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Power law relaxation kinetics in multistep reversible reaction
Chemical Physics Letters 369 (2003) 643–649 www.elsevier.com/locate/cplett
Power law relaxation kinetics in multistep reversible reaction Sujata Paul, Gautam Gangopadhyay
*
S.N. Bose National Centre for Basic Sciences, J.D. Block, Sector-III, Salt Lake, Kolkata 700098, India Received 19 April 2002; in final form 27 November 2002
Abstract The relaxation kinetics of the diffusion influenced reversible one step reaction A þ A (kk11 ) A2 , consecutive two step reaction A þ A (kk11 ) A2 (kk22 ) C and two step parallel reaction A þ A (kk11 ) A2 , A (kk22 ) C in one dimension have been studied. All these two step reactions show a power law ðt1=2 Þ asymptotics as in the one step case although one of the rate equation is linear. In both the two step cases, such power law kinetics can be observed if all the steps of a reaction are reversible. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The theory of bimolecular reactions in solution has a rich history beginning with SmoluchowskiÕs original work in 1917 [1]. Since then the diffusionlimited chemical reactions have attracted considerable attention [2–13]. When the reaction is diffusion controlled, competition between reacting particles introduces corrections in their diffusive motion. This should be manifested in the long time dependence of the reactive concentrations. The fact that in bimolecular reactions in d dimension, the concentrations approach equilibrium not exponentially but rather as td=2 was first discovered by using simple physical arguments involving spatial concentration fluctuations [3]. The basic idea is that, since the initial and equilibrium spatial fluctuations are different, a redistribution of particles through
*
Corresponding author. Fax: +91-33-2335-3477. E-mail address: [email protected] (G. Gangopadhyay).
diffusion must occur before equilibrium can be reached. Since the relaxation due to reaction proceeds exponentially, the final approach to equilibrium is determined by the diffusion of particles. This slow asymptotic decay was subsequently obtained by using many theoretical approaches [4–7], through experiment [8] and simulations [9–11]. These studies have established that the ultimate approach of the reaction to equilibrium is a power law. Recently the diffusion influenced reversible reaction A þ B () C has been studied by Gopich et al. [12]. They have presented a simpler unified derivation of the long time asymptotics of the reversible reaction in the frame work of fluctuation theory. Their procedure can be readily used to treat more complex bimolecular reaction. The purpose of this work is to present a complete solution of the asymptotics [12] of A þ A () A2 for uniform random initial concentrations. As most of the reactions proceed through some intermediate steps, we have also considered a consecutive two step reaction [2] A þ A ()
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(02)01986-3
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S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
A2 () C where one of the rate equations is linear. If we ignore the first step of the reaction, the relaxation to equilibrium is exponential. So the question is whether all the reactants should relax to equilibrium with a power law or not. Similarly we have also analyzed the case of a two step parallel reaction. We have started with a set of reaction-diffusion equation in one dimension that describes concentration fluctuations at equilibrium. These are obtained by adding diffusion terms and random forces to the conventional nonlinear rate equation of chemical kinetics [12,13]. To obtain the relaxation behaviour of the concentration to its equilibrium value, we have exploited the Onsager Regression Hypothesis [14]. Normally one linearizes the reaction diffusion equation about equilibrium [14–16] to obtain the relaxation matrix but in this approach quadratic fluctuations of the concentrations from their equilibrium values are considered as a small perturbation. In one dimension this nonlinearity gives rise to the t1=2 asymptotic relaxation of the concentrations to equilibrium.
have zero mean and are statistically uncorrelated with the initial concentrations. The equilibrium concentrations Ae and ðA2 Þe obey the mass action law Ke A2e ¼ ðA2 Þe , where Ke ¼ k1 =k1 is the equilibrium constant. In absence of diffusion and noise, the reaction parts of Eqs. (1) and (2) satisfy
2. One step reaction
where v is a column vector with components (2,)1) and fðq; tÞ is a column vector with the random forces ðfA ; fA2 Þ in q space as components. The matrix KðqÞ contains both reaction and diffusion terms as 0 4k1 Ae 2k1 2 DA KðqÞ ¼ þq : ð5Þ 0 DA2 2k1 Ae k1
Let us consider a dimerisation reaction, A þ A (kk11 ) A2 , where Aðr; tÞ and A2 ðr; tÞ be the local random concentrations of A and A2 particles. The particles are weakly interacting and the concentration fluctuations due to diffusion in one dimension in the presence of chemical reaction are described by the following system of approximate equations, oAðr; tÞ ¼ DA r2 A 2k1 A2 þ 2k1 A2 þ fA ðr; tÞ; ð1Þ ot oA2 ðr; tÞ ¼ DA2 r2 A2 þ k1 A2 k1 A2 þ fA2 ðr; tÞ: ot ð2Þ The first term on the right-hand side of Eqs. (1) and (2) describes the diffusion of A and A2 particles with diffusion coefficients DA and DA2 . The subsequent terms describe the chemical reaction with rate constants k1 and k1 in the forward and backward direction, respectively. Finally, the last terms fi , represent the noise in the system. For the present purposes, we need only to assume that they
o ½Aðr; tÞ þ 2A2 ðr; tÞ ¼ 0; ot i.e., a linear combination of concentrations, namely, ðA þ 2A2 Þ is constant in time. Let the one-dimensional Fourier transform of the concentration fluctuation from equilibrium, cðq; tÞ be a column vector with components Z c1 ¼ dr eiqr ½Aðr; tÞ Ae ; Z ð3Þ iqr c2 ¼ dr e ½A2 ðr; tÞ ðA2 Þe : Eqs. (1) and (2) can be rewritten in one-dimension as Z ocðq; tÞ dq0 ¼ KðqÞc k1 v c1 ðq q0 ; tÞ ot 2p c1 ðq0 ; tÞ þ fðq; tÞ; ð4Þ
In what follows, we use the formalism which is based on the analysis of the two time correlation functions of the equilibrium fluctuations of the concentrations as formulated in Gopich et al. [12] by taking quadratic nonlinearity as a small perturbation. We consider the ð2 2Þ matrix of correlation function of concentration fluctuation with elements hcj ðq; tÞcyl ðq0 ; 0Þi, hcðq;tÞcy ðq0 ;0Þi ¼ eKðqÞt hcðq;0Þcy ðq0 ;0Þi k1 eKðqÞt Z ih i dq00 Dh Kðqq00 Þt 00 v cðq q00 ;0Þ eKðq Þt cðq00 ;0Þ e 1 1 2p E y 0 c ðq ;0Þ : ð6Þ
S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
Here h i denotes the ensemble average over the noise and equilibrium (Poissonian) initial conditions, and ‘‘*’’ R t denotes the time convolution f ðtÞ gðtÞ ¼ 0 f ðt t0 Þgðt0 Þ dt0 . Note that the random forces are averaged out and do not contribute here. The properties of the correlation function for a uniform (Poisson) equilibrium distributions are hci ðq1 ; 0Þcyj ðq2 ; 0Þi ¼ dij ð2pÞdðq1 q2 ÞCie ; hci ðq1 ; 0Þcj ðq2 ; 0Þcyl ðq3 ; 0Þi ¼ dij dil ð2pÞdðq1 þ q2 q3 ÞCie ;
645
DA k1 þ 4Ae k1 DA2 ; k1 þ 4Ae k1
DA2 k1 þ 4Ae k1 DA D2 ¼ k1 þ 4Ae k1
D1 ¼
and k0 ¼ Tr½Kð0Þ . From Eq. (9) we can write, Rð0; tÞ ¼ eKð0Þt Z t 2u1 ð0; sÞ 2u2 ð0; sÞ ds; k1 eKð0ÞðtsÞ u1 ð0; sÞ u2 ð0; sÞ 0
ð7Þ
ð12Þ
where Cie represents the equilibrium concentration of the ith species. Using Eq. (7), Eq. (6) reduces to
where uj ð0; sÞ is defined in Eq. (10). Using the spectral decomposition of exp½KðqÞt in Eq. (11) and evaluating the integrals in Eq. (10), we find that as t ! 1, " # 2 1 k1 k1 R11 ¼ pffiffiffiffiffiffiffiffiffiffiffi ð13Þ t1=2 ; 8pD1 ð4Ae k1 þ k1 Þ3
hcðq; tÞcy ðq0 ; 0Þi ¼ Rðq; tÞhcðq; 0Þcy ðq0 ; 0Þi;
ð8Þ
"
where the Relaxation matrix is given by Rðq; tÞ ¼ eKðqÞt k1 eKðqÞt v uT ðq; tÞ:
ð9Þ
R22 ¼
# 2 1 2k1 k1 pffiffiffiffiffiffiffiffiffiffiffi t1=2 : 8pD1 ð4Ae k1 þ k1 Þ3
ð14Þ
The row vector uT ðq; tÞ has the components Z dq0 Kðqq0 Þt Kðq0 Þt
e e : ð10Þ uj ¼ 1j 1j 2p
According to the regression hypothesis, coarse grained deviation of concentrations from equilibrium values designated by cðq; tÞ, with components ci ðq; tÞ obey
Now, to find out the asymptotic behaviour of the Relaxation matrix, it is convenient to represent the exponentials in Rðq; tÞ as a sum over eigenvalues ki ðqÞ of matrix KðqÞ as
ci ðq; tÞ hci ðq; tÞci ðq; 0Þi ¼ : 2 ci ðq; 0Þ hci ðq; 0Þ i
exp½KðqÞt ¼ P expðKd tÞP ¼
2 X
Xi exp½ki ðqÞt ;
From Eq. (8) one can write Rðq; tÞ ¼
1
ð11Þ
i¼1
where P is a matrix of order ð2 2Þ whose columns are the right eigenvectors of the KðqÞ matrix. Here Kd is the diagonalized matrix of KðqÞ matrix i.e., Kd ¼ P1 KðqÞP. Note that X1 ¼ Y1 P1 and X2 ¼ Y2 P1 , where Y1 and Y2 are ð2 2Þ matrices. In Y1 only the first column and in Y2 only the second column are nonzero, and these columns are the eigenvectors of KðqÞ matrix with eigenvalues k1 and k2 , respectively. In the q ! 0 limit, the eigenvalues of KðqÞ are k1 ¼ q2 D1 and k2 ¼ k0 þ q2 D2 , where
ð15Þ
hcðq; tÞcy ðq0 ; 0Þi : hcðq; 0Þcy ðq0 ; 0Þi
ð16Þ
Comparing Eqs. (15) and (16) and using Eqs. (13) and (14) one obtains, " # 2 1 k1 k1 c1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi t1=2 c1 ð0Þ; 8pD1 ð4Ae k1 þ k1 Þ3 ð17Þ " c2 ðtÞ ¼
#
2 1 2k1 k1 pffiffiffiffiffiffiffiffiffiffiffi t1=2 c2 ð0Þ: 8pD1 ð4Ae k1 þ k1 Þ3
ð18Þ
Therefore, we observe that the asymptotics consist of one term which is a consequence of one zero eigenvalue of the stability matrix K(0). This asymptotic power law relaxation was previously
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obtained by Rey and Cardy [17] by a completely different argument and is also supported by [12]. If DA ¼ DA2 , Eqs. (17) and (18) coincides with the result obtained by Rey and Cardy [17] exactly. Although the exponent of the power law is universal, as it comes from some conservation law, the amplitude of the power law is model-dependent [12,17]. Amplitudes depend also on the initial condition of the concentrations.
3. Consecutive reaction Here we consider a two step consecutive reaction A þ A (kk11 ) A2 (kk22 ) C. Again our starting point is the following approximate reaction diffusion equations in one dimension, oAðr; tÞ ¼ DA r2 A 2k1 A2 þ 2k1 A2 þ fA ðr; tÞ; ot ð19Þ oA2 ðr; tÞ ¼ DA2 r2 A2 þ k1 A2 þ k2 C ot ðk1 þ k2 ÞA2 þ fA2 ðr; tÞ; oCðr; tÞ ¼ DC r2 C þ k2 A2 k2 C þ fC ðr; tÞ; ot
ð20Þ ð21Þ
ðA2 Þe k1 ¼ 2 Ae k1
dq0 c1 ðq q0 ; tÞc1 ðq0 ; tÞ 2p ð23Þ
where v is a column vector with components (2,)1,0) and fðq; tÞ is a vector with the random forces ðfA ; fA2 ; fC Þ in q space as components. The matrix KðqÞ contains both reaction and diffusion terms as 0 1 4k1 Ae 2k1 0 KðqÞ ¼ @ 2k1 Ae ðk1 þ k2 Þ k2 A 0 k2 k2 0 1 DA 0 0 þ q2 @ 0 DA2 0 A: ð24Þ 0 0 DC As in previous section, we have to consider a ð3 3Þ matrix of correlation functions of concentration fluctuations with elements hcj ðq; tÞcyl ðq0 ; 0Þi. The form of the Relaxation matrix is Z t Rð0; tÞ ¼ eKð0Þt k1 dseKð0ÞðtsÞ 0 0 1 2u1 ð0; sÞ 2u2 ð0; sÞ 2u3 ð0; sÞ B C @ u1 ð0; sÞ u2 ð0; sÞ u3 ð0; sÞ A; 0 0 0
where uj ð0; sÞ is given by Z dq0 KðqÞs Kðq0 Þs
e uj ¼ e : 1j 1j 2p
ð26Þ
Now, we use the spectral decomposition of the exponential described by
and Keð2Þ
Z
ð25Þ
where Aðr; tÞ; A2 ðr; tÞ and Cðr; tÞ be the local random concentrations of A; A2 and C particles. The equilibrium concentrations Ae ; ðA2 Þe and Ce obey the mass action law as Keð1Þ ¼
ocðq; tÞ ¼ KðqÞc k1 v ot þ fðq; tÞ;
exp½KðqÞt ¼ P expðKd tÞP1 Ce k2 ¼ ¼ ; ðA2 Þe k2 Keð1Þ ; Keð2Þ
ð22Þ
are the equilibrium constants. In where absence of diffusion and noise, for this reaction scheme one can find that a linear combination of concentrations is time invariant i.e., A þ 2A2 þ 2C ¼ constant. From Eqs. (19)–(21), one can write the quantities of microscopic fluctuation of concentrations from equilibrium in one dimensional Fourier space as
¼
3 X
Xi exp½ki ðqÞt ;
ð27Þ
i¼1
where P is a matrix of order ð3 3Þ whose columns are eigenvectors of the KðqÞ matrix. Kd is the diagonalized matrix of KðqÞ matrix i.e., Kd ¼ P1 KðqÞP, with X1 ¼ Y1 P1 ; X2 ¼ Y2 P1 and X3 ¼ Y3 P1 , where Y1 ; Y2 and Y3 are ð3 3Þ matrices. In Y1 only the first column, in Y2 only the second column and in Y3 only the third column are nonzero, and these columns are the eigenvectors of
S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
KðqÞ matrix with eigenvalues k1 ; k2 and k3 , respectively. In the q ! 0 limit, the eigenvalues of KðqÞ are k1 ¼ q2 D1 ; k2 ¼ q1 þ q2 D2 and k3 ¼ q2 þ q2 D3 , where the effective diffusion coefficients are 1 X D1 ¼ ka ðDb þ Dc Þ 2k0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ka2 ðDb Dc Þ þ 2ka kb ðDc Da ÞðDc Db Þ ; ð28Þ D2 ¼ D3 ¼
h
1 q ðD1 q1 q 2 2
X
h
X
1 q ðD1 q2 q 1 1
Da Þ þ Da Þ þ
X
i ka D a ;
X
i ka D a :
ð29Þ
" R22 ¼ " R33 ¼
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
Here D1 can be written explicitly as, 1 D1 ¼ 4Ae k1 ðDA2 þ DC Þ þ ðk1 þ k2 ÞðDC þ DA Þ 2k0 2 2 þ k2 ðDA þ DA2 Þ þ ð4Ae k1 Þ ðDA2 DC Þ 2 þ ðk1 þ k2 Þ2 ðDC DA Þ2 þ k2 ðDA DA2 Þ2
þ 2ð4Ae k1 Þðk1 þ k2 ÞðDC DA ÞðDC DA2 Þ þ 2ðk1 þ k2 Þðk2 ÞðDA DA2 ÞðDA DC Þ 1=2 þ 2ðk2 Þð4Ae k1 ÞðDA2 DA ÞðDA2 DC Þ ; ð31Þ where k0 is the sum of the diagonal elements of the matrix K(0). ka , Da are defined by the diagonal elements of KðqÞ in the reaction part and in the diffusion part, respectively. The summations are taken over all permutations of a, b and c. Here q1 and q2 are the nonzero eigenvalues of the matrix K(0) and are given by q1;2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Tr½Kð0Þ ðTr½Kð0Þ Þ2 4f4Ae k1 k2 þ ð4Ae k1 þ k1 Þk2 g 2
ð32Þ
and the third eigenvalue of the K(0) matrix is zero. Therefore, from Eq. (25) in the limit t ! 1 " # k1 ðk2 þ k2 Þ ðk1 k2 Þ2 1=2 pffiffiffiffiffiffiffiffiffiffiffi ; ð33Þ R11 ¼ t 3 8pD1 ðq1 q2 Þ
# t1=2 ;
ð34Þ
t1=2 :
ð35Þ
#
So the macroscopic deviation of the concentrations from equilibrium assume the power law asymptotics as " # 2 k1 ðk2 þ k2 Þ ðk1 k2 Þ 1=2 pffiffiffiffiffiffiffiffiffiffiffi c1 ðtÞ ¼ c1 ð0Þ; ð36Þ t 8pD1 ðq1 q2 Þ3 "
ð30Þ
647
c2 ðtÞ ¼ " c3 ðtÞ ¼
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
# t1=2 c2 ð0Þ;
ð37Þ
t1=2 c3 ð0Þ:
ð38Þ
#
Here also we observe that the asymptotics consist of one term which is a consequence of one zero eigenvalue of the stability matrix K(0), where, q1 q2 ¼ ½4Ae k1 k2 þ ð4Ae k1 þ k1 Þk2 and D1 is given by Eq. (31). Let us consider some special cases, Case I. For k2 ¼ 0 with k1 6¼ 0, i.e., the second step is irreversible but the first step is reversible, one can find q1 q2 ¼ 4Ae k1 k2 and
4Ae k1 DA2 þ ðk2 þ k1 ÞDA D1 ¼ : 4Ae k1 þ k2 þ k1 In this case each numerator in Eqs. (36)–(38) becomes zero, i.e., there is no power law of t1=2 asymptotic decay. Case II. Similarly, for the case when k1 ¼ 0 but k2 6¼ 0, i.e., when the first step is irreversible but the second step is reversible, one obtains q1 q2 ¼ ½4Ae k1 ðk2 þ k2 Þ and
4Ae k1 DA2 þ k2 DC þ k2 DA D1 ¼ : 4Ae k1 þ k2 þ k2 In this case also the numerators in Eqs. (36)–(38) become zero, so there will be no such power law relaxation.
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S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
Case III. If the whole reaction scheme is irreversible, i.e., if say k1 ¼ 0 and k2 ¼ 0 then also all the numerators are zero but the denominators are nonzero and hence there will be no power law relaxation as in Eqs. (36)–(38). So only for the reversible cases one obtains the asymptotic relaxation of t1=2 power law. If one of the step is irreversible no such power law relaxation exists. Note that in the total calculation we have considered the eigenvalues of the stability matrix KðqÞ in the limit q ! 0, and in the eigenvalues higher powers of q are neglected, because we are interested only in the slowest decay. We have not considered the other relatively faster relaxation mechanism, so from this analysis it is not possible to say what kind of relaxation will happen for the irreversible case which may be different from the exponential relaxation. Also in this analysis we have assumed an equilibrium Poissonian distribution which is strictly valid for reversible case. In other words, if the amplitude of relaxation is zero, it does not necessarily mean that the relaxation is exponential. However, to obtain the power law relaxation the amplitude has to be nonzero which is possible only when all the steps of the reactions are reversible. It can be shown that in absence of the last step of the reaction, i.e., in the limit k2 ; k2 ! 0, the results in Eqs. (36)–(38) reduce exactly to that of the one step case. When the last step is absent i.e., in the limit k2 ; k2 ! 0 one obtains from Eq. (31),
4Ae k1 DA2 þ k1 DA D1 ¼ : 4Ae k1 þ k1 For k2 ¼ 0, one can have, ðq1 q2 Þ3 ¼ ð4Ae k1 þ 3 3 k1 Þ k2 and therefore the amplitudes relax as c1 ðtÞ ¼ ¼
c2 ðtÞ ¼ ¼
2 3 k1 k1 k2
pffiffiffiffiffiffiffiffiffiffiffiffi c1 ð0Þ 3 8pD1 t ð4Ae k1 þ k1 Þ3 k2 2 k1 k1
pffiffiffiffiffiffiffiffiffiffiffiffi c1 ð0Þ; ð4Ae k1 þ k1 Þ3 8pD1 t
2 3 k1 k1 k2 2 c ð0Þ 3 3 pffiffiffiffiffiffiffiffiffiffiffiffi 2 ð4Ae k1 þ k1 Þ k2 8pD1 t 2 2k1 k1 pffiffiffiffiffiffiffiffiffiffiffiffi c2 ð0Þ ð4Ae k1 þ k1 Þ3 8pD1 t
ð39Þ
and c3 ðtÞ ¼ 0: Therefore, Eqs. (39) and (40) exactly correspond to Eqs. (17) and (18), respectively.
4. Parallel reaction Next we consider a two-step parallel reaction, A þ A (kk11 ) A2 ; A (kk22 ) C. The approximate reaction diffusion equations in one dimension are given by oAðr; tÞ ¼ DA r2 A 2k1 A2 þ 2k1 A2 k2 A ot þ k2 C þ fA ðr; tÞ; oA2 ðr; tÞ ¼ DA2 r2 A2 þ k1 A2 k1 A2 þ fA2 ðr; tÞ; ot oCðr; tÞ ¼ DC r2 C þ k2 A k2 C þ fC ðr; tÞ: ot ð41Þ The corresponding differential equation for cðq; tÞ vector can be written as in Eq. (23), where the stability matrix is, 0 1 4k1 Ae þ k2 2k1 k2 KðqÞ ¼ @ 2k1 Ae k1 0 A k2 0 k2 0 1 DA 0 0 þ q2 @ 0 DA2 0 A: ð42Þ 0 0 DC Here also one linear combination of concentrations, ðA þ 2A2 þ CÞ remains constant in time when there is no diffusion and noise. In absence of diffusion, KðqÞ matrix has one eigenvalue which is zero and the other two nonzero eigenvalues are q1;2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Tr½Kð0Þ ðTr½Kð0Þ Þ 4fð4Ae k1 þ k1 Þk2 þ k2 k1 g : 2
ð43Þ
ð40Þ
Following the prescription in Section 3, here one can identify the column matrix v has the components (2,)1,0). Therefore, the macroscopic deviation of the reactants assume the power law asymptotics as,
S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
" c1 ðtÞ ¼ " c2 ðtÞ ¼ " c3 ðtÞ ¼
k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
# t1=2 c1 ð0Þ;
2k1 ðk2 þ k2 Þ ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3 k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
2
ð44Þ
# t1=2 c2 ð0Þ;
ð45Þ
# t1=2 c3 ð0Þ;
ð46Þ
where q1 q2 ¼ ð4Ae k1 þ k1 Þk2 þ k2 k1
ð47Þ
and D1 ¼
1 ð4Ae k1 þ k2 ÞðDA2 þ DC Þ þ k1 ðDC þ DA Þ 2k0 n þ k2 ðDA þ DA2 Þ þ ð4Ae k1 þ k2 Þ2 ðDA2 DC Þ2 2
þ ðk1 Þ ðDC DA Þ
2
We are grateful to the unknown referee for critical remarks.
þ 2ð4Ae k1 þ k2 Þðk1 ÞðDC DA ÞðDC DA2 Þ
þ 2ðk2 Þð4Ae k1 þ k2 ÞðDA2 DA ÞðDA2 DC Þ
which is nonlinear. In absence of diffusion when atleast one linear combination of concentrations is time invariant, it gives a zero eigenvalue of the stability matrix. This zero eigenvalue corresponds to a slow dynamical mode in the system which when couples to nonlinearity, gives rise to the power law relaxation as observed in [12]. It is difficult to conclude by this approach about the relaxation mechanism when one of the steps is irreversible. Hence a systematic approach using fluctuation theory to deal with the irreversible cases [18,19] are in order which will be discussed elsewhere. This method is working well for simple reversible reactions and is straight forward to extend in three dimensions.
Acknowledgements
2 þ k2 ðDA DA2 Þ2
þ 2ðk1 Þðk2 ÞðDA DA2 ÞðDA DC Þ
649
o1=2
:
References ð48Þ
Here k0 is the sum of the diagonal elements of K(0) matrix. Similarly, by the straight forward arguments as given in Section 3, one can show if any one of the step is irreversible or if the whole reaction scheme is irreversible, all the amplitudes of tð1=2Þ in Eqs. (44)–(46) become zero. So reversibility in every steps of the reaction is necessary to obtain the power law relaxation as given in Eqs. (44)–(46). Similarly, if we drop the second step of this reaction scheme, i.e., k2 ; k2 ! 0, arguments provided at the end of previous section holds and Eqs. (44)–(46) reduce exactly to the one step reaction, A þ A () A2 , for arbitrary diffusion coefficients of the reactants.
5. Conclusion From these reaction schemes we observe that an asymptotic power law relaxation kinetics occurs when all the steps of a reaction are reversible and there should be atleast one rate equation
[1] M. Smoluchowski, Z. Phys. Chem. 92 (1917) 129. [2] J. Keizer, J. Am. Chem. Soc. 112 (1990) 22. [3] Ya.B. Zeldovich, A.A. Ovchinnikov, JETP Lett. 26 (1977) 440. [4] I.V. Gopich, N. Agmon, Phys. Rev. Lett. 84 (2000) 2730. [5] N. Agmon, I.V. Gopich, J. Chem. Phys. 112 (2000) 2863. [6] K. Kang, S. Redner, Phys. Rev. A 32 (1985) 435. [7] S.F. Burlatsky, A.A. Ovchinnikov, G.S. Oshanin, J. Exp. Theor. Phys. 68 (1989) 1153. [8] D. Huppert, S.Y. Goldberg, A. Masad, N. Agmon, Phys. Rev. Lett. 68 (1992) 3932. [9] N. Agmon, A.L. Edelstein, J. Chem. Phys. 100 (1994) 4181. [10] A.L. Edelstein, N. Agmon, J. Phys. Chem. 99 (1995) 5389. [11] A.L. Edelstein, N. Agmon, J. Mol. Liq. 64 (1995) 241. [12] I.V. Gopich, A.A. Ovchinnikov, A. Szabo, Phys. Rev. Lett. 86 (2001) 922. [13] E.L. Elson, D. Magde, Biopolymers 13 (1974) 1. [14] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, 1986. [15] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, SpringerVerlag, Berlin, 1985. [16] B.J. Berne, R. Giniger, Biopolymers 12 (1973) 1161. [17] P.A. Rey, J. Cardy, J. Phys. A 32 (1999) 1585. [18] J.L. Spouge, Phys. Rev. Lett. 60 (1988) 871. [19] T.O. Masser, D. ben-Avraham, Phys. Rev. E 64 (2001) 062101.
Power law relaxation kinetics in multistep reversible reaction Sujata Paul, Gautam Gangopadhyay
*
S.N. Bose National Centre for Basic Sciences, J.D. Block, Sector-III, Salt Lake, Kolkata 700098, India Received 19 April 2002; in final form 27 November 2002
Abstract The relaxation kinetics of the diffusion influenced reversible one step reaction A þ A (kk11 ) A2 , consecutive two step reaction A þ A (kk11 ) A2 (kk22 ) C and two step parallel reaction A þ A (kk11 ) A2 , A (kk22 ) C in one dimension have been studied. All these two step reactions show a power law ðt1=2 Þ asymptotics as in the one step case although one of the rate equation is linear. In both the two step cases, such power law kinetics can be observed if all the steps of a reaction are reversible. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction The theory of bimolecular reactions in solution has a rich history beginning with SmoluchowskiÕs original work in 1917 [1]. Since then the diffusionlimited chemical reactions have attracted considerable attention [2–13]. When the reaction is diffusion controlled, competition between reacting particles introduces corrections in their diffusive motion. This should be manifested in the long time dependence of the reactive concentrations. The fact that in bimolecular reactions in d dimension, the concentrations approach equilibrium not exponentially but rather as td=2 was first discovered by using simple physical arguments involving spatial concentration fluctuations [3]. The basic idea is that, since the initial and equilibrium spatial fluctuations are different, a redistribution of particles through
*
Corresponding author. Fax: +91-33-2335-3477. E-mail address: [email protected] (G. Gangopadhyay).
diffusion must occur before equilibrium can be reached. Since the relaxation due to reaction proceeds exponentially, the final approach to equilibrium is determined by the diffusion of particles. This slow asymptotic decay was subsequently obtained by using many theoretical approaches [4–7], through experiment [8] and simulations [9–11]. These studies have established that the ultimate approach of the reaction to equilibrium is a power law. Recently the diffusion influenced reversible reaction A þ B () C has been studied by Gopich et al. [12]. They have presented a simpler unified derivation of the long time asymptotics of the reversible reaction in the frame work of fluctuation theory. Their procedure can be readily used to treat more complex bimolecular reaction. The purpose of this work is to present a complete solution of the asymptotics [12] of A þ A () A2 for uniform random initial concentrations. As most of the reactions proceed through some intermediate steps, we have also considered a consecutive two step reaction [2] A þ A ()
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(02)01986-3
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S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
A2 () C where one of the rate equations is linear. If we ignore the first step of the reaction, the relaxation to equilibrium is exponential. So the question is whether all the reactants should relax to equilibrium with a power law or not. Similarly we have also analyzed the case of a two step parallel reaction. We have started with a set of reaction-diffusion equation in one dimension that describes concentration fluctuations at equilibrium. These are obtained by adding diffusion terms and random forces to the conventional nonlinear rate equation of chemical kinetics [12,13]. To obtain the relaxation behaviour of the concentration to its equilibrium value, we have exploited the Onsager Regression Hypothesis [14]. Normally one linearizes the reaction diffusion equation about equilibrium [14–16] to obtain the relaxation matrix but in this approach quadratic fluctuations of the concentrations from their equilibrium values are considered as a small perturbation. In one dimension this nonlinearity gives rise to the t1=2 asymptotic relaxation of the concentrations to equilibrium.
have zero mean and are statistically uncorrelated with the initial concentrations. The equilibrium concentrations Ae and ðA2 Þe obey the mass action law Ke A2e ¼ ðA2 Þe , where Ke ¼ k1 =k1 is the equilibrium constant. In absence of diffusion and noise, the reaction parts of Eqs. (1) and (2) satisfy
2. One step reaction
where v is a column vector with components (2,)1) and fðq; tÞ is a column vector with the random forces ðfA ; fA2 Þ in q space as components. The matrix KðqÞ contains both reaction and diffusion terms as 0 4k1 Ae 2k1 2 DA KðqÞ ¼ þq : ð5Þ 0 DA2 2k1 Ae k1
Let us consider a dimerisation reaction, A þ A (kk11 ) A2 , where Aðr; tÞ and A2 ðr; tÞ be the local random concentrations of A and A2 particles. The particles are weakly interacting and the concentration fluctuations due to diffusion in one dimension in the presence of chemical reaction are described by the following system of approximate equations, oAðr; tÞ ¼ DA r2 A 2k1 A2 þ 2k1 A2 þ fA ðr; tÞ; ð1Þ ot oA2 ðr; tÞ ¼ DA2 r2 A2 þ k1 A2 k1 A2 þ fA2 ðr; tÞ: ot ð2Þ The first term on the right-hand side of Eqs. (1) and (2) describes the diffusion of A and A2 particles with diffusion coefficients DA and DA2 . The subsequent terms describe the chemical reaction with rate constants k1 and k1 in the forward and backward direction, respectively. Finally, the last terms fi , represent the noise in the system. For the present purposes, we need only to assume that they
o ½Aðr; tÞ þ 2A2 ðr; tÞ ¼ 0; ot i.e., a linear combination of concentrations, namely, ðA þ 2A2 Þ is constant in time. Let the one-dimensional Fourier transform of the concentration fluctuation from equilibrium, cðq; tÞ be a column vector with components Z c1 ¼ dr eiqr ½Aðr; tÞ Ae ; Z ð3Þ iqr c2 ¼ dr e ½A2 ðr; tÞ ðA2 Þe : Eqs. (1) and (2) can be rewritten in one-dimension as Z ocðq; tÞ dq0 ¼ KðqÞc k1 v c1 ðq q0 ; tÞ ot 2p c1 ðq0 ; tÞ þ fðq; tÞ; ð4Þ
In what follows, we use the formalism which is based on the analysis of the two time correlation functions of the equilibrium fluctuations of the concentrations as formulated in Gopich et al. [12] by taking quadratic nonlinearity as a small perturbation. We consider the ð2 2Þ matrix of correlation function of concentration fluctuation with elements hcj ðq; tÞcyl ðq0 ; 0Þi, hcðq;tÞcy ðq0 ;0Þi ¼ eKðqÞt hcðq;0Þcy ðq0 ;0Þi k1 eKðqÞt Z ih i dq00 Dh Kðqq00 Þt 00 v cðq q00 ;0Þ eKðq Þt cðq00 ;0Þ e 1 1 2p E y 0 c ðq ;0Þ : ð6Þ
S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
Here h i denotes the ensemble average over the noise and equilibrium (Poissonian) initial conditions, and ‘‘*’’ R t denotes the time convolution f ðtÞ gðtÞ ¼ 0 f ðt t0 Þgðt0 Þ dt0 . Note that the random forces are averaged out and do not contribute here. The properties of the correlation function for a uniform (Poisson) equilibrium distributions are hci ðq1 ; 0Þcyj ðq2 ; 0Þi ¼ dij ð2pÞdðq1 q2 ÞCie ; hci ðq1 ; 0Þcj ðq2 ; 0Þcyl ðq3 ; 0Þi ¼ dij dil ð2pÞdðq1 þ q2 q3 ÞCie ;
645
DA k1 þ 4Ae k1 DA2 ; k1 þ 4Ae k1
DA2 k1 þ 4Ae k1 DA D2 ¼ k1 þ 4Ae k1
D1 ¼
and k0 ¼ Tr½Kð0Þ . From Eq. (9) we can write, Rð0; tÞ ¼ eKð0Þt Z t 2u1 ð0; sÞ 2u2 ð0; sÞ ds; k1 eKð0ÞðtsÞ u1 ð0; sÞ u2 ð0; sÞ 0
ð7Þ
ð12Þ
where Cie represents the equilibrium concentration of the ith species. Using Eq. (7), Eq. (6) reduces to
where uj ð0; sÞ is defined in Eq. (10). Using the spectral decomposition of exp½KðqÞt in Eq. (11) and evaluating the integrals in Eq. (10), we find that as t ! 1, " # 2 1 k1 k1 R11 ¼ pffiffiffiffiffiffiffiffiffiffiffi ð13Þ t1=2 ; 8pD1 ð4Ae k1 þ k1 Þ3
hcðq; tÞcy ðq0 ; 0Þi ¼ Rðq; tÞhcðq; 0Þcy ðq0 ; 0Þi;
ð8Þ
"
where the Relaxation matrix is given by Rðq; tÞ ¼ eKðqÞt k1 eKðqÞt v uT ðq; tÞ:
ð9Þ
R22 ¼
# 2 1 2k1 k1 pffiffiffiffiffiffiffiffiffiffiffi t1=2 : 8pD1 ð4Ae k1 þ k1 Þ3
ð14Þ
The row vector uT ðq; tÞ has the components Z dq0 Kðqq0 Þt Kðq0 Þt
e e : ð10Þ uj ¼ 1j 1j 2p
According to the regression hypothesis, coarse grained deviation of concentrations from equilibrium values designated by cðq; tÞ, with components ci ðq; tÞ obey
Now, to find out the asymptotic behaviour of the Relaxation matrix, it is convenient to represent the exponentials in Rðq; tÞ as a sum over eigenvalues ki ðqÞ of matrix KðqÞ as
ci ðq; tÞ hci ðq; tÞci ðq; 0Þi ¼ : 2 ci ðq; 0Þ hci ðq; 0Þ i
exp½KðqÞt ¼ P expðKd tÞP ¼
2 X
Xi exp½ki ðqÞt ;
From Eq. (8) one can write Rðq; tÞ ¼
1
ð11Þ
i¼1
where P is a matrix of order ð2 2Þ whose columns are the right eigenvectors of the KðqÞ matrix. Here Kd is the diagonalized matrix of KðqÞ matrix i.e., Kd ¼ P1 KðqÞP. Note that X1 ¼ Y1 P1 and X2 ¼ Y2 P1 , where Y1 and Y2 are ð2 2Þ matrices. In Y1 only the first column and in Y2 only the second column are nonzero, and these columns are the eigenvectors of KðqÞ matrix with eigenvalues k1 and k2 , respectively. In the q ! 0 limit, the eigenvalues of KðqÞ are k1 ¼ q2 D1 and k2 ¼ k0 þ q2 D2 , where
ð15Þ
hcðq; tÞcy ðq0 ; 0Þi : hcðq; 0Þcy ðq0 ; 0Þi
ð16Þ
Comparing Eqs. (15) and (16) and using Eqs. (13) and (14) one obtains, " # 2 1 k1 k1 c1 ðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi t1=2 c1 ð0Þ; 8pD1 ð4Ae k1 þ k1 Þ3 ð17Þ " c2 ðtÞ ¼
#
2 1 2k1 k1 pffiffiffiffiffiffiffiffiffiffiffi t1=2 c2 ð0Þ: 8pD1 ð4Ae k1 þ k1 Þ3
ð18Þ
Therefore, we observe that the asymptotics consist of one term which is a consequence of one zero eigenvalue of the stability matrix K(0). This asymptotic power law relaxation was previously
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obtained by Rey and Cardy [17] by a completely different argument and is also supported by [12]. If DA ¼ DA2 , Eqs. (17) and (18) coincides with the result obtained by Rey and Cardy [17] exactly. Although the exponent of the power law is universal, as it comes from some conservation law, the amplitude of the power law is model-dependent [12,17]. Amplitudes depend also on the initial condition of the concentrations.
3. Consecutive reaction Here we consider a two step consecutive reaction A þ A (kk11 ) A2 (kk22 ) C. Again our starting point is the following approximate reaction diffusion equations in one dimension, oAðr; tÞ ¼ DA r2 A 2k1 A2 þ 2k1 A2 þ fA ðr; tÞ; ot ð19Þ oA2 ðr; tÞ ¼ DA2 r2 A2 þ k1 A2 þ k2 C ot ðk1 þ k2 ÞA2 þ fA2 ðr; tÞ; oCðr; tÞ ¼ DC r2 C þ k2 A2 k2 C þ fC ðr; tÞ; ot
ð20Þ ð21Þ
ðA2 Þe k1 ¼ 2 Ae k1
dq0 c1 ðq q0 ; tÞc1 ðq0 ; tÞ 2p ð23Þ
where v is a column vector with components (2,)1,0) and fðq; tÞ is a vector with the random forces ðfA ; fA2 ; fC Þ in q space as components. The matrix KðqÞ contains both reaction and diffusion terms as 0 1 4k1 Ae 2k1 0 KðqÞ ¼ @ 2k1 Ae ðk1 þ k2 Þ k2 A 0 k2 k2 0 1 DA 0 0 þ q2 @ 0 DA2 0 A: ð24Þ 0 0 DC As in previous section, we have to consider a ð3 3Þ matrix of correlation functions of concentration fluctuations with elements hcj ðq; tÞcyl ðq0 ; 0Þi. The form of the Relaxation matrix is Z t Rð0; tÞ ¼ eKð0Þt k1 dseKð0ÞðtsÞ 0 0 1 2u1 ð0; sÞ 2u2 ð0; sÞ 2u3 ð0; sÞ B C @ u1 ð0; sÞ u2 ð0; sÞ u3 ð0; sÞ A; 0 0 0
where uj ð0; sÞ is given by Z dq0 KðqÞs Kðq0 Þs
e uj ¼ e : 1j 1j 2p
ð26Þ
Now, we use the spectral decomposition of the exponential described by
and Keð2Þ
Z
ð25Þ
where Aðr; tÞ; A2 ðr; tÞ and Cðr; tÞ be the local random concentrations of A; A2 and C particles. The equilibrium concentrations Ae ; ðA2 Þe and Ce obey the mass action law as Keð1Þ ¼
ocðq; tÞ ¼ KðqÞc k1 v ot þ fðq; tÞ;
exp½KðqÞt ¼ P expðKd tÞP1 Ce k2 ¼ ¼ ; ðA2 Þe k2 Keð1Þ ; Keð2Þ
ð22Þ
are the equilibrium constants. In where absence of diffusion and noise, for this reaction scheme one can find that a linear combination of concentrations is time invariant i.e., A þ 2A2 þ 2C ¼ constant. From Eqs. (19)–(21), one can write the quantities of microscopic fluctuation of concentrations from equilibrium in one dimensional Fourier space as
¼
3 X
Xi exp½ki ðqÞt ;
ð27Þ
i¼1
where P is a matrix of order ð3 3Þ whose columns are eigenvectors of the KðqÞ matrix. Kd is the diagonalized matrix of KðqÞ matrix i.e., Kd ¼ P1 KðqÞP, with X1 ¼ Y1 P1 ; X2 ¼ Y2 P1 and X3 ¼ Y3 P1 , where Y1 ; Y2 and Y3 are ð3 3Þ matrices. In Y1 only the first column, in Y2 only the second column and in Y3 only the third column are nonzero, and these columns are the eigenvectors of
S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
KðqÞ matrix with eigenvalues k1 ; k2 and k3 , respectively. In the q ! 0 limit, the eigenvalues of KðqÞ are k1 ¼ q2 D1 ; k2 ¼ q1 þ q2 D2 and k3 ¼ q2 þ q2 D3 , where the effective diffusion coefficients are 1 X D1 ¼ ka ðDb þ Dc Þ 2k0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ka2 ðDb Dc Þ þ 2ka kb ðDc Da ÞðDc Db Þ ; ð28Þ D2 ¼ D3 ¼
h
1 q ðD1 q1 q 2 2
X
h
X
1 q ðD1 q2 q 1 1
Da Þ þ Da Þ þ
X
i ka D a ;
X
i ka D a :
ð29Þ
" R22 ¼ " R33 ¼
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
Here D1 can be written explicitly as, 1 D1 ¼ 4Ae k1 ðDA2 þ DC Þ þ ðk1 þ k2 ÞðDC þ DA Þ 2k0 2 2 þ k2 ðDA þ DA2 Þ þ ð4Ae k1 Þ ðDA2 DC Þ 2 þ ðk1 þ k2 Þ2 ðDC DA Þ2 þ k2 ðDA DA2 Þ2
þ 2ð4Ae k1 Þðk1 þ k2 ÞðDC DA ÞðDC DA2 Þ þ 2ðk1 þ k2 Þðk2 ÞðDA DA2 ÞðDA DC Þ 1=2 þ 2ðk2 Þð4Ae k1 ÞðDA2 DA ÞðDA2 DC Þ ; ð31Þ where k0 is the sum of the diagonal elements of the matrix K(0). ka , Da are defined by the diagonal elements of KðqÞ in the reaction part and in the diffusion part, respectively. The summations are taken over all permutations of a, b and c. Here q1 and q2 are the nonzero eigenvalues of the matrix K(0) and are given by q1;2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Tr½Kð0Þ ðTr½Kð0Þ Þ2 4f4Ae k1 k2 þ ð4Ae k1 þ k1 Þk2 g 2
ð32Þ
and the third eigenvalue of the K(0) matrix is zero. Therefore, from Eq. (25) in the limit t ! 1 " # k1 ðk2 þ k2 Þ ðk1 k2 Þ2 1=2 pffiffiffiffiffiffiffiffiffiffiffi ; ð33Þ R11 ¼ t 3 8pD1 ðq1 q2 Þ
# t1=2 ;
ð34Þ
t1=2 :
ð35Þ
#
So the macroscopic deviation of the concentrations from equilibrium assume the power law asymptotics as " # 2 k1 ðk2 þ k2 Þ ðk1 k2 Þ 1=2 pffiffiffiffiffiffiffiffiffiffiffi c1 ðtÞ ¼ c1 ð0Þ; ð36Þ t 8pD1 ðq1 q2 Þ3 "
ð30Þ
647
c2 ðtÞ ¼ " c3 ðtÞ ¼
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
2k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
# t1=2 c2 ð0Þ;
ð37Þ
t1=2 c3 ð0Þ:
ð38Þ
#
Here also we observe that the asymptotics consist of one term which is a consequence of one zero eigenvalue of the stability matrix K(0), where, q1 q2 ¼ ½4Ae k1 k2 þ ð4Ae k1 þ k1 Þk2 and D1 is given by Eq. (31). Let us consider some special cases, Case I. For k2 ¼ 0 with k1 6¼ 0, i.e., the second step is irreversible but the first step is reversible, one can find q1 q2 ¼ 4Ae k1 k2 and
4Ae k1 DA2 þ ðk2 þ k1 ÞDA D1 ¼ : 4Ae k1 þ k2 þ k1 In this case each numerator in Eqs. (36)–(38) becomes zero, i.e., there is no power law of t1=2 asymptotic decay. Case II. Similarly, for the case when k1 ¼ 0 but k2 6¼ 0, i.e., when the first step is irreversible but the second step is reversible, one obtains q1 q2 ¼ ½4Ae k1 ðk2 þ k2 Þ and
4Ae k1 DA2 þ k2 DC þ k2 DA D1 ¼ : 4Ae k1 þ k2 þ k2 In this case also the numerators in Eqs. (36)–(38) become zero, so there will be no such power law relaxation.
648
S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
Case III. If the whole reaction scheme is irreversible, i.e., if say k1 ¼ 0 and k2 ¼ 0 then also all the numerators are zero but the denominators are nonzero and hence there will be no power law relaxation as in Eqs. (36)–(38). So only for the reversible cases one obtains the asymptotic relaxation of t1=2 power law. If one of the step is irreversible no such power law relaxation exists. Note that in the total calculation we have considered the eigenvalues of the stability matrix KðqÞ in the limit q ! 0, and in the eigenvalues higher powers of q are neglected, because we are interested only in the slowest decay. We have not considered the other relatively faster relaxation mechanism, so from this analysis it is not possible to say what kind of relaxation will happen for the irreversible case which may be different from the exponential relaxation. Also in this analysis we have assumed an equilibrium Poissonian distribution which is strictly valid for reversible case. In other words, if the amplitude of relaxation is zero, it does not necessarily mean that the relaxation is exponential. However, to obtain the power law relaxation the amplitude has to be nonzero which is possible only when all the steps of the reactions are reversible. It can be shown that in absence of the last step of the reaction, i.e., in the limit k2 ; k2 ! 0, the results in Eqs. (36)–(38) reduce exactly to that of the one step case. When the last step is absent i.e., in the limit k2 ; k2 ! 0 one obtains from Eq. (31),
4Ae k1 DA2 þ k1 DA D1 ¼ : 4Ae k1 þ k1 For k2 ¼ 0, one can have, ðq1 q2 Þ3 ¼ ð4Ae k1 þ 3 3 k1 Þ k2 and therefore the amplitudes relax as c1 ðtÞ ¼ ¼
c2 ðtÞ ¼ ¼
2 3 k1 k1 k2
pffiffiffiffiffiffiffiffiffiffiffiffi c1 ð0Þ 3 8pD1 t ð4Ae k1 þ k1 Þ3 k2 2 k1 k1
pffiffiffiffiffiffiffiffiffiffiffiffi c1 ð0Þ; ð4Ae k1 þ k1 Þ3 8pD1 t
2 3 k1 k1 k2 2 c ð0Þ 3 3 pffiffiffiffiffiffiffiffiffiffiffiffi 2 ð4Ae k1 þ k1 Þ k2 8pD1 t 2 2k1 k1 pffiffiffiffiffiffiffiffiffiffiffiffi c2 ð0Þ ð4Ae k1 þ k1 Þ3 8pD1 t
ð39Þ
and c3 ðtÞ ¼ 0: Therefore, Eqs. (39) and (40) exactly correspond to Eqs. (17) and (18), respectively.
4. Parallel reaction Next we consider a two-step parallel reaction, A þ A (kk11 ) A2 ; A (kk22 ) C. The approximate reaction diffusion equations in one dimension are given by oAðr; tÞ ¼ DA r2 A 2k1 A2 þ 2k1 A2 k2 A ot þ k2 C þ fA ðr; tÞ; oA2 ðr; tÞ ¼ DA2 r2 A2 þ k1 A2 k1 A2 þ fA2 ðr; tÞ; ot oCðr; tÞ ¼ DC r2 C þ k2 A k2 C þ fC ðr; tÞ: ot ð41Þ The corresponding differential equation for cðq; tÞ vector can be written as in Eq. (23), where the stability matrix is, 0 1 4k1 Ae þ k2 2k1 k2 KðqÞ ¼ @ 2k1 Ae k1 0 A k2 0 k2 0 1 DA 0 0 þ q2 @ 0 DA2 0 A: ð42Þ 0 0 DC Here also one linear combination of concentrations, ðA þ 2A2 þ CÞ remains constant in time when there is no diffusion and noise. In absence of diffusion, KðqÞ matrix has one eigenvalue which is zero and the other two nonzero eigenvalues are q1;2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Tr½Kð0Þ ðTr½Kð0Þ Þ 4fð4Ae k1 þ k1 Þk2 þ k2 k1 g : 2
ð43Þ
ð40Þ
Following the prescription in Section 3, here one can identify the column matrix v has the components (2,)1,0). Therefore, the macroscopic deviation of the reactants assume the power law asymptotics as,
S. Paul, G. Gangopadhyay / Chemical Physics Letters 369 (2003) 643–649
" c1 ðtÞ ¼ " c2 ðtÞ ¼ " c3 ðtÞ ¼
k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
# t1=2 c1 ð0Þ;
2k1 ðk2 þ k2 Þ ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3 k1 k2 ðk1 k2 Þ pffiffiffiffiffiffiffiffiffiffiffi 8pD1 ðq1 q2 Þ3
2
2
ð44Þ
# t1=2 c2 ð0Þ;
ð45Þ
# t1=2 c3 ð0Þ;
ð46Þ
where q1 q2 ¼ ð4Ae k1 þ k1 Þk2 þ k2 k1
ð47Þ
and D1 ¼
1 ð4Ae k1 þ k2 ÞðDA2 þ DC Þ þ k1 ðDC þ DA Þ 2k0 n þ k2 ðDA þ DA2 Þ þ ð4Ae k1 þ k2 Þ2 ðDA2 DC Þ2 2
þ ðk1 Þ ðDC DA Þ
2
We are grateful to the unknown referee for critical remarks.
þ 2ð4Ae k1 þ k2 Þðk1 ÞðDC DA ÞðDC DA2 Þ
þ 2ðk2 Þð4Ae k1 þ k2 ÞðDA2 DA ÞðDA2 DC Þ
which is nonlinear. In absence of diffusion when atleast one linear combination of concentrations is time invariant, it gives a zero eigenvalue of the stability matrix. This zero eigenvalue corresponds to a slow dynamical mode in the system which when couples to nonlinearity, gives rise to the power law relaxation as observed in [12]. It is difficult to conclude by this approach about the relaxation mechanism when one of the steps is irreversible. Hence a systematic approach using fluctuation theory to deal with the irreversible cases [18,19] are in order which will be discussed elsewhere. This method is working well for simple reversible reactions and is straight forward to extend in three dimensions.
Acknowledgements
2 þ k2 ðDA DA2 Þ2
þ 2ðk1 Þðk2 ÞðDA DA2 ÞðDA DC Þ
649
o1=2
:
References ð48Þ
Here k0 is the sum of the diagonal elements of K(0) matrix. Similarly, by the straight forward arguments as given in Section 3, one can show if any one of the step is irreversible or if the whole reaction scheme is irreversible, all the amplitudes of tð1=2Þ in Eqs. (44)–(46) become zero. So reversibility in every steps of the reaction is necessary to obtain the power law relaxation as given in Eqs. (44)–(46). Similarly, if we drop the second step of this reaction scheme, i.e., k2 ; k2 ! 0, arguments provided at the end of previous section holds and Eqs. (44)–(46) reduce exactly to the one step reaction, A þ A () A2 , for arbitrary diffusion coefficients of the reactants.
5. Conclusion From these reaction schemes we observe that an asymptotic power law relaxation kinetics occurs when all the steps of a reaction are reversible and there should be atleast one rate equation
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