J. Phys. Chem. 1990, 94, 251-258 and by assuming that the effect of excess energy on each of these quantum yields is the same. The point in Figure 9 for TMPD has been corrected on this basis. Results (not shown) obtained by using the same chemical systems and 248-nm light,S1which increases the excess energy, are consistent with the curve drawn in Figure 9 and support the leveling off above 2.5 eV. The foregoing analysis of the effect of excess energy is qualitative, because structure is observed in the photoionization quantum yields crffi) of solutions of TMPD" and anthracenes2 as a function of wavelength; however, the general trend of a sharp increase in yield with excess energy followed by a leveling off is also seen in the latter two studies. It is of interest to note that the magnitudes of thefiff, values for single-photon ionization of TMPD in isooctane observed by Choi et aI.lI are within a factor of 2 of our value of 0.039 (at 2.7-eV excess energy) for excess energies 0.4-1.7 eV lower than we used. Their value at 2.7 eV is about an order of magnitude lower than 0.039, so our result suggests that the decrease infffi observed by Choi et al. above 7-eV excitation energy may be an artifact. The general question of how the excess energy available in the photoionization event is related to the distance traveled by the ejected electron and if it depends on whether the process involves one or two photons remains to be completely answered. Holroyd et a1.Is and Bottcher and SchmidtI6J7argued that their results Sauer, M. C., Jr.; Schmidt, K. H.; Lu,Y.Unpublished results. (52) Tweeten, D. W.; Lipsky, S.J . Phys. Chem. 1989, 93, 2683. (51)
25 1
on structure in the photoconductivity spectra indicate changes in the energy of the ejected electron as a function of the Rydberg level reached in the excited ion, whereas Tweeten and L i p ~ k y ~ ~ have presented arguments based on their recent results that vibrationally excited states of the positive ion are responsible for the structure.
Summary The two-photon nature of the process at 308 nm has been established, and overall quantum yields of free ions for the ionization of various solutes in several nonpolar liquids have been measured. From the overall quantum yields and measurements of light absorption, the quantum yields of free ions from the absorption of the second photon by the excited states have been measured. The quantum yield results have been combined with photoconductivity decay dynamics to yield information on the distribution of anion-cation separation distances and the primary quantum yield for ionization of the excited states. The results obtained here were compared with the results of single-photon ionization in similar chemical systems and were found to be generally consistent. An $-exponential distribution is consistent with the results of this study and several other photoionization studies, in contrast to the case of radiolysis, where an exponential distribution is indicated. Registry No. TMPD, 100-22-1; SF,, 2551-62-4; anthracene, 120-12-7; fluoranthene, 206-44-0; coronene, 191-07-1.
Stochastic Modeling of Fast Kinetics in a Radiation Track N. J. B. Green, M. J. Pilling, Physical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ, U.K.
S. M. Pimblott,* Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556
and P. Clifford Mathematical Institute, St. Giles, Oxford, OX1 3LB, U.K. (Received: February 27, 1989; In Final Form: July 21, 1989)
A method is presented for calculating the time-dependent G values and product yields formed in a fast electron track. The calculations are based on an efficient stochastic simulation technique-the independent reaction times (IRT) method-which has previously been developed and validated in detail. The IRT method combines the reactants in pairs and selects reaction times randomly from the appropriate marginal reaction time distributions. Reactions are counted sequentially, starting with the shortest reaction time and limiting subsequent consideration to pairs in the set of remaining particles. The technique is efficient and accurate and is capable of dealing with ionic species, partially diffusion-controlled reactions, and reactive products. Our previous calculations based on the IRT method have considered only reactions of small numbers of particles in isolated spurs. A new simulation protocol is presented that efficiently handles the extended distribution and the large number of reactive particles encountered in a radiation track while making use of any isolation that exists on appropriate timescales. The simulated G values at zero time and 0.1 p s for a model 22-MeV electron track differ significantly from experimental values, and the discrepancy is ascribed to deficiencies in the initial distribution.
1. Introduction
to ionizing radiation, reactive species When a liquid is are generated in an initially nonhomogeneous distribution. Traditionally, for low LET radiation, this distribution has been described as consisting predominantly of isolated clusters known as spurs, which contain small numbers of reactive species.' More extended regions of energy deposition occur at track ends or in secondary tracks ( h a y s ) . ' A significant proportion of the
chemical reactions following radiolysis occurs at early times when the nonhomogeneous distribution has not yet relaxed by diffusion. In Previous papers we have demonstrated that conventional deterministic t i w " of these early processes, based upon macroscopic kinetics, are inappropriate and that stochastic models must be For example, we showed that, although the (1) Mozumder, A.; Magee, J . L. Radio(. Res. 1966, 28, 215.
0022-3654/90/2094-025 1$02.50/0 0 1990 American Chemical Society
252
The Journal of Physical Chemistry, Vol. 94, No. I , 1990
decays of the primary radical species are modeled acceptably by deterministic models, the growth, the yields, and the ratios of the products are modeled very Since product yields have been widely employed to infer properties of the initial spatial distribution, this observation is of considerable significance. A description based upon isolated spurs is itself idealized. The primary particle loses its energy in a series of discrete events, generating local distributions of reactive species. These distributions may differ significantly from the isolated Gaussian spurs on which our previous calculations have been based. Stochastic techniques have been developed that simulate the energy deposition events along a radiolysis track.*-I0 This paper describes the use of such a track as the basis for the development of an improved stochastic treatment of short-time kinetics in irradiated liquids. Three stochastic models have been proposed for the modeling of spur kinetics, a Monte Carlo (MC) simulation, the independent reaction times (IRT) method, and the master equation (ME). Of these models, the MC method is the most accurate, but is computationally too slow to apply to an extended system containing many particles and requiring many realizations. The master equation has the virtue of not requiring stochastic simulation, but would be far too unwieldy to set up for such a large system and has not been developed sufficiently to be able to account for secondary reactions, charge, and incomplete diffusion control. The IRT method, however, is much faster than the MC method and can be realized for an arbitrary configuration of many particles; in addition methodology has been developed for the modeling of secondary reactions. This paper therefore uses the IRT method to model kinetics from a computer-generated track. Similar calculations have been reported by Zaider and Brenner,"J2 but they have not described the techniques used to deal with the last two of these effects, and the method they use for modeling reactions between ions has some deficiencies and no tests have been reported. The main body of this paper brings together and synthesizes the techniques for modeling spur kinetics reported for a number of isolated and idealized systems in recent papers.&' New methods are also developed here, notably for modeling partially diffusion-controlled reactions. These methods are applied with an extensive kinetic scheme to a computer-generated track, and computational techniques for applying the IRT model efficiently to an extended system are reported. Such an approach automatically includes the effects of spur overlap, which have not been examined by use of model distributions. In consequence the kinetics may be modeled to longer times. The validity of the independent pairs approximation in systems where spurs overlap is the subject of current research, which we hope to report shortly. 11. The Independent Reaction Times (IRT) Model
The primary aim of the present work is to develop a technique for modeling the short-time kinetics in irradiated liquids that is computer efficient and may be applied to the analysis of exper(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. 1982,86, 1318. (4) Clifford, P.; Green, N . J. B.; Pilling, M. J. J. Phys. Chem. 1985, 89, 925. ( 5 ) Clifford, P.; Green, N . J. B.; Oldfield, M . J.; Pilling, M. J.; Pimblott, S. M. J. Chem. SOC.,Faraday Trans. 1 1986, 82. 2673. (6)Clifford, P.; Green, N . J. B.; Pilling, M. J.; Pimblott, S. M . J . Phys. Chem. 1987, 91, 4417. (7) Clifford, P.; Green, N . J. B.; Pilling, M. J.; Pimblott. S. M.; Burns, W. G. Radiar. Phys. Chem. 1987, 30, 125. (8) Zaider, M.; Brenner, D.J.; Wilson, W. E. Radiat. Res. 1983, 95, 231. (9) Kaplan, I . G.: Miterev, A . M.; Sukhonosov, V . Ya. Radial. Phys. Chem. 1986, 27, 8 3 . (10) (a) Turner, J. E.; Paretzke, H. G . ; Hamm, R. N.; Wright, H . A,; Ritchie, R. H . Radiat. Res. 1982, 92, 47. (b) Turner, J. E.; Magee, J. L.; Wright, H . A.; Chatterjee, A.; Hamm, R. N. Radiar. Res. 1983, 96, 437. (c) Paretzke, H . G.; Turner, J. E.; Hamm, R. N.; Wright, H . A,; Ritchie, R. H . J . Chem. Phys. 1986,84, 3182. (d) Bolch, W. E.; Turner, J. E.; Yoshida, H.; Jacobson, K . B.; Hamm, R. N.; Wright, H . A.; Ritchie, R. H.; Klots, C. E. Report ORNL/TM-10851, 1988. ( 1 1 ) Zaider, M.; Brenner, D. J. Radiat. Res. 1986, 100, 245. (12) Brenner, D. J.; Zaider, M. Radial. Prot. Dosim. 1985, 13, 127
Green et al.
-Pair r,,lnm __ 1+2
13
1t3
21
1+4
10
2+3
11
2+4
17
3+4 ~
40
j
2
18
-
1
Figure 1. Independent reaction times methodology. A reaction time is generated independently for each of the six reactive pairs from the mrrect marginal distribution function for the pair. After reaction of pair (2 t 3) only the time for pair ( 1 + 4 ) is considered further.
imental data. The most promising stochastic approach is based upon the independent reaction times (IRT) method in which the particles are taken pairwise in all possible combinations and a reaction time tR is chosen for each pair from the marginal distribution function of reaction times appropriate to the initial separation of that pair. The approach which is then adopted may be appreciated by reference to Figure 1. The ensemble of reaction times is examined and the minimum time (tR)min is taken to define the first reactive pair and their time of reaction. The two particles are reacted and all other times for pairs containing one or the other reacted particle are eliminated unless the product is reactive. The process is repeated until no potentially reactive pairs remain or until a cutoff time is reached. The method is extremely efficient, allowing many realizations to be performed, so that good statistics may be established on the time-dependent G values. The technique has been extensively validated by comparison with detailed random flight which provide a realistic description of the diffusion/reaction process on a time scale above a few picoseconds. In the remainder of this section we review briefly the technique developed for the implementation of the IRT method. Thus far, it has been applied mainly to isolated Gaussian spurs but also to some other idealized configurations. (i) Zero Time Reaction. The IRT simulation starts from an initial configuration in which the coordinates of the reactants are specified (e.g., particle positions are sampled from a Gaussian distribution or one resulting from the Monte Carlo simulation of a particle track). The initial interparticle separations are then tested for "overlap": if the pair separation is less than the appropriate reaction distance, then the overlapping particles are allowed to react and are replaced by the corresponding products. The approach required to model zero time reaction for species that react slower than the diffusion-controlled rate is described in section iv below. (ii)Neutral Species. For neutral reactants, the evolution of the separation of a pair of particles is described in terms of a probability density, p, which, assuming that the liquid is isotropic, obeys a backward equation of the typeI3-l5
where r is the initial separation of the pair and D'their relative diffusion coefficient. The probability that the pair has reacted by the time t , denoted W(r,R,t),where R is the reaction distance for the pair, obeys the same backward e q ~ a t i o n . ' ~ - W ' ~ is determined by applying the appropriate boundary conditions; for diffusion-controlled reactions the inner boundary condition corresponds to instantaneous reaction on encounter: W(R,R,t) = I
f 2 0
The time-dependent reaction probability may generally be expressed in the form ~~
(13) Kolmogorov, A. N. Marh. Ann. 1931, 104, 415. (14) Goel, N . S.; Richter-Dyn, N . Srochasric Models in Biology, Academic Press: New York, 1974. (15) Ripley, B. D. In/.Statist Reu 1983, 51, 301.
Fast Kinetics in a Radiation Track
The Journal of Physical Chemistry, Vol. 94, No. I, 1990 253
W(r,R,t) = W , W
(2)
where W , is the probability of ultimate reaction (Le., W , = W(r,R,t+m)) and 19* is the time-dependent reaction probability conditioned on ultimate reaction. For diffusion-controlledreactions between neutral particles W, = R/r
(3)
and
w = erfc
(4)
2(D’t)II2
The reaction times, tu, required for the IRT method are obtained by the inversion method,I5 i.e., by generating a random number U (uniform on (0,l)) and inverting (2)-(4), giving tu =
I
-
4 0 ’ erfc-’r (- UR/ W,)
J‘
k,,, = 4nR2v
U < W,
u 1 w,
tu=
(sa)
--]
a2w + (2r + rc) aw r2
(1 1)
whence the inner boundary condition for the backward equation becomes20
(5b)
The error function may be inverted by using a standard algorithm.I6 References 3-5 describe the testing and validation of the IRT method for neutral particles reacting at the diffusioncontrolled rate. (iii) Ionic Species. When the reactants are charged, their relative diffusive motion is biased by the interparticle Coulomb force and the backward equation for the geminate pair reaction probability now becomes -aw =D‘[--g at
The success of the IRT method, given the many-body ( > 2 ) Coulomb forces that are present in dense spurs, is remarkable. The general approach even describes accurately spur reactions in low-permittivity solvents such as alkanes, where the forces are much greater and of longer range.’* In this case reaction times must be generated from a numerical solution for W(r,R,t). (iv) Partially Diffusion-Controlled Reactions. When particles do not react instantaneously on encounter, the inner boundary condition must be changed and the reaction time and first passage time distributions are no longer equivalent. Collins and Kimball first addressed this problem by setting the rate of reaction to k , d ( R ) ,where p(R) is the pair probability density at the “reaction distance”, R , and k,,, is the second-order rate constant for the activation process.19 k,,, may be used to define a “velocity of reaction”:
ar
(6)
where r is the initial pair separation. W(r,R,t)is again decomposed in the same way as ( 2 ) with, for neutral particles
and w* = erfc ( b ) - exp(a2
(14)
where Rfea= v R 2 / ( R v + D ’ )
where r, is the Onsager distance: r, = z l z 2 e 2 / ( 4 ~ ~ o t r k B T )
+ 2ab) erfc ( a + 6)
a = - -R v ; D’(
(7)
z , e and z2e are the charges on the ionic reactants,
to is the permittivity of free space, t, is the relative permittivity of the medium, k B is the Boltzmann constant, and 7’is the temperature. (Note that in the convention used here r, < 0 for an attractive force.) An exact, closed form for W(r,R,t) in charged systems is not available, but we have demonstrated that, for high-permittivity solvents such as water, an excellent approximation may be devised which retains the form of ( 2 ) with6J7
and b = -r-R 2( D ’t)1/2
The effective reaction distance, R i f f ,may be used to link v to the experimental bimolecular rate constant k,: k2 = 4 ~ D ‘ R : f=f
and
4nD’R’v (Rv D‘)
+
Green has demonstrated that (14) can also be applied to partially diffusion-controlled reactions between ions,21 with a=
where reffand Reffare defined by the natural scale Xeff
= -,/(I
- exp(r,/x))
4R2a
7 C r
(10)
with x = r and R , respectively. The equivalence of ( 3 , 4 ) and (8, 9), once the natural distance scale has been applied, leads to identical inversion procedures for both neutral and ionic reactions with the exception that the distances between ions must first be transformed. We demonstrated in ref 6 that it is important to use the transformed scale for the interparticle distance as well as for the reaction radius. If only the transformed reaction distance is used (e&. from k = 4nD‘Reff), then significant errors will occur for reactants initially with short separations. Such distances have a disproportionate influence on spur kinetics. The validation of the approximation for ions (eq 8 and 9) and of its use within the IRT method have been described eIsewhere.6*” (16) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1970. (17) Clifford, P.; Green, N . J. B.; Pilling, M. J. J . Phys. Chem. 1984.88, 4171.
(F )
t ‘I2
sinh2 ( r c / 2 R )
and b = r,[coth (rc/2r]- coth l r , / 2 R J ] / 4 ( D ’ t ) ’ / Z (20)
where a =v
+ r,D’/[R2(1- exp(-r,/R)j]
(21)
W , is now equal to RrLff/reff with reffdefined via the natural distance scale (eq 10) and R’Lfrby R’Lff = - r c / [ l - exp(r,/R)(l
+ D‘r,/vR2)]
(22)
The steady-state rate constant corresponds, once more, to ( 1 8) (18) Green, N . J. B.; Pilling, M. J.; Pimblott, S. M.; Clifford, P. J . Phys. Chem., in press. (19) Collins, F. C.; Kimball, G. E. J . Colloid. Sci. 1949, 4 , 425. (20) Sano, H.; Tachiya, M. J . Chem. Phys. 1981, 75, 2870. (21) Green, N . J. B. Chem. Phys. Lett. 1984, 107, 485.
254
The Journal of Physical Chemistry, Vol. 94, No. I , 1990
Green et al.
with R i f f replaced by R”eff of (22). While these equations for the reaction probability are accurate, they cannot be inverted to provide reaction times in a sufficiently facile manner for incorporation in an efficient IRT procedure. However, few reactions in radiation chemistry are far from the diffusion-controlled limit and it is usually sufficiently accurate to parameterize the first passage time distribution function using R and D‘ values that are compatible with the experimental steady-state rate constant, i.e., by setting k2 equal to 4nRD’or 4nR,ffD‘. An important exception arises for oppositely charged ions and concerns, in particular, the reaction between e,- and H,+. A suitable approximation, validated by comparison with the detailed equations given above, is provided by setting
vr + c
w
W Lc
0 0
z
v) + c
w* = erfc {(reff- R’:ff)/2(D’f)1/2J
01 > Lc
but with W , given by the correct R”eff/reff. Three parameters are required to characterize a partially diffusion-controlled reaction, R , D’, and v, which are related to the experimental rate constant, k2,via (18) and (22). Independent estimates of D’ are usually available, but this leaves some ambiguity in the choice of R and v. The approach adopted is to set R to a “reasonable” value (see below), leaving u to be determined from kz. Some ambiguity is also encountered in determining zero Haq+,which is not fully diffusion-contime reaction for ea; trolled, since particles in contact do not necessarily react. Several procedures were investigated, and the method adopted was the simple device of counting reaction if the initial separation was less than or equal to R. A fuller discussion of partially diffusion-controlled reactions, and of their use in the IRT method, will be given elsewhere.22 ( u ) Reactive Products. Some reactions lead to products which can themselves react further, e.g.
+
OH
+ OH
-
H 0 2 + eaqHO2
+H
H202 H02-
HzO2
Since the IRT method does not follow explicitly the trajectories of the reactive species but simply determines the times of reaction, special procedures are required to account for the subsequent reaction of reactive products. Three approaches have been devised and reported in detail e l ~ e w h e r e .These ~ incorporate varying degrees of spatial information and can be summarized as follows: ( I ) The time approach uses a rescaling of the original reaction time to account for the different diffusion coefficient of the product. A small independent time is added if the reaction distance of the product is smaller than that of its parent. (2) The diffusion approach generates a new distance independently for each potentially reactive pair incorporating the new product particle. Each distance is sampled from the correct distribution that would pertain if the pair were there in isolation, and reaction times are generated for each new distance in the same way as at zero time. (3) The position approach generates a new position for the product from the correct distribution and then generates new positions independently for every other reactive particle. Reaction times for the new particle are generated, but those not involving the new particle are left unchanged. A hybrid of methods 1 and 2 was found to be the most computationally efficient and accurate. It is described in ref 5 and is employed here. 111. Track Simulation
The remainder of this paper is concerned with the simulation of kinetics from a Monte Carlo radiation track of the type described earlier. The track used in the present simulation was provided by Zaider and Brenner and comprises a list of events simulated to occur in the initial 250 pm of a 22-MeV electron (22) Green, N . J . B.; Pimblott, S. M. Manuscript in preparation.
0
0
z
io
Ob
do
do
io
,bo
Separationhm
Figure 2. Variation in separation between energy loss along (a) the section of low LET electron track and (b) the IO-keV &ray.
track. Separations between successive primary events along the main track are exponentially distributed as shown in Figure 2. There are 750 primary events which account for the loss of about 40 keV. The track section contains a &ray of about 10 keV energy, whose event separations are also approximately exponentially distributed. The simulation depends on cross sections which were taken from vapor-phase data with the density scaled to that of liquid water. This approximation has a number of shortcomings as it ignores the possibility of collective excitations and the energy loss distribution will be in error. However, recently, B r e r ~ n e r ~ ~ has attempted to account for these effects and it is hoped that these improvements will yield model tracks that are closer to reality. Events on the track are identified as of four main types: r H 2 0 *
L
2
H
+
0’
H
+
e-
+
OH
.
-
(F1)
2H
+
H’
+ OH
0
+
+ e4- (F4)
The fragmentation probabilities for a given energy loss are also taken from the gas-phase cross sections. After application of these reactions, which are assumed to be effectively instantaneous, the track section consists of approximately 9000 potentially reactive particles. Fragmentations F3 and F4 suggest the initial production of the 0 atom. The involvement of this species in the radiation chemistry of water was postulated by Allen,24and it has been included in previous kinetic calculations.2s 0 atoms produced in the singlet state undergo a very rapid reaction with water molecules in the gas phase O(lD)
+ H 2 0 -.+2 0 H
while the triplet species O(3P) is much less reactive. A similar affinity is to be expected in the liquid, with the reaction of the singlet being effectively instantaneous. In our calculations, we have assumed that 0 does not undergo solvolysis to produce OH radicals. This assumption may be inappropriate, however; as the (23) Brenner, D. J. Radial. Phys. Chem. 1988, 32, 157. (24) Allen, A. 0. Radiat. Res. Suppl. 1964, 4, 54. (25) Kuppermann, A. In Physical Mechanisms in Radiation Biology; Cooper, R. D., Woods, R. W., Eds.; US. Atomic Energy Commission: Springfield, VA, 1974; p 155.
The Journal of Physical Chemistry, Vol. 94, No. 1, 1990 255
Fast Kinetics in a Radiation Track TABLE I: Track Species and Their Diffusion Coefficients species eH H+ 0 OH OHH2
D/
m2 s-'
species
4.5 7.0 9.0 2.0 2.8 5 .o 5.0
0 2
02H20 H02 H02H202
D/iO-9 m2 s-l
IV. IRT Simulation of Track Kinetics (i) Chemical Model. Analysis of the chemistry of a track requires the formulation of a chemical reaction scheme. The model used here is based upon that of Burns and Marsh.26 Twelve reactive species are involved and are listed in Table I together with their diffusion coefficients. A 22-reaction scheme is employed, as shown in Table 11, with their steady-state (homogeneous conditions) rate coefficients. The rate coefficients and diffusion coefficients were used to estimate effective reaction distances assuming diffusion control, i.e.
k = 4rR,ffD'
(234
k = 2rRefrD'
(23b)
for reactions between different species and the same species, respectively. All reactions from the scheme of Burns and Marsh with Reff< 0.01 nm have been omitted from the simulation as being too slow to be of importance. For reactions between uncharged particles or between an ion and an uncharged particle the reaction distance is set equal to Refp For reactions between ions R is found by inverting (10):
+ rc/Reff) (24) This procedure breaks down if, for rc < 0 (Le., attraction), Reff R = rc/ln (1
< -rc. This behavior is found because under these conditions -rc is a lower bound on Rerp Thus the reaction cannot be fully diffusion-controlled and a boundary velocity must be found. This occurs only for two reactions in the scheme. For reaction R2 a
-
+ Haq+
H
-
no.
2.1 2. l 2.8 2.0 1.4 1.4
initial yield of 0 atoms is small, the errors introduced are unlikely to affect the kinetics significantly. The rate coefficients we have used for the reactions of 0 were taken from the compilation of Burns and Marsh26 and are those measured for the gas-phase reactions. The track structure simulation provides the location of the events which constitute the track and of the electrons degraded to about 0.4 eV kinetic energy. In order to simulate kinetics, the IRT model requires the positions of the reactive particles after thermalization and solvation. Indeed, it is a major purpose of analyzing kinetics to obtain these parameters. Unfortunately, these processes are the least well understood of the primary processes of radiation chemistry and it is, at present, necessary to take a prescription. Thus fragment positions were sampled from Gaussian distributions of standard deviation 0.75 nm. This prescription was taken from the theoretical study of S ~ h w a r z , ~who ' used the deterministic prescribed diffusion technique to model the kinetics of a distribution of spurs similar to that found along a high-energy electron track. Schwarz varied the width of a spur according to the cube root of the number of dissociations, N , Le., a(N) = N1I3a(l).The value of 0.75 nm, which we use, was obtained by Schwarz for a( 1) by comparing the calculated and the observed effects of a scavenger, NOT, upon the yield of HzOz. The positions of the solvated electrons were taken to be those given by the track structure simulation, and the energy loss by an electron from subexcitation levels (0.4 eV) to thermal was not assumed to contribute toward the spatial thermalization distribution.
ea;
TABLE 11: Reaction Scheme for the Radiolysis of Water k l 1o 1 O mol-'
(R2)
(26) Burns, W. G.;Marsh, W. R.J . Chem. SOC.,Faraday Trans. 1 1981, 77, 197. (27) Schwarz, H. A. J . Phys. Chem. 1969, 73, 1928.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
dm' SKI 0.5 2.4 2.5 3.0 1.9 1.3 2.0 1.2 1.o 2.0 2.0 1.9 2.0 2.0 14.3 5.0 2.2 2.0 2.0 0.45 1.2 1.2
reaction
+
e' e20H- + H2 e-+Ht-H e- + H OH- H, e-+OH-+OHe - + 02-02e- + 02- OH- + H 0 2 e-+H02-HOT e- + H 2 0 2 OH- O H H+H-H2 H+O-OH H+OH-H20 H 024HO2 H+O,-HOT H H 0 2 4 H202 H++OH--H20 H++0Y-HO2
- + - +
+ +
0 + 0 - 0 2
O+OH-H02 O+HO2-0H+02 OH OH H202 OH 0 2 OH- 0 2 O H + HO2 --c 0 2 H 2 0
+ +
-
+ +
+
Rlnm 0.34 0.50 0.29 0.54 0.38 0.54 0.41 0.29 0.19 0.29 0.27 0.28 0.29 0.29 1.23 0.50 1.46 0.55 0.66 0.21 0.32 0.33
v1ms-l 4.19
18.12
partially reflecting boundary condition is needed. A value of 0.5 nm was chosen for R, based on the analysis of Hart and Anbar,28 and inversion of (22) then gave a boundary velocity of 4.2 ms-I. The rate of reaction R16 is very close to, but probably slightly H+
-
+ 02-
H02
below, the lower bound of the diffusion-controlled limit, k D = 4rD'lrcl. A boundary velocity of 18.1 ms-' was calculated assuming a reaction distance of 0.5 nm, and these parameters were used in the IRT simulation. This reaction, however, is relatively unimportant, and the effect is small. The use of a radiation boundary condition to model partially diffusion-controlled reactions is not always correct. For example in the H H recombination reaction it is likely that the spin state of the encountering pair determines whether the pair can react. If reaction does not occur, it is unlikely to occur on subsequent encounters unless some event intervenes to change the spin state as the time scale of the hyperfine interaction is of the order of I O ns. We have not made any attempt to model spin effects in this paper except insofar as they are taken into account in the bulk rate constant. The effect of spin on spur kinetics is the subject of current research. Solvolysis reactions have not been included in the scheme but may legitimately be neglected. The fastest of these is
+
eaq-
+ H20
-
H
+ OH-
which would be easy to include in the IRT simulation because the solvent is present in excess so that the solvolysis is a pseudo-first-order process with exponentially distributed reaction times: w23(t) = 1 - exp(-kZ3[H20]t)
-
However, taking kZ3= 16 dm3 mol-' s-' and [H20] = 55 mol dm-3 gives a mean reaction time of 1 ms, demonstrating that solvolysis reactions are unimportant on the short time scales considered here (t 5 10 ns) and may therefore be neglected. Use of a time-dependent rate constant with an appropriate boundary velocity
does not alter this conclusion. The kinetic scheme of Table I1 coupled with the techniques for sampling reaction times described in section I1 permit the IRT method to be implemented for the 9000 particles in the idealized track as described below. (28) Hart, E. J.; Anbar, M. The Hydrated Electron; Wiley: New York, 1970.
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The Journal of Physical Chemistry, Vol. 94, No. I , 1990
( i i ) Implementation. In previous investigations3-’ the IRT method was applied to isolated Gaussian spurs. For a given N-particle spur all (1/2)N(N - I ) pairs were examined and reaction times sampled. ln the present investigation this approach is neither necessary nor desirable as it would waste a large amount of computer time generating reaction times for pairs whose reaction probabilities are close to zero, because of their large initial separations. Many particles will react with their neighbors to form inert products at early times, well before these unlikely events can take place. The simulation protocol has therefore been reformulated in such a way as to increase the computational efficiency without loss of accuracy. The basic idea is to determine a cutoff distance r l such that up to a given prescribed time, t l , any pairs whose separations are greater than rl are extremely unlikely to react. Pairs are then only examined if their separations are less than rl and the IRT method is realized up to time t l . During this period many particles will react, greatly reducing the number of pairs to be considered for reaction beyond t , . A new cutoff time t2 is introduced, with a corresponding cutoff distance r2,and all remaining pairs whose initial separations lie within r2 are examined for this second phase. The procedure is repeated until the simulation is complete. A second simplification is that rather than treat each particle in turn (to find all particles within r“),the localized nature of the clusters of radicals and ions centered on each energy-loss event can be used. In each cluster the particles are initially located close to their parent event. As long as rl >> 6,the typical cluster size, the intercluster separation can be used instead of the interparticle distances. Thus, rather than determine which particle separations lie within r l , we determine which events on the primary and its daughter tracks lie within r l . In order to determine a value of r , from t l , we consider the conditioned reaction time distribution function W*(r,R,t)= erfc
(=I
l o g ( t i m e / ps)
Figure 3. Predictions of the IRT simulation for the kinetics of the section of low LET electron track described in section III.
TABLE 111: Comparison of Simulated with Experimental C Values2’ species simuld yield/ 100 eV exDtl vield/100 eV 1.70 2.42 2.08 3.00 0.70 0.5 1
2.70 3.15 0.60 2.85 0.45 0.75
2(D’t)‘I2
which gives the probability that a neutral pair has reacted by time t given that its initial separation is r a n d that it will eventually react. r , is determined in such a way that at time t , W* will be less than 0.01 for all pairs separated by more than r l . This is achieved by using the maximum values of R and D’that can be found from Tables I and I1 and inverting (27) for rl with a w* value of 0.01, i.e. r l = R,,,
Green et al.
+ 2(D’,,,tl)l~*
erfc-] (0.01)
(28)
Note that since (28) is an increasing function of both R,,, and Dfmax,the inclusion of the maximum values of these quantities leads to the safest estimate of r l . Note also that the value of W at t I will be smaller than 0.01 by a factor of less than Rmax/rl for any given reaction. The formula for neutral particles, (27), has been used because at the distances considered, Coulombic effects are negligible. In realizing track kinetics by this method program efficiency is very sensitive to the choice of the intermediate cutoff times t , , t2, ..., tmx. The optimal choice keeps the number of reactive pairs in play approximately constant at each stage. However, the simulated kinetics are independent of the cutoff times if these times are chosen correctly. Exhaustive tests have been conducted to ensure that this condition is met. (iii) Results. Figure 3 shows the results of the IRT simulation for the kinetics of the track section described in section 111. The maximum cutoff time tmax= 0.1 ps was employed with intermediate cutoff times t I and t 2 of 100 and 1000 ps, respectively. The first cutoff time was chosen such that reaction between particles in adjacent clusters separated by just over r , was very unlikely to occur before that time. In other words, the separation rl must be much larger than typical intracluster distances. The choice of 100 ps for t , yields a value for rl that is about 7 times larger than the standard deviation of the Gaussian distribution used to describe the initial configuration. The second cutoff time t 2 is less critical and can be selected to optimize the speed of
computation. Calculations were performed using several different sets of cutoff times, and those reported above were found to be optimal. One thousand realizations were performed to obtain satisfactory statistics even for minority species such as 02-.One realization of the kinetics of the complete track requires approximately 75 s of’ CPU time on a Digital VAX 11/780 computer. Simulated G values (at 0.1 ps) are compared with experimental C values for electron radiolysis in Table Agreement is not particularly good. In addition, comparisons with the time-dependent G values of Jonah et al.30331show that the predicted kinetics from the track are about an order of magnitude too fast. However, we are confident that the IRT method has correctly described the kinetics of the simulated track used as an initial condition. Any discrepancies in the results lie in errors in this initial particle distribution. The main errors arise from the fact that the electron positions have been taken directly from the track simulation, in which they have been degraded down to a fixed cutoff energy. The electron distribution is therefore probably too narrow, which will contribute to the overestimated rate. In addition, spur widths for the other species have been taken directly from ref 27 and give a similar overestimate when employed in deterministic models. Another reason for the overestimate is that the track section employed in the simulation contained a IO-keV h a y . This &ray contains about one-fourth of the reactive particles in the track section and includes many overlapping spurs. As a result the track section is not typical of a 22-MeV electron track. The section of track was chosen, however, because it provides a (29) (a) Schwarz, H. A. Radiat. Res. Suppl. 1964, 4, 89. (b) Anbar, M. In Fundamental Processes in Radiation Chemistry; Auslws, P., Ed.; WileyInterscience: New York, 1968. (c) Draganic, I. G.; Nenadovic, M. T.; Draganic, Z. D. J . Phys. Chem. 1969, 73, 2564. (d) Pikaev, A. K.; Kabakchi, S . A,; Zansokhova, A. A. Faraday Discuss. 1972, 63, 112. (30) Jonah, C. D.; Matheson, M . S.; Miller, J. R.; Hart, E. J. J . Phys. Chem. 1976, 80, 1267. (31) Jonah, C D.; Miller, J . R. J . Phys. Chem. 1977, 81, 1974.
The Journal of Physical Chemistry, Vol. 94, No. 1 , 1990 257
Fast Kinetics in a Radiation Track TABLE I V Comparison of Initial G Values Yielded by Monte Carlo Track Simulation with Results Obtained by Back-Extrapolating Exwrimental Data
species ea< Hag+ H OH
0
simuld init yield/100 eV
4.10 4.10 4.18 6.73 0.78
H2
simuld yield allowing for reaction/lOO eV
exptl vield“/100 eV
3.89 3.95 4.19 6.53 0.67
4.7 4.7 0.8 6.0
0.02
0.25
“References 24, 27, and 28.
good test of the technique. The obvious remedy would be to select a larger or more typical track section and to expand the spurs so as to reduce the encounter rates. This optimization of parameters is not within the scope of the present paper, whose aim is rather to describe the methodology. Such an expansion may, however, bring the final yields of molecular hydrogen and hydrogen peroxide further out of line with experiment because of the stochastic number effect of the small spurs.7 In addition, we reiterate here that it is not surprising that the predictions of the model for the solvated electron and O H radical decays are close to those of Schwarz, since the same spur parameters were used. The main differences with the deterministic models lie in the product yields, which are significantly different from those by Schwarz. There is not enough information in the experimental radical decays, therefore, to allow us to distinguish between stochastic and deterministic models; however, if the radical decays and the product yields are taken together, an experimental distinction should be possible. V. Discussion The aim of the present work is to develop a fast and efficient technique for modeling the kinetics in a radiation track. It should be emphasized that this paper represents the first step toward this aim. The purpose of this paper is to bring together all the various techniques that are required for such a model and to demonstrate the feasibility of the approach. In the light of the approximate initial conditions used, it would be an error to place any weight on the failure of the particular simulation to agree with experiment at this stage. All the methods used in the IRT simulation to generate reaction times, which are summarized in section 11, and described in more detail in ref 3-7, have been extensively tested on model systems and found to be accurate by comparison with full random flight simulations, which have in turn been validated against known analytic and numerical solutions for single pairs. For this reason we conclude that the failure of the simulation to agree with experimental results arises either from a breakdown of the currently accepted theory of diffusion-limited kinetics, which seems unlikely, or from the failure of the simulated track, and the prescription used for the fragment distributions, to represent accurately the initial conditions pertaining after radiolysis. The track simulation itself is not above suspicion. The use of cross sections and fragmentation patterns appropriate to the gas phase is difficult to justify and has been criticized in section 111. In more recent work Brenner2j has discussed modifications to allow for a single collective excitation at 23 eV and estimated branching ratios for the various fragmentation pathways. However, the track used in this study did not contain these improvements. An estimate of the extent of the problems with the track simulation can be obtained from the initial G values, which are a function of the track itself and independent of any kinetic model. These are compared with “experimental” values in Table IV. The lack of agreement may indicate one of two things: either the track simulation is in error or a great deal happens between the formation of the track and spur kinetics. It is our opinion that both of these problems are present in this work. The major difficulty is associated with this “zero time” condition, which must contribute to the experimentally observed yield of nonscavengeable hydrogen. Either the initial G values must
be rescaled to agree with experiment, as has been done by Brenner,23or a detailed model of the dynamics of the primary events is necessary. The early kinetics, which we have attempted to model in this paper, are critically dependent on the initial interparticle distances. Reactions can arise either within clusters or between clusters. In the track considered here typical event separations are of the order of 100 nm in the main track and 10 nm in the &ray (see Figure 2). Thus most of the reaction in the first 10 ns occurs within isolated clusters and is essentially independent of the track structure. The subsequent kinetics depends on cluster overlap, but in order to model this correctly the clusters must yield the correct numbers of escaping particles of each type. These yields depend on the earlier intracluster reactions and thus the geometries of the clusters. Therefore, between the track simulation and the kinetics there is a critical missing stage in the current work. This stage would describe the distribution of the radical fragments and solvated electron in water at about 1 ps. In particular any recombination of these fragments before solvation must be accounted for. At present this is done in an arbitrary way by reacting any particles found to overlap at zero time in the distributions generated. The track simulation does not include this zero time reaction, or static quenching, explicitly, and the amount of zero time reaction occurring in the kinetic simulation depends entirely on the initial distribution chosen to overlay the track. On the other hand, the experimental yields of Jonah et al.30331extrapolate back to a “zero time” yield after the zero time reaction has occurred. The initial simulated yield and the extrapolated experimental yield should therefore not be expected to agree until the zero time reaction has been correctly accounted for in the simulation, and this cannot be done without a satisfactory initial distribution or model of presolvation dynamics. Significant progress is now being made on electron thermalization in ~ a t e r , ~ which ~ - ~ *should enable more realistic initial conditions to be produced when the fundamental processes are more fully understood. In the present work we have used the spur parameters recommended by S c h w a r ~ .However, ~~ these have been obtained by fitting experimental results using a deterministic model and are therefore open to question in the light of more recent theoretical development^.^^ This is a cause of the poor comparison of the simulated yields with the experimental values shown in Table 111. However, although in principle the simulations would be optimized with respect to the experimental yields, there seems little point at this stage in view of the poor initial G values in the simulated track (notably that for H). Other groups have also attempted to simulate kinetics from track structure calculations. All these studies must be subject to the general reservations discussed above. The Oak Ridge g r o ~ p ’have ~ , ~simulated ~ electron tracks with a fragmentation scheme based on F1 and F2 and H2O H2 + 0
-
They used a random-walk technique to simulate the kinetics. Originally the simulation model used a time rescaling to make the method computationally feasible.Iob This approach appears to have been replaced in more recent s t u d i e ~ ’which ~ . ~ ~employ a varying time step and model reaction during a jump using an approximate bridging process. No report has been made on checks of the validity of either method, but Green et al., in a simulation of the geminate recombination of H / O H pairs, have shown that unacceptable artifacts were introduced by the former a p p r 0 a ~ h . j ~ Terrisol et al.37 have also performed random-walk simulations. (32) Migus, A.; Gauduel, Y . ;Martin, J. L.; Antonetti, A. Phys. Reo. Lett. 1987, 58, 1559.
(33) Lu, H.; Long, F. H.; Bowman, R. M.; Eisenthal, K. B. J . Phys. Chem. 1989, 93, 27.
(34) Goulet, T.; Jay-Gerin, J.-P. J . Phys. Chem. 1988, 92, 6871. (35) Green, N. J. B.; Pilling, M. J.; Pimblott, S. M. Radial. Phys. Chem. 1989, 34, 105. (36) Turner, J. E.; Hamm, R. N.; Wright, H. A.; Ritchie, R. H.;Magee, J. L.; Chatterjee, A.; Bolch, W. E. Radial. Phys. Chem. 1988, 32, 503. (37) Terrissol, M.; Beaudre, A.; Caudrelier, V. Radiation Research; Fielden, E. M., Fowler, J. F., Hendry, J. H., Scott, D., Eds.; Taylor and Francis: Philadelphia, 1987; Vol. I.
J . Phys. Chem. 1990, 94, 258-262
258
Neither group has provided for the effects of electrostatic forces in reactions between ions, although Clifford et aL6 have shown that these are important. Zaider and Brenner have also used the IRT method to simulate both electron and proton They used an earlier version of the IRT method in conjunction with their own track structure calculations but made no comment on the problems presented by reactive products, ions, and partially diffusion-controlled reactions. Since they have discussed neither the techniques nor their validation, it is impossible to comment further on their results. Brenner has recently reported simulations of scavenging kinetics, using the same early version of the IRT method.23 However, the results he reports still require a rescaling of the initial G values to bring them into line with experiment. A detailed analysis of the IRT method in scavenging studies will be reported elsewhere together with a discussion of pH effects on radiolysis kinetics.38
VI. Conclusions This paper has concentrated on the application of the IRT method to the simulation of the short-time kinetics in a radiation (38) Green, N. J. B.; Pilling, M. J.; Pimblott, S. M. Manuscript in preparation.
track. The results of the simulation are not readily compared with experimental data, because of shortcomings in the radiation track itself. Once realistic methods are available to assess the excitation cross sections in liquid water and to describe the subsequent presolvation dynamics, then the techniques described here should provide a suitable vehicle for a realistic kinetic simulation. The long-term aim of this research is the provision of a simulation technique for the analysis of experimental data referring to short-time kinetics. It is unlikely that a technique could start from a track section of the type discussed above; rather it would refer to a suitably simplified initial distribution of clusters such as spurs and short tracks. The realization of the kinetics resulting from these simplified distributions must, in turn, be checked against the kinetics of realistic tracks, which will require simulations of the type reported here. Work is in progress on such a description, and we are currently examining techniques for simulating reaction from initially separated Gaussian spurs and in the dense regions found at track ends. Acknowledgment. We thank Drs. W. G. Burns and G. V. Buxton for helpful discussions. This work was supported by the Office of Basic Energy Sciences of the U S . Department of Energy and AERE, Harwell, U.K. This is Document No. NDRL-3174 from the Notre Dame Radiation Laboratory.
Competltlve Ionic Hydration I nvolvhg Outer-Shell Solvent: Temperature Dependence J. Lee Picosecond and Quantum Radiation Laboratory, P . 0. Box 4260, Texas Tech University, Lubbock, Texas 79409 (Received: March 21, 1989; In Final Form: July 11, 1989)
Retardation of proton dissociation from the excited state of I-naphthol (p&* = 0.4) by added LiCI, MgCI2, and CaCI2 in aqueous solutions is studied as a function of salt concentrationand temperature. The observed “salt effects” cannot be satisfactorily explained by using the bulk properties of electrolyte solutions such as water activity. It is found that near room temperature the proton dissociation is strongly affected by an intermediate hydration region of an ion, which couples the “primary solvation shell” to the bulk region of the solvent. Two concepts are proposed to help classify these imperfectly defined regions: a “relative hydration energy” AE, which is based on the activation energy of kdia,and a temperature-dependent”kinetic solvation number” no, which defines the number of water molecules so strongly bound to the ion that they cannot participate in the proton dissociation process. The analysis of AE as a function of salt concentrationand of n,as a function of both salt concentration and temperature reveals the nature of the important solvation structure of an electrolyte at the molecular level. When comparisons with gas-phase results are made, both types of data indicate a gradual change of hydration energy with increasing hydration number, rather than discrete sets of solvent shell energies.
I. Introduction The Debye-Hiickel theory for electrolyte solutions was first introduced in 1923.’ Since then, chemists and biochemists, on the basis of this theory, have treated these solutions as structureless liquid entities. Averaged bulk properties, such as dielectric constant,2 water activity,’ and steady-state X-ray structure! have been used by theoreticians and experimentalists alike to characterize the solution chemistry of these systems. In spite of valuable past applications of these continuum theories, fast chemical reactions in solution and in biological systems have shown marked deviations from them in some cases.5 A molecular model of these systems seems to be demanded. The most important reason that a molecular level description of these solutions is important and necessary is that they are no longer continuous and structureless in terms of fast chemical ( 1 ) Debye, P.; Hiickel, E. Phys. 2. 1923, 24, 185.
(2) Hasted, J. B. Aqueous Dielectrics; Chapman and Hall London, 1973. (3) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd revised ed.; Butterworths: London, 1965. (4) Lawrence, R. M.; Kruh, R. F. J . Chem. Phys. 1967, 47, 4758. (5) Simon, J . D. Acc. Chem. Res. 1988, 21, 128.
0022-3654/90/2094-0258$02.50/0
reactions. Experiments carried out on a short time scale, such etc., as NMR,6 picosecond or femtosecond laser spectros~opies,~ become effective tools for revealing the dynamical and structural differences between various electrolyte solutions at the molecular level. Meanwhile, developing a new version of solution theory demands a revolutionary new approach if the modern ultrafast techniques are to gain their maximum utility in the promotion of our understanding of the chemical reactions of ions in solution. Using molecular probes to study fast chemical reactions in solution, such as cis-trans isomerization,8 solvent ~ r i e n t a t i o n , ~ proton dissociation,I0 and electron ejection,” has revealed a great (6) Struis, R. P. W. J.; de Bleijser, J.; Leyte, J. C. J. Phys. Chem. 1987, 91, 6309. (7) Fleming, G. R. Chemical Applications of Ultrafast Spectroscopy;
Oxford University Press: New York, 1986. (8) Shank, C . V.; Ippen, E. P.; Teschke, 0.;Eisenthal, K. B. J. Chem. Phys. 1977, 69, 5547. (9) Ben-Amotz, D.; Scott, T. W. J. Chem. Phys. 1987, 87, 3739. (IO) Harris, C. M.; Selinger, B. K. J . Phys. Chem. 1980, 84, 891. Laws, W. R.; Brand, L. J. Phys. Chem. 1979, 83, 795. ( 1 1 ) Robinson, G. W.; Robbins, R. J.; Fleming, G. R.; Morris, J. M.; Knight, A. E. W.; Morris, R. J. S. J. Am. Chem. SOC.1978, 100, 7145.
0 1990 American Chemical Societv
