Relative diffusion coefficients

Figure C2.7.13. Schematic representation of diffusion and reaction in pores of HZSM-5 zeolite-catalysed toluene disproportionation the numbers are approximate relative diffusion coefficients in the pores 1131.

A number of metals, such as copper, cobalt and h on, form a number of oxide layers during oxidation in air. Providing that interfacial thermodynamic equilibrium exists at the boundaries between the various oxide layers, the relative thicknesses of the oxides will depend on die relative diffusion coefficients of the mobile species as well as the oxygen potential gradients across each oxide layer. The flux of ions and electrons is given by Einstein s mobility equation for each diffusing species in each layer... 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

The theoretical results described have implications for the design of experimental approaches for the study of transfer processes across the interface between two immiscible phases. The current response in SECMIT is clearly sensitive to the relative diffusion coefficients and concentrations of a solute in the two phases and the kinetics of interfacial transfer over a wide range of values of these parameters. 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

Table 3.15 Relative diffusion coefficients of disperse dyes [113]...

Table 3.21 Relative diffusion coefficients of basic dyes on basic-dyeable polyester and...

Distance distributions can be determined when the interchromophoric distance does not change significantly during the donor lifetime. Otherwise, energy transfer is enhanced by translational diffusion of the donor and acceptor moieties towards each other. Then, information on relative diffusion coefficients of chain ends in oligopeptides (Katchalski-Katzir et al., 1981) or polymers can be obtained. 

Fig. 14. The outer-sphere relaxivity at zero magnetic field as a function of relative diffusion coefficient for S = 1. Reproduced with permission from Kruk, D. Nilsson, T. Kowalewski, J. Mol. Phys. 2001, 99,1435-1445. Copyright 2001 Taylor and Francis Ltd (http //www.tandf.co.uk/journals/tf/00268976.html).

The crossover concentration calculated for polystyrene with this expression is in the range of the reported values(12,13) Finally, the above discussion points out that c is not a sharp dividing line It is therefore not physically meaningful to scale the concentration axis in a plot of D/Dq versus concentration divided by the crossover concentration Equation 13 demonstrates that it is better to plot the relative diffusion coefficient vs the weight concentration of the polymer, as was done by Munch et al (12) ... 

Until the geminate pairs start to mix, i.e., at relatively short times r monomolecular kinetics reads n(t) = n(0)u>(t), with n(0) = nA(0) = ne(0) being initial particle concentration. The distinctive feature of this stage is the linearity of the recombination kinetics n(t) with respect to the irradiation dose n(0). 

The distinctive feature of the kinetics sought for is its dependence on the relative diffusion coefficient D = Da + D% only. In turn, the time derivative of equation (3.2.6) reads (using also equation (3.2.7))... 

Moreover, for the immobile reactants A, the exact solution yields a = 1 (d > 1) rather than (5.2.1) This argues for the relative diffusion coefficient k = Da/(Da + DB) as one of the key parameters of this kinetics. However, at present there are no rigorous estimates in the general case 0 k 1 (see, e.g., [47] and discussion below). 

This is indeed the reactive flux for a trap-particle pair with a relative diffusion coefficient D = Da + DB. 

As it was mentioned above, up to now only the dynamic interaction of dissimilar particles was treated regularly in terms of the standard approach of the chemical kinetics. However, our generalized approach discussed above allow us for the first time to compare effects of dynamic interactions between similar and dissimilar particles. Let us assume that particles A and B attract each other according to the law U v(r) = — Ar-3, which is characterized by the elastic reaction radius re = (/3A)1/3. The attraction potential for BB pairs is the same at r > ro but as earlier it is cut-off, as r ro. Finally, pairs AA do not interact dynamically. Let us consider now again the symmetric and asymmetric cases. In the standard approach the relative diffusion coefficient D /D and the potential 1/bb (r) do not affect the reaction kinetics besides at long times the reaction rate tends to the steady-state value of K(oo) oc re. 

Both equations (7.2.16) and (7.2.18) have the same dependence on the relative diffusion coefficient, D — D + D, but different dependence on the elastic interaction between defects. However, in both cases the stronger similar defect attraction, the lower is the critical dose rate. In the mesoscopic approach this effect is less pronounced (logarithmic vs. linear dependence) and here pc is considerably higher. It seems that this approach is able to detect only those mesoscopic-size aggregates which are already well-developed - unlike the microscopic formalism able to detect even the marginal aggregation effects. 

The behaviour of the correlation function X% (r, t) is defined by the auto-catalytic reaction stage the probability to find some particle B near another B is rather high if they are reproduced by a division. For the short relative distances r the function X% (r,t) has a singularity which is, however, weakly pronounced, i.e., particles B are quasi-randomly distributed in space. For a chosen parameter k = 0.02 the relative diffusion coefficients is large Db =2(1 - k), Db Dp,. The aggregates emerging under reproduction of B s are spread out rapidly due to the diffusion. 

Figure 11.3 DNA size dependence of relative diffusion coefficients in cytoplasm (Dcyto/Dw) and nucleus (Dnuc/Dw). Apparent diffiisional coefficients in the cytoplasm (Dcyt) relative to aqueous solution (Dw) were determined from 50% recovery times (t1/2) using an experimentally determined calibration relation of t1/2 versus D as described in Kao et al., 1993. Each point is the mean SE for 5-15 independent measurements for DNA diffusion in cytoplasm (open circles) or nucleus (filled circles). For comparison, I )/l )w values for microinjected FITC-dextrans are shown. Reproduced with permission from Lukacs etal., 2000.

DNA size dependence of relative diffusion coefficients in cytoplasm (Dcyt0/Dw) and 196... 

At an O/W interface, however, molecule B in the W phase can approach only from the W-phase side to molecule A staying in contact with the 0-phase side of the interface. It is here assumed that molecule B reacts with molecule A just when it reaches the reaction surface , i.e., the part of spherical surface of the radius of tab around the centre of A which bulges out to the W phase (see the shadow part in Figure 8.8b). The diffusion-controlled molar flux of B towards the reaction surface, (,ep would be obtained by analogy of Equation (16). However, the relative diffusion coefficient Dab in Equation (16) should be replaced by the absolute diffusion coefficient of B in the W phase (D ), because in this case, molecule A is regarded as staying at the interface for a reaction with B. Consequently, can be expressed by... 

Table 6-2. Solvent effect on the geminate singlet and triplet recombination yields ( (t)s and (t>r) and the yield of the ion radicals ((()i) for Ks = 6.176 A/ns and Kt = 18.53 A/ns. Relative diffusion coefficients defined by D = D(Py) + D(DMA) and dielectric constants (e) for various solvents are employed by the values at 25 °C. (Reproduced from Ref. [27] by permission from The American Institute of Physics)...

The conclusion can be drawn that the traditional approach [30] corresponds to a zero-level approximation, i.e., a cut-off of the hierarchy at m = 1 thus linearising (5.2.9) for gi and neglecting all integral terms in wi (equation (5.2.7)) which describe competition of several B s for some reactant A. It is not surprising that the standiird critical exponent a = 1 obtained from equations (5.2.5) and (5.2.9) is independent of the relative diffusion coefficient, since k enters into equation (5.2.6) only for m 2. Therefore, in an approximation at the next level we have to consider the equation for the correlation function p2 at least. This level of approximation could be acceptable (at least to reproduce equation (5.2.1)) if the reaction for 7 0 remains weakly non-ideal (the g is nearly multiplicative). 

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