Smoluchowski limit, validity

Though Kramers expression describes a somewhat complicated viscosity dependence and hence is difficult to apply to extract microviscosity, the relation becomes very simple at very high viscosity (Smoluchowski s limit). Since many organized assemblies possess very high microviscosity it is reasonable to assume that the Smoluchowski limit is valid for them. We will demonstrate that assuming the Smoluchowski limit, the microviscosities of some organized assemblies can be estimated quite accurately. 

Finally, we would like to elaborate the proposed protocol of the high-friction map, eqn (13.17). Its construction is based purely on the thermodynamic consideration, eqn (13.15), validated by the central limit theorem. Therefore it may offer a general rule to obtain the Smoluchowski limit to any phase-space dynamics under study. The protocol proposed in this chapter is based on the fact that the map is universal at formal level and is therefore obtainable with thermodynamic consideration. It means the Smoluchowski dynamics can be taken care of by the related Fokker-Planck equation, upon the universal map being carried out. It is worth pointing out that the resultant diffusion operator in eqn (13.18) clearly originates from only the Hamiltonian part of the... 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

On the contrary, under diffusion control the nonstationary quenching lasts almost the whole lifetime and its contribution into quantum yield is dominant at least for large concentrations. Some insights into the concentration dependence of the corresponding Stem-Volmer constant, k(c), one can obtain, using in Eqs. (3.10) and (3.4), the simplest Smoluchowski expression (3.23), which is valid at any time in a limit of diffusional control (when ko ko). For this limit the analytic solution was given in Ref. [54]... 

The first conclusion regarding figures 4.26 and 27 is that for xa /(xa) approaches 1.5 (fig. 4.26) whereas for xa -4 0 /(xa) -4 1.0 (fig. 4.27). In other words, in the appropriate limits the Helmholtz-Smoluchowski [4.3.4) and Huckel-Onsager equations (4.3.5) are retrieved. So it is concluded that these two limiting laws also remain valid when the double layer is polarized. This is an extension of Morrison s result, quoted in connection with 14.3.20). 

The solution chemistry of nonaqueous solvents is very different from that of water-rich mixed solvents. pH measurement in nonaqueous solvents is difficult or impossible. Salts often show a limited degree of dissociation and limited solubility (see [132] for solubility of salts in organic solvents). Ions that adsorb nonspecifically from water may adsorb specifically from nonaqueous solvents, and vice versa. Therefore, the approach used for water and water-rich mixed solvents is not applicable for nonaqueous solvents, with a few exceptions (heavy water and short-chain alcohols). The potential is practically the only experimentally accessible quantity characterizing surface charging behavior. The physical properties of solvents may be very different from those of water, and have to be taken into account in the interpretation of results. For example, the Smoluchowski equation, which is often valid for aqueous systems, is not recommended for estimation of the potential in a pure nonaqueous solvent. Surface charging and related phenomena in nonaqueous solvents are reviewed in [3120-3127], Low-temperature ionic liquids are very different from other nonaqueous solvents, in that they consist of ions. Surface charging in low-temperature ionic liquids was studied in [3128-3132]. 

In summary then, we expect the usual sink Smoluchowski description to be valid for a slow (on the order of the diffusion rate) reaction. The usual description also entails neglect of the operator character of D, (high friction limit) and assumes that velocity correlations relax rapidly so that the 2 = 0 limit of D, can be taken. The coupling between the center of mass and relative motion is also neglected in the usual formulations. These latter conditions reduce (9.38) to (now in /-space)... 

The atbove-described Debye-Smoluchowski model is subject to severe limitations (Wilemski and Fixroam, 1973). First, the choice of the coordinate system is not self-evidently valid. Second, the mutual diffusion coefficient is assumed to be constant, even for short separation distances between A and B, where some vairiations are expected. Third, the diffusion ecjuation is only valid for low concentrations. A last limitation is due to the method of describing the reaction process in which it is... 

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