Determinant, diffusivity matrix

This expression does not determine the mixing model uniquely. However, by specifying that the diffusion matrix in the resulting FP equation must equal the conditional joint scalar dissipation rate,88 the FP model for the molecular mixing term in the form of (6.48)... 

Values of a diffusion coefficient matrix, in principle, can be determined from multicomponent diffusion experiments. For ternary systems, the diffusivity matrix is 2 by 2, and there are four values to be determined for a matrix at each composition. For quaternary systems, there are nine unknowns to be determined. For natural silicate melts with many components, there are many unknowns to be determined from experimental data by fitting experimental diffusion profiles. When there are so many unknowns, the fitting of experimental concentration... 

For the dissolution of a crystal into a melt, if one wants to predict the interface melt composition (that is, the composition of the melt that is saturated with the crystal), the dissolution rate, and the diffusion profiles of all major components, thermodynamic understanding coupled with the diffusion matrix approach is necessary (Liang, 1999). If the effective binary approach is used, it would be necessary to determine which is the principal equilibrium-determining component (such as MgO during forsterite dissolution in basaltic melt), estimate the concentration of the component at the interface melt, and then calculate the dissolution rate and diffusion profile. To estimate the interface concentration of the principal component from thermod5mamic equilibrium, because the concentration depends somewhat on the concentrations of other components, only... 

Tracer diffusivities are often determined using the thin-source method. Self-diffusivities are often obtained from the diffusion couple and the sorption methods. Chemical diffusivities (including interdiffusivity, effective binary diffusivity, and multicomponent diffusivity matrix) may be obtained from the diffusion-couple, sorption, desorption, or crystal dissolution method. 

The methods of determination of the reaction matrix [AT] are considered in Refs. 167, 181, 183, 184 and 186. Another important matrix parameter entering into the linearized film mass transport equation is the multicomponent diffusion matrix /). The latter results from the transformation of the Maxwell-Stefan Eqs. (1) to the form of the generalized Fick s law (83). Matrix [D] is generally a function of... 

The first step is to determine the matrix of Fick diffusion coefficients [ >]. The arithmetic average mole fractions will be needed in the evaluation of the [D] these average mole fractions are... 

The reader should note that the microscale model is used to determine the nonzero terms in B, and thus for the following discussion B can be assumed to be known. Using matrix notation and the properties of the Wiener process (Gardiner, 2004), a symmetric N x N diffusion matrix D can be defined by... 

R", and D is the n X n diffusion matrix. We focus on front solutions and illustrate how to determine the front velocity for KPP kinetics. Let us write, for simplicity. 

Meanwhile we have an opportunity to determine the rule numerically, and thus perhaps to show that it does not exist or that it does or does not depend sensitively on the choice of local reaction kinetics or on the diffusion matrix. Even if that hope fails we will at least obtain clearly described experimental data against which to check the level of approximation in putative analytical solutions. 

Very high conversion - the polymerization medium is a glassy matrix. Most chains are immobile and reaction diffusion is the rate-determining diffusion mechanism. New chains are rapidly terminated or immobilized. Initiator efficiencies are very low. 

The purpose of this chapter is to provide the reader with robust, easily applied methods for estimating diffusivities within the dominant natural phases (air, water, solid matrix, oil, etc.) for a range of environmentally and geochemically relevant compounds. The types of compounds considered include atomic (Hg) and diatomic (O2, CI2) elements, simple molecules (H2S, CO2, NO), organic contaminants (polynuclear aromatic hydrocarbons, polychlorinated biphenyls, chlorinated solvents), as well as dissolved cationic (Pb +, Cr +) and anionic (Br , SO ) species. Examples showing the application of estimation methods are provided and the results of different methods are compared to measured values from the literature. When available, estimates regarding the accuracy of theoretical and empirical methods are included. Experimental methods for determining diffusivities in air, water, porous... 

It is difficult to determine diffusivities within a reacting solid matrix. 

Membra.ne Diffusiona.1 Systems. Membrane diffusional systems are not as simple to formulate as matrix systems, but they offer much more precisely controlled and uniform dmg release. In membrane-controlled dmg deUvery, the dmg reservoir is intimately surrounded by a polymeric membrane that controls the dmg release rate. Dmg release is governed by the thermodynamic energy derived from the concentration gradient between the saturated dmg solution in the system s reservoir and the lower concentration in the receptor. The dmg moves toward the lower concentration at a nearly constant rate determined by the concentration gradient and diffusivity in the membrane (33). 

The alternative rate-determining process to diffusion is die transfer of atoms across tire particle-matrix interface. In this case there is a rate constant for... 

Note in passing that the common model in the theory of diffusion of impurities in 3D Debye crystals is the so-called deformational potential approximation with C a>)ccco,p co)ccco and J o ) oc co, which, for a strictly symmetric potential, displays weakly damped oscillations and does not have a well defined rate constant. If the system permits definition of the rate constant at T = 0, the latter is proportional to the square of the tunneling matrix element times the Franck-Condon factor, whereas accurate determination of the prefactor requires specifying the particular spectrum of the bath. 

The kinetic requirements for a successful application of this concept are readily understandable. The primary issue is the rate at which the electroactive species can reach the matrix/reactant interfaces. The critical parameter is the chemical diffusion coefficient of the electroactive species in the matrix phase. This can be determined by various techniques, as discussed above. 

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