Diffusivity effective medium approximation

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

Figure 4.9 Estimation of reduced diffusion coefficient by effective medium approximation. Reduced diffusion coefficient for solutes of radius 3.4 and 10 nm in a solution...

Figure 4.13 Estimation of reduced diffusion coefficient by effective medium approximation. Combination of steric and hydrodynamic effects on reduced diffusion coefficient. The solid lines represent hydrodynamic effect for probe radii of 3.4 and 10 A calculated using Brinkman s equation (see Figure 4.9). The dashed lines represent the combined steric and hydrodynamic effect using Equation 4-40 for the steric effect.

An analytical description of the diffusivity of molecules in pore systems with open and obstructed windows may be based on the effective medium approximation of percolation systems [156,157]. Lowest-order effective medium approximation yields for the diffusivity [149,156,158,159]... 

Fig. 17 Intracrystalline self-diffusivity of methane ( 2 molecules per supercage, at 25 °C) as a function of the amount of co-adsorbed molecules per window . The solid lines are predictions based on the effective medium approximation of percolation theory with / denoting the ratio of the transition rates through blocked and open windows. From [158] with permission...

Kapoor, A., and Yang, R.T., Surface diffusion on energetically heterogeneous surfaces An effective medium approximation approach, Chem. Eng. Sci., 45(11). 3261-3270 (1990). 

The inclusion of 2 extends the previously reported procedure of Relaxation Spectrum Analysis (44). in this form can include contributions from static disorder such as porosity (45), random mixture of conductor and insulator that can be described by the effective medium approximation at percolation (46), or an interface that can be described by a fractal geometry (47). It can also include contributions from dynamic disorder such as diffusion. To provide one specific example if originates from diffusion capacitance in the semiconductor, then r is the minority carriers diffusion time, n = 0.5 and... 

A. Horner. Self-diffusion in metallic glasses Approximation of the effective medium and molecular simulation. PhD thesis, Stuttgart University, 1993. In German. 

These reactions generate electrochemical impedances due to charge transfer, gas or solid state diffusion, etc. Since these impedances appear specifically at the boundaries between dissimilar phases, the composites cannot be fully described by simple effective medium models, even if these impedances are approximated by linear resistive elements. As pointed out by several authors, in the mixture of electronic and ionic phases there are clusters connected to (i) both current collector and electrolyte, (ii) only to the electrode, and (iii) isolated clusters. Clusters of all three types are visible in Figure 4.1.14. 

Burganos, V.N., and Sotirchos, S.V., Diffusion in pore networks Effective medium theory and smooth field approximation, AlChE J., 33(10), 1678-1689 (1987),... 

Chapter 15 - It was shown, that the reesterification reaction without catalyst can be described by mean-field approximation, whereas introduction of catalyst (tetrabutoxytitanium) is defined by the appearance of its local fluctuations. This effect results to fractal-like kinetics of reesterification reaction. In this case reesterification reaction is considered as recombination reaction and treated within the framework of scaling approaches. Practical aspect of this study is obvious-homogeneous distribution of catalyst in reactive medium or its biased diffusion allows to decrease reaction duration approximately twofold. 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

The mean-field theory has a number of shortcomings, including the approximations of a mean concentration around all particles and the establishment of spherically symmetric diffusion fields around every particle, similar to those that would exist around a single particle in a large medium. The larger the particles total volume fraction and the more closely they are crowded, the less realistic these approximations are. No account is taken in the classical model of such volume-fraction effects. Ratke and Voorhees provide a review of this topic and discuss extensions to the classical coarsening theory [8]. 

In one important respect, this derivation is not quite complete. Just as there are two ways in which the encounter complex A -B can be formed, so there are two ways in which it can react. Because the average reaction time is comparable to the time taken for the steady state to be set up, only a certain fraction w of the excited molecules will obey the Stem-Volmer equation. The remaining (1 —h ) reacts immediately after excitation and so does not contribute to the relative fluorescence yield. Put another way, if a molecule of A has a B within the reaction distance when it is excited, it may react immediately and so will not fluoresce. As may be predicted, the effect of this transient excess reactivity is more important the harder it is for A and B to diffuse apart, i.e., the greater the viscosity of the medium, and the more efficient is the reaction. Thus < >/( >o = W(1+ 2< b < o)> the stationary rate coefficient may be evaluated if w is known. The latter can be calculated from the expression w = exp(— VoCj,), where is a characteristic reaction volume surrounding A and w represents the probability that no B molecule will be found inside this space. Vjy is a function of the diffusion coefficients of A and B, the mean lifetime of A in the absence of B(xo) and the effective encounter distance. In most cases approximate values of w can be calculated and then, by successive approximations, the stationary rate coefficients and encounter distances which best lit the data are computed. 

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