Collective diffusion, process

The kinetics of swelling is successfully described as a collective diffusion process. Tanaka et al. (Tanaka et al. 1973) developed a theory for the dynamics of polymeric gels. They realized that the polymer chains are cormected by chemical bonds and a gel has to be treated as a continuum. In addition, the network behaves as an assembly of springs due to their entropy elasticity. 

Interface instabilities, known as myelins, are an example of exotic nonequilibrium behavior present during dissolution in a number of surfactant systems. Although much is known about equilibrium phase behavior much still remains to be understood about nonequilibrium processes present in surfactant dissolution. In this chapter nucleation and growth, self and collective diffusion processes and nonlinear dynamics and instabilities observed in various polymeric systems are reviewed. These processes play an important role in our understanding of myelin instabilities. Kinetic maps and the concept of the free energy landscape provide a useful approach to rationalize some of the more complex behavior sometimes observed. 

Although much evidence exists for the occurrence of the former mechanism [130], the distinction between both pathways is far from obvious, explaining the contestation of the existence of such species [131]. However, recent diffusive light scattering (DLS) data showed the existence of fast relaxation attributed to a collective diffusion process of 2-nm precursors, viz. nanoslab-like species [130]. 

In this MCT model, all the dynamic processes are intrinsically included and coupled. The fast vibration, the slow structural relaxation, and the intermediate decay appear as solutions of the equations of motion. The vibrational dynamics is characterized by two frequency parameters. Cl and w, of about 1 THz, in reasonable agreement with the mean firequeney of the vibrational modes found in previous studies. The crossover from the fast to the slow processes correctly describes the intermediate relaxation. This model does not require extra over-damped vibrational modes, as required by the previous interpretations. According to this interpretation, the intermediate relaxation is simply the merging of the local vibrational dynamies into the slow collective diffusive process. This merging is not trivial, as it has already been proved in glass-former liquids, because the two processes are strongly coupled. 

A number of special processes have been developed for difficult separations, such as the separation of the stable isotopes of uranium and those of other elements (see Nuclear reactors Uraniumand uranium compounds). Two of these processes, gaseous diffusion and gas centrifugation, are used by several nations on a multibillion doUar scale to separate partially the uranium isotopes and to produce a much more valuable fuel for nuclear power reactors. Because separation in these special processes depends upon the different rates of diffusion of the components, the processes are often referred to collectively as diffusion separation methods. There is also a thermal diffusion process used on a modest scale for the separation of heflum-group gases (qv) and on a laboratory scale for the separation of various other materials. Thermal diffusion is not discussed herein. 

Several differences from that of an integrated circuit can be noted. First of all, two (2) electrlced contacts must be established across the bulk of the silicon wafer. When light strikes the surface of the solcU cell, its absorption within the silicon bulk releases electrons which are collected as a current. Also shown is the p-n junction. However, the actual silicon disc is only about 350 pm. in thickness. Diffusion processes are used, as a matter of practicality, to form both the p-layer and the n-layer. Thus, the... 

Hence the top grid pattern is usualty widely spaced but not the extent that the electrical contact layer will have difficulty in collecting the current produced by the cell s other active layer. Cleau ly, the silicon disc needs to be heated as well during the process to aid the diffusion process. Note that the surface will be rieh in diffusing species and that the density of species declines within the interior What happens is that once the ion contacts the silicon surface, it "hops from site to site into the interior of the bulk of the silicon matrix. 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

Glade, N., Demongeot, J., and Tabony, J. (2004). Micrombules self-organization by reaction-diffusion processes causes collective transport and organization of cellular particles. BMC Cell Biol, 5, 5-23. 

When aerosols are in a flow configuration, diffusion by Brownian motion can take place, causing deposition to surfaces, independent of inertial forces. The rate of deposition depends on the flow rate, the particle diffusivity, the gradient in particle concentration, and the geometry of the collecting obstacle. The diffusion processes are the key to the effectiveness of gas filters, as we shall see later. 

The process of mathematical fitting is error-prone, and especially two different issues have to be considered, the first one dealing with the boundary conditions of the fitting procedure itself A pure diffusion process is considered here as the only transport mechanism for fluorine in the sample. A constant value for the diffusion constant D, invariant soil temperatures and a constant supply of fluorine (e.g. a constant soil humidity) are assumed, the latter effect theoretically resulting in a constant surface fluorine concentration for samples collected at the same burial site. In mathematical terms, Dt is influenced by the spatial resolution of the scanning beam, the definition of the exact position of the bone surface, which usually coincides with the maximum fluorine concentration, and by the original fluorine concentration in the bulk of the object, which in most cases is still detectable. A detailed description on... 

Table 4 contains a collection of diffusion coefficients determined experimentally for a variety of adsorbate systems. It shows that the values may vary considerably, which is of course due to the specific bonding of the adsorbate to the surface under consideration. Surface diffusion plays a vital role in surface chemical reactions because it is one factor that determines the rates of the reactions. Those reactions with diffusion as the rate-determining step are called diffusion-limited reactions. The above-mentioned photoelectron emission microscope is an interesting tool to effectively study diffusion processes under reaction conditions [158], In the world of real catalysts, diffusion may be vital because the porous structure of the catalyst particle may impose stringent conditions on molecular diffusivities, which in turn leads to massive consequences for reaction yields. 

Diffusion control of termination is presently beyond any doubt. Various authors differ, however, in their opinion concerning the type of diffusion and the quantitative contribution of diffusion in the collection of processes manifested as the gel effect. Attempts to explain the Norrish-Tromsdorff effect on a molecular level are still of empirical character [8, 12, 36-38],... 

The extent of collection of minority carriers from the region beyond the depletion layer is dictated by the diffusion process. A diffusion length, L, can be defined ... 

Gotlib, Yu. Ya., Svetlov, Yu. E. The theory of vibrational and rotational diffusive processes in diains of rotators and polymer chains. In collected articles Mekhanizmy relaksatsionnykh yavlenii v tverdykh telakh (Mechanisms of relaxation phenomena in solids), Moscow Nauka 1972 (pp- 215-219)... 

As shown in later sections, the coefficient of diffusion is a function of particle size, with small particles diffusing more rapidly than larger ones. For a polydisperse aerosol, the concentration variable can be set equal to nj( ipr r, t)d(Jp) or h(iv, r, /) dv, and both sides of (2.3) can be divided by d(dp) or dv. Hence the equation of diffusion describes the changes in the particle size distnbntion with time and position as a result of diffusive processes. Solutions to the diffusion equation for many different boundary conditions in the absence of flow have been collected by Carslaw and Jaeger ( 959) and Crank (1975). 

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