Nonlinear diffusion coefficient

A nonlinear diffusion coefficient may cause the generation of patterns and a long-wavelength instability. Consider a two-dimensional reaction-diffusion system for the bacteria density B(r,t) with a nonlinear diffusion term, and nutrient density N(r,t) with a linear diffusion term... 

In order to model the transport phenomena in polymeric materials, Lefebvre et derived a nonlinear diffusion coefficient based on the concept of free volume. According to this theory, the diffusion coefficient D for a polymeric material above its glass transition temperature is given by... 

The first approach to obtain exact time characteristics of Markov processes with nonlinear drift coefficients was proposed in 1933 by Pontryagin, Andronov, and Vitt [19]. This approach allows one to obtain exact values of moments of the first passage time for arbitrary time constant potentials and arbitrary noise intensity moreover, the diffusion coefficient may be nonlinear function of coordinate. The only disadvantage of this method is that it requires an artificial introducing of absorbing boundaries, which change the process of diffusion in real smooth potentials. 

The second major difference found in vapor-liquid extraction of polymeric solutions is related to the low values of the diffusion coefficients and the strong dependence of these coefficients on the concentration of solvent or monomer in a polymeric solution or melt. Figure 2, which illustrates how the diffusion coefficient can vary with concentration for a polymeric solution, shows a variation of more than three orders of magnitude in the diffusion coefficient when the concentration varies from about 10% to less than 1%. From a mathematical viewpoint the dependence of the diffusion coefficient on concentration can introduce complications in solving the diffusion equations to obtain concentration profiles, particularly when this dependence is nonlinear. On a physiced basis, the low diffiisivities result in low mass-transfer rates, which means larger extraction equipment. 

The cell sizes are expected to exceed any molecular (atomic) scale so that a number of particles therein are large, Ni(f) 1. The transition probabilities within cells are defined by reaction rates entering (2.1.2), whereas the hopping probabilities between close cells could easily be expressed through diffusion coefficients. This approach was successfully applied to the nonlinear systems characterized by a loss of stability of macroscopic structures and the very important effect of a qualitative change of fluctuation dispersion as the fluctuation length increases has also been observed [16, 27]. In particular cases the correlation length could be the introduced. The fluctuations in... 

Fig. 15. Diffusion of dilute, aqueous acetaminophen into a long, swollen cylinder of 10x4 PNIPAAm gel at 25 °C. The diffusion coefficient is extracted from a nonlinear least squares curve fit of the exact solution for diffusion into a cylinder of infinite length immersed in a well-stirred solution of finite volume to the data [123, 149]...

The analytical solutions to Fick s continuity equation represent special cases for which the diflusion coefficient, D, is constant. In practice, this condition is met only when the concentration of diffusing dopants is below a certain level ( 1 x 1019 atoms/cm3). Above this doping density, D may depend on local dopant concentration levels through electric field effects, Fermi-level effects, strain, or the presence of other dopants. For these cases, equation 1 must be integrated with a computer. The form of equation 1 is essentially the same for a wide range of nonlinear diffusion effects. Thus, the research emphasis has been on understanding the complex behavior of the diffusion coefficient, D, which can be accomplished by studying diffusion at the atomic level. 

On the other hand, most weak nonlinearities can be associated with the dependence of specific heats (Cp) upon temperature, or of diffusion coefficients upon concentrations, etc. 

Section IIA summarizes the physical assumptions and the resulting mathematical descriptions of the "concentration-dependent (5) and "dual-mode" ( 13) sorption and transport models which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In Section IIB we elucidate the mechanism of the "non-ideal" diffusion in glassy polymers by correlating the phenomenological diffusion coefficient of CO2 in PVC with the cooperative main-chain motions of the polymer in the presence of the penetrant. We report carbon-13 relaxation measurements which demonstrate that CO2 alters the cooperative main-chain motions of PVC. These changes correlate with changes in the diffusion coefficient of CO2 in the polymer, thus providing experimental evidence that the diffusion coefficient is concentration dependent. 

Nonlinear, pressure-dependent solubility and permeability in polymers have been observed for over 40 years. Meyer, Gee and their co-workers (5) reported pressure-dependent solubility and diffusion coefficients in rubber-vapor systems. Crank, Park, Long, Barrer, and their co-workers (5) observed pressure-dependent sorption and transport in glassy polymer-vapor systems. Sorption and transport measurements of gases in glassy polymers show that these penetrant-polymer systems do not obey the "ideal sorption and transport eqs. (l)-(5). The observable variables,... 

Wonders and Paul (15) report that a nonlinear least-squares fit of the dual-mode expression [eq. (16) in the preceding chapter] to the permeability versus pressure data, for C02 in polycarbonate, gives and values of 4.78 x 10 8 and 7.11 x 10 9 cm2/sec, respectively. The broken curve in Fig. 2 was calculated from the dual-mode sorption coefficients of Fig. 1 and the values of the diffusion coefficients given above. 

In order to obtain values for the diffusion coefficient at spherical electrodes, a logarithmic plot of the current versus time would lead to nonlinear dependence (see Eq. 2.147). In this case a plot of the current versus 1 / fl is more appropriate (see inner curve in Fig. 2.15) and this plot also allows the determination of the electrode radius by combining the values of the slope (FAsc Q JD/n) and intercept (FAJ)c ())lrK... 

In summary, although the construction of micro-ITIES is, in general, simpler than that of microelectrodes, their mathematical treatment is always more complicated for two reasons. First, in micro-ITIES the participating species always move from one phase to the other, while in microelecrodes they remain in the same phase. This leads to complications because in the case of micro-ITIES the diffusion coefficients in both phases are different, which complicates the solution when nonlinear diffusion is considered. Second, the diffusion fields of a microelectrode are identical for oxidized and reduced species, while in micro-ITIES the diffusion fields for the ions in the aqueous and organic phases are not usually symmetrical. Moreover, as a stationary response requires fDt / o (where D is the diffusion coefficient, r0 is the critical dimension of the microinterface, and t is the experiment time), even in L/L interfaces with symmetrical diffusion field it may occur that the stationary state has been reached in one phase (aqueous) and not in the other (organic) at a given time, so a transient behavior must be considered. 

The overall mass-transfer rates on both sides of the membrane can only be calculated when we know the convective velocity through the membrane layer. For this, Equation 14.2 should be solved. Its solution for constant parameters and for first-order and zero-order reaction have been given by Nagy [68]. The differential equation 14.26 with the boundary conditions (14.28a) to (14.28c) can only be solved numerically. The boundary condition (14.28c) can cause strong nonlinearity because of the space coordinate and/or concentration-dependent diffusion coefficient [40, 57, 58] and transverse convective velocity [11]. In the case of an enzyme membrane reactor, the radial convective velocity can often be neglected. Qin and Cabral [58] and Nagy and Hadik [57] discussed the concentration distribution in the lumen at different mass-transport parameters and at different Dm(c) functions in the case of nL = 0, that is, without transverse convective velocity (not discussed here in detail). 

Errors and confusion in modelling arise because the complex set of coupled, nonlinear, partial differential equations are not usually an exact representation of the physical system. As examples, first consider the input parameters, such as chemical rate constants or diffusion coefficients. These input quantities, used as submodels in the detailed model, must be derived from more fundamental theories, models or experiments. They are usually not known to any appreciable accuracy and often their values are simply guesses. Or consider the geometry used in a calculation. It is often one or two dimensions less than needed to completely describe the real system. Multidimensional effects which may be important are either crudely approximated or ignored. This lack of exact correspondence between the model adopted and the actual physical system constitutes the basic problem of detailed modelling. This problem, which must be overcome in order to accurately model transient combustion systems, can be analyzed in terms of the multiple time scales, multiple space scales, geometric complexity, and physical complexity of the systems to be modelled. 

The general solutions of the fundamental systems of nonlinear equations [Eq. (2)] will be of the type wherein the state variables are dependent both on time and space, which will manifest in the form of wave propagation. Coupling between several parts of the system will be transmitted through the generalized diffusion coefficient D. If the associated transport process proceeds on a time scale comparable to or slower than the period of the temporal oscillation, macroscopic wave propagation phenomena are to be expected, as, for example, realized with the Belousov-Zhabotinsky... 

Nanocrystalline systems display a number of unusual features that are not fully understood at present. In particular, further work is needed to clarify the relationship between carrier transport, trapping, inter-particle tunnelling and electron-electrolyte interactions in three dimensional nan-oporous systems. The photocurrent response of nanocrystalline electrodes is nonlinear, and the measured properties such as electron lifetime and diffusion coefficient are intensity dependent quantities. Intensity dependent trap occupation may provide an explanation for this behaviour, and methods for distinguishing between trapped and mobile electrons, for example optically, are needed. Most models of electron transport make a priori assumptions that diffusion dominates because the internal electric fields are small. However, field assisted electron transport may also contribute to the measured photocurrent response, and this question needs to be addressed in future work. 

Here, Ds and Dd are the coefficients representing the Soret and Dufour effects, respectively, Du is the self-diffusion coefficient, and Dik is the diffusion coefficient between components / and k. Equations (7.149) and (7.150) may be nonlinear because of, for example, reference frame differences, an anisotropic medium for heat and mass transfer, and temperature- and concentration-dependent thermal conductivity and diffusion coefficients. 

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