Diffusion on a Solid Surface: Anomalous is Normal

2018

Diffusion is a natural or artificial process that governs many phenomena in nature. The most known diffusion is the Brownian or normal motion, where the mean-square-displacement of the tracer (diffusive particle among others) increases as the square-root of time. It is not the case, however, for complex systems, where the diffusion is rather slow , because at small-scales, these media present an heterogenous structure. This kind of slow motion is called subdiffusion , where the associated mean-square-displacement increases in time, with a non trivial exponent, alpha, whose value is between 0 and 1. In this review paper, we report on new trends dealing with the study of the anomalous diffusion in Condensed Matter Physics. The study is achieved using a theoretical approach that is based on a Generalized Langevin Equation . As particular crowded systems, we choose the so-called Pickering emulsions (oil-in-water), and we are interested in how the dispersed droplets (protected by small s...

2005

Surface transport and diffusion at low damping reveals a number of unexplained or at least not systematically explained behaviors. Here we present two problems that require further study for a full understanding. One involves motion on a random surface, the unresolved issue being the parameter regimes leading to subdiffusive, diffusive, and superdiffusive motions at intermediate times. The other involves the

Journal of Physical Chemistry B, 2004

The dynamics of Brownian particles diffusing across a one-dimensional, incoherent stochastic potential of mean force in the Smoluchowski regime has been intensely investigated by several groups. In recent work, we have developed a phenomenological equation of motion that extends this representation throughout the friction regime and, in particular, extends it to the low-friction regime relevant to surface diffusion. Resonant activation is observed throughout; it is manifested by a peak in the transport as a function of the correlation time in the potential fluctuations. The phenomenological equation of motion has now been utilized to probe the dynamics on a variety of one-and two-dimensional surfaces in order to provide a qualitative description of the fundamental factors that govern the surface hopping events of an adsorbate weakly bound to a metal surface. The primary focus is placed on differences that may arise when the substrate is modeled using a oneor two-dimensional potential of mean force, thereafter the effects of spatially coherent or incoherent barrier heights are also addressed. The two-dimensional behavior can be adequately described by the direct product of two separable one-dimensional analogues, as might naïvely be expected, provided the lattice spacing is sufficient to decouple the two degrees of freedom. Coherency between the barriers affects the rates to a smaller degree and is significant only when the barriers are strongly correlated in time.

Physical Review E, 2013

We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean-squared displacement ( r 2 (t) ∼ t γ with 0 < γ ≤ 1.5) but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub-and super-diffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent γ. Simulations based on the Master Equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.

• Diffusion of non-interacting Brownian particles through a deformed substrate. • Distinguish between substrate potential and interaction potential. • Deformed and non deformed substrate. • Deformability parameter advantages for studying the diffusion mechanism. • Effect of deformability parameter and friction on the diffusion process. a b s t r a c t We study the diffusion mechanism of non-interacting Brownian particles through a deformed substrate. The study is done at low temperature for different values of the friction. The deformed substrate is represented by a periodic Remoissenet–Peyrard potential with deformability parameter s. In this potential, the particles (impurity, adatoms.. .) can diffuse. We ignore the interactions between these mobile particles consider them merely as non-interacting Brownian particles and this system is described by a Fokker–Planck equation. We solve this equation numerically using the matrix continued fraction method to calculate the dynamic structure factor S(q, ω). From S(q, ω) some relevant correlation functions are also calculated. In particular, we determine the half-width line λ(q) of the peak of the quasi-elastic dynamic structure factor S(q, ω) and the diffusion coefficient D. Our numerical results show that the diffusion mechanism is described, depending on the structure of the potential, either by a simple jump diffusion process with jump length close to the lattice constant a or by a combination of a jump diffusion model with jump length close to lattice constant a and a liquid-like motion inside the unit cell. It shows also that, for different friction regimes and various potential shapes, the friction attenuates the diffusion mechanism. It is found that, in the high friction regime, the diffusion process is more important through a deformed substrate than through a non-deformed one.

Journal of The Royal Society Interface

The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modelling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can also be somewhat alternative to each other, e.g. continuous time random walk and fractional Brownian motion. To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modelling of velocity dynamics. The complexity of the medium is parametrized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particle&#39;s dynamics. We prove that, for pro...

Physica A: Statistical Mechanics and its Applications, 2006

We study the stability of the Maxwell-Boltzmann (i.e., Gaussian) distribution for the density of states at equilibrium, against an arbitrary choice of the friction in the Langevin equation. We find that this distribution is Gaussian, if and only if the friction is Lipschitz continuous. In particular, we argue that the origin of the exponential (instead of Gaussian) velocity distribution (PDF) of particles when the viscous friction is replaced by the Coulomb friction in the Langevin equation with white noise is due to the non-Lipschitz continuity of the Coulomb friction, a feature of solid friction. The use of the Fokker-Planck equation to determine the exponential PDF is justified, since the subset on which the friction is not continuous is of zero probability ðv ¼ 0Þ. The application to the motion of granular gases is discussed.

The Journal of Chemical Physics, 2002

We study the diffusion of Brownian particles with a short-range repulsion on a surface with a periodic potential through molecular dynamics simulations and theoretical arguments. We concentrate on the behavior of the tracer and collective diffusion coefficients D T () and D C (), respectively, as a function of the surface coverage . In the high friction regime we find that both coefficients are well approximated by the Langmuir lattice-gas results for up to Ϸ0.7 in the limit of a strongly binding surface potential. In particular, the static compressibility factor within D C () is very accurately given by the Langmuir formula for 0рр1. For higher densities, both D T () and D C ()show an intermediate maximum which increases with the strength of the potential amplitude. In the low friction regime we find that long jumps enhance blocking and D T () decreases more rapidly for submonolayer coverages. However, for higher densities D T ()/D T (0) is almost independent of friction as long jumps are effectively suppressed by frequent interparticle collisions. We also study the role of memory effects for many-particle diffusion.

2016

We investigate random walk of a particle constrained on cells, where cells behave as a lattice gas on a two dimensional square lattice. By Monte Carlo simulation, we obtain the mean first passage time of the particle as a function of the density and temperature of the lattice gas. We find that the transportation of the particle becomes anomalously slow in a certain range of parameters because of the cross over in dynamics between the low and high density regimes; for low densities the dynamics of cells plays the essential role, and for high densities, the dynamics of the particle plays the dominant role.

Advances in Physics, 2002

We review in this article the current theoretical understanding of collective and single particle diffusion on surfaces and how it relates to the existing experimental data. We begin with a brief survey of the experimental techniques that have been employed for the measurement of the surface diffusion coefficients. This is followed by a section on the basic concepts involved in this field. In particular, we wish to clarify the relation between jump or exchange motion on microscopic length scales, and the diffusion coefficients which can be defined properly only in the long length and time scales. The central role in this is played by the memory effects. We also discuss the concept of diffusion under nonequilibrium conditions. In the third section, a variety of different theoretical approaches that have been employed in studying surface diffusion such as first principles calculations, transition state theory, the Langevin equation, Monte Carlo and molecular dynamics simulations, and path integral formalism are presented. These first three sections form an introduction to the field of surface diffusion. Section 4 contains subsections that discuss surface diffusion for various systems which have been investigated both experimentally and theoretically. The focus here is not so much on specific systems but rather on important issues concerning diffusion measurements or calculations. Examples include the influence of steps, diffusion in systems undergoing phase transitions, and the role of correlation and memory effects. Obviously, the choice of topics here reflects the interest and expertise of the authors and is by no means exhaustive. Nevertheless, these topics form a collection of issues that are under active investigation, with many important open questions remaining.