Sink Smoluchowski equation

In Section III, we described how the Smoluchowski equation could be used in conjunction with boundary conditions or sink terms to describe chemical reactions. We now return to (9.11) and consider its relation to sink Smoluchowski equation. 

It is clear that if all reactive contributions to the third term in the brackets of (9.11) are neglected, then 

This equation has the form of the sink Smoluchowski equation discussed 

there are terms of the form 5S(12)[z —0L (12 z)] 5S(12) o. From its definition in (9.12), we see that 55(12) is the deviation of the reactive operator from its velocity average. The correlation function above characterizes the time evolution (Laplace transformed) of these fluctuations. If the chemical reaction is slow, we expect that perturbations of the velocity distribution induced by the reaction will be small hence such contributions may be safely neglected in this limit. This argument may be made more formal using limiting procedures analogous to those described in Section V. In principle, one may also use this term to introduce a modification to in S(/- 2) due to velocity relaxation effects. This will lead to some effective reactive collision frequency in place of k p. 

Second, there are cross terms 5S(12)[z —gL (12 z)] v (,. These correspond to a coupling between reaction and diffusion, which arises from the perturbation in velocity space induced by the reaction (or the reverse process). To examine this term in more detail, we consider the simple model for T f in (6.18), and neglect the reactive terms in the propagator. Then 

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An equation with the form of the sink Smoluchowski equation may now be derived by introducing an operator that projects out the velocities of the solute molecules. This is just the pair version of the derivation presented in the previous section. 

Thus the configuration space correlation function P(r 2, r li t) satisfies the sink Smoluchowski equation under the conditions discussed above. 

The second term has the same structure as the term neglected in the derivation of the sink Smoluchowski equation in Section IX.C. Therefore, we make the same approximation here to make connection with the configuration space theories. In this approximation, Akj becomes... 

The operator on the left-hand side of (10.26) is exactly the operator that appears in the sink Smoluchowski equation (3.12). The problem is now identical to that considered in Section III and Appendix A, and the same results as the Smoluchowski equation are obtained. As an illustration, we give some details of the solution for the case g(r) = 0(r —o b) i i Appendix E. There we show that... 

This equation has been used by Sundstrom and coworkers [151] and adapted to the analysis of femtosecond spectral evolution as monitored by the bond-twisting events in barrierless isomerization in solution. The theoretical derivation of Aberg et al. establishes a link between the Smoluchowski equation with a sink and the Schrodinger equation of a solute coupled to a thermal bath. The reader is referred to this important work for further theoretical details and a thorough description of the experimental set up. It is sufficient to say here that the classical link is established via the Hamilton-Jacobi equation formalism. By using the standard ansatz Xn(X,t)= A(X,i)cxp(S(X,t)/i1l), where S(X,t) is the action of the dynamical system, and neglecting terms in once this... 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

New difficulties arise when we try to take into account the dynamical interaction of particles caused by pair potentials U(r) mutual attraction (repulsion) leads to the preferential drift of particles towards (outwards) sinks. This kind of motion is described by the generalization of the Smoluchowski equation shown in Fig. 1.10. In terms of our illustrative model of the chemical reaction A + B —> B the drift in the potential could be associated with a search of a toper by his smell (Fig. 1.12). An analogy between Schrodinger and Smoluchowski equations is more than appropriate indeed, it was used as a basis for a new branch of the chemical kinetics operating with the mathematical formalism of quantum field theory (see Chapter 2). 

The problem may also be formulated in an alternate but equivalent way using a Smoluchowski equation with sink terms. ... 

Strictly speaking, the Smoluchowski model is not apphcable to free-radical termination reactions. The derivation of this model comprises the assumption of an A + B reaction in which one of the reactants is present in large excess a so-called single-sink model. Termination, however, represents a multi-sink problem in which, all A and B species can react with each other. Nevertheless, the Smoluchowski equation has proved to be an accurate description for the termination kinetics of small radials [20, 31]. 

This analysis concerns for instance the formation of the twisted intramolecular charge transfer state, the photoisomerization processes, etc... in solution for which the intramolecular motion is related with the solvent motion. The unimolecular reaction -the passage from the reactant well to the product well in the Kramers treatment- is modeled by a sink term depending on the reaction coordinate. In the high-viscosity case the motion is governed by the modified Smoluchowski equation [6 ]... 

We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

Note that the particle diffusion term is ignored, just like particle dispersion due to SGS motions (this was found justified in a separate simulation). The shape of the sink term in the right-hand term of this equation is due to Von Smoluchowski (1917) while the local value of the agglomeration kernel /i0 is assumed to depend on the local 3-D shear rate according to a proposition due to Mumtaz et al. (1997). 

Consider the simplest situation of Figure 9.4a, where an absorbing spherical sink of radius R is at the center of the coordinate system and polymer chains are present in the solution around the sink. Following the classical theory of Smoluchowski (Chandrasekhar 1943), we assume that the sink absorbs the polymer chains as soon as the center-of-mass of a polymer chain approaches the surface of the sink. We identify the capture rate of polymer chains as the steady-state net flux of polymer chains into the absorbing sink. Let the initial number concentration of the polymer chains be cq. The polymer concenuation is continuously maintained as co at distances far from the sink. The polymer chains undergo only diffusion and there are no other convective contributions. The continuity equation for the number concentration of polymer chains in three... 

Noyes [8], Wilemski and Fixman [11] have pointed out that it is not strictly correct to apply Smoluchowski [1] or radiation [7] boundary conditions to the diffusion equation to model bimolecular chemical reactions. Both Teramoto and Shigesada [12] and Wilemski and Fixman [11] have proposed a modified diffusion equation by introducing a sink term to represent the reaction rate at a set of relative phase-space coordinates of two reacting species. Let a simple diffusive process be described as... 

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