Glenn T. Evans and Marshall Fixman
1544
Atom Recombination in Liquids on a Picosecond Timescale Glenn T. Evans and Marshall FSxman* Department of Chemistry, Yale University, New Haven, Connecticut 06520
(Received February 19, 1976)
Publication costs assisted by Yale university
Diffusion-controlled recombination in a liquid is considered using (1)the standard Fick’s law model for diffusion and (2) an improvement of Fick’s law which includes an effective spatial dependence in the relative diffusion constant. The results are viewed in light of the recent picosecond optical experiments of Eisenthal et al.
A. Introduction The problem of diffusion-controlled recombination of spherical bodies in a liquid is a time-honored one originally considered by Smoluchowskila and solved to desired accuracy by Noyeslh in the mid 1950’s. Evidence from chemically induced dynamic nuclear polarization (CIDNP)2 in the late 1960’s and, more recently, picosecond optical studies3 have refocused our attention to a more detailed solution to the problem. An improved theory describing the time dependent recombination rate of spherical bodies in a normal liquid must account for: (1)the relative translational motion of the spheres subject to the constraint that the spheres do not overlap, owing to hard core repulsion; and (2) the chemical reaction of the spheres to form a recombination product, occurring a t a separation slightly greater than the sum of the hard sphere radii. In terms of the equation of motion for the relative two body distribution function, P (r, t ) ,the independent translational motion of the spheres can be accomplished by including a function of the Laplacian of the relative separation, and the chemical reaction achieved by the inclusion of a spherically symmetric sink. Thus, the equation of motion for P(r, t ) will have the f 0 r m ~ 9 ~
been applied to chemically induced dynamic nuclear polarization (CIDNP). From the rather sparse CIDNP evidence4b it seems adequate to neglect the explicit sink term in eq 1,and simply calculate P ( r = a’, t ) ,which is the probability density of attaining the separation, a’, required for chemical reaction. Allowing a’ a amounts to solving eq 1 subject to the reflecting sphere boundary condition and evaluating the solution at this reflecting separation, r = a. Aside from the implementation of a boundary condition, this is basically the physical content of the Noyes theorylb of diffusion-controlled recombination. In the Appendix we give a stronger demonstration that neglecting K ( r ) does not affect the form of the time dependence of the solutions. The motivation for a more detailed study of recombination dynamics arises primarily from the recent work of Eisenthal et al.4 wherein picosecond optical methods are utilized in the direct detection of the time dependence of the diffusioncontrolled recombination of iodine atoms. The work presented in section B of this paper will invoke a Fick’s law representation for the flux, J, and will be applied to the interpretation of Eisenthal’s data. In section C we shall improve Fick’s law by including higher order spatial derivatives in the flux and comment on the timescale requirements necessary for the detection of the new effects.
-
dP(r, t ) -- -V J - K(r)P(r, t ) a
dt
B. Fick’s Law Recombination
where the flux, J, is
J
=
-V(f(P(r, t ) ) )
Ob)
f is yet an unspecified function which includes diffusion constants and possible refinements of a diffusional theory. The hard core of the spheres is enforced by restricting the spherically averaged normal component of J to vanish at the hard core separation, r = a. Jr,=a
dsn.
J
=0
(2)
The sink will operate only if the spheres are in contact. This process can be accommodated by the use of a delta function on a sphere of radius, a’, in which a’ is infinitesimally greater than a. Thus, K ( r )will have the form4s5 K ( r ) = K6(r - ~ ’ ) / 4 7 r a ’ ~
(IC)
The strength of this pseudopotential, K , has been discussed elsewhere.4a The solution of an equation similar to eq 1is known4cfor the simple case of Fick’s law translational diffusion and has The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
1. Basic Theory. The equation of motion for the two body distribution function, Po(r, t ) , describing the relative translational diffusion of two spheres is
(3)
So(r, t ) is a point source So(r, t ) = 6(r - ro)6(t)/4kro2 and D1 the pair diffusion constant. The source in eq 3 was chosen in order that the solution, Po(r,ro, t ) ,has the property of a Green’s function. Equation 3 will be solved subject to the initial condition, PO(r, ro, t = 0) = 0; the reflecting sphere boundary condition using J = -DIVPo in eq 2 ; and the requirement that P(r, ro, t ) vanishes a t r = a ~ . Equation 3 can be solved by Laplace transformation with respect to time, thereby generating a linear second-order differential equation with constant coefficients which is easily handled. After inverse Laplace transformation, we find the time-dependent solution evaluated at the collisional separation, r = a, to be
Atom Recombination in Liquids
1545
PO(r = a, ro, t ) = (4ar0D1~,)-~([~,/rrt]~/~ X exp(-(ro - ~ ) ~ / 4 D l-t )exp((r0 - a ) / a t/Ta) erfc ( [ t / ~ , ] l / ~ (ro - a)[4D1t]-1/2)) (4)
+
+
with T , = a2/D1. A term, 4 ~ a D 1 may , be factored from the denominator of eq 4 ensuring that the time integral of Po(a, ro, t) evaluated at infinite time equals aha, as discussed in the Appendix. 2. Application to the Iodine Recombination Experiment. For the case considered by Eisenthal et the source function, S(r, t ) has contributions, S1 and Sa, from: (1) direct photolysis of 1 2 which produces an isotropic distribution of I atoms with relative initial separation rl after the light pulse att = O S l ( r , t ) = 6(r - r1)6(t) S1(0)/4ar1~
(5)
(2) indirect I atom formation a t the relative separation r2 from the collisional deactivation of a molecular iodine excited state, called Iz*.
Sz(r, t ) = 6 ( r - r2) exp(-hRt) S2(0)/4Tr&R
(5a)
Sl(0) and Sz(0) are the initial concentrations of the 12 excited states which dissociate immediately and after a time delay, respectively, kR is the rate constant for the first-order decay of Iz*.The total source, S ( r , t ) , is the sum of the formation rates Sl(r, t ) and &(r, t). Denote the solution to eq 3, using the source function in eq 5, by Ps(r, t ) . In terms of the Green's function, Po(r= a, ro, t), Ps(r = a, t ) is given by
Ps(r = a, t ) =
J td t l 1dr S ( r , t l ) Po(r= a, r, t - tl)
is) The total concentration of I 2 resulting from I atom recombination will be denoted by C,(t) and is related to the time integral of eq 6, evaluated a t a collisional separation, r = a C,(t) = X L ' d t ' Ps(r = a, t')
(7)
The parameter, X, represents the fraction of I atoms at r = a which react to form molecular 12. The origin of this parameter X is clarified in the Appendix. The concentration of the excited state of 12, also detected by the probe pulses, is simply C d t ) = exp(-h,t) SA01
(8)
The transmittance of the light beam, T(t), follows from Beer's law T ( t ) = exp(-QVl(t)
+ Cz(t)J)
(9)
where Q contains the path length of the cell and the extinction coefficient. From eq 6-9 the transmittance can be calculated as a function of time, S1(0),S2(0),k,, Q, r1, r2, A, and 7,. The parameter, T,, can be estimated using the Stokes-Einstein relation for the iodine atom diffusion constant, Do, and the iodine atom radius, r1 r, = 2r1~IDo= l Z i r r ~ ~ ( ~ / k T )
Using rI = 1.35 A and r ) = 0.89 cP, we calculate 7, = 20 ps. With Green's function, eq 4, and the expression for the source, eq 5 , the integrals in eq 7 can be performed c i ( t ) = X(Si(0)+ Sz(0)lh(x, E )
- XSz(0)
Lx
dxl d x l , 4 exp(-Kc(x - X I ) )
(10)
with (1
+
E)
I: I
h(x, t) = erfc - e / &
1 - exp(x + t) erfc 6+ -e/*]
I
+
(1
E)
g(x, e) = (
2
(Ila)
U G ) exp(-t2/4x) 1 2
- exp(x + t ) erfc & + - e / & ] I
(llb)
and x = t/Ta; K = hrTa; E = (r1 - a ) / a = (r2 - a)/a. As a simplification, the initial separation of the iodine atom pairs, used in eq 5, have been chosen to be equal, r1 = r2. The effect of the CZ(t) part of T(t) is to increase T for times of the order of kr-l. For times longer than h,-l, the Cz contribution can be simply ignored. Equation 10 for C,(t) has contributions from a K dependent and a K independent part. For K >> 1,we can integrate eq 10 by parts generating an asymptotic expansion in K - ~ ,
Numerical computation indicates that for K > 4 and for times h,t > 1, the K dependent corrections can be neglected. This arises because of the inverse K dependence and, more importantly, the function g decreases much faster than h. For the case, K = 2 (hr = 1011 s-1, T~ = 20 ps), a 3% error in underestimating T is obtained at 40 ps by neglecting the K dependent part. The error rapidly diminishes with increasing time. Therefore, on the basis of these remarks, we can treat I atom recombination for times longer than 40 ps as if the source were a delta function in time and we are assured that the curvature in T i s not due to the reaction kinetics. For times long compared to h,-l, we have the result T ( t ) = exp(-Q,h(t/.r,,
t))
(13)
T is a function of three variables: T , was calculated above; @ is a collection of constants Q, = Q(Sl(0) + Sz(0))X; and t, a dimensionless number which measures the deviation of the initial relative separation of the radicals from the case of tangent radicals. In comparing the diffusion theory to Eisenthal's experimental results, we shall fit the 800-ps point in Figure 1,using @ as the adjustable parameter, fixing T , at 20 ps. In Figure 1 we represent three cases: initially tangent radicals, ro = 2r- or t = 0; radicals having the separation, ro = 4rr or E = 1;and finally a 50-50 mixture of the two previous cases. All three cases required a different value of to fit the long time asymptote. Clearly, the 50-50 mixture of initial conditions better accommodates the data. A closer fit could be achieved by further experimenting with the initial conditions, however, we think the plotted results serve to underline the physics involved in the iodine atom pair dynamics. We therefore forego detailed curve fitting owing to the equivocal nature of the distance distribution which would result from such a procedure. Alternatively, one could choose T , = 80 ps and thereby fit the data assuming initially tangent radicals. It is apparent that there are enough parameters which may be varied within reasonable limits and accommodate the experimental results. One is forced to conclude that the data from this experiment does not mitigate against a simple diffusional description for iodine recombination. The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
Glenn T. Evans and Marshall Fixman
1546 I
I
I
I
I
I
I
I
where Do is the one particle, Stokes-Einstein diffusion coefficient, and D’,’the super-Burnett c~efficient,~ is related to the fourth moment of the position variable. To fix the value of D’, we shall consider the k-space single particle equationlo generalized to include a wave vector dependent diffusion constant, D(k).
-at
I
0
I
I
I
I
100
200
300
I
400 500 TIME (Psec)
I
600
1
700
800
Figure 1. Probe transmission, Il//o for iodine in CC14. In all cases T~ = 20 ps: (A) d, = 1.81 and E = 0; (6)a = 2.50 and a 50-50 mixture of t = Oand E = 1; (C) d, = 3.96andt = 1.
1. General Considerations. In picosecond experiments, it is possible that the direct temporal dependence of D is significant. Likewise, due to the shortness of the timescale, the distances probed by the translating radicals are the order of a few molecular diameters, and on this distance scale, the fluid will not appear continuous, as required by the simple diffusional models. It is appropriate to ask under what conditions will the spatial and temporal characteristics of D modify the simple model of Fick’s law Brownian motion. The criteria for the frequency breakdown of the diffusion equation for single particle motion is that the timescale be comparable to p-l ( = particle masdfriction constant). For times of the order of p-’, the one particle distribution function or singlet density is coupled to the particle velocity and to higher moments of the velocity.’ For I atoms in room temperature CC14, p-’ is very short s) and, consequently, the momentum degrees of freedom relax essentially instantaneously on a timescale of a few picoseconds. The equation of motion for the momentum relaxed singlet density is just the ordinary diffusion equation. To sample the frequency dependent behavior of the diffusion process, the solvent viscosity would have to be increased by at least two orders of magnitude in order to shift P-l into the tens of picosecond regime. The situation is not nearly as clear regarding the spatial dependence of D . As a simple model, we will allow the I atoms to execute independent trajectories being correlated only by the initial conditions in the equation of motion for the two body distribution function. Because of the assumed independence of the particle trajectories, the two body distribution function can be derived from the equations of motion of the one particle function. If the singlet density, u(r1, t ) ,is Markovian, it will obey a Chapman-Kolmogorov-Smoluchowski equations
+ tl) = 1 d r ’ u(r1 - r’, t ) u(r’, t l )
~
+D o ’o ~ ~ (~c * ~ )
The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
D ( k ) = Do/(l
+ Dok2To)
(164
D ( h ) = D o ( l -Dok2T0..
.I
(16b)
Inserting eq 16b into eq 15 and comparing this result with the k space analogue of eq 14 indicates that D’ = 0 0 ~ ~Therefore, 0 . for the case of independent particle diffusion, the effect of a spatially dependent diffusion constant is to recast eq 1 into the form
aP/at = D1V’P
+ D2V2(V2P)
(17)
where D1 = 2D0, and Dz = 2Do2T0.In the following section we shall solve eq 17 exactly and illustrate the effect of the correction term, D2, on the time dependence of the relative translational motion. 2. Solution of t h e Improved Diffusion Equation. The equation of motion for the two particle density can be rewritten as
dPldt = -C J
(18)
where the flux, J, is
J
-{DlOP
+ D2V(V2P)J
(19)
The reflecting sphere boundary condition, eq 2, restricts the flux on the spherical surface, r = a. The initial condition is a spherically symmetric poiut source a t ro.
P ( r , t = 0 ) = 6(r - ro)/4rro2
a 5 ro
(20)
Since eq 17 is a fourth order differential equation, two additional boundary conditions are required. We shall use the constraints for n = 0 , l J rznP(r, t ) d r < a (21) These conditions restrict P ( r , t ) to be volume normalizable and to have a finite second moment. After performing the Laplace time transform of eq 17, defining P(r, s ) by
(14a)
If a(r, t ) evolves slowly in time and space, we can expand the left-hand side of eq 14 in a powerpries in time and the right-hand side in a power series in the spatial variable. Retaining the first-order time derivative and the second and fourth spatial derivatives (first and third vanish by symmetry) yields the equation
w a t =D
D(k) can be measured directly from the incoherent, inelastic scattering of thermal neutrons. From the neutron scattering experiments it is found that for large k, 0(1A-’), D(k) is ‘ explicitly wave vector dependent. A model for D ( k ) proposed by Egelstaffll which has had considerable success in accommodating neutron scattering line widths is the so-called jump mode1,ll wherein
In this version of a jump model, the time constant, TO, is interpreted as the time spent vibrating prior to a translation jump. For small k, D ( k ) can be expanded
C. Refinements in t h e Theory of Recombination Dynamics
u(r1, t
t , - -k2D(h)o(k, t )
(14b)
P ( r , s)
= Jo
d t exp(-st)P(r, t )
(22)
the resulting fourth-order linear differential equation with constant coefficients can be solved by standard means (taking the usual precautions regarding’the continuity and appropriate discontinuity of P ( r , s) and its derivative across the position of the delta function source).In terms of the Laplace transform variable, the solution for P ( r , s ) evaluated at the
Atom Recombination in Liquids
1547
collisional separation is P ( r = a, s) = exp(-A(ro - a ) ) /{4Tro&(1
+ a N ( 1 + D2A2/D1))
(23)
In this intermediate regime, P(a, t) increases relative to Po(a, t ) ,crosses Po(a,t ) and, for longer times, bounds PO(a, t ) from above. Finally, for very long times, t >> T,, both P(a, t ) and Po(a,t ) decay as
with
+
- 1)1/2 (24)
A = (D1/2D2)1/2{[1
One would expect that the tail would be unaffected by the extra diffusion constant. Likewise, the role of the spatial dependence in D is to retard motion for small separations, whereupon, P(a, t ) will fall off more slowly than Po(a,t ) . In order to maintain time normalization, the P(a, t ) must lie below Po(a,t ) for very short times. It is notable that the be( T , + 70/2)P(a,t ) = (U2) exp(-t/2~0)g(t/2~0,6 = 0) havior in the intermediate range, TO > T O , the second g(x, e ) is given by eq l l b and WI(Y)has been tabulated for moment decays as t-5/2and therefore does not affect the t-3/2 small arguments.13 Although the integral in eq 25 could not tail of the zeroth moment. be performed analytically for arbitrary T O and T , i t could be The moment calculation indicates that the analytic results approximated in the limit of TO/T,
