Position dependence, diffusion

The monovariate Fokker-Planck equation with a position dependent diffusion coefficient D x),... 

Position dependent diffusivity and/or catalytic activity. (Kasaoka and Sakata Corbett and Luss Becker and Wei [141,142,143].)... 

Various extensions to DDFT have been proposed (see below). The generalization to mixtures of different particle types is straightforward. A background flow and a position-dependent diffusion constant can also be included. 

Currently, analytical approaches are still the most preferred tools for model reduction in microfluidic research community. While it is impossible to enumerate all of them in this entry, a generalized analytical model on species transport under pressure-driven flow in rectangular microchaimel with arbitrary aspect ratio will be discussed [10]. The nonuniform velocity profile along the cross section of the microchaimel under the pressure-driven flow results in unique and complex species transport phenomena including Taylor dispersion, heterogeneous transport rate, and position-dependent diffusion scaling law. 

The diffusion equation given by Fick s second law where Cj x,t) is the position dependent diffusible hydrogen concentration, t is time, D is the diffusion coefficient, and x is distance through the foil has been solved for the initial conditions Cr[x, f = 0) = 0 and for the boundary conditions Cr(0, t = ) = C , Cr L, f = ) = 0. In other words, the conditions are that before the experiment is started, the foil contains no hydrogen and, at any time after the start of the experiment, the concentration, Cr, on one side (x = 0) is a constant, Q, and on the other side (x = L) it is equal to zero. The time f = 0, in this case, corresponds to the time at which the concentration is switched from 0 to Q. The solution of Eq 47 with the above initial and boundary conditions yields a series, which, to a good approximation, can be given by its first term ... 

This section is concerned with measurement techniques of the diffusivity and solubility from which the permeability can easily be calculated. In the following analysis we restrict ourselves to the measurement of constant values of D. Concentration- and position-dependent diffusivities are analyzed in Crank and Park (1968) and Crank (1975). Generally, the techniques are for permeability, steady-state and time lag techniques and for diffusivity, sorption and desorption kinetics and concentration-distance curves. For self-diffusivity in polymer melts the techniques are (Tirrell, 1984) nuclear magnetic resonance, neutron scattering, radioactive tracer, and infrared spectroscopy. 

The additional deterministic contribution from the position-dependent diffusivity is known... 

It is now seen that only the resistance to the mass transfer term for the stationary phase is position dependent. All the other terms can be used as developed by Van Deemter, providing the diffusivities are measured at the outlet pressure (atmospheric) and the velocity is that measured at the column exit. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

Fig. 23. Position dependence of the effective collective diffusion constant normalized by the collective diffusion constant of spherical gels. D0 = (K + 4ji/3)/f. At the boundary, the values for sphere, cylinder, and disk are 1, 2/3, and 1/3, respectively...

Up to now, only hydrodynamic repulsion effects (Chap. 8, Sect. 2.5) have caused the diffusion coefficient to be position-dependent. Of course, the diffusion coefficient is dependent on viscosity and temperature [Stokes—Einstein relationship, eqn. (38)] but viscosity and temperature are constant during the duration of most experiments. There have been several studies which have shown that the drift mobility of solvated electrons in alkanes is not constant. On the contrary, as the electric field increases, the solvated electron drift velocity either increases super-linearly (for cases where the mobility is small, < 10 4 m2 V-1 s-1) or sub-linearly (for cases where the mobility is larger than 10 3 m2 V 1 s 1) as shown in Fig. 28. Consequently, the mobility of the solvated electron either increases or decreases, respectively, as the electric field is increased [341— 348]. 

To show this connection, consider an ion-pair as above (Sect. 2.1). Not only may the ion-pair diffuse and drift in the presence of an electric field arising from the mutual coulomb interaction, but also charge-dipole, charge-induced dipole, potential of mean force and an external electric field may all be included in the potential energy term, U. Both the diffusion coefficient and drift mobility may be position-dependent and a long-range transfer process, Z(r), may lead to recombination of the ion-pair. Equation (141) for the ion-pair density distribution becomes... 

While eqn. (211) is a bit complex, the similarity to the more familiar diffusion equation [e.g. eqns. (43), (44), (158) or (197)] is apparent. The diffusion coefficient has to be replaced by a position-dependent tensor which couples (or connects) the motion of one of the particles, e.g. the particle k, with that of the other particles, e.g. a particle j. When these particles are a long way apart, the solvent between them can be squeezed out easily. As the particles approach, this is no longer true because the two particles block certain directions for escape of the solvent as the particles approach. Increasingly, the solvent has to be squeezed out of the way in a direction perpendicular to that of the approach of the particles and this causes the solvent to impede the particle approach more and more effectively. When j = k, the effect on the same particle of its own motion is negligible. Hence, the diffusion coefficient tensor elements, Tjj = kBT/%, are the same as the diffusion coefficient for the particle... 

In this Appendix, the equivalence of the diffusion equation treatment and the molecular pair analysis is proved (see Chap, 8, Sect. 3.2) for the situation where there is a potential energy E/(r) between the reactants and the diffusion coefficient is tensorial and position-dependent. This Appendix is effectively a generalisation of the analysis of Berg [278]. The diffusion equation has a Green s function G(r, f r0, t0) which satisfies eqn. (161)... 

Now let us discuss the evaluation of the chemical diffusion constant If from the transient behavior.3 58210 For this discussion we assume sufficiently small signals, as to be able to neglect stoichiometry dependencies and thus positional dependencies of the transport parameters creon, crjon, and Tf. 

Following the work of Amovilli and Mennucci [21] a model for repulsion interactions at diffuse interfaces has been developed. Since the repulsion energy depends on the solvent density it is then natural to replace the constant density p with a position-dependent density p(z). The first attempt made use of p(z) in the final expression for the repulsion energy [17]. Such a model has subsequently been improved by a derivation of a new repulsion expression [18]. 

The cavitation expression may also be extended to diffuse interfaces, by weighting each contribution coming from the tesserae with an appropriate position-dependent function. 

Figure 2.38 Interfacial solvation (a) a solvated molecule embedded in a cavity lying on top of a metal surface (b) a solvated molecule at the diffuse interface characterized by position-dependent properties, e.g. permittivity e(z).

The results are reported in Fig. 2.1 in which z is the coordinate perpendicular to the interface. The limit values (z - oo) refer to bulk cyclohexane and bulk water in the latter case, the result obtained for the solvated supermolecule is also reported (in red). In the same graph, we also report the position-dependent value of the dielectric constant used to mimic the diffuse interface. 

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