MATERlALS SCIENCE & ENGlMEERtWG A ELSEVIER Materials Science and Engineering A217/218 (1996) 381-383 Percolation process in a model granular system in two dimensions S. Toyofuku, T. Odagaki Department of Physics, Kyushu Unitiersity, Fukuoka 812-81, Japan Abstract The role of grain shape on aggregation is studied on the basis of a percolation theory. It is demonstrated that the coverage at the percolation point depends strongly on projections of grains. Ke.vwords: Percolation process; Grain shape; Aggregation; Coverage 1. Introduction Granular materials have been attracting wide atten- tion because of their unusual phenomena such as giant magnetoresistance [I]. Since the properties of granular materials depend strongly on their structure, it is im- portant to characterize the structure of these systems. In this paper, we study the structure of a model granular system in two dimensions using the percola- tion theory [2]. We investigate how the size and shape of grains affect the percolation process using a Monte Carlo simulation and show that the shape of grains plays an important role in determining the critical percolation coverage (CPC). We also discuss the depen- dence of the critical percolation grain density (CPGD) on the grain size. 2. Model Grains of the same size and shape are randomly distributed on a square lattice. We characterize a two- dimensional grain by the generation n, where a grain consists of lattice points which are located within the nth nearest neighbour distance t;t from the center. Fig. 1 shows grains up to the 10th generation. In our model system, overlapping of grains is permitted except when they occupy exactly the same position. This assump- tion, similar to that in the continuum percolation, may represent the situation when the grain is grown on a surface. 3. Results and discussion We study 300 x 300 and 500 x 500 square lattices with periodic boundary conditions in the simulation and distribute grains randomly using a pseudo-random number generator. The number of grains per lattice site is denoted by pi and the fraction of lattice points covered by grains by pg. In Fig. 2, pg is plotted as a function of pi for n = 0, 1,. . .,8. The percolation is judged by the existence of a cluster joining all sides of the system. Closed circles in Fig. 2 denote the percola- tion point for each generation n which in turn def?nes CPGD, p;(n), and CPC, p;(n), Needless to say, p;(O) = p;(O), being very close to 0.593. This corresponds to the critical percolation concentration for the site process on the square lattice. . .:. .. . ::: 2. . . .. . . .. . . n=O 1 2 3 rn = 0.00 1 .oo 1.41 2.00 l *. . ... * ::::: ::::: ::::: . . . . ..** ... ..... . .. .... . ... *.. l :::::* . . . 4 5 6 7 2.24 2.83 3.00 3.16 .*... A.. . ..’ . . . . . . . s...... .***. . . . . . . . . . . . . . . .*..... . . . . . . . ..a . . . . l , . . . . . . . . . . . . . . . . . . . . . . . . ..* . . . . . l :::::* l :::::* . . . . . . . . . . . . . . . . . .a... . . . 8 9 10' 3.61 4.00 4.12 Fig. 1. Grains for the 0th to the 10th generation. Grains for n = 1, 6, 9 have projections on the surface. 0921-5093(96/$15.00 0 1996 - Elsevier Science S.A. All rights reserved PII SO921-5093(96)10336-l