单团簇生长模型中平行正方形的阻塞与渗流
Jamming and percolation of parallel squares in single-cluster growth model
摘要：
本文研究了单簇生长模型中平行正方形的阻塞和渗流问题。Leath-Alexandrowicz方法用于从活性种子位点生长簇。正方形晶格的位置被相同大小的$k \times k$正方形（E-问题）或$k \times k$和$m \times m$（$m \leqslant k$）正方形的混合（M-问题）占据。假设较大的$k \times k$方块为有源（导电），较小的$m \times m$方块为阻塞（不导电）。对于相等大小的$k \times k$正方形（E-问题），在$k\rightarrow\infty$的极限内获得干扰浓度的$p_j = 0.638 \pm 0.001$值。该值明显大于先前报道的随机顺序吸附模型，$p_j = 0.564 \pm 0.002$。观察到逾渗阈值$p_{\mathrm{c}}$（即，活性$k \times k$正方形面积与渗滤点处$k \times k$正方形总面积的比率）随着$k$的增加而增加。对于$k \times k$和$m \times m$平方的混合（M-问题），在$m$的固定值下，随着$k$的增加，$p_{\mathrm{c}}$的值显著增加，并且在$k\geqslant 10m$时接近1。这反映了在M-问题中，当存在较小的阻塞正方形时，可以有效地抑制较大的活动正方形的渗流。
英文摘要：
This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size $k \times k$ squares (E-problem) or a mixture of $k \times k$ and $m \times m$ ($m \leqslant k$) squares (M-problem). The larger $k \times k$ squares were assumed to be active (conductive) and the smaller $m \times m$ squares were assumed to be blocked (non-conductive). For equal size $k \times k$ squares (E-problem) the value of $p_j = 0.638 \pm 0.001$ was obtained for the jamming concentration in the limit of $k\rightarrow\infty$. This value was noticeably larger than that previously reported for a random sequential adsorption model, $p_j = 0.564 \pm 0.002$. It was observed that the value of percolation threshold $p_{\mathrm{c}}$ (i.e., the ratio of the area of active $k \times k$ squares and the total area of $k \times k$ squares in the percolation point) increased with an increase of $k$. For mixture of $k \times k$ and $m \times m$ squares (M-problem), the value of $p_{\mathrm{c}}$ noticeably increased with an increase of $k$ at a fixed value of $m$ and approached 1 at $k\geqslant 10m$. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.
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