J. Phys. Chem. 1992, 96, 8074-8019
8074
Consecutive Captures of Electrons and Positrons in the Positron Spur by Organic Halogenated Compounds and Halide Detachment in Polar Solvents F. Didierjean, J. Ch. AbM, and G.Duplltre* Laboratoire de Chimie Nuclgaire et Physicochimie des Rayonnements. 67037 Stasbourg Cgdex 2, France (Received: April 6, 1992; In Final Form: June 10, 1992)
The existence of fast successive captures of quasi-free electrons and then of positrons, previously evidenced with HgC12in various polar solvents, is confiied in methanol and in dioxane for a variety of organic halogenated compounds (RX= CCL, CHC13,CH2C12,CH212,and CHJ) by combining the two independent techniques of lifetime s p c c t ” y and Doppler broadening (DBARL). In methanol, the DBARL parameters characteristicof the [=-e+] bound states are significantly different from those related to the corresponding halide ion bound state, [X-e’], indicating that no fast halide detachment occurs from RX- after electron capture by RX. For a given halo compound in a specified solvent, the constants for the inhibition of positronium formation by the solute and for the positron bound-state formation are very close. On the basis of a simple probabilistic approach, it is shown that this indicates that only one among the spur electrons is able to reach the positron to form positronium.
Introduction Although it has been extensively studied in the past decade, positronium (Ps) formation in condensed matter is still a very important topic, not only on fundamental grounds, particularly as regards the implication of fast radiolysis processes, but also because of the numerous potential applications of Ps as a probe of the physicochemical environment, which demand a precise knowledge of its behavior in matter.’ Regarding Ps formation in liquids, the spur model, which probasically that Ps is formed by the reaction of the positron (e’) with one of the electrons (e-) released by ionization of the medium at the end of the e+ track, is now firmly established. In polar solvents, a salient argument in favor of the model is the strong correlation found between the reactivity of solutes with the precursors of the solvated electrons and their ability to suppress Ps formati~n.~.~ However, important features of the positron spur are still unknown, such as its size or the number and distribution of the reactive radiolytic species. Recently, experiments have been performed with HgC12 as a solute in various solvents, showing the existence of consecutive e-, then e+ captures in the spur.4 Besides their intrinsic interest, these results also gave information on the process of halide detachment after e- capture and should be useful to unravel some of the unknown features of the spur, particularly its extent and the number of electrons. Therefore, it appeared interesting to assess the general character of such consecutive reactions, which would seem diffcult to occur on the very short time scale involved, and thus gain more insight on the positron spur and fast radiolytic reactions. To this end, a variety of organic halogenated compounds (CC14, CHCI3, CHzC12,CH212,and CH31) was studied in polar protic (methanol) and nonprotic (dioxane) solvents. As in the previous work: precise information on any positron bound states present in the solutions was sought by combining two independent techniques, the Doppler broadening (DBARL) and lifetime spectroscopy (LS). Experimental Section
The solvents, methanol (MeOH) and dioxane from Merck, and the solutes, from Prolabo, were pure grade and used as received, except CH31and CH212. The latter compounds, whose reddish color denoted the presence of free iodine, were purified by reduction and extraction of iodine in the presence of a saturated NaHS03 aqueous solution. The two phases were vigorously agitated, then decanted, and separated, and the aqueous phase was changed several times, until complete bleaching of the organic phase, which was finally distilled under reduced pressure. All measurements were carried out at 294 K. The positron source, 7.4 X lo5 Bq of 22Nadiffused in a thin glass foil, was immersed in the solution and the containing ampule was degassed by the usual freezethaw technique, before sealing. The ampules
TABLE I: LS and DBAlU Parameters for the Pure Solvents lop,Io3,
fwhm,
solvent ns 96 keV MeOH 3.30 22.0 2.61 dioxane 2.90 55.0 2.25
r,,
r29
*
*
l-39
keV keV keV 0.83 0.06 2.63 0.02 1.82 & 0.04 0.75 h 0.06 2.70 h 0.02 2.10 0.04
*
were wrapped in an aluminum foil, to prevent any photochemical reaction on the halogenated compounds. Two different fast coincidence circuits were used for the LS measurements in MeOH and in dioxane, respectively equipped with organic (NE111) or with inorganic (BaF2)scintillators. The time resolutions, given by the full width at half-maximum (fwhm) of the 6oCoprompt curve, were 330 and 210 ps for the organic and inorganic scintillators, respectively. The spectra were analyzed in terms of three components, with lifetimes T i (or decay rate constants, Xi = I/ri) and intensities, Zi. Subscripts 1-3 are to be ascribed to singlet positronium (pPs), free positrons, and triplet positronium (0-Ps), respectively. The will denote values in the pure solvents. The experimental errors were within 0.2% for 4 and 0.02 ns for r3. The DBARL data were collected using either a Ge(Li) or an intrinsic Ge detector, with energy resolutions, given by the fwhm photopeak, of 1.43 and 1.33 keV, respectively. of the 514-keV These served for the experiments in MeOH and in dioxane, respectively. The results will be expressed in terms of the fwhm of the experimental annihilationpeak, with a reproducibility within 0.01 keV. After correction for the detector resolution, the annihilation peak can be resolved into a sum of Gaussians characterizing the momentum distributions of the various positron states, with intensities Z? and fwhm’s, ria5 Subscripts 1-3 relate to the same species as in LS while subscript 4 refers to any positron bound state involving a solute molecule. Knowing the ri)sfor the pure solvent and the ZiD either experimentally derived or calculated on the basis of the LS data, it is possible to predict the variation of fwhm with solute concentration. Comparing with the experimental plot then allows a good cross-checking of the processes witnessed through either LS or DBARLe4?’ Resuits and Discussion Pure Solvents. Table I gives the various parameters derived from LS and DBARL. As in previous w0rk.9,~a broad component, possibly attributable to incompletely thermalized positrons, appears in the deconvoluted DBARL spectra, with a small intensity of 1.0 and 1.7% and r = 5.7 and 8.5 keV for MeOH and dioxane, respectively. The various values given in Table I for MeOH and the LS results for dioxane are in good agreement with previous determination~.~*~.’
0022-3654/92/2096-8014$03.00/00 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 8015
The Positron Spur
TABLE [I: LS Parameters for Various Halogenated Compounds in Methrad a d in Dioxane
solvent methanol
solute
cc4
CHCl, CH2C12 WI2
CH31 CCl, CHCl3 CHzC12
dioxane
k', mol-' dm3ns-l
k, mol-' dm3
0 0 0 0.10 0.02 0.20 0.02
5.3 0.1 1.40 0.05 0.6 0.2 5.8 0.1 1.40 0.05 15.4 f 0.5 1.3 0.3 3.4 0.2
0.09
0.05 0.03
* * *
** 0.02 * 0.01 0.01
*
5
10
15
k C dio x ane 1( m o I- d m )
Figure 2. Correlation between the inhibition constants, k (mol-' dm'), of the chloro compounds in MeOH and in dioxane: CC14 (O), CHC13 (o), and CH2Clz(A).
-
0.1
1 .o
0.5
-
c
z
0.05
;
2.
0-m \
c
2. 0.05
- 2
> 0
0.025
0.25
0.5
0.75
C(mol
Figure 1. Variation of l/13m(%-I) with concentration C (mol dm-3) of (a) CCl, (0)and CHCl,, (0) in MeOH; (b) CHzIz(A)and CHJ (V) in MeOH; (c) CCl, (O), CHC13(m) and CHzClz(A) in dioxane. The solid linea are calculated using eq 3. 1. o
LS Results. No significant changes are observed in 73 for the chloro compounds in MeOH, while 7, decreases smoothly with increased solute concentration, C, in all other caw. The variations of A3 with C (not reported) are linear, allowing to derive the quenching reaction rate constant, k', according to A3 = A', + k% (1) The derived values, given in Table 11, are rather low, at the limit of being indicative of an effective reaction with Ps. It is possible that such low reaction rate constants would in fact merely reflect an increase in the probability of pick-off annihilation by Ps due to an increase in the availability of outer molecular electrons in the presence of the solutes. In MeOH, the variations with C of 1/13for C C 4 and CHC1, are linear (Figure l), while CH2Clzappears to have little influence on Ps formation, so that the inhibiting action of this solute was determined by making repeated measurements of Z3at C = 1 mol dm-, only. For the other cases, where quenching is present, the measured intensity must be corrected according to the following equation:z 13mr =
13[1- k%/(Ao2 - AO,)]
(2)
With this (slight) correction made, the variations of l/Z3mrwith Care also liiear (Figure 1). It thus appears that all solutes behave as "total" inhibitors2of Ps formation, and the inhibition constants, k, displayed in Table 11, can be derived from I, (or 4"') = Ioj/(l
+ kC)
(3)
A good correlation has been found previously in water, between the Ps inhibition constants of numerous solutes and their reaction rate constants with the solvated electron, kS,*and an even better correlation, between k and the scavenging constants K37= 1 representative of the reactivity of the solutes with the non solvated
2.0
C ( m o l dm+)
Figure 3. Variation of fwhm (keV) with concentration C (mol dm-9 of (a) CC4 (0)and (b) CHC13(0)in MeOH. The broken and solid linea are calculated supposing the absence or the presence of a positron bound state involving a solute molecule, respectively.
electron^.^ Although less documented, similar results have also appeared concerning methanol.4v8 These results confirmed that Ps would be formed on a very short time scale, before electron solvation occurs. All studied solutes are well-known as efficient electron scavengersgand, as was the case for HgC12 in various polar solvents, including methanol? this property is certainly at the origin of the Ps inhibition. This conclusion is confirmed by comparing the ratios of the various constants, for the couple HgC12/CC14in MeOH (the units are 1O'O mol-' dm3 s-l for k,, and mol-' dm3 for k and K37): ks(HgC12)4/k,(CCl~)10= 1.46/1.7 0.86 K~7(HgClz)4/K,7(CC14)"
3.56/6.25
0.57
or K37(HgClz)4/K37(CC14)12 = 3.56/4.17 = 0.85 k(HgC12)4/k(CC14) = 4.5/5.3 = 0.85 Obviously, there is a strong correlation between the various constants for the solutes in the same solvent. Similarly, a correlation appears between the inhibition constants of the chlorocompounds (Table 11) in MeOH and in dioxane, as illustrated in Figure 2. DBARL Results. For the chloro compounds, fwhm increases with increased concentration, as shown in Figures 3 and 5. This is qualitatively expected in the case of Ps inhibition, as the Ps components which are suppressed represent the two narrowest
8076 The Journal of Physical Chemistry, Vol. 96, No. 20, 1992
Didierjean et al.
TABLE IIk DBARL Parameters for Various Hdogemted Compouads in Methud d Dioxane r4?bkeV solvent solute K4,0d mol-' dm3 methanol CC4 2.24 f 0.03/2.29 f 0.03 4.7 f 1/5.3 f 1 CHCI, 2.21 i 0.03/2.25 f 0.04 0.2 f 0.1/0.2 f 0.1 CH2I2 2.05 f 0.03/2.07 f 0.04 3.9 f 1/4.1 f 1 CH31 2.05 f 0.03/2.00 f 0.04 1.2 f 0.5/0.9 f 0.5 c12.35 i 0.05c I1.76 f 0.06d dioxane CC14 2.33 f 0.03/2.32 f 0.04 20 f 5 CHCI, 2.30 f 0.03/2.28 i 0.04 14 f 4 CH2C12 2.30 f 0.03/2.30 f 0.04 3.5 f 1 Values derived by resolving the spectra.
2 . 8 --I
'
I
I
I
Fitting fwhm.
1
err.
(a)
0
From ref 6.
p,mx ad 41/40 100/100 71/77 100/100 100 100 100
From ref 13.
1
err.+
I
" h 1 V 2.71r2hd
2.65
0.5
1 .o
C(mol d ~ n - ~ )
Figure 4. Variation of fwhm (keV) with concentration C (mol dm-3) of (a) CH212(A)and (b) CH31 (V)in MeOH. For the broken and solid lines, see Figure 3.
components in the DBARL spectra (Table I). For the iodo compounds, however, fwhm increases fvst but then decreases at high enougb concentrations (Figure 4). As was the case for HgC12,' this behavior denotes the presence of an additional e+ bound state, besides pPs and 0-Ps, as otherwise fwhm could increase only when Ps formation is inhibited. Owing to the very low reaction rate constants (k'in Table 11), these cannot account for the observed maxima in fwhm. Quantitatively, if only Ps inhibition due to electron scavenging and some weak Ps oxidation reaction, in the case of the iodocompounds, were present, the variations of fwhm with C can be completely calculated, using the LS data from Table I1 together with the r, values in Table I.2,4 This is illustrated by the broken lines in Figures 3-5. As for HgClz," the calculated curves increase much more rapidly than the experimental data, confirming the presence of an additional e+ bound state. Effectively, resolving the deconvoluted DBARL spectra shows the existence of a fourth component in all solutions, with reasonably constant r4values (Table 111). The intensities, Z4D, associated with this component increase with C as shown in Figure 6,allowing us to derive the bound-state formation constants, K4 (table 111), according to the following empirical e q ~ a t i o n : ~ 14D = (100 - 13D)P4"""K4C/(1 + K4c> (4) where the first term represents the proportion of free e+ available to form the bound state at each concentration, and PqmXis an efficiency factor?-' Because of the high values obtained for r4 in the case of the chloro compounds in dioxane, closer to r2as compared to methanol (Tables I and 111), the derived values for 14Dwere not very reliable. An alternative method to derive the bound-state parameters and applicable to all cases,is to fit directly the experimenta1fwhm values. The results, given in Table 111, show excellent agreement
1 .o
0.5
C ( m o l dm-3)
of fwhm (keV) with concentration C (mol dm-') of (a) CC14 (O), (b) CHC13 (m), and (c) CH2C12(A) in dioxane. For the broken and solid lines, see Figure 3. Figure 5. Variation
1 .o
0.5
2 .o
1 .o
C(mol dm-3)
F i p e 6. Variation of 14D(8) with concentration C (mol dm-') of (a) C C 4 (0)and CHC13 (0).(b) CHJ2 (A)and CHJ (V) in Md3H. The solid lines are calculated using eq 4.
The Journal of Physical Chemistry, Vol. 96, No. 20, 1992 8077
The Positron Spur
Using the kl values from Table I1 ands k2 = 1.1 mol-' dm3, cqs 5 and 6 lead to the variations shown as broken lines in Figure 7. These are too slow as compared to the data and, as previously proposed for some mixtures with HgC12,4a plausible explanation to this discrepancy might arise from the implication of electrontransfer reactions between solute molecules, such as RX-
I
I
1 .o
0.5 C(mol dm-3)
Figsn 7. Variation in MeOH of IdD (5%) with concentration C (mol dm-3)of NH4N03in 0.35 mol dm-3 of CC1, (0)and in 1.0 mol dm-)of CHC13 (0). The broken and solid lines are calculated s u p p i n g the absence (eq 5 ) or the presence (eq 7) of charge transfer between the two solutes, respectively.
with those derived by resolving the deconvoluted spectra, which gives confidence for the values thus obtained in the case of dioxane. The variations of fwhm with C calculated using the various data from Tables 1-111, shown as solid lines in Figures 3-5, are the same whatever the method to derive the bound-state parameters and are, of course, in excellent agreement with the experimental plots. It thus appears that the LS results indicate that Ps formation is inhibited by quasi-free electron scavenging, while the DBARL data show the presence of an additional positron bound state, whosc intensity increases with increased solute concentration. The correlation between the inhibition constants and the electron scavenging properties of the solutes implies that the electron is necessarily captured first: if the positron were to react first with the solutes, this would automatically result in Ps inhibition, independently of any further reaction with an electron, and the Ps inhibition process would not be correlated at all with the latter reaction. As for HgC12 therefore: the following scheme of reactions must be considered, where RX stands for the halogenated solute: RX
+ -
+ e-
RX-(or X-)
competing with
-
RX
+ S-
(IV)
In this case,denoting by krv the constant expressive of this transfer, the probability P12in eq 5 should be modified as4
I
next
+S
+ X-)
(1)
[RX-e+] (or [X-e+])
(11)
RX- (or R'
e+loc
e+
+ e-
-
Ps
(111)
where e+locstands for the localized, not fully solvated positron.2 The validity of the above scheme can be assessed by adding an electron scavenger, S,to the solutions: if the bound-state is formed by direct capture of the positron by the solute, as is the case with the halide ions? this addition should result in an increase of the bound-state formation probability, as previously verified experimentally., Conversely, if the bound state arises from a secondary reaction, as proposed in reactions I and 11, the addition of S,competing with RX for the electron capture, reaction I, should promote a decrease in Mixtures of an electron scavenger with halo compounds were therefore studied, as reported in the following. Mixed Solutions in MeOH. The solutes chosen were CCl,, at 0.35mol dm-3, and CHC13, at 1 mol dm-3, while the cosolute was NH4N03, which reacts reasonably well with the quasi-free electrons in this solvent.* As shown in Figure 7 , adding this cosolute results in a marked decrease in for the chloro compounds Quantitatively, the variation of (subscript 1) with cusolute concentration (subscript 2) should obey eq 4 modified as f01lows:~ I,D = (100- ZID - Z~D)P~"""P~2K4C/(1 &C) (5)
+
where the term P I 2= klC1/(l + klCl + k2C2)expresses the competition between the two electron scavenging solutes, and the first term represents the availability of free positrons, with, in extension to eq 3 (i = 1 or 3 ) :
zp = P i / ( l + klCl + k,C*)
(6)
Piz kiCi/(1 + kiCi + kzC2 + ~ I v C I C ~ ) (7) The solid lines in Figure 7 are obtained by using eqs 5-7 with kIv = 7 and 3 molT2dm6 for the mixtures of NH4N03with CCl, and CHC13, respectively. Positron Bound State and Halide Detachment. The I", values given in Table I11 should be characteristic of the bound state formed by the reaction of e+ with either X- or RX-(reaction 11). As f& HgC12: the nature of the latter ions can therefore be specified by comparing the r4values in Table I11 with those measured independentlyfor the corresponding halide ions, Cl- and I-, also included in Table III!J3 The comparison cannot be made in dioxane, because the halides are not soluble. Taking account of the experimental errors, it may be seen that the iodo compounds lead to r4values which are quite distinct from r4(I-), showing that there is no halide detachment before bound-state formation occurs. The results are less clear-cut for the chloro compounds; however, the closeness of the values for r4(CC14) and r4(CHC13) when compared to r4(Cl-) strongly suggests that these are different enough to conclude that there is no halide detachment in this case too. In pulse radiolysis experiments, it is usually found that electron capture leads to halide detachment.14J5 However, the time scale involved in these measurements expands over nanoseconds or more, while the positron bound state is most probably formed with the localized, not fully solvated positrons? on a (sub)picosecond time scale. On this basis, the present results would set a lower limit to halide detachment from halo compounds in MeOH, roughly given by the solvation time of the electrons in this solvent, at about 17 p.l6 The possibility of having long-lived radical anions in nonpolar solvents has been evoked in positron chemistry works, such as for CH3CI-, which would have a lifetime of about 30 ns in cyclohexane." However, direct comparison with our results appears to be unsafe, because of the strong influence that polar solvent molecules must have on the stability of such anions by contrast with the nonpolar solvent. The closeness of the r4values for a series of halo compounds in a specified solvent appears to be a general rule. Thus, one may define the following average values: 2.22 f 0.03,2.31 f 0.03, and 2.05 f 0.03 keV for RCl in MeOH and in dioxane and for RI in MeOH, respectively. This shows that the electron momentum distribution probed by the positron in the various compounds is, to a large extent, not sensitive to the exact nature of the moiety, R, to which the halogen is bound. Therefore, the positron is most probably rather tightly bound to the negatively charged halogen atom in RX-. On the other hand, the change observed when passing from MeOH to dioxane in the case of the chloro compounds indicates that solvent molecules can intervene in the final form of the bound state, at the level of the charged end. Considering the very short times at which the e+ bound states are formed, it is likely that the solvent molecules would play a role after this formation only. In the case of the free halide ions on the contrary, these are initially solvated and thus possess a different net charge than RX- produced radiolytically. Number of E l e ~ t r oParticipating ~ in Ps Formation. By comparing Tables I1 and 111, it appears that the Ps formation inhibition constants, k,and the e+ bound-state formation constants, K4, are very similar for a given solute in a specified solvent. As for HgC12,4 this stresses the common origin to the two processes, as expressed by reaction I, and has the potential of giving some information on the distribution and number of the active species in the positron spur.
8078 The Journal of Physical Chemistry, Vol. 96, No. 20, 19192 It is out of the scope of this paper to develop complex deterministic or stochastic calculations on the e+ spur reactions as are currently made regarding the electron.'* However, a simple probabilistic approach can give useful information on the gross properties of the e+spur and has been frequently used in positron works to establish semiempirical expressions such as eqs 3 and 4.13.19 In the present case, a simplified scheme of the spur processes can be written as follows: annihilation
solvation, annihilation
d4
d2
e+&-
e+
-t- dl
Ps
ol
(S-e+)
ne-
02
solvation and recombination
S-
(v)
annihilation
The u's represent the timsaveraged rate constants, n denotes the number of electrons participating in Ps formation, and S is the solute. To illustrate how the experimentally derived values for k and K4 can give information on n, the above scheme of reactions will be considered essentially on the electron side, starting on the hypothesis that the various rate constants are independent, which implies that ul and u3 are small. In the absence of solute, the sum of the independent contributions of the n electrons to Ps formation gives the total yield of Ps as lop,
= nul/(ul + .2)
(8)
At concentration C of solute, the yield becomes Ips ~ u ~ / ( u I + ~2 + u ~ C ) Zoh/(l k9C)
+
(9)
+
where k9 = u3/(ul u2). Equation 9 identifies with eq 3 and, therefore, b,with the inhibition constant, k. Similarly, the yield of e+ bound-state with S-is given by I(S-e+) =f(e+)nu3C/(ul
+ u2 + u3C9 = f(e+)nk9C/(1 + k9C) (10)
wheref(e+) represents the fraction of positrons available for the bound-state formation: Ae') = 1 - Zk By comparing eq 4, with Pgmax = 1, and eq 10, it may be seen that these are identical provided that n = 1. Thus, it is only if the average number of electrons per spur participating in Ps formation is one that k = k9 = K4 as is experimentally found in this work and previ~usly.~ If n is larger than 1, the yield of e+ bound-state, Z4, increases much more rapidly with C than the Ps yield, Z3, decreases. Equations 8-10 are valid only if ul and u3 are small, but they allow to point out clearly the influence of the number of available electrons, n, on the measurable constants, k and K4. In a more general case, although not taking account of the discrete nature of the number of solute molecules in a spur, the Ps yield in the neat solvent is IOp,
= 1 - [l - U ' / ( U 1
+ u2)]*
(11)
where the term in brackets represents the probability for no reaction of the n electrons with e+. In the presence of a solute, the equivalent to eq 9 is Ip, = 1 - [l - U l / ( U , u2 qc)]=
+ +
and the bound-state yield is Z(S-e+) =Ae+)[l - 1/(1
+ k12C)"l
(13) For n = 1, eqs 12 and 13 identify with eqs 3 and 4, respectively. For n > 1, it may be numerically verified that eq 12 behaves very closely like eq 3, almost independently of the value of n. To a
Didierjean et al. very good approximation, eq 12 identifies with eq 3 provided that kl2 = kn[l
- (1 - Zo~)'/"]/Iop,
(14)
As for eq 9,parameter n thus appears to have little influence on the expression of the inhibition of Ps formation by the solute. On the contrary, for n > 1, the bound-state yield, eq 13, increases much more rapidly with C than the Ps yield decreases, so that one cannot find experimentally that k E K4. From the preceding, it appears that the average number of electrons available to form Ps is close to 1. The total number of electrons produced per positron spur might be higher; however, since we are dealing with polar solvents, the rapidity of the solvation process for both e- and e+ apparently plays a major role. Due to the short times involved, only that electron which happens to be closest to e+ has a chance to reach this particle and form Ps, while the diffusive motion of any additional, farther electrons is rapidly lowered upon solvation, making these ineffective to produce Ps. A different situation may be anticipated in the nonpolar solvents, where solvation can be totally absent: in this case, due to a lower dielectric constant,the attractive force bet" e+ and the spur electrons is increased over much larger distances and much longer times. Conelmion The present results c o n f i i the existence of very fast successive captures, within the positron spur, of electrons, then of the positron by halogenated compounds in polar solvents. They also strongly suggest the possibility of fast scavenged electron transfer from a solute molecule to another. Considering the very short times at which these processes occur, with an upper limit at a few picoseconds, these results have the potential of providing a rather tight framework to any theoretical calculations. Likewise, the similarity between the constants derived for the Ps inhibition and for the e+ bound-state formation implies information on the dimensions of the positron spur and on the number of species playing an effective role; in particular, only one of the spur electrons would contribute to Ps formation. It would be most interesting to study some nonpolar solvents by combining quantitatively, as in this work, two independent techniques. Examining the influence of temperature should also provide useful information on the common origin of Ps inhibition and e+ bound-state formation, as the former process is well-known to be little dependent on temperature. The information gathered on the positron bound states also raises some interesting questions to be solved in future. Thus, the implication of solvent molecules in the final form of the bound state and the role of the moiety to which the halogen atoms are attached should be further explored by extending the variety of solutes and of solvents, whether polar or not, alone or mixed.
References and Notes (1) Positron and Positronium Chemistry;Studies in Physical and Theoretical Chemistry; Schrader, D. M., Jean, Y. C., Eds.; Elsevier: Amsterdam, 1988; Vol. 57. (2) AbW, J. Ch.; DuplPtre, G.;Maddock, A. G.; Talamoni, J.; Haessler, A. J. Imrg. Nucl. Chem. 1981,43, 2603. (3) Duplltre, G.;Jonah, C. D. Radiar. Phys. Chem. 1985, 24, 557. (4) Didierjean, F.;DuplPtre, G.; AbW, J. Ch. Radiat. Phys. Chem. 1991,
38, 413. (5) DuplPtre, G.; Ab&, J. Ch.; Maddock, A. G.; Haessler, A. J . Chem. Phys. 1986, 72, 89. (6) Talamoni, J.; AbW, J. Ch.; Duplltre, G.; Haessler, A. Radiat. Phys. Chem. 1983.21, 431. (7) Mogensen, 0. E.;Jacobsen, F. M. Chem. Phys. 1982, 73, 223. (8) DuplPtre, G.;Jonah, C. D. J. Phys. Chem. 1991, 95, 897. (9) Buxton, G.; Greenstock, C. L.;Helman, W. P.; Ross,A. B. J. Phys. Chem. Re/. Data 1988.17, 513. (IO) Razem, D.; Hamill, W. H. J. Phys. Chem. 1978,82, 1347. (1 1) Lam, K.Y.; Hunt, W. J. Inr. J . Radiar. Phys. Chem. 1975, 7 , 317. (12) Jonah, C. D. Private communication. The value measured without any corrections for C3,(CCl,) in MeOH is 0.18 mol dm-', in good agreement with the value from Lam and Hunt, 0.16 mol dm". After correcting for the
decay of the solvated electron during the exciting pulse however, the value turns out to be 0.24 mol dm-? which appears to be in much better agreement with our data. (13) Talamoni, J.; Duplitre, G.; AbM, J. Ch.; Haessler, A. J. Chem. Phys. 1984,83,471. (14) Gremlich, H. U.; Blhlcr,
R. E. J. Phys. Chem. 1983, 87, 3267.
8079
J. Phys. Chem. 1992, 96, 8079-8084 ( 1 9 Nazhat. N. B.: Asmus. K. D. J. Phvs. Chem. 1973. 77. 614. (16j Wang, Y.; Crawford, ‘M. K.; Mciuliffe, M. J.; Eisenthal, K. B. Chem. Phys. Lerr. 1980, 74, 160. (17) Wikander, G.; Mogensen, 0. E.; Pedersen, N. J. Chem. Phys. 1983,
77. 159.
(18) Green,N. J. B.; Pilling, M. J.; Pimblott, S. M. Rudiur. Phys. Chem. 1989, 34, 105. (19) Beling, C . D.; Smith, F. A. Chem. Phys. 1983, 81, 243.
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Influence of External Steady Source Structure on Particle Distributions and Kinetics of Diffusion-Limited Reactions I: A A 0 Simulations
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L. Li and R. Kopelman* Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055 (Received: May 7, 1992)
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The effect of a steady external source and its correlation structure on the particle organization patterns and the kinetics of A + A 0 reactions at steady state was studied by Monte Carlo simulations. The simulation results are compared with existing theoretical predictions for the reaction order and the self-orderingof particles. The spatial organization of the system is described by the interparticle distance (‘gap”) distribution. The pair correlation length of particles in the external source alters the particle distribution at the steady state and affects the macroscopicrate law. The most correlated external s o u m result in a random particle distribution at steady state, while random source structurm result in the most correlated particle distributions.
Introduction Theoretical computer simulation^,^.^ and experi m e n t ~have ~ . ~ all demonstrated nonclassical kinetics (anomalous kinetics) for the one-species, A + A 0, diffusion-limited steady-state annihilation in low dimensions, when particles randomly land on a one-dimensional lattice. From both simulation3 and theory’ the effective reaction order, X,was found to be 3 on a one-dimensional system, rather than the value 2, derived in classical kinetic theory for a binary reaction. The order X for a steady-state reaction is simply defined as
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R = kpSX (1) where pa is the steady state density, k is the rate constant, and R is the number of particles created per unit time per lattice site. Moreover, it has been suggested2q6s that at the steady state particles are distributed nonrandomly on the lattice. The nonrandom particle distribution is referred to as the spatial organization, or self-organization of the particles. To characterize this spatial organization, many methods have been developed. They can be classified into two kinds: a single “characteristic” length, such as a typical size of the depletion zone around each particle,2-6and the entire particle The particle distribution can be described by a depletion zone distribution, i.e., an interparticle distance distribution (IPDD, or “gap”), or by a nearest-neighbor distance distribution (NNDD). The IPDD is used here to compare simulation results with theoretical predictions on one-dimensional systems, whereas the NNDD is more easily generalized to systems with higher dimensionalities.12 Most of the investigations on binary diffusion-limited reactions are based on the assumption that particles land randomly on a lattice, which may not be applicable to some real chemical systems. For example, two atoms may land at adjacent sites as a result of a dissociative adsorption of diatomic molecules, and solitons and antisolitons may be created in pairs on a polymer (e.g., tranr-polyacetylene).13Thus, some theoretical approaches have taken into consideration the correlation in the external particle source. MCZ’ has derived the rate law of A + A 0 steady-state reactions on onedimensional systems for the case in which particles are produced in pairs (geminately) at the nearest-neighbor lattice sites. The reaction order is found to be 2. Recently, Clsment et al. have also proposed a kinetic equation for the external source with correlations.“ We note that much work has been done on source correlation for the A A OsJ5-” and A B OI5-l9 botch reactions,as well as A B 0 steady-state rea~tions.~~J*”
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For the last case, a separate paper is in p r e p a r a t i ~ n .Here, ~ ~ we limit ourselves to a correlated steady-state A source. We present here a study of the influence of the steady external source on the particle distribution (i.e., depletion m e distribution) and the kinetics laws of binary reactions by Monte Carlo simulations with the emphasis on the one-dimensional A A 0 diffusion-limited system. The effects of a variety of details of the source were investigated, such as production of a pair of particles separated by a fmed length (the pair correlation length in the source, 6). We fmd that the particle distribution at steady state does change with correlation length in the source. The IPDDs are compared with the NNDDs, and both these distributions are used to describe the spatial organization depletion zone of the particles. The simulation results are compared with recent theory.l4 This concerns reaction orders, rate CoeficientS, and various measures of depletion zones and self-organization.
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Defuritions and Method of Computation Our simulations were camed out on a one-dimensional lattice with a periodical boundary condition. The simulations were performed on the IBM 3090-400/vm at the University of Michigan. The uniform pseudorandom number generator FUNIF with MTS (Michigan Terminal System) was used. For each simulation, 3-20 runs were done under specific conditions. The Monte Carlo method was used. Particles execute a simple random walkzs.26and experience landing, diffusing, and reacting on a lattice3 Each walker is forced to move randomly to one of the nearest-neighbor sites during each time step.27 When two particles occupy the same site, a reaction occur^, and both particles are removed from the system. Since our emphasis is on the external source,only the detailed description of the particle landing processes is given here. Other aspects of the diffusion simulation have been reported in ref 26. In a steady-state reaction, there is a constant influx of particles from a source throughout the reaction. Two components of the landing process were considered: the reaction during the landing and the spatial distribution. When a particle lands at a site occupied by another particle, either the particles react or the landing particle tries to land at another site again immediately. The first case is called landing without vertical reaction, and the second case is called landing with vertical reaction. Since at very low density, there is no difference between landing without vertical reaction and landing with vertical reaction, only the random landing with vertical reaction is studied here.
0022-3654/92/2096-8079%03.00/00 1992 American Chemical Society
