Percolation of aligned rigid rods on two-dimensional square lattices

Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices

We investigate the percolation transition of aligned, overlapping, anisotropic shapes on lattices. Using the recently proposed lattice version of excluded volume theory, we show that shape-anisotropy leads to some intriguing consequences regarding the percolation behavior of anisotropic shapes. We consider a prototypical anisotropic shape-rectangle-on a square lattice and show that, for rectangles of width unity (sticks), the percolation threshold is a monotonically decreasing function of the stick length, whereas, for rectangles of width greater than two, it is a monotonically increasing function. Interestingly, for rectangles of width two, the percolation threshold is independent of its length. We show that this independence of threshold on the length of a side holds for d-dimensional hypercubiods as well as for specific integer values for the lengths of the remaining sides. The limiting case of the length of the rectangles going to infinity shows that the limiting threshold value is finite and depends upon the width of the rectangle. This "continuum" limit with the lattice spacing tending to zero only along a subset of the possible directions in d dimensions results in a "semicontinuum" percolation system. We show that similar results hold for other anisotropic shapes and lattices in different dimensions. The critical properties of the aligned and overlapping rectangles are evaluated using Monte Carlo simulations. We find that the threshold values given by the lattice-excluded volume theory are in good agreement with the simulation results, especially for larger rectangles. We verify the isotropy of the percolation threshold and also compare our results with models where rectangles of mixed orientation are allowed. Our simulation results show that alignment increases the percolation threshold. The calculation of critical exponents places the model in the standard percolation universality class. Our results show that shape anisotropy of the aligned, overlapping percolating units has a marked influence on the percolation properties, especially when a subset of the dimensions of the percolation units is made to diverge.

Standard and inverse site percolation of triangular tiles on triangular lattices: Isotropic and perfectly oriented deposition and removal

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of triangular tiles of side k (k-tiles) on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, k-tiles are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing k-tiles [composed by k(k+1)/2 monomers] from the lattice. Two schemes are used for the depositing and removing process: the isotropic scheme, where the deposition (removal) of the objects occurs with the same probability in any lattice direction; and the anisotropic (perfectly oriented or nematic) scheme, where one lattice direction is privileged for depositing (removing) the tiles. The study is conducted by following the behavior of four critical concentrations with the size k: (i) [(ii)] standard isotropic (oriented) percolation threshold θ_{c,k} (ϑ_{c,k}), which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k} (ϑ_{c,k}) is reached by isotropic (oriented) deposition of k-tiles on an initially empty lattice; and (iii) [(iv)] inverse isotropic (oriented) percolation threshold θ_{c,k}^{i} (ϑ_{c,k}^{i}), which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i} (ϑ_{c,k}^{i}) is reached after removing isotropic (completely aligned) k-tiles from an initially fully occupied lattice. The obtained results indicate that (1)θ_{c,k} (θ_{c,k}^{i}) is an increasing (decreasing) function of k in the range 1≤k≤6. For k≥7, all jammed configurations are nonpercolating (percolating) states and, consequently, the percolation phase transition disappears. (2)ϑ_{c,k} (ϑ_{c,k}^{i}) show a behavior qualitatively similar to that observed for isotropic deposition. In this case, the minimum value of k at which the phase transition disappears is k=5. (3) For both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(ϑ)=0.5. Thus, a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and ϑ_{c,k}+ϑ_{c,k}^{i}=1), which has not been observed in other regular lattices. (4) Finally, in all cases, the jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the orientation (isotropic or nematic) or the size k considered. In addition, the accurate determination of the critical exponents ν, β, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and five (three) for isotropic (nematic) deposition scheme, has the same universality class as the standard percolation problem.

Universal Critical Behavior of Percolation in Orientationally Ordered Janus Particles and Other Anisotropic Systems

We combine percolation theory and Monte Carlo simulation to study in two dimensions the connectivity of an equilibrium lattice model of interacting Janus disks which self-assemble into an orientationally ordered stripe phase at low temperature. As the patch size is increased or the temperature is lowered, clusters of patch-connected disks grow, and a percolating cluster emerges at a threshold. In the stripe phase, the critical clusters extend longer in the direction parallel to the stripes than in the perpendicular direction, and percolation is thus anisotropic. It is found that the critical behavior of percolation in the Janus system is consistent with that of standard isotropic percolation, when an appropriate spatial rescaling is made. The rescaling procedure can be applied to understand other anisotropic systems, such as the percolation of aligned rigid rods and of the q-state Potts model with anisotropic interactions.

Percolation thresholds of randomly rotating patchy particles on Archimedean lattices

We study the percolation of randomly rotating patchy particles on 11 Archimedean lattices in two dimensions. Each vertex of the lattice is occupied by a particle, and in each model the patch size and number are monodisperse. When there are more than one patches on the surface of a particle, they are symmetrically decorated. As the proportion χ of the particle surface covered by the patches increases, the clusters connected by the patches grow and the system percolates at the threshold χ_{c}. We combine Monte Carlo simulations and the critical polynomial method to give precise estimates of χ_{c} for disks with one to six patches and spheres with one to two patches on the 11 lattices. For one-patch particles, we find that the order of χ_{c} values for particles on different lattices is the same as that of threshold values p_{c} for site percolation on these lattices, which implies that χ_{c} for one-patch particles mainly depends on the geometry of lattices. For particles with more patches, symmetry becomes very important in determining χ_{c}. With the estimates of χ_{c} for disks with one to six patches, using analyses related to symmetry, we are able to give precise values of χ_{c} for disks with an arbitrary number of patches on all 11 lattices. The following rules are found for patchy disks on each of these lattices: (1) as the number of patches n increases, values of χ_{c} repeat in a periodic way, with the period n_{0} determined by the symmetry of the lattice; (2) when mod(n,n_{0})=0, the minimum threshold value χ_{min} appears, and the model is equivalent to site percolation with χ_{min}=p_{c}; and (3) disks with mod(n,n_{0})=m and n_{0}-m (m<n_{0}/2) share the same χ_{c} value. The results can be useful references for studying the connectivity of patchy particles on two-dimensional lattices at finite temperatures.

Standard and inverse site percolation of straight rigid rods on triangular lattices: Isotropic and perfectly oriented deposition and removal

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, linear k-mers (particles occupying k consecutive sites along one of the lattice directions) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by randomly removing sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with size k: (i) [(ii)] standard isotropic[oriented] percolation threshold θ_{c,k}[ϑ_{c,k}], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ_{c,k}[ϑ_{c,k}] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold θ_{c,k}^{i}[ϑ_{c,k}^{i}], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i}[ϑ_{c,k}^{i}] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θ_{c,k}, ϑ_{c,k}, θ_{c,k}^{i}, and ϑ_{c,k}^{i} are determined for a wide range of k (2≤k≤512). The obtained results indicate that (1)θ_{c,k}[θ_{c,k}^{i}] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k=11, and finally increases and asymptotically converges towards a definite value for large segments θ_{c,k→∞}=0.500(2) [θ_{c,k→∞}^{i}=0.500(1)]; (2)ϑ_{c,k}[ϑ_{c,k}^{i}] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers ϑ_{c,k→∞}=0.5334(6) [ϑ_{c,k→∞}^{i}=0.4666(6)]; (3) for both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(ϑ)=0.5. Thus a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and ϑ_{c,k}+ϑ_{c,k}^{i}=1) which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems, and (4) in all cases, the critical concentration curves divide the θ space in a percolating region and a nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.

Percolation of aligned rigid rods on two-dimensional triangular lattices

The percolation behavior of aligned rigid rods of length k (k-mers) on two-dimensional triangular lattices has been studied by numerical simulations and finite-size scaling analysis. The k-mers, containing k identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The connectivity analysis was carried out by following the probability R_{L,k}(p) that a lattice composed of L×L sites percolates at a concentration p of sites occupied by particles of size k. The results, obtained for k ranging from 2 to 80, showed that the percolation threshold p_{c}(k) exhibits a increasing function when it is plotted as a function of the k-mer size. The dependence of p_{c}(k) was determined, being p_{c}(k)=A+B/(C+sqrt[k]), where A=p_{c}(k→∞)=0.582(9) is the value of the percolation threshold by infinitely long k-mers, B=-0.47(0.21), and C=5.79(2.18). This behavior is completely different from that observed for square lattices, where the percolation threshold decreases with k. In addition, the effect of the anisotropy on the properties of the percolating phase was investigated. The results revealed that, while for finite systems the anisotropy of the deposited layer favors the percolation along the parallel direction to the alignment axis, in the thermodynamic limit, the value of the percolation threshold is the same in both parallel and transversal directions. Finally, an exhaustive study of critical exponents and universality was carried out, showing that the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered.

Jamming and percolation of k^{3}-mers on simple cubic lattices

Jamming and percolation of three-dimensional (3D) k×k×k cubic objects (k^{3}-mers) deposited on simple cubic lattices have been studied by numerical simulations complemented with finite-size scaling theory. The k^{3}-mers were irreversibly deposited into the lattice. Jamming coverage θ_{j,k} was determined for a wide range of k (2≤k≤40). θ_{j,k} exhibits a decreasing behavior with increasing k, being θ_{j,k=∞}=0.4204(9) the limit value for large k^{3}-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}≈3/2. However, the obtained results for the percolation threshold θ_{p,k} showed that θ_{p,k} is an increasing function of k in the range 2≤k≤16. For k≥17, all jammed configurations are nonpercolating states, and consequently, the percolation phase transition disappears. The interplay between the percolation and the jamming effects is responsible for the existence of a maximum value of k (in this case, k=16) from which the percolation phase transition no longer occurs. 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 3D random percolation, regardless of the size k considered.

Jamming and percolation for deposition of k^{2}-mers on square lattices: A Monte Carlo simulation study

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.

Continuum percolation of polydisperse rods in quadrupole fields: Theory and simulations

We investigate percolation in mixtures of nanorods in the presence of external fields that align or disalign the particles with the field axis. Such conditions are found in the formulation and processing of nanocomposites, where the field may be electric, magnetic, or due to elongational flow. Our focus is on the effect of length polydispersity, which-in the absence of a field-is known to produce a percolation threshold that scales with the inverse weight average of the particle length. Using a model of non-interacting spherocylinders in conjunction with connectedness percolation theory, we show that a quadrupolar field always increases the percolation threshold and that the universal scaling with the inverse weight average no longer holds if the field couples to the particle length. Instead, the percolation threshold becomes a function of higher moments of the length distribution, where the order of the relevant moments crucially depends on the strength and type of field applied. The theoretical predictions compare well with the results of our Monte Carlo simulations, which eliminate finite size effects by exploiting the fact that the universal scaling of the wrapping probability function holds even in anisotropic systems. Theory and simulation demonstrate that the percolation threshold of a polydisperse mixture can be lower than that of the individual components, confirming recent work based on a mapping onto a Bethe lattice as well as earlier computer simulations involving dipole fields. Our work shows how the formulation of nanocomposites may be used to compensate for the adverse effects of aligning fields that are inevitable under practical manufacturing conditions.

Diffusion-driven self-assembly of rodlike particles: Monte Carlo simulation on a square lattice

The diffusion-driven self-assembly of rodlike particles was studied by means of Monte Carlo simulation. The rods were represented as linear k-mers (i.e., particles occupying k adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size L×L up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, L, was varied from 128 to 2048, and periodic boundary conditions were applied along both x and y axes, while the length of the k-mers (determining the aspect ratio) was varied from 2 to 12. The k-mers oriented along the x and y directions (k_{x}-mers and k_{y}-mers, respectively) were deposited equiprobably. In the course of the simulation, the numbers of intraspecific and interspecific contacts between the same sort and between different sorts of k-mers, respectively, were calculated. Both the shift ratio of the actual number of shifts along the longitudinal or transverse axes of the k-mers and the electrical conductivity of the system were also examined. For the initial random configuration, quite different self-organization behavior was observed for short and long k-mers. For long k-mers (k≥6), three main stages of diffusion-driven spatial segregation (self-assembly) were identified: the initial stage, reflecting destruction of the jamming state; the intermediate stage, reflecting continuous cluster coarsening and labyrinth pattern formation; and the final stage, reflecting the formation of diagonal stripe domains. Additional examination of two artificially constructed initial configurations showed that this pattern of diagonal stripe domains is an attractor, i.e., any spatial distribution of k-mers tends to transform into diagonal stripes. Nevertheless, the time for relaxation to the steady state essentially increases as the lattice size growth.

Monte Carlo simulation of evaporation-driven self-assembly in suspensions of colloidal rods

The vertical drying of a colloidal film containing rodlike particles was studied by means of kinetic Monte Carlo (MC) simulation. The problem was approached using a two-dimensional square lattice, and the rods were represented as linear k-mers (i.e., particles occupying k adjacent sites). The initial state before drying was produced using a model of random sequential adsorption (RSA) with isotropic orientations of the k-mers (orientation of the k-mers along horizontal x and vertical y directions are equiprobable). In the RSA model, overlapping of the k-mers is forbidden. During the evaporation, an upper interface falls with a linear velocity of u in the vertical direction and the k-mers undergo translation Brownian motion. The MC simulations were run at different initial concentrations, p_{i}, (p_{i}∈[0,p_{j}], where p_{j} is the jamming concentration), lengths of k-mers (k∈[1,12]), and solvent evaporation rates, u. For completely dried films, the spatial distributions of k-mers and their electrical conductivities in both x and y directions were examined. Significant evaporation-driven self-assembly and orientation stratification of the k-mers oriented along the x and y directions were observed. The extent of stratification increased with increasing value of k. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of p_{i}, k, and u.

Electrical conductivity of a monolayer produced by random sequential adsorption of linear k-mers onto a square lattice

The electrical conductivity of a monolayer produced by the random sequential adsorption (RSA) of linear k-mers (particles occupying k adjacent adsorption sites) onto a square lattice was studied by means of computer simulation. Overlapping with predeposited k-mers and detachment from the surface were forbidden. The RSA process continued until the saturation jamming limit, p_{j}. The isotropic (equiprobable orientations of k-mers along x and y axes) and anisotropic (all k-mers aligned along the y axis) depositions for two different models-of an insulating substrate and conducting k-mers (C model) and of a conducting substrate and insulating k-mers (I model)-were examined. The Frank-Lobb algorithm was applied to calculate the electrical conductivity in both the x and y directions for different lengths (k=1 - 128) and concentrations (p=0 - p_{j}) of the k-mers. The "intrinsic electrical conductivity" and concentration dependence of the relative electrical conductivity Σ(p) (Σ=σ/σ_{m} for the C model and Σ=σ_{m}/σ for the I model, where σ_{m} is the electrical conductivity of substrate) in different directions were analyzed. At large values of k the Σ(p) curves became very similar and they almost coincided at k=128. Moreover, for both models the greater the length of the k-mers the smoother the functions Σ_{xy}(p),Σ_{x}(p) and Σ_{y}(p). For the more practically important C model, the other interesting findings are (i) for large values of k (k=64,128), the values of Σ_{xy} and Σ_{y} increase rapidly with the initial increase of p from 0 to 0.1; (ii) for k≥16, all the Σ_{xy}(p) and Σ_{x}(p) curves intersect with each other at the same isoconductivity points; (iii) for anisotropic deposition, the percolation concentrations are the same in the x and y directions, whereas, at the percolation point the greater the length of the k-mers the larger the anisotropy of the electrical conductivity, i.e., the ratio σ_{y}/σ_{x} (>1).

Percolation of heteronuclear dimers irreversibly deposited on square lattices

The percolation problem of irreversibly deposited heteronuclear dimers on square lattices is studied. A dimer is composed of two segments, and it occupies two adjacent adsorption sites. Each segment can be either a conductive segment (segment type A) or a nonconductive segment (segment type B). Three types of dimers are considered: AA, BB, and AB. The connectivity analysis is carried out by accounting only for the conductive segments (segments type A). The model offers a simplified representation of the problem of percolation of defective (nonideal) particles, where the presence of defects in the system is simulated by introducing a mixture of conductive and nonconductive segments. Different cases were investigated, according to the sequence of deposition of the particles, the types of dimers involved in the process, and the degree of alignment of the deposited objects. By means of numerical simulations and finite-size scaling analysis, the complete phase diagram separating a percolating from a nonpercolating region was determined for each case. Finally, the consistency of our results was examined by comparing with previous data in the literature for linear k-mers (particles occupying k adjacent sites) with defects.

Jamming and percolation in generalized models of random sequential adsorption of linear k-mers on a square lattice

The jamming and percolation for two generalized models of random sequential adsorption (RSA) of linear k-mers (particles occupying k adjacent sites) on a square lattice are studied by means of Monte Carlo simulation. The classical RSA model assumes the absence of overlapping of the new incoming particle with the previously deposited ones. The first model is a generalized variant of the RSA model for both k-mers and a lattice with defects. Some of the occupying k adjacent sites are considered as insulating and some of the lattice sites are occupied by defects (impurities). For this model even a small concentration of defects can inhibit percolation for relatively long k-mers. The second model is the cooperative sequential adsorption one where, for each new k-mer, only a restricted number of lateral contacts z with previously deposited k-mers is allowed. Deposition occurs in the case when z≤(1-d)z(m) where z(m)=2(k+1) is the maximum numbers of the contacts of k-mer, and d is the fraction of forbidden contacts. Percolation is observed only at some interval k(min)≤k≤k(max) where the values k(min) and k(max) depend upon the fraction of forbidden contacts d. The value k(max) decreases as d increases. A logarithmic dependence of the type log(10)(k(max))=a+bd, where a=4.04±0.22,b=-4.93±0.57, is obtained.

Site trimer percolation on square lattices

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.

The structure of adsorbed cyclic chains

In order to determine the structure of polymer films formed of cyclic chains (rings) we developed and studied a simple coarse-grained model. Our main goal was to check how the percolation and jamming thresholds in such a system were related to the thresholds obtained for linear flexible chains system, i.e., how the geometry of objects influenced both thresholds. All atomic details were suppressed and polymers were represented as a sequence of identical beads and the chains were embedded to a square lattice (a strictly 2D model). The system was athermal and the excluded volume was the only potential introduced. A random sequential adsorption algorithm was chosen to determine the properties of a polymer monolayer. It was shown that the percolation threshold of cyclic chains was considerably higher than those of linear flexible chains while the jamming thresholds for both chain architectures are very similar. The shape of adsorbed cyclic chains was found to be more prolate when compared to average single chain.

Impact of defects on percolation in random sequential adsorption of linear k-mers on square lattices

The effect of defects on the percolation of linear k-mers (particles occupying k adjacent sites) on a square lattice is studied by means of Monte Carlo simulation. The k-mers are deposited using a random sequential adsorption mechanism. Two models L(d) and K(d) are analyzed. In the L(d) model it is assumed that the initial square lattice is nonideal and some fraction of sites d is occupied by nonconducting point defects (impurities). In the K(d) model the initial square lattice is perfect. However, it is assumed that some fraction of the sites in the k-mers d consists of defects, i.e., is nonconducting. The length of the k-mers k varies from 2 to 256. Periodic boundary conditions are applied to the square lattice. The dependences of the percolation threshold concentration of the conducting sites p(c) vs the concentration of defects d are analyzed for different values of k. Above some critical concentration of defects d(m), percolation is blocked in both models, even at the jamming concentration of k-mers. For long k-mers, the values of d(m) are well fitted by the functions d(m)∝k(m)(-α)-k(-α) (α=1.28±0.01 and k(m)=5900±500) and d(m)∝log(10)(k(m)/k) (k(m)=4700±1000) for the L(d) and K(d) models, respectively. Thus, our estimation indicates that the percolation of k-mers on a square lattice is impossible even for a lattice without any defects if k⪆6×10(3).

Reentrant disordered phase in a system of repulsive rods on a Bethe-like lattice

We solve exactly a model of monodispersed rigid rods of length k with repulsive interactions on the random locally tree-like layered lattice. For k≥4 we show that with increasing density, the system undergoes two phase transitions: first, from a low-density disordered phase to an intermediate density nematic phase and, second, from the nematic phase to a high-density reentrant disordered phase. When the coordination number is four, both phase transitions are continuous and in the mean field Ising universality class. For an even coordination number larger than four, the first transition is discontinuous, while the nature of the second transition depends on the rod length k and the interaction parameters.

Percolation of linear k-mers on a square lattice: from isotropic through partially ordered to completely aligned states

Numerical simulations by means of Monte Carlo method and finite-size scaling analysis have been performed to study the percolation behavior of linear k-mers (also denoted in publications as rigid rods, needles, sticks) on two-dimensional square lattices L × L with periodic boundary conditions. Percolation phenomena are investigated for anisotropic relaxation random sequential adsorption of linear k-mers. Especially, effect of anisotropic placement of the objects on the percolation threshold has been investigated. A detailed study of the behavior of percolation probability R(L)(p) that a lattice of size L percolates at concentration p in dependence on k, anisotropy, and lattice size L has been performed. A nonmonotonic size dependence for the percolation threshold has been confirmed in the isotropic case. We propose a fitting formula for percolation threshold, p(c) = a/k(α)+blog(10)k+c, where a, b, c, and α are the fitting parameters depending on anisotropy. We predict that for large k-mers (k >/≈ 1.2 × 10(4)) isotropically placed at the lattice, percolation cannot occur, even at jamming concentration.

Percolation in random sequential adsorption of extended objects on a triangular lattice

The percolation aspect of random sequential adsorption of extended objects on a triangular lattice is studied by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps on the lattice. Jamming coverage θ{jam}, percolation threshold θ{p}, and their ratio θ{p}/θ{jam} are determined for objects of various shapes and sizes. We find that the percolation threshold θ{p} may decrease or increase with the object size, depending on the local geometry of the objects. We demonstrate that for various objects of the same length, the threshold θ{p} of more compact shapes exceeds the θ{p} of elongated ones. In addition, we study polydisperse mixtures in which the size of line segments making up the mixture gradually increases with the number of components. It is found that the percolation threshold decreases, while the jamming coverage increases, with the number of components in the mixture.