Kirkendall effect: vacancy concentration and pair interaction energy

[always_learning]

TL;DR Summary

Relationship between vacancy concentration and pair interaction energy in a core-shell binary spherical nanoparticle on a f.c.c. lattice - where does it come from?

I am reading the following article on Kirkendall effect leading to the Formation of a hollow binary alloy nanosphere: a kinetic montecarlo study. I am unable to understand or find in references the reasoning to obtain equation (1):

Let us consider a kinetic Monte Carlo model of a core-shell binary spherical nanoparticle on a rigid f.c.c. lattice with lattice parameter $a$, where $N_A = 45995$ atoms of the faster diffuse species A are located in the core with radius $r_c \sim 14a$ whilst $N_B = 30434$ atoms of the slower diffuse species B form the shell with external radius $r_s \sim 16.58a$. We assume that the three nearest neighbour pair-interaction energies are equal $\phi_{AA}=\phi_{BB}=\phi_{AB}$ but the effective attempt frequencies $\nu_{A0}$ and $\nu_{B0}$ of A and B atoms for exchange with a vacancy are strongly different $\nu_{A0} / \nu_{B0} =10^3$ and do not depend on the local arrangement of a vacancy. This model has a zero enthalpy of mixing. Furthermore, it has been shown [12,14-16] that with an appropriate choice of reduced pair-interaction energy $\phi/kT$ and using the equation:
$$-6\phi/kt \sim \frac{1}{(1-2c_v^{eq})} \ln{\left(\frac{1-c_v^{eq}}{c_v^{eq}}\right)} \quad (1)$$
for a pure element f.c.c. system it is possible to fulfil two requirements necessary for our numerical simulation. These are a reasonable calculation speed and a value of the equilibrium vacancy composition $c_v^{eq}$ sufficiently close to the vacancy composition at the melting temperature $T_m$ of a typical crystalline solid. For the case of very small $c_v^{eq}$ Eq. (1) will correctly describe an equilibrium vacancy composition in a model of a binary alloy with equal pair-interaction energies.
In this case, the vacancy formation energy will not depend on the atomic distributions and the additional configurational entropy term for vacancies appearing in a binary alloy will be negligible (for details see [18]). Following these reasons, the reduced pair-interaction energy $\phi/kT = -1.5$ was chosen using Eq. (1) to provide an equilibrium vacancy composition of $c_v \sim 1.24×10^{-4}$ in pure A and B as well as in a binary A-B alloy of f.c.c. systems.

 

[Fred Wright]

This is just an observation and I'm probably wrong, but I find equ. (1) highly suggestive of the Ostwald-Freundich equation.