Lattice diffusion equation

Diffusion Equations Related to Fundamental Vibrations of the Lattice... 

Eggels and Somers (1995) used an LB scheme for simulating species transport in a cavity flow. Such an LB scheme, however, is more memory intensive than a FV formulation of the convective-diffusion equation, as in the LB discretization typically 18 single-precision concentrations (associated with the 18 velocity directions in the usual lattice) need to be stored, while in the FV just 2 or 3 (double-precision) variables are needed. Scalar species transport therefore can better be simulated with an FV solver. 

With the additional assumption of the Poisson distribution for the probability of n jumps in a time t (mean time between jumps r), Equations 8 and 9 can be used in existing lattice diffusion theories to calculate relaxation times (approximately, i.e., within 10%) (16,18). 

They have calculated the continuous diffusion equation (3.2.30) with U(r) = -a/r3 for several kinds of nn F, H centres in the crystalline lattice. Figure 3.9 demonstrates well that both defect initial separation and an elastic interaction are of primary importance for geminate pair recombination kinetics. The 3nn defects are only expected to have noticeable survival probability. Its magnitude agrees well with equation (3.2.60). 

The process of introducing impurities into silicon is called predeposition. Chemical predeposition is described in terms of a solution to the diffusion equation. Predeposition by ion implantation is described in terms of ion penetration into silicon, distributions of implanted impurities, lattice damage, etc. 

Therefore, diffusivity is basically the product of the lattice vibration frequency, vacancy concentration, and activated-lattice concentration (equation 26). 

In the case of YTZP, on which a large number of studies have been performed, the data could be fitted to a constitutive equation, which is identical to that found in metals when lattice diffusion is the rate-controlling mechanism 29... 

The transfer of the adspecies within macrodistances is described with the aid of diffusion equations. The concentrations of the adspecies therein is characterized by the local degrees of lattice coverage 6(x) with adspecies / in the vicinity of the point with the coordinates a. In the absence of external fields the diffusion equation for each component has the form [154,155] ... 

The kinetic equations for the volume phase of the solid body are equations of the diffusion type (63). Much attention has been given to them in the literature [154,155], therefore here will be reminded only those aspects of the theory of mass transfer for which the lattice-gas model has been used. These are problems involved in the construction of expressions for the diffusion the coefficients and boundary conditions of the diffusion equations. 

Both categories of transport discussed above involve the motion of defects relative to an otherwise ordered array of ions, so the transport is referred to as defect transport. The concentrations of such defects (as contrasted with the total atom concentrations within the oxide lattice) are the important quantities for mass transport, charge transport, and space—charge effects, so it is the species defect concentration which appears in the diffusion equation. Likewise, the diffusion coefficients... 

We use our previous results for the return and recombination probabilities of the vacancy, and consider the random walk of the tracer-vacancy pair on the bond lattice. Let p(r, re) denote the probabihty that the tracer-vacancy pair is at position r at instance n, where re counts the number of moves the tracer-vacancy pair has already made. Since the subsequent moves of the pair are independent, we can write an effective diffusion equation for the evolution of p(r, n) ... 

Figure 5 Multiscale approach to understand rate of CO2 diffusion into and CH4 diffusion out of a structure I hydrate, (left) Molecular simulation for individual hopping rates, (middle) Mesoscale kinetic Monte Carlo simulation of hopping on the hydrate lattice to determine dependence of diffusion constants on vacancy, CO2 and CH4 concentrations, (right) Macroscopic coupled non-linear diffusion equations to describe rate of CO2 infusion and methane displacement. Graph from Stockie. ...

The term "disruptive quencher" is applied to the case in which all exciton contacts with the quencher results in completely effective quenchingIt is not at all clear that this property is applicable to Cu quenching. The case of a non-dlsruptive quencher is much harder to analyze and does not lead to a convenient expression for the excitation decay, analogous to eqn. (18). Based on classical diffusion equations it seems plausible that the disruptive quencher model is applicable so long as the quenching rate at the quenched site is of the same order of magnitude as the transfer rate from that lattice site to neighboring sites. ... 

For translational long-range jump diffusion of a lattice gas the stochastic theory (random walk, Markov process and master equation) [30] eventually yields the result that Gg(r,t) can be identified with the solution (for a point-like source) of the macroscopic diffusion equation, which is identical to Pick s second law of diffusion but with the tracer (self diffusion) coefficient D instead of the chemical or Fick s diffusion coefficient. 

If we assume that the variation of the function P(m,f) over distances of the order of the lattice constant is small, then eqn (14.27) reduces to the diffusion equation. Indeed, assuming that the vector m varies continuously, we obtain... 

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