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Calculation of the rate constants of diffusion-controlled chemical reactions for reagents of arbitrary shape
Chemical Physics 300 (2004) 253–266 www.elsevier.com/locate/chemphys
Calculation of the rate constants of diffusion-controlled chemical reactions for reagents of arbitrary shape N.I. Chekunaev
*
Department of Matter Structure, N.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Ulitsa Kosygina 4, Moscow 117334, Russian Federation Received 15 October 2003; accepted 11 February 2004
Abstract A method of describing small-size reactant diffusion in the presence of any amount of arbitrary located sinks is developed. The propagation function of mobile reagent (MR) in such a system is found. The developed method was used to calculate rate constants of bimolecular reactions of MRs with absorbents (traps) having arbitrary shape. The procedure of calculation of the rate constant has been reduced to integral equation for flux density towards a trap which is MRsÕ absorber. If small parameters exist, the expansion in powers of these parameters is possible. The bimolecular rates were calculated for traps of different shape. The equation was obtained which permits to determine the asymptotic time dependence of rate constants. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Rate constants; Diffusion-controlled reactions
1. Introduction The problem of calculation of chemical reactions rate constants is of great interest in chemical physics. The diffusion-controlled reactions (when the diffusive approaching of reagents to each other is a limiting stage of reaction) are widely spread in chemistry. Usually, it is believed that at close contact the fusion of reagents takes place. In other words, there is a definite boundary upon reaching which, reaction definitely occurs between reagents. For reagents of spherical shape this boundary is a sphere with radius equal to the sum of reagents radiuses R ¼ R1 þ R2 . In this case, as was found by Smoluchowski [1], the rate constant equals to K ¼ 4pDR;
ð1Þ
where D is a diffusion coefficient. Eq. (1) is valid in case of ordinary Gaussian diffusion which takes place in ordered systems. In disordered systems, the diffusion has dispersive character [2]. In this case, the reagentsÕ transport is described by modified diffusion equation [3,4]. The rate constant of spherical reagents in disordered systems was calculated in [5] taking into account the electrostatic interaction. In many cases, one has to take into account non-spherical shape of reagents. The necessity to know the rate constant for non-spherical reagents arises when considering adsorbing of diffusing molecules by surface of the cell of microorganism [6]. The cell carries on its surface specific receptors for molecules of definite species. The receptor can be idealized as a circular patch. So in [6] the cell was approximately considered as a set of circular adsorbing patches on spherical reflecting surface of the cell. In [7] was calculated association rate of irregular bodies such as enzymes, which have specific distribution of positive and negative groups in them. In [8] were numerically calculated association rates for diffusion-limited site-specific association process for non-spherical molecules. This process is typical to enzymatic
*
Tel.: +95-9397275; fax: +95-1378318. E-mail address: [email protected] (N.I. Chekunaev).
0301-0104/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.02.008
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reactions, attachment of ligands to receptors on cell surfaces, antigen-antibody reactions, etc. Effect of molecular shape on the rate of diffusion-controlled reactions was considered in [9]. On the other hand, when studying chemical reactions, it is often important to take into account the influence of immovable reagentsÕ (trapsÕ) concentration on reactions rates [10–12]. This influence is originated by mutual screening of participating reagents. To find the dependence of the rate constant on traps concentration, one needs to know timedependent rate constants for a single trap, and for two, three (and so on) separated traps [11,12]. Therefore, it would be useful to find the rate constants of chemical reactions of mobile reagents with a combination of two, three and more absorbing reagents. In [13] was considered a particular case of combination of two congruent spherical reagents which are sinks for mobile reagents. As it is clear from the aforesaid, it is interesting also to calculate rate constants which are more complicated than for reagents with a spherical shape. The present paper is dedicated to rate constants calculation when the reaction occurs at direct contact of reagents. The case of ordinary Gaussian diffusion will be considered first and then the results in the case of dispersive transport will be stated. In Section 2, a method to solve the problem of diffusion transfer of a mobile point reagent in the environment of arbitrary located point traps will be developed. In Section 3, this method will be applied to calculate rate constants of a small-size mobile reagent absorbed by another reagent of any shape considered as a set of point traps. The application of this method to a simple case of spherical absorbing boundary will be considered. Then, the method will be applied to calculation of rate constants in cases of combination of two and three spherical absorbing traps with different radiuses. Also will be considered the case of toroidal trap. The solution will be found by expanding in powers of small parameters – quotients of spheres radiuses divided by distances between their centers in case of spheresÕ combination and quotient of cross-section radius divided by ring radius in case of tore.
2. An exact solution of reagent diffusion problem in the presence of N neutral point traps Suppose neutral centers 1 (traps) are located at points l1 ; l2 ; l3 ; . . . ; lN of lattice. These traps absorb random walking mobile reagents (MRs) and the latter as a result ‘‘die’’. In most cases, when the concentration of mobile reagents is small, one can neglect the fact that the trap cannot catch the mobile reagent more than once. Then, it is possible to ignore the fact that after MR absorption the trap disappears. In this case, the problem of MR migration is reduced to determining the probability that the MR is at a point x at the time t after the start on condition that MR has visited no one of centers l1 ; l2 ; l3 ; . . . ; lN , because MR ‘‘dies’’ after first visit and cannot ‘‘die’’ twice (see Fig. 1). For further purposes, we will need to use the GreenÕs function P 0 ðx; tÞ, determined in the random walk theory [14], which is the probability of MR being at point x at time t, under the condition that it started from origin x ¼ 0 at time t ¼ 0 in a lattice without traps. Let MR start at a point l0 6¼ l1 ; . . . ; lN at time moment t ¼ 0. We define the propagation function Pflg ðl0 ; x; tÞ to be the probability that MR which started at point l0 will be at a point x at the time moment t and visited no points flg ¼ l1 ; . . . ; lN (and therefore not being ‘‘died’’). In further consideration, we will need also to use function Fflg ðl0 ; x; tÞ which is the probability density that MR started at point l0 at time t ¼ 0, reaches the point x for the first time at time t and visited no points l1 ; . . . ; lN . Obviously, the flux (i.e., amount of MRs reached the trap lN per unit time) equals to Fflg ðl0 ; lN ; tÞ. The decrease rate of MR survival probability Ps to time moment t equals total MRsÕ flux to all traps. Then ð2Þ dPs =dt ¼ Fflg ðl0 ; l1 ; tÞ þ Fflg ðl0 ; l2 ; tÞ þ þ Fflg ðl0 ; lN ; tÞ : Let us note the main properties of the propagation function Pflg ðl0 ; x; tÞ. First, because of translational invariance violation in the system, this function depends not on difference jl0 xj but individually on both arguments l0 and x. Second, at x ¼ l1 ; l2 ; . . . ; lN its value becomes zero because traps are sinks for MRs and hence the concentration (proportional to P ) of MRs approaches to 0 close to these sinks. Then, one may write the following equations: Pflg ðl0 ; l1 ; tÞ ¼ 0; Pflg ðl0 ; l2 ; tÞ ¼ 0; ..................; Pflg ðl0 ; lN ; tÞ ¼ 0:
ð3Þ
Now, we will find the functions Pflg ðl0 ; x; tÞ and Fflg ðl0 ; x; tÞ. In the absence of traps ðflg ¼ ;Þ the equation Pf;g ðl0 ; x; tÞ ¼ P 0 ðl0 x; tÞ is valid. If there is one trap l1 , then evaluating Pflg ðl0 ; x; tÞ we have to exclude all the paths from point l0 to x, at which MR visited l1 at least once. Then 1 Here, ‘‘neutral centers’’ means that the problem is not complicated by existence of local electric fields. It is supposed also that traps do not distort lattice, so that MRÕs hops rates are the same as in a perfect lattice.
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
255
Fig. 1. The process of diffusion of mobile reagent from the point l0 toward point x in the presence of absorbing traps l1 ; l2 ; l3 ; . . . ; lN .
Pflg ðl0 ; x; tÞ ¼ P 0 ðl0 x; tÞ
Z
t
F ðl0 ; l1 ; sÞP 0 ðl1 x; t sÞ ds:
ð4Þ
0
Here, we subtract the probability that MR visited point l1 at least once at any time moment between 0 and t. After the Laplace transform, we have P ðl0 ; x; sÞ ¼ P 0 ðl0 x; sÞ F ðl0 ; l1 ; sÞP 0 ðl1 x; sÞ:
ð5Þ
The Laplace transform of function f ðtÞ is determined by the integral Z 1 f ðtÞ est dt: f ðsÞ ¼ L½f ðtÞ ¼ 0
(Below the image and original of Laplace transform will be denoted equally which will not lead to any confusion. Also we will omit parameter s for short, writing it if necessary). Since according to (3) at x ¼ l1 , the equation P ðl0 ; l1 ; sÞ ¼ 0 ¼ P 0 ðl0 l1 ; sÞ F ðl0 ; l1 ; sÞP 0 ð0; sÞ; is valid, then we have F ðl0 ; l1 ; sÞ ¼ P 0 ðl0 l1 ; sÞ=P 0 ð0; sÞ:
ð6Þ
This equation at l1 6¼ l0 coincides with that obtained in [14] formulae. For N traps similarly to (5) we have P ðl0 ; xÞ ¼ P 0 ðl0 xÞ Fflg ðl0 ; l1 ÞP 0 ðl1 xÞ Fflg ðl0 ; lN ÞP 0 ðlN xÞ:
ð7Þ
Putting to zero the left side in (7) at x ¼ l1 ; l2 ; . . . ; lN according to (3), we obtain the simultaneous equations (we omit often repeated subscripts {l}) P 0 ðl0 l1 Þ ¼ F ðl0 ; l1 ÞP 0 ð0Þ þ F ðl0 ; l2 ÞP 0 ðl2 l1 Þ þ þ F ðl0 ; lN ÞP 0 ðlN l1 Þ; P 0 ðl0 l2 Þ ¼ F ðl0 ; l1 ÞP 0 ðl0 l2 Þ þ F ðl0 ; l2 ÞP 0 ð0Þ þ þ F ðl0 ; lN ÞP 0 ðlN l2 Þ; ...............................................................;
ð8Þ
P 0 ðl0 lN Þ ¼ F ðl0 ; l1 ÞP 0 ðl0 lN Þ þ F ðl0 ; l2 ÞP 0 ðl2 lN Þ þ þ F ðl0 ; lN ÞP 0 ð0Þ; or in matrix form
P P 0 ðl0 li Þ ¼ k F ðl0 ; lk ÞAki : Aki ¼ P 0 ðlk li Þ
0
ð8 Þ
The solution of Eq. (80 ) is X Fflg ðl0 ; lk Þ ¼ ðA1 Þkj P 0 ðl0 lj Þ;
ð9Þ
j
and explicitly
0 P ðl0 l1 Þ 0 P ðl0 l2 Þ .. . P 0 ðl0 lN Þ Fflg ðl0 ; l1 Þ ¼ 0 0 P ð0Þ P ðl1 l2 Þ .. . P 0 ðl1 lN Þ
P 0 ðl2 l1 Þ P 0 ð0Þ .. .
.. .
P 0 ðl2 lN Þ P 0 ðl2 l1 Þ P 0 ð0Þ .. .
.. .
P 0 ðl2 lN Þ
P 0 ðlN l1 Þ P 0 ðlN l2 Þ .. . P 0 ð0Þ . . . etc: P 0 ðlN l1 Þ P 0 ðlN l2 Þ .. . P 0 ð0Þ
0
ð9 Þ
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Eqs. (7) and (90 ) determine the Laplace image of the propagation function Pflg ðl0 ; x; sÞ. Particular cases N ¼ 1: F ðl0 ; l1 Þ ¼
P 0 ðl0 l1 Þ ; P 0 ð0Þ
ð10Þ
that coincides with (6). N ¼ 2: Fflg ðl0 ; l1 Þ ¼
Fflg ðl0 ; l2 Þ ¼
P 0 ðl0 l1 ÞP 0 ð0Þ P 0 ðl0 l2 ÞP 0 ðl1 l2 Þ 2
½P 0 ð0Þ ½P 0 ðl1 l2 Þ
2
P 0 ðl0 l2 ÞP 0 ð0Þ P 0 ðl0 l1 ÞP 0 ðl1 l2 Þ 2
½P 0 ð0Þ ½P 0 ðl1 l2 Þ
2
;
ð11Þ
:
If we use exact expressions for Laplace transform of the propagation function P 0 ðx; sÞ, then the formulas (9) and (90 ) give an exact result. An exact expression of the generating function P 0 ðx; sÞ in d dimensions is given in [14] as a ddimensional integral. In case of three dimensions, the function P 0 ðx; sÞ can be calculated in two limits: rffiffiffiffi X s 0 exp jxj P ðx; sÞ ¼ jxj r; ð12Þ at s m ¼ s1 0 ; 4pDr D rffiffiffi 1 s 0 u0 u1 P ð0; sÞ ¼ ð13Þ ; at s m ¼ s1 0 ; m m where X is a unit cell volume of the lattice, r is an average hop distance, s0 is an hopping time, D ¼ mr2 =2 is a diffusion coefficient, u0 and u1 are the constants dependent on the lattice type, which have been calculated in [14]. After inverse Laplace transform, we can get the time-dependent propagation function P 0 ðx; tÞ at t s0 and jxj r ! 2 jxj 3=2 0 exp P ðx; tÞ ¼ Xð4pDtÞ ; ð14Þ 4Dt which differs from GreenÕs function of diffusion equation by the factor X, because it is normalized per one particle but not per volume unit. Calculation of determinants in (90 ) is possible in the one-dimensional case. In this case, P 0 ðx; tÞ and its Laplace transform P 0 ðx; sÞ are equal rffiffiffiffi x2 s 1=2 1=2 0 0 P ðx; tÞ ¼ ð4pDtÞ exp exp jxj : ð15Þ ; P ðx; sÞ ¼ ð4DsÞ D 4Dt Here, the factor X was omitted, because of its cancellation in formulas. It is seen that P 0 ðli lk ; sÞ can be written as the product of two factors pffiffiffiffiffiffiffiffi P 0 ðli lk ; sÞ ¼ ð4DsÞ1=2 exp jli lk j s=D ¼ f1 ðli Þf2 ðlk Þ: ð16Þ The determinants in (90 ) can be easily calculated when their elements are of (16) type. Let MR started at point l0 have two neighbors, l1 and l2 , on left and right sides correspondingly (l1 < l0 < l2 Þ. After some simplifications, the determinants (90 ) can be reduced to P 0 ðl0 l1 Þ P 0 ðl2 l1 Þ P 0 ðl0 l2 Þ P 0 ðl2 l1 Þ P 0 ðl0 l2 Þ P 0 ðl0 l1 Þ P 0 ð0Þ P 0 ð0Þ ; Fflg ðl0 ; l2 Þ ¼ : Fflg ðl0 ; l1 Þ ¼ ð17Þ 0 P 0 ð0Þ P 0 ðl2 l1 Þ P 0 ðl2 l1 Þ P ð0Þ P 0 ðl1 l2 Þ P 0 ðl1 l2 Þ P 0 ð0Þ P 0 ð0Þ This equation is identical to (11), which is valid in the case of two traps. This is easy to explain. In the one-dimensional case, the MR is located at point l0 being between two neighbor traps, l1 and l2 , which are positioned left and right to l0 correspondingly. It is clear that mobile reagent cannot reach any more distant traps and inevitably MR will be absorbed either by trap in l1 or by trap in l2 . Therefore, the existence of other traps except l1 and l2 will be insignificant for MR transport. The total MRsÕ flow toward traps l1 and l2 is equal to
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
Fflg ðl0 ; l1 Þ þ Fflg ðl0 ; l2 Þ ¼
P 0 ðl0 l1 Þ þ P 0 ðl0 l2 Þ : P 0 ð0Þ þ P 0 ðl1 l2 Þ
Substituting here Eq. (15) and using (2), one can find the fraction Ps ðtÞ of survived MRs by equation 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 3 þ exp jl exp jl l j s=D l j s=D 0 1 0 2 dPs 5; ¼ L1 4 pffiffiffiffiffiffiffiffi dt 1 þ exp l s=D
257
ð18Þ
ð19Þ
where l ¼ jl2 l1 j is the size of diffusion area, L1 indicates an inverse Laplace transform. After averaging over initial locations l0 between l1 and l2 , we have " # " rffiffiffiffi rffiffiffiffi# 1 dPs D l s 8X 1 1 2 1 tanh ¼ L : ð20Þ ¼ L l s 2 D p2 k¼0 ð2k 1Þ2 þ sl2 =p2 D dt Here was used the expansion of hyperbolic tangent in rational fractions [15] 1 px 4x X 1 tanh ¼ : 2 p k¼1 ð2k 1Þ2 þ x2 Making inverse Laplace transform in (20) and integrating with respect to t, one can find the fraction Ps of survived MRs to the time moment t " # 1 8X 1 ð2k 1Þ2 p2 D Ps ðtÞ ¼ 2 exp t : ð21Þ p k¼1 ð2k 1Þ2 l2 Naturally, this coincides with the result of a diffusive decay in one-dimensional interval of length l with zero boundary conditions.
3. The application of the method to calculation of chemical reactions rate constants Now, we shall apply the obtained equations to calculation of the rate constants of diffusion-controlled bimolecular chemical reactions. The rate constant K is determined from the equation K¼
IðSÞ ; cð1Þ
ð22Þ
where IðSÞ is MRsÕ total flow toward surface S of absorbing reagent, cð1Þ is the concentration of MRs in infinity. Taking cð1Þ ¼ 1, then K ¼ I. The rate constant K depends on time, approaching to its stationary value at large times. The formulae for KðtÞ can be obtained at large times t L2r =D, where Lr is a characteristic size of absorbing reagent. To find the rate constants at these times, one has to use Laplace images of used functions at limit s ! 0. To calculate rate constant K we will use Eq. (80 ). If the absorbing reagent (trap) has size much grater than the lattice constant, it can be considered as a continuous body. Then the trap of any shape can be represented as a continuous set of point traps, located on the reagentÕs surface A. In this case, the sum in Eq. (80 ) transforms into integral. As a result, it becomes an integral equation Z 0 P ðl0 gÞ ¼ F ðl0 ; nÞP 0 ðn gÞ dSn ; ðn; g 2 AÞ: ð23Þ SA
Substituting (12) into (23) we may take the factor X ¼ 1, because of its cancellation. After evaluating F ðl0 ; nÞ from this equation, it becomes possible to find the total flow I to the trap Z F ðl0 ; nÞ dSn : ð24Þ I¼ SA
Here, dSn is a surface element of the trap A. The first two terms of GreenÕs function Laplace image expansion in series of s1=2 are pffiffi 1 s 0 pffiffiffiffi : ð25Þ P ðr; s ! 0Þ ¼ 4pDr 4pD D
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We emphasize that the second term of expansion in series of s1=2 in Eq. (25) does not depend on r. This allows deducing important relations for dependence of rate constant on time (see below). Now, we average in (23) over initial positions l0 of MRs. To do this, we have to multiply both sides of Eq. (23) by MRÕs concentration cðl0 Þ and then integrate with respect to all values of l0 over the whole volume out of the surface SA . Putting cðl0 Þ ¼ const ¼ 1, we have
Z Z Z 0 F ðl0 ; nÞ dV0 P 0 ðn gÞ dSn : P ðl0 gÞ dV0 ¼ ð26Þ SA
The integral in the left side can be easily found. It equals to rffiffiffiffi Z 1 Z 1 s 1 0 1 P ðl0 gÞ dV0 ffi r exp r 4pr2 dr ¼ þ : s!0 4pD Lr D s
ð27Þ
For our consideration only the first term in the sum is significant. We denote the integral with respect to dV0 in the right side of (26) as F ðnÞ which is a flux density to the trap Z F ðnÞ ¼ F ðl0 ; nÞ dV0 : ð28Þ Then the integral equation for F ðnÞ is valid Z 1 F ðnÞP 0 ðn gÞ dSn : ¼ s SA
ð29Þ
After solving this integral equation and calculating the total MRsÕ flow I to the trap, it is possible to find the rate constant Z K¼I ¼ F ðnÞ dSn : ð30Þ SA
Substituting the expansion (25) of P 0 into (29), we deduce the expression pffiffi Z Z s 1 1 F ðnÞ dSn pffiffiffiffi F ðnÞ dSn : ¼ s 4pD SA rn rg 4pD D SA Using (30) we have pffiffi Z s 1 1 F ðnÞ dSn : pffiffiffiffi I ¼ þ s 4pD D 4pD SA rn rg
ð31Þ
ð32Þ
The solution of Eq. (32) without second addend in the left side corresponds to stationary ðt ! 1Þ solution F ðnÞ ¼ Fst ðnÞ=s, where Fst ðnÞ is solution of the integral equation Z 1 F ðnÞ st dSn : 1¼ ð33Þ 4pD S rn rg A
When the in the left side of Eq. (32) is present, the be found by substituting 1=s by pffiffiffiaddend ffi pffiffiffican ffi pffiffi second pffiffi solution 1=s þ I s=ð4pD DÞ. Then, the expression F ðnÞ ¼ Fst ðnÞ=s þ Fst ðnÞI s=ð4pD DÞ is valid. Applying (30) we have Z pffiffi 1 1 pffiffiffiffi : I¼ F ðnÞ dSn ¼ Ist þ sIst I ð34Þ s 4pD D SA Evaluating I from this equation and expanding the result in powers of s1=2 , we find 1 pffiffi Ist Ist 1 1 Ist pffiffiffiffi pffiffiffiffi : 1 s ffi Ist I¼ þ pffiffi s s s 4pD D 4pD D Making inverse Laplace transform, we find asymptotic time dependence of the rate constant Kst pffiffiffiffiffiffiffiffi : KðtÞ ¼ Kst 1 þ 4pD pDt
ð35Þ
ð36Þ
Here, Kst is the rate constant at t ! 1 The expression (36) is valid for traps of any shape. This result corresponds with the result obtained in [16] when calculating steric factor for reactions with anisotropic molecules. Now, we come to calculation of the stationary rate constant Kst of a bimolecular chemical reaction. We will consider the traps with shapes of most interest.
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
259
3.1. Point traps First consider a single point trap located at l1 . The flux of MR started at l0 on it is equal to F ðl0 ; l1 Þ. Averaging over the lattice with the use of (6), we have for s m " # X NMR X NMR N MR m 0 pffiffiffiffiffiffiffi 1 ; IðsÞ ¼ F ðl0 ; l1 ; sÞ ¼ P ðl; sÞ ¼ ð37Þ Ntot l Ntot P 0 ð0; sÞ l N tot sðu0 u1 s=mÞ 0
where NMR is a number of mobile reagents (random walkers), Ntot is a total number of lattice points. In the sum the point l0 ¼ l1 was excluded, as MR cannot start from the point occupied by trap. Summing in (37), MRs were assumed to be conserved X l
1 P 0 ðl; sÞ ¼ : s
ð38Þ
Introducing traps concentration c ¼ NMR =V ¼ NMR =XNtot , we have after expanding in powers of s rffiffiffiffi! mX 1 u1 1 NMR IðsÞ ¼ c : þ u0 s u0 ms Ntot Making inverse Laplace transform and putting c ¼ 1, we find the time-dependent rate constant for single point trap at t s0 u1 ð1Þ ð1Þ k ðtÞ ¼ kst 1 þ pffiffiffiffiffiffiffi ; ð39Þ u0 pmt ð1Þ
where kst ¼ mX=u0 is a stationary rate constant for point trap. The equation for the stationary rate constant can be written formally in the form of Smoluchowski formulae (1) ð1Þ
kst ¼ 4pDRtrap
with Rtrap ¼
X : 2pr2 u0
ð40Þ
Though at distances of order of lattice constant the diffusion equation does not work, the diffusion coefficient D is present in (40). Notice that Rtrap is not real trap size, but an effective radius. Let us consider now the combination of two point traps located at points l1 and l2 , separated by distance L. Averaging flux on them as in (37) with use of Eqs. (11), (18), and (38), we have NMR X P 0 ðl0 l1 ; sÞ þ P 0 ðl0 l2 ; sÞ NMR 1 ¼2 IðsÞ ¼ 1 : P 0 ð0; sÞ þ P 0 ðl1 l2 ; sÞ Ntot I Ntot s½P 0 ð0; sÞ þ P 0 ðL; sÞ 0
Here, in the sum over l0 were excluded points l0 ¼ l1 and l0 ¼ l2 , as the MR cannot start from them. For small s and L r we can use (12) and (13), then ( ) NMR m pffiffiffiffiffi 1 : IðsÞ ¼ 2 pffi Ntot s½u0 u1 ms þ 2prX2 L exp Lr 2 ms It is possible to make inverse Laplace transform, when Lr1 ðs=mÞ1=2 1 (or t L2 =mr2 L2 =D) by the use of expanding in series of s " rffiffiffiffi# 2mX 1 2u1 1 NMR IðsÞ ¼ c : þ 2 Ntot ðu0 þ 2prX2 LÞ s u0 þ 2prX2 L ms Then the time-dependent rate constant is equal to ! 2u1 1 ð2Þ ð2Þ k ðtÞ ¼ kst 1 þ pffiffiffiffiffiffiffi ; u0 þ 2prX2 L pmt
ð41Þ
where the stationary rate constant for two point traps combination is ð1Þ
ð2Þ
kst ¼
2mX 2kst ¼ : X u0 þ 2pr2 L 1 þ kstð1Þ 4pDL
ð42Þ
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3.2. A spherical absorbing trap Here, we shall demonstrate the use of the method in the simplest case of a spherical shape of absorbing reagent with radius R. Because of spherical symmetry, the value Fst ðnÞ does not depend on variable n. Therefore, it can be moved out of integral with respect to dSn in (33). Then Fst ðnÞ ¼ R
1 : 0 ðn g; s ¼ 0Þ dS P n SA
ð43Þ
Note that the integral in denominator does not depend on g 2 SA because of the spherical symmetry. Then, according to (30) the rate constant equals to Kst ¼ I ¼ R
4pR2 : P 0 ðn g; s ¼ 0Þ dSn SA
ð44Þ
The integral in the right side of (44) can be easily calculated (see Fig. 2). Since below we will need to use this integral when g is out of the surface of the sphere A at a distance l > R from the center, so we will calculate this integral at l P R Z Z Z p dSn 2pR2 sin a da R2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : ð45Þ P ðn g; s ¼ 0Þ dSn ¼ ¼ 2 2 lD SA SA 4pDRng 0 4pD l þ R 2lR cos a Substituting this expression into (44) and setting l ¼ R, we get well-known result (1). 3.3. Absorption by trap, having shape of combination of two spheres Let the absorbing reagent have the shape of two spheres A and B having radiuses R1 and R2 , which centers are separated by a distance L > R1 þ R2 . In that case (S ¼ SA þ SB ) the Eq. (23) transforms into two simultaneous integral equations for values F ðl0 ; n 2 AÞ and F ðl0 ; n 2 BÞ to be determined Z Z P 0 ðl0 gA Þ ¼ F ðl0 ; nÞP 0 ðn gA Þ dSn þ F ðl0 ; nÞP 0 ðn gA Þ dSn gA 2 A; S S ZA ZB ð46Þ 0 0 P ðl0 gB Þ ¼ F ðl0 ; nÞP ðn gB Þ dSn þ F ðl0 ; nÞP 0 ðn gB Þ dSn gB 2 B; SA
SB
where gA;B are the arbitrary points on spheres A and B, correspondingly. Averaging over initial positions l0 , we get equations for functions Fst ðnÞ analogically to (33) Z Z 1¼ Fst ðnÞP 0 ðn gA ; s ¼ 0Þ dSn þ Fst ðnÞP 0 ðn gA ; s ¼ 0Þ dSn ; ð47aÞ SA
1¼
Z
SB
Fst ðnÞP 0 ðn gB ; s ¼ 0Þ dSn þ SA
Z
Fst ðnÞP 0 ðn gB ; s ¼ 0Þ dSn :
ð47bÞ
SB
Now as opposed to the spherical case F ðnÞ depends on n. To solve Eqs. (47a) and (47b) exactly is rather difficult. But it is possible to find expansion in powers of parameters a1 ¼ R1 =L and a2 ¼ R2 =L. Now, we will show the procedure. Let us integrate both sides of Eq. (47a) over variable gA
Fig. 2. A diagram of integration in (45).
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
261
Fig. 3. A diagram of integration in (47). The point nA is determined by angles h1 and u1 . The point nB is determined by angles h2 and u2 (is not shown in diagram).
and (47b) over variable gB on surfaces of spheres A and B, correspondingly. According to (45) the first integral of function P 0 ðn gÞ in the right side of (47a) equals R1 =D and second integral equals R21 =ðl1 ðnÞDÞ (see Fig. 3). In the right side of (47b) vice versa, the first integral equals R22 =ðl2 ðnÞDÞ and second R2 =D. Integrals in the left sides of Eqs. (47a) and (47b) are equal to 4pR21 and 4pR22 , correspondingly. Then Eqs. (47a) and (47b) transform into Z
R21 D
Z
Fst ðnÞ dSn ; SB l1 ðnÞ
ð48aÞ
Z Z R22 Fst ðnÞ R2 dSn þ Fst ðnÞ dSn ; D SA l2 ðnÞ D SB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1;2 ðnÞ ¼ L2 þ R21;2 2LR1;2 cos h1;2 :
ð48bÞ
4pR21 ¼
R1 D
Fst ðnÞ dSn þ
SA
4pR22 ¼
At L R1;2 , the values l1;2 ðnÞ in (48a) and (48b) approximately can be replaced by L. Then, these equations become R1 R2 I1 þ 1 I2 ; D LD 2 R R2 4pR22 ¼ 2 I1 þ I2 ; LD D 4pR21 ¼
where I1 ¼
Z
Fst ðnÞ dSn
and
ð49Þ
I2 ¼
Z
SA
Fst ðnÞ dSn ; SB
are the total flows of MRs toward spheres A and B, correspondingly. Solving Eq. (49) and keeping first terms of expansion in series of parameters a1 and a2 , we find I1 ¼ 4pDR1 ð1 a1 Þ=ð1 a1 a2 Þ; I2 ¼ 4pDR2 ð1 a2 Þ=ð1 a1 a2 Þ;
ð50Þ
Kst ¼ I1 þ I2 ¼ 4pDLða1 þ a2 2a1 a2 Þ=ð1 a1 a2 Þ: As it is seen from (50), the bimolecular rate constant K is less than the sum of individual rate constants K1 ¼ 4pDR1 ¼ 4pDLa1 and K2 ¼ 4pDR2 ¼ 4pDLa2 , i.e., rate constants are not additive. This result is quite natural, since the existence of one trap near another leads to lowering of MRsÕ flow to the second trap because of MRsÕ absorption on the first trap. The expression (50) for K is an expansion in powers of a1;2 and has the accuracy of order a31;2 . To find next terms of this expansion, one has to expand F ðnÞ in Eqs. (47a) and (47b) in powers of cosn h. As it is shown in Appendix A, the term proportional to cosn h affects only terms in expansion of K having power greater than an1;2 . Details of calculation are given in Appendix A. The result of the expansion up to the fifth order is a1 þ a2 a1 a2 ð2 þ a31 þ a32 Þ þ O a61;2 : Kst ¼ 4pDL ð51Þ 1 a1 a2 ð1 þ a21 þ a22 Þ þ O a61;2 For two spheres with equal radiuses ða1 ¼ a2 ¼ aÞ, this result coincides with the expansion in powers of of the exact solution for this case, obtained in [13].
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3.4. The absorption by combination of three spherical traps In case of a trap having the shape of three spheres A, B and C with radiuses R1 , R2 , and R3 , Eq. (23) transforms into the equation similar to (48). When condition Lik R1;2;3 (Lik is the distance between the centers of spheres i and k) is fulfilled, the equation for values of flows I1 ; I2 and I3 toward spheres A, B and C by analogy with (49) has the form R1 R2 R2 I1 þ 1 I2 þ 1 I3 ; D L12 D L13 D 2 R R2 R2 4pR22 ¼ 2 I1 þ I2 þ 2 I3 ; L21 D D L23 D 2 2 R R R3 4pR23 ¼ 3 I1 þ 3 I2 þ I3 ; L31 D L32 D D Kst ¼ I1 þ I2 þ I3 : 4pR21 ¼
ð52Þ
To solve these equations one would not find any significant difficulties, but the result is rather cumbersome therefore we will not write it here. 3.5. The trap of toroidal shape The process of cracksÕ growth in mechanically stressed polyethylene at enhanced temperatures was considered in [17]. If the inhomogeneities of ÔvacancyÕ type are present in a polymer, they can be absorbed by cracks, which are present in a polymer. This leads to the growth of cracks. Near the cracks, which are stress concentrators, there are very high local stresses rloc , much greater than applied r0 . The value rloc increases toward cracksÕ tips (mouths). At reaching the area with high local tensions, the vacancies begin to drift toward the cracksÕ mouths and are eventually absorbed by them. In the volume of material disk shaped cracks can propagate. Then, near disk-shaped cracks there is an area (whose shape one can roughly take for a tore) at reaching which the vacancies are absorbed by cracks with high probability (see Fig. 4). Outside of this area, transport of vacancies is described by a diffusion equation. In light of this, the calculation of the rate constant of toroidal trap would be of interest. To calculate the rate constant, we shall use Eqs. (30) and (33). Then we have Z 1 dSn ; 1¼ ð53Þ Fst ðnÞ 4pD ST rn rg where the integration is made over the tore surface ST with radius R of a ring and radius r of a cross-section. The solution of the integral Eq. (53) is easier to find when the condition R r is fulfilled. In this case, one can approximately take Fst ðnÞ to be independent on n and then take it out of the integral. The remaining integral is not difficult to integrate Z dSn 8R 4pr ln : ð54Þ r ST rn rg Substituting this expression into (53), we find 1 8R Fst D r ln : r
ð55Þ
Fig. 4. The process of absorption of mobile ÔvacancyÕ V by the mouth of the disk-shaped crack in a polymer loaded by tensile stress r0 . The area of overstress r r0 , where diffusion motion changes into drift flow, is shown.
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
After putting (55) into (30), the resultant rate constant is
8R 2 Kst ¼ 4p DR ln : r
263
ð56Þ
One can find the next terms of the expansion of the rate constant in powers of (r=R) and lnðR=rÞ by expansion F ðnÞ in powers of cos h in (53). We have to note that there is an electrostatic analogy [6], according to which if diffusion coefficient D is replaced by electric constant e0 then expression for rate constant K turns into electric capacitance of isolated conductor of the same size and shape. The exact expression for tore capacity was quoted in [18]. In the limit r R, that expression fully corresponds with (56).
4. Conclusion The method developed here allows to describe the process of diffusive motion of mobile reagent in the presence of numbers of point traps (sinks). The calculation of the bimolecular rate constants of reagents with arbitrary shape is possible with the use of this method. The problem is reduced to the solution of the integral equations, like (29) for flux density F ðnÞ of mobile reagent on the surface of absorbing reagent and then to the calculation of integral (30). An important feature of the present method is that the flux density F ðnÞ is present in expressions only in integrals over traps surfaces. One does not need to know the density distribution of MRs in the space out of trapsÕ surface. This significantly simplifies calculations. If there are any small parameters, the integral equations can be solved with the help of the expansion in powers of these parameters. The solution was found for traps, having both the shape of the combination of spheres (Eqs. (51) and (52)) and tore (Eq. (56)). Such traps can be both immovable chemical reagents and different inhomogeneities like cracks or pores in materials with high degree of dispersity. Having found the stationary rate constant Kst , one can find also the dependence of the rate constant on time at approaching to stationary value (Eq. (36)). In this consideration, the reaction of absorbing of mobile reagent by fixed absorbing reagent (sink) was considered. In reality the results can be applied to a wider area. If both reagents are mobile, the result can be used when substituting the diffusion coefficient by the sum of reagentsÕ diffusion coefficients (if the rotational diffusion is slow enough). If both reagents have finite sizes the problem is reduced to the problem considered above, but with changed absorbing surface. In a disordered system, when the timesÕ distribution of MRsÕ hops from one center to another center is present, the motion of reagent is described by a modified diffusion equation [3,4]. In that case the solution can be obtained by substituting s in (35) by s ! m0 ðs=m0 Þb ;
ð57Þ
where m0 is a frequency factor, 0 < b 6 1 is a dispersion parameter (b ¼ 1 corresponds to the ordinary Gaussian diffusion in ordered system). The rate constant in disordered systems can be obtained by performing this substitution into (35). After inverse Laplace transform, we get # " b1 1=2 b=21 ðm0 tÞ K0 m0 ðm0 tÞ pffiffiffiffiffiffi : KðtÞ ¼ K0 ð58Þ þ CðbÞ 4pCðb=2ÞD0 D0 Here, K0 differs from Kst by substitution of D ! D0 , where D0 is a diffusion coefficient over the so-called diffusion cluster (see [3,4]).
Appendix A. The expansion of the rate constant K in series of a1 and a2 in the case of an absorbing reagent with the shape of a combination of two spheres To obtain the explicit solution of integral Eq. (46), we will expand function F ðnÞ in powers of cos h (owing to axial symmetry F ðnÞ ¼ F ðcos hÞ) Fst ðn 2 AÞ ¼ FA ðnÞ ¼ a0 þ a1 cos h þ a2 cos2 h þ a3 cos3 h þ a4 cos4 h þ ; Fst ðn 2 BÞ ¼ FB ðnÞ ¼ b0 þ b1 cos h þ b2 cos2 h þ b3 cos3 h þ b4 cos4 h þ
ðA:1Þ
Restricting to fourth order of expansion in cos h, we obtain equation for g 2 A (for g 2 B the equation is similar)
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4pD ¼
Z
a0 þ a1 cos h þ a2 cos2 h þ a3 cos3 h þ a4 cos4 h pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX R1 2 1 cos h cos hg sin h sin hg cos u Z b0 þ b1 cos h þ b2 cos2 h þ b3 cos3 h þ b4 cos4 h ffi dX: þ R22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 L2 þ R1 þ R2 2LðR1 cos hg þ R2 cos hÞ þ 2R1 R2 ðcos h cos hg sin h sin hg cos uÞ R21
ðA:2Þ
Here, h is an angle between On and the axis linking the centers of spheres (h ¼ h1 for sphere A and h ¼ h2 for sphere B (see Fig. 3)), dX ¼ sin h dh du. Parameters a0 ; . . . ; a4 and b0 ; . . . ; b4 have to be determined. The total flow of mobile particles toward both spheres equals Z Z I ¼ I1 þ I2 ¼ R21 FA ðcos hÞ dX þ R22 FB ðcos hÞ dX: Since
Z
dX ¼ 4p;
Z
4 cos h dX ¼ p; 3 2
Z
4 cos h dX ¼ p; 5 4
Z
cos h dX ¼
Z
cos3 h dX ¼ 0;
then using expansion (A.1), we have I ¼ 4pR21 ða0 þ a2 =3 þ a4 =5Þ þ 4pR22 ðb0 þ b2 =3 þ b4 =5Þ:
ðA:3Þ
The first integral in (A.2) can be directly calculated with the use of identity Z 2p Z p cosn h sin h du dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jn ¼ 1 cos h cos #g sin h sin hg cos u 0 0 Z Z p n ð1Þ on 2p du sin h dhð1 X cos h Y sin h sin uÞn1=2 ; ðX ¼ cos hg ; Y ¼ sin hg Þ: ¼ ðn 12Þðn 32Þ; . . . ; 12 X n 0 0 The latter integral can be calculated using formulae [15] Z 2p Z p Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du f ðX cos h þ Y sin h cos uÞ sin h dh ¼ 2p f X 2 þ Y 2 t dt: 0
0
0
With the help of these equations we have pffiffiffi pffiffiffi 1 pffiffiffi 4 1 2 J0 ¼ 4p 2; J1 ¼ 4p 2 cos hg ; J2 ¼ 4p 2 þ cos hg ; 3 15 5 pffiffiffi 4 pffiffiffi 16 1 8 1 3 2 4 cos hg þ cos hg ; J4 ¼ 4p 2 cos hg þ cos hg : þ J3 ¼ 4p 2 35 7 105 105 9 The first integral on the right-hand side of (A.2) equals to R1 pffiffiffi ða0 J0 þ a1 J1 þ a2 J2 þ a3 J3 þ a4 J4 Þ: 2
ðA:4Þ
Then, we expand the expression on the right side of (A.2) in a series of small parameters a1 ¼ R1 =L and a2 ¼ R2 =L 2 1=2 L þ R21 þ R22 2LðR1 cos hg þ R2 cos hÞ þ 2R1 R2 cos hg cos h 2R1 R2 sin hg sin h cos u 1 1 1 2 3 2 2 3 2 cos hg cos h ¼ L 1 þ a1 cos hg þ a2 cos h þ a1 þ a2 þ 2a1 a2 cos hg cos h 2 2 2 2 3 3 9 3 3 5 3 3 5 3 2 2 cos hg cos hg þ a2 cos h cos h þ a1 a2 cos hg cos h cos h þ a1 2 2 2 2 2 2 3 15 35 15 35 2 9 2 4 3 2 4 4 3 2 4 þ a1 a2 cos hg cos h cos hg þ a1 cos hg þ cos hg þ a2 cos h þ cos h 2 2 8 4 8 8 4 8 þ a31 a2 10 cos3 hg cos h 6 cos hg cos h þ a1 a32 10 cos hg cos3 h 6 cos hg cos h
9 9 27 2 2 3 2 2 2 2 þ a1 a2 cos hg cos h : ðA:5Þ cos hg cos h þ 2 2 2 2
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
265
In (A.5) the terms proportional to cos u were omitted, as their integral with respect to u equals 0. At the same time, cos2 u was substituted by its average value equal to 1/2. Substituting (A.4) and (A.5) into (A.2) and integrating over dh du, we obtain 4 16 1 4 1 8 1 a4 þ a1 þ a3 cos hg þ a2 þ a4 cos2 hg þ a3 cos3 hg 4pD ¼ 4pR1 a0 þ a2 þ 15 105 3 35 5 105 7
1 4pR22 1 3 1 1 1 1 2 2 a2 a21 a2 b1 þ 1 a21 þ a41 b0 þ þ a4 cos4 hg þ a a2 9 2 8 3 2 3 6 1 15 2 L 1 2 1 3 2 1 4 2 1 3 a2 a21 a2 þ a32 b3 þ þ a41 a21 a22 b2 þ þ a2 a21 b4 þ a1 a31 b0 8 5 5 10 35 5 35 10 2 2 1 1 2 2 6 8 a1 a2 2a31 a2 b1 þ a1 a31 þ a1 a22 b2 þ a1 a2 a31 a2 þ a1 a32 b3 þ 3 3 2 5 5 5 35
1 3 3 12 3 2 15 4 3 2 1 2 6 2 2 5 4 2 a1 a1 þ a1 a2 b4 cos hg þ a a1 b0 þ a1 a2 b1 þ a þ a a a b2 þ 5 10 35 2 1 4 2 2 1 5 1 2 4 1
9 2 3 2 36 2 2 3 4 5 3 10 3 5 3 1 3 2 3 a þ a a a b4 cos hg þ a b0 þ a1 a2 b1 þ a1 b2 þ 2a1 a2 b3 þ a1 b4 þ a1 a 2 b3 þ 10 10 1 35 1 2 4 1 2 1 3 6 2
35 4 35 7 a1 b0 þ a41 b2 þ a41 b4 cos4 hg : cos3 hg þ ðA:6Þ 8 24 8 In (A.6) the terms of order a61 and a62 and higher were omitted, as the expansion is up to the fifth order. It will be seen nþ1 nþ1 from (A.12) that coefficients have order an a1;2 a0 , bn a1;2 b0 . Equating the terms proportional to cos0 hg , cos1 hg , 2 3 4 cos hg , cos hg , cos hg in the left and right sides, we obtain (omitting terms of higher order than a51;2 ) cos0 hg :
D 4 16 1 2 3 4 1 1 2 ¼ a1 a0 þ a2 þ a4 þ a2 b0 1 a1 þ a1 þ a2 b1 þ b2 ; L 15 105 2 8 3 3
cos1 hg :
0 ¼ a1 cos2 hg : 0 ¼ a1
1 4 a1 þ a3 3 35
1 8 a2 þ a4 5 105
3 2 1 þ a22 b0 a1 a31 þ b1 a1 a2 þ b2 a1 ; 2 3 3
þ a22 b0
3 2 15 4 a1 a1 ; 2 4
ðA:7Þ
ðA:8Þ
ðA:9Þ
cos3 hg : 1 5 0 ¼ a1 a3 þ a31 a22 b0 ; 7 2 4 cos hg :
ðA:10Þ
1 35 0 ¼ a1 b4 þ a41 a22 b0 : 9 8
ðA:11Þ
For g 2 B equations for coefficients can be obtained by substitutions in (A.7)–(A.11) a1 $ a2 and ai $ bi . Thus, we have 10 simultaneous equations with 10 variables a0 ; . . . ; a4 , b0 ; . . . ; b4 . The solution is 315 3 2 35 15 135 3 2 1 7 9 a1 a2 b0 ; a3 ¼ a21 a22 b0 ; a2 ¼ a1 a22 þ a1 a2 b0 ; a1 ¼ a22 b0 þ a21 a22 b0 þ a21 a32 a0 : a4 ¼ 8 2 2 4 3 2 2 ðA:12Þ 315 2 3 a a a0 ; b4 ¼ 8 1 2
35 b3 ¼ a21 a22 a0 ; 2
b2 ¼
15 2 135 2 3 a a a0 ; a1 a2 þ 2 4 1 2
1 7 9 b1 ¼ a21 a0 þ a21 a22 a0 þ a31 a22 b0 ; 3 2 2
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N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
where a0 and b0 satisfy simultaneous equations: 7 5 27 D=L ¼ a0 a1 a21 a32 þ b0 a22 a21 a22 þ a41 a22 ; 2 2 8 5 27 7 D=L ¼ a0 a21 a21 a22 þ a21 a42 þ b0 a2 a31 a22 : 2 8 2
ðA:13Þ
Solving Eq. (A.13) and substituting the solution into (A.12) and then into (A.3), we find the rate constant Kst a1 þ a2 a1 a2 ð2 þ a31 þ a32 Þ þ O a61;2 : Kst ¼ I ¼ 4pDL ðA:14Þ 1 a1 a2 ð1 þ a21 þ a22 Þ þ O a61;2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Smoluchowski, Z. Phys. Chem. 92 (1917) 129. H. Scher, E.W. Montroll, Phys. Rev. B 12 (1975) 2455. N.I. Chekunaev, Yu.A. Berlin, V.N. Fleurov, J. Phys. C 15 (1982) 1219. N.I. Chekunaev, Yu.A. Berlin, V.N. Fleurov, J. Phys. C 16 (1983) 2569. Yu.A. Berlin, N.I. Chekunaev, V.I. Goldanskii, Radiat. Phys. Chem. 37 (1991) 407. H.C. Berg, E.M. Purcell, Biophys. J. 20 (1977) 193. K. Sharp et al., J. Phys. Chem. 91 (1987) 3624. Ying-Ta Wu, J.M. Nitsche, Chem. Eng. Sci. 50 (1995) 1467. A.I. Shushin, A.V. Barzykin, Biophys. J. 81 (2001) 3137. B.U. Felderhof, J.M. Deutch, J. Chem. Phys. 64 (1976) 4551. A. Szabo, R. Zwanzig, N. Agmon, Phys. Rev. Lett. 61 (1988) 2496. I.V. Gopich, A.M. Berezhkovskii, A. Szabo, J. Chem. Phys. 117 (2002) 2987. R. Samson, J.M. Deutch, J. Chem. Phys. 67 (1977) 847. E.W. Montroll, G.H. Weiss, J. Math. Phys. 6 (1965) 167. I.S. Gradshtein, I.M. Ryzhik, Table of Integrals, Series and Products, fifth ed., Academic Press, San Diego, CA, 1995. A.I. Shustin, J. Chem. Phys. 110 (1999) 12044. A.M. Kaplan, N.I. Chekunaev, V.G. NikolÕskii, Russ. Polymer News 4 (1999) 16. Hand Buch der Physik, vol. XVI, Springer, Berlin, 1958, p. 133.
Calculation of the rate constants of diffusion-controlled chemical reactions for reagents of arbitrary shape N.I. Chekunaev
*
Department of Matter Structure, N.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Ulitsa Kosygina 4, Moscow 117334, Russian Federation Received 15 October 2003; accepted 11 February 2004
Abstract A method of describing small-size reactant diffusion in the presence of any amount of arbitrary located sinks is developed. The propagation function of mobile reagent (MR) in such a system is found. The developed method was used to calculate rate constants of bimolecular reactions of MRs with absorbents (traps) having arbitrary shape. The procedure of calculation of the rate constant has been reduced to integral equation for flux density towards a trap which is MRsÕ absorber. If small parameters exist, the expansion in powers of these parameters is possible. The bimolecular rates were calculated for traps of different shape. The equation was obtained which permits to determine the asymptotic time dependence of rate constants. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Rate constants; Diffusion-controlled reactions
1. Introduction The problem of calculation of chemical reactions rate constants is of great interest in chemical physics. The diffusion-controlled reactions (when the diffusive approaching of reagents to each other is a limiting stage of reaction) are widely spread in chemistry. Usually, it is believed that at close contact the fusion of reagents takes place. In other words, there is a definite boundary upon reaching which, reaction definitely occurs between reagents. For reagents of spherical shape this boundary is a sphere with radius equal to the sum of reagents radiuses R ¼ R1 þ R2 . In this case, as was found by Smoluchowski [1], the rate constant equals to K ¼ 4pDR;
ð1Þ
where D is a diffusion coefficient. Eq. (1) is valid in case of ordinary Gaussian diffusion which takes place in ordered systems. In disordered systems, the diffusion has dispersive character [2]. In this case, the reagentsÕ transport is described by modified diffusion equation [3,4]. The rate constant of spherical reagents in disordered systems was calculated in [5] taking into account the electrostatic interaction. In many cases, one has to take into account non-spherical shape of reagents. The necessity to know the rate constant for non-spherical reagents arises when considering adsorbing of diffusing molecules by surface of the cell of microorganism [6]. The cell carries on its surface specific receptors for molecules of definite species. The receptor can be idealized as a circular patch. So in [6] the cell was approximately considered as a set of circular adsorbing patches on spherical reflecting surface of the cell. In [7] was calculated association rate of irregular bodies such as enzymes, which have specific distribution of positive and negative groups in them. In [8] were numerically calculated association rates for diffusion-limited site-specific association process for non-spherical molecules. This process is typical to enzymatic
*
Tel.: +95-9397275; fax: +95-1378318. E-mail address: [email protected] (N.I. Chekunaev).
0301-0104/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.02.008
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N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
reactions, attachment of ligands to receptors on cell surfaces, antigen-antibody reactions, etc. Effect of molecular shape on the rate of diffusion-controlled reactions was considered in [9]. On the other hand, when studying chemical reactions, it is often important to take into account the influence of immovable reagentsÕ (trapsÕ) concentration on reactions rates [10–12]. This influence is originated by mutual screening of participating reagents. To find the dependence of the rate constant on traps concentration, one needs to know timedependent rate constants for a single trap, and for two, three (and so on) separated traps [11,12]. Therefore, it would be useful to find the rate constants of chemical reactions of mobile reagents with a combination of two, three and more absorbing reagents. In [13] was considered a particular case of combination of two congruent spherical reagents which are sinks for mobile reagents. As it is clear from the aforesaid, it is interesting also to calculate rate constants which are more complicated than for reagents with a spherical shape. The present paper is dedicated to rate constants calculation when the reaction occurs at direct contact of reagents. The case of ordinary Gaussian diffusion will be considered first and then the results in the case of dispersive transport will be stated. In Section 2, a method to solve the problem of diffusion transfer of a mobile point reagent in the environment of arbitrary located point traps will be developed. In Section 3, this method will be applied to calculate rate constants of a small-size mobile reagent absorbed by another reagent of any shape considered as a set of point traps. The application of this method to a simple case of spherical absorbing boundary will be considered. Then, the method will be applied to calculation of rate constants in cases of combination of two and three spherical absorbing traps with different radiuses. Also will be considered the case of toroidal trap. The solution will be found by expanding in powers of small parameters – quotients of spheres radiuses divided by distances between their centers in case of spheresÕ combination and quotient of cross-section radius divided by ring radius in case of tore.
2. An exact solution of reagent diffusion problem in the presence of N neutral point traps Suppose neutral centers 1 (traps) are located at points l1 ; l2 ; l3 ; . . . ; lN of lattice. These traps absorb random walking mobile reagents (MRs) and the latter as a result ‘‘die’’. In most cases, when the concentration of mobile reagents is small, one can neglect the fact that the trap cannot catch the mobile reagent more than once. Then, it is possible to ignore the fact that after MR absorption the trap disappears. In this case, the problem of MR migration is reduced to determining the probability that the MR is at a point x at the time t after the start on condition that MR has visited no one of centers l1 ; l2 ; l3 ; . . . ; lN , because MR ‘‘dies’’ after first visit and cannot ‘‘die’’ twice (see Fig. 1). For further purposes, we will need to use the GreenÕs function P 0 ðx; tÞ, determined in the random walk theory [14], which is the probability of MR being at point x at time t, under the condition that it started from origin x ¼ 0 at time t ¼ 0 in a lattice without traps. Let MR start at a point l0 6¼ l1 ; . . . ; lN at time moment t ¼ 0. We define the propagation function Pflg ðl0 ; x; tÞ to be the probability that MR which started at point l0 will be at a point x at the time moment t and visited no points flg ¼ l1 ; . . . ; lN (and therefore not being ‘‘died’’). In further consideration, we will need also to use function Fflg ðl0 ; x; tÞ which is the probability density that MR started at point l0 at time t ¼ 0, reaches the point x for the first time at time t and visited no points l1 ; . . . ; lN . Obviously, the flux (i.e., amount of MRs reached the trap lN per unit time) equals to Fflg ðl0 ; lN ; tÞ. The decrease rate of MR survival probability Ps to time moment t equals total MRsÕ flux to all traps. Then ð2Þ dPs =dt ¼ Fflg ðl0 ; l1 ; tÞ þ Fflg ðl0 ; l2 ; tÞ þ þ Fflg ðl0 ; lN ; tÞ : Let us note the main properties of the propagation function Pflg ðl0 ; x; tÞ. First, because of translational invariance violation in the system, this function depends not on difference jl0 xj but individually on both arguments l0 and x. Second, at x ¼ l1 ; l2 ; . . . ; lN its value becomes zero because traps are sinks for MRs and hence the concentration (proportional to P ) of MRs approaches to 0 close to these sinks. Then, one may write the following equations: Pflg ðl0 ; l1 ; tÞ ¼ 0; Pflg ðl0 ; l2 ; tÞ ¼ 0; ..................; Pflg ðl0 ; lN ; tÞ ¼ 0:
ð3Þ
Now, we will find the functions Pflg ðl0 ; x; tÞ and Fflg ðl0 ; x; tÞ. In the absence of traps ðflg ¼ ;Þ the equation Pf;g ðl0 ; x; tÞ ¼ P 0 ðl0 x; tÞ is valid. If there is one trap l1 , then evaluating Pflg ðl0 ; x; tÞ we have to exclude all the paths from point l0 to x, at which MR visited l1 at least once. Then 1 Here, ‘‘neutral centers’’ means that the problem is not complicated by existence of local electric fields. It is supposed also that traps do not distort lattice, so that MRÕs hops rates are the same as in a perfect lattice.
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
255
Fig. 1. The process of diffusion of mobile reagent from the point l0 toward point x in the presence of absorbing traps l1 ; l2 ; l3 ; . . . ; lN .
Pflg ðl0 ; x; tÞ ¼ P 0 ðl0 x; tÞ
Z
t
F ðl0 ; l1 ; sÞP 0 ðl1 x; t sÞ ds:
ð4Þ
0
Here, we subtract the probability that MR visited point l1 at least once at any time moment between 0 and t. After the Laplace transform, we have P ðl0 ; x; sÞ ¼ P 0 ðl0 x; sÞ F ðl0 ; l1 ; sÞP 0 ðl1 x; sÞ:
ð5Þ
The Laplace transform of function f ðtÞ is determined by the integral Z 1 f ðtÞ est dt: f ðsÞ ¼ L½f ðtÞ ¼ 0
(Below the image and original of Laplace transform will be denoted equally which will not lead to any confusion. Also we will omit parameter s for short, writing it if necessary). Since according to (3) at x ¼ l1 , the equation P ðl0 ; l1 ; sÞ ¼ 0 ¼ P 0 ðl0 l1 ; sÞ F ðl0 ; l1 ; sÞP 0 ð0; sÞ; is valid, then we have F ðl0 ; l1 ; sÞ ¼ P 0 ðl0 l1 ; sÞ=P 0 ð0; sÞ:
ð6Þ
This equation at l1 6¼ l0 coincides with that obtained in [14] formulae. For N traps similarly to (5) we have P ðl0 ; xÞ ¼ P 0 ðl0 xÞ Fflg ðl0 ; l1 ÞP 0 ðl1 xÞ Fflg ðl0 ; lN ÞP 0 ðlN xÞ:
ð7Þ
Putting to zero the left side in (7) at x ¼ l1 ; l2 ; . . . ; lN according to (3), we obtain the simultaneous equations (we omit often repeated subscripts {l}) P 0 ðl0 l1 Þ ¼ F ðl0 ; l1 ÞP 0 ð0Þ þ F ðl0 ; l2 ÞP 0 ðl2 l1 Þ þ þ F ðl0 ; lN ÞP 0 ðlN l1 Þ; P 0 ðl0 l2 Þ ¼ F ðl0 ; l1 ÞP 0 ðl0 l2 Þ þ F ðl0 ; l2 ÞP 0 ð0Þ þ þ F ðl0 ; lN ÞP 0 ðlN l2 Þ; ...............................................................;
ð8Þ
P 0 ðl0 lN Þ ¼ F ðl0 ; l1 ÞP 0 ðl0 lN Þ þ F ðl0 ; l2 ÞP 0 ðl2 lN Þ þ þ F ðl0 ; lN ÞP 0 ð0Þ; or in matrix form
P P 0 ðl0 li Þ ¼ k F ðl0 ; lk ÞAki : Aki ¼ P 0 ðlk li Þ
0
ð8 Þ
The solution of Eq. (80 ) is X Fflg ðl0 ; lk Þ ¼ ðA1 Þkj P 0 ðl0 lj Þ;
ð9Þ
j
and explicitly
0 P ðl0 l1 Þ 0 P ðl0 l2 Þ .. . P 0 ðl0 lN Þ Fflg ðl0 ; l1 Þ ¼ 0 0 P ð0Þ P ðl1 l2 Þ .. . P 0 ðl1 lN Þ
P 0 ðl2 l1 Þ P 0 ð0Þ .. .
.. .
P 0 ðl2 lN Þ P 0 ðl2 l1 Þ P 0 ð0Þ .. .
.. .
P 0 ðl2 lN Þ
P 0 ðlN l1 Þ P 0 ðlN l2 Þ .. . P 0 ð0Þ . . . etc: P 0 ðlN l1 Þ P 0 ðlN l2 Þ .. . P 0 ð0Þ
0
ð9 Þ
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Eqs. (7) and (90 ) determine the Laplace image of the propagation function Pflg ðl0 ; x; sÞ. Particular cases N ¼ 1: F ðl0 ; l1 Þ ¼
P 0 ðl0 l1 Þ ; P 0 ð0Þ
ð10Þ
that coincides with (6). N ¼ 2: Fflg ðl0 ; l1 Þ ¼
Fflg ðl0 ; l2 Þ ¼
P 0 ðl0 l1 ÞP 0 ð0Þ P 0 ðl0 l2 ÞP 0 ðl1 l2 Þ 2
½P 0 ð0Þ ½P 0 ðl1 l2 Þ
2
P 0 ðl0 l2 ÞP 0 ð0Þ P 0 ðl0 l1 ÞP 0 ðl1 l2 Þ 2
½P 0 ð0Þ ½P 0 ðl1 l2 Þ
2
;
ð11Þ
:
If we use exact expressions for Laplace transform of the propagation function P 0 ðx; sÞ, then the formulas (9) and (90 ) give an exact result. An exact expression of the generating function P 0 ðx; sÞ in d dimensions is given in [14] as a ddimensional integral. In case of three dimensions, the function P 0 ðx; sÞ can be calculated in two limits: rffiffiffiffi X s 0 exp jxj P ðx; sÞ ¼ jxj r; ð12Þ at s m ¼ s1 0 ; 4pDr D rffiffiffi 1 s 0 u0 u1 P ð0; sÞ ¼ ð13Þ ; at s m ¼ s1 0 ; m m where X is a unit cell volume of the lattice, r is an average hop distance, s0 is an hopping time, D ¼ mr2 =2 is a diffusion coefficient, u0 and u1 are the constants dependent on the lattice type, which have been calculated in [14]. After inverse Laplace transform, we can get the time-dependent propagation function P 0 ðx; tÞ at t s0 and jxj r ! 2 jxj 3=2 0 exp P ðx; tÞ ¼ Xð4pDtÞ ; ð14Þ 4Dt which differs from GreenÕs function of diffusion equation by the factor X, because it is normalized per one particle but not per volume unit. Calculation of determinants in (90 ) is possible in the one-dimensional case. In this case, P 0 ðx; tÞ and its Laplace transform P 0 ðx; sÞ are equal rffiffiffiffi x2 s 1=2 1=2 0 0 P ðx; tÞ ¼ ð4pDtÞ exp exp jxj : ð15Þ ; P ðx; sÞ ¼ ð4DsÞ D 4Dt Here, the factor X was omitted, because of its cancellation in formulas. It is seen that P 0 ðli lk ; sÞ can be written as the product of two factors pffiffiffiffiffiffiffiffi P 0 ðli lk ; sÞ ¼ ð4DsÞ1=2 exp jli lk j s=D ¼ f1 ðli Þf2 ðlk Þ: ð16Þ The determinants in (90 ) can be easily calculated when their elements are of (16) type. Let MR started at point l0 have two neighbors, l1 and l2 , on left and right sides correspondingly (l1 < l0 < l2 Þ. After some simplifications, the determinants (90 ) can be reduced to P 0 ðl0 l1 Þ P 0 ðl2 l1 Þ P 0 ðl0 l2 Þ P 0 ðl2 l1 Þ P 0 ðl0 l2 Þ P 0 ðl0 l1 Þ P 0 ð0Þ P 0 ð0Þ ; Fflg ðl0 ; l2 Þ ¼ : Fflg ðl0 ; l1 Þ ¼ ð17Þ 0 P 0 ð0Þ P 0 ðl2 l1 Þ P 0 ðl2 l1 Þ P ð0Þ P 0 ðl1 l2 Þ P 0 ðl1 l2 Þ P 0 ð0Þ P 0 ð0Þ This equation is identical to (11), which is valid in the case of two traps. This is easy to explain. In the one-dimensional case, the MR is located at point l0 being between two neighbor traps, l1 and l2 , which are positioned left and right to l0 correspondingly. It is clear that mobile reagent cannot reach any more distant traps and inevitably MR will be absorbed either by trap in l1 or by trap in l2 . Therefore, the existence of other traps except l1 and l2 will be insignificant for MR transport. The total MRsÕ flow toward traps l1 and l2 is equal to
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
Fflg ðl0 ; l1 Þ þ Fflg ðl0 ; l2 Þ ¼
P 0 ðl0 l1 Þ þ P 0 ðl0 l2 Þ : P 0 ð0Þ þ P 0 ðl1 l2 Þ
Substituting here Eq. (15) and using (2), one can find the fraction Ps ðtÞ of survived MRs by equation 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 3 þ exp jl exp jl l j s=D l j s=D 0 1 0 2 dPs 5; ¼ L1 4 pffiffiffiffiffiffiffiffi dt 1 þ exp l s=D
257
ð18Þ
ð19Þ
where l ¼ jl2 l1 j is the size of diffusion area, L1 indicates an inverse Laplace transform. After averaging over initial locations l0 between l1 and l2 , we have " # " rffiffiffiffi rffiffiffiffi# 1 dPs D l s 8X 1 1 2 1 tanh ¼ L : ð20Þ ¼ L l s 2 D p2 k¼0 ð2k 1Þ2 þ sl2 =p2 D dt Here was used the expansion of hyperbolic tangent in rational fractions [15] 1 px 4x X 1 tanh ¼ : 2 p k¼1 ð2k 1Þ2 þ x2 Making inverse Laplace transform in (20) and integrating with respect to t, one can find the fraction Ps of survived MRs to the time moment t " # 1 8X 1 ð2k 1Þ2 p2 D Ps ðtÞ ¼ 2 exp t : ð21Þ p k¼1 ð2k 1Þ2 l2 Naturally, this coincides with the result of a diffusive decay in one-dimensional interval of length l with zero boundary conditions.
3. The application of the method to calculation of chemical reactions rate constants Now, we shall apply the obtained equations to calculation of the rate constants of diffusion-controlled bimolecular chemical reactions. The rate constant K is determined from the equation K¼
IðSÞ ; cð1Þ
ð22Þ
where IðSÞ is MRsÕ total flow toward surface S of absorbing reagent, cð1Þ is the concentration of MRs in infinity. Taking cð1Þ ¼ 1, then K ¼ I. The rate constant K depends on time, approaching to its stationary value at large times. The formulae for KðtÞ can be obtained at large times t L2r =D, where Lr is a characteristic size of absorbing reagent. To find the rate constants at these times, one has to use Laplace images of used functions at limit s ! 0. To calculate rate constant K we will use Eq. (80 ). If the absorbing reagent (trap) has size much grater than the lattice constant, it can be considered as a continuous body. Then the trap of any shape can be represented as a continuous set of point traps, located on the reagentÕs surface A. In this case, the sum in Eq. (80 ) transforms into integral. As a result, it becomes an integral equation Z 0 P ðl0 gÞ ¼ F ðl0 ; nÞP 0 ðn gÞ dSn ; ðn; g 2 AÞ: ð23Þ SA
Substituting (12) into (23) we may take the factor X ¼ 1, because of its cancellation. After evaluating F ðl0 ; nÞ from this equation, it becomes possible to find the total flow I to the trap Z F ðl0 ; nÞ dSn : ð24Þ I¼ SA
Here, dSn is a surface element of the trap A. The first two terms of GreenÕs function Laplace image expansion in series of s1=2 are pffiffi 1 s 0 pffiffiffiffi : ð25Þ P ðr; s ! 0Þ ¼ 4pDr 4pD D
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We emphasize that the second term of expansion in series of s1=2 in Eq. (25) does not depend on r. This allows deducing important relations for dependence of rate constant on time (see below). Now, we average in (23) over initial positions l0 of MRs. To do this, we have to multiply both sides of Eq. (23) by MRÕs concentration cðl0 Þ and then integrate with respect to all values of l0 over the whole volume out of the surface SA . Putting cðl0 Þ ¼ const ¼ 1, we have
Z Z Z 0 F ðl0 ; nÞ dV0 P 0 ðn gÞ dSn : P ðl0 gÞ dV0 ¼ ð26Þ SA
The integral in the left side can be easily found. It equals to rffiffiffiffi Z 1 Z 1 s 1 0 1 P ðl0 gÞ dV0 ffi r exp r 4pr2 dr ¼ þ : s!0 4pD Lr D s
ð27Þ
For our consideration only the first term in the sum is significant. We denote the integral with respect to dV0 in the right side of (26) as F ðnÞ which is a flux density to the trap Z F ðnÞ ¼ F ðl0 ; nÞ dV0 : ð28Þ Then the integral equation for F ðnÞ is valid Z 1 F ðnÞP 0 ðn gÞ dSn : ¼ s SA
ð29Þ
After solving this integral equation and calculating the total MRsÕ flow I to the trap, it is possible to find the rate constant Z K¼I ¼ F ðnÞ dSn : ð30Þ SA
Substituting the expansion (25) of P 0 into (29), we deduce the expression pffiffi Z Z s 1 1 F ðnÞ dSn pffiffiffiffi F ðnÞ dSn : ¼ s 4pD SA rn rg 4pD D SA Using (30) we have pffiffi Z s 1 1 F ðnÞ dSn : pffiffiffiffi I ¼ þ s 4pD D 4pD SA rn rg
ð31Þ
ð32Þ
The solution of Eq. (32) without second addend in the left side corresponds to stationary ðt ! 1Þ solution F ðnÞ ¼ Fst ðnÞ=s, where Fst ðnÞ is solution of the integral equation Z 1 F ðnÞ st dSn : 1¼ ð33Þ 4pD S rn rg A
When the in the left side of Eq. (32) is present, the be found by substituting 1=s by pffiffiffiaddend ffi pffiffiffican ffi pffiffi second pffiffi solution 1=s þ I s=ð4pD DÞ. Then, the expression F ðnÞ ¼ Fst ðnÞ=s þ Fst ðnÞI s=ð4pD DÞ is valid. Applying (30) we have Z pffiffi 1 1 pffiffiffiffi : I¼ F ðnÞ dSn ¼ Ist þ sIst I ð34Þ s 4pD D SA Evaluating I from this equation and expanding the result in powers of s1=2 , we find 1 pffiffi Ist Ist 1 1 Ist pffiffiffiffi pffiffiffiffi : 1 s ffi Ist I¼ þ pffiffi s s s 4pD D 4pD D Making inverse Laplace transform, we find asymptotic time dependence of the rate constant Kst pffiffiffiffiffiffiffiffi : KðtÞ ¼ Kst 1 þ 4pD pDt
ð35Þ
ð36Þ
Here, Kst is the rate constant at t ! 1 The expression (36) is valid for traps of any shape. This result corresponds with the result obtained in [16] when calculating steric factor for reactions with anisotropic molecules. Now, we come to calculation of the stationary rate constant Kst of a bimolecular chemical reaction. We will consider the traps with shapes of most interest.
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259
3.1. Point traps First consider a single point trap located at l1 . The flux of MR started at l0 on it is equal to F ðl0 ; l1 Þ. Averaging over the lattice with the use of (6), we have for s m " # X NMR X NMR N MR m 0 pffiffiffiffiffiffiffi 1 ; IðsÞ ¼ F ðl0 ; l1 ; sÞ ¼ P ðl; sÞ ¼ ð37Þ Ntot l Ntot P 0 ð0; sÞ l N tot sðu0 u1 s=mÞ 0
where NMR is a number of mobile reagents (random walkers), Ntot is a total number of lattice points. In the sum the point l0 ¼ l1 was excluded, as MR cannot start from the point occupied by trap. Summing in (37), MRs were assumed to be conserved X l
1 P 0 ðl; sÞ ¼ : s
ð38Þ
Introducing traps concentration c ¼ NMR =V ¼ NMR =XNtot , we have after expanding in powers of s rffiffiffiffi! mX 1 u1 1 NMR IðsÞ ¼ c : þ u0 s u0 ms Ntot Making inverse Laplace transform and putting c ¼ 1, we find the time-dependent rate constant for single point trap at t s0 u1 ð1Þ ð1Þ k ðtÞ ¼ kst 1 þ pffiffiffiffiffiffiffi ; ð39Þ u0 pmt ð1Þ
where kst ¼ mX=u0 is a stationary rate constant for point trap. The equation for the stationary rate constant can be written formally in the form of Smoluchowski formulae (1) ð1Þ
kst ¼ 4pDRtrap
with Rtrap ¼
X : 2pr2 u0
ð40Þ
Though at distances of order of lattice constant the diffusion equation does not work, the diffusion coefficient D is present in (40). Notice that Rtrap is not real trap size, but an effective radius. Let us consider now the combination of two point traps located at points l1 and l2 , separated by distance L. Averaging flux on them as in (37) with use of Eqs. (11), (18), and (38), we have NMR X P 0 ðl0 l1 ; sÞ þ P 0 ðl0 l2 ; sÞ NMR 1 ¼2 IðsÞ ¼ 1 : P 0 ð0; sÞ þ P 0 ðl1 l2 ; sÞ Ntot I Ntot s½P 0 ð0; sÞ þ P 0 ðL; sÞ 0
Here, in the sum over l0 were excluded points l0 ¼ l1 and l0 ¼ l2 , as the MR cannot start from them. For small s and L r we can use (12) and (13), then ( ) NMR m pffiffiffiffiffi 1 : IðsÞ ¼ 2 pffi Ntot s½u0 u1 ms þ 2prX2 L exp Lr 2 ms It is possible to make inverse Laplace transform, when Lr1 ðs=mÞ1=2 1 (or t L2 =mr2 L2 =D) by the use of expanding in series of s " rffiffiffiffi# 2mX 1 2u1 1 NMR IðsÞ ¼ c : þ 2 Ntot ðu0 þ 2prX2 LÞ s u0 þ 2prX2 L ms Then the time-dependent rate constant is equal to ! 2u1 1 ð2Þ ð2Þ k ðtÞ ¼ kst 1 þ pffiffiffiffiffiffiffi ; u0 þ 2prX2 L pmt
ð41Þ
where the stationary rate constant for two point traps combination is ð1Þ
ð2Þ
kst ¼
2mX 2kst ¼ : X u0 þ 2pr2 L 1 þ kstð1Þ 4pDL
ð42Þ
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3.2. A spherical absorbing trap Here, we shall demonstrate the use of the method in the simplest case of a spherical shape of absorbing reagent with radius R. Because of spherical symmetry, the value Fst ðnÞ does not depend on variable n. Therefore, it can be moved out of integral with respect to dSn in (33). Then Fst ðnÞ ¼ R
1 : 0 ðn g; s ¼ 0Þ dS P n SA
ð43Þ
Note that the integral in denominator does not depend on g 2 SA because of the spherical symmetry. Then, according to (30) the rate constant equals to Kst ¼ I ¼ R
4pR2 : P 0 ðn g; s ¼ 0Þ dSn SA
ð44Þ
The integral in the right side of (44) can be easily calculated (see Fig. 2). Since below we will need to use this integral when g is out of the surface of the sphere A at a distance l > R from the center, so we will calculate this integral at l P R Z Z Z p dSn 2pR2 sin a da R2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : ð45Þ P ðn g; s ¼ 0Þ dSn ¼ ¼ 2 2 lD SA SA 4pDRng 0 4pD l þ R 2lR cos a Substituting this expression into (44) and setting l ¼ R, we get well-known result (1). 3.3. Absorption by trap, having shape of combination of two spheres Let the absorbing reagent have the shape of two spheres A and B having radiuses R1 and R2 , which centers are separated by a distance L > R1 þ R2 . In that case (S ¼ SA þ SB ) the Eq. (23) transforms into two simultaneous integral equations for values F ðl0 ; n 2 AÞ and F ðl0 ; n 2 BÞ to be determined Z Z P 0 ðl0 gA Þ ¼ F ðl0 ; nÞP 0 ðn gA Þ dSn þ F ðl0 ; nÞP 0 ðn gA Þ dSn gA 2 A; S S ZA ZB ð46Þ 0 0 P ðl0 gB Þ ¼ F ðl0 ; nÞP ðn gB Þ dSn þ F ðl0 ; nÞP 0 ðn gB Þ dSn gB 2 B; SA
SB
where gA;B are the arbitrary points on spheres A and B, correspondingly. Averaging over initial positions l0 , we get equations for functions Fst ðnÞ analogically to (33) Z Z 1¼ Fst ðnÞP 0 ðn gA ; s ¼ 0Þ dSn þ Fst ðnÞP 0 ðn gA ; s ¼ 0Þ dSn ; ð47aÞ SA
1¼
Z
SB
Fst ðnÞP 0 ðn gB ; s ¼ 0Þ dSn þ SA
Z
Fst ðnÞP 0 ðn gB ; s ¼ 0Þ dSn :
ð47bÞ
SB
Now as opposed to the spherical case F ðnÞ depends on n. To solve Eqs. (47a) and (47b) exactly is rather difficult. But it is possible to find expansion in powers of parameters a1 ¼ R1 =L and a2 ¼ R2 =L. Now, we will show the procedure. Let us integrate both sides of Eq. (47a) over variable gA
Fig. 2. A diagram of integration in (45).
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
261
Fig. 3. A diagram of integration in (47). The point nA is determined by angles h1 and u1 . The point nB is determined by angles h2 and u2 (is not shown in diagram).
and (47b) over variable gB on surfaces of spheres A and B, correspondingly. According to (45) the first integral of function P 0 ðn gÞ in the right side of (47a) equals R1 =D and second integral equals R21 =ðl1 ðnÞDÞ (see Fig. 3). In the right side of (47b) vice versa, the first integral equals R22 =ðl2 ðnÞDÞ and second R2 =D. Integrals in the left sides of Eqs. (47a) and (47b) are equal to 4pR21 and 4pR22 , correspondingly. Then Eqs. (47a) and (47b) transform into Z
R21 D
Z
Fst ðnÞ dSn ; SB l1 ðnÞ
ð48aÞ
Z Z R22 Fst ðnÞ R2 dSn þ Fst ðnÞ dSn ; D SA l2 ðnÞ D SB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1;2 ðnÞ ¼ L2 þ R21;2 2LR1;2 cos h1;2 :
ð48bÞ
4pR21 ¼
R1 D
Fst ðnÞ dSn þ
SA
4pR22 ¼
At L R1;2 , the values l1;2 ðnÞ in (48a) and (48b) approximately can be replaced by L. Then, these equations become R1 R2 I1 þ 1 I2 ; D LD 2 R R2 4pR22 ¼ 2 I1 þ I2 ; LD D 4pR21 ¼
where I1 ¼
Z
Fst ðnÞ dSn
and
ð49Þ
I2 ¼
Z
SA
Fst ðnÞ dSn ; SB
are the total flows of MRs toward spheres A and B, correspondingly. Solving Eq. (49) and keeping first terms of expansion in series of parameters a1 and a2 , we find I1 ¼ 4pDR1 ð1 a1 Þ=ð1 a1 a2 Þ; I2 ¼ 4pDR2 ð1 a2 Þ=ð1 a1 a2 Þ;
ð50Þ
Kst ¼ I1 þ I2 ¼ 4pDLða1 þ a2 2a1 a2 Þ=ð1 a1 a2 Þ: As it is seen from (50), the bimolecular rate constant K is less than the sum of individual rate constants K1 ¼ 4pDR1 ¼ 4pDLa1 and K2 ¼ 4pDR2 ¼ 4pDLa2 , i.e., rate constants are not additive. This result is quite natural, since the existence of one trap near another leads to lowering of MRsÕ flow to the second trap because of MRsÕ absorption on the first trap. The expression (50) for K is an expansion in powers of a1;2 and has the accuracy of order a31;2 . To find next terms of this expansion, one has to expand F ðnÞ in Eqs. (47a) and (47b) in powers of cosn h. As it is shown in Appendix A, the term proportional to cosn h affects only terms in expansion of K having power greater than an1;2 . Details of calculation are given in Appendix A. The result of the expansion up to the fifth order is a1 þ a2 a1 a2 ð2 þ a31 þ a32 Þ þ O a61;2 : Kst ¼ 4pDL ð51Þ 1 a1 a2 ð1 þ a21 þ a22 Þ þ O a61;2 For two spheres with equal radiuses ða1 ¼ a2 ¼ aÞ, this result coincides with the expansion in powers of of the exact solution for this case, obtained in [13].
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3.4. The absorption by combination of three spherical traps In case of a trap having the shape of three spheres A, B and C with radiuses R1 , R2 , and R3 , Eq. (23) transforms into the equation similar to (48). When condition Lik R1;2;3 (Lik is the distance between the centers of spheres i and k) is fulfilled, the equation for values of flows I1 ; I2 and I3 toward spheres A, B and C by analogy with (49) has the form R1 R2 R2 I1 þ 1 I2 þ 1 I3 ; D L12 D L13 D 2 R R2 R2 4pR22 ¼ 2 I1 þ I2 þ 2 I3 ; L21 D D L23 D 2 2 R R R3 4pR23 ¼ 3 I1 þ 3 I2 þ I3 ; L31 D L32 D D Kst ¼ I1 þ I2 þ I3 : 4pR21 ¼
ð52Þ
To solve these equations one would not find any significant difficulties, but the result is rather cumbersome therefore we will not write it here. 3.5. The trap of toroidal shape The process of cracksÕ growth in mechanically stressed polyethylene at enhanced temperatures was considered in [17]. If the inhomogeneities of ÔvacancyÕ type are present in a polymer, they can be absorbed by cracks, which are present in a polymer. This leads to the growth of cracks. Near the cracks, which are stress concentrators, there are very high local stresses rloc , much greater than applied r0 . The value rloc increases toward cracksÕ tips (mouths). At reaching the area with high local tensions, the vacancies begin to drift toward the cracksÕ mouths and are eventually absorbed by them. In the volume of material disk shaped cracks can propagate. Then, near disk-shaped cracks there is an area (whose shape one can roughly take for a tore) at reaching which the vacancies are absorbed by cracks with high probability (see Fig. 4). Outside of this area, transport of vacancies is described by a diffusion equation. In light of this, the calculation of the rate constant of toroidal trap would be of interest. To calculate the rate constant, we shall use Eqs. (30) and (33). Then we have Z 1 dSn ; 1¼ ð53Þ Fst ðnÞ 4pD ST rn rg where the integration is made over the tore surface ST with radius R of a ring and radius r of a cross-section. The solution of the integral Eq. (53) is easier to find when the condition R r is fulfilled. In this case, one can approximately take Fst ðnÞ to be independent on n and then take it out of the integral. The remaining integral is not difficult to integrate Z dSn 8R 4pr ln : ð54Þ r ST rn rg Substituting this expression into (53), we find 1 8R Fst D r ln : r
ð55Þ
Fig. 4. The process of absorption of mobile ÔvacancyÕ V by the mouth of the disk-shaped crack in a polymer loaded by tensile stress r0 . The area of overstress r r0 , where diffusion motion changes into drift flow, is shown.
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
After putting (55) into (30), the resultant rate constant is
8R 2 Kst ¼ 4p DR ln : r
263
ð56Þ
One can find the next terms of the expansion of the rate constant in powers of (r=R) and lnðR=rÞ by expansion F ðnÞ in powers of cos h in (53). We have to note that there is an electrostatic analogy [6], according to which if diffusion coefficient D is replaced by electric constant e0 then expression for rate constant K turns into electric capacitance of isolated conductor of the same size and shape. The exact expression for tore capacity was quoted in [18]. In the limit r R, that expression fully corresponds with (56).
4. Conclusion The method developed here allows to describe the process of diffusive motion of mobile reagent in the presence of numbers of point traps (sinks). The calculation of the bimolecular rate constants of reagents with arbitrary shape is possible with the use of this method. The problem is reduced to the solution of the integral equations, like (29) for flux density F ðnÞ of mobile reagent on the surface of absorbing reagent and then to the calculation of integral (30). An important feature of the present method is that the flux density F ðnÞ is present in expressions only in integrals over traps surfaces. One does not need to know the density distribution of MRs in the space out of trapsÕ surface. This significantly simplifies calculations. If there are any small parameters, the integral equations can be solved with the help of the expansion in powers of these parameters. The solution was found for traps, having both the shape of the combination of spheres (Eqs. (51) and (52)) and tore (Eq. (56)). Such traps can be both immovable chemical reagents and different inhomogeneities like cracks or pores in materials with high degree of dispersity. Having found the stationary rate constant Kst , one can find also the dependence of the rate constant on time at approaching to stationary value (Eq. (36)). In this consideration, the reaction of absorbing of mobile reagent by fixed absorbing reagent (sink) was considered. In reality the results can be applied to a wider area. If both reagents are mobile, the result can be used when substituting the diffusion coefficient by the sum of reagentsÕ diffusion coefficients (if the rotational diffusion is slow enough). If both reagents have finite sizes the problem is reduced to the problem considered above, but with changed absorbing surface. In a disordered system, when the timesÕ distribution of MRsÕ hops from one center to another center is present, the motion of reagent is described by a modified diffusion equation [3,4]. In that case the solution can be obtained by substituting s in (35) by s ! m0 ðs=m0 Þb ;
ð57Þ
where m0 is a frequency factor, 0 < b 6 1 is a dispersion parameter (b ¼ 1 corresponds to the ordinary Gaussian diffusion in ordered system). The rate constant in disordered systems can be obtained by performing this substitution into (35). After inverse Laplace transform, we get # " b1 1=2 b=21 ðm0 tÞ K0 m0 ðm0 tÞ pffiffiffiffiffiffi : KðtÞ ¼ K0 ð58Þ þ CðbÞ 4pCðb=2ÞD0 D0 Here, K0 differs from Kst by substitution of D ! D0 , where D0 is a diffusion coefficient over the so-called diffusion cluster (see [3,4]).
Appendix A. The expansion of the rate constant K in series of a1 and a2 in the case of an absorbing reagent with the shape of a combination of two spheres To obtain the explicit solution of integral Eq. (46), we will expand function F ðnÞ in powers of cos h (owing to axial symmetry F ðnÞ ¼ F ðcos hÞ) Fst ðn 2 AÞ ¼ FA ðnÞ ¼ a0 þ a1 cos h þ a2 cos2 h þ a3 cos3 h þ a4 cos4 h þ ; Fst ðn 2 BÞ ¼ FB ðnÞ ¼ b0 þ b1 cos h þ b2 cos2 h þ b3 cos3 h þ b4 cos4 h þ
ðA:1Þ
Restricting to fourth order of expansion in cos h, we obtain equation for g 2 A (for g 2 B the equation is similar)
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4pD ¼
Z
a0 þ a1 cos h þ a2 cos2 h þ a3 cos3 h þ a4 cos4 h pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dX R1 2 1 cos h cos hg sin h sin hg cos u Z b0 þ b1 cos h þ b2 cos2 h þ b3 cos3 h þ b4 cos4 h ffi dX: þ R22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 L2 þ R1 þ R2 2LðR1 cos hg þ R2 cos hÞ þ 2R1 R2 ðcos h cos hg sin h sin hg cos uÞ R21
ðA:2Þ
Here, h is an angle between On and the axis linking the centers of spheres (h ¼ h1 for sphere A and h ¼ h2 for sphere B (see Fig. 3)), dX ¼ sin h dh du. Parameters a0 ; . . . ; a4 and b0 ; . . . ; b4 have to be determined. The total flow of mobile particles toward both spheres equals Z Z I ¼ I1 þ I2 ¼ R21 FA ðcos hÞ dX þ R22 FB ðcos hÞ dX: Since
Z
dX ¼ 4p;
Z
4 cos h dX ¼ p; 3 2
Z
4 cos h dX ¼ p; 5 4
Z
cos h dX ¼
Z
cos3 h dX ¼ 0;
then using expansion (A.1), we have I ¼ 4pR21 ða0 þ a2 =3 þ a4 =5Þ þ 4pR22 ðb0 þ b2 =3 þ b4 =5Þ:
ðA:3Þ
The first integral in (A.2) can be directly calculated with the use of identity Z 2p Z p cosn h sin h du dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jn ¼ 1 cos h cos #g sin h sin hg cos u 0 0 Z Z p n ð1Þ on 2p du sin h dhð1 X cos h Y sin h sin uÞn1=2 ; ðX ¼ cos hg ; Y ¼ sin hg Þ: ¼ ðn 12Þðn 32Þ; . . . ; 12 X n 0 0 The latter integral can be calculated using formulae [15] Z 2p Z p Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du f ðX cos h þ Y sin h cos uÞ sin h dh ¼ 2p f X 2 þ Y 2 t dt: 0
0
0
With the help of these equations we have pffiffiffi pffiffiffi 1 pffiffiffi 4 1 2 J0 ¼ 4p 2; J1 ¼ 4p 2 cos hg ; J2 ¼ 4p 2 þ cos hg ; 3 15 5 pffiffiffi 4 pffiffiffi 16 1 8 1 3 2 4 cos hg þ cos hg ; J4 ¼ 4p 2 cos hg þ cos hg : þ J3 ¼ 4p 2 35 7 105 105 9 The first integral on the right-hand side of (A.2) equals to R1 pffiffiffi ða0 J0 þ a1 J1 þ a2 J2 þ a3 J3 þ a4 J4 Þ: 2
ðA:4Þ
Then, we expand the expression on the right side of (A.2) in a series of small parameters a1 ¼ R1 =L and a2 ¼ R2 =L 2 1=2 L þ R21 þ R22 2LðR1 cos hg þ R2 cos hÞ þ 2R1 R2 cos hg cos h 2R1 R2 sin hg sin h cos u 1 1 1 2 3 2 2 3 2 cos hg cos h ¼ L 1 þ a1 cos hg þ a2 cos h þ a1 þ a2 þ 2a1 a2 cos hg cos h 2 2 2 2 3 3 9 3 3 5 3 3 5 3 2 2 cos hg cos hg þ a2 cos h cos h þ a1 a2 cos hg cos h cos h þ a1 2 2 2 2 2 2 3 15 35 15 35 2 9 2 4 3 2 4 4 3 2 4 þ a1 a2 cos hg cos h cos hg þ a1 cos hg þ cos hg þ a2 cos h þ cos h 2 2 8 4 8 8 4 8 þ a31 a2 10 cos3 hg cos h 6 cos hg cos h þ a1 a32 10 cos hg cos3 h 6 cos hg cos h
9 9 27 2 2 3 2 2 2 2 þ a1 a2 cos hg cos h : ðA:5Þ cos hg cos h þ 2 2 2 2
N.I. Chekunaev / Chemical Physics 300 (2004) 253–266
265
In (A.5) the terms proportional to cos u were omitted, as their integral with respect to u equals 0. At the same time, cos2 u was substituted by its average value equal to 1/2. Substituting (A.4) and (A.5) into (A.2) and integrating over dh du, we obtain 4 16 1 4 1 8 1 a4 þ a1 þ a3 cos hg þ a2 þ a4 cos2 hg þ a3 cos3 hg 4pD ¼ 4pR1 a0 þ a2 þ 15 105 3 35 5 105 7
1 4pR22 1 3 1 1 1 1 2 2 a2 a21 a2 b1 þ 1 a21 þ a41 b0 þ þ a4 cos4 hg þ a a2 9 2 8 3 2 3 6 1 15 2 L 1 2 1 3 2 1 4 2 1 3 a2 a21 a2 þ a32 b3 þ þ a41 a21 a22 b2 þ þ a2 a21 b4 þ a1 a31 b0 8 5 5 10 35 5 35 10 2 2 1 1 2 2 6 8 a1 a2 2a31 a2 b1 þ a1 a31 þ a1 a22 b2 þ a1 a2 a31 a2 þ a1 a32 b3 þ 3 3 2 5 5 5 35
1 3 3 12 3 2 15 4 3 2 1 2 6 2 2 5 4 2 a1 a1 þ a1 a2 b4 cos hg þ a a1 b0 þ a1 a2 b1 þ a þ a a a b2 þ 5 10 35 2 1 4 2 2 1 5 1 2 4 1
9 2 3 2 36 2 2 3 4 5 3 10 3 5 3 1 3 2 3 a þ a a a b4 cos hg þ a b0 þ a1 a2 b1 þ a1 b2 þ 2a1 a2 b3 þ a1 b4 þ a1 a 2 b3 þ 10 10 1 35 1 2 4 1 2 1 3 6 2
35 4 35 7 a1 b0 þ a41 b2 þ a41 b4 cos4 hg : cos3 hg þ ðA:6Þ 8 24 8 In (A.6) the terms of order a61 and a62 and higher were omitted, as the expansion is up to the fifth order. It will be seen nþ1 nþ1 from (A.12) that coefficients have order an a1;2 a0 , bn a1;2 b0 . Equating the terms proportional to cos0 hg , cos1 hg , 2 3 4 cos hg , cos hg , cos hg in the left and right sides, we obtain (omitting terms of higher order than a51;2 ) cos0 hg :
D 4 16 1 2 3 4 1 1 2 ¼ a1 a0 þ a2 þ a4 þ a2 b0 1 a1 þ a1 þ a2 b1 þ b2 ; L 15 105 2 8 3 3
cos1 hg :
0 ¼ a1 cos2 hg : 0 ¼ a1
1 4 a1 þ a3 3 35
1 8 a2 þ a4 5 105
3 2 1 þ a22 b0 a1 a31 þ b1 a1 a2 þ b2 a1 ; 2 3 3
þ a22 b0
3 2 15 4 a1 a1 ; 2 4
ðA:7Þ
ðA:8Þ
ðA:9Þ
cos3 hg : 1 5 0 ¼ a1 a3 þ a31 a22 b0 ; 7 2 4 cos hg :
ðA:10Þ
1 35 0 ¼ a1 b4 þ a41 a22 b0 : 9 8
ðA:11Þ
For g 2 B equations for coefficients can be obtained by substitutions in (A.7)–(A.11) a1 $ a2 and ai $ bi . Thus, we have 10 simultaneous equations with 10 variables a0 ; . . . ; a4 , b0 ; . . . ; b4 . The solution is 315 3 2 35 15 135 3 2 1 7 9 a1 a2 b0 ; a3 ¼ a21 a22 b0 ; a2 ¼ a1 a22 þ a1 a2 b0 ; a1 ¼ a22 b0 þ a21 a22 b0 þ a21 a32 a0 : a4 ¼ 8 2 2 4 3 2 2 ðA:12Þ 315 2 3 a a a0 ; b4 ¼ 8 1 2
35 b3 ¼ a21 a22 a0 ; 2
b2 ¼
15 2 135 2 3 a a a0 ; a1 a2 þ 2 4 1 2
1 7 9 b1 ¼ a21 a0 þ a21 a22 a0 þ a31 a22 b0 ; 3 2 2
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where a0 and b0 satisfy simultaneous equations: 7 5 27 D=L ¼ a0 a1 a21 a32 þ b0 a22 a21 a22 þ a41 a22 ; 2 2 8 5 27 7 D=L ¼ a0 a21 a21 a22 þ a21 a42 þ b0 a2 a31 a22 : 2 8 2
ðA:13Þ
Solving Eq. (A.13) and substituting the solution into (A.12) and then into (A.3), we find the rate constant Kst a1 þ a2 a1 a2 ð2 þ a31 þ a32 Þ þ O a61;2 : Kst ¼ I ¼ 4pDL ðA:14Þ 1 a1 a2 ð1 þ a21 þ a22 Þ þ O a61;2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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