Monte Carlo Simulations of the Metal-Directed Self-Assembly of Y-Shaped Positional Isomers

Abstract

The rational fabrication of low-dimensional materials with a well-defined topology and functions is an incredibly important aspect of nanotechnology. In particular, the on-surface synthesis (OSS) methods based on the bottom-up approach enable a facile construction of sophisticated molecular architectures unattainable by traditional methods of wet chemistry. Among such supramolecular constructs, especially interesting are the surface-supported metal–organic networks (SMONs), composed of low-coordinated metal atoms and $\pi$-aromatic bridging linkers. In this work, the lattice Monte Carlo (MC) simulation technique was used to extract the chemical information encoded in a family of Y-shaped positional isomers co-adsorbed with trivalent metal atoms on a flat metallic surface with (111) geometry. Depending on the intramolecular distribution of active centers (within the simulated molecular bricks, we observed a metal-directed self-assembly of two-dimensional (2D) openwork patterns, aperiodic mosaics, and metal–organic ladders. The obtained theoretical findings could be especially relevant for the scanning tunneling microscopy (STM) experimentalists interested in a surface-assisted construction of complex nanomaterials stabilized by directional coordination bonds.

1. Introduction

In arts and crafts, the openwork ornamentation made by cutting out holes with well-defined geometrical shapes in a preselected material (fabric, clay, wood, etc.), is particularly popular. Thanks to the holes, a beam of light is introduced into the artistic composition, which shapes the object with a play of chiaroscuro and subtly emphasizes its fragile internal structure [1]. Interestingly, many one-atom-thick molecular assemblies, such as surface-supported metal–organic networks (SMONs), exhibit a non-trivial, openwork structure [2,3,4,5,6,7]. This is due to the presence of empty cavities (nanopores) within their structure that could differ in shape, spatial distribution, and chirality [8,9,10]. Moreover, the openwork supramolecular architectures stabilized by directional ligand$\to$metal coordination bonds are not only aesthetically pleasing but could also find practical applications in nanotechnology (as selective molecular sieves, atomically defined adsorbents, surface protective coatings, etc.), nanotribology, and heterogeneous catalysis [11,12,13,14].

To date, numerous molecular precursors with a different symmetry and predefined shape, size, and intramolecular distribution of functional groups have been synthesized and successfully applied as building blocks of a plethora of SMONs with tailorable morphology, porosity, and functions [2,4,5,9,11,15,16,17,18,19]. Among them, especially prominent is a family of small star-shaped molecules with $C_3$-symmetrical backbones comprising interconnected phenyl moieties. In recent decades, such carbon-based molecular bricks made possible the on-surface synthesis (OSS) of diverse metal–organic networks [4,13,16,18,19,20,21,22], sophisticated planar tessellations [23], ladders [24], nanoribbons, and small isolated aggregates [25,26]. However, much less attention so far has been paid to the analogous carbon-based molecules with a reduced symmetry [27,28,29]. For these molecular building blocks, the outcome of metal-directed self-assembly is usually hard to predict. It is especially true in the case of conformationally flexible molecules with high structural heterogeneity, which could exist in different chemical forms, such as tautomers, conformers, enantiomers, etc. [30,31].

In our previous work, we demonstrated that the tripodal ligands with properly assigned interaction centers could be valuable building blocks of the novel, metal–organic architectures [32]. However, the performed computer simulations were limited only to the molecular bricks with $C_3$-symmetrical backbones. Therefore, in this contribution, we extend our theoretical investigations on a family of Y-shaped positional isomers whose arms differ in length and are equipped with active centers (functional groups) embedded in para and/or meta positions. To that end, we have applied a classical lattice Monte Carlo (MC) method, which allows for a precise investigation of complex molecular assemblies [33,34,35]. Moreover, the lattice MC simulations, based on discrete intermolecular potentials, enable us to avoid tedious, time-consuming force calculations [36,37]. The main objective of our work is to extract the chemical information encoded in the studied molecules and exploit the gained hints to steer the morphology of surface-confined patterns, comprising Y-shaped bridging linkers, and trivalent metal atoms. The obtained results could be especially useful in the context of surface-assisted fabrication of novel openwork nanomaterials, metal–organic ladders, and aperiodic mosaics with untrivial topologies.

2. Materials and Methods

To explore the surface-supported self-assembly of positional isomers A–J (see Figure 1) the coarse-grained Monte Carlo (MC) simulations were performed in the $NLT$ ensemble, where $N=l+m$ is the total number of adsorbed entities (where $l=600$ is the number of Y-shaped ligands, and $m=l$ metal atoms), $L$ stands for the linear size of the simulated system ($L=130$), and $T$ is temperature. The MC computer calculations were performed on a rhombic fragment of flat triangular lattice comprising $L\times L$ energetically equivalent adsorption sites (vertices of a triangular lattice) at low surface coverage, defined as $\theta =\left(5l+m\right)L^{-2}$ (if not stated otherwise). In the adopted model, the two-dimensional (2D) triangular lattice represents an atomically smooth crystalline $\mathrm{M}\left(111\right)$ surface with no defects and corrugations, where $\mathrm{M}=\mathrm{Cu}, \mathrm{Ag},$ or $\mathrm{Au}$. The distance between two neighboring adsorption sites on a triangular lattice was arbitrarily set to $1$ and, for the sake of simplicity, expressed in dimensionless units. To minimize the influence of negative edge-effects on the morphology of the resulting molecular patterns, imposed were periodic boundary conditions in all planar directions of a 2D lattice. Additionally, to avoid the formation of unwanted point structural defects (missing ligands/metal atoms, self-interstitial defect, etc.), the annealing procedure was applied. Namely, the simulated overlayer was linearly cooled down from the initial temperature, $T=1.001$, to the target temperature $T=0.001$ in ${10}^3$ equal intervals $dT=0.001$, comprising $6N·{10}^3$ MC steps each (one MC step is a single displacement/in-plane rotation trial of randomly selected entity adsorbed on a triangular lattice).

The MC simulations started with a random distribution of $l$ tripodal ligands and $m$ trivalent metal atoms on a triangular lattice in an optimal $l/m=1:1$ stoichiometric ratio. Then, in the next step, the investigated metal–organic overlayer was energetically equilibrated in a series of $6N·{10}^6$ MC steps. For this purpose, one component of the adsorbed overlayer was selected at random, and its total potential energy $E_1$ was determined. Then, the chosen ligand/metal atom was displaced within the 2D lattice or rotated in place (only if the Y-shaped molecule was selected) by a magnitude of acute angle $\pm$60. If this operation was successful, its total potential energy $E_2$ in a new position $\mathsf{\Omega }$ was calculated. Otherwise, the tested entity was left in the original configuration on a lattice, and the above outlined procedure was successively iterated. The ligand/metal atom displacement attempt was accepted/rejected based on the Metropolis MC criterion [38,39] involving a calculation of transition probability $p=min\left\{1,exp\left(-\Delta E/k_{\mathrm{B}}T\right)\right\}$, where $\Delta E=E_2-E_1$ is the potential energy difference between the new and the old position of the probed entity, $k_{\mathrm{B}}$ denotes the reduced Boltzmann constant ($k_{\mathrm{B}}=1$ in our model), and $T$ is a system temperature. In the case where $E_2\le E_1$, the new position $\mathsf{\Omega }$ was accepted unconditionally. Otherwise, the random number $r\in \left(0,1\right)$ was generated and compared with a transition probability $p$. If a random number $r$ satisfied inequality $r\le p$, the new position $\mathsf{\Omega }$ was accepted. In the latter case ($r>p$), the tested configuration was rejected, and the ligand/metal atom was moved back to the original position on a triangular lattice.

In the adopted theoretical model, each of the $\pi$-aromatic positional isomers A–J is represented by a group of in-plane interconnected hard spheres (segments) with properly assigned active centers (black arrows in Figure 1), while metal atoms are modeled as single flat segments. In our coarse-grained approach, a single gray segment corresponds to one phenyl ring of real surface isomer A–J, while segment–segment connections represent C–C sigma bonds linking aromatic moieties of the studied molecules. Additionally, each gray segment present within the simulated linker A–J covers only one adsorption site on a triangular lattice (localized, monolayer adsorption). The same is true in the case of a single metal atom. Contrary to the classical Langmuir adsorption model, the desorption of components of the investigated metal–organic overlayers is prohibited ($NLT$ ensemble). Moreover, lateral interactions between physisorbed Y-shaped ligands and metal atoms on the adjacent adsorption sites of a triangular lattice are possible—the in-plane ligand$\to$metal interactions were assumed to be short-ranged (limited to the neighboring adsorption sites on a lattice), anisotropic, equal in strength ($\omega =-1.0$, negative value means attraction), and reversible. We assumed that a single coordination bond could only be formed if the metal atom $m$ and linker $l$ met on a triangular lattice in a proper spatial configuration; one of the outer segments of the Y-shaped linker and the metal atom cover two adjacent adsorption sites (vertices of a triangular lattice), and interaction directions (black arrows) assigned to them are oppositely directed ($\to \leftarrow$) and collinear. To solely extract the chemical information encoded in the geometric shape and the intramolecular distribution of active centers of the positional isomers A–J, we assumed that the potential energy of an anisotropic coordination bond prevails over other interactions possible in the studied metal–organic overlayers. For this purpose, all potential energies associated with the ligand$\leftrightarrow$surface, metal$\leftrightarrow$surface, metal$\leftrightarrow$metal, and ligand$\leftrightarrow$ligand side-to-side interactions were set to zero. Additionally, to compare the structural properties of the ordered 2D metal–organic patterns, we introduced the structural parameter called density$\rho$. This dimensionless parameter was defined as the ratio of occupied adsorption sites inside a unit cell of arbitrary selected phase and the total area $A$ of this unit cell, that is $\rho =\left(5x+y\right)A^{-1}$, where $x$ and $y$, respectively, denote the number of Y-shaped linkers and metal atoms within a unit cell.

3. Results and Discussion

Initial MC simulations were conducted for the $C_1$-symmetric positional isomer A, the arms of which are functionalized exclusively in para positions. Such intramolecular distribution of identical active centers makes linker A achiral, which is manifested in a relatively simple morphology of the emerging supramolecular patterns. However, even such a trivial functionalization scheme combined with a reduced symmetry of molecule A directed the surface-confined self-assembly into two polymorphic superstructures with distinct morphological properties such as density $\rho$, pore size/shape, and their spatial distribution (see Figure 2a,b).

As seen in Figure 2a, the first polymorph $\left(\rho =0.371\right)$ is characterized by a rhombic unit cell $\left(7\times 7, \angle 60/{120}^{\mathrm{o}}\right)$ comprising three linkers A and three metal atoms in a 1:1 stoichiometric ratio. The cavities present within the corresponding network are periodically distributed and arranged into a flower-like pattern with a central hexagonal void (7 adsorption sites) and six petals (12 adsorption sites each) with the shape of a truncated equilateral triangle. In comparison to the discussed phase, the brick wall-shaped polymorphic network shown in Figure 2b has a $~15\%$ higher density ($\rho =0.433$). Its parallelogram unit cell $\left(2\sqrt{3}\times \sqrt{19}, \angle 67/{113}^{\mathrm{o}}\right)$ contains only one linker A and one metal atom. Moreover, all Y-shaped ligands A present within this homogenous superstructure are oriented unidirectionally. The adjacent pores are also oriented along the same direction of a triangular lattice and comprise 10 adsorption sites each. Although the discussed 2D polymorphs are almost energetically equivalent, the surface-confined self-assembly of a heteroporous network illustrated in Figure 2a is privileged due to entropic reasons. Namely, after a visual inspection of 50 independent system replicas, it revealed that the two-porous phase was formed selectively in $~66\%$ of them. This result could be reasonably explained by comparing the number of spatial orientations of linkers A within the polymorphic phases—in the superstructure shown in Figure 2a each bridging linker A could be incorporated into the growing metal–organic network in three different ways, whereas in the case of a uniporous network (Figure 2b), only a unidirectional arrangement of ligands A could result in formation of a defect-free overlayer. Interestingly, similar crystalline polymorphs were observed experimentally in the case of 3,3′,5,5′-tetrabromo-2,2′,4,4′,6,6′-hexafluorobiphenyl (Br₄F₆BP) molecules [40], brominated tetrathienoanthracene [41], as well as a theoretical model [42]. In the light of the above-listed scientific reports, the discussed two-directional self-assembly is no exception but rather the next example of a dichotomous formation of complex metal–organic patterns.

Contrary to the achiral molecule A, the next studied positional isomer B is functionalized at one (S)-meta and two para positions. As shown in Figure 3a, this intramolecular distribution of active centers (functional groups G) allows for a highly selective formation of the openwork metal–organic network with huge cavities (55 adsorption sites) surrounded concentrically by smaller pores comprising 12 and 2 adsorption sites each. The gear-like shaped voids have diameter $d=8$ (corner-to-corner) and a unique set of teeth differing in length, that is ${\left(\mathrm{1.1.3}\right)}^6$. In the adopted notation, an individual number denotes the length of a single tooth, while the number sequence corresponds to their order of occurrence at the edges of the largest pore. The characteristic feature of the heteroporous phase is a relatively low density $\rho =0.344$ and an exceptionally large rhombic unit cell $\left(11\times 11, \angle 60/{120}^{\mathrm{o}}\right)$, comprising six molecules B and six metal atoms in a 1:1 stoichiometric ratio.

Embedding an active center on the second short arm of tritopic molecule B in the (S)-meta position gives a subsequent positional isomer, denoted by the letter C. In this molecule, the (S)-meta active center is pointed towards its long arm, monofunctionalized in the para position. Interestingly, even such a tiny structural modification strongly affects the topology of the resulting supramolecular network. As illustrated in Figure 3b, the infinite metal–organic phase exhibits basic structural features of the adsorbed overlayer from Figure 3a. The gear-like voids ${\left(\mathrm{1.1.2}\right)}^6$ of the considered superstructure encompass 31 adsorption sites and are separated by the adjacent small pores comprising 12 and 3 free adsorption sites. Additionally, with a relatively small diameter of the largest pores $\left(d=6\right),$ a substantial density follows, increasing the crystalline structure ($\rho =0.416$) as well as the side length reduction of its unit cell $\left(10\times 10, \angle 60/{120}^{\mathrm{o}}\right)$. Surprisingly, a displacement of the para active center attached to the short arm of linker C to the (S)-meta position (within the same outer segment) directed the metal-assisted self-assembly into the subsequent open porous network (see Figure 4a) with vast cavities, characteristic for the overlayer illustrated in Figure 3a. However, within the discussed adlayer, the gear-like voids ${\left(\mathrm{1.1.2}\right)}^6$ are separated by alternatively arranged small pores with a star-shaped (four adsorption sites) and needle-like geometry (three adsorption sites). The rhombic unit cell of the corresponding supramolecular structure contains six surface isomers D and six metal atoms in a 1:1 stoichiometric ratio.

In contrast to the above-discussed molecular assemblies, in the case of isomer E, which arms are functionalized in different positions—in para/(R)-meta/(S)-meta mode counting anticlockwise—we did not observe an emerging open porous phase with translational symmetry. Instead, the isolated metal–organic ladders were obtained. In the exemplary snapshot shown in Figure 4b, each ladder comprises alternately arranged small pores comprising four and two adsorption sites, while the metal atoms incorporated into its structure are arranged in a zig-zag fashion (blue line in the magnified fragment of Figure 4b). Although the metal–organic ladders are randomly distributed on a triangular lattice, all of them are pointed unidirectionally. Such a preferred spatial configuration is not surprising, as it provides unperturbed elongation (metal-directed growth) of individual ladders.

As shown in Figure 5a, also in the case of positional isomer F, the obtained metal–organic phase differs largely in morphology from the supramolecular architectures depicted in Figure 2 and Figure 4a. Namely, its main feature is the lack of translational order. As shown in the magnified fragment of Figure 4a, the multiporous phase is built of three types of empty cavities marked in different colors: pink (10 adsorption sites), chalk (7 adsorption sites), and light-green (4 adsorption sites). Interestingly, the basic morphological properties of the considered overlayer are also conserved at a two times higher surface coverage (see Figure 5b).

The aperiodicity of the overlayers shown in Figure 5a,b originates exclusively from a non-regular distribution of bridging linkers F, whereas trivalent metal atoms occupy vertices of a well-defined triangular lattice with a primitive unit cell (see Figure 6b). Another unique feature of the aperiodic mosaics from Figure 5a,b are the randomly distributed flower-like rosettes with closed and open walls. As shown in the magnified fragment of Figure 6a, the central point of a closed hexagonal rosette is a windmill-like three-fold coordination node surrounded by two coordination shells comprising a periodical array of metal atoms and molecules F. However, this perfect spatial arrangement of bridging linkers/metal atoms could be easily perturbed without generating any point structural defects. Therefore, in the mosaic superstructure, many open-walled rosettes could be easily identified. Within the aperiodic metal–organic network from Figure 5a, the most abundant are irregular pores marked in chalk color ($~$50%), while the light-green and pink elongated cavities are almost equally populated and constitute $~$27% and $~$23% of all empty pores present in the considered overlayer (averages from five independent system replicas).

Even after the cursory visual inspection of the aperiodic mosaics illustrated in Figure 5a,b, it becomes clear that the metal-directed self-assembly of molecules F could produce also other metal–organic patterns with a highly periodic distribution of bridging linkers F and metal atoms. Small fragments of such artificially created connections are shown in Figure 7. Especially interesting among them are the overlayers composed of flower-like rosettes (Figure 7a,b). The first of them ($\rho =0.533$) is built of adjacent hexagonal rosettes with central windmill-like coordination nodes oriented clockwise (see Figure 7a). The second polymorph, with a huge rhombic unit cell ($\sqrt{234}\times \sqrt{234} \angle 60/{120}^{\mathrm{o}}$), also comprises closed hexagonal rosettes, but some of their central windmill-like nodes are oriented anticlockwise (Figure 7b). Moreover, one could imagine a morphologically similar phase (with equal density $\rho$), where the windmill-like coordination nodes are randomly oriented. The third artificial overlayer, shown in Figure 7c, resembles an open porous network obtained for molecule C (see Figure 3b). However, within this phase, the gear-like cavities ${\left(\mathrm{1.1.2}\right)}^6$ with a diameter $d=6$ (corner-to-corner), are surrounded concentrically by smaller pores comprising four and six adsorption sites. The last crystalline pattern is homogenous in terms of porosity ($\rho =0.533$) and comprises only irregular voids containing seven adsorption sites each (Figure 7d).

The next studied molecule, denoted by the letter G, is characterized by a highly symmetrical distribution of active centers (see Figure 8). Namely, the long arm of anchor-like ligand G is equipped with an active center in the para position, whereas the remaining two shorter arms (equal in length) are functionalized in meta positions with opposite in-plane orientations: (R) and (S). Consequently, both meta active centers are pointed to the para functionalized arm of $C_1$-symmetric molecule G. Such a unique functionalization scheme allows this isomer to self-assemble with trivalent metal atoms into a highly aperiodic superstructure with no structural defects (see Figure 8a). As shown in the magnified fragment of Figure 8a, the complex mosaic phase is composed of six different cavities. Among them are kite-like voids (○, ○–○, ○–○–○) encompassing 6, 12, and 18 free adsorption sites; elongated saw-like pores ● (nine adsorption sites), and ●–● (15 adsorption sites); and small needle-like voids, comprising only three adsorption sites. As in the case of the mosaic overlayers depicted in Figure 5a,b, the aperiodicity of the considered metal–organic network originates only from a non-regular distribution of anchor-like linkers G, while the isolated metal atoms occupy vertices of a periodic triangular lattice with primitive unit cell $\left(2\sqrt{3}\times 2\sqrt{3}, \angle 60/{120}^{\mathrm{o}}\right)$. Surprisingly, at high surface coverage ($\theta ~0.53$), some fraction of molecules G is engaged in the formation of metal–organic ladders coexisting with an aperiodic phase (see Figure 8b). As shown in a magnified fragment of Figure 8b, the metal–organic ladders are close-packed and oriented along the same direction of a triangular lattice. However, the adjacent ladders comprising small needle-like voids do not create a periodic crystalline phase, as some of them are translated in-plane with respect to the others. A closer visual inspection of an aperiodic superstructure shown in Figure 8a reveals that similar ladder-like metal–organic connections are also present in its structure. However, the lengths of these ordered connections are relatively small; most of the metal–organic ladders incorporated into the phase illustrated in Figure 8a comprise only from two to five parallel oriented linkers G.

Unlike the aperiodic mosaics created by the positional isomers F and G, the supramolecular pattern obtained for molecule H is highly regular. As seen in Figure 9a, the open porous network comprises three types of nanocavities. Among them are large gear-like shaped voids ${\left(\mathrm{1.2.2}\right)}^6$ with a diameter $d=2\sqrt{13}$ (corner-to-corner), hexagonal pores (seven adsorption sites), and truncated rhombus-like small voids (four adsorption sites). Inside the rhombic unit cell $\left(\sqrt{111}\times \sqrt{111}, \angle 69/{111}^{\mathrm{o}}\right)$ of the crystalline overlayer, there are six molecules H and six metal atoms. Moreover, the multiporous phase has low density $\left(\rho =0.346\right)$, comparable with the metal–organic array composed of surface isomers B mixed with trivalent metal atoms in a 1:1 molar ratio (see Figure 3a). A very similar topology corresponds also to the defect-free overlayer comprising linkers I. As shown in Figure 9b, the infinite metal–organic phase ($\rho =0.381$) is composed of large voids ${\left(\mathrm{1.2.2}\right)}^6$ with diameter $d=2\sqrt{13}$ (corner-to-corner), windmill-like pores (nine adsorption sites), and very small cavities encompassing only two adsorption sites. The rhombic unit cell of the considered network $\left(\sqrt{109}\times \sqrt{109}, \angle 60/{120}^{\mathrm{o}}\right)$ contains six ligands I and six metal atoms in a 1:1 stoichiometric ratio, which agrees perfectly with the overall ligand/metal proportion in the system under study.

For the last examined molecule (denoted as J), with arms functionalized selectively in (S)-meta positions, we observed the formation of two polymorphs differing in terms of porosity. As seen in Figure 10a,b, both metal–organic networks are very similar to those obtained for achiral linker A (see Figure 2a,b). Namely, the multiporous overlayer shown in Figure 10a is characterized by a rhombic unit cell $\left(\sqrt{37}\times \sqrt{37}, \angle 60/{120}^{\mathrm{o}}\right)$ comprising three ligands J and three trivalent metal atoms in a 1:1 stoichiometric ratio. Within the open porous crystalline structure ($\rho =0.562$), the periodically distributed windmill-like pores (nine adsorption sites) could be easily identified. Their edges are decorated by adjacent small voids enclosing four and six free adsorption sites. The second brick wall-like polymorph with a parallelogram unit cell $\left(3\times 4, \angle 60/{120}^{\mathrm{o}}\right)$ is built of unidirectionally oriented voids including six adsorption sites each (see Figure 10b). Interestingly, such close-packed arrangement of relatively small voids makes this metal–organic superstructure the densest among all 2D crystals discussed in this work $\left(\rho =0.577\right)$.

4. Conclusions

Exploiting the MC simulations, we examined the metal-directed self-assembly of flat $\pi$-aromatic molecules A–J with a reduced symmetry. The obtained results indicate that the dichotomous self-assembly is a unique feature of tritopic molecules A and J, in which terminal segments are functionalized selectively in para/(S)-meta positions. For the positional isomers with a more complex functionalization pattern (linkers B, C, D, H, and I), we observed the formation of metal–organic networks with large gear-like shaped voids decorated by smaller cavities, differing in size and shape. Interestingly, only in the case of the simulated system comprising linkers E, did we observe the formation of isolated metal–organic ladders instead of an open porous phase. However, the most intriguing supramolecular patterns emerged in the adsorbed systems comprising ligands F and G mixed with trivalent metal atoms. Namely, these simple building blocks were able to create complex mosaics with an aperiodic distribution of bridging linkers and a highly periodic arrangement of metal atoms.

To conclude, the performed calculations indicate that a subtle modification of intramolecular distribution of active centers embedded in termini of Y-shaped molecules A–J strongly affects the final topology of self-assembled overlayers, where anisotropic coordination bonds play a decisive role. In this context, the tritopic positional isomers investigated herein could be considered valuable building blocks of novel surface-confined supramolecular architectures.