Abstract
This is part II in a series of two papers that introduce a general expression for the tracer diffusivity in complex, periodic energy landscapes with $M$ distinct hop rates in one-, two-, and three-dimensional diluted systems (low coverage, single-tracer limit). While Part I [Gos\'alvez et al., Phys. Rev. B 93, 075429 (2016)] focuses on the analysis of diffusion in systems where the end sites of the hops are located symmetrically with respect to the hop origins (symmetric hops), as encountered in many ideal surfaces and bulk materials, this report (Part II) presents a more general approach to determining the tracer diffusivity in systems where the end sites can be located asymmetrically with respect to the hop origins (asymmetric hops), as observed in reconstructed and/or chemically modified surfaces and/or bulk materials. The obtained diffusivity formulas for numerous systems are validated against kinetic Monte Carlo simulations and previously reported analytical expressions based on the continuous-time random walk (CTRW) method. The proposed method corrects some of the CTRW formulas and provides new expressions for difficult cases that have not been solved earlier. This demonstrates the ability of the proposed formalism to describe tracer diffusion.
Highlights
The ability of different adsorbates to diffuse quickly or slowly on a given substrate is described by the low coverage diffusion coefficient, DTθ ≈0 = 1 2α lim t →∞|r(t) − r(0)|2 t (1)where α = 1,2,3 is the number of dimensions, r(t) designates the position of the diffusing particle at time t, and · is an ensemble average
The obtained diffusivity formulas for numerous systems are validated against kinetic Monte Carlo simulations and previously reported analytical expressions based on the continuous-time random walk (CTRW) method
Tracer diffusion has been previously studied using the continuous-time random walk (CTRW) formalism [2,3], where the diffusivity is obtained in reciprocal space by finding the longest-living eigenvalue of a matrix equation that results from Fourier and Laplace transformations of the master equation
Summary
The ability of different adsorbates to diffuse quickly or slowly on a given substrate is described by the low coverage diffusion coefficient (or tracer diffusivity [1]), DTθ ≈0. (i) The derivation of a generalized expression for the low coverage diffusivity for systems with asymmetric hops and, possibly, boundary crossings (Sec. II). (ii) The derivation of an alternative procedure (method M-2) in order to determine the effective hop rates for the terminal and crosser sites based on a one-dimensional (1D) analysis (iii) The numerical validation of all the derived formulas for the low coverage diffusivity against KMC simulations of the corresponding random walks, followed by a comparison to previous studies based on the continuous-time random walk formalism (Sec. V) and the conclusions of the study (Sec. VI). Appendixes 2 and 3 [14] provide examples for systems with and without boundary crossings, respectively, and Appendix 4 [14] considers additional systems for comparison to previous studies
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