Dynamic equilibrium solid-state diffusion

One possibility that would lead to larger inferred porosities for the U-series was introduced by Qin (1992, 1993), who proposed that the retained melt was only in complete equilibrium with the surface of minerals and that solid-state diffusion limited the re-equilibration of the retained melt with the solid. In other respects, this model is identical to the ACM model of Williams and Gill (1989). Qin introduced a specific microscopic melting/diffusion model for spherical grains and coupled it to the larger-scale dynamic melting models. The net affect of this... 

In this section, we develop an atomistic picture to understand solid-state diffusion in more detail. At the most fundamental level, a solid-state diffusion coefficient D is a measure of the intrinsic rate of the hopping process by which atoms/molecules can move from one site to another in a solid medium. Even in the absence of any driving force, hopping of atoms from site to site within the lattice still occurs at a rate that is characterized by the diffusivity. Of course, without a driving force, the net movement of atoms is zero, but they are still exchanging lattice sites with one another. This is another example of a dynamic equilibrium, compare it to the dynamic reaction equilibrium processes that we discussed in Chapter 3. 

Theoretical studies of the properties of the individual components of nanocat-alytic systems (including metal nanoclusters, finite or extended supporting substrates, and molecular reactants and products), and of their assemblies (that is, a metal cluster anchored to the surface of a solid support material with molecular reactants adsorbed on either the cluster, the support surface, or both), employ an arsenal of diverse theoretical methodologies and techniques for a recent perspective article about computations in materials science and condensed matter studies [254], These theoretical tools include quantum mechanical electronic structure calculations coupled with structural optimizations (that is, determination of equilibrium, ground state nuclear configurations), searches for reaction pathways and microscopic reaction mechanisms, ab initio investigations of the dynamics of adsorption and reactive processes, statistical mechanical techniques (quantum, semiclassical, and classical) for determination of reaction rates, and evaluation of probabilities for reactive encounters between adsorbed reactants using kinetic equation for multiparticle adsorption, surface diffusion, and collisions between mobile adsorbed species, as well as explorations of spatiotemporal distributions of reactants and products. 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Fig. 2B, one such curve published by Lin et al. [8]. The theoretical solid curve was obtained by these authors using a scheme similar to the one just described - solution of the Ward-Tordai equation together with the Frumkin isotherm and substitution in the equation of state to calculate the surface tension. Note that the parameters a, P and a can be fitted from independent equilibrium measurements, so the dynamic surface tension curve has only one fitting parameter, namely the diffusion coefficient, D. As can be seen, the agreement with experiment is quite satisfactory. However, when the adsorption is not diffusion-limited, such a theoretical approach is no longer applicable, as will be demonstrated in the next section. 

The accumulation is a dynamic process that may turn into a steady state in stirred solutions. Besides, the activity of accumulated substance is not in a time-independent equilibrium with the activity of analyte in the bulk of the solution. All accumulation methods employ fast reactions, either reversible or irreversible. The fast and reversible processes include adsorption and surface complexation, the majority of ion transfers across liquid/liquid interfaces and some electrode reactions of metal ions on mercury. In the case of a reversible reaction, equilibrium between the activity of accumulated substance and the concentration of analyte at the electrode surface is established. It causes the development of a concentration profile near the electrode and the diffusion of analyte towards its surface. As the activity of the accumulated substance increases, the concentration of the analyte at the electrode surface is augmented and the diffusion flux is diminished. Hence, the equilibrium between the accumulated substance and the bulk concentration of the analyte can be established only after an infinitely long accumulation time (see Eqs. II.7.12-II.7.14 and II.7.30). The reduction of metal ions on mercury electrodes in stirred solutions is in the steady state at high overvoltages. Redox reactions of many metal ions, especially at solid... 

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