Surface Diffusion in Systems of Interacting Brownian Particles

Abstract

The paper reviews recent results on diffusive phenomena in two-dimensional periodic potential. Specifically, static and dynamic properties are investigated by calculating different correlation functions. Diffusion process is first studied for one-dimensional system by using the Fokker–Planck equation which is solved numerically by the matrix continued fraction method in the case of bistable potential. The transition from hopping to liquid-like diffusion induced by variation of some parameters is discussed. This study will therefore serve to demonstrate the influence of this form of potential. Further, an analytical approximation for the dc-conductivity is derived for a wide damping range in the framework of the Linear Response Theory. On the basis of this expression, calculations of the ac conductivity of two-dimensional system with Frenkel–Kontorova pair interaction in the intermediate friction regime is performed by using the continued fraction expansion method. The dc-conductivity expression is used to determine the rest of the development. By varying the density of mobile ions we discuss commensurability effects. To get information about the diffusion mechanism, the full width at half maximum λω(q), of the quasi-elastic line of the dynamical structure factor S(q,ω) is computed. The calculations are extended up to large values of q covering several Brillouin zones. The analysis of λω(q) with different parameters shows that the most probable diffusion process in good two-dimensional superionic conductors consists of a competition between a back correlated hopping in one direction and forward correlated hopping in addition to liquid-like motions in the other direction.

On a Stochastic Theory of Ion Transfer at Intermediate Damping

“…In addition to these times, a substantial role is played by time τ d , which characterizes the process of spatial diffusion. For the diffusion of ions in solid electrolytes, this scale is usually connected with the value τ d ~ 10 -10 s (or slower), which far exceeds the period of atom vibrations in a rigid sublattice ( 10 -13 s) [11]. Going from a "mechanical" scale ( t ≤ τ f ) to a rougher scale is represented in the following manner [10]: the first to change (at τ f Ӷ t Ӷ τ u ) is the character of the particle's behavior, whereas the shifts depend on the initial conditions; then (at t ӷ τ u ) the initial conditions are "discarded" and the average square of a shift of a Brownian particle is described by the Einstein formula 〈 ( r -r 0 ) 2 〉 = 6 D r ( t -t 0 ) , where D r is a diffusion coefficient and where r 0 and t 0 refer to the initial instant.…”

“…However, as it is impossible to calculate accurately a potential relief, one can take into account only pairwise additive interactions [12]. Consequently, one can assume that [11] …”

“…Given concrete potentials, the ions' dynamics may be investigated using, for example, molecular dynamics methods through correlation functions, from which one can subsequently obtain both the static ionic conductance and the dependence of the conductance on the alternating-field frequency [12]. Such models have made it possible to describe a large body of experimental data for solid electrolytes (see recent review [11]). However, this path unavoidably involves labor consuming numerical calculations.…”

“…The potential in formula (1) can also be the effective single-particle potential V = V eff , which reflects simultaneously the role played by potentials V (1) and V (2) . For example, for the Brownian particles interacting in a two-dimensional periodic field, the effective potential may be represented by bistable potential V (1) = V 0 cos(q 0 x) + V 1 cos(2q 0 x), where q 0 = 2π/a (a is the lattice constant) [11]. The effective potential, which is seen by silver ions in RbAgI 5 , contains two different barriers indeed [13].…”

“…formulated in this fashion, the effective potential is not associated with the memory effects, which introduce no marked contribution to the diffusion properties of a many-particle system in the case of large friction [11]. Actually, the shape of potential V(r) is selected in each particular case from considerations of convenience.…”