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Bridging the Gap between the Ultrafast and the Ultraslow
journal of MOt.F-~U LAR
IaQUIDS ELSEVIER
Journal of Molecular Liquids, 64 (1995) 241-245
Bridging the Gap between the Ultrafast and the Ultraslow Arieh L. Edelstein and Noam Agmon
Department of Physical Chemistry and the Fritz Haber Research Center, The Hebrew University, Jerusalem 9190~, Israel. Received 10 September 1994; accepted 27 June 1995
ABSTRACT
A Brownian simulation of a microscopic pseudo-unimolecular reversible reaction in one-dimension is reported over a huge time range for varying values of the dissociation parameter. The simulation shows how the time behavior of the approach to equilibrium of the binding probability, which is predominantly power-law for ultrafast dissociation, changes to a faster decay law as dissociation slows down. In terms of the mea~-field approximations which are compared with the simulations, the ultrafast dissociation limit is characterized as "superposition dominated" whereas the ulraslow limit is "convolution dominated". Bimolecular chemical reactions in solution are traditionally treated by chemical kinetics. For reactions proceeding in a single elementary step, the chemical rate equations predict an exponential decay of molecular concentrations. This agrees with experimental data of slow, "rate limited" reactions. For fast irreversible reactions the rate equation description was found to be inaccurate. Such "diffusion limited" reactions need to be treated by the theory of diffusion influenced reactions [1]. They are characterized by a non-equilibrium distribution of reacting particles leading to a time-dependent pair recombination rate coefficient, k(t). In simple cases the many-body recombination probability is determined by pair dynamics, hence by k(t). Consider, as a concrete example, the pseudo-unimolecular recombination reaction A + B ~ AB with a single A molecule and many diffusing B particles of diffusion coefficient D and concentration c. The particles are non-interacting, except that A and B react at a contact distance, a, with an association rate parameter ~ . If A is static and the diffusion space is infinite (the "thermodynamic limit"), the Smoluchowski theory is exact and the survival probability of an unreacted A molecule is simply [2]
Sir~(t[eq) = exp[--c
k(t') dr'].
(i)
In our notation, Si,~ (tleq) is the many-body survival probability for an irreversible reaction with initially equilibrated B particles.
0167-7322/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00858-6
242
The expression for k(t) depends on dimensionality, d. In three dimensions [2]
k(t) = 4
Da,tj [1 + "rao s
(2a)
where aell =- a ~ / (47rDa + ~ ) and ? =- 1/(a - a , l l ). For fast, diffusion limited reactions a , f l ~ a and 7 a e l / i s very large. For slow reactions a , l f --* ~ / 4 r D << 1 so that k(t) -* ~ . In this limit S~,(t[eq) = e x p ( - c ~ t ) , the conventional kinetic result. In the general case, k(t) --* 4~rDa~ll as t ~ oo so that the long time decay of the survival probability is always exponential. In one dimension one may set without loss of generality a = 0, obtaining k(t) = ~ e x p ( x ~ 2 t / D ) e r f c ( x , ~ t ~ ) .
(2b)
This expression is valid for B's located on one side of the origin. As t --* oo, k(t) vanishes so that only one "limiting" case exists. When the reaction is reversible, A + B ~ AB, both association (~a) and dissociation (~d) rate parameters are non-zero. Since twice as many rate parameters are involved the number of limiting cases is doubled. What these limiting cases are and how they are bridged is not fully understood, although intensive research is currently invested in reversible diffusion-influenced reactions [3-11]. Here we focus on bridging the gap between fast and slow reversible diffusion-influenced reactions, whereas a fuller exposition of theory and simulations is presented elsewhere [12]. In three dimensions both rate parameters may be large or both may be small compared with diffusion. These represent the "diffusion" or "reaction" control limits. In addition, one rate parameter may be large and the other small, or vice versa. Experimentally, reversible chemical reactions in the diffusion control limit have been observed [3,4]. In proton transfer to water from excited hydroxypyrene-trisulfonate (HPTS) both dissociation and recombination rate parameters are large. The approach to equilibrium exhibits a power law behavior, t -d/2, where d = 3 for reactions in three dimensions. In one dimension, nd may be either large or small relative to ~a so that only two limiting cases are expected. For this reason and for computational ease we have chosen to investigate d = 1 first. This limit typically provides the most stringent test on the meanfield approximations discussed below. These approximations are in turn easily applied to three-dimensional systems even in the presence of an interaction potential. We have simulated 1000 randomly-moving, non-interacting particles on the half-line with a static trap at the origin (Fig. 1, symbols). The trap may bind at most one particle, and this introduces correlations between the particles, making the problem difficult. The particles are initially randomly distributed on the interval [0, L]. L =2,500 is sufficiently large to ensure achievement of the thermodynamic limit. The particle concentration is thus c =_ N / L = 0.4. The diffusion coefficient, D, and the association parameter, ~ , were arbitrarily set to unity. This is equivalent to choosing time and distance scales. The dissociation parameter, ~;d, was varied by 4 orders of magnitude as denoted in Fig. 1. In each case we have monitored the approach to equilibrium over 5 orders of magnitude in time. The short time behavior is taken from lattice random walks. The long time behavior was calculated using a novel Brownian dynamics algorithm [10]. This is an off-grid (continuous space) method that utilizes the exact one-particle solution in propagating a
243
many-particle trajectory. Since for d = 1 the single-particle solution is known analytically, computation time is reduced by the use of large time steps without a sacrifice in accuracy. To cover an extensive time range such as in Fig. 1, several simulations with different time-steps are combined. As t ~ o0, the survival probability tends to the equilibrium limit 8,q = 1/(1 + cK, q), where K ~ -- ~a/~a is the equilibrium constant. To convert to the experimentally relevant initial condition of one bound AB molecule, $(t[*), we use the rigorously exact "Generalized Mass Action Law" [6]
,.%q
8(tl*)
-
[8(tleq)
=
8~q]/cK¢q.
-
(3)
This is the "deviation from equilibrium" shown in Fig. 1.
0
1 A
-3
~" 0
E 0
1
0.1 0.01
-6
oooooo~
0.001
e,,
0
-9
nu im
> -12
0 "0
c
SA -15 "18
CA 1
-2
I
f
0
,
r
I
2
,
'
r
a
4
I
6
I, ,
~
,)I,
~,P~'
8
10
In t FIG. 1. Approach to equilibrium in ultrafast and ultraslow reversible reactions in one dimension. See text for details. The simulations are compared with two approximations, the "Superposition Approximation" (SA, bold curves) and the "Convolution Approximation" (CA, thin curves). In the SA, the many-body survival probability for initially randomly distributed particles is given by [7,11]
SsA(tleq)
= exp[-cP(*;t)],
(4)
244
where (for d = 1) P(*; t) - 1 - foL[1 -- P(x; t)] dx and P(x; t) obeys an effectively single-particle diffusion equation, only with the non-linear boundary condition D O-fi(z; t)/Ox]==o = x~ P(O; t) - ,~d [exp(cP(*; t)) - 1]/c.
(5)
The initial condition is P(z; 0) = 1. For small cP(*; t), Eq. (5) reduces to the linear backreaction boundary condition characterizing a geminate pair [3]. P(*; t) is small when K~q is small, so that the SA becomes exact for small e K , q. This is reasonable because particle-particle correlations are introduced only by an occupied trap, and these become weak for low trap occupancy, justifying the approximation invoked [7]. Comparison with the Browniem simulations in Fig. 1 indeed shows good agreement with the SA in the large xd limit, which may therefore be termed "superposition dominated". For smaller ~d the SA agrees with the data only at short times. Interestingly, however, it predicts a universal t -~/~ approach to equilibrium which seems always to be followed. This leads to a sharp crossover to the asymptotic behavior at long times. The CA is shown as thin curves in Fig. 1. It involves solving numerically the convolution relation [6,7] Sc~(tl*)
=
"d /o'[1
-
ScA(t']*)],gi,r(t
-
t'la)dt'
-
(6)
Here 8~**(t[a) is the survival probability for irreversible recombination, =
(7)
given that one particle was initially at contact and the remaining particles were equilibrated. S~rr(t[eq) is given by Eq. (1) and k(t) depends on d, see Eq. (2). The physical justification to Eq. (6) is as follows: At time t, a particle is bound with probability [1 - S(t[*)]. The dissociation probability between t and t + dt is xd [1 - S(t[*)]. This produces a particle B at the contact distance, a, with the remaining particles assumed to be randomly distributed. The bound state is subsequently regenerated if one of the free particles binds irreversibly. This process is described by S~r~(t[a), hence the convolution. The approximation is in the assumption that by the time a particle dissociates all other particles have attained their equilibrium distribution. This is best justified for slow dissociation, when the reaction becomes "convolution dominated" (Fig. 1). The CA (thin curves) agrees with the simulation over an extended period of time and only at extremely long times a switchover to the universal power-law decay is observed. Based on the above results, one may expect to observe the following limits in three dimensional reversible reactions. When both rate parameters are large, the reaction is diffusion limited and "superposition dominated". As ted decreases (or cx~ increases), the reaction becomes "convolution dominated" exhibiting faster initial decay to equilibrium over a wider time window. Equation (2a) suggests that when ~d ---*0, in three-dimensions, the long-time decay becomes exponential. This may correspond to HPTS in methanolwater mixtures [8], where dissociation is inhibited by dielectric destabilization of the ionic products, and explain the success of "convolution kinetics" for monomer-excimer kinetics [5]. Finally, if x~ is also small compared with D, k(t) ~ n~, $ ~ ( t [ a ) = S~r~(t[eq) = exp(-c~,t) so that Eq. (6) reduces to a familiar rate equation
245
dS(tl,)ldt
=
,
,5(tl*)]
-
(8)
These predictions should still be tested in three-dimensional simulations. This will bridge over the spectrum of kinetic behaviors ranging from ultrafast to ultraslow reactions. Monitoring over such a huge range of parameters is a technical challenge for an ultra-accurate experimental setup.
REFERENCES
[1] S. A. Rice, in Diffusion-Limited Reactions, Vol. 25 of Comp. Chem. Kinet., edited by C. H. Bamford, C. F. H. Tipper, and R. G. Compton (Elsevier, Amsterdam, 1985). [2] A. Szabo, J. Phys. Chem. 93, 6929 (1989). [3] E. Pines, D. Huppert, and N. Agmon, in Ultrafast Phenomena VI, Vol. 48 of Chemical Physics, edited by T. Yajima, K. Yoshihara, C. B. Harris, and S. Skionoya (Springer Verlag, Berlin, Germany, 1988), pp. 517-519. [4] A. Masad, S. Y. Goldberg, D. Huppert, and N. Agmon, in Ultrafast Phenomena VIII, Vol. 55 of Chemical Physics, edited by J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail (Springer Verlag, Berlin, 1993), pp. 664-666. [5] J. Vogelsang and M. Hauser, J. Phys. Chem. 94, 7488 (1990). [6] N. Agmon and A. Szabo, J. Chem. Phys. 92, 5270 (1990). [7] A. Szabo, J. Chem. Phys. 95, 2481 (1991). [8] N. Agmon, D. Huppert, A. Masad, and E. Pines, J. Phys. Chem. 95, 10407 (1991), erratum, ibid. 96, 2020 (1992). [9] D. Huppert, S. Y. Goldberg, A. MasaA, and N. Agmon, Phys. Rev. Lett. 68, 3932 (1992). [10] A. L. Edelstein and N. Agmon, J. Chem. Phys. 99, 5396 (1993). [11] N. Agmon and A. L. Edelstein, J. Chem. Phys. 100, 4181 (1994). [12] A. L. Edelstein and N. Agmon, J. Phys. Chem. 99, 5389 (1995).
IaQUIDS ELSEVIER
Journal of Molecular Liquids, 64 (1995) 241-245
Bridging the Gap between the Ultrafast and the Ultraslow Arieh L. Edelstein and Noam Agmon
Department of Physical Chemistry and the Fritz Haber Research Center, The Hebrew University, Jerusalem 9190~, Israel. Received 10 September 1994; accepted 27 June 1995
ABSTRACT
A Brownian simulation of a microscopic pseudo-unimolecular reversible reaction in one-dimension is reported over a huge time range for varying values of the dissociation parameter. The simulation shows how the time behavior of the approach to equilibrium of the binding probability, which is predominantly power-law for ultrafast dissociation, changes to a faster decay law as dissociation slows down. In terms of the mea~-field approximations which are compared with the simulations, the ultrafast dissociation limit is characterized as "superposition dominated" whereas the ulraslow limit is "convolution dominated". Bimolecular chemical reactions in solution are traditionally treated by chemical kinetics. For reactions proceeding in a single elementary step, the chemical rate equations predict an exponential decay of molecular concentrations. This agrees with experimental data of slow, "rate limited" reactions. For fast irreversible reactions the rate equation description was found to be inaccurate. Such "diffusion limited" reactions need to be treated by the theory of diffusion influenced reactions [1]. They are characterized by a non-equilibrium distribution of reacting particles leading to a time-dependent pair recombination rate coefficient, k(t). In simple cases the many-body recombination probability is determined by pair dynamics, hence by k(t). Consider, as a concrete example, the pseudo-unimolecular recombination reaction A + B ~ AB with a single A molecule and many diffusing B particles of diffusion coefficient D and concentration c. The particles are non-interacting, except that A and B react at a contact distance, a, with an association rate parameter ~ . If A is static and the diffusion space is infinite (the "thermodynamic limit"), the Smoluchowski theory is exact and the survival probability of an unreacted A molecule is simply [2]
Sir~(t[eq) = exp[--c
k(t') dr'].
(i)
In our notation, Si,~ (tleq) is the many-body survival probability for an irreversible reaction with initially equilibrated B particles.
0167-7322/95/$09.50 © 1995 - Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00858-6
242
The expression for k(t) depends on dimensionality, d. In three dimensions [2]
k(t) = 4
Da,tj [1 + "rao s
(2a)
where aell =- a ~ / (47rDa + ~ ) and ? =- 1/(a - a , l l ). For fast, diffusion limited reactions a , f l ~ a and 7 a e l / i s very large. For slow reactions a , l f --* ~ / 4 r D << 1 so that k(t) -* ~ . In this limit S~,(t[eq) = e x p ( - c ~ t ) , the conventional kinetic result. In the general case, k(t) --* 4~rDa~ll as t ~ oo so that the long time decay of the survival probability is always exponential. In one dimension one may set without loss of generality a = 0, obtaining k(t) = ~ e x p ( x ~ 2 t / D ) e r f c ( x , ~ t ~ ) .
(2b)
This expression is valid for B's located on one side of the origin. As t --* oo, k(t) vanishes so that only one "limiting" case exists. When the reaction is reversible, A + B ~ AB, both association (~a) and dissociation (~d) rate parameters are non-zero. Since twice as many rate parameters are involved the number of limiting cases is doubled. What these limiting cases are and how they are bridged is not fully understood, although intensive research is currently invested in reversible diffusion-influenced reactions [3-11]. Here we focus on bridging the gap between fast and slow reversible diffusion-influenced reactions, whereas a fuller exposition of theory and simulations is presented elsewhere [12]. In three dimensions both rate parameters may be large or both may be small compared with diffusion. These represent the "diffusion" or "reaction" control limits. In addition, one rate parameter may be large and the other small, or vice versa. Experimentally, reversible chemical reactions in the diffusion control limit have been observed [3,4]. In proton transfer to water from excited hydroxypyrene-trisulfonate (HPTS) both dissociation and recombination rate parameters are large. The approach to equilibrium exhibits a power law behavior, t -d/2, where d = 3 for reactions in three dimensions. In one dimension, nd may be either large or small relative to ~a so that only two limiting cases are expected. For this reason and for computational ease we have chosen to investigate d = 1 first. This limit typically provides the most stringent test on the meanfield approximations discussed below. These approximations are in turn easily applied to three-dimensional systems even in the presence of an interaction potential. We have simulated 1000 randomly-moving, non-interacting particles on the half-line with a static trap at the origin (Fig. 1, symbols). The trap may bind at most one particle, and this introduces correlations between the particles, making the problem difficult. The particles are initially randomly distributed on the interval [0, L]. L =2,500 is sufficiently large to ensure achievement of the thermodynamic limit. The particle concentration is thus c =_ N / L = 0.4. The diffusion coefficient, D, and the association parameter, ~ , were arbitrarily set to unity. This is equivalent to choosing time and distance scales. The dissociation parameter, ~;d, was varied by 4 orders of magnitude as denoted in Fig. 1. In each case we have monitored the approach to equilibrium over 5 orders of magnitude in time. The short time behavior is taken from lattice random walks. The long time behavior was calculated using a novel Brownian dynamics algorithm [10]. This is an off-grid (continuous space) method that utilizes the exact one-particle solution in propagating a
243
many-particle trajectory. Since for d = 1 the single-particle solution is known analytically, computation time is reduced by the use of large time steps without a sacrifice in accuracy. To cover an extensive time range such as in Fig. 1, several simulations with different time-steps are combined. As t ~ o0, the survival probability tends to the equilibrium limit 8,q = 1/(1 + cK, q), where K ~ -- ~a/~a is the equilibrium constant. To convert to the experimentally relevant initial condition of one bound AB molecule, $(t[*), we use the rigorously exact "Generalized Mass Action Law" [6]
,.%q
8(tl*)
-
[8(tleq)
=
8~q]/cK¢q.
-
(3)
This is the "deviation from equilibrium" shown in Fig. 1.
0
1 A
-3
~" 0
E 0
1
0.1 0.01
-6
oooooo~
0.001
e,,
0
-9
nu im
> -12
0 "0
c
SA -15 "18
CA 1
-2
I
f
0
,
r
I
2
,
'
r
a
4
I
6
I, ,
~
,)I,
~,P~'
8
10
In t FIG. 1. Approach to equilibrium in ultrafast and ultraslow reversible reactions in one dimension. See text for details. The simulations are compared with two approximations, the "Superposition Approximation" (SA, bold curves) and the "Convolution Approximation" (CA, thin curves). In the SA, the many-body survival probability for initially randomly distributed particles is given by [7,11]
SsA(tleq)
= exp[-cP(*;t)],
(4)
244
where (for d = 1) P(*; t) - 1 - foL[1 -- P(x; t)] dx and P(x; t) obeys an effectively single-particle diffusion equation, only with the non-linear boundary condition D O-fi(z; t)/Ox]==o = x~ P(O; t) - ,~d [exp(cP(*; t)) - 1]/c.
(5)
The initial condition is P(z; 0) = 1. For small cP(*; t), Eq. (5) reduces to the linear backreaction boundary condition characterizing a geminate pair [3]. P(*; t) is small when K~q is small, so that the SA becomes exact for small e K , q. This is reasonable because particle-particle correlations are introduced only by an occupied trap, and these become weak for low trap occupancy, justifying the approximation invoked [7]. Comparison with the Browniem simulations in Fig. 1 indeed shows good agreement with the SA in the large xd limit, which may therefore be termed "superposition dominated". For smaller ~d the SA agrees with the data only at short times. Interestingly, however, it predicts a universal t -~/~ approach to equilibrium which seems always to be followed. This leads to a sharp crossover to the asymptotic behavior at long times. The CA is shown as thin curves in Fig. 1. It involves solving numerically the convolution relation [6,7] Sc~(tl*)
=
"d /o'[1
-
ScA(t']*)],gi,r(t
-
t'la)dt'
-
(6)
Here 8~**(t[a) is the survival probability for irreversible recombination, =
(7)
given that one particle was initially at contact and the remaining particles were equilibrated. S~rr(t[eq) is given by Eq. (1) and k(t) depends on d, see Eq. (2). The physical justification to Eq. (6) is as follows: At time t, a particle is bound with probability [1 - S(t[*)]. The dissociation probability between t and t + dt is xd [1 - S(t[*)]. This produces a particle B at the contact distance, a, with the remaining particles assumed to be randomly distributed. The bound state is subsequently regenerated if one of the free particles binds irreversibly. This process is described by S~r~(t[a), hence the convolution. The approximation is in the assumption that by the time a particle dissociates all other particles have attained their equilibrium distribution. This is best justified for slow dissociation, when the reaction becomes "convolution dominated" (Fig. 1). The CA (thin curves) agrees with the simulation over an extended period of time and only at extremely long times a switchover to the universal power-law decay is observed. Based on the above results, one may expect to observe the following limits in three dimensional reversible reactions. When both rate parameters are large, the reaction is diffusion limited and "superposition dominated". As ted decreases (or cx~ increases), the reaction becomes "convolution dominated" exhibiting faster initial decay to equilibrium over a wider time window. Equation (2a) suggests that when ~d ---*0, in three-dimensions, the long-time decay becomes exponential. This may correspond to HPTS in methanolwater mixtures [8], where dissociation is inhibited by dielectric destabilization of the ionic products, and explain the success of "convolution kinetics" for monomer-excimer kinetics [5]. Finally, if x~ is also small compared with D, k(t) ~ n~, $ ~ ( t [ a ) = S~r~(t[eq) = exp(-c~,t) so that Eq. (6) reduces to a familiar rate equation
245
dS(tl,)ldt
=
,
,5(tl*)]
-
(8)
These predictions should still be tested in three-dimensional simulations. This will bridge over the spectrum of kinetic behaviors ranging from ultrafast to ultraslow reactions. Monitoring over such a huge range of parameters is a technical challenge for an ultra-accurate experimental setup.
REFERENCES
[1] S. A. Rice, in Diffusion-Limited Reactions, Vol. 25 of Comp. Chem. Kinet., edited by C. H. Bamford, C. F. H. Tipper, and R. G. Compton (Elsevier, Amsterdam, 1985). [2] A. Szabo, J. Phys. Chem. 93, 6929 (1989). [3] E. Pines, D. Huppert, and N. Agmon, in Ultrafast Phenomena VI, Vol. 48 of Chemical Physics, edited by T. Yajima, K. Yoshihara, C. B. Harris, and S. Skionoya (Springer Verlag, Berlin, Germany, 1988), pp. 517-519. [4] A. Masad, S. Y. Goldberg, D. Huppert, and N. Agmon, in Ultrafast Phenomena VIII, Vol. 55 of Chemical Physics, edited by J.-L. Martin, A. Migus, G. A. Mourou, and A. H. Zewail (Springer Verlag, Berlin, 1993), pp. 664-666. [5] J. Vogelsang and M. Hauser, J. Phys. Chem. 94, 7488 (1990). [6] N. Agmon and A. Szabo, J. Chem. Phys. 92, 5270 (1990). [7] A. Szabo, J. Chem. Phys. 95, 2481 (1991). [8] N. Agmon, D. Huppert, A. Masad, and E. Pines, J. Phys. Chem. 95, 10407 (1991), erratum, ibid. 96, 2020 (1992). [9] D. Huppert, S. Y. Goldberg, A. MasaA, and N. Agmon, Phys. Rev. Lett. 68, 3932 (1992). [10] A. L. Edelstein and N. Agmon, J. Chem. Phys. 99, 5396 (1993). [11] N. Agmon and A. L. Edelstein, J. Chem. Phys. 100, 4181 (1994). [12] A. L. Edelstein and N. Agmon, J. Phys. Chem. 99, 5389 (1995).