Radial diffusion, 517 pairs

Figure 14. Histograms for the number of the saddle points with respect to the value of the barrier height. The barrier energy is normalized by the depth of the pair potential e. (a) The data of the first- and second-order saddles for the noncompact A47B20 are shown by a thick line and a thin line, respectively, (b) The data of the first- and second-order saddles for the compact A47B2o are also shown by a thick line and a thin line. The barrier height for the surface diffusion is denoted by a thin dotted line, while that for radial diffusion is shown by a thick dotted line.

An approximate analytical solution has been developed to calculate the exit concentration from a continuously recirculating facilitated transport liquid membrane system. The system is modeled as a series of SLM-CSTR pairs. The solution allows for two-dimensional transport (axial convective and radial diffusive) and laminar flow. The solution allows one to estimate the effect of a change in system variables on the operating performance. Comparison with experimental data was very good. 

Fig. 1.3-15 Radial atomic pair distribution function G(r) of an amorphous material. Its shape can be deduced from diffuse scattering...

A radial distribution function can be determined by setting up a histogram for various distances and then looking at all pairs of molecules to construct the diagram. Diffusion coefficients can be obtained by measuring the net distances... 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

Fig. 1.19. The radial pair correlation function of the steady-state overlayer generated by the A + B -> 0 annihilation reaction, with no particle diffusion. Averaged over five simulations.

In the case of a single test particle B in a fluid of molecules M, the effective one-dimensional potential f (R) is — fcrln[R gBM(f )]. where 0bm( ) is th radial distribution function of the solvent molecules around the test particle. In this chapter it will be assumed that 0bm( )> equilibrium property, is a known quantity and the aim is to develop a theory of diffusion of B in which the only input is bm( )> particle masses, temp>erature, and solvent density Pm- The friction of the particles M and B will be taken to be frequency indep>endent, and this should restrict the model to the case where > Wm, although the results will be tested in Section III B for self-diffusion. Instead of using a temporal cutoff of the force correlation function as did Kirkwood, a spatial cutoff of the forces arising from pair interactions will be invoked at the transition state Rj of i (R). While this is a natural choice because the mean effective force is zero at Rj, it will preclude contributions from beyond the first solvation shell. For a stationary stochastic process Eq. (3.1) can then be... 

If the reaction coordinate is reduced to a simple radial separation of the cage pair, then the transition structure for diffusive separation (td) can be identified with the cage radius (R, Figure 1). 

And of course statistic accumulation parts of the code have to be changed accordingly. Energy, pair radial distribution function (PRDF), or diffusion coefficient should be calculated separately for molecules in each of the... 

The environmental radial distribution function (RDF) for Ni is plotted with a solid curve in fig. 83. The ordinary RDF computed from the interference function, Qj Q), which is shown in fig. 84, is shown as a dotted curve in fig. 83. Although six partial RDFs are overlapped in the ordinary RDF, the environmental RDF for Ni is only the sum of the three partial RDFs of Ni-Ni, Ni-Mg and Ni-La pairs. In terms of three constituent elements of different sizes, the first peak of the ordinary RDF consists of some broad diffuse peaks. It is almost impossible to determine an atomic distance and a coordination number for each atomic pair in the nearest-neighbor region from the ordinary RDF. On the other hand, the first peak in the environmental RDF for Ni becomes an... 

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