Influence of multipolar correlations and surface imperfections on the

7876

J . Phys. Chem. 1989, 93, 7876-7887

that water of hydration occurs in different hydration shells and two water molecules are more tightly bound to AOT than the rest of the water of hydration. Six water molecules in the hydration shell of AOT-Na’ are sufficiently perturbed so that freezing is prevented. In contrast to AOT reversed micelles, in aquaus AOT dispersions forming a smectic phase even these six water molecules freeze when the dispersion is cooled at -40 o c . The difference in the thermal behavior of the two AOT systems is ascribed to

supercooling taking place in the submicroscopic droplets of AOT reversed micelles. Acknowledgment. We acknowledge the technical assistance of J. Stauble and E. Blochliger. The NMR measurements were carried out by F. Bangerter, and we are indebted to Dr. P.Skrabal for useful discussions. This work was supported by the ETH. Registry No. AOT, 577-1 1-7; isooctane, 540-84-1.

Influence of Multipolar Correlations and Surface Imperfections on the Efficiency of Dlffuslon-Controlled Reactive Processes on Molecular Organizates and Colloidal Catalysts Joseph B. Mandeville,+David E. Hurtubise,* Richard Flint,$ and John J. Kozak* Department of Chemistry, Franklin College of Arts and Sciences, University of Georgia, Athens, Georgia 30602 (Received: January 20, 1989; In Final Form: May 9, 1989)

A stochastic-mechanical lattice model is introduced and extensive calculations are performed to assess the role of multipolar correlations and surface imperfections in influencing the efficiency of encounter-controlled reactive processes on the surface of a molecular organizate or colloidal catalyst particle. The change in the diffusion-controlled rate constant kDin the presence and absence of a down-range biasing potential u(r) [specifically the quantity kD(v= u(r))/k,(u = O)] for a reactant pair subject to a short-range chemical (or cage) effect is studied as a function of system size, concentration of defects, medium temperature, and dielectric constant. For d = 2 dimensional surfaces of Euler characteristicx = 2 (“shells”)or x = 0 (“planes”), if the length parameter r characterizing the spatial extent of the system is larger than the (generalized) Onsager length s, the long-range attractive Coulombic potential is, as expected, more effective in enhancing the reaction efficiency than the short-range (angle-averaged) ion-dipole or dipoledipole potentials. However, both for “shells” and “planes” when the size of the system is decreased or the d = 2 reaction space otherwise restricted (so that the lengths r a n d s are of comparable magnitude), not only can short-range attractive potentials compete in effectiveness with long-range potentials but also, for a given potential, the interplay between these two scale parameters can produce inversions in the reaction efficiency. The role of repulsive potentials in influencing the reaction efficiency is straightforward: the reaction time increases with increase in the strength of the potential with the influence on the reaction rate (much) more pronounced than for attractive potentials. Finally, the relevance of our study to two experimental systems [the dismutation reaction of Br2- ion radicals in aqueous solution in the presence of CTAB micelles and the photochemical production of H2 from water via the reaction of the reduced radical cation of methyl viologen with a colloidal platinum catalyst (Pt,)] is brought out.

I. Introduction The surface of a colloidal catalyst particle or molecular organizate (such as micelle, vesicle, or cell) is not smooth and continuous, but rather differentiated by the geometry of the constituents and, if the surface composition is not homogeneous, often organized into domains or clusters. To explore the consequences of such structure on the efficiency of surface-mediated, encounter-controlled reactive processes, we developed recently a lattice-based stochastic model (described below) sufficiently general that the influence of spatial structure on the dynamics could be assessed in detail.’,2 We considered a polyhedral surface characterized by dimension d = 2 and Euler characteristic x = 2 [where x is defined by x = F - E + V, with F, E, and V the number of faces (cells), edges, and vertices of the polyhedral complex], with N distinct sites on the surface organized into an array defined locally by the site valency u (or connectivity) of the resulting network. Given this structure we considered a target molecule A anchored to the surface at one of the N sites, a coreactant B free to migrate among the N - 1 satellite sites and then analyzed the dynamics of the diffusion-controlled irreversible reaction, A + B C. This was done by formulating a stochastic master equation for each geometry considered [in ref l and 2 we focused on the polyhedral ---+

Department of Physics, University of Illinois, Urbana, IL 61801. *Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556.

0022-3654/89/2093-7876$01.50/0

surfaces defined by the 5 Platonic solids and 16 Archimedean solids] and solved this equation numerically for two classes of initial conditions. Specifically, we determined the survival probability p ( t ) versus time t of the diffusing coreactant B and in addition calculated the first four moments of the probability distribution function describing the process. From the consequent evolution curves, we extracted the individual relaxation times to and from these and the associated moments we were able to identify and quantify the separate influences of the variables N a n d u on the kinetics. We found that for fixed N, the time to decreased with increase in the (global or local) valency u and, secondly, for a given local symmetry ( u fixed), the to increased with N in (almost) all cases. The purpose of the present study is to broaden this earlier discussion in two ways. The first is to determine the extent to which surface imperfections can influence the efficiency of an encounter-controlled reactive process relative to the situation where the surface is free of defects. Our second objective will be. to assess the role of multipolar correlations in modulating the efficiency of surface-mediated reactive processes; in addition to considering short-range forces (viz. chemical or cage effects) we shall also consider the diffusing coreactant B and the target molecule A to (1) Politowicz, P. A.; Kozak, J. J. Proc. Natl. Acod. Sci. U S A . 1987,84, 81 75-8179. (2) Politowicz, P. A.; Garza-Lopez, R. A,; Hurtubise, D. E.; Kozak, J. J. J . Phys. Chem. 1989, 93, 3728-3735.

0 1989 American Chemical Society

Diffusion-Controlled Reactive Processes

The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 7877

interact via down-range (attractive/repulsive) ion-ion, (angleaveraged) ion-dipole, and (angle-averaged) dipole-dipole forces. Specifically, we shall quantify the change in the diffusion-controlled rate constant kD in the presence and absence of a biasing potential u(r) by calculating the ratio kD(u = u(r))/kD(u= 0 ) as a function of system size, concentration of defects, medium temperature, and dielectric constant. By comparing these data with those obtained in an earlier study,3 we will also address an issue of long-standing interest in chemical kinetics, viz. the extent to which kinetic processes on planar surfaces (x = 0) (with both reactants confined to a finite domain) can be used to interpret kinetic processes on the (finite) surface of a catalyst particle (x = 2), and vice v e r ~ a . ~ ~ ~ In the following section, the formulation of the stochastic mechanical model developed here is described with the results generated in the study of the model summarized in section 111. The generalizations that can be extracted from these data are presented in section IV and the significance of our results in the interpretation of encounter-controlled, reactive processes on d = 2 dimensional surfaces of different topology (x = 2 versus x = 0) is established in section V. Finally, the relevance of our study to two experimental systems, the first involving a problem in micelle kinetics and the second involving photochemical water cleavage, is brought out. The regime of parameter space (r, s) where the theoretical effects uncovered here can be observed in these two experimental systems is specified. 11. Formulation To account for the discretized surface structure induced by the geometry and organization of the atomic/molecular constituents of a colloidal catalyst particle (or molecular organizate), reaction spaces of differing structure can be represented by the surfaces of different polyhedral lattices. In particular, in ref 1 and 2 we focused on the domain and/or channel structures defined by the surfaces of the 5 Platonic solids' and 16 of the Archimedean solids2 Even if one considers explicitly such a discretized surface structure, however, the assumption that the surface itself is so wonderfully regular that no spatial imperfections exist is clearly open to question. The problem then is: Given the irreversible reaction A + B C, where A is a stationary target molecule anchored to the surface and B is a coreactant diffusing on that surface, how is the reaction efficiency modulated by the presence of surface imperfections? A natural way of addressing this question is to design a surface on which imperfections are interspersed in a more-or-less regular way and then to examine how the probability distribution function governing the reaction-diffusion process changes as a function of the (relative) number of defect sites. This is the plan of attack implemented in the present study. We consider a catalyst particle/molecular organizate for which surface imperfections (defects) characterized locally by the (minimum) valency v = 3 are distributed symmetrically on a surface characterized globally by the (intermediate) coordination v = 4; in particular, we require that each defect site be separated by a common distance Lifrom all nearest-neighbor defect sites. This information can be organized in terms of a lattice model by positioning sites of valency v = 3 at the vertices of a generalized Cartesian shell, the majority of whose sites are characterized by a valency v = 4 (see Figure 1). The influence of the u = 3 defect sites in modulating the surface-mediated reactive process can be determined by maintaining constant the absolute number of defect sites and then studying the reaction efficiency as a function of the total number N of surface sites. To summarize our approach, we choose the defining lattice to be a Cartesian shell (or, more prosaically, the surface of a stacked set of checker boards) of edge length Li, a structure which is topologically equivalent (homeomorphic) to a sphere,6 i.e. both

-

(3) Mandeville, J. B.; Golub, J.; Kozak, J. J. Chem. Phys. Lett. 1988, 143, 117-122. (4) Somorjai, G . A. Chemistry in Two Dimensions: Surfaces; Cornell University Press: Ithaca, NY, 198 I . ( 5 ) Castleman, A. W., Jr.; Keesee, R. G. Science 1988, 241, 36.

Figure 1. Diffeomorphic distortion of a surface of arbitrary shape into a Cartesian shell. The sites 1 and 2 are sites of uniform (local) valency Y = 4 while site 3 represents a defect site of (local) valency Y = 3; the dotted lines represent virtual excursions in the neighborhood of sites 1 , 2, and 3 (see text, sections I1 and IV).

surfaces are characterized by dimension d = 2 and Euler characteristic x = 2. Corresponding to different settings of L there will be different values of N , the total number of surface sites (including the site at which the target molecule A is anchored). For the set of lattices considered here the correspondence is ( L , N) = (3, 26), ( 5 , 98), (7, 218), and (9, 386). In contrast to a planar Cartesian lattice ( d , x) = (2, 0) where for periodic boundary conditions each site is of local valency u = 4, the vertex sites of a Cartesian shell will be of valency v = 3 while all remaining facial and edge sites will be of valency v = 4. Relative to the common valency of the N - 8 sites of valency v = 4, the 3-fold coordination of a vertex site then defines a local imperfection in the geometry of the surface. By virtue of the Cartesian geometry adopted here, the eight sites of local coordination v = 3 will be distributed symmetrically on the surface, equidistant apart and maximally separated. Given these specifications, the percentage of imperfections relative to the total number N (= 26, 98, 216, 386) of surface sites is 30.875, 8.275, 3.775, and 2.1%. Consider now a surface reaction, A B C, between a stationary target molecule A (reaction center) and a diffusing coreactant B. Our objective is to determine how the reaction efficiency changes as a function of the location of the target molecule and to assess how that efficiency is modulated by the presence of surface imperfections. We consider here three limiting cases (see Figure 1). In the first case, we anchor the target molecule at a site on the surface farthest removed from any site of valency v = 3, i.e. at the centrosymmetric site of a face. This location will be referred to in our later discussion as site 1. A second case specifies the location of the target molecule at a site ( ~ 2 somewhat ) closer to (a pair of) defect sites, viz. at the midpoint of an edge. And, in a third case, we anchor the target molecule at a site (03) of valency v = 3, i.e. at one of the surface imperfections. For a Cartesian shell of edge length L, the distance between the target molecule and the nearest surface defect in each case would be L/2'I2 = 0.71L, L / 2 = 0.5L and L for cases (sites) 1, 2, and 3, respectively.

+

-

(6).Henle, M. A . Combinatorial Introduction to Topologv;Freeman: San Francisco, 1979.

1878

The Journal of Physical Chemistry, Vol. 93, No. 23, 1989

Although some calculations will be reported here for the Cartesian reaction space specified in the preceding paragraph, a generalization must be introduced to account more realistically for the motion of a diffusing atom/molecule on a surface. Two observations pertain. First, the image one has of a particle diffusing on a surface is not that of a rigidly constrained, two-dimensional motion but rather that of a quasi-three-dimensional motion with the particle “skipping” from site to site across the surface. Unless a diffusing coreactant B is physically bound to a membrane (say, via a hydrophobic tail) or unless physisorption at the surface of a catalyst restricts significantly the particle’s motion, one anticipates that for systems at finite temperature small excursions normal to the surface are not only possible but probable. Secondly, although one can imagine, for example, a reactant on a micellar surface moving from “head group to head group” or a molecule on a platinum catalyst transiting from “atom to atom”, given the geometrical volumes characterizing the molecules or atoms comprising the surface, it is reasonable to suppose that the channel structure which networks the surface is more appropriately described as a series of “hills and valleys” across which the diffusing coreactant must pass. To acknowledge these realities, we have augmented the lattice model described above by permitting “up-down” excursions by the diffusing coreactant at each site of the lattice while, at the same time, assuming that surface forces are sufficiently important that the coreactant remains entrained in the vicinity of the surface. In our model, the “up-down” excursions at each site of the two-dimensional lattice are “virtual” in the sense that such displacements simply reset the molecule at the original lattice site before the reactant undergoes a random excursion to a neighboring site. We shall denote this expanded reaction space by the notation v = 4 + 2 in order to stress the fact that the particle’s motion is spatially two-dimensional (with Y = 4 pathways available to the diffusing coreactant at each accessible site) but is augmented by two additional degrees of freedom (which do not change the Cartesian position of the reactant on the lattice). In this notation then the eight vertex (defect) sites will be characterized by the valency: Y = 3 3. For regular (e.g. sites 1 and 2 ) and defect (e.g. site 3) sites on the Cartesian shell, virtual excursions are denoted by the dotted lines in Figure 1. The augmented lattice model described in the previous paragraph was motivated by an underlying physical picture in which it was recognized that random thermal fluctuations (or solvent buffeting) can perturb the motion of a diffusing coreactant. Of course, at finite temperatures, this motion will also be strongly influenced by the multipolar character of the partners in the reaction, A B C. It is important therefore to consider the role of potential correlations in influencing the efficiency of the underlying diffusion-reaction process. Accordingly, rather than assuming that the motion of the diffusing coreactant B from a site i to a nearest-neighbor site j is completely “random”, i.e. characterized by an equal a priori probability, we construct a finite temperature (local) partition function to describe the probability of a multipolar coreactant moving through a field of sites in the vicinity of a multipolar target molecule. Specifically, we define the probability of moving from i - - j in the next step to be

Mandeville et al. molecule, the diffusing molecule reacts with the target molecule in the next step. The Boltzmann factor for the case of interacting ionic species A and B may be expressed as eXp(-pU) = eXp{-ZAZB [S(C,C) / R ] I-’] = exp[-W(c,c)l-’] where

-

p ( i / j ) = exp[-b(u, - q)I/s,

(1)

where q, =

2 exp[-p(ui - u , ) ~ , i p ( i / j j

J=

1

= I

J=1

and /3 = 1 / k T . Here, u k , the potential sensed by the diffusing particle at the site k , is reciprocally dependent on the distance rk from the reaction center (target molecule) v k 0: rk-P(r,p > 0) (2) To account for quantum-chemical effects which may become important at short range or cage effects operative in the immediate vicinity of the reaction center, we impose the constraint that, upon arriving at a site which is a nearest-neighbor to the reaction center, i.e. one separated by a single lattice spacing R from the target

= [1/(4ag)crI(e2/kT)

S(C,C)

w(c,C) = ~ . A . ~ B [ S ( C , C ) / R ]

(4a)

(4b)

and

I =r/R (5) In our subsequent discussion lattice dimensions will be expressed in terms of this reduced length 1 (rather than L ) . Also, s is a length, often called the Onsager length, which gives the distance apart at which the mutual electrical potential energy of a pair of singly charged, point charges has the magnitude of the thermal energy; k is Boltzmann’s constant, e is the magnitude of the electronic charge, zi is the signed magnitude of the charge of species i, to is the permittivity of free space, and cr is the dielectric constant of the medium at the temperature T . For comparison, we also consider the shorter-range, ion-dipole and dipole-dipole potentials, but shall bypass the explicit consideration of orientational effects, i.e. we consider only the angle-averaged versions of these potentials. The corresponding factor for angle-averaged ion-dipole interactions is exP(-b) = e x p l + z ~ ~ c L ~ ~ [ s ( c , l l ) / R ] ~ 1 - 4 ) = exP[W(c,P)l-41 where S(C,M)

= ( 1 /3)’/41[1 /(4*e0)c,I(ed/kT)l”~

(6a)

w(C,M) =

(6b)

and

+

+

(3)

~A~FB~[S(C,CL)/R]~

while the expression for angle-averaged dipole-dipole interactions is exp(-pv) = eXpbA2pB2[ s ( p , p )/ R16161 = explW(11,WI where S(P,P)

= (2/3)”61[1/(4aco)crI(~/kT)1”3

(7a)

and w(p&)

= pA2pB2[S(p&)/R]6

(7b)

In these expressions, pi is the magnitude of the dipole moment of species i and d is the unit dipole (in D). We now state what has been calculated in this paper. Let ( n ) c be the average number of steps required for reaction (trapping) on a d = 2, x = 2 lattice (the previously specified Cartesian shell having N total sites of which eight are defect sites of valency Y = 3 3 and the remaining N - 8 sites are of valency Y = 4 + 2 ) whereon the particle’s random motion is subject only to a nearest-neighbor, short-range chemical/cage effect. Let ( n ) c j p be the corresponding quantity calculated assuming the particle is subject to the combined influences of this short-range effect as well as a longer-ranged biasing potential (thereby taking into account that at finite temperature the target molecule may exert down-range correlations on the motion of the diffusing coreactant). In the theory of finite Markov processes, the mean walklength (n) is just the first moment of the underlying probability distribution function governing the process and, in turn, this moment is related to the reciprocal of the smallest eigenvalue of the N X N transition matrix G i .in the stochastic master equation formulation of the problem)-9 viz. dpi N - = -CGijpi(t) (i = 1, 2, ..., N )

+

dt

j=l



TABLE I: The Ratio ( n ) c / ( n ) c l p k! [v = v(r)]/kdv = 01 for Diffusion-Controlled Reactive Processes on a Shell [ d = 2, x = 2, Y = 4 with the Target Molecule Positioned at Site 1 N = 218 N = 386 potential W N = 26 N = 98 0.00 16.039 43 113.046 2 330.091 7 686.702 7 N

ion-ion: attractive

-1.0 -2.0 -4.0 -9.0

1.466 759 2.007 430 3.143 744 5.092 9 18

1.622 793 2.340629 3.849 244 7.288 732

1.644 879 2.338 337 3.699 077 6.984 056

1.639 512 2.296 168 3.528 923 6.452 202

ion-dipole: angle averaged

-1 .o -2.0 -4.0 -9.0

1.734 544 2.551 156 3.771 245 5.065 530

1.480 970 1.848 247 2.277 829 2.926 431

1.372 549 1.619 137 1.878 724 2.234 840

1.3 14 891 1.509 896 1.707 616 1.969 872

dipole-dipole: angle averaged

-1 .o -2.0 -4.0 -9.0

1.698 802 2.424015 3.346 533 4.231 946

1.414 096 1.698 467 1.952 594 2.185 456

1.315 924 1.507 940 1.666 623 1.809 385

1.265 666 1.417733 1.539 378 1.648 260

ion-ion: repulsiveb

+1.0 +2.0 +4.0

1.577 858 2.678 729 9.428 134

1.869 370 4.025 476 25.437 13

1.991 869 4.8 12 856 41.206 9 1

2.040 575 5.215 229 52.33704

+ 21,

OThe values for W = 0 are the values of ( n ) c . bThe values for the ion-ion repulsive case are for the ratio ( n ) c l p / ( n ) c .

-

TABLE 11: The Ratio ( n ) c / ( n ) c l p k d v = v(r)]/kdv = 01 for Diffusion-Controlled Reactive Processes on a Shell [ d = 2, x = 2, Y = 4 with the Target Molecule Positioned at Site 2 potential W N = 26 N = 98 N = 218 N = 386 0.0" 18.22670 119.721 3 343.309 7 709.1 15 6

ion-ion: attractive

-1.0 -2.0 -4.0 -9.0

1.511 556 2.085 120 3.186913 4.848 313

1.657 95 1 2.395 636 3.856 388 6.980 960

1.668 975 2.381 026 3.739 137 6.791 599

1.656 741 2.328 850 3.574068 6.376 187


-1.0 -2.0 -4.0 -9.0

1.683 283 2.326 640 3.137 21 3 4.060 253

1.450 417 1.780980 2.163516 2.733 185

1.354 887 1.585 326 1.827 154 2.159 040

1.302 555 1.487 347 1.673 696 1.920 205


-1 .o -2.0 -4.0 -9.0

1.633 682 2.1 85 97 1 2.769 902 3.290 953

1.381 709 1.633 61 3 1.860097 2.093 085

1.299 651 1.478 069 1.624 910 1.761 005

1.255 009 1.399 028 1.513 573 1.616 637

ion-ion: repulsiveb

+1.0 +2.0 +4.0

1.664 3 16 3.030001 12.27093

1.956 989 4.499 06 1 33.02200

2.053 732 5.192 171 49.479 87

2.085 530 5.51 1490 60.038 08

+ 21,

"The values for W = 0 are the values of ( n ) c . bThe values for the ion-ion repulsive case are for the ratio ( n ) c l p / ( n ) c .

where p i ( r ) is the survival probability of the diffusing coreactant at time t . From the structure of the general solution to the above system of linear equations, viz. N

pi(t)= m=l

(where the A, are coefficients determined by the initial conditions and the A,, are the eigenvalues of the C matrix above), it is evident that the smallest eigenvalue (related to the zero-mode relaxation time of the system) may be interpreted as an effective first-order rate constant descriptive of the dynamical process. This interplay has been examined theoretically in great detail in our previous study2 and hence in the present discussion we simply state that the ratio ( n ) c / ( n)c,p can be placed in correspondence with the ratio kD(v= v(r))/kD(v= 0 ) of encounter-controlled rate constants in the presence and absence of a governing potential, i.e. we write (n)c kD(v = v ( r ) ) (n)c/p k d v = 0)

--

Although the moments ( n ) c and (n)c,p (as well as the higher(7) Montroll, E. W.; Shuler, K. E. Adu. Chem. Phys. 1958, No. 1, 361. (8) Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Systems; Wiley: New York, 1977. (9) Haken, H. Synergetics; Springer-Verlag: Heidelberg, West Germany, 1977.

order moments) of the probability distribution function describing the process can be calculated via brute force Monte Carlo simulation, we have shown in our recent work"'-15 that numerically precise values of these quantities can be calculated very efficiently for finite lattices using methods based on the theory of finite Markov processes, and it is the latter approach that has been mobilized in this paper. 111. Results

Extensive calculations of ( n ) cand (n)c/p have been performed for the lattice shells (I, N ) = (3, 26), ( 5 , 98), (7, 218), and (9, 386) and for several classes of potential functions: attractive and repulsive ion-ion potentials, angle-averaged ion-dipole, and angle-averaged dipole4ipole potentials. In order to study correlation effects in tandem with the role of surface imperfections in influencing the diffusion-reaction process, each series of potential calculations on each shell has been performed assuming the target molecule is positioned at site 1 (the centrosymmetric site on a face), Walsh, C. A,; Kozak, J. J. Phys. Reu. Lett. 1981, 47, 1500-1502. Walsh, C. A.; Kozak, J. J. Phys. Reu. B 1982, 26, 4166-4189. Politowicz, P. A.; Kozak, J. J. Phys. Reo. B 1983, 28, 5549-5569. Musho, M. K.; Kozak, J. J. J . Chem. Phys. 1984, 81, 3229-3238. (14) Politowicz, P. A,; Kozak, J. J.; Weiss, G. H. Chem. Phys. Lett. 1985,

(10) (11) (12) (13)

120, 388-392. (15) Politowicz, P. A,; Kozak, J. J. Chem. Phys. Lett. 1986, 127, 257-262.

7880 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989

Mandeville et al.

-

TABLE III: The Ratio ( ~ ) ~ / ( n ) ~ , kpd v = v(r)]/kdv = 01 for Diffusion-Controlled Reactive Processes on a Shell [d = 2, x = 2, with the Target Molecule Positioned at Site 3 ( v = 3 + 3)

Y

=4

+ 21,

~~

potential

W

0.0'

N = 26 22.200 00

N = 98 155.1942

N = 218 451.126 2

N = 386 935.901 1

ion-ion: attractive

-1 .o -2.0 -4.0 -9.0

1.553 6 16 2.177 634 3.370 64 1 5.229 273

1.731 295 2.564936 4.203 892 7.662 669

1.731 576 2.530 100 4.066 262 7.487 359

1.708 847 2.454 7 15 3.860732 7.023 937


-1.0 -2.0 -4.0 -9.0

1.670 549 2.268 703 3.014 142 4.028 250

1.482 937 1.841 252 2.243 627 2.809 528

1.369 645 1.613 189 1.870 157 2.222 454

1.310584 1.502 219 1.697 168 1.957 702


-1.0 -2.0 -4.0 -9.0

1.611762 2.107881 2.608 592 3.142 021

1.411473 1.689 767 1.935 997 2.167 141

1.311 242 1.499 314 1.654955 1.797 687

1.260 974 1.409 564 1.528 373 1.635 630

ion-ion: repulsiveb

+1.0

1.729 530 3.292 145 14.545 95

2.078 096 5.105 346 42.15379

2.153243 5.715 660 59.068 30

2.166582 5.948 7 16 69.001 51

+2.0 +4.0

'The values for W = 0 are the values of (n)c. bThe values for the ion-ion repulsive case are for the ratio ( n ) c l p / ( n ) c .

-

TABLE I V The Ratio ( n ) c / ( n ) c l p k d v = v(r)]/kdv = 01 for Diffusion-Controlled Reactive Processes on a Planar Surface [ d = 2, x = 0, Y = 41, Based on Data Reported in Reference 3 and Table V N = 386 potential W N = 26 N = 98 N = 218 0.0" 9.219 777 67.700 24 201.964 4 425.053 9

ion-ion: attractive

-1.0 -2.0 -4.0 -9.0

1.391 154 1.787 242 2.402 556 2.990 735

1.776 017 2.803 936 4.551 864 8.097 794

1.779 454 2.677 292 4.496 260 8.633 868

1.759 279 2.602 908 4.285 815 8.309 281


-1.0 -2.0 -4.0 -9.0

1.616821 2.103 636 2.525 992 2.698 955

1.550 838 1.989 637 2.504 757 3.247 95 1

1.417 109 1.706 3 17 2.020 344 2.458 964

1.346 673 1.569 492 1.801 817 2.1 18951


-1.0 -2.0 -4.0 -9.0

1.608 911 2.091 896 2.5 14 508 2.820 957

1.470683 1.810665 2.123796 2.458 316

1.351 692 1.574 436 1.763 845 1.961 053

1.291 102 1.462 942 1.603 261 1.747 643

ion-ion: repulsiveb

+1.0 +2.0 +4.0

1.494 181 2.421 101 15.922 66

2.105 847 5.146 036 39.799 17

2.185389 5.790446 58.109 05

2.206 670 6.068 929 68.650 89

'The values for W = 0 are the values of

( t ~ ) bThe ~ . values

TABLE V: Coefficients in the Representation: Domain) W A 0.0 0.31831794E+OO 0.1 0.33815154E+00 0.2 0.36063770E+00 0.4 0.41535472E+OO 0.9 0.6391 5472E+OO 1 .o 0.7061 7877E+00 2.0 0.23558832E+01 4.0 0.43697584E+02 5.0 0.19428837E+03 9.0 0.62709983E+05 0.25452648E+06 10.0

for the ion-ion repulsive case are for the ratio ( n ) c l p / ( n ) c .

( n ) c j p= [ N / ( N- 1)NANIn N

B -0.80498842E+00 -0.8440258OE+OO -0.89029230E+00 -0.10107418E+01 -0.1 58 5 3740E+0 1 -0.1 7752355E+01 -0.73904525E+01 -0.18903 148E+03 -0.90509829E+03 -0.33075795E+06 -0.13622237E+07

site 2 (the midpoint of an edge), and site 3 (a vertex site). These data are reported in Tables 1-111. We note that the data in these ~ ;remaining tables for the case W = 0 are for the moment ( r ~ )the data in these tables are for the ratio ( n ) ~ / ( n ) c , Thus ~ . our original data for the (n)c,p in each case can be recovered, if desired, by using the ratio results in conjunction with the W = 0 values. Since a principal objective of this study will be to contrast and distinguish the efficiency of encounter-controlled reactive processes on d = 2, x = 2 shells versus d = 2, x = 0 planar surfaces (subject to periodic/confining boundary conditions), we report in Table IV data for processes on planar reaction spaces; these data can

+ B N + C + D I N ] for Repulsive Ion-Ion

Correlations (Planar

C

D

adj R2

0.28839093E+OI 0.23864359E+01 0.1 8544020E+Ol 0.69810041E+OO -0.23579299E+01 -0.28708674E+01 0.92685175E+Ol 0.16309363E+04 0.96598775E+04 0.48957800E+07 0.2095901 1E+08

-0.1401 8805E+01 0.29574918E+01 0.77634674E+01 0.1 8808388E+02 0.55 148894E+02 0.63762670E+02 0.12424541E+03 -0.71917337E+04 -0.491 14245E+05 -0.29386372E+08 -O.I2799284E+09

1.oooo 1.oooo

1 .oooo 1 .oooo

.oooo

1 1.oooo 1.oooo 1 .oooo

0.9999 0.9984 0.9975

be compared, N by N a n d potential by potential, with the results presented in Tables 1-111. These planar data were constructed from the analytic (parametric) expressions reported in our previous study which, in turn, were constructed from moment data on ( n)c and ( n ) c j pfor 1 X 1 square-planar lattices (v = 4) with a centrosymmetric trap (reaction center) (viz., 10 odd lattices ranging from 1 = 3 to 1 = 21 subject to periodic/confining boundary conditions) for 11 values of 1 Wl in the range 0 IIWl I45,and for reactants interacting via attractive ion-ion, angle-averaged ion-dipole, and angle-averaged dipole-dipole potentials. Thus the data reported in Table IV are interpolated values of ( n ) c and (n)c,p, calculated using expressions whose validity extends over



TABLE VI: Coefficients in the Representation: (n)clp = [ N / ( N - 1)XANIn N + B N + C + D I N ] for Attractive Ion-Ion Correlations (Cartesian Shell) site W A B C D R2

0.00

1

0.0 -1 .o -2.0 -4.0 -9.0 0.0" 0.0 -1 .o -2.0 -4.0 -9.0 0.00 0.0 -1.0 -2.0 -4.0 -9.0

2

3

0.47794072E+OO 0.47794025E+00 0.3 1961416E+00 0.25782471E+OO 0.19978334E+00 0.13947041 E+OO 0.478361 70E+00 0.47965955E+00 0.32673513E+OO 0.26367868E+00 0.20071685E+00 0.13 IO55 10E+00 0.63808489E+OO 0.63808679E+00 0.44126282E+OO 0.35337365E+OO 0.26048042E+OO 0. I6050940E+OO

0.42025659E+00 -0.1079741 5E+01 -0.84401 318E+00 -0.7888 1213E+00 -0.71 253824E+00 -0.58370962E+00 0.47 197800E+00 -0.10368406E+01 -0.86722424E+00 -0.81395171E+OO -0.7 117 1322E+00 -0.51815722E+00 0.6 1018989E+00 -0.1389821 7E+01 -0.12490299E+01 - 0 . 1 1615698E+01 -0.96460403E+OO -0.64245043E+00

-0.10633147E+01 0.2936722OE+Ol 0.901 63348E+01 0.1031 1079E+02 0.100761 14E+02 0.1 1150059E+02 0.39518035E+00 0.48514847E+01 0.10801 232E+02 O.l2026232E+O2 0.1 1475769E+02 0.99083955E+01 -0.201 40435E+01 0.29861537E+01 0.34312884E+02 O.I6702093E+02 O.I5721606E+02 O.l222048OE+02

0.43870727E+01 0.18855994E+01 -0.94431431E+02 -0.10295236E+03 -0.9277 1447E+02 -0.12375971E+03 0.13323284E+01 -0.26004194E+02 -0.1 1275808E+03 -0.12466302E+03 -0.1 1634455E+03 -0.10200459E+03 0.145121 l7E+02 O.l1510348E+O2 -0.14242935E+03 -0.17246552E+03 -0.16573351 E+03 -0.13082035E+03

1.oooo 1.oooo 1.oooo 1 .oooo 1.oooo 1 .oooo 1 .oooo 1.oooo 1.oooo

1.oooo 1 .oooo 1.oooo 1.oooo 1 .oooo

1 .oooo 1 .oooo

1 .oooo 1 .oooo

'coefficients assuming no cage/chemical effect. TABLE VI1: Coefficients in (Cartesian Shell) site W 1 -1.0 -2.0 -4.0 -9.0

2

3

the Representation: ( n )c,p = [ N / ( N- 1)IAN In N

+ BN + C + D / N ] for Angle-Averaged Ion-Dipole Correlations

B

C

D

R2

0.47361354E+00 0.4669521 3E+00 0.45537 122E+00 0.43761360E+00

-0.14970082E+01 -0.16359340E+01 -0.17039702E+01 -0.17365839E+01

O.I0088523E+02 0.1 1778008E+02 0.12108846E+02 0.12028521 E+02

-0.62268761E+02 -0.7 1608340E+02 -0.5956 1944E+02 -0.234834778+02

1.oooo 1.oooo 1.oooo 1.oooo

-1.0 -2.0 -4.0 -9.0

0.474 14657E+OO 0.46822909E+OO 0.45882921E+OO 0.44641 135E+00

-0.14457473E+01 -O.l5895482E+01 -0.16717252E+01 -0.17400488E+01

0.1 1240009E+02 O.l2908794E+02 0.132501 7 1E+02 0.13857273E+02

-0.88508684E+02 -0.96508306E+02 -0.79731314E+02 -0.54998825E+02

1 .oooo

-1.0 -2.0 -4.0 -9.0

0.63707915E+00 0.63605836E+00 0.63380409E+OO 0.62795699E+00

-0.197901 15E+01 -0.22187430E+01 -0.24014072E+Ol -0.257241OOE+Ol

0.1 1635248E+02 0.1 5734198E+02 0.2016971 1E+02 0.2664471 1E+02

-0.35628482E+02 -0.65487339E+02 -0.1 1286578E+03 -0.19909486E+03

1.oooo 1.oooo 1.oooo 1.oooo

A

1.oooo

1 .oooo 1.oooo

TABLE VIII: Coefficients in the Representation: ( t ~ ) , -=/ ~ [ N / ( N - 1)XANIn N + BN + C + D / N l , for Angle-Averaged DipoleDipole Correlations (Cartesian Shell) site W A B C D R2 1 -1 .o 0.47769425E+00 -0.14648904E+01 0.85307813E+Ol -0.47603042E+02 1 .oooo O.I0504980E+02 -0.67295041 E+02 1 .oooo -2.0 0.47617716E+00 -0.161 12127E+01 -4.0 0.47326677E+00 -O.l6956043E+O 1 0.1 1623622E+02 -0.78521868E+02 1 .oooo 0.1 2608054E+02 -0.8534453 1 E+O2 1 .oooo -9.0 0.46815337E+00 -0.17437990E+01

2

-2.0 -4.0 -9.0

0.479 16429E+00 0.47690773E+00 0.47205688E+00 0.46304 105E+00

-0.14207608E+01 -0.15622530E+01 -0.163301 67E+01 -O.l6526315E+01

0.10642009E+02 0.1247657 1 E+02 0.12658253E+02 0.1 1038327E+02

-0.92683939E+02 -0.1 1023353E+03 -0.10038130E+03 -0.51 192463E+02

I .oooo I .oooo 1 .oooo 1 .oooo

-1.0 -2.0 -4.0 -9.0

0.63835393E+00 0.63857347E+00 0.63874371E+00 0.63862598E+OO

-0.19073154E+01 -0.21 176977E+01 -0.22577748E+01 -0.23675 154E+01

0.89745 160E+O I 0.1 1663998E+02 0.13899501E+O2 0.16558669E+02

-0.56061 979E+0 I -0.14843806E+02 -0.29188751 E+02 -0.60003762E+02

1 .oooo 1 .oooo 1 .oooo 1 .oooo

-1

3

.o

a much wider range of ( N , W) values. We had not reported previously for the planar case data for repulsioe ion-ion correlations; these new data are presented in parametric form in Table V so as to allow ( n)c/p to be determined for values of N other than those obtained for the 10 odd lattices, 1 = 3 to 1 = 21, for which the Markovian calculations were performed. The functional form adopted for the parametric representation was chosen as in ref 3 to mimic the asymptotic analytic form determined by MontrollI6

( n ) = -(AN In N 4- BN N- 1

+ C+D/N)

(8)

in his study of the d = 2 dimensional random walk on planar lattices in the absence of a chemical/cage effect or a biasing (16) Montroll,

E. W. J . Math. Phys. 1969, 10, 753-765.

potential. The goodness of the fit in each case is indicated by the Rad,*value reported in the tables. dimilarly, to assist the experimentalist in applying the results obtained in this study to shells characterized by values of N other than those considered explicitly in our calculations, we report in Tables VI-IX parametric representations of the data reported in Tables 1-111. Thus, the ratio kD(c = c ( r ) ) / k D ( = c 0 ) for target molecule concentrations CTin the range 0.026 I CT I 0.39 can be determined directly for (attractive and repulsive) ion-ion, (angle-averaged) ion-dipole, and (angle-averaged) dipole-dipole potentials for four values of W, viz. IkVl = 1.0, 2.0, 4.0, and 9.0. Since, as we shall see later. the coefficient A in the representation (8) dominates the behavior of (n)c,p in the limit of large system size ,V, we give in Table X the parametric behavior of this coefficient as a function of the potential strength parameter W. Coupled with the values of A given in Tables VI-IX, one can determine in the limit of large system size the consequences as

7882 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 TABLE IX: Coefficients in (Cartesian Shell) site W 1 1.O 2.0 4.0 9.0 2

3

Mandeville et al.

the Representation: (n),-jp = [ N / ( N - 1)IANIn N

+ BN + C + D / N ] , for Repulsive Ion-Ion Correlations

A O.l0633769E+Ol 0.36443702E+Ol 0.68494745E+02 0.91 326207E+05

B -0.26943875E+01 -0.1255 1401E+02 -0.32646148E+03 -0.52035063E+06

C

D

-0.77968157E+01 0.37 17 1966E+02 0.45077282E+04 0.12166202E+08

0.3 1476275E+03 0.56577960E+03 -0.43590266E+05 -0.16552871E+09

2.0 4.0 9.0

O.l0245837E+O 1 0.34320741E+01 0.67788504E+02 0.10090736E+06

-0.2255828 1E+01 -0.10332078E+02 -0.302 19457E+03 -0.55460298E+06

-0.10442273E+02 -0.6353 1646Ef01 0.33359241E+04 0.11490188E+08

0.29821842E+03 0.97128835E+03 -0.26 161401E+05 -O.l4578668E+09

I .o 2.0 4.0 9.0

0.13 1 10767E+01 0.4316443OE+Ol 0.8941 6455E+02 0.12 1 12424E+06

-0.24907847E+01 -0.1 1107094E+02 -0.3735 1995E+03 -0.6291 1366E+06

-0.31711179E+02 -0.88492649E+02 0.30675015E+04 0.10383228E+08

0.58053928E+03 0.21294943E+04 -0.161 19899E+05 -0.1 1095347E+09

1 .o

site 1 2 3

A, 0.47794072E+00 0.478361 70E+00 0.63808489E+OO

W + A W 2 + A I W 3+ A 4 W' A, A, 0.23262248E+OO 0.89 14824E-0 1 0.1 577 164E-0 1 0.8 117471E-01 0.1422484E-01 0.21940285E+00 0.9881 356E-01 0.27953346E+00 0.1709202E-01

averaged

I 2 3

0.47794072E+OO 0.478361 70E+00 0.63808489E+00

0.21 10755E-02 0.2446 192E-02 0.102 1649E-02


1 2 3

0.47794072E+OO 0.47836170E+OO 0.63808489E+OO

1

0.47794072E+00 0.478361 70E+00 0.63808489E+00

TABLE X: Coefficients in the Representation: A = A

ootential ion-ion: attractive ion-dipole: angle

ion-ion:

repulsive

2 3

R* 1.oooo

1 .oooo 1 .oooo 1 .oooo

1.oooo 1 .oooo 1 .oooo 1 .oooo 1 .oooo

I .oooo 1.oooo 1 .oooo

+A A,

-0.85800E-03 0.2839418-01 -0.843356E-02 0.28203524E+03 0.3 1240 17 5E+03 0.3739 1296E+03

-0.2821 228E-02 -0.23001 3E-02 0.29995E-04

-0.644877E-03 -0.567351E-03 0.15269E-04

-0.40074E-04 -0.361 56E-04 0.1 184E-05

1 .oooo 1.oooo 1.oooo

-0.137435E-02 0.2418 184E-01 -0.1 109949E-01

-0.28753E-03 0.597894E-02 -0.3 16060E-02

-0.1765E-04 0.40240E-03 -0.22563E-03

1 .oooo 1.oooo 1 .oooo

0.494956648+03 0.54808509E+03 0.65615269E+03

0.24809822E+03 0.274701 35E+03 0.328995178+03

1.oooo 1.oooo

0.35762256E+02 0.39564226E+02 0.47428430E+02

6ool

regards reaction efficiency of simultaneous variations in the control parameters ( N , W).

IV. Discussion Although a planar surface subject to periodic boundary conditions and a shell are both of topological dimension d = 2, their Euler characteristic x is different, respectively x = 0 for the plane and x = 2 for the shelL6 Locally, however, the neighborhoods of a site on a planar lattice of uniform valency u = 4 and at a regular (non defect) site on a Cartesian shell are similar (see later discussion). Hence one anticipates that trends (with respect to system size, the presence of short- and long-range forces, and the interplay of these factors) which emerged in our earlier study3 of correlation effects on reaction efficiency in monolayer domains (d = 2, x = 0), should emerge here as well. Of course, the presence of defect sites may well preclude any exact correspondence in behavior; placing a target molecule at a vertex (site 3) will certainly change the local neighborhood of the trap vis-a-vis the case where the target molecule is anchored to a planar lattice of uniform valency. The larger question then is to determine the extent to which this (defect) perturbation of the system's geometry influences the reaction efficiency at target sites somewhat displaced from the (here, eight) lattice imperfections. The successful fitting of the data in Tables 1-111 with an expression of the form (8) (see Tables VI-IX) suggests at the very least that the strong monotonicity of the increase in ( n ) with respect to system size N , uncovered by MontrollI6 in his study of uncorrelated random walks on planar periodic lattices, should pertain here as well. Displayed in Figure 2 is the first moment ( n)c,p of the probability distribution function versus system size: specifically, the mean walklength of a coreactant diffusing on a Cartesian shell with a target molecule anchored at sites 1, 2 , and 3 is plotted versus the edge length of the lattice for the case where the biasing potential is an attractive ion-ion potential of strength W = -1. The anticipated strong monotonicity with respect to N is clearly evident in this plot as is the fact that for each system size the longest walklength (or, equivalently, the longest reaction time) is that for which the target molecule is anchored at site 3, Le. one of the vertex (defect) sites of the Cartesian shell. It is

R= 1 .oooo 1.oooo 1.oooo

A*

0.919318E-03 0.82641OE-03 0.98985 1E-03

1

.oooo

500

-I

/

4ooj

j";i c

" 200

-I

100

O f I

1

1

I

I

1

3

5

7

9

II

LATTICE EDGE LENGTH Figure 2. The mean walklength (n)cjp versus the edge length of the lattice for a target molecule anchored at the center of a face (solid line), the midpoint of an edge (dashed line), or a vertex position (hyphenated line). The governing potential is an attractive ion-ion potential of strength parameter W = -1.

also evident that positioning the reaction center away from a defect site (of valency v = 3 3) has about the same effect on the overall reaction efficiency whether the target molecule is positioned at site 1 or site 2 , with the influence of the lattice imperfection felt a bit more at the (closer) site 2 than at site 1. These same overall trends are observed when one considers stronger ion-ion potentials ( W = -4) (see Figure 3) or angle-averaged ion-dipole or angleaveraged dipole-dipole forces of different strengths (see Figure 4 for W = -1 and Figure 5 for W = -4). Turning to the data displayed in Tables I-IV, we now examine in more detail the trends observed with respect to system size N and potential strength parameter W when one considers a co-

+



2501

I

I

I

I

7001

4

500

01

I

I

I

I

I

I

3

5

7

9

II

LATTICE EDGE LENGTH Figure 3. The same conventions as in Figure 2 except that the governing potential is an attractive ion-ion potential of strength parameter W = -4.

“1

I

/

600

molecule concentrations CT< 0.003. It is also clear from the data reported in these tables that (with one exception) the trends uncovered for angle-averaged ion-dipole and dipole-dipole correlations exhibit no such turnover, i.e. the ratio ( n ) c / ( n ) c , p decreases monotonically with increase in system size for each setting of W (the single exception occurs for processes on a planar surface for the case of ion-dipole correlations parametrized by the largest value of W (=-9), where a turnover occurs in the 100). vicinity of N A similar effect was uncovered in our earlier study’ of diffusion-reaction processes between correlated reactants on monolayer domains where we noted that the interplay between the strength and range of the potential (as calibrated by the Onsager length s) and the system size (as scaled by the (reduced) length I ) is apparently very discriminating in the regime of intermediate 1. In fact, in ref 3 we found that even ion-dipole and dipole-dipole interactions could lead to “crossover” behavior in the reaction efficiency, noted above for Coulombic interactions in the regime of small I but sufficiently large s (or W). A second major point which emerges from an examination of these data is that for the smallest systems considered ( N = 26) [for which the target molecule concentration CTis highest (CT O.OS)], both angle-averaged ion-dipole and angle-averaged dipole-dipole correlations have a comparable or greater influence on the ratio (n),--( n)c [and hence on the ratio of rate constants k~(= u u(r))/kD(u= as does the Coulombic potential. This rather surprising behavior disappears once system sizes N > 98 are considered; there one finds, in accordance with one’s intuition, that Coulombic correlations do indeed have a more pronounced effect on the reaction efficiency than either ion-dipole or dipole-dipole correlations, regardless of the value taken for the potential strength parameter W. A plausible explanation for this behavior in small systems (see also the discussion in ref 3) is that a long-range attractive potential will vary but slowly over systems of small domain size and hence the net force (the negative gradient of the potential) experienced by the diffusing coreactant will be smaller than that associated with a more short-range potential (for which the corresponding gradient will change more rapidly over small distances). Whereas attractiue correlations between a diffusing coreactant and a target molecule result in profiles of (n)c,p versus N which are concave upward, the curves describing the change in reaction efficiency with respect to system size for repulsive potentials (see Figures 6 and 7 where we consider Coulombic potentials of strength parameter W = +1 and +4, respectively) are concave downward. The behavior displayed by curves of ( n)clp versus N for repulsive potentials suggests that repulsive interactions play

-

400 v

,,i 0 I

3

5

7

9

II

LATTICE EDGE LENGTH Figure 4. The same conventions as in Figure 2 except that the governing potential is an attractive, angle-averaged dipole-dipole potential of strength parameter W = -1.

reactant on a surface whose random motion is subject to the combined influences of a long-range attractive potential as well as a short-range chemical/cage effect. By scaling the mean walklength ( n ) c in the absence of a biasing potential to the moment (n)C/p,the latter calculated assuming that both shortand long-range effects are at play, we obtain directly the dependence of the ratio kD(u = u(r))/kD(u = 0) on the parameters N , W (see discussion in preceding section). We shall comment in the next section on the difference in the magnitude of the ratios reported in Tables 1-111 for the Cartesian shell versus those given in Table IV for the planar surface (a consequence, at least in part, of the fact that the former data were generated assuming a local valency of u = 4 + 2 for the regular sites of the lattice and v = 3 3 for the defect sites, whereas a common valency u = 4 was assumed for all sites of the square planar lattice), and focus here only on a discussion of the qualitatiue trends which can be extracted from the data. Considering first the case of attractive Coulombic interactions, it is seen from the data in Tables I-IV that the ratio (n)c-(n)clp for each setting of W first increases and then decreases with increase in system size, with the turnover in this behavior occurring (always) for values of the system size N C 386, i.e. for target

+

Figure 5. The same conventions as in Figure 2 except that the governing potential is an attractive, angle-averaged dipole-dipole potential of strength parameter W = -4.

-

d)]

7884 The Journal of Physical Chemistry, Vol. 93, No. 23, 1989 34

30

26 P

\

22 t

v

0

-0

I .8

14 1.2 1

I

I

I

1

1

3

5

7

9

II

LATTICE EDGE LENGTH Figure 6. The same conventions as in Figure 2 except that the governing potential is a repulsive ion-ion potential of strength parameter W = +1.

1

5.0 1

4.6

3.8 0

/

n

1

2’61 2.2

L.”

1

I

I

I

I

I

3

5

7

9

II

Mandeville et a]. V. Reactive Processes on Surfaces In this section we deal explicitly with the general question of whether and to what extent encounter-controlled reactive processes taking place on the surface of molecular organizate or colloidal catalyst can be interpreted using experimental evidence obtained from kinetic studies performed on monolayers, and vice versa. Although “planes” are not topologically equivalent to “spheres” (or “shells”) (the Euler characteristic of these two d = 2 surfaces being x = 0 and x = 2, respectively), we demonstrated in an earlier contribution] that in the absence of correlations between reactants, there is a numerical degeneracy in the values calculated for the stochastic determinants of the motion of a coreactant diffusing on a planar lattice subject to periodic boundary conditions and on the polyhedral surfaces defined by certain Platonic solids. Specifically, calculations of the mean walklength ( n )(a measure of the mean reaction time) before trapping (reaction) of a coreactant diffusing on a surface having a single stationary reaction center, performed for the three Platonic solids, (d, N , v) = (2, 4,3), ( 2 , 8, 3), and ( 2 , 20, 3) [respectively, the tetrahedron, the hexahedron, and the icosahedron], were found to be numerically consistent with those performed for finite, planar hexagonal lattices ( d , N , v) = (2, N , 3) with a centrosymmetric trap and subject to periodic/confining boundary conditions.12 We noted that it was in this sense that comparisons of reactivity between nonpolar species confined to these two topologically distinct types of d = 2 dimensional surfaces could be made precise. Some practical consequences of this degeneracy were suggested in ref 2 in reference to the recent discussion of Castleman and KeeseS who, drawing upon the experimental studies of Morse et al.,” Parks et al.,’* Reents et al.,19 Ruatta et a1.,2O and Zakin et noted analogies between reactions on metal clusters and on surfaces in such processes as catalysis, adsorption, and etching. Since such experimental studies often involve multipolar reactants, however, it is fair to say that only inferences could be drawn from the trends found in ref 1 and 2 supporting these analogies (inasmuch as in these earlier studies the reactants were assumed to be nonpolar and uncorrelated). Given that a main thrust of the present study has been to examine the influence of multipolar correlations on the reaction efficiency, we now examine whether the previously reported, phenomenologically based analogies5 can be supported by using the results obtained in the more general stochastic mechanical model studied here. To proceed, note that, given the Montroll representation (8), in the limit of large system size, by L‘Hospital’s rule, the following relationship is exact:

LATTICE EDGE LENGTH Figure 7. The same conventions as in Figure 2 except that the governing potential is a repulsive ion-ion potential of strength parameter W = +4. a relatively greater role in influencing the reaction efficiency in small organizates. It should be stressed, however, that for all system sizes considered here the influence of repulsive correlations on the reaction efficiency is much more pronounced than is the case for attractive interactions. Upon comparing the results displayed graphically for (corresponding) attractive and repulsive Coulombic potentials (see Figures 2 and 6 or Figures 3 and 7), one notices that the ordinate in Figures 2 and 3 is (n)c,p whereas the ordinate in Figures 6 and 7 is the logarithm of that quantity. Despite these differences, one again finds that the most significant effect on the reaction efficiency is realized when the target molecule is situated at a defect site. Also consistent with the behavior found earlier is that the relative ordering of the curves displayed for a target molecule anchored at sites 1 and 2 is the same for repulsive as for attractive potentials. Finally, considering the effect of repulsive correlations on the ratio ( n ) c l p / ( n ) c ,the data displayed in Tables I-IV show clearly that this ratio increases monotonically with respect to system size N (for fixed w)and potential strength parameter W (for fixed N ) . There are no exceptions. For both surface topologies, ( d , x) = ( 2 , 2) and (2,0), repulsive correlations simply prolong the length of time it takes for the encounter-controlled reaction to occur, period.

where A is the coefficient of the leading term in the expansion (8). For a common setting of v and W, then, the ratio of the coefficients A should be unity if the topological degeneracy noted in our earlier study of surface-mediated reactive processes involving uncorrelated reactants is to be sustained. In fact, we shall find that in the limit of large N this speculation is essentially correct for all cases studied here involving angle-averaged ion-dipole and angle-averaged dipole-dipole interactions, but is satisfied only in the limit of small W for ion-ion potentials. We now elaborate this statement. First of all, we must make clear the difference between the results reported in our earlier study (ref 3) of correlation effects on reaction efficiency in monolayer domains (x = 0) versus those (17) Morse, M. D.; Geusic, M. E.; Heath, R. E.; Smalley, R. E. J. Chem. Phys. 1985.83, 2293. (18) Parks, E. K.; Liu, K.; Richtsmeier, S.C.; Pobo, L. G.;Riley, S. J. J . Chem. Phys. 1985,82, 5470. (19) Reents, Jr., W. D.; Mujsoe, A. M.; Bondybey, V. E.; Mandich, M . L. J . Chem. Phys. 1987, 86, 5568. (20) Ruatta, S. A.; Hanky, L.; Anderson, S.L. Chem. Phys. Lett. 1987, 137, 5.

(21) Zaken, M. R.; Cox, D. M.; Whetten, R. L.; Trevor, D. L.; Kaldor, A. Chem. Phys. Lett. 1987, 135, 223.


Diffusion-Controlled Reactive Processes reported here. In this earlier study, the local uniform valency of each site on the lattice was v = 4. In the present study, to account for random “up-down” fluctuations in the reactant’s motion on the surface (owing to the presence of thermal noise, solvent buffeting or the like), the uniform valency of the N - 8 regular sites of the shell was taken to be v = 4 2, while the valency of the (eight) defect sites was set at v = 3 3. Now, if one returns to the problem of diffusion-controlled reactive processes on planar surfaces subject to periodic boundary conditions (Euler characteristic x = 0) and introduces such “up-down” fluctuations in the particle’s motion, one can prove (via straightforward inspection of the structure of the fundamental matrix N of the Markovian theory) that

+ +

exactly. Hence, if the previously reported coincidence in ( n ) values calculated for surface-mediated, reactive processes involving nonpolar/uncorrelated reactants on ( d = 2) surfaces of different topologies (x = 2 versus x = 0) is to be sustained when one considers “up-down” fluctuations in the particle’s motion, the ratio A(SHELL; v = 4 2, W)/A(PLANE; v = 4, W) should be exactly 3/2 in the limit W - 0. Using the values of the coefficients A reported in ref 3 in tandem with the results reported in this study (Tables V-VIII), one finds (see Table XI) that for sites 1 and 2 the ratio of these coefficients is essentially 1.50 in the limit W 0. Moreover, we find the surprising result that for angle-averaged ion-dipole and dipole-dipole correlations, this value (3/2) is (essentially) realized for values of W spanning the range: -9 I W I 0. It is also evident that the ratio calculated for site 3 is somewhat displaced from this limiting ratio of 3/2, a reflection of the fact that the local valency of a defect site is v = 3 3 rather than v = 4 + 2 (for which the proof of the limiting result 3/2 was carried out). There is an interesting insight that can be extracted from the data reported in Table XI. Notice that for attractive ion-ion potentials, the consequence of increasing the potential strength parameter W is that the ratio (10) shifts away from the W = 0 limiting value of 3/2 to (larger) values in the neighborhood of 2.00. Since the ratio (10) calculated for the defect site 3 in the limit W - 0 is also -2.0, this suggests that spatial imperfections associated with sites of valency lower than the global valency (and in the absence of reactant correlations) play the same role as attractive ion-ion correlations (with the target molecule away from a defect site) in mediating the efficiency of encounter-controlled reactive processes on ( d , x) = (2, 2) surfaces. Also of interest in Table XI are values calculated for the ratio A(SHELL; v = 4 2, W = O)/A(SHELL; v = 4 2, W # 0). Since finite-range potential correlations should have no significant influence on the dynamics if the target molecule is embedded in an infinitely large reaction space, this ratio should tend to unity in the limit N 0 3 . Here we find for angle-averaged ion-dipole and angle-averaged dipole-dipole correlations that this limiting ratio is essentially realized, but that the ratio changes systematically with increase in W for attractive ion-ion correlations. The reason that the ion-ion case yields values greater than unity is that the coefficients A in the representation (8) were determined empirically by fitting the data generated in our Markovian study. In particular, the maximum shell size considered here had N = 386 total sites, a system size which is (apparently) not large enough so that finite range effects associated with the potential are washed out; or, put the other way around, the angle-averaged ion-dipole and dipole-dipole potentials are sufficiently short ranged that the asymptotic limit of unity is already realized for system sizes in 386. the vicinity of N

+

TABLE XI: Limiting Ratios

potential ion-ion: attractive

site 1


-

+

+

+

-

-

VI. Relevance to Experimental Studies The objective of the research reported in this paper was to quantify the role of potential correlations and surface imperfections in influencing the efficiency of diffusion-limited reactive processes on molecular organizates and colloidal catalysts. Moreover, we sought to place the study of such surface-mediated processes within


-2.0 -4.0 -9.0 0.0 -1 .o -2.0 -4.0 -9.0 0.0 -1.0 -2.0 -4.0 -9.0

A(SHELL; v = 6, W)/. A(PLANE, v = 4, W) 1.50 1.55 1.62 1 .I9 2.25 1.51 1.59 1.66 1.80 2.1 1 2.00 2.14 2.23 2.34 2.59

0.0 -1.0 -2.0 -4.0 -9.0 0.0 -1.0 -2.0 -4.0 -9.0 0.0 -1.0 -2.0 -4.0 -9.0

1.50 1.50 1.49 1.48 1.46 1.51 1.50 1.49 1.49 1.49 2.00 2.02 2.03 2.05 2.10

1 .oo

0.0 -1.0 -2.0 -4.0 -9.0 0.0 -1.0 -2.0 -4.0 -9.0

1.50 1.50 1.49 1.49 1.48 1.51 1 .so 1.50 1.48 1.47 2.00 2.00 2.00 2.01 2.02

1 .oo 1 .oo 1 .oo

W 0.0 -1 .o

0.0

-1.0 -2.0 -4.0 -9.0

A(SHELL; = 6, W = 0)/ A(SHELL; v = 6, W # 0) 1.00 1.50

Y

1.85 2.39 3.43 1 .oo 1.46 1.81 2.38 3.65 1 .oo 1.45 1.81 2.45 3.98

1.01 1.02 1.05 1.09 1 .oo

1.01 1.02 1.04 1.07 1 .oo 1 .oo 1 .oo 1.01 1.02

1.01 1.02 1 .oo 1 .oo 1 .oo 1.01 1.03 1 .oo 1 .oo 1 .oo 1 .oo 1 .oo

the framework of our previous stochastic-mechanical studies of encounter-controlled reactive processes on monolayer domain^.^^^^ Taken together, the results described in these studies demonstrate unequivocally that trends in the reaction efficiency for irreversible processes A + B C taking place on d = 2 dimensional surfaces of different topologies [(d,x) = (2, 2) for “shells” and (2,O) for “planes”], where the motion of a diffusing coreactant B in the neighborhood of a stationary target molecule A is biased by an attractive down-range potential and subject to a short-range chemical or cage effect, are fully in accord with one’s intuition ifthe spatial extent of the reaction space is sufficiently large; in this case our results quantify the degree of enhancement in the reaction efficiency. However, we also found that if the reaction space is of restricted spatial extent, one obtains results that are, at first sight, surprising. For example, for both “planes” and “shells”, we find that short-range attractive potentials can have a comparable or greater effect on the reaction efficiency as attractive potentials of longer range (e.g. Coulombic potentials) if the system size is small. We also noted that the ratio kD(u = u(r))/k,(u = 0) of diffusion-controlled rate constants in the presence and absence of a biasing potential can increase and then decrease (asymptotically to unity) with increase in the spatial

-

(22) Politowicz, P. A,; Kozak, J. J. Langmuir 1985, I , 429-443.

7886 The Journal of Physical Chemistry, Vol. 93, No. 23, 19619

Mandeville et al.

extent of the system, depending on the strength and range of the governing (attractive) potential function. Insofar as we know, these two points have never been taken into account (or at least quantified) in discussions of experimental data derived from the study of kinetic processes taking place on the surface of a colloidal catalyst particle or molecular organizate (a micellar vesicular, or cellular system) ( d , x) = (2,2) or on a planar catalyst support or fatty-acid monolayer, ( d , x) = (2, 0 ) . The rather striking effects noted above (in the regime of small system size) are a consequence of the interplay between the strength and range of the attractive potential (as scaled by the (generalized) Onsager length s) and the spatial extent of the system (see our later discussion for concrete estimates here). On the other hand, for reactants interacting via purely repulsive potentials, the results obtained are free of subtleties and totally predictable. Repulsive correlations simply prolong the length of time it takes for the irreversible reaction A B C to occur. In fact, the only surprise here is that from a strictly quantitative point of view, repulsive correlations play a relatively more important role in influencing the reaction efficiency than do attractive correlations (compare Figures 2 and 6 or Figures 3 and 7). Before proceeding, it is probably worth reiterating the philosophy underlying the research reported in this paper. The basic strategy has been to associate with a given (discretized) reaction space a (polyhedral) lattice and then to work out (numerically) the stochastic determinants characterizing the irreversible process A B C. It is important to stress that once the model had been defined, no further (analytical or numerical) approximations are introduced in the calculations; the results obtained are numerically precise (see the discussion in ref 10-15). Hence, the question of the relevance of the results obtained here to experiment reduces to the question of whether the model itself has any semblance of reality. Lest the reader think that the Cartesian shells discussed in this paper, the polyhedral surfaces studied in ref 1 and 2 or the planar Cartesian lattices studied in ref 3 are far too rigid representations of reality (Le. “real” molecular organizates or colloidal catalysts) to be of any practical relevance, let us review the concept of a diffeomorphism. The idea here is that if a geometrical figure is subject to all sorts of twisting, pulling, and stretching and yet can be continuously deformed into, say, a sphere, then that figure is topologically equivalent to a sphere. Hence, at the most fundamental level, what we have characterized in the present study (and in our earlier work, ref 1-3) is the extent to which kinetic p r m s e s on surfaces of a given “form” are driven by the underlying topology of that surface (see Figure I ) . This is certainly not to say that the structural details of a particular surface are unimportant in influencing the kinetics. It is just that from a topological point of view, the most fundamental characteristic of that surface is its Euler characteristic x and hence our objective has been to sort out the importance of this topological invariant first before considering in detail other structural properties of the surface. This emphasis on the underlying topology of the surface has an immediate and very practical consequence as regards programs of experiment in which results derived from studying kinetic processes on fatty acid monolayers (x = 0) are used to interpet (or infer) what happens on the surface of a cellular system (x = 2) or in which studies of the dynamics of catalytic reactions on planar supports (x = 0) are linked to colloidal catalyst dynamics (x = 2). The point is that although “planes“ and ”shells” are characterized globally by different Euler characteristics, locally these two surfaces may be quite similar; certainly, for a common valency, the neighborhoods of points on these two surfaces will be indistinguishable. This, in turn, explains why the overall trends uncovered in this work on the influence of a biasing potential on the reaction efficiency for irreversible processes on Cartesian shells are qualitatively consistent with those found earlier in our study of the same irreversible reaction assumed to take place on a planar lattice. It is also the reason why our earlier s t u d i e ~ of ~ ~dif-~~

fusion-controlled reactive processes on monolayer domains of variable valency (viz., v = 3,4, and 6) have immediate (qualitative) relevance to the understanding of reactive processes on colloidal catalyst particles or molecular organizates, although calculations similar to those reported here would have to be performed to project out quantitative differences in the reaction efficiency. Turning to the practical side of things, we now discuss how the results reported in this paper can be used in the interpretation of experimental studies. Recall that the parameter W calibrates the role of multipolar correlations in influencing the efficiency of the underlying diffusion-reaction process; as is evident from the definitions, eq 4b, 6b, and 7b, Walso encompasses information on the temperature and dielectric constant of the medium. For a given multipolar reactant pair, an increase in W (which can be realized by decreasing the temperature and/or the system’s dielectric constant) means that the electrostatic energy of interaction between two reactants is playing a relatively more important role than nonspecific thermal effects in determining the stochastic behavior of the system. Hence, the range of Wvalues considered in this paper [0 IIWl I91 allows one to consider changes in the relative balance of potential and kinetic results which span (nearly) an order of magnitude. To get some sense of what changes in the host medium would be required to produce an order of magnitude change in W, consider a singly charged reactant pair. In this case, the associated parameter s (the classical Onsager length) is characteristically -7 A for an aqueous medium at room temperature; on the other hand, decreasing the dielectric constant to a value more typical of a hydrocarbon environment would yield values of s in the vicinity of 70 A. Such dramatic changes in s are much more difficult to realize if variations in the ambient temperature (only) are considered; for the case of singly charged reactants in an aqueous medium, an increase/decrease in temperature of 10 ‘C relative to room temperature produces a change in the value of s of only -4%. For charge types other than 1-1 [for example, coreactants which are triply charged (“3-3 electrolytes”)], larger values of IWl [say, IWl = 91 are understood to mean that electrostatic forces play a significant role in driving the underlying stochastic process at much higher temperatures. For multipolar reactant pairs other than ionic ones, the (generalized) Onsager distance at which nonspecific thermal effects begin to overwhelm the (specific) correlation effects is much smaller; thus, the value of s calculated for (unit) ion-dipole or (unit) dipole-dipole reactant pairs embedded in an aqueous environment at room temperature is (about) an order of magnitude smaller (-0.7 A) than the value calculated for (unit) ion-ion pairs. To assess the interplay between effects associated with system size versus correlation effects, one can phrase the discussion in terms of either the lengths r and s or the corresponding reduced variables 1 and W (see eq 3-7). For the special case where the characteristic lattice spacing R is set at 1 A, the magnitudes of r and 1 and s and Ware obviously the same. In order to discuss conveniently the physical-chemical implications of the trends reported here, we choose this setting of R so as to make immediate use of the estimates of s noted in the preceding paragraph. (We hasten to add, however, that for a specific experimental system characterized by a different (effective) lattice spacing R, the results reported in the tables and figures can be translated immediately into the “real” lengths (r, s) for that system via simple rescaling.) To obtain an estimate of the distance r (and hence I ) which would characterize a typical organizate system, we refer to an earlier studyZ5of radical decay kinetics induced by a micellar system, viz., the dismutation reaction of Br,- ion radicals in aqueous solution in the presence of micellar cetyltrimethylammonium bromide (CTAB) by pulse radiolysis techniques. Under the ionic strength conditions relevant to that study, it was estimated that a CTAB micelle of 221 molecules would have a net charge of 5 5 units or, in other words, that - 5 5 free sites on the CTAB surface would be available to the Br, radical cations

(23) Staten, G.J.; Musho, M. K.; Kozak, J. J. Langmuir 1985,1,443-452. (24) Politowicz, P. A,; Kozak, J. J. Langmuir 1988, 4, 305-320.

(25) Frank, A. J.; Gratzel, M.; Kozak, J . J. J . Am. Chem. SOC.1976, 98, 33 17-332 1 .

+

+

-

-


Diffusion-Controlled Reactive Processes for adsorption followed by surface diffusion. If these 55 free sites were distributed regularly on a surface with 221 regularly spaced sites overall, the separation of the free sites is essentially twice the separation of the adjacent head groups. The latter distance is -7 8, for the CTAB aggregates studied in ref 25, and hence the separation between adjacent free sites is 14 8,. The number 55 of free sites Is well within the range of system sizes considered in this study (as is the overall number 221 for that matter). If one assumes, to a first approximation, that the diffuse layer in the vicinity of the micellar surface has an aqueous dielectric character, then under room temperature conditions s 7 A, a length which is approximately the same as the headgroup spacing and (about) half the estimated separation between the available free sites on the micellar surface. Although these estimates of the two lengths r and s are admittedly very rough, it is also the case that they are of the same order of magnitude and certainly in the range where their interplay can influence quantitatively the consequent encounter-controlled reactive process. Or, put the other way around, by varying the ionic strength conditions (so as to change the effective dielectric constant in the vicinity of the micellar surface), changes in the observed first-order rate constant for the process

-

-

Br2- + Br2-

kl

Br3- + Br-

( k 2 = 2.1 X lo6 s-I) should certainly be observable in a pulse radiolysis experiment. As a second example of the type of experimental system to which the results reported in this study are relevant, we cite a recent contribution26 in which a kinetic model for the photochemical production of hydrogen from water as mediated by a colloidal catalyst was formulated and then examined via computer simulation. In particular, an indepth analysis of the reactions of the reduced radical cation of methyl viologen (MV’+) with a colloidal platinum catalyst (Pt,) MV’+

+ H+

MV2+

+ Y2H2

was carried out and, in the mechanism elaborated, several steps in the reaction scheme which involved diffusion and reaction of intermediates on the catalyst surface were found to influence critically the production of molecular hydrogen. Since the fraction f, of platinum atoms which are surface sites can easily be calculated from the relation2’ = 1 - (1 - 2 ~ , , / ~ , ) 3 where here R, is the radius of the platinum particle and RPt = 1.39 8, is the radius of one platinum atom, predictions could be made regarding the dependence of the overall reaction rate on catalyst particle size. (For the particle radii R, = 15, 25, and (26) Sassoon, R. E.; Lenoir, P. M.; Kozak, J. J. J . Phys. Chem. 1986,90, 4654-4663. Lenoir, P. M.; Sassoon, R. E.; Kozak, J. J. J. Phys. Chem. 1988, 92, 2526-2536. (27) Kittel, C. Elementary Statistical Physics; John Wiley & Sons, Inc.: New York, 1958.

100 8, considered in ref 26 (values of R, spanning the range of particle sizes reported in the earlier experimental work), the theoretical number of surface sites is (approximately) 577, 1733, and 3020; of course, the actual number of surface sites available at any given stage in the reaction will be much smaller owing to the presence of adsorbed intermediates, adsorbed counterions, and the associated polymer stabilizer.) A pronounced decrease in the rate of reaction on increasing the size of the catalyst particle (an effect found experimentally in the studies cited in ref 26) was observed and quantified in the computer simulations on the model introduced in ref 26 and was attributed to a corresponding reduction in the number of reactive surface sites at constant Pt atom concentration. Also influencing the variation of order with respect to platinum concentration were factors which changed the dielectric environment of the catalyst particle, e.g. the pH and the ionic strength. As is plainly evident from the results reported in Tables 1-111 of the present study, the efficiency of a surfacemediated, encounter-controlled reactive process is very sensitive to the number of surface sites; moreover, the interplay between the range of the potential correlating the reactants and the spatial extent of the reaction space is very discriminating indeed and strongly dependent on the dielectric character of the catalyst’s environment. While this qualitative agreement is certainly very encouraging, what is needed here (as in the preceding example) is to go beyond this stage and to translate these qualitative statements into specific quantitative predictions on the dependence of the observed rate constants on the experimental conditions. It is at this point that we can underscore the utility (and flexibility) of the lattice-based, stochastic approach advanced here (and in our earlier work). For although some general conclusions might well emerge from a study of surface-mediated, encounter-controlled reactive processes using a continuum model and an approach based on a Fickian second-order partial differential equation subject to various boundary conditions, the present lattice model provides a framework for incorporating directly the structural details of a “real” (i.e. differentiated) surface and the attendant stochastic mechanical theory provides the avenue for calculating the governing rate constants under a variety of experimental conditions. Studies focusing on the two experimental problems noted above are presently under way, and it is hoped that the results obtained, apart from casting light on the underlying reaction schemes, will also be of use in designing new organizate/catalyst systems whereon the conversion of reactants to products may be optimized. At the very least, we believe the results of the present study will be of assistance in providing an overall theoretical framework for interpreting previously reported, qualitative trends in data generated in the experimental study of kinetic processes on micellar, vesicular, or cellular systems or on colloidal catalyst particles. Acknowledgment. The work described herein was based on studies done by J.B.M., D.E.H., and R.F. as part of an undergraduate research project at the University of Notre Dame. During this period, J.J.K. was partially supported by the Office of Basic Energy Sciences of the Department of Energy with funds provided to the Notre Dame Radiation Laboratory.

Influence of multipolar correlations and surface imperfections on the

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