Entropy-driven phases at high coverage adsorption of straight rigid rods on two-dimensional square lattices

Extending Tensor Network Methods Beyond Pairwise Interactions in Adsorption Systems

Accurate modeling of complex physical systems often requires accounting for many-body interactions. Traditional statistical physics methods, such as Monte Carlo, transfer matrix, cluster approximations, and others, face significant computational challenges. This study introduces a unified tensor algorithm that efficiently incorporates interactions up to the third nearest neighbor. We applied our algorithm to a system of 1,3,5-tris(4-pyridyl)benzene and copper on Au(111). Many-body interactions were considered in two ways: by expressing them through pairwise interactions and by explicitly considering DFT energies for each many-body configuration. This led to both quantitative and qualitative differences in the results. The most significant difference is the lower thermal stability of the "superflower" phase and its subsequent replacement by a disordered structure with higher density. The developed unified tensor algorithm opens up new possibilities for the accurate modeling of complex systems taking into account many-body interactions.

Exclusion statistics for structured particles on topologically correlated states. II. Multicomponent lattice gases

Statistical thermodynamics of particles having a spectrum of topological correlated states and observing statistical exclusion is developed to describe mixtures of species of arbitrary size and shape. A generalized statistical distribution is obtained through a configuration space ansatz recently introduced for single species accounting for the multiple exclusion statistical phenomena, where spatially correlated particle states can be simultaneously excluded by more than one particle. Statistical exclusion on a correlated states spectrum is characterized by exclusion statistical parameters β_{cij} which are self-consistently determined within the multiple exclusion from thermodynamic boundary conditions. Self-exclusion and cross-exclusion frequency functions e_{ij}(n) and average cumulative exclusion functions G_{ij}(n) are introduced to characterize the state exclusion spectrum as density varies. Haldane's statistics and Wu's distribution for statistically independent excluding species are recovered in the limit of uncorrelated states for single species as well as for mixtures of self- and cross-excluding species with constant mutual statistical exclusion. The multiple exclusion statistics formalism is applied to the k-mer problem on a square lattice rationalized as a mixture of two differently oriented self-excluding and cross-excluding pseudospecies. An isotropic-nematic and a high-density nematic-isotropic (disordered) phase transitions is predicted only for k≥7. The isotropic-nematic transition is continuum as expected, but the high-density transition results in a first-order one. The formalism provides phase coexistence lines and the chemical potential dependence of the low- and high-density branches in the nematic regime. The theoretical approach to lattice gases presented in this work offers a unique general framework applicable to mixtures of entropy-complex lattice gases. From this framework, k-mer phase transitions can be reproduced, and significant configuration features can be derived from the state exclusion spectrum functions.

Exclusion statistics for structured particles on topologically correlated states. I. Single species lattice gases

A statistical thermodynamics description of particles having a set of spatially correlated states with statistical exclusion is developed. A general approximation for the density of states is presented from a state-counting ansatz recently introduced accounting for the multiple state exclusion statistical phenomena as a consequence of state spatial correlations. The multiple exclusion statistics is characterized by an exclusion correlation constant g_{c} which is consistently determined within the formalism from proper thermodynamic limits. The analytical form of g_{c} is given in terms of the Lambert function from the particle-lattice geometry. A generalized statistical distribution is obtained reducing to Haldane's statistics and Wu's distribution in the limiting case of particles on a set of spatially uncorrelated states. The problem of hard rods (k-mers) on a square lattice is studied with this formalism. From the entropy density dependence of the isotropic (I) and fully oriented nematic (N) phases, the approximation predicts two transitions, I→N and high-coverage N→I (disordered), only for k≥7 with the entropy at saturation matching to the known value from a Monte Carlo (MC) simulation. Critical coverage of both transitions is given for k=7 to k=20 in the first and second orders of approximations, in qualitative and quantitative agreement with results from MC simulations. State exclusion frequency e(n) and exclusion average G(n) functions are introduced and given in terms of the chemical potential to obtain a thermodynamic characterization of the state exclusion evolution on density. Results of chemical potential and state exclusion are shown for ideal lattice gases of k-mers, squares, and rectangles on a square lattice. Analytical results are compared with fast-relaxation grand canonical MC simulations.

Entropy-driven phases at high coverage adsorption of straight rigid rods on three-dimensional cubic lattices

Combining Monte Carlo simulations and thermodynamic integration method, we study the configurational entropy per site of straight rigid rods of length k (k-mers) adsorbed on three-dimensional (3D) simple cubic lattices. The process is monitored by following the dependence of the lattice coverage θ on the chemical potential μ (adsorption isotherm). Then, we perform the integration of μ(θ) over θ to calculate the configurational entropy per site of the adsorbed phase s(k,θ) as a function of the coverage. Based on the behavior of the function s(k,θ), different phase diagrams are obtained according to the k values: k≤4, disordered phase; k=5,6, disordered and layered-disordered phases; and k≥7, disordered, nematic and layered-disordered phases. In the limit of θ→1 (full coverage), the configurational entropy per site is determined for values of k ranging between 2 and 8. For k≥6, MC data coincide (within the statistical uncertainty) with recent analytical predictions [D. Dhar and R. Rajesh, Phys. Rev. E 103, 042130 (2021)2470-004510.1103/PhysRevE.103.042130] for very large rods. This finding represents the first numerical validation of the expression obtained by Dhar and Rajesh for d-dimensional lattices with d>2. In addition, for k≥5, the values of s(k,θ→1) for simple cubic lattices are coincident with those values reported in [P. M. Pasinetti et al., Phys. Rev. E 104, 054136 (2021)2470-004510.1103/PhysRevE.104.054136] for two-dimensional (2D) square lattices. This is consistent with the picture that at high densities and k≥5, the layered-disordered phase is formed on the lattice. Under these conditions, the system breaks to 2D layers, and the adsorbed phase becomes essentially 2D. The 2D behavior of the fully covered lattice reinforces the conjecture that the large-k behavior of entropy per site is superuniversal, and holds on d-dimensional hypercubical lattices for all d≥2.

Tensor network construction for lattice gas models: Hard-core and triangular lattice models

The representation of complex lattice models in the form of a tensor network is a promising approach to the analysis of the thermodynamics of such systems. Once the tensor network is built, various methods can be used to calculate the partition function of the corresponding model. However, it is possible to build the initial tensor network in different ways for the same model. In this work, we have proposed two ways of constructing tensor networks and demonstrated that the construction process affects the accuracy of calculations. For demonstration purposes, we have done a brief study of the 4 nearest-neighbor (NN) and 5NN models, where adsorbed particles exclude all sites up to the fourth and fifth nearest neighbors from being occupied by another particle. In addition, we have studied a 4NN model with finite repulsions with a fifth neighbor. In a sense, this model is intermediate between 4NN and 5NN models, so algorithms designed for systems with hard-core interactions may experience difficulties. We have obtained adsorption isotherms, as well as graphs of entropy and heat capacity for all models. The critical values of the chemical potential were determined from the position of the heat capacity peaks. As a result, we were able to improve our previous estimate of the position of the phase transition points for the 4NN and 5NN models. And in the model with finite interactions, we found the presence of two first-order phase transitions and made an estimate of the critical values of the chemical potential for them.

Entropy of rigid k-mers on a square lattice

Using the transfer matrix technique, we estimate the entropy for a gas of rods of sizes equal to k (named k-mers), which cover completely a square lattice. Our calculations were made considering three different constructions, using periodical and helical boundary conditions. One of those constructions, which we call profile method, was based on the calculations performed by Dhar and Rajesh to obtain a lower limit to the entropy of very large chains placed on the square lattice. This method, so far as we know, was never used before to define the transfer matrix, but turned out to be very useful, since it produces matrices with smaller dimensions than those obtained using the usual approach. Our results were obtained for chain sizes ranging from k=2 to k=10 and they are compared with results already available in the literature. In the case of dimers (k=2) our results are compatible with the exact result. For trimers (k=3), recently investigated by Ghosh et al., also our results were compatible, with the same happening for the simulational estimates obtained by Pasinetti et al. in the whole range of rod sizes. Our results are also consistent with the asymptotic expression for the behavior of the entropy as a function of the size k, proposed by Dhar and Rajesh for very large rods (k≫1).

Orientational phase transition in monolayers of multipolar straight rigid rods: The case of 2-thiophene molecule adsorption on the Au(111) surface

Monte Carlo simulations and finite-size scaling theory have been carried out to study the critical behavior and universality for the isotropic-nematic (IN) phase transition in a system of straight rigid pentamers adsorbed on a triangular lattice with polarized nonhomogeneous intermolecular interactions. The model was inspired by the deposition of 2-thiophene molecules over the Au(111) surface, which was previously characterized by experimental techniques and density functional theory. A nematic phase, observed experimentally by the formation of a self-assembled monolayer of parallel molecules, is separated from the isotropic state by a continuous transition occurring at a finite density. The precise determination of the critical exponents indicates that the transition belongs to the three-state Potts universality class. The finite-size scaling analysis includes the study of mutability and diversity. These two quantities are derived from information theory and they have not previously been considered as part of the conventional treatment of critical phenomena for the determination of critical exponents. The results obtained here contribute to the understanding of formation processes of self-assembled monolayers, phase transitions, and critical phenomena from novel compression algorithms for studying mutual information in sequences of data.

Entropy of fully packed rigid rods on generalized Husimi trees: A route to the square-lattice limit

Although hard rigid rods (k-mers) defined on the square lattice have been widely studied in the literature, their entropy per site, s(k), in the full-packing limit is only known exactly for dimers (k=2) and numerically for trimers (k=3). Here, we investigate this entropy for rods with k≤7, by defining and solving them on Husimi lattices built with diagonal and regular square-lattice clusters of effective lateral size L, where L defines the level of approximation to the square lattice. Due to an L-parity effect, by increasing L we obtain two systematic sequences of values for the entropies s_{L}(k) for each type of cluster, whose extrapolations to L→∞ provide estimates of these entropies for the square lattice. For dimers, our estimates for s(2) differ from the exact result by only 0.03%, while that for s(3) differs from best available estimates by 3%. In this paper, we also obtain a new estimate for s(4). For larger k, we find that the extrapolated results from the Husimi tree calculations do not lie between the lower and upper bounds established in the literature for s(k). In fact, we observe that, to obtain reliable estimates for these entropies, we should deal with levels L that increase with k. However, it is very challenging computationally to advance to solve the problem for large values of L and for large rods. In addition, the exact calculations on the generalized Husimi trees provide strong evidence for the fully packed phase to be disordered for k≥4, in contrast to the results for the Bethe lattice wherein it is nematic.