Nonlinear diffusion

The metal ion in electroless solutions may be significantly complexed as discussed earlier. Not all of the metal ion species in solution will be active for electroless deposition, possibly only the uncomplexed, or aquo-ions hexaquo in the case of Ni2+, and perhaps the ML or M2L2 type complexes. Hence, the concentration of active metal ions may be much less than the overall concentration of metal ions. This raises the possibility that diffusion of metal ions active for the reduction reaction could be a significant factor in the electroless reaction in cases where the patterned elements undergoing deposition are smaller than the linear, or planar, diffusion layer thickness of these ions. In such instances, due to nonlinear diffusion, there is more efficient mass transport of metal ion to the smaller features than to large area (relative to the diffusion layer thickness) features. Thus, neglecting for the moment the opposite effects of additives and dissolved 02, the deposit thickness will tend to be greater on the smaller features, and deposit composition may be nonuniform in the case of alloy deposition. 

Substituting Equations 3-83a and 3-83b into Equation 3-78 leads to a complicated nonlinear diffusion equation for [H20t], which must be solved numer-... 

Helfferich F. and Plesset M.S. (1958) Ion exchange kinetics, a nonlinear diffusion problem. /. Chem. Phys. 28, 418-425. 

Roering J.J., Kirchner J.W., and Dietrich W.E. (1999) Evidence for nonlinear, diffusive sediment transport on hillslopes and implications for landscape morphology. Water Resources Res. 35, 853-870. 

To evaluate the demixing process under the nonisoquench depth condition, they carried out computer simulations of the time dependent concentration fluctuation using the Cahn-Hilliard nonlinear diffusion equation. 

A somewhat similar situation occurs in one-dimensional multi-ionic systems with local electro-neutrality in the absence of electric current. It will be shown in Chapter 3 that in this case again the electric field can be excluded from consideration and the equations of electro-diffusion are reduced to a coupled set of nonlinear diffusion equations. 

The next level is that of one-dimensional electro-diffusion with local electro-neutrality in the absence of an electric current. This is the realm of nonlinear diffusion to be treated in Chapter 3. A still higher level of the same hierarchy is formed by the nonlinear effects of stationary electric current, passing in one-dimensional electro-diflFusion systems with local electro-neutrality. A few typical phenomena of this type will be studied in Chapter 4. The treatment of Chapter 4 will lay the foundation for the discussion of the effects of nonequilibrium space charge characteristic of the fourth level to be treated in Chapter 5. 

In this chapter we shall treat some particular instances of the system (3.1.15) and the related phenomena. Thus in 3.2, we shall concentrate upon binary ion-exchange and discuss the relevant single nonlinear diffusion equation. It will be seen that in a certain range of parameters this equation reduces to the porous medium equation with diffusivity proportional to concentration. Furthermore, it turns out that in another parameter range the binary ion-exchange is described by the fast diffusion equation with diffusivity inversely proportional to concentration. It will be shown that in the latter case some monotonic travelling concentration waves may arise. 

Before we turn to this issue, we would like to substantiate the above discussion of basic features of nonlinear diffusion with some examples based upon the well-known similarity solutions of the Cauchy problems for the relevant diffusion equations. Similarity solutions are particularly instructive because they express the intrinsic symmetry features of the equation [6], [28], [29], Recall that those are the shape-preserving solutions in the sense that they are composed of some function of time only, multiplied by another function of a product of some powers of the time and space coordinates, termed the similarity variable. This latter can usually be constructed from dimensional arguments. Accordingly, a similarity solution may only be available when the Cauchy problem under consideration lacks an explicit length scale. Thus, the two types of initial conditions compatible with the similarity requirement are those corresponding to an instantaneous point source and to a piecewise constant initial profile, respectively, of the form... 

Equation (3.3.10a) represents a proper nonlinear diffusion equation with effective diffusivity Deg, defined by (3.3.10b)... 

J. G. Berryman, Evolution of stable profile for a class of nonlinear diffusion equations with fixed boundaries, J. Math. Phys., 18 (1977), pp. 2108-2115. 

M. A. Herrero, A Limit Case in Nonlinear Diffusion, to appear. 

Preliminaries. In the previous chapter we dealt with locally electro-neutral time-dependent electro-diffusion under the condition of no electric current in a medium with a spatially constant fixed charge density (ion-exchangers). It was observed that under these circumstances electrodiffusion is equivalent to nonlinear diffusion with concentration-dependent diffusivities. 

Disks fall into the second category of electrode, at which nonlinear diffusion occurs. The lines of flux to a disk electrode (Figure 12.2B) do not coincide with the simple geometries for which we derived Fick s second law, and the diffusion problem must therefore be expressed in two dimensions. Note that a line passing through the center of the disk and normal to the plane of the disk is a cylindrical axis of symmetry, so it is sensible to choose the radial distance from this axis as one of the coordinates for the problem. Diffusion along this radial coordinate, r, is described by Equation 12.7. Also, diffusion along the coordinate, x, normal to the plane of the electrode is described by Equation 12.4. Thus, the form of Fick s second law that must be solved for the disk is ... 

The equivalency found between the behavior of hemisphere and that of disk electrodes also exists between cylinder and band electrodes [29]. Diffusion to a cylinder electrode is linear and described by Equation 12.7, while diffusion to a band is nonlinear. A plane of symmetry passes through the center of the band and normal to its surface, so the nonlinear diffusion process can be broken down into two planar components, one in the direction parallel to the electrode surface, x, and the other in the direction perpendicular to the electrode surface, y. So Fick s second law for a band electrode is... 

As noted earlier, nonlinear diffusion to disks is predicted to produce nonuniform current densities across the electrode surface. Similarly, the current den-... 

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. 

Weickert J 1997 A review of nonlinear diffusion filtering In Scale-Space Theory in Computer Vision (eds. ter Haar Romeny B, Florack L, Koenderink J and Viergever M), pp. 3-28. Springer-Verlag, Berlin. 

Weickert J, ter Haar Romeny B and Viergever MA 1998 Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing 7(3), 398-410. 

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. 

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... 

Liou and Bruin, An Approximate Method for the Nonlinear Diffusion Problem with a Power Relation between the Diffusion Coefficient and Concentration. 

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