Chapter 15: Binary System Solution Models

Binary System Solution Models significantly expands upon foundational concepts in the thermodynamics of condensed binary alloys, progressing beyond basic solution models to account for complex, non-ideal behaviors. The review confirms that the ideal solution model assumes no inter-constituent interactions, defining the Gibbs free energy of mixing (G of mixing) solely by its temperature and the entropy of mixing (S of mixing), predicting a completely random distribution of atoms above zero Kelvin. The regular solution model incorporates interaction energy (defined as alpha times the concentration of A times the concentration of B), which is equivalent to the excess Gibbs free energy or enthalpy of mixing (H of mixing). If the interaction parameter alpha is positive, the alloy tends toward clustering and decomposition below a critical temperature (T sub C). If alpha is negative, the solution favors ordering. To accurately model systems where properties are not symmetrical about the 50% composition, the sub-regular solution model is introduced, which allows the interaction parameter alpha to be composition-dependent (e.g., a plus b times X B​ ). This empirical flexibility allows for the H of mixing to change signs across compositions, enabling the modeling of systems that display both ordering and decomposition tendencies. The statistical basis of the regular solution model is explored using a near-neighbor approach, showing that ideal behavior requires the energy of an A-B bond to be the average of the A-A and B-B bond energies. When interaction energies are strong, the assumption of random mixing fails, necessitating a compromise between minimizing enthalpy (through ordering or clustering) and maximizing entropy (random mixing). To address this non-randomness, the Long-Range Order (LRO) parameter, eta, is incorporated as a thermodynamic variable, quantifying the degree of atomic arrangement from completely disordered (eta equals zero) to fully ordered (eta equals one). This order parameter forms the basis of the Landau theory of phase transitions, which models the excess Gibbs free energy (G excess) as a power series expansion in eta. The two-four Landau model (using eta squared and eta to the fourth power) is symmetrical and describes a second-order (continuous) transition at T sub C, characterized by zero latent enthalpy and a finite discontinuity in the excess heat capacity. The two-three-four model (adding an asymmetric eta cubed term) describes a first-order transition at the equilibrium temperature T 0​ , which results in thermodynamic hysteresis between the instability temperature for the disordered phase on cooling (T i​ minus) and the instability temperature for the ordered phase on heating (T i​ plus). Finally, the two-four-six model (with negative coefficient for the fourth power term) also produces a first-order transition despite being symmetrical. A general limitation noted for these Landau models is that they predict a nonzero slope for the order parameter versus temperature curve as temperature approaches zero Kelvin, which violates the Third Law of Thermodynamics.