Atomistic Treatment of Diffusion

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

The treatment so far has been phenomenological and therefore the dependence of the diffusion coefficient on factors such as temperature and type of ion can be theoretically understood only by an atomistic analysis. The quantity D can be understood in a fundamental way only by probing into the ionic movements, the results of which show up in the macroscopic world as the phenomenon of diffusion. What are these ionic movements, and how do they produce diffusion The answering of these two questions will constitute the next topic. 

This is the Einstein relation. It is probably the most important relation in the theory of the movements and drift of ions, atoms, molecules, and other submicroscopic particles. It has been daived ho e in an atomistic way. It will be recalled that in the phenomenological treatment of the diffusion coefficient (Section 4.2.3), it was shown that... 

The two standard approaches in any treatment of kinetics [28] are to explain the system in terms of me thermodynamic driving forces (namely, VjJ.) or in terms of the fnndamental rate eqnations. The rate equations can be fnrther subdivided into an atomistic, or microscopic, approach that accounts for individual molecules as they go through the various processes (adsorption, desorption, diffusion, capture, and release) or a phenomenological, or macroscopic, explanation that looks for correlations and the so-called scaling laws over large distances (much larger than the lattice spacing). 

Integration of KMC simulations with larger-scale models such as finite element continuum approaches can enhance the impact of a KMC simulation. In many cases, certain components of an electrochemical device can be effectively approximated by a bulk continuum treatment (e.g., ionic diffusion through the bulk of an electrolyte) while other components of a device that involve interfacial transport require discrete treatments like either Butler-Volmer empiricism or an atomistic treatment to capture important details related to the distribution of charge in the double layer. 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

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