RESUMEN
Irreversible multilayer adsorption of semirigid k-mers on one-dimensional lattices of size L is studied by numerical simulations complemented by exhaustive enumeration of configurations for small lattices. The deposition process is modeled by using a random sequential adsorption algorithm, generalized to the case of multilayer adsorption. The paper concentrates on measuring the jamming coverage for different values of k-mer size and number of layers n. The bilayer problem (n≤2) is exhaustively analyzed, and the resulting tendencies are validated by the exact enumeration techniques. Then, the study is extended to an increasing number of layers, which is one of the noteworthy parts of this work. The obtained results allow the following: (i) to characterize the structure of the adsorbed phase for the multilayer problem. As n increases, the (1+1)-dimensional adsorbed phase tends to be a "partial wall" consisting of "towers" (or columns) of width k, separated by valleys of empty sites. The length of these valleys diminishes with increasing k; (ii) to establish that this is an in-registry adsorption process, where each incoming k-mer is likely to be adsorbed exactly onto an already adsorbed one. With respect to percolation, our calculations show that the percolation probability vanishes as L increases, being zero in the limit Lâ∞. Finally, the value of the jamming critical exponent ν_{j} is reported here for multilayer adsorption: ν_{j} remains close to 2 regardless of the considered values of k and n. This finding is discussed in terms of the lattice dimensionality.
RESUMEN
Percolation and jamming of k×k square tiles (k^{2}-mers) deposited on square lattices have been studied by numerical simulations complemented with finite-size scaling theory and exact enumeration of configurations for small systems. The k^{2}-mers were irreversibly deposited into square lattices of sizes L×L with L/k ranging between 128 and 448 (64 and 224) for jamming (percolation) calculations. Jamming coverage θ_{j,k} was determined for a wide range of k values (2≤k≤100 with many intermediate k values to allow a fine scaling analysis). θ_{j,k} exhibits a decreasing behavior with increasing k, being θ_{j,k=∞}=0.5623(3) the limit value for large k^{2}-mer sizes. In addition, a finite-size scaling analysis of the jamming transition was carried out, and the corresponding spatial correlation length critical exponent ν_{j} was measured, being ν_{j}≈1. On the other hand, the obtained results for the percolation threshold θ_{c,k} showed that θ_{c,k} is an increasing function of k in the range 1≤k≤3. For k≥4, all jammed configurations are nonpercolating states and, consequently, the percolation phase transition disappears. An explanation for this phenomenon is offered in terms of the rapid increase with k of the number of surrounding occupied sites needed to reach percolation, which gets larger than the sufficient number of occupied sites to define jamming. In the case of k=2 and 3, the percolation thresholds are θ_{c,k=2}(∞)=0.60355(8) and θ_{c,k=3}=0.63110(9). Our results significantly improve the previously reported values of θ_{c,k=2}^{Naka}=0.601(7) and θ_{c,k=3}^{Naka}=0.621(6) [Nakamura, Phys. Rev. A 36, 2384 (1987)0556-279110.1103/PhysRevA.36.2384]. In parallel, a comparison with previous results for jamming on these systems is also done. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the ordinary random percolation, regardless of the size k considered.
RESUMEN
The deposition of one-dimensional objects (such as polymers) on a one-dimensional lattice with the presence of impurities is studied in order to find saturation conditions in what is known as jamming. Over a critical concentration of k-mers (polymers of length k), no further depositions are possible. Five different nematic (directional) depositions are considered: baseline, irreversible, configurational, loose-packing, and close-packing. Correspondingly, five jamming functions are found, and their dependencies on the length of the lattice, L, the concentration of impurities, p=M/L (where M is the number of one-dimensional impurities), and the length of the k-mer (k) are established. In parallel, numeric simulations are performed to compare with the theoretical results. The emphasis is on trimers (k=3) and p in the range [0.01,0.15], however other related cases are also considered and reported.
RESUMEN
Percolation of site trimers (k-mers with k=3) is investigated in a detailed way making use of an analytical model based on renormalization techniques in this problem. Results are compared to those obtained here by means of extensive computer simulations. Five different deposition possibilities for site trimers are included according to shape and orientation of the depositing objects. Analytical results for the percolation threshold p(c) are all close to 0.55, while numerical results show a slight dispersion around this value. A comparison with p(c) values previously reported for monomers and dimers establishes the tendency of p(c) to decrease as k increases. Critical exponent ν was also obtained both by analytical and numerical methods. Results for the latter give values very close to the expected value 4/3 showing that this percolation case corresponds to the universality class of random percolation.