J . Phys. Chem. 1990, 94, 8315-8322 stirred solution with unchanged flow conditions in the inner compartment. The diffusion coefficient in the membrane has the same value as D, obtained in the experiments described before. This indicates that the membrane surface is uniformly accessible for the diffusing penetrant.
Conclusion The hydrodynamic characteristics of a new steady-state diffusion cell has been investigated with the help of limiting current measurements. The thicknesses of the diffusional boundary layers in both compartments have been determined experimentally. These values were in good agreement with the values calculated with the formalism describing the hydrodynamics at the surface of a rotating disk electrode. Steady-state diffusion measurements were carried out with p-aminoazobenzene as penetrant and cellophane foils as membrane. From the results it can be concluded that the mass transfer in the solutions is quantitatively described by the equations used. The relation between the experimentally determined overall diffusion coefficient Dcxpand the diffusion coefficient in the membrane D , varies between D,,,/D, = 0.8 and Dcxp/Dm= 0.4, depending on the stirring rate. These values and the fact that the thicknesses of the boundary layers are dependent on the diffusion coefficient of the penetrant indicates that it is a basic requirement for a quantitative interpretation of permeation measurements to take the convective diffusion in the solutions into account. Glossary b tortuosity factor C concentration (mol dm-’) concentration in the free solution (mol dm-’) C, concentration at the electrode surface (mol dm-’) ce concentration in the membrane (mol dm-’) Cm D diffusion coefficient (m2 s-I) diffusion coefficient in the solution (m2 s-I) 0,
Dm
d F I
I Ilim
J I
Ri P p, P 4e
4m R ri re *s1
sc T t
u’ X Z
6 A ?d Y
P
$ W
*dl
8315
diffusion coefficient in the membrane (m2 s-I) distance between the stirrer and the electrode (m) Faraday constant (A s mol-’) electric current (A) ionic strength (mol dm-’) limiting current (A) diffusional flux (mol m-2 s-I) thickness of the membrane (m) flux of the component i (mol m-2 s-!) porosity of the membrane (referred to the weight of the substrate) (dm’ kg-I) porosity of the membrane (referred to the volume of the substrate) hydrostatic pressure ( N surface area of the electrode (m2) surface area of the membrane (m2) universal gas constant (J K-’ mol-’) rate of reaction of the component i (mol m-* s-l) radius of the electrode (m) radius of the stirrer (m) Schmidt number temperature (“C) time (s) velocity of the flow (m s-I) Cartesian coordinate charge number thickness of the diffusional boundary layer (m) Laplace operator diffusional overpotential (V) kinematic viscosity (m2 s-I) density (kg dm-’) electrical potential angular velocity (s-I) effective angular velocity (s-l)
Subscripts
exP 1
2
experimentally determined values outer solution of the diffusion cell inner solution of the diffusion cell
Kinetic Changes Arising from Surface Defects and Inhomogeneities Roberto A. Garza-Gpez, J. K. Rudra,+ Russell Davidson,* and John J. Kozak* Department of Chemistry and Franklin College of Arts and Sciences, University of Georgia, Athens, Georgia 30602 (Received: March 13, 1990; In Final Form: June 4, 1990)
We present an analytic and numerical study of the role of (local) defects and (larger scale) spatial inhomogeneities in influencing the efficiency of diffusion-controlled reactive processes taking place on surfaces. We consider finite domains of integral and fractal dimensionality and study both unconstrained and biased flows of a coreactant on a surface having a single, stationary reaction center. Both for the Euclidean and fractal surfaces studied here (respectively,the triangular lattice and the Sierpinski gasket), we find that the higher the (local) valency of the site at which the reaction is anchored, the faster the kinetic process. This conclusion pertains both to the case of unconstrained flow and to (almost all) cases where a uniform bias is imposed on the motion of the diffusing coreactant. The influence of larger scale spatial inhomogeneities (arising from the presence of (self-similar) domains excluded to the diffusing coreactant) is studied and such domains are found to have a significant effect on the reaction efficiency; when compared with processes on the corresponding “space-filling” regular (Euclidean) lattice, the reaction efficiency is always lower. Depending on the initial conditions imposed on solving the stochastic master equation for the problem, our results show for reaction spaces of both integral and fractal dimensionality that the decay profile can have a decidedly nonexponential character on short time scales, an effect which is interpreted geometrically. Finally, we use the results generated in this study to quantify the relative efficiency of two possible sieving mechanisms in the aluminosilicate, mordenite.
I. Introduction Consider a coreactant migrating across a surface on which a reaction center is anchored at a particular location. There are To whom correspondence should be addressed. ‘Permanent address: Department of Physics, Xavier University, New Orleans, LA. *Permanent address: Department of Economics, Queen’s University, Kingston, Ontario K7L 3N6, Canada.
many physical problems where the (otherwise random) motion of a diffusing species (atom, molecule) is subject to a bias, either because of multipolar interactions between reaction partners or because of the presence of an external field or a concentration gradient. Potential correlations are, of course, a sensitive function of the interparticle separation, and the separate roles of attractive and repulsive potentials (of the form u(r) i f ss,> 0 ) in influencing the reaction efficiency between a diffusing coreactant (X) and a stationary target molecule ( Y )on surfaces of integral
0022-3654/90/2094-83 15$02.50/0 0 1990 American Chemical Society
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The Journal of Physical Chemistry, Vol. 94, No. 21, 1990
dimension but of different topologies have been studied in our recent work; we quantified the effectiveness of attractive ion-ion, (angle-averaged) ion-dipole, and (angle averaged) dipole-dipole interactions and repulsive ion-ion interactions in modulating the reaction efficiency on d = 2 dimensional planar surfaces] (Euler characteristic, x = 0) and Cartesian shells2 (x = 2). In the latter study,2 we considered simultaneously the role of surface defects. Specifically, we quantified the difference in reaction efficiency when a target molecule is positioned at a regular versus a defect site on the surface of a colloidal particle or molecular organizate, represented as a Cartesian shell. Formally, the connectivity (or valency) u of a lattice specifies the number of pathways (or channels) intersecting at a particular point on the surface; in the study,* regular sites were characterized by a valency u = 4 whereas defect sites by v = 3. For this geometry, a target molecule positioned at a defect site is somewhat less accessible to a diffusing coreactant than if it were located at a regular site of the lattice; both for attractive and for repulsive interparticle correlations, this translated into a measurably lower reaction efficiency at the defect versus regular site. [See Figures 2-7 in ref 2.) In the present contribution, we shall broaden these earlier studies in several ways. First, independent of interparticle correlations, a diffusing coreactant may at every point in the reaction space experience a bias owing to the presence of an external field or concentration gradient. We would like to quantify the consequences of this perturbation on the coreactant's motion, but in a sufficiently general way that the influence of surface inhomogeneities on the reaction efficiency can be assessed as well. The surface inhomogeneities considered in this study are of two principal types. First, we will quantify the difference in reaction efficiency when a reaction center is positioned at a regular versus defect site on the surface (both in the presence and absence of a uniform bias on the motion of the diffusion coreactant). This kind of spatial inhomogeneity is local, i.e., a defect site is a local imperfection in the surface which can be specified by the site valency. Clearly, however, there can also be present larger scale inhomogeneities. For example, the surface of a cell may (will) have transmembrane (and other) proteins that would break the translational symmetry and interfere significantly with the lateral motion of a diffusing coreactant (and hence influence the reaction efficiency in a diffusion-controlled process). How to deal with these "excluded regions" systematically (Le., formulating the problem so that the influence on the reaction efficiency of excluded regions of different spatial extent can be studied and contrasted with what would be expected if the surface were free of such spatial inhomogeneities) will be a principal objective of this paper. To study the effects noted above, we consider a finite planar network of two dimensionalities, a regular Euclidean triangular lattice (interior valence u = 6, but with boundary defect sites of valence u = 4 and 2) of dimension d = 2 and a fractal lattice (Sierpinski gasket) (principal valence Y = 4, with u = 2 defect sites) of dimension d = 1.584 962 .... The punctuated structure of the second lattice [see Figure I ] effectively restricts the diffusing coreactant to a subset of the overall (space filling) reaction space, thereby providing the opportunity to study systematically the influence of longer range spatial inhomogeneities on the dynamics. Overall, then, the primary objective of this paper is to study diffusion-controlled reactive processes on this lattice and on the associated triangular one both in the presence and in the absence of an external (uniform) bias on the motion of the diffusing coreactant. There is. however, an interesting practical question that can be addressed by using the numerical results generated in this study. The zeolite mordenite has a channel structure which allows for two possible sieving mechanisms (for small molecules or rare gases) depending on whether side pockets lining the channels are blocked ( I ) Mandeville, J. B.; Golub,J.; Kozak, J. J Chem. Phys. Left. 1988, 143, 117-122. (2) Mandeville, J. B.; Hurtubise, D. E.; Flint, R.; Kozak. J. J. J . Phys. Chem. 1989. 93, 7876-7887.
Garza-L6pez et al. 1
. ........... N.6
__
- ....... ... N = 1 5
115 ' - - -N
=
123
Figure 1. Diagram of the fractal lattice (the Sierpinski gasket) studied here. The companion ("space-filling") Euclidean lattice is a triangular lattice of (interior) valency six.
or not. Our study allows an estimate (a lower bound) to be constructed for the difference in flow through this aluminosilicate for the case of d = 1 dimensional flow (only) versus the case where lateral excursions of the diffusing species into the side pockets are possible. 11. Analytic Results
We consider the finite, planar network displayed in Figure 1, a surface of fractal dimensionality d = 1.584962.... known as the Sierpinski gasket. A planar surface of integral dimensionality d = 2 (the triangular lattice) can be generated by completing the internal lattice structure displayed in the figure. The formal question is the following: Given that the trajectory of the diffusing coreactant is initiated at some site on a finite reaction space of N sites overall, what is the time required (on average) for the molecule to reach the stationary reaction center? Let pi(t) be the probability that the molecule has reached site i at time t and let G, be an A' X N matrix describing the probability that the diffusing coreactant has migrated from location i to location j in the network. The time dependence of this process can be explored by defining a stochastic master equation
for the specific geometry assumed to characterize the structure of the system. It is well-known that the solution of the above system of linear equations is of the f ~ r m ~ - ~ IV
pi(t)
=
m= I
aime-xm'
(2)
where the a, are coefficients determined by the initial conditions on the problem and the A, are the eigenvalues of the C matrix. Thus, for a given initial condition, the efficiency of reaction can be monitored by determining the time required for the molecule to reach, say, site 1. It is this efficiency that has been (3) Montroll, E. N.; Shuler, K. E. Adu. Chem. Phys. 1958, I , 361-399. (4) Nicolis, G.; Prigogine, 1. Self-Organirafion in Noneguilibrium Sysrems, Wiley: New York, 1977. ( 5 ) Haken, H. Synergetics; Springer-Verlag: Heidelberg, West Germany.
1977.
Kinetic Changes Arising from Surface Defects
The Journal of Physical Chemistry, Vol. 94, No. 21, 1990 8317
TABLE I: Mean Walk Length Data, ( n ) ( ( n),) case 10 case I I b case 111‘
lattices defined by overall N = 15,42, 123 and for the sequence of triangular lattices defined by N = 15, 45, 153. The results of these calculations are presented in Table I; examination of the data listed in parenthesis under case I shows that the relationship 3 is satisfied (exactly). One can generalize the above analysis and calculate for both lattices the mean number of steps ( n ) before trapping (reaction) when the motion of the diffusing coreactant is initialized at any nontrapping site, again assuming that the flow is guided (biased) by a uniform concentration gradient or external field. For a triangular lattice, the number of vertices on level m is ( m 1 ) . Thus, the total number N of vertices for this geometry is
N = 6
3.200 (4.000)‘
9.200 (lO.OOO)e
1.800 (3.000)’
case IVd 3.800 (5.000)’
Sierpinski Gasket: d = 1.584962... N = 15 N = 42
N = 123
5.714 (8.000)c 10.927 (16.000)‘ 21.508 (32.000)‘
43.429 (50.000)‘ 21 1.561 (250.000)‘ 1047.31 ( 1250.00)‘
5.286 (9.000)’ 17.098 (28.000)’ 59.926 (92.000)’
17.714 (25.000)’ 85.3414 (I25.000)’ 420.262 (625.000)’
+
Triangular Lattice: d = 2 N = 15
5.714 (8.000)c
N = 45 N = 153
10.9091 (16.000)c 21.4737 (32.000)6
44.674 (49.707)‘ 223.812 (247.581)‘ 1118.414 (1 21 8.20)‘
5.09 1 (8.300)’ 15.186 (22.574)’ 48.503 (61.875)’
m
17.723 (23.220)’ 85.285 (l08.936)f 413.724 (510.996)’
N =
(m m=O
+ 1)
= y2(m
+ l)(m + 2)
The total number N’of nontrapping (satellite) vertices is therefore N ’ = N - 1 =y2(m+3)
‘Vertex site ( I ) target, biased flow (see text). bVertex site ( I ) target, unconstrained flow. Midpoint-base site target, biased flow (see text). Midpoint-base site target, unconstrained flow. e Initialized at midpoint-base site. ’Initialized at vertex site ( 1 ) .
Averaging over all nontrapping sites, the average number ( n ) of steps taken by a diffusing coreactant before trapping is “‘L,
calculated here (by calculating p ( t ) ) , and the results of our numerical study of the system of eq l subject to a variety of initial temporal and spatial constraints are reported in the following sect ion. Before examining the results of these numerical calculations, however, it is instructive to develop some formal results. Suppose the target molecule is anchored at a defect site, say site 1 (valency Y = 2) in Figure 1 , and the motion of the diffusing coreactant is initialized at the lattice site farthest removed from the target site; actually, on the discrete (regular and fractal) spaces considered here, any point on the base of Figure 1 will satisfy this condition and for definiteness we choose the midpoint site 115 (valency u = 4). Moreover, we wish to impose a uniform bias on the motion of the diffusing coreactant; Le., owing to the presence of a uniform external field or concentration gradient, the motion will be constrained in such a way that the molecule can move only laterally or toward the target. On inspection of the figure it becomes clear that every vertex has either two upward connections and two sideways connections or one upward connection and one sideways connection. In either case, the ratio of upward to sideways connections is always equal to unity. As a result, at each time step the molecule either transits one level closer to vertex 1 or stays at the same level. The ratio of the probabilities of these two motions is always the same, a conclusion valid both for the Sierpinski gasket and for the associated regular triangular lattice. Given these constraints, the migration of a molecule through the lattice can now be regarded as a classical random walk relative to a fixed point. In this new frame of reference, the particle moves (on average) 1 / 2 a level spacing in a single step i; a step forward of 1 / 2 a level spacing and a step backward of 1/2 a level spacing are equally likely. Inasmuch as the above analysis is valid both for regular and fractal lattices, the probability of surviving to time t should be the same for both lattice geometries. Specifically, for a molecule positioned initially at the midpoint of the base, if the particle moves (on average) 1/2 step upward in a single time step and there are m levels separating the particle initially from the trap (defined to be at level m = 0), the average number ( n ) of steps required for trapping (reaction) in either case is
3
( m + 3)
\ ’ I
The validity of the result (4) can be checked against the numerical evidence presented in Table I under “triangular” for case I . [For m = 2, 4, 8, 16 the eq 4 gives 3.2, 5.714, 10.9091, 21.4737, respectively .] A similar sort of analysis shows that, for the same initial condition [diffusion of the coreactant proceeding from any nontrapping (satellite) work in the network with the subsequent flow biased in the direction of the reaction center, site 1 1 , the average number ( n ) of steps before trapping on the Sierpinski gasket is m
(n) = C
m=l
(number of vertices on level m ) (2m) N‘
(5)
This result can be checked numerically (using the (analytical/ numerical) method developed in our earlier work based on the theory of finite Markov processesb”) for the sequence of fractal
The results obtained for N = 6, 15, 42, and 143 [respectively, for m = 2, 4, 8, 16 (5) gives 3.2, 5.714, 10.92682, and 21.508 191 are in accord with the numerical evidence presented under “Sierpinski” in case I of Table I. The above analytic/numerical results for ( n ) were calculated assuming that the diffusing coreactant initiated its motion at a single site (the midpoint site of the base for each geometry considered) or at any site i (then averaging over all nontrapping (satellite) sites of the regular/fractal lattice). In both calculations, the subsequent motion of the coreactant was biased in the direction of the reaction center (site 1). The full dynamics of the decay can be calculated via numerical solution of the stochastic master equation (1). For the largest networks considered here (the N = 123 Sierpinski gasket and the N = 153 triangular lattice), these results are displayed in Figure 2 for the two cases noted above. The profiles in the upper right of this figure correspond to a flow initiated from the single (midpoint base) site while the ones in the lower left are those for which initialization is possible at any satellite site. In both cases, the circles and triangles specify results generated for the Sierpinski gasket and the triangular lattice, respectively. Two features of these evolution curves are immediately noticeable. First, for both initial conditions, trapping (reaction) on the fractal lattice is distinctly slower than reaction on the triangular one. At first sight, this result would appear to be anomalous inasmuch as the space-filling triangular lattice has more sites than does the SierpinsG gasket (fi= 153 versus N = 123, respectively) and one would expect the average walk length ( n ) (and hence the characteristic relaxation time) to increase with increase in the number N of sites. In fact, in his study of random walks on lattices of integral dimensionality subject to periodic boundary conditions,
(6) Walsh, C. A.; Kozak, J. J. Phys. Reu. Lett. 1981, 47, 1500-1502. (7) Walsh, C. A.; Kozak, J. J. Phys. Reu. E . 1982, 26, 4166-4189. (8) Politowicz, P. A.; Kozak. J. J. Phys. Reu. E . 1983, 28, 5549-5569.
(9) Musho, M. K.; Kozak, J. J. J . Chem. Phys. 1984, 81, 3229-3238. (10) Politowicz, P. A.; Kozak, J. J.; Weiss, G . H.Chem. Phys. Lett. 1985, 120. 388-392.
( n ) = 2m
(3)
8318
The Journal of Physical Chemistry, Vol. 94, No. 21, 1990
Garza-Mpez et al.
B
-1
0
Figure 3. Same conventions as in Figure 2 except that, following activation, there is no restriction on the directionality of the coreactant's motion.
Figure 2. Survival probability p ( r ) versus (reduced) time t for a reactive event monitored at the (vertex) site I . The motion of the diffusing coreactant is initialized either at a single site (here, midpoint of base) (upper right figure) or (averaged over) all nontrapping (satellite) sites, with the consequent motion such that the molecule moves either 'right/left" or "toward" site I . The upper (circles) and lower (triangles) curves in each of these figures refer to reaction spaces modeled as a Sierpinski gasket and a regular triangular lattice, respectively.
Montroll" proved analytically that ( n ) = N ( N + 1)/6 in d = 1, ( n ) N In N in d = 2 and (n) N in d = 3. From these results it is evident that for lattices of a given (integral) dimensionality, ( n ) indeed increases with increase in N but for fixed N , {n)decreases with increase in the dimensionality d . Moreover, for fixed ( N , d ) there is a further effect arising from the connectivity (or valency) of the reaction space. Montroll proved that, for fixed ( N , d), ( n ) increases with decrease in the valency v. For example. in d = 2 he showed that the leading term in the asymptotic expansion for (n),viz. N ( n ) -{ A , N In NJ N-1
-
N
-
is A, = 3 6 / 4 7
A , = l/s A, = fi/27
for
=3
for
u
u
=4
for
u
=6
I n the present context, note that all interior sites of the fractal lattice have valency u = 4 whereas all interior sites of the triangular one have valency u = 6. On this basis, one would expect the average walk length ( n ) to be smaller on the triangular lattice than on the Sierpinski gasket. Thus, increasing both d and v tends to counteract the increase in ( n )arising from the greater number of vertices at each level m (>2) (and hence the overall N) on the triangular versus the fractal lattice. Qualitatively, the results displayed in Figure 2 can be understood in the following way: for a given metric (lattice spacing) increasing d and v places more satellite sites closer to the reaction center. These general results, ~ .also ~ ~ consistent with already noticed in our earlier ~ o r k , 'are ( I I ) Montroll. E. W. J . Math. Phys. 1909, 10, 753-765
the theoretical insight of Gefen, Aharony, and Alexander14 that diffusion on percolation networks is slower than on Euclidean ones, viz., the mean-square displacement of a random walker is given in their work by with 0 = 0.8 in dimension d = 2 for percolating networks versus 8 = 0 for the Euclidean case. The second notable feature of these evolution curves is the pronounced "shoulder" effect seen on short time scales, particularly for the case where the flow is initiated from a site farthest removed from the reaction center. The appearance of these shoulders is related to the fact that, for a particle initiating its motion at a specific site somewhere in the lattice, there is a minimum time required for the coreactant to reach the reaction center; this time is proportional to the length of the shortest path, and hence the reactive event cannot occur until (at least) that interval of time has expired. This effect is quite analogous to the one observed in computer simulations of Boltzmann's H function calculated for two-dimensional hard disks.I5 Starting with disks on lattice sites with an isotropic velocity distribution, there is a time lag (a horizontal shoulder) in the evolution of system owing to the time required for the first collision between two hard particles to occur. It is also analogous to the "horizon effect" in astronomy,16 a consequence of the finite velocity of light, Le., the fact that the observable region of the universe is limited in extent by the distance light has traveled during the time interval since the singularity. Having developed a qualitative and quantitative understanding of the similarities/differencesin the dynamics generated for these initial conditions and spatial constraints, we now examine the consequences of imposing a broader range of conditions. ~~
(12) Abowd, G. D.; Garza-Lbpez, R. A.; Kozak, J. J. Phys. Lett. A 1988, 127. 155-159. (13) Garza-L6pez, R. A.; Kozak, J. J. Phys. Reo. A. 1989.40.7325-7333, (14) Gefen, Y . ;Aharony, A.; Alexander, S. Phys. Reti. Lett. 1983, 50, 77-80. There is now a well-developed literature on this general class of problems; see, for example: (a) Weiss, G.; Rubin, R. J . Ado. Chem. Phys. 1983, 52, 363-505. (b) Blumen, A,; Klafter, J.; Zumofen, G . In Optical Spectroscopy of Glasses, Zschokke, I., Ed.; D. Reidel: Dordrecht, 1986; pp 199-265. ( c ) Havlin, S.; Ben-Avraham, D. Adu. Phys. 1987, 36, 695-798. (d) Drake,, J . M.; Klafter, J. Phys. Today 1990, 43, 46-55. ( I 5 ) Prigogme, I.; George, C.; Henin, F.; Rosenfield, L. Chem. Scr. 1973, 4. 5-32. ( 1 6 ) Silk, J . The Big Bang, Freeman: San Francisco, 1980.
The Journal of Physical Chemistry, Vol. 94, No. 21, 1990 8319
Kinetic Changes Arising from Surface Defects 111. Numerical Results In the previous section, we explored the consequences of imposing a strong directionality on the motion of the diffusing coreactant; i.e., motion of the coreactant was either lateral or toward a reaction center situated at a defect site of the lattice. The consequences of relaxing this motional constraint are displayed in the data given in case 11 of Table I and in the companion profiles in Figure 3. There, following initiation of the flow, no spatial restrictions were placed on the subsequent migration of the diffusing molecule. From an examination of these data vis a vis those presented in case I and Figure 2 two observations follow. First, the mean walk length ( n ) and time scale characterizing the (eventual) reaction at the target site for the case of an unconstrained flow are significantly longer relative to the case of a focused flow. The diffusion-reaction process is much less efficient. Second, on the extended time scale characterizing the case of unconstrained flow, the pronounced shoulder effect noted in Figure 2 is suppressed for both choices of initial conditions; the evolution profiles displayed in Figure 3 have a canonical exponential form. Thus, both the qualitative and quantitative characteristics of the decay profiles change vis a vis the cases studied in section 11. The calculations reported thus far refer to the case where the target molecule is anchored at a defect site ( v = 2) of the reaction space. I t is of interest to assess the generality of the conclusions drawn from these data when the location of the target molecule is switched to a site of valency Y = 4. Accordingly, companion calculations for the problem studied here were performed for the case where the target molecule was positioned at a variety of interior and exterior locations. Consider first positioning the target molecule at the midpoint base site. Entries for case Ill in Table I can be compared directly with corresponding entries for case I; in both cases it is assumed that the motion of the diffusing coreactant is biased, Le., in its trajectory the molecule suffers either a lateral displacement or migrates in the direction of the reaction center. Our calculations show that, if the diffusing coreactant is injected at the vertex (defect) site and migrates toward a reaction center situated at the centrosymmetric site on the opposite boundary, the associated mean walk length is systematically longer than if the coreactant is injected at the midpoint base site and migrates toward a reaction center placed at the vertex [compare data in parentheses under case 111 versus case I, respectively]. On the other hand, for unconstrained (unbiased) flow, exactly the opposite conclusion pertains [compare data in parentheses under case IV versus case 11, respectively]. Note that these generalizations are valid for both the fractal and regular lattices studied here. Finally, if one relaxes the constraint that the coreactant is injected into the system at a specific site (data in parenthese in Table 1) and instead assumes that the coreactant can initiate its motion at any site i of the reaction space, some interesting new effects arise. For the case of unconstrained (unbiased) motion, the results calculated for the overall ( n ) (i.e., the walk length calculated by averaging over all trajectories initiated from all satellite sites i of the defining lattice) show that placing the reaction center at the midpoint base site ( v = 4) results in a much more efficient diffusion-reaction process tha:i placing the trap at a defect site ( v = 2) (here the vertex site 1 ) . On the other hand, biasing the motion of the diffusing coreactant (e.g., by switching on a uniform external field) leads to the same conclusion only if N I 15. For system sizes N > 15, for biased flows, the most efficient diffusion-reaction process overall (e.g., averaged over all possible initial starting points) is the one for which the target molecule is anchored at the defect (vertex) site v = 2. Clearly in this case there is a "crossover" in reaction efficiency for system sizes N 15.
-
For the largest fractal and regular lattices considered in this study [ N = 123 and N = 153, respectively], conclusions drawn from an examination of the time scales characterizing the evolution curves displayed in Figure 2 versus Figure 4 for biased flows and in Figure 3 versus Figure 5, for unconstrained ones, are consistent with those developed from an examination of the ( n ) (for N >
8
Figure 4. Survival probability p ( t ) versus (reduced) time r for a reactive event monitored at the (midpoint) base site 1 1 5 . The motion of the diffusing coreactant is initialized either at a single site (top vertex) (upper right figure) or (averaged over) all possible nontrapping (satellite) sites (lower left figure), with the consequent motion such that the molecules move either "right/left" or "toward" site 115. The upper (circles) and lower (triangles) curves in each of these figures refer to reaction spaces modelled as a Sierpinski gasket and a regular triangular lattice, respectively.
? 0
?
-1
Figure 5. Same conventions as in Figure 4 except that, following activation, there is no restriction on the directionality of the coreactant's motion.
15) [as they must, since in the limit of large N it can be shown that ( n ) = vX1-', where XI is the smallest eigenvalue of the stochastic master equation ( l ) ] . Specifically, for a biased flow, the diffusing coreactant is certainly trapped within t < 20 when the target molecule is anchored at the vertex defect site 1 of valency v = 2, whereas when the target is positioned at the midpoint base site ( v = 4) the trapping is within t < 150. Conversely, for totally unconstrained flows the coreactant is trapped within t C 100 when
8320 The Journal of Physical Chemistry, Vol. 94, No. 21, 1990 the target molecule is anchored at the defect site 1, but is trapped within t < 50 when the target is positioned at a site of higher local coordination ( v = 4 ) . IV. Conclusions The first of conclusions that can be drawn from the results presented in this paper is that for surfaces of integral or fractal dimension, in the absence of any external bias, the diffusion-reaction process is more efficient when the target molecule is anchored at a site of higher valency. This conclusion is consistent with the one which emerged in an earlier study1’ of diffusioncontrolled reactive processes taking place on surfaces of Euler characteristic x = 2 (Le,, surfaces topologically equivalent (homeomorphic) to a sphere). There we considered a target molecule anchored at one of the N surface sites of a polyhedral solid and a coreactant free to migrate among the N - 1 satellite sites. We formulated a stochastic master equation for each surface geometry considered and for different choices of initial conditions determined the survival probability p ( t ) versus t for the diffusing coreactant. In all cases, we found that, for fixed N , the characteristic relaxation time (related to the mean walk length of the coreactant on the surface before trapping) decreased with increase in the (local or global) valency u of the site on which the target molecule was positioned. This result was also consistent with a lattice-based study of reactivity at terraces, ledges, and kinks on a (structured) surface.l* There the reaction X+Y
2
[XY]*-.Z
between a diffusing coreactant X and a stationary target molecule Y was studied as a function of the degree of reversibility s of reaction. Changes in the reaction efficiency, as gauged by the value of the mean walk length ( n ) before trapping (reaction), were quantified by using methods developed in refs 6-10 for various settings of s, the probability that the above reaction proceeds at once to completion with formation of the product Z [or, conversely, a probability ( 1 - s) that the diffusing coreactant X, upon confronting the target molecule Y, forms an excited-state (activated) complex, but one that eventually falls apart with regeneration of the species X (which subsequently commences its random walk on the surface)]. These calculations demonstrated convincingly that in the absence of specific energefic effects (e.g., those associated with the extra number of “frec bonds” at a defect site of valency on a regular lattice), the reaction efficiency was always higher the more highly coordinated was the site to which the reaction center was anchored. Specifically, the reaction efficiency increased as one positioned the reaction center at sites of valency u = 3, u = 4, and u = 5 on a reaction space characterized overall by a valency u = 4. Also consistent with the general result above is the one that emerged in a lattice-based study of sequestering and the influence of domain structure on excimer formation in spread monolayers.19 There, changes in lifetime of a diffusing, excited-state monomer (probe molecule) in a monolayer, studied experimentally by monitoring the excimer-monomer steady-state photoexcitation of the probe, were examined in a variety of situations with the overall conclusion being that the lifetime of the probe molecule was always longer on domains of lower local coordination (valency). Finally, we consider the influence of symmetry-breaking potentials in modulating the efficiency of diffusion-controlled reactive processes on surfaces. In the recent contribution,2 we reported that, for topological surfaces characterized by x = 2, when one assumed that the motion of the diffusing coreactant was influenced by the presence of a down range (attractive or repulsive) potential u ( r ) of the form u(r) f r - s (s > 0) centered on the active site (reaction center. target molecule), the efficiency of reaction was
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(17) Politowicz, P. A.; Garza-Lbpez, R . A,; Hurtubise, D. E.; Kozak, J. J. J . Phys. Chem. 1989, 93, 3728-3735. ( 1 8 ) Staten, G .J.; Musho, M. K.; Kozak, J. J. Langmuir 1985,1,443-452. (19) Politowicz. P. A.; Kozak, J. J . Langmuir 1988, 4, 305-320.
Garza-Ltipez et al. measurably lower if the target molecule was positioned at a defect site on the surface. Put the other way around, the higher the (local or global) valency of the target site, the smaller the characteristic trapping time required for the underlying diffusion-reaction process to occur. This result is consistent with the results reviewed above for processes taking place in the absence of any short- or long-range biasing potential. In the present contribution, we considered a more global symmetry-breaking effect, viz., we imposed a uniform bias on the motion of the diffusing coreactant at each and every point of the reaction space, thereby correlating the motion of the coreactant with respect to the target molecule. Here, we found [see Table I] that, for system sizes N I 15, the results obtained were consistent with those uncovered in ref 2 and in the studies cited above. Both for fractal and Euclidean surfaces, upon averaging over all possible trajectories, placing the target molecule at a defect site (IJ = 2) resulted in longer trapping times (and associated walk lengths) than if the target were anchored at a site of valency (IJ = 4). However, a new feature appeared when a uniform bias was imposed on the motion of the diffusing coreactant for diffusionreaction processes taking place on extended reaction spaces ( N > 15). Here, the focusing effect of this (global) constraint leads to an enhanced efficiency of reaction if the target molecule is positioned at a defect site (here, u = 2 ) . This “crossover” effect in reaction efficiency with increase in system size emerges both for surfaces of fractal ( d = 1.58) and integral ( d = 2 ) dimensionality. Since this is the only case in the lattice models studied to date where, in the absence of any specific energetic effects at a given defect site, one finds that anchoring the target molecule at a site of lower coordination (relative to the overall lattice valency) leads to an enhancement in the reaction efficiency, further discussion of this case will be instructive. V. Discussion In this section we focus attention on calculations performed to assess the role of a uniform bias in influencing the efficiency of reaction on a structured surface. In particular, we wish to understand more thoroughly the (only) case uncovered in this study where anchoring the target molecule at a site of lower (rather than higher) valency leads to an enhancement in the reaction efficiency. In case 1, two factors are at play in focusing the flow to the site (1) of lower valency. First there is the bias imposed on the particle’s motion; and second, there is a geometrical constraint, i.e., the fact that the reaction space available to the diffusing coreactant contracts systematically as the molecule moves (irreversibly) from site to site closer to the trap. We seek to disentangle these two factors in order to determine whether the enhanced efficiency at the site of lower valency is driven by the biasing effect, the geometrical effect or, perhaps, a synergetic interplay of both factors. It is already known (see cases 11 and IV in Table I) that, if one turns off the potential, the results obtained are consistent with those reported in earlier studies; Le., in a diffusion-controlled reactive process, the reaction efficiency is higher the greater the number of channels intersecting at the reaction center. However, it remains to be determined whether, in the presence of a biasing potential, positioning the target molecule at a site of lower valency ( u = 2 ) , set in a reaction space not characterized by the constrictive geometry of case I, the reaction efficiency remains higher at a site of lower valency. To investigate this point, suppose that molecules were injected, one at a time, at the defect site 1 in the reaction space, Figure 1. If these molecules were regarded as “hard spheres” and if gravity were acting downward in Figure 1, we would be dealing with an apparatus known as a Galton board,20after Francis Galton who constructed the first one (based on a hexagonal geometry). Such an apparatus is often displayed in museums to demonstrate visually how one produces a Gaussian distribution starting with h ( - 1000) balls initialized at the vertex (here site 1 ) and exiting (20) See: Kac, M . Mathematics in the Modern World W. H. Freeman: San Francisco. 1968; pp 165-1 74.
The Journal of Physical Chemistry, Vol. 94, No. 21, 1990 8321
Kinetic Changes Arising from Surface Defects 8
Figure 7. Staggered configuration of side pockets along a d = 1 dimensional channel.
TABLE II: Mean Walk Length Data (i- j ) d = 1 channel
N = 3; ( n ) = 4.000 N = 5 ; ( n ) = 16.000 N = 9; ( n ) = 64.000 N = 17; ( n ) = 256.000
N N
fractal side pocket = 6; ( n ) = 10.000 = 15; ( n ) = 50.000 = 42; ( n ) = 250.000
Euclidean site pocket
N = 6; ( n ) = 10.000 N = 15; ( n ) = 50.029 N N = 45; ( n ) = 250.892 N = 123; ( n ) = 1250.000 N = 153; ( n ) = 1233.94
‘1* I d
i.8
i-d
i4
i-2
i
i+2
i+4
i 4
id
TARGET HOLECULE SITE
Figure 6. A plot of the factor P = ( n ) (midpoint base site)/(n) (base site) versus exit location for the Sierpinski gasket (circles) and the Euclidean triangular lattices (triangles). For the Sierpinski gasket as one moves pointwise from the midpoint base site (where ( n ) = 59.926) to the corner vertex site, the ( n ) values are 61.934, 67.959, 78.000, 92.057, 110.131, 132.221, 158.328, and 188.451. For the triangular lattice, the corresponding ( n ) values are 48.503 (midpoint),50.842, 57.823, 69.342, 85.227, 105.250, 129.131. 156.550, and 187.154.
at one of the sites defining the lower boundary of Figure 1. In the calculations reported here, based on the theory of finite Markov m molprocesses,6-’0 the results correspond to initializing N ecules at the vertex site l and determining ( n ) , and the overall average ( n ) for molecules being trapped (exiting the reaction space) at a basal site i. Thus, a plot can be constructed for the ( n ) (midpoint base normalized quantity (probability), P site)/(n) (base site) versus exit location, both for the fractal lattice and the “space filling” Euclidean one. As displayed in Figure 6, this plot yields the (anticipated) Gaussian profile of probability theory,20with the maximum in the profile centered at the midpoint site on the base. (Note that the relative width of the curve generated for the Sierpinski gasket is somewhat broader than for the triangular lattice, reflecting the longer residence time of the diffusing coreactant on the fractal surface.) From an examination of Figure 6, one finds (either for the regular or fractal lattice) that the mean walk length ( n ) , of a diffusing coreactant injected at the vertex site 1 and flowing downward (under the uniform bias) is smallest at the midpoint base exit site ( u = 4) and increases ( P decreases) systematically as one moves away from that site. In particular, values of ( n ) , calculated for the left/right vertex sites on the base yield values 0.31 for the Sierpinski gasket and P 0.25 for the of P triangular one. Hence, consistent with the results obtained in all previous studies, once again the reaction efficiency is found to be higher if the target molecule is positioned at a site of higher valency (the midpoint base site has a valency u = 4 while the corner vertex sites have a valency u = 2). It is only when one “inverts” the process, i.e. injects molecules at the base of the Figure 1 and monitors the flows at the (top) vertex site 1 , that one calculates an enhanced efficiency of reaction for a target molecule anchored at a site of lower valency, and then only if N 1 15. Thus we conclude that it is only when the motion of the diffusing particle is influenced simultaneously by a global focusing effect (the bias) and a systematic contraction in the reaction space that the “crossover” effect noted in section 111 emerges.
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1 dimensional flow is contrasted with a process in which the diffusing coreactant also has access to an expanded ( 1 < d I2 ) reaction space at periodic (spatial) intervals. Processes involving energy transfer in uranyl-exchanged zeolites have previously been discussed from the perspective of fractal geometry by El-Sayed et aL2’ Mordenite22is a zeolite which, in its hydrogen form, is characterized by an internal system of d = 1 dimensional (wide) channels. Each channel is lined by side pockets which can be entered through 8-ring windows of free dimension 2.9 X 5.7 A. These side pockets provide a pathway for molecules to pass from one channel into (either of two) others. Given the size of these windows it is known that larger molecules (like benzene) cannot enter these side pockets; hence, larger molecules are confined to the wide channels. Smaller molecules, on the other hand, can occupy both the wide channels and the side pockets, and hence can migrate from one channel to another. In the sodium form the 8-ring of this aluminosilicate (Na8[A18Si40096].24H20), windows are effectively blocked by Na+ ions; in this case, all diffusion is effectively one-dimensional, regardless of the size of the diffusing species. In considering diffusion-controlled processes in mordenite, the question arises on the extent to which the time scale for diffusion changes when the diffusing species can migrate along the channels (only) versus the case where the species has access to the side pockets. The further calculations reported in this section will provide a quantitative assessment of the difference in these two “sieving” mechanisms. Computer graphics displays of the wide channels of m ~ r d e n i t e ~ ~ show that the side pockets are staggered along the channel;24a cross-sectional representation of this configuration is sketched in Figure 7. In terms of this representation, one possible trajectory of a diffusing species would be along the d = 1 channel (horizontal as drawn) with random, left/right excursions characterizing the particle’s motion. Diffusion into the side pockets would delay passage along the channel by providing an expanded reaction space into which the diffusing species could migrate. To estimate the extension in time scale resulting from the latter possibility, we model the reaction spaces defined by the side pockets by the integral and fractal spaces considered previously in this paper. With respect to the question posed above, then, we imagine that a molecule is injected at point i, leaves eventually at point j , with the time scale for the restricted ( d = 1 ) motion to be compared with the time scale calculated assuming the diffusing species can migrate into the side pockets. These two processes can also be studied as a function of the spatial extent of the side pockets. ~
VI. Application: Diffusion in Mordenite Although a variety of diffusion-controlled processes taking place on the surface of a supported catalyst or a molecular organizate (e.g., cell, fatty acid monolayer at the air-water interface, ...) can be analyzed by using the results reported in this study, we focus attention here on (one aspect of) diffusion through zeolites. In particular, we treat a problem in which strictly channeled d =
(21) (i) Yang, C. 1.; El-Sayed, M. A. J . Phys. Chem. 1987,9/, 444C-4443. (ii) Yang, C. L.; Evesque, P.; El-Sayed, M. A. In Molecular Dynamics in Restricted Geometries; Klafter, J . , Drake, J . M., Eds.; Wiley: New York, 1989; pp 371-386. (22) Barrtr, R. M. In Inclusion Compounds; Atwood, J . L., Davies, J . E. D., MacNicol, D. D., Eds.; Academic Press: New York, 1984; Vol. I , pp
19 1-248. (23) Newsam, J. M. Science 1986, 231, 1093-1099. (24) See ref 22, p 201.
8322
J . Phys. Chem. 1990, 94, 8322-8325
Presented in Table I1 are the results of our calculations of the average walk length of a diffusing species migrating from the entry point i to the exit p o i n t j for the two cases elaborated above. The data show that the time scale for diffusion ( i - j ) can be extended by factors of 2.5-4.9, depending on the spatial extent of the reaction space assumed to characterize each side pocket. Since the model developed above has omitted consideration of excursions perpendicular to the plane of Figure 7, a degree of freedom which would dilate further the time scale, we conclude that the sieving of small molecules (or rare gases) through the hydrogen versus sodium form of mordenite should be characterized by markedly
different time scales. Moreover, if the diffusing species is subject to a uniform bias in its motion (arising, for example, from a uniform concentration gradient developed in the interior wide channels in the direction i - J ) , the difference in diffusion times for the two processes becomes even more pronounced. For the largest reaction spaces considered in our model [ N = 123 for the Sierpinski gasket and N = 153 for the triangular lattice], the data summarized in the caption of Figure 6 show that the factor 4.9 calculated for the case of unbiased flow (see above) more than doubles (188/16 12) when the motion of the diffusion coreactant is directionally assisted.
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Concentratlon Dependence of the Micelle Size of Hexaoxyethylene Glycol Decyl Ether As Revealed by Gel Filtration Chromatography Noriaki Funasaki,* Sakae Hada, and Saburo Neya Kyoto Pharmaceutical University, Misasagi, Yamashina- ku, Kyoto 607, Japan (Received: March 19, 1990)
Frontal gel filtration chromatograms (GFC) of hexaoxyethylene glycol decyl ether (C,,Es) have been obtained with five kinds of gel, different in pore size, at 25 OC. Among these gels, Sephadex '3-10 with small pores gives the most easily analyzable data, since both small and large micelles are excluded from the pores. The concentration dependence of centroid volumes determined from GFC patterns is used to estimate monomer concentrations C, of Cl0E6. Analysis of the concentrationdependence of C, by a multiple-equilibrium model for micelle formation provides micellar weight-average aggregation numbers n, as a function of total CioE6concentration C. The n, value increases with an increase in C and is similar to that obtained from a n analysis of Debye plots of light scattering data. Derivative GFC patterns also suggest the formation of small micelles at low concentrations. The utility of the GFC method in investigations of surfactants has been demonstrated.
Introduction To consider almost all properties of a surfactant in aqueous solution, we need to estimate the monomer (intermicellar) concentration, C,, of the surfactant. Since this concentration above the critical micelle concentration (cmc) cannot be easily determined, it is often assumed to equal the cmc. We have already reported the principle of gel filtration chromatography (GFC) for determining CI of surfactant., The dependence of CI on the total surfactant concentration C is related to the aggregation number of micelles. If we can determine C1as a function of C very accurately, we may be able to estimate the micellar aggregation number as a function of C. I n 1964, Balmbra et al. reported the effects of C and temperature on the micellar size for homogeneous hexaoxyethylene glycol monoalkyl ethers (C,E6).2 They found that the excess turbidity vs Cgraphs were linear up to a concentration above the cmc (as determined by surface tension measurements). Therefore, Debye plots of these data showed negative slopes at low concentrations and roughly horizontal portions at high concentrations. They suggested that micellar weight-average aggregation numbers n, for these surfactants increase with an increase in C. Becher considered that this increase is ascribed to the relative increase of micellar concentration to C,.3 Recently, the large amount of research on this problem has been carried out by using static and dynamic light scattering, nuclear magnetic resonance, static and dynamic neutron scattering, fluorescence decay and quenching, sedimentation velocity, viscosity, GFC, electron paramagnetic resonance, and vapor pressure osm ~ m e t r y . ~Nevertheless, -~ it is still a matter of controversy whether or not micellar size of nonionic surfactants increases with either C or t e m p e r a t ~ r e . ~ - ~ (1) Funasaki, N.; Hada, S.; Neya, S. J. Phys. Chem. 1988, 92, 7112. (2) Balmbra R . R.; Clunie, J. S.: Corkill, J. M.; Goodman, J. F. Trans. Faraday Soc. 1964, 60, 979. ( 3 ) Becher, P. Narure 1965, 206, 61 1
0022-3654/90/2094-8322$02.50/0
I n our previous studies, we used gels having large pores (Sephadex G-200) and showed the coexistence of small and large micelles of CI2E5'and C,2E6.6 For CIoEsand CloEBwe used Sephadex G-25 and encountered some difficulties in estimating elution volumes of small micelles.' In this work, to overcome these difficulties, we use several gels different in pore size and establish GFC methodology for determination of accurate values of C,. From these C, data we will estimate micellar aggregation numbers and show that the micellar size of CIoE6increases with an increase in C.
Theoretical Basis Multiple-Equilibrium Model for Micelle Formation.Io*'l For the sake of simplicity, we now consider an aqueous solution of a nonionic surfactant whose activity coefficient is unity regardless of total surfactant concentration C. When the i-mer Ai is formed from the ( i - I)-mer and monomer, the stepwise association constant ki can be given by
(4) Zulauf, M.; Weckstrom, K.;Hayter, J. B.; Degiorgio, V.; Corti, M. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum: New York, 1986; Vol. 4, p 131. (5) Magid, L. J. In Nonionic Surfactants. Physical Chemistry; Schick, M. J.. Ed.; Marcel Dekker: New York. 1987; p 677. (6) Funasaki, N.; Hada, S.; Neya, S. J. Phys. Chem. 1988,92, 3488 and
references cited therein. ( 7 ) Funasaki, N.; Hada, S.; Neya, S. Bull. Chem. Sor. Jpn. 1989.62.2485 and references cited therein. (8) Brown, W.; Johnsen, J.; Stilb, P.; Lindman, B. J. Phys. Chem. 1983, 87. 4548. (9) Brown, W.; Rymden, R.; van Stam, J.; Almgren, M.; Svensk, G. J . Phys. Chem. 1989, 93, 2512. (IO) Corkill, J. M.; Goodman, J. F.;Walker, T.; Wyer, J. Proc. R. SOC. London 1969, A3I2, 243. ( I I ) Mukerjee, P. J . Phys. Chem. 1972, 76, 565.
0 1990 American Chemical Society
