Thermodynamics of Binary bcc and fcc Phases for Exclusive Second-Neighbour Pair Interactions Using Cluster Variation Method: Analytical Solutions

Abstract

Background

In the framework of cluster expansion—cluster variation methods (CE - CVM), the equilibrium values of the microscopic state variables called correlation functions (CFs) can be obtained by minimization of the Helmholtz energy. This involves solving the nonlinear equilibrium equations using numerical techniques. As an exception, Guggenheim obtained an analytical solution for the pair CF in the pair approximation of quasi-chemical theory for a binary alloy.

Results

In this communication, analytical solutions for the CFs for the case of exclusive second neighbour pair interactions in binary A2 & B32 phases and A1 & L11 phases respectively under tetrahedron and tetrahedron–octahedron approximations of CVM are obtained as functions of the corresponding energy coefficient, temperature, and composition.

Results obtained by results in the previous section and a few new results

Further, analytical expressions for the phase boundaries in phase separating and ordering systems have been obtained. Apart from these, thermodynamic quantities such as heat capacities have also been evaluated. In addition, the effect of composition and temperature dependence of the energy coefficient on the phase boundary is also discussed.

Application

For the selected composition, the solution of CF reduces to a simple rational function. Taking this as a guide, the coefficients of the polynomials used to approximate the CFs are expressed as rational functions, which was discussed elsewhere.

Appendix

At the phase boundary,

$$\begin{aligned} \begin{aligned} \varDelta \psi&=-x_{\text{A}}x_\text{B}\left( \frac{\partial }{\partial \xi } \left( \frac{\partial G_{B32}^{\text{mix}}/\text{R}T}{\partial x_\text{B}}\right) _\xi \right) _{x_\text{B}} \frac{{ {d}}\xi }{{{d}}x_\text{B}}\\&=-\frac{x_{\text{A}}x_\text{B}}{2\xi } \left( \frac{\partial }{\partial \xi } \left( \frac{\partial G_{B32}^{\text{mix}}/\text{R}T}{\partial x_\text{B}}\right) _\xi \right) _{x_\text{B}} \frac{{ {d}}\xi ^2}{{ {d}}x_\text{B}} \end{aligned} \end{aligned}$$

(58)

At the phase boundary, the differential terms in the numerator as well as the denominator in Eq. (58) vanish, making their ratio indeterminate. It can be evaluated by considering

$$\begin{aligned} \frac{{ {d}}\xi ^2}{{ {d}}x_\text{B}}=2\xi \frac{{ {d}}\xi }{{{d}}x_\text{B}}=-2\xi \left( \frac{\partial }{\partial x_\text{B}} \left( \frac{\partial G_{B32}^{\text{mix}}}{\partial \xi }\right) _{x_\text{B}}\right) _\xi \Big / \left( \frac{\partial ^2 G_{B32}^{ \text{mix}}}{\partial \xi ^2}\right) _{x_\text{B}} \end{aligned}$$

(59)

The derivatives in Eq. (44) are evaluated from Eq. (17) and substituted in Eq.  (44) which after simplification becomes

$$\begin{aligned} \frac{{{d}}\xi ^2}{{{d}}x_\text{B}}=-\frac{4u_0 \xi ^2 \left( 1+\left( u_0^2-\xi ^2\right) \left( 1-\eta _2\right) +\eta _2-4 \text{X}/3\right) }{\left( 1-u_0^2\right) \left( \eta _2+u_0^2\left( 1-\eta _2\right) -2\text{X}/3\right) +\xi ^2\left( u_0^2\left( 1-\eta _2\right) -\eta _2+2\text{X}/3\right) } \end{aligned}$$

(60)

where

$$\begin{aligned} \text{X}=\sqrt{\eta _2+\left( 1-\eta _2\right) \left( u_0^2-\eta _2\xi ^2\right) } \end{aligned}$$

(61)

The terms which are independent of $\xi ^2$ in the denominator of RHS in Eq. (60) together are expanded around $u_0$ corresponding to its value at the phase boundary, namely $u_0 = u_{0b}$ (given in Eq. (28)) which yields

$$\begin{aligned} \begin{aligned}&\left( 1-u_{0b}^2\right) \left( \eta _2+u_{0b}^2\left( 1-\eta _2\right) +2\text{X}_b/3\right) -2u_{0b}\left( u_0-u_{0b}\right) \left( u_{0b}^2-\left( 1-u_{0b}^2\right) \left( 1-2\eta _2\right) \right. \\&\quad \left. +\frac{1-\left( 3u_{0b}^2-2\eta _2 \xi ^2\right) \left( 1-\eta _2\right) -3\eta _2}{3\text{X}_b}\right) +\text{O}\left( \left( u_0-u_{0b}\right) ^2\right) \end{aligned} \end{aligned}$$

X$_b$ in the above expression corresponds to the value of X at $u_0$ = $u_{0b}$. Expanding the terms which are independent of ($u_0 - u_{0b}$) in the above expression together around $\xi$ = 0 and with substitution of boundary condition corresponding to $u_0$ from Eq. (28) yields

$$\begin{aligned} \frac{5}{18}\xi ^2\eta _2+\text{O}\left( \xi ^4\right) +\frac{5}{9} u_{0b}\left( u_0-u_{0b}\right) +\text{O}\left( \left( u_0-u_{0b}\right) ^2\right) \end{aligned}$$

Substituting the above expression in Eq. (60) and cancelling $\xi ^2$ from the numerator and denominator

$$\begin{aligned} \frac{{{d}}\xi ^2}{{{d}}x_\text{B}}=-\frac{4u_0 \left( 1+\left( u_0^2-\xi ^2\right) \left( 1-\eta _2\right) +\eta _2-4 \text{X}/3\right) }{-\frac{13}{18}\eta _2+u_0^2\left( 1-\eta _2\right) -2\text{X}/3+\text{O}\left( \xi ^2\right) +\frac{5}{9}u_{0b}\frac{\left( u_0-u_{0b}\right) }{\xi ^2}+\frac{\left( u_0-u_{0b}\right) }{\xi ^2}\text{O}\left( \left( u_0-u_{0b}\right) \right) } \end{aligned}$$

(62)

In the limit of $u_0 \rightarrow u_{0b}$ and $\xi \rightarrow 0$ corresponding to the phase boundary,

$$\begin{aligned} \left. \frac{\left( u_0-u_{0b}\right) }{\xi ^2}\right| _{u_0 \rightarrow u_{0b},\xi \rightarrow 0}=\frac{du_0}{{{d}}\xi ^2}=\frac{1}{\left( \frac{{{d}}\xi ^2}{du_0}\right) }=\frac{1}{\left( \frac{{{d}}\xi ^2}{2{{d}}x_\text{B}}\right) } \end{aligned}$$

(63)

Substituting from Eq. (63) in Eq. (62), we obtain

$$\begin{aligned} \frac{{{d}}\xi ^2}{{{d}}x_\text{B}}=-\frac{4u_{0b}}{\frac{4+27u_{0b}^2}{18\left( 1-u_{0b}^2\right) }+\frac{u_{0b}}{\left( \frac{{{d}}\xi ^2}{2{{d}}x_\text{B}}\right) }} \end{aligned}$$

(64)

The above equation can be solved for ${{d}}\xi ^2/{{d}}x_\text{B}$ in terms of $u_{0b}$ to yield

$$\begin{aligned} \frac{{{d}}\xi ^2}{{{d}}x_\text{B}}=\frac{54u_{0b}\left( -1+u_{0b}^2\right) }{4+27u_{0b}^2} \end{aligned}$$

(65)

Substituting from Eq. (65) in Eq. (43) and utilizing the boundary equation, Eq. (28), we get

$$\begin{aligned} \varDelta \psi =\frac{135u_0^2}{2\left( 4+27u_0^2\right) }=\frac{15\left( 4-9\eta _2\right) }{2\left( 16-31\eta _2\right) } \end{aligned}$$

(66)