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

Percolation of straight slots on a square grid

This work is devoted to the emergence of a connected network of slots (cracks) on a square grid. Accordingly, extensive Monte Carlo simulations and finite-size scaling analysis have been conducted to study the site percolation of straight slots with length l measured in the number of elementary cells of the grid with the edge size L. A special focus was made on the dependence of the percolation threshold p_{C}(l,L) on the slot length l varying in the range 1≤l≤L-2 for the square grids with edge size in the range 50≤L≤1000. In this way, we found that p_{C}(l,L) strongly decreases with increase of l, whereas the variations of p_{C}(l=const,L) with the variation of ratio l/L are very small. Consequently, we acquire the functional dependencies of the critical filling factor and percolation strength on the slot length. Furthermore, we established that the slot percolation model interpolates between the site percolation on square lattice (l=1) and the continuous percolation of widthless sticks (l→∞) aligned in two orthogonal directions. In this regard, we note that the critical number of widthless sticks per unit area is larger than in the case of randomly oriented sticks. Our estimates for the critical exponents indicate that the slot percolation belongs to the same universality class as standard Bernoulli percolation.

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.

Random sequential adsorption of self-avoiding chains on two-dimensional lattices

Random sequential adsorption of extended objects deposited on two-dimensional regular lattices is studied. The depositing objects are chains formed by occupying adsorption sites on the substrate through a self-avoiding walk of k lattice steps; these objects are also called "tortuous k-mers." We study how the jamming coverage, θ_{j,k}, depends on k for lattices with different connectivity (honeycomb, square, and triangular). The dependence can be fitted by the function θ_{j,k}=θ_{j,k→∞}+B/k+C/k^{2}, where B and C are found to be shared parameters by the three lattices and θ_{j,k→∞} (>0) is the jamming coverage for infinitely long k-mers for each of them. The jamming coverage is found to have a growing behavior with the connectivity of the lattice. In addition, θ_{j,k} is found to be higher for tortuous k-mers than for the previously reported for linear k-mers in each lattice. The results were obtained by means of numerical simulation through an efficient algorithm whose characteristics are discussed in detail. The computational method introduced here also allows us to investigate the full-time kinetics of the surface coverage θ_{k}(t) [θ_{j,k}≡θ_{k}(t→∞)]. Along this line, different time regimes are identified and characterized.

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.

Jamming and percolation of linear k-mers on honeycomb lattices

Numerical simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of elongated objects deposited on two-dimensional honeycomb lattices. The depositing particle is modeled as a linear array of length k (so-called k-mer), maximizing the distance between first and last monomers in the chain. The separation between k-mer units is equal to the lattice constant. Hence, k sites are occupied by a k-mer when adsorbed onto the surface. The adsorption process starts with an initial configuration, where all lattice sites are empty. Then, the sites are occupied following a random sequential adsorption mechanism. The process finishes when 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. Jamming coverage θ_{j,k} and percolation threshold θ_{c,k} were determined for a wide range of values of k (2≤k≤128). The obtained results shows that (i) θ_{j,k} is a decreasing function with increasing k, being θ_{j,k→∞}=0.6007(6) the limit value for infinitely long k-mers; and (ii) θ_{c,k} has a strong dependence on k. It decreases in the range 2≤k<48, goes through a minimum around k=48, and increases smoothly from k=48 up to the largest studied value of k=128. Finally, the precise determination of the critical exponents ν, β, and γ indicates that the model belongs to the same universality class as 2D standard percolation regardless of the value of k considered.

Irreversible multilayer adsorption of semirigid k-mers deposited on one-dimensional lattices

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 L→∞. 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.

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.

Percolation phase transition by removal of k^{2}-mers from fully occupied lattices

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of k×k square tiles (k^{2}-mers) from square lattices. The process starts with an initial configuration, where all lattice sites are occupied and, obviously, the opposite sides of the lattice are connected by occupied sites. Then the system is diluted by removing k^{2}-mers of occupied sites from the lattice following a random sequential adsorption mechanism. The process finishes when the jamming state is reached and no more objects can be removed due to the absence of occupied sites clusters of appropriate size and shape. The central idea of this paper is based on finding the maximum concentration of occupied sites, p_{c,k}, for which the connectivity disappears. This particular value of the concentration is called the inverse percolation threshold and determines a well-defined geometrical phase transition in the system. The results obtained for p_{c,k} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤4. For k≥5, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed sites is reached. The jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of 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 four, has the same universality class as the standard percolation problem.

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.

Standard and inverse bond percolation of straight rigid rods on square lattices

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bond percolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty. Then, linear bond k-mers (sets of k linear nearest-neighbor bonds) are randomly and sequentially deposited on the lattice. Jamming coverage p_{j,k} and percolation threshold p_{c,k} are determined for a wide range of k (1≤k≤120). p_{j,k} and p_{c,k} exhibit a decreasing behavior with increasing k, p_{j,k→∞}=0.7476(1) and p_{c,k→∞}=0.0033(9) being the limit values for large k-mer sizes. p_{j,k} is always greater than p_{c,k}, and consequently, the percolation phase transition occurs for all values of k. In the case of inverse percolation, the process starts with an initial configuration where all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides of the lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removing linear bond k-mers from the lattice. The central idea here is based on finding the maximum concentration of occupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold p_{c,k}^{i}, and determines a geometrical phase transition in the system. On the other hand, the inverse jamming coverage p_{j,k}^{i} is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bonds of appropriate size. It is easy to understand that p_{j,k}^{i}=1-p_{j,k}. The obtained results for p_{c,k}^{i} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤18. For k>18, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed bonds p_{j,k}^{i} is reached. In terms of network attacks, this striking behavior indicates that random attacks on single nodes (k=1) are much more effective than correlated attacks on groups of close nodes (large k's). Finally, the accurate determination of critical exponents reveals that standard and inverse bond percolation models on square lattices belong to the same universality class as the random percolation, regardless of the size k considered.

Particle morphology effects in random sequential adsorption

The properties of the random sequential adsorption of objects of various shapes on a two-dimensional triangular lattice are studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps, whereby the size of the objects is gradually increased by wrapping the walks in several different ways. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the jamming density θ_{J} and on the temporal evolution of the coverage fraction θ(t). Our results suggest that the order of symmetry axis of a shape exerts a decisive influence on adsorption kinetics near the jamming limit θ_{J}. The decay of probability for the insertion of a new particle onto a lattice is described in a broad range of the coverage θ by the product between the linear and the stretched exponential function for all examined objects. The corresponding fitting parameters are discussed within the context of the shape descriptors, such as rotational symmetry and the shape factor (parameter of nonsphericity) of the objects. Predictions following from our calculations suggest that the proposed fitting function for the insertion probability is consistent with the exponential approach of the coverage fraction θ(t) to the jamming limit θ_{J}.

Percolation and jamming of linear k-mers on a square lattice with defects: Effect of anisotropy

Using the Monte Carlo simulation, we study the percolation and jamming of oriented linear k-mers on a square lattice that contains defects. The point defects with a concentration d are placed randomly and uniformly on the substrate before deposition of the k-mers. The general case of unequal probabilities for orientation of depositing of k-mers along different directions of the lattice is analyzed. Two different relaxation models of deposition that preserve the predetermined order parameter s are used. In the relaxation random sequential adsorption (RRSA) model, the deposition of k-mers is distributed over different sites on the substrate. In the single-cluster relaxation (RSC) model, the single cluster grows by the random accumulation of k-mers on the boundary of the cluster (Eden-like model). For both models, a suppression of growth of the infinite (percolation) cluster at some critical concentration of defects d(c) is observed. In the zero-defect lattices, the jamming concentration p(j) (RRSA model) and the density of single clusters p(s) (RSC model) decrease with increasing length k-mers and with a decrease in the order parameter. For the RRSA model, the value of d(c) decreases for short k-mers (k<16) as the value of s increases. For k=16 and 32, the value of d(c) is almost independent of s. Moreover, for short k-mers, the percolation threshold is almost insensitive to the defect concentration for all values of s. For the RSC model, the growth of clusters with ellipselike shapes is observed for nonzero values of s. The density of the clusters p(s) at the critical concentration of defects d(c) depends in a complex manner on the values of s and k. An interesting finding for disordered systems (s=0) is that the value of p(s) tends towards zero in the limits of the very long k-mers, k→∞, and very small critical concentrations d(c)→0. In this case, the introduction of defects results in a suppression of k-mer stacking and in the formation of empty or loose clusters with very low density. On the other hand, denser clusters are formed for ordered systems with p(s)≈0.065 at s=0.5 and p(s)≈0.38 at s=1.0.

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.

Dimer covering and percolation frustration

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidently been considered. Here, we propose a procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattices on the remaining bonds connecting the dimers. We obtain p_{c}=0.367713(2) and p_{c}=0.235340(1) for random coverings of the square and the triangular lattices, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this paper, has an unusually low percolation threshold and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this paper, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly coordinated networks.

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).