Percolation of randomly distributed growing clusters

arXiv:1004.1526v2 [cond-mat.stat-mech] 6 Jul 2010 Percolation of randomly distributed growing clusters N. Tsakiris,1 M. Maragakis,1 K. Kosmidis,1 and P. Argyrakis1 1Department of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece (Dated: September 9, 2018) We investigate the problem of growing clusters, which is modeled by two dimensional disks and three dimensional droplets. In this model we place a number of seeds on random locations on a lattice with an initial occupation probability, p. The seeds simultaneously grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The probability that such a system will result in a percolating cluster depends on the density of the initially distributed seeds and the dimensionality of the system. For very low initial values of p we find a power law behavior for several properties that we investigate, namely for the size of the largest and second largest cluster, for the probability for a site to belong to the finally formed spanning cluster, and for the mean radius of the finally formed droplets. We report the values of the corresponding scaling exponents. Finally, we show that for very low initial concentration of seeds the final coverage takes a constant value which depends on the system dimensionality. PACS numbers: 64.60.ah, 61.43.Bn, 05.70.Fh, 81.05.Rm I. INTRODUCTION Percolation theory has drawn a continuous interest from the scientific community for several years [1]-[3]. It has been studied in a wide variety of systems ranging from lattices [4] to complex networks [3]. The fields that percolation applies are as diverse as electromagnetism [5]- [6], chemistry [7], materials science [8], geology [9], social systems [10], wireless networks [11] and many more. In chemistry and materials science it is of major importance for the movement of liquids or gases in porous media. Problems in this area relate to the leakage in seals [12] and the gas permeability in cement paste [13]. Various algorithms have been used to simulate the phase transformation kinetics. In many pattern forma- tion models several small spherical seeds are nucleated at a constant rate (homogeneous nucleation). Seeds can also initiate on defects in the case of heterogeneous nucle- ation. From the simulation point of view the defects are considered as points in the lattice representing the seeds. The seeds once formed are in a metastable phase and grow at a constant velocity as long as there is adequate available material for adsorption. Additionally, several models exist that do not allow the adsorption of a new particle in contact with or overlap- ping with an already adsorbed one. An example is the random sequential adsorption (RSA) model [14]. This model has been extensively used for colloid and globu- lar protein adsorption in heterogeneous surfaces. In such systems discrete lattice sites can act as adsorption sites with attractive short range interactions [15]. The jam- ming coverage and the structure of the particle monolayer as a function of the site coverage and the particle/site size ratio have been studied. Models studying pattern formation ranging in between these two cases have not been used extensively. An- drienko proposed [16] the idea of disks and droplets grow- ing at a constant rate on random initial sites over the lattice and stopping once they come in contact. In the so called “Touch and Stop” model the droplets grow at a constant rate in all directions (circular in 2D, spherical in 3D). The main characteristic of this model is located in the notion that the droplets stop growing after two or more of them come in contact. This can be due to several reasons. In material science it is possible to have a strong surface tension that inhibits the nuclei from taking any shape other than that of a circular or spherical one. Addition- ally, a significant interacting force between the substrate and the forming droplet can prevent two or more discs from coalescing in the time scale needed for the growth of other islands. This problem also relates to the well studied Apollo- nian packing problem [17] for circles and spheres. In fact it can be considered as a random version of packing with various discrete sizes, where growth velocity is constant but not infinitely large. The Touch and Stop model has also been studied in some variations (random insertion of seeds in time) as a packing limited growth problem [18]. II. MODEL DESCRIPTION The system used can be described as follows. Ini- tially, lattices of 106 sites (1000 × 1000 for 2D, and 100×100×100 for 3D) are randomly populated with seeds of singular size in a non overlapping way. The initial oc- cupation probability of these sites is p. At every time step all seeds are investigated once as to the possibility of growing in size instantaneously in all neighboring sites. Investigation sequence is random in order. Each seed is allowed to grow its peripher

…(Full text truncated)…

This content is AI-processed based on ArXiv data.