Diffusion constant, collective

In addition, it is also of interest to consider the collective diffusion constant D which measures how a local excess of coverage spreads out. 

Fig. 32. Collective diffusion constant D plotted versus coverage at two temperatures where the square lattice gas of Fig. 31 is ordered near 0= 1/2 (arrows denote the coverages at which the order-disorder transitions occur for = O.S). The mean-field...

Sect. 6 a Aa T Do Fsh T X A M R De final radius of cylindrical gel after swelling displacement relaxation time for swelling collective diffusion constant shear energy trace of the strain tensor u,k swelling rate ratio total change of the radius of the gel longitudinal modulus ratio of the shear modulus to the longitudinal modulus effective collective diffusion constant... 

Therefore, the effective collective diffusion constant for an infinitely long cylinder is 2/3 of that of a spherical gel. 

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...

Figure 23 shows the effective collective diffusion constant normalized by the diffusion constant for a spherical gel as a function of the position from the center of the cylinder. From r/a = 0, i.e., on the cylinder axis, De/D0 decreases gradually and approaches the value 2/3 and 1/3, respectively for a cylindrical and disk gels. 

The results obtained by the present mechanical measurements are also consistent with the previous experimental results of the dynamic light scattering studies of the collective diffusion coefficient of gels and the rheological studies of the shear modulus of gels. The studies published by different researchers indicate that the concentration dependence of the collective diffusion constant of the polymer networks of gel and that of the elastic modulus are well represented by the following power law relationships [2, 3, 5]... 

Figure 8.4 Collective diffusion constant for a mixture of polystyrene and poly(vinyl methyl ether)., SANS Data , FRES data. Reproduced with permission from ref. 22.

If we omit the complication introduced by the gel modes, the collective diffusion constant might be calculated using a Kawasaki-Ferrell approximation (see equations 42 and 84), which always produces a hydrodynamic correlation length proportional to the static correlation length Marginal solvents and the crossover between good and 9 solvents have been reported in detail in ref. 112 and in the review of Schaefer and Han. ... 

Ifihe Bath relaxation con start t, t, is greater than 0.1 ps. yon should be able Lo calculate dyriani ic p roperlies, like time correlation fun c-tioris and diffusion constants, from data in the SNP and/or C.SV files (sec "Collecting Averages from Simulations"... 

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

How is the process affected by changes in conditions The diffusion coefficient is an inverse function of pressure, while concentration varies directly with pressure. The result is that weight collected is constant with respect to changes in... 

Let us examine the critical dynamics near the bulk spinodal point in isotropic gels, where K + in = A(T — Ts) is very small, Ts being the so-called spinodal temperature [4,51,83-85]. Here, the linear theory indicates that the conventional diffusion constant D = (K + / )/ is proportional to T — Ts. Tanaka proposed that the density fluctuations should be collectively convected by the fluid velocity field as in near-critical binary mixtures and are governed by the renormalized diffusion constant (Kawasaki s formula) [84],... 

Nevertheless, the characteristic time constant is roughly proportional to the square of the typical size and to the inverse of collective diffusion coefficient D which is given by the modulus divided by the friction. The porous structures presented here are one of the solutions to achieve a high response material. 

T = (Dq2) 1 is the collective diffusion time constant, DT the thermal diffusion coefficient. In Eq. (18), the low modulation depth approximation c( M c0, resulting in c(x,t)(l-c(x,t)) c0(l-c0)y has been made, which is valid for experiments not too close to phase transitions. Eqs. (16) and (20) provide the framework for the computation of the temperature and concentration grating following an arbitrary optical excitation. 

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... 

A more mechanistic approach, Instantaneous Normal Mode (INM) theory [122], can be used to characterize the collective modes of a liquid. Ribeiro and Madden [123] applied this theory to a series of fused salts, including both noncoordinating and coordinating species. They found that the INM analysis provided a good estimate of the diffusion constants for noncoordinating fused salts. For coordinating ions, however, the situation was complicated by the existence of transient, quasimolecular species. While a more detailed analysis is possible [124], the spectrum becomes sufficiently complicated that it would be difficult to characterize specific motions in the system. 

Other important examples which exhibit both confinement and diffusion in the classical dynamics of their cyclic collective coordinates are the positronium [20] and the excitonic [14] atom. Because of the comparable masses of the two particles in both cases the mean CM velocity as well as the diffusion constant are orders of magnitude larger than the corresponding values of the hydrogen atom. 

In order to estimate the first contribution from the elastic adsorption, a C gel picture is used, in which the gel is a collection of adjacent blobs of radius that has a characteristic relaxation time Ti / >coop, where Dcoop r/bnrji is the cooperative diffusion constant of the gel, T is temperature and t] is the viscosity of the solvent [80]. Each blob is associated with a partial polymer chain (the polymer chain between two next-neighboring cross-linking points). The scaling theory relates this molecular structure of the gel with its elastic modulus E by the equation [80]. 

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