Kirkendall effect: vacancy concentration and pair interaction energy

In summary, the equation (1) is a mathematical statement that suggests that the vacancy formation energy in a binary alloy with equal pair-interaction energies is not dependent on the atomic distributions.
  • #1
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.
 
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