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The in uence of relative motion peculiarities of reactants in solutions on the rate of physico-chemical processes A.A. Kipriyanov a , O.Yu. Kulyatina , A.B. Doktorov b; ∗ b Institute

a Novosibirsk State University, Novosibirsk, 630090, Russia of Chemical Kinetics and Combustion, the Russian Academy of Sciences, Novosibirsk, 630090 Russia

Received 15 May 2000

Abstract The quasistationary rate constant of irreversible reactions of the type of trapping by diusing traps excitations, electrons or particles migrating by stochastic jumps of arbitrary length in condensed matters has been calculated. It has been shown that the two-scaled nature of the relative translational mobility in reactions proceeding near the contact of reactants gives rise to new kinematic mechanisms of such reactions. These mechanisms are characterized by unusual c 2001 Elsevier Science dependence of the reaction rate constant on microscopic parameters. B.V. All rights reserved.

1. Introduction Rather dilute systems of interacting particles are characterized by a small density parameter = 43 R3 [C] 1 ;

(1.1)

where R is the sum of eective radii of two interacting particles (for example, van der Waals radii), [C] are their concentrations. In the systems the frequency of binary encounters (collisions) is the most important kinetic parameter = k[C] ∗ Corresponding author. Fax: +7-3832-342350. E-mail address: [email protected] (A.B. Doktorov).

c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 3 2 2 - 8

(1.2)

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which plays a decisive role in both relaxation, and plausible chemical conversions. Here k is the frequency constant independent of concentration. As is known [1], in rare ed gases k = w ;

(1.3)

2

where = R is the gas-kinetic cross section, w is the mean heat relative velocity. Though dilute liquid or solid solutions of reacting particles resemble a rare ed “gas” of reactants, the frequency of binary encounters is quite dierent. According to the currently existing ideas [2], in condensed matters impurity particles move by random walks, i.e., the relative displacement is the jumps occurring with the frequency −1 0 , with the mean step being of a nite size 0 . The frequency constant of dilute particles has been calculated in Smoluchowsky’s coagulation theory [3] which postulates that 2 relative random walks result in continual diusion (0 → 0; 0 → 0; D = 600 = const) k = 4DR :

(1.4)

Then these concepts were extended to the theory of reactions in liquid solutions [4]. If the approach of particles is described as a continual diusion, in principle, no question arises of how it is organized. Diusion description of space migration is universal, and is always justi ed in macroscopic scales. However, it can cease to be valid when the distances between partners are reduced to molecular sizes. Such is the case with rare ed gases where diusion results from a sequence of free travels, and the reaction starts and ends within one of them when their length is large as compared to the reaction zone size. Similarly, in condensed matter where migration is by random walks of particles a single step is sucient if it sharply promotes the reaction. This is just what happens when an electron or exciton is trapped in liquid and solid solutions. These concepts correspond to the hopping mechanism of reaction [5] when 4 3 1 R : (1.5) 3 0 The above constant diers from diusion one by the dependence on electron (or exciton) migration rate and constants (specifying R) that de ne their transfer to acceptor. The hopping mechanism of trapping the electron moving in the liquid has been experimentally identi ed [6], and successfully interpreted, according to Eq. (1.5). The analog of the phenomenon has also been encountered in studies of electron conduction in chalkogeneous glassy semiconductors [7]. Though migration mechanisms are physically dierent, all constants given in Eqs. (1.3), (1.4) and (1.5) may be put in a universal form corresponding to ergodic concepts v (1.6) k= ; where v is the “reaction volume” (the interaction zone volume), and is the mean residence time of reactants in it. In all cases under consideration, for penetrable sphere model v = 43 R3 , = 43 R= w for rare ed gases, and = R2 =3D or = 0 in condensed solutions for diusion or hopping crossing of the reaction zone, respectively. k=

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Of course, the intermediate, the so-called mixed mechanism may take place between the cases of diusion and hopping reaction mechanisms. It corresponds to the values of the mean lengths of 0 that are intermediate in comparison with the reaction zone size. In this case calculation of the mean residence time in the reaction zone is more dicult than that for extremely small and extremely large values of 0 . Besides, the reaction zone de nition itself has a clear meaning only in the model of rigid spheres with well-de ned boundaries. Both physical interactions in physical kinetics, and microscopic rates of elementary chemical conversion in condensed solutions have real space dependence. Thus, generally speaking, one has always to do with the determination of eective cross sections or radii of eective spheres (dependent on mobility). That is why much attention is given to the cases where at any 0 the problem can be solved rigorously without resort to “black” sphere notions. Such a program was rst realized in Ref. [8] for migration mechanism peculiar to the electron self-stabilizing in the liquid (the Torrey model [9]). It was used to interpret the quenching rate estimates in migration of excitations over impurity centers in solids [10]. However, the above situations do not exhaust all possible motions of reactants in condensed solutions. The encounter rate constant is de ned by the relative motion of partners; only in rare ed gases taking account of this fact gives just a numerical coecient in the de nition of the frequency of encounters. In condensed solutions the cases are possible where rapid migration for long distances results in trapping of excitations or electrons by heavy particles with limited mobility moving by diusion. In solids this phenomenon was encountered when considering the concentration decay kinetics of autolocalized holes (Vk -centers) in alcalino-haloid crystals [11,12], and in studies of photocurrents in varizone semiconductors [13,14]. One of the limiting cases of such a motion is migration in two parallel independent channels with essentially dierent characteristic scales of migration. This type of migration has also been experimentally identi ed in studies of excess electron transport properties in nonpolar liquids [15]. Such a case of the so-called two-scaled migration was theoretically treated in Ref. [16] in the frame of partially penetrable rigid sphere model (the step approximation of the rate of the elementary event of transfer). One of the partners was assumed to move by continual diusion, while the other one — by in nitely large jumps. The present contribution deals with a more general case of spatially dependent rate of the elementary event of trapping (by diusing traps) excitations or electrons that migrate by stochastic jumps of arbitrary length in condensed matters. Section 2 is devoted to a mathematical statement of the problem. We dwell brie y on some points from the kinetic theory of elementary irreversible reactions necessary for the calculation of the rate constant. In Section 3 the concept of the attendant reaction pair is introduced on the basis of the operator technique. In such a pair free motion depends solely on the motion by stochastic jumps of one of the reactants in the pair, while diusion motion of the other partner speci es the reaction operator of the pair. Owing to the restatement performed, in Section 4 a small parameter for near-contact reactions typical for chemistry is used (the smallness of the decay scale of the elementary event rate as compared to the sum of Van der Waals radii of reactants), and the asymptotically

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exact expression for the rate constant in the framework of the Torrey model is derived. The rate constant corresponding to the two-scaled migration mechanism that appears in the obtained expression is to be found. Calculation of this rate constant for arbitrary dependence of the elementary event rate is done in Section 5. In Section 6 the limiting kinematic mechanisms are considered, expressions for the rate constant calculation given by a quadrature are presented, and the limits of their realization are established. The obtained results are shown to be independent of a particular method of the hopping mobility organization. 2. Statement of the problem The Markovian kinetics of bulk bimolecular irreversible reactions proceeding by formal patterns A + B → B and A + B → C obeys the macroscopic equations, @t [A]t = −k[B][A]t and @t [A]t = @t [B]t = −k[B]t [A]t ;

(2.1)

respectively [17,18]. Here [A]t and [B]t are the mean concentrations of A and B reactants, and the kinetic coecient k is the Markovian (stationary) rate constant. Eqs. (2.1) agree completely with the law of mass action of formal chemical kinetics [19]. One of the main goals of the microscopic theory of elementary reactions in liquid solutions (including the present contribution) is the development of rate constant calculation methods based on microscopic concepts of chemical reaction course. Traditional description takes account of a twofold eect of a solvent. The theory of an elementary event considers the in uence of a solvent on dynamic rearrangement of reactants that results in irreversible chemical conversion. According to the theory of migration-in uenced reactions, the presence of a solvent makes the translational motion of reactants a stochastic process. Based on these notions, one can formulate a mathematically closed procedure of the Markovian rate constant calculation [17,20]. However, the realization of the procedure in a general form is rather dicult. The performed studies have revealed the limiting mechanisms of the reaction that (1) are easy to interpret physically, (2) allow the expressions for the Markovian rate constant to be given by a quadrature. These are: diusion, hopping and mixed mechanisms [8]. In the traditional approach the Markovian rate constant is expressed in terms of the microscopic values of the reaction pair as [17,20] Z k = dr U (r)n(r) : (2.2) Here U (r) is the local rate of the elementary event which depends on the relativeposition vector between A and B reactants. The probability density n(r) of nding the reactants with the relative-position vector r in the reaction pair apart from the elementary event rate depends on the nature of their translational motion. Assume that the reactant of A type is light; it moves translationally by stochastic jumps. The light reactant’s partner on encounter is a heavy reactant of B type. Its translational motion is

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described by continual diusion with the coecient DB . Under the assumptions made, the equation for the probability density takes the form [5,16] Z 1 (2.3) − U (r)n(r) − [n(r) − dr0 fA (r|r0 )n(r0 )] + DB r n(r) = 0 : A The equation should be complemented by the boundary conditions n(r → ∞) = 1;

@r n|r=R0 = 0 :

(2.4)

The rst condition agrees with the idea that the reaction cannot aect the distribution of reactants for large distances. The second condition represents the absence of the diusion ux through the sphere of the radius R0 . Therefore, the radius R0 is the distance of the closest approach of reactants. In order of magnitude it is equal to the sum of Van der Waals radii of reactants. Thus, we accept the widely used description of reactants by the model of rigid spheres. Eq. (2.3) de nes the stationary evolution of the reaction pair in relative coordinates. The rst term in the left-hand side is responsible for the pair decay due to chemical conversion. The second and the third terms describe the evolution caused by the translational motion of A and B reactants, respectively. The time A is the mean time between successive jumps of A reactant. The function fA (r|r0 ) is the density of the conditional probability to nd A reactant at point r after a jump from point r0 . The explicit form of fA (r|r0 ) depends on the type of the hopping mobility organization. At the rst stage of the investigation it is convenient to consider a speci c mechanism of the hopping mobility organization. The most physically clear models of the hopping mobility organization are formulated in the framework of the Torrey representation [9], according to which A reactant may be found in two stages. In the ground state it is at rest. Within the mean time A it goes into the excited state, moves rapidly for some time, and then is captured by the ground state. The travelling time distribution is assumed to be Poisson. Under these assumptions, the density of the conditional probability of a jump for A reactant obeys the equation [5] fA (r|r0 ) = A2 r fA (r|r0 ) + (r − r0 ) :

(2.5)

Here r is the Laplace operator, and the value A is the most probable length of A reactant jumps in laboratory reference system. Note that the conditional probability density fA describes the result of A reactant jump in the presence of B reactant. That is why fA (r|r0 ) depends parametrically on the coordinate of B. This dependence arises in establishing the boundary conditions for Eq. (2.5) that are necessary for unambiguous de nition of their solutions. Place B reactant at rest in the origin of coordinates. Then the boundary condition corresponding to the absence of diusion ux through the sphere of the radius R0 in the excited state is of the form [21] @r fA (r|r0 )|r=R0 = 0 :

(2.6)

At large distances and times the translational motion of A reactant by stochastic jumps turns into diusion [5,22]. The coecient DA of such a macrodiusion is expressed in

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terms of the mean square displacement A in one jump Z 2 2 dr (r − r0 )2 fA (r|r0 ) : DA = A =(6A ); where A = lim r0 →∞

(2.7)

In the framework of the Torrey representation it is easily established [5] that the mean square displacement A is related to the length A as 2 A = 6A2 :

(2.8)

Thus the macrodiusion coecient of A reactant is readily expressed in terms of the parameters of the model under consideration DA = A2 =A :

(2.9)

According to the foregoing, the mathematical statement of the problem consists in the calculation of the rate constant k by recipe (2.2) using the solution of Eq. (2.3). To develop physical approximation, formulate the problem in terms of the operator technique [21]. Thus, Eq. (2.3) takes the form (Vˆ + LˆA + LˆB )n(r) = 0 :

(2.10)

The reaction operator Vˆ and the free motion operators LˆA and LˆB are de ned by the kernels V (r|r0 ) = −U (r)(r − r0 ) LA (r|r0 ) = −A−1 [(r − r0 ) − fA (r|r0 )];

LB (r|r0 ) = DB r (r − r0 ) ;

(2.11)

where (x) is the Dirak delta-function. Introduction of the resolvent operator ˆ Vˆ + LˆA + LˆB ) = −1ˆ (Vˆ + LˆA + LˆB )gˆ = g(

(2.12)

gives formal solution of Eq. (2.10). The unit operator 1ˆ is de ned by the kernel 1(r|r0 ) = (r − r0 ). The equation is complemented by the boundary conditions @r g(r|r0 )|r=R0 = 0;

lim g(r|r0 ) = 0

r→∞

(2.13)

that ensure the ful lment of the boundary conditions for the density n(r) derived using the resolvent g. ˆ Eq. (2.10) is recast as (Vˆ + LˆA + LˆB )(n(r) − 1) = −Vˆ |1(r)i :

(2.14)

Here 1(r) is a unit function equal to unity for all accessible values of the argument r, i.e., r¿R0 . In the derivation of Eq. (2.14) we have taken into account that the function 1(r) is a static contour of operators LˆA and LˆB [22] LˆA |1(r)i = LˆB |1(r)i = 0 :

(2.15)

The function n(r) − 1 goes to zero with r → ∞. Thus, we can operate on both sides of Eq. (2.14) by the resolvent operator g, ˆ and use Eq. (2.12). This gives the desired formal solution of Eq. (2.10) n(r) = 1 + gˆVˆ |1(r)i :

(2.16)

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In the operator technique the introduction of the free resolvent operator gˆ0 is always useful. It describes the reaction pair evolution in the absence of chemical conversion (Vˆ ≡ 0), and is de ned as (LˆA + LˆB )gˆ0 = gˆ0 (LˆA + LˆB ) = −1ˆ

(2.17)

with allowance for appropriate boundary conditions similar to Eq. (2.13). Eqs. (2.12) and (2.17) give the relation between the resolvent gˆ and free resolvent gˆ0 gˆ = gˆ0 + gˆ0 Vˆ gˆ = gˆ0 + gˆVˆ gˆ0 :

(2.18)

The above equations are analogous to the Lippmann–Schwinger equations for the resolvent of the scattering theory [23]. Eqs. (2.17) make it possible to obtain the integral equation for the density n(r). Operating on both sides of Eq. (2.14) by the free resolvent operator, we have n(r) = 1 + gˆ0 Vˆ |n(r)i :

(2.19)

In terms of the operator technique Eq. (2.2) for the rate constant takes the form Z Z k = − dr Vˆ |n(r)i = − dr(Vˆ + Vˆ gˆVˆ )|1(r)i : (2.20) Note that the expression in circular brackets is a stationary t-matrix of the reaction pair [22]. Eqs. (2.12) – (2.20) provide standard operator description of the reaction pair.

3. Attendant reaction pair The operator representation allows one to develop several mathematically identical descriptions of the reaction pair evolution. Further such descriptions may be used to develop physical approximation. Above all we are interested in the case where diusion motion of B reactants does not aect considerably the process of reactants moving apart by macrodistances, but has an essential in uence on the rate constant value. This in uence may be due to the fact that the motion of B reactants by small displacements controls the delivery to the reaction zone from its vicinity. The reactants are delivered to the vicinity primarily by A reactant motion. Thus, the in uence of the diusion motion of B reactant may be interpreted as the distortion of the reactivity, while the change in the relative distance in the pair depends solely on the motion of A reactant. To obtain a mathematical description of the reaction pair adequate to the above physical notions, we introduce the free motion resolvent operator of A reactant by the equations LˆA gˆ0A = gˆ0A LˆA = −1ˆ :

(3.1)

They should be complemented by the boundary conditions of the type of Eqs. (2.13). Eqs. (2.17) and (3.1) yield gˆ0 = gˆ0A + gˆ0 LˆB gˆ0A = (1ˆ + gˆ0 LˆB )gˆ0A :

(3.2)

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Substitute this expression for the free resolvent operator into Eq. (2.18), and introduce new operators ˆ gˆatt = (1ˆ + gˆ0 LˆB )−1 g;

Vˆatt = Vˆ (1ˆ + gˆ0 LˆB ) :

(3.3)

We have gˆatt = gˆ0A + gˆ0A Vˆatt gˆatt = gˆ0A + gˆatt Vˆatt gˆ0A :

(3.4)

Being structurally similar to Eqs. (2.18), the above equations describe the evolution of some attendant pair in which translational migration is determined by the motion of A reactant only. Thus, the operator gˆatt appearing in Eq. (3.3) has the meaning of the resolvent operator of this pair, and obeys the operator equations (Vˆatt + LˆA )gˆatt = gˆatt (Vˆatt + LˆA ) = −1ˆ

(3.5)

that follow from (3.1) and (3.4). We identify the operator Vˆatt involved in Eq. (3.3) with the reaction operator of the attendant pair, and call it the attendant reaction operator. Unlike Vˆ (see (2.11)), it is a nonlocal operator, i.e., there exists no concept of the local rate of the elementary event for the attendant pair. Nevertheless, similar to the operator Vˆ , the operation of Vˆatt on nonnegative basic function ’(r) gives the nonpositively de ned function Vˆatt |’(r)i60 :

(3.6)

Since this inequality plays an important role in the subsequent discussion, it will receive particular attention. First the de nition of the attendant reaction operator is recast as Vˆatt = A−1 Vˆ gˆ#0 where gˆ#0 = A (1ˆ + gˆ0 LˆB ) :

(3.7)

gˆ#0

appearing in Eqs. (3.7) has all imporInequality (3.6) is valid because the operator tant properties of the free resolvent operator. In particular, the inequality gˆ#0 |’(r)i¿0

(3.8)

equivalent to inequality (3.6) is valid. Inequality (3.8) is most easily proved in the case where A reactant moves by in nitely large jumps (the two-scaled migration mechanism [16]). It follows from Eqs. (2.17), (2.11) and (3.7) that gˆ#0 |’(r)i = −A gˆ0 LˆA |’(r)i = gˆ0 [’(r) − fˆA |’(r)i] ;

(3.9)

where the operator fˆA is de ned by the kernel fA (r|r0 ). In the limit A → ∞ one can neglect the second term in square brackets in Eq. (3.9) that describes the return of A reactant to point r in the process of random walks [5]. Obviously, the free resolvent operator has the required property. Thus, when A → ∞ inequality (3.8) is proved. In the case of a less jump length A , it is necessary to allow for the return of A reactant to point r. The return is maximum in the diusion limit = LˆA ' DA r =. Then, in view of Eqs. (2.17), DB DA A gˆ0 (LˆA + LˆB ) |’(r)i = ’(r)¿0 : (3.10) gˆ#0 |’(r)i = A 1ˆ + DA + DB DA + DB To study the case of arbitrary jump length of A reactant, we obtain the equation for the operator gˆ#0 and use a speci c way of the hopping mobility organization in the

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framework of the Torrey representation. Substitution of Eqs. (3.2) into Eq. (2.17), in view of Eqs. (3.1) and the de nition of the operator gˆ#0 , gives the conjugate operator equations 1 1 1 1 (3.11) − 1ˆ + gˆ0A LˆB gˆ#0 = gˆ#0 − 1ˆ + gˆ0A LˆB = −1ˆ : A A A A Formally they are similar to Eqs. (3.1). Making allowance for the explicit form of the operator LˆA , we rewrite Eqs. (3.1) as 1 1 1 1 (3.12) − 1ˆ + fˆA gˆ0A = gˆ0A − 1ˆ + fˆA = −1ˆ : A A A A However, there is an important dierence between these equations. To nd it out, take account of Eqs. (3.12) and put Eqs. (3.11) in the form 1 1 1 1 − 1ˆ + LˆB + gˆ0A fˆA LˆB gˆ#0 = gˆ#0 − 1ˆ + LˆB + gˆ0A fˆA LˆB = −1ˆ : A A A A (3.13) Now use the property exhibited by the operator fˆA in turning to the hopping mobility in the framework of the Torrey representation, i.e., DB ˆ = − DB 1ˆ : gˆ0A fˆA LˆB = 2 gˆ0A (fˆA − 1) DA A

(3.14)

Here we employ Eqs. (2.5) and (3.12). Substituting Eq. (3.14) into Eq. (3.13) yields the desired operator equations for the calculation of the operator gˆ∗0 (further it will stand for the operator gˆ#0 for the Torrey model) 1 ˆ 1 ˆ ∗ ∗ ˆ ˆ (3.15) − ∗ 1 + LB gˆ0 = gˆ0 − ∗ 1 + LB = −1ˆ ; A A where the parameter ∗A has the dimensionality of time. It is de ned as ∗A =

DA A : DA + DB

(3.16)

The operator equations (3.15) coincide with the equations for the free resolvent operator in the two-scaled migration mechanism accurate to rede nition of the mean time between successive jumps of in nite length (with ∗A instead of A ) [16]. Thus, the operator gˆ∗0 is the free resolvent operator of such a two-scaled pair (hereinafter called the “asterisk” pair), and the validity of inequality (3.8) at all values of A is obvious. Now we can discuss the important dierence between free resolvents g0A (r|r0 ) and g0∗ (r|r0 ) =i:e:; g0# (r|r0 )=. In any two-scaled pair the jump of in nite length of A reactant interrupts the encounter of reactants, and prevents space correlations arising on the encounter from extending for macroscopically long distances. Formally this means that, unlike g0A (r|r0 ), the free resolvent g0∗ (r|r0 ) has no slowly decreasing diusion asymptotics at r → ∞, and has a nite norm Z Z (3.17) drg0∗ (r|r0 ) = dr0 g0∗ (r|r0 ) = ∗A :

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The above equalities are obtained by integrating Eqs. (3.15). Thus, in the case of the hopping mobility organization in the framework of the Torrey representation the kernel of the attendant reaction operator is expressed in terms of the well-studied free resolvent of the two-scaled reaction pair ∗ Vatt (r|r0 ) = −U (r)−1 A g0 (r|r0 ) :

(3.18)

This is just what enables us to develop physical approximation for the calculation of the near-contact reaction rate constant in the next sections. The explicit form (3.18) of the attendant reaction operator makes it possible to study the scale of the reactivity distortions in the attendant pair due to diusion mobility of B reactant. First note that the decay of the kernel g0∗ (r|r0 ), for example, as the function r, has the scale l∗ of diusion displacement for the two-scaled pair life time [16]. i.e., p (3.19) l∗ = DB ∗A : So if A reactant moves by continual diusion (A → 0), the parameter l∗ → 0, and the kernel g0∗ (r|r0 ) is delta-shaped. To nd it explicitly, neglect the operator LˆB in Eqs. (3.15). In view of Eq. (3.16), Eq. (3.18) gives DA (r − r0 ) : (3.20) DA + DB Thus, in the case at hand Vˆatt retains the character of the reaction operator Vˆ but with renormalized rate of the elementary event. Additional factor (compare with Eq. (2.11)) appears due to retention of the action accumulated by the reactant when in the reaction zone [5,8]. At any nonzero jump length of A reactant the parameter l∗ 6= 0, and the attendant operator becomes nonlocal. According to Eq. (3.18), the reaction zone radius increases by the value l∗ . Such an expansion of the zone is related to the fact that the time A between successive jumps is enough for B reactant to get into the reaction zone from its vicinity by diusion. Additional B reactant motion by small displacements increases the probability of reaction due to increasing number (statistical weight) of paths through the reaction zone. Obviously, no increase is observed if A reactant moves diusionally as well. This is seen from Eq. (3.20). One of the most important characteristics of the reaction operator is the integral rate of chemical conversions, the so-called intrinsic constant Z ∞ Z dr U (r) = − dr Vˆ |1(r)i : (3.21) kr = Vatt (r|r0 ) = −U (r)

R0

The intrinsic constant for the attendant reaction operator is easily calculated by its explicit expression (3.18), in view of Eq. (3.17) Z DA att kr : (3.22) kr = − dr Vˆatt |1(r)i = DA + DB The discrepancy between intrinsic constants kr and kratt is observed solely in the stationary theory presented here. In the appendix the non-Markovian (time-dependent) theory is developed, and the time representation of the attendant pair is constructed. In the framework of such an approach any nite time interval is not enough for diusion

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to move B reactant to in nity. That is why only space distortion of the reactivity is observed, not the change in its integral value, i.e., in the non-Markovian theory intrinsic constants kr and kratt coincide. As is known [18], the stationary theory is the long-term limit of the non-Markovian theory. In this limiting case, there exists a nite probability for B reactant to move to in nity by diusion. This is what accounts for the disagreement between intrinsic constants kr and kratt observed in Eqs. (3.21) and (3.22). General character of the above explanations requires that Eq. (3.22) should hold both in the framework of the Torrey representation, and at arbitrary organization of the hopping migration of A reactants. Thus, generalization of Eq. (3.17) to the case of the general operator gˆ#0 is needed. As follows from de nition (3.7), the desired generalization requires that the operation of gˆ0 LˆB on the nonnormalized state |1(r)i from the left and from the right be de ned in a proper way. The uncertainty of such an operation is associated, for example, with the fact that, according to Eqs. (2.15), the operation of LˆB on the function |1(r)i gives zero which can be considered as the nonnormalized state C1 |1(r)i with the constant C1 = +0. The operation of gˆ0 on the function |1(r)i gives the state C2 |1(r)i with the constant C2 = ∞. Thus, the uncertainty of the type 0 · ∞ is observed. In the framework of the operator technique such uncertainties manifest themselves as follows. The nonnormalized state is represented as the limit of the sequence of basic functions |1(r)i = lim |(R − r)i :

(3.23)

R→∞

Here (x) is the Heaviside step function. According to Eq. (3.23), the operation of gˆ0 LˆB on the nonnormalized state, for example, from the right should be considered as gˆ0 LˆB |1(r)i = lim gˆ0 LˆB |(R − r)i = − R→∞

DB |1i : DA + DB

(3.24)

To establish the second equality, we use gˆ0 LˆB |(R − r)i = 4DB R2 @R g0 (r|R) :

(3.25)

and the fact that the asymptotic behavior of the free resolvent is of diusion type [22], i.e., g0 (r|R) ∼ g0D (r|R) ≡ [4(DA + DB )|r − R|]−1 ∼ [4R(DA + DB )]−1 : (3.26) R→∞

R→∞

The operation of gˆ0 LˆB on the nonnormalized state from the left is calculated similarly Z DB ˆ dxg0 (x|r) LˆB = − h1(r)| : (3.27) h1(r)|gˆ0 LB = lim R→∞ DA + DB v(R)

The integral in Eq. (3.27) is taken inside the sphere of the radius R. If the radius of the sphere increases unboundedly, the integral also increases unboundedly at any xed r. This means that the main contribution into the integral is given by the domain of large values of x, and the free resolvent g0 (x|r) may be replaced by its diusion asymptotics g0D (x|r). The second equality in Eqs. (3.27) takes into account that the operator LˆB describes diusion, and, therefore, it is reverse to the free diusion resolvent operator

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gˆD 0 . Thus, de nition (3.7), in view of Eqs. (3.24) and (3.27), yields the desired generalization of Eqs. (3.17) to the case of arbitrary organization of the hopping mobility of A reactant Z Z # (3.28) drg0 (r|r0 ) = dr0 g0# (r|r0 ) = ∗A : Note that in the derivation of Eqs. (3.28) we have established the validity of formal operator technique approach to the determination of the operation of operators on nonnormalized states which are easy to interpret physically in the problem under study. The use of the operator technique has made it possible to introduce the concepts of the resolvent and the reaction operator of the attendant pair. Based on the new notions, de ne the density natt (r) of the probability to nd the reactants in the attendant pair at the relative-position r (the attendant density) natt (r) = 1 + gˆatt Vˆatt |1(r)i :

(3.29)

This de nition is a natural generalization of Eq. (2.16) that expresses the density n(r) in the reaction pair in terms of the operator technique elements. Inequality (3.6) ensures the positivity of the quantity introduced. Note that, unlike the reaction pair, the density natt (r) does not coincide with the long-term limit of the corresponding value in the non-Markovian (time) consideration of the attendant pair evolution (see the appendix). The introduced attendant density may be related to the density n(r) in the reaction pair. To establish this relation, operate on both sides of Eq. (3.29) by the operator 1ˆ + gˆ0 LˆB from the left, and use Eqs. (3.24). We have (1ˆ + gˆ0 LˆB )|natt (r)i =

DA (1ˆ + gˆVˆ )|1(r)i : DA + DB

(3.30)

Comparison of the above equality with Eq. (2.16) shows n(r) =

DA + DB ˆ (1 + gˆ0 LˆB )|natt i : DA

(3.31)

With this relation in Eq. (2.20), and in view of de nition (3.3) for the attendant reaction operator, we obtain the recipe for the calculation of the rate constant via the introduced attendant quantities Z DA + DB (3.32) dr Vˆatt |natt (r)i : k =− DA In the perturbation theory with respect to reactivity =natt (r) = 1(r)= Eq. (3.32) gives the relation between intrinsic constants kr and kratt established in Eq. (3.22). Note that in the derived recipe the expression of the rate constant in terms of pair quantities diers formally from recipe (2.20). Formula (3.32) involves an extra factor expressed in terms of the coecient of A reactant macroscopic diusion that results from A reactant hopping movement for macroscopic distances. Nevertheless, the expression for the rate constant in terms of the asymptotic behavior of the function natt (r) at r → ∞ agrees with that for the density n(r). Owing to this, the function natt (r) has been treated as the probability density in the attendant pair. To establish the asymptotic behavior of

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natt (r), we need the equation for this function. To derive it, use Eqs. (3.4) in de nition (3.29). So natt (r) = 1 + gˆ0A Vˆatt |natt (r)i :

(3.33)

The above equation is similar to Eq. (2.19). Consider the behavior of the attendant density in the range of large values of r. For the free resolvent of A reactant we use the asymptotic diusion estimate analogous to estimate (3.26) g0A (r|r0 ) ∼ (4DA r)−1 :

(3.34)

r→∞

In view of Eq. (3.32), Eq. (3.33) gives Z natt (r) ∼ 1 + (4DA r)−1 dr0 Vˆatt |natt (r0 )i = 1 − r→∞

k 4r(DA + DB )

:

(3.35)

This asymptotic behavior coincides exactly with the asymptotic behavior for the density n(r) in the reaction pair [5,22]. Thus, outside the reaction zone the attendant density natt (r) is equal to the density n(r) up to the rst two terms of the asymptotic expansion. Therefore, the expressions for the rate constant in terms of the asymptotic behavior of pair densities in attendant and reaction pairs are of the same formal form k = lim 4R2 (DA + DB )@R n(R) = lim 4R2 (DA + DB )@R natt (R) : R→∞

R→∞

(3.36)

Due to the similarity of the density behavior in the reaction and attendant pairs, investigation of near-contact (in terms of the attendant pair) reactions may be based on the approach developed in the description of near-contact reactions in the case where the relative motion of partners in the reaction pair may be characterized by a single space parameter – the mean square length of the jump [21,24]. The rst step in the realization of such an approach is the consideration of the reaction in free motion described in the frame of the Torrey representation. This is the subject of the next section. 4. Near-contact reactions The examination of a speci c mechanism of the hopping mobility organization of A reactant (in the framework of the Torrey representation) forms the basis for the development of the approximation for the description of the attendant pair evolution. This approximation is the adaptation of the approach to the description of the evolution of the reaction pair where the relative motion of partners has been considered in the framework of the Torrey representation [9], i.e., it has been characterized by a single eective length, which determines the mean space displacement. In this section the expression for near-contact reaction rate constant will be derived under the assumptions made in the present contribution, i.e., when two radically dierent space scales are needed to adequately describe the relative motion of partners in the reaction pair. In this section and further only the reactions with isotropic rate of the elementary event =U (r) = U (r)= will be dealt with. To describe them, it is sucient to use the values averaged over the orientation angles of the radius-vector, and thus dependent

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solely on its absolute value [5,25]. Since no anisotropic reactions will be considered below, the averaged values will be denoted by the same symbols as their nonaveraged prototypes. A wide class of the so-called near-contact reactions is the subject of a considerable literature. For these reactions the characteristic decay scale of the elementary event rate is small as compared to the closest approach distance Z ∞ U (r)

R0 : dr (4.1) ≡ U (R0 ) R0 Here and below U (r) is considered a monotonically decreasing function, so its maximum value is attained at point R0 . This restriction is used for convenience, and is not crucial. The results obtained below are easily generalized to the case of arbitrary dependence U (r). In this paper we con ne the discussion to near-contact reactions. The rst step is the calculation of the free resolvent of A reactant in the frame of the Torrey model. This calls for solving the Fredholm integral equation of the second type [26] −1 ˆ − −1 A g0A (r|r0 ) + A fA g0A (r|r0 ) = −(r − r0 )=(4rr0 ) :

(4.2)

It is obtained from operator equations (3.12) by averaging over the orientation of the vector r. Equations for the averaged operator fˆA are derived similarly from Eq. (2.5) ˆ fˆA = fˆA (A2 r − 1) ˆ = −1ˆ : (A2 r − 1)

(4.3)

Here r is the Laplace operator without angular part, and the averaged unit operator 1ˆ has the kernel (r − r0 )=(4rr0 ). Owing to operator equations (4.3), the integral equation (4.2) may be reduced to a dierential equation. For this purpose, operate ˆ in view of on the left- and right-hand sides of Eq. (4.2) by the operator (A2 r − 1), Eqs. (4.3). Thus ˆ − r0 )=(4rr0 ) : DA r g0A (r|r0 ) = (A2 r − 1)(r

(4.4)

It is easily seen that the solution of this equation is the sum of the hopping- and J D and g0A , respectively diusion-free resolvents of A reactant, g0A J D (r|r0 ) + g0A (r|r0 ) ; g0A (r|r0 ) = g0A

(4.5)

where J (r|r0 ) g0A

A (r − r0 ) = ; 4rr0

D g0A (r|r0 )

1 = 4DA

(r − r0 ) (r0 − r) + r r0

: (4.6)

So in the framework of the Torrey representation the free resolvent of A reactant is just the sum of free resolvents of the two limiting descriptions of the hopping mobility: hopping (A → ∞) and diusion (A → 0; A → 0; A2 =A = DA = const). At the next stage of the development of the near-contact reaction theory we use D (r|r0 ) within the reaction zone. the smallness of the variation of the free resolvent g0A

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Thus, the expression for the rate constant will be given by a quadrature. Substituting Eq. (4.5) into Eq. (3.33) brings the equation for the density natt (r) into the form D ˆ Vatt |natt (r)i : (1ˆ − Vˆatt A )natt (r) = 1 + gˆ0A

(4.7)

Introduce the inverse operator Qˆ ≡ (1 − Vˆatt A )−1 =

∞ X

(Vˆatt A )l :

(4.8)

l=0

When applied to both sides of Eq. (4.7), it gives D ˆ ˆ Vatt |natt (r)i : + Qˆ gˆ0A natt (r) = Q|1(r)i

(4.9)

In the second term of Eq. (4.9) integration over the initial (right) argument of the resolD (r|r0 ) is eectively limited by the reaction zone region. By virtue of inequality vent g0A D (4.1), the change in the resolvent g0A (r|r0 ) may be neglected, and the decoupling in Eq. (4.9) may be performed Z D ˆ ˆ 0A + Q|g (r|R0 )i dr Vˆatt |natt (r)i : (4.10) natt (r) = Q|1(r)i As the calculation of the rate constant calls for the knowledge of the decay rate density of the attendant pair in the reaction zone, operate on both sides of Eq. (4.10) by the operator Vˆatt , and allow for de nition (3.32) for the rate constant. This gives ˆ − Vˆatt |natt (r)i = Vˆatt Q|1(r)i

DA D k Qˆ Vˆatt |g0A (r|R0 )i : DA + DB

(4.11)

ˆ Now it is necessary to We have used the commutativity of operators Vˆatt and Q. D calculate the value Vˆatt |g0A (r|R0 )i. According to Eq. (3.18), this calls for the calculation of the function DB ∗ D (r − R0 ) ∗ D ∗ D (r − R0 ) : (4.12) i≡ gˆ gˆ gˆ0 |g0A (r|R0 )i = gˆ0 gˆ0A | 4rR0 DA 0 0B 4rR0 This is easily done taking account of two important facts following from Eq. (3.15): D is the limit of the free resolvent g0∗ at ∗A → ∞; sec rst, the diusion resolvent g0B ∗ L ond, the resolvent g0 coincides formally with the free resolvent g0B (s) (the propagator Laplace transform) of diusion motion (with the diusion coecient DB ) at s = 1=∗A . D and g0∗ are related by [22] Thus, the resolvents g0B 1 ∗ D D gˆ gˆ = gˆ0B − gˆ∗0 : ∗A 0 0B

(4.13)

Substituting Eq. (4.13) into the right-hand side of Eqs. (4.12) gives the result representable as 1 DB ∗ D ∗ ∗ ˆ ∗ (r − R0 ) ˆ ˆ V gˆ |1(r)i − A V gˆ0 : (4.14) V gˆ0 |g0A (r|R0 )i = DA 4R0 DB 0 4rR0 We have used the normalization condition (3.17) for the resolvent g0∗ , and neglected the D in the reaction zone region when integrating variation of the values of the resolvent g0B

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the resolvent over the nite (left) argument. In view of the explicit expression for Vˆatt (see Eq. (3.18)) valid in the framework of the Torrey model, Eq. (4.14) takes the form 1 DB A ˆ (r − R0 ) D (r|R0 )i = Vˆatt |1(r)i − Vatt : (4.15) Vˆatt |g0A 4R0 DA DA + DB 4rR0 Substitution of Eq. (4.15) into Eq. (4.11) followed by integration of both sides of the equality in view of de nition (3.32) for the rate constant yields the algebraic equation for k. The solution of the equation may be given by a quadrature

where

1 1 1 1 1 + ; = + ≡ k kv 4R0 (DA + DB ) kv kD

(4.16)

R ˆ −(DA + DB )=DA dr Vˆatt Q|1(r)i : kv = R ˆ ˆ 1 + DB A =(DA + DB ) dr Vatt Q|((r − R0 )=4rR0 )i

(4.17)

Eq. (4.16) shows that the near-contact reaction may be represented as two sequential independent processes: diusion approach of reactants (with relative diusion coecient) from macroscopic distances to some vicinity of the reaction zone, and the entry into the reaction zone followed by the reaction. The rate constant kv describes exactly the second process. Further in this section we shall establish the relation between the rate constant kv and the “asterisk” pair evolution, thus re ning the concepts of the reaction zone vicinity and the evolution of reactants in it (see Eq. (4.16)). First the numerator in Eq. (4.17) is shown to coincide with the rate constant kts∗ for the “asterisk” pair, i.e., Z Z DA + DB ˆ (4.18) ≡ − dr Vˆ |n∗ (r)i : dr Vˆatt Q|1(r)i kts∗ = − DA The second equality in Eqs. (4.18) introduces the function n∗ (r) =

1 ∗ gˆ Q|1(r)i : ∗A 0

(4.19)

According to Eqs. (2.20), the above function is to be the density of the probability to nd the reactant at the distance r in the “asterisk” pair. To prove this, form the equation this function obeys. Using the operator identity Qˆ = 1ˆ + Vˆatt A Qˆ :

(4.20)

following from Eqs. (4.8) and normalization (3.17), we obtain n∗ (r) = 1 + gˆ∗0 Vˆ |n∗ (r)i :

(4.21)

Comparison of the above equation with Eq. (2.19) shows that the function n∗ (r) is the probability density in the “asterisk” pair. Therefore, the right-hand side of Eqs. (4.18) does de ne the rate constant in the ensemble of such pairs. To nd the denominator of Eq. (4.17), introduce the “asterisk” pair resolvent following the logic pattern of Section 2. Equation of the form of Eq. (2.10) for the density

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n∗ (r) is obtained by operating on both sides of Eq. (4.21) by the inverse operator [gˆ0∗ ]−1 (see Eqs. (3.15)). Thus DB r n∗ −

1 ∗ (n − 1) + Vˆ n∗ = 0 : ∗A

(4.22)

The second term de nes the A reactant getting out of or into point r by the jumps of in nite length occurring at the mean intervals of time ∗A . The “asterisk” pair resolvent operator gˆ ∗ is introduced according to Eq. (2.12) that in the case at hand takes the form 1 ˆ 1 ˆ ∗ ∗ ˆ ˆ (4.23) V − ∗ 1 + DB r gˆ = gˆ V − ∗ 1 + DB r = −1ˆ : A A As A reactant jump is in nite, space correlations between reactants arising on the encounter are nite. So operation of gˆ ∗ (just as gˆ0∗ ) on the nonnormalized state is de ned, while for the density n∗ (r) there exist two equivalent expressions in terms of the “asterisk” pair resolvent 1 n∗ (r) = 1 + gˆ ∗ Vˆ |1(r)i = ∗ gˆ ∗ |1(r)i : A

(4.24)

Comparison of Eqs. (4.19) with (4.24) establishes a useful (for further studies) relationship between the operators of the “asterisk” pair resolvents and Qˆ gˆ ∗ = gˆ0∗ Qˆ :

(4.25)

With the above relation in the denominator of Eq. (4.17), and in view of Eq. (3.18), we obtain the desired representation of the rate constant kv in terms of the “asterisk” pair evolution kv =

1+

DB DA +DB

R

kts∗ : dr Vˆ |g∗ (r|R0 )i

(4.26)

Thus, the knowledge of the “asterisk” pair evolution is the necessary and sucient condition for the calculation of the vicinity rate constant kv . On the other hand, the evolution of reactants in the reaction zone vicinity reduces to the evolution of the “asterisk” pair, i.e., to entering the reaction zone or its vicinity by in nite jump of A reactant with the subsequent reaction in the process of continual diusion (with the coecient DB ) during the mean time ∗A , until the next in nite jump of A reactant interrupts the vicinity evolution. The described physical picture agrees with the course of chemical reaction in the framework of the two-scaled migration mechanism up to the substitution of A for ∗A [16]. The above mechanism has been qualitatively studied by step approximation of the dependence of the elementary event rate U (r) on the distance r between reactants. The analytical expression for the rate constant kts has been derived. It may be used to qualitatively investigate the speci c features of the behavior of kts∗ , with ∗A in place of A . However, though covering principal features of the phenomenon, the step approximation adds extra degrees of degeneration to the problem, thus making some important general points vague. In the next section a more delicate approach will be developed to study the peculiarities of the behavior of kts

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(or kts∗ ). Restricting ourselves to migration control region (when each encounter results in the reaction with the probability of unity), we calculate the rate constant kts (or kts∗ ) for a rather wide class of functions U (r). Note that Eq. (4.26) is not very convenient in this case. For migration control region the following expression for the vicinity rate constant will be more suitable DA + DB R : (4.27) kv = kts∗ DA + DB (dr=∗A )g∗ (r|R0 ) It is obtainedR from Eq. (4.26) with the use of Eqs. (4.23). According to Kac [27], the integral dr g∗ (r|R0 ) is the mean life time of the reactant in the zone and its vicinity provided that the start was from the distance R0 . According to Eqs. (3.17), in the absence of reaction it is ∗A . Obviously, in migration control region the mean time is zero. Thus, the vicinity rate constant kvM in migration control region is de ned by the reduced expression DB M ∗ ∼ kts 1 + : (4.28) kv U (R0 )→∞ DA 5. Two-scaled migration mechanism According to Eqs. (4.26) – (4.28), calculation of the vicinity rate constant kv calls for that of the rate constant kts∗ for the ensemble of “asterisk” pairs. The probability density n∗ (r) of such pairs obeys Eq. (4.22) that coincides with the equation for the density nts (r) in the two-scaled migration mechanism up to the substitution of A for ∗A [16] ˆ DB r nts − −1 A (nts − 1) + V nts = 0 : So calculations of the rate constant Z kts = − dr Vˆ |nts (r)i :

kts∗

(5.1)

and the rate constant (5.2)

for the two-scaled migration mechanism are mathematically equivalent problems. However, physically, calculation of the rate constant kts is preferable. The obtained results are interpreted in the terms independent of model assumptions of A reactant hopping mobility organization. On the other hand, in the case of arbitrary dependence U (r) (though under restriction (4.1)) there is no reason to expect that the problem will be solved exactly. Eq. (5.1) cannot be generally solved by quadratures. According to the foregoing, this section will be devoted to the development of the approach to approximate calculation of the rate constant in the framework of the two-scaled migration mechanism for arbitrary dependence U (r). In the next section the obtained results will form the basis for revealing the limiting kinematic mechanisms and deriving the resulting (free of model assumptions of the hopping mobility organization) expression for the vicinity rate constant. In migration control limit, when the reaction starts as soon as the reactant nds itself in the reaction zone, the reaction in the frame of the two-scaled migration mechanism

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proceeds in two independent parallel ways in accordance with the two ways of reactant entering the zone. The reaction induced by immediate delivery to the zone from the in nity in a single jump will be called the hopping channel. The reaction induced by the entry into the zone by diusion displacement from the zone vicinity during the time A will be called the diusion channel. Accordingly, in migration control region the rate constant kts is to be represented as the sum of rate constants taken over either channel kts = kj + kd :

(5.3)

Representation (5.3) will serve as a basis for the approximation under development, i.e., it will be used at arbitrary intensity of the reaction. Note that recipe (5.2) is unsuitable for the calculation of the rate constant, since the density nts (r) is not representable as the sum of densities of the corresponding channels. However, the deviation of the density from unity due to the reaction m(r) = 1 − nts (r)

(5.4)

exhibits the required property. This ensures formal ful lment of additivity property (5.3) for the rate constant for arbitrary intensity of the reaction. So the recipe for the calculation of the rate constant kts (see Eqs. (5.1) and (5.2)) is conveniently rewritten in terms of the deviation m(r). Substituting Eq. (5.1) into Eq. (5.2), in view of the boundary conditions (2.4), yields the desired rate constant representation Z −1 drm(r) : (5.5) kts = A On substitution of m(r) for n(r), according to (5.4), into (5.1), we have the dierential equation for the calculation of the deviation [DB r − −1 A − U (r)]m(r) = −U (r) :

(5.6)

To single out physically correct solution, complement the above equation by the boundary conditions @r m|r=R0 = 0;

m(r → ∞) = 0

(5.7)

that follow from the corresponding boundary conditions for the probability density. According to representation (5.3), the deviation m(r) is represented as the sum m(r) = mj (r) + md (r)

(5.8)

with each term satisfying the boundary conditions of the form (5.7). The rst term describes the reaction by the hopping channel. It seems easy to derive the explicit expression for it from Eq. (5.6) putting formally DB = 0. However, generally speaking, the obtained deviation mj (r) for arbitrary function U (r) will not meet the required boundary condition at the closest approch of reactants. To proceed further, reduce the class of admissible functions U (r) to the class of functions U (r) that obey the additional condition @r U (r)|r=R0 = 0, except restriction (4.1). Thus, the deviation U (r)A (5.9) mj (r) = 1 + U (r)A

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derived from Eq. (5.6) by the above formal procedure satis es the required boundary condition @r mj |r=R0 = 0. The explicit formation of the class of functions U (r) may be carried out in a number of ways. The method based on the operator related to operators used in the present article will serve as an example. Each elementary event rate U (r) de ned by physical statement of the problem is represented as the limit of the sequence in the reduced class of functions of the form U (r) = gˆ |U (r)i

(5.10)

with → 0. The kernel of the operator gˆ is given by the equation 2 r g (r|r0 ) − g (r|r0 ) = −(r − r0 )=(4rr0 )

(5.11)

similar to Eqs. (3.15) for the “asterisk” pair free resolvent. Ful lment of the required boundary condition for the function U (r) at the closest approach of reactants is provided by the following boundary conditions for the kernel g (r|r0 ): @r g (r|r0 )|r=R0 = 0;

lim g (r|r0 ) = 0 :

r→∞

(5.12)

It is easily shown that at small the asymptotic behavior of the kernel g (r|r0 ) is of the form |r − r0 | r + r0 − 2R0 1 exp − + exp − : (5.13) g (r|r0 ) ∼ →0 8rr0 As is seen from Eqs. (5.12) and (5.13), at small the operator gˆ is close to unit operator. That is why at r − R0 the function U (r) diers slightly from U (r). The main dierences are associated with the region r − R0 . ; they provide the ful lment of the required boundary condition for the function U (r). Obviously, at rather small the distinctions between functions U (r) and U (r) do not aect the rate constant value. This is just what ensures the correctness of using the class of functions U (r) instead of that of U (r). √ In the migration control region at small values of diusion displacements l = DB A for the time A p l ≡ DB A (5.14) the hopping reaction channel is eective. Z Z −1 dr mj (r) = dr kts ' kj = lim A →0

U (r) : 1 + U (r)A

At large values of diusion displacements p l = DB A

(5.15)

(5.16)

the hopping channel is ineective. The reactants are primarily delivered into the zone by diusion from the zone vicinity. To calculate the rate constant for this case, we obtain the equation for the deviation md (r). Substitution of Eq. (5.8) in Eq. (5.6), in view of Eq. (5.9), yields (−−1 A + DB r )md = U md − DB r mj :

(5.17)

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The above equation should be complemented by the boundary conditions of the form (5.7). For the rate constant that coincides with diusion channel rate constant in the case of migration control, there exist two equivalent representations Z Z (r) = − lim (5.18) drm drU (r)md (r) : kts ' kd = lim −1 d A →0

→0

The second recipe for the calculation of the constant kd is more convenient. Its integration domain is eectively limited by the reaction zone boundary. To apply it, we need the integral form of Eq. (5.17). It is obtained by introducing the free resolvent of the two-scaled migration of the reaction pair ts −1 ˆ ˆ ˆ ts ˆ (LˆB − −1 A 1)gˆ0 = gˆ0 (LB − A 1) = −1 :

(5.19)

The explicit form of the averaged resolvent g0ts (r|r0 ) has been calculated in Ref. [16], and is de ned by |r − r0 | R0 − l r + r0 − 2R0 A ts exp − + exp − : (5.20) g0 (r|r0 ) = 8rr0 l l R0 + l l Under condition (5.16), variations of the resolvent g0ts (r|r0 ) values are shown to be small for all points r and r0 within the reaction zone. Below we use the smoothness of the function g0ts (r|r0 ). The desired integral form is obtained by operating on both sides of Eq. (5.17) by gˆts0 , in view of Eqs. (5.19). So md (r) = DB gˆts0 |r mj (r)i − gˆts0 |U (r)md (r)i :

(5.21)

Multiply both sides of the equality by the rate U (r) of the elementary event, and perform integration Z (5.22) drU md ≡ h1|U md i = DB hU |gˆts0 |r mj i − hU |gˆts0 |U md i : Here the well-known Dirak designations of a scalar product are employed. If the above-mentioned smoothness of the function g0ts (r|r0 ) is taken into account, then the following decoupling (estimate) of the matrix element is valid at small Z Z hU |gˆts0 |U md i ' drU (r)g0ts (r|R0 )h1|U md i ' −kd g0ts (R0 |R0 ) drU (r) : (5.23) If the deviation md (r) were a positive value, then the validity of Eq. (5.23) would be evident. However, as is seen from Eqs. (5.18), the function md (r) may change sign. Thus, decoupling (5.23) is a more delicate property. To prove it, represent the matrix element as () ts ts hU |gˆts0 |U md i = −1 A hU |gˆ0 |mj i − hU |gˆ0 |U nts i :

(5.24)

Here de nitions (5.4), (5.8) and (5.9) are used, and the density nts (r) of the reaction pair with the rate U (r) of the elementary event is denoted by nts() (r). Since functions mj (r) and nts() (r) are positive, each of the terms in sum (5.24) may be calculated up

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to the leading term using the decoupling of the type (5.23). For rather small , we have Z (5.25) hU |gˆts0 |U md i ' drU (r)g0ts (r|R0 )[kj − kts ] : The estimate is valid, if the terms in square brackets do not compensate each other. Otherwise, the next term (after the leading one) of the expansion in the smoothness of the function g0ts (r|r0 ) should be taken into consideration. In the case in question kj kts ' kd , thus estimate (5.25) is valid, and agrees with estimate (5.23). Note that the decoupling of the type (5.23) is not suitable for the estimation of the rst matrix element on the right-hand side of Eq. (5.22), since it gives zero result. To obtain correct estimate, use Eqs. (5.20), and represent the matrix element under consideration as ts DB hU |gˆts0 |r mj i = −1 A hU |gˆ0 |mj i − hU |mj i :

(5.26)

Further the rst term will be estimated, and shown to be much less than the second term, i.e., DB hU |gˆts0 |r mj i ' −hU |mj i :

(5.27)

For the rst term the estimate of the type (5.23) is valid up to the leading term Z ts −1 ts hU | g ˆ |m i ' g (R |R )k drU (r) : (5.28) −1 0 j 0 0 j 0 A A To nd the value of the resolvent g0ts (R0 |R0 ), use its explicit form (5.20) g0ts (R0 |R0 ) = A =(4R0 l(R0 + l)) :

(5.29)

To simplify the comparison of the terms on the right-hand side of Eq. (5.26), apply a speci c form of the elementary event rate, namely, the “step” approximation Ust (r) = U0 (R0 + − r) :

(5.30)

Note that the function Ust (r) belongs to the reduced class of functions U (r), since @r Ust |r=R0 =0. For the elementary event rate Ust (r) the integrals involved in Eqs. (5.26) and (5.28) are easily calculated. This gives the desired relation between the terms on the right-hand side of Eq. (5.26) under restrictions (4.1) and (5.16) ts −1 A hU |gˆ0 |mj i =

R0 hU |mj i hU |mj i : l(l + R0 )

(5.31)

The validity of estimate (5.27) follows. Substitution of estimates (5.23) and (5.27) for matrix elements into Eq. (5.22) transforms it, in view of Eqs. (5.18), into the algebraic equation for kd . Solving this equation gives R drU (r)lim→0 mj (r) R∞ : (5.32) kd = A 1 + l(RR00+l) drU (r) R0 The obtained rate constant of the diusion channel coincides with the rate constant kts of the two-scaled migration solely in the limit of migration control and condition

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(5.16). If the reactivity decreases, kts should be calculated by Eq. (5.3) with the rate constant kj de ned in Eqs. (5.15). (Note that the de nition is not restricted by (5.14)). We have R Z ∞ drU (r) R0 U (r)A c R∞ 1+ dr : kts ' kts ≡ A l(R0 + l) R0 1 + U (r)A 1 + l(RR00+l) drU (r) R0 (5.33) The derivation of the above expression is based on the concepts of the hopping and diusion channels, thus herein after this approximation will be called channel. The rate constant obtained in the channel approximation will be marked by the upper index “c”. If condition (5.16) is met, the second term in square brackets may be neglected. As a result, Eq. (5.33) reduces to R drU (r) R∞ : (5.34) ktsc ∼ ktsD ≡ A l 1 + l(RR00+l) drU (r) R0 The expression describes the rate constant kts at arbitrary reactivity in the range of large (as compared to ) values of diusion displacement of reactant. So it is easily obtained immediately from Eqs. (5.1) and (5.2) (without representation (5.3)) by the method similar to that employed in the calculation of the diusion channel rate constant. The use of Eq. (5.33) in the region l is quite reasonable, since the reactant is chie y delivered to the reaction zone by a jump from the in nity, with kd kj , with the details of the calculation of kd being insigni cant. At arbitrary reactivity the two-scaled migration rate constant kts coincides with the hopping rate constant Z U (r) ktsc ∼ ktsJ ≡ kj = dr : (5.35) l 1 + U (r)A The validity of Eq. (5.33) in the region ∼ l will be studied with the “step” rate of the elementary event de ned in Eq. (5.30) as an example. In this case all integrals in Eq. (5.33) are easily found. We have " −1 # zÿ2 4R20 z c 1+z 1+ (5.36) kts = A 1 + z 1 + ÿ that involves the dimensionless parameters z = U0 A , = R0 = and ÿ = =l. The exact calculation of the two-scaled migration rate constant for the “step” rate of the elementary event has been done in Ref. [16]. For near-contact reactions the value kts has been shown to be given by the asymptotically exact expression z(1 + ÿ) 4R20 z 1+ where kts = A 1 + z (1 + z)ÿ2 (1 + ) =

√ ÿ √ 1 + zthÿ 1 + z : 1 + ÿ

(5.37)

Comparison of Eqs. (5.36) with (5.37) demonstrates that at z 1 and z 1, and arbitrary ratio between and l the channel approximation is valid. To establish the

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Fig. 1. The dependence of the ratio between ktsc and kts on the parameters z and ÿ at = 100 for the “step” rate of the elementary event.

Fig. 2. The dependence of the ratio between ktsc and kts for the “step” rate of the elementary event, and = 100 for characteristic cross sections of Fig. 1: (a) on the parameter ÿ at z = 1; 6 and (b) on the parameter z at ÿ = 1.

channel approximation inaccuracy in the vicinity of the point z ∼ 1 and ÿ ∼ 1, numerical calculations have been performed. The results are given in Figs. 1 and 2 for R0 = 100 ( = 100). The analysis shows that the maximum discrepancy between the rate constant calculated in the channel approximation and the exact rate constant does not exceed 18%, and is observed at z = 1; 6 and ÿ = 1. The region where the accuracy of the channel approximation is not less than 6% is given by the inequalities z ¡ 0; 1; z ¿ 100; ÿ ¡ 0; 1;

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329

Fig. 3. The dependence of the ratio between ktsc and kts on the parameters z and ÿ at = 100 for the exponential rate of the elementary event.

ÿ ¿ 10. Fig. 3 presents the results of numerical calculation for the exponential rate of the elementary event U (r) = U0 exp[ − (r − R0 )=] :

(5.38)

The ratio of the rate constant ktsc calculated in the channel approximation to the rate constant kts obtained by solving Eq. (5.1) numerically with the subsequent numerical integration by recipe (5.2) is plotted on the vertical axis. Comparison between Figs. 1 and 2 points to the similarity of the parametric dependences of the channel approximation inaccuracy for the “step” and exponential rates of the elementary event. Thus, outside the vicinity of the point z ∼ 4, ÿ ∼ 1 the channel approximation accuracy may be regarded quite acceptable. The next section is devoted to revealing the limiting kinematic mechanisms based on the channel approximation. In the range of their realization the channel approximation gives the asymptotically exact results for the vicinity rate constant. Finally, the expressions for the near-contact reaction rate constant will be given by a quadrature in the framework of the kinematic mechanisms.

6. Kinematic reaction mechanisms This section deals with the limiting kinematic reaction mechanisms peculiar to the problem under study. The limits of their realization will be established, and the expressions for the calculation of the rate constant will be given by a quadrature in the frame of these mechanisms. With the “step” rate of the elementary event as an example, we show that in all kinematic mechanisms calculation of the rate constant may well be done using just the two-scaled reaction pair evolution (not the “asterisk” pair one) that is obviously independent of the model assumptions of A reactant hopping mobility organization. The extension of the above substitution to the entire variation

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range of the problem parameters may result in the inaccuracy of about 6% in the rate constant calculation at some boundaries of the kinematic mechanism realization areas. Thus, in this section we discard model assumptions of the hopping mobility organization used in the derivation of the expression for the near-contact reaction rate constant (see Eq. (4.16)). The rate constant of migration controlled reactions k

∼

U (R0 )→∞

kM

(6.1)

is proportional to the frequency of binary encounters of reactants in solution. Its calculation is most interesting from the standpoint of general physics. On the other hand, the value k M depends most signi cantly on the kinematic details of the approach of reactants. Thus, revealing the limiting kinematic mechanisms will be based on the analysis of the explicit expression for k M . Most easily it may be obtained for the “step” rate of the elementary event. Since, according to Eq. (4.16), the following equality takes place: 1 1 1 = M + kM kv 4R0 (DA + DB )

(6.2)

only the vicinity rate constant in the migration control area is to be calculated. One can use Eq. (4.28), in view of Eq. (5.36) (or Eq. (5.37)), with ∗A in place of A , and l∗ instead of l. We have (l∗ )2 DB DB 4R20 l∗ M ∗ + : (6.3) lim k = 1 + 1+ kv = 1 + DA U0 →∞ ts DA ∗A R0 Eq. (6.3) involves three terms in brackets on the right. Each of the terms has the range of values of parameters R0 , and l∗ where it is dominant. Accordingly, in the problem in question there exist three limiting kinematic reaction meachanisms. Before we start exploring them one by one, it is pertinent to make a comment of a general character. It is seen that in the range of rather large values of DA =DA → ∞= the second term in Eq. (6.2) may be neglected, and the two-scaled migration mechanism k M ' kvM ' ktsM is √ realized, with ∗A ' A . As the jump length A = DA A decreases (with the value of A being the same), the second term in Eq. (6.2) increases monotonically, while the rst one decreases monotonically. Almost always the second term becomes dominant even in the region DA DB . So with further decrease in A (to the region where A ∗A ), the decay rate of the rst term does not aect the rate constant value, therefore, the use of A instead of ∗A is justi ed. Further the above reasoning is shown to be valid for two of the three kinematic mechanisms. Note that for the third mechanism the rate constant does not depend on ∗A at all. The vicinity hopping mechanism. In the region DA DB (A l) the rst term in brackets on the right in Eq. (6.3) is dominant if l ; with ∗A ' A , and the vicinity migration controlled rate constant being de ned by the expression kvM '

4R20 A

(l ) :

(6.4)

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√ As A decreases, the terms in Eq. (6.2) become equal at A = R0 l, i.e., in the region DA DB . Thus, according to the foregoing, one can always use A instead of ∗A , and apply Eq. (6.4) over the entire variation range of the parameter A . Since under restriction l the asymptotically correct equality kts ' ktsJ takes place, the mechanism revealed in Eq. (6.4) will further be called the vicinity hopping mechanism. Generalization of Eq. (6.4) to the case of arbitrary dependence U (r) and arbitrary reactivity is based on Eqs. (5.35) and (4.26), with the second term in the denominator of Eq. (4.26) being neglected. Thus Z U (r) (l ) : (6.5) kv ' ktsJ = dr 1 + U (r)A The vicinity quasione-dimensional diusion mechanism. In the area DA DB the second term in brackets on the right in Eq. (6.3) is dominant given l R0 ; with ∗A ' A , and the vicinity migration controlled rate constant being de ned by the expression r DB 4R20 l M 2 = 4R0 ( l R0 ) : (6.6) kv ' A A The dependence of the rate constant on both large-scaled motion parameters and small-scaled motion ones is due to the quasione-dimensionality of the problem under the restrictions used [16]: diusion walks near the in nite plane may be interrupted by the in nite length jump only. (Going to in nity by diusion is impossible). This kinematic mechanism is called the quasione-dimensional diusion mechanism [16]. With √ decreasing A , the terms in Eq. (6.2) become equal at A = R0 l l, i.e., in the region DA DB . Thus, as before, ∗A may always be replaced by A , and Eq. (6.6) may be used over the whole variation range of the parameter A . Generalization of Eq. (6.6) to the case of arbitrary form of U (r) and arbitrary reactivity is based on Eq. (5.35) and Eq. (4.26), with the second term in the denominator of Eq. (4.26) being neglected. So r #−1 Z −1 " DB 1 1 1 2 = drU (r) + 4R0 = + M ( l R0 ) : (6.7) kv A kr kv The above result shows that in the mechanism under study the vicinity evolution consists of two sequential independent processes: diusion delivery of reactants to the reaction zone from its vicinity, and the reaction in the zone proceeding at the rate kr . The vicinity diusion mechanism. In the region DA DB the third term in brackets on the right in Eq. (6.3) is dominant if R0 l; with ∗A ' A , and the vicinity migration controlled rate constant being de ned by the expression kvM ' 4R0 DB

(R0 l) :

(6.8)

Though structurally similar to the diusion description, this result diers essentially from the generally accepted expression. The relative diusion coecient (DA + DB ) is replaced by B reactant diusion coecient which in the case at hand is much less than the diusion coecient DB DA +DB ' DA . Just the delivery to the reaction zone from

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its vicinity is the control process, not the approach of reactants from macrodistances. As A decreases, the terms in Eq. (6.2) become equal at A ∼ R0 =DA ∼ DB =. Obviously, the dierence between ∗A and A should be taken into account. However, it is easily seen from Eqs. (6.3) and (6.2) that the rate constant does not depend on these parameters at all k M ' 4R0 DB

(R0 l) :

(6.9)

In other words, the large-scaled motion of A reactant does not aect the rate constant value. When A reactant moves by small jumps (diusionally), DA DB , so this motion may be neglected. Thus, just as in the mechanisms considered above, the use of A instead of ∗A in the vicinity diusion mechanism is justi ed. According to Eq. (6.3), in the mechanism in question at DA . DB the vicinity rate constant kvM depends essentially on the macroscopic parameter DA . This means that the concept of the vicinity evolution ceases to have its microscopic meaning, i.e., the spatial size of the reaction zone vicinity increases up to the size of a correlation pair that is formed from reactants on their encounters in solution. Thus, representation of the near-contact reaction as two sequential processes (the approach and the vicinity evolution) is very relative. Only the observed rate constant is physically meaningful. According to the foregoing, we generalize Eq. (6.9) to the case of arbitrary intensity and space dependence U (r) of reactants. For this purpose, use Eqs. (4.16), (4.26) and (5.34). Calculation of the integral in Eq. (4.26) is performed by the method similar to that employed in the calculation of the diusion channel rate constant, i.e., on the basis of the integral (see Eq. (2.18)) form of Eq. (5.19), with the subsequent making allowance for the smoothness of the free resolvent. We have Z −1 1 1 1 + [4R0 DB ]−1 = + M : (6.10) = d rU (r) k kr k Thus, in the vicinity diusion mechanism the near-contact reaction may be represented as two sequential independent processes: diusion approach (with the diusion coecient DB ) from macrodistances followed by the entry into the reaction zone, and the reaction with intrinsic constant kr . When considering all three kinematic mechanisms, we have established that in the area of their realization the substitution of A for ∗A is justi ed. Assume that the validity of such a substitution extends to the entire variation range of the problem parameters. Then, according to Eq. (4.26), the vicinity rate constant will be de ned by the expression kts kv ' R DB 1+ dr Vˆ |gts (r|R0 )i (DA + DB ) DA + DB kr + 4R0 l(R0 + l)=A : (6.11) = kts (DA + DB ) 4R0 l(R0 + l) DA kr + DA A In the derivation of the second equality for the calculation of the integral in Eq. (6.11) the approach has been used similar to that applied to the calculation of

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333

Fig. 4. The dependence of the ratio between k(∗A ) and k(A ) on the parameters ÿ and q = DB =DA = l2 =A2 at = 100 for the “step” rate of the elementary rate at (a) z = 104 , and (b) z = 102 .

the diusion channel rate constant in Section 5. The accuracy of approximation (6.11) has been tested for the case of the “step” rate of the elementary event by comparing the rate constants calculated with the use of the “asterisk” pair evolution k(∗A ) and the two-scaled pair evolution k(A ). The results are given in Fig. 4. Maximum inaccuracy is observed in the case of migration control (z → ∞) at the boundary between realization areas of the vicinity diusion mechanism and the vicinity quasione-dimensional diusion mechanism l ≈ R0 =ÿ ≈ 1= at A ≈ l. It does not exceed 6%, and does not decrease with increasing parameter = R0 =. This means that crossing the boundary depends essentially on the hopping mobility organization of A reactant, and there exists no asymptotically exact universal expression for the rate constant that is valid for any hopping mobility organization. As the reactivity value decreases, the accuracy of approximation (6.11) increases (see Fig. 4(b)). Thus, the accuracy of approximation (6.11) seems acceptable. Neglecting the above inaccuracy (about 6%), one can treat expression (6.11) as a universal expression for the vicinity rate constant valid for any hopping mobility organization of A reactant. Though obtained in the frame of the Torrey representation, it does not depend on speci c features of a particular way of the hopping mobility organization. In the general case, expression (3.14) is invalid, and passing from Eqs. (3.13) to Eqs. (3.15) is impossible. However, approximation (6.11) can well be derived without this procedure. Substitution of A for ∗A used in the derivation of Eqs. (6.11) allows one just to neglect the third term in circular brackets in Eqs. (3.13). Eq. (3.13) are reduced to Eqs. (5.19), and the operator gˆ#0 ' gˆts0 . In other words, substitution of A for ∗A is equivalent to ignoring speci c features of the hopping mobility organization in the framework of the Torrey model.

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7. Conclusion In experimental studies of excess electron drift in nonpolar liquids in the presence of external electric eld it has been established [15] that adequate description of electron behavior calls for the consideration of just two states: localized and quasifree. Most of the time the electron is in the localized state. Migration is related to tunnel transition to the neighboring localized state arising from heat uctuations in the liquid. The experiment for n-hexane shows the value of such transitions to be less than 10 A; migration may be described by continual diusion model. During activation the electron can go from the localized state into quasifree one, and remain in it for a rather short time (by two (or more) orders less than in the localized state). The electron does not enter into the reaction from the qusifree state, however, this time is enough for Thus, the validity of translational migration it to move for distances up to 200 A. description of an excess electron in the framework of stochastic walking model has been ascertained. It is also established that the conditional probability density of a jump in this model is de ned by two essentially dierent space scales, i.e., electron motion is of two-scaled nature. In processing of experimental data on excess electron trapping by acceptors in nonpolar liquids obtained in migration control limit the so-called reaction radius has been introduced R=

k : 4D

(7.1)

Here k is the experimentally found rate constant of the irreversible reaction A + e− → A− , and D is the experimentally obtained coecient of macroscopic electron diusion. Mobility of A acceptors has been neglected, since at room temperatures their diusion coecient is ∼ 10−5 sm2 =s, and the diusion coecient of the excess electron in the localized state is Dloc ∼ 2×10−3 sm2 =s (ignoring electron displacements in the quasifree state). When the reaction radius was much less than the acceptor one (R R0 ), it was considered that the hopping mechanism of the reaction was observed for which [5] kM =

4 [(R0 + )3 − R30 ]=loc : 3

(7.2)

The mean residence time loc of the electron in the localized state determined by activation depends strongly on temperature. Estimation made in Ref. [15] gives loc ∼ 10−11 –10−12 s. In paper [15] the acceptors have been used for which ' 5R0 . Thus, electron trapping reaction is of remote character, and expression (7.2) for the hopping mechanism reduces to kM =

43 : 3loc

(7.3)

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Comparison of Eqs. (7.1) and (7.3), in view of Eq. (2.7), yields the relation between the observables and the mean-square length of electron jump 23 2 : = R

(7.4)

This leads to estimate ' 70 A. There is every reason to believe that the above procedure of experimental evidence processing is only applicable in the limiting case R R0 . In the general case it has been recommended to use the rate constant dependence on instead of Eq. (7.2). Paper [15] presents numerical calculation of this dependence. For conditional probability density of a jump the function is used the decay of which is characterized by a single space parameter (as in the Torrey model), i.e., in this case diusion displacement of the electron in the localized state is also neglected. As is seen from Ref. [28], for remote reactions application of such an approach is justi ed. However, the present contribution shows that for near-contact reactions taking into account the two-scaled nature of electron motion is of fundamental importance, since it gives rise to two new limiting mechanisms of the reaction (vicinity quasione-dimensional diusion mechanism and vicinity diusion mechanism). Ignoring these mechanisms may lead to relative errors of dozens times. For near-contact reactions the hopping rate constant is de ned by kM =

4R20 : loc

(7.5)

By analogy with the preceding expression, (7.5) gives the relation between the observables and the mean-square length of electron jump 6R2 2 = 0 : R

(7.6)

However, the condition R R0 is not sucient for the validity of estimate (7.6). It merely ensures that the two-scaled migration mechanism is observed. In the framework of this mechanism, the rate constant is given by the expression (similar to Eq. (6.3)) l2 4R20 l : (7.7) 1+ + kM = loc R0 Thus, when R R0 the relation between the observables and the mean-square length of electron jump is l l2 6R2 2 1+ + : (7.8) = 0 R R0 √ for characteristic values of observables That is why we have l = Dloc loc ∼ 14 A 2 from paper [15]. At ∼ 1 A possible relative error in nding by the stationary rate constant value may be of a factor of 14.

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Acknowledgements The authors are grateful to A. Neufeld for numerical calculations in the determination of the channel approximation accuracy for exponential rate of elementary event. The Russian Foundation of Fundamental Research (Project 99-03-33155) is thanked for support.

Appendix The appendix deals with the development of the time representation of the attendant pair based on the non-Markovian theory. The problem of formulating the stationary theory that is the long-term limit of the non-Markovian approach (description) [18] is considered. For the reaction pair there exists two equivalent possibilities of choosing physical quantities on the long-term limits of which the stationary theory is developed. These are either the non-Markovian probability density n(r; t), or the time operator technique elements (time analogs of the operators given in Section 3). For the attendant pair the realization of the above possibilities results in dierent stationary theories. The rst possibility is shown to lead to the development of the closed recipe for the calculation of the Markovian rate constant in the framework of stationary representations. Development of the stationary theory on the basis of the second possibility is the subject of Section 3. As is known [18], the Markovian (stationary) rate constant k is the long-term limit of the non-Markovian rate constant K(t) k = lim K(t) = lim sK L (s) : t→∞

s→0

(A.1)

Here we use the Laplace transform K L (s) of the rate constant K(t) de ned by the expression Z ∞ dt e−st K(t) : (A.2) K L (s) = 0

Further all Laplace transforms of time functions will be denoted similarly, i.e., by the upper index L. The Laplace transform of the rate constant may be calculated by the recipe [18] Z (A.3) K L (s) = dr U (r)n L (r) : In view of (A.1), it gives Eq. (2.2). For the case of diusion motion of B reactant the Laplace transform n L (r) of the non-Markovian density n(r; t) obeys the equation Z L [n (r) − dr0 fA (r|r0 )n L (r0 )] + DB r n L (r) : sn L (r) − 1 = −U (r)n L (r) − −1 A (A.4)

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The above equation is complemented by the boundary condition @r n L (r) r=R0 = 0 quite similar to condition (2.4). Since the density n(r) is the long-term limit of the density n(r; t), n(r) = lim n(r; t) = lim sn L (r) : t→∞

(A.5)

s→0

Eq. (A.4) easily gives Eq. (2.3). According to Eq. (2.11), the recipe for the calculation of K L (s) is reformulated in the operator technique terms using the operators Vˆ ; LˆA ; LˆB . Then Eq. (1.4) takes the form (s − LˆA − LˆB − Vˆ )n L (r) = 1 :

(A.6) L

To solve it formally, we introduce the resolvent operator gˆ that is the Laplace transform of the reaction pair propagator [22] (s − LˆA − LˆB − Vˆ )gˆ L = gˆ L (s − LˆA − LˆB − Vˆ ) = 1ˆ :

(A.7)

L

Unlike the case of the resolvent operator g, ˆ operation of gˆ on the nonnormalized state L is de ned. Thus, operation of gˆ on the left- and right-hand sides of Eq. (A.6) yields the desired formal solution n L (r) = gˆ L |1(r)i :

(A.8)

Comparison between Eqs. (A.8) and (2.16) shows that the obtained solution of the non-Markovian (time) problem is formally more simple than the solution of the Markovian (stationary) problem. In the operator technique terms recipe (A.3) is Z Z L L ˆ (A.9) K = − dr V |n (r)i = − dr Vˆ gˆ L |1(r)i : The free resolvent operator gˆ0L is introduced by the conjugate equations derived from Eqs. (A.7) at Vˆ ≡ 0 [22] (s − LˆA − LˆB )gˆ0L = gˆ0L (s − LˆA − LˆB ) = 1ˆ :

(A.10)

With Eq. (A.10) in Eq. (A.7), we obtain the relation between the resolvent and free resolvent operators gˆ L = gˆ0L + gˆ0L Vˆ gˆ L = gˆ0L + gˆ L Vˆ gˆ0L :

(A.11)

Note that at s → 0 Eqs. (A.7), (A.10) and (A.11) turn into their stationary analogs given in Section 2. The above operator technique provides conventional time description of the reaction pair evolution [22]. The time description of the attendant pair is carried out by analogy with the stationary description presented in Section 3. The free resolvent operator of A reactant is de ned by equation L L = gˆ0A (s − LˆA ) = 1ˆ : (s − LˆA )gˆ0A

(A.12)

Thus the free resolvent operator of the reaction pair is represented as L : gˆ0L = (1ˆ + gˆ0L LˆB )gˆ0A

(A.13)

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Substituting Eq. (A.13) into Eq. (A.11) yields the following de nitions for the resolvent L L and the reactivity operator Vˆatt in the attendant pair operator gˆatt L = (1ˆ + gˆ0L LˆB )−1 gˆ L ; gˆatt

L Vˆatt = Vˆ (1ˆ + gˆ0L LˆB ) :

(A.14)

In terms of these operators the attendant pair evolution is expressed as L L L ˆL L L L ˆL L Vatt gˆatt = gˆ0A Vatt gˆ0A : = gˆ0A + gˆ0A + gˆatt gˆatt

(A.15)

The equations are similar to Eqs. (A.11), and this justi es the above interpretation of the operators introduced. As the operation of gˆ0L on the nonnormalized state is de ned, the intrinsic constants L for the operators Vˆ and Vˆatt coincide. In view of Eqs. (2.15), we have Z Z Z L kr = − dr Vˆatt |1(r)i = − dr Vˆ (1ˆ + gˆ0L LˆB )|1(r)i = − dr Vˆ |1(r)i : (A.16) Thus, in the time attendant pair only spatial distortion of the reactivity is observed, not the change in its integral value. For the same reason, the recipe for the calculation of the rate constant Laplace transform via the attendant pair quantities is formally the same. Eqs. (A.9) give Z Z L L |1(r)i : K L (s) = − dr Vˆ (1 + gˆ0L LˆB )(1 + gˆ0L LˆB )−1 gˆ L |1(r)i = − dr Vˆatt gˆatt (A.17) Eqs. (A.14) – (A.17) provide the time description of the attendant pair evolution. One remark should be made before we pass from the time description of the attendant pair evolution to stationary description. When considering the reaction pair, we presented two ways of constructing the stationary theory. The rst method is based on the calculation of n L (r) followed by the calculation of the stationary density n(r) as a long-term limit n(r) = lim snL (r) = lim sgˆ L |1(r)i : s→0

s→0

(A.18)

This density should be used in recipe (2.2). The second way is based on the calculation of the stationary resolvent g(r|r0 ) by Eqs. (2.18) (or Eqs. (2.12)) derived from Eqs. (A.11) (or Eqs. (A.7)) by passing to the limit s → 0. Then the stationary density is calculated by Eq. (2.16). Both methods were equivalent. Realization of similar methods for the attendant pair leads to dierent stationary theories. This is related to incommutativity of passing to the limit s → 0 and the operation of the attendant reaction operator on the nonnormalized state, (compare Eqs. (A.16) with Eqs. (3.20) and (3.21)), i.e., L L ˆ ˆ 6 lim Vatt |1(r)i ≡ Vˆatt |1(r)i : (A.19) lim Vatt |1(r)i = s→0

s→0

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339

L Since the state gˆatt |1(r)i is also nonnormalized, passing to the limit (A.1) in Eq. (A.17) proves to be nontrivial, i.e., Z Z L L L ˆ 6 − dr Vˆatt lim sgˆatt |1(r)i : (A.20) k = − dr lims→0 Vatt sgˆatt |1(r)i = s→0

The stationary theory developed in Section 3 is the realization of the second method. Realization of the rst one results in the introduction of the function L |1(r)i n˜att (r) ≡ lim sgˆatt s→0

(A.21)

(compare with Eq. (A.18)). Though it is impossible to use this function in the integral recipe (3.31) for the calculation of the Markovian rate constant via stationary values of the attendant pair, the knowledge of n˜att (r) is sucient for the calculation of k. With L L , we express the Eqs. (A.15) in de nition (A.21), in view of the equality Vˆ gˆ L = Vˆatt gˆatt function n˜att (r) in terms of the reaction pair stationary density n˜att (r) = 1 + gˆ0A Vˆ |n(r)i :

(A.22)

The above expression makes it possible to easily explore the behavior of the function n˜att (r) in the range of large values of r. Using the asymptotic estimate (3.34) and recipe (2.20) for the free resolvent of A reactant, we have n˜att (r) ∼ 1 − r→∞

k : 4DA r

(A.23)

Thus, just as for the densities n(r) and natt (r), the knowledge of the coecient in the second term of the asymptotic expansion of the function n˜att (r) is sucient for nding the Markovian rate constant. However, no closed equation for the calculation of the function n˜att (r) exists. References [1] J.W. Bond Jr., K.M. Watson, J.A. Welch Jr., Atomic Theory of Gas Dynamics, Addison-Wesley, Reading, MA, 1968, 555 p. [2] Ya.I. Frenkel, Kinetic Theory of Liquids, Nauka, Leningrad; 1975. p. 592 (in Russian). [3] M.V. Smoluchowski, J. Phys. Chem. 92 (1916) 129. [4] F.C. Collins, G.E. Cimbal, J. Colloid Sci. 4 (1949) 437. [5] A.B. Doktorov, A.A. Kipriyanov, A.I. Burshtein, Zh. Eksp. Teor. Fiz. 74 (1978) 1184 (Sov. Phys. JETP 47 (1978) 623). [6] B.S. Yakovlev, S. Vanin, A.A. Balakin, KhVE 16 (1982) 139 (in Russian). [7] L.E. Stys, M.G. Foigel, Fiz. Tekh. Polyprovodn. 15 (1981) 761 (in Russian). [8] A.I. Burshtein, A.B. Doktorov, A.A. Kipriyanov, V.A. Morozov, S.G. Fedorenko, Zh. Eksp. Teor. Fiz. 88 (1985) 878 (in Russian). [9] H.C. Torrey, Phys. Rev. 52 (1953) 962. [10] I.A. Bondar, A.I. Burshtein, A.V. Kutikov, A.P. Metsentseva, V.V. Osino, V.P. Sakun, V.A. Smirnov, I.A. Sherbakov, Zh. Eksp. Teor. Fiz. 81 (1981) 96 (in Russian). [11] E.D. Aluker, Yu.P. Rulev, V.A. Stankevich, S.A. Chernov. Izv. AN USSR. Fiz. 38 (1974) 1230 (in Russian). [12] E.D. Aluker, R.G. Deich, F.V. Pirogov, S.A. Chernov. Fiz. Tv. Tela, 22 (1980) 3689 (in Russian). [13] A.I. Gazyk, F.P. Kesamanly, V.F. Kovalenko, I.E. Maronchuk, G.P. Peka, Fiz. Tekh. Poluprovodn. 16 (1982) 1333 (in Russian).

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[14] M.B. Kerami, G.V. Tsarepkov, Fiz. Tek. Poluprovodn. 17 (1983) 777 (in Russian). [15] B.S. Yakovlev, Properties and chemical reactions of excess electron in nonpolar organic matters, Ph.D. Theses, Chernogolovka,1979 (in Russian). [16] A.A. Kipriyanov, A.A. Karpushin, Khim. Fiz. 7 (1988) 60 (in Russian). [17] T.R. Waite, Phys. Rev. 107 (1957) 463. [18] A.A. Ovchinnikov, S.F. Timashev, A.A. Belyy, Kinetics of Diusion Controlled Chemical Processes, Nova Science, Commack, NY, 1989, 239 p. [19] H. Eyring, S.H. Lin, S.M. Lin, Basic Chemical Kinetics, John Wiley, New York, 1980. [20] N.N. Tunitzky, H.S. Bagdasaryan, Opt. Spektr. 15 (1963) 100 (in Russian). [21] A.B. Doktorov, A.A. Kipriyanov, Khim. Fiz. 1 (1982) 599 (in Russian). [22] A.A. Kipriyanov, A.B. Doktorov, Physica A 230 (1996) 75. [23] J. Taylor, The Scattering Theory, the Quantum Theory of Nonrelativistic Collisions, Wiley, New York, 1972. [24] A.B. Doktorov, A.A. Kipriyanov, KhimFiz. 1 (1982) 794 (in Russian). [25] A.A. Kipriyanov, I.V. Gopich, A.B. Doktorov, Physica A 255 (1998) 347. [26] G. Korn, T. Korn, Mathematical Handbook, McGraw-Hill, New York, 1968. [27] M. Kac, Some Probability Problems in Physics and Mathematics, Nauka, Moskva; pp. 1967-154 (in Russian). [28] O.Yu. Kulyatin, Bachelor’s Theses, Novosibirsk State University, 1998 (in Russian).

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