J. Phys. Chem. 1982, 86, 1322-1327
1322
Monte Carlo Simulation of Diffusion and Reaction in Radiation-Induced Spurs. Comparisons with Analytic Models Peter Clifford, Mathemtical Instttute, Oxford OX1 3L8, U n M Kingdom
Nicholas J. 6. Green, and Michael J. Pllllng* myslcal Chemlstry Laboratory, Oxford OX1 302. Unifed Klngdom (Received:AprH 30, 198 1)
Reaction in radiation-induced spurs, containing one type of radical, is simulated by using random walks on a cubic lattice. The simulations are shown to agree satisfactorily with analytical results for the evolution of a single-particle density function and for the reaction between two particles. For spurs containing more than two particles, no exact analytical treatments are available. Results of simulations are presented for systems initially containing up to six particles. The simulations are compared with prescribed diffusion approximations, both with and without time-dependent rate constants, and the approximationsof prescribed diffusion are shown to be inadequate. Satisfactory agreement, however, is obtained with the stochastic independent pair model developed in the previous paper.
1. Introduction
High-energy radiation is deposited in liquids in a series of discrete events and produces regions of locally high radical concentration. These “spurs” evolve by diffusion and radical-radical reaction to yield, at times greater than -100 ns in water a t room temperature, a homogeneous distribution of surviving radicals and product molecules. The majority of radiation chemical experiments have concentrated on the homogeneous regime, but many steady-state experiments using high scavenger concentrations’+ and, more recently, time-resolved pulse radiolysis experiments using picosecond have probed the evolution of the spurs and deduced initial radical yields. Mathematical have been developed to describe the results of such experiments and deduce the nature of the initial spur distributions. In these models the time evolution and the residual radical yields are of primary importance in matching experiment and theory. In the past models have relied on the use of deterministic rate equations and averaged concentrations; in addition, most employ the prescribed diffusion approximation or a modification. In the previous paper,16 (1) Kuppermann, A. In “RadiationReesarch”; Silini, G., Ed.;NorthHolland Publishing Co.: Amsterdam, 1967; p 212. (2) Buxton, G. V. Radiat. Res. Rev. 1968, 1 , 209. (3) Schwarz, H. A. J. Phys. Chem. 1969, 73, 1928. (4) Balkas, T. 1.; Fender, J. H.; Schuler, R. H. J. Phys. Chem. 1970, 74, 4491.
(5) Freeman, G. R. Actions Chim. Biol. Radiat. 1970, 14, 73. (6) Rzad, S. J.; Fender, J. H. J. Chem. Phys. 1970,52,5395. (7) Hunt, J. W. Adu. Radiat. Chem. 1976, 5, 185. (8) Singh, A.; Chaae, W. J.; Hunt, J. W. Faraday Discuss. Chem. SOC. 1977,63, 28. (9) Bums, W. G.; May, R.; Buxton, G. V.; Tough, G. S.Faraday Discuss. Chem. SOC.1977,63, 47. (10) Trumbore, C. N.; Short, D. R.; Fanning, J. E.; Olson, J. H. J. Phys. Chem. 1978,82,2762. (11) Samuel, A. H.; Magee,J. L. J. Chem. Phys. 1963,21,1080. (12) Monchick, L.; Magee, J. L.; Samuel, A. H. J. Chem. Phys. 1957, 26,935. (13) Mozumder, A.; Magee, J. L. Radiat. Res. 1966,18, 215. (14) Kuppermann, A. In ‘Physical Mechanieme in Radiation Biology”; Washington, DC, 1975; USAEC 721001, p 155. (15) Clifford, P.; Green, N. J. B.; Pilling, M. J. J. Phys. Chem., pre-
ceding paper in this issue.
0022-3654/82/2086-1322$01.25/0
we criticized the deterministic approach and showed how it differs from a more realistic stochastic approach. We also criticized the prescribed diffusion assumption as taking no account of correlation effects which arise from particles having a nonzero size, and we proposed a model based on the assumption of pair independence which correctly takes account of size and does not require the use of ad hoc time-dependent rate constants. On the other hand, neither the independent pair approach nor the prescribed diffusion approach accounts for competition effects between sinks for the same reactive particle or the effect on the correlation of distances caused by the presence of many particles. When considering homogeneous kinetics, we are accustomed to dealing with large numbers of reactive particles which react individually or in a well-defined pairwise manner. Thus, treatments of diffusion-controlled reactions consider motion into a sink and assume implicitly that there is no competition between Theories of geminate r e c o m b i n a t i ~ n consider ~ ~ - ~ ~ the relative motion of two particles and, once again, each recombination is assumed to occur independently of others. These theories break down when more than two particles are involved, as in a spur. Furthermore, each spur evolves in a different manner from that of its fellows. Although the averaged distribution may be Gaussian, for example, each spur has its own initial distribution of radical-radical distances, and these lead to different probabilities of recombination and also, when more than one radical is involved, to different product yields. Except by coincidence, the time evolution and long-time asymptotes when averaged over many spurs will differ from those obtained by considering an averaged spur. There is, however, no a priori way of estimating the errors involved in using either a prescribed diffusion theory (16) Collins, F. C.; Kimball, G. E. J. Colloid Sci. 1949, 4, 425. (17) Waite, T.R. Phys. Rev. 1967,107,463. (18) Waite, T.R.J. Chem. Phys. 1968,28, 103. (19) Waite, T.R. J. Chem. Phys. 1960,32,21. (20) Noyes, R. M. h o g . React. Kinet. 1961,1, 129. (21) Noyes, R. M. J. Chem. Phys. 1964,22, 1349. (22) Noyes, R. M. J. Am. Chem. SOC.1956, 77, 2042. (23) Noyes, R. M. J . Am. Chem. Soc. 1966, 78, 5486. (24) Berg, 0. G. Chem. Phys. 1978, 31, 47.
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 8, 1982 1323
Monte Carlo Simulation of Spur Reactions
of an "average" spur or the independent pair stochastic theory, which does not consider an average spur but is inherently capable of calculating not only mean values of N (and its asymptote) but also standard deviation^.^^ A basic model must therefore be set up which recognizes the discrete nature of the processes involved. Because of the complexity of the problem, the method of analysis of such a model must be numerical. This paper takes a preliminary look at the problem by examining the behavior of spurs containing a single type of radical. The basic model may be readily extended to describe the behavior of more complex spurs. It is used in the present paper to assess the validity of analytic approximations including prescribed diffusion and the independent pair approximation, developed in the previous paper. 2. Numerical Simulation of the Basic Model The calculations seek to determine the spatial evolution and decay of radicals in a large number of independent spurs and then to average this behavior to obtain a record comparable with that obtained in an idealized experiment which could follow the averaged radical yield to short times. The diffusion of a particle was modeled by a random walk on a cubic lattice in which all of the x , y, and z coordinates were changed by fl at each time step. This corresponds to examining a stochastic process in which the probability of finding the particle at the point ( i j , k ) (having started at the origin at time zero) after n steps is
(11
If the initial distribution is Gaussian with variance %, then the probability distribution is described by eq 1 with n replaced by n + n,. The root mean square jump distance is 4 3 and therefore the diffusion coefficient (D= l / & n ( I 2 ) ) is 1 / 2 m if the jump frequency, m, is unity and (P)is the mean square jump distance. It might be argued that attempting to model diffusion by motion constrained to a body-centered cubic lattice with a constant jump distance is a poor approximation. It amounts to modeling three independent Gaussians with three independent binomials. Although this is a good approximation for a large number of steps ( n > lo), it breaks down for small numbers, i.e., at short times; however, it is precisely in this time regime that Fick's laws fail to hold because the predicted particle velocity diverges and it becomes necessary to employ higher-order equation^.^' It must be borne in mind, however, that comparisons between the present and classical calculations for less than 10 jumps are questionable, although the comparisons which will be made in section 3c involve initially extended distributions where no > 100 so that the model is acceptable as a diffusion model at all times. Single-particle simulations were performed to establish that there was no directional bias in the random walks, and the development of the radical distribution was compared with the appropriate Gaussian, eq 2, and the optimal
p(r,n) = (27r(n+ n0))-3/2 exp[-r2/2(n + no)] (2) random number generator was established on this basis. Reaction in a tweparticle spur can be solved analytically and thus provides an ideal vehicle for testing the validity of the reaction model. The simulations followed the (25) McQuarrie, D. A. J. Appl. h o b . 1967, 4, 413. (26) Chandrasekhar, S. Rev. Mod. Phys. 1943,15, 1. (27) Cox,D. R.; Miller, H. D. 'The Theory of Stochastic Processes"; Chapman and Hall: London, 1965; p 225.
0.8r
00 IO0
IO'
IO2
lo'
Time
I0 '
Flgure 1. Comparison of simulatlon of two-particle system (initial dtstance, 15 units; mutual dmuskn coeffldent, 'I2)with analytic solutkn, eq 3.
motion of an ensemble of single particles generated in an initial &function distribution (typically at (15,0,0)) into a spherical sink of radius 10 units, centered on the origin. Absorption occurred with unit probability on crossing the boundary of the sink (corresponding to the Smoluchowski boundary condition)20and the first passage time was recorded for lo00 simulations. A problem arises in comparing the first passage time distribution function (the probability that fiist passage has occurred by time t ) with its analytic form F,,(t) = ( a / r o )erfc [(ro- ~ ) / ( 4 D ' t ) ' / ~ ] (3) where D'is the relative diffusion coefficient (= 2 0 , ro is the initial separation of the particle and the center of the sink,and a is the radius of the sink. Becaw a cubic lattice has been used, absorption does not necessarily occur at exactly the radius a but sometimes occurs at a distance less than a because not all of the lattice points in the sink which can be reached in a single jump from outside the sphere lie on its surface. The problem may be reduced by using a finer mesh, but we have attempted to allow for this artifact of the model with reasonable mesh sizes by finding the mean distance at which reaction can occur. This was achieved by calculating the mean of the distances to all points which lie inside or on the limiting sphere and from which it is possible to jump outside in one step. For a sphere nominally 10 units in radius, the effective radius, a', is 9.263, and this value was employed in eq 3 when comparing analytic and simulated first passage time distributions. The simulated data were compared with the analytic function by means of statistical goodness-of-fit tests. Three different tests,28 Kolmogorov-Smirnov, Cram&-Von Mises, and Kuiper, were employed, and all showed good agreement when the optimal random number generator was used, even at short times. A typical comparison is shown in Figure 1. In the simulations of spurs containing more than two radicals, the initial position of each particle was generated from three independent binomials (eq 1 with n = no,the initial variance). The position of each particle was then followed with time (number of jumps) and reaction was assumed to occur whenever two particles approached within 10 units. Every realization was followed until reaction was complete or to loo00 jumps, and the number of reactions was recorded as a function of time; lo00 realizations were performed and the transients then averaged. Figure 2 shows transients generated in this manner for three- and six-particle spurs with different values of no. 3. Comparison with Prescribed Diffusion Models One of the main aims of this paper is to establish the extent of the error made by the prescribed diffusion ap~~
(28) 'Biometrika Tables for Statisticians"; Cambridge University Press, Cambridge, England, 1972; Vol. 2, pp 117 and 359.
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The Journal of phvscal Chemhtry, Vol. 86, No. 8, 7982
Clifford et al.
3b- - - - _ _ _ _ _ _
the probability that Y,+, is zero. At each step Y, may change by &2 each with a probability of ll4,or by 0 with a probability of Each step of Y, is thus identical with two steps of a process, Z, for which the increments are k l , each with a probability of l / > Formally
I
i
IO
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1000 1oOoo Time
u = P(Yn+, = 0) = P(Z2(n+no) = 0)
(8)
but 2, is mathematically indistinguishable from a simple, one-dimensional random walk, X,,which follows a binomial law; hence, u may be equated to the probability that X2(n+,)= 0, which is given by the center term in the corresponding binomial. Provided n no is large enough, this may be evaluated by using Stirling's approximation. This is valid for all of our simulations as no > 100.
+
0;
i
10
100
1000TimelOOOO
Flpun 2. Comparisons of simulations of three-and six-partlcle spurs (inrtial Gaussians, standard devietkn, 10 units; mutual diffusion coefficient, 1) with simple prescribed dmuskn models: (-) simulation; (- - -) stochastic rate ~ u a t b n ; (. .) Samuel and kgee; (- -) ~ ~ z u m d and e r Magee.'
'
-
proximation. We must first establish a basis on which comparisons may be made between simulations on a lattice and continuum models. 3a. Simple Prescribed Diffusion Model. The basic equation of continuum prescribed diffusion is d(N)/dt = -2kf((N))1p2 dV
(4)
where ( N ) is the expectation number of radicals remaining in the spur, k is the rate constant for reaction, and p is the time-dependent distribution function p = [47rD(t
+ t0)]-3/2exp{-P/[4D(t + to)]}
(5)
Thie is equivalent to eq 2 if 2Dt, = no, the variance of the initial Gaussian. The function f((N)) is either 1/2(N)2 or l12(N)(N- 1) depending on the model used."J3 Substituting eq 5 into eq 4, we find -d(N)/dt = 2kf((N))/[&D(t
+ to)]'/'
(6)
3b. Prescribed Diffusion on a Lattice. We now need to develop an equivalent prescribed diffusion model on a cubic lattice to facilitate comparison with the simulations. The continuum model depends on evaluating the density of the probability of finding two particles independently in the same volume element dV. On the lattice, the probability that two particles, whose positions are independent samples from the product of the same three binomial distributions of initial variance n,are at the same point after n steps is given by the square of eq 1with n replaced by n + no, summed over all lattice points. The summation corresponds to the integral over space in eq 4. This may be separated into the product of three independent and identical sums. We denote this by u3
The probability of finding two particles at the same point after n jumps is u3; however, as in the continuum prescribed diffusion model, we want the density of this probability and so we must divide by the volume of the unit cube on the lattice which this point represents. As only alternate integers are allowed, the span of the process is 2 in each dimension and the volume of this elementary cube is 8 units. Hence, the equation corresponding to eq 4, which describes the continuum model, is -d(N)/dn = Y4kf((N))u3
Since there is one jump per unit time in lattice units n t, and substituting from eq 9 for ,'a we find -d(N)/dt = 2kf((N))/[4~(t+
(11)
This equation may be compared with eq 6. We are looking at the case D = lism(P) = 1/2 and so eq 6 reduces to eq 11. The continuum prescribed diffusion model is, therefore, equivalent to that based on lattice prescribed diffusion to within the accuracy of Stirling's approximation. In the simulation n + no > 100 and so the two approaches are equivalent to within 0.1% and we can, in consequence, make a fair comparison between continuum prescribed diffusion and the lattice simulations. 3c. Comparison between Simulation and Prescribed Diffusion. We now move on to compare the simulated transients with those predicted by the simplest prescribed diffusion models. Following Mozumder and Magee,13 steady-state diffusion-controlled rate constants (k = 27ra 'D? were used initially to generate the prescribed diffusion transients (eq 11)with a' = 9.263 and D' = 1 (D' is the coefficient of relative diffusion). The time dependence of (N) is thus given by eq 12, where No is the
( N ) = No/V + NoQ(t)I
f ( ( N ) )=
+
W2
( N ) = No/[No(1 - No)e-Q(t)]
f ( ( N ) )= ( N ) ( N - 1)
where E' denotes summation over alternate i. (This procedure is necessary because only alternate points in a given coordinate direction are available to a particle at any one time since all three coordinates change at each step.) u is equivalent to the probability that two random variables, sampled independently from the same binomial distzibution, are equal. We define a new stochastic process, Y,, which is the difference between two samples taken at random from a binomial distribution; u is thus equal to
(10)
(12a) ( 12b)
number of particles initially present in the spur and where Q(t) = (to-'/' - (t + to)-'/2)4k/(8~D)3/2. Figure 2 compares the transients for No = 3 and 6, calculated from eq 12, with those obtained by simulation. The most obvious discrepancies occur at short times and arise from initial correlation between the particles on the lattice. In the simulations the initial positions of the particles were independently generated from Gaussians centered on the origin and rounded to the nearest lattice point. There is a nonzero probability of generating two particles close enough to react at zero time. This corre-
The Joumal of Physlcel Chemistry, Vol. 86, No. 8, 1982 1325
Monte Carlo Simulation of Spur Reactions
IO
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Time
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*I4 I
I
It
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F i g w 3. Comparisons of simulations of three- and six-particle spws withshplepresabeddmusknmodelsaWoqforreduced(N,): (-) slmutatbn; (-
- -) stochastic rate equation;
(. .) Samuel and Ma-
gee; l l (-. -) Mzumder and ~ a g 9 e . l ~
sponds to a correlation, which is built into the simulations, but which prescribed diffusion is unable to recognize.29 The precise form of the correlation in a real spur will depend on the dynamics of the energy deposition process; nevertheless, the same primary act leads to the eventual production of all of the radicals in the spur and there will undoubtedly be some form of correlation which will not be accounted for by prescribed diffusion. In order to overcome this problem and for the purpose of making a comparison which is favorable to the prescribed diffusion model, No was reduced in eq 12 to the value of (N) pertaining immediately after time zero in the simulations. This value is nonintegral and less than the initial number of particles. The transients for this case are shown in Figure 3. Although the agreement is much improved, neither solution shows a rapid enough rate of reaction at short times, whereas eq 12a predicts too rapid a rate at long times. Figure 3 also shows transients from the stochastic version of prescribed diffusion proposed in the previous paper.ls Since No must be integral for this model, the transients were generated by taking an average of spurs with a range of No values, weighted as observed in the simulation at zero time, so as to generate the required expectation value of No. Even with this favorable comparison the simple prescribed diffusion models still show poor agreement with the simulations. In particular they underestimate the rate of reaction at short times. This arises because of the use of averaged concentrations and the consequent inability to take account of the individual evolution of the distribution of radical-radical distances. Radicals generated in close proximity react rapidly, and this leads to a depletion of short radical-radical distances as time proceeds and a consequent reduction in the reaction rate which is initially greater than that estimated from a simple consideration of averaged concentrations. This effect is similar to that found for diffusion-controlledreactions with random initial distributions, where time-dependent rate constants are invoked, thus permitting the use of bulk concentrations in the rate equations. Indeed, the more recent numerical applications of prescribed diffusion to spur kinetics have employed these time-dependent rate constants. The next section examines their application and compares the results with the simulations. (29) Czapski, G.;Peled, E.J. Phye. Chem. 1973,77, 893. (30)Mozumder, A.; Magee, J. L.Int. J. Radiat. Phys. Chem. 1976,7,
83.
i
10
100
1000T,"DIOOOO
Flgure 4. Comparisons of simulations of three- and six-particle spurs with prescribed dmuskn modelsalbwhg W"d& rate constants and reduced ( N o ) : (-) slmulatkm; (---) stochastic rate equation; (.-a)
Samuel and Magee;
(-e-)
Morumder and Magee.l3
3d. Refined Prescribed Diffusion Theories. Attempts have been made, notably by Schwarz? to overcome the problems of short-time correlation by using time-dependent rate constants. k ( t ) = 2ruD71 + ~/(?rD't)'/~]
(13)
The function Q(t)in eq 12 should now be replaced by q(t), where if D' = 2 0
q ( t ) = ~(2?rD)-'/~[to-~/~ - (t + to)-'/2] + ~ ~ t ' / ~ / [ 2 ~ D+ t oto)'/'] (t
(14)
Figure 4 shows transients generated from this solution compared with the simulations. Again ( N o )is reduced to account for zero time reaction. Agreement is again improved and is almost exact for three particles (although comparison with two and four particles shows this to be fortuitous). Schwan's model3 also allows an extra refinement. This is an increased rate of widening of the Gaussian prescription arising from relatively faster reaction in the center than at the edges. The Gaussian prescription is p
= (7rb2)-3/2exp(-?/b2)
(15)
and from this and the rate equation two differential equations may be obtained d(N)/dt = -2k(t)(N)2/(2?rb2)3/2
(16)
2k(t)(N)b2[p- ( 2 ~ b ~ ) - ~ / ~ ] db2_ -40(17) dt
[-y2 + r2/b2]
The right-hand side of eq 17 contains a singularity at r2 = ( 3 / 2 ) b 2and hence an approximation is made by replacing the second term with its value at r = 0 and multiplying by a factor fitted from experimental results, which Schwarz finds to be 0.67. Thus we replace eq 17 with db2/dt = 40
+ 0.8126k(t)b2(N)(2?rb2)-3/2 (18)
The rate law clearly corresponds to that of eq 12a, and so we expect the predicted rate to be faster than that given by a stochastic rate law. The simultaneous solution of eq 16 and 18 is complicated by the divergence of k ( t ) at short times, but the solution is defined and can be found asymptotically. Results are shown in Figure 5 for three and six particles with reduced values of ( N o ) . The model is seen to deviate from the simulations in the manner expected. Attempts to fit a stochastic rate law into this model have so far failed because of numerical instabilities
1326
The Journal of Physlcal Chemistry, Vol. 86,No. 8, 1982
i
Clifford et ai.
I
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' L
OO
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Figure 6. Comparison of simulations of three- and six-particle spurs with Schwarz reduced ( N,) : (-) simulation; (- -1 Schwarz
-
"
OO
I
IO
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10000
Time
with respect to the divergence of k ( t ) at short times. Although the use of time-dependent rate constants improves the fit, it is still far from perfect. There is, however, a more fundamental objection to their use, since they are strictly applicable only to reactions in which there is a random initial distribution with the concentration at r maintained at the bulk value. Thus, they are not designed for use with Gaussian or &function initial distributions, and any improvement in the fit resulting from their use is qualitative in the sense that the original model did not take any account of the short-time depletion of small radical-radical distances, whereas a decreasing reaction cross section makes some allowance for it.
-
Q)
4. Stochastic Independent Pair Model The stochastic independent pair model was derived in the previous paper'&by convolving the first passage time distribution function for a diffusing pair with the initial pair distance density function. This results in an expression for the probability that reaction has occurred by time t of
n,(t) =
(?r~'to)-l/2ae-42/4D(t+to)-
( t :to)/z
This expression is exact for a two-particle spur which is drawn from an initial Gaussian distribution. It has the advantage that it does not need any externally fitted parameters or arbitrary time-dependent rate constants but depends only on the reactive distance, the relative diffusion coefficients, and the initial standard deviation of the Gaussian (spur width). This expression is treated in a stochastic manner by assuming all pairs to be independent and results in the equation
alternate
where (1- 2 ~ ) r(No+ 1)r['/w0 - N + 111 AN = 2~ r(N0- N i)r[f/z(No N 1)1
+
+ +
This equation may be compared with simulated transients, as shown in Figure 6 for three and six particles. The transients are an average of 4000 random walks divided into four equal groups, and the error bars join the furthest
Figwe 6. Comparison of simulations of three- and six-particle spurs with independent pair model, eq 20, and numerical random first passage time simulations from eq 21 (see text): (I) simulation; (-) random passage times; (- - -1 independent pair model (eq 20).
simulated points. The model c8n be seen to be a very good fit for three particles, but poor for six (although a great improvement on prescribed diffusion). The model is obviously a good approximation to the correct mathematical form, although the asymptote is predicted incorrectly and agreement gradually deteriorates in the transition from three to six particles. In order to ascertain the cause of this disagreement, which is particularly marked at zero time, one must examine the simulations more closely. If we allow a = 10 in eq 20 instead of the reduced value of 9.26 as discussed previously, we obtain extremely good agreement at time zero and poor agreement elsewhere; thus, the simulations are behaving as though the particles have a reaction distance of 9.26 at all times except zero, where the distance is 10. This phenomenon arises because particles displaced by up to 10 units at time zero react then, whereas the reaction distance, a, must be reduced to 9.26 at all other times. The suggestion is that, provided the initial distribution of particles in the spur is taken account of, the subsequent development of the spur will be adequately described by the independent pair approximation. This explanation is verified by the following Monte Carlo simulations. The coordinates of the particles are generated at random from a normal distribution and all '/&W- 1)distances calculated. Those that react at zero time with a reaction distance 10 are removed and all of the other distances are used to generate the corresponding random first passage times from the distribution function (with a ' = 9.26)
(21)
The minimum of these times is taken, and the particles which gave rise to that distance are removed, as are all other times arising from these particles. This is repeated until reaction is complete or the remaining particles are deemed to have escaped. This procedure is much quicker than the full simulations, and correspondingly more realizations are possible. The transients so evaluated are also shown in Figure 6 and are seen to agree with the full simulations very well. We conclude, therefore, that the disagreement between the independent pair equation (eq 20) and the simulations is indeed introduced merely by the initial distributions which we have chosen to use and that
J. Phys. Chem. 1982, 86, 1327-1332
the independent pair model is therefore an acceptable approximation to the diffusion and reaction in a system containing a few particles with initial Gaussian distributions. A similar idea underlies the model for the kinetics of charge recombination and scavenging in systems containing a number of isolated pairs proposed by Freeman and c o - ~ o r k e r s . ~ l The - ~ ~ independent pair model described above extends the range of systems covered by showing how to deal with a few-particle system with stochastic methods and allowing a distribution of reaction times for each initial separation distance. 5. Conclusions
This paper describes the developent of a Monte Carlo study of reaction in a radiation-induced spur based on a model involving motion on a three-dimensional lattice. The model is a good approximation to the diffusive motion and reaction of neutral radicals for all times at which Fick's (31) Freeman, G. R.; Fayadh, J. M. J. Chem. Phys. 1965,43,86. (32) Freeman, G. R. J. Chem. Phys. 1965,43,93. (33) Freeman, G. R. J. Chem. Phys. 1967,46,2822.
1327
lam apply. These simulations are necessary for the testing of analytical or partly analytical approximations of spur reactions. Comparison of the simulations with extant treatments based on prescribed diffusion shows that the latter are unable to reproduce any zero-time correlation between the radicals and also underestimate the rate at short times. The latter problem is alleviated by the use of time-dependent rate constanta although there is no exact theoretical justification for their use. The independent pair approximation, on the other hand, shows good agreement with the simulations for three- to six-particle spurs suggesting that it may be appropriate for the modeling of the majority of radiation-induced spurs, where the number of radicals per spur is generally34 less than 10. Important effecta which have not yet been examined are those arising from (i) different types of radical, (ii) reactive products, (iii) Coulomb interactions, and (iv) specific correlations arising from the mode of energy deposition. We are currently examining the first three effects by using both Monte Carlo simulations and the independent pair model.
-
(34) Magee, J. L.; Chatterjee, A. Radiat. Phys. Chem. 1980, 15, 125.
Interaction of Ethene, 2-Methyipropene, and Benzene with the Na+ Ion. 1. Quantum Chemical Study of Gas-Phase Complexes Joachlm Sauer and Detlef Deinlnger Cenbgl Institute of m y s h l Chemisby. Acedemy of SClences of the W ,DDR- 1199 &rlln-AdkKshof, German Democretic Republic, and N M Labatmy, Depertmnt of ~ y s l c s Karl , Merx University, DDR-701 Le@&, &man Lknocratic Republlc (Received:June 3, 1981)
The complexes of ethene, 2-methylpropene, and benzene with Na+ have been investigated by means of ab initio SCF calculations. For the hydrocarbon, the 4-31G basis set and, for the cation, a reoptimized STO-3G basis set have been used. The interaction energy equals -49, -60, and -78 kJ/mol, respectively. Discussion of the errors involved in the calculations and comparison with related experiments show that these values are too low by about 20%. The arrangement of the cation above the plane of the *-electron system is found to be the most stable one. This is the optimum structure for the electrostatic ion-quadrupole contribution to the interaction energy. The charge transfer to the cation proves to be very small and amounts to a few hundreths of an electron. These characteristics are in agreement with what is known for the cation-n donor complexes. On the basis of the ab initio results, the capability of CNDO/2 with such complexes is assessed.
1. Introduction
The knowledge of the interaction between alkali ions and molecules is a prerequisite for understanding such important processes as separations using zeolites (molecular sieves),' heterogeneous catalysis on surfaces of solids: and ion complexing in biomole~ules.~Moreover, the comparison of affinities toward different cations on the one hand and with proton affinities on the other hand is of general chemical interest.4~~ Whereas only a few gas-phase complexes of Na+ (with Hz06 and with NH3') have been observed, the situation (1) Breck, D. W. 'Zeolite Molecular Sieves";Wiley New York, 1974. (2) Fomi, L.; Invemizzi, R. I d . Eng. Chem. Process Res. Dev. 1973, 12, 455. (3) Laszlo, P. Angew. Chem. 1978, 17, 254. (4) Staley, R K.; Eieauchamp, J. L. J. Am. Chem. SOC.1976,97,5920. (5) Kollman, P.; Rothenberg, St. J. Am. Chem. SOC.1977,99, 1333. (6) Woodin, R. L.; Eieauchamp, J. L. J. Am. Chem. SOC.1978,100,601. ( 7 ) Castleman, A. W., Jr.; Holland, P. M.; Lindsay, D. M.; Peterson, K. I. J. Am. Chem. SOC.1978,100, 6039. 0022-365418212086-1327$01.25/0
with zeolites is very different: in recent years a lot of information about the interaction of Na+ with olefins and aromatics within zeolites has been gathered from various spectrocopic (IRand Raman,"'O NMFt,l1-I4 UV15), neutron scattering,16 and thermodynamic"J8 experiments. (8) FBrster, H.; Seelemann,R. J. Chem. SOC., Faraday Trans. I 1978, 74, 1435. (9) Carter, J. L.; Yates, D. J. C.; Lucchesy, P. J.; Elliott, J. J.; Kevorkian, V. J. Phys. Chem. 1966, 70, 1126. (10) Freeman, J. J.; Unland, M. L. J. Catal. 1978,54, 183. (11) Hoffmann, W.-D. 2.Phys. Chem. (Leipzig) 1976,257,315. (12) Lechert, H.; Wittern, K.-P. Ber. Bunsenges. Phys. Chem. 1978, 82,1054. (13) Michel, D.; Meiler, W.; Pfeifer, H. J. Mol. Catal. 1975, 1 , 85. (14) Pfeifer, H. Phys. Rep. 1976,26,293. (15) Unland, M. L.; Freeman, J. J. J. Phys. Chem. 1978, 82, 1036. (16) Wright, C. J.; Riekel, C. Mol. Phys. 1978, 36, 695. (17) Bezue, A. G.; Kiselev, A. V.; SedlaEek, Z.; Pham Quang Du Trans. Faraday SOC.1971,67,468. (18) Schirmer, W.; Tha", H.; Stach, H.; Lohae, U. Spec. Pub1.Chem. SOC. 1980, No. 33, 204.
0 1982 American Chemical Society
