7278 The Journal of Physical Chemistry, Vol. 94, No. 18, I990 7.0
4.0
r I
1
W
I
02
t
1
0.4
0.6
0.8
1
1.o
6 Figure 3. Plot of the statistical entropy S (per unit volume, in units of Boltzmann’s constant) as a function of the statistical order parameter 6 for two (representative) concentrations C,. The upper curve displays results for nine configurations characterized by CT = 6/256 while for the (different) set of nine configurations noted on the lower curve, C, = 16
1256
Going beyond these qualitative statements, it is of interest to determine whether changes in concentration CTare relatively more (or less) important than changes in the statistical order parameter 6 in influencing the magnitude of the statistical entropy. An answer to this question can be gleaned from the profiles displayed in Figure 3. By inspection, we see that the distributions I 2 0 and ABDE are characterized by (essentially) the same value of the coefficient 6, viz. 0.5 13 and 0.5 15, respectively. However, the concentration CTof reaction centers for the distribution ABDE is a factor of 2.7 greater than for 120; this leads to a reduction of 21% in the statistical entropy on going from the configuration 120 to ABDE (or a factor of 3.4 in the value of (n)). Alternatively, one can fix the concentration CTand consider changes in the statistical entropy resulting from a similar (quantitative) change in the coefficient 6. For the common setting CT= 0.0625 [lower curve on Figure 31, we note that the statistical order parameter 6 changes by a factor of 2.6 on going from the configuration B4V to BFG; this is accompanied by a decrease of 18% in the statistical entropy (or a factor of 2.6 in the value of (n)). The percentage change in the statistical entropy is similar enough in the two cases (21% versus 18%) that one can assert (to a fair approximation) that corresponding quantitative changes in the control parameters (CT, 6) produce similar changes in the entropy of the underlying diffusion-controlled reactive process. This last conclusion is of interest in light of experimental studies on heterogeneous catalytic systems that exhibit ‘aging”, Le., changes in the net turnover of reactant to product with time. This aging may be a consequence of catalyst poisoning, i.e., a change in the effective concentration CT of active sites. However, as illustrated above, similar (quantitative) changes in the reaction efficiency can also be realized by a redistribution of reaction centers (accompanied by a concomitant change in the statistical order parameter 6) with time. That is, on some time scale, an initial, semirandom distribution of active sites may, depending on the nature of adsorbate/substrate interactions, either sequester into smaller clusters (in which case, df < Si) or, alternatively, become more randomized (6, > q). In either case, one will move along the curve of S versus 6, with the different S values reflecting a diffusion-controlled reactive process governed by different system geometries (and hence characterized by different turnover numbers and effective (or pseudo-first-order) rate constants). It is our expectation that this perspective, now quantified (Tables 1-IV and the representative cases displayed in Figure 3), can cast light on the interpretation of experiments on the catalytic properties of bimetallic systems (e.g., the studies of Sinfelt et aLI3on Ni/Cu alloys) and on catalyst dea~tivation.’~Moreover, it is known that surface recognition events in cellular systems may be regulated by changes in the sequestering of receptors: processes at synapsesIs and those involving cell-adhesion molecules CAMS)'^ come immediately to mind. These experiments are now open to (quantitative) analysis given the results presented in this paper. We now consider some consequences of the fact that highly
Politowicz and Kozak configured distributions of reaction centers can often be decomposed into a number of disjoint (geometrically uncoupled) subsystems. The question is: If, for each of the subsystems comprising the overall assembly, one calculates the statistical entropy associated with an irreversible (or quasi-reversible) process taking place on the reaction space defined by each subsystem, is the entropy S for the composite system a simple additive function of the entropies (Si,S,, ...) calculated for the individual subsystems? That is, is A S = S - (Si+ Sj + ...) = 0, thereby implying that diffusion-controlled reactive processes taking place on highly configured systems can be thought of as comprising statistically-independent events? The unequivocal conclusion drawn from our data (see Table V) is that this is clearly not the case: AS is always negative and, in fact, satisfies an invariance relation that has quite important physical consequences. To illustrate this point in a specific case, consider the binary distribution of monomers and hexamers defined by the case AD. For N = 256, the site classification for this case is given by T, 20, 50 (see Table 11 and/or the illustration in Figures 3 and 16 of I), CT = 0.027 34, and 6 = 0.61 2. This distribution can be realized by the direct union of the precursor systems T, I IO, A 0 [T @ 110 @ A 0 = AD] or stepwise via the union of subsystems of intermediate complexity [T I1 0 = 11, I1 @ A 0 = AD; T @ A 0 = A, A @ I 1 0 = AD; A 0 @ 110 = ADO, ADO @ T = AD]. The entropy difference between the final-state AD and the three precursor states T, 110, A 0 can be determined directly from the tables and (in units of Boltzmann’s constant and per unit volume) has the value T @ 1 1 0 @ A 0 = AD AS = S A D - (ST + Silo + SAo) = -1 5.696 As noted above, the compound state AD can also be realized starting from the precursor states T, 110, and A 0 via a pathway through the intermediate state 1 1 , viz. T @ 110 = I1 AS,’= SI1 - (ST Silo) = -7.815
+
I 1 @ A 0 = AD
AS; = S A D - (Si1 + S A O ) = -7.88 1
with, overall
AS = AS1’
+ AS,‘ = -15.696
Similarly, consideration of the possible intermediate states A and ADO leads to the decompositions T @A0 =A ASl“ = S A - (ST S A O ) = -8.429
+
A
I 1 0 = AD
AS; = S A D - ( S A
+ Silo) - 7.267
with
AS = AS]”+ AS, = -15.696 A 0 @ 110 = ADO
ASI”’ = SADO - (SAO+ Silo) = -7.861
ADO @ T = AD
AS*’”= S A D - ( S A N
+ ST) = -7.329
with
AS = ASI”’
+ AS?
= -15.696
As is evident from these data, although the overall change in AS
is invariant with respect to the set of possible intermediate con(13) (i) Sinfelt, J . H.; Carter, J. L.; Yates, D. J. J . Catal. 1972, 34, 283. (ii) Sinfelt, J . H. Ace. Chem. Res. 1977, I O , 15. (iii) Sinfelt, J. H. In Many-Body Phenomena at Surfaces; Langreth, D., Suhl, H., Eds.; Academic Press: New York, 1984. (14) See, for example: Caralysr Deactivation; Delmon, B., Froment, G . F., Eds.; Elsevier Scientific Publishing: Amsterdam, 1980. ( 1 5 ) (i) Kandel, E. R.; Schwarz, J. H. Principles of Neural Science; Elsevier North-Holland: New York, 1981. (ii) Kuffler, S.W.; Nicholls, J. G.; Martin, A. R. From Neuron to Brain, 2nd ed.; Sinauer Associates: Sunderland, MA, 1984. (16) (i) Edelman, G.M.; Gally, J . A. In The Neurosciences; Schmitt, F. O., Ed.; Rockefeller University Press: New York, 1970. (ii) Edelman, G. M. Immunol. Rev. 1987, 100, 1 I . (iii) Jan, S.-S.; Crossin, K. L.; Hoffman, S.; Edelman, G.M. Proc. Nail. Acad. Sci. U.S.A. 1987.84, 7977. (iv) Edelman, G. M. Topobiology: An Introduction to Molecular Embryology; Basic Books: New York, 1988.
The Journal of Physical Chemistry, Vol. 94, No. 18, 1990 1219
Reaction Efficiency in Surface Catalysis figurations that might be identified (in the example, the subsystems I I , A, and ADO), the entropy changes characterizing the indiuidual stages realized in achieving the final configuration (state) are certainly different. The practical implications of this last result are immediate. Since different subsystems may be thought of as different stages in the activation (or deactivation) of a heterogeneous catalyst, one finds in case AD, for example, that laying down (activating) monomers first (T, I1 0)and then adding (activating) hexamers (AO) has different dynamical consequences than laying down (or activating) hexamers first and then adding (activating) monomeric reaction centers individually (T) or in sets (IIO), even though the overall change in the statistical entropy of the process will be the same irrespective of the sequencing. The first scenario (monomers first, then hexamers) highlights the importance of the intermediate monomeric distribution 110, whereas the second unfolds from the hexameric configuration AO. For the N = 256 unit cell, diffusion-controlled reactive processes on the system I IO are characterized by the parameter set {CT,(n),6, S] = (1.172 X IO-*, 176.496,0.960, 6.1731,whereas for A 0 we have (CT, (n), 6, S1 = { 1.172 X 798.910,0.212, 7.683). From the relationship (3), these data show immediately that for irreversible processes taking place on the composite distribution AD (monomers, hexamers) with ICT, ( n ) , 6, S] = (2.734 X IO-', 98.69879, 0.612, 5.5921, the slowest (rate-determining) step is the one associated with reactive events involving the highly sequestered, hexameric clusters. Thus, the approach developed in this paper can provide a practical method for predicting beforehand (quantitatively) the relative statistical efficiency of different reaction pathways. Of course, in the highly configured multiplets one expects to find in actual experimental situations, a variety of possible decompositions into simpler subsystems can be realized. To assist the experimentalist in applying our approach to these more-articulated configurations, the statistical (and dynamical) characteristics of processes taking place on reaction spaces characterized by a range of "elementary" n-mer configurations have been reported in this paper. Finally, the strict monotonicity of the results displayed in Figure 3 for S versus 6 for each concentration has a further practical consequence. Suppose it is of interest to determine the dynamics induced by a distribution of active sites nor considered explicitly in this work. Such a distribution may be one suggested by the results of experimental or quantum-chemical studies on the stability and properties of small clusters. Estimating 6 for the multiplet configuration of interest allows an immediate estimate of the statistical entropy S. But, given the relationships laid down in section 11, one can use eq 8 to calculate ( n ) for this new distribution and eq 3 to calculate XI (and hence estimate the pseudo-first-order rate constant k for the underlying diffusioncontrolled reactive process). Realization of the latter program is the basis for a principal conclusion of this study: From the results assembled here and the theoretical strategy proposed, one can calculate (or, at least, estimate) the statistical and dynamical characteristics associated with reactive processes carried out on (nearly) arbitrary nonrandom constellations of reaction centers distributed in/on a d = 2 reaction space.
VI. Concluding Remarks
To conclude this paper we wish to comment briefly on the relationship between the stochastic approach developed here and the standard method for treating encounter-controlled reactive processes in condensed media. In the physical problem under study, a (single) coreactant is assumed to diffuse randomly on a d = 2 dimensional surface in/on which there are embedded nonrandom constellations of (stationary) reaction centers.'* In phenomenological theories of such processes modelled by a Fickian equation of the form
ac/at = D V C +
kc
(10)
(where C is the concentration of reactant), there is a natural separation between the diffusional component (scaled by the diffusion coefficient D) and the chemical component (specified
by the rate constant k ) . In formulating the stochastic master equation ( I ) (biased or unbiased) diffusion between neighboring sites in the reaction space is dealt with by assigning appropriate (nonzero) values to the elements of the transition matrix C,; the irreversibility1' associated with a reactive encounter at the reaction center is then handled by assigning a null probability to subsequent excursions of the coreactant away from the reaction center. In solving the parabolic partial differential equation (IO), one must specify two spatial (and one temporal) boundary conditions, one of which must account for the behavior of C(r,t) on the specific cluster geometry considered. In the stochastic approach ( l ) , geometrical complications introduced by considering clusters of reactive sites distributed in arbitrary ways are handled by specifying the fate of the diffusing coreactant at each and every point of the (discretized) reaction space. (From a computational point of view and especially for complicated geometries, the latter approach is much easier to implement.) Analytically, it can be proved that in the limit of large systems @e., N large), the decay p ( t ) in the long-time limit is dominated by the smallest eigenvalue XI of the G matrix of the stochastic master equation (where, as well, ( n ) = uXI-'); in this limit, p ( t ) Thus, in terms of the classical theory of diffusion-controlled reactive processes, the rate law that best describes the reactive events studied in this paper is that for a pseudo-first-order process, i.e., C(r) e-&'. Finally, we wish to comment on the "structuredness" of the system, as portrayed by the statistical order parameter 6, and then to specify the concentration regime where the effects uncovered in this paper are likely to be of experimental significance. In the limiting case where singlet reaction centers are assumed to be symmetrically distributed (specifically, where a fixed distance separates each and every pair of reaction centers), the quantity 6 = ( n ) M / ( n is, ) by definition, unity. Deviations from unity, Le., values of 6 < I , then gauge the statistical consequences of sequestering reaction centers into nonrandom distributions (clusters) of target atoms/molecules. In our study, the surface concentrations (mole fractions) of the atoms/molecules comprising the stationary reaction centers (relative to the total number of surface sites in the lattice unit considered) range from 3.0 X ICT 5 1 .O X IO-' (see Tables I-IV). The concentration of the diffusing coreactant (again relative to the total number of surface sites) will be C, 3X Our calculations show that the effect on the reaction efficiency for values of 6 < 1 will be the most pronounced in the concentration regime CT< IO-'. Moreover, these calculations show convincingly that as one approaches this upper concentration limit a "saturation effect" sets in, Le., the results become more and more insensitive to the configuration/distribution of reaction centers. The conclusion is that unless one is dealing with a relatively low concentration of reaction centers (viz. C, < IO-'), there should be no (experimentally significant) difference in the reaction kinetics exhibited by two systems, one with a perfectly ordered array of reaction centers (6 = 1) and one with an array of reaction centers sequestered into clusters (6 1). On the other hand, it is precisely the concentration regime CT< 0.1 that characterizes the experimental problems in inorganic and biophysical catalysis cited earlier in the text. Thus, it may be hoped that one's intuitive understanding of the role of the entropy in influencing diffusion-controlled reactive processes taking place in such spatially structured systems can now be placed on a quantitative footing.
-
-
-
(17) We remark that the irreversibility associated with this scenario Fan be relaxed and the consequences as regards the reaction efficiency can be explored. One considers the possibility that the diffusing coreactant will form an excited-state complex at the (stationary) reaction center; a probability s is then assigned that the reaction will proceed to completion, with a probability (1 - s) that the excited-state complex will fall apart with the coreactant continuing its random motion in reaction space. Some consequences of adopting this more general approach were explored in 1. (18) The analytic/numerical difficulties associated with working out the dynamics of multiple (even two) diffusing species are quite formidable and probably can be handled at this time only by a full-scale Monte Carlo simulation. Recently, however, J. K. Rudra and one of us (J.J.K.) have made some progress in analyzing stochastic processes involving rwo diffusing species and one reaction center; this work is under development and will be reported when completed.
