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Evidence for Spur Overlap in the Pulse Radiolysis of Water James E. Fanning, Jr., Conrad N. Trumbore,* P. Glenn Barkley, and Jon H. Olsont Department of Chemistry and Department of Chemical Engineering, University of Delaware, Newark, Delaware 19711 (Received December 16, 1974; Revised Manuscript Received April 18, 1977)
The kinetics of the decay of hydrated electron absorbance in pure water has been studied as a function of time and radiation dose delivered by 20-ns pulses of 15-MeV electrons. In order to identify intraspur reactions of the hydrated electron, values of the function 1[-d(e,,-)/(e,,-)]/dtJ, (= Q t ) were investigated as a function of pulse dose and time. Qt represents the fraction of hydrated electrons disappearing at any given time, t , following the pulse. Whereas Q has the same functional form as a first-orderrate constant, it is not a constant, but decreases with increasing time following the pulse. However, the functional form of Qt us. time remains the same at low pulse doses (37-380 rads) for relatively long periods of time (up to 1 ys). The dose independence of Q is interpreted as evidence for predominantly intraspur hydrated electron decay. A plot of Q t vs. the function [ (time)1/2(dose)1i3] yields a relatively sharp transition from dose independent hydrated electron decay (interpreted as arising predominantly from intraspur reactions) to dose dependent decay (both interspur and intraspur decay). This transition region is interpreted as the onset of significant spur overlap. In the range 0-8000 rads per 20-11s pulse, an empirical constant: (approximate time of detectable o ~ e r l a p ) ~ i ~ ( d o s=e110 ) ~ ’f~ 10 ns’’’ rads1i3is obtained which correlates the onset of experimentally observable spur overlap. Based upon these results, implications regarding spur sizes and spatial distributions of spurs and the need to include spur overlap considerations in modeling studies are developed.
Introduction Much of the experimental evidence in support of the spur model in aqueous radiation chemistry has been inDespite difficulties with the spur model, experimental evidence which has appeared to conflict with the model has been accommodated by some small, but significant, modifications in initial parameters. Experiments utilizing ultrashort radiation pulses and time resolution have provided new challenges to the spur model. While certain of these challenges, most notably the results of Hunt and co-w~rkers,~ have been answered by Schwarz,’ the results of Jonah et a1.6 appear to require modifications of certain spur model parameters. What is needed to test any modified spur model is an experimental indication of the distributions of the hydrated electron and of other reactive intermediates as a function of space and time following the pulse. Toward this end, we have investigated the hydrated electron decay in pure water as a function of time and of dose delivered by 20-ns pulses of 15-MeV electrons. We assume that the distribution function for nearestneighbor interspur distances at the time of spur formation contains a relatively small fraction of near-zero distances. If such is the case, a t low doses, the bulk of the newly formed spurs will be far enough apart that electron decay which arises predominantly from intraspur reactions can be observed. Thus, a t low enough pulse doses, the measured electron decay rate for the “average” spur should be independent of pulse dose; that is, a rate which is directly proportional only to the number of spurs present. At higher pulse doses (and for longer times at lower pulse doses), it is presumed that spurs begin to overlap in large enough numbers to detect deviations from the independent, intraspur electron decay established a t lower doses and thus provide rough experimental criteria for the average “size” of spurs at different times. At large enough pulse doses and a t long enough times, homogeneity of all species is approached, and homogeneous treatment of the *Address correspondence to this author at the Department of Chemistry. 1’ Department of Chemical Engineering. The Journal of Physical Chemistry, Vol 81, No. 13, 1977
kinetics then can be compared and contrasted with the nonhomogeneous kinetics of the spur.
Experimental Section The experimental work reported here was performed a t the Chemistry Division of the Argonne National Laboratory, Argonne, 111. Most of the techniques employed in this work are described el~ewhere.~” Water, which was a t least triply distilled, was obtained from three different stills at three different laboratories with no variation in results. Three gases were used for saturating water. The minimum gas purities were 99.95% for hydrogen and 99.995% for helium and argon. During bulk saturation of samples helium and hydrogen were passed through a liquid nitrogen tr?P for further purification. The syringe technique’, was used to exchange the desired gas for the air initially present. Random checks on the oxygen concentration in these solutions showed values less than 2 pM, usually between 0.2 and 1.0 pM. (See supplementary material for additional details.) The absorbance of the hydrated electron was followed as a function of time a t 633 nm in most experiments. In a number of experiments, the wavelength employed was 560 nm and after correction for the differences in extinction coefficient,” the kinetic data were in very good agreement with those taken a t 633 nm. Silica cells of 2.0 and 4.0 cm length and 2.0 cm diameter with Suprasil end windows were used for sample irradiation. The percent absorption and time values were obtained from oscilloscope traces. The center of the trace line was used for this measurement, and in areas where random noise produced fluctuations of the line a smooth line was drawn through the noise. At the beginning of the scan the trace changed most rapidly with time, and the noise was largest. Thus the errors are greatest in the first few points of each trace. The data from each experiment were recorded from the oscilloscope trace and are available.12 Three typical traces are shown in Figure 1 for doses of 140,700, and 970 rads. As expected, the figure shows that the data do not correlate with either a conventional first- or second-order empirical rate function in the time range of the experiments. Ac-
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Spur Overlap in the Pulse Radiolysis of Water I O
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Figure 2. Comparison of cumulative radial probabilities of (a) finding electrons in a giiven spur and (b) of finding a randomly distributed nearest-neighbor spur center. Calculations based upon formula derived in the Appendix arid on an assumed hydrated electron distribution function (see text) with a maximum at 4 nm from the spur center ( p = O), a pulse dose or 2000 rads, and 100 eV/spur. p is the dimensionless radius of a spheire enclosing a given spur. p = r / r 0 where r is the Nbeing the number of spurs per unit volume. radius and r, = (3/45~N)''~, Curve a represents the accumulated probability of finding (immediately following an irradiation pulse) all the hydrated electrons belonging to a given spur within the volume of a.sphere of radius p. Curve b represents the accumulated probability of finding the center of another (nearest neighbor) spur within the sphere of radius p. This curve is interpreted to mean that there are relatively few spurs which overlap immediately following the pulse and that the earliest hydrated electron decay is predominantly intraspur.
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Figure 1. Typical hydrated electron decay kinetic plots. (a) In absorbance as a function of time of gas-saturated, pure water, pH 7.0, irradiated with a 20-ns, 15-MeV electron pulse. The light intensity was monitored at 560 or 633 nm. The pulse doses were as follows; (05 140 rads ( e / = 5.00 X IO5 L mol-'); (W) 700 rads ( e / = 1.276 X 10 L mol-'); (A)970 rads (e/ = 1.276 X lo5 L mol-'); (b) reciprocal absorbance vs. time following pulse for same solutions as in (a), Curves in (a) and (b) are least-squares fits to data points. Parameters derived from these fitted curves were used to obtain data for Figures 3-5.
cordingly, an alternative interpretation of these data is needed to bring the data set into a common framework. The concentration history of the solvated electron was converted into point rate data for the bulk of the runs by differentiation of an empirical function fit to the data by a Marquardt non-linear least-squares computation routine (typical estimated error in the derivative is 3%). At early times during very high pulse dose experiments (a minority of cases), graphical differentiation was used (typical estimated error of 5%). Fits to the data using the empirical function are shown in Figure 1. We believe the error estimates for differentiation are conservative. The details of the data reduction are given in ref 12.
Results In treating our data, we assume that radiation energy is initially deposited in such a manner that randomly distributed, nonhomogeneous groups of radicals, called are formed in the pulse radiolyzed water. Among the species ultimately populating these spurs are the hydrated electron, the hydroxyl radical, the hydrogen ion, and the hydrogen atom. At low pulse dose levels and early times following the pulse, we postulate (see the Appendix for detailed reasoning) that the average interspur distance is large when compared with the size of
the individual spur regions so that the probability for very early interspur interaction (overlap) is finite, but low, and consequently the overall kinetics initially are dominated by intraspur reactions. Diffusion causes the spurs to mix, and ultimately homogeneous kinetics are observed. The following is an effort to demonstrate experimentally and quantify certain aspects of these transformations from predominantly intraspur hydrated electron decay to homogeneous hydrated electron reaction. The analysis shows the importance of gathering statistically meaningful data in the nanosecond range using a wide variation of pulse dose to extend our understanding of the transition to homogeneous kinetics. Figure 2 shows two cumulative distributions which illustrate our basic ideas regarding spur overlap. The cumulative probability distribution for the electron is based on a shewed Gaussian distribution (see later discussion) with a probability maximum at 4 nm from the spur center. The cumulative distribution function for the center of the nearest-neighbor spur is based upon the development found in the Appendix using a 2000-rad dose with 100 eV/spur average energy deposition. The curves imply that a sinall fraction of the spurs do overlap initially, but that most of the hydrated electron reactions occur within isolated spurs initially. The majority of radicals in individual spurs mix by diffusion before reactions with species in other spurs can occur. The detailed kinetics of decay of the hydrated electron in the period from 60 to 900 ns after the pulse is quite complex and i13 not the major topic of this paper. However, the hydrated electron decay data obtained in our experiments can be arranged to yield a simple test of the above postulates. If the decay of the hydrated electron initially is controlled by intraspur reactions, the fraction of electrons decaying at any specific time, { [-d(eJ/(eJ] /dt), E Qt, should be independent of pulse dose for a given time, t , following the radiation pulse. The value of Qt decreases with increasing time following the pulse, indicating that The Journal of Physical Chemistry, Vol. 81, No. 13, 1977
Trumbore et al. I I I
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Flgure 4. Test of spur model parameters Qt and the product (time of spur overlap)"*(puise dose)"3 according to the model of Kenney and Walker12 in the pulse radiolysis of pure water. Breaks in curve at a value of t'"(do~e)'/~ 110 z t 10 ns'" radiI3 yield a useful indicator for a given dose of the time at which experimentally detectable spur overlap is most significant. Before this time, hydrated electron decay is shown to be predominantly intraspur by the nearly horizontal lines for each of the times listed. Lines in the positive slope region are regression fits.
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Figure 3. Time-pulse dose dependence of Of, the instantaneous /dt). Curve (fractional) hydrated electron decay rate (Of3 [-d(ea;)/(ea;)] fined data from absorbance measurements similar to those represented in Figure 1 are differentiated to yield Q,data. The points represent mean vaiues of Qfover the pulse dose ranges indicated on the figure. The numbers in parentheses represent the total number of experiments performed in the dose range indicated. Data in the 37-364-rad dose range showed no statistically observable trends and are shown as a single curve.
the intraspur kinetics are not first order. Accordingly, a plot of Qt vs. pulse dose with time as a fixed parameter should have two regions. At low pulse dose the value of Qt should be independent (constant) of pulse dose. Behavior of this type suggests that the bulk of the hydrated electrons in spurs are reacting independently in isolated spurs. At still larger pulse doses, for a fixed time following the pulse, the spurs begin to interact (overlap) at the time value chosen; the relative decay rate (vs. no spur overlap) increases. Figure 3 is a plot of Q t , the kinetic test parameter for intraspur hydrated electron decay, vs. time for a range of doses. There is substantial scatter in these sets of curves. ow eve^, the general features of the curves clearly are: (a) am asymptotic region for times >500 ns in which the curves approach a nearly constant value; (b) the asymptotic value of Q for times >500 ns increases with dose; and (c) for times in the range 60-150 ns an asymptote with a large, negative slope from which the curves depart more readily with increasing dose. The existence of two types of asymptotes requires clarification. The relationship between the dose and corresponding time a t which the decay mode changes is expected by diffusional kinetic a r g ~ m e n t s to ' ~ be directly related to the functional form (time)''2(dose)1'3. This form is based upon the treatment of Menney and Walker,14 who considered the time needed for spherical spurs, spaced evenly throughout the solution, to reach homogeneity. The factor (time)"' arises from the diffusion equation where the characteristic diffusion length is proportional to (Dt)':' (t is the time, D , the diffusion coefficient). The factor The Journal of Physical Chemistry, Vol. 81, No. 13, 1977
(dose)'/3 relates, through the number of spurs formed, to the average distance between them. Kenney and Walker14 proposed that the product of (time)1/2(dose)'/3should be a constant. Accordingly, values of Qt in Figure 3 are plotted vs. (time)1/2(dose)'/3on Figure 4 for each of five times following the pulse, from 60-140 ns (beyond 140 ns there are insufficient data to define adequately the zero slope region value). On Figure 4 the lines are averages of the points along the zero slope region and regression fits along the positive slope region (including data at 5700 rads, but not including the more limited data at 1600 rads). The intersection of the two lines for each time yields a constant that relates the time-dose values at which the decay mode apparent1 changes. The average value of this constant is 110 n~'''rad'/~ with an approximate error of 10%. This constant specifically applies to 20-11s pulses of 15-MeV electrons of less than 8000 rads total dose. However, given the relatively small changes in LET with energy for high energy electron^,'^ this constant should have order-ofmagnitude applicability over a relatively broad range of electron energies. The spherical spur model can also be used to estimate the time required for spurs to mix by diffusion sufficiently to yield homogeneous kinetics. Qualitatively, spurs can be viewed as a set of points in random array for which the least sophisticated model is a cubic lattice scaled by the average spur separation distance. The time required to achieve homogeneous kinetics in the model can be estimated from simple diffusional relaxation. The time required to obtain homogeneous kinetics of the spurs should be roughly proportional to the diffusion time. Discussion and Conclusions The above data for solute-free water provide valuable experimental information about the timing of events in aqueous radiation chemistry. The data in Figure 4 demonstrate the importance of both the pulse dose and the time following the pulse in determining the degree to which interspur reactions are important in causing the disappearance of hydrated electrons. These findings offer research workers a method of determining quantitatively the limited amount of time available to observe predominantly intraspur hydrated electron decay processes, a time which is strongly dependent upon pulse dose. The
Spur Overlap in the Pulse Radiolysis of Water
use of the Qt parameter a t very early times in pulse radiolysis studied at higher LET'S for pulses of fast electrons and other accelerated particles offers a tool for studying the hypotheses of overlapping spurs forming cylindrical tracks, blobs, and short tracks.16 The aqueous, electron-pulsed system can be viewed as the interaction of spurs of differing sizes which have a distribution of distances from their nearest-neighbor spurs. The time required for experimentally detectable overlap interrelates the spur size and separation distribution function, though it does not reveal the mathematical nature of either. The experimental determination of constraints on these two functions is a major contributioh of this work. Kenney and WalkerI4 report observing the attainment of homogeneity in electron pulse irradiated water. A direct comparison of their results is not possible since they worked with much lower energy and therefore higher LET-electron pulses and at doses of 39,57, and 68 krads, whereas the data here are for
