J. Phys. Chem. 1993,97, 10380-10384
10380
Kinetics of Diffusion-Limited Adsorption on Fractal Surfaces Alon Seri-Levy and David Avnir’
-
Department of Organic Chemistry and the F. Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel Received: January 4, 1993; In Final Form: May 14, 1993
Delahay’s equation for adsorption kinetics on flat electrodes is extended to fractally rough morphologies. The distance of diffusion term in Delahay’s equation, where 3)is the diffusion coefficient and t is time, where D is the surface fractal dimension. The generalized equation is verified by is generalized to the use of Monte Carlo computer simulations. The results are applicable not only to electrochemistry but also to any chemical heterogeneous system which is governed by Henry’s law of adsorption.
Introduction
In a series of papers published by Delahay and his colleagues,l-3 notably with Trachtenberg,’ a basic problem in electrochemistry has been solved (cf. also ref 4), namely, understanding the kinetic behavior of diffusion-controlled, monolayer-limited adsorption on an electrode surface.s Delahay’s result goes beyond electrochemistry and is in effect applicable to any heterogeneous chemical system which is governed by Henry’s law of adsorption: Delahay’s equation (eq 5) was tailored to flat electrodes only.’ In practice, however, solid electrode surfaces have some roughness, either because of technical problems in creating ideal flatness or due to buildup of roughness after repetitive use of the electrodes or because electrodes are built rough and porous in the first place7 due to the specific advantages such morphologies may offer.8 Since these, and not ideally flat, surfaces are the rule,g there is an evident need to include nontrivial morphologies in theoretical modelings. Fractal geometry has proven to be a useful tool for the analysis of molecular events in heterogeneous environments9-12 and, for relevance to Delahay’s study, in ele~trochemistry.~J’J3-20 Derivation of the Adsorption Kinetics Equation We consider the problem of initially homogeneously distributed adsorptives, C(x,O)= C*, where Cis the adsorptive concentration and x is the distance from the surface, diffusing under semiinfinite diffusion conditions (limvm C(x,t) = C*) toward an exposed electrode. Following Delahay,’~~ the surface concentration, r,of the adsorbed molecules at time t is obtained by integrating the flux of the adsorptives a t the surface from time 0 to t
where 2) is the diffusion coefficient. One can solve eq 1 by assuminglJ that the adsorption process is monolayer limited and that the initial concentration is low enough, such that the adsorption obeys Henry’s law. In that model the equilibrium concentration of the adsorbed molecules, res, is simply proportional to the adsorptive concentration
rCs= r,Kc
(2) where rmis the monolayer concentration (the number of adsorbed molecules in complete monolayer on a unit surface area) and K is the adsorption constant
K = kA,/r,v,D, (3) Here, k counts the number of collisions per unit time per unit area per unit pressure, v1 is the frequency of oscillation of an adsorbed molecule in a direction normal to the surface, A1 is the probability of a molecule being adsorbed upon collision with the
l’O
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 loo00
Time (reaction steps)
Figure 1. Rates of establishing equilibrium according to eq 6b for, from top to bottom, D = 1.00, D = 1.32, D = 1.52, and D = 1.99.
surface, and D I is the probability of desorption. Adding the boundary condition
r(t)= b’c(o,t)
(4)
where b’ = r,K, the solution of eq 1
We now extend eq 5 to include both smooth and fractally rough morphologies. To do so, we recall that, for the case of a flat surface ( D = 2), thevolume above the surface, V,whichundergoes diffusional depletion, is proportional to the thickness r, of that layer, which in turn is proportional to (2)t)W. For a fractal surface, that volume has been termed “neighboring volume”.21 It is defined as the volume within a boundary, each point on which is at distance r from the surface. The expression for that generalized volume is V(r) = k’fi3-D)= k’(Dt)(s-D)/z where k’is constant.20*22Consequently, we replace the term (2)t)lI2 in eq 5 with (2)t)(3-D)lZ. This leads to
where b is a constant of units (length)(3-D) (which enter through the prefactor k’).21 Equation 6a is our proposed general equation for diffusion-controlled adsorption, which takes into account surface geometry effects, at the limits indicated above. (Note that eq 5 is retrieved by substituting D = 2 in eq 6a.) We show below the correctness of eq 6a for two-dimensional diffusion (from an area to a line). Here, the neighboring area is proportional to
0022-365419312097-10380%04.00/0 0 1993 American Chemical Society
Kinetics of Adsorption on Fractal Surfaces
The Journal of Physical Chemistry, Vol. 97, No. 40, 1993 10381
-
0.10
h
E
b
l
,
/
0
0
500
,
,
,
,
,
/
,
1
1000 1500 2000 2500 3000 3500 4000 4500 5000
Time (reaction steps) Figure 3. Establishing equilibrium by a mixing process for the Koch curve with D = 1.45,initial concentration 1 X 10-4, and a reservoir of 623 X lo4 pixels. of eq 6b. It is evident that the kinetics of adsorption is highly sensitive to the surface morphology. In the next sections we show that the prediction given by eq 6b is indeed observed.
d
Figure 2. (a) Koch curve with D = 1.32. (b) Random fractal line with D = 1.31. (c)KochcurvewithD= 1.45. (d)KochcurvewithD= 1.52. r’z-D),
and eq 5 generalizes to
Equation 5 is retrieved by substituting D = 1. Finally, it should be noted that the generalization of (Z)t)l/2 to (2)t)(3-D)/2 was shown to be correct for the case of Eley-Rideal catalytic23924 and electrochemical processes,13as was suggested by de Gennes in ref 20. In Figure 1 we show the behavior (for b = 20 and 3 = l / 4 )
Simulation Details Five lines were taken: a straight line (100 pixels long) as a reference case; a fractal Koch-type25 curve with a box-counting dimension21bof D = 1.32 (Figure 2a, 10 207 pixels long); a random fractal line with a similar dimension of D = 1.3 1 (Figure 2b, 5612 pixels long; construction details are given in ref 23); and two additional Koch curves with D = 1.45 (Figure 2c, 24 400 pixels long) and D = 1.52 (Figure 2d, 49 167 pixels long). Each of these lines was placed in a periodic reservoir. Adsorptive molecules, each occupying 1 pixel, were initially homogeneously distributed in the reservoir. The initial reservoir concentrations are detailed below, and it will be shown that the semiinfinite condition is satisfactorily approached. At each time step, all the reservoir molecules (adsorptives and adsorbed molecules) were chosen in a random order and treated according to the following rules: If the chosen molecule is an adsorptive adjacent to an unoccupied surface site, it is adsorbed on the surface with probability A , = 1. If the chosen molecule is an already adsorbed molecule, desorption is attempted with probability Dl = 0.05. If desorption is successful the molecule becomes unadsorbed, and if not, the molecule remains adsorbed. At each time step, r(t) is counted and the next time step begins by randomly moving all the unadsorbed molecules a distance of 1 pixel, in one of four directions. An attempt to move into an occupied site will not be successful, and the molecule will remain in its place. Each simulation was repeatedly executed, and the values of r(t)were averaged for each t . The calculation of ralwas accelerated by replacing the relatively slow diffusion procedure in the previous algorithm with a mixing procedure; Le., each new time step begins with the homogeneous redistribution of the unadsorbed reservoir molecules. (Note that this procedure is possible when one deals with an equilibrium state.) F,wastakenas thevalueofI’(t),afterthelatterremained constant for at least 4000 time steps. An example is given in Figure 3. As mentioned above, a desorption probability of Dl = 0.05 was taken. From this, b is calculated as follows: eq 3 reduces in our simulations to K = l/D1. This is so because rm= 1, A I = 1, v1 = 1 (one desorption attempt per time step), and k = 1. (The number of molecules adjacent to a unit surface area per unit concentration is exactly one, and all of them collide with the surface.) Since b = r,K and I’, = 1, b = l/Dl; Le. in our simulations, b = 20. Finally, as shown in a previous report23 in this type of simulation, 2) =
10382 The Journal of Physical Chemistry, Vol. 97, No. 40, 1993
Seri-Levy and Avnir
1.o
0.8
0.6 \
n
*
W
L
0.4
0.2
0
I
0
I
I
I
I
t
I
I
I
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time (reaction steps)
0.5
0.4
0.3 \
n
e! h
0.2
0.1
o f
0
I
I
I
1
I
I
I
I
1
50
100
150
200
250
300
350
400
450
x102
Time (reaction steps)
5
The Journal of Physical Chemistry, Vol. 97, No. 40, 1993 10383
Kinetics of Adsorption on Fractal Surfaces
Figure 4. (left page, top) Theoretical prediction (solid line) and simulation results (colored dots, see simulation conditions in the text) for diffusionlimited adsorption on a smooth line. Figure 5. (left page, bottom) Theoretical predictions (-, D = 1.32, 1.35; - - -,D = 1.28, 1.31; - - -, D = 1.45, 1.47;- - -, D = 1.52) and simulation results (colored dots, see simulation conditions in the text) for diffusion-limitedadsorption on fractal curves with D = 1.32, D = 1.31, D = 1.45, and D = 1.52, respectively.
0.32
..
0.24
0.08
0
0
I
I
I
100
200
300
I
I
I
I
I
I
400
500
600
700
800
900
1000
Time (reaction steps) Figure 6. Magnification of the first 1000 reaction steps for Figure 5 (-, D = 1.32; - - -,D = 1.28; - - -, D = 1.45).
Simulation Results We begin our simulations with diffusion and adsorption on a straight line for which the surface concentration, r(t),can be calculated directly from eq 5 . This test case allows us to examine the accuracy of our algorithm under various initial conditions. In Figure 4, r(t)vs t is plotted for three different initial conditions. The solid line is the theoretical prediction of eq 5 and of eq 6b for D = 1. The red dots are the simulation results for b = 20 and for an initial concentration of 1 X 10-3 adsorptives/pixel in a reservoir of 1 X 1O6 pixels. It is seen that the fit is much improved by doubling the size of the reservoir to 2 X 106 pixels, while keeping the concentration unchanged (green dots), and even further improved by also decreasing the initial concentration to 1 X 1 V adsorptives/pixel (blue dots), thus approaching conditions of an infinite reservoir and keeping KC
