J . Phys. Chem. 1989, 93, 3728-3735
3728
more compact, thereby resulting in lower compressibility. The adiabatic compressibility coefficient of aqueous micelle solutions of CTAB containing these solubilizates is seen to increase with increasing solubilizate concentrations (Figure 6). This increase may arise because of the decrease in the degree of "structured" water as a result of transfer of the solubilizates from the aqueous to the micellar phase, this change being more than compensated by the loss of free space in the micelle interior upon solubilization. In conclusion, we find that CTAB micelles undergo a postmicellar transition that takes place around 0.28 mol kg-' CTAB in water. The solubilization behavior of CTAB micelles formed below and above the transition concentration is significantly different. It appears that both BHT and BHA are preferentially solubilized in the micellar aggregates. BHA solubilized in 0.10 m CTAB appears to be located at the micelle-water interfacial region at low solubilizate concentrations and penetrates into the inner part of the micelle with increasing surfactant and solubilizate concentration and temperature. BHT is more hydrophobic in nature than BHA and resides in the hydrocarbon-like environment
of the micelle. When solubilization in small ellipsoidal micelles (0.10 m CTAB) occurs, this hydrophobic molecule initially occupies the outer region of the nonpolar part of the micelle and moves into the inner part of the micelle with increasing solubilizate concentration. In 0.30 m CTAB it appears to be solubilized in a uniform site within the rod-shaped micelle. Acknowledgment. We are grateful to the Natural Sciences and Engineering Research Council of Canada for their financial support. The authors thank one of the reviewers for helpful suggestions concerning terminology. Registry No. CTAB, 57-09-0; 2,6-di-tert-butyl-4-methoxyphenol, 128-37-0;2-terr-butyl-4-methoxyphenol,121-00-6; 3-tert-butyl-4-methoxyphenol, 88-32-4.
Supplementary Material Available: Tables of ultrasonic velocity, density, apparent molar volume, apparent molar compressibility, and adiabatic compressibility coefficient data for CTAB with water, BHT, and BHA at 25,35, and 45 OC (8 pages). Ordering information is given on any current masthead page.
Influence of Surface Structure on the Kinetics of Diffusion-Controlled Reactive Processes on Molecular Organizates and Colloidal Catalysts Philip A. Politowicz, Research School of Chemistry, Australian National University, Canberra, Australia ACT 2601
Roberto A. Garza-Lbpez, David E. Hurtubise,+and John J. Kozak* Department of Chemistry, Franklin College of Arts and Sciences, University of Georgia, Athens, Georgia 30602, and Radiation Laboratory and Department of Chemistry,! University of Notre Dame, Notre Dame, Indiana 46556 (Received: September 6 , 1988)
At the molecular level, the surface of a colloidal catalyst particle or molecular organizate (such as a cell or vesicle) 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 clusters or domains. In this paper, we develop a model to study the influence of such structure on the efficiency of encounter-controlled,surface-mediated reactive processes. We focus on the surface structures defined by a number of classic figures, the five Platonic solids and (here) 16 Archimedean solids; in each case, the attendant polyhedral surface is characterized by dimension d = 2 and Euler characteristic x = 2, with N distinct locations (sites) on the surface organized into an array defined locally by the site valency v (or connectivity)of the consequent network. Given this structure, we consider a target molecule A anchored to the surface at one of the N sites and a coreactant B free to migrate among the N - 1 satellite sites and study the dynamics of the diffusion-controlled irreversible reaction A + B C by formulating a stochastic master equation for each surface geometry considered and solving this equation numerically for two classes of initial conditions. Specifically, we determine the survival probability p ( t ) versus time t of the diffusing coreactant B and calculate the first four moments of the underlying probability distribution function. From the consequent evolution curves we extract the relaxation times to, and from these and the associated moments we are able to disentangle the separate influences of the variables N a n d v on the kinetics. In all cases, we find that, for fixed N, the time io decreases with increase in the (local or global) valency u. Secondly, for a given local symmetry (v fixed), we find that to increases with N in (almost) all cases; the single exception occurs when the number of domains of triangular symmetry begins to dominate the overall surface structure.
-
I. Introduction In many diffusion-controlled reactive processes in which the reactants are confined to the surface of a particle, the species are not free to diffuse freely over the surface. For example, in cellular systems, although it is known that lateral diffusion over the surface of the encompassing medium may be quite free, the diffusing species must nonetheless negotiate transmembrane (and other) 'To whom correspondence should be addressed at the University of Georgia. 'Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556. $This work was initiated at the University of Notre Dame, Notre Dame, Indiana. This is Document No. NDRL-3122 from the Notre Dame Radiation Laboratory.
0022-365418912093-3728$01.50/0
proteins whose presence bifurcates the surface into domains connected by channels. In colloidally dispersed catalytic systems, randomly dispersed "islands" of catalytic activity on the surface of a particle are believed to govern the turnover and overall kinetic response of the system. These and many other examples led two of us to investigate in a previous work' the consequences of developing a model wherein the presence of individual sites connected by "pathways" or "channels" to a reaction center (the consequence of having a surface broken up into domains or differentiated into clusters) was built into the formulation of the problem from the very outset. ( 1 ) Politowicz, P. A.; Kozak, J. J. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 8175.
0 1989 American Chemical Society
Molecular Organizates and Colloidal Catalysts That is, rather than assuming that all sites are accessible to a diffusing coreactant so that a rotational diffusion equation can be written down
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3729 variables (N,u) in influencing the dynamics of d = 2 surfacemediated processes. We also consider a generalization of the above program wherein all satellite sites surrounding the target molecule are assumed to have affixed molecules that can, with fractional probability s (0 5 s < l ) , undergo a quasi-reversible reaction, A B s [AB]* C, with the diffusing coreactant. Analytic expressions are obtained for the dependence of the first moment of the distribution function (the mean walklength (n)) on the probability s, from which one can extract the dependence of the relaxation time to on s. Some implications of our numerical and analytical results in interpreting experimental data on the kinetics of surface-mediated, reactive processes on metal clusters, colloidally mediated photochemical water cleavage systems, and morphogenetically developing cells are presented in the concluding section.
+
(where 8 is an angle defined with respect to an underlying spherical polar coordinate system and D is an associated rotational diffusion coefficient) and solved subject to a variety of initial conditions, C(8,t=O) and C(8=8o,t), a procedure followed by Bloomfield and PragerZin their study of diffusion-controlled reactions on spherical surfaces, we developed a lattice model whose elaboration allowed us to explore in a preliminary way some consequences of relaxing the free diffusion approximation. In the approach taken in ref 1, we considered a regular lattice structure wrapped on the surface of a particle homeomorphic to a sphere. In particular, we considered the connectivity and attendant domain structure defined by the five Platonic solids. If we denote by d the dimensionality of the surface of the particle, by N the number of sites on the surface, and by u the number of channels (or “reaction pathways”) accessible to a diffusing coreactant at each site on the surface, then the surface topology of the five cases considered in ref 1 can be specified by the triple {d,N,u);thus, the tetrahedron would be (2,4,3),the octahedron (2,6,4),the hexahedron (2,8,3],the icosahedron (2,12,5),and the dodecahedron (2,20,3).The specific problem solved in ref 1 was to compute, using the theory of finite Markov processes, the average number of displacements (the average walklength ( n ) ) required before a particle diffusing on the surface reacted with a target molecule, the latter anchored at a fixed site on the surface. (Since all sites N on the surface of a given Platonic solid are characterized by the same valency u and local symmetry (the defining angles and bond lengths), any site can be specified as the reaction center without loss of generality.) It was found that the mean walklength ( n ) increased with increase in the number of site surrounding the target molecule. Thus, if one assigns a common jump time T for the site-to-site transitions on the underlying lattice, the observed increase in ( n ) with increase in N suggests that the mean reaction time also increases with increase in the number of surface lattice sites. Whereas this trend seems intuitively reasonable, its generality is open to question for reasons we shall now describe. In calculating ( n ) for the five Platonic solids as a function of increasing N , it is to be noticed that the valency u in the triple (d,N,u)also changes (irregularly) with increase in N (from u = 3 to 4 to 3 to 5 tb 3 for the five members of the series). Now, it is known from the theory of stochastic p r o c e s s e ~ ~(see - ~ section 11) that the relation between the mean walklength ( n ) and the zero-mode relaxation time of a stochastic system is mediated by the valency (or number of reaction channels) of the underlying reaction network. Thus, there exists the possibility that the dynamics of the diffusion-controlled reaction process, as calibrated by a characteristic relaxation time and determined by solving explicitly the stochastic master equation for the problem, may not of necessity be characterized by the same regularity as that which is inferred by studying the N dependence of ( n ) . To deal with this concern, we generalize the study presented in ref 1 in two directions. First, we determine (numerically) the exact dynamics of the reaction-diffusion model described above and study changes in reactivity as a function of (d=2,N,u)= (N,u). Second, we broaden the class of surface domain structures considered by studying the efficiency of diffusion-controlled reactive processes on the surfaces of the so-called Archimedean solids, of which 16 examples are considered in this paper. This allows us to study in a much more comprehensive way the interplay of the (2) Bloomfield, V. A,; Prager, S . Biophys. J . 1979, 27, 447. (3) Montroll, E. W.; Shuler, K. E. Adu. Chem. Phys. 1958, 1 , 361. (4) N,xolis, G.; Prigogine, I. SelJOrganizafion in Nonequilibrium Sysiems; Wiley: New York, 1977. (5) Haken, H. Synergelics; Springer-Verlag: Heidelberg, West Germany,
1977.
-
11. Surface Dynamics The time dependence of diffusion-controlled reactive processes can be studied by formulating a stochastic master equation
for the specific geometry characterizing the reaction space of the system. Here, pi(?) is the probability of realizing a particular state i and GI, is an N X N matrix that describes the transition probabilities between the states of the system. As is shown in ref 3-5, the general solution of the above system of linear equations is of the form N
pi([)
=
m=l
aj,e-Xmt
(3)
where the a,, are coefficients determined by the initial conditions and the A, are the eigenvalues of the G matrix above. The eigenvalue spectrum {A,) has been determined for each of the Platonic structures and for two of the Archimedean solids, the rhombic dodecahedron, and the truncated octahedron. Complementary to this program of calculation, we have also determined the first four moments of the probability distribution function describing the diffusion-controlled reactive processes studied here; data on the above systems were presented in ref 1, and data on 14 additional Archimedean structures are presented in section 111. One can prove that, for N large, the reciprocal of the smallest eigenvalue of the G matrix (the zero-mode relaxation time) is related to the first moment (the mean walklength ( n ) ) of the distribution function; this relationship will be exploited in the ensuing discussion. Two classes of initial conditions were considered in this study, and these will now be described. Consider first a coreactant diffusing in an ambient environment (e.g., a solution) and colliding with a colloidal particle (or cellular assembly) at some site k on its surface. From that moment ( t = 0) on, the coreactant is assumed to diffuse randomly on the surface (only) from site to site until it reacts irreversibly with a target molecule anchored at one site. The number of discrete steps in this trajectory will be some number, say k , . Note, however, that the trajectory followed by the diffusing coreactant starting from that initial site k is not unique; a variety of possible paths on the surface will be accessible to the coreactant, and each will be characterized by a walklength k,. The most probable or average walklength will be the statistical average of all possible paths from site k to the reaction center, and this number will be denoted ( n ) k . Of course, there is nothing sacred about the particular site k; the coreactant may collide with the surface at any of the N - 1 satellite sites defining the polyhedral assembly. The corresponding, overall average walklength characterizing all possible flows from all possible satellite sites will be denoted (n). Relative to the system of eq 2, the first class of initial conditions is specified by assigning p k ( t = O ) = 1 with pi(t=O) = 0 for all i # k . The second class of conditions is characterized by assigning pk(t=O) = ( N - l)-I for all k (excluding the trap or target site). In more descriptive language, the first class of initial conditions
Politowicz et al.
3730 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 1.07
TABLE I: Eigenvalue Spectrum for Specific Polyhedral Geometries
figure tetrahedron
0.8
N 4
i i ”
u 3
0
4
8
12
16
6
4
[p(O) =
1
0.763932 4.00000 5.23607 6.00000
3
1 2 3 4 5
1 2 1 2 1
0.354249 2.00000 3.00000 4.00000 5.64575
icosahedron
I2
5
1 2 3 4 5 6
1
2 1 4 1 2
0.388565 2.76393 3.803 11 6.00000 6.81082 7.23607
dodecahedron
20
3
1 2 3 4 5 6 7 8 9 10
( N - 1)-’].
I .o
3 4 5 6 7 8
Y t
Q
>+ - 0.6 i m
0.4
truncated octahedron
0
a: Q
0.2
0.0
0
4
8
I2
16
20
TIME t Figure 2. Conventions here are the same as in Figure 1 except that we assume the diffusing coreactant initiates its motion at the site on the surface farthest removed from the reaction center (Le., p ( 0 ) = 1 at that site).
corresponds to evolution from a “pure state” while the second class relates to evolution from a “manifold of states”. Displayed in Figures 1 and 2 are the survival probabilities p ( t ) of a coreactant diffusing on a polyhedral surface with a single center (target molecule) anchored at a site of valency v = 3 (the N = 4 tetrahedron, the N = 8 hexahedron, the N = 20 dodecahedron, the N = 24 truncated octahedron), v = 4 (the N = 6 octahedron), or v = 5 (the N = 12 icosahedron). The two curves for the case N = 14 refer to a rhombic dodecahedron; this polyhedron has eight sites of valency u = 3 and six sites of valency v = 4 and hence an average (or “fractal”) valency of i~= 3.4286. The two curves for N = 14 correspond to having the reaction center positioned a t a site of valency v = 3 or v = 4. In the language of the preceding paragraph, the profiles of Figure 2 correspond to evolution from a “pure state”, specifically from trajectories of the coreactant initiated from the site on the polyhedral surface farthest removed from the reaction center, while those in Figure 1 correspond to evolution from a “manifold of states”, i.e., from trajectories initiated from any (and all) possible satellite sites of the system.
24 3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1
4 1
3 1 3 1 2
1 1.92583 3 3.00000 2 4.00000 1 5.07417 2 5.56155 1 6.81994 1 2 1 1 1 2 1 2 1 2 1 2 1 1 1 2 1
0.100647 0.763932 1.11238 2.00000 2.53758 3.00000 4.10625 5.00000 5.14314 5.23607
1 2
figure N Y i j X,C rhombic 14 3.4286b 1 1 0.180060 dodecahedron 2 2 1.43845
0.8
2
I 2 1
8
20
Figure 1. Survival probability p ( t ) of a coreactant diffusing on a polyhedral surface with a single reaction center (target molecule) anchored at a site of valency Y = 3 (the N = 4 tetrahedron, N = 8 hexahedron, N = 20 dodecahedron, N = 24 truncated octahedron), u = 4 (the N = 6 octahedron), or Y = 5 (the N = 12 icosahedron). The two curves for the case N = 14 refer to a rhombic dodecahedron ( 3 = 3.4286) with a reaction center positioned at a site of valency u = 4 ( 0 )or u = 3 (0). As initial conditions, we assume an equal a priori probability of the diffusing coreactant being at any of the N - 1 satellite sites at the time
=0
1 2 3 4
hexahedron
TIME t
f
1.00000 4.00000
2
octahedron
A,
1 2
1
i j 1 2 2 3 1 4 4 5 1 6 1 7 2 8 1 1
0.225533 1.43845 2.16663 3.00000 4.00000 4.83337 5.56155 6.77447
0.0760907 0.585786 0.848668 1.26795 1.49030 2.00000 2.26288 2.58579 3.00000 3.41421 3.73712 4.00000 4.50970 4.73205 5.15133 5.41421 5.92391
“Degeneracy of the eigenvalue Xi. ”he rhombic dodecahedron has eight sites of valency u = 3 and six sites of valency u = 4. CTrappositioned at a site of valency u = 3. dTrap positioned at a site of valency 1
= 4.
The eigenvalue spectrum for each of the polyhedral geometries considered in Figures 1 and 2 is given in Table I, and in Table I1 we show the correspondence, noted above, between the smallest eigenvalue XI of the stochastic master equation (2) and the average walklength (n). In the limit of large N, the formal relationship is
( n ) = vX1-l
(4)
so that the correspondence involves the valency v of the surface considered. The extent to which the single eigenvalue A, dominates the decay can be gauged by fitting the profiles displayed in Figure
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3731
Molecular Organizates and Colloidal Catalysts TABLE 11: Correspondence between ( n ) , vX1-', and v&-' figure N U (n) 4 3 3.0000 tetrahedron octahedron 6 4 5.200 hexahedron 8 3 8.2857 icosahedron 12 5 12.7273 14 3.4286" 14.6154b rhombic dodecahedron
dodecahedron truncated octahedron
20 24
18,7692' 28.8417 37.9752
3 3
A,
lA-1
%d
1.000 00 0.763 93 0.35425 0.388 57 0.225 53 0.18006 0.10065 0.076091
3.00000 5.2361 8.4686 12.8679 15.2022 19.04 14 29.8071 39.4266
0.00 0.69 2.21 1.10 4.01 1.45 3.35 3.82
x
uX,-l
%'
1.003 91 0.76830 0.35372 0.391 33 0.225 5 5
2.9883 5.2063 8.4813 12.7770 15.2009
-0.39 0.12 2.36 0.39 4.01
0.101 24 0.076 907
29.6340 39.0079
2.74 2.72
OThe rhombic dodecahedron has eight sites of valency u = 3 and six sites of valency u = 4. bTrap positioned at a site of valency v = 4. CTrap positioned at a site of valency u = 3. d % = [ ( u X - l - ( n ) ) / ( n ) ]X 100. '% = [(ux-' - ( n ) ) / ( n ) ]X 100. 1 to a two-term polynomial of the form In p ( t ) = In si - Xt and determining i;via a least-squares procedure. The encompasses information on the set {A,) for each polyhedron (see Table I) and the close correspondence between A, and (Table 11) clearly shows the dominance of the eigenvalue AI in driving the decay of the system of eq 2. It is evident from the structure of the formal solution (3) to the system (2) that A, may be interpreted as an effective first-order rate constant (or A1-I as a characteristic relaxation time) of the system. From these calculations, two general features emerge: ( I ) the correspondence (eq 4) between the random walk characteristic ( n ) and the dynamical parameter AI-' is quite close, even for the smallest systems, and (2) the decays on all surfaces studied are effectively dominated by the single eigenvalue AI. An examination of the results displayed in Tables I and I1 and Figures 1 and 2 reveals several interesting features. As is evident from the data on ( n ) for the Platonic solids, the mean walklength increases with increase in the number N of sites on the polyhedral surface. Since the bonds (and angles) defining the surface of a given Platonic solid are all the same, if one assumes a common bond length (or metric) for all the Platonic solids and a common jump time T between adjacent sites on the surface, the trend in the data on ( n ) versus N seems perfectly sensible. It is in examining the full dynamics of the decay (see Figures 1 and 2) that one realizes there is more to the story. While the evolution curves displayed in Figures 1 and 2 indeed show a general slowing of the decay with increase in N , there is an obvious inversion of order in the curves describing the evolution on the hexahedral ( N = 8) versus the octahedral ( N = 12) surface. The factor which has been overlooked in understanding the ordering of the decay curves is the valency v, Le., the number of channels at each site available to the diffusing coreactant in its migration across the surface. It was this factor that was necessary to make precise the relationship between ( n ) and Al-l (see eq 4), and it will be a principal objective in the remaining discussion of this paper to quantify precisely how the valency influences the overall decay process. First of all, it is clear that, for afixed valency v, the ordering of the decay profiles displayed in Figures 1 and 2, specified by A1-I as a function of N , is in a one-to-one correspondence with the order of increase of ( n ) with respect to N, for the tetrahedron (N,v) = (4,3), the hexahedron (8,3), and the dodecahedron (20,3), polyhedra characterized by a common valency v = 3, both measures of the efficiency of the underlying diffusion-controlled reactive process are entirely consistent. The data for the truncated octahedron (with N = 24 and u = 3) are also consistent with the trends noted above for the three Platonic solids [(N,v) = (4,3), (8,3), and (20,3)] although a qualification enters here. The angles of the truncated octahedron (N,u) = (24,3) (second member of the second row in Figure 3) are not all the same; hence, if the results on this polyhedral (Archimedean) surface are to be compared with those for the Platonic surfaces, one must assume that the nearest-neighbor jump times T are the same irrespective of this difference (see later text). A second indicator that the channel structure of the surface is a critical determinant of the dynamics emerges upon examination of the data on the rhombic dodecahedron. This figure has eight sites of valency Y = 3 and six sites of valency u = 4. As is seen from the data in Figures 1 and 2 and Tables I and 11, switching the target molecule from a site of valency u = 3 to a
x
\
.3
.3
v r 4
v .5
v .3
v =4
v =5
Y
N.30
N.60
v .3
Figure 3. Diagrams of 11 Archimedean solids considered in this paper. The Platonic figure ( N p ) = ( 1 2 3 , the icosahedron, is also included.
site of valency v = 4 decreases the average walklength ( n ) (from ( n ) = 18.8 to 14.6) and decreases the time scale over which the diffusing coreactant may be expected to react with the (fixed) target molecule (for both sets of initial conditions). In order to set the stage for the study reported in the following section, it is useful to summarize the results reported above in a more symbolic way. The three characteristics which may be used to characterize and distinguish the surface structures of the Platonic and Archimedean solids considered here are the dimensionality d (which for all systems studied in this paper is d = 2), the number N of surface sites, and the site valency v ; we shall systematically suppress in this work possible differences in the bond lengths {li]and bond angles {4i)(see, however, later text). The trends uncovered for the Platonic solids N = 4, 8, and 20 indicate that holding d and v constant in the triple {d,N,vJwhile increasing N leads to stochastic and dynamic system responses that are both self-consistent and intuitively reasonable. Secondly, the study of the N = 8 hexahedron and the N = 12 icosahedron indicates that, for fixed dimensionality d , the interplay between the variables N a n d v in the triplet {d,N,v)may lead to results that are, at first site, counterintuitive. And, thirdly, the data on the rhombic dodecahedron, where both d and N are held constant, suggest that the variable v in the triple (d,N,v]is responsible for the differences observed in calculations on the N = 8 and N = 12 Platonic surfaces. With this emerging pattern before us, we shall explore systematically (in section 111) the influence of channel structure (as coded by the valency v) on the underlying stochastic process.
3732
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
TABLE 111: Analytic Dependence of ( n ) i on (s,NJ N normalization 4 n4=1+2s
(n)i
(n), (n)
6
n6=1+4s-s2
8
n8 = 2 i14s - 7s2
12
nI2 = 2
+ 24s + s2 - 2s'
(n), (n), (n), (n)
20
nzo = 2
Politowicz et al.
+ 60s + 67s2 - 50s' + 2ss
(n), (n), (n)4
(n)5 (n)
S
f
s=o
O
N- 1 N- 1
3n4-' 3n4-'
(22 + 6s - 3s2)nI2-l (28 - s - 2s2)ni2 5(6 - s)n12 5/11(56 4s - 5s2)n12-I
+
+ + + +
(38 + 82s - 33s2 - 8s' 2s4)nzo-I (54 + 87s - 63s2 + 3s4)nzo-' (64 68s - 57s2 5s3 s4)n2(I (68 + 55s - 42s2 - 2s' 2s4)n2(' (70 + 47s - 30s2 - 10s3 + 4s4)n2c' 1/19(1096 1388s - 975s2 - 10s' 4 0 ~ ~ ) n , ~ - '
+
+
+
N- 1 ' / d N - 2) N + 12 N + 14 N + 15 '/19(19N 168)
+
TABLE IV: Characteristics of Archimedean Solids
12
3 4 5
24
34 3 4 5
30 36
4
B
3
A
48 60
3 3" 3 4 5
108
3
D C
E B A 120
3
16.418 13.455 12.727 45.130 37.975 32.723 31.091 44.690 89.208 85.258 95.21 1 156.20 116.65 103.53 98.466 474.10 467.89 459.37 456.27 433.01 306.92
1.0147 0.9832 0.9762 1.0385 1.0221 1.0105 1.0072 1.0165 1.0608 1.0569 1.0307 1.0397 1.0256 1.0220 1.0205 1.0701 1.0692 1.0726 1.0669 1.0671 1.0304
2.0241 2.0105 2.0089 2.0227 2.01 57 2.01 12 2.0101 2.0114 2.0343 2.0345 2.0147 2.0172 2.01 15 2.0101 2.0096 2.0354 2.0354 2.0381 2.035 1 2.0374 2.0115
6.1049 6.0446 6.0376 6.0995 6.068 1 6.0482 6.0435 6.0492 6.1521 6.1542 6.0638 6.0749 6.0498 6.0436 6.0410 6.1567 6.1573 6.1694 6.1564 6.1676 6.0497
5.47 3.36 2.55 15.0 12.7 8.18 6.22 11.2 29.7 28.4 31.7 52.1 38.9 25.9 19.7 158 156 153 152 144 102
"Left-hand member in row in Figure 3. Before proceeding, however, we note that, in all the calculations reported in Tables I and I1 and Figures 1 and 2, it is assumed that a single target molecule is anchored at one site on the surface. One can, of course, relax this assumption. In a concentration regime high enough such that a target molecule may be assumed to occupy all available sites of the assembly, the efficiency of reaction of a coreactant that strikes the surface (at t = 0) and proceeds to migrate across the surface will obviously be enhanced. To calibrate these differences relative to the situation considered previously in this paper, suppose that we maintain the original target site as a "deep trap", Le., a site at which the diffusing species reacts with unit probability upon reaching the target molecule. Suppose, however, that at each of the satellite sites the diffusing species can also react with probability s (0 I s < 1); Le., the reaction at the satellite sites is quasi-reversible. In our earlier study, we displayed graphically the behavior of ( n ) versus s for each of the Platonic solids (see Fig. 3 in ref 1). We have since developed analytic expressions for the dependence of the individual ( n ) k (the mean walklength from a site which is a kth nearest neighbor to the deep trap site) and the overall ( n )on the variables { N , s ] ,and these are displayed in Table 111. These expressions gauge the consequences of satellite site participation on the overall diffusion-controlled reactive process and, by virtue of the relationship 4, also scale the effective first-order rate constants
characterizing the decay from each of the nearest-neighbor sites k. 111. Influence of Surface Structure on Reaction Dynamics
In this section we examine further how changes in the reaction efficiency can be induced by changes in the surface structure of the molecular organizate or colloidal particle. To provide some systemization to this study, we focus on that class of structures referred to as the Archimedean solids. The one assumption made in the present discussion (an assumption which may be relaxed easily in subsequent work) is that the jump time T between adjacent sites is the same regardless of differences in the length of the channel (or bond length) separating the sites or the angular orientation of a local group of sites. Apart from this qualification, however, we believe the results reported here are quite general. Consider first the 11 Archimedean solids displayed in Figure 3 (along with the icosahedron). Recorded in Table IV are the first four moments of. the probability distribution function characterizing the overall decay, calculated assuming that class of initial conditions wherein the diffusing coreactant can initiate its random motion on the surface from any site with equal a priori probability. For purely exponential decay, the second moment , third moment (the skewness (the relative width ( u 2 / ( n ) ) ' l Z )the rl),and the fourth moment (the kurtosis y2) must have the values
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
Molecular Organizates and Colloidal Catalysts v.3
t
r
i
a
nhexagon y n k
g
i
3733
e
hexagon
N
48
N
120
Figure 4. Two Archimedean solids characterized by the valency
Y
=3
1 , 2, and 6 , respectively. From the data recorded in Table IV, we can conclude that the evolution curves are essentially exponential. Given the interplay between the valency v, the first moment of the distribution function (the mean walklength ( n ) ) , and the zero-mode relaxation time (the reciprocal of the smallest eigenvalue of the C matrix, A-I) demonstrated in the previous section, we use the calculated mean walklength and the valency to reconstruct the characteristic relaxation time to. For the structures considered in Table 11, this assignment is in error by (at most) a few percent; hence, we anticipate only slight differences between toand the relaxation time determined via solution of the stochastic master equation ( 2 ) . Upon correlating changes in the characteristics toand ( n ) with changes in the surface structure, the following trends emerge. Considering first the series (N,v) = (1 2,3), ( 1 2,4), and ( 1 2,5), it is seen that both the walklength ( n ) and the characteristic relaxation time to decrease with increase in v. This trend is also clearly displayed in the series ( N , v ) = (24,3), (24,4), and (24,5) and in the series ( N p ) = (60,3),(60,4), and (60,5). Note that in the latter two series there are actually two structures consistent with the settings (N,v) = (24,3) and (N,v) = (60,3) but that, despite the differences in ( n ) (and t o ) observed between the members of a pair considered at fixed N (a point we shall return to later), both structures induce dynamics that are consistent with the trends noted above. The second major trend observed is that, for a fixed setting of v, both the average walklength ( n ) and the relaxation time increase with increase in N . This trend is clearly seen in examining the data on the three figures for which v = 5, the four figures for which v = 4 , and the seven structures displayed in Figures 3 and 4 for the case u = 3 . Having established these broad trends, we now examine the data for the pair of figures ( N , v ) = (24,3) and the pair (N,v) = (60,3). Differences in ( n ) and to are seen in each case, and the feature common to the ordering observed is that the figure with triangular regions on its surface produces values of ( n ) and to that are systematically larger than those for which no triangular regions are present. Since the presence of triangular domains (or clusters of sites) seems to drive the ( n ) to higher values (perhaps because the diffusing particle gets trapped in a localized corner of reaction space), one can imagine a situation wherein the presence of many triangular domains might drive these ( n ) values sufficiently high that the dominant role of the variable N in the series ( N p ) = ( N , 3 ) ,say, might be compromised. To explore whether for fixed u a compression or inversion in ( (n),to)values with increase in N can ever be realized, we have constructed three generically related figures wherein the number of triangular domains is systematically increased. The first is the figure ( N p ) = (12,3) displayed in Figure 3 and in facial projection in Figure 5 . This figure is usually referred to as the singly truncated tetrahedron. The doubly truncated tetrahedron ( N J ) = (60,3) and the triply truncated tetrahedron ( N , v ) = (108,3) are shown in facial projection in Figures 6 and 7, respectively. What one finds in examining the data laid down in Table IV is that whereas the trend of increasing { ( n ) , t owith ) increasing N is preserved in these three figures, the values of ( ( n ) , t o )for N = 108 are actually larger than those calculated for ( N , v ) = (120,3),the figure displayed in Figure 3. What this shows is that if the number N of the surface sites of two figures is roughly comparable (in the present example, the difference is IO%), a n inversion in the broad trend noted above can occur if the percent of triangular domains relative to all other
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triangle
triangle
Figure 5. Facial projection of the Archimedean solid (N,u) = (12,3), the truncated tetrahedron (see Figure 3). All possible locations of the single reaction center (A) are topologically equivalent. dodecagon
h
e
x
triangle1
a
g
G
n
;
r
;
l
e
dodecagon
hexagon
hexagon
Figure 6. Facial projection of the Archimedean solid ( N , v ) = (36,3), the doubly truncated tetrahedron. The two symmetry-distinct locations accessible to a (fixed) reaction center (target molecule) are designated as positions A and B. The other polyhedral faces adjoin the projection displayed at the “starred” positions; their specification is similar to that displayed for the upper dodecahedral face.
L tetracosagon face (24-gon)
Figure 7. Facial projection of the Archimedean solid ( N , Y )= (108,3), the triply truncated tetrahedron. The five symmetry-distinct locations accessible to a (fixed) reaction center (target molecule) are designated positions A, B, C, D, and E. Triangular faces adjoin the projection displayed at the “starred” positions, as indicated in the upper portion of the diagram.
surface domain structures increases. As is evident from Figures 5-7, this percentage increases from 50% to 60% to 64% as one
3734 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 considers sequentially the singly, doubly, and triply truncated tetrahedron. A second insight which follows from the data generated by using the configurations displayed in Figures 6 and 7 is that, even for a given {d,N,v),sites of different local symmetry can have { ( n ) , t o ] values that are slightly different. Notice the data recorded for sites A and B for the figure (N,v) = ( 3 6 , ~ )and the data for the sites A, B, C, D, and E for the figure (N,v) = (108,3). These differences are not great, a difference of -4% for the two sites A and B in Figure 6 and a maximum difference of 8.7% for the five distinct sites of Figure 7, but they do suggest that the relative efficiency of reaction on the surface of a molecular “organizate” or colloidal catalyst can be fine-tuned depending on the site to which the target molecule is attached. We shall comment further on this point in the following section.
IV. Discussion We have studied in this paper the influence of surface structure on the kinetics of surface-mediated, diffusion-controlled reaction processes. The motivation for this study springs from the realization that, in studying such processes at the molecular level, the inherent discreteness of the surface does matter and what we have sought to do here is to identify the factors at play in influencing the efficiency of reaction under these circumstances and to quantify their importance. The dynamical study presented in section 11, wherein the stochastic master equation was solved numerically for a variety of geometries (the five Platonic solids, the rhombic dodecahedron, and the truncated octahedron) and for two classes of initial conditions (diffusion initiated from a single site or, with equal a priori probability, from all satellite sites) revealed the necessity of disentangling the influence of the two variables, N a n d v. The further documentation of the moment structure of the underlying probability distribution function for the variety of polyhedral surfaces presented by 14 additional Archimedean solids refined our understanding of this interplay of variables and allowed us to develop the following generalizations. First, for a given N in the triple {d=Z,N,v),increasing the number Y of reaction channels available to a diffusing coreactant at each site results in a decrease in the first moment of the underlying probability distribution function (the mean walklength ( n ) ) and a concomitant decrease in the characteristic relaxation time t o of the system. This relaxation time may be placed in correspondence with the (reciprocal of an) effective (or pseudo) first-order rate constant k characterizing the reaction of a diffusing coreactant with a stationary target molecule (or reaction center), so the relevance to many experimental situations is immediate (see later text). Secondly, for a given v in the triplet {d=2,N,v),increasing the number ( N - 1) of satellite sites available to a diffusing coreactant leads to an increase in ( n ) and an increase in the characteristic relaxation time to of the system. The single exception to this conclusion was generated when we purposefully increased the number of triangular domains relative to domains of higher symmetry so that, for N values differing by -lo%, an inversion in the above trend was elicited. Increasing the number of triangular domains apparently prolongs the occupancy of the diffusing species in “pockets” of the overall reaction space accessible to the coreactant. It is now important to address the question of whether the above generalizations would hold if variations in the bond lengths I, and bond angles q5i were taken into account. Whereas this question can only be settled by direct numerical calculation, we believe the overall behavior with respect to (NJ) would persist for the following reason. All figures considered here are topologically equivalent to a sphere;6 i.e., they are characterized by surfaces of dimension d = 2 and Euler characteristic x = 2 (where x is defined as x = F - E + V, with F, E, and V the number of faces (6). Henle, M. A Combinatorial Introduction to Topology; Freeman: San Francisco, 1979.
Politowicz et al. (cells), edges, and vertices of the polyhedral complex). In fact, severe distortions of these figures, wherein the I, and 4, may become quite deformed, also result in figures that are topologically equivalent to a sphere. So, although we may expect the numerical results to change somewhat as one accounts for variations in {I,,di), to the extent that the overall trends with respect to (N,v) are driven by the topology of the surface as a whole, we believe the qualitative results obtained here will be sustained. There is an interesting way in which topological equivalence has found its way into recent discussions of reactivity on metal clusters, ensembles of atoms whose geometries are thought to mimic the surface structure of some of the Archimedean solids, e.g., the CWLacluster discovered by Smalley et a].’ and christened “buckminsterfullerene”. Castleman and Keesee,*upon reviewing the experimental studies of Morse et al.,9 Parks et a1.,I0 Reents et al.,” Ruatta et al.,I2 and Zakin et aI.,l3 have noted analogies between reactions on clusters and reactions on surfaces (with reference to catalysis, adsorption, and etching). In interpreting these analogies, however, it is important to keep in mind that “planes” are not topologically equivalent to “spheres”; the Euler characteristic for a surface is x = 0, and there is no way it can be deformed (diffeomorphically transformed) into a sphere with x = 2. However, it was demonstrated in ref 1 that in calibrating reaction efficiency there can occur an “accidental” topological degeneracy between planar lattices subject to periodic boundary conditions and certain Platonic solids. Specifically, calculations of the mean walklength ( n ) performed on the three Platonic solids (N,v) = (4,3), (8,3), and (20,3) were found to be entirely consistent with those performed on planar hexagonal lattices (N,v) = (N,3) subject to periodic boundary condition~.’~It is in this sense that comparisons of reactivity on these two topologically distinct types of two-dimensional surfaces can be made precise. The results obtained in this study have many other resonances with earlier theoretical and experimental work in the literature. For example, on the theoretical side, the different character of the decay curve, p ( t ) versus t, near t = 0 found when one assumes evolution from a “pure state” versus a “manifold of states” was noticed by Zwanzig over 30 years ago15and uncovered as well in a study of the exact dynamics of an excited two-level atom in a radiation field.I6 In exactly solved Hamiltonian models, a “shoulder” in the decay curve emerges when evolution is from a pure state (e.g., in ref 16 from a single level of the excited atomic system). The model of diffusion-controlledreactions on spherical surfaces considered by Bloomfield and Prager2 may be regarded as a limiting case of the discrete model studied here; viz., they considered a d = 2 (spherical) surface (Euler characteristic x = 2) for which one has passed (simultaneously) to the limits N m, v m . It is of some theoretical interest to make contact with their work, a problem which will be pursued in a subsequent contribution, but for the moment we wish to stress the relevance of our results to the underlying physical problems that motivated our study. The delicate interplay between the variables N and v uncovered in our study suggests how a monodisperse catalyst, with all particles characterized by a fixed number N of surface sites, can yield measurably different turnover numbers depending only on
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(7) See the discussion in: Chem. Eng. News 1985, 63, 20-22. (8) Castleman, Jr., A. W.; Keesee, R. G. Science 1988, 241, 36. (9) Morse, M. D.; Geusic. M. E.; Heath, R. E.; Smalley, R. E. J . Chem. Phys. 1985, 83, 2293. (10) Parks, E. K.; Liu, K.; Richtsmeier, S. C.; Pobo, L. G.; Riley, S. J . J . Chem. Phys. 1985, 82, 5470. (11) Reents, Jr., W. D.; Mujsoe, A. M.; Bondybey, V. E.; Mandich, M. L. J . Chem. Phys. 1987,86, 5568. (12) Ruatta, S. A,; Hanley, L.; Anderson, S . L. Chem. Phys. Lett. 1987, 137. 5 . ( 1 3 ) Zakin, M. R.; Cox, D. M.; Whetten, R. L.; Trevor, D. L.; Kaldor, A. Chem. Phys. Lett. 1987, 135, 223. (14) Politowicz, P. A,; Kozak, J. J. Phys. Reo. B 1983, 28, 5549. (1 5) Zwanzig, R. W. Statistical Mechanics of Irreversibility. In Lectures in Theoretical Physics (Boulder, 1960); Brittin, W. E., Downs, B. W., Downs, ~~
J., Eds.; Interscience: New York, 1961. (16) Davidson, R.; Kozak, J. J. J . Math. Phys. 1978, 19, 1074.
J . Phys. Chem. 1989, 93, 3735-3740
3735
the channel structure linking the target molecule to other regions of the catalytic surface. This effect was demonstrated both for surfaces characterized by a c m m o n valency [e.g., the topologies ( N , Y )= (24,3), (24,4), ( 2 4 3 and ( N P ) = (60,3), (60,4), (60,5); figures diagrammed in Figure 31 and for surfaces of variable valency (recall the data on the rhombic dodecahedron in Table 11). Such differences will, of course, be washed out in the conm, Y m, which is why we believe that retinuum limit N action-diffusion models based on the formulation and solution of a Fickian equation of the form ( 1)2~’7 are inherently incapable of accounting for such differences in catalytic activity. One can see this lack of discrimination in continuum models in a slightly different way by noting that even on a discretized surface if all sites N become catalytically active then, as may be 1) seen from Fig. 3 of ref 1 or from the limiting behavior (s of the analytic results presented in Table 111, differences in the reaction efficiency induced by different surface geometries essentially vanish. The inverse of this behavior, Le., considering what happens when one passes from the limit s = 1 to values s 0, may be relevant to the understanding of one facet of colloidally mediated, photochemical water cleavage reactions, a problem under study by many experimental groups.’* The variation of reaction order of the fast decay of the methylviologen cation radical MV” with respect to platinum as a function of size and pH was attributed to the fractional coverage of the colloidal catalyst particle by adsorbed species.19 Given relation 4 and the
systematic variation of ( n ) with respect to s (Table HI),the interpretation presented in ref 19 seems entirely consistent with the behavior s 0 in the model considered here, although calculations of the (hi]would be necessary to establish this correspondence quantitatively. Finally, turning to a consideration of molecular “organizates” (cells, vesicles, etc.), the intriguing possibility suggested by our study is the means by which a morphogenetically developing cell may control the efficiency of surface-mediated reactions.20*21For a receptor molecule positioned at a site of common valency Y, the sequence ( N y ) = (12,3), (24,3), (48,3), (60,3), and (120,3) (see Figures 3 and 4) or the series of singly, doubly, or triply truncated tetrahedra (displayed in facial projection in Figures 6-8) show how dramatically the time scale of a given surface reaction can change as the surface bifurcates with growth into domains of varying geometrical structure. Since the domain structure of the surface (and attendant channel patterns) can be induced by the incorporation of transmembrane or membrane-bound proteins, it may be that the distribution of such proteins at particular stages of cell growth is coupled to the control of surface-mediated reactions at that stage of development. At any rate, the (numerically precise) data reported in this study leave little doubt that the kinetic behavior of a surface-mediated reaction induced by the interplay of the variables ( d , N , u }is far more subtle and interesting m) would than studies based on continuum models ( N m, u have led us to suspect.
(17) Ark, R. The Mathematical Theory of Dijfusion and Reaction in Permeable Catalysts; Clarendon Press: Oxford, 1975; Vols. 1 and 2. (18) See list of references in: (a) Gratzel, M. Acc. Chem. Res. 1981, 14, 376. (b) Kiwi, J.; Kalyanasundaram, K.; Gratzel, M. Visible Light Induced Cleavage of Water into Hydrogen and Oxygen in Colloidal Microheterogeneous Systems; Springer-Verlag: Heidelberg, West Germany, 1981; Struct. Bonding (Berlin) 1982, 49, 37 (c) Fendler, J. H. J . Phys. Chem. 1985, 89,
(19) Sassoon, R. E.; Lenoir, P. M.; Kozak, J. J. J . Phys. Chem. 1986, 90, 4654. Lenoir, P. M.; Sassoon, R. E.; Kozak, J. J. J . Phys. Chem. 1988, 92, 2526. (20) A summary and discussion of the recent literature on lateral diffusion rates and surface-mediated reactive processes on biological membranes is given in: McCloskey, M. A,; Poo, M. J . Cell Biol. 1986, 102, 88. (21) Sackmann, E. In Biophysics; Hoppe, W., Lohmann, W., Marki, H., Zuegler, H., Eds.; Springer-Verlag: Heidelberg, West Germany, 1983; pp 425-451.
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2730.
“
Ideal” Behavior of the Open Circuit Voltage of Semiconductor/Liquid Junctions Mary L. Rosenbluth and Nathan S. Lewis* Division of C h e m i s t r y and Chemical Engineering,‘ California I n s t i t u t e of Technology, Pasadena, California 91 125 (Receiued: S e p t e m b e r 12, 1988)
Using simple bimolecular kinetic expressions for interfacial charge-transfer rates at semiconductor/liquidjunctions, we present an analysis of the dependence of the open circuit voltage, V,, on the solution redox potential E(At/A). It is found that if E ( A + / A )is varied by using a series of redox couples with identical heterogeneous rate constants and the concentrations [A] and [A’] are held constant, V, should change as E ( A + / A )is varied. However, if E ( A + / A )is manipulated for a given redox system by changing [A’] and holding [A] constant, “ideal” behavior predicts no change in V, at different solution redox potentials. These considerationsare of importance both in diagnosing the presence of Fermi level pinning at semiconductor electrodes and in the design of photoelectrochemical cells for use in solar energy conversion.
Introduction One of the key parameters of a semiconductor/liquid junction is the open circuit voltage ( V , ) . This parameter defines the maximum Gibbs free energy that can be obtained from a given semiconductor/liquid junction cell under constant temperature and light intensity conditions. The response of V, to changes in solution redox potential, E(At/A), is a key factor in diagnosing whether the semiconductor/liquid junction is “ideal”’ or exhibits efficiency limitations resulting from Fermi level pinning, carrier inversion, et^.^,^ Consequently, there have been numerous studies of the behavior of V ,vs E(At/A) for aqueous and nonaqueous *Author to whom correspondence should be addressed. ‘Contribution No. 7834.
0022-3654/89/2093-3735$01 .50/0
based semiconductor/liquid systems.4-’’ In practice, however, there are two different methods for changing the solution redox (1) (a) Gerischer, H. In Physical Chemistry, An Advanced Treatise; Eyring, H., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1973; Vol. IXA, p 463. (b) Gerischer, H. Ado. Electrochem. Electrochem. Eng. 1961, 1 , 139. (c) Morrison, S. R. Electrochemistry ut Semicanductor and Oxidized Metal Electrodes; Plenum Press, New York, 1980. (d) Memming, R. Elecrroanal. Chem. 1979, Z l , 1. (e) Myamlin, V. A.; Pleskov, Yu. V. The Electrochemistry of Semiconductors; Plenum Press, New York, 1967. (f) Pleskov, Yu. V.; Gurevich, Yu. Ya. Semiconductor Electrochemistry; Consultants Bureau: New York, 1986. (2) (a) Bard, A. J.; Bwarsly, A. B.; Fan, F. R. F.; Walton, E. G.; Wrighton, M. S. J . A m . Chem. SOC.1980, 102, 3671. (b) Bard, A. J.; Fan, F. R. F.; Gioda, A. S.;Nagasabrumanian, G.;White, H . S . Faraday Discuss. Chem. SOC.1980, 70, 19.
0 1989 American Chemical Society
