both the liquid and solid -phase Cu-Ni mixtures at the melting temperature of pure Cu (1085 C), assuming that there is Energy H A C B HT TS T H B G T Figure 5C.1 A schematic of the enthalpy (H T ) for an ideal mixture of A and B atoms that have enthalpy H A and H B ; the total entropy of the mixture of atoms (S T ) times the temperature (T), such as 1000 K; and the GFE of mixtures of atoms (G T ) as a function of the atom percent of B atoms (C B ).

10 W-160 CHAPTER 5 Energy G L C Ni G s Figure 5C.2 A schematic of the GFE of liquid (G L ) and solid (G S ) as a function of the atom percent of Ni atoms (C Ni ) for Cu-Ni mixtures at 1085 C. ideal mixing in both the liquid and the solid phases. The general shape of both the liquid and solid GFE curves is similar to that shown in Figure 5C.1; however, the enthalpy and the entropy of the liquid and the solid mixtures are different, because the bonding and atom configuration is different in the liquid than in the solid. At the melting temperature of pure Cu (1085 C), the GFE for pure liquid and solid Cu are equal, as we discussed in Section 5.3; this is shown in Figure 5C.2. Also, at 1085 C for all chemical compositions with a Ni atom percent greater than 0, the solid -phase GFE curve (G S ) is lower in energy than the liquid phase GFE curve (G L ), as shown in Figure 5C.2, because at 1085 C all alloys with any amount of Ni are solid phase, as shown in the phase diagram in Figure 5.7a. At a temperature of 1300 C, pure Cu is liquid and pure Ni is solid, because the melting temperature of pure Cu is 1085 C, and for pure Ni it is 1455 C. At 1300 C and 0% Ni, the G L curve must be below the G S curve, whereas for 100% Ni the G S curve must be below the G L curve. Therefore at 1300 C the G S and G L curves must cross somewhere between 0% and 100% Ni, as shown in Figure 5C.3. For an alloy mixture of 50 atom percent Cu and 50 atom percent Ni at 1300 C, solid and liquid are in equilibrium, as shown in Figure 5.7a. As a result, the chemical potential of the Ni atoms in both the liquid and the solid phases must be equal. Also, the chemical potential of the Cu atoms in the liquid and solid phases must be equal, but it is not equal to the chemical potential of the Ni atoms. The equality of chemical potentials for atoms in two phases that are in equilibrium is stated in Equation 5.7, and this general equation is rewritten for Ni in liquid and solid as shown in Equation 5C.3. S N i 5 L N i 5C.3 The chemical potential is defined in Equation 5.6, and for Ni in solid and liquid the equation is presented in Equation 5C.4. S N i 5 GS N N i 5 GL N N i 5 L N i 5C.4 In Figure 5C.3, L is proportional to the slope of the liquid GFE N i (GL ) versus chemical composition plot, and S is proportional to the slope of the solid -phase GFE N i (GS ) versus chemical composition plot, because the x axis is N Ni /N times 100. If the chemical potentials are equal, the slopes of the plots for the GFE versus the chemical composition must also be equal. One way to ensure this is to have a common tangent, as shown in Figure 5C.3. It can be shown that the two-phase mixture with chemical compositions given by the points of common tangency results in the minimum total GFE for alloys in the two-phase region, even though the common tangent points are not necessarily at the minimum of the individual G L and G S plots. At equilibrium in the two-phase region of solid plus liquid, the chemical compositions of the liquid (C L ) and solid N i (CS ) phases are given by the chemical composition of the contact points of N i

11 Phase Transformations and Phase Diagrams W-161 Energy C Ni L C Ni S C Ni G L G S Figure 5C.3 A schematic of the GFE of liquid (G L ) and solid (G S ) for Cu-Ni mixtures at 1300 C, as a function of atom percent Ni atoms (C Ni ), showing that the equilibrium chemical compositions of the liquid (C L ) and solid N i ) phases are equal to the compositions of the common tangents. (C S N i the common tangent, as shown in Figure 5C.3. The liquid phase and the solid phase do not have the original chemical composition of 50 atom percent Ni at 1300 C, because the chemical potential of the Ni atoms in the liquid and solid must be equal, and the same argument holds for the Cu atoms. Appendix 5D Eutectic Phase Diagrams and Gibbs Free Energy The GFE versus chemical composition plot of alloys of Ag and Cu can show why the Ag-Cu eutectic phase diagram in Figure 5.8a is significantly different from the Cu-Ni phase diagram in Figure 5.7a. Cu and Ag do not strongly bond to each other. The weak bonding results in a positive enthalpy of mixing a Cu atom into a Ag crystal (DH ); this is the enthalpy required to remove a Ag atom from the Ag crystal cu and replace it with a Cu atom. The value of the enthalpy of the phase (H ) is the enthalpy of pure Ag (H Ag ) plus D H times the number of Cu atoms in the Ag crystal (N cu C u ); in Figure 5D.1 this is plotted as a function of the chemical composition (C C u ). In Figure 5D.1 the value of the enthalpy of the phase (H ) H H Ag H(ideal) H Energy C Cu H Cu TS T Figure 5D.1 A schematic showing that the GFE of the phase ( ) in Ag-Cu alloys is equal to the enthalpy of the phase (H ) that has a positive enthalpy of mixing, minus the temperature times the total entropy (2TS T ) plotted as a function of the atom percent of Cu (C Cu ). The GFE of the phase ( ) is equal to the enthalpy of the phase (H ) minus the temperature times the total entropy (2TS T ). The temperature is approximately 770 C. Also shown is the enthalpy for an ideal mixture (dashed line).

12 W-162 CHAPTER 5 is the enthalpy of pure Cu (H C u ) plus the positive enthalpy of mixing a Ag atom into Cu (DH Ag ) times the number of Ag atoms in the Cu crystal (N Ag ). In Figure 5D.1 the enthalpy for Cu is lower than for Ag, because Cu melts at a higher temperature than Ag does. H(ideal) is the enthalpy of mixing ideal atoms, such as the inert gas atoms, shown as the dashed line in Figure 5D.1. The GFE of the phase ( ) is the sum of the enthalpy of the phase minus the temperature times the total entropy (2TS T ), as shown in Equation 5D.1. 5 H 2 TS T 5D.1 is plotted as a function of chemical composition in Figure 5D.1. The total entropy has the shape of the mixing entropy as discussed in Appendix 5C. The GFE of the phase ( ) is found in a similar way. The values of and are not plotted significantly past their minima, because we will see that when and phases are both present, the GFE of the plus phases follows the common tangent line between the and plots. The Ag-Cu phase diagram in Figure 5.8a shows that the maximum amount of Cu that can be substituted into the phase and the maximum amount of Ag that can be substituted into the phase both increase with temperature up to 780 C. The amount of Cu that minimizes the GFE of the phase is given by Equation 4.16 by inserting D H, and a similar calculation can be made for the amount of Ag cu that minimizes the GFE of the phase by substituting the value of D H into Equation According Ag to Equation 4.16, as the temperature approaches absolute zero, the chemical composition that results in a minimum of the GFE for substitution of Cu into Ag approaches zero, and it is the same for the substitution of Ag into Cu. As temperature is increased, the chemical composition corresponding to the minimum GFE increases according to Equation 4.16, and this explains why the solubility of Cu into Ag and Ag into Cu increases with temperature for temperatures up to 780 C. However, the solvus lines in the Ag-Cu phase diagram in Figure 5.8 do not necessarily correspond to the chemical composition having the minimum GFE. In the Ag-Cu phase diagram in Figure 5.8a, the solvus lines are plots of the chemical composition of the common tangent points to the GFE plot of the and phases ( and ), as is demonstrated in Figure 5D.2 for 700 C. The equilibrium chemical compositions of the and phases in the two-phase region 1 are where the chemical potentials of the Cu atoms in the and phases are equal to each other. The chemical potential of the Cu atoms in the and phases is proportional to a slope of the plot of and versus C C u. A common tangent has equal slopes at each T = 700 C Energy 1 9 C Cu 96 Figure 5D.2 A schematic of the GFE for the and phases in Ag-Cu alloys, as a function of the atomic percent of Cu (C Cu ) at 700 C, showing that the equilibrium chemical compositions of the and phases are given by the points of common tangency.

13 Phase Transformations and Phase Diagrams W-163 tangent point. The points of common tangency and versus composition plots give the equilibrium chemical compositions of the and phases. It can be shown that these chemical compositions also minimize the energy of the combined and phases. The chemical compositions of the phases participating in the eutectic reaction in Figure 5.8a at 780 C are also explained with GFE versus composition plots. At 700 C, G L is higher than and at all chemical compositions, because there is no liquid for any original chemical composition of Ag-Cu, as shown in Figure 5.8a. Therefore, the liquid GFE is not shown in Figure 5D.2. As is shown in Equation 5D.1, there is a negative TS T term in the GFE for each phase. The liquid is disordered and the solid is ordered. The entropy of the liquid (S L ) is greater than that of the solid (S L. S S ). As the temperature increases, the 2TS L term decreases more rapidly than the solid 2TS S term. For this reason, as temperature increases, G L decreases relative to and, and G L eventually touches the common tangent line drawn between and. All three phases have a common tangent at 780 C, as shown in Figure 5D.3. At 780 C for chemical compositions between 13 and 95 atom percent, the chemical potentials of Cu in the liquid,, and phases are all equal, and the three phases are in equilibrium. The chemical compositions of these three phases in equilibrium are equal to the chemical compositions at the points of common tangency to the GFE curves or at 40, 13, and 95 atom percent Cu, respectively. These chemical compositions are shown in the Ag-Cu phase diagram in Figure 5.8a, where the singlephase-fields of liquid,, and touch the eutectic isothermal line at 780 C. In the Ag-Cu phase diagram, for increasing temperatures above 780 C, there is a decrease in the solubility of Cu in and of Ag in. The eutectic phase diagram has ears. Equation 4.16 predicts that the atomic percent of Cu in the phase that minimizes the GFE of the phase and the atomic percent of Ag in the phase that minimizes the GFE of the phase should both increase with temperature. Why does the Ag-Cu phase diagram at temperatures above 780 C appear to be inconsistent with the minimum in GFE of the phase and phase? Figure 5D.4 shows hypothetical GFE curves for a temperature such as 900 C. As temperature is increased, the liquid phase GFE becomes more negative relative to the phase and phase GFE, and the chemical composition of the points of common tangency of the liquid GFE curve to the -phase and -phase GFE curves wrap around toward 0 atom percent Cu for the phase and 100 atom percent Cu for the phase. This results in the equilibrium amounts of Cu in the phase and Ag in the phase approaching 0 as the melting temperature of the solid is approached. It is the requirement of equal chemical potentials for each type of atom in the liquid and solid phases that are in equilibrium that forces the ears onto the eutectic phase diagram. The GFE versus composition plots in Figure 5D.4 show each phase that is present in the Ag-Cu phase diagram in Figure 5.8a, and their equilibrium chemical composition at 900 C, depending upon the original chemical composition (C 0 ) of the alloy. For original chemical compositions from 0 to 6 atom percent Cu, T = 780 C Energy G L L 1 L C Cu Figure 5D.3 A schematic of the GFE for,, and liquid (L) phases in Ag-Cu alloys, as a function of the atom percent of Cu (C Cu ) at 780 C, showing that the equilibrium chemical compositions of these three phases are given by the three points of common tangency.

14 W-164 CHAPTER 5 T = 900 C Energy G L L 1 L L C Cu Figure 5D.4 A schematic of the GFE for,, and liquid phases in Ag-Cu alloys, as a function of the atomic percent of Cu (C Cu ) at 900 C, showing that the equilibrium chemical compositions of the and liquid phases are given by the points of common tangency to the -phase and liquid-phase GFE curves, and similarly for the chemical compositions for equilibrium between the and liquid phases. is lower in energy than for any other plot, and there is only solid phase present in the phase diagram for these original chemical compositions. Original chemical compositions between 6 and 13 atom percent Cu are between the points of common tangency to the and G L versus composition plots in Figure 5D.4; therefore, phase and liquid phase are present with chemical compositions of 6 and 13 atom percent Cu, respectively. For original chemical compositions from 13 to 69 atom percent Cu, there is only liquid, because the G L versus composition plot is lower in energy than the and versus composition plots. Original chemical compositions from 69 to 95 atom percent Cu are between the points of common tangency to the G L and versus composition plots; therefore there is liquid phase plus phase present, and the chemical composition of the liquid and are 69 and 95 atom percent Cu, respectively. For original chemical compositions greater than 95 atom percent Cu, the versus composition plot is lower in energy than for any other of the plots; therefore, there is only solid phase present in the phase diagram in Figure 5.8a. Appendix 5E Phase Diagrams with Compounds and Gibbs Free Energy The GFE of the,,, and liquid phases as a function of temperature and composition in the figures in Appendix 5E explains why the Ga-As phase diagram in Figure 5.14 is different from the eutectic and solid-solution phase diagrams in Figures 5.8 and 5.7. As shown in Figure 5E.1 for 25 C, there is a GFE plot for the compound GaAs ( ) in addition to the GFE for the ( ) and ( ) phases. The versus composition plot is very narrow at the chemical composition 50 atom percent As. This is consistent with the phase diagram, which shows that GaAs has a chemical composition of only the 50 atom percent As. Only this composition of GaAs is stable. The versus composition plot is shown as having the lowest GFE at 25 C, because it has the highest melting temperature of 1238 C. has the highest energy because it has the lowest melting temperature of C. At 25 C for original chemical compositions greater than 0 and less than 50 atom percent As, the phase diagram in Figure 5.14 shows that the and phases are in equilibrium. Figure 5E.1 shows that this two-phase region corresponds to the region between the common tangent points to and versus composition plots. The equilibrium chemical

15 Phase Transformations and Phase Diagrams W-165 T = 25 C Energy 1 C As 1 Figure 5E.1 A schematic of the GFE for,, and phases in Ga-As alloys, as a function of the atomic percent of As (C As ) at a temperature of 25 C. composition of the and phases corresponds to the chemical compositions at the points of common tangency to the to and versus composition plots at 0 and 50 atom percent As, respectively. For original chemical compositions of 50 atom percent, the phase diagram shows that only the phase is present. The chemical composition for the single-phase region of corresponds to the tip of the plot at 50 atom percent, and the plot is below and at 50 atom percent As. The Ga-As phase diagram in Figure 5.14 shows that for original chemical compositions greater than 50 and less than 100 atom percent As at 25 C, there is a two-phase region of plus. Figure 5E.1 shows that this two-phase region corresponds to the region between the common tangent points to the and versus composition plots. The equilibrium chemical composition of the phase and phase corresponds to the chemical compositions at the points of common tangency of the and versus composition plots at 50 and 100 atom percent As, respectively. According to Figure 5.14, at 800 C for chemical compositions from pure Ga to 6 atom percent As, only liquid exists, and the liquid chemical composition is equal to the original chemical composition, because there is only one phase present. Figure 5E.2 shows the GFE versus composition plots at 800 C, and it includes a liquid phase GFE plot (G L ). G L is the lowest energy GFE plot in the region between 0 atom percent As and the tangent point between the liquid and -phase GFE versus composition plots at 6 atom percent As. T = 800 C Energy L G L C As 6 L 1 1 Figure 5E.2 A schematic of the GFE for,,, liquid, and vapor phases in Ga-As alloys, as a function of the atom percent of As (C As ) at a temperature of 800 C.

16 W-166 CHAPTER 5 T = 1000 C Energy C As L G L 14 L 1 L 1 87 L Figure 5E.3 A schematic of the GFE for,,, liquid, and vapor phases in Ga-As alloys, as a function of the atomic percent of As (C As ) at a temperature of 1000 C. According to Figure 5.14, at 800 C for original chemical compositions with more than 6 but less than 50 atom percent As, the chemical composition of the liquid is 6 atom percent As and the phase is 50 atom percent As. These chemical compositions correspond to the points of common tangency to the G L and plots in Figure 5E.2. For the original chemical composition of 50 atom percent As, there is only solid GaAs ( phase), because at 50 atom percent has the lowest energy. According to Figure 5.14, at 800 C for original chemical compositions with As greater than 50 and less than 87 atom percent, GaAs ( phase) is in equilibrium with the phase, whose chemical compositions are 50 atom percent As and 100 atom percent As, respectively. These chemical compositions correspond to the points of common tangency to the and versus composition plots in Figure 5E.2. For chemical compositions of more than 87 atom percent As, the phase diagram is uncertain. The phase diagram in this uncertain region is not justified with Gibbs free energy plots except to show that the vapor free energy plot is less than the liquid and solid free energy plot for pure As. According to Figure 5.14, at 1000 C for chemical compositions from 0 to 14 atom percent As, only liquid exists, and the liquid chemical composition is equal to the original chemical composition, because there is only one phase present. In Figure 5E.3 this corresponds to the region where G L has the lowest energy, between 0 and 14 atom percent As, where the common tangency between liquid and the phase begins. At 1000 C, for original chemical compositions greater than 14 atom percent As but less than 50 atom percent As, there is a mixture of liquid that has a chemical composition of 14 atom percent As and GaAs ( phase) that has a chemical composition of 50 atom percent As. These chemical compositions correspond to the points of common tangency to G L and at 14 and 50 atom percent As, respectively. For the original chemical composition of 50 atom percent As, there is only solid GaAs ( phase), because at 50 atom percent As, is the lowest GFE plot. At 1000 C, for original chemical compositions with As greater than 50 atom percent and less than 87, there is a mixture of GaAs ( phase) with liquid containing 87 atom percent As. These chemical compositions correspond to the points of common tangency to the and G L lots at 50 and 87 atom percent As, respectively. For chemical compositions of more than 87 atom percent As, the phase diagram is uncertain. The phase diagram in this uncertain region is not justified with Gibbs free energy plots except to show that the vapor free energy plot is less than the liquid and solid free energy plot for pure As. At the composition of 50 atom percent As, the GaAs ( phase) melts at 1238 C, as shown in the phase diagram in Figure Figure 5E.4 shows that at 1238 C the G L and equal and have a common tangent at the chemical composition of 50 atom percent As. For any higher temperature only the liquid phase is present, because the liquid GFE plot is lower than any solid GFE versus composition plots. The melting of GaAs at temperatures greater than 1238 C is a congruent melting point.

17 Phase Transformations and Phase Diagrams W-167 T = 1238 C Energy C As G L L L Figure 5E.4 A schematic of the GFE for,,, liquid, and vapor phases in Ga-As alloys, as a function of the atomic percent of As (C As ) at a temperature of 1238 C. Intermediate compounds, such as GaAs, shown in the Ga-As phase diagram in Figure 5.14, and Al 3 Ni, shown in the Al-Ni phase diagram in Figure 5.15, have very sharp GFE versus composition plots, as shown for GaAs in Figure 5E.1. However, intermediate phases, such as AlNi in Figure 5.15, have much broader GFE versus composition plots as a function of composition. Appendix 5F Miscibility Gaps in Metal and Ceramic Binary Phase Diagrams In the phase diagrams discussed above, we assumed that different atoms in the liquid metals and ceramics always form continuous-liquid solutions. However, this is not always the case. It is not unusual for liquids not to mix; oil and water are an example of two immiscible liquids. It is known that copper (Cu) and tungsten (W) liquid metals do not mix at any temperatures. This observation, along with the different densities and melting temperatures, makes production of a Cu-W alloy difficult. Cu-W electrodes are used in spot-welding applications. Cu-W composites are produced by powder metallurgy techniques where powders of the metals are mixed and pressed at high temperatures to sinter the powder particles together. Sintering is covered in Chapter 13, on processing materials. When different atoms mix a high temperature and segregate at lower temperatures, this is called a miscibility gap. Miscibility gaps can occur in liquids as well as in solids. Nickel (Ni) and gold (Au) form a continuous-solid solution at high temperatures, such as 900 C, as shown in Figure 5F.1. The phase 1 has the FCC structure of Au with substitutional atoms of Ni, and the 2 phase has the FCC structure of Ni with substitutional atoms of Au. At 900 C it is possible to continuously change from the 1 phase to the 2 phase by changing the atomic percent of Ni. But the Au and Ni segregate at lower temperatures, primarily because they have a 16% difference in lattice parameter of nm and nm, respectively. The miscibility gap in Figure 5F.1 is the line with a maximum at 810 C and a chemical composition of 41.7 weight percent Ni. Inside the miscibility gap, the atoms separate into the two phases 1 and 2, and the chemical compositions of the 1 and 2 phases are determined by the intersections of a tie line at the specified temperature with the two sides of the miscibility gap. The lever rule is used to determine the atom fractions of each phase. A miscibility gap occurs if the enthalpy of separate A and B atoms or molecules is more negative than the GFE of A and B randomly mixed. There is no mixing entropy when the A and B do not mix; therefore, the enthalpy of separate atoms is equal to the GFE of the separate atoms.

18 W-168 CHAPTER 5 Atomic percent nickel C 1300 Liquid C Temperature C C a C 41.7 (Au, Ni) Au Weight percent nickel Ni Figure 5F.1 The gold-nickel (Au-Ni) phase diagram demonstrating a miscibility gap in a solid. (Based on Binary Alloy Phase Diagrams 2nd ed. Editor Massalski, T.B., ASM International (1990), p. 403.) The microstructure of alloys inside the miscibility gap can be from individual grains of the two final phases to a layered structure of the two phases, depending upon factors such as the GFE of the phases involved, the mobility of the atoms, and the interface energies. Miscibility gaps also occur in solid ceramic systems; for example, there is a miscibility gap in solid mixtures of TiO 2 and SnO 2, which have the same crystal structure. If the miscibility gap is for two liquids L 1 and L 2, then L 1 and L 2 replace 1 and 2 in the discussion above. Lithium (Li) and sodium (Na) liquids have a miscibility gap, as shown in Figure 5F.2, in the region labeled L 1 1 L 2. For temperatures and compositions inside the miscibility gap, the liquid separates into the two liquids L 1 and L 2, where L 1 is rich in Li and L 2 is rich in Na. The chemical compositions of L 1 and L 2 inside the miscibility gap are given by the intersection of a tie line drawn through the miscibility gap at the temperature of interest with the line of the miscibility gap. The single-phase liquid (L) outside the region labeled L 1 L can be continuously varied in composition from pure Li to pure Na. 1 2 A miscibility gap occurs in the Cu-Ni phase diagram in Figure 5.7a for temperatures below C and for compositions centered on 65.5 atom percent Ni. Homework Problem 5F.4 shows that the GFE of substituting 67 atom percent Ni into Cu at 200 C is positive in agreement with this phase diagram. If the GFE change is positive, this substitution is not spontaneous. In addition, there is a dashed line in the phase diagram centered on the point 65.5 atom percent Ni and 345 C called the spinodal. The spinodal corresponds to compositions where d 2 Gyd 2 C B 5 0. The spinodal is the composition where the curvature in the plot of G versus C B changes from curvature down to curvature up (or the reverse); in other words, at the inflection point in the GFE versus composition plots. In between the spinodal lines, the separation of and phases occurs throughout the entire material by wave like changes in composition as a result of coordinated diffusion. The formation of the and phases is periodic, which

19 Phase Transformations and Phase Diagrams W Atomic percent sodium L 250 L 1 + L 2 Temperature C C C (Li) C 97.8 C (Na) Li Weight percent sodium Na Figure 5F.2 The lithium-sodium (Li-Na) phase diagram, demonstrating a miscibility gap in a liquid. (Based on American Society for Metals Online Handbook Vol. 3, ASM International, Metals Park, OH (1992).) differs from the standard nucleation and growth transformation where the nucleation is a local event that results from local diffusion. Appendix 5G Ordering in Metal and Ceramic Binary-Phase Diagrams It is possible to have a metal or ceramic solid solution at elevated temperatures, and an ordered intermediate phase or compound at lower temperatures, as shown in the iron (Fe)-platinum (Pt), phase diagram in Figure 5G.1. At the temperatures between 1350 C and 1394 C, all original chemical compositions shown are a solid solution of Fe and Pt of the FCC ( ) structure, with Fe and Pt atoms randomly mixed onto the lattice sites. For an alloy with the original chemical composition of 35 atom percent Fe and 65 atom percent Pt, there is a transformation upon cooling to the FePt 3 structure at approximately 1350 C. The FePt 3 is an ordered simple cubic structure with iron atoms at the corners of the cube and platinum atoms at the face-centered positions. The transformation of solid solution (disordered) to FePt 3 (ordered) is a disorder-order transformation. Disorder-order transformation is observed at a temperature when the GFE of the ordered A x B y compound becomes more negative than the GFE of a random mixture x A-type atoms and y B-type atoms. The ordered A x B y compound does not have a mixing entropy of mixing, because the atoms are ordered not randomly mixed. There are also disorder-order transformations at 50 atom percent platinum to form FePt and at 25 atom percent platinum to form Fe 3 Pt.

20 W-170 CHAPTER Weight percent platinum C L C 1519 C Temperature C ( Fe) 1394 C ( Fe, Pt) ~1300 C ~50 ~1350 C ~ ( Fe) Fe 912 C Magnetic transformation 770 C 1 (Fe3Pt) 2 3 (FePt) (FePt 3 ) Atom percent platinum Figure 5G.1 The iron-platinum (Fe-Pt) phase diagram. (Based on Binary Alloy Phase Diagrams 2nd ed. Editor Massalski, T.B., ASM International (1990), p ) Pt Appendix 5H Shape-Memory Polymers Shape-memory polymers (SMPs) are designed to return to a shape as a result of some stimulus. SMPs are usually made from two polymeric materials. One polymer is elastic, meaning its deformation is recovered, as in a spring or an elastic rubber band. Elastic polymers are discussed in Chapter 6. The other polymer, called the transition polymer, responds to a stimulus that changes the polymer from compliant to rigid. The stimulus can be heat, light (IR to UV), chemicals, or electromagnetic fields. Figure 5H.1 presents an example of a temperature-sensitive SMP to demonstrate how SMPs function. The spring like material is an elastic polymer for all temperatures under consideration, such as a thermosetting polymer. The elastic material does not have to be in a shape of a spring in the actual material. The transition material is sensitive to heat. A thermoplastic polymer is an example of a temperature-sensitive polymer. At a low temperature, the transition polymer is rigid, and at high temperature, the transition polymer is easily deformed. In Figure 5H.1, blue represents low temperatures and red represents high temperatures. In Figure 5H.1a the SMP is at a low temperature. The SMP is heated and then stretched in Figure 5H.1b by an applied force (F). The transition polymer is easily deformed at a high temperature. The stretched SMP is then cooled in Figure 5H.1c, and the transition polymer is rigid. The rigid transition polymer makes the deformation permanent. The force can be removed without recovery of the deformation. The deformation is recovered by heating the SMP to a high temperature where the transition polymer is easily deformed, and the elastic polymer recovers to its original shape in Figure 5H.1d.

21 Phase Transformations and Phase Diagrams W-171 F (a) (b) (c) (d) Figure 5H.1 A shape-memory polymer (SMP) to which different temperatures are applied in series. Blue represents a low temperature, and red represents a high temperature. (a) The SMP at a low temperature and in its original shape. (b) The SMP is heated and the transition polymer deforms easily along with the elastic polymer as a result of the applied force F. (c) The material is cooled to a low temperature, and then the applied force is removed. The transition polymer is rigid at the low temperature, and the deformation created in (b) is locked into the polymer. (d) The SMP is heated to a high temperature and the transition polymer deforms easily, contracting back into its original shape. Appendix Homework Problems Problem 5E.1 Problem 5F.1 For the lead-tin phase diagram in Figure 5.11a, draw possible Gibbs free energy as a function of composition diagrams that could lead to the phases present and their chemical compositions at the temperatures of 150 C, 183 C, 200 C, and 250 C. A gold-nickel alloy is produced that is 50 weight percent nickel. This alloy is first annealed at 900 C to homogenize the alloy. The alloy is then cooled to 500 C and held until equilibrium chemical compositions are reached. At both 900 C and at 500 C: (a) What phases are present? (b) What is the chemical composition of each phase? (c) What is the atom fraction of each phase? Problem 5F.2 A lithium-sodium alloy is produced that is 9 weight percent sodium. The alloy is initially heated to 200 C, and then slowly cooled to 50 C so that chemical compositions are equal to the equilibrium values. (a) At 200 C, what phases are present, what are their chemical compositions, and what is the weight fraction of each phase? (b) Specify the reaction type and the phase changes upon cooling through C. (c) At 150 C, what phases are present, what are their chemical compositions, and what is the weight fraction of each phase? (d) Specify the reaction type and the phase changes upon cooling through C for this alloy. (e) At 50 C, what phases are present, what are their chemical compositions, and what is the weight fraction of each phase?

22 W-172 CHAPTER 5 Problem 5F.3 A lithium-sodium alloy is produced that is 7 weight percent sodium. The alloy is initially heated to 350 C, and then slowly cooled to 50 C so that chemical compositions are equal to the equilibrium values. (a) (b) (c) (d) At 350 C, what phases are present, what are their chemical compositions, and what is the weight fraction of each phase? At 200 C, what phases are present, what are their chemical compositions, and what is the weight fraction of each phase? Specify the reaction type and the phase changes that occur upon cooling through C. At 150 C, what phases are present, what are their chemical compositions, and what is the weight fraction of each phase? (e) At 50 C, what phases are present, what are their chemical compositions, and what is the weight fraction of each phase? (f ) What is the weight fraction of proeutectic phase present at 50 C? You can assume that there is no change in composition or microstructure between C and 50 C. (g) Describe the microstructure that you would expect to observe in this alloy at room temperature. Problem 5F.4 If 33 atom percent Cu is mixed into a Ni crystal at a temperature of 473 K (200 C), determine the change in the GFE relative to segregated Cu and Ni for 1 mole of total atoms if the enthalpy of solution for Cu into Ni is 0.11 ev/atom, according to data in Hultgren et al. (a) What is the GFE change for the mixing? (b) Based upon your result, should it be possible to mix 33 atom percent Cu into Ni at 473 K? Explain your answer and compare this result to the phase diagram in Figure 5.7. Problem 5G.1 An iron-80 atom percent platinum alloy is produced and initially heated to 1400 C, and then it is slowly cooled to 600 C to allow for equilibrium chemical compositions. (a) (b) At 1400 C, what phases are present, what are their chemical compositions, and what is the atom fraction of each phase? At 600 C, what phases are present, what are their chemical compositions, and what is the atom fraction of each phase?