Diffusion Equation Approach

From Eq. (B-113) the segment density distribution p(ir) and the adsorption isotherm are calculated as follows  

Moreover, the number of polymers adsorbed per unit area, C, is given by 

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This power law decay is captured in MPC dynamics simulations of the reacting system. The rate coefficient kf t) can be computed from — dnA t)/dt)/nA t), which can be determined directly from the simulation. Figure 18 plots kf t) versus t and confirms the power law decay arising from diffusive dynamics [17]. Comparison with the theoretical estimate shows that the diffusion equation approach with the radiation boundary condition provides a good approximation to the simulation results. 

Recently, Steiger and Keizer [259b] have discussed the theory of reactions between anisotropically reactive species in considerable detail. They illustrated their analysis by using a diffusion equation approach to solve for the rate coefficient for reaction between species which displayed dipolar reactivity. The rate coefficient was reduced by approximately 15% from the Smoluchowski value [eqn. (19)]. Berdnikov and Doktorov [259c] have also analysed the rate of reactions between a spherical reactant having a reactive site, which is a spherical shell of semi-angle 60, and a spherical symmetric reactant. Again, these reactants were not allowed to rotate. Approximate analytic expressions were obtained for the rate coefficient, which was a factor feit less than the Collins and Kimball expression, where... 

There are several criticisms of the diffusion equation approach to radical pair recombination [5]. In particular, the treatment of the radical... 

Noyes [269, 270] and, more recently, Northrup and Hynes [103] have endeavoured to incorporate some aspects of the caging process into the Smoluchowski random flight or diffusion equation approach. Both authors develop essentially phenomenological analyses, which introduce further parameters into an expression for escape probabilities for reaction, that are of imprecisely known magnitude and are probably not discrete values but distributed about some mean. Since these theories expose further aspects of diffusion-controlled processes over short distances near encounter, they will be discussed briefly (see also Chap. 8, Sect. 2.6). 

The diffusion equation analysis is discussed in Sect. 2. It has been used very much more frequently in studies of diffusion-limited reactions rates than the analysis based on molecular pair behaviour, which is discussed in Sect. 3. This is probably because the diffusion equation approach is rather more direct, clear and versatile than the molecular pair analysis (furthermore, time-dependent Green s functions are required for the molecular pair approach). Besides, the probability that a molecular pair will reencounter one another is often derived from a diffusion equation analysis in any case and under these circumstances the two approaches are identical. 

Returning to the survival probability, in Fig. 57, the kinetic theory and diffusion equation [cf. eqn. (132)] predictions are compared. Three values of the activation rate coefficient are used, being 0.5, 1.0 and 2.0 times the Smoluehowski rate coefficient for a purely diffusion-limited homogeneous reaction, 4ttoabD. With a diffusion coefficient of 5x 10 9 m2 s1 and encounter distance of 0.5 nm, significant differences are noted between the kinetic theory and diffusion equation approaches [286]. In all cases, the diffusion equation leads to a faster rate of reaction. In their measurements of the recombination rate of iodine atoms in hydrocarbon solvents, Langhoff et al. [293] have noted that the diffusion equation analysis consistently predicts a faster rate of iodine atom recombination than is actually measured. Thus there is already some experimental support for the value of the kinetic theory approach compared with the diffusion equation analysis. Further developments cannot fail to be exciting. 

As a first example of the diffusion equation approach, we consider a very simple problem the absorption of small (point) particles A, which are di-lutely dispersed in a viscous continuum, by a collection of large stationary... 

The equations in this form are especially suitable for comparison with diffusion equation approaches, which are most often applied to the irreversible reaction case. The reaction scheme we have selected is also convenient because B clearly plays the role of the sink in the diffusion equation approaches, and a rather direct comparison with these methods is possible. 

Seki, K., Bagchi, B., TacMya, M. Dynamics of barrierless and activated chemical reactions in a dispersive medium within the fractional diffusion equation approach. J. Phys. Chem. B 112(19), 6107-6113 (2008). http //dx.doi.org/10.1021/Jp076753q... 

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