3412
J . Phys. Chem. 1994,98, 3412-3416
Cage Effect on the Diffusion-Controlled Recombination Kinetics? Ilya Rips Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel Received: October 21, 1993; In Final Form: December 29, 1993”
The effect of the spherically symmetric caging field on the asymptotic time behavior of the geminate recombination (GR) kinetics is explored. Time-dependent solution of the Smoluchowski equation with the Coulomb potential and the spherically symmetric linear caging potential is derived. The asymptotic kinetics is characterized by the nonuniversal exponent, which increases linearly with the field for weak fields. The deviation from the universal diffusion-controlled kinetics k ( t ) r-3/2 is due to the “finite volume” effects. The model can mimic the many-body effects on GR kinetics. Its possible use for the interpretation of the recent experimental data on the p H effect on the long-time GR kinetics is discussed.
-
1. Introduction
The kinetics of bimolecular reactions is largely determined by the reactants’ motion. In condensed phase the mean-free path is short, so that the motion of the reacting particles is diffusive. This results in the diffusion-controlled kinetics.’ The geminate recombination (GR) reactions, in which the reactants are produced from the same parent molecule by ionization or dissociation, are of particular interest.2.3 This interest is associated both with the experimental ubiquity of G R reactions in radiation chemistry2 and photochemistry3 and with the fact that they allow simple theoretical description in terms of the pair distribution function. The role of the many-body effects has become recently one of the most popular topics in the study of diffusion-controlled reactions.” A number of theoretical studies and numerical simulations dealing with these effects have been published. The simplest way to account for the many-body effects on the reaction kinetics is via the introduction of the effective (self-consistent) potential in the Smoluchowski equation. The latter describes the temporal evolution of the distribution function within the single pair approximation, provided that the mean-free path of the reacting particles is short enough.8 The solutions of the Smoluchowski equation for the simple “nonpathological” potentials are well-known.9 For the potentials decreasing with the distance (nondivergent at infinity), the reaction rate is characterized by the well-known universal power law decay: k(t) 0: t d / 2 where d is the Euclidean dimension for d L 2 and the spectral dimension for d < 2. If. on the other hand. the Darticle motion is restricted to a finite volume (perfect cage), the reaction kinetics is exponential: k ( t ) 0: exp(-yt). In the particular case of the charged particle recombination with the parent ion, the potential in which the motion occurs is the superposition of the Coulomb potential and the effective potential, which mimics the effect of the surrounding particles on the selected one, Due to the internal symmetry ofthe problem, this effective potential is also spherically symmetric. The effect of the separating electric field on the diffusioncontrolled G R kinetics of charged particles has been studied first by 0nsager.Io H e has found the steady-state solution of the corresponding Smoluchowski equation and obtained an expression for the yield of G R as a function of the field. The model has been employed as a theoretical framework for the description of the experiments on photogeneration of free charge carriers in semiconductors and insulators. Hong and Noolandill obtained an exact solution of the problem of 3D continuum diffusion in I
t
.
It is my pleasure to dedicate this paper to Joshua Jortner’s 60th anniversary. Abstract published in Aduance ACS Abstracts, March 15, 1994.
0022-3654/94/2098-3412$04.50/0
the Coulomb potential and of the time-dependent version of the Onsager problem.12 They have studied the effect of the electric field on the kinetics of the G R reaction. The most interesting physical effect of the electrical field predicted by the theory is the critical dependence of the asymptotic long-time kinetics on the field strength. The theory predicts the power law decay of the asymptotic kinetics: k(t) 0: ?43/2-fi) for the fields lower than the critical one. On the other hand, for the fields exceeding the critical one, the asymptotic kinetics is exponential: k ( t ) a exp(-At). The aim of this paper is to explore the effect of the spherically symmetric constant caging field on the long-time geminate recombination kinetics. It is shown that the caging field destroys the universality of the kinetics. The actual dependence of the exponent in the power law on the caging field is determined. The paper is organized as follows. In section 2 the model is formulated. The solutions of the Smoluchowski equation with the appropriate boundary conditions are derived, and their asymptotic long-time behavior is analyzed. These solutions are employed in section 3 to derive explicit results for the asymptotic time-dependent rate and the survival probability. The kinetics is shown to be nonuniversal, and the dependence of the exponent on the magnitude of the caging field is determined. Finally, section 4 contains the discussion of the physical implications of the results and their relevance for the description of the manybody effects in geminate recombination kinetics. 2. Formalism 2.1. The Model. The description of the geminate recombination process below will be based on the single-particle approximation. Within the approximation the motion of the charged particle (e.g. proton or electron) can be described in terms of the continual diffusion in a spherically symmetric potential’ The potentia’ is given by the superposition Of the Coulomb potential of the parent ion and the linear attractive potential (uniform field), which accounts for the many-body effects. For simplicity, the parent ion is assumed to be fixed in the center. Furthermore, at the initial time moment the particle distribution is assumed spherically symmetric. The temporal evolution of the probability density ~ ( x , T that ) the particle is at thedistancex from thecenter at time i is described in terms Of the Smoluchowski equation:
The dimensionless variables for the distance 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3413
Diffusion-Controlled Recombination Kinetics x
2.2. Formal Solution. Now let zl(z)(x;s) be the linearly independent solutions of the corresponding homogeneous equation
2r/rc
and for the time 7 14
Dtlrf
will be employed throughout the paper. Here r, e2/ekT is the Onsager distance,”Jt is the static dielectric constant of the medium, and D is the diffusion constant. The total potential in which the particle moves V V(x) can be recast in the form
with the following asymptotic behavior:
and
vi-=-v(x) kT
x
+ 2 fx
The first term represents the Coulomb field of the parent ion, while the second one accounts for the caging potential cf> 0) or the separating cf< 0) potential. Below we shall focus on the case of the caging potential. Its main physical effect consists in preventing the particle from escaping to infinity. The dimensionless field, f, is defined as
Then the Green’s function can be recast in the form11
where W(z1,zz) is the Wronskian:
eFrc 4kT
fl-
Fqz,;z,) = z’l(x;s) z,(x;s) - z~(x;s)zl(x;s)
To complete the definition of the model, one has to specify the initial condition and the boundary condition. At the initial time moment the particle is assumed to be uniformly spread over the sphere of radius XO:
1 p(x;O) = ,6(x 4UXo
- xo)
(3)
and x< 5 min(x;xo); x>
z,(x;s)
Z,(x;s)
+ C,z,(x;s)
(12)
and the coefficient Cz is determined from the imposed boundary condition11.12 z’l(a;s)
Finally, the boundary condition expresses the flux,jr(a;7), through the reaction sphere of radius a in terms of the reaction rate, K: jr(U;7) = -Kp(a;7)
max(x;xo). Furthermore
+ (7-Z-K 1 1
+f)zl(W)
c, = -
(13)
(4)
- -
The solutions of the homogeneous equation with the field, eq 10, with the appropriate limiting behavior for x 0 and x can be expressed in terms of the solutions without the field cf= 0). This is done in the following way. Let Go(x,x’;s) be the Green’s function
Recasting the solution in the form
d2Go(x,x’;s)
dx2
the function h(x;r) can be shown to satisfy the equation
+ -x1 dG,(x,x’;s) - [’ + + s] Go(x,x’;s) = dx x4 xz 1 --~(x-x? X’
(14)
Then the solutions of eq 10 are also solutions of the integral equations:
with
g = 114 + 2f
(7)
Laplace transformation with respect to the time variable
zi(x;s) = y,(x;s)
+ 2f
Jam
dx’G,(x,x’;s)z,(x’;s)
( i = 1,2)
(15)
$(x;s) = JOm d 7 exp(-sr) h(x;7)
Here y ~ ( ~ ) ( x ; are s ) the linearly independent solutions of the homogeneous equation
together with the initial condition, eq 3, reduces the problem to finding the Green’s function (fundamental solution) of the secondorder differential equation
dzk + - -1-dk [$+$-y+s] dx2
xdx
with the boundary condition
k = - G 61( x - x o )
(8)
This equation can be reduced to the Mathieu equation. Its solutions with the appropriate asymptotic behavior can be found in ref 11. The Green’s function Go(x,x’;s) can be recast in the form
3414 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
with Y(s)= xwly1;y2)and O(x - x’) being the Heaviside function. The Green’s function is continuous for x’+ x with Go(x;x;s) = 0. Its first derivative has a discontinuity
a
-G ax
a + 0,~ 0 ; s )- -Go(~o
x
and y2(x;s) in this limit, eqs 19 and 20, we obtain z,(x;s) = s-’/2[u1(x)
+ s’u,(x)] +
1 - 0 ~ 0 ;=~-)-
ax
O( 0
Rips
XO
Substitution of the Green’s function into eq 15 results in zi(x;s) = yi(x;s) + $,(x;s)
~axdx’y,(x’;s)zi(x’;s) -
y,(x;s) ~oxdx’y2(x’;s)zi(x’;s)j
( i = 1,2) (18)
Thus eqs 1 1 and 12together with eq 15 allow the Green’s function for the Smoluchowski equation with the caging field to be expressed in terms of the known solutions in the absence of the field. 2.3. Asymptotic Behavior. For the analysis of the long-time asymptotic kinetics, we require the solutions zl(x;s) and z ~ ( x ; s ) in thelimitsl/zx-O. Thesolutionsofthe homogeneousequation, y1(2)(x;s),have the following form in this limit (see ref 12): y,(x;s), = K,(l/X)
E
uo(x)
In the limit s
-
0 we can look for the solution in the form
+
+
z,(x;s) = s-’/’[v1(x) sYu2(x) 0(s2”)]
(28)
Formal substitution of this Ansatz into eq 27 leads to the integral equations for the determination of the functions U ~ ( ~ ) ( X ) :
(19) and
and y,(x;s)
s-’/2[ul(x) + syuz(x)+ O(S In s)]
N
(20)
In these expressions u ~ ( x= ) A(v)zu(i/x) = r ( i
+ v)r(v)22Y-1zu(i/~)
u,(x) = A(--u)Lu(l/x)
(21)
with I&) and &(x) being the modified Bessel functions and r(x) the Euler gamma-function. The characteristic exponent, u, is given in the limit s 0 by
-
v
= g1/2= (1/4 + 2f)’/’
(22)
It should be pointed out that in the small-s expansion of yz(x;s), eq 20, terms of the order -8 In s have been omitted. Thus it is valid only if the field is sufficiently weak, namely, f < 3/8. For the evaluation of the rate expression for the particular case of the absorbing boundary condition, we shall require the solution zz(x;s), which decays exponentially for x -.
-
z,(x;s) = y,(x;s)
For small values of the field these integral equations can be solved perturbatively. Within the first-order perturbation theory in the field we obtain
and
+ ~Y(s> t v , ( x ; s ) ~ a x d x ’ y l ( x ’ ; s ) z t ( x ’-; s ) y , (x;s) s,”dx’yz(x?s)z2(x’;s)j (23)
The Wronskian in the limit s1I2 x
