Ordered Assemblies of Triangular-Shaped Molecules with Strongly Interacting Vertices: Phase Diagrams for Honeycomb and Zigzag Structures on Triangular Lattice

Equilibrium structure of a dense trimesic acid monolayer on a homogeneous solid surface: from atomistic simulation to thermodynamics

A general methodology for determining the thermodynamic characteristics of rigid organic crystals on the atomistic level is presented. The proposed approach is based on a combination of grid interpolation of the precalculated intermolecular potential and kinetic Monte Carlo simulation of the gas-crystal system with an explicit interphase. The two-phase system is stabilized in a wide range of external parameters with an imposed external potential and damping field. The damping field reduces the intermolecular potential at the edges of the crystals and turns it off in the gas phase. To determine the thermodynamic characteristics of a crystal the conditions of equality of chemical potentials in coexisting phases are used. The intermolecular pairwise potential can be calculated on the atomistic or quantum level. In the kinetic Monte Carlo simulations, a grid interpolation of the precalculated potential is performed on each iteration of the algorithm. We have applied the approach to the thermodynamic analysis of a dense monolayer of trimesic acid on a homogeneous surface. The calculated free energy and entropy for the dense "superflower" and filled chicken-wire phases obey the Gibbs-Duhem equation, which confirms the thermodynamic consistency of our approach. Using the proposed approach, we have revealed that the dense "superflower" phase becomes metastable at zero pressure and 470-500 K. Under these conditions, the filled chicken-wire structure with partially released hexagonal cages is thermodynamically favourable. The proposed approach is a potentially universal tool for the thermodynamic analysis of crystals formed by "rigid" organic molecules of any complexity on the atomistic level.

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.

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.

Surface-Confined Self-Assembly of Asymmetric Tetratopic Molecular Building Blocks

Surface-confined self-assembly of functional molecular building blocks has recently been widely used to create low-dimensional, also covalent, superstructures with tailorable geometry and physicochemical properties. In this contribution, using the lattice Monte Carlo simulation method, we demonstrate how the structure-property relation can be established for the 2D self-assembly of a model tetrapod molecule with reduced symmetry. To that end, a rigid functional unit comprising a few interconnected segments arranged in different tetrapod shapes was used and its self-assembly on a triangular lattice representing a (111) crystal surface was simulated. The results of our calculations show strong dependence of the structure formation on the molecular symmetry, in particular on the (pro)chiral nature of the building block. The simulations predicted the formation of unusual ordered racemic networks with unique aperiodic spatial distribution of the surface enantiomers. Molecular symmetry was also found to have significant influence on the enantiopure self-assembly which resulted in the Kagome and brickwall networks and other less ordered extended superstructures with parallelogram pores. The theoretical findings of this contribution can be relevant to designing and on-surface synthesis of molecular superstructures with predefined geometries and functions. In particular, the predicted molecular architectures can stimulate experimental efforts to fabricate and explore new nanostructures, for example graphitic, having the composition and geometry proposed in our study.

On-surface self-assembly of tetratopic molecular building blocks

Self-assembly of functional molecules on solid substrates has recently attracted special attention as a versatile method for the fabrication of low dimensional nanostructures with tailorable properties. In this contribution, using theoretical modeling, we demonstrate how the architecture of 2D molecular assemblies can be predicted based on the individual properties of elementary building blocks at play. To that end a model star-shaped tetratopic molecule is used and its self-assembly on a (111) surface is simulated using the lattice Monte Carlo method. Several test cases are studied in which the molecule bears terminal arm centers providing interactions with differently encoded directionality. Our theoretical results show that manipulation of the interaction directions can be an effective way to direct the self-assembly towards extended periodic superstructures (2D crystals) as well as to create assemblies characterized by a lower degree of order, including glassy overlayers and quasi one-dimensional molecular connections. The obtained structures are described and classified with respect to their main geometric parameters. A small library of the tetratopic molecules and the corresponding superstructures is provided to categorize the structure-property relationship in the modeled systems. The results of our simulations can be helpful to 2D crystal engineering and surface-confined polymerization techniques as they give hints on how to functionalize tetrapod organic building blocks which would be able to create superstructures with predefined spatial organization and range of order.

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.

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.

Theoretical Modeling of Surface Confined Chiral Nanoporous Networks: Cruciform Molecules as Versatile Building Blocks

Patterning of solid surfaces with functional organic molecules has been a convenient route to fabricate two-dimensional materials with programmed architecture and activities. One example is the chiral nanoporous networks that can be created via controlled self-assembly of star-shaped molecules under 2D confinement. In this contribution we use computer modeling to predict the formation of molecular networks in adsorbed overlayers comprising cruciform molecular building blocks equipped with discrete interaction centers. To that end, we employ the Monte Carlo simulation method combined with a coarse-grained representation of the adsorbed molecules which are treated as collections of interconnected segments. The interaction centers within the molecules are represented by active segments whose number and distribution are adjusted. Our particular focus is on those distributions that produce prochiral molecules able to occur in adsorbed configurations being mirror images of each other (surface enantiomers). We demonstrate that, depending on size, aspect ratio, and intramolecular distribution of active sites, the surface enantiomers can co-crystallize or segregate into extended homochiral domains with largely diversified nanosized cavities. The insights from our theoretical studies can be helpful in designing 2D chiral porous networks with potential applications in enantioselective adsorption and asymmetric heterogeneous catalysis.

Phase transition properties of the Bell-Lavis model

Using Monte Carlo calculations we analyze the order and the universality class of phase transitions into a low-density (honeycomb) phase of a triangular antiferromagnetic three-state Bell-Lavis model. The results are obtained in a whole interval of chemical potential μ corresponding to the honeycomb phase. Our results demonstrate that the phase transitions might be attributed to the three-state Potts universality class for all μ values except for the edges of the honeycomb phase existence. At the honeycomb phase and the low-density gas phase boundary the transitions become of the first order. At another, honeycomb-to-frustrated phase boundary, we observe the approach to the crossover from the three-state Potts to the Ising model universality class. We also obtain the Schottky anomaly in the specific heat close to this edge. We show that the intermediate planar phase, found in a very similar antiferromagnetic triangular Blume-Capel model, does not occur in the Bell-Lavis model.

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.

Two-dimensional self-assembly of a symmetry-reduced tricarboxylic acid

Investigations of the self-assembly of simple molecules at the solution/solid interface can provide useful insight into the general principles governing supramolecular chemistry in two dimensions. Here, we report on the assembly of 3,4',5-biphenyl tricarboxylic acid (H3BHTC), a small hydrogen bonding unit related to the much-studied 1,3,5-benzenetricarboxylic acid (trimesic acid, TMA), which we investigate using scanning tunneling microscopy (STM) and density functional theory (DFT) calculations. STM images show that H3BHTC assembles by itself into an offset zigzag chain structure that maximizes the surface molecular density in favor of maximizing the number density of strong cyclic hydrogen bonds between the carboxylic groups. The offset geometry creates "sticky" pores that promote solvent coadsorption. Adding coronene to the molecular solution produces a transformation to a high-symmetry host-guest lattice stabilized by a dimeric/trimeric hydrogen bonding motif similar to the TMA flower structure. Finally, we show that the H3BHTC lattice firmly immobilizes the guest coronene molecules, allowing for high-resolution imaging of the coronene structure.

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

The statistical three-state model is proposed to describe the ordering of triangular TMA molecules into flower phases. The model is solved on a rescaled triangular lattice, assuming following intermolecular interactions: exclusion of any molecules on nearest neighbor sites, triangular trio H-bonding interactions for molecules of the same orientation on next-nearest neighbor sites, and dimeric H-bonding interactions for molecules of different ("tip-to-tip") orientations on third-nearest neighbor sites. The model allows us to obtain the analytical solution for the ground state phase diagram with all homologous series of flower phases included, starting with the honeycomb phase (n=1) and ending with the superflower structure (n=∞). Monte Carlo simulations are used to obtain the thermodynamical properties of this model. It is found that phase transitions from disordered to any of the flower phases (except n=1) undergo via intermediate correlated triangular domains structure. The transition from the disordered phase to the intermediate phase is, most likely, of the first order, while the transition from the intermediate to the flower phase is definitely first order phase transition. The phase diagrams including low-temperature flower phases are obtained. The origin of the intermediate phase, phase separation, and metastable structures are discussed.