The partially reflecting boundary condition

Collins and Kimball [4] suggested that the chemically activated process which leads to the formation of products from the encounter pair occurs at a rate proportional to the probability that the encounter pair exists. Defining an encounter pair as a pair of reactants which lie within a distance of R to (R + 8R) of one another, and since the probability that B is within this range of distances about A is p R), then the rate of reaction of encounter pairs is feactP( )- act is the second-order rate coefficient for the reaction of A and B when they are almost in contact and close enough to react with each other. It is the rate coefficient for reaction between A and B if the rate of diffusion were infinitely rapid. It has unit of dm mol s . From eqn. (7), the rate at which B diffuses towards A to form encounter pairs is 47r(f + 5/ ) D(3p/3r) R+5jj. For sufficiently small 5R (e.g. 0.01 nm), the term in Si is unimportant and this becomes the diffusive flux to the encounter separation 4irR D dp/dr) ji from Fick s first law. Providing the probability of A and B existing as an encounter pair rapidly reaches a steady-state value, the rate of formation and rate of reaction of the encounter pairs may be equated, i.e. 

This is called the partially reflecting or radiation boundary condition by analogy to the heat conduction equation analyses. It is also variously called a mixed, an inhomogeneous or the Robbins boundary condition because it mixes the value of the dependent variable p and the first 

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, However, 

The width of the encounter pair reactivity zone, bR, is to be considered small. There is no reason for this choice, save convenience. Probably rather larger widths would be more appropriate following work on gas-phase collision kinetics or long-range transfer processes (Chap. 4). In such circumstances, the partially reflecting boundary condition is no longer suitable and other techniques have to be used (see Chap. 8 Sect. 2.4 and Chap. 9 Sect. 4). 

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Calculation of the electric field dependence of the escape probability for boundary conditions other than Eq. (11b) with 7 = 0 poses a serious theoretical problem. For the partially reflecting boundary condition imposed at a nonzero R, some analytical treatments were presented by Hong and Noolandi [11]. However, their theory was not developed to the level, where concrete results of (p(ro,F) for the partially diffusion-controlled geminate recombination could be obtained. Also, in the most general case, where the reaction is represented by a sink term, the analytical treatment is very complicated, and the only practical way to calculate the field dependence of the escape probability is to use numerical methods. 

To solve the diffusion equation (9) or (10) for the density p(r, f) with the random initial condition (3), the outer boundary condition (4) and the partially reflecting boundary condition (22) is straightforward. Again, the solution follows from eqn. (12), but the Laplace transform of eqn. (22) is... 

Finally, for completeness, the Green s function corresponding to a pair of reactants initially formed with separation r0 and subjected to the partially reflecting boundary condition, is quoted (Pagistas and Kapral [37], Naqvi et al. [38]. 

To find the second-order rate coefficient for the reaction of A and B subject to the encounter pair reacting with a rate coefficient feact, the method developed in Sect. 3.7 can be used. Using eqn. (19), the rate coefficient, k(t), can be defined in terms of the diffusive current of B towards the central A reactant. But the partially reflecting boundary condition (22) equates this to the rate of reaction of encounter pairs. The observed rate coefficient is equal to the rate at which the species A and B could react were diffusion infinitely rapid, feact, times the probability that A and B are close enough together to react, p(R). 

Thus the partially reflecting boundary condition reduces the effective encounter distance by a factor of fcact (4nRD + fcact) 1 for both the steady-state and transient terms in the rate coefficient. 

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 - ... 

Furthermore, the initial and outer boundary conditions are effectively identical [eqns. (3), (4) and (165)] as are also the partially reflecting boundary conditions [eqns. (46) and (165)]. This can be shown by substituting p by exp — p p in the boundary conditions (165). Consequently, the relationship between the survival probability of an ion-pair at a time t0 after they were formed at time t and separation r and the density distribution of an initial (time t0) homogeneous distribution of the majority ion species around the minority ionic species, p(r, f f0), is an identity. 

Comparing eqns. (170) and (171) shows that the density distribution for the steady-state formation, recombination and scavenging, pss(r cs r0), is closely related to the Laplace transformed (time-dependent) density distribution for recombination and escape. The partially reflecting boundary condition [eqn. (46)] with p replaced by p or pss... 

In this section, the various methods which have been developed to treat chemical reaction rates between solutes in solution are discussed, with specific concern for those reactions where the rate of reaction of encounter pairs is of comparable magnitude to the rate of diffusive formation of encounter pairs. Some of the detailed comments on the partially reflecting boundary condition are discussed, the effects of angular variation of the reaction rate and the possibility of using a sink term to represent chemical reaction rather than a boundary condition are presented. Such comments are contrasted with the relatively few instances where experimental data has been obtained for the rate of the concomitant chemical reaction. Recently, attention has been given to a development of aspects of gas-phase reaction rate theory to be applied to reactions in liquids. 

This rate coefficient is approximately 30% less than the Smoluchoswki value. Finally, it might reasonable have been assumed that the rate of reaction of encounter pairs is not much larger than the diffusion-limited rate. Incorporating the partially reflecting boundary condition gives a steady-state rate coefficient of... 

It is interesting to substitute the partially reflecting boundary condition form of h(t) of eqn. (194) into the rate expression deduced by Noyes [eqn. (191)]. Integrating (191) from — °° to t, leads to a rate coefficient [278]... 

Keizer [455a, 498] has applied non-equilibrium statistical mechanics to the calculation of the reaction rate between two species which can both diffuse with mutual diffusion coefficient ) and encounter distance R. The partially reflecting boundary condition can be incorporated, but in the limit of fast reaction of encounter pairs for identical species... 

The Collins and Kimball [4] analysis of reaction between two species uses the partially reflecting boundary condition... 

Finally, on integrating the delta function and using the partially reflecting boundary condition, noting there is no flux over the outer boundary... 

Fig. 5. Density distribution from eqn. (23) for a diffusion coefficient of D = 10 s and encounter distance R = 0.5 nm. The partially reflecting boundary condition (22) is used with = lO an infinite time as in Fig. 1.

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