Linear K-mers Research Articles

Inverse percolation by removing straight semirigid rods from bilayer square lattices.

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing semirigid rods from a L×L square lattice that contains two layers (and M=L×L×2 sites). 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 removing groups of k consecutive monomers according to a generalized random sequential adsorption mechanism. The study is conducted by following the behavior of two critical concentrations with size k: (1) jamming coverage θ_{j,k}, which represents the concentration of occupied sites at which the jamming state is reached, and (2) inverse percolation threshold θ_{c,k}, which corresponds to the maximum concentration of occupied sites for which connectivity disappears. The obtained results indicate that (1) the jamming coverage exhibits an increasing dependence on the size k-it rapidly increases for small values of k and asymptotically converges towards a definite value for infinitely large k sizes θ_{j,k→∞}≈0.2701-and (2) the inverse percolation threshold is a decreasing function of k in the range 1≤k≤17. For k≥18, all jammed configurations are percolating states (the lattice remains connected even when the highest allowed concentration of removed sites is reached) and, consequently, there is no nonpercolating phase. This finding contrasts with the results obtained in literature for a complementary problem, where straight rigid k-mers are randomly and irreversibly deposited on a square lattice forming two layers. In this case, percolating and nonpercolating phases extend to infinity in the space of the parameter k and the model presents percolation transition for the whole range of k. The results obtained in the present study were also compared with those reported for the case of inverse percolation by removal of rigid linear k-mers from a square monolayer. The differences observed between monolayer and bilayer problems were discussed in terms of vulnerability and network robustness. Finally, the accurate determination of the critical exponents ν, β, and γ reveals that the percolation phase transition involved in the system 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.

Inverse percolation by removing straight rigid rods from square lattices in the presence of impurities

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing straight rigid rods from square lattices contaminated with non-conducting impurities. The presence of impurities provides a more realistic approach to the deposited monolayer, which usually presents inhomogeneities due to the irregular arrangement of surface and bulk atoms, the presence of various chemical species, etc. The process starts with an initial configuration, where all lattice sites are occupied by an impurity (with a concentration ) or a conducting particle (with a concentration ). Then, the system is diluted by randomly removing linear k-mers (linear clusters of k consecutive conducting particles) from the lattice. The impurities remain fixed in its position and cannot be removed. The central idea of this paper is based on finding the maximum concentration of conducting sites (minimum concentration of empty sites) for which the connectivity disappears. This particular value of the concentration is called inverse percolation threshold, and determines a well-defined geometrical phase transition in the system. On the other hand, the inverse jamming coverage 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-neighbour sites of appropriate size. The dependence of percolation and jamming thresholds on the concentration of defects was investigated for different values of k, ranging from 1 to 120. The obtained results show that the behaviour of the system is significantly affected by the presence of impurities. In addition, the nature of the jamming and percolation transitions was studied. In the first case, the corresponding spatial correlation length critical exponent was measured, being . This value coincides with previous calculations of this exponent for the standard random sequential adsorption of linear k-mers on square lattices. Critical exponents were also calculated for the percolation phase transition, showing that the universality class corresponding to ordinary percolation is preserved regardless the values of k and considered.

Impact of packing fraction on diffusion-driven pattern formation in a two-dimensional system of rod-like particles

Pattern formation occurring in a two-dimensional system of rod-like particles has been simulated using a lattice approach. The rod-like particles were modelled as linear k-mers of two mutually perpendicular orientations (kx - and ky-mers) on a square lattice with periodic boundary conditions (torus). Two different models of random sequential adsorption were used to produce an initial homogeneous and isotropic distribution of k-mers with different packing fraction values. By means of the Monte Carlo technique, translational diffusion of the k-mers was simulated as a random walk, while rotational diffusion was ignored, so, the kx- and ky-mers were considered as individual species. The system tended towards a well-organised nonequilibrium steady state in the form of diagonal stripes for relatively long k-mers (k ≥ 6) and moderate packing fractions (in the interval pdown < p < pup, where both the critical packing fractions pdown and pup were dependent on k).

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.

Effect of aging on electrical conductivity of two-dimensional composite with rod-like fillers

Two-dimensional composite with rod-like fillers has been simulated using a lattice approach. The fillers were assumed to be conductors, whilst the substrate (host matrix) was assumed to be an insulator. Rod-like fillers were considered as linear k-mers of two mutually perpendicular orientations (kx- and ky-mers) on a square lattice with periodic boundary conditions applied along both directions of the lattice. A random sequential adsorption was used to produce an initial homogeneous and isotropic distribution of fillers. By means of the Monte Carlo (MC) technique, translational diffusion of the fillers was simulated as a random walk, while rotational diffusion was ignored. At each given MC step, the system under consideration was transformed into a random resistor network (RRN); the Frank–Lobb algorithm was applied to calculate the electrical conductivity of such RRNs. Diffusional relaxation of the system leads to a kind of aging. For example, when concentration of fillers exceeds the percolation threshold, the aging leads to decreasing of the electrical conductivity and, in some cases, the inhibition of percolation was observed.

Pattern formation in a two-dimensional two-species diffusion model with anisotropic nonlinear diffusivities: a lattice approach

Diffusion in a two-species two-dimensional system has been simulated using a lattice approach. Rodlike particles were considered as linear k-mers of two mutually perpendicular orientations (kx- and ky-mers) on a square lattice. These kx- and ky-mers were treated as species of two kinds. A random sequential adsorption model was used to produce an initial homogeneous distribution of k-mers. The concentration of k-mers, p, was varied in the range from 0.1 to the jamming concentration, pj. By means of the Monte Carlo technique, translational diffusion of the k-mers was simulated as a random walk, while rotational diffusion was ignored. We demonstrated that the diffusion coefficients are strongly anisotropic and nonlinearly concentration-dependent. For sufficiently large concentrations (packing densities) and , the system tends toward a well-organized steady state. Boundary conditions predetermine the final state of the system. When periodic boundary conditions are applied along both directions of the square lattice, the system tends to a steady state in the form of diagonal stripes. The formation of stripe domains takes longer time the larger the lattice size, and is observed only for concentrations above a particular critical value. When insulating (zero flux) boundary conditions are applied along both directions of the square lattice, each kind of k-mer tries to completely occupy a half of the lattice divided by a diagonal, e.g. kx-mers locate in the upper left corner, while the ky-mers are situated in the lower right corner (‘yin–yang’ pattern). From time to time, regions built of kx- and ky-mers exchange their locations through irregular patterns. When mixed boundary conditions are used (periodic boundary conditions are applied along one direction whereas insulating boundary conditions are applied along the other one), the system still tends to form the stripes, but they are unstable and change their spatial orientation.

Jamming and percolation in random sequential adsorption of straight rigid rods on a two-dimensional triangular lattice

Monte Carlo simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of linear k-mers (also known as rods or needles) on a two-dimensional triangular lattice of linear dimension L, considering an isotropic RSA process and periodic boundary conditions. Extensive numerical work has been done to extend previous studies to larger system sizes and longer k-mers, which enables the confirmation of a nonmonotonic size dependence of the percolation threshold and the estimation of a maximum value of k from which percolation would no longer occur. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system is not affected, having the same universality class of the ordinary random percolation.

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.

Influence of defects on the effective electrical conductivity of a monolayer produced by random sequential adsorption of linear [formula omitted]-mers onto a square lattice

The effect of defects on the behaviour of electrical conductivity, σ, in a monolayer produced by the random sequential adsorption of linear k-mers (particles occupying k adjacent sites) onto a square lattice is studied by means of a Monte Carlo simulation. The k-mers are deposited on the substrate until a jamming state is reached. The presence of defects in the lattice (impurities) and of defects in the k-mers with concentrations of dl and dk, respectively, is assumed. The defects in the lattice are distributed randomly before deposition and these lattice sites are forbidden for the deposition of k-mers. The defects of the k-mers are distributed randomly on the deposited k-mers. The sites filled with k-mers have high electrical conductivity, σk, whereas the empty sites, and the sites filled by either types of defect have a low electrical conductivity, σl, i.e., a high-contrast, σk∕σl≫1, is assumed. We examined isotropic (both the possible x and y orientations of a particle are equiprobable) and anisotropic (all particles are aligned along one given direction, y) deposition. To calculate the effective electrical conductivity, the monolayer was presented as a random resistor network and the Frank–Lobb algorithm was used. The effects of the concentrations of defects dl and dk on the electrical conductivity for the values of k=2n, where n=1,2,…,5, were studied. Increase of both the dl and dk parameters values resulted in decreases in the value of σ and the suppression of percolation. Moreover, for anisotropic deposition the electrical conductivity along the y direction was noticeably larger than in the perpendicular direction, x. Phase diagrams in the (dl,dk)-plane for different values of k were obtained.

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.

Percolation and jamming in random sequential adsorption of linear k-mers on square lattices with the presence of impurities

Percolation and jamming of linear k-mers (particles occupying k adjacent sites) on two-dimensional square lattices with impurities have been studied by numerical simulations and finite-size scaling analysis. The model offers a simplified representation of the problem of percolation in amorphous solids, where the presence of defects in the system is simulated by introducing a fraction of imperfect bonds ρ, which are considered forbidden for deposition. The dependence of percolation and jamming thresholds on the concentration of defects was investigated for different values of k, ranging from 2 to 64. The results obtained show that: for each fixed value of k, percolation can occur when ρ is smaller than a certain value ; and in the range , the percolation threshold is practically independent of the fraction of defects. The behavior of as a function of k indicates that the percolation of linear k-mers on square lattices is impossible even for an ideal lattice if . Critical exponents were also calculated to show that the universality class corresponding to ordinary percolation is preserved.

Influence of anisotropy on percolation and jamming of linear k-mers on square lattice with defects

By means of the Monte Carlo simulation, we study the layers produced by the random sequential adsorption of the linear rigid objects (k-mers also known as rigid or stiff rods, sticks, needles) onto the square lattice with defects in the presence of an external field. The value of k varies from 2 to 32. The point defects randomly and uniformly placed on the substrate hinder adsorption of the elongated objects. The external field affects isotropic deposition of the particles, consequently the deposited layers are anisotropic. We study the influence of the defect concentration, the length of the objects, and the external field on the percolation threshold and the jamming concentration. Our main findings are (i) the critical defect concentration at which the percolation never occurs even at jammed state decreases for short k-mers (k < 16) and increases for long k-mers (k > 16) as anisotropy increases, (ii) the corresponding critical k-mer concentration decreases with anisotropy growth, (iii) the jamming concentration decreases drastically with growth of k-mer length for any anisotropy, (iv) for short k-mers, the percolation threshold is almost insensitive to the defect concentration for any anisotropy.

Percolation of polyatomic species on a simple cubic lattice

In the present paper, the site-percolation problem corresponding to linear k-mers (containing k identical units, each one occupying a lattice site) on a simple cubic lattice has been studied. The k-mers were irreversibly and isotropically deposited into the lattice. Then, the percolation threshold and critical exponents were obtained by numerical simulations and finite-size scaling theory. The results, obtained for k ranging from 1 to 100, revealed that (i) the percolation threshold exhibits a decreasing function when it is plotted as a function of the k-mer size; and (ii) the phase transition occurring in the system belongs to the standard 3D percolation universality class regardless of the value of k considered.

Phase behavior of attractive k-mers on two-dimensional lattices: a study from quasi-chemical approximation

The phase behavior of attractive linear rigid k-mers on two-dimensional lattices was studied by theoretical and simulation calculations in the framework of the lattice-gas model. Combining (i) the analytical expression for the partition function of non-interacting k-mers, and (ii) a generalization of the classical quasi-chemical approximation (QCA) in which the adsorbate can occupy more than one adsorption site, the main thermodynamic functions of the system were explicitly obtained. It was found that, for temperatures below a certain condensation temperature, the system undergoes a first-order phase transition which is observed as a clear discontinuity in the adsorption isotherms. The transition was studied in detail by calculating the temperature–density phase diagrams. Comparisons with analytical data from Bragg–Williams approximation and Monte Carlo simulations were performed in order to test the validity of the theoretical model. The results obtained allowed us not only to analyze the effect of introducing the lateral interactions by following the configuration-counting procedure of the QCA, but also to discuss the consequences of choosing the configurational factor associated to adsorption of non-interacting k-mers from different models developed to treat the multisite occupancy adsorption problem.

Statistical thermodynamics of straight rigid rods with nonadditive lateral interactions: Theory and Monte Carlo simulations

The statistical thermodynamics of straight rigid rods of length k (k-mers) with nonadditive lateral interactions was developed on a generalization in the spirit of the lattice-gas model and the classical Bragg-Williams approximation (BWA) and the quasichemical approximation (QCA). The new theoretical framework is obtained by combining (i) the exact analytical expression for the partition function of noninteracting linear k-mers adsorbed in one dimension and its extension to higher dimensions, and (ii) a generalization of BWA and QCA in which the adsorbate can occupy more than one adsorption site. The traditional assumption of a strictly pairwise additive nearest-neighbors interaction is replaced by a more general one, namely that the bond linking a certain atom with any of its neighbors depends considerably on how many of them are actually present (or absent) on the sites in the first coordination shell of the atom. The coverage and temperature dependence of the Helmholtz free energy, chemical potential, configurational entropy, and differential heat of adsorption are given. The formalism (i) reproduces the classical results for monomers, (ii) leads to the exact statistical thermodynamics of nonadditive interacting k-mers adsorbed in one dimension, and (iii) provides a close approximation for two-dimensional systems, taking into account multisite occupancy and nonadditive lateral interactions. Comparisons with Monte Carlo simulations are performed in order to test the validity of the theoretical model. Significant quantitative differences are shown and discussed. In all cases, the QCA appears to be the more accurate approach.

Critical behavior of long linear k-mers on honeycomb lattices

Monte Carlo (MC) simulations, finite-size scaling and theoretical analysis have been carried out to study the critical behavior of long linear particles of length k ( k -mers) on honeycomb lattices. A nematic phase, characterized by a big domain of parallel k -mers, is separated from the isotropic state, by a continuous transition occurring at a finite density θ c . Our study allowed: (1) to determine the minimum value of k ( k min ), which allows the formation of the nematic phase, being k min = 11 ; (2) to predict the dependence of θ c on k , being θ c ( k ) ∝ k − 1 ; and (3) to obtain the critical exponents, which indicate that the transition belongs to the 2D three-state Potts universality class.