Atomic Models for Diffusivities

In the dissociative mechanism, a substitutional solute atom enters an interstitial site, leaving a vacancy, V, behind according to the reaction 

These reactions are reversible. This dual-site occupancy leads to complicated solute diffusion behavior and has been described for several solute species in Si [4, 6. 8]. There is no compelling evidence that the ring mechanism in Fig. 8.1 contributes significantly to diffusion in any material. 

Atomic models for the diffusivity can be constructed when the diffusion occurs by a specified mechanism in various crystalline materials. A number of cases axe considered below. 

Diffusion of Solute Atoms by the Interstitial Mechanism in the B.C.C. Structure. The 

Because each interstitial site has four nearest-neighbors, the jump rate, F, is given by 41, where T has the form of Eq. 7.25.3 If a is the lattice constant for the b.c.c. unit cell in b.c.c. Fe, then r = a/2 and Eq. 8.3 yields 

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One of the simplest models for diffusion is that of the random movement of atoms. The model is generally called a random (or drunkard s) walk.4 A random walk produces a path that is governed completely by random jumps (Fig. S5.3). That is, each individual jump is unrelated to the step before and is governed solely by the probabilities of taking the alternative steps. The application of random walks to diffusion was first made by... 

Figure 6.13 Schematic illustration of a lattice model for diffusion of Ag atoms on Pd doped Cu(100). The diagrams on the right show the fourfold surface sites in terms of the four surface atoms defining the site. The section of the surface shown on the left includes two well separated Pd atoms in the surface. The Pd atoms are located at the centers of the two grey squares in the diagram on the left.

Microscopic and mechanistic aspects of diffusion are treated in Chapters 7-10. An expression for the basic jump rate of an atom (or molecule) in a condensed system is obtained and various aspects of the displacements of migrating particles are described (Chapter 7). Discussions are then given of atomistic models for diffusivities and diffusion in bulk crystalline materials (Chapter 8), along line and planar imperfections in crystalline materials (Chapter 9), and in bulk noncrystalline materials (Chapter 10). 

A united-atom model for [C4mim][PF6] and [C4mim][N03] was developed in the framework of the GROMOS96 force field [71]. The equilibrium properties in the 298-363 K temperature range were validated against known experimental properties, namely, density, self-diffusion, shear viscosity, and isothermal compressibility [71]. The properties obtained from the MD simulations agreed with experimental data and showed the same temperature dependence. 

And, thus, the macroscopic diffusivity can be obtained from a consideration of the random atomic motions. The importance of this and the previous derivation is that uq and t are both parameters that can be easily extracted from a KMC simulation and thus diffiisivities can be obtained that can be used to compare with experimentally determined values or that can be used to calibrate less easily measured parameters, such as atom-electrolyte interactions or interspecies bonding, that are used to determine the energy barriers in the local bond-breaking model for diffusion and dissolution. 

The proton-jump model for diffusion is based upon the premise that the structure of liquid water derives from an icelike hydrogen-bonded network in addition to the two normal OH bonds in water each oxygen atom is linked to neighboring ones via hydrogen bonds. The linkage is imperfect and there are defects in the network. These imperfections, known... 

Discovery of the hydrated electron and pulse-radiolytic measurement of specific rates (giving generally different values for different reactions) necessitated consideration of multiradical diffusion models, for which the pioneering efforts were made by Kuppermann (1967) and by Schwarz (1969). In Kuppermann s model, there are seven reactive species. The four primary radicals are eh, H, H30+, and OH. Two secondary species, OH- and H202, are products of primary reactions while these themselves undergo various secondary reactions. The seventh species, the O atom was included for material balance as suggested by Allen (1964). However, since its initial yield is taken to be only 4% of the ionization yield, its involvement is not evident in the calculation. 

The simplest and most basic model for the diffusion of atoms across the bulk of a solid is to assume that they move by a series of random jumps, due to the fact that all the atoms are being continually jostled by thermal energy. The path followed is called a random (or drunkard s) walk. It is, at first sight, surprising that any diffusion will take place under these circumstances because, intuitively, the distance that an atom will move via random jumps in one direction would be balanced by jumps in the opposite direction, so that the overall displacement would be expected to average out to zero. Nevertheless, this is not so, and a diffusion coefficient for this model can be defined (see Supplementary Material Section S5). 

The random-walk model of diffusion can also be applied to derive the shape of the penetration profile. A plot of the final position reached for each atom (provided the number of diffusing atoms, N, is large) can be approximated by a continuous function, the Gaussian or normal distribution curve2 with a form ... 

The random-walk model of diffusion needs to be modified if it is to accurately represent the mechanism of the diffusion. One important change regards the number of point defects present. It has already been pointed out that vacancy diffusion in, for example, a metal crystal cannot occur without an existing population of vacancies. Because of this the random-walk jump probability must be modified to take vacancy numbers into account. In this case, the probability that a vacancy is available to a diffusing atom can be approximated by the number of vacant sites present in the crystal, d], expressed as a fraction, that is... 

The simplest model for an atomic assembly is to consider the atoms as hard spheres with a radius a. Computer simulations have been used to describe the physical behaviour of such assemblies as the number density is changed. At low number densities the assembly is a fluid and the hard spheres diffuse in a gaseous fashion. There are three degrees of freedom corresponding to kBT(2 for each orthogonal translational direction. At intermediate densities the motion of an individual sphere becomes more complex. Some of the time it will move inside a transient cage of... 

In this subsection, we are not concerned with simulations which study the motion of single adsorbate atoms for realistic choices of the corrugation potential, but again restrict attention to simplified lattice gas models, where diffusion events are modelled by stochastic hops of adatoms from one lattice site to the next The description of the dynamics hence again is done... 

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