Dynamic scaling of diffusion-limited reactions over fractal surfaces: computer simulation
Introduction
Many important processes in physics and chemistry, such as adsorption and catalytic reaction, take place over gas–solid interfaces, which can be mimicked by fractal-like surfaces. With those heterogeneous phenomena, the geometry of the environment plays an important role in their dynamic behaviors. Due to its importance in both basic research and practical applications, the study of geometric heterogeneity have attracted extensive attention and propelled intensive research activities in this field [1], [2]. Diffusion-limited reaction (DLR) [3] is particularly important, since it is an elementary step of some complex reaction mechanisms in condensed phases. Typical examples, such as reaction of free nitric-oxide with erythrocytes [4], photolysis of ozone in solid nitrogen producing N2O complex with oxygen at a low temperature [5], and mechanism of folding of the dimeric core domain of E. coli Trp repressor [6], can be described by DLR process. Understanding its dynamic behavior provides us further information to interpret the underlying principles of those complicated reaction systems.
A series of investigations have been devoted to DLR processes. Given et al. [7] estimated the rate constant for macromolecules by first-passage algorithm in order to eliminate the need of explicit diffusing trajectory. Recently, dimensional crossover of diffusion-limited annihilation reaction in a quasi-one-dimensional lattice was also studied by Lee [8] and density was found to be a scaling function in the time domain. Geometric effects of the reaction environment on heterogeneous catalytic reactions were studied by Lee et al. [9], [10], [11] by performing Monte Carlo simulation of DLR process over fractal surface. Multi-fractal characters were revealed in the resulting reaction probability distribution. Due to the lacking of the time parameter in the above analysis, the available information was limited to a description of the process at quasi-equilibrium state. A new analyzing algorithm including the time parameter is adopted in this work in order to investigate the dynamic behavior of DLR over fractal surface.
Scaling relations with trivial or even non-trivial exponents are usually found when the dynamic behavior of interested systems is governed by stochastic processes involving non-equilibrium many-body effect. In dendritic growth process, the dendrite tip curvature radius and solidification velocity were determined by scaling analysis [12], [13]. It was also applied on the growth of platinum films on Si (1 0 0) and revealed that the values of roughness exponents were in good agreement with those from the theoretical approach [14]. Dynamic scaling of most surface growth models were usually investigated by Monte Carlo simulation with or without introducing several different options, such as diffusion or surface tension, in order to mimic the Brownian motion of particles in reality [15], [16], [17]. Brownian motion [18] is the basic behavior of molecules, for example, the precipitating particles in the growth model or the reactants in a reacting matrix. It is intuitive to apply dynamic scaling analysis on the reacting system to obtain more information about its dynamic behavior.
Diffusion-limited aggregation (DLA) model, established first in 1981 by Witten and Sander [19], has provided a fundamental model to investigate complicated phenomena such as the effect of uniform drift and surface diffusion in electrodeposition [20], [21], surface thermodynamics and crystal morphology [22], and the growth mechanism in shear flow [23]. DLA is adopted in this work to imitate a catalyst’s surface, which is often found irregular and fractal-like. Of course, the catalysts are not necessarily fractal. However a better opportunity can be achieved to quantitatively examine the effect of geometrical heterogeneity on the catalytic process when the fractal geometry is adopted.
In the previous studies of DLR over a DLA surface [9], [10], [11], emphasis was laid upon the multi-fractal characters involved in the fluctuation of the reaction probability distribution caused by the geometric heterogeneity. In this paper, dynamic scaling is introduced in exploring the time-dependent effect involved in DLR over a DLA surface. Results are presented and discussed in the third section. Comparisons are also made with the theoretical results of surface roughening in the growth model [24].
Section snippets
Methods
To mimic the irregular surface involved in the heterogeneous catalytic reaction, DLA is obtained by modified version of Witten and Sander’s model [25]. Besides the conventional random walk procedure of deposing particles in this modified model, acceptable radii of the launching boundary can be chosen in order to optimize the cluster growth rate. The boundary, from which particles are released, is an equal-event boundary, i.e. a diamond-shaped boundary, in order to obtain an isotropic DLA
Result and discussions
The simulation results in Fig. 2 show the dependence of fluctuation of reaction probability on r∑. Three separated regions can be found in this graph. Initially σ depends on r∑, followed with a transition region and then σ becomes independent of r∑ in the last region, as r∑ approaches to infinity.
In the first region, the scaling relation between σ and r∑ iswith β=−0.46±0.02. The negative exponent obtained here is different from the positive value for the surface growth model. It is
Conclusions
Computer simulations are performed on Eley–Rideal DLR occurring over fractal surfaces of DLA in order to examine the effect of geometrical heterogeneity. The dynamic behavior of the model catalytic reaction is governed by two order parameters, α and β with scaling function in different time domain, respectively. The scaling relation of σ with respect to time and surface size is established in Eq. (7) and is marked with negative exponents. The tendency of the reaction probability towards a more
Acknowledgements
The authors thank Professor W. Rudzinski for a helpful discussion and the National Science Council, Taiwan for financial support.
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