Surface diffusion of particles adsorbed on an inhomogeneous lattice with two non-equivalent sites

We investigate surface diffusion in a system of particles with the nearest neighbor pairwise lateral interaction adsorbed on a two-dimensional inhomogeneous lattice of square symmetry with deep and shallow sites. General analytical expressions for the chemical and jump diffusion coefficients have been derived in case of strong inhomogeneity. The expressions are valid for the inhomogeneous lattices with different geometries and dimensionalities. We have calculated the coverage dependencies of the tracer, jump and chemical diffusion coefficients for different temperatures using the real-space renormalization group (RSRG) method and compared the data with the numerical results obtained by the MC simulations. The coincidence between the data obtained by these quite different methods is rather good.     

Diffusion in heterogeneous lattices

We present here the results of our investigations of particle diffusion over different heterogeneous lattices with deep and shallow adsorption sites. A general analytical expression for chemical diffusion coefficient has been derived for a number of inhomogeneous lattices of different dimensionality and symmetry. We have calculated coverage dependencies of diffusion coefficients. The analytical data have been compared with the numerical data obtained by the kinetic Monte Carlo simulations. Almost perfect agreement between the respective results has been found.     

Diffusion of interacting particles in one-dimensional heterogeneous systems

The equilibrium and transport properties of interacting ad-particles on bivariate heterogeneous chains are studied by combining analytical and simulation approaches. Heterogeneity is introduced in the way of patches of shallow and deep adsorbing sites distributed in a deterministic alternating way. Adsorption isotherms and mean-square fluctuations of the surface coverage, as well as the jump and collective diffusion coefficients, are calculated for different values of lateral interactions between ad-particles and substrate heterogeneity. In addition, different elementary jump mechanisms are introduced and their influence in the coverage dependence of the collective diffusion coefficient is investigated.     

Diffusion of particles on a heterogeneous surface: A case of adsorption-induced surface reconstruction

The influence of surface reconstruction on diffusion of particles adsorbed on the surface is investigated in the framework of symmetrical four-position model. The analytical expressions for free energy and diffusion coefficients are obtained assuming the lateral interaction between particles is negligibly small.The critical behavior of the system is described by the Ising spin model. The coverage dependencies of the tracer, jump and chemical diffusion coefficients are calculated for some representative temperatures. The dependencies show clearly strong influence of the surface reconstruction on the thermodynamic and kinetic phenomena: diffusion coefficients become anisotropic on the reconstructed surface. To check the analytical results we have used Monte Carlo simulations of the diffusion on this lattice.     

Theoretical description of adatom migration in two-dimensional highly-ordered states

Analytical expressions for chemical, jump, and tracer diffusion coefficients are obtained for interacting lattice gases on a square lattice. Strongly repulsive nearest neighbor interactions cause the formation of a highly-ordered c() state in the vicinity of half coverage. It is shown that only strongly correlated successive adatom jumps contribute to the particle flow. This allows to describe the adatom kinetics by considering an almost ideal lattice gas of defects. Two types of defects are considered, adatoms in the empty sublattice and vacancies in the filled sublattice of the c() ordered state. The diffusion equations for these defects are developed considering the generation and recombination of defects. In addition we have considered adatom transport caused by the motion of defect pairs (dimers). Dimer transport mechanism prevails in the high coverage region. The characteristic features of the various diffusion coefficients near half coverage are analyzed and discussed. The theory is compared with the results of sophisticated Monte-Carlo simulations which have been executed with the use of a fully parallelized algorithm on a Cray T3E (LC784-128). The agreement between theoretical and MC results is excellent if the motion of dimers at is taken into account. Received 24 June 1998     

Jumps in icosahedral quasicrystals

Atomic jumps in icosahedral (AlCu)Li quasicrystals and related structures have been studied by molecular dynamics simulations. In quasicrystalline structures jumps exists with jump vectors much shorter than an average nearest neighbor distance. This is a consequence of the phasonic degree of freedom. The jumps therefore are called flips and the sites connected by the jump vector are denoted alternative positions. We find that the atoms in the quasicrystal structures studied here do not flip to alternative positions as proposed and observed in decagonal or dodecagonal quasicrystals but jump to sites which are at least an ordinary interatomic distance apart. Furthermore we observe two diffusion regimes: below about 55% of the melting temperature only small (AlCu) atoms carry out ring processes whereas at higher temperatures both kinds of atoms contribute to long-range diffusion. Received 21 July 1999     

Diffusion of particles over strongly anisotropic surface with traps

We investigate surface diffusion in a system of particles adsorbed on a two-dimensional strongly anisotropic lattice. There are two kinds of the lattice sites - ordinary sites and deep traps. Particles adsorbed in the ordinary sites can migrate over the surface, but particles adsorbed in traps are immobile. These particles do not move over the surface and they obstacle also the mobile particles migration (surface defects). Using kinetic Monte Carlo simulations we obtained coverage dependencies of the tracer, jump, and chemical diffusion coefficients. The coefficients are rather sensitive to the defect concentration. Even small admixture of the defects decreases drastically the fast diffusion. The effect is rather specific: strong dependence of the pre-exponential factor on the defect concentration and almost independent activation energy. The defect influence on the slow diffusion is weak. It results in strong decreasing of the surface diffusion anisotropy with the defect concentration. Such unusual behavior of the diffusion coefficients was observed in many experimental investigations of the surface diffusion of lithium, cesium, potassium, and strontium over strongly anisotropic W(1 1 2) and Mo(1 1 2) planes. It was shown that this specific behavior arises exclusively due to the surface anisotropy, and does not depend on the lateral interaction between the particles.     

Lineshapes in quasi-elastic scattering from species hopping between non-equivalent surface sites

We present an analytic model applied to quasi-elastic scattering from an adsorbed surface species undergoing jump diffusion between adsorption sites described by a Bravais lattice combined with a basis of multiple points. The model allows for hops between adsorption sites which are both symmetrically and energetically inequivalent. We give results for 1-D hopping, which are applicable to a species jumping between the top and bridge sites along the [11?0] direction on an fcc-(110) surface or for jumps along a step edge. In 2-D, results for hopping between fcc and hcp hollow sites and between the bridge sites of an fcc-(111) surface are presented. These examples give characteristic signatures which will allow these forms of motion to be recognized in experimental data and will enable the underlying physical parameters to be extracted by comparison with the analytical forms derived here.     

Pricing on electricity market based on coupled-continuous-time-random-walk concept

In this paper we propose a model of electricity market based on the forward rate dynamics described by a diffusion with jumps as a generalization of the classical diffusion approach. We consider jump components resulting from a coupled continuous-time random walk (CTRW) with jump lengths proportional to the corresponding inter-jump time intervals. In the framework of the model we derive a formula for the EURO-price of a standard European call option, showing applicability of CTRW processes for pricing of financial instruments. The result, obtained by an advance theory of semimartingales, is an essential extension of the pricing formula derived in the classical diffusion model of the forward rate dynamics. It indicates an influence of both, the continuous and the jump parts of the forward rate process on the option price.     

Adatom diffusion on a square lattice. Theoretical and numerical studies

A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The minimal relative error in determining the critical values of the interaction parameters is equal to . Using the RSRG method, we have explored phase diagrams of both a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We also have investigated the influence of the attractive and repulsive interactions on both the thermodynamic properties of the lattice gas and the diffusion of adsorbed particles over surface. We have calculated adsorption isotherms and coverage dependences of the pair correlation function, isothermal susceptibility and the chemical diffusion coefficient. In addition, we have included in our analysis the interaction of the activated particle in the saddle point with its nearest neighbors. We have also used Monte Carlo (MC) technique to calculate these dependences. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG approach can be applied to characterize the thermodynamic and kinetic properties of systems of particles with strong lateral interactions. Received 1st September 1998 and Received in final form 8 March 2000     

Optimal disorder for segregation in annealed small worlds

We study a model for microscopic segregation in a homogeneous system of particles moving on a one-dimensional lattice. Particles tend to separate from each other, and evolution ceases when at least one empty site is found between any two particles. Motion is a mixture of diffusion to nearest-neighbour sites and long-range jumps, known as annealed small-world propagation. The long-range jump probability plays the role of the small-world disorder. We show that there is an optimal value of this probability, for which the segregation process is fastest. Moreover, above a critical probability, the time needed to reach a fully segregated state diverges for asymptotically large systems. These special values of the long-range jump probability depend crucially on the particle density. Our system is a novel example of the rare dynamical processes with critical behaviour at a finite value of the small-world disorder.     

Diffusion of adsorbates on random alloy surfaces

In this work the diffusion of non-interacting adsorbates on a random AB alloy surface is considered. For this purpose a simple cubic (sc), body-centered cubic (bcc) or face-centered cubic (fcc) auxiliary metal lattice is introduced. The auxiliary lattice is truncated parallel to its (100) plane in such a way that the fourfold hollow positions of the metal surface form a regular net of adsorption sites with square symmetry. The adsorption energy of each adsorption site is determined by its own environment, i.e. by the numbers of direct A or B neighbors. The Monte-Carlo method has been utilized to simulate surface diffusion of adsorbates on such energetically heterogeneous alloy surfaces and to calculate the tracer, jump and chemical diffusion coefficients. The chemical diffusion coefficient was calculated via two different approaches: the fluctuation and the Kubo-Green method. The influence of energetical heterogeneities on the surface diffusion is largely pronounced at low temperatures and low surface coverages, where most of the adatoms are trapped by deep adsorption sites. It was found that at low temperatures the sequential occupation of the different types of adsorption sites can be observed. Received: 24 October 1997 / Accepted: 17 December 1997     

Quantitative study of diffusion jumps in atomistic simulations of model gas–polymer systems

Microscopic mechanisms underlying the diffusion of particles in polymeric and other systems include ‘jumps’ that are said to provide for a substantial contribution to the overall particle displacement. Such jumps have been observed in molecular simulations and experimentally, leading to important qualitative conclusions. An efficient method has been proposed for the identification and quantitative processing of jumps, and successfully employed in simulations of gas–liquid alkane systems. In the present work, the same method is applied in equilibrium Molecular Dynamics simulations of methane-like molecules dispersed in polymer-like alkanes, at atmospheric pressure and temperature well above the polymer melting point. The systems studied were prepared and equilibrated and a linear diffusion regime was confirmed by means of various criteria. The occurrence of distinct jump events was clearly revealed and their average length and frequency were calculated. In this way, a random-walk-type diffusion coefficient, D _s,?jumps, of the penetrants, was obtained and found to be lower than the overall diffusion coefficient D _s,?MSD calculated by the mean square displacement method. This is a strong indication that the overall diffusion is a combination of longer jumps with other microscopic mechanisms such as smoother and more gradual displacements effected upon the diffusing particle by its surroundings.     

Structure and size distribution of percolating clusters. Comparison with gelling systems

Dynamic properties of Brownian particles immersed in a periodic potential with two barriers V₁ and V₂ (symmetric bistable potential) are studied by using the Fokker-Planck equation which we solve numerically by the matrix continued fraction method. This study will therefore serve to demonstrate the influence of this form of potential, which is of great interest for superionic conductors and for many other solid systems, on the diffusion process. Thus, we have calculated the full width at half maximum (FWHM) ) of the quasi-elastic line of the dynamic structure factor, for a large range of values of the wave-vectors q. Our results show clearly that, by varying the ratio of the barriers strictly between and 1, the Fokker-Planck equation describes a diffusive process which has some characteristic of jump and liquid-like regimes. While in the limit cases, i.e. when tends to or 1, the diffusion process can be described only by a simple jump motion. However, the jump-lengths corresponding to each limit case are not equal. In general the change of the ratio is found to have a significant effect on the character of the diffusive motion. We have also performed Fokker-Planck dynamics calculations of the diffusion coefficient in a bistable potential. We have found a good agreement between numerical calculations and analytical approximation results obtained in the high friction limit. Received 25 May 1998 and Received in final form 15 November 1998     

Contribution a la theorie atomique de la diffusion chimique

Here we present an analysis and a development of the atomic theory of chemical diffusion as proposed by Manning for a binary system a/b.The general expression for the flux of a tracer in a concentration gradient is first established. This expression of the flux is identified with that deduced in the phenomenological theory. Thus a relationship between the partial correlation factors of vacancies with each of the a and b species is obtained.The effect of “vacancy flow” can be described in terms of these correlation factors. Thus the vacancy flow on species A leads to a correlation of the vacancy jumps with species B and vice versa.We shall see that the Nernst-Einstein equation can be extended to the case of chemical diffusion and that the ratio of the intrinsic diffusion coefficients is equal to the ratio of the mean jump frequencies W_A and W_B.Also, the activation energies of intrinsic diffusion coefficients are related very simply to the activation enthalpies of atomic jumps.In conclusion, we shall see that chemical diffusion in a binary system a/b can be completely described if either the thermodynamic factor and the coefficients of self diffusion, or the thermodynamic factor and the coefficients of intrinsic diffusion are known as functions of the concentration.     

Silicon diffusion in molybdenum disilicide: correlation effects

Correlation factors for silicon diffusion by a vacancy mechanism in the silicon sublattice of the tetragonal MoSi₂ structure have been calculated by combining an analytical and a Monte Carlo approach. The ratio of the silicon diffusivity perpendicular to the tetragonal axis to that parallel to the tetragonal axis is also deduced. An effect of forward correlation of tracer atom jumps in the silicon sublattice with the corresponding partial correlation factor of 1.5 appears at small frequencies of silicon atom jumps along the tetragonal axis with respect to the jump frequencies in the silicon layer perpendicular to the tetragonal axis of the MoSi₂ structure. The anisotropy of silicon diffusion in MoSi₂ measured by Salamon et al. is explained in terms of correlation effects of silicon diffusion on its own sublattice.     

Surface diffusion of k-mers in one-dimensional systems

The surface diffusion of interacting k-mers is studied both through analytical and Monte Carlo simulation methods in one-dimensional systems. Adsorption isotherms, jump diffusion coefficients and collective diffusion coefficients are obtained for attractive and repulsive k-mers, showing a variety of behaviors as a function of the size of particles, k. The following main results are found: (a) diffusion coefficients increase with k for non-interacting particles; (b) for fixed k, diffusion coefficients increase as the interaction energy increases from negative (attractive) to positive (repulsive) values; (c) for attractive interactions diffusion coefficients increase with k in the whole range of coverage; (d) for repulsive interactions diffusion coefficients decrease with k up to moderately high coverage and increase with k at high coverage. Results are rationalized in terms of the behavior of the vacancy probability distribution.