A Model for Randomly Correlated Deposition

Transcription

1 A Model for Randomly Correlated Deosition B. Karadjov and A. Proykova Faculty of Physics, University of Sofia, 5 J. Bourchier Blvd. Sofia-116, Bulgaria ana@hys.uni-sofia.bg Abstract: A simle, discrete, arametric model is roosed to describe conditional (correlated) deosition of articles on a surface and formation of a connecting (ercolating) cluster. The surface changes sontaneously its roerties (hase transition) when a ercolating cluster aears. The arameter k is included in the robability (k) of articles to stick together and form a cluster on the surface. The case k=1 corresonds to the ordinary D ercolation on a square lattice. Thus the ercolation threshold is controlled by the k-value: the larger k the higher threshold for ercolation. The growth model seen from the ercolation oint of view allows us to describe several interesting alications in addition to irreversible aggregation in the resence of a reulsive force, k>1. For examle, the occuied lattice sites might reresent regions of secific magnetization in an otherwise disordered medium. Then the whole system is ordered or not according to the concentration of the deosited articles. Object-oriented code is develoed for the Monte Carlo art of the calculations. I. Introduction One way to study non-equilibrium interfaces is to construct discrete models and investigate them using comuter simulations. The formation of a real surface is influenced by a large number of factors, and it is almost imossible to consider all of them. The hoe is, however, that there is a small number of basic laws determining the morhology and the dynamics of the growth [1]. The action of these basic laws can be described in microscoic detail through discrete growth models - models that mimic essential hysics byassing some of the less essential details. Sometimes the interface growth is of interest because the surface must change its roerties under given circumstances. For examle, this is the critical hase transition. The critical change of conductance of covered surfaces could be considered in the frame of the ercolation theory []. The conductivity is related to the ercolation robability of the branching model, e.g. a network with all but a fraction of conducting links removed. Percolation and hase transitions are well understood statistical henomena [3]. The geometrical connectivity in the ercolation roblem [4] roved to be useful in understanding The resent aer resents a discrete, arametric model of random correlated deosition (RCD) of articles onto a clean surface, e.g. an emty square lattice x. The deosited articles stick conditionally together - a arameter k determines the conditional robability - and form aggregates (clusters), which might connect two oosite sides of a finite surface for a given article concentration c. The value of c is controlled by k. The most accurate value known for the unconditional ercolation, k=1, is c = (13) obtained in Monte Carlo simulations of 7 billion samles of configurations obtained in a square lattice wraed as a torus no boundary conditions [6]. The goal of our roject is to relate the ercolation threshold c (k) to the reulsive force, e.g. k-value and to study the change of the fractal dimension D f of the connecting (ercolating for ) cluster as a function of k. The classical ercolating cluster is characterized with D f = Error! = (3) [6].

2 II. The model Each simulation starts with an emty (clean) surface a square lattice with x cells each labeled with (i,j), i,j = 1,,3,. A article can stick (be deosited) to an emty, randomly selected cell deending on the state of the neighbouring cells. The rules for deosition are shown in Fig.1.!!! 0 = 1!! 1 = 1 k! 1 = k Fig.1 The model for correlated deosition of articles on a square surface. The emty circle indicates the new article which would be deosited with a robability 1 if there were no neighbours at the trial cell, one neighbour, or two neighbours along a diagonal. If three or two neighbours in u-down or left-right ositions were there, then the robability was 1/k. The most reulsive are four neighbours and the robability for deosition is 1/k. This model simulates a reulsive surface with a force deending on the local environment. Once a article is deosited it is not allowed to diffuse. This is like the dust on the comuter screen coated with a secial anti-dust liquid. It is obvious that k=1 corresonds to the usual ercolation on a square lattice each article is deosited indeendently on the status of the surrounding cells; reulsion is mimicked with k > 1, and the attraction with k<1. III. Results We have studied the influence of k >1 on the ercolation threshold, the fractal dimension and critical exonent for various sizes = For а comarison we resent the data for k=1 as well. We have obtained 0.598±0.000 which is worse that the value reorted in [6] robably due to the aroach (Figure ) we have followed for threshold determination. A finite-size lattice has a connecting rather than ercolating (infinite) cluster. Formation of a connecting cluster (u-down and/or/xor left-right) changes the surface roerties: clean becomes dirty, the isolator becomes a conductor. At low concentrations the robability of formation of a connecting cluster is small it is zero for a number of articles smaller than the lattice side. Following [7] we accet that the mean cluster size distribution is P ( ) = Φ σ c x y 1, Φ( x) = e dy. Figure illustrates the idea of π

3 determination the threshold c for a small lattice size, =50. The disersion σ limits the accuracy of the c determination P = Φ((- c )/σ) P c = ±0.000 σ = ± σ σ Figure Determination of the ercolation threshold from the simulations on a 50x50 square lattice, k=1. The convergence of the threshold found for finites size lattices towards its actual value for has been studied for various k. Recently a romising aroach for finite-size corrections has been ublished [8]. Figure 3 a,b,c shows the size deendence of the ercolation threshold C comuted from c( ) = c( ) 1 with an uncertainty σ ( ) 1 1 [9]. ν ν

4 k= c ( )=0.598± c () c /ν k= c ( )=0.607± c () c /ν Fig.3a Determination of the ercolation threshold for the case of k=1 (the uer art of the figure, this is the usual ercolation) and k=. The reulsion causes a shift towards higher threshold values.

5 k= c ( )=0.604± c () c /ν k= c ( )=0.631±0.0 c () 0.65 c /ν Fig. 3b Determination of the ercolation threshold for the case of k= (the uer art of the figure) and k=. The reulsion causes a shift towards higher threshold values.

6 k= c ( )=0.6338±0.0 c () 0.65 c /ν k=4 c ( )=0.6417± c () c /ν Fig. 3c Determination of the ercolation threshold for the case of k=3 (the uer art of the figure) and k=4. The reulsion causes a shift towards higher threshold values.

7 The table summarizes the results for the ercolation threshold c, its disersion, the critical exonent, and the fractal dimension as functions of the reulsive force, k. The fractal dimension has been comuted using a modification [10] of the box-counting technique [11]. K c σ 100 ν D f ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.00 et us comute the relative change of the threshold and its disersion: indicates the absolute change of the quantity for successive k-values; δ is the uncertainty. We get for the ercolation

8 threshold δ c ~10, and for the disersion δσ 100 ~ 1, i.e. the effect is comarable with its c σ100 error. However, a detailed analysis of the simulation data oints at a decrease of the width in the vicinity of the ercolation. IV. Conclusions The resence of a reulsive k>1 force causes increase of the ercolation threshold. However the critical exonent agrees with the well-known value (1.33) for D ercolation and the fractal dimension is in accordance with the value for the ordinary ercolation. This means that the only role of the reulsive force is to delay the ercolation if we consider the rocess as a function of time for a fixed concentration value. Acknowledgment This work has been financed by a secial grant (F-3/) of the Ministry of Education and Science, Bulgaria, to suort research at the universities. References: [1] Manna S. S. and Jan N., 1991, J. Phys. A [] Stinchcombe, R., 1983, Dilute Magnetism, in Domb, C., and ebowitz, J.., Phase Transitions and Critical Phenomena, Volume 7, Academic Press, [3] Stauffer D and Aharony A 1991 Introduction to Percolation Theory nd edn. (ondon: Taylor and Francis). [4] Stanley H E and Ostrowsky N (eds.) 1990 Correlations and Connectivity (Dordrecht: Kluwer). [5] M.E.J. Newman and R.M. Ziff, Phys. Rev. E64, (001); Phys. Rev. ett. 85, 4104 (000). [6] B. M. Smirnov, The roerties of fractal clusters, Phys. Re. 188, 1 (1990). [7] A.. Efros, Physics and Geometry of Disorder (in Russian htt://edu.ioffe.ru/edu/efros), eq.6. [8] P.M.C. de Oliveira1, R.A. N obrega1, and D. Stauffer, Brazilian Journal of Physics, vol. 33, no. 3, Setember,. [9] D. Stauffer, Scaling theory of ercolation clusters, Physics Reorts 54 1 (1979). [10] B. Karadjov, Master thesis, University of Sofia (1996); in rearation for ublication. [11] B.B. Mandelbrot, The Fractal Geometry of Nature. New York, Freeman, 1983.