J. Phys. Chem. 1989, 93, 8025-8031 elimination occurs and is consistent with our proposed matrix mechanism.
Conclusion Photosensitization of saturated hydrocarbons with Hg(3P) has been demonstrated in cryogenic matrices. The principal products result from molecular elimination, H2 + CH2CHF from E F and HX-C2H4hydrogen-bonded A complex from ECl and EBr, though the mechanism is probably not concerted. Weaker product bands have provided further strong evidence of metal atom insertion into C-Cl and C-Br bonds. This process is unique to the matrix photochemistry where the cage effect and low-temperature environment may act to stabilize the haloethylmetal compounds thus formed. In agreement with previous work, no insertion into C-H and C-F bonds has been observed. A dependence on the electronic structure of the reactant molecule
8025
is strongly suggested as an explanation for the selectivity of the insertion process. Finally, simple matrix statistics have suggested that the sensitizer and acceptor need not be in close proximity for elimination processes to occur. For ethyl chloride, energy transfer over distances as large as 27 A, presumably in several smaller steps, is consistent with the experimental results.
Acknowledgment. This work was supported by the Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division, of the U S . Department of Energy under Contract No. DE-AC03-76SF00098. Registry No. H3CCH2CI,75-00-3; H3CCH2F,353-36-6; H3CCH2Br, 74-96-4; Hg; 7439-97-6; HCI-C2H4, 54419-95-3; CZH,CI, 75-01-4; HCI-C2H2, 52218-20-9; H,CCH2HgCl, 107-27-7;HBr-C2H4, 5441999-7; H,CCH2HgBr, 107-26-6; C2H3F,75-02-5; H3CCH21,75-03-6; 1,2-difluoroethane,624-72-6; 1,2-dichloroethane,107-06-2.
Stochastic Models of Dlffusion-Controlled Ionic Reactions in Radiation-Induced Spurs. 2. Low-Permittivity Solvents Nicholas J. B. Green,* Michael J. Pilling, Physical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, United Kingdom
Simon M. Pimblott,* Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556
and P. Clifford Mathematical Institute, St. Giles, Oxford OX1 3LB, United Kingdom (Received: March 13, 1989; In Final Form: June 15, 1989)
The diffusion-controlled recombination of small clusters of ions in low-permittivity solvents is considered. A Monte Carlo simulation technique used previously for high-permittivity solvents is introduced and discussed briefly. The independent reaction times (IRT) simulation method is described, and its implementation for the kinetics of ion clusters is detailed. The IRT approximationis tested against the full Monte Carlo simulation for a variety of random and nonrandom initial configurations and against alternative analytic theories. We find the IRT method to be surprisingly accurate under these stringent conditions.
1. Introduction In a number of recent papersI4 we have demonstrated the necessity for a stochastic treatment of the diffusion-controlled kinetics of small clusters of reactive particles, such as occur in radiation tracks. During these investigations we formulated three alternative models of the diffusion and reaction processes. The first method is a simulation technique that models the trajectories of the diffusing particles using a formalism equivalent to the diffusion equation for the system of interest; particles react when they encounter each other. This type of simulation, if performed correctly and for a sufficient number of realizations, gives an accurate solution to the many-body diffusion equation for the cluster but is expensive in computational resources. The second model, the IRT method, is a much more rapid simulation tech(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.; Oldfield, M. J.; Pilling, M. J.; Pimblott, S. M. J . Chem. SOC.,Faraday Trans. I 1986,82, 2673. (4) Clifford, P.; Green, N. J. B.; Pilling, M. J.; Pimblott, S. M. J . Phys. Chem. 1987, 91, 4417.
0022-365418912093-8025$01.50/0
nique that relies on the independent pairs approximation: reaction times for each pair are generated independently from the corresponding interparticle separations. The third theory is an analytic model that requires the formulation and solution of a master equation to describe all the possible reactions in the cluster. In our earlier studies we investigated the kinetics of clusters of r a d i ~ a l s l -and ~ of ions in solvents with high relative permittivities, such as waters4 In this paper we turn our attention to clusters of ions in nonpolar solvents with dielectric properties similar to those of alkanes. The Monte Carlo simulation technique requires little modification for these systems. However, the IRT method requires a method for simulating geminate recombination times from the correct marginal probability distribution. In our previous work this could be done either using the exact analytic solution (for radicals) or using an accurate, approximate analytic solution with the same structure (for ions in water). For lowpermittivity solvents no such convenient approximation is known except at very small initial distance^,^ and an alternative method must be used. Furthermore, in low-permittivity media, where the Coulombic interion forces extend over large distances, many-ion (5) Green, N. J. B.; Pilling, M. J.; Clifford, P. Mol. Phys., in press.
0 1989 American Chemical Society
8026
The Journal of Physical Chemistry, Vol. 93, No. 24, 1989
interactions are significant and the description of reactions in a cluster using pair reaction probabilities may be inappropriate. The IRT approximation might therefore be expected to fail for these systems. Conventional deterministic theories of radiation chemistry@ are not widely used in these systems and have already been shown to have serious shortcomings.I4 It is generally considered that ion clusters degrade rapidly to single pairs, and hence the observable kinetics can be described by the theory of geminate recombination.I0 However, in light of recent simulation results of Bartczak and H ~ t n m e l , ’ l - ’this ~ view is an oversimplification and there are important multipair effects to be discovered. Unfortunately, simulation is currently too computationally expensive to use routinely in these investigations, and for this reason we wish to test more approximate descriptions, such as the I R T model against simulation results, to determine the extent of their reliability and utility. We also consider the first-passage time approach of WilliamsI4 and Freemanl5-I7 which only applies to single-pair clusters. In section 2 we describe briefly the Monte Carlo technique for simulating diffusive trajectories and compare it with the similar technique of Bartczak and Hummel. In section 3 we discuss the problem of geminate recombination and in particular the numerical technique we employ to generate reaction times for the IRT method. The Monte Carlo simulation and the method to be used for generating reaction times in the IRT model are tested against numerical solutions for the recombination of an isolated pair. In section 4 we describe the IRT method and its implementation. We also compare the results of the IRT approximation with those from the Monte Carlo simulations for several different types of initial configurations. Comparison is also made with the firstpassage time approach of Williams14 and Freeman.15-17 Finally, we discuss systems where the IRT method works well and systems where it fails. 2. Monte Carlo Simulation It has been established in the mathematical literature for some time now that if the spatial probability distribution function of a particle obeys a diffusion equation, then the path of the particle evolves according to a stochastic differential equation.’* Thus, if a particle at position R has a diffusion coefficient D and feels an average force F, then its infinitesimal change in position in a time period dt is given by
F
+
dR = -D dt (2D)’I2dW (1) kT The first term is a deterministic “drift” term describing the influence of the force F. The second term describes the random force of Brownian motion and is the “derivative” of a Wiener process in three dimensions. A suitable rule must be introduced to interpret dW, which must be constructed as the limiting increment for a sequence of random processes converging to a Wiener process. Two constructions can be distinguished by reference to a process in which successive normally distributed jumps are made with a given rate. In the Ito interpretation the random variable of the jump process remains constant until the jump occurs and then jumps instantaneously to a new value. In the Stratonovich interpretation the random variable is interpolated linearly between the two values. The choice of interpretation only has a substantive effect when the diffusion coefficient depends upon the configuration, which is not the case here. Further (6) Samuel, A. H.; Magee, J. L. J. Chem. Phys. 1953, 21, 1080. (7) Schwarz, H. A. J. Phys. Chem. 1969, 73, 1928. ( 8 ) Mozumder, A. J. Chem. Phys. 1971, 55, 3020. (9) Burns, W. G.; Sims, H. E.; Goodall, J. A. B. Radiat. Phys. Chem. 1984, 23, 143. (IO) Hummel, A. Ado. Radiat. Chem. 1974, 4 , 1 . (1 1) Bartczak, W.; Hummel, A. Radiat. Phys. Chem. 1986, 27, 71. (12) Bartczak, W.; Hummel, A. J. Radioanal. Nucl. Chem. 1986, 101, 299. (13) Bartczak, W.; Hummel, A. J. Chem. Phys. 1987, 87, 5222. (14) Williams, Ff. J . Am. Chem. SOC.1964,86, 3954. (15) Freeman, G. R.; Fayadh, J. M. J. Chem. Phys. 1965, 43, 86. (16) Freeman, G. R. J. Chem. Phys. 1967, 46, 2822. (17) Freeman, G. R. Ado. Chem. Ser. 1968, No. 82, 339.
Green et al.
0 0
I 0
2 0
3.0
4.0
log (time / ps)
log ( t i m e l p s )
P
a
rc = -29.0 nm
0
rc =
w -0 . 0 0.0
1.0
- 2 . 3 nm 3.0
2.0
4.0
w
0.0 0.0
1.0
2.0
3.0
4.0
log ( t i m e l p s )
log ( t i m e l p s )
Figure 1. Geminate recombination of an ion pair in intermediate- and low-permittivity solvents; (0)numerical solution of the backward equation for W(ro,r);(-) Monte Carlo simulation ( l o 5 realizations). (a, r, = -1.7 nm) Methanol: D‘= 1.0 X 10” m2 s-l; initial separation, ro = 2.1 nm; reaction distance, R = 0.5 nm. (b, rc = -2.3 nm) Ethanol: D’ = 1.0 X m2 s-I; initial separation, ro = 2.1 nm; reaction distance, R = 0.5 nm. (c, r, = -13.0 nm) Diethyl ether: D’= 1.0 X m2 s-’; initial separation, ro = 2.1 nm;reaction distance, R = 0.5 nm. (d, r, = -29.0 nm) Hexane: D’= 1.O X m2 s-I; initial separation, ro = 2.0 nm; reaction distance, R = 1.0 nm.
detailed and more rigorous discussion can be found in the text by Arnold.Is The stochastic differential equation is “solved” in the stochastic sense that a trajectory is simulated from it. To achieve a solution, it is necessary to discretize the time. Several methods of discretization are possible1g-21but we choose the simplest, described by Rumelin,2° which is a generalization of the Euler method for ordinary differential equations
F
bR = -Dbt kT
+ (2D6t)’/’N(0,1)
in which 6t represents the discrete time interval and N ( 0 , l ) is a vector of independent normally distributed random numbers of mean zero and unit variance. For uncharged particles, the force F is zero and the process is therefore a scaled Wiener process.22 Under these conditions, the discretized stochastic differential equation, eq 2 , is exact and the trajectories can be sampled exactly. When interparticle factors are present, however, this is no longer true, and the discretization represents an approximation that converges to the true stochastic differential equation in the limit of small 6t. Operationally, 6 t must be sufficiently small that F is effectively constant throughout the time step. When the forces are Coulombic, eq 2 becomes
zizjlrcl bRi = -C-rijDibt J
rij3
+ (2Dibt)1/2N(0,1)
(3)
for particle i , where ri, is the vector from particle i (charge zi) to particle j (2,) and IrJ is the Onsager distance, at which the Coulomb potential energy of interaction between two singly charged ions is equal to * k T . The method used for calculating 6 t and other details of the implementation have been discussed (18) Arnold, L. Stochastic Differential Equations; Wiley: New York, 1971. (19) Milshtein, G. N. Theory Probab. I r s Appl. (Engl. Transl.) 1982, 23, 396. (20) Riimelin, W. SIAM J. Numer. Anal. 1982, 19, 604. (21) Pardoux, E.; Talay, D. Acra Appl. Math. 1985, 3, 23. (22) Green, N. J. B.; Pilling, M. J.; Pimblott, S. M. Radiat. Phys. Chem. 1989, 34, 105.
Diffusion-Controlled Ionic Reactions previously4 and will be considered only briefly here. The time step is calculated in such a way that the interparticle forces are unlikely to change appreciably during a jump: For each particle the time step is calculated such that the contribution to the drift vector from each component force in the summation of eq 3 will change by less than 10% with 95% confidence. The smallest of these time steps is used subject to a minimum cutoff. The method is efficient but is possibly open to criticism as interparticle forces reinforce in complicated ways; however, tests run with more stringent criteria give the same results. In applying the Monte Carlo simulation to cluster reactions of radicals, or of ions in water, it was necessary to consider the possibility of encounter and reseparation during a time step, using the Brownian bridge encounter p r ~ b a b i l i t y . ~A*similar ~ ~ ~ ~ approach ~ could be employed for alkanes, since 6t has been chosen so that the drift vector is approximately constant, and the distance between the particles behaves locally like a Wiener process with constant drift. However, because interparticle forces are so strong in alkanes, it is not necessary to allow for reseparation because particles sufficiently close to encounter during a step will almost certainly encounter and therefore react within the next few time steps. A typical comparison of the simulation results against a numerical solution for a single pair is shown in Figure 1. The simulation technique of Bartczak and Humme11*-13is very similar to that employed here. The main differences are as follows: (i) They use a uniform discretization of the random force in the stochastic differential equation, as suggested by Milshtein,19 rather than the normal (Le., Gaussian) discretization. For small time steps both discretizations will converge to the true sample paths. (ii) They do not use a Brownian bridge interpolation, as we do. Although, as argued above, it is not necessary here, it is very important to do this for radical systems and for those with weak forces. (iii) They use a small fixed time step, whereas we allow the time step to vary with the interparticle separations. The variable time step method is safer when the two particles are very close together (i.e., it gives smaller time steps) but may be less safe in more extended configurations of large numbers of ions. It is certainly more rapid to compute as it enables relatively large time steps to be taken when the particles are far apart. Results of the two methods have been compared previously.zz In all cases analyzed so far they have given identical kinetics. Schmidt et a1.25*z6 have also employed a simulation technique, but their fixed time step of 40 ps is too long to reproduce kinetics accurately and can lead to particles jumping through each other and oscillating in position.
3. Geminate Recombination An essential part of the IRT method which is discussed in section 4 is the recipe for generating a random reaction time for a pair of particles from their initial separation. As stated earlier, either exact solutions or good analytic approximations to the time-dependent geminate recombination probability are available for all the problems analyzed thus However, except for rather large4 or for very small5 initial distances, this is not the case for low-permittivity solvents. We therefore require a numerical method that can be implemented rapidly and accurately, and for this reason we choose to interpolate in a look-up table whose preparation and use will now be described. The time-dependent geminate recombination probability, W, from an initial interparticle separation ro obeys a backward diffusion which is adjoint to the usual forward Debye-Smoluchowski equation29 a w / a t = D'[V,Z w - vow.voq (4) Clifford, P.; Green, N. J. B. Mol. Phys. 1986, 57, 123. Green, N. J. B. Mol. Phys. 1988,65, 1399. Schmidt, K. H. Chem. Phys. Lett. 1983, 103, 129. Sauer, M. C.; Schmidt, K. H.; Liu, A. J. Phys. Chem. 1987,91,4876. Kolmogorov, A. N. Math. Ann. 1931, 104, 415. (28) Onsager, L. Phys. Reu. 1938, 54, 554. (29) Debye, P. W. Trans. Electrochem. SOC.1942, 82, 265.
(23) (24) (25) (26) (27)
The Journal of Physical Chemistry, Vot. 93, No. 24, 1989 8027 where D'is the relative diffusion coefficient of the pair, U is the dimensionless potential energy in units of kT, and the operator Vo operates in the space of the initial interparticle vector r,. After application of the usual requirements of spherical symmetry and substitution of the Coulomb potential U = rc/ro,eq 4 becomes
with the convention that rc is negative for an attractive force and positive for a repulsive force. The backward equation ( 5 ) can be taken through a number of transformations, as discussed p r e v i o ~ s l y . ~Firstly, ~ a dimensionless coordinate system is chosen
x = 2r0/rc
T
= 4D't/r?
which is the system introduced originally by Umberger and LaMer3' and used by Hong and N o ~ l a n d i . ~The ~ equation becomes
(7) Separations of interest are generally in the region to lo-* m and times of interest from to lo4 s. For a typical nonpolar solvent of Ircl = 29 nm and with a relative diffusion coefficient mz S-I, these regions of interest translate to 7 X of 1.5 X < x < 7 X lo-* and 7 X < T < 7 X lo-'. The numerical solution must therefore encompass at least these ranges of distance and time. A second transformation is to separate out the infinite time asymptote Wm(ro) which is known analytically:
where R is the encounter distance. For low-permittivity solvents W, is very close to zero for ions of like charge (rc > 0) and very close to the Onsager formulaZ8for ions of unlike charge (rc < 0) W, = 1 - exp(r,/ro)
(9)
If Wm(ro) is factored out of W(ro,t),then the remaining function increases with time from 0 to 1 and is a proper probability distribution function V ( r 0 , t ) .
W O =J )Wm(ro)W*(ro,t)
(10)
W* represents the reaction probability as a function of time conditioned on ultimate reaction33and in dimensionless coordinates obeys the backward equation
aw
-=37
azw + -(x 2 ax2
xz
aw
- coth ( l / ~ ) ) -
ax
(11)
which is even in x.33 Many attempts have been made to solve the Debye-Smoluchowski equation, either exactly or approximately. Of these, the most complete is the series solution of Hong and N o ~ l a n d for i~~ the Laplace transform of W. Calculation of the series is computationally nontrivial, and a numerical inversion of the Laplace transform is still necessary. A number of simpler approximations have been p r ~ p o s e d ,but ~ ~ even , ~ ~ here the Laplace transforms cannot be inverted analytically, apart from the lowest orders of approximation, when they yield essentially the same approximation as ref 30 which only holds, at reasonable times and separations, for high-permittivity solvents. Numerical treatments have also (30) Clifford, P.; Green, N. J. B.; Pilling, M. J. J. Phys. Chem. 1984,88, 4171. (31) Umberger, J.; LaMer, V. J. Am. Chem. SOC.1945,67, 1099. (32) Hong, K. M.; Noolandi, J. J. Chem. Phys. 1978, 68, 5163. (33) Clifford, P.; Green, N. J. B.; Pilling, M. J. Chem. Phys. Lert. 1982, 92, 101. (34) Pedersen, J. B.; Sibani, P. J. Chem. Phys. 1981, 75, 5368. (35) Raaen, S.; Hemmer, P. C. J . Chem. Phys. 1982, 75, 5368.
8028 The Journal of Physical Chemistry, Vol. 93, No. 24, 1989
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Figure 3. Recombination probability of a Gaussian ion pair: ( 0 )variable time step Monte Carlo simulation using normally distributed random flights with bridge interpolation;(m) fixed time step Monte Carlo simulation using uniformly distributed random flights;'l-13 (-) IRT simulation; (---) kinetics predicted by Freeman's single-pair model.ls-"
0
4. The Independent Reaction Times (IRT) Model The basis of the IRT method has already been discussed several and the only difference between previous implementations and that used in the present study is the method of generating reaction times, which has been discussed in section 3. We will therefore only describe the method briefly here. The basis of the I R T method is the independent pairs approximation, which is also implicit in the usual theory of diffusion-controlled rate coefficient^.^^ In the IRT method a slightly less strict approximation is made as the initial configuration is always geometrically realizable, whereas in other methods the independence assumption is extended to zero time.22 An initial configuration of particles is generated, either at preassigned positions or at random positions sampled from a model distribution. All the distances between potentially reactive particles are calculated (in this case all the cation-anion distances). For each such distance, taken in turn, a random reaction time is generated for a geminate pair with that initial separation, according to the method outlined in section 3. This is done independently for each pair, irrespective of the positions of any other particles. It is this independence which forms the basic approximation of the IRT method. If the configuration contains No cations and No anions initially, then at this stage NJ distances have been calculated along with an associated reaction time for each distance. The minimum time in the ensemble is found, and the corresponding pair is removed from further consideration. All the 2(N0 - 1) pairs containing either of the reacted particles are knocked out, leaving (No- 1)2 remaining times. The minimum of these times is found and the corresponding particles are removed and so on until no further reactions can occur. One realization of the IRT simulation therefore involves the generation of No2random numbers and a few searches through an array. The method is obviously much faster than the full simulation but involves the compromise of the IRT approximation. It is the purpose of this paper to present comparisons between the IRT simulation and the full Monte Carlo simulation for the purpose of testing the approximation as stringently as possible. These comparisons are made in the next section. 5. Results and Discussion Most of the configurations of ions considered in this paper are sampled from Gaussian distributions. In order to check our procedures for such configurations, we present in Figure 3 results for a single Gaussian pair. It will be seen that agreement between the IRT method, the Monte Carlo simulation, and a simulation following the methodology of Bartczak and H ~ m m e l " - ' ~is very satisfactory, but the kinetics modeled by the first-passage time approach of using the deterministic approximation of Williams,I4 is unsatisfactory and will not be considered further. (36) (37) (38) (39)
Ludwig, P. K. J . Chem. Phys. 1969, 50, 1787. Hummel, A.; Infelta, P. P. Chem. Phys. Letr. 1974, 24, 559. Delaire, J. A.; Croc, E.; Cordier, P. J . Phys. Chem. 1981,85, 1549. Pimblott, S.M. D. Phil. Thesis, Oxford University, 1988.
(40) Clifford, P.; Green, N. J . B.; Pilling, M. J. J . R. Star. SOC.E 1987, 49,
266.
(41) Green, N. J. B. Chem. Phys. Lett. 1984, 107, 485.
The Journal of Physical Chemistry, Vol. 93, No. 24, 1989 8029
Diffusion-Controlled Ionic Reactions
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Figure 4. Comparison of IRT simulation and Monte Carlo simulation for the kinetics of a two-ion-pair spur: ( 0 )variable time step Monte
Carlo simulation using normally distributed random flights with bridge interpolation; (-) IRT simulation. Initial positions from concentric spherical Gaussians. r, = 29.0 nm; D+ = D-= 5.0 X lo4 m2s-'; reaction distance, R = 1.0 nm. (a, top) u+ = 1.0 nm; u- = 1.0 nm. (b, bottom) u+ = 1.0 nm; u- = 8.0 nm.
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Figure 5. Comparison of IRT simulation and Monte Carlo simulation for the kinetics of a three-ion-pair spur: ( 0 )variable time step Monte
Carlo simulation using normally distributed random flights with bridge interpolation; (-) IRT simulation. Details as in Figure 4. Figure 4 shows a comparison for the first nontrivial case, that of a two-pair Gaussian spur in hexane. The cations and anions are distributed randomly from spherical Gaussian probability densities of standard deviation u+ and u-, respectively. Two situations are considered: in one of these both cation and anion distributions have standard deviations of 1 nm, packing all four ions into a compact volume. The reason for choosing such a tight configuration was to attempt to maximize the effects of the interacting strong Coulombic forces and to provide a stringent test of the IRT approximation. The excellent agreement seen in Figure 4 is therefore encouraging. The second configuration considered is one where the width of the anion distribution has been increased
Figure 6. Comparison of IRT approximation with Monte Carlo simulation for nonrandom initial configurations: (0)variable time step Monte Carlo simulation using normally distributed random flights with bridge interpolation;(-) IRT simulation. Initial positions as shown in figure. r, = 29.0 nm; D+ = D- = 5.0 X lo4 m2 s-*; reaction distance, R = 1.O nm.
to 8 nm. It might be expected that this configuration would result in faster reaction than the IRT model could predict, because the central cation distribution will initially have a net charge of +2 and so will attract the anions more strongly than a single cation could. However, once again Figure 4 shows, surprisingly, that this is not a significant effect. A similar study of a three-pair cluster is shown in Figure 5 . The differences are clearly greater than in the two-pair case, but only amount to 3% at worst. The unexpected success of the IRT approximation for ions in low-permittivity solvents clearly needs further investigation. For this reason we examined a number of nonrandom configurations in which the initial positions of four ions were fixed. Results are shown in Figure 6. Each of the initial configurations has been chosen so that, in some respect, the Coulombic interion forces reinforce each other strongly. The IRT model treats each pair independently and so ignores this reinforcement. Even for these configurations, the IRT method does not go badly wrong, although the errors would not be acceptable if these configurations were physically important. The results shown in Figures 4 and 5 demonstrate that they do not play an important role in Gaussian spurs.
8030 The Journal of Physical Chemistry, Vol. 93, No. 24, 1989
Green et al.
0.05 A
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lation for Gaussian clusters with different spatial distributions, but the same distribution of distances: (0)variable time step Monte Carlo simulation using normally distributed random flights with bridge interpolation; (-) IRT simulation. Initial positions from concentric spherical Gaussians. r, = 29.0 nm; D+ = D-= 5.0 X l@ m2 s-'; reaction distance, R = 1.0 nm. (a, top) u+ = 3.0 nm; u- = 4.0 nm. (b, bottom) u+ = 0.0
5.0
2.5
)
(
Figure 7. Comparison of IRT approximation with Monte Carlo simu-
m
10.0
7.5
Cation separation / nm Figure 8. IRT simulation of escape yield as a function of cation separation and time for two-pair clusters. Geminate anions positioned on spherical shells at a separation of 5.0 nm from their partners. r, = 29.0 nm; D+ = D- = 5.0 X IO4 m2 s-I; reaction distance, R = 1.0 nm.
nm; u- = 5.0 nm.
n 776 ".L I I
Another interesting investigation suggests itself. The IRT model depends only on the distances in the initial configuration; in a Gaussian configuration the marginal distribution of each distance depends only on the sum u + ~ u - ~ ,and all the distances are identically distributed. Figure 7 shows results for two configuu - ~ ) =~ 5.0 / ~ nm, but u + and ~ are rations for which different: in one case u+ = 3 nm and u- = 4 nm, and in the other case u+ = 0 and u- = 5 nm. In the former system the two sets of results agree very well, whereas in the latter system the IRT method slightly underestimates the reaction rate. Interestingly, both methods give the same free-ion yield. The small discrepancy may result from the pathologically unstable initial configuration where the cations are coincident and so repel each other considerably. Bartczak and H ~ m m e l " - ' ~have simulated similar systems, but with varying numbers of single-pair clusters. One intriguing finding they report is that, for a two-pair situation where the electron-cation distribution is a spherical &function, the free-ion yield, expressed as a function of the separation of the cations, goes through a maximum. The reasons for this are not entirely clear, but it is of obvious interest to investigate whether the IRT method can describe a similar anomaly. Yields at several different times are shown in Figure 8, using the same type of initial distribution as Bartczak and Hummel. The figure shows that the IRT method does describe the observed maximum and that the position of the maximum is independent of time. A similar study of two Gaussian pairs as a function of cation-cation separation is shown in Figure 9. It will be seen that the hump has almost been washed out by the extra randomness in the initial distribution. Use of increasingly narrow Gaussians for each anion distribution progressively reinstates the hump.
0.250
+
t
0.225
+
6. Conclusions In this paper we have reported investigations of the recombination kinetics of ion clusters in low-permittivity solvents. The main result of the investigation is the surprising accuracy of the IRT approximation under these stringent conditions. We have reported a wide range of tests of the approximation for various initial configurations, and although it is possible to find configurations where the approximation is unacceptable, these seem to
I
a n a A
n
t
0.01 ps
a 4
a
a
a a a
a
a
0.200 n
z v
0.125
0.lOOb
0.0751
0.050 0.0
2.5
5.0
7.5
10.0
12.5
15.0
Cation separation / nm Figure 9. IRT simulation of escape yield as a function of cation separation and time for two-pair clusters. Geminate anions positioned from a spherical Gaussian of standard deviation 5.0 nm,centered on their partners. Details as in Figure 8.
be rare in the random configurations usually considered in radiation chemistry. It should be borne in mind, however, that the work here is subject to a number of basic approximations, which do not hold for all alkane-like solvents. Most notably, and in common with Bartczak and Hummel,I1-l3Freeman,"-17 and we have assumed that particle motion is a diffusion process with linear response to the Coulomb potential. Shin and K a ~ r a used 1 ~ ~the kinetic theory of particle motion to study the dynamics of a reactive radical pair and to examine the breakdown of the diffusion approximation. More recently, T a ~ h i y has a ~made ~ ~ ~a study of ~
(42) Shin, K. J.; Kapral. R.J . Chem. Phys. 1978,69, 3685. (43) Tachiya, M. Chem. Phys. Lert. 1987, 127, 55. (44) Tachiya, M. Radial. Phys. Chem. 1988, 32, 37. (45) Tachiya, M.; Schmidt, W. F. J . Chem. Phys. 1989, 90, 2471.
J. Phys. Chem. 1989, 93, 8031-8037
8031
sion-controlled ion recombination kinetics in low-permittivity media. This finding should facilitate the investigation of kinetics in the radiation chemistry of low-mobility alkanes as the I R T simulation can be realized much more rapidly than the full random flight simulations, which are at present the only realistic alternative. The surprising success of the independent pairs approximation under these conditions merits further investigation but also suggests the possibility of an analytic or numerical theory which, being based on the same approximation, should be equally successful. Such a description is part of our current research, and we hope to be able to describe it in a future publication.
the diffusion approximation by performing classical simulations of electron random flights with a large mean free path. For ion-ion recombination and electron-ion recombination in low-permittivity solvents, the mean free path is low and the diffusion approximation is probably valid. The second approximation is the implicit assumption that there is a linear response to interparticle forces. Ion mobility is field-dependent and hence so is the relative diffusion coefficient. The effect has been discussed by Mozumder,& Baird:' and Rice4*but is not usually considered to be important in geminate recombination and generally Subject to these limitations, however, we have shown that the IRT simulation is a remarkably successful description of diffu-
Acknowledgment. The authors thank Dr. A. Mozumder for suggesting the study of nonrandom configurations. The research described was in part funded by S.E.R.C., U.K.A.E.A. Harwell, and the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-3177 from the Notre Dame Radiation Laboratory.
(46) Mozumder, A. J. Chem. Phys. 1976, 65, 3798. (47) Baird, J. K.; Anderson, V. E.; Rice, S. A. J . Chem. Phys. 1977,67, 3842. (48) Rice, S. A. Diffusion-LimitedReactions; Elsevier: Amsterdam, 1985.
-
-
Temperature Dependence of Termolecular Association Reactions N,+ 4- 2N, N4+ 4- N, and 0,' 4- 20, 0,' 4- 0, Occurring in Free Jet Expansions below 20 K L. K. Randeniya, X. K. Zeng, R. S. Smith, and M. A. Smith* Department of Chemistry, University of Arizona, Tucson, Arizona 85721 (Received: March 30, 1989; In Final Form: June 19, 1989)
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A new experimental technique that combines supersonic expansions and laser ionization mass spectrometry has been used to measure the termolecular association rate coefficients, k3, for the gas-phase reactions Nz+ + 2Nz N4++ N2 and 02+ 2 0 2 04+ + O2below 20 K. The continuity of the proposed inverse temperature law of k3 = CT" was observed down to 4 K for both reactions. Statistical analysis of our results along with the higher temperature data obtained by other authors gives the values of 1.92 0.15 and 1.86 0.1 5 for n for the reactions of N2+and 02+, respectively. The results of phase space calculations are presented for these reactions and excellent agreement between theory and experiment is obtained.
+
-
*
*
Introduction Except in those instances where cluster ions are generated by the ionization of neutral clusters or via surface bombardment/ sputtering techniques, they are produced via a series of association reactions. The dynamics of ion-molecule association at the low temperatures and pressures that prevail in the interstellar medium has become an interesting astrophysical problem.I4 The radiative association reaction, which is thought to be fast at low temperatures, has been invoked by many authors to explain the abundance of a variety of molecules that have been observed in the interstellar medium. The measurement of rate coefficients of the related termolecular association process at low pressures and temperatures is expected to contribute to the advancement of these models. The study of the rates of ternary association reactions is particularly interesting since at low pressures the third-order rate coefficient follows an inverse temperature dependence given by
Inaccuracies can occur whenever the temperature range of measurements is fairly small or the measurements are not made in the low pressure limit where k3 is independent of the pressure of the third body. We have developed a technique that enables the measurement of ternary association rates for reactions occurring in the core of a supersonic expansion. This technique is unique since it extends the low-temperature limit to approximately 4 K. The method also permits the study of these reactions in the low-pressure limit which is otherwise obtained with difficulty by more conventional techniques. In this paper we report the study of the following reactions:
+
N2+ 2N2
N4+
+ N2
(2)
02+ + 2 0 2 04++ O2 (3) The above reactions have been the subject of much of the research work done to date on ternary association reactions. Both reaction rates have been studied down to about 20 K, yet for reaction 3 only lower limits to the rate coefficients have been obtained at temperatures below 70 K.8-" Extensive theoretical work has been done on the temperature dependence of the ternary association rate coefficients and has been largely successful in explaining the observed behavior for a wide range of association However, since low-
where C and n are constants for a given reaction. The value of n for a variety of such reactions has been found to vary from 0.4 to about 6 with n increasing with the complexity of the r e a ~ t a n t s . ~ ' A fair amount of controversy surrounds the range of experimental values reported for n for many association reactions. Williams, D. A. Astrophys. Lett. 1972, 10, 17. Black, .I.H.; Dalgarno, A. Astrophys. Lett. 1984, 15, 79. Herbst, E.; Klemperer, W. Astrophys. J. 1973, 185, 505. Herbst, E.; Leung, C. M. Astrophys. J . 1986, 310, 378. (5) Castleman, A. W., Jr.; Keesee, R. G. Chem. Reu. 1986, 86, 589. (6) Adams, N. G.; Smith, D. In Reactions of Small Transient Species; Fontijn, A., Clyne, M. A. A., Eds.; Academic Press: London, 1983; p 31 1. (7) Liu,S.; Jarrold, M. F.; Bowers, M. T. J. Am. Chem. Soc. 1985, 89, (1) (2) (3) (4)
(8) Van Koppen, P. A. M.; Jarrold, M. F.; Bowers, M. T.; Bass, L. M.; Jennings, K. R. J . Chem. Phys. 1984, 81, 288. (9) Bohringer, H.; Arnold, F. J. Chem. Phys. 1982, 77, 5534. (10) Bohringer, H.; Arnold, F.; Smith, D.; Adams, N. G. Int. J . Mass. Spectrom. Ion Phys. 1983, 52, 25. ( 1 1) Rowe, B. R.; Dupeyrat, G.; Marquette, J. B.; Gaucherel, P. J . Chem. Phys. 1984, 80, 4915. (12) Bates, D. R. J. Phys. B 1979, 12, 4135.
3127.
0022-3654/89/2093-8031$01.50/0
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0 1989 American Chemical Societv
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