Abstract

We consider the percolation model with nucleation and simultaneous growth of multiple finite clusters, taking the initial seed concentration \(\rho \) as a tunable parameter. Growing objects expand with constant speed, filling the nodes of the triangular lattice according to rules that control their shape. As growing objects of predefined shape, we consider needle-like objects and “wrapping” objects whose size is gradually increased by wrapping the walks in several different ways, making triangles, rhombuses, and hexagons. Growing random walk chains are also analyzed as an example of objects whose shape is formed randomly during the growth. We compare the percolation properties and jamming densities of the systems of various growing shapes for a wide range of initial seed densities \(\rho < 0.5\). To gain a basic insight into the structure of the jammed states, we consider the size distribution of deposited growing objects. The presence of the most numerous and the largest growing objects is recorded for the system in the jamming state. Our results suggest that at sufficiently low seed densities \(\rho \), the way of the object growth has a substantial influence on the percolation threshold. This influence weakens with increasing \(\rho \) and ceases near the value of the site percolation threshold for monomers on the triangular lattice, \(\rho _\text {p}^* = 0.5\).

Graphical Abstract

中文翻译：

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颗粒形状控制的种子生长模型中的渗流和干扰特性

摘要

我们考虑具有多个有限簇的成核和同时生长的渗透模型，将初始种子浓度\(\rho \)作为可调参数。生长的物体以恒定的速度膨胀，根据控制其形状的规则填充三角形格子的节点。作为预定义形状的增长对象，我们考虑针状对象和“包裹”对象，这些对象的大小通过以几种不同的方式包裹步道而逐渐增加，制作三角形、菱形和六边形。生长的随机游走链也被分析为在生长过程中随机形成形状的对象的示例。我们比较了各种生长形状系统的渗透特性和干扰密度，用于各种初始种子密度\(\rho < 0.5\). 为了对阻塞状态的结构有一个基本的了解，我们考虑了沉积的生长物体的尺寸分布。为处于干扰状态的系统记录最多数量和最大生长对象的存在。我们的结果表明，在足够低的种子密度\(\rho \)下，物体生长的方式对渗透阈值有很大的影响。这种影响随着\(\rho \)的增加而减弱，并在三角形晶格上单体的位点渗透阈值附近停止，\(\rho _\text {p}^* = 0.5\)。

图形概要