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Kinetic Scaling Behavior of the Two-Species Annihilation Reaction with Input Ezequiel V. Albano* Instituto de Fisica de Líquidos y Sistemas Biologicos, (IFLYSIB), CCT La Plata, CONICET, UNLP, Calle 59 Nro, 789 (1900) La Plata, Argentina and Departamento de Fisica, Facultad de Ciencias Exactas, UNLP, La Plata, Argentina ABSTRACT: We present an extensive simulation study of the kinetic behavior of the two-species annihilation reaction of the type A + B f 0, with input of species that takes place immediately after each reaction event. Simulations are performed by using lattices of length L in d = 1 dimension. Two different types of processes are considered: (i) the locally conservative kinetic (LCK) case, which involves the conservation of the densities of both types of particles during the whole reaction, and (ii) the so-called globally conservative kinetic (GCK) case where the total density of particles still remains constant, but after each reaction event, the type of particle to be introduced into the system is selected at random with the same probability. By starting from a random distribution of particles, it is found that the reaction rate, given by the number of reaction events per unit of time and length, decreases as a power law of the time according to Rate tβ, with β = 1/2 and β = 1/4 for the GCK and LCK cases, respectively. It is found that the GCK never leads to the occurrence of a steady state, and the fluctuations of the density difference between different types of species in the lattice grow as Æγ2(t)æ tδ, where δ = 1 is an exponent. However, for the LCK case, we observe that after a crossover time of the order of τ Lz, where z = 2 is a dynamic exponent, the systems reach stationary regimes, such that Ratestat FX, where F is the density of the species, and X = 3 is the (anomalous) reaction order. Our simulation results not only confirm some existing analytical predictions but also, in many kinetic scaling aspects, go beyond the present knowledge addressing new and interesting theoretical challenges.
1. INTRODUCTION The study and understanding of heterogeneous reactions is a challenging topic in several branches of science, including chemical kinetics, surface and solid state physics, surface physical chemistry, biological and ecological systems, etc.14 Within this broad context, the behavior of even simple reactions, involving only few species, is still not very well understood. A typical example is the annihilation reaction between different species that leads to inert products, e.g., the two-species diffusion-limited reaction of the type13,518 A þ Bf0
ð1Þ
which is also of interest in chemical kinetics and solid-state reactions: e.g., electronhole, soliton-antisoliton, and defect antidefect recombination.16 Further potential interest arises for the understanding of matterantimatter annihilation in the early universe12,13 and catalytic reactions.19 Also, this type of reaction occurring in low-dimensional and fractal media exhibits a rich physical behavior including segregation of the reactants5,6 and anomalous reaction orders.3 The usefulness of the two-species diffusion-limited reaction model is greatly increased when one relaxes the restriction that the reaction must be irreversible. By allowing for back reactions, the model becomes suitable for the description of other systems where the onset of stationary states and the kinetic approach to that regime are relevant features of interest.9 Within this context, the reaction model with inputs of the reactants has also been studied numerically12 and by means of analytical approaches.1416 r 2011 American Chemical Society
In the present article, we are interested in the kinetic scaling behavior of the reaction given by eq 1 under steady-source conditions, such that the total density of species is conserved, namely, FA ðt ¼ 0Þ þ FB ðt ¼ 0Þ ¼ FA ðtÞ þ FB ðtÞ ¼ constant
ð2Þ
where FA(t) and FB(t) are the densities of species A and B at time t, respectively. Two different situations are studied: one involving the conservation of the densities of both types of particles during the whole process, namely, FA(t) = FA(t = 0) and FB(t) = FB(t = 0), which will be referred to as the locally conservative kinetic (LCK) case; and also, the so-called globally conservative kinetic (GCK) case where eq 2 still holds, but after each reaction event, the type of particle to be introduced into the system is selected at random with the same probability. Then, the densities of each type of species may change during the reaction process so that one may have FA(t) 6¼ FA(t = 0) and FB(t) 6¼ FB(t = 0). The manuscript is organized as follows: in section 2 we provide a brief overview of previous theoretical and numerical results, section 3 gives details on the simulation method, and section 4 is devoted to the presentation and discussion of the results. Finally, our conclusions are stated in Section 5. Received: August 29, 2011 Revised: October 28, 2011 Published: November 01, 2011 24267
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2. PREVIOUS RESULTS AND THEORETICAL BACKGROUND The transient behavior of the two-species diffusion-limited reaction, as given by eq 1 and in the absence of sources, has received considerable attention due to the fact that, most likely, it is the simplest case where fluctuations in the local concentration of the reactants play a prevailing role in the emerging kinetic behavior.1,2,58,11 According to eq 1, the concentration difference of both species is conserved so that FA(t) FB(t) = FA(0) FB(0) t constant. If the initial concentrations are different, the mean-field approach predicts an exponential decrease of the minority species.1,2 However, for FA(0) = FB(0), one expects an algebraic decay of the form F 1/t. Let us now briefly review the effect caused by spatial fluctuations in the concentrations.1,2,58,11 For the sake of simplicity, let us assume that both species have the same diffusion constant (D) and that the initial concentrations are equal. For a random initial distribution of particles, a region of the space of linear size l would initially have NA = FA(0)ld ( (FA(0)ld)1/2 particles of type A and equivalently for B-particles. The second term accounts for the local fluctuation in the number of particles, at a scale of the order of l. By assuming classical diffusion, one has that for a time t, such that Dt = l1/2, all the particles within the considered region would have enough time to react with each other. In the absence of fluctuations, the concentrations of both species will vanish. However, at the end of the reaction process, one still has a number of particles of the majority species of the order of (F(0)ld)1/2. Then, the concentration of either A- or B-particles at time t is actually given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffi FA, B ð0Þ FA, B ð3Þ Dt d=4 According to the above scaling reasoning, one would expect the formation of alternating domains of A- and B-particles. This segregation of the reactants, known as the Toussaint-Wilczek effect,6 causes the slowdown of the kinetics since the reaction actually takes place along the boundary between domains. A careful analysis reveals that for d g 4, the domains are unstable so that dc = 4 is the upper critical dimension above which the mean-field approach holds. In an early work, Anacker and Kopelman12 studied, by means of numerical Monte Carlo simulations, the two-species annihilation reaction with a steady source for the LCK case on the linear lattice, the planar Sierpinsky gasket, and the simple cubic lattice. They concluded that the behavior of the reaction is completly different than the transient case in the absence of a source. In fact, they reported the observation of segregation of the reactants in d = 1, but a steady state is not obtained during the observation time, i.e., 106 Monte Carlo steps. However, a steady state is observed in the (finite) Sierpinsky gasket and on the cubic lattice. It is worth mentioning that segregation of the reactants is observed in the Sierpinsky gasket in contrast to the case of the cubic lattice so that the Toussaint-Wilczek effect is lost in d = 3 due to the presence of a source of particles, i.e., in a lower dimension than in the transient case. Apart from numerical simulations, the description of the time evolution of the concentration of reacting particles (Fi, where i = 1, 2, ..., N, identifies the type of particle) can be achieved by formulating a kinetic rate equation ∂Fi ðr, tÞ ¼ F½Fi ðr, tÞ ∂t
ð4Þ
where F is a function. This description, known in physical chemistry as the law of mass action, states that the rate of a chemical reaction is proportional to the concentration of the reacting species. Within this framework, ben-Avraham et al.14 formulated eq 4 for the two-species annihilation reaction with random input as follows: ∂FA, B ðr, tÞ ¼ DΔFA, B ðr, tÞ kFA ðr, tÞFB ðr, tÞ þ ηA, B ðr, tÞ ∂t ð5Þ where the first term represents the diffussion process with a diffusion constant D, the second term accounts for the reaction with a rate constant k, and ηA,B are stochastic variables representing the input of species A and B. By assuming the same diffusion coefficient for both species, one has that the density difference given by γ(r,t) = FA(r,t) FB(r,t) obeys a linear differential equation, namely, ∂γðr, tÞ ¼ DΔγðr, tÞ þ ηγ ðr, tÞ ∂t
ð6Þ
where ηγ(r,t) = ηA(r,t) ηB(r,t). It is worth mentioning that eq 6 is independent of the explicit form of the reaction term. This is a remarkable advantange in view of the fact that, particularly in lowdimensional media, diffusion-controlled reactions may be anomalous in the sense that the effective reaction order could be different than that expected according to the classical rate theory.3 Equation 6 can be solved for both the LCK and GCK cases. For the case of independent A and B inputs (GCK), the quantity of interest is the standard deviation of the particle number difference γ2(r,t),14 which in the case of an infinite volume is expected to behave as Æγ2 ðr, tÞæ t ð2 dÞ=2 , d < 2
ð7Þ
where Æ...æ denotes an ensemble average. Note that here we only refer to the solution valid for d = 1 that is relevant to our subsequent numerical study. So, the prediction is that the difference in concentrations never reaches a stationary state. However, for finite volumes, the long-time behavior expected for any dimension is given by Æγ2 ðtÞæ t δ
ð8Þ
where δ = 1 is an exponent. These results are consistent with the fact that the particle number difference is described by a simple random walk constrained only by the spatial diffusion.14 When the volume is finite, the excess particles are constrained to stay inside the system and Æγ2æ grows linearly with t as in a pure random walk fashion. In contrast, for an infinite volume, diffusion controls the growth of Æγ2æ by spreading local buildups of excess particles, and for d = 1, such a growth is slower than linear. However, when the number of particles is conserved (LCK case), one has that for d < 2 the stationary regime may not be achieved if the spatial correlation of the inputs is not localized enough. For example, in d = 1, a stationary state can exist only if the mean particle-input distance is finite. Also, in finite volumes, a stationary state is found in any dimension for the conserving input.14 So, the fact that the stationary state has not been observed in the simulations reported in ref 12 may be due to the use of a short observation time. The A + B f 0 reaction with input has also been studied by Yi-Cheng15 and Lindenberg et al.16 by using analytical approaches. 24268
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These studies are focused on the identification of the suitable spatial dimension for the observation of segregation of the reactants. By working with the density difference γ(r,t), Yi-Cheng15 concludes that segregation phenomena can occur in two dimensions or below. In a subsequent paper, Lindenberg et al.16 also deal with the sum of densities since the use of γ(r,t) only is not meaningful because for the observation of segregation one requires a large relative excess in the density of species. Therefore, these authors arrive at different results and conclude that in finite systems segregation occurs for d = 1, is marginal for d = 2, and does not occur at all for d = 3. Also, in infinite systems it is expected to observe segregation in d = 1 and d = 2, while it should be marginal in d = 3.
3. SIMULATION METHOD The diffusion-controlled annihilation reaction between two species with input, as given by eq 1, is studied by means of Monte Carlo simulations in dimension d = 1, by using samples of length L and assuming periodic boundary conditions. Initially, a fixed number of particles NA = NB is distributed at random on the sample. In order to implement the diffusion-reaction process, a particle is selected at random and a jumping attempt is performed with the same probability in each direction. Jump trials into sites occupied by the same species are disregarded, while a reaction event takes place when a particle attempts to jump to a site occupied by a different type of particle. After a successful reaction event, a pair of particles is introduced into the sample at randomly selected (empty) positions. For the case of the LCK one particle of each type is introduced, while for the GCK case the type of particle to be introduced is selected at random with the same probability. The re-entrance of particles just after reaction is a simulation trick introduced in order to avoid the temporal fluctuations of both the source flux and the concentration of the reactants. We computed the rate (R) as the number of reaction events per unit of time and length. One Monte Carlo time step (MCS) involves NA + NB randomly selected attempts so that each particle may be chosen once, on average. Typical simulations are perfomed for samples of length 103 e L e 105; averages are taken over a number 50 e Ns e 500 of different runs; and simulation times are of the order of 106109 MCS. Also, different densities of species FA = NA/L and FB = NB/L are considered within the range 5 104 e FA,B e 1/2. For the case of the GCK, a relevant magnitude to be evaluated is the fluctuation of the concentration difference between species given by γ2 ðtÞ ¼ ƽFA ðtÞ FB ðtÞ2 æ
ð9Þ
See also eqs 7 and 8, where Æ...æ means ensemble averages.
4. RESULTS AND DISCUSSION In order to gain insight into the coarsening process taking place upon reaction, it is convenient to analyze time-dependent snapshot configurations. Figure 1a corresponds to a typical snapshot configuration showing the time evolution of the coarsening process upon the annihilation reaction when the density of each type of particle is conserved, i.e., the LCK case. Here, one observes that for early times the system slowly but monotonically coarsens up to, e.g., t ≈ 104 MCS. During this time interval, the rate of reaction decreases (see Figure 1c). However, during the long-time regime, one observes the formation of mainly two large domains corresponding to A- and B-rich
Figure 1. (a,b) Typical snapshot configurations of the coarsening process, obtained for samples of size L = 900 (horizontal axis), such that the vertical axis corresponds to the time evolution of the system. Aand B-particles are shown in blue and yellow, respectively. Results were obtained by keeping the total density FA + FB = 1/3 constant. For panel a, one has the LCK case so that the density of each type of particle is conserved, and one has FA = FB = 1/6, while for panel b after each reaction event, the particles are introduced into the system at random, i. e., one has the GCK case. (c) Loglog plot of the reaction rate versus time as obtained for cases a and b given by the upper and lower curves, respectively.
regions, and the coarsening process becomes stabilized, as does the reation rate (see Figure 1c). When the particles are introduced into the system at random (Figure 1b, corresponding to the GCK case), one observes a rather quick coarsening process where the reaction rate decreases (see Figure 1c), which leads to the formation of a solid cluster of one of the species with the same probability (species of type A in the case of Figure 1b). The time required for the prevalence of one species depends on the fluctuations taking place in each particular run, e.g., t ≈ 103 in Figure 1b. Of course, at this late stage the reaction stops and a sationary regime of the reaction rate as in the case of Figure 1a is no longer observed. Now let us focus our attention on extensive simulation results. Figure 2a shows loglog plots of the rate of reaction versus time as obtained for the GCK case and three different (total) densities of species. In all cases, the rate decreases as a power law of the form Rate tβ, with slope β = 1/2, and the data show superimposition for the same density but a systematic shift in the vertical axis when different densities are considered. That : shift is consistent with eq 5, i.e., F(t) kFA(t)FB(t) 2 kFT (t), (see Figure 2b) where the total density FT = FA + FB as well as the average densities of each species (taken over different runs) are conserved, despite the fact that each individual density is no longer constant due to the NCK nature of the process. So, by rescaling the results according to Rate/FTX versus t, where X = 2 is the mean-field reaction order, one obtains an excellent data collapse as shown in Figure 2b. 24269
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Figure 3. Loglog plots of the rate versus time as obtained by starting from fully segregated species A and B and assuming the GCK process and the following parameters (from top to bottom): NA,B = 104, L = 5 104; NA,B = 104, L = 1 105; and NA,B = 5 103, L = 5 104. The dashed (full) line with slope 1/2 (1/4) has been drawn for the sake of comparison.
Figure 2. (a) Loglog plots of the rate of reaction versus time as obtained for the GCK case and different (total) densities of particles, as shown in the panel. Data were obtained for lattices of size L = 5 104 (two upper symbols) and L = 105 (three lower symbols). (b) Rescaled data already shown in panel a, i.e., R/FT2 versus time. The dashed line with slope β = 1/2 has been drawn for the sake of comparison. (c) Loglog plots of the standard deviation of the particle number difference γAB2(t) versus time as obtained for FT = 2/5 and by using lattices of size L = 105. The full line with slope δ = 1 has been drawn for the sake of comparison.
However, the fluctuations in the concentration difference between species diverge with exponent δ = 1, as shown in Figure 2c. This result is in full agreement with theoretical predictions14 (see also eq 8) and is due to the fact that when the volume is finite, the excess particles are constrained to stay
inside the system, and Æγ2æ grows in a pure random walk fashion. Because of these large fluctuations, in finite systems the sample becomes ultimately covered by one of the two competing species, with the same probability. So, at this stage the reaction stops and one does not need to continue the simulation. Consequently, all results shown in Figure 2 corespond to data obtained for time intervals where both types of species remain present in the sample. Because of this shortcoming, we are constrained to study the behavior of the system for rather large total densities of particles (2/5 e FT e 1/2), but we are still confident that our conclusions would also hold in the low-coverage regime in view of the agreement with the theoretical results. So, concerning the GCK process, our simulation results not only agree with the prediction of eq 8 with δ = 1 but also go beyond the theoretical calculation by predicting a power-law decrease of the rate of reaction (with exponent β = 1/2) as well as its dependence on the second power (X = 2) of the total density as in a typical mean-field second-order reaction. Since by starting from a random initial distribution of the reactants, the GCK process does not arrive at a stationary state with a well-defined segregation of the species in essentially two dominant clusters, it is also interesting to study the behavior of the system for different initial conditions. In particlular, we selected a fully segregated initial condition such that all species A (B) are randomly placed on the left-hand side of the sample in possitions xiA e L/2 with i = 1, 2, 3, ..., NA (right-hand side of the sample in positions xiB > L/2 with i = 1, 2, 3, ..., NB). This type of initial condition for the A + B f 0 reaction in the absence of sources has also been studied by other authors but focusing the interest on the behavior of the interface between A- and B-rich domains, see e.g., refs 3, 20, and 21 and references therein. By choosing this condition, one observes an early time initial increase of the rate (see Figure 3), which after a crossover time starts to decrease. The initial increase with slope close to 1/2 is due to the fact that after each reaction event occurring at the border between the two initial clusters, the particles are randomly introduced into the bulk of these clusters leading to a speeding up of the rate. Subsequently, when this mechanism is no longer the dominant one, i.e., for a time of the order of the crossover time, the rate starts to decrease, as typically occurs in the GCK process but in this case with an exponent close to 1/4 (see Figure 3). 24270
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Figure 4. (a) Loglog plots of the rate versus time as obtained for FA = FB = 1/100 constant and lattices of different size L, as listed in the figure. (b) Loglog plot of the data already shown in panel a but after proper rescaling of the axis according to LRate t/L2. The upper right-hand side inset shows a loglog plot of Rstat versus L, and the straight line has a slope of 1. The lower left-hand side inset shows a loglog plot of τ versus L, and the straight line has slope z = 2.
This behavior is due to the fluctuations in the number of particles of different types that end, in finite samples, with only one kind of particle present in the lattice. Pointing now our attention to the LCK process with initially randomly distributed species, we grouped the simulation results in two sets: (i) those obtained by keeping the number of species constant, while the system size is varied, and (ii) those corresponding to the constant density of particles that is obtained by varying both NA,B and L, accordingly. Figure 4a shows loglog plots of the rate of reaction versus time as obtained by keeping FA = FB = 1/100 constant. Here, one observes a rapid drop of the rate that ultimately reaches a stationary state at a size-dependent crossover time τ(L). A fit of the loglog plot of τ(L) versus L, as shown in the lower left-hand side inset of Figure 4b, yields a power-law behavior of the form τ Lz, with a dynamic exponent z = 2. However, the size-dependent stationary rate decreases as Rstat L1, as shown in the upper right-hand side inset of Figure 4b. Since the rate involves the number of reaction events per unit of time and length, these results simply show that for data taken at a constant density of species, the number of reaction events per unit of time is independent of the system size. Summing up, all these results suggest the scaling behavior of the data given by F1Rate LRate t/L2, which is nicely proved by the data collapse shown in the main panel of Figure 4b.
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Figure 5. (a) Loglog plots of the rate versus time as obtained for samples of different size, as indicated in the figure, and taking NA,B = 100. (b,c) Plots of the crossover time (τ) versus L and of the stationary rate (Ratestat) versus the total density of species, respectively. (d) Loglog plot of the data already shown in panel a but after proper rescaling of the axis according to F3Rate t/L2.
It should be recognized that we are unable to accurately determine the slope of the power-law decrease of the rate at early times, i.e., before the crossover time, due to the presence of a systematic curvature. This topic will be addressed in detail in the following paragraphs when analyzing data obtained at the same density of species. Figures 5a and 6a show loglog plots of the rate of reaction versus time as obtained for different system sizes but always keeping the number of species NA,B = 100 and NA,B = 50, respectively. As in the previous examples, one observes here an initial drop of the rate that, after some size-dependent crossover time, is followed by a stationary regime where the rate remains stationary (Ratestat) but depends on the density. Subsequently, and for NA,B = 100, we have just plotted the size dependence of τ (Figure 5b) and the density dependence of Ratestat (Figure 5c), where power-law dependences with exponents z = 2 and X = 3 are obtained, respectively. The same power-law dependences and exponents are also obtained for NA,B = 50, which are not shown here for the sake of space. It is interesting to discuss the fact that the measured reaction order X = 3 is anomalous in the sense that it departs from the (textbook) mean-field value given by XMF = 2. This result also validates the analytical calculations performed in ref 14 because 24271
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Figure 7. Linear log plots of the effective decay exponent βeff versus 1/L obtained for a different number of particles, as listed in the figure. The inset shows the dependence of the average value of βeff versus the number of particles (N). The dashed line indicates the extrapolation to the L f ∞ and N f 0 limits and has been drawn in order to guide the eye.
Figure 6. (a) Loglog plots of the rate as a function of time as obtained for a fixed number of species NA,B = 50 and lattices of different length L, as listed in the figure. (b) Loglog plot of the data already shown in panel a but after proper rescaling of the axis according to F3Rate t/L2.
by working with the difference in the concentration of species, as already discussed in the context of eqs 5 and 6, the theoretical results become independent of the reaction order. However, the exponent X = 3 is the well-known anomalous reaction order of the diffusion-controlled reaction of the same type of species, i.e., A + A f 0, see e.g., ref 3. On the basis of the assumption that the type of particle, either A or B, is irrelevant when the coarsening process becomes stabilized, it is possible to outline some heuristic arguments in order to explain the obtained reaction order. In fact, at this late stage one has few reactions occurring at the border between the A- and B-rich segregated regions. Then, after each reaction event, a pair of particles is introduced at random into the sample. Then, one that has A- and B-particles introduced into the respective segregated regions are irrelevant for further reaction. In contrast, the main contribution to the rate arises from A-and B-particles introduced into B- and A-rich regions. Under this circumstance, each of the newly introduced particles is a random walk embedded in a set of random walkers of different type. Following the reasoning of Kopelman et al.,2225 the number of distinct sites visited by those particles (SN) behaves as SN tds/2, where ds is the spectral dimension, and the visitation efficiency (ɛ), which is proportional to the rate constant K, is given by K E ∂SN =∂t t ðds =2Þ 1
ð10Þ
which leads to an anomalous reaction order X = 1 + 2/ds = 3 when the spectral dimension in d = 1, namely, ds = 1, is considered.
All these results obtained by keeping the number of species constant but varying the system size, as well as data corresponding to NA,B = 10 and NA,B = 20 (not shown here for the sake of space), suggest the scaling behavior of the data given by F3Rate t/L2, which is nicely proven by the data collapses shown in Figures 5d and 6b. However, we observe that in all cases, the power-law decrease of the rate exhibits a systematic curvature. While the early time behavior seems to be compatible with a decay exponent close to β = 1/2 mostly due to stochastic effects of the initial (random) distribution of species, we focused our interest on the long-time behavior, i.e., a time regime such that the rate still decreases but has not reached the stationary state, which is characterized by an effective decay exponent βeff. Figure 7 shows that plots of βeff versus 1/L as obtained for different values of NA,B exhibit some interesting features: data corresponding to fixed values of NA,B show a sligthly decreasing trend when L f ∞, and however, the determined values of the exponents βeff clearly decrease as NA,B also decreases. So, the inset of Figure 7 shows the dependence of the average value of βeff, as determined for each set of curves shown in the main panel, as a function of NA,B t N. The clear trend observed allows us to conjecture that the properly extrapolated (L f ∞ and N f 0) value of the true exponent should be β = 1/4, as shown with the aid of the dashed line.
5. CONCLUSIONS On the basis of the results obtained by means of extensive numerical simulations of the diffussion-controlled annihilation reaction between two types of particles with an input source, we have confirmed two early theoretical predictions: (i) for the LCK case in finite systems, the reaction reaches a (nonequilibrium) stationary regime, and (ii) for the GCK case, the fluctuation of the density difference between different particles grows with time with an exponent δ = 1 (finite systems). Going one step further, our results allow us to outline very interesting conclusions that shed light on the understanding of the kinetic scaling behavior of the reaction: (i) for the GCK case, the rate of reaction decreases monotonically with an exponent βGCK = 1/2 until the sample becomes fully covered by a single (dominant) species, and the reaction stops forever (finite systems); (ii) also, 24272
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(23) Anacker, L. W.; Kopelman, R.; Newhouse, J. S. J. Stat. Phys. 1984, 36, 591–602. (24) Anacker, L. W.; Kopelman, R. J. Chem. Phys. 1984, 81, 6402–6403. (25) Anacker, L. W.; Parson, R. P.; Kopelman, R. J. Phys. Chem. 1985, 89, 4758–4761.
’ AUTHOR INFORMATION Corresponding Author
*Fax: 0054-221-4257317. E-mail: ealbano@iflysib.unlp.edu.ar.
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