J . Phys. Chem. 1987,91, 4417-4422
4417
Stochastic Models of Diffusion-Controlled Ionic Reactions in Radiation-Induced Spurs. 1. High-Permittivity Solvents Peter Clifford, Mathematical Institute, St. Giles. Oxford OX1 3LB, UK
Nicholas J. B. Green,* Radiation Laboratory, University of Notre Dame, Indiana 46556
Michael J. Pilling, and Simon M. Pimblott Physical Chemistry Laboratory, Oxford OX1 3QZ. UK (Received: March 2, 1987)
Alternative theories for the kinetics of diffusion-controlledreactions between ions in radiation-induced spurs are considered for solvents of high permittivity, such as water. A Monte-Carlo (MC) technique for simulating the paths of the diffusing ions and their encounters is developed and shown to provide an accurate description of the time-dependent reaction probability for the case of a single pair, by comparison with numerical solutions of the Debye-Smoluchowski equation. The MC simulations are then used to test the application of more approximate theories for the kinetics of multipair spurs. Two models are examined: (i) prescribed diffusion, which is conventionally applied in radiation chemistry, and (ii) the independent reaction times (IRT) model in which reaction times are associated independently with each ion pair. The IRT model is shown to describe the kinetics better than prescribed diffusion for a two-pair Gaussian, showing time-dependent reaction probabilities that are close to those obtained from the MC simulations. Good agreement is also found for the regular tetrahedron, where the effects of competition are maximized. The performance of both approximate models improves as the number of pairs in an initially Gaussian spur is increased and prescribed diffusion copes better with ionic reactions than it does with reactions between neutral species. Finally, a comparison between ionic and neutral spurs demonstrates that it is necessary to include charge in any model of spur kinetics involving ions, even in a high-permittivity solvent such as water.
I. Introduction In a series of recent papersI4 we have developed a theory of diffusion-controlled reactions in systems where the initial distributions of the reactants are clustered. The aim has been to provide realistic models of the kinetics in systems where the clusters are effectively isolated, such as the tracks formed by the deposition of energy in liquids by high-energy radiation. The small clusters of reactants in radiation tracks are called spurs in radiation chemistry. We have been particularly interested in clusters that contain only a few particles (typically 2-6), since the usual deterministic theories of kinetics can no longer be applied in this regime. Thus far we have investigated (i) clusters containing a single type of radical, where the particles react pairwise;’q2(ii) clusters containing a single type of radical and a scavenger, which is initially homogeneously di~tributed,~ and (iii) clusters containing several types of radical with a range of general reaction schemes describing the painvise reaction between radicals and their reaction with product molecule^.^ The purpose of this paper is to report the generalization of these models to include reactions between ions. Since the main application of the models is in radiation chemistry, and water is of central importance in this field, the paper will deal exclusively with clusters of ions in solvents of high relative permittivity, where the Coulomb forces between ions are weak. This restriction allows us to use certain closed approximations for the geminate recombination probability as a function of time, which we have reported r e c e n t l ~ . We ~ will describe elsewhere6 an alternative numerical approach, which may be used for liquids of lower relative permittivity, where these closed approximations are not applicable. Throughout this paper we assume that the reactions in the clusters are truly diffusion-controlled; partially diffusion-controlled reactions will be the subject of a separate investigation. We start the paper with a short section on the isolated ion pair, which has received wide theoretical attention. We consider the theoretical framework which is the basis for the description of geminate recombination and we review methods of solution and
* Author
to whom correspondence should be addressed.
0022-3654/87/2091-4417$01.50/0
approximation within this framework. Based on these approximations we demonstrate that it is necessary to consider ionic forces, even in solvents of high relative permittivity, such as water. Section I11 describes a Monte Carlo technique for simulating the paths of the diffusing ions. As in our previous investigations the Monte Carlo simulations must be realized for many clusters in order to obtain statistical significance, but, once these realizations have been completed, the results are used as our measure of “reality”, against which we compare the other more approximate theories. The Monte Carlo technique is validated for a single pair by comparison with numerical solutions of the backward Debye-Smoluchowski equation for the geminate pair recombination probability as a function of time. In section IV we compare the results of the Monte Carlo simulations with the results of two approximate theories: (a) the independent pairs approximation, which is the model we have developed and validated for neutral particle^;^-^ (b) prescribed diffusion, which is the model conventionally employed in radiation c h e m i ~ t r y . ~ -We ~ also show a comparison between the independent pairs approximation and the Monte Carlo simulation for an initial configuration where four ions are placed at the vertices of a regular tetrahedron, the situation where interdistance competition is expected to be maximized. Finally we compare the results with the corresponding free-radical clusters to show the extent of the effects of incorporating the Coulomb interaction. (1) Clifford, P.; Green, N. J. B.; Pilling, M. J. J . Phys. Chem. 1982, 86, 1318.
(2) Clifford, P.; Green, N. J. B.; Pilling, M. J. J . Phys. Chem. 1982, 86, 1322. (3) Clifford, P.; Green, N. J. B.; Pilling, M. J. J . Phys. Chem. 1985, 89, 925. (4) Clifford, P.; Green, N. J. B.; Oldfield, M. J.; Pilling, M. J.; Pimblott, S. M. J . Chem. SOC.,Faraday Trans. I 1986, 82, 2613. ( 5 ) Clifford, P.; Green, N. J. B.; Pilling, M. J. J . Phys. Chem. 1984, 88, 4111.
(6) Clifford, P.; Green, N. J. B.; Pilling, M. J.; Pimblott, S. M., manuscript in preparation. (7) Mozumder, A. J . Chem. Phys. 1971, 55, 3020. (8) Mozumder, A. J . Chem. Phys. 1971, 55, 3026. (9) Schwarz, H. A. J . Phys. Chem. 1969, 73, 1928.
0 1987 American Chemical Society
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The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
11. Geminate Ion Recombination
The diffusion equation is generally believed to provide a good description of the motion of reactive particles on time scales longer than the velocity correlation time in the liquid. Reactions in radiation-induced clusters occur on time scales in the range 10 ps to I O ns, so that models may be legitimately based on the diffusion equation. The solvent is treated as a continuum whose only effects on the reaction process are to supply the random force which drives the Brownian motion and to mediate any long-range forces between the ions. The diffusion of a particle in a force field is described by the Debye-Smoluchowski equationlo
d4 at = V D { V q + & V
{
a
(2r - r h ar r2
TABLE I: Ratio of the Value of Eq 9 to That of Eq 8 for Various Values of r and P rlnm 1% I/PS 1.0 2.0 3.0 4.0 5.0 1 0.7391 0.6227 0.5838 0.5627 0.5484 2 0.7475 0.6367 0.6022 0.5854 0.5754 3 0.7482 0.6377 0.6033 0.5865 0.5766 4 0.7483 0.6378 0.6034 0.5866 0.5767 5 0.7483 0.6378 0.6034 0.5866 0.5767 ~
~~~
“ R = 0.5 nm, rc = -0.7 nm, D’= 1.0 X IO-* mz s-’ TABLE 11: Ratio of the Value of Eq 10 to That of Eq 8 for Various Values of r and P
d
where q denotes the spatial probability density of the particle, D denotes its diffusion coefficient (tensor), and Udenotes its potential energy. The same equation is also used to describe the relative diffusion of an isolated pair in relative coordinates if D is replaced by D’, the relative diffusion coefficient, and U represents the potential energy of the pair. The reactions in which we are interested occur in solution, where the diffusion is isotropic and the Coulomb interaction is spherically symmetrical. Given these simplifications, the diffusion coefficient becomes a constant and the recombination process becomes independent of angle in the absence of external fields so that the problem is reduced to a one-dimensional diffusion. The marginal radial density of the motion, irrespective of angle, may be obtained by integration of q over all directions. The final simplification is to assume that the bulk dielectric constant of the solvent applies at the short distances involved. The forward equation for the radial density function p becomes - - D ‘ - - a2p at ar2
Clifford et al.
)
(2)
where rc, the Onsager distance, is the separation at which the Coulomb interaction of the pair equals kT.” In addition to the forward equation (2), the radial density also obeys the adjoint or backward equation, which describes how p depends on the initial separation r0.’2-14The time-dependent recombination probability, W(ro,t), also obeys the same backward equation (3).15
1% tlps 1 2 3 4 5
rlnm
1.0
2.0
3.0
4.0
5.0
1.5656 1.4060 1.3920 1.3907 1.3905
1.3689 1.1981 1.1864 1.1853 1.1852
1.3362 1.1335 1.1224 1.1213 1.1212
1.3443 1.1022 1.0912 1.0902 1.0901
1.3702 1.0839 1.0727 1.0717 1.0716
‘R = 0.5 nm, rc = -0.7 nm, D’= 1.0 X IO-* m2 SC’. however, is straightforward to find. The natural distance scale of the radial process’* is given by reff
= rc/ [ e x ~ ( r c / r )- 11
(6)
and the ultimate recombination probability is given, in terms of this effective distance scale, by (7) where Reff(= rc/[exp(rc/R) - I ] ) is the effective reaction distance. Although no closed form is available for W(r,t), we have recently demonstrated that there is an excellent approximate solution which holds for liquids such as water, where r, is smalL5 This approximation may be expressed in terms of the effective scale by the equation
This equation may be compared with the expression for the recombination probability for neutrals
(3)
In order to solve the backward equation for the recombination probability it is necessary to apply boundary conditions. For an isolated pair in an infinite state space (R,m) the outer boundary condition is naturalI6 and an absorbing boundary condition is applied at the inner, reactive boundary, R: W(R,t) = 1
(t
> 0)
(4)
The initial condition which must also be applied is clearly W(ro,O) = 0
(5)
The solution of the time-dependent problem, as formulated above, is not possible in closed form, although a series solution has been obtained by Hong and Noolandil’ in Laplace transform space. The infinite time limit of the recombination probability, (10) Debye, P. Trans. Electrochem. SOC.1942, 82, 265. (1 1) Onsager, L. Phys. Reu. 1938, 54, 554. (12) Kolmogorov, A. N. Mafh. Ann. 1931, 104, 415. (13) Clifford, P.; Green, N. J . B.; Pilling, M. J. Chem. Phys. Lerf. 1982, 91, 101. (14) Sano, H.: Tachiya, M. J . Chem. Phys. 1981, 75, 2870. (15) Goel, N. S.; Richter-Dyn, N. Stochastic Models in Biology; Academic: New York, 1974. (16) Feller, W. Ann. Math. 1952, 55, 468. (17) Hong, K. M.: Noolandi, J . J . Chem. Phys. 1978, 68, 5163.
Indeed one of the techniques employed in the derivation of eq 8 was that of analogy with the neutral case, with suitable adjustments of the distance scale and the reaction d i s t a n ~ e .We ~ are now in a position to judge whether charge effects are likely to be important for geminate recombination in water. It is clear that for large r, reffapproaches r asymptotically so that if r and R are sufficiently large then the approximation of eq 8 approaches the solution for neutrals, eq 9. If we were to ignore the effects of charge completely, then, for a given reaction distance, we would apply eq 9 instead of eq 8. The effects of this approximation are illustrated in Table I. In this table the ratio of the values of eq 9 to eq 8 is shown for several values of time and initial separation. The reaction distance was taken to be 0.5 nm and the Onsager distance to be -0.7 nm (the sign denoting attraction), and the relative diffusion coefficient, D’, was 1.0 X IO-* m2 s-l. The agreement is seen to be very poor and to deteriorate as r increases. This is because reffapproaches r but Reffis far from R. It is clearly necessary to take some account of charge effects. It is frequently the case, however, that models of ion recombination in clusters make use of a formula for the diffusion of neutrals, but use an experimental rate constant, which already takes account of Reff. (18) Karlin, S.; Taylor, H. M. A Second Course in Stochastic Processes; Academic: London, 1981.
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987 4419
High-Permittivity Solvents This approximation corresponds to the use of the formula
W ( r , t ) i=
Reff r erfc
I Reff t
(4D’t)Ii2 -
In Table I1 the ratio of the values of eq 10 and eq 8 is shown for the same parameters as in Table I. It is now seen that if the reaction distance is adjusted to be compatible with the rate constant for homogeneous reaction, 4*ReffD’, then the use of formula 10 becomes asymptotically correct at large initial separations. However, the error is still of the order of 7%, even for the relatively large initial separation of 5.0 nm. Substantially larger errors are incurred if eq 10 is used to describe reaction from the small initial separations typically found in radiation-induced spurs. We believe that this order of error is unacceptable and that more explicit note of charge effects should be taken in the description of the radiation chemistry of water than is currently the case.9
-1.0 -0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
L o g (t/PS) 0.50F
1
I
I
I
I
I
I
1.5
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2.5
3.0
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111. Monte Carlo Simulation
The Monte Carlo simulation of free-radical ~ y s t e m s lis. ~relatively straightforward because the Green’s function for the diffusion of a radical is Gaussian; in consequence, the trajectories can be simulated by sampling normally distributed random variables for the increments in each coordinate and arbitrarily large time steps can be accommodated. When Coulomb forces are included, however, the situation becomes more problematical for two reasons: (i) distinct particles no longer diffuse independently because they exert equal and opposite forces on each other; (ii) even for a two-particle system the Green’s function for the diffusion has no known closed form” and so cannot be used as a basis for simulation. When more than two particles are present there is no known Green’s function. We can still simulate the diffusion process, however, by making use of the theory of stochastic differential equations. The spatial density of a diffusing particle spreads according to eq 1, where U is the total potential energy of the particle in the field provided by the other particles. Equation 1 describes a diffusion process in three dimensions whose infinitesimal mean (drift vector) is DVUJkT and whose infinitesimal variance-covariance matrix is (20) 1.” This means that the evolution of the trajectory of the particle over an infinitesimal period of time is prescribed by the stochastic differential e q ~ a t i o n ’ ~ ~ ~ ~ d r = - D - V U / k T dt
+ (20)1/2dW,
(1 1)
where dW3 represents the increment of a three-dimensional Wiener process and can be interpreted in either the Ito or the Stratonovich sense.1a,20The stochastic differential equation is only exact in the limit of infinitesimal dt, but it may be approximately solved in a stochastic sense (Le., by simulating sample paths) by using a time-discretized method. Methods for the time discretization of stochastic differential equations are basically generalizations of numerical methods for solving ordinary differential equations. Discussions of these methods and of the orders of error incurred will be found in ref 21-23. We choose the simplest numerical method, the Euler method, which corresponds to the Ito interpretation of the stochastic differential equation.22 Equation 1 is discretized to give
6r i= -D-VU/ kT 6t
+ ( 2 0 6t)1/2N3(0,1)
(12)
where N3(0,1) denotes a vector of three independent normally distributed random variables. each with zero mean and unit (19) Gikhman, I. I.; Skorokhod, A. V. Stochastic Differential Equations; Springer: Berlin, 197 1 . (20) Arnold, L. Stochastic Differential Equations; Wiley: New York, 1971. (21) Milshtcin, G. N. Theory Probab. Its Appl. (Engl. Transl.) 1978, 23, 396. (22) Rumelin, W. SIAM J . Numer. Anal. 1982, 19, 604. (23) Pardoux, E.; Talay, D.Acta Applicandae Mathematica 1985, 3 , 23.
0.40 0.451
0.05 0.00 0.0
0.5
1.0
4.0
Log(t/ps)
Figure 1. Geminate recombination of an ion pair: (0) MC simulation with bridge interpolation (5000 realizations); (-) numerical solution of the backward equation for W(r,t). (a, top) r = 6.5 X m; (b, bottom) r = 1.5 X m;D’= 1.0 X m2 s-I; R = 5.0 X m. Error bars represent two standard errors of the mean.
variance. The discretization is adequate if the time step 6t is sufficiently small that the infinitesimal parameters (mean and variance) do not change significantly during the course of the time step. In the present application, the time step for a given configuration is calculated by noting that the strongest force in the cluster will arise from the pair separated by the shortest distance. 6t is then equated to the time for which the drift arising from the minimum distance has a 95% chance of changing by less than 10% and may be calculated from the discretized stochastic differential equation for that distance formulated as if the pair were in isolation:
6r = D(2r - r c ) / r 2 6 t
+ ( 2 0 6t)1/2N(0,1)
(13)
The time step is calculated such that the drift term varies by 10% if the normally distributed random number A’ takes the value 2 . The occurrence of reaction is modelled in two ways: (a) if the particles are in a reactive configuration at the end of a time step then a reaction occurs and the time is noted; (b) it is possible that particles may diffuse together and reseparate during a time step: reactions of this type were missed by our earliest simulations,’ but may be allowed for, to a good approximation, by working out the reaction probability for each pair given the separation at the start and at the end of the time step. This use of the Brownian bridge process has been reported p r e v i o u ~ l y . ~It~is~ ~still appropriate here without any modification because, if the solution is convergent to a true sample path with the correct law, then during the time step the diffusion process is indistinguishable from a three-dimensional Wiener process with constant drift, and the separation converges to a one-dimensional Wiener process with (24) Clifford, P.; Green, N. J. B. Mol. Phys. 1986, 57, 123
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
4420
Clifford et a].
A v)
x [z
a
+
a r
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0.001 0.0
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Log(t/ps) I
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A
vi x LL
m
+
+a u0
2 V
V.”
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2.5
3.0
3.5 4.0
0.0 0.5 1.0
1.5 2.0 2.5 3.0 3.5 4.0 Log(t/ps)
Log(t/ps)
Figure 2. Comparison of IRT simulation and Schwarz model for Scheme I: (0) MC simulation (5000 realizations); (-)
IRT simulation (los realizations); (---) Schwarz model, steady-staterate constants: Schwarz model, time-dependent rate constants. (a, left panels) Two-pair spur; m2 s-l; RA+, = RA+B= R B + B = 5 X (b, right panels) three-pair spur. (Gaussian spurs, uA = uB = 1 nm; D, = DB = 5 X m.) (.e
e)
constant drift, which is exactly the case we have already analy~ed.’~ Since the time discretization is an approximation we must confirm that the process we are simulating has indeed converged to the diffusion of interest. We do this by simulating geminate pair recombination. The pair is started from a fixed initial distance and is allowed to diffuse until reaction occurs or until some fixed cutoff time (usually 10 ns) is attained. The reaction time is noted and the realization is repeated with a different random number sequence. After a large number of realizations the reaction times are pooled to give a cumulative distribution function for the recombination probability. This simulated distribution function is then compared with a numerical solution of the backward equation for that initial configuration. A typical comparison is shown in Figure 1, from which it is deduced that the time discretization used is acceptable.
IV. Multiparticle Clusters There are very few models of diffusion-controlled kinetics of ions in few-body systems, and for those that are available the approximations have not been tested. The development of the Monte Carlo technique outlined in section 111 enables these models to be tested for the first time. The first attempts to allow for the spatial inhomogeneities inherent in a cluster were by Jaffez5and Lea26 and form the basis of the prescribed diffusion model of Mozumder.’.* In this model the approximation is made that the cluster can be characterized by a singlet density, or concentration, which is Gaussian at all times, and whose variance increases linearly with time. Reaction is incorporated with a local rate proportional to the product of the concentrations of the relevant species and to a characteristic rate constant, which is set equal ( 2 5 ) Jaffe, G.Ann. Phys. IV 1913, 42, 303. (26) Lea, D. E. Proc. Camb. Philos. SOC.1934, 30, 80
to that which would apply under steady-state conditions. Essentially a macroscopic equation has been applied to a microscopic system. The effects of charge in this model are incorporated only through the rate The approximations have been discussed at length in previous and do not need further attention here. In contrast to the established approach, we have advocated the use of models which explicitly recognize pair correlations but neglect higher order correlations, by formulating the problem in terms of pair distances and assuming that these distances evolve independently.2 There is no obvious way in which the effects of this assumption on the kinetics can be assessed, except by comparison of the predicted kinetics with Monte Carlo simulations. We have established that the independent pairs approximation is remarkably good for free-radical systems, but the assumption has still not been tested for ionic systems, where there is a stronger correlation between distances through long-range forces. The remainder of this section is concerned with tests of this nature. The particular form of the independent pairs model which we will test here is the independent reaction times simulation. This is not an analytic formulation, but rather a very efficient simulation method, in which the independent pairs approximation is made. Realization of the simulation for a given initial configuration proceeds in the following way. (A more detailed description will be found in ref 4.) The interparticle distances in the cluster are calculated from the initial configuration and considered in turn. If the pair under consideration were isolated, then its recombination probability would be well approximated by eq 8, which gives the probability distribution of the reaction time of that pair in isolation. It is straightforward to generate a random number with exactly the correct distribution function from a uniform random ( 2 7 ) Hummel, A. Adu. Radiur. Chem. 1974, 5 , 1
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
High-Permittivity Solvents
4421
TABLE 111: Ten-nanosecond Product Yields for Scheme I" A, Single Pair prescribed diffusionb 0.0339 prescribed diffusionc 0.05 18 Monte Carlo simulation 0.000 (0) IRT simulation 0.000 (0)
AB
B2
0.2862 0.2931 0.376 (3) 0.376 (3)
0.0339 0.0518 0.000 (0) 0.000 (0)
prescribed diffusionb prescribed diffusionC Monte Carlo simulation IRT simulation
Two Pairs 0.1053 0.1554 0.090 (2) 0.089 (2)
0.8106 0.7950 0.884 (4) 0.911 (4)
0.1053 0.1554 0.089 (2) 0.086 (2)
prescribed diffusionb prescribed diffusion' Monte Carlo simulation IRT simulation
Three Pairs 0.1945 0.2826 0.213 (3) 0.211 (2)
1.4106 1.3492 1.463 (6) 1.484 (4)
0.1945 0.2826 0.209 (3) 0.210 (2)
0.0 0.5
1.0
1.5
3.0 3.5 4.0
2.0 2.5
L o g ( t / ps)
"All reaction distances 0.5 nm, diffusion coefficients 5 X IO4 m2 s-I. Figures in parentheses represent two standard errors of the mean in the third decimal place for the number of simulations performed. bPrescribed diffusion: Schwarz model: steady-state rate constant. Prescribed diffusion: Schwarz model,9 time-dependent rate constant.
number U, on (O,l), by inverting the function of eq 8 for t and setting W = U. This procedure is repeated independently for each distance in the cluster, yielding an ensemble of associated distances and times. The independent association of a reaction time with each distance is the application of the independent pairs approximation. The minimum time in the cluster is found, the corresponding particles are removed, and the reaction time is noted. The minimum time for the remaining ensemble of particles is found and the second reaction is deemed to occur at this time. The procedure is repeated until all particles have reacted or all remaining reaction times are infinite. Thus the entire cluster history can be realized by using only one set of random numbers. The IRT model has been realized by using the approximation of eq 8 for the pair recombination probability for a number of idealized ionic systems. Figure 2 shows a comparison of the Monte Carlo simulations with the I R T model and with prescribed diffusion for a spur in which the initial particle positions are independently sampled from a three-dimensional spherically symmetrical normal distribution of standard deviation 1.O nm. The reaction distances are all 0.5 nm and all reactions are possible according to Scheme I. The products are not removed from the m2 s-l, simulation but are allowed to diffuse with D = 5 X because their charges affect the subsequent diffusion of the other ions. Figure 2 shows comparisons of the numbers of each type of reaction occurring in a two-pair cluster and in a three-pair cluster. It will be seen that the I R T model produces errors of the order of 3%, which is worse than was found for neutral species4 but which is a distinct improvement on prescribed diffusion, either with steady-state rate constants or with approximate time-dependent rate c o n s t a n t ~ . ~ *However, ,~~ prescribed diffusion is better than was found for the neutral case in that the product ratios are closer to the simulation (see Table 111). As would be expected, agreement between all the approximate models and the Monte Carlo simulation is better for a three-pair cluster than for a two-pair cluster.
SCHEME I A+
+ A+
-
X
LT (c
0
z V
..
0.0 0.5
1.0
1.5
2.0
2.5
3.0
35
40
L o g ( t / ps)
F w e 3. Comparison of IRT simulation and Schwarz model for Scheme 11: (0) MC simulation (4000 realizations); (-) IRT simulation (lo5 realizations); (---) Schwarz model, steady-state rate constants; (-..) Schwarz model, time-dependent rate constants. (a, top) Two-pair spur; (b, bottom) three-pair spur. (Gaussian spurs, uA = uB = 1 nm; DA = DB = 5 X IO+ m2 s-I; R = 5 X m.) Error bars represent two standard errors of the mean.
0.2
t
0.0 0.0
/ 1.0
2.0
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4.0
L o g ( i Ips)
Az2+
A+ + B-+ AB
B- + B-
A In
B22-
The same clusters have also been analyzed by using a different reaction scheme (Scheme 11), consisting of a single cross-reaction. Comparisons are shown in Figure 3, where the same pattern of (28) Rice, S. A.; Butler, P. R.; Pilling, M. J.; Baird, J. K. J. Chem. Phys. 1979, 70, 4001. (29) Green, N. J. B. Chem. Phys. Lett. 1984, 107, 485.
Figure 4. Recombination of tetrahedrally disposed ions: (0) MC simulation (5000 realizations); (-) IRT simulation ( lo5 realizations). (Initial distribution of ions, regular tetrahedron of side 2'/2 nm; N A = N B = 2; DA = D , = 5 X m2 SC'; R A + A = R A + B = R B + B = 5 X m.) Error bars represent two standard errors of the mean.
agreement will be observed, although the Schwarz version of prescribed diffusion performs very well here; indeed for the three-pair cluster it agrees with the simulations more closely than does the I R T model, for reasons which are not clear. SCHEME 11 A+ + Bproducts
-
J . Phys. Chem. 1987, 91, 4422-4428
4422
A
In 1
a: U
+
Q c
0 0
z
i
V
0.0051# I
’
0.000~
0.0
I2L
0.5
I
1
1
1.0
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I
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I
00
I
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1.5 2.0 2.5 Log ( t I p s )
3.0
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10
1.5 2.0 2.5 Log (t Ips)
3.0
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,
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This configuration was chosen to maximize the effects of competition between initially equal distances and hence to provide a more stringent test of the approximation than the more random distributions usually assumed. It is clear that this type of competition is not a very important effect in diffusion-controlled kinetics. Finally, for the sake of comparison, Figure 5 presents the results of a two-pair Gaussian cluster using Scheme I, where the charges have been turned off and the same effective reaction distances used. The inclusion of charge has two main effects: (a) the product ratios are altered significantly, and (b) the time scale of the kinetics is speeded up slightly. The conclusion that charge effects should be included in radiation chemical calculations is therefore reinforced.
3.5
4.0
Figure 5. Recombination in a two-pair spur without the effects of charge: (0)MC simulation (lo5 realizations); (-) IRT simulation (lo5 realimz s-I; zations). (Gaussian spurs, U, = UB = 1 nm; D, = DB = 5 X RA+, = = 2.3 X m; RA+B= 9.3 X lo-’’ m.) Error bars
represent two standard errors of the mean.
For the comparison in Figure 4 we return to Scheme I. In this system four particles have been started on the vertices of a regular tetrahedron of edge length 21/2nm. While prescribed diffusion is unable to deal with this initial, non-Gaussian, configuration, the IRT model is still within a few percent of the simulation results.
V. Conclusions (i) The independent reaction times model provides an accurate description of the kinetics occurring in clusters of ions in water. It is superior to prescribed diffusion in that (a) it provides a more realistic description of both the time-dependent reaction probabilities and the final product yields and (b) it is capable of handling non-Gaussian initial distributions; in particular the particles of the I R T model form a real initial configuration. (ii) It is important to recognize charge effects in models of ionic reactions in water. This conclusion has been demonstrated both by consideration of the time-dependent reaction probability for a pair, W(r,t)(eq 8), and by explicit modelling of reaction in twoand three-pair spurs (Figures 2 and 5 ) . (iii) The present application of the independent reaction times model depends on the availability of an approximate analytic form for W ( r , t ) . SNo such approximation, of sufficient accuracy on the time scales and distances of interest, is available for lowpermittivity solvents. W e have recently developed a technique for realiz,ing the I R T model for such systems, in which W(r,t)is obtained by interpolation between a set of numerical solutions of the backward equation. This will be the subject of a future paper.6
Acknowledgment. The research described herein has been supported by the Science and Engineering Research Council (N.J.B.G.), Jesus College, Oxford (N.J.B.G.), UK Atomic Energy Authority, AERE, Harwell (S.M.P.), and the Office of Basic Energy Sciences of the Department of Energy. This is document No. NDRL-2953 from the Notre Dame Radiation Laboratory. Registry No. HzO, 7732-18-5
Linear Solvation Energy Relationships. A Scale Describing the “Softness”of Solventst Y. Marcus* Department of Inorganic and Analytical Chemistry, The Hebrew University of Jerusalem, 91 904 Jerusalem, Israel (Received: March 13, 1987) The w-scale of solvent softness is defined as the difference between the mean of the Gibbs free energies of transfer of sodium and potassium ions from water to a given solvent and the corresponding quantity for silver ions (in kJ mol-’) divided by 100. Values are tabulated for 34 solvents. The p-scale is compatible with a combination of the vibration frequencies of the C-I bond in ICN and the 0-H bond in phenol, providing semiquantitative p-values for a further 60 solvents. The p-scale can be used in linear solvation energy correlations, for instance for the transfer of ions into soft solvents, such as N,N-dimethylthioformamide, tetrahydrothiophene, or pyridine.
Introduction Solvents are characterized by many properties, such as their polarity, their ability to donate or accept electrons or hydrogen *Present address: Department of Chemistry, University of California at Irvine, Irvine, CA 92717. ?Presentedin part at the 10th International Conference on Non-Aqueous Solvents, ICNAS X,Leuven, Belgium, Aug. 1986. 0022-3654/87/2091-4422$01.50/0
bonds to or form solutes, their self-cohesiveness and structuredness, etc. These properties govern the solubilities of solutes and their distribution between a given solvent and a reference one, the kinetics of reactions taking place in the solvents, and many other thermodynamic, spectroscopic, and transport properties of the solutes. Generally, more than one property of the solvent is pertinent, and a multivariable regression brings out the contri0 1387 American Chemical Society
