J. Phys. Chem. 1996, 100, 19049-19054
19049
Rapid Chemical Reactions in Two Dimensions: Spatially Nonlocal Effects† Andrzej Molski,| Sebastian Bergling,‡ and Joel Keizer*,‡,§,⊥ Institute of Theoretical Dynamics, Department of Chemistry, and Section of Neurobiology, Physiology and BehaVior, UniVersity of California, DaVis, California 95616, and Department of Physical Chemistry, Adam Mickiewicz UniVersity, 60-780 Poznan` , Grunwaldzka 6, Poland ReceiVed: June 4, 1996X
Using a recent generalization of fluctuation theory that includes spatially nonlocal effects, we examine the influence of nonlocal chemical reactivities on rapid chemical reactions in two dimensions. We focus attention of the irreversible reaction A + A f products at low density, exploring the time dependence of the radial distribution of pairs of reactants, its asymptotic time dependence, and steady states in the presence of inputs. As in the case of three dimensions, we find that the nonlocal theory removes inherent problems with the local theory that occur at short times, gives the expected logarithmic divergence in the absence of inputs, but has well-behaved steady state properties in the presence of inputs. Thus, the nonlocal theory appears to provide a systematic method for treating rapid reactions in membranes and on surfaces at higher densities.
1. Introduction The theory of rapid chemical reactions in membranes has a checkered history. The standard Smoluchowski approach, which works so well in three dimensions, diverges when applied to steady states in two dimensions, and a number of alterations to the standard theory have been developed to circumvent this difficulty.1-3 We have shown that molecular fluctuation theory also can be used to calculate steady state rate constants in two dimensions,4-6 and this theory has been applied to several problems in biophysics.4,7,8 Recently, simulations of the steady state reactivity for hard hexagons in two dimensions have been carried out in which reacting pairs representing identical molecules are replaced at random positions after reaction.8 Those simulations confirm the low-density predictions of the fluctuation theory at steady state but show significant deviations at higher densities. The fluctuation theory of rapid reactions was originally formulated as a spatially local theory; i.e., it relies on transport equations with rate and transport coefficients that are true constants. This is a special case of fluctuation theory that is valid in the long wave vector limit.9-11 Despite the success of the local theory in removing the divergence of the Smoluchowski theory for steady states in two dimensions, the local theory encounters problems when applied to time-dependent problems.12 These problems occur at short times in the diffusion-controlled limit and are caused by molecular length scale kinetic and interaction terms that are missing in the nonlocal theory. In three dimensions the spatially nonlocal version of fluctuation theory eliminates these problems.12 The success of the nonlocal fluctuation theory in three dimensions suggests that it may be useful for treating rapid reactions in two-dimensional systems, i.e., on membranes or surfaces. In three dimensions the nonlocal theory provides * Address correspondence to J. Keizer at the Institute of Theoretical Dynamics (
[email protected]). † This paper is dedicated to John Ross on the occasion of his 70th birthday. John was one of the first people to take an interest in our application of fluctuation theory to rapid chemical reactions, and we hope that he will continue to find our current work of interest. ‡ Institute of Theoretical Dynamics. § Department of Chemistry. ⊥ Section on Neurobiology, Physiology and Behavior. | Department of Physical Chemistry. X Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01613-9 CCC: $12.00
corrections to the Smoluchowski theory that become relevant at higher densities. Comparable corrections may be especially important in two dimensions since the steady state rate coefficient is intrinsically dependent on the density in two dimensions.4,7,8 In fact, it is the lack of density dependence in the Smoluchowski theory that is the cause of the logarithmic divergence.4 Here we treat the bimolecular association reaction A + A f products in two dimensions, presenting the full nonlocal theory for the density-dependent radial distribution function. Higherorder density-dependent terms make these equations difficult to solve in two dimensions, and as a first step we concentrate on the lowest order terms in the density. We explore several nonlocal reactivity functions and show that the nonlocal theory removes the difficulties associated with the local theory at short times. The Smoluchowski reactivity in the absence input terms gives an equation for the radial distribution function that is equivalent to a time-dependent version of the Smoluchowski theory. In the absence of inputs we show that asymptotically in time the radial distribution function in both the local and nonlocal theories has a logarithmic spatial dependence with an amplitude that vanishes like 1/ln(t). This divergence is removed by adding inputs to the local theory, which yields an asymptotic steady state distribution similar to that in the local theory. Our results suggest that the spatially nonlocal theory should be useful for treating higher order density effects for rapid reactions confined to membranes and surfaces. 2. Nonlocal Theory for A + A f Products Both the local and nonlocal implementations of fluctuation theory rely on the radial distribution function and its connection to density fluctuations to calculate rate constants. As the basis of the theory has been discussed frequently before,5,6 we touch here only on the high points. According to statistical mechanics, the radial distribution function, g(|r - r′|, t) for pairs of identical particles (in this case A), is related to the density-density correlation function via11
〈δF(r,t) δF(r′,t)〉 ) F(r,t) δ(r - r′) + F(r,t) F(r′,t) × (g(|r - r′|, t) - 1) (1) where δF(r,t) is the instantaneous fluctuation of the local number density of A from its average value, F(t), which may be © 1996 American Chemical Society
19050 J. Phys. Chem., Vol. 100, No. 49, 1996
Molski et al.
dependent on time but is taken independent of position. The rate constant for an elementary bimolecular reaction like A + A f products is obtained from the integral of the intrinsic reactivity function, k°(r,x), weighted by the radial distribution of pairs, i.e.
k(t) ) ∫dx k°(|r - x|) g(|r - x|, t)
(2)
The intrinsic reactivity is simply the bimolecular rate constant for a pair of particles separated by |r - x|. Statistical nonequilibrium thermodynamics11 is then used to obtain stochastic partial differential equations to describe the density fluctuations and, finally, to derive an equation satisfied by the radial distribution function. Here we treat the irreversible reaction A + A f products, which we write explicitly as a collection of nonlocal process, each with rate constants k°(x - x′):
A(x) + A(x′) f products
(3)
Written in this nonlocal form, the stoichiometric coefficients for the reaction have the form -δ(|r - x|) - δ(|r - x′|).12 Neglecting the coupling of fluctuations in the energy density to those in the number density, statistical nonequilibrium thermodynamics11,12 provides the following spatially nonlocal expression for the rate of change of the average of the density of A:
∂F(r,t)/∂t ) ∇‚(DA(∇F(r,t) - F(r,t) ∇c1(r,t))) 2 ∫dx k°(|r - x|) g(|r - x|, t) F(r,t) F(x,t) (4) The first term on the right-hand side of eq 4 is the usual lowdensity Fickian diffusion term. The second term provides corrections that arise from nonideality of the solution and produce dynamic correlations among the A molecules. The third term expresses the nonlocal character of the chemical reaction term and describes the net loss of A molecules at r due to reactions with neighboring A molecules at x. In the local theory the second term in eq 4 is absent, and the reaction term is replaced with its spatially local expression, -2k(t)F(t)2. Indeed, for a system that is spatially uniform even the nonlocal eq 4 reduces to
dF(t)/dt ) -2k(t) F(t) + I(t) 2
(5)
in which the observed rate coefficient, k(t), defined in eq 2 appears and I(t) represents an external input of A molecules. Fluctuations around the average satisfy a linearized stochastic version of eq 4, which we write symbolically as
∂δF(r,t)/∂t ) ∫K(r,x) δF(x,t) + ˜f (r,t)
(6)
where the integral term represents all the terms coming from the linearization of eq 4 around the average in eq 5. The final term is the purely random, Gaussian contribution to the rate of change of δF(r,t). On average ˜f (r,t) vanishes, and its correlation function, γ(r,r′,t), can be obtained from standard expressions using the elementary processes for diffusion and reaction and their stoichiometric coefficients:11,12
〈f˜(r,t) ˜f (r′,t)〉 ) γ(r,r′,t) ) 2k(t) F(t)2 δ(|r - r′|) + 2F(t)2 k°(|r r′|, t) g(|r - r′|, t) - 2DAF(t)∇2δ(|r - r′|) (7)
Using eq 7, the explicit form for eq 6, eq 1, and a good deal of algebra, it is possible to obtain the partial integrodifferential equation solved by the radial distribution function. This procedure is independent of dimension, and we refer the interested reader to our previous work12 for details. The final result is
∂g(r,t)/∂t ) 2DA∇‚(∇g(r,t) - ∇c(r)) - 2k°(r) g(r,t) 2I(t)(g(r,t) - 1)/F(t) - 4F(t) ∫ dx k°(|r - x|)g(|r -
x|, t)(g(|r - r′ - x|, t) - 1) - 2DAF(t)∇2 ∫ dx c(|r x|)(g(|r - x|, t) - 1) + 4F(t) ∫ dx k°(|r - x|)g(|r x|, t)c(|r - r′ - x|) + 4F(t) k(t) c(|r - r′|) +
4F(t)2 k(t) ∫ dxc(|r - r′|)(g(|r′ - x|, t) - 1) +
4F(t)2 ∫ dx k°(|r - x|) g(|r - x|, t) ∫ dx′ c(|x -
x′|)(g(|r′ - x′|, t) - 1) (8)
Here c(r) is the local equilibrium direct correlation function and r ) |r - r′|. The first line on the right-hand side of eq 8 contains all the terms that are independent of density as well as the term that involves the random input of A molecules, while the second line contains the only density-dependent term that derives exclusively from the nonlocal chemical reactivity. The remaining terms (lines 3-8) involve the direct correlation function and arise from nonlocal activity corrections to the transport coefficients. Note that eq 8 depends on the solution, F(t), of eq 5, which in turn depends on the solution of g(r,t) from eq 8. Thus, except in specific cases, i.e., when all densitydependent terms in eq 8 are neglected, one must solve both eqs 8 and 5 simultaneously. Since the manipulations leading to eq 8 do not depend on spatial dimension, it is valid for one and three dimensions as well. In the presence of a random input of A molecules a nonvanishing input source strength, I(t), appears on the righthand side of eq 5. The term, -2I(t)(g(r,t) - 1)/F(t), on the second line of eq 8 comes from the random input of single particles as described previously.12,13 3. Low Density Limit As a first step in solving the complete equation for the radial distribution function in the nonlocal theory, we focus here on three issues in the low density limit: (1) the dependence of the rate coefficient on time for several different reactivity functions, k°(r), (2) the asymptotic time dependence of the radial distribution function in the absence of inputs, and (3) the steady state radial distribution function in the presence of inputs. At low density only the first four terms of eq 8 survive. Furthermore, at low density the direct correlation function can be written c ) exp(-u/kBT)g with u(r) the A - A intermolecular potential, kB Boltzmann’s constant, and T the absolute temperature.12 This then gives
∂g(r,t)/∂t ) 2DA∇‚(∇g(r,t) - exp(-u(r)/kBT)g(r,t)) 2k°(r) g(r,t) - 2I(t)(g(r,t) - 1)/F(t) (9) where we have added the input term that appears when single particles are added randomly to replace pairs removed by reaction. For purposes of comparison, we also consider the low density limit of the local theory.21 In the absence of inputs this gives
Rapid Chemical Reaction in Two Dimensions
J. Phys. Chem., Vol. 100, No. 49, 1996 19051
∂g(r,t)/∂t ) 2DA∇2g(r,t) - 4F(t) k(t) (g(r,t) - 1) 2k(t) δ(r) (10) and in the presence of inputs
∂g(r,t)/∂t ) 2DA∇2g(r,t) - 8F(t) k(t)(g(r,t) - 1)- 2k(t) δ(r) (11) We have examined two different nonlocal reactivity functions: the Smoluchowski reactivity,4
k°(r) ) (k°/2πR)δ(r - R)
(12)
where R is the encounter radius, and the disk reactivity14-16
k°(r) ) τ-1Θ(R - r)
(13)
where Θ is the Heaviside function and τ is the lifetime of an A - A pair at a relative distance r e R. The Smoluchowski reactivity can be used to construct two different spatially nonlocal models: The usual Smoluchowski-Collins-Kimball model in which a hard-sphere potential, exp(-u(r)/kBT) ) Θ(R - r), is used. In this case the second and third terms on the right-hand side of eq 9 are equivalent to the usual Smoluchowski equation with the radiation boundary condition12
4πRDA
(∂g∂r)
R+
) 2k°g(R,t)
(14)
where the derivative is evaluated as r approaches R from above. Alternatively, we can choose the potential to vanish and obtain the circle model for which the final terms in eq 9 can be replaced by
4πRDA
(∂g∂r)
R+
- 4πRDA
(∂g∂r)
R-
) 2k°g(R,t)
(15)
The third model set of equations that we solve is the disk model, which uses the disk reactivity with u(r) ) 0. In the nonlocal versions of the Smoluchowski-Collins-Kimball and circle models g(r,t) satisfies
∂g(r,t)/∂t ) 2DA∇2g(r,t) - 2I(t)(g(r,t) - 1)/F(t) (16) supplemented with the boundary conditions in eqs 14 and 15, while for the disk model we have
∂g(r,t)/∂t ) 2DA∇2g(r,t) - τ-1Θ(R - r) g(r,t) 2I(g(r,t) - 1)/F(t) (17) 3.1. Time-Dependent Rate Coefficients. We have solved for the Laplace transform of the radial distribution function, gˆ (r,s) ) ∫∞0 dt exp(-st)g(r,t), for the spatially nonlocal Smoluchowski-Collins-Kimball, circle, and disk models based on the equations developed in the previous section. The details of the solution of the transformed equations are given in the Appendix. Using the Stehfest algorithm,17 which we have previously used in three dimensions,12 it is possible to invert the Laplace transform to obtain g(r,t). Since we are interested in the time-dependent rate coefficient, we have used eq 2 instead to obtain the Laplace transform of k(t), i.e., kˆ (s). Figure 1 shows the results of the calculations for the scaled rate coefficient, κ(θ)/κ° ) k(t)/k°, versus the scaled time, θ ) 2DAt/R2, for the nonlocal Smoluchowski-Collins-Kimball, circle, and disk models, all in the absence of inputs. The local version of the theory for the Smoluchowski reactivity is shown for comparison. The three sets of curves represent increasing
Figure 1. Scaled rate coefficient κ(θ)/κ° ) k(t)/k°, obtained by numerically inverting the Laplace transform using the Stehfest algorithm,17 is shown as a function of the scaled time, θ ) 2DAt/R2. Four different calculations are presented: the circle reactivity for the nonlocal theory (dotted line), the Smoluchowski-Collins-Kimball reactivity for the local theory (dashed line), the Smoluchowski-Collins-Kimball reactivity for the nonlocal theory (full line), and the disk reactivity for the nonlocal theory (dash-dot line). Three values of the scaled intrinsic rate constant (κ° ) 0.1, 1.0, and 10) are used to illustrate the differences and similarities among the four models. For the disk reactivity k° ) R2/2τDA.
values of the scaled intrinsic reactivity κ° ) k°/2πDA (0.1, 1.0, and 10), where for the disk model κ° ) R2/2τDA. Note that when the intrinsic reactivity is relatively small (upper set of curves κ° ) 0.1), all four calculations produce quite similar results. Even the local theory, which does not get the correct initial slope, is comparable to the other calculations for t g R2/ DA. Significant differences at short times begin to appear in the middle set of curves of Figure 1 (κ° ) 1.0), illustrating that the details of the reactivity function have a significant impact on short time behavior for reactions that are not diffusion controlled. In fact, for membranes these initial times could easily be as long as a tenth of a millisecond, since lateral diffusion constants in membranes are notoriously small.18 The lower set of curves (κ° ) 10) illustrates what happens as the diffusion-controlled limit is approached. The local theory breaks down completely at short times, while κ(θ) in the nonlocal circle, SmoluchowskiCollins-Kimball, and disk models all approach each other relatively rapidly. Thus, as is the case in three dimensions,12 the nonlocal theory eliminates the unphysical short-time behavior of the local theory in two dimensions. 3.2. Asymptotic Behavior for Large Times. The Laplace transform of the radial distribution function can be used to obtain the behavior of the rate coefficient asymptotically in time. We have carried out this explicitly for the nonlocal SmoluchowskiCollins-Kimball model and for the local theory with the Smoluchowski reactivity. In the former case (cf. eq 37) one has19
gˆ (r,s) )
(
)
κ°K0(Fxs) 1 1s xsF0K1(F0xs) + κ°K0(F0xs)
(18)
19052 J. Phys. Chem., Vol. 100, No. 49, 1996
Molski et al.
where we use the quantities defined in the Appendix, eq 30. For the local theory we use eq 48 and the relationship
gˆ (r,s) ) 1/s - 2κ°K0(Fxs) gˆ (R,s)
(19) gss(r) ) 1 - (kss/πDA)K0(r/η)
that follows from earlier work21 to obtain
gˆ (r,s) )
simplified eq 26 using the steady state condition 2kssFss2 ) I. The first of these equations is easily solved using Fourier transforms:4
(
)
2κ°K0(Fxs) 1 1s 1 + κ°K0(F0xs)
(20)
Equation 26, which must be solved with the radiation boundary condition at R, is recognized as a Bessel equation with the wellknown solution19
Using the asymptotic formula for s f 0
K0(x) ∼ ln(2/x) - γ, K1(x) ∼ 1/x
(21)
Equation 18 asymptotically becomes
gˆ (r,s) )
(
ln(2/Fxs) - γ
)
1 1 - κ° + O(Fxs) s 1 + κ°(ln(2Fxs) - γ)
(
) 2 ln(r/R) +
)
(( ) )
1 1 1 +O κ° s ln(F2s) ln(F2s)
2
(22) (23)
Using this and the fact that the inverse Laplace transform of 1/(s ln s) is asymptotically -1/ln(t)20 in eq 23, it follows that for large t
( (Rr ) + κ°1 ) ln(2D1 t/R )
g(r,t) ∼ 2 ln
2
(24)
A
Similar manipulations using eq 20 also yield this result, and it is easy to see comparing eq 36 with eq 49 that the circle model also has this asymptotic time dependence. This confirms the impression in Figure 1 that the circle, local, and nonlocal Smoluchowski-Collins-Kimball models all have the same asymptotic behavior, which is reminiscent of the logarithmic divergence inherent in the usual Smoluchowski theory in two dimensions. 3.3. Steady States with Inputs. In the presence of a nonvanishing, constant input term, I, the local theory gives rise to a finite steady state radial distribution function in one, two, and three dimensions.4 In one and two dimensions, however, one must be careful not to neglect the lowest order terms in the densitysotherwise, the linear and logarithmic divergences that typify the usual Smoluchowski theory will occur. In fact, even in three dimensions there is a divergence problem in the Smoluchowski theorysalthough in that case it is only the total number fluctuations that diverge. These problems do not occur in fluctuation theory due to the appearance of a correlation length of the form η ) (DA/4kssFss)1/2, where the superscripts represent the steady state. It is the divergence of the correlation length at low density that causes all of these divergence problems in the Smoluchowski theory, which ignors the bimolecular lifetime of the reactant pair.4 To check that the nonlocal theory in the presence of constant inputs correctly includes these correlations, we have used the local equation eq 10 with the Smoluchowski reactivity and the corresponding nonlocal eq 16 to obtain the radial distribution function at steady state. This gives following equations
∇2gss(r) - 4(Fsskss/DA)(g(r,t) - 1) - (kss/DA)δ(r) (25) and
∇2gss(r) - 2(kssFss/DA)(gss(r) - 1) ) 0
(26)
for the local and nonlocal theories, respectively, where we have
(27)
gss(r) ) 1 -
κ°K0(r/x2η) (R/x2η)K1(R/x2η) + κ°K0(R/x2η)
(28)
Notice that neither of these equations provides a complete solution for the radial distribution function since both require a knowledge of kss, which must be obtained by solving the equation kss ) k°gss(r). As we have shown previously,4 it is easy to solve this equation by iteration, and it leads to finite values for both kss and gss(r). Thus, we see that in the presence of inputs the nonlocal theory has a well-defined asymptotic steady state solution in two dimensions. 4. Concluding Remarks Rapid chemical reactions have been documented to occur in a variety of model and biophysical membranes. The poreforming peptide, melittin, and other small polypeptide molecules appear to associate via bimolecular association reactions like those treated here.8,24 Whether or not these reactions are rate limiting for the formation of pores is not known. Nonetheless, the great varity of rapid bimolecular processes occurring in biomembranes means that it is important to have a consistent theory of the effects of diffusion and other transport processes on reaction rates. Moreover, the facts that diffusion of biomacromolecules in biomembranes is slow and that high concentrations of reactants are typical mean that a theory that can deal with both time-dependent and density-dependent reaction rates is required to interpret experimental data. Here we have explored the low density limit of a spatially nonlocal description of rapid reactions rates based on molecular fluctuation theory. We have shown that the theory removes two of the problems associated with other theories of rapid reactions in 2 dimensions. (1) It provides a consistent explanation of the short-time kinetics of rapid reactions by including molecular length scale interactions, and (2) by including correlations due to the density of particles, it provides a divergence-free description of the long-time behavior of the reaction rate. The theory has the further advantage that it is not restricted to a particular functional form for the spatial dependence of reaction rate on separation. Although we have restricted ourselves here to the reaction A + A f products, many of the results derived here are easily extended for other reaction mechanisms. For example, with minor changes all of the results in the Appendix can be applied to the reaction A + B f products and so would be applicable to excimer formation in membranes.25 In addition to the analysis of experimental data, the nonlocal theory can be used to analyze numerical simulations. The theory has already been used to treat simulations in one dimension,22,23 and it should prove useful as well in understanding the density dependence of simulations in two dimensions.8 It is our intention to analyze the higher order density-dependent terms in eq 8 in an effort to understand simulations with association and other bimolecular reaction.
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J. Phys. Chem., Vol. 100, No. 49, 1996 19053
Acknowledgment. This work was supported by a generous grant to J.K. from the Research Committee at UC Davis and continuing support from NSF Grant BIR 9300799. Appendix. Laplace Transformed Equations
kˆ (s) ) k°gˆ (R,t) )
k°F0K1(F0xs) κ°xsK0(F0xs) + F0sK1(F0xs)
or in terms of the dimensionless quantities in eq 35:
Circle Model. The circle model uses the Smoluchowski reactivity without a hard core potential. Therefore, unlike the standard Smoluchowski-Collins-Kimball approach, g(r,t) does not vanish for r < R. From eqs 2 and 12 the rate coefficient becomes
k(t) ) k°g(R,t)
(29)
κˆ (z) )
κ°K1(xz)
(38)
κ°xzK0(xz) + zK1(xz)
Disk Model. This model uses the disk reactivity in eq 13 and the radial distribution function satisfies eq 17. The rate coefficient is defined by
k(t) ) 2πτ-1 ∫0 rg(r,t) dr
Defining
R
F)
(37)
(39)
0
R k r , F0 ) , hh ) 2πDAR 2D 2D x A x A
(30)
For this model eq 31 is valid for r > R, whereas for r < R we have
and using the initial condition g(F,0) ) 1, the Laplace transform, hˆ (F,s), of h(F,t) ) g(F,t) - 1 satisfies (cf. Eq 16 with I ) 0):
τ-1 d2hˆ 1 dhˆ + - (s + 2τ-1)hˆ ) 2 F dF s dF
d2hˆ 1 dhˆ + - shˆ ) 0 dF2 F dF
with h ) g - 1. The inhomogeneity in eq 40 is removed by the substitution
(31)
The boundary condition (14) becomes
(∂F∂hˆ )
F 0+
-
(∂F∂hˆ )
F0-
hh s
(32)
(41)
d2w 1 dw + - (s + 2τ-1)w ) 0 2 F dF dF
(42)
to obtain
The solution of eq 31 can be expressed in terms of the modified Bessel functions I0 and K0 as
hˆ ) A(s) K0(Fxs), F > F0; hˆ ) B(s) I0(Fxs), F < F0 (33) where the coefficients A(s) and B(s) can be found using the continuity of hˆ at F0 and the boundary condition (32). Using the Wronskian I0(x)K′0(x) - I′0(x)K0(x) ) -1/x, where the prime denotes differentiation, one gets for the Laplace transform, kˆ (s), of the molecular rate coefficient
kˆ (s) ) k°gˆ (R,s) )
w 2τ-1 + s(s + 2τ-1) s + 2τ-1
hˆ ) -
- hhhˆ )
k° s [1 + F0hhK0(F0xs) I0(F0xs)]
(34)
It is useful to introduce dimensionless variables for the time, distance, and rate coefficients:
θ ) t/τD, τD ) R2/2DA, κ° ) k°/2πDA, κ(θ) ) k(t)/2πDA (35)
As in the circle model, the solution to this equation can be written hˆ ) A(s)K0(Fxs) for F > F0 and w ) B(s)I0
(Fxs + 2τ-1) for F < F0, with the coefficients A(s) and B(s) determined from the continuity of hˆ and its spatial derivative at F0. For F < F0 this gives
gˆ (F,s) )
B(s) )
(43)
s + 2τ-1
2τ-1K′0(q) sK′0(q) I0(q1) - xs(s + 2τ-1)K0(q) I′0(q1)
(44)
with q ) F0xs and q1 ) F0xs + 2τ-1. Equation 44 can be further rearranged using K′0(λx) ) -λK1(λx) and I′0(λx) ) +λI1(λx), and the fact that ∫ xI0(λx) dx ) xI1(λx)/λ, to give
kˆ (s) )
(36)
where z ) τDs. Smoluchowski-Collins-Kimball Model. This model uses the Smoluchowski reactivity in eq 12 with a hard-sphere repulsion. The radial distribution function satisfies eq 16 with the radiation boundary condition eq 14. The rate coefficient is given by eq 29, which after Laplace transformation using the initial condition g(r,0) ) 1 for r > R and zero otherwise gives19
1 + B(s) I0(q1)
where
In this notation the Laplace transform of the dimensionless rate coefficient κ(θ) is
κ° κˆ (z) ) z[1 + κ°K0(xz)I0(xz)]
(40)
2πDAτ-1F02 -1
s + 2τ
(
1+
)
2B(s) I1(q1) q1
(45)
The dimensionless rate coefficient is found, using the dimensionless lifetime γ ) 2DAτ/R2, to be
κˆ (z) )
γ-1 {1 + [4γ-1K1(xz)I1(xz + 2γ-1)]/ -1 z + 2γ
[xz xz + 2γ-1[xzK1(xz)I0(xz + 2γ-1) + xz + 2γ-1 × K0(xz)I1(xz + 2γ-1)]]} (46)
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Molski et al.
Local Theory: Smoluchowski Reactivity. In the limit of low density and in absence of inputs eq 11 with the Smoluchowski reactivity becomes
∂g(r,t)/∂t ) 2DA∇2g(r,t) - 2k°δ(r)
1 s(1 + κ°Ko(F0xs))
(48)
Thus, in terms of the dimensionless quantities in eq 35
κˆ (z) )
κ° z(1 + κ°Ko(xz))
Keizer, Keizer, Keizer, Keizer,
J. J. Phys. Chem. 1982, 86, 5052. J. Acc. Chem. Res. 1985, 18, 235. J. Chem. ReV. 1987, 87, 167. J.; Ramirez, J; Peacock-Lopez, E. Biophys. J. 1985, 47,
49.
(47)
with the initial condition g(r,0) ) 1. Using the spatial Fourier transform, inverting, and then Laplace transforming for r ) R gives21
g˜ (R,s) )
(4) (5) (6) (7)
(49)
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