6136
J. Phys. Chem. 1994,98, 6136-6143
Diffusion-Kinetic Theories for LET Effects on the Radiolysis of Water Simon M. Pimblott' and Jay A. LaVerne Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 Received: December 2, 1993; In Final Form: February 18, 1994"
Diffusion-kinetic methods are used to investigate the effects of incident particle linear energy transfer (LET) on the radiolysis of water and aqueous solutions. Chemically realistic deterministic diffusion-kinetic calculations examining the scavenging capacity dependences of the scavenged yield of eaq- and of OH demonstrate that the scavenged yields are related to the underlying time-dependent kinetics in the absence of the scavenger by a simple Laplace transform relationship. This relationship is also shown to link the effect of an eaq-scavenger on the formation of H2 with the time dependence of H 2 production in the absence of the scavenger. The simple Laplace relationship does not work well when applied to HzOz formation in high-LET particle tracks even though such a relationship is valid with low-LET particles. It is found that while the secondary reaction of
HzO2 with e,, can be neglected in low-LET particle radiolysis, it is of considerable significance in the tracks produced by high-LET particles. The increased importance of this reaction with increasing LET is the major reason for the failure of the Laplace relationship for H202. It is also the cause of the larger yields of OH compared to eaq-at long times (small scavenging capacities). Analytic analyses using simple models show that the asymptotic time dependence of the kinetics of high-LET tracks is very different from that for low-LET tracks but is analogous to that for geminate pair recombination in two dimensions.
1. Introduction The fast chemistry that occurs in radiolysisreflectsthe structure of the track of the primary radiation particle. Tracks produced in radiolysis with low linear energy transfer (LET) particles, like fast electrons, are characterized by primary energy losses that are well separated and produce essentially isolated clusters of ionization and excitation events. As a result, the chemistry of fast electron radiolysis is that of spatially isolated entities, known as spurs,' which contain only small numbers of reactive ions and radicals. The distance between energy transfers from the incident particle decreases as the particle's LET increases.2 Consequently, the primary ionization and excitation events created by highLET particles, e.g. a heavy ion or a low-energy electron, may overlap one another to give tracks that might be visualized as having long cylindrical or cigar shaped geometry with large numbers of reactive entities in close proximity. The chemical consequences of the different initial spatial distributions of the radiation-induced reactants in low-LET and in high-LET radiolysis are significant,3but they are poorly understood. Quantitative studies of high-LET radiolysis are limited and not of universal application. Hence, it is virtually impossible to extrapolate to the yields in one system from the known experimental yields in another. There is only a small amount of experimental information available on the radiation chemical kinetics of high-LET particle tracks. A number of scavenger experiments probing the yields of the radicals e,, 4,s and OH610 and of the molecular products Hz and HzOz have been p e r f ~ r m e d , ~ . ~ -and ~ J Ithere - ~ ~ are some limited data on the time-resolved determination of the yield of ea,,- I4-l7 and of OH.'* This paper considers diffusion-kinetic theories for elucidating the chemical consequences of track structure with the goal of providing a better understanding of the experimentaldata. One problem of particular significance,which is discussed at length, is the relationship between the chemistry of solutions of scavengers and the underlying kinetics of water. The aims of this paper are to help determine which factors are of most importance in the radiation chemical kinetics of highLET particle tracks, to probe how thesefactorsaffect the reactions of the radiation-inducedspecies,and to analyze how the chemistry e
Abstract published in Aduance ACS Abstracts, June 1, 1994.
0022-3654/94/2098-6136$04.50/0
of high-LET particle tracks differs from that of low-LET radiation. Simple models are used to provide qualitative answers to these complicated questions and then to formulate a series of problems for future study using more complex theoretical techniques. The following section discusses the different diffusion-kinetic models utilized. First, the recombination of a geminate pair in two- and three-dimensionalsystems, and the effect of a scavenger on the chemistry of these systems, is considered. As the geminate pair is the prototypical nonhomogeneous chemical system, this analysis provides insight into the expected behavior of more complex spatially nonhomogeneous systems with a large number of reactive particles. The modeling of an idealized single-species cylindrical high-LET particle track employing the prescribed diffusion approximation is then described, and the solution obtained is used to provide information that aids in the elucidation of the asymptotics of the decay kinetics in water and in the discussion of the form of the scavenger concentration dependence of yields. Finally, a deterministic diffusion-kinetic method for modeling the radiation chemical kineticsof an idealizedcylindrical track or of the track core of high-LET particles in water and in aqueous solutionsis considered. In section 4, the results obtained using the various diffusion-kinetic techniques are discussed and the deterministic diffusion-kinetic model is used to investigate the accuracy of a Laplace transform relationship between the concentration dependence of yields in aqueous scavenger experiments and the time-dependent kinetics of water. The final section of the paper is a summary of the conclusions. 2. Diffusion-Kinetic Track Models 2.1. Radical Pair Kinetics. The simplest spatially nonhomogeneous chemical system is the geminate pair. The recombination kineticsof a geminate pair are generally modeled using the adjoint to the diffusion equation,lg which describes the time-dependent survival probability of the pair, R, as a function of the initial pair separation, r,
an at = D'[V2fl - V Q V v ] where D'is the relative diffusion coefficient of the pair, U is the potential energy of interaction of the pair (in units of ~ B with T 0 1994 American Chemical Society
Diffusion-Kinetic Theories for LET Effects
The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6137
kg being the Boltzman constant and Tthe absolute temperature), and V operates in the initial coordinate space. A scavenger reaction is included by adding a further term, which depletes the survival probability with a first-order rate constant s = k [ S ] ,to the equation
a@ = D’[V2@- V@ V v ] - s 9 at Here CP is the time-dependent survival probability in the presence of a scavenger of scavenging capacity s, k is the second-order rate coefficientfor thescavengingreaction, and [SI is theconcentration of scavenger. The time-dependent survival probability of a pair, Q ( t ) , is related to the probability of ultimately scavenging one member of the pair, CP(s,t = m), by a simple Laplace transform relation, @(s,t = m) = sJomQ(t)exp(-st) dt
(3)
In theory, this relation between the kinetics in the absence of a scavenger and the yield of scavengerreaction provides a formidable technique for radiation chemical kinetics.2G22 Experimental information about the former is very limited, while there is an extensive literature on scavenger experiments.22 The solutions of the adjoint equations for the diffusioncontrolled reaction of a geminate pair of neutrals in a threedimensional system are well known:23924
Q(t) = 1 -; erfc a (2% 3 ( s , t = a) = 1
- ar exp( -(r - a)&,)
(4b)
Here a is the encounter or reaction distance of the pair and r is its initial separation. Notice that the equation for the scavenged yield corresponds to that suggested by Humme120 to describe the scavenger concentration dependence of scavenged yields in the fast electron radiolysis of hydrocarbons, i.e. G(s) = GW
+ (Go- e’)( 1-e x p ( - 6 ) )
(5)
where Geafand Goare the spur escape and the initial yields of ion pairs in the absence of scavenger, respectively, and a! is a fitting parameter. The ion recombination kinetics in the fast electron radiolysis of a low permittivity solvent such as a hydrocarbon are not like those of a pair of neutrals, as there is a strong Coulombic interaction between the ions. Furthermore, the observed chemistry reflects a distribution of initial spatial separations. Nevertheless, both in hydrocarbons and in aqueous solution the longtime asymptotic of the time dependence of the decay kinetics ( t 1 /and 2 )the small scavenging capacity asymptotic of the scavenged yield dependence SI/^) are the same.2s Clearly, the theory of geminate pair recombination provides some insight into the description of radiation chemical kinetics. The pseudo-two-dimensional symmetry of high-LET particle tracks would imply that their chemistry is analogous to the geminate recombination of a pair diffusing in two dimensions. The solutionof the adjoint equations for two-dimensionaldiffusion gives the survival probability of a geminate pair in the absence of a scavenger as26
9(s,t = m) = 1 -
Ko(r dsTD’)
(7)
KO(-)
The two equations are, of course, linked by the Laplace relation given earlier, eq 3. The integral for Q ( t ) is more complicated than the corresponding equation for the three-dimensional system, but it can be evaluated using straightforward analytic and numerical t ~ c h n i q u e s . ~ *The ~ ~ ~longtime dependence of the geminate survival probability is Q(t
-
a) = In(r/u)/(Z1 ln(4D’t)
where 7 is Euler’s constant, while the small scavenging capacity limit for the scavenged yield is @(s
-
0,t = 03) = -ln(r/u)/(i ln(r%/D’)
+ y)
(9)
Both of the asymptotic dependences demonstrate a very slow approach to complete geminate reaction. The absence of escape is characteristic of systems of low dimensionality and contrasts the finite escape yield in three-dimensional systems. 2.2. Sie-SpeciesTracks. Perhaps the most basic treatment of high-LET particle radiolysisconsidersthechemistry of a section of track in which the LET is constant.’O This analysis gives the tracksegment or differential yields at a well-defined LET. Several simple descriptions for the local structure of the radiation track have been considered in the literature, e.g. a cylinder with an axially homogeneous concentration and a radial concentration pr0file~0.~~ or a linear arrangement of spherically symmetric, overlappingspurs.32 Because of the geometric symmetry of these model tracks, they are amenable to approximate analytic or numeric analysis for a number of reaction schemes. These simple idealized systemsprovide insight into the effectsof track structure on radiation chemistry. For instance, using a conventional deterministic model, the reaction kinetics of a track made up of a single type of radical are described by the diffusion reaction equation
ac = DV’C - 2kc2 - sc at where c is the position-dependent concentration of the reactive radical with diffusioncoefficient D. This equation is easily solved using the prescribed diffusion approximation. For a cylindrical track with a Gaussian radial profile33of standard deviation u, the time-dependent kinetics in the absence of a scavenger (s = 0) are well known:32 G(t) =
e(1 + A In((? + to)/to))-’
(1 1)
where A = kGOLI40QxD and to = u2/2D,and where GO is the initial yield of radicals per 100 eV of energy deposited34and L is the LET. The limiting long-time asymptotic of this equation has the same (ln(t))-l dependence observed earlier for a twodimensional geminate pair (eq 8). The solution of eq 10 for the scavenged yield is
E,(st where Jo and KO are Bessel functions27and the probability of scavenging one member of the pair in a system of scavenging capacity, s = k[S],isu-26
- y)
+ sto)])-’dt
(12)
where El(x) is the exponential integral.27 This result appears to bevery different from the Laplace transform prediction for C(s); however, in the limit that the scavenging capacity, s, is small, the scavenged yield is
6138 The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 C(s
-
0) = s @ c e x p ( - s r ) ( l
+ A ln((t + t0)/t0))-* dt
Pimblott and Laverne 1 .o
I
I
I
I
(13)
This limit is the appropriate Laplace transform result. Furthermore, since ln(x) Ix - 1, it is straightforward to show that eq 13 has an asymptotic (ln(s))-l dependence, Le. the same as that found for the geminate pair. 2.3. Multispecies Track. The chemistry of water and aqueous solutions following irradiation with high-LET particles has received considerably less attention than fast electron radiolysis: only limited experimental data are available,"8 and theoretical analyses are very crude.3S-38 As mentioned in the previous subsection,thekineticsof a high-LET particle trackare frequently described using "cylindrical track" models. The trackof a particle can be pictured as made up of two component^:^.^^ (i) the track core which is due to the energy loss eventsof the primary radiation particleand low-energysecondary electrons,and (ii) the penumbra resulting from 6 rays which are tracks of energetic secondary electrons. Obviously this type of model is a simplification; however, a more complete spatial description requires a detailed Monte Carlo track structure c a l ~ u l a t i o n .The ~ ~ distributions of the energy losses, and therefore of the radiation-induced reactants, in the two track regions arevery different. In particular, 6 rays, being energetic electrons, can be thought of in terms of isolated spurs and blobs and are therefore distinctly different from the trackcore. Consequently,thechemistry in the penumbra is very different from that in the core. This difference can have a significant effect on the observed chemical yields because a substantial fraction of the radiation particle energy may be dissipated by energetic secondary electrons.44 Those studies that incorporate the chemistry of the track core and the penumbra consider the two regions to be spatially isolated. This a p proximation is not strictly valid as reactants from the track core must eventually diffuse outward and envelope the penumbra. There is also the computational problem of the separation of the track into core and penumbra. This article is concerned with general problems in high-LET particle radiation chemistry,and with how this chemistry compares with that found in fast electron radiolysis. In the following calculations, the kinetics of a high-LET cylindrical track similar to that expected in a particle track core are considered. The simplified system is modeled using a deterministicdiffusion-kinetic method similar to that developed for modeling spur chemistry.22.45 The chemistry of a sectionof track with a specifiedLET is modeled using a set of diffusion-reaction equations. Each equation describes the kinetics of one of the radiation-induced species and has the form
where ci is the spatially nonhomogeneousconcentration of species i, which has diffusion coefficient D,. The terms in the equation represent (i) the diffusion of the reactive species, (ii) its reaction with other track reactants or with scavengers, and (iii) its production by the reaction of other radiation-induced species. The initial concentrations of the radiation-induced reactants are assumed to be Gaussian33
ci(r) = GO' exp(-?/2a2) 200*c72 with r and u being the distance from the center of the track and the standard deviation of the radial concentration profile, respectively. The set of equations aresolved by dividing the track into radial shells that are sufficiently thin so the concentrations of the radiation-induced reactants are homogeneous. Diffusion is then allowed between adjacent shells, and reaction takes place
0.2 '
0.01
id
'
io3
'
IO'
'
lo5
'
roe
'
lo7
'
ion
'
'
roo role
'
I
lott
ro12
Figure 1. Comparison of the geminate pair reaction kinetics for two- and three-dimensional systems. Initial separation, r = 1.O nm; reaction distance, a = 0.5 nm; relative diffusion coefficient, D' = 5.0 X 10-9 mz s-l. (Top) Timedependenceofthegeminaterecombinationintheabsence of a scavenger. (Bottom) Scavenging capacity dependence of the scavenged yield.
according to deterministic rate laws. The result is a set of coupled differential equations which is solved using the FACSIMILE algorithm.& The kinetic calculations reported in this paper consider the chemistry of an idealized cylindrical track. Appropriately scaled, they can be used in a general description of the entire track of a high-LET particle, or more correctly, the contribution due to the track core. The initial radii, the initial C values, and the diffusion coefficients of the radiation-induced reactants are the same as those used in earlier studies of spur kinetics; however, a more extensive reaction scheme" was employed here because of the extended time frame over which reaction occurs. The radii are assumed to be independent of LET; however, in a more quantitative treatment a dependence of the radii on LET will probably be necessary. 3. Results and Discussion Kinetic calculations for geminate radical pairs and for singlespecies tracks obviously do not allow for a complete or a quantitative description of the radiation chemistry of water. However, the simple analytic equations for the time and the scavenging capacity dependences of these systemsprovide insight into the chemical kinetics of two important geometrical configurations encountered in radiation chemistry. Figure 1 shows time-dependent survival probability and the scavenging capacity dependence of the scavenged yield of a simple geminate pair for two-dimensionaland threedimensional systems. This comparison demonstrates a significant change in the diffusive relaxation and reaction of the nonhomogeneously distributed particles, and this
Diffusion-Kinetic Theories for LET Effects
The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6139
.
Oa a
10‘0
109
10-8
10’
10.8
time (s)
Figure 2. Effect of scavengingcapacity on the scavenged yield of radical from an idealized single-speciescylindrical track. Initial concentration profile of reactants is Gaussian with standard deviation, u = 1.0 nm; LET, L = 40 eV/nm; reaction distance, u = 0.5 nm; diffusion coefficient, D = 5.0 X m2s-]. The points are the results of prescribed diffusion , the dashed line is the best nonlinear leastcalculations for G ( k [ S ] ) and squaresfit of eq 10 to these points. The solid line is the Laplace transform prediction of G ( k [ S ] )obtained from the time-dependentdecay kinetics in the absence of the scavenger.
change results in a dramatic increase in the amount of recombination reaction with time in the two-dimensional system. A detailed examination of eqs 4a and 6 for the survival probabilities shows the kinetics of three-dimensional systems have a long-time time dependence that varies as t1/*, while that of a twodimensional system varies as (ln(r))-I. In addition, in the asymptotic limit there is a finite amount of escape in a threedimensional system, whereas all geminate pairs eventually react in the two-dimensional case. It is to be expected that in the radiolysis of water the very slow asymptotic time dependence of the two-dimensional geometry would not be experimentally observable48especially since the tracks of high-LET particles are not truly two dimensional. The most desirable (useful) data from the radiolysis of water are the time dependences of the radiation-induced species. These data contain the most direct information about the fundamental physical and chemical processes occurring in radiolysis. Although it is possible to obtain the temporal variation of the yields of the radicals e,, and OH produced by fast electrons, the corresponding yields of molecular products have not been measured directly. There have been very few measurements of the time dependences of the radiation-induced species in the radiolysis of water with heavier ions because of the experimental difficulties. The experiments have all utilized relatively high-energy light ions, that is, ions with LETS that were not very high. Extensive theoretical studies have shown that the time dependences of the radiation-induced species in the fast electron radiolysis of water can be obtained from measurements of the effects of scavenging capacity on scavenged radical or molecular product yields using an inverse Laplace transform relationship. With high-LET particles, rather than measure time-dependent yields, it would be much easier to measure scavenged yields as a function of scavenger concentration and then, if appropriate, to convert these results via an inverse Laplace transform to the temporal dependences in water. The scavenging capacity dependence of G(s) for a singlespecies cylindrical track is shown in Figure 2 and is compared with the yield predicted using the Laplace transform relation, eq 3. There is somediscrepancy between the two sets of calculations for the scavenging capacity dependent yields; however, it is of similar magnitude to those found earlier, using both stochastic and deterministic methods, for spurs comprised of a single type of radi~a1.~~950 The differences, and hence the errors, associated with the Laplace transform analysis are due to cooperative scavenging effects: the scavenging of one radical affects the
Figure3. Effect of LETon the time-dependentdecayofe,q-inanidealized cylindrical track. The parameters used in the calculations are given in Tables 1 and 2. The solid lines are calculationsfor the high-LETparticle radiolysis of water, and the dashed line refers to fast electron radiolysis.z2 The dotted line and the dash-dotted line denote calculations for aerated water and for water in which k(e,,- + H202) = 0, respectively, for an LET of 100 eV/nm.
subsequent chemistry of a second radical that would have normally reacted with it.49J1.52 Thereis no a priori geometric consideration that forbids the use of Laplace transform techniques in water radiolysis with high-LET particles. In practice, inverse Laplace transforms of experimental yields are never taken. The scavenged yields are fitted to a function for which the inverse Laplace transform is analytically known. A detailed treatment of the spur chemistry produced in water by fast electrons demonstrated the usefulness of the prescribed diffusion approximation for predicting the functional form of the equation used to fit the scavenged yields.49350 Also included in Figure 2 is the nonlinear least-squares best fit of the calculations for G(s) (eq 10) to the function
varying the parameters a! and j3. Equation 16 is simply Go times eq 7. This empirical function accurately reproduces the predictions of eq 10 with a! = 0.494 ns and j3= 0.412ns. The good fit of eq 16 and the limiting long-time dependence of G(t) reinforce the analogy between the two-dimensional geminate problem and high-LET particle track kinetics. While eq 16 clearly has the correct form for use as an empirical equation for describing the scavenging capacity dependence of scavenged yields, it involves the ratio of two Bessel functions. Numerical algorithms for these functions are available, but a best-fit calculation is not straightforward, as the fitting is intractable numerically when only limited data incorporating “experimental” scatter are available. In practice, a simpler function for the dependence of scavenged and molecular product yields on scavenging capacity is desirable. Similar agreement to that presented for the cylindrical track segment is also found when a track made up of a string of overlapping spurs is considered. The effect of LET on the time-dependent decay kinetics of eaqin a model cylindrical track is examined in Figure 3 for the full set of reactions given in Table 2. It is apparent that the chemistry is strongly dependent on the particle LET, which determines the initial density of the radiation-induced particles. Initially (time I1 ns) reaction is rapid; however, at longer times the decay rate is slower. This two-stage time-dependent chemistry is a reflection of the two-dimensional nature of the modeled system. For comparison, the modeled decay of e,, in fast electron radiolysis is also included in the figure.22.53 At short times (50.1 ns) the
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The Journal of Physical Chemistry, Vol. 98, No. 24, 1994
Pimblott and Laverne
TABLE 1: Parameters Used in Modeling the Radiolysis of Water and Aqueous Solutionsu diffusion coeff radial width of initial yield swcies m4-9 track. u (nm) (molec/100 eVI
c
I d
IO'
~
4.5
9.0 7.0 2.8
5.0
2.3 0.85 0.85 0.85 0.85
4.78 4.78 0.42 5.50 0.15
r=Onm
100
IO' '
102
5.0
2.1 2.0 2.2
TABLE 2 Reaction Scheme Used in Modeling the Radiolysis of Water and Aqueous Solutions22 reaction k (1010 M-l ~ - 1 ) ~ ' eaq-+ eqHz+OH+OH0.55 eaq- + Haq+ H 2.3 eaq-+ H H2 + OH2.5 esq- + OH OH3.0 eaq-+H202 OH+OH1.1 Haq++OHH20 14.3 H+H + Hz 0.78 H+OH Hz0 2.0 H + H202 OH + H20 0.009 OH + OH H202 0.55 eaq-+ 0 2 021.9 H+Oz H02 2.1 Haq++ 0 2 HO2 3.8
--
+
r = 4.6 nm
I 0-3
y r
I
4.6 nm
I
I0't 10.51 100
'**-.*.I
.*,*.-.I
IO'
lo2
..*.-*.I
.
.**J
. . . . . a . f i
1o3
1o5
10'
LET (eV/nm)
Figure 4. Effect of LET on the effective %- concentration at various radial distances, r, in the idealized cylindrical track; standard deviation, u(%-) = 2.3 nm. The solid lines refer to cylindrical tracks while the dashed lines denote the concentrations in the prototypical spur.
4
+
--. 4
4
kinetics of fast electron radiolysis are very similar to those in a high-LET particle track core of LET = 10 eV/nm, but at longer times the two sets of decay kinetics diverge. While the nonhomogeneous chemistry of fast electron radiolysis shows a definite escape yield (G" = 2.55 molecules/100 eV),ZZthere is no limiting escape yield in high-LET particle radiolysis. The similarity between the kinetics at short times is easily understood when the "effective initial concentrations" of the reactants are considered. Figure 4 demonstrates the influence of LET on the effective concentration at a number of radial distances from the center of the cylindrical track and makes comparisons with the effective concentrations at these distances from the center of the prototypical spherical spur used to model fast electron radiolysis. The concentrations of e,, in the track core match those in the spur at a LET of approximately 10 eV/nm. The divergence between the cylindrical track and spur kinetics at longer times (>1 ns) and the lack of an escape yield in high-LET particle radiolysis reflect the difference in the dimensionality of the kinetics. Figure 5 shows the effect of LET on the scavenging capacity dependence of the scavenged yield of eaq-. For each LET considered the scavenged yield of eaq-increases as the scavenging capacity of the solution is increased; however, the scavenged yield for a particular scavenging capacity depends upon the LET of the radiation. As was observed in Figure 4, there is no limiting escape yield (low scavengingcapacity yield) in high-LET particle radiolysis. Also shown in the figure are the scavenged yields of eaq-qredicted by the Laplace transform treatment of the decay kinetics given in Figure 4. The agreement between the true scavenged yields and the Laplace transform predictions is excellent. This good agreement is unexpected as the radiation chemical kinetics are very nonlinear,49especially in high-LET particle tracks, and it would not be unreasonable for eq 3 to break down under these conditions.54The agreement suggests that the chemistry observed in the high-LET particle radiolysis of aqueous solutions reflects the underlying kinetics in the absence of the scavenger. Consequently, experimental scavenger data can be used to imply the time dependence of yields in the absence of the scavenger. Such time-dependent information is not generally available directly for high-LET particles.
10'
10'
io8
10'
ioe
ios
ioio
ioi1
k [SI(s.7 Figure 5. Effect of LET on the scavenging capacity dependence of the scavenged yield of %- in an idealized cylindricaltrack. The parameters used in the calculations are given in Tabla and 2. The filled points are the results of FACSIMILE calculations for G ( k [ S ] )while the solid lines are the Laplace transform predictions of G ( k [ S ] )obtained from the time-dependent decay kinetics in the absenceof the scavenger (cf. Figure 3). The open points and the dashed line are for fast electron radiolysis.22
The effect of LET on the formation kinetics of HZin pure water and on the scavenging capacity for ea,,- dependence of the Hz yield in aqueous solutions is modeled in Figure 6. In water, as the LET of the cylindrical track increases, the yield of Hz increases. This increase in Hz is due to an increase in the relative yield of e,,,- ea,,- reaction compared to that of %q- + OH reaction and reflects the change in the density of the nonhomogeneously distributed radiation-induced reactants with LET. (Remember that the spatial relaxations of e, and of OH are different as D(eaq-) >> D(OH).) Increasing the scavenging capacity for %results in a decrease in the production of H1. Naturally, this decrease occurs over the same range of scavenging capacity as the increase in the yield of scavenged eaq-observed in Figure 5. Comparison of the modeled dependence of the Hz yield on scavenging capacity with the dependence predicted using the Laplace transform relation shows good agreement. This agreement implies that the scavenging capacity dependence of the HZ yield reflects the underlying formation kinetics of Hz for radiolysis with particles of high LET, as well as low LET. The modeled decay kinetics of OH and the scavengingcapacity dependence of the scavenged yield of OH in water are shown in Figure 7. The effect of LET is qualitatively similar to that observed in Figures 3 and 5 for e,; however, the kinetics predicted for LET = 10 eV/nm are not similar to those for fast electron radiolysis. This differencesimply reflectsthe fact that the effective
+
Diffusion-Kinetic Theories for LET Effects
Figure 6. Effect of LET on the chemistry of H2 in an idealized cylindrical
The Journal of Physical Chemistry, Vo1. 98, No. 24, 1994 6141
WI (s.7 Fig11re7. Effect of LET on the chemistryof OH in an idealizedcylindrical
track. The parameters used in the calculations are given in Tables 1 and 2. (Top) Time-dependentformationof H2. Thesolid lines arecalculations for the high-LET radiolysis of water, and the dashed line refers to fast electron radiolysis. (Bottom) Scavenging capacity dependence of the yield of H2. The filledpoints are the resultsof FACSIMILEcalculations for G ( k [ S ] )while the solid lines are the Laplace transform predictions of G ( k [ S ] )obtained from the time-dependent formation kinetics in the absence of the scavenger (cf. part a). The open points and the dashed line are for fast electron radiolysis.22
track. The parameters used in the calculations are given in Tables 1 and 2. (Top) Time-dependent decay of OH. The solid lines are calculations for the high-LET particle radiolysis of water, and the dashed line refers to fast electron radiolysis. (Bottom) Scavengingcapacity dependenceof the scavenged yield of OH. The filled points are the results of FACSIMILEcalculationsforG(R[S])whilcthesolidlinesarttheLaplace transformpredictionsof G ( R [ S ]obtainedfrom ) thctime-dependent decay kinetics in the absence of the scavenger (cf. part a). Theopen points and the dashed line are for fast electron radiolysis.=
radialconcentrations of OH in the twosystems are different. The agreement between the Laplace transform prediction for the scavenging capacity dependence of the scavenged yield of OH is much poorer than that found for e,,,- (in Figure 9,but it is still sufficientlyaccurate to suggest that experimental scavenger data can be used to obtain an estimate for the underlying kinetics of OH in the absence of the scavenger. A small discrepancybetween the two different analyses was also observed in the earlier treatment offast electronradiolysis.22 Theorigins ofthedifference are discussed later in this manuscript. The production of H202 in water in the model cylindrical track is examined in Figure 8. The time-dependent formation kinetics and the scavenging capacity dependences of the H202 yield are somewhat similar to those shown in Figure 6 for H2. For instance, the yield of H202 increases with LET because of the increase in the yield of OH + OH reaction compared to that of e,, + OH. Notice, however, that the time-dependent yield of H202 reaches a maximum and then decreases at longer time. Also included in Figure 7a are a calculation for water with an enq-scavenger of capacity 4.8 X 106 s-1, cf. aerated water, and a calculation in which the rate constant for the intratrack eaq; + H202 reaction was set to zero. In the former case, there is still a small decrease in the yield of Hz02on the nanosecond time scale; however, the continued decay at longer times is prevented. When the eaq-+ H202 reaction is removed from the reaction scheme, there is no decrease predicted in the H202 yield at long times. This detailed
examination of the modeled chemistry shows that the decrease in H202 yield is the result of the secondary reaction of H202 with eaq-. This reaction becomes significant in high-LET particle radiolysis because the amount of the nonhomogeneous reaction is much larger than in low-LET radiolysis. The reaction eaqH202 results in the production of an OH radical, and this production of OH by intratrack reaction is the origin of the small discrepancies in the yields observed in Figure 7. While the Laplace transform treatment of the decay kinetics of enq-and O H and of the formation kinetics of H2 agree well with the modeled scavenged yields in the calculations reported here, the agreement shown in Figure 9 for H202 is poor. There are significant discrepancies in the predictions for H202 at high scavenging capacities for OH. These discrepancies also result from the secondary reaction ens- + H202 already discussed. In the earlier studies of spur kinetics,22.55 the Laplace transform relationship was also found to hold for Hz02 as well as for e?q-, H2, and OH. In fast electron radiolysis, the e, + H202 reaction has a negligible role on the observed kinetics. Figure 9 examines the effect of the secondary reaction enq-+ H202 on the scavenging capacity dependence of the H202 yield for a cylindrical track of LET = 100 eV/nm. The inclusion of an eaq-scavenger to prevent this reaction at longer times increases the yield of H202 for low scavenging capacities; however, there is still some discrepancy between the true and the Laplace transform yields at high scavenging capacities. It is only when the eaq-+ H202 reaction
+
6142 The Journal of Physical Chemistry, Vol. 98, No. 24, 1994
Pimblott and LaVerne
LET = io5 eV/nm
4
h
--3 -0
10
-
A
deaerated
r ”
A
c7 lo3
io4
io5
ioL)
io7
io8
lo9
iolo
10”
k [SI(5.7
Figure 9. Significanceof the reaction %- + H202 on the yield of H202
0
x
1.5
\ 1
102 A
. -
last electron
..
lo3
io4
lo5
lo6
10’
lo6
lo9
1olo
10”
k [SI(s-? Figure 8. Effect of LET on the chemistry of H202 in an idealized
cylindrical track. The parameters used in the calculations are given in Tables 1 and 2. (Top) time-dependent formation of H202. The solid linesare calculationsfor the high-LETradiolysisof water, and the dashed line refers to fast electron radiolysis. (Bottom) Scavenging capacity dependence of the yield of H202. The filled points are the results of ) thesolid lines are theLaplace FACSIMILEcalculationsfor G ( k [ S ] while transform predictions of G ( k [ S ] )obtained from the time-dependent formation kinetics in the absence of the scavenger (cf. part a). The open points and the dashed line are for fast electron radiolysis.22 is prevented completely (by setting the rate constant to zero in the calculation) that the true and the Laplace transform results agree. The chemistry of spatially nonhomogeneous systems depends critically on thedistribution of the reactive transients. A complete quantitative description of the chemistry of tracks of high-LET particles will require a detailed description of the track structure and in particular the differences between the two regions of the track the core and the penumbra. In the calculations reported for aqueous systems, a simple cylindrical track was considered. This uncomplicated model provides a useful qualitative description of the effects of LET on the spatial distribution of the reactants, using a radial concentration profile that is independent of the LET and particle type. It also provides a rigorous examination of the effects of LET on nonhomogenous kinetics and the validity of the Laplace transform relation between G(t) and G(s) for spatially extended systems. For a quantitative treatment of highLET particle radiolysis, a more detailed picture of the initially nonhomogeneous spatial distribution is necessary which incorporates the influences of the properties of the radiation particle on the physical and the physicochemical processes of radiolysis. 4. Summary
Analytic models and deterministic calculations have been used to examine the way that LET might effect the radiation chemistry of aqueous solution. Calculations for an idealized cylindrical
in the high-LET particle radiolysis of aqueous solutions. The points are the results of FACSIMILE calculations for G ( k [ S ] )while the lines are the Laplace transform predictions of G ( k [ S ] )obtained from the timedependent formation kinetics in the absence of the scavenger. The solid line and the filled triangles are calculations for the high-LET particle radiolysis of water, the dotted line and the open circles are for aerated water, and the dashed line and the open squares denote calculations for water in which k(eaq- + H202) = 0. track support the use of a Laplace transform relationship to link yields in scavenger experiments with the underlying timedependent kinetics in the absence of the scavenger for eaq-,OH, and H2.The relationshipcannot be used todescribe the formation of H202 because of a complicating secondary reaction. In lowLET radiolysis, secondary reactions like eaq-+ Hz02do not have a significant impact on the observed chemistry; however, in highLET particle radiolysis, the secondary reactions of the molecular product Hz02 cannot be ignored. Analytic results obtained from simple track models show that the limiting long-time kinetics associated with high-LET particle tracks have time dependences whichvary as (ln(r))-l, These long-time kinetics areconsiderably slower than those found for low-LET particle radiolysis which have a t 1 1 2dependence. Naturally, the predictions of the simple models suggest a significant reduction in the long-time yields of radicals in high-LET particle radiolysis compared to radiolysis with particles of low LET. Acknowledgment. The work described herein was supported by the Office of Basic Energy Sciences of the U.S.Department of Energy. This is contribution NDRL-3663 from the Notre Dame Radiation Laboratory. References and Notes (1) Mozumder, A.; Magee, J. L. J . Chem. Phys. 1966,45, 3332. ( 2 ) Chatterjee, A.; Magee, J. L. In Radiation Chemistry. Principles and Applications;Farhataziz,Rodgers, M. A. J., Eds.;VCH Publishers: New York, 1987. (3) Allen, A. 0. The Radiation Chemistry of Water and Aqueous Solutions; Van Nostrand: New York, 1961. (4) Appleby, A.; Schwarz, H . A. J . Phys.
Chem. 1969, 73, 1937. (5) Laverne, J. A.; Yoshida, H. J . Phys. Chem. 1993, 97, 10720. (6) Schwarz, H. A.; Caffrey, J. M., Jr.; Scholes, G. J. Am. Chem. SOC. 1959,81, 1801. (7) Lefort, M.; Tarrago, X . J . Phys. Chem. 1959, 63, 833. (8) Anderson, A. R.; Hart, E . J. Radiat. Res. 1961, 14, 689. (9) Burns, W. G.; Sims, H. E. J. Chem. SOC.Faraday Trans. 1981,177, 2803. (10) Laverne, J. A. Rodiat. Res. 1989, 118, 201. ( 1 1 ) Bibler, N. E. J. Phys. Chem. 1975, 79, 1991. (12) Schuler, R. H.; Allen, A. 0. J. Am. Chem. SOC.1957, 79, 1565. (13) Baverstwk, K.F.; Cundall, R. B.; Burns, W. G. In Proc. 3rd Tihany
Symp. on Radiation Chemistty;D o h , J., Hedvig, P., Eds.;Akademiai Kiado: Budapest, 1972; p 1133. (14) Burns, W. G.; May, R.; Buxton,G. V.; Tough, G. S . Faraday Discuss. Chem. SOC.1977,63,47. (15) Burns, W. G.; May, R.; Buxton, G. V.; Wilkinson-Tough, G. S . J . Chem. SOC.Faraday Trans. I 1981, 77, 1543.
Diffusion-Kinetic Theories for LET Effects (16) Rice, S.A.; Playford, V. J.; Burns, W. G.; Buxton, G. V. J . Phys. E . 1982, I S , 1240. (17) Sauer, M. C., Jr.;Schmidt, K.H.;Hart,E. J.;Naleway,C.A.; Jonah, C. D. Radiat. Res. 1977, 70, 91. (18) Sauer, M. C., Jr.; Jonah, C. D.; Schmidt, K. H.; Naleway, C. A. Radiat. Res. 1983, 93, 40. (19) Onsager, L. Phys. Rev. 1938, 51, 554. (20) Hummel, A. J . Chem. Phys. 1968,48, 3268; 1968,49,4840. (21) Warman, J. M.;Asmus, K.-D.; Schuler, R. H. J. Phys. Chem. 1969, 73. 93 1. '(22j Laverne, J. A.; Pimblott, S. M. J. Phys. Chem. 1991, 95, 3196. (23) Rice, S. A. In Diffusion-1imiredReaction.v.ComprehensiwChemical Kinetics Vol. 2% Bamford, C.H., Tipper, C. F. H., Compton, R. G., Eds.; Elsevier: Amsterdam, 1985. (24) Tachiya, M. J . Chem. Phys. 1979, 70, 238. (25) Green, N. J. B.; Pimblott, S. M. J . Chem.SOC.Faraday Tram. 1993, 89. -1299. (26) Carslaw, H. S.; Jaeger, J. C. Conductionof Hea?inSolids;Clarendon Press: Oxford, 1959. (27) Abramowitz, M.; Stegun, I. A. HandbwkofMathematicalFunctionF; Dover: New York, 1970. (28) Jaeger, J. C. J . Math. Phys. 1955, 34, 316. (29) Bolton,C.E.;Grbesa,J.I.;Green,N. J.B.;Pimblott,S.M. Submitted I
for publication. (30) Jaffe, G. Ann. Phys. IV 1913,42, 344. (31) Samuel, A. H.; Magee, J. L. J . Chem. Phys. 1953, 21, 1080. (32) Ganguly, A. K.! Magee, J. L. J . Chem. Phys. 1956, 25, 1955. (33) A variety of different profiles have been employed in modeling radiation chemical kinetics (Freeman, G. R. Adu. Chem. Ser. 1968,82,339); however, the Gaussian, considered here, is the most common used. The use of a different profile, for instance an exponential, would not influence the general results obtained in, or the conclusions drawn from, the calculations. It would, of course, affect detailed predictions. (34) Following convention, throughout this article radiation chemical yields are given in the units molecules per 100 eV.
The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6143 (35) Naleway, C. A.; Sauer, M. C., Jr.; Jonah, C. D.; Schmidt, K. H. Radiat. Res. 1919, 77, 47. (36) Katz, R.; Huang, G.-R. Radiat. Phys. Chem. 1989, 33, 345. (37) Magee, J. L.; Chatterjee, A. J. Phys. Chem. 1980.84, 3529; 1980, 84, 3537. (38) Kapoor, S.; Gopinathan, C. J. RadioanaI. Nucl. Chem. 1991, 150, 3. (39) Mozumder, A.; Chatterjee, A.; Magee, J. L. In Aduances in Chemistry Series Vol.81; Gould, R.F.,Ed.; American Chemical Society: Washington, DC, 1968. (40) Zaider, M.; Brenner, D. J.; Wilson, W. E. Radiat. Res. 1983,95,231. (41) Turner, J. E.; Hamm, R. N.; Wright, H. A.; Ritchie, R. H.; Magee, J. L.; Chatterjec, A.; Bolch, W. E. Radia?. Phys. Chem. 1988, 32, 503. (42) Kaplan, I. G.; Miterev, A. M.; Sukhonosov, V. Ya. Radiat. Phys. Chem. 1990,36,493. (43) Terrissol, M.; Beaudre, A. Radia?. Prof.Dosim. 1990, 31, 175. (44) Xapsos, M.A. Radiar. Res. 1992, 132, 282. (45) Bums, W. G.; Sims, H. E.; Goodall, J. A. B. Radiaf.Phys. Chem. 1984, 23, 143.
(46)C h a m , E. M.; Curtis, A. R.; Jones,I. P.; Kirby, C. R. Report AERE-R 8775: A E R E Hanvell. 1977. (47) Buxton, G. V.: Greenstock, C. L.; Helman, W. P.; Ross, A. B. J . Phys. Chem. Ref. Data 1988, 17, 513. (48) Gosele, U. M. Prog. Reac?.Kiner. 1984, 13, 65. (49) Green, N. J. B.; Pimblott, S.M. Mol. Phys. 1991,74,795; 1991,74, 811. (50) Pimblott, S. M.; Laverne, J. A. J. Phys. Chem. 1992, 96, 746. (51) Pimblott, S. M.;Laverne, J. A. J. Phys. Chem. 1992, %, 8904. (52) Laverne, J. A,; Pimblott, S. M. J . Chem. SOC.Faraday Trans. 1993, 89, 3527. (53) The predicteddecay of the hydrated electron is somewhat at variance
with the experimentally observed decay. The discrepancy is minor and of no consequence in the present discussion. (54) Mozumder, A.; Magee, J. L. Radiat. Phys. Chem. 1975, 77, 83. (55) Pimblott, S.M.; Laverne, J. A. Radiat. Res. 1992, 129, 265.
