Smoluchowski limit

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

In the Smoluchowski limit the reaction is by definition the slow coordinate, such that y(kr) y(0) = dL yW Lrand Agii fO). Though the time-dependent friction... 

The Kramers equation (2.1) in the high-viscosity limit yields the Smoluchowski limit (see also Eqs. 4.171 and 4.178)... 

Unfortunately, even for isomerization reactions it has been shown that the Smoluchowski limit fits neither for alkane54 nor for alcohol solutions47 and that a frequency-dependent friction should be used in the vicinity of 6C.55 51... 

It is known as the Helmholtz-Smoluchowski (limiting) law. after Helmholtz, who derived an expression similar to 14.3.41 but without e in it. and Smoluchowski who Improved Helmholtz s expression. 

The same result is obtained from the Smoluchowski limit of Eq. (2.3),... 

The friction (more generally, molecule-solvent interaction) is taken in the Kramers model to be a constant, independent of the position along the reaction coordinate. As seen, generalization to position-dependent friction is trivial in the Smoluchowski limit. In many systems position-dependent friction should be considered also in the underdamped case. An obvious example is desorption where the dissociating particle ceases to feel the thermal bath as it draws further away from the surface. 

The viscosity rj dependence of the rate k in some of the studied systems (stilbene in alkanes, DPB in alkanes, and DODCI in alcohols ) is weaker (that is, /c 0 < a < 1) than predicted by the Kramers theory in the Smoluchowski limit. In others (stiff stilbene in alkanes, stilbene in alcohols, and DPB in alcohols ) the usual dependence was observed. These... 

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. 

Further Complications. The theory as discussed in this section, when applied to dilute suspensions (since at not very small cp the diffusion distance of the particles is smaller than presumed) of small particles (say, < 1 pm) under quiescent conditions (see Section 13.2.3), has been successful in predicting a number of observations, at least for the first few aggregation steps. This concerns (a) the reaction being second order (b) the value of t0.s being inversely proportional to N0, (c) the distributions of multiplets obtained and (d) the effect of viscosity on the aggregation rate. Also the absolute rate is often well predicted (within a factor of two or three) if it is sure that fast aggregation occurs this rate is often called the Smoluchowski limit. However, the values of the capture efficiency often deviate markedly from the prediction, even for smooth spherical particles. 

In a similar phenomenological approach to unimolecular reactions involving large-amplitude motion, the effect of friction on the rate constant can be described by a simple transition formula between the high-pressure limit of the rate constant at negligible solvent viscosity and the so-called Smoluchowski limit of... 

In the Smoluchowski limit the reaction is by definition the slow coordinate, such that... 

The Smoluchowski limit. /According to the Schaaf-D6Jardin numerical analysis, this limit of negligible interfacial bulk concentration during the first part of the adsorption process should be reached as early as t <10 Tq. A priori, if we look at the orders of magnitude mentioned previously, such a limit should not apply to a great number of systems, t/ arious experiments performed at the liquid-air interface, seem to be described by the t law (1,38). ... 

This is also sometimes referred to as Brownian dynamics or Smoluchowski dynamics. The transition from Langevin to Brownian dynamics is typically referred to as the Kramers to Smoluchowski limit. ... 

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