Symmetric diffusion

Diffusion into a sphere represents a three-dimensional situation thus we have to use the three-dimensional version of Fick s second law (Box 18.3, Eq. 1). However, as mentioned before, by replacing the Cartesian coordinates x,y,z by spherical coordinates the situation becomes one-dimensional again. Eq. 3 of Box 18.3 represents one special solution to a spherically symmetric diffusion provided that the diffusion coefficient is constant and does not depend on the direction along which diffusion takes place (isotropic diffusion). Note that diffusion into solids is not always isotropic, chiefly due to layering within the solid medium. The boundary conditions of the problem posed in Fig. 18.6 requires that C is held constant on the surface of the sphere defined by the radius ra. 

Therefore, the steady-state radially-symmetric diffusion equation becomes... 

In the mean-field approximation, each particle develops a spherically symmetric diffusion field with the same far-field boundary condition fixed by the mean concentration, (c). This mean concentration is lower than the smallest particles ... 

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 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. 

For multicomponent diffusion, and the mass flow expressions, we mainly use Fick s law and the Maxwell-Stefan forms. Using the symmetric diffusivity, in length2/time, we have... 

There are n(n -1)/2 independent diffusivities Ay, which are also the coefficients in a positive definite quadratic form, since according to the second law of thermodynamics, the internal entropy of a single process never decreases. In terms of these symmetric diffusivities, the mass flow becomes... 

Assumption 2. The second additional assumption states that the radical diffusion flux—i.e., the number of radicals entering a particle per unit surface area per unit time—is independent of the radius of the particle. This will lead to too small a number of particles because it can be derived from Picks law that, in symmetrical diffusion into a sphere, the current is proportional to the radius of the particle and therefore the diffusion flux is inversely proportional to that radius. In other words, the smaller the particles the more efficient they are in capturing radicals. N calculated on the basis of this assumption is given by ... 

Stone2 has summarized a generalization of the solution for this simple linear problem, due to Pattle3 and Pert4, which is extremely useftd in the analysis of thin-film problems. This is the development of similarity solutions for the (/-dimensional symmetric diffusion equation with a diffusivity that depends on the concentration,... 

Consider an example of such calculation [36] for a spherically-symmetric diffusion double layer, formed on a non-conducting spherical particle of radius a. A spherical double layer of thickness dr contains the charge... 

For symmetrical diffusion, the diffusion equation can be solved analytically [84] to give the following solution. 

The discussion so far centered on proton diffusion in an infinite space. Hence, a spherically symmetric diffusion (Smoluchowski) equation in three dimensional space has been employed in the data analysis. An inner boundary condition (at the contact distance) has been imposed to describe reaction, but no outer boundary condition. Almost all of the interesting biological applications [4] involve proton diffusion in cavities and restricted geometries. These may include the inner volume of an organelle, the water layers between membranes or pores within a membrane. 

Fig. 11. a) The direct transmittance (%) versus the sample tilt angle (degree) of the symmetric diffuser. The different composites may be seen from the different symbol of the skeleton. Sample 1 PEGDMA, 4.0 g Irgl84, 0.2 g Sample 2 PEGDMA, 4.0g BMIMBF4, l.Og Irgl84, 0.2 g. b) Photo of the diffusion pattern of the commercial particle-type diffuser, c) Photo of the diffusion pattern of the diffuser fabricated with sample 2. 

Fig. 12. The direct transmittance (%) versus the sample tilt angle (degree) of the symmetric diffuser. The different composites may be seen from the different symbol of the skeleton.

Fig. 13. The cross section optic images of the diffusers based on sample 2. (a) symmetric diffuser, (b) asymmetric diffuser.

The model is evaluated with experimental data obtained at Gaz de France and Sandia National Laboratories where a non-confined bluff-body stabilized burner has been investigated [4]. The axi-symmetric diffusion-flame consists of a methane-jet on the centerline surrounded by a co-axial air-flow. Except of swirl the burner has all features which are typical for industrial configurations such as walls and re-circulation-zones. Contours of the mean temperature are shown in Fig. 4.1. For comparison, calculations based on the standard eddy-dissipation-model (EDM) as it is commonly implemented in commercial CFD-codes are included. Only the PDF-approach is able to reproduce the experimental temperature-fields, whereas the standard EDM overestimates the temperature. 

The circularly symmetrical diffusion of ad-atoms to kink sites has also been considered [12] as well as diffusion processes at screw dislocations [13], and further details may be found, for example, in Vetter s book [14]. 

The EPRLL programs also allow for non-Brownian rotational diffusion, which implies a discrete, step motion of the spin probe. Two limiting models are available (i) jump diffusion, and (2) approximate free diffusion. In currently available implementations of the SEE fine shape calculation, non-Brownian models may not be used with an orienting potential, and only with the assumption of an axially symmetric diffusion tensor. For these reasons, and since Brownian motion is usually an... 

It is the presence of the non-zero elements mi2 which brings about a coupling of the memory function equations for the correlation functions with different m subscripts (but a fixed j superscript). It can be seen that the matrix of "memory constant" is diagonal whenf (y these symmetric diffusers, the equations for each m decouple and one recovers the well-known result ... 

Solid electrodes utilized in electrochemical investigations are often made of wire and present a small cylinder with radius Tq and height h. It is commonly assumed that symmetric diffusion flows are directed toward side surface, whereas... 

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