Statistical model for self-assembly of trimesic acid molecules into homologous series of flower phases

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.

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.

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.

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.

Thermodynamics of self-assembled molecular layers of trimesic acid from fields-supported kinetic Monte Carlo simulation

A technique has been developed for calculating the thermodynamic characteristics of rigid self-assembled organic adsorption layers and the parameters of polymorphic transitions using two types of external fields and the kinetic Monte Carlo method.

Thermodynamics, EOS, and heat capacity in molecular modeling of self-assembled molecular layers

Self-assembled monolayers (SAMs) on solid surfaces represent a rapidly developed class of non-autonomous phases widely used in organic electronics, sensors, catalysis, and other applications. In many cases, the same organic molecules form various stable and metastable polymorphous structures that can transform to each other at certain parameters. A high rigidity of SAMs extremely complicates the evaluation of the chemical potential using standard methods based on thermodynamic integration. This study presents results of molecular modeling of two-dimensional structures of tripod-shaped molecules associated with the trimesic acid (TMA) molecules. A technique used here is based on a recently developed method of external fields imposed on an elongated simulation cell in the framework of a kinetic Monte Carlo algorithm. These fields are the external potential and a damping field that reduces the intermolecular potential and affects the system similar to the increase in temperature. Equations of state (EOS) for several TMA polymorphs have been obtained with the conventional Monte Carlo simulation. It was shown that, in each case, only one constant links the chemical potential obtained with the external field method and the EOS at any temperature and pressure. The heat capacities of SAMs at constant volume and pressure were also determined as functions of temperature and compressibility of the structure at given degrees of freedom. The approach can be used as a general tool for modeling and evaluation of thermodynamic properties of various rigid structures, including SAMs of functional organic molecules.

Dynamic Supramolecular Template: Multiple Stimuli-Controlled Size Adjustment of Porous Networks

Dynamically switchable porous networks offer exciting potential in functionalizing surfaces. The structure and morphology of the networks can be controlled by applying external stimuli. Here, a dynamic supramolecular template assembled by 1,3,5-tris(4-carboxyphenyl)benzene (BTB) is successfully achieved at the liquid-solid interface by applying two external stimuli simultaneously. Upon varying the concentration of BTB solution together with switching the polarity of the sample bias, self-assembled monolayers (SAMs) undergo phase transitions twice: an immediate transition from a compact structure to a macroporous (honeycomb) structure as a response to the change in the electric field and a fast-changing transition from the macroporous to a microporous (oblique) structure. With saturated BTB solution, however, the initial compact structure can only transform into the oblique structure after switching the polarity of the sample bias without the appearance of a honeycomb structure. The different phase transitions suggest that the dynamic supramolecular template can only survive at a specific concentration range and is obtainable by performing multiple stimuli simultaneously. Interestingly, introducing a guest molecule to the system can adjust the phase transition process and effectively stabilize the honeycomb structure of BTB. The flexibility associated with the porous networks renders it a dynamic supramolecular template for guest binding.

Coverage-Controlled Superstructures of C3 -Symmetric Molecules: Honeycomb versus Hexagonal Tiling

The competition between honeycomb and hexagonal tiling of molecular units can lead to large honeycomb superstructures on surfaces. Such superstructures exhibit pores that may be used as 2D templates for functional guest molecules. Honeycomb superstructures of molecules that comprise a C₃ symmetric platform on Au(111) and Ag(111) surfaces are presented. The superstructures cover nearly mesoscopic areas with unit cells containing up to 3000 molecules, more than an order of magnitude larger than previously reported. The unit cell size may be controlled by the coverage. A fairly general model was developed to describe the energetics of honeycomb superstructures built from C₃ symmetric units. Based on three parameters that characterize two competing bonding arrangements, the model is consistent with the present experimental data and also reproduces various published results. The model identifies the relevant driving force, mostly related to geometric aspects, of the pattern formation.

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

We have theoretically studied the features of self-assembly in organic adsorption layers where both “molecule–surface” and “molecule–molecule” interactions are anisotropic.

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.

Phase behaviour of self-assembled monolayers controlled by tuning physisorbed and chemisorbed states: A lattice-model view

The self-assembly of molecules on surfaces into 2D structures is important for the bottom-up fabrication of functional nanomaterials, and the self-assembled structure depends on the interplay between molecule-molecule interactions and molecule-surface interactions. Halogenated benzene derivatives on platinum have been shown to have two distinct adsorption states: a physisorbed state and a chemisorbed state, and the interplay between the two can be expected to have a profound effect on the self-assembly and phase behaviour of these systems. We developed a lattice model that explicitly includes both adsorption states, with representative interactions parameterised using density functional theory calculations. This model was used in Monte Carlo simulations to investigate pattern formation of hexahalogenated benzene molecules on the platinum surface. Molecules that prefer the physisorbed state were found to self-assemble with ease, depending on the interactions between physisorbed molecules. In contrast, molecules that preferentially chemisorb tend to get arrested in disordered phases. However, changing the interactions between chemisorbed and physisorbed molecules affects the phase behaviour. We propose functionalising molecules in order to tune their adsorption states, as an innovative way to control monolayer structure, leading to a promising avenue for directed assembly of novel 2D structures.

Antiferromagnetic triangular Blume-Capel model with hard-core exclusions

Using Monte Carlo simulation, we analyze phase transitions of two antiferromagnetic (AFM) triangular Blume-Capel (BC) models with AFM interactions between third-nearest neighbors. One model has hard-core exclusions between the nearest-neighbor (1NN) particles (3NN1 model) and the other has them between the nearest-neighbor and next-nearest-neighbor particles (3NN12 model). Finite-size scaling analysis reveals that in these models, the transition from the paramagnetic to long-range order (LRO) AFM phase is either of the first order or goes through an intermediate phase which might be attributed to the Berezinskii-Kosterlitz-Thouless (BKT) type. The properties of the low-temperature phase transition to the AFM phase of the 1NN, 3NN1, and 3NN12 models are found to be very similar for almost all values of a normalized single-ion anisotropy parameter, 0 < δ < 1.5. Higher temperature behavior of the 3NN12 and 3NN1 models is rather different from that of the 1NN model. Three phase transitions are observed for the 3NN12 model: from the paramagnetic phase to the phase with domains of the LRO AFM phase at T(c), from this structure to the diluted frustrated BKT-type phase at T(2), and from the frustrated phase to the AFM LRO phase at T(1). For the 3NN12 model, T(c) > T(2) > T(1) at 0 < δ < 1.15 (range I), T(c) ≈ T(2) > T(1) at 1.15 < δ < 1.3 (range II), and T(c) = T(2) = T(1) at 1.3 < δ < 1.5 (range III). For the 3NN1 model, T(c) ≈ T(2) > T(1) at 0 < δ < 1.2 (range II) and T(c) = T(2) = T(1) at 1.2 < δ < 1.5 (range III). There is only one first-order phase transition in range III. The transition at T(c) is of the first order in range II and either of a weak first order or a second order in range I.