Statistical theory of diffusion in concentrated alloys

Variational Principle for Mass Transport

“…A brief, albeit incomplete list of methods to compute transport coefficients from atomic mechanisms include stochastic methods like kinetic Monte Carlo [11][12][13][14][15], master-equation methods like the self-consistent mean-field method [16,17] and kinetic mean-field approximations [18][19][20], path probability methods for irreversible thermodynamics [21][22][23], Green function methods [24][25][26][27],…”

“…For vacancy-mediated diffusion, the dilute Green function [26,27] and matrix methodology [24,25] work in a restricted state space {χ} where only one solute and vacancy are present, and then effectively construct a full basis in that state space. Finally, self-consistent mean-field [16,17] and kinetic mean-field [18][19][20] work with a cluster expansion of chemistryand direction-independent basis functions {φ χ,n } that are products of site occupancies for different chemistries. It should be noted that these latter two methods derive their solution for the parameters θ n using a ladder of n-body correlation functions on which they invoke "closure approximations" for higher order correlation functions; in a variational framework, such closure approximations become unnecessary.…”

Unexpected variations in the kinetics of solid solution alloys due to local interactions

“…There have been numerous studies on solid-state diffusion based on first-principles calculations of either dilute alloys [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] or non-dilute alloys [35][36][37][38][39][40][41][42][43][44][45][46]. A brief, albeit incomplete, list of atomistic methods that are typically being used to estimate diffusion properties in solid-state systems can be primarily categorized into two main groups: (i) analytical approaches -which include master-equation methods like the self-consistent meanfield method [31,35] and kinetic mean-field approximations [47][48][49], Green function methods [28,33,34], five-frequency model [32] etc., and (ii) numerical approaches -which involve either use of kinetic Monte Carlo (kMC) [7, 39-43, 45, 50] or Monte Carlo (MC) simulations [40,46].…”

A Ranking Scheme for Mass-Transport Predictions

“…A brief, albeit incomplete, list of methods to compute transport coefficients from atomic mechanisms includes stochastic methods like kinetic Monte Carlo calculations [11][12][13][14][15], master-equation methods like the self-consistent mean-field method [16,17] and kinetic mean-field approximations [18][19][20], path probability methods for irreversible thermodynamics [21][22][23], Green function methods [24][25][26][27], and Ritz variational methods [28][29][30]. The different approaches all have different computational and theoretical complexity and rely on different approximations, which may or may not be controlled.…”

“…For vacancymediated diffusion, the dilute Green function [26,27] and matrix methodology [24,25] work in a restricted state space f χg, where only one solute and vacancy are present, and then effectively construct a full basis in that state space. Finally, self-consistent [16,17] and kinetic mean-field [18][19][20] methods work with a cluster expansion of chemistry-and direction-independent basis functions fϕ χ;n g, which are products of site occupancies for different chemistries. It should be noted that these latter two methods derive their solution for the parameters θ n using a ladder of n-body correlation functions on which they invoke "closure approximations" for higher order correlation functions; in a variational framework, such closure approximations become unnecessary.…”