Analysis of the methods for the derivation of binary kinetic equations in the theory of fluorescence concentration quenching.

Analysis of the methods for the derivation of binary kinetic equations in the theory of fluorescence concentration quenching A. B. Doktorov Citation: The Journal of Chemical Physics 141, 104104 (2014); doi: 10.1063/1.4894285 View online: http://dx.doi.org/10.1063/1.4894285 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Quenching of highly vibrationally excited pyrimidine by collisions with C O 2 J. Chem. Phys. 128, 054304 (2008); 10.1063/1.2825599 Rate kernel theory for pseudo-first-order kinetics of diffusion-influenced reactions and application to fluorescence quenching kinetics J. Chem. Phys. 126, 214503 (2007); 10.1063/1.2737045 Theoretical analysis and computer simulation of fluorescence lifetime measurements. I. Kinetic regimes and experimental time scales J. Chem. Phys. 121, 562 (2004); 10.1063/1.1756577 Rotationally resolved quenching and relaxation of CH (A 2 Δ,v=0,N) in the presence of CO J. Chem. Phys. 116, 2757 (2002); 10.1063/1.1436110 Kinetic theory of bimolecular reactions in liquid. I. Steady-state fluorescence quenching kinetics J. Chem. Phys. 108, 117 (1998); 10.1063/1.475368

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THE JOURNAL OF CHEMICAL PHYSICS 141, 104104 (2014)

Analysis of the methods for the derivation of binary kinetic equations in the theory of fluorescence concentration quenching A. B. Doktorov Voevodsky Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia and Physics Department Novosibirsk State University, Novosibirsk 630090, Russia

(Received 9 April 2014; accepted 14 August 2014; published online 10 September 2014) In the framework of unified many-particle approach the familiar problem of fluorescence concentration quenching in the presence of pumping (light pulse) of arbitrary intensity is considered. This process is a vivid and the simplest example of multistage bulk reaction including bimolecular irreversible quenching reaction and reversible monomolecular transformation as elementary stages. General relation between the kinetics of multistage bulk reaction and that of the elementary stage of quenching has been established. This allows one to derive general kinetic equations (of two types) for the multistage reaction in question on the basis of general kinetic equations (differential and integro-differential) of elementary stage of quenching. Relying on the same unified many-particle approach we have developed binary approximations with the use of two (frequently employed in the literature) many-particle methods (such as simple superposition approximation and the method of extracting pair channels in three-particle correlation evolution) to the derivation of non-Markovian binary kinetic equations. The possibility of reducing the obtained binary equations to the Markovian equations of formal chemical kinetics has been considered. As an example the exact solution of the problem (for the specific case) is examined, and the applicability of two many particle methods of derivation of binary equations is analyzed. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894285]

is studied2, 3

I. INTRODUCTION

Fluorescence concentration quenching of excited particles A∗ well studied in the literature both theoretically and experimentally1 is a classic example of bulk diffusion influenced irreversible pseudo-monomolecular reaction proceeding at constant concentration [B] of the quencher B by the scheme A∗ + B → A + B.

(1.1)

Fluorescence of excited particle A∗ results from spontaneous deactivation into the ground state (of the particle A) with the mean time τ A , and particles A go into excited state under the action of pumping at time dependent rate J(t) (in the most general statement of the problem) 1/τA

A∗ −−−−−→A,

J (t)

A−−−−−→A∗ .

(1.2)

Two methods for experimental investigation of fluorescence quenching are known. The first method consists in powerful pulse pumping giving rise of fluorescent exited particles which are deactivated both due to spontaneous radiation, and as a result of radiationless decay caused by the interaction with the quencher molecules. Most commonly, concentration dependence of relative quantum yield of fluorescence 0021-9606/2014/141(10)/104104/22/$30.00

∞ η= 0

dt t S (t) exp − τA τA

⎞ ⎛ 1 1 1 ⎠ (1.3) ⎝or S L ≡ ≡ 1 1 + kq τA [B] τA + k [B] q τA to find the quenching constant kq .1, 4, 5 Here S (t) =

[A∗ ]t [A∗ ]0

(S (0) = 1)

(1.4)

is reaction quenching kinetics (1.1) (without regard to spontaneous decay and pumping), [A∗ ]t is concentration of excited particles A∗ at the instant of time t, SL (s) is the Laplace transform of the kinetics. Here and below the Laplace transform is marked by the upper index L. In modern experiments the process kinetics itself can be observed. The second method of experimental examination is to use constant pumping J(t) = J = const. In essence, in this case the stationary part of excited particles is studied6–8

141, 104104-1

Pst∗ = lim P ∗ (t) ≡ t→∞

1 τA

J , + J + kQ [B]

(1.5)

© 2014 AIP Publishing LLC


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where kQ is the quenching constant (with allowance for pumping), or the kinetics P ∗ (t) =

[A∗ ]t [A∗ ]t = , [A]0 [A∗ ]0 + [A]0

(1.6)

i.e., time variation of the part of excited particles where [A]0 = [A∗ ]0 + [A]0 is the total initial concentration of excited and unexcited particles A. The part of unexcited particles is [A]t [A]t = ∗ . (1.7) P (t) = 0 [A] [A ]0 + [A]0 Obviously (see below), P ∗ (t) + P (t) = 1.

(1.8)

Besides at J (t) = J ≡ τ ∗ = const processes (1.2) may A be considered as reversible reaction of monomolecular transformation, and quenching reaction as a whole—as a model of multistage reaction with stages (elementary reactions) of bulk irreversible bimolecular reaction (1.1) (for example, catalytic transformation) and reversible monomolecular transformation (1.2) (for instance, cis-trans isomerization). Diffusion influenced multistage reactions play an important role in a wide field of research and technologies, so the development of the theory of such reactions is growing in importance.9–12 From this point of view, consistent theoretical treatment of the problem based on adapting rigorous many-particle methods of non-equilibrium statistical mechanics to chemical reacting systems in liquid solutions13–15 is of interest. In particular, this is related to the fact that in some cases simple superposition approximation widely used in the derivation of binary kinetic equations of bulk reactions (for example, in treating elementary reversible reactions) proves to be inapplicable (see review article15 ). Though the employment of another approach, namely, the method of extracting pair channels in three-particle correlation evolution eliminates problems in the theory of elementary reversible reactions, substantial difficulties remain in the presence of additional elementary stage, we mean monomolecular transformation.16 Such a treatment is the main goal of the present work where general relation between the kinetics of multistage reaction considered and that of elementary stage of quenching (1.1) is established in the frame of unified many-particle approach. This makes it possible to obtain general kinetic equations (of two forms) for the multistage reaction under discussion on the basis of general kinetic equations (differential or integrodifferential) of elementary stage of quenching. Application of two forms of kinetic equations is related to the fact that the use of the above different method of deriving binary kinetic equations gives equations of these two different types. Relying on the employed unified many-particle approach we have developed the binary approximation using the above two many-particle methods (such as simple superposition approximation7, 8, 17, 18 and the method of extracting pair channels in three-particle correlation evolution14, 15, 19 ) to the derivation of non-Markovian binary kinetic equations. The possibility of reducing the obtained binary equations to the Markovian equations of formal chemical kinetics is studied. 1

To clarify the meaning and to establish the criteria of the applicability of the approximations used (which are of general character and do not refer to any specific model), we consider, as an example, the so-called “target model” described in the literature that enables one to obtain the exact solution of the problem, develop binary approximation on its basis (by expanding the kinetics in small density parameter), and to establish the applicability criteria of the above two methods for the construction of such an approximation. Section II of the contribution is devoted to exact manyparticle consideration of the problem that allows one to relate it to the problem of the theory of irreversible reactions. Section III deals with kinetic equations of multistage reaction based on two generally accepted forms (differential and integro-differential) of kinetic equation for elementary stage of quenching. Apart from general consideration, the example of diffusion controlled contact quenching reaction is given that admits exact solution of the many-particle problem in the context of the known “target model.” In Section IV the concept of binary approximation is formulated, and the best known methods for the derivation of nonMarkovian binary kinetic equations of the multistage reaction under study are analyzed. The main results are given in Sec. V. II. EXACT MANY-PARTICLE DESCRIPTION

Traditionally (with rare exception)5, 15 in the theory of diffusion-influenced reactions by the reacting system one means a many-particle system of chemically reacting particles (reactants) dissolved in chemically inert continuous medium. In such a consideration molecular structure of a solvent is neglected, and the motion of reactants in a continuous medium is treated as a random process of the change of their coordinates resulting in the approach of reactants to distances sufficient for elementary event of chemical conversion to occur. For many-particle description of such reacting system, introduce a set of distribution functions of reactants F(K, N) (A∗K , AN , BM , t) (the so-called Fock space20 ). Each of these functions is the probability density that at the moment of time t the macroscopic volume contains K excited particles A∗ at points A∗K = {A∗1 , A∗2 , . . . A∗K }, N unexcited particles A at points AN = {A1 , A2 , . . . AN }, and M particles of the quencher B at points BM = {B1 , B2 , . . . BM } (M does not change in the course of quenching reaction). Due to spontaneous decay and excitation (pumping), the number of excited and unexcited particles A∗ and A varies with time. It is assumed that the coordinate of a new excited particle remains the same as the coordinate of the initial unexcited particle from which it was formed as a result of excitation. Correspondingly, the coordinate of a new unexcited particle formed due to spontaneous decay is also the same as that of the initial excited particle. Then the evolution of the system is described by a set of Liouville balance equations determining time variation of distribution functions (by virtue of transitions between subensembles differing in the number of excited and unexcited


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particles)

K N M ∂ − Lˆ A∗k − Lˆ A − Lˆ B F (K,N) (A∗K , AN , B M , t) n m ∂t k=1 n=1 m=1 =−

N K (K,N) ∗K 1 (K+1,N−1) ∗K F (A , AN , B M , t) + F (A An , AN /An , B M , t) τA τA n=1

−N J (t)F (K,N) (A∗K , AN , B M , t) + J (t)

K

F (K−1,N+1) (A∗K /A∗k , AN A∗k , B M , t)

k=1

−

K,M

Vk,m F (K,N) (A∗K , AN , B M , t) +

with the given initial conditions F (K,N) (A∗K , AN , B M , 0) = F0(K,N) (A∗K , AN , B M ). Here Lˆ A∗k , Lˆ A , Lˆ B are functional operators of stochasn m tic molecular motion of reactants dissolved in continuous chemically inert medium (usually diffusion motion operators), and the terms in the left-hand side of Eq. (2.1) involving these operators describe the distribution functions variation due to molecular motion. The mean time τ A defines the rate of spontaneous transition of an excited particle to unexcited one, and the pumping intensity J(t) the rate of transitions from the ground state to the excited state. Quantities Vk,m = V (Bm − A∗k ) and Vn,m = V (Bm − An ) are elementary rates of excitation quenching by particles B causing the particle to go into unexcited state. The corresponding terms in the righthand side of Eq. (2.1) describe going out of the sub-ensemble and coming into it from another sub-ensemble owing to these processes. Designations AN /An , A∗K /A∗k , AN A∗k , and A∗K An correspond to sets of coordinates {A1 , A2 , . . . An − 1 , An + 1 , . . . AN }, {A∗1 , A∗2 , . . . A∗k−1 , A∗k+1 , . . . A∗K }, {A1 , A2 , . . . AN , A∗k }, and {A∗1 , A∗2 , . . . A∗K , An }, respectively. Now introduce the Reduced Distribution Functions (RDFs) in the thermodynamic limit (K, N → ∞, → ∞)21 q

r

ϕp,q,r (A , A , B , t) ∞ dA∗p+1 . . . dA∗K dAq+1 . . . dAN = T − lim ... (K − p)! (N − q)! K,N ×

dBr+1 . . . dAM (K,N) ∗K (A , AN , B M , t). F (M − r)!

Vn,m F (K+1,N−1) (A∗K An , AN /An , B M , t)

(2.1)

n,m

k,m

∗p

N,M

(2.2)

Using this definition, one can derive infinite hierarchies for RDFs from Liouville equations. Equations for RDFs ϕ 1, 0, r (A∗ , Br , t) and ϕ 0, 1, r (A, Br , t) are of interest for further discussion. To obtain the closed set of equations, the coordinate A∗ of excited particle in the first function will be put equal to the coordinate A of unexcited particle (as in the second function). Besides, we consider the motion of excited particle the same as that of unexcited one, i.e., Lˆ A∗ = Lˆ A . Using designations ϕA∗ ,r (A, B r , t) and ϕ A, r (A, Br , t) for these RDFs,

we have

r ∂ ˆ ˆ LB ϕA∗ ,r (A, B r , t) − LA − m ∂t m=1 1 ϕ ∗ (A, B r , t) + J (t)ϕA,r (A, B r , t) τA A ,r r r − Vm ϕA∗ ,r (A, B , t)− dBr+1 Vr+1 ϕA∗ ,r+1 (A, B r+1 , t),

=−

m=1

r

(2.3)

∂ Lˆ B ϕA,r (A, B r , t) − Lˆ A − m ∂t m=1 =

1 ϕ ∗ (A, B r , t) − J (t)ϕA,r (A, B r , t) τA A ,r r + Vm ϕA∗ ,r (A, B r , t)+ dBr+1 Vr+1 ϕA∗ ,r+1 (A, B r+1 , t), m=1

where the rates of elementary reaction between excited particle at the point A and the mth particle of the quencher at the point Bm are introduced Vm = V (Bm − A). Quantities ϕA∗ ,r (A, B r , t) and ϕ A, r (A, Br , t) are independent of distribution functions of more than one particle A∗ and A. Thus in the system under study many-particle effects of particles A∗ and A do not manifest themselves, and the problem reduces to the examination of the evolution of one particle A∗ and one particle A in the ensemble of particles B. For simplicity, in formulating the initial conditions we shall consider spatially homogeneous systems, though the homogeneity of distribution of particles A∗ and A∗ is not of crucial importance, since it makes no influence on the result for variation of quantities averaged over the volume (see below). Then neglecting, as usual, initial correlations in distributions of reactants, we obtain ϕA∗ ,r (A, B r , 0) = [A∗ ]0 [B]r ,

ϕA,r (A, B r , 0) = [A]0 [B]r . (2.4)


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Now introduce the distribution function ϕr (A, B r , t) = ϕA∗ ,r (A, B r , t) + ϕA,r (A, B r , t)

(2.5)

with the initial conditions

relation between the corresponding RDFs ⎛ ⎞ t t ϕA∗ ,r (A, B r , t) = P ∗ (0) exp ⎝− − J (z) dz⎠ τA 0

ϕr (A, B r , 0) = [A∗ ]0 [B]r + [A]0 [B]r ≡ [A]0 [B]r , (2.6)

× φA∗ ,r (A, B , t) ⎛ ⎞ t t t − τ + J (τ ) exp ⎝− − J (z) dz⎠ τA r

where [A]0 = [A∗ ]0 + [A]0 , just as in Eq. (1.6), is the total initial concentration of excited and unexcited particles A that is obviously time independent. Indeed, the sum of two equations of (2.3) yields the equation for the function ϕ r (A, Br , t)

(2.7)

which, in view of initial conditions (2.6), has the solution ϕr (A, B r , t) = ϕr (A, B r , 0) = [A]0 [B]r = const.

(2.8)

This makes it possible to establish the relation between distribution functions ϕA,r (A, B r , t) = [A]0 [B]r − ϕA∗ ,r (A, B r , t).

(2.9)

−

(2.10)

In the absence of spontaneous decay and pumping ( τ1 = 0, J (t) = 0) the kinetics of irreversible quenching A reaction A∗ + B → A + B (reaction of A + B → B type) is determined by the distribution function φA∗ ,r (A, B r , t) that satisfies the equation following from Eq. (2.10)

r ∂ ˆ ˆ − LA − LB φA∗ ,r (A, B r , t) m ∂t m=1 −

r

Vm φA∗ ,r (A, B r , t) −

dBr+1 Vr+1 φA∗ ,r+1 (A, B r+1 , t).

m=1

is the initial part of excited particles A. III. KINETIC EQUATIONS AND THE REACTION KINETICS

Evidently, the introduced quantities P∗ (t) (1.6), P(t) (1.7), and S(t) (1.4) are related to distribution functions ϕA∗ ,0 (A, t), ϕ A, 0 (A, t), and φA∗ ,0 (A, t) as 1 dA P ∗ (t) = lim ϕA∗ ,0 (A, t) , 0 [A] υ→∞ υ υ

1 lim P (t) = [A]0 υ→∞

ϕA,0 (A, t)

dA , υ

(3.1)

r

As the initial condition for the function φA∗ ,r (A, B , t), we choose the condition (2.12)

differing from the initial condition (2.4) for the function ϕA∗ ,r (A, B r , t) in that here the initial concentration [A∗ ]0 of excited particles is replaced by total concentration [A]0 of particles A. Using Eqs. (2.10) and (2.11), we easily establish the

φA∗ ,0 (A, t)

dA . υ

υ

Then, in view of Eqs. (2.5) and (2.8) (at r = 0), we have Eq. (1.8), and Eq. (2.13) (at r = 0) immediately gives the desired relation between the kinetics P∗ (t) of the multistage reaction under study and the kinetics S(t) of elementary stage of quenching ⎛ ⎞ t t P ∗ (t) = P ∗ (0) exp ⎝− − J (z) dz⎠ S (t) τA t + 0

(2.11)

φA∗ ,r (A, B r , 0) = [A]0 [B]r

(2.14)

P ∗ (0) =

1 S (t) = lim [A]0 υ→∞

dBr+1 Vr+1 ϕA∗ ,r+1 (A, B r+1 , t).

[A∗ ]0 [A∗ ]0 = [A]0 [A∗ ]0 + [A]0

where

υ

Vm ϕA∗ ,r (A, B r , t)

m=1

−

(2.13)

A. General relations

The use of this relation in the first part of Eq. (2.3) gives the closed equation describing space-time evolution of the distribution function of excited particles

r ∂ − Lˆ A − Lˆ B ϕA∗ ,r (A, B r , t) m ∂t m=1 1 + J (t) ϕA∗ ,r (A, B r , t) + J (t) [A]0 [B]r =− τA r

× φA∗ ,r (A, B , t − τ )dτ, r

r ∂ − Lˆ A − Lˆ B ϕr (A, B r , t) = 0, m ∂t m=1

τ

0

⎛

0

t −τ J (τ ) exp ⎝− − τA

× S (t − τ ) dτ.

t

⎞ J (z) dz⎠

τ

(3.2)

In the specific case P∗ (0) = 0 similar relation has been obtained in Ref. 8 on the basis of more concise (not rigorous in the general case) derivation relying on the application of complete distribution function in particles B. However, in the present work reproduction and generalization of the result based on RDF hierarchies are of crucial importance, since exactly these hierarchies are used to derive binary kinetic equations.


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In the particular case J(t) = J = const most interesting for our investigation, Eq. (3.2) reduces to 1 P ∗ (t) = P ∗ (0) exp − + J t S (t) τA t +J 0

1 exp − + J τ S (τ ) dτ, τA

(3.3)

the kinetic equation for P∗ (t) d ∗ P (t) = −k (t) [B]P ∗ (t) + k2 (t) [B] dt 1 − P ∗ (t) + J (t) (1 − P ∗ (t)), τA

(3.9)

where t

or, for Laplace transform of the kinetics, 1 J ∗L ∗ L S s+ +J . P (s) = P (0) + s τA

k2 (t) =

J (τ ) (k (t) − k (t − τ )) 0

⎛

(3.4)

For stationary value of the quantity P∗ (t) we have from Eq. (3.3) or (3.4) 1 ∗ ∗ ∗L L +J , Pst = lim P (t) = lim s P (s) = J S t→∞ s→0 τA

t −τ × exp ⎝− − τA

t−τ J (z) dz − [B]

τ

⎞ k (z) dz⎠ dτ,

0

(3.10) or, at J(t) = J = const: t

(3.5) which, with allowance for Eq. (1.5), gives the definition of the quenching constant kQ 7, 8 quite similar to that of the quenching constant kq (see Eq. (1.3)) 1 1 L . (3.6) +J = 1 S τA + J + kQ [B] τ

t

k2 (t) = J

(k (t) − k (τ )) 0

⎛

× exp ⎝−

1 + J τ − [B] τA

τ

⎞ k (z) d z⎠ d τ.

0

(3.11)

A

Definition for the quenching constant kq follows from Eq. (3.6) at J → 0. This agrees with the known fact that the quenching constant kq defined in terms of quantum yield under pulse pumping coincides with the quenching constant kQ obtained at low stationary pumping. The derived general relation (3.2) enables one to calculate the complete kinetics P∗ (t), if the kinetics S(t) of the elementary stage is known. However, of interest are the kinetic equations based on relation (3.2). Their specific form depends on the form of kinetic equation for S(t). For the quenching reaction at hand, two types of kinetic equations are known. We shall consider them below. B. Differential rate equations

Differential rate equation of Smoluchowski type22 is the best known equation of elementary irreversible quenching reaction A∗ + B → A + B d S (t) = −k (t) [B] S (t) . dt

(3.7)

The kinetics of elementary irreversible reaction defined by this equation is ⎞ ⎛ t (3.8) S (t) = exp ⎝−[B] k (τ ) dτ ⎠ . 0

Note that in the general case the value k(t) can depend on concentration [B]. Time differentiation of relation (3.2) and the use of Eqs. (3.2), (3.7), and (3.8) in the expression obtained lead to

At the initial instant of time the rate constant k(t) is equal to the reaction constant defined by the elementary conversion rate, with time it normally decreases reaching its steady-state value from above. In this case, as is seen from Eq. (3.11), the constant k2 (t) is always negative thus increasing the decay rate of excitations as compared to the rate determined by quenching reaction (the first term in Eq. (3.9)) and spontaneous deactivation (the third term in Eq. (3.9)). In full accordance with physical meaning, the obtained differential kinetic equation (3.9) involves inhomogeneous terms determined by pumping. However, from the standpoint of the consideration of the process as two-stage chemical reaction containing irreversible bimolecular and reversible monomolecular transformations, it is a set of homogeneous differential equations obtained from Eq. (3.9) (with the aid of Eq. (1.8)) describing the transitions between excited and unexcited states of the particle A which is generally accepted d ∗ P (t) = −k1 (t) [B]P ∗ (t) + k2 (t) [B]P (t) dt 1 − P ∗ (t) + J (t) P (t) , τA d P (t) = k1 (t) [B]P ∗ (t) − k2 (t) [B]P (t) dt 1 + P ∗ (t) − J (t) P (t) , τA

(3.12)

where the kinetic coefficients k1 (t) = k (t) − k2 (t)

(k1 (t) + k2 (t) = k (t)) (3.13)


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(just as the terms involving pumping) have the meaning of rate constants. In this case Eq. (1.8) is treated as the relation of complete material balance. If the first two terms in the right-hand sides of each of equations in set (3.12) are considered as referring to elementary stage of quenching, it is seen that such a stage is affected by pumping and excitation deactivation processes. Only in the case of low pumping, when the value of the constant k2 (t) is negligibly small, such an influence is insignificant. In the Markovian limit (when transient stage is complete) Eqs. (3.12) take the form of traditional formal chemical kinetics equations that formally describe transitions between states in the presence of pumping and deactivation d ∗ P (t) = − k1 [B]P ∗ (t) + k2 [B]P (t) dt 1 − P ∗ (t) + J (t) P (t) , τA d P (t) = k1 [B]P ∗ (t) − k2 [B]P (t) dt 1 + P ∗ (t) − J (t) P (t) , τA

1 s + L (s) [B]

or

L (s) [B] =

(3.17) where SL (s) is the Laplace transform of kinetics (3.8). Thus integro-differential kinetic equation (3.16) with the kernel (3.17) is absolutely equivalent to differential kinetic equation (3.7). However, relation (3.17) of the kinetic coefficient (t) (the kernel in Eq. (3.16)) and the kinetic coefficient k(t) (the reaction rate constant in Eq. (3.7)) is via the Laplace transforms, and it is rather complicated. Performing time differentiation of relation (3.2), employing Eq. (3.16) in the expression obtained, and changing the integration order, we derive, in view of Eq. (3.2), the kinetic equation for P∗ (t) d ∗ P (t) = − dt

t

J ( t| τ ) [B] P ∗ (τ ) d τ

−

1 ∗ P (t) + J (t) 1 − P ∗ (t) , (3.18) τA

where

⎛

t −τ J ( t| τ ) = (t − τ ) exp ⎝− − τA

t→∞

(k1 + k2 = k,

k = lim k (t)).

(3.15)

t→∞

However, they have an important specific feature, namely, negativity of the coefficient k2 which, strictly speaking, cannot be treated as the reaction rate constant. However, it is readily seen that total kinetic coefficient k2 [B] + J at P(t) is positive and defines total transformation rate of reactant A into A∗ .

C. Integro-differential equations

In the most general case, the kinetics of elementary irreversible reaction A∗ + B → A + B can be described both by differential equation (3.7), and by integro-differential equation (retarded equation) d S (t) = − dt

t

This is established using the technique of projection operators.26 However, the kernel (τ ) of kinetic equation (3.16) depends essentially on concentration, and its calculation is a separate, and, in the general case, difficult problem. Nevertheless, given the quenching kinetics, the kernel of Eq. (3.16) may be obtained in the form of the Laplace transform. Application of the Laplace transformation to Eq. (3.16)

τ

1 + J (t − τ ) , = (t − τ ) exp − τA (3.20) 1 +J τ . J (τ ) = (τ ) exp − τA

As in the case of differential theory, inhomogeneous equation (3.20) can be transformed, with the aid of Eq. (1.8), into a set of homogeneous kinetic equations (accepted in the field of chemical kinetics) describing the transitions between excited and unexcited states of the particle A d ∗ P (t) = − dt

t −0

J (z) d z⎠ .

J ( t| τ ) ≡ J (t − τ )

− (3.16)

⎞

For J(t) = J = const, we have

−0

(t − τ ) [B] S (τ ) d τ .

t

(3.19)

(τ ) [B] S (t − τ ) d τ

≡−

1 − s, S L (s)

−0

k1 = lim k1 (t) , t→∞

S L (s) =

(3.14)

where

k2 = lim k2 (t)

gives

d P (t) = dt

t

t

J ( t| τ ) [B] P ∗ (τ ) d τ

−0

1 ∗ P (t) + J (t) P (t) , τA

(3.21)

J ( t| τ ) [B] P ∗ (τ ) dτ

−0

+

1 ∗ P (t) − J (t) P (t) . τA

So for integro-differential formulation (3.16) of kinetic equation for elementary stage of quenching, kinetic equations of the multistage reaction at hand (apart from the terms defining spontaneous deactivation and pumping) contain the only


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J. Chem. Phys. 141, 104104 (2014)

term that describes the quenching stage (unlike differential formulation). This term preserves the integral character of the description of elementary stage of quenching (but with a different, in accordance with Eq. (3.19), kernel), and this is an essential advantage of the formalism. At first glance the presence of the exponential term in the modified kernel (3.19) (or (3.20)) decreases the value of this kinetic coefficient as compared to its value (t − τ ). This contradicts the conclusion made for differential formulation of the problem. The reason is that the kinetic equation kernel is a particular object (defined on a class of generalized functions) to which the above conclusion regarding the decrease in kinetic coefficient cannot be applied, so it increases, as it should be. We shall analyze this fact in further consideration of exactly solvable “target model.” At J(t) = J = const for the Laplace transforms of kernels in Eqs. (3.28) and (3.32), we have 1 L L +J . (3.22) J (s) = s + τA As for the Laplace transform of the kinetics P∗ (t), in the case under discussion it can easily be expressed in terms of the Laplace transform of the kernel (t), both by general formulae (3.4), (3.17), and (3.22), and by direct Laplace transformation of kinetic equation (3.18) with allowance for Eq. (3.22) 1 J . P ∗L (s) = P ∗ (0) + s s + 1 + J + L s + 1 + J [B] τ τ A

dius a with diffusing (diffusion coefficient D) point particles B. Quenching reaction is considered instantaneous when the particle B hits the surface of the particle A∗ (“black” sphere model). In this case18, 22

a2 3ξ k k (t) = 4π aD 1 + ≡k+ . (3.26) π Dt π [B] t For convenience, we introduce the following quantities: k = kD = 4π aD,

The Laplace transform of the quenching kinetics (3.28) S L (s) = S0L (s + k[B]) ,

S0L (s) =

s→0

1 τA

1 1− s

+ J + L

1 τA

(3.29)

3ξ k[B] 3ξ k[B] 3ξ k[B] exp erfc , s πs πs

where 2 erfc(x) = 1 − erf (x) = √ π

Stationary part of excited particles is as follows:

(3.27)

is

A

J

4 3 π a [B]. 3

The first quantity is the steady-state reaction rate constant (for diffusion controlled reaction under study—diffusion constant) which we shall call the Markovian constant.14 The second quantity is the density parameter of reactants. So we have for kinetics (3.8) 2 S (t) = exp −k [B] t − √ 3 ξ k[B] t . (3.28) π

(3.23)

Pst∗ = lim s P ∗L (s) =

ξ=

∞ exp(−t 2 )dt x

+ J [B]

.

(3.24) So, according to Eq. (1.5), the expression for the quenching constant is 1 kQ = L +J . (3.25) τA

is a complementary error function of complex variable.25 Using Eq. (3.28) in Eq. (3.3) yields the kinetics J 2 3 ξ k[B] t P ∗ (t) = P ∗ (0) − 2 exp −R 2 t − √ R π J (1 − F (t)) , R2 where R = τ1 + J + k[B], and +

(3.30)

A

√

F (t) = D. Diffusion controlled contact quenching reaction

1. Differential rate equations

The results obtained in Sec. III are absolutely general and do not refer to any particular model of a chemical system. As a concrete example, in this section we consider a specific, the so-called “target model”23, 24 corresponding to the reaction of immobile particles A∗ with point particles B at arbitrary concentration [B]. Besides, we put J(t) = J = const. We shall use this model to confirm general derivations and approximations developed in the present work. For this model Eq. (3.8) (where k(t) does not depend on concentration [B]) is the exact many-particle kinetics for arbitrary concentration [B]. To make the value k(t) more concrete, we examine diffusion controlled reaction of immobile spherical particles A of the ra-

√ √ 3ξ k[B] 3ξ k[B] 3ξ k[B] erf R t + exp √ R π R2 πR √ 3ξ k[B] . (3.31) − erf √ πR

Using Eqs. (3.29) in Eq. (3.4) gives the Laplace transform of the kinetics J S0L (s + R 2 ). (3.32) P ∗L (s) = P ∗ (0) + s Stationary part of excited particles is as follows: Pst∗ = lim P ∗ (t) = lim sP ∗L (s) = t→∞

=

J R2

s→0

1−

√ 3ξ k[B] , √ πR

J (1 − F (∞)) R2 (3.33)


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J. Chem. Phys. 141, 104104 (2014)

where

Eq. (3.29), in view of Eq. (3.17)

(x) =

√ √ π x exp(x 2 )erfc (x) , (x) ≈ π x at x → 0, (3.34)

that corresponds to the result obtained in Ref. 6. Correspondingly, the quenching constant kQ defined according to Eq. (1.5) is J 1 kQ [B] = ∗ − +J Pst τA √ 3ξ k[B] k[B] + (R 2 − k[B]) √ πR = . (3.35) √ 3ξ k[B] 1− √ πR The rate constant k2 (t) calculated with the use of Eqs. (3.11) and (3.26) is 3ξ k[B] 1 k2 (t) [B] = −J F (t) − πt R2

3ξ k[B] t × 1 − exp −R 2 t − 2 − F (t) . π (3.36) It has a steady-state value k2 (the Markovian rate constant) k2 [B] = lim k2 (t) [B] = −J F (∞) = −J t→∞

√ 3ξ k[B] . √ πR (3.37)

As mentioned above, the rate constant k2 (t) and, accordingly, its Markovian value k2 are negative, i.e., pumping (due to the term involving k2 (t) in kinetic equation (3.9) or (3.12)) leads to the increase in deactivation rate of excited states. Of course, such an increase does not exceed the pumping rate determined by the last terms in the right-hand sides of kinetic equation (3.9) or (3.12). Note that all the results for the well-studied model are for the first time presented here in the form that explicitly separates out the density parameter, and this is essential for our investigation. By analogy, more general “target model” for the specific reaction proceeding at a finite rate is considered in the Appendix.

2. Integro-differential equations

In the literature the “target model” has always been treated in the framework of the differential theory. For the first time, in this subsection we consider it in the context of integro-differential theory. For the “target model” of diffusion controlled contact quenching reaction under consideration, one can find the Laplace transform of the kernel of integro-differential kinetic equation (3.18) (or (3.21)) using

3ξ k[B] k[B] + s π (s + k[B]) L (s) [B] = . 3ξ k[B] 1− π (s + k[B])

(3.38)

Note that with s → ∞ along real axis (see Eq. (3.34)) L (s) [B] ≈ k[B] 1 + 3 ξ 1 − 2 π √ + 3ξ k[B] s → ∞, (3.39) i.e., the Laplace transform of the kernel does not satisfy the property of ordinary Laplace transformation (tending to zero at s → ∞ along real axis), and is the Laplace transformation of generalized function.27 These functions are defined on the extended time axis −∞ < t < ∞ and complete coordinate configuration space (time derivative on the axis is designated as ∂ t ). Extension of the time interval of the measured quantities defined at t ≥ 0 to the entire time axis is performed by multiplying it by the Heaviside functions θ (t) (θ (t) = 1 at t > 0 and θ (t) = 0 at t < 0). The quantities thus defined are equal to zero at t < 0 and coincide with the measured quantities at t > 0. When t = 0 their values become uncertain, and initial values appear as sources in appropriate differential equations for these functions. Just because of this uncertainty the lower bounds of time integration in Eqs. (3.18) and (3.21) are put equal to −0. IV. BINARY APPROXIMATION A. General remarks

In this section we consider the problem of the development of binary approximation for bulk multi-stage chemical reaction involving bimolecular irreversible reaction and reversible monomolecular transformation of reactants as elementary stages. The binary approximation corresponds to taking into account just pair encounters of reactants in solution (neglecting triple encounters and encounters of a higher order). Obviously, the necessary validity condition of such a theory is the density parameter smallness. As for sufficient conditions, they will be established in the framework of two different general approaches to the derivation of binary kinetic equations. To make the development of such an approximation more clear, in Sec. IV B we begin with the examination of exactly solvable many-particle model and demonstrate derivations of binary kinetic equations by expanding the kinetic values into a series in terms of density parameter both in the framework of differential and integro-differential formulations. In Sec. IV C we shall use two familiar general methods for the derivation of binary kinetic equations: simple superposition decoupling and the method of binary channel extraction in the evolution of three-particle correlations. The first method gives the differential form of binary kinetic equations, while the second one—integro-differential form. These approaches are general and are based on general equation (2.10) for RDF. They do not refer to a definite model of the reaction system. Contact “target model” is used solely for clarity and demonstration of applicability conditions of the above two methods


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J. Chem. Phys. 141, 104104 (2014)

for the derivation of binary kinetic equations of the multistage reaction under study. B. Diffusion controlled contact quenching reaction

1. Differential formulation of the problem

Approximation of binary encounters of reactants in solution (binary approximation) will be developed, first of all, for “target model” of diffusion controlled contact quenching reaction with the rate constant defined by Eq. (3.26). Evidently, such an approximation is valid on condition of smallness of the density parameter ξ of reactants (see Eq. (3.27)) ξ 1.

(4.1)

The approximation is developed neglecting the terms of the first and higher orders in ξ . We use the same designations of quantities in the binary approximation, as in general kinetics. For binary kinetics of quenching, Eq. (3.28) gives 2 3 ξ k[B] t , (4.2) S (t) = exp (−k [B] t) 1 − √ π and for its Laplace transform (see also Eq. (3.29)) 3ξ k[B] 1 L L L S (s) = S0 (s + k[B]) , S0 (s) = 1− . s s (4.3) It follows from Eqs. (4.2) and (4.3) that the binary approximation is valid over a restricted time range (binary kinetics interval)14, 15, 28 k[B]t 1 ξ (s 3ξ k[B]) . (4.4) A limited interval of binary kinetics follows at least from the fact that at rather long times the so-called fluctuation kinetics is to take place29, 30 the calculation of which is impossible in the binary approximation. The binary kinetics of the multistage reaction at J(t) = J = const 2 J ∗ ∗ 1− √ P (t) = P (0) − 2 3 ξ k[B] t exp(−R 2 t) R π √ √ 3ξ k[B] J erf(R t) , (4.5) + 2 1− R R is obtained either on substitution of Eq. (4.2) in Eq. (3.3), or follows immediately from Eqs. (3.30) and (3.31), taking into account that in the binary approximation √ √ 3ξ k[B] erf(R t) F (t) ≈ R √ √ 3ξ k[B] 3ξ k[B] × F (∞) = 1 . ≈ √ R πR (4.6) Its Laplace transform is (see Eqs. (3.32) and (4.3)) 1 3ξ k[B] J ∗L ∗ P (s) = P (0) + 1− . (4.7) s s + R2 s + R2 Note that time restriction (4.4) of the binary kinetics refers just to the first term in Eq. (4.5) obtained from the first

term in Eq. (3.30). This term decreases exponentially and under condition (4.1) almost goes to zero at the upper bound of the binary interval. The possibility of expanding the function F(t) (3.31) in Eq. (3.30) into a series in terms of density parameter has no time restrictions. Due to this, at t → ∞ Eq. (4.5), or at s → 0 Eq. (4.7) yield correct value of stationary part of excitations √ 3ξ k[B] J ∗ ∗ ∗L , Pst = lim P (t) = lim s P (s) = 2 1 − t→∞ s→0 R R (4.8) and the quenching constant kQ defined both by general Eq. (1.5), and by Eq. (3.35), in view of Eq. (4.6), is kQ [B] = k[B] + 3ξ k[B]R

2 1 a + J + k[B] ≡ k[B] 1 + . (4.9) τA D In the second formula Eq. (4.9) we explicitly separated out (using Eq. (3.27)) the concentration dependence of the quenching constant showing the deviation from Stern-Volmer law (equivalent to concentration independence of the quenching constant). The constant k2 (t) is derived from Eq. (3.36) in view of Eq. (4.6) J 3ξ k[B] k2 (t) [B] = − R √ 1 × erf(R t) − √ (1 − exp(−R 2 t)) , R πt (4.10) and has a steady-state value (the Markovian constant) in this approximation J k2 [B] = lim k2 (t) [B] = − 3ξ k[B]. (4.11) t→∞ R √ Proportionality of this constant to ξ is to evidence that its formation is completely related to non-stationary (nonMarkovian) part of elementary stage of quenching in the absence of pumping. Note that, despite the smallness of the density parameter ξ , absolute values of rate constants k2 (t) or k2 under intensive pumping are not small as compared to rate constants k(t) or kof elementary stage √ of quenching, since these values increase in proportion to J with J → ∞. 2. Integro-differential statement of the problem

In the binary approximation (on condition that (4.1) is satisfied) the expansion in the parameter ξ yields, with the required accuracy, the expression for the Laplace transform of kernel (3.38) of kinetic equation (3.16) 3ξ k √ L (s) = eL (s + k[B]) , eL (s) = k + s. [B] (4.12) The behavior of this transform at s → ∞ along real axis is in full agreement with Eq. (3.39), and it corresponds to the


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A. B. Doktorov

J. Chem. Phys. 141, 104104 (2014)

Laplace transform of generalized function, i.e.,14, 15 (t) = e (t) exp (−k[B] t) , 3ξ k θ (t) ∂ √ , e (t) = kδ (t) + π [B] t t

(4.13)

where δ(t) is delta-function, and ∂ t denotes the derivative of generalized functions. Note that multiplication of the kernel e (t) by exponent to obtain the kernel (t) does not affect the first positive term in the second part of Eq. (4.13), but decreases the second negative term (in the interval t > 0) thus increasing the kinetic coefficient instead of diminishing it, as might be expected from general consideration. Representing the rate constant of elementary quenching stage (3.26) as the generalized function K(t) on the extended time axis, one has the relation e (t) = ∂t K (t) , K (t) = θ (t) k (t) ,

(4.14)

by virtue of their equivalence. However, this means that in the binary approximation integro-differential kinetic equations (3.21) with kernel (3.20) can be transformed into differential form (3.12). Following literature,14, 15, 31 we shall call such a transformation of integro-differential equations a transition to Regular Form of kinetic equations. To perform the above transformation (at J(t) = J = const), introduce the state vector P(t), the reaction matrix J (τ ), and the matrix Q of transitions due to dissipation and pumping P (t) = Q=

P ∗ (t) , P (t)

t K (t) = −0

k=

(τ ) =

0 , 0 (4.18)

.

Write Eqs. (3.21) (J(t) = J = const, in a matrix form d P (t) = [B] dt

− J (τ ) J (τ )

J (t|τ ) = J (t − τ ))

t J (t − τ ) P (τ ) d τ + Q P (t)

−0

t

∞ e (τ ) dτ,

J

−1/τA J 1/τA − J

eL (s) = s K L (s) = s k L (s) , i.e., in the binary approximation there is a simple relation between the kernel of integro-differential equation and the quenching rate constant in equivalent differential equation. So

e (τ ) dτ ≡ eL (0) . (4.15)

−0

−0

The distinctive feature of generalized function integration deserves attention. Such an integral of ordinary function diverges at the lower limit, the integral of generalized function is equal to zero at τ = −0. The integral from −0 to ∞ of the second term in the second part of Eq. (4.13) is equal to zero. Equations (3.17) and (4.12) for the Laplace transform of the quenching kinetics give S L (s) = S0L (s + k[B]) , (4.16) 3ξ k[B] 1 1 S0L (s) = ≈ 1− . √ s s s + 3ξ k[B]s In deriving the second expression for S0L (s) (coinciding with Eq. (4.3)) we used condition (4.4) of a limited nature of the binary kinetics time interval. Under the same assumptions, general formulae (3.4) and (3.5) yield Eqs. (4.7) and (4.8) for the Laplace transform of complete kinetics and stationary part of excitations, respectively. As for the quenching constant, general formula (3.25) and Eq. (4.12) immediately give the same result 1 3ξ k R + J = eL (R 2 ) = k + kQ = L τA [B] 3ξ k 1 =k+ + J + k[B] , (4.17) [B] τA as Eq. (4.9) of the differential theory. Thus all the results obtained in the context of binary differential and integrodifferential theories coincide completely, just as it should be

J (τ ) P (t − τ ) d τ + Q P (t) .

≡ [B]

(4.19) Introducing the following rate constant matrices:

−k1 (t) k2 (t) , k1 (t) − k2 (t)

−k1 k2 , K = lim K (t) = t→∞ k1 − k2

K (t) =

(4.20)

we also bring differential non-Markovian (3.12) and Markovian (3.14) kinetic equations into a matrix form d P (t) = (K (t) [B] + Q) P (t) , dt

(4.21)

d P (t) = (K[B] + Q) P (t) . dt Now use the time shift rule14, 15, 31 in Eq. (4.19) P (t − τ ) = R−1 (τ ) P (t)

(P (t) = R (τ ) P (t − τ )) , (4.22)

which implies that the evolution of the vector P(t) on the interval [t − τ , t] in the limits of binary approximation accuracy has the Markovian character with some Markovian constants k1 and k2 that are to be calculated after transformation of Eq. (4.19) to its regular form (4.21). The Markovian evolution is specified by the matrix R(τ ) defined by the second part


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A. B. Doktorov

J. Chem. Phys. 141, 104104 (2014)

of Eq. (4.21), so 1 + J + (k1 + k2 )[B] ⎛ 1 (1/τ +J +(k1 +k2 )[B])τ ⎜ k2 [B] + J + τ + k1 [B] e A ⎜ A ×⎜ ⎜

⎝ 1 + k1 [B] 1 − e(1/τA +J +(k1 +k2 )[B])τ τA

R−1 (τ ) = exp (− ((K[B] + Q) τ )) =

1 τA

(J + k2 [B]) 1 − e(1/τA +J +(k1 +k2 )[B])τ 1 + k1 [B] + (J + k2 [B])e(1/τA +J +(k1 +k2 τA

⎞

⎟ ⎟ ⎟. (4.23) ⎟ ⎠ )[B])τ

Employing Eq. (4.22) in Eq. (4.19), in view of Eqs. (4.18), (3.20), and (4.23), we arrive at the differential equation (4.21) with the matrix K(t) (4.20) where (at t > 0) ⎫ ⎧ t t ⎬ ⎨ 1 1 k1 (t) = 1 (τ ) e−(1/τA +J )τ dτ + + k1 [B] (τ ) e(k1 +k2 )[B]τ dτ , k2 [B] + J ⎭ τA + J + (k1 + k2 )[B] ⎩ τ −0

A

k2 (t) =

1 τA

J + k2 [B] + J + (k1 + k2 )[B]

t

(4.24)

(τ ) e(k1 +k2 )[B]τ − e−(1/τA +J )τ dτ.

−0

With allowance for the first part of Eq. (4.13) t k1 (t) + k2 (t) =

−0

quantity k2 . From the same Eq. (4.24) we also find k2 (t) k2 (t) = J

(τ ) e(k1 +k2 )[B]τ dτ

−0

≡

e

(τ )e(k1 +k2 −k)[B]τ dτ .

(4.25)

With t → ∞ Eq. (4.25) gives the closed equation for the quantity k1 + k2 which, in view of the second part of Eq. (4.15), yields the same relation k1 + k2 = k between Markovian rate constants as general differential theory (see Eq. (3.15)). From Eq. (4.25) and the first part of Eq. (4.15) one can also derive a more general property (3.13) k1 (t) + k2 (t) = k(t) (at t > 0). We also introduce the designations k0 (t) =

k2 = J

To calculate the quantity k0 (t) (4.26), it is convenient to find its Laplace transform using Eq. (4.12) k0L

1 1 L (s) = e s + + J + k[B] s τA √ 3ξ k s + R 2 1 L k 2 . ≡ e (s + R ) = + s s [B] s

−0

k0 = lim s

≡

e (τ ) e

−(1/τA +J +k[B])τ

s→∞

dτ ,

−0

∞ k0 = lim k0 (t) = t→∞

(4.26)

≡

e (τ ) e−(1/τA +J +k[B])τ dτ .

−0

At t → ∞ the second part of Eq. (4.24) with allowance for the established properties gives the closed equation for the

k0L

(s) = k +

3ξ k R. [B]

(4.29)

Recovering the original of k0L (s),25 we have

(τ ) e−(1/τA +J )τ dτ

−0

∞

(4.28)

This immediately gives the stationary value k0 of the quantity k0 (t)

(τ ) e−(1/τA +J )τ dτ

t

k − k0 . 1/τA + J + k0 [B] (4.27)

t −0

t

k (t) − k0 (t) , 1/τA + J + k0 [B]

k0 (t) = k +

3ξ k [B]

√ 1 exp(−R 2 t) + R erf(R t) . √ πt (4.30)

The use of Eqs. (3.26), (4.30), and (4.29) in Eqs. (4.27) in the binary approximation reproduces results (4.10) and (4.11) of the differential theory.


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A. B. Doktorov

J. Chem. Phys. 141, 104104 (2014)

C. Many-particle methods for the derivation of binary kinetic equations

1. General remarks

In this subsection we shall analyze, on the basis of general RDFs hierarchies (4.11), the applicability of two familiar methods for the derivation of binary kinetic equations: (a) simple superposition approximation7, 8, 18 and (b) method of extracting binary channels in the evolution of three-particle correlations.14, 15, 19 These methods have been used in the literature in studies of elementary reactions. We apply them to the consideration of multistage reaction of fluorescence concentration quenching in the presence of spontaneous decay and intensive pumping. First, note that simple superposition approximation based on decoupling of three-particle RDF in hierarchies for these functions, for elementary irreversible quenching reaction A∗ + B → A + B, leads to Eq. (3.7) of the differential theory with concentration independent rate constant coinciding with the rate constant for the ϕA∗ ,0 (A, t) = [A∗ ]t ,

“target model” (at the same relative diffusion coefficients). Thus the kinetics defined by this equation formally coincides with the exact many-particle kinetics for the “target model.”23, 24 We try to elucidate the question of whether this fact takes place for multistage reaction. Besides, we examine the applicability of this approximation for the derivation of non-Markovian binary kinetic equations in the presence of intensive pumping and spontaneous decay. The applicability of the method of binary channel extraction in the evolution of three-particle correlations is studied with the same aim. 2. Simple superposition approximation

Superposition approximation may be used solely for spatially homogeneous chemical systems,7, 8, 18 so they will be the only subject of our investigation from the very beginning. In such systems all RDFs remain unchanged when the origin of coordinates is displaced (for example, to the point A). Then

ϕA∗ ,1 (A, B1 , t) = ϕA∗ ,1 (B1 − A, t) ≡ ϕA∗ ,1 (q, t)

ϕA∗ ,2 (A, B1 , B2 , t) = ϕA∗ ,2 (B1 − A, B2 − A, t) ≡ ϕA∗ ,2 (q, q , t)

(q = B1 − A), (q = B2 − A).

(4.31)

From Eq. (2.10) we have for the first two hierarchies (r = 0, 1) d 1 [A∗ ]t = − V (q) ϕA∗ ,1 (q, t)dq − + J (t) [A∗ ]t + J (t) [A]0 , dt τA (4.32) ∂ − Lˆ q + V (q) ϕA∗ ,1 (q, t) = − V (q ) ϕA∗ ,2 (q, q , t) dq ∂t 1 − + J (t) ϕA∗ ,1 (q, t) + J (t) ϕ (q) [A]0 [B], τA

where Lˆ q is the functional operator describing relative motion of particles, V (q) = V B1 − A ≡ V1 . Note that in Eqs. (2.4) and (2.6) at r = 1 we used more general initial conditions ϕA∗ ,1 (q, 0) = ϕ (q) [A∗ ]0 [B],

ϕA,1 (q, 0) = ϕ (q) [A]0 [B],

ϕ1 (q, 0) = ϕ (q) [A]0 [B],

(4.33)

where ϕ(q) is the so-called static contour (i.e., the initial equilibrium spatial distribution in a pair of reactants in the thermodynamic limit (for instance, in the presence of force interaction)). In simple superposition approximation18

ϕA∗ ,1 (q, t)ϕA∗ ,1 q , t ϕA∗ ,2 (q, q , t) ≈ , [A∗ ]t (4.34) ϕA∗ ,1 (q, 0)ϕA∗ ,1 (q , 0) ∗ 2 ϕA∗ ,2 (q, q , 0) ≈ = ϕ (q) ϕ(q )[A ]0 [B] , [A∗ ]t i.e., it preserves the required initial condition for three-particle RDF. Application of decoupling (4.34) of three-particle RDF leads to the closed set of equations. Setting ϕA∗ ,1 (q, t) = g(q, t)[A∗ ]t [B], kJ (t) = V (q) g(q, t)d q, (4.35) in view of Eq. (1.6), it can be brought into equivalent form analogous to the form obtained in Ref. 8

d ∗ 1 P (t) = −kJ (t) [B] P ∗ (t) − P ∗ (t) + J (t) 1 − P ∗ (t) , dt τA

∂ J (t) (ϕ (q) − g(q, t)) , − Lˆ q + V (q) g(q, t) = ∗ ∂t P (t)

(4.36)

but with different initial condition g(q, 0) = ϕ(q) (in Ref. 8 g(q, 0) = 0), according to definition (4.35) and conditions (4.33). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.2.176.242 On: Mon, 01 Dec 2014 08:35:50

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A. B. Doktorov

J. Chem. Phys. 141, 104104 (2014)

We introduce pair distribution function n(q, t) in the absence of pumping which obeys the equation following from the second part of Eq. (4.36) at J(t) = 0: ∂ ˆ − Lq + V (q) n(q, t) = 0 (4.37) ∂t

that provides the fulfillment of the required initial condition for g(q, t). It follows from Eq. (4.39) using definitions (4.35) and (4.38) of the rate constants kJ (t) and k(t), respectively, that ⎞ ⎛ t t (z) J J (τ ) d z⎠ + kJ (t) = k (t) exp ⎝− ∗ P (z) P ∗ (τ )

with the same initial condition n(q, 0) = ϕ(q). As is known,4, 14, 15 the rate constant of elementary irreversible reaction of quenching in Eq. (3.7) is k (t) = V (q) n(q, t)d q. (4.38)

0

⎛ × exp ⎝−

⎛ × exp ⎝−

t

P ∗ (t) =− ln ∗ P (τ )

t +

(4.40)

1 + J (z) + kJ (z) [B] d z τA

J (z) d z, P ∗ (z)

(4.41)

τ

obtained in Ref. 8 from the first part of Eq. (4.36) d (by integration from τ to t of the expression dz ln P ∗ (z) 1 d ∗ ≡ P ∗ (z) dz P (z)). Expressing the last integral in the righthand side of Eq. (4.41) in terms of the other two terms and employing this value in Eq. (4.40), we have

τ

Note that Eq. (4.39) differs from the expression derived in Ref. 8 in that it involves the first term, and it is just this term

⎛ kJ (t) P ∗ (t) = k (t) P ∗ (0) exp ⎝−

+

t τ

J (z) d z⎠ n (q, t − τ ) d τ. (4.39) P ∗ (z)

t

J (z) d z⎠ k (t − τ ) d τ. P ∗ (z)

Now use the relation

0

⎞

t τ

It is seen that at J(t) = 0, 1 / τ A = 0 Eqs. (4.36)–(4.38) do give the differential rate equation (3.7). The solution of the second part of Eq. (4.36) is expressed via the solution of Eq. (4.37) as ⎞ ⎛ t t J (z) J (τ ) ⎠+ d z g(q, t) = n(q, t) exp ⎝− P ∗ (z) P ∗ (τ ) 0

0

⎞

t 0

⎛ J (τ ) exp ⎝−

⎞ 1 + J (z) + kJ (z) [B] d z⎠ k (t − τ ) d τ. τA

t τ

0

⎞ 1 + J (τ ) + kJ (τ ) [B] d τ ⎠ τA

(4.42)

With the solution of linear inhomogeneous differential equation (the first part of Eq. (4.36)) ⎛ ⎞ t 1 P ∗ (t) = P ∗ (0) exp ⎝− + J (τ ) + kJ (τ ) [B] d τ ⎠ τA 0

t + 0

⎛ J (τ ) exp ⎝−

t τ

⎞ 1 + J (z) + kJ (z) [B] d z⎠ d τ τA

(4.43)

in Eq. (4.42)), we arrive at kJ (t) P ∗ (t) = k (t) P ∗ (t) − k˜2 (t) , where t k˜2 (t) =

⎛ J (τ ) (k (t) − k (t − τ )) exp ⎝−

0

Thus, with Eq. (4.44) taken into account, the kinetic equation (4.36) for the quantity P∗ (t) takes the form of exact kinetic equation (3.9) of the differential theory. However, the defini-

t τ

⎞ 1 + J (z) + kJ (z) [B] d z⎠ d τ. τA

(4.44)

(4.45)

tion of the rate constant k˜2 (t) (4.45) differs from definition k2 (t) (3.10) in that instead of the rate constant k(z) of elementary quenching stage (in the integrand in Eq. (3.10)), the


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J. Chem. Phys. 141, 104104 (2014)

integrand in Eq. (4.45) involves the rate constant kJ (z) with allowance for pumping and spontaneous decay. In particular, this means that for the problem in question superposition decoupling fails to reproduce the result of exactly solvable “target model,” as it was for elementary quenching reaction A∗ + B → A + B. Besides, finding the quantity kJ (z) is a difficult problem of self-consistent solution of Eq. (4.44) with allowance for Eqs. (4.45) and (4.43). In the general case this quantity can be found only under the following conditions: 4π 3 R [B] 1 (4.46a) 3 eff (where Reff is the effective radius of quenching14, 15 ) and 2 Reff 1 1. (4.46b) + max J (t) τA D ξ=

In this case, the value k˜2 (t) in Eq. (4.44) (see Eqs. √ (4.10) and (4.11)) is a small correction of the order of ξ to the value k(t)P∗ (t). Then Eqs. (4.44) and (4.45) may be solved by iteration method with the first term in Eq. (4.44) used as a zero approximation; this gives kJ (t) ≈ k(t). Besides, in the binary approximation it is necessary to put k(t) ≈ k. Substitution of this value in the integrands in the exponent in Eqs. (4.45) and (3.10) reproduces general expression for the constant k2 (t) in the binary approximation t k˜2 (t) ≈ k2 (t) =

J (τ ) (k (t) − k (t − τ )) 0

1 + k[B] (t − τ ) × exp − τA ⎞ t − J (z) d z⎠ d τ, τ

k˜2 (t) ≈ k2 (t) = J

k (t) 1 − exp(−R 2 t) 2 R

t −

(4.47)

with the value of the quenching constant kQ that, according to Eqs. (4.44) and (4.45) at t → ∞ and Eq. (1.5) in the general case, is determined, as is shown in Ref. 8, from “selfconsistent” relation

kQ = J + τA−1 + kQ [B] k L J + τA−1 + kQ [B] , (4.48) where kL (s) is the Laplace transform of rate constant (4.38). Of course, the solution of such an equation is only possible for specific models. Further we shall find this value for the “target model” of diffusion controlled contact quenching reaction so as to establish the conditions of its applicability.

3. The method of binary channel extraction in the evolution of three-particle correlations

The method under consideration14, 15, 19 can also be applied to initially inhomogeneous systems. Though basic equation (2.10) was derived using solution (2.8) of Eq. (2.7) corresponding to homogeneous systems (by virtue of initial conditions (2.6)), it is easily seen that taking into account the inhomogeneity of distributions does not affect basic equations for quantities averaged over volume. This is also valid for equations for RDFs because of Eq. (2.7). As the method at hand leads to integro-differential form of kinetic equations, we shall consider, from the very beginning, the values on extended time axis, and space and time coordinates will be treated as the coordinates of a unified (for example, for one particle, four-dimensional) space. As already mentioned, in this case, initial values appear as sources in appropriate differential equations for these functions. Besides, we employ operator technique. For instance, we shall describe the reaction proceeding at the elementary rate Vm (Bm − A) by the integral operator Vˆm defined by its kernel Vm (A, Bm , t|A0 , B0m , t0 ) = −Vm (Bm − A)δ(A − A0 )δ(Bm − B0m )δ(t − t0 ).

⎞

(4.49)

k (τ ) × exp(−R 2 τ )dτ ⎠

0

(J (t) = J = const) .

However, note that just one condition, namely, the density parameter smallness, should be met to obtain result (3.10) in general differential theory. The requirement of the fulfillment of additional condition to find the quantity kJ (t) is the main demerit of superposition approximation for the multistage reaction in question. The problem of finding k˜2 (t) (4.45) if condition (4.46b) is violated in the case of constant pumping J(t) = J = const can be slightly simplified by employing the steadystate value kJ = lim kJ (t) in Eq. (4.45) (as in Eq. (4.47)). By t→∞ virtue of the first part of Eq. (4.36) and Eq. (1.5), it coincides ϕA∗ ,0 (A, t) = π1,0 (A, t),

ˆ (defined by the In general, the action of some operator G kernel G(A, B, t|A0 , B0 , t0 )) on some function of variables A, B, and t consists in integration over variables A0 ,B0 , and t0 , and this gives the function of variables A, B, and t ˆ f = ϕ(A, B, t) ϕ=G = G(A, B, t|A0 , B0 , t0 )f (A0 , B0 , t0 ) dA0 dB0 dt0 . (4.50) Then, we pass from RDFs hierarchies (2.10) to hierarchies of Correlation Patterns (CP)19 widely employed in non-equilibrium statistical mechanics.21 For the lower order RDFs (r = 0, 1, 2), we have

ϕA∗ ,1 (A, B1 , t) = π1,0 (A, t)[B] + π1,1 (A, B1 , t),

ϕA∗ ,2 (A, B1 , B2 , t) = π1,0 (A, t)[B] 2 + π1,0 (A, t)π0,2 (B1 , B2 , t)

(4.51)

+ π1,1 (A, B1 , t)[B] + π1,1 (A, B2 , t)[B] + π1,2 (A, B1 , B2 , t). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 152.2.176.242 On: Mon, 01 Dec 2014 08:35:50

104104-15

A. B. Doktorov

J. Chem. Phys. 141, 104104 (2014)

Using Eqs. (4.51) in Eq. (2.10) accurate to three-particle correlations, we obtain the closed set of equations 1 (∂t − Lˆ A ) π1,0 (A, t) = δ (t) [A∗ ]0 − + J (t) π1,0 (A, t) τA + J (t) [A]0 + dB1 Vˆ1 (π1,1 (A, B1 , t) + [B] π1,0 (A, t)), (∂t − Lˆ A − Lˆ B )π1,1 (A, B1 , t) = − 1

1 + J (t) π1,1 (A, B1 , t) τA

+ Vˆ1 (π1,1 (A, B1 , t) + [B] π1,0 (A, t)) +

(∂t − Lˆ A − Lˆ B

1

dB2 Vˆ2 (π1,2 (A, B1 , B2 , t)

+ [B] π1,1 (A, B1 , t)), 1 ˆ − LB ) π1,2 (A, B1 , B2 , t) = − + J (t) π1,2 (A, B1 , B2 , t) 2 τA

(4.52)

+ (Vˆ1 + Vˆ2 ) π1,2 (A, B1 , B2 , t) + Vˆ1 π1,1 (A, B2 , t)[B] + Vˆ2 π1,1 (A, B1 , t)[B]. ˆ 2J (two-particle propagator) the kernel of which is the Green function obeying the equation Now introduce the operator G 1 + J (t) − Vˆ1 G2J (A, B1 , t|A0 , B01 , t0 ) = δ(A − A0 )δ(B1 − B01 )δ(t − t0 ). (4.53) ∂t − Lˆ A − Lˆ B + 1 τA ˆ 2 . Its kernel satisfies Eq. (4.53) with 1 / τ A In the absence of spontaneous decay and pumping, this operator is denoted as G = J(t) ≡ 0, and, evidently, ⎛ ⎞ t ⎜ t − t0 ⎟ G2J (A, B, t|A0 , B0 , t0 ) = G2 (A, B, t|A0 , B0 , t0 ) exp ⎝− − J (τ ) dτ ⎠ . (4.54) τA t0

ˆ 2 specify T-operators of the reaction pair in the presence and in the absence of decay and pumping, ˆ 2J and G Propagators G respectively, ˆ 2J Vˆ1 , TˆJ = Vˆ1 + Vˆ1 G

ˆ 2 Vˆ1 . Tˆ = Vˆ1 + Vˆ1 G

(4.55)

The kernels TJ (A, B, t|A0 , B0 , t0 ) and T(A, B, t|A0 , B0 , t0 ) of these operators called T-matrices define the kernels M1J (A, t|A0 , t0 ) and M1 (A, t|A0 , t0 ) of one-particle mass operators Mˆ 1J and Mˆ 1 , respectively, as well as the averaged T-matrices TJ (t|t0 ) and T(t|t0 ) necessary for further calculations M1J (A, t|A0 , t0 ) = TJ (A, B, t|A0 , B0 , t0 )dB dB0 , TJ (t|t0 ) = M1 (A, t|A0 , t0 )dA ≡ TJ (A, B, t|A0 , B0 , t0 )dA dB dB0 , (4.56) T (A, B, t|A0 , B0 , t0 )dBdB0 , M1 (A, t|A0 , t0 ) = T (t|t0 ) = M1 (A, t|A0 , t0 )dA ≡ T (A, B, t|A0 , B0 , t0 )dA dB dB0 , and, according to Eqs. (4.54) and (4.49),

⎛

t

⎞

⎜ t − t0 ⎟ − J (τ )dτ ⎠ , M1J (A, t|A0 , t0 ) = M1 (A, t|A0 , t0 ) exp ⎝− τA t0 ⎛ ⎞ t ⎜ t − t0 ⎟ TJ (t|t0 ) = T (t|t0 ) exp ⎝− − J (τ )dτ ⎠ . τA

(4.57)

t0

As is known,14, 15, 19 the averaged T-matrix T(t|t0 ) defines the kernel e (t|t0 ) = −T(t|t0 ) of integro-differential equation (3.16) of Integral Encounter Theory (IET) of elementary

quenching reaction A∗ + B → A + B derived from Eqs. (4.52) neglecting all three-particle correlation patterns. This kernel completely corresponds to the kernel e (t − t0 ) ≡ e (t


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J. Chem. Phys. 141, 104104 (2014)

− t0 |0) in Eqs. (4.13) (obtained for diffusion controlled reaction in the framework of “target model”), and, in a rather general case, is defined by Eqs. (4.14) with constant k(t) specified in Eq. (4.38). Of course, relation (4.15) between the rate constant k(t) of differential rate equation and IET kernel is preserved. By virtue of Eqs. (4.57), relation (3.19) following from general relation (3.2) takes place in IET. However, IET applicability time range does not cover the whole time interval of binary approximation,28 thus modification of the theory is needed based on Eqs. (4.52) making allowance for three-particle correlations.15 To develop such a Modified Encounter Theory (MET),14, 15, 19, 28 one should simplify Eqs. (4.52) so as to avoid overestimation of binary approximation accuracy. For this purpose, we use the method19 of extracting pair channels in the evolution of three-particle correlations which is similar to the Faddeev reduction in quantum three-body theory.32 The method consists in that for the quantity Vˆ2 π1,2 in the integrand in the right-hand side of the second equation we can restrict ourselves to the evolution of the correlation pattern π 1, 2 (defined by the third equation in Eqs. (4.52)) when taking account of the reaction interaction between particle A and particle B2 , while the motion of particle B1 may be considered free, determined by free oneˆ 1 with the kernel obeying the equation particle propagator G

∂t − Lˆ B G01 (B1 , t|B01 , t0 ) = δ(B1 − B01 )δ(t − t0 ). 1

(4.58) Then we have from the third equation (4.52) ˆ 3J Vˆ2 π1,1 (A, B1 , t), Vˆ2 π1,2 (A, B1 , B2 , t) ≈ [B] Vˆ2 G (4.59) ˆ 3J is where the kernel of the three-particle propagator G

Substituting expression (4.61) in the first part of Eq. (4.52) and passing to survival probability (3.1) yield Eq. (3.18) with the kernel J (t|t0 ) = −Teff (t|t0 ) ≡− Teff (A, B, t|A0 , B0 , t0 )dAdBdB0 , (4.64) expressed in terms of the effective pair T-operator kernel ˆ eff Vˆ1 , Têff = Vˆ1 + Vˆ1 G ˆ 0eff Têff = Vˆ1 + Têff G ˆ 0eff Vˆ1 . Têff = Vˆ1 + Vˆ1 G

The second and the third equations in Eqs. (4.65) include ˆ 0 with the kernel specified by free effective propagator G eff Eq. (4.62) with Vˆ1 ≡ 0 which can be represented (by direct substitution with allowance for Einstein-Smoluchowski equation) as G0eff (A, B1 , t|A0 , B01 , t0 ) = G1J (A, t|A0 , t0 )G01 (B1 , t|B01 , t0 ),

= δ(A − A0 )δ(t − t0 ).

ˆ eff Vˆ1 π1,0 (A, t), π1,1 (A, B1 , t) ≈ [B] G

(4.61)

ˆ eff is the effective propagator the kernel of which is where G the solution of the equation 1 ˆ ˆ ˆ ˆ ∂t − L A − L B + + J (t) − V1 − M2 [B] 1 τA × Geff (A, B1 , t|A0 , B01 , t0 ) = δ(A − A0 )δ(B − B01 )δ(t − t0 ).

(4.67)

In binary approximation one should make use of point approximation for the kernel M1 (A, t|A0 , t0 ). In the absence of pumping and decay this approximation19 implies that ∞ T (t |τ ) dτ δ(t − t0 )δ(A − A0 )

M1 (A, t|A0 , t0 ) ≈ 0

G3J (A, B1 , B 2 , t|A0 , B01 , B02 , t0 )

Substituting expression (4.59) in the second part of Eq. (4.52) and solving the deduced closed equation for π 1, 1 (A, B1 , t) with the use of the Green function, we have

(4.62)

(4.66)

where 1 ˆ ˆ ∂t − LA + + J (t) − M1J [B] G1J (A, t|A0 , t0 ) τA

≡ −kδ(t − t0 )δ(A − A0 ).

= G2J (A, B2 , t|A0 , B02 , t0 ) G01 (B1 , t|B01 , t0 ). (4.60)

(4.65)

(4.68)

However, in the presence of pumping and decay the quantity M1J (A, t|A0 , t0 ) contains exponential decay at the rate 1 / τ A + J(t) (see Eqs. (4.57)) that should be eliminated by the substitution G1J (A, t|A0 , t0 )

⎛

⎜ t − t0 = G1 (A, t|A0 , t0 ) exp ⎝− − τA

t

⎞ ⎟ J (τ ) dτ ⎠ .

t0

(4.69) Substituting Eq. (4.69) in Eq. (4.67), we arrive at the equation in the absence of decay and pumping (∂t − Lˆ A − Mˆ 1 [B])G1 (A, t|A0 , t0 ) = δ(A − A0 ) δ(t − t0 ), (4.70)

Equation (4.62) involves two-particle mass operator Mˆ 2 . In view of the definition of one-particle mass operator Mˆ 1 , its kernel

where point approximation (4.68) can be performed. Then the solution of Eq. (4.70) is

M2 (A, B1 , t|A0 , B01 , t0 ) = M1J (A, t|A0 , t0 )G01 (B1 , t|B01 , t0 ).

G1 (A, t|A0 , t0 ) = G01 (A, t|A0 , t0 ) exp(−k[B](t − t0 )),

(4.63)

(4.71)


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J. Chem. Phys. 141, 104104 (2014)

following general form: t

J + k2 [B] e (τ ) 1 − exp −R 2 τ dτ k2 (t) = 2 R

where G01 (A, t|A0 , t0 ) is the free propagator kernel for particles A obeying the equation similar to Eq. (4.58). Combining Eqs. (4.71), (4.69), and (4.66), we have

−0

G0eff (A, B1 , t|A0 , B01 , t0 ) =

with the corresponding expression (4.47) of general binary differential theory. Using Eq. (4.14) in the integrand in Eq. (4.74), and integrating by parts, we get the expression coinciding with the second part of Eq. (4.47), on condition that the value k2 [B] as compared to the value J in the co-factor in front of the integral √ in Eq. (4.74) is neglected. This is justified, since √ k2 [B] ∼ J ξ , and the value k2 (t)[B] itself is of the order of ξ , so such a simplification results in neglecting the value of the order of ξ ; this is consistent with the binary approximation. In conclusion of this section we shall bring Eq. (4.19) into regular form (4.21) with allowance for time dependence of the pumping rate. However, note that this calls for the assumption of the existence of steady-state (Markovian) value of the rate constant k2 (t), i.e., the existence of the limit at t → ∞ in the first expression of Eq. (4.47), so certain requirements are imposed on the time dependence J(t). If time dependence of pumping is taken into account, exponential representation (4.23) of time transformation matrix is inapplicable. Besides, the kernel J (t|τ ) in Eq. (4.19) is not time difference function. In this case, according to the time shift rule,

G02 (A, B1 , t|A0 , B01 , t0 ) ⎛

1 ⎜ × exp ⎝− k[B] + τA

t (t − t0 ) −

⎞ ⎟ J (τ ) dτ ⎠ ,

t0

(4.72) where G02 (A, B1 , t|A0 , B01 , t0 )=G01 (A, t|A0 , t0 )G01 (B1 , t|B01 , t0 ) is the kernel of free two-particle propagator. Employing Eq. (4.72) in Eqs. (4.65), we obtain the relation between operators Têff and Tˆ which gives the relation between the kernel J (t|t0 ) (4.64) and the IET kernel e (t|t0 ) ⎛ 1 ⎜ (t − t0 ) J (t|t0 ) = e (t − t0 ) exp ⎝ − k[B] + τA t −

⎞ ⎟ J (τ ) dτ ⎠ .

(4.73)

P (τ ) = R−1 (t |τ ) P (t)

t0

1 + k1 [B] τA

t

!x

(1/τA +J (z)+(k1 +k2 )[B])dz

⎜1 + dxeτ ⎜ ⎜ τ ⎜ ⎜ t !x ⎝ (1/τA +J (z)+(k1 +k2 )[B])dz 1 − + k1 [B] dxeτ τA

(P (t) = R (t |τ ) P (τ )) . (4.75)

−1

Note that in the absence of decay and pumping the derived MET kernel (t|t0 ) = (t − t0 |0) corresponds to the first expression in Eq. (4.13) deduced on the basis of exactly solvable many-particle problem. In the presence of pumping the kernel J (t|t0 ) satisfies exact relation (3.19). Thus general basic relations (3.13)–(3.15) and (3.19), the first part of Eqs. (4.12) and (4.13), Eqs. (4.14), (4.15), and (4.24)–(4.27), and matrix kinetic equation (4.19) obtained using exactly solvable model retain their form in the framework of binary modified encounter theory at any rates of spontaneous decay and pumping, depending on time. Now compare expression (4.24) for the rate constant k2 (t) in binary integro-differential theory which, in view of the relation between MET and IET kernels and Eq. (3.15), has the ⎛

(4.74)

The matrix R (t|τ ) can easily be found, taking into consideration that it is a solution of time differential matrix equation d R (τ ) = (K[B] + Q (τ )) R (τ ) . (4.76) dτ In the interval from t to τ Eq. (4.76), in view of K (4.20) and Q(τ ) (4.18), yields two sets of equations one of which relates the matrix element R11 (τ ) to R21 (τ ), and the other relates R22 (τ ) to R12 (τ ). Besides, properties R11 (τ ) + R21 (τ ) = 1 and R12 (τ ) + R22 (τ ) = 1 take place; owing to these properties, each set of equations gives inhomogeneous differential equation for each element. Solving them under initial conditions R11 (t) = R22 (t) = 1, R12 (t) = R21 (t) = 0, we have the desired matrix R−1 (t|τ ) ⎞ !x (1/τA +J (z)+(k1 +k2 )[B])dz ⎟ J (x) + k2 [B] d x e τ − ⎟ ⎟ τ ⎟. t x ⎟ !

(1/τA +J (z)+(k1 +k2 )[B])dz ⎠ τ J (x) + k2 [B] d x e 1+ t

τ

(4.77)

τ

With values J (.t|τ ) (4.73) (t0 ≡ τ ) used as matrix elements of the matrix J (t|τ ) (see Eq. (4.20)), we obtain for the elements !t of the matrix K (t) = J (t |τ ) R−1 (t |τ ) dτ (see Eq. (4.20)) −0

⎞ t !x (1/τA +J (z)+(k1 +k2 )[B])dz 1 ⎝1 + ⎠dτ, k1 (t) = e (t − τ ) e τ + k1 [B] dxeτ τA τ 0 ⎛ t ⎞ !t !x t − (1/τA +J (z)+k[B])dz

(1/τA +J (z)+(k1 +k2 )[B])dz ⎝ ⎠dτ. k2 (t) = e (t − τ ) e τ J (x) + k2 [B] d x e τ t

0

−

!t

⎛

(1/τA +J (z)+k[B])dz

(4.78)

τ


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J. Chem. Phys. 141, 104104 (2014)

Calculating the sum of rate constants (4.78) and using the identity ⎛ x ⎞ !t t

(1/τA +J (z)+(k1 +k2 )[B])dz 1/τA + J (z) + (k1 + k2 )[B] dz⎠ d x ≡ e τ ((1/τA + J (z) + (k1 + k2 )[B])) exp ⎝ − 1, τ

τ

we arrive at Eq. (4.25) and therefore Eqs. (3.15) and (3.13). So it is sufficient to calculate the rate constant k2 (t) by the second formula in Eq. (4.78). As mentioned above, in the binary approximation one can ignore the value k2 (t)[B] as compared to J(t). This gives ⎛ t ⎛ ⎞ ⎞ t t

1/τA + J (z) + k[B] dz⎠ d x ⎠dτ. (4.79) k2 (t) = e (t − τ ) ⎝ J (x) exp ⎝− τ

0

x

Employing the relation following from Eq. (4.14) e (t − τ ) = − ∂τ K (t − τ ) , K (t − τ ) = θ (t − τ ) k (t − τ ) ,

(4.80)

we obtain, as a result of integration by parts, the first formula from Eq. (4.47).

model” at hand, i.e., the binary kinetics in the framework of this method is determined by Eq. (4.5) which, if quantities (4.81) and parameters (4.83) are used, yields the expression √ √ √ Qc (x) = 1 − (1 − 2 γ x) exp(−x) − π γ erf( x). (4.84)

4. Comparison between results coming from different approximations

In this subsection we compare numerically exact kinetics (3.30) for the “target model” of diffusion controlled contact quenching reaction at J(t) = J = const with the kinetics obtained for this model in the framework of the two developed approximations to derive binary kinetic equations of two stage reaction in question. We consider the case of the absence of excited particles (P∗ (0) = 0) at the initial moment of time, i.e., investigate the accumulation kinetics of excited particles under the action of pumping. Let us introduce the reduced dimensionless quantities Q∗ (t) =

J+ R ∗ P (t) ≡ J 2

τA−1 J

+ k[B]

(4.81)

The first quantity is a share of excited particles in units J /R 2 , and the second one is the rate of pumping and deactivation in quenching rate units. It follows from Eq. (3.30) that in this case the quantity Q∗ (t) ≡ Q(x) √ Q (x) = 1 − exp(−x − 2 γ x) √ √ √ √ − π γ {erf( x + γ ) − erf( γ )} (4.82) is the function of parameters γ =

3ξ , π (λ + 1)

x = (λ + 1) k[B]t.

3 kQ = k 1 + ξ + 2

P ∗ (t) ,

J + τA−1 . λ= k[B]

Evidently, it is the expansion of expression (4.82) in the √ parameter γ accurate to the terms of the order of γ . To obtain the kinetics in the framework of simple superposition approximation, it is required that the quantity kJ (t) be calculated which is necessary for the calculation of the quantity k˜2 (t) (4.45). On the assumption that at small density parameters in such a calculation one can use the steady-state value kJ ≡ kQ , Eq. (4.45) gives the result of paper8 that may be represented as

(4.83)

The first parameter is the reduced density one, and the second parameter is time in units ((λ + 1)k[B])−1 . It is of interest that the parameter γ can be a small quantity even at not small values of the density parameter if the value λ is rather large. As mentioned above, general method of extracting pair channels gives the results coinciding with those for the “target

2 9 2 1 a + ξ . J+ + k[B] τA D 4 (4.85)

At arbitrary values of the parameter ξ this result does not coincide with general expression (3.35) for exactly solvable problem. So simple superposition approximation fails to give formal solution of exactly solvable problem for the reaction under consideration, unlike the case with elementary reaction of quenching. However, at small values of the parameter ξ the quenching constant coincides with result (4.9) of exactly solvable model. It is seen that the quantity kQ coincides with the rate constant k (this is necessary for the coincidence of Eqs. (4.45) and (4.47)) only at conditions (4.46b) restricting pumping and deactivation rates. However, the deviation of the quantity kQ in Eq. (4.45) from the quantity k at high pumping and deactivation rates does not mean that the kinetics in superposition approximation at small parameters ξ differs essentially from the exact one. To calculate such a kinetics, let us find the quantity k˜2 (t) (4.45) using J(t) = J = const and kQ (t) = kQ from Eq. (4.85). The solution of Eq. (3.9), with the quantity k2 replaced by the quantity k˜2 (t) (4.45) (in view of


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A. B. Doktorov

J. Chem. Phys. 141, 104104 (2014)

FIG. 1. Reduced accumulation kinetics for γ = 0.01. Solid line: exact kinetics. Dashed curve: kinetics in superposition approximation. Dashed-dotted curve: kinetics in channel extraction approximation.

Eq. (4.85)), gives the expression √ Qs (x) = 1 − (1 − 2 γ x) exp (−x) −

√ πγ erf( αx) α

1 √ π exp (−x) − 1− γ α α−1 √ × erf (α − 1)x − 2Dowson( x) , (4.86)


Figs. 1–3. As is seen, both approximations Qc (x) and Qs (x) give good agreement with the exact solution. Moreover, as the density parameter increases, the superposition approximation turns out to be more accurate. However, an essential disadvantage of the superposition approximation is the necessity of finding the quantity kQ (t) (or the steady-state value kQ which can be determined under constant pumping only). In the general case calculation of kQ (t) is possible solely at conditions (4.46b) that restrict pumping and spontaneous decay rates.

where Dowson (y) is the Dowson integral y Dowson(y) = exp(−y ) 2

exp(t 2 ) dt

V. SUMMARY

(4.87)

0

and

α =1+

π π πγ 1 + γ + γ. 4 2

(4.88)

The behavior of the kinetics Q(x), Qc (x), and Qs (x) at different values of the reduced density parameters is shown in


The well-known problem of fluorescence concentration quenching determined by spontaneous decay of excited molecules in the presence of pumping of arbitrary intensity which, in the general case, is time dependent, is analyzed in the frame work of unified many-particle consideration of a reacting system. From the point of view of chemical kinetics, such a reaction is a multistage reaction with one elementary stage being irreversible bimolecular (pseudo-monomolecular) quenching reaction, and another—reversible monomolecular transformation reaction. So the analysis of the problem in the frame of unified many-particle treatment of the reacting system is of interest both for the investigation of fluorescence, and from the standpoint of examining the methods for the derivation of non-Markovian binary kinetic equations of multistage reactions. Using the above unified approach we have generalized the exact relation between the kinetics of multistage reaction under study and that of the elementary stage of quenching obtained in the literature using a simplified technique. On its basis and with the aid of kinetic equations for elementary bimolecular stage, general kinetic equations of multistage reaction under study have been derived both in differential and equivalent integro-differential forms. For clarity we considered the exact solution of the problem at arbitrary concentration of a quencher for “target model.” The model describes the quenching reaction of immobile particles A (excited A∗ and unexcited ones) with


104104-20

A. B. Doktorov

mobile point particles B of arbitrary concentration. For the first time, kinetic equations of this model are formulated both in differential and in a more general integro-differential form. This allowed us (by way of expansion in density parameter) to derive both differential rate equations, and integro-differential (retarded) binary kinetic equations, as well as to establish their equivalence, and the possibility of reduction to equations of formal chemical kinetics. An interesting result of such a reduction is the appearance of extra negative “rate constant” in the set of equations of formal chemical kinetics. From the standpoint of formal chemical kinetics this is related to the change in the course of elementary stage of quenching under intensive pumping. The value of this constant is proportional to the square root of the density parameter. So it is evident that its formation is associated with non-stationary (non-Markovian) part of elementary stage of quenching in the absence of pumping. Proportionality of this constant to a small parameter does not mean its smallness as compared to the rate constant of elementary rate of quenching in the absence of pumping, since it is proportional to a square root of intensive pumping rate, and can become essential. In the framework of unified many-particle approach we have analyzed the applicability of two most widely used general methods for the derivation of non-Markovian binary kinetic equations, namely, simple superposition approximation and the method of extracting pair channels in three-particle correlation evolution. As is shown, application of simple superposition approximation results in differential equations which, in contrast to equations in the absence of pumping, are correct at small density parameter, just as it should be. At small density parameters, such an approximation for the “target model” of diffusion controlled contact reaction gives the kinetics consistent with the exact solution of the problem. However, a significant demerit of the method consists in the necessity of finding the constant kJ (t), and in the general case this is possible solely at not very high pumping and spontaneous decay rates. It is demonstrated that, unlike this technique, the method of extracting pair channels in three-particle correlation evolution leading to integro-differential (retarded) binary kinetic equations is free of this demerit, and is valid on the only condition of density parameter smallness. ACKNOWLEDGMENTS

The author is grateful to the Russian Foundation of Basic Research for financial support (Project No. 12-03-00058). The author is also thankful to Dr. Alexey A. Kipriyanov, Professor N. N. Lukzen, and Dr. Alexander A. Kipriyanov for useful discussions. APPENDIX: DIFFUSION INFLUENCED CONTACT QUENCHING REACTION

As mentioned above, general kinetics of elementary irreversible quenching reaction in the framework of the “target model”23, 24 is given by Eq. (3.8) where the reaction rate constant k(t) follows from Eqs. (4.38) and (4.37). Sometimes, at small concentrations [B] of a quencher the expansion of the

J. Chem. Phys. 141, 104104 (2014)

exponent into a series to the first order terms in concentration is used:4 t S (t) ≈ 1 − [B] k (τ ) dτ . (A1) 0

However, such an approximation does not agree with binary approximation. Determination of the Laplace transform (3.29) of the kernel of integro-differential equation corresponding to"kinetics (A1), according to Eq. (4.14), leads to the kernel e (t) of Integral Encounter Theory (IET) which has a limited time interval of applicability. Correct transition to the binary approximation based on the density parameter smallness consists in the representation of the rate constant k(t) as k (t) ≈ k + (k (t) − k) ≡ k + k (t) ,

(A2)

that separates the Markovian value of the rate constant k (steady-state reaction rate constant) (see Eqs. (3.15) and (4.15)) and its non-Markovian part k(t).14, 15, 28 It is of interest that over the entire time range the value k(t) is not small as compared to the value k (see, for example, Eq. (3.16)). However, when examining the binary kinetics obtained by the expansion of just the exponent containing the non-Markovian part of the rate constant, we have ⎞ ⎛ t S (t) ≈ exp (−[B] k t) ⎝1 − [B] k (τ ) dτ ⎠ . (A3) 0

At small density parameter the second term between the brackets really proves to be small in comparison with unity over the whole range of binary kinetics (see, for instance, Eqs. (4.2) and (4.4)). According to Eq. (4.4), finding the Laplace transform (3.29) of the kernel of integro-differential equation corresponding to kinetics (A3) results in the Laplace transform of the kernel L (s) = eL (s + k[B]) (see the first part of Eq. (4.12)), or the kernel (t) = e (t)exp ( − [B] k t) (the first part of Eq. (4.13)) of Modified Encounter Theory (MET). For illustration, consider a more general problem of diffusion influenced contact quenching reaction. It differs from the problem studied above in the fact that at contact quenching proceeds not at an infinite rate but at a finite one which is specified by the reaction constant kr .33 In this case4, 33 √ kr 2 k (t) = k 1 + exp(α t)erfc(α t) , kD (A4) √ kr k 2 k (t) = exp(α t)erfc(α t), kD where diffusion constant kD = 4π aD (3.17), and the Markovian (steady-state) rate constant k = lim k (t) = t→∞

kD kr ≡ 4 π D aeff . kD + kr

(A5)

Further we introduce the effective radius and the density parameter, respectively, aeff = a

kr , kD + kr

ξ=

4 3 π a [B]. 3 eff

(A6)


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A. B. Doktorov

J. Chem. Phys. 141, 104104 (2014)

The parameter α appearing in Eq. (A4) is kr kr D D kr k[B] 1+ ≡ . (A7) α= ≡ 2 a2 kD kD aeff kD 3 ξ

Though we use the density parameter that is assumed to be small in the binary approximation, the value k(t) is not proportional to this parameter, and, as already stated, is not small as compared to k. However, the binary kinetics is of the form

√ t kr k[B] 2 S (t) ≈ exp (−[B] k t) 1 − exp(α t)erfc α t − 1 + 2α 2 kD α π

≡ exp (−[B] k t) 1 −

exp(α 2 t)erfc 3ξ k[B] α

√

αt −1

t +2 , π

and its Laplace transform is S L (s) = S0L (s + k[B]), where ⎛ ⎞

k[B] 3ξ k[B] 1 1 k k[B] ⎠, S0L (s) = 1 − √ r √ ≡ ⎝1 − √ k s s s kD s α + s k[B] + D k 3ξ k[B]s

(A8)

(A9)

r

i.e., actually, the second terms between the brackets in Eqs. (A8) and (A9) are proportional to the square root of the density parameter and are small over the whole time range (4.4) of binary kinetics. As for the term proportional to the square root of the density parameter in the denominator of the second term in Eq. (A9), it is not small. Correspondingly, in the binary approximation the Laplace transform of IET kernel (in terms of which the Laplace transform of MET kernel is expressed) is 2 sa kr kD 1 + D eL (s) = sk L (s) = 2 sa kr + kD 1 + D ⎛

⎞ √ 3ξ k[B]s ⎠ . (A10) ≡ k ⎝1 + √ k[B] + kD k 3ξ k[B]s r The Laplace transform k2L (s) of the constant k2 (t) (4.48) at J(t) = J = const and its steady-state value k2 are expressed " through Le (s) as follows: J L e (s) − eL (s + R 2 ) , 2. sR J k2 = 2. eL (0) − eL (R 2 ) R √ 3ξ k[B] Jk =− . √ R k[B] + kD R k 3ξ k[B]

k2L (s) =

(A11)

r

The deduced expressions enable one to apply all general results of the work to contact diffusion influenced reaction considered in the binary approximation.

1 V.

M. Agranovich and M. D. Galanin, Electron Excitation Energy Transfer in Condensed Matter (North-Holland, Amsterdam, 1982). 2 Th. L. Nemzek and W. R. Ware, J. Chem. Phys. 62, 477 (1975). 3 M. Sikorski, E. Kristowiak, and R. P. Steer, J. Photochem. Photobiol. A 117, 1 (1998). 4 A. I. Burshtein, Adv. Chem. Phys. 114, 419 (2000); 129, 105 (2004). 5 S. A. Rice, in Comprehensive Chemical Kinetics, Diffusion-Limited Reaction Vol. 25, edited by C. H. Bamford, C. F. H. Tripper, and R. G. Kompton (Elsevier, Amsterdam, 1985). 6 A. Szabo, J. Phys. Chem. 93, 6929 (1989). 7 J. Sung, K. J. Shin, and S. Lee, J. Chem. Phys. 101, 7241 (1994). 8 W. Naumann and A. Szabo, J. Chem. Phys. 107, 402 (1997). 9 A. V. Popov, E. Gould, M. A. Salvitti, R. Hernandex, and K. M. Solntsev, Phys. Chem. Chem. Phys. 13, 14914 (2011). 10 D. V. Dodin, A. I. Ivanov, and A. I. Burshtein, J. Chem. Phys. 138, 124102 (2013). 11 N. Agmon and M. Gutman, Nat. Chem. 3, 840 (2011). 12 S. Lee, C. Y. Son, J. Sung, and S.-H. Chong, J. Chem. Phys. 134, 121102 (2011). 13 A. A. Kipriyanov, I. V. Gopich, and A. B. Doktorov, Physica A 255, 347 (1998). 14 A. B. Doktorov and A. A. Kipriyanov, J. Phys.: Condens. Matter 19, 065136 (2007). 15 A. B. Doktorov, in Recent Research Development in Chemical Physics, edited by S. G. Pandalai (Transworld Research Network, Kerala, India, 2012), Vol. 6, Chap. 6, pp. 135–192. 16 N. N. Lukzen, A. B. Doktorov, and A. I. Burshtein, Chem. Phys. 102, 289 (1986). 17 L. Monchick, J. L. Magee, and A. H. Samuel, J. Chem. Phys. 26, 935 (1957). 18 T. R. Waite, Phys. Rev. 107, 463 (1957). 19 A. A. Kipriyanov, O. A. Igoshin, and A. B. Doktorov, Physica A 268, 567 (1999). 20 M. Doi, J. Phys. A 9, 1465 (1976). 21 R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (John Wiley & Sons, New York, 1978), Vols. 1 and 2. 22 M. von Smoluchowski, J. Phys. Chem. 92, 129 (1916). 23 I. Z. Steinberg and E. Katchalski, J. Chem. Phys. 48, 2404 (1968). 24 M. Tachiya, Radiat. Phys. Chem. 21, 167 (1983). 25 G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill Company, New York, 1968). 26 R. Kapral, Adv. Chem. Phys. 48, 71 (1981). 27 V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1981) (in Russian).


104104-22 28 A.

A. B. Doktorov

A. Kipriyanov, I. V. Gopich, and A. B. Doktorov, Chem. Phys. 187, 241 (1994). 29 G. Oshanin, A. A. Ovchinnikov, and S. F. Burlatsky, J. Phys. A: Math. Gen. 22, L977 (1989). 30 I. V. Gopich, A. A. Ovchinnikov, and A. Szabo, Phys. Rev. Lett. 86, 922 (2001).

J. Chem. Phys. 141, 104104 (2014) 31 A.

A. Kipriyanov, A. A. Kipriyanov and A. B. Doktorov, J. Chem. Phys. 133, 174508 (2010). 32 E. Schmid and H. Ziegelmann, The Quantum Mechanical Three-Body Problem (Pergamon Press, Braunschweig, 1978). 33 F. C. Collins and G. E. Kimball, J. Colloid Interface Sci. 4, 425 (1949).


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