The discussion centers on the Kirkendall effect in the context of a kinetic Monte Carlo model for a core-shell binary spherical nanoparticle on a face-centered cubic (f.c.c.) lattice. The model features 45,995 atoms of the faster diffusing species A in the core and 30,434 atoms of the slower diffusing species B in the shell. The effective attempt frequencies for atom exchange with a vacancy differ significantly, with a ratio of 1000. Equation (1) is derived to relate the reduced pair-interaction energy to the equilibrium vacancy composition, demonstrating its applicability in simulating vacancy behavior in binary alloys.
PREREQUISITESResearchers and students in materials science, particularly those focusing on alloy behavior, vacancy dynamics, and kinetic Monte Carlo simulations in solid-state physics.
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.