Quasichemical Model of a Ternary Solution

A quasichemical model is one type of statistical models of solutions. It is more realistic than the ideal and regular models. The solutions in a quasichemical model have a finite enthalpy change and nonrandom atomic configuration of mixing. Two improvements to the previous ternary interstitial quasichemical model have been made in this work. The new model is more general and rigorous, which means it could deal with more solutions and the predictions from the model should be more accurate. Besides, the consistency between the current quasichemical model and the previous binary and ternary models has been approved. Introduction In a solution, atoms prefer like or unlike atoms as their neighbours, if the binding energy can be lowered. A completely random solution can only be found at very high temperature where heat overwhelms any tendency for ordering or clustering. Therefore, the ideal and regular solution models are not realistic because both of them assume a random distribution of atoms. The quasichemical model has a better approach to deal with the non-zero Enthalpy and Entropy change. The name of the model comes from two mass action equations derived from the model, which describes the number of various pairs in the equilibrium state. The binary quasichemical model was firstly introduced by Guggenheim in his book ‘Mixtures’. McLellan and his colleagues applied this method on the Fe-C binary and then the Fe-X-C (where X represents any substitutional solute atom) ternary system. Those models can be used to interpret the Thermodynamic data from the experiment and to predict the carbon-carbon interaction energy in ferrite. To make a quasichemical model, the major work is to construct the partition function Ω of the system, because most other thermodynamic functions can be easily deduced from the partition function. For example, Helmholtz free energy F can be obtained by the equation Ω − = ln kT F where k is the Boltzmann constant and T is the absolute temperature. Previous ternary quasichemical model The solution Alex and McLellan concerned consists of Nv solvent atoms (v), Nu substitutional solute atoms (u) and Ni octahedral interstitial solute atoms (i) (Fig.1). The empty squares represent the empty interstitial gaps (e). Figure 1 Schematic diagram of the Fe-X-C solution Parameter definitions v N Number of solvent atoms u N Number of substitutional solute atoms i N Number of interstitial solute atoms ε Pair energy. Use subscripts v, i, u, and e to indicate the solvent, interstitial, substitutional solute atoms and the empty interstitial gaps respectively 1 W Coordination number, the number of nearest-neighbouring interstitial sites around any given interstitial site 2 W Coordination number, the number of nearest-neighbouring main lattice atoms around any given interstitial site β Number of interstitial sites per one main lattice atom 1 1λ W Number of e i − pairs 2 2λ W Number of u i − pairs i θ ( ) u v i N N N + u θ ( ) u v u N N N + Solvent Atoms (v) Substitutional Solute Atoms (u) Interstitial Solute Atoms (i)