Photoprocesses on Fractal Surfaces

353

PHOTOPROCESSES ON FRACTAL SURFACES A.

SERI-LEVY, J. SAMUEL, D. FARIN, and D. AVNIR

1. INTRODUCTION THE FRACTAL APPROACH TO PROBLEMS OF COMPLEX GEOMETRY.

The initial interest in Fractal Geometry stemmed mainly from the smking similarity between computer-generated objects, as formed by applying the rules of this geometry (1,2), and "real" objects as found in many natural or man-made objects. The algorithms used for creating the fractal objects are usually quite simple and involve an iterative construction procedure (Fig. 1). Because of this iterative procedure, the objects thus obtained have the property of being self-similar, i.e., that various magnifications of the object look similar. This. in turn, results in a simple relation between the magnification power and a measurable geometric feature of the object, say its length. The relation has the form of a power-law: length

0

magnificationD

Dl

where D, the fractal dimension, carries information on the degree of geomemc irregularity of the object. Let us look at two extremes for the case of a line: if it is a straight smooth line then magnifying it by a factor of two will double the number of, say pixels, necessary to present it; the D value in eq. [ 11 is 1, i.e., the familiar dimension of a line. Let us assume now that one has a line which is so convoluted and irregular that it actually fills the plane. In this case the relation between length and magnification will be through D=2 (actually D+2) in eq. [ 11. One can see therefore that the 1cDR range provides a measure for the degree of line irregularity. (Readers interested in a rigorous discussion which is beyond the intuitive picture presented here are referred to ref's 1.2). In order to test whether fractal geometry is applicable for problems of heterogeneous chemistry, one has to translate eq. [l] to actual experimental techniques in chemistry.

On a first level one can in principle use eq. [ 11 directly by microscopy image analysis techniques, for instance by measuring the fractal dimension of an irregular boundary line of an object (3). However, the accumulated experience in our laboratory has been that the real geometry as reflected, e.g., in the image of an object need not coincide with the effective geometry as "seen" by a specific chemical process. So while very useful at least for comparative purposes, we shall not discuss image analysis techniques here (these are reviewed in ref. 4) but concentrate only on effective geometries for (photo)chemical interactions. In order to do so, we first generalize eq. [ 11and write it in the form: (molecule-surface interaction parameter) = k (resolution of measuremmt)b

121

354

D = LOG 9 / LOG 5

=: 1.365

1

Fig. 1: The iterative process in forming a fractal line is shown. Each of the straight line segments of the top line (the generator) is replaced by a smaller version of the generator. Magnifying the generator by a factor of 5 reveals for each of the line segments 9 smaller new segments.

355

where p. as we shall see below, is a simple function of the effective D for the process. The generalization of eq. [2] over eq. [ 13 is through defining "magnification" as any procedure in which one observes a property or a process with a set of yardsticks varying in size, i.e.. one observes how a measurable property characteristic of molecule-surface interactions changes with the resolution of observation. Four types of yardstick-sets are discussed below: (a) The cross sectional area of a molecule interacting with a surface. (b) The size of an interacting particle (of an, e.g., dispersed metal catalyst). (c) The static intermolecular distances between adsorbed molecules. (d) The diffusional distances of molecules moving from the bulk volume (of pores) to the reactive surface. Obviously, use of the approach of q.121 can be justified only if it is indeed applicable for the analysis of chemical process on surfaces, and if p can serve for characterization of the effective geometry details and their role in affecting the process. We found that the yardstick sets (a) and (b) are quite general in heterogeneous chemistry (5.6); methods (c),(d), however, are still at an exploratory stage but some interesting results related to surface photochemistry are already at hand and are described below. In the following Sections we concentrate on results obtained in our laboratories. The applications of fractal geometry to problems of photochemistry and photophysics is spreading fast, and some examples of recent studies in other laboratories are collected in ref. 7.

2. THE USE OF MOLECULES AS YARDSTICKS. THE MOLECULAR ACCESSIBILITY OF A SUR-

FACE (89).

The first yardstick for the resolution analysis we discuss is the molecule interacting with the surface. For an irregular surface, the smaller the molecule is. the finer are the surface details it can probe and the larger will be the apparent surface area, A, as determined from monolayer coverage. In principle one can use the sensitivity of A or of n, the monolayer value. to changes in the size of the molecule (its radius, r, or its cross sectional area, a), as a measure for the degree of surface irregularity and molecular accessibility. A flat surface will be equally accessible to all sizes of adsorbed molecules, but the more irregular the surface is, the faster A or n will drop with increasing Q or r; and if the irregularity is fractal. then the sensitivity of A or n to changes in Q or r is given directly by (10):

n = b-D'

[3a1

where Da is the fractal dimension of the accessible surface for molecular interactions. The prefactor k, here and in all the equations below, has the value of the "property" (eq. [2]) for a

356

yardstick of unit length; its units change, of course, from equation to equation. For most cases one finds (Sa,6a,6b,6e-g,61,10-12,44) 1cDaS3.Notice that for the classical two dimensional surface, D,=2, A becomes independent of U. The consequences of the A-u dependency for photoprocesses of adsorbates on surfaces have been discussed in great detail in ref. 9; only the main points and conclusions are summarized here: (a) The Concept of Reaction Area. The N2-BET surface area (or any other value of A obtained with a small molecule) which serves routinely in surface photochemistry studies (13) is applicable only for smooth surfaces, for which the accessibility towards N2 and towards the larger photolabile molecule are equal. If this is not the case, then the use of the N2-BET area leads to erroneous values of intermolecular distances (too large), of the area occupied by an adsorbed molecule (too large) and of rate constants (too high). For several cases (13) it has been shown (9) that interpretations of experimental results may change, if the accessibility factor is neglected. For a bimolecular process, e.g., the collisional energy transfer: Q + B * +(QB)* + P there are three distinct available areas: The area accessible for Q. the area accessible for the excited molecule B* (which need not be equal to the area accessible to the ground state B), the area accessible for the product molecules, P, and, most important, the area accessible to the encounter complex (QB)*. Since the latter is the largest species in the reaction scheme, the area accessible to it is the smallest. In other words, regardless of the size of Q and B, the reaction is limited only to what we call the reaction area, AR:

AR = k N ~ ( Q B )

[41

This is perhaps the most pronounced effect of a reaction on a surface: Q and B cannot react along any point of their diffusional trails which pass in AQ or AB, but only when they collide in AR. Estimation of U(QB) is not a trivial task; perhaps the best approximate method for doing it is to assume (T(QB) = GQ + CSB or ~ ( Q B ) ap, and then to calculate these u valuesby one of the several methods available for that purpose, as reviewed in ref. 8. (b) The Distance between Two Adsorbed Molecules. There are three principal types of distances between two adsorbed molecules: First, the "aerial" distance through the bulk (solid and pore volume) of the material. This distance may be relevant for non-collisional energy transfers (radiative and non-radiative; see Section 4). Second, the shortest distance on the surface. For molecules of the same size, this distance, d, is given by: d = (NC)11'2

[51

where N is Avogadro's number and C is the concentration in units of moles/m2 of the available

357

surface area, A (and not the N2-BET area). In the case of two different molecules (say, a donor, B, and an acceptor, Q). then for a flat surface (13d.14): dQB = ( 2 / 3 ) d ~ ~

[61

On an irregular surface we encounter the following, perhaps unexpected phenomenon: Since AQ#AB, then

This is another unique feature of heterogeneous chemishy, compared to homogeneous situations. The third distance, perhaps the most relevant to reactions on surfaces, is the actual distance traversed by a diffusing molecule. This is a very complex issue which we only begin to understand. The diffusional distance reflects not only the geometric considerations made above, but also the facts that the surface is energetically heterogeneous, and that the diffusion is some combination of movements which follow closely the surface features, and of jumps from pore-wall to pore-wall and from one tip to the next. Obviously this diffusional distance is also a function of the temperature and of the solvent interfaced with the solid. Furthennore, since different types of connectedness can yield the same D value, this textural characteristic is an additional parameter to be considered (the fracton or spectral dimension '(15)). In view of this complex picture, what is then the practical advise? Under the current state of art, the best one can do is to get a preliminary estimate of d from eq's [4]-[6]; the direct observation of actual diffusional process in disordered systems, is still in its infancy. For some recent studies see ref. 16.17. (c) Estimation of Molecule-Surface Parameters From the Fractal Dimension and the N2BET Value. The recommended practice for the estimation of these parameters is to perform an actual measurement of the available surface area. If it has been established, however, that the surface is fractal, then these parameters can be estimated, using D, and the N2-BET value. An example given in Fig. 2; for many others see ref. (9). (d) The Application of the Accessibility Problem to Photochemical Studies. The correct estimation of surface concentrations and intermolecular distances is of course critical for the interpretation of bi-molecular photoprocesses. Applying the concepts described above, we have re-analysed a number of recent surface photochemistry studies (13). in which the N2-BET value was taken as a starting point. We showed that if the surfaces employed in these studies have different accessibilities to N2 and to the larger organic adsorbates, then conclusions (on static vs dynamic processes, for instance) change and can actually be reversed (9). Examples included the photodimerization of 9-cyanophenantrene, the chemiluminescent oxidation of adsorbed fatty acids, the triplet energy transfer from benzophenone to naphtalene and the pyrene excimerization in a pyrene-derivatized surface, all of these on porous silica surfaces for which molecular accessibility strongly depends on molecular size (18).

358

b2.0

Y)

C

t L

0

0.72-

d

\ a

Fig. 2: The ratio of apparent surface areas, A1/As, as seen by large, 01, and small, 0,. molecules. or the ratio of adsorbed moles, nl/n,. at the same surface concentrations (moles/available m2)as a function of 01/Gs.

2HzO + 2 W +

hv ___)

R

2w+2 Hz + 20H- + 2MV+2

h

8 P .$

‘3

2 0.90

~=1.93fo.08

-

3

3

3 8

3 0.60 -

0.30

2.00

I

2.20

I

2.40

2.b

2.60

Fig. 3: The dependence of the initial H2 evolution rate on the hydrodynamic radius of polyvinylalcohol supported Pt catalyst.

359 3. THE USE OF PARTICLE SIZES AS YARDSTICKS PHOTOCATALYSIS ON DISPERSED METALS.

Rather than measuring area with a set of molecules, one can pcrPorm an equivalent resolution analysis, namely, to take only one yardstick and to measure the area of the object as a function of its size (19). Both procedures amount to an observation at different magnifications. For instance, the area, A' per one spheroidal object of size 2R is given by: A' = k RDr

[81

(where the subscript r is added to emphasize the different technique by which D was measured). Theomically, for fractal objects D, = De,but for highly porous materials this need not coincide

(20). Not all of the surface sites which are available for the formation of a physisorbed blanket, necessarily participate in a (catalytic) reaction. In other words, in the case of reactions with a surface, the reaction area. AR. defined in Section 2, may be further diminished by chemical (and not only geomemcal) selectivities. The reactive sub-set of sites has a distinctive geometry of its own, and a recent finding we made is that not only is eq. [81 obeyed in many materials (6b,6g,Sh,6j,6n,60,11,12,18,2I ,23a24), but so do also their conjugated reactive areas (22-24):

n, = k RDR

[91

in which na is the number of active sites, and DR,defined as the reaction dimension, is a characteristic parameter of the heterogeneousreaction which quantifies the sensitivity of the process to the particle size. From the relation between Dr (eq. 181) and DR (eq.[9])one can elucidate useful information on the location of the reactive sub-set (23). For instance, if for a porous object DR a Dr,then one can assume a screening effect, i.e., that only the outer perifery of the particle reacts. The simplest way to count na is to use the surface equivalent of the mass-law (Wenzel law (23)), i.e., the assumption that the reaction rate, a, moletime-', is first order in n,: a = na

[I01

or: a=kRDR If the mass of the particle scales like R3 (20), then in units of mole time-'. gr-': ag = k RDR-3

[Ilal

As an example we re-analyse (25) the results of the study of Kiwi and Gratzel (26)on the

photocatalytic hydrogen production from water in the system Pt/dimethylviologen/EDTA, in

360

which the Pt was dispersed in polyvinyl alcohol (PVA). The activity was measured as a function of the hydrodynamic radius of the Pt-containing PVA. Analysis of the data according to eq. [lla] clarifies some of the debate over the correct interpretation, as appeared in ref. 27. The D~=1.93iO.08value obtained for this case, (Fig. 3). is quite close to D=2.0, the scaling exponent of an area of a sphere. This suggests that in the Pt-PVA particle, the only Pt crystallites which participate in the reaction are those which are exposed at the outer bounds of the polymer, whereas all of the Pt crystallites which are trapped inside are stericaly hindered and become inaccessible for the large organic reactants. Perhaps the most widely used dispersed photocatalyst is Ti02. We have analysed (25) a number of studies in which Ti02 particle size effect was recorded and found that the sensitivity of activity on particle size can be characterized by a typical DR value. Some examples: 1. Harada and Ueda have studied the photocatalytic decomposition activity of watedmethanol mixtures on particles of 5% Pfli02, generating H2 and C02 (29). They found sensitivity to particle size in the range 80-350A. and presented it by a straight line on logarithmic axes with a slope of -1.6 (Fig. 3 in ref. 28). A slope of that value means DR = 3-1.6 = 1.4, i.e., that only a sub-set of all surface sites is active in that reaction. 2. In a similar study, Sakata et al studied the photodecomposition of watedethanol (29). The preparation and size of the particles are quite different than the previous example. A DR value of 2.62 f 0.02 over three and a half orders of magnitude, was obtained for this case (Fig. 4). It seems unlikely that this high DR value reflects a highly irregular surface morphology, because the particles were prepared by crushing of large crystals (29); cleavage along crystal imperfections during the crushing procedure should not produce more than crystal-face imperfections (unless aggregation is also involved in the procedure). However, as discussed in detail elsewhere (23,30.31). a situation of DR > Dr is possible, although in such cases the pattern of distribution of active sites is not fractal. 3. Anpo et al have studied Ti02 sizeeffects on the photocatalytic hydrogenation of methylacetylene with water (32) and found a dependency of the quantum-yield on Ti02 anatase particle size. The relation between the Ar-BETsurface area and particle size obeys eq. [l la], (Fig. 3, but (Table 11 in ref. 32) with Dr=l.59rH).07, smaller than the possible minimal value for this case, Dr=2.0. Dr<2 values are usually interpreted as indicating adsorption which does not coat the whole surface. For the case of anatase we do not believe that on a single particle there exists a preferred sub-set for Ar adsorption; instead we think that the low Dr value indicates loss of surface area by aggregation, namely that the contact areas between the particles become inaccessible to the Ar. One cannot therefore use the nominal particle size as reflecting the size of the aggregate (25). However, some useful information can be extracted as follows: Since the quantum yield. 4, reflects the overall reactions rate, we apply eq. [1 la] for the relation to particle size: # = k RDR-3

[llbl

and obtain (from Table I1 in ref. 32) D~=1.67iO.16(Fig. 5). The similarity between the Dr and

361

3.20 1

1.80

'

2.ko

3.50

I

3.40 4.60 log particle size (A)

5.30

6.h

Fig. 4:The dependence of H2 evolution rate on Pr/ri02 particle size for the photodecomposition of ethanol-water.

3.20

-

P8

2.60

-1.48

2

-

b

t

s

t;

3. &

-2.08 ," 8 Y

2.00

1.40

~~

1.50

1.82

2.15

2.48

-~

2.80

Q

-LA.-

log particle size (A)

Fig. 5: The dependence of the quantqm yield of the photocatalytic reactions of CX3CCH with water, A , and of anatase-type Ti02 surface area, o, on Ti02 particle size.

362

the DR values suggests that the nominal R values used in both cases reflect the same effective size, rhar the structure of the aggregate in H20 solution (the photochemical reaction) and in rhe dry (the area measurement) are quire similar. Furthermore, it also means that the geometrically accessible surface and the reactive surface are quite the same (structure insensitivity). These conclusions can also be reached from another point of view: Elsewhere we have shown (23) that the relation between D,, DR and the BET area, A, is given by: v = k A m = k R@R-3)/(Dr-3)

[I21

or: O=kAm

in which m is the reaction order in surface area. It is seen that m values close to one mean that the reactivity and area are linearly related, or that DR D,. Indeed analysis of the relation between 0 and A (Table I1 in ref. 32) gives (eq. [13], Fig. 6) m4.94M.10. 4. In another study of Anpo et al (33), the photocatalytic activity of alkenes hydrogenations and isomerizations on Ti-A1 binary metal oxides was determined as a function of the anatase Ti02 particle size. A typical result is shown in Fig. 7 (data from Table 1 and Fig. 4 in ref. 33; the cluster of the end three points 97, 98.5 and 100% Ti02. which are actually not binary catalysts are not included). The extremely low D~d.26k0.12obtained for propene hydrogenation is typical of all other reactions in this study (25). The authors concluded in their study that the binary catalyst acts at the periphery of a Ti02 layer surrounded by Al2O3. This would lead to DR'I, (as we found for the case of the photoassisted water-gas shift reaction (25)). The DR-d values indicate that the number of active sites changes very little with panicle size. which in turn is possible if only comers or fractions of edges of the crystallite are active (30,31). Furthermore, in terms of fractal geometry, D R < values ~ can be interpreted as originating from a a fractal Cantor set distribution of active sites (31). Since anatase is inactive in the specific reactions studied (33). we suggest that the reactive sub-set is composed of sites where the A1203 interferes with the ordered crystalline structure of anatase, and that these imperfect sites are the active ones.

-

4. THE USE OF INTERMOLECULAR DISTANCES BETWEEN ADSORBATES AS YARDSTICKS:

ELECTRONIC ENERGY TRANSFER (34).

The shape of the decay profile of an excited donor is determined, amongst other parameters, by the distribution profile of the surrounding acceptors. Thus, the classical three dimensional Forster equation for non-collisional, one step electronic energy transfer (ET), had to be modified for the case of a two dimensional arrangement of donors and acceptors (35). This has been generalized recently, to include not only two and three dimensional acceptor distributions, but also D-dimensional distributions (36):

363

4-

-2.60 1.30

1

3.10

Fig. 6: The dependence of the quantum yield of the photocatalytic reaction of (JH3CCH wlth water on anatase-typeTi02 surface area.

Photocatalytic Pmpane Hydrogenation on Ti@&&

$

1

\.

“-4.2&to.12 -n-

0 0.25

-0.25 I 1.85

1.45

2.b log particle size (A)

I

2.15

2.b

Fig. 7: The dependenceof the photocatalytic activity (mole/lu/gr) for propene hydrogenation on TiOz/Al203 particle size.

in which S(t) is the survival probability, T, is the fluorescence life time of the donor (in the absence of acceptors), D characterizes the fractal distribution of the acceptors around the donor, and the prefactor y is: y=

(h/rdID

r (1-D/6)

[I51

in which BA is the degree of surface coverage of the acceptors. is the critical Forster radius, rd is the radius of the donor molecule and r is the J? dismbution function. The prefactor tells actually, how much of 8 A is at a distance % in the fractal environment. Several experimental studies have shown that decay profiles in disordered environments can be satisfactorily fitted to eq. [14] by adjusting the two parameters y and D (37). The two parameter fitting procedure to eq. [14] is not devoid of the classical difficulty in fitting models to decay profiles: One can design several alternative models which will produce the desired decay curve. Indeed, fractal models based on fits to eq. [I41 have been questioned, and Euclidean models put forward (38,39). Other supportive evidence is therefore necessary for the attachments of models to specific decay profiles (see below). Recently we simplified the two-parameter fitting situation of eq. 1141, by converting it to a one parameter fitting procedure (40).We did it by taking the original fractal considerations that led to eq. [14] one additional step, namely by expressing the unknown 8A as a function of the known N2-BET value and of D as explained in Section 2. The y prefactor becomes then: y = (nA/nNz) (2xwz2/D) ( % / r N ~ ) ~(1-6/D) r

[I61

where nA is the amount of adsorbed acceptor, nN2 the monolayer value of N2 and I - N ~its radius

(2.27A). Substituting eq. [16] in eq. [14] gives the final single parameter (D) model. The approach was applied to three pairs of donors/acceptors adsorbed on silicas with various pore size distributions (40-42); The three pairs were rhodamine 6G (6G.donor) and malachite green (MG,acceptor) (42); rhodamine B (RF3, donor) and MG (40); and R6G/R6G in a depolarization experiment (41). all at various concentrations. Typical decay profiles are shown in Fig. 8, and the experimental results collected in Table 1. The main conclusion is that the fractal dimension of the distribution profile of acceptors around a donor is inversely dependent on the pore size. It is also important to notice that the same D values are obtained with all three donor/acceptor pairs. We interpret these D values as reflecting the geometry of the support as seen by an adsorbed molecule, and in particular that these D values are the surface fractal dimensions for adsorption, for the following reasons: (a) The fact that the D values were found to be insensitive to the different values of the three pairs and to the concentrations employed, is in keeping with the scale-invariance of the fractal model. (b) In a number of studies (21,43) it has been shown that for the same material, higher

365

Table 1 D of various silicas from energy transfer data.

Porediameter

i4

S

D

D

D

(m2/gr)

RB/h4G

RGGMG

200

2.08k0.02

2.08

So00

3

2.05fo.02

2500

8

2.23M.02

loo0

20

2.31f0.02

500

50

2.36M.02

200

150

2.3M.03

2.35

2.30

100

320

2.51fo.03

2.51

2.50

60

500

2.82iO.03

2.18

2.11

RGGBGG

-~

Aerosil

Time ( n s ) Fig. 8: Typical decay cuyes of "%orbed rhodamine 6G (donor) on silica-60 without (a) and with (b) 0.85 x 10- molec/A malachite green (acceptor). The experimental data (dots) were fitted (solid line) to eq's [14],[16].

366

surface areas are linked to higher D values. (c) Other molecule/swface interaction studied have revealed similar low D values (-2) for Aerosil (12). and high D values (2.8-3.0)for Si-60 (18,44). Since the ET experiment described here is also based on molecule-surface interactions on these materials, these independent results seem to comborate each other and to indicate that they refer to the same effective geometry (which need not coincide with the one seen by scattering techniques. (Small angle x-ray scattering probes all of the surface, including the closed and bottle-neck pores).

In conclusion of this Section we re-iterate that analysis of ET decay protiles can be only suggestive as to the underlying mechanism and environmental geometry, and that independent additional experimental results are needed.

5. THE USE OF DIFFUSIONAL DISTANCES AS YARDSTICKS THE INTERACTION OF AN EXCITED

STATE WITH A (CTALYTIC) FRACTAL WALL.

The fourth type of yardstick we use for fractal resolution analysis is the diffusional distance that a molecule traverses from the bulk until it reacts with a (catalytic) fractal wall (the Eley-Rideal mechanism). The distance is measured by following the kinetics of a diffusion controlled reaction, i.e., the yardstick is actually the rime it takes a molecule to react. The use of this type of yardstick can be understood in the following fashion (5445-48): We start again with the basic property of a fractal surface, as given in q.[3a] n = r-D

[3bl

where n here is the number of (partially overlapping) "phantom" spheres centered on the surface. On multiplying both sides by 3 we see that the volume, V, of a blanket of thickness r that covers the surfaces scales as [49]

V=&

[I71

and from eq. [3b]:

V = kr3-D We now replace V and r with two other measurable quantities. V is expressed in terms of the number of reactive molecules, B,, that at t 4 are contained and evenly distributed in it: V = kvBv

r 191

The B molecules react with the wall so that the number of active sites, W, does not change during the reaction.

B+W-+P

POI

367

For a photochemical reaction, B is in the excited state, and we further assume that its natural decay is negligible compared to its reaction ume. P is either a product (for a catalytic wall) or ground state B (for a quenching wall). After time t, all B molecules within a distance r(t) from the wall have reacted, and so one can rewrite eq. [ 191 in terms of P(t):

WaI

P(t) = V(t)/k, The time it takes a B molecule located at a distance r from the wall to reach it is given by: r = (Kt)'l2

[211

where K is the diffusion constant. Substituting V and r in eq. [18] with eq's [19a].[21] one gets: p(t) = k*t<3-D)/2

t221

(or ),t beyond Like for physical fractal objects, here too one will find in practice an r, which the object ceases to be fractal. P(t) which is formed ufrer ,t comes from outside the distances, it will take the B molecules t d / K (eq. 1211) to fractal region. From these o r reach the fractal object (5b): P(t) = k*t"2

t

* tmax

I231

One can also visualize this situation as "seeing" the fractal object from a large distance, i.e., at very low resolution, which amounts to D R = i~n eq. [22]. In practice, in photochemical experiments one excites all of the reservoir, B ., We confine ourselves to the case where the volume, .,V in which this initial reservoir is contained is within the fractal domain, i.e., that eq. [IS] and eq. [22] hold:

*

Bmax = k tmax

(3-D)/2

To study the effects that changes in D induce in the kinetic behaviour, reference is made to some standard conditions. as follows: The amount of unreacted B at time t (these are the molecules at the corresponding distance r and further) is

Since t/t,,
368

Fig. 9: The fractal lines used for the simulations am a) D11.61 (above), b) 1.50 (follows next), c) 1.34 (Fig, l), d) 1.00 (a straight line). The indicated D values are the actual D obtained by the box-counting technique. The analytical values are respectively a) 1.613 b) 1.500 c) 1.365, d) 1.OOO.

...

is equivalent to eq. [22] since k**=k*, i.e.,

as can be seen immediatelyfrom eq. [%I. It is important to notice (51.52) that also

or in terms of r, the prefactor is

and r- for the pnfactor, or the minimal values: r& is the size of the interacting molecule (a pixel), and therefore B , h is simply the monolayer value of the surface as determined by that yardstick. So one can either use B,

We have performed some two dimensional Monte-Carlo simulations of the above reaction scheme. The two dimensional analogue of eq's [22],[26] is P(t) = Bmm - B(t) = k*t(2-DR)/2

POI

in which the subscript R emphasizes, as in Section 3 that the D value is obtained from the analysis of a reaction, so that the DR refers to the effective geometry for the reaction. Fig's 1 and 9 show the fractal interfaces on which the simulations were carried out. Fig. 10 shows the DR values obtained for these fractal interfaces under similar starting conditions (constant value of the initial amount of interacting molecuIes in units of molecule/mength (mo~ec/pixe~~)). Fig. 11 demonstrates, by following B(t), that the reaction is faster the higher D is. Fig. 12 shows that the accessibility for reactioddiEusion is not equal for all surface points, but is determined by geometry. It is seen that there is a "screening" effect: The hidden zones participate in the reaction to a significantly lesser degree than the exposed ones. Elsewhere (51) we discuss k*@R) in detail, we analyse the superposition of spontaneous decay of B at t=+ and we describe in detail the technical aspects of the simulation.

Acknowledgments: Supported by the US-IsraelBinational Foundation, by the Belfer Foundation and by the Aronberg Foundation. We thank P.Heifer for useful discussions, especially for

310

-0.30

z

9 rn

z

-1.00

g

8u

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1 2.50

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I

4.30

UXI TIME (readonsteps)

Fig. 1 0 The conversion (P(t)/B,) of reaction [20] a8 a function of time for the fractal lines of Fig. 9. The nsulting DR values arc indicated. ('Iticy arc slightly lower than the measured D values, Fig. 9).

37i

. 8

3.60

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100.

xld

150.

200.

TIME (reaction steps)

Fig. 11: The decay profiles of excited state B in reaction 1201 as a function of time. Notice that the reaction is faster the higher D is. THE COLLECTION of REACTIVE SITES AFTER 20,OOO REACTION STEPS

Fig. 1 2 The collection of reactive sites on the Dr1.50 (D~”1.45) fractal (Fig. 1) after 20,000 reaction steps. Notice that the exposed zones tue diffusional more reactive than the hindered zones.

372

important clarifications regarding Section 5.

REF'ERENCES

1 2 3 4 5

6

7 8 9 10 11 12 13

14 15 16 17 18 19

B.B. Mandelbrot, The Fractal Geometry of Narure, Freeman, San-Francisco, 1982. H. Takayasu. Fractals, Asakura-Shoten,Toyou, 1986. D. Farin. S. Peleg, D. Yavin and D. Avnir, Lungmuir, 1 (1985) 399. B.H. Kaye, Powder Technol., 46 (1986) 245. For recent reviews see: a) D. Farin and D. Avnir, in: K.K. Unger et al (Ed's), Characterizarion of Porous Solids, Elsevier, Amsterdam, 1988, pp. 421-432. b) P. Pfeifer. in: P. Laszlo (Ed.), Preparative Chemistry Using Supporred Reagents, Academic Press, NewYork. 1987, p. 13. a) J.J. Fripiat and H. Van Damme. Bull. SOC.Chim. Belg., 94 (1985) 825. b) H. Van Damme and J.J. Fripiat, J. Chem. Phys., 82 (1985) 2785. c) M. Czemichowski, R. Erre and H. Van Damme, INSERM Symposia on Silicosis and Mixed Dust Pneumoconiosis. Paris, 1986. d) H. Van Damme, P. Levitz, F. Bergaya, J.F. Alcover, L. Gatineau and J.J. Fripiat, J. Chem. Phys.,85 (1986) 616. e) J. Schriider. FRG, hivate Communication. 1985. f ) F.M. Gasparini and S . Mhlanga. Phys. Rev. B, 33 (1986) 5066. g) C. Fairbridge, H.S. Ng and A.D. Palmer, Fuel, 65 (1986) 1759. h) S.H. Ng, C. Fairbridge and B.H. Kaye, fungmuir, 3 (1987) 340. i) C. Fairbridge, A.D. Palmer, S.H. Ng and E. Furimsky, Fuel, 66 (1987) 688. j) D.H. Everett, Private Communication; D.H. Everett. D. Phil. Thesis, Oxford, 1942. k) G. Guo, Y. Chen, Y. Tang,X.Cai and S. Lin, Reprints of the IUPAC Symp., Churacterizarion of Porous Solids, FRG, 1987, p. 77. 1) H. Frank, H. Zwanziger and T. Welsch. Fres. Z . Anal. Chem., 326 (1987) 153. m) H. Spindler, P. Szargan and M. Kraft, Z. Chem., 27 (1987) 230. n) R.R. Mather, Scottish College of Textiles, Private Communication, 1987. 0 ) D.F.R. Mildner. R. Rezvani, P.L. Hall and R.L. Borst, Appl. Phys. Lett., 48 (1986) 1314. p) W.I. Friesen and R.J. Mikula, J. Colloid Interface Sci., 120 (1987) 263. q) Ref's 20, 24, M a , 44b, below. r) S.B. Ross, D.M. Smith, A.J. Hurd and D.W.Schaefer, Lungmuir. 4 (1988) 977. s) D.L. Stenmr, D.M. Smith and A.J. Hurd. preprint, 1988. t) K. Nakanishi and N. Soga, J. Non-Crysr.Solids, 100 (1988) 399. a) A. Takami and N. Mataga. J. Phys. Chem., 91 (1987) 618. b) I. h s a d and R. Kopelman, J. Phys. Chem., 91 (1987) 265. c) Ref's 16.17.36-39, A.Y. Meyer. D. Farin and D. Avnir, J. Am. Chem. SOC.,108 (1986) 7897. D. Avnir, J. Am. Chem. SOC., 109 (1987) 2931. D. Avnir and P. Pfeifer, Nouv. J. Chim., 7 (1983) 71. D. Avnir, D. Farin and P. Pfeifer, Nature (London), 308 (1984) 261. D. Avnir. D. Farin and P. Pfeifer, J . Chem. Phys., 79 (1983) 3566. a) R.K. Bauer, R. Bornstein, P. de-Mayo, K. Okada, M. Rafalska, W.R. Ware and C.K. Wu. J. Am. Chem. Soc., 104 (1982) 4635. b) P. de-Mayo, L.V. Natarajan and W.R. Ware, J. Phys. Chem., 89 (1985) 3526; D. Avnir, P. de-Mayo and I. Ono, J. Chem. SOC. Chem. Commun., (1978) 1109. c) A.W. Adamson and V. Slawson, J. Phys. Chem., 85 (1981) 116. d) N.J. Tumo, M.B. Zimmt and I.R. Gould, J. Am. Chem. Soc., 107 (1985) 5826; H. Lochmuller. A.S. Colborn, M.L. Hunnicutt and J.M. Harris, J. Am. Chem. SOC., 106 (1984) 4077; H. Lochmuller, A.S. Colborn, M.L. Hunnicutt and J.M.Harris, Anal. Chem., 55 (1983) 1344. N.J. TUKO and M.B. Zimmt, private communication. S. Alexander and R, Orbach, J. P hys. Left., 43 (1982) 2625. W.D. Dozier, J.M. Drake and J. Klafter, Phys. Rev. Letr., 56 (1986) 197. R. Kopelman. S. Praus and J. Prasad, Phys. Rev. Lett., 56 (1986) 1746. D. Farin, A. Volpert and D. Avnir, J. Am. Chem. Soc., 107 (1985) 3368 & 5319. P. Pfeifer and D. Avnir, J. Chem. Phys., 79 (1983) 3558; 80 (1984) 4573.

373

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

36 37

38 39 40 41 42 43 44 45 46 47 48 49 50

a) H. Van Damme, P. Levitz, L. Gatineau, J.F. Alcover and J.J. Fripiat, J. Colloid Interface Sci., 122 (1988) 1. b) M. Ben-Ohoud, F. Obrecht, L. Gatineau, P. Levitz and H. Van Damme, J. Colloid Interface Sci., in press, 1988. D. Avnir, D. Farin and P. Pfeifer. J. Colloid Interface Sci., 103 (1985) 112. P. Pfeifer, Chimia, 39 (1985) 120. a) D. Farin and D. Avnir, J. Phys. Chem., 91 (1987) 5517. b) M. Silverberg. D. Farin, A. Ben-Shaul and D. Avnir, Ann, Isr. Phys. SOC.,8 (1986) 451. E. Ignatzek. P.J. Plath and U. Hundorf, 2.Phys. Chem. (Leiprig), 268 (1987) 859. D. Farin, J. Kiwi and D. Avnir, Manuscript in preparation. J. Kiwi and M . Gratitzel, J. Am. chem. SOC., 101 (1979) 7214. P. Keller and A. Moradpour, J . Am. Chem. SOC., 102 (1980) 7193. H. Harada and T. Ueda,Chem. Phys. Lett., 106 (1984) 229. T. Sakata, T. Kawai and K. Hashimoto. Chem. Phys. Lett., 88 (1982) 50. D. Farin and D. Avnir, J. Am. Chem. Soc., 110 (1988) 2039. D. Farin and D. Avnir, in: M.J. Phillips and M. Ternan (Ed's). Proceedings of the 9th Int. Congress on Catalysis, Vol 3, Characterization and Metal Catalysts, Chem Inst. Canada, Ottawa, 1988, pp. 998-1005. M. Anpo. T. Shima. S. Kodama and Y . Kubokawa, J. Phys. Chem., 91 (1987) 4305. M. Anpo, T. Kawamura, S. Kodama, K. Maruya and T. Onishi, J. Phys. Chem., 92 (1988) 438. In collaboration with D. Pines-Rojanski and D. Huppert. A.G. Tweet, W.D. Bellamy and G.L.Gaines, J. Chem. Phys., 41 (1964) 2068; M. Hauser, U.K.Klein and U. Gosele, 2.Phys. Chem., 10191976) 255; P.K. Wollerand B.S. Hudson, Biophys. J., 28 (1979) 197; R.E. Dale, J. NOVTOS, S. Roth, M. Eididin and L. Brand, in: G.S.Beddard and M.E. West (Ed's), Fluorescent Probes, Academic, London, 1981. 3. Klafter, A. Blumen and G.Zumofen, J. Lwninesc., 31/32 (1984) 627. U. Even, K. Rademan, J. Jortner, N. Manor and R. Reisfeld, Phys. Rev. Lett., 52 (1984) 2164, U. Even, K. Rademan, J. Jortner, N. Manor and R. Reisfeld, Phys. Rev. Lett., 58 (1987) 285; D. Rojanski, D. Huppert, H.D. Bale, X. Dacai. P.W. Schmidt, D. Farin, A. Sen-Levy and D. Avnir. Phys. Rev. Lett., 56 (1986) 2505; A. Takami and N. Mataga, J. Phys. Chem., 91 (1987) 618; Y. Lin, M.C. Nelson and D.M. Hanson, J. Chem. Phys., 86 (1987) 158. P. Levitz and J.M. Drake, Phys. rev. Len., 58 (1987) 686. C.-L. Yang, M.A. El-Sayed and S.L. Suib, J . Phys. Chem., 91 (1987) 4440. D. Pines-Rojanski, D. Huppert and D. Avnir, Chem. Phys. Len., 139 (1987) 109. D. Pines and D. Huppert, J. Phys. Chem., 91 (1987) 6569. D. Pines, D. Huppert and D. Avnir, J. Chem. Phys., 92 (1988) 4734. A.J. Hurd. D.W. Schaefer and J.E. Martin, Phys. Rev. A, 35 (1987) 2361; A. Le-Mehaute, C. Crepy, A. Hurd, D. Schaefer, J. Wilcoxon and S . Spooner, Compt. Rend. Acad. Sc. Paris, 304 (1987) 491. a) S.V. Christensen and H. Topsoe, Haldor Topsoe Co., Denmark, Private Communication, 1987. b) J.M. Drake, P. Levitz and S. Sinha. Mat. Res. SOC.Symp. Proc., 73 (1986) 305. c) D. Farin and D. Avnir, J. Chromatogr., 406 (1987) 317. P.-G. de-Gennes, C.R . Hebd. Sceances Acad. Sci., Ser 11,295 (1982) 685. P. Pfeifer, D. Avnir and D. Farin. J. Star. Phys., 36 (1984) 699; 39 (1985) 263. P. Pfeifer and A. Salli. submitted 1987. L. Nyikos and T. Pajkossy, Electrochim. Acta, 31 (1986) 1347. S. Peleg, J. Naor, R. Hartley and D. Avnir, IEEE Trans. Panern Anal. Mach. Intelligence, 6 (1984) 518. For some recent applications of this relation to the problem of scattering from fractal surfaces see: P. Pfeifer and P.W. Schmidt, Phys. Rev. Left., 60 (1988) 1345.

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51 A. Sen-Levy, P. Heifer and D. Avnir, to be published. 52 P. Heifer, Private Communication.