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.

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

Polymers are frequently deposited on different surfaces, which has attracted the attention of scientists from different viewpoints. In the present approach polymers are represented by rigid rods of length k (k-mers), and the substrate takes the form of an L×L square lattice whose lattice constant matches exactly the interspacing between consecutive elements of the k-mer chain. We briefly review the classical description of the nematic transition presented by this system for k≥7 observing that the high-coverage (θ) transition deserves a more careful analysis from the entropy point of view. We present a possible viewpoint for this analysis that justifies the phase transitions. Moreover, we perform Monte Carlo (MC) simulations in the grand canonical ensemble, supplemented by thermodynamic integration, to first calculate the configurational entropy of the adsorbed phase as a function of the coverage, and then to explore the different phases (and orientational transitions) that appear on the surface with increasing the density of adsorbed k-mers. In the limit of θ→1 (full coverage) the configurational entropy is obtained for values of k ranging between 2 and 10. MC data are discussed in comparison with recent analytical results [D. Dhar and R. Rajesh, Phys. Rev. E 103, 042130 (2021)2470-004510.1103/PhysRevE.103.042130]. The comparative study allows us to establish the applicability range of the theoretical predictions. Finally, the structure of the high-coverage phase is characterized in terms of the statistics of k×l domains (domains of l parallel k-mers adsorbed on the surface). A distribution of finite values of l (l≪L) is found with a predominance of k×1 (single k-mers) and k×k domains. The distribution is the same in each lattice direction, confirming that at high density the adsorbed phase goes to a state with mixed orientations and no orientational preference. An order parameter measuring the number of k×k domains in the adsorbed layer is introduced.

Hard core lattice gas with third next-nearest neighbor exclusion on triangular lattice: One or two phase transitions?

We obtain the phase diagram of the hard core lattice gas with third nearest neighbor exclusion on the triangular lattice using Monte Carlo simulations that are based on a rejection-free flat histogram algorithm. In a recent paper [Darjani et al., J. Chem. Phys. 151, 104702 (2019)], it was claimed that the lattice gas with third nearest neighbor exclusion undergoes two phase transitions with increasing density with the phase at intermediate densities exhibiting hexatic order with continuously varying exponents. Although a hexatic phase is expected when the exclusion range is large, it has not been seen earlier in hard core lattice gases with short range exclusion. In this paper, by numerically determining the entropies for all densities, we show that there is only a single phase transition in the system between a low-density fluid phase and a high density ordered sublattice phase and that a hexatic phase is absent. The transition is shown to be first order in nature, and the critical parameters are determined accurately.

Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases

We introduce a rejection-free, flat histogram, cluster algorithm to determine the density of states of hard-core lattice gases. We show that the algorithm is able to efficiently sample low entropy states that are usually difficult to access, even when the excluded volume per particle is large. The algorithm is based on simultaneously evaporating all the particles in a strip and reoccupying these sites with a new appropriately chosen configuration. We implement the algorithm for the particular case of the hard-core lattice gas in which the first k next-nearest neighbors of a particle are excluded from being occupied. It is shown that the algorithm is able to reproduce the known results for k=1,2,3 both on the square and cubic lattices. We also show that, in comparison, the corresponding flat histogram algorithms with either local moves or unbiased cluster moves are less accurate and do not converge as the system size increases.

Porous Honeycomb Self-Assembled Monolayers: Tripodal Adsorption and Hidden Chirality of Carboxylate Anchored Triptycenes on Ag

Molecules with tripodal anchoring to substrates represent a versatile platform for the fabrication of robust self-assembled monolayers (SAMs), complementing the conventional monopodal approach. In this context, we studied the adsorption of 1,8,13-tricarboxytriptycene (Trip-CA) on Ag(111), mimicked by a bilayer of silver atoms underpotentially deposited on Au. While tripodal SAMs frequently suffer from poor structural quality and inhomogeneous bonding configurations, the triptycene scaffold featuring three carboxylic acid anchoring groups yields highly crystalline SAM structures. A pronounced polymorphism is observed, with the formation of distinctly different structures depending on preparation conditions. Besides hexagonal molecular arrangements, the occurrence of a honeycomb structure is particularly intriguing as such an open structure is unusual for SAMs consisting of upright-standing molecules. Advanced spectroscopic tools reveal an equivalent bonding of all carboxylic acid anchoring groups. Notably, density functional theory calculations predict a chiral arrangement of the molecules in the honeycomb network, which, surprisingly, is not apparent in experimental scanning tunneling microscopy (STM) images. This seeming discrepancy between theory and experiment can be resolved by considering the details of the actual electronic structure of the adsorbate layer. The presented results represent an exemplary showcase for the intricacy of interpreting STM images of complex molecular films. They are also further evidence for the potential of triptycenes as basic building blocks for generating well-defined layers with unusual structural motifs.

Entropy of fully packed hard rigid rods on d-dimensional hypercubic lattices

We determine the asymptotic behavior of the entropy of full coverings of a L×M square lattice by rods of size k×1 and 1×k, in the limit of large k. We show that full coverage is possible only if at least one of L and M is a multiple of k, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a k×k square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large k, we show that the entropy per site S_{2}(k) tends to Ak^{-2}lnk, with A=1. We conjecture, based on a perturbative series expansion, that this large-k behavior of entropy per site is superuniversal and continues to hold on all d-dimensional hypercubic lattices, with d≥2.

Simulations of the 2D self-assembly of tripod-shaped building blocks

We introduce a molecular dynamics (MD) coarse-grained model for the description of tripod building blocks. This model has been used by us already for linear, V-shape, and tetratopic molecules. We wanted to further extend its possibilities to trifunctional molecules to prove its versatility. For the chosen systems we have also compared the MD results with Monte Carlo results on a triangular lattice. We have shown that the constraints present in the latter method can enforce the formation of completely different structures, not reproducible with off-lattice simulations. In addition to that, we have characterized the obtained structures regarding various parameters such as theoretical diffraction pattern and average association number.

Self-Assembled Two-Dimensional Nanoporous Crystals as Molecular Sieves: Molecular Dynamics Studies of 1,3,5-Tristyrilbenzene-Cn Superstructures

Due to their unique geometry complex, self-assembled nanoporous 2D molecular crystals offer a broad landscape of potential applications, ranging from adsorption and catalysis to optoelectronics, substrate processes, and future nanomachine applications. Here we report and discuss the results of extensive all-atom Molecular Dynamics (MD) investigations of self-assembled organic monolayers (SAOM) of interdigitated 1,3,5-tristyrilbenzene (TSB) molecules terminated by alkoxy peripheral chains Cn containing n carbon atoms (TSB3,5-Cn) deposited onto highly ordered pyrolytic graphite (HOPG). In vacuo structural and electronic properties of the TSB3,5-Cn molecules were initially determined using ab initio second order Møller-Plesset (MP2) calculations. The MD simulations were then used to analyze the behavior of the self-assembled superlattices, including relaxed lattice geometry (in good agreement with experimental results) and stability at ambient temperatures. We show that the intermolecular disordering of the TSB3,5-Cn monolayers arises from competition between decreased rigidity of the alkoxy chains (loss of intramolecular order) and increased stabilization with increasing chain length (afforded by interdigitation). We show that the inclusion of guest organic molecules (e.g., benzene, pyrene, coronene, hexabenzocoronene) into the nanopores (voids formed by interdigitated alkoxy chains) of the TSB3,5-Cn superlattices stabilizes the superstructure, and we highlight the importance of alkoxy chain mobility and available pore space in the dynamics of the systems and their potential application in selective adsorption.

Alternative characterization of the nematic transition in deposition of rods on two-dimensional lattices

We revisit the problem of excluded volume deposition of rigid rods of length k unit cells over square lattices. Two new features are introduced: (a) two new short-distance complementary order parameters, called Π and Σ, are defined, calculated, and discussed to deal with the phases present as coverage increases; (b) the interpretation is now done beginning at the high-coverage ordered phase which allows us to interpret the low-coverage nematic phase as an ergodicity breakdown present only when k≥7. In addition the data analysis invokes both mutability (dynamical information theory method) and Shannon entropy (static distribution analysis) to further characterize the phases of the system. Moreover, mutability and Shannon entropy are compared, and we report the advantages and disadvantages they present for their use in this problem.

Computer modeling of 2D supramolecular nanoporous monolayers self-assembled on graphite

Nano-porous two-dimensional molecular crystals, self-assembled on atomically flat host surfaces offer a broad range of possible applications, from molecular electronics to future nano-machines. Computer-assisted designing of such complex structures requires numerically intensive modeling methods. Here we present the results of extensive, fully atomistic simulations of self-assembled monolayers of interdigitated molecules of 1,3,5-tristyrilbenzene substituted by C6 alkoxy peripheral chains (TSB3,5-C6), deposited onto highly-ordered pyrolytic graphite. Structural and electronic properties of the TSB3,5-C6 molecules were determined from ab initio calculations, then used in Molecular Dynamics simulations to analyze the mechanism of formation, epitaxy, and stability of the TSB3,5-C6 nanoporous superlattice. We show that the monolayer disordering results from the competition between flexibility of the C6 chains and their stabilization by interdigitation. The inclusion of guest molecules (benzene and pyrene) into superlattice nanopores stabilizes the monolayer. The alkoxy chain mobility and available pore space defines the systems dynamics, essential for potential application.

Phase transitions in a system of hard Y-shaped particles on the triangular lattice

We study the different phases and the phase transitions in a system of Y-shaped particles, examples of which include immunoglobulin-G and trinaphthylene molecules, on a triangular lattice interacting exclusively through excluded volume interactions. Each particle consists of a central site and three of its six nearest neighbors chosen alternately, such that there are two types of particles which are mirror images of each other. We study the equilibrium properties of the system using grand canonical Monte Carlo simulations that implement an algorithm with cluster moves that is able to equilibrate the system at densities close to full packing. We show that, with increasing density, the system undergoes two entropy-driven phase transitions with two broken-symmetry phases. At low densities, the system is in a disordered phase. As intermediate phases, there is a solidlike sublattice phase in which one type of particle is preferred over the other and the particles preferentially occupy one of four sublattices, thus breaking both particle symmetry as well as translational invariance. At even higher densities, the phase is a columnar phase, where the particle symmetry is restored, and the particles preferentially occupy even or odd rows along one of the three directions. This phase has translational order in only one direction, and breaks rotational invariance. From finite-size scaling, we demonstrate that both the transitions are first order in nature. We also show that the simpler system with only one type of particle undergoes a single discontinuous phase transition from a disordered phase to a solidlike sublattice phase with an increasing density of particles.

Sequential nested assembly at the liquid/solid interface

Studying the stepwise assembly of a four component hybrid structure on Au(111)/mica, the pores of a hydrogen bonded bimolecular network of 3,4,9,10-perylenetetracarboxylic diimide (PTCDI) and 1,3,5-triazine-2,4,6-triamine (melamine) were partitioned by three and four-armed molecules based on oligo([biphenyl]-4-ylethynyl)benzene, followed by the templated adsorption of either C₆₀ fullerene or adamantane thiol molecules. The characterisation by ambient scanning tunneling microscopy (STM) reveals that the pore modifiers exhibit dynamics which pronouncedly depend on the molecular structure. The three-armed molecule 1,3,5-tris([1,1'-biphenyl]-4-ylethynyl)benzene (3BPEB) switches between two symmetry equivalent configurations on a time scale fast compared to the temporal resolution of the STM. Derivatisation of 3BPEB by hydroxyl groups substantially reduces the switching rate. For the four-armed molecule configurational changes are observed only occasionally. The observation of isolated fullerenes and small clusters of adamantane thiol molecules, which are arranged in a characteristic fashion, reveals the templating effect of the trimolecular supramolecular network. However, the fraction of compartments filled by guest molecules is significantly below one for both the thermodynamically controlled adsorption of C₆₀ and the kinetically controlled adsorption of the thiol with the latter causing partial removal of the pore modifier. The experiments, on the one hand, demonstrate the feasibility of templating by nested assembly but, on the other hand, also pinpoint the requirement for the energy landscape to be tolerant to variations in the assembly process.

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.

Synthesis of new dithia[3.3]parapara- and metapara-cyclophane based tectons: toward an universal surface-confined 2D/3D molecular binding motif

A series of new paraphenylene-based 2D and 3D tectons has been designed for supramolecular self-assembly on both HOPG and Au. Several versatile chemical strategies have been developed to reach the target tectons, bearing either metaparacyclophane or paraparacyclophane cores, functionalized or not, allowing the obtention soon of more complex 3D tectons bearing functional unit such as chromophores. Moreover, preliminary STM results show that these compounds can successfully self-assemble both on HOPG and Au(111) substrates at liquid-solid interface, encouraging us in finding an universal surface-confined 2D/3D molecular binding motif. This feature combined with the preliminary results of the self-assembly on the plasmonic substrate Au(111) open-up opportunities in the field of Nanoscience.

On-surface synthesis of two-dimensional imine polymers with a tunable band gap: a combined STM, DFT and Monte Carlo investigation

Two-dimensional polymers are of great interest for many potential applications in nanotechnology. The preparation of crystalline 2D polymers with a tunable band gap is critical for their applications in nano-electronics and optoelectronics. In this work, we try to tune the band gap of 2D imine polymers by expanding the conjugation of the backbone of aromatic diamines both laterally and longitudinally. STM characterization reveals that the regularity of the 2D polymers can be affected by the existence of lateral bulky groups. Density functional theory (DFT) simulations discovered a significant narrowing of the band gap of imine 2D polymers upon the expansion of the conjugation of the monomer backbone, which has been confirmed experimentally by UV absorption measurements. Monte Carlo simulations help us to gain further insight into the controlling factors of the formation of regular 2D polymers, which demonstrated that based on the all rigid assumption, the coexistence of different conformations of the imine moiety has a significant effect on the regularity of the imine 2D polymers.

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.

Bestowing structure upon the pores of a supramolecular network

Trigonal molecules compartmentalise the pores of a honeycomb network of 3,4:9,10-tetracarboxylic diimide (PTCDI) and 1,3,5-triazine-2,4,6-triamine (melamine). Extending the 1,3,5-tri(phenylene-ethynylene)benzene core by a phenyl group allows for a well-defined accommodation of the molecule into two symmetry equivalent positions in the pore. The corresponding styryl or phenylene-ethynylene derivatives exceed the pore size and, thus, impede pore modification.