Shape-interaction dualism: unraveling complex phase behavior in triangular particle monolayers.

This paper examines the effect of finite attractive and repulsive interactions on the self-assembly of triangular-shaped particles on a triangular lattice. The ground state analysis of the lattice model has revealed an infinite sequence of ordered structures, a phenomenon referred to as the 'devil's staircase' of phase transitions. The model has been studied at finite temperatures using both the transfer-matrix and tensor renormalization group methods. The concurrent use of these two methods lends credibility to the obtained results. It has been demonstrated that the initial ordered structures of the 'devil's staircase' persist at non-zero temperatures. Further increase of the attraction between particles or a decrease of the temperature induces the appearance of subsequent ordered structures of the 'devil's staircase'. The corresponding phase diagram of the model has been calculated. The phase behavior of our model agrees qualitatively with the phase behavior of trimesic acid adsorption layer on single crystal surfaces.

Tensor renormalization group study of orientational ordering in simple models of adsorption monolayers.

A simple lattice model of the orientational ordering in organic adsorption layers that considers the directionality of intermolecular interactions is proposed. The symmetry and the number of rotational states of the adsorbed molecule are the main parameters of the model. The model takes into account both the isotropic and directional contributions to the molecule-molecule interaction potential. Using several special cases of this model, we have shown that the tensor renormalization group (TRG) approach can be successfully used for the analysis of orientational ordering in organic adsorption layers with directed intermolecular interactions. Adsorption isotherms, potential energy, and entropy have been calculated for the model adsorption layers differing in the molecule symmetry and the number of rotational states. The calculated thermodynamic characteristics show that entropy effects play a significant role in the self-assembly of dense phases of the molecular layers. All the results obtained with the TRG have been verified by the standard Monte Carlo method. The proposed model reproduces the main features of the phase behavior of the real adsorption layers of benzoic, terephthalic, and trimesic acids on a homogeneous surface of metal single crystals and graphite.

Triangles on a triangular lattice: Insights into self-assembly in two dimensions driven by shape complementarity.

A series of models for reversible filling of a triangular lattice with equilateral triangles has been developed and investigated. There are eight distinct models that vary in the set of prohibitions. In zeroth approximation, these models allow one to estimate the influence of the particles' shape and complementarity of their pair configurations on the self-assembly of dense monolayers formed by reversible filling. The most symmetrical models were found to be equivalent to hard-disk models on the hexagonal lattice. When any contact of hard triangles by vertices is prohibited, the dense monolayers are disordered, and their entropy tends to the constant. If only one pair configuration is prohibited, the close-packed layer appears through the continuous phase transition. In other cases, the weak first-order transition resulting in the self-assembly of close-packed layers is observed.

Tensor renormalization group study of hard-disk models on a triangular lattice.

High accuracy and performance of the tensor renormalization group (TRG) method have been demonstrated for the model of hard disks on a triangular lattice. We considered a sequence of models with disk diameter ranging from a to 2sqrt[3]a, where a is the lattice constant. Practically, these models are good for approximate description of thermodynamics properties of molecular layers on crystal surfaces. Theoretically, it is interesting to analyze if and how this sequence converges to the continuous model of hard disks. The dependencies of the density and heat capacity on the chemical potential were calculated with TRG and transfer-matrix (TM) methods. We benchmarked accuracy and performance of the TRG method comparing it with TM method and with exact result for the model with nearest-neighbor exclusions (1NN). The TRG method demonstrates good convergence and turns out to be superior over TM with regard to considered models. Critical values of chemical potential (µ_{c}) have been computed for all models. For the model with next-nearest-neighbor exclusions (2NN) the TRG and TM produce consistent results (µ_{c}=1.75587 and µ_{c}=1.75398 correspondingly) that are also close to earlier Monte Carlo estimation by Zhang and Deng. We found that 3NN and 5NN models shows the first-order phase transition, with close values of µ_{c} (µ_{c}=4.4488 for 3NN and 4.4<µ_{c}<4.5 for 5NN). The 4NN model demonstrates continuous yet rapid phase transition with 2.65<µ_{c}<2.7.

Phase diversity in an adsorption model of an additive binary gas mixture for all sets of lateral interactions.

Complete analysis of phase behavior of an adsorption model of a binary gas mixture on a square lattice was carried out for all possible sets of lateral interactions between nearest adsorbed molecules of the same type and no interaction between adsorbed molecules of different types. The model was completely investigated in the ground state, and it was shown that the phase behavior of the system is conserved at finite temperatures by means of a transfer matrix method.

Cross-impact of surface and interaction anisotropy in the self-assembly of organic adsorption monolayers: a Monte Carlo and transfer-matrix study.

Using a simple lattice gas model we study the features of self-assembly in adsorption layers where both "molecule-surface" and "molecule-molecule" interactions are anisotropic. Based on the example of adsorption layers of mono-functional organic molecules on the heterogeneous surface with strip-like topography, we have revealed plenty of possible self-assembled structures in this simple system, such as discrete, linear, zigzag, chess board-like, two-dimensional porous and close-packed patterns. However, the phase behavior of the adsorption layer is much richer, if the interactions between functional and non-functional parts of adjacent adsorbed molecules have comparable strength and opposite signs. It is demonstrated that filling of the strips composed of relatively "strong" adsorption sites with the increase of chemical potential can be non-monotonic. This effect is associated with surface anisotropy and results from the changing of the driving force of the self-assembly process - interactions between the adsorbed molecule and the surface dominate at low surface coverages, but intermolecular forces prevail at higher ones. Additionally, when the width of the strip composed of "strong" adsorption sites is two or more times greater than that of the adsorbed molecule, a local assembly of the ordered phases on the "strong" adsorption sites is observed. Our results suggest strategies for controlling the self-assembly in experiments involving mono-functional organic molecules on a strip-like heterogeneous surface.

Generalized lattice-gas model for adsorption of functional organic molecules in terms of pair directional interactions.

A generalized lattice-gas model that takes into account the directional character of pair interactions between the lattice sites is proposed. It is demonstrated that the proposed model can be successfully used to deeply understand the self-assembly process in adsorption monolayers of functional organic molecules driven by specified directional interactions between such molecules (e.g., hydrogen bonding). To illustrate the idea, representative cases of the general model with different numbers of identical functional groups in the chemical structure of the adsorbed molecule are investigated with Monte Carlo and the transfer-matrix methods. The model reveals that the phase behavior of the adsorption systems considered can be characterized as a hierarchical self-assembly process. It is predicted that in real adsorption systems of this type, the energy of hydrogen bonding sufficiently depends on the mutual orientation of the adsorbed molecules.

Model of homonuclear dimer adsorption in terms of two possible molecule orientations with respect to surface: square lattice.

A model of homonuclear dimer adsorption in terms of two possible molecule orientations with respect to the surface on a square lattice has been constructed and studied. The dimers can occupy one or two sites on the lattice. The thermodynamics of this system has been studied by transfer-matrix and Monte Carlo methods. The phase diagram has been constructed. It was shown that in the vicinity of the tricritical point the coverage as a function of chemical potential possesses a minimum. This phenomenon seems to be the common one for molecules with several ways of adsorption.