Smoluchowski boundary condition

Fig. 10. Plots of the pair density distribution versus distance for (a) rc = 0 (Smoluch-owski solution), (b) rc = 0.7 nm, (c) rc —0.7 nm. The distributions are shown for times of 10 12 to 10 6 s in decadic intervals. The calculations refer to a random initial condition and to the Smoluchowski boundary condition, i.e. p(ft,f) = 0, ft = 0.5 nm, D = 10 8 m2 s 1.

Fig. 11. Plots of the rate coefficient as time for a random initial distribution and the Smoluchowski boundary condition, p(R,t) = 0 for t > 0. In order of decreasing rate coefficient, the plots refer to rc = —22.4 nm, — 5.6nm, —2.8 nm, — 1.4nm, — 0.7 nm, + 0.7 nm and + 1.4 nm. -----, Numerical calculations ----, approximate analy-...

To solve eqn. (214) subject to ths usual Smoluchowski boundary conditions requires two integrations following the approach of Mozumder [315]. Deutch and Felderhof [70] showed that the rate coefficient was... 

As usual, the Smoluchowski boundary condition on the density and escape probability adequately described the fact that, once an anion and cation form an encounter pair, the chance of separation is negligible,... 

Monchick [525] has used the telegrapher s equation to describe chemical kinetics. Rice [484] solved the field-free form of the telegrapher s equation for the Smoluchowski boundary conditions, supplemented by n f=Q = 0, to find the rate coefficient as... 

This means that no recombination occurs unless the particles approach each other to within the critical distance ro which is associated with the unavoidable fall into the recombination sphere. It can be shown that for instant recombination, cto —> oo, w(r < ro, t) = 0, which results in the generally accepted Smoluchowski boundary condition... 

The black sphere approximation permits us to obtain the most simple and physically transparent results for the kinetics of diffusion-controlled reactions. We should remind that this approximation involves a strong negative correlation of dissimilar particles at r ro, where Y(r, t) = 0, described by the Smoluchowski boundary condition... 

Due to the instant recombination all the dissimilar particles with relative distances r ro disappear, which results in the Smoluchowski boundary condition... 

The standard boundary conditions reflect weakening of the correlations at large relative distances (oo,t) = Y(oo, t) = 1. We impose also the Smoluchowski boundary condition,... 

Initial and boundary conditions are necessary for solving this equation. Generally, it is assumed that the bulk density distribution is random, so that w(r oo, t) = l for all t. Two kinds of boundairy conditions have been taken at r — R i) in the so-called Smoluchowski boundary condition, it is assumed that once B species are inside the sphere of radius R,they react immediately with A, and thus... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

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 ... 

The Smoluchowski-Levich approach discounts the effect of the hydrodynamic interactions and the London-van der Waals forces. This was done under the pretense that the increase in hydrodynamic drag when a particle approaches a surface, is exactly balanced by the attractive dispersion forces. Smoluchowski also assumed that particles are irreversibly captured when they approach the collector sufficiently close (the primary minimum distance 5m). This assumption leads to the perfect sink boundary condition at the collector surface i.e. cp 0 at h Sm. In the perfect sink model, the surface immobilizing reaction is assumed infinitely fast, and the primary minimum potential well is infinitely deep. 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

When the distance between each A reactant is very large compared with that between each pair of B reactants, at a point about midway between a pair of A reactants, the concentration of B reactants is unlikely to be significantly affected by the presence of the A reactants. Smoluchowski suggested that such B reactants are effectively an infinite distance from the A reactants under discussion. By effectively an infinite distance is meant perhaps 1000 times the molecular diameter or encounter distance R. In this region, the concentration of B reactants at any time during the reaction is very close to the initial concentration, i.e. [B](1000iZ) [B]0 for all time (t > 0). From the definition of the density distribution, eqn. (2), this boundary condition as r - °° is... 

The weakness of this boundary condition is being able to justify a large enough distance to be comparable with an infinite distance, or perhaps 1000R. In practice, this would require B to be in excess over A by about 109 times A more reasonable approach to this outer boundary condition would be to require that there be no loss or gain of matter over this boundary, as there is an approximately equal tendency for the B reactants to migrate towards either A reactant upon each side of the boundary. The proper incorporation of this type of boundary condition into the Smoluchowski model leads to the mean field theory of Felderhof and Deutch [25] and is discussed further in Chap. 8 Sect. 2.3 and Chap. 9 Sect 5. 

The Smoluchowski reactive (or inner) boundary condition [eqn. (5)] is implied in the partially reflecting boundary condition [eqn. (22)]. When reaction between A and B at the encounter separation is very fast compared with the rate of diffusive approach of A and B, kaci -> °°. However, the concentration gradient remains finite, so that the density of B about reactant A tends to zero. 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

In order to solve for the survival and recombination probabilities, p and q in eqn. (126), it is necessary to solve eqn. (122) for p(r, f]r0, f0) and use eqn. (123) to find p or eqn. (125) for q. Again, the boundary and initial conditions are required. Before the pair is formed (f < and ttf is slightly less than f0), the density p is zero, of necessity. The boundary conditions are closely related to the Smoluchowski conditions [eqns. (5), (22), (46) and (47)]. As the radicals approach each other they have a probability of reacting, which can be related to an effective second-order rate coefficient, fcact> f°r the activation-limited process of recombination by... 

Because the diffusive flux is enhanced by this drift of a charge under the influence of the coulomb potential [as represented in eqn. (142)], the partially reflecting boundary condition (127) has to be modified to balance the rate of reaction of encounter pairs with the rate of formation of encounter pairs [eqn. (46)]. However, the rate of reaction of ion-pairs at encounter is usually extremely fast and the Smoluchowski condition, eqn. (5), is adequate. The initial and outer boundary conditions are the same as before [eqns. (131) and (128), respectively], representing on ion-pair absent until it is formed at time t0 and a negligibly small probability of finding the ion-pair with a separation r - ... 

If the rate of reaction of encounter pairs is comparably fast to the rate of formation of encounter pairs, Collins and Kimball [4] suggested that the slowness of the chemical reaction rate could be incorporated into the theory of diffusion-limited reaction rates by modifying the Smoluchowski [3] boundary condition, eqn. (5), to the partially reflecting boundary... 

When there are two or more reactants diffusing throughout space, the motion of each reactant influences that of all the others due to the solvent being squeezed from between the approaching reactants. The effect of this hydrodynamic repulsion on the rate of a diffusion-limited reaction was discussed in Chap. 8, Sect. 2.5. In this section, this discussion is amplified. First, the nature of the hydrodynamic repulsion is discussed further and then a general diffusion equation for many particles is derived. The two-particle diffusion equation is selected and solved subject to the usual Smoluchowski initial and boundary conditions to obtain the rate coefficient. Finally, this is compared with the rate coefficients in the absence of hydrodynamic repulsion and from experiments. 

On the right-hand side, the terms represent the rate of loss A and all m quencher molecule density by diffusion of A and of each of the Q quencher molecules and, finally, by reaction of each quencher with A. The boundary conditions on the density n are different from those of the Smoluchowski model. No loss of any particle can occur on. the outer surface (Vnj , -> 0 etc.), i.e. a closed system such as a glass beaker Where the quencher and fluorophor can interpenetrate each other, there is no net... 

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