Diffusion-Controlled Reactions

Ann. Rev. Phys. Chern. 1983. 34:493-524 Copyright © 1983 by Annual Reviews Inc. All rights reserved

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Annu. Rev. Phys. Chem. 1983.34:493-524. Downloaded from www.annualreviews.org Access provided by CONRICYT EBVC and Econ Trial on 09/23/15. For personal use only.

REACTIONS D an iel F. Calef and J. M. D eutch Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 INTRODUCTION

The purpose of this article is to review some recent developments on the theory of diffusion-controlled reactions. The importance of the subject to chemistry, biology, and engineering is illustrated by consideration of the prototype reaction mechanism ko

A + B � [AB] kr

kr -+

products.

1.

This mechanism indicates the reversible association of reactants A and B into a reaction intermediate [AB] in which the reactants are placed into proximity and relative orientation that permits the chemical transforma tion to products to occur. Use of the steady-state approximation for [AB] leads to the following effective rate coefficient for the reaction

kDkr kerf=� . r+ kr

2.

It is apparent that the kinetic rate kD influences the effective rate of reaction and that in certain circumstances (kr � kr) it becomes the rate limiting step [keff = kD]. Accordingly, the rate coefficient kD' which describes the mass transport rate at which reactants encounter each other, is of central concern to solution kineticists. There is little doubt that the subject of diffusion-controlled reactions has broad application beyond this prototype example. Diffusion-controlled reactions play a prominent role in heterogeneous catalysis (after all, the perfeot catalyst leads to mass transport limitations), polymer chain growth kinetics, colloid or crystal growth, and biophysical chemistry. 493 0066-426X/83/1 101-0493$02.00


494

CALEF & DEUTCH

The authors of this review are n9t qualified to survey recent develop ments in all these related areas. Rather the review contains only a selection of subjects that bears on the theory of diffusion-controlled reactions. The criteria employed for selection of these subjects are substantial recent activity and/or interest to the authors. It is hoped that the reader after completing this review will have an appreciation of the present status of the theory of diffusion-controlled reactions, the range of application of the concept to physical and chemical phenomena, and current outstanding research issues. The class of reactions in which the mass transport step is rate determining is referred to as diffusion-controlled reactions because diffusion is the transport process in solution that determines the encounter between the reacting pair. The classic work of Srrlbluchowski (1) established the steady state rate of diffusion-controlled reactions by considering the rate of coagulation of a pair of reacting molecules by diffusion. In three dimen sions, assuming spherical symmetry, one seeks the steady state flux 3. from the diffusion equation 1 8 2' r J(r) = O. r2 8r

4.

This equation is solved subject to the boundary conditions that the reactants vanish into products when they touch P(RA + RB) 0 (RA and RB are the radii of species A and B) and that the reactants are continuously replenished so that at large relative separation r, P(r) tends to the product of the mean concentrations of species A and B. The now classic result is =

5.

One expects that each diffusion coefficient will be related to its friction coefficient according to the Einstein relation D = k T/� and that the friction coefficient will be given by Stokes law � = 6nl1oR for each species where 110 is the solvent shear viscosity. Thus kD oc 1101; the indicated viscosity dependence is frequently taken as evidence for a diffusion-controlled reaction although application of this viscosity dependence as a diagnostic for diffusion-controlled reaction is not always a simple matter. There have been many notable extensions of these classic ideas of diffusion control. For example the influence of direct intermolecular forces between the species u(r) was introduced by Kramers (2) and Debye (3); one finds kD

=

4n(DA +DB)

[f

OO

RA+RB

J

eU(r)/r2 dr

-

l

.

6.

DIFFUSION-CONTROLLED REACTIONS

495

Several authors (4-7) have noted that the diffusion of particles toward each other is impeded by the hydrodynamic effect that arises from the need to force the solvent out of the path of mutual approach. The modification to ko from this hydrodynamic effect is (5, 6) ko

=

4n(DA+DB)

[f

OO

KA+KB

ur

e

( )jr 2

]

d-1(r) dr

-l

7.


where d(r) is the radial part of the reduced relative pair diffusion tensor �(r) d(r) = r'

[f � -

]

(DA +DB)-l§"(r) 'r.

8.

The tensor §"(r) contains the physical effect on the relative motion of the hydrodynamic resistance introduced by the continuum solvent. Frequently, one adopts (7) the Oseen form for §"(r) ff(r)

=

1

--[.;1 +rr] 8nY/or

9.

for all relative separations although it is strictly valid only in the limit [(RA + Ru)/r] � 1. Inclusion of these hydrodynamic effects has a significant impact on the predicted rate coefficient and on other phenomena (8) that are sensitive to relative diffusion. Attention has also been given to the influence of convective flow on the rate of diffusion-controlled reactions. In laminar flow at uniform velocity v the classical result is due to Levich (9); consult Ref. (10) for more recent work. The Levich result for a reacting sink of radius R has a non-analytic dependence on the Peclet number (vRjD) ko

=

4nDR[1 +O.64(vRjD)1/3].

10.

An interesting calculation for the rate of coagulation in turbulent flow has been presented by Delichatsios & Probstein (11). Significant work has also been undertaken to examine the effect of relaxing the steady state assumption of the classical theory. These generalizations take various forms, from replacing the absorbing boundary condition with a mixed or radiative boundary condition to examining the temporal evolution toward the steady state. Early work in this direction was done by Collins & Kimball (12) and by Noyes (13). The latter author also began an important inquiry concerning the differences to be expected from continuous space-time descriptions of diffusion and discrete-lattice random walk descriptions. We also mention a line of development, particularly important for macromolecular systems, in which the assumption of uniform reactivity on the spherical surface is relaxed. Sole & Stockmayer (14) began this line of

496

CALEF & DEUTCH


inquiry, which has been extended (15, 16, 17) in a number of directions, including to buried active (18) sites. A recent approach is based on direct consideration of a coupled translation and rotation diffusion equation (19). This model of asymmetric reactivity appears to be relevant to the decomposition kinetics of protein surface disulfide bonds (20). Finally we note the recent appearance of an excellent book by van Kampen (21) that critically examines the foundations of stochastic methods, including the diffusion equation, in physics and chemistry. CONCENTRATION DEPENDENCE OF THE DIFFUSION CONSTANT

Before proceeding to discuss the concentration dependence of diffusion controlled reactions, it is necessary to review briefly the state of knowledge concerning the concentration dependence of diffusion coefficients in solutions containing (spherical) macroparticles that do not react. In recent years there has been a great deal of attention given to this important many body system and an entire review article could legitimately be devoted to this subject. Our more modest purpose here is to provide the reader with some background in order to provide the basis for subsequent discussion of reacting systems. The dynamics of a many-Brownian-particle system is normally described by the N-particle Smoluchowski equation (22) 8P(RN,t) at

=

N

N

i�l j�l Vj· p'&jj. [VjP+ P(VP)P]

11.

where p(RN, t) is the configurational probability density of determining the N-particle positions R N (R1, , R N); U is the effective potential acting between the macro-particles, P = llkB T and P.&jj is the effective diffusion coefficient acting between the pair of particles (i,j). The physical basis of this model is many-particle Brownian motion. The Smoluchowski equation is presumed to describe the diffusive motion of the N-particltis in a background solvent, normally assumed to be a continuum fluid described by low Reynolds number hydrodynamics. The equation is only valid on a time long compared to momentum relaxation so that a local equilibrium description in coordinate space is applicable. Relaxation of this assumption requires consideration of a many-particle Fokker-Planck equation in which many-particle pair friction coefficients arise (23-26). The dynamics of an individual particle in this system will be influenced by the presence of effective direct interactions contained in U and by diffusion. In the absence of particle interactions one would adopt the usual inde=

• • •


497

pendent particle diffusion coefficient Do, according to

kBT

kBT 67r:lJoa

Do = -- = --


�

12.

where ( is the individual particle friction coefficient given by Stokes Law. In fact when two particles are executing relative motion in an incom pressible low Reynolds number fluid, there is a hydrodynamic force (27) that increases the effective friction and modifies the relative pair diffusion. This modification arises from the interaction of the Stokes velocity field that each particle is attempting to establish. In the limit in which the pair of particles is well separated, one obtains the well-known (22) perturbation which involves the Oseen tensor Eq. 9, 13. This expression reveals the long-range nature of the hydrodynamic interaction since Y(r) r-1 for large separations. This long-range charac ter foreshadows the fact that the concentration dependence in this many particle system can be quite complicated and, in particular, lead to non analytic concentration dependence for pertinent transport coefficients. There will, of course, be corrections to the effective pair diffusion coefficient that will arise in a more complete hydrodynamic calculation in which the particles are not treated as point particles and finite size effects are taken into account. These corrections lead to modifications to �ij in powers of (particle size/particle separation). The two key quantities that one wishes to extract from the N-particle Smoluchowski equation are the effective mutual diffusion coefficient Dc and the effective self-diffusion coefficient Ds. �

The Mutual Diffusion Coefficient The mutual diffusion coefficient is identified as the transport coefficient that appears in the dynamical equation for the average particle concentration c(r, t) which is defined by

c(r,t) =

it1 f dRN(j(r-Rj)P(RN,t)

14.

or, in terms of spatial Fourier transforms

c(k, t)

=

f

dr eik.rc(r,t) =

Jl f

dRN eik.RiP(RN, t).

15.

In the limit of long time and large wavelengths (k � 0) one expects an

498

CALEF & DEUTCH

average diffusion equation of the form


16. which can be obtained by a variety of high technology many body methods. (See references cited in Table 1 .) In general, Dc will depend upon the form of the direct interparticle potential and the level of approximation introduced for the hydrodynamic interaction. In the linear regime, the mutual diffusion coefficient will be a function of the equilibrium concentration Co (NIV); for example for hard spheres Dc Dc(cjJ) where cjJ is the volume fraction cjJ = [(4nI3)Na3 IV] and a is the macroparticle radius. According to linear response theory, Dc can also be extracted from the equilibrium time correlation function for the dynamic structure factor (see Eg. 30 below). The mutual diffusion coefficient should be identified with the diffusion coefficient that appears in the transport equations of nonequilibrium thermodynamics for two component fluids (28). A recent analysis of the relationships of the molecular and nonequilibrium thermodynamic formu lations for Dc has been given by Schurr (29). In noneguilibrium thermody namics, the mutual diffusion coefficient appears as the product of an eqUilibrium thermodynamic factor and a kinetic factor (29), =

Dc

=

l (��)T.�.

=

17.

where fl s is the chemical potential ofthe solvent. In Eq. 17, the kinetic factor has been written in a form, in analogy to Stokes law, that defines a mutual friction coefficient !c. The thermodynamic factor can be identified with the static coherent structure factor (see Eg. 28) according to

(��)T'�S

=

kB T

I(!��

S ( k, t

=

)

0) .

1 8.

Thus a theory that attacks the calculation of Dc directly will include equilibrium and kinetic contributions. Of course both the thermodynamic factor and the friction coefficient are separate functions of concentration. A variety of theoretical treatments (27, 37) lead to the following general expression for Dc in the long wavelength limit Dc

Do

=

kBT

S(O) +cS(0)k2

f

dr k'§"(r)'k(g(r)-l)

1 9.

where g(r) is the equilibrium pair correlation function for the macro particles in solution and §" is the Oseen tensor or perhaps a more sophis ticated version of the hydrodynamic interactions. The important point to


499

note is that if one excludes the effect of hydrodynamics and sets:T = 0 one obtains the result (37)

D Dc = o

20.


S(O)"

In the case of hard spheres the first order result will be D c = Do(1 +8.

32

.


503

These authors quote a privately communicated value calculated by G. K. Batchelor of IlCs 2.68, which is considerably larger than the theoretical values included in Table 1. The work of Vrij and co-workers suggests a fairly successful ability to predict at least the low concentration dependence of Dc and Ds. =


Ion ic Systems Most of the macromolecular and colloidal systems of interest biologically and chemically involve charged species. The presence of long-range Coulomb forces in ionic systems greatly complicates theoretical calcu lation. The motion of small ions in solution has been discussed in this series by Wolynes (56). From the small ion studies, one would expect that under standing the diffusion of macroions, which can be very highly charged, at finite concentration would be difficult. But it is just these systems that are of greatest experimental interest. Certain simplifications are possible when the size and charge of the

macroions are much greater than those of counterions and solvent molecules. The counterions will have much larger diffusion coefficients than the macroions, and hence the charge cloud surrounding the macroion can be regarded as relaxing "instantaneously." The electrostatic part of the interaction between two macroions can be approximated by the Debye H iickel form

u (r) =

Uo

e-rcr

-

r

33.

where" is the inverse of the Debye screening length,

Lc"q;

,, = 4n-a skBT

34.

where Ca. and qa. are the concentration and charge of the counterion Ilc, and B the dielectric constant of the solvent. This formulation assumes that a Debye cloud is formed about each macroion. In fact the counterion distribution will itself be perturbed from its ideal low concentration distribution by the presence of other macroions. However, with this approximation for the potential one may employ the N-particle Smolu chowski equation to compute the mutual and self-diffusion coefficients. The effective pair potential now depends on several parameters, and so the resulting diffusion coefficients cannot be as neatly tabulated as for hard spheres. Hence our remarks must be more qualitative. The self-diffusion coefficient has been discussed recently by several authors (40, 45,57). Marqusee & Deutch (45) found that to lowest order in the density of macroions (co), with a potential of the form ofEq. 33 and no

504

CALEF & DEUTCH

)

hydrodynamic interactions, that


(1

_ (Puo)2nco . 35. 3K These authors also stressed that the frequency-dependent diffusion coeffi cient can be important in understanding macroions. Klein & Hess (57, 58) have derived a similar result for the static self-diffusion coefficient. Ohstuki & Okano (40) have included hydrodynamic interactions, in a calculation of hard spheres with additional Debye-Hiickel repulsions using the sophisti cated, Felderhof (36) form for the hydrodynamic interaction tensor. The resulting expressions must be evaluated numerically. Qualitatively, their results show that including hydrodynamic interactions further decreases the self-diffusion coefficient, and the decrease is of the same magnitude as that due to direct interactions. For the mutual diffusion coefficient, in the absence of hydrodynamic interactions, Eq. 20 holds Ds

=

D0

36. and using the Debye-Hiickel potential, to lowest order in the concentration, one obtains Dc

=

(

Do 1 +

4ncopuo

K2

)

37.

which has a different functional dependence on temperature, ionic strength, and the Debye length than the corresponding self-diffusion coefficient CEq. 35). Hydrodynamic interactions have been found to be less important for the mutual diffusion coefficient. Ohstuki & Okano (40), again using Felderhof's (36) hydrodynamic interaction tensor fOf hard spheres with additional Debye-Hiickel repUlsions, find a small, but significant decrease in the mutual diffusion coefficient. This is in qualitative agreement with the experimental results of Weissman & Marque (59). Dorshow & Nicoli (60) have compared the prediction for Dc employing the hydrodynamic interaction of Felderhof(36) to data on bovine serum albumin and obtained excellent agreement.

Different Physic al Situations The preceding discussion has been concerned with the diffusion o f macro particles in a suspension in which the particles are moving relative to one another in a passive hydrodynamic medium. One finds that the leading concentration correction term is I1Is ,...., ¢; we refer to this situation as Case 1.


505

One may also consider the situations of a periodic (Case 2) or random (Case 3) array of jixed centers through which there is a flow of an incompressible fluid at low Reynolds number. In these cases the self-friction coefficient corresponds to the drag felt by a representative sphere in the array. It is known (61) that for the periodic array (Case 2) N 1/3, while for the random array (Case 3) 111 1/2 (62). The nonanalytic concentra tion dependence that emerges in Case 2 and Case 3 compared to Case 1 is due to the long-range perturbations induced in the velocity flow field by the various fixed centers. This velocity field perturbation once again, may be described by the Oseen interaction Eq. 9. We are not aware of a simple physical argument to explain the startling difference between the concen tration dependence in Case 2 and Case 3. We note that the results presented hold for three dimensions and may not be valid in other dimensions; in particular d 2. Finally, a fourth situation may be considered in which ajixed random (or perhaps periodic) array of spheres is present in a passive medium and a small solute species is diffusing through the medium. In this case, which we refer to as Case 4, the spheres act as obstacles for the diffusing solute species. Case 4 is particularly pertinent for biological applications; for example, diffusion of solute species in protein solutions. The predictions for the modified self-diffusion of the solute species in such blocked systems are of the form '"


'"

=

_ D(l-y]}

ko 2-

2

3¢

48. 49.

For small concentration the MC result for k(¢) reduces to the small concentration result of FD. However, the coefficient of the first concentra tion correction to the diffusion coefficient found by MC and also by BZ differs from that obtained by FD. Both MC and BZ find a factor of 2, while FD give a factor of 3. We return to this interesting point below. Tokuyama & Cukier (88) have developed a dynamical scaling approach that is designed to explore the transition from the exact averaged equation for the concentration (which includes nonlocal effects in both space and time) to the local form Eq. 46 as a function of sink concentration. Both Muthukumar (89) and Cukier (90) have developed effective medium approaches that permit examination of the behavior of both the effective rate and diffusion coefficients over the entire concentration range. One should also note that all the results presented above are for three dimensional systems [except for (84)] and that a strong dimensionality dependence should be expected in this type of dynamical system. Cukier (91) has recently applied the effective medium approach to two-dimensional surface diffusion, which is of great practical interest. His results exhibit an interesting dependence on the pertinent parameters, which are the surface fraction of reacting sites and the square root of the ratio of the absorption/desorption rate to the surface diffusion rate. The dependence is similar to that found in a model investigated by Prager & Frisch (92). We return to the vexing point concerning the coefficient of the first concentration correction to the effective diffusion coefficient in this reacting system

D

=

Do(1 +a¢).

50.



509

FD found 0( = 3, whereas both BZ and MC determined 0( = 2. This difference might not seem significant except for the fact that it reveals some inadequacy in our understanding of the fundamental physics ofthis reactive system. Felderhof, Deutch & Titulaer (FDT) (93) have reexamined this question, employing the local field electrostatic method introduced by FD. These authors determine 0( = 5/2; the difference between this value and the value 0( = 3 found by FD is attributable to the inclusion of "quadrupole" contributions in the electrostatic method. More importantly, FDT argue that the scattering method introduces an unphysical assumption that solute particles can be created within the reactive sinks. If particles are present within the sinks there is instantaneous absorption that influences the first order concentration correction to the diffusion coefficient. The source of this difficulty is that there is an essential correlation between the initial solute concentration and the location of the sinks; namely, solute species can only be located outside the reactive traps. The scattering approach or other methods that assume statistical independence of sink location and the initial solute spatial distribution must be handled delicately.

Experimental Verification The question of experimental verification of the (nonanalytic) concen tration dependence discussed above is of substantial interest. However, the theory has been developed for static random traps (Case 4) rather than relative diffusion between reactants (Case 1); this makes application to solutions problematic. Case 4 does not correspond to excitation transport in an impurity doped solid because the continuous diffusion picture discussed here differs from the discrete random walk on a periodic lattice (94, 95) in many respects. The transition from a discrete to a continuous space description when an absorbing boundary is present has been studied by several authors, notably van Kampen & Oppenheim (96-98; D. C. Torrey, private communication). One method that has been suggested for experimentally testing the concentration dependence of the diffusion-controlled rate coefficient is fluorescence quenching. Baird, McCaskill & March (100) have developed a theory for the concentration dependence of the Stern-Volmer coefficient k(Q) that describes the deviation ofthe fluorescence I(Q) offluorophores as a function of the concentration of quenching molecules Q, 1(0)

I(Q)

-1

=

Qk(Q).

51.

According to the theories developed above one would expect (if diffusion is the rate-determining step for fluorophore-quencher interaction) the

510

CALEF & DEUTCH

structure k(Q) aO+alQl/2 =

52.

rather than a more conventional virial expression


k(Q)

=

ao +a2Q.

53.

Baird & Escott (101) have reviewed data on fluorescence quenching for a number of solutions in order to evaluate deviations from the Stern-VoIrner law. These authors find that the expression (Eq. 52) is as capable as Eq. 53 in explaining the available data, although the evidence is not completely conclusive. Other applications of fluorescence to the study of diffusion-controlled reactions have been explored. For example, van der Auweraer et al (102) have considered the fluorescence quenching kinetics (at the pair level) that arise when the fluorescent probe molecule and quencher are confined to a micellar surface. Heisel & Miehe ( 103) have examined a model in which energy transfer between the probe-quencher diffusing pair is distance dependent

k( op(r, t) -at = D t72 v -p - r)p.

54.

This model resembles the pair Smoluchowski equation description intro duced by Northrup & Hynes (104, 105), which includes the influence of the intermolecular potential u(r) and distance dependent diffusion coefficient

op ot

=

V.�(r).[Vp+(Vu)pp]-k(r)p.

55.

Agmon & Hopfield ( 106) have studied the properties of the diffusion equation, Eq. 54, by an eigenfunction method in order to describe the transient kinetics of ligands bound to proteins.

�esearch Issues We have described recent theoretical developments that bear on the concentration dependence of diffusion controlled reactions. The central point of these theoretical developments is that the anticipated concentra tion dependence is nonanalytic in the concentration; this reflects the long range character of the diffusion field that forms around reactive sinks. The state of knowledge concerning this physical situation is not entirely satisfactory and more work remains to be done. First, and foremost, there is the need to establish experimentally for one or a series of chemical systems in solution the nature of the concentration dependence for the effective rate


511

coefficient. Second, there are a number of theoretical questions that remain only partially answered. These questions include the following: 1. the influence of chemical reaction on diffusion in the reacting system;


2.

the difference (at higher concentration) between the physical situations in which the reacting species are moving relative to one another (Case 1) and those in Which the reactive cel1ters are fixed (Case 4); 3. the dependence of both the effective rate coefficient and diffusion coefficient on concentration at shorter wave length disturbances and higher frequency; 4. the influence of back reaction on the reactive dynamics studied here as well as temperature effects that arise from locally generated heats of reaction; 5. a more careful treatment of ionic systems in which the reacting species bear charge. Finally, we mention the important subject of nonlinear chemical mechan isms in which mass transport plays a role. This problem deserves the attention of the physical chemist for two reasons: (a) the importance of diffusion-controlled reactions in systems of chemical interest and (b) the need to revise fundamentally the traditional view of the microscopic basis of diffusion-controlled encounters when concentration effects must be taken into account. FOKKER-PLANCK EQUATION IN DIFFUSION-CONTROLLED REACTION

The diffusion equation is valid for timescales much larger than the momentum relaxation time of the macroparticle. Clearly, to discuss shorter times a description that includes explicitly the particle's momentum should be adopted. There will also be a characteristic length scale associated with this timescale (the distance a particle travels before collisions have erased the memory of its initial momentum), and processes occurring on that length scale will also be inaccurately described by the diffusion equation. This may be called a boundary layer effect. In this section we briefly discuss attempts to move beyond the diffusion equation to include momentum relax�tion in diffusion-controlled reactions. One 'approach is to use kinetic theory and treat collisions between the "diffu�ing" 'reactants and solvent molecules. In condensed phases this calculation can become involved. This approach has been recently reviewed by Kapral (107). A second approach departs from the Fokker-Planck equation (FPE) for the Brownian particle's phase space probability distr�bution in both position and velocity. This distribution function

512

CALEF & DEUTCH

f(r, o f

V,


ot

t) will obey a FPE (ld) F

[kBT]

+v'VJ + ;'Vv/= �Vv' vf + --;;;- Vv/

56.

where m is the mass of the Brownian particle, F is the external force, and � is the friction coefficient. Equation 56 is for a single Brownian particle, but a similar equation can be derived for the N-particle distribution function for interacting Brownian particles. Titulaer (26) has discussed the role of hydrodynamic interaction in an N-particIe FPE and procedures for reducing this equation to a Smoluchowski equation (SE). Hess (108) has studied the role of direct interactions in the N-particle FPE in order to delimit the validity of the simpler SE. We restrict our attention here to the single particle FPE. Several authors (104, 109-1 1 5) have applied the FPE to diffusion controlled reactions in solution. The chemical reaction can be introduced either by the use of modified boundary conditions, or by adding an inhomogeneous source/sink term to Eq. 56 (108). The boundary condition for irreversible reaction (or complete absorption) of a particle at a surface S at the FPE level is

reS

f(r, v, t) = 0

v . i>0

57.

where r is a vector normal outward from the surface. Equation 57 simply says the probability is zero of observing a particle moving away from the surface. Harris (109) and Burschka & Titulaer (1 10) have studied the one dimensional FPE, in the absence of an external force, with Boundary Condition (57). Both groups considered the steady state situation, o

of = _ v f at ax

[ kBTav]

Vf + + �� av

with absorption at x

f(x

=

O,V)

=

0

=

m

Of

=

0

58.

0

V>O

59.

with an additional condition far from x = 0 to maintain the steady state. Although Eq. 58 is the simplest possible FPE with chemical reaction, it is not possible to find an exact solution. The approximate solutions found by the two groups, using different approximation schemes, give the same qualitative picture. We can compare these results to those from the diffusion equation with an absorbing boundary condition in one dimen sion. The FPE leads to a boundary layer near x = O. Away from x = 0, the


513

concentration c(X)

=

f

60.

A(x+Xm).

61.

f(x, v) d v

is seen to behave as


c(X)

=

Here xm is the "Milne extrapolation length," the point to the left of the origin at which the concentration obeys the DE level boundary condition C( x� O. Both groups find -

Xm

=

�

1.45

(kT ' .Jm[i

62.

Equation 61 is also the solution to the diffusion equation with the radiation boundary condition oc(X) ox

=

�c(x) Xm

63.

which is used to model partially absorbing sinks. Other aspects of this total distribution f(x, v) have also been discussed by these authors. Harris has extended his calculations to higher dimensions, studying the steady state absorption in two dimensions (111) (absorbing circles) :and three dimensions (112) (absorbing spheres). In these problems, an additional length scale, the radius of the trap, enters the problem, and the results are less easily summarized. However, the boundary layer and Milne extrapolation length are again observed. In three dimensions, other authors have also addressed this problem (113). Burschka & Titulaer have extended their calculation by considering partially absorbing walls (114) and the presence of an external field (115), both for one-dimensional systems. Partially absorbing walls can be modeled by generalizing the boundary condition to uf(u,O)

=

f�

00

where a wall scattering kernel choice

u + and specular reflection of particles with lower velocity. Since rapidly -


514

CALEF & DEUTCH

moving particles are absorbed, a build-up of concentration is observed in the boundary layer. The Milne extrapolation length is also seen to increase with increasing u+, as would be expected from the relation Eq. 63 between the partially absorbing boundary condition and Xm• When a constant external force is added (1 1 5), driving particles towards the absorbing barrier, Burschka & Titulaer find that Xm is decreased from its field-free value. This result implies that the partially absorbing boundary condition that would be applied to the Smoluchowski equation for the particle density is dependent on the field strength. These studies indicate that there is still much to be learned about the relationship between the boundary conditions for Fokker-Planck and Smoluchowski/Diffusion equations. OTHER APPLICATIONS OF DIFFUSION CONTROLLED REACTION THEORY

In this section we briefly survey some other recent interesting applications of the theory of diffusion-controlled reactions. References are intended to provide the reader with directions to pertinent recent work and do not include comprehensive coverage of the vast literature on these subjects.

Mean First Passage Times The calculation of a "mean first passage time" (MFPT) is a time-honored method of estimating the decay rate for a diffusive process [see Reference (116) for early applications]. The simplicity of this method, and the recent presentation of exact expressions for the MFPT ( 1 1 7-121), have sparked considerable interest. The MFPT is defined as � =

Loo (��) t

dt

66.

where N(t) is the total probability of being in a given region N(t)

=

L

P(r, t) dr.

67.

The MFPT is a time-averaged decay time. If the total population decayed as a simple exponential N(t)

=

e-I/t

68.

the MFPT prescription would then give the exact decay time. Higher moments of the decay time can be similarly defined (98).

515


The MFPT i s particularly convenient for situations that can be modeled b y Brownian particles moving in an external potential, ob eying a single particle Smoluchowski equation

oP(r, t) -ot-

=

])

[

u(r) V · D(r) (VP(r, t) + P(r, t) V k T B

69.

.


Here D(r) is a position dependent diffusion coefficient and u(r) is the external potential. Of course, it is necessary to specify both boundary and initial conditions. Most applications of the MFPT have considered d-dimensional spherically symmetric systems, which reduce Eq. 69 to oP(r, t) ot

-- = r

[

{

]} {

OP l -d O d - l p ou - r D(r) - + or or kB T or

=

}

l O _d- l ' - r -d r J(r t) or '

70.

where j(r, t) is the particle flux. A pertinent set of boundary conditions is a reflecting wall at r = R and a partially absorbing wall at r = a < R, j(r = R, t) j(r

=

a, t)

=

=

0 - KP(r

=

a, t).

71.

The resulting expression for the MFPT can be found by simple integration (120),

72.

where e - u(r)/kBT

73.

is the equilibrium distribution in the absence of any absorption (K = 0) and P(y, O) is the initial condition. This type of formula can also be derived for other boundary conditions and for systems obeying a master equation (120).

The accuracy of the MFPT approximation has been discussed by Szabo,

Schulten & Schulten ( 1 22), who find the MFPT will be a good approxi mation to the full solution of Eq. 69 even for reasonably short times. These

authors have also discussed applying MFPT to diffusive barrier crossing, a common model for the dynamics of a "reaction coordinate."


516

CALEF & DEUTCH

Weaver has discussed the application of the MFPT to a variety of situations. Karplus & Weaver (123) used MFPT results in modeling protein folding. Weaver has also shown how the effective diffusion coefficient of a Brownian particle in a periodic potential can be calculated using MFPT arguments (124) and commented on the relevance of this to diffusion in biological membranes ( 125). Other investigators have explored applications of the MFPT approach. Sano & Tachiya ( 1 26) have studied Eq. 69 on spherical surfaces using MFPT, as a model for chemical reactions occurring on micellar surfaces. Mozumber (127) has generalized Deutch's results (102) in order to investigate geminate recombination of ionic pairs and neutral fragments produced in photo- or radiation chemistry of condensed media. Rigos & Deutch (128) have studied a simple model of explosive chemical reactions using the MFPT. Because of the possibility of determining simple analytic forms, the MFPT method will undoubtedly prove to be of continuing utility in a wide variety of applications.

Simu lation o f Diffusion-Co n trolled Colloid Growth Recen�ly there has been an amazingly interesting set of simulations of diffusion-controlled particle growth. The basic idea of these simulations undertaken by Witten & Sander ( 129), Rikvold (1 30), and Meakin ( 1 3 1 ) is to place a nucleating center at the origin of a coordinate system and to permit solute particles to diffuse inwardly in a spherically symmetric fashion toward the origin. Successive solute particles that reach the origin stick to the cluster and form a random, irregular shape. The particles on the cluster do not rearrange their surface structure as might be expected to occur in actual physical situations due to surface forces. The purpose of the simulations is to examine the cluster shapes that result as a function of the number of particles N. Note that these shapes result from kinetic con siderations and not from equilibrium considerations derivable from a free energy. One might expect that the cluster would grow in a spherically symmetric manner in which the relationship between number of particles in a cluster N and the size or characteristic radius of the cluster R is given by 74. where D d the dimensionality of the system. It turns out that the observed growth (129-1 3 1 ) is tremendously asymmetric-indeed the clusters re semble dendritic growth. The surprising result is that the quantity D, the so called Hausdorf dimension, turns out to be in d = 2, 3, 4, ( 1 3 1) =

D = '7d

with

'7 = 5/6

75.


517

which reflects the greater surface t o volume ratio ofthe observed articulated cluster. Recently Muthukumar (1 32) has presented a theoretical argument that yields

D=

d2 + 1 d+ l

-

in remarkably close agreement to Eq. 75. The phenomenon is related to the instability mechanisms that lead to pattern formation in crystal growth Annu. Rev. Phys. Chem. 1983.34:493-524. Downloaded from www.annualreviews.org Access provided by CONRICYT EBVC and Econ Trial on 09/23/15. For personal use only.

(1 33).

It is clear that this observed behavior of asymmetric growth will require · revision of conventional views concerning coIIoidal growth (when surface rearrangement mechanisms are not operative) and, by analogy, to certain models of interfacial growth. An example is provided by Deutch & Meakin (134), who considered the consequences of these observed shapes on the predicted rate laws according to the conventional diffusion picture. For the conventional picture with D = d (Eq. 74) one predicts linear growth in the square of the particle radius 76.

for all dimensions d 2: 3. On the other hand, the relation for the asymmetric clusters D = Yfd with Yf < 1 predicts the relation with the exponent IX dependent on dimension according to 1X(d) = 2 + (Yf - l)d.

77.

78.

Above the critical dimension determined by the relation lX(dc) = 0 the growth of the cluster will be exponential R(t) '" exp [const.(t)]. Note that in three dimensions the predicted (1 35) growth rate R3/2(t) '" t varies noticeably from the square rate law presented in Eq. 76.

Gated Diffusion-Co ntrolled Reaction s It has been suggested that the binding of ligands t o proteins may deviate from the diffusion reaction rate limit due to internal motions of the protein that shield the active site from reaction (1 36). McCammon and co-workers have investigated simplified models that explore this interesting possibility (136-1 38).

The simplest model consists of imagining a single center that changes with time in either an oscillatory or random manner to permit reaction. This motion can be considered to be equivalent to a time-dependent boundary condition at the surface of the sink where c(R, t) = 0 for those periods oftime when the reactive gate is "open" and 8cj8r(R, t) = 0 for these

518

CALEF & DEUTCH

periods of time when the reactive gate is "closed." There evidently is a transient time after each switching of the gate as the concentration profile attempts to respond to the newly established (absorbing or reflecting) boundary conditions. This simple spherically symmetric model with time dependent boundary conditions has not been solved exactly. Northrup, Zarrin & McCammon (1 37) have considered the model oc at


-

=

- ( ) - ks ( ) ( , )i5( 1 a oc r2D or r2 or -

htcr t

-

r

R)

79.

where h(t) is a prescribed gate-switching function. If the gate is always open, permitting reaction, the steady state rate is found, by a method analogous to that employed by Collins & Kimball ( 12), to be ko

=

keqko/( keq + ko)

80.

where 81.

Northrup et a l next assume that the gate i s opened for a time 'ro and then closed for a time 're' They solve the model for the limiting case when the gate is closed for a time long compared to the characteristic diffusion time 'rn (R /D) required to return the concentration profile to its equilibrium value ( 'rc � 'ro). Their result for three dimensions is =

2

82.

k = ['ro('rO + !e) - l]k(!o)

where k(!o) = kot(k6Ikok/�{2(n!o) - 1/2

- h -1 - 2 (f o)

[1

exp(p !o)erfc(PFo) ] }

83.

with

fi = [(ko + ko)/ko0] and erfc(x) is the complementary error function integral. In the limit !o � 'ro, K(!o) � keq since the concentration profile differs negligibly from its equilibrium value. In the opposite limit 'ro � 'ro, the reaction iate may be approximated by ko while the gate is' opened and K(!o)' � ko. An alternative formulation to this problem of gated diffusion-controlled reactions is to consider a stochastic diffusion equation in which. the system is described by coupled-diffusion equations DV2 PO - kaP o + kbPe

=

0 84.


519

i n the steady state limit. Here Po(PJ denotes the probability distribution of solute species around the center when it is open (closed). The boundary conditions accompanying this set of stochastic differential equa�ions are p�(R) = 0 for the closed state (reflecting the solute species) and for the open state 4nR2Dp'o(R)

=

keqPo(R).


This problem has been considered by Szabo et al ( 1 38) for a sphere as well as disk and cylinder. For the sphere the authors find k

=

kOkeqkaZ(ka + kb)

ka[keq + kDZ(ka + kb)] + kbZ(ka + kb) (ko + keq)

85.

where Z(x)

=

1 + (X''Co) 1/2.

At first glance one might assume that this stochastic formulatioI) is identical to the problem of the random boundary conditions mentioned above, but this is probably not the case and certainly the equivalence has not been demonstrated. The importance of this work is its potential applicability to problems of biological interest and the extension of the theory of diffusion-controlled reactions to situations in which the intrinsic reactivity fluctuates with time.

Ionic Diffusion-Controlled Recombin ation An important application of diffusion-controlled reactions concerns the recombination or escape of a pair of ions interacting via the Coulomb potential 2 _ - Z+Z_e -1 _ - (kBT) rC 86. u(r) r r 4ne where Z + (Z _) are the valence ofthe positive (negative) ion, e is the electron charge, and e is the solvent dielectric constant. An article on this subject appeared in last year's Annual Review of Physical Chemistry (1 39). The theoretical treatment of this problem is based on the Diffusion Equation 1 (3) OP at

=

[ ( )pJ

Vu DV ' VP - k BT

87.

subject to the initial condition and the boundary condition appropriate for recombination P(r, 0)

=

b(r - r0), P(R, t)

=

0

88.

520

CALEF & DEUTCH

also p( 00, t) = O. Considerable progress has been made in recent years on this general time-dependent problem (140-142), although exact solutions for the general case at finite R are not available. For the case of steady state, Tachiya (143) has demonstrated that the probability w(ro)

=

lim

t� oo

f

v

P(r, t) dr

89.


satisfies the adj oint equation V2w(r) - (V w) (VU)/kB T 0

=

0

90.

with associated boundary conditions w(R) 0 and w( (0) = 1 ; radiative boundary conditions have also been treated (144). For the case of a Coulomb potential (Eq. 86) one recovers the classical result of Onsager (1 45) =

w(r)

=

91.

exp( - reM

in the limit R � O. A related problem concerns the recombination probability in the presence of homogeneously distributed scavengers that are assumed (without derivation) to satisfy an equation of the form V2 w(r) - [(Vw) o(Vu)/kB T] - (y/D)w

=

0

92.

where y is proportional to the scavenger concentration (146, 147). The methods required to solve this problem involve singular perturbation theory and involve complex and nonlinear dependence on the parameter (147 ; see also 148). Considerable attention has been given in the literature to the problem of ionic recombination in the presence of an electric field (145, 146, 149). More recently the formulation of Eq. 90 has been applied to a single mobile electron in the field of N fixed cations (142, 1 50), which closely resembles the model of diffusion/reaction system described in the section on Con centration Effects in Chemically Reacting Systems. Pair ionic recombination in dilute and concentrated systems, both in the presence and absence of electric fields, will continue to be a subject of active research. CONCLUDING REMARKS

In this review we have attempted to provide a snapshot of the status of our understanding of diffusion-controlled reactions. We have devoted par ticular attention to theoretical developments and selected several recent


521


applications that are noteworthy and illustrative of the ubiquitous nature of diffusion-controlled reactions. Most chemists regard the chemical encounter as primarily a matter of bond rearrangements when molecules have attained appropriate relative positions and orientations. The principal point that we would hope to leave with the reader is the following : The mass transport process that leads to these chemically active configurations is critical to the understanding of chemical kinetics and involves a variety of subtle physical effects. ACKNOWLEDGMENTS This work has been supported in part by the Defense Advanced Research Projects Agency.

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