Institute of Metals Division - Analytical Representation of Certain Phase Boundaries

Using an expression for the free energy of a homogeneous phase as a function of composition, a relationship is derived which interrelates the phase boundaries extending from the allotropic transformation of the solvent metal. Comparisons between observed and calculated phase boundaries in the systems Fe-Ni, Fe-Mn, Fe-W, Ti-Mo, and Ti-Cb are presented. IT is the purpose of this paper to illustrate how the general free energy equations for two phases in equilibrium may be used to derive an expression which interrelates the phase boundaries extending from the allotropic transformation of the solvent metal. The free energy per gram atom of a disordered solid solution may, for many purposes, be represented by the equation:' F = ½NZ [C-V,, + (1-C) VbB + 2C (l-C)V] + K(T) +RT[C-lnC+ (l-C)ln(l-C) ] [1] where N is Avogadro's number; Z, the coordination number; C, the concentration of the solute element in atomic percent; the subscripts refer to concentrations at phase mixture boundaries; VAA, VBB, and V,, are the bond energies between like and unlike adjacent atoms, respectively, at the interatomic spacing characteristic of the phase. The bond energies are usually assumed to be independent of small changes in the lattice parameter occasioned by solute additions and variations in temperature; V = VAB ---------------—— ; K(T) is the sum of the heat con- tent and heat entropy; R, the molar gas constant; and T, the temperature in degrees Kelvin. The free energy of a homogeneous phase is there- fore the sum of an internal energy, a specific heat, and a configurational entropy term, respectively. The compositions of two phases in equilibrium at a given temperature are governed by the minimum total free energy principle. Graphically, this is represented by the points of tangency of the tangent common to the free energy vs. concentration curves of the two phases. Analytically, this may be expressed as: ( df) = (df) [2] dc dc Expanding both sides using eq 1, the following equation is obtained after convenient readjustment of the terms: RT In[(ca)(1-c)]-½N (Vaa-Vbb)(Za-Zb) +NV [Za(l-2Ca)-Z,(l-2C,)] [3] The following assumptions have been made at this point: 1—The specific heats of the two phases are independent of composition variations within the range under consideration. Accordingly, the K(T) terms disappear on differentiation. 2— (Vaa)o = (VAA); (VBB). = (VBB); Va = V, Theyjustification for these assumptions can lie only in the successful application of the resultant equation. It may be noted that Zener2 has used assumption 1 in the calculation of the boundaries limiting the (a+j3) phase mixture in the Cu-Zn system.