Numerical simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of straight semirigid rods adsorbed on two-dimensional square lattices. The depositing objects can be adsorbed on the surface forming two layers. The filling of the lattice is carried out following a generalized random sequential adsorption (RSA) mechanism. In each elementary step, (i) a set of $k$ consecutive nearest-neighbor sites (aligned along one of two lattice axes) is randomly chosen and (ii) if each selected site is either empty or occupied by a $k$-mer unit in the first layer, then a new $k$-mer is then deposited onto the lattice. Otherwise, the attempt is rejected. The process starts with an initially empty lattice and continues until the jamming state is reached and no more objects can be deposited due to the absence of empty site clusters of appropriate size and shape. A wide range of values of $k$ ($2\le k\le 64$) is investigated. The study of the kinetic properties of the system shows that (1) the jamming coverage ${\theta }_{j,k}$ is a decreasing function with increasing $k$, with ${\theta }_{j,k\to \infty }=0.7299\left(21\right)$ the limit value for infinitely long $k$-mers and (2) the jamming exponent ${\nu }_j$ remains close to 1, regardless of the size $k$ considered. These findings are discussed in terms of the lattice dimensionality and number of sites available for adsorption. The dependence of the percolation threshold ${\theta }_{c,k}$ as a function of $k$ is also determined, with ${\theta }_{c,k}=A+Bexp(-k/C)$, where $A={\theta }_{c,k\to \infty }=0.0457\left(68\right)$ is the value of the percolation threshold by infinitely long $k$-mers, $B=0.276\left(25\right)$, and $C=14\left(2\right)$. This monotonic decreasing behavior is completely different from that observed for the standard problem of straight rods on square lattices, where the percolation threshold shows a nonmonotonic $k$-mer size dependence. The differences between the results obtained from bilayer and monolayer phases are explained on the basis of the transversal overlaps between rods occurring in the bilayer problem. This effect (which we call a “cross-linking effect”), its consequences on the filling kinetics, and its implications in the field of conductivity of composites filled with elongated particles (or fibers) are discussed in detail. Finally, the precise determination of the critical exponents $\nu , \beta$, and $\gamma$ indicates that, although the increasing in the width of the deposited layer drastically affects the behavior of the percolation threshold with $k$ and other critical properties (such as the crossing points of the percolation probability functions), it does not alter the nature of the percolation transition occurring in the system. Accordingly, the bilayer model belongs to the same universality class as two-dimensional standard percolation model.

• Received 11 June 2020
• Accepted 13 July 2020

DOI:https://doi.org/10.1103/PhysRevE.102.012153