Trimesic Acid Molecule in a Hexagonal Pore: Central versus

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Trimesic Acid Molecule in a Hexagonal Pore: Central versus Non-Central Position Andrius Ibenskas, Mantas Šim#nas, Kasparas Jonas Kizlaitis, and Evaldas E. Tornau J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 23 Jan 2019 Downloaded from http://pubs.acs.org on January 24, 2019

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The Journal of Physical Chemistry

Trimesic Acid Molecule in a Hexagonal Pore: Central versus Non-Central Position ˇ enas,‡ Kasparas Jonas Kizlaitis,‡ and Evaldas E. Andrius Ibenskas,∗,† Mantas Sim˙ Tornau† †Semiconductor Physics Institute, Center for Physical Sciences and Technology, Saul˙etekio 3, LT-10257 Vilnius, Lithuania ‡Faculty of Physics, Vilnius University, Saul˙etekio 9, LT-10222 Vilnius, Lithuania E-mail: [email protected] Abstract Self-assembly of trimesic acid (TMA) molecules into the honeycomb structure with filled pores and resulting host-guest chemistry are studied by the density functional theory (DFT) and Monte Carlo (MC) simulations. The DFT calculations demonstrate that a guest TMA molecule prefers a non-central position in a relaxed hexagonal pore formed of six TMA molecules, and it is binded by two intermolecular interactions. The symmetric central position of the guest molecule is energetically favorable only in the honeycomb structure which is compressed by more than 3%. Based on the estimated host-guest dimeric interactions, a model is proposed to identify the conditions for central and non-central positioning of TMA molecule within the pore during their ordering into the honeycomb structure with partly filled pores. The MC simulations reveal that increase of the molecule-substrate interaction in the center of the pore or interactions of the central molecule with the cage molecules has a significant effect in preserving the central position of the guest molecule. However, if these interactions are

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not too strong, the non-central position is favored due to multiplicity of non-central arrangements in a hexagonal cage.

Introduction Self-assembly of the ordered supramolecular structures has become an area of the intense research activity over the past decade. 1,2 The two-dimensional molecular networks comprising sizable cavities can be used as templates for engineering the host-guest systems sustained by the non-covalent interactions. The interest in these structures is fueled by their unique functional properties, such as scalability, chemical and chiral selectivity of the guest binding, and responsiveness to external stimuli (heat and electric fields). 3 The host networks can be specifically tailored to immobilize the guest molecules or metal clusters of a particular size. For example, the trimesic acid (TMA) molecules, possessing a threefold symmetry, assemble into the planar honeycomb (HON) framework on various substrates. 4–8 The nanometer-sized pores of the TMA HON phase can accommodate a wide variety of the guest species: coronene, 9–14 fullerenes C60 and C84 , 15–17 oligothiophene macrocycle complex with C60 , 18 heterocirculenes, 19 chiral molecule 5-amino[6]helicene, 20 Ag and Zn nitrates, 21 as well as bismuth nanoclusters. 22 The pore size does not change upon adsorption of suitably-matched guests, though it might slightly depend on the type of underlying substrate. Some insight into the host-guest binding might be gained from the position and orientation of the guest molecules within the pores of the host network, as imaged by the scanning tunneling microscopy (STM). However, small and round-shaped guest species, such as coronene, usually have a blurry appearance in the STM images. Even the triangular-shaped TMA molecules often are not resolved clearly when acting as guests in the TMA HON phase at the liquid-solid interface 23–25 and under ultra-high vacuum (UHV) conditions, 26,27 due to their fast rotation inside the pores. 8

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Interestingly, several experiments show two non-equivalent guest TMA positions within a HON pore. The non-central (NC) arrangement has been detected on highly oriented pyrolytic graphite (HOPG) 28,29 and Au(111), 30 and also implied by a combined STM and molecular mechanics analysis. 25,26 In this arrangement, the guest TMA molecule lies close to the edge of the pore and forms the hydrogen bonds with its two closest neighbors. The central (C) positioning of the guest TMA molecules was observed on a silver substrate in UHV, 31,32 possibly due to the smaller HON pores compared to the pores on HOPG. The C arrangement was also determined in the TMA hexamers which exhibited shrinking due to the co-adsorption of large foreign molecules in the TMA monolayer. 33 Despite these studies, it is not exactly known how small the HON nanopores should be, in order for a guest TMA molecule to stay in the central position. The guest arrangement may also be affected by the molecule-substrate interactions which are relatively strong on metallic surfaces. Some of these issues can be addressed by the computational modeling. Previously, the adsorption of TMA molecules on HOPG, graphene and several metallic surfaces has been studied by the density functional theory (DFT) methods. The calculations for an isolated TMA molecule on graphene 34 yielded the adsorption energy between -25.4 (on hollow site) and -27.3÷-28.2 kcal/mol (on different atom sites) making the diffusion barrier about 2 kcal/mol which is roughly 10% of the double H-bond energy. In comparison, the values -30.7 and -32.7 kcal/mol were found for a single-molecule adsorption on a bridge site of Ag(111) and Au(111) surfaces, respectively. 35 A study of TMA assembly on Cu(100) 36 has indicated that an energy difference between adsorption on a 4-fold hollow (preferred) and the atom sites was about 4.6 kcal/mol, approximately the same as the diffusion barrier for a similar compound, terephthalic acid, on Cu(111) 37 and Cu(001). 38 Therefore, the corrugation of the adsorption energy landscape on copper substrate corresponds to 20% of the double Hbond energy. For the TMA molecules in the HON phase, the molecule-graphene interaction energy was calculated to be -21 kcal/mol, about 1 kcal/mol higher than on HOPG, and it is weakly dependent on the adsorption site. 6 A slightly larger value (-25.7 kcal/mol) was

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obtained by a large-scale molecular mechanics study of non-commensurate TMA HON phase on graphene. 39 The DFT-calculated adsorption energy of the TMA HON structure in three different unit cells commensurate with the Ag(111) substrate varied between -23.1 and 27.4 kcal/mol per molecule. 35 In general, the adsorption energy of TMA molecules on the mentioned substrates slightly exceeds the double H-bond energy in the HON phase, while the diffusion barrier can constitute up to one fourth of this value. An energy gain due to the inclusion of a guest coronene molecule inside the TMA HON pore was calculated in Ref., 40 but similar DFT studies involving the guest TMA molecule are still lacking. The self-assembly, chirality and ordering properties of large supramolecular structures at finite temperatures have also been described by the phase transition models and studied by Monte Carlo (MC) simulations. 41–50 Several models including the Bell-Lavis model 51,52 have been modified and employed for the simulations of the self-assembly of the TMA and other similar molecules into the hexagonal structures. 42,43,47 In these models it was assumed that the ordering is driven by a strong dimeric H-bond interaction occurring between the carboxyl groups of the two TMA molecules which occupy the adjacent or other nearby sites of the triangular lattice. In general, these models can also be used to analyze the filling of the pores with coronene 43 and other guest molecules including TMA itself. 48,52 However, they are not refined enough to distinguish between the C and NC guest arrangements, because in the simplest case only one lattice site is available within every pore for the adsorption. Herein, to study the position of the molecule in a pore, we propose a lattice model with the dimeric interaction formed between two TMA molecules which are separated by a much larger distance on the lattice (six lattice constants). In this model each molecule can be in one of the two orientations. The intermolecular interactions are estimated by the DFT calculations in the fully optimized and then separately in slightly compressed HON pores. The obtained interaction values are used in the MC simulations. We show that even a small variation of the adsorption energy across the lattice sites may be sufficient to shift the

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preference from the NC towards the C position.

Computational Details All DFT calculations were conducted using ORCA 3.0.2 program package. 53 The geometries of the TMA molecular structures were optimized with the hybrid B3LYP functional 54,55 and the 6-31G(d,p) basis set augmented by the Grimme's atom-pairwise dispersion correction (D3 version). 56,57 All computations were carried out in the gas-phase. For dimeric interactions, we also calculated the basis set superposition error (BSSE) 58,59 which accounts for the fact that the monomers partly share their basis functions when the dimer energy is calculated. The results were found to be sensitive to the inclusion of the BSSE correction: the absolute value of the TMA-TMA dimeric interaction was found to notably decrease, especially under compression of the system. For MC calculations, we used the triangular lattice of 72 × 72 sites with periodic boundary conditions and employed the Kawasaki dynamics, where the diffusion of molecules was simulated for fixed value of molecular concentration. Ideal tiling by the HON phase on this lattice at the intermolecular TMA-TMA distance of 6a corresponds to N = 96 molecules. We have chosen N = 120 for our calculations of the HON structure with partly occupied pores and used the following MC procedure. First, the initial interaction energy Ei of a randomly selected molecule was calculated. Then an empty site, where the molecule is supposed to jump, was randomly found, the molecular state after the jump randomly chosen and the final energy Ef calculated. To avoid freezing of the system in a metastable state, we employed the version in which the molecule can jump to any non-occupied site at an arbitrary distance. Then the Metropolis procedure was performed: the new state was accepted, if ∆E = Ef − Ei < 0, or accepted with the probability ∼ exp(−∆E/kB T ), if ∆E > 0. Our simulations were performed by cooling the system starting from high temperature and random initial molecular configurations. Then the temperature was lowered until fully ordered

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HON structure was obtained, followed by further cooling to very low values of temperature. The total number of MC steps per site was at least 105 .

Results DFT First, we performed the DFT geometry optimization without any constraints of the isolated, successively larger fragments of the hydrogen-bonded hexagonal network of the TMA molecules: dimer, tetramer, and hexamer (Figure 1). Although we did not employ the periodic boundary conditions, the hexamer retained nearly perfect hexagonal symmetry. The mean energy of the double hydrogen bond was the same for all three clusters: E2 = -23.3 kcal/mol. All clusters also revealed the same bonding geometry. The obtained OH· · · O bond length was 0.26 nm which is in the range of typical experimental 60,61 and calculated 25,62 values for the carboxylic acids. We found that the H· · · O distance in the hydrogen bonds was 0.159 nm, as compared to 0.162 nm reported by other DFT 34 and molecular mechanics 25 studies. The O-H· · · O=C angle was 124.6◦ in comparison to 126.4◦ . 25 The nearest-neighbor intermolecular distance was 0.955 nm for all three optimized structures presented in Figure 1. We estimated the TMA molecular lattice constant to be dTMA = 1.654 nm from the distance between the second nearest neighbors in the hexamer. This value is in a good agreement with the experimental 4–6,12,21,25,62–64 and DFT 6,64 results. A fragment of the HON phase with the guest TMA molecule was prepared by inserting an additional TMA molecule in the cage of the optimized TMA hexamer. A constraint-free geometry optimization led to buckling and twisting of the 7-molecule cluster. The freezing of all benzene ring carbons, except for the inner molecule, was found to be insufficient to prevent the guest molecule from moving out of the hexamer plane. Therefore, further DFT calculations were carried out with all atoms restricted to the plane. The stability of each final configuration was verified by the subsequent DFT optimization with all constraints 6


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E2 = -23.3 kcal/mol

E6 = -139.7 kcal/mol

E4 = -69.8 kcal/mol

Figure 1: Optimized structures of dimer, tetramer and hexamer TMA clusters and their respective enthalpies of association, as calculated by the DFT.

A

C

B 1

6

2

7 5

3 4




Figure 2: Most favorable molecular arrangements and corresponding energies of the filled TMA hexagon as calculated by the DFT. Molecules of the hexagon cage are numbered 1 to 6, and the inner molecule is labeled 7. Black circle indicates the center of the hexagon cage.

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removed, during which the molecules remained in the lowest-energy planar configuration. The lowest-energy (E7 = -157.8 kcal/mol) configuration of the filled TMA hexagonal pore is presented in Figure 2A. It is similar to the one predicted in the molecular mechanics simulations 25,26 of the filled TMA HON phase. The center of the inner molecule lies 0.15 nm away from the pore center. Note that the position of the inner molecule was also NC in the starting configuration during this calculation. The energy difference of 18.1 kcal/mol, as compared to the open TMA hexagon (Figure 1), must be attributed mainly to the H-bonds formed between the inner molecule 7 and its two closest neighbors, denoted as 4 and 5 in Figure 2A. The 4-7 and 5-7 OH· · · O bond lengths are 0.291 nm and 0.273 nm, and the H· · · O distances are 0.194 nm and 0.175 nm, respectively, while the O-H· · · O=C angles are 159.7◦ and 138.0◦ . Therefore, the 5-7 interaction should be stronger than the 4-7 one. The third carboxyl group does not establish a proper H-bond. In order to estimate all available dimeric interactions in the cluster, we used the following procedure. All possible TMATMA dimers (six of the cage molecules and six involving interactions of the cage molecules with the inner molecule) were isolated from the structure shown in Figure 2A. Then we performed a separate DFT optimization of each dimer allowing a partial relaxation. All carbon atoms were held fixed. The interaction energy was calculated for each TMA-TMA dimer as an energy difference between the partially relaxed dimer and two fully optimized monomers. The obtained data (Table 1) confirms that the inner molecule participates mostly in two interactions. Their energies are -11.5 and -5.9 kcal/mol. Other interactions between the inner and cage molecules are very small or negligible. The dimeric interactions in the hexagon walls are almost the same as in the empty-pore hexagon, with the average value of -23.0±0.3 kcal/mol, where the uncertainty is based on a standard deviation. Two less stable arrangements are shown in Figure 2B,C. They were obtained by optimizing the initial configurations, in which inner molecule was placed in the center of the TMA hexagon. The starting orientations of the inner molecule were different for 2B and 2C. For the latter, we used a previously suggested configuration 31 as input in our DFT optimization.

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Table 1: Interaction energies (kcal/mol) for dimers of the filled TMA hexagon. Structure and numbers 1-7 refer to Figure 2A.

Dimer

Energy

Dimer

Energy

1-2 2-3 3-4 4-5 5-6 6-1

-23.2 -23.3 -22.6 -23.0 -22.6 -23.2

1-7 2-7 3-7 4-7 5-7 6-7

0 0 -2.3 -5.9 -11.5 0

The optimized 2B and 2C geometries have very similar interaction energies, and they both indicate that the inner molecule could not be sustained in the center of the pore. The bonding arrangements in Figure 2B,C are quite different from the one in Figure 2A. In 2B and 2C, the guest molecule is a little closer to the hexagon center. It forms the strongest H-bonds with two cage molecules (3 and 5 in Figure 2B,C) which are the next-nearest neighbors to each other. The corresponding OH· · · O bond lengths are larger than in the 2A configuration. However, the third carboxyl group of the guest molecule in 2B and 2C is still too far from the cage molecules and apparently cannot form the H-bond. For all cases, the optimized filled TMA hexagon is slightly deformed in comparison with the starting (unfilled hexagon) configuration. However, in a real network the deformation is expected to be very small, since each hexagonal cage is surrounded by other cages of the HON network. In order to see the effect of this deformation on the final energy, we fixed the benzene rings of the hexagon cage as in the unfilled configuration. The inner molecule was initially placed in the center position. The optimized configuration was nevertheless NC and very similar to the one shown in Figure 2A (E7 = -157.2 kcal/mol). We also considered a situation of an inner molecule strongly bound to a site located in the center of the hexagonal cage. This setup mimics the binding of the inner molecule to a surface site. During DFT optimization the central molecule was allowed only to rotate, while the center of its benzene ring (held rigid) always coincided with the honeycomb cage

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center. The benzene rings of the six neighboring molecules were also held fixed, while the -COOH groups of all molecules were allowed to relax. Such a cluster after optimization has interaction energy of -151.6 kcal/mol, and the bonding geometry of the central molecule is rather similar to the one presented in the NC position of Figure 2C. The energy difference, as compared to the lowest-energy NC TMA cluster, is 5.6 kcal/mol. This value corresponds to the minimal additional energy required to sustain the guest molecule in the C position, rather than NC. It may be attributed to the change of the interaction between the inner molecule and the cage and the interaction responsible to preferential binding of the inner molecule in the pore center. It has been proposed that the C position of TMA molecule in a hexagonal TMA cage may be favored on Ag/Cu(111) and Ag(111) due to the smaller honeycomb cage size than on HOPG. 31 The dTMA values are 1.60 nm on silver substrate (Ref., 35 also derived from molecular density given in Ref. 31 ) and 1.65 nm on HOPG. 63,64 Most likely, dTMA is generally smaller on some metal surfaces, though the difference may fall within the uncertainty of an STM measurement. 22,27,62 The gas-phase value of dTMA obtained in this and other DFT calculations approximately coincides with the experimental dTMA /HOPG value. The artificial reduction of dTMA from its optimal value in the gas-phase DFT calculations is known to cause an energy penalty. 34,35,64 If the TMA hexagon due to interaction with the metal substrate is reduced in size, the bonding distances between the inner TMA molecule and its closest TMA neighbors can vary, thereby increasing the strength of the respective interactions. This may bias the energy balance towards the C position of the TMA molecule. In order to test this hypothesis, we performed a DFT optimization of a compressed TMA hexagon. The initial configuration was prepared by reducing all the TMA-TMA nearest-neighbor distances of the relaxed TMA hexamer. As a result, E2 has decreased from -23.3 (uncompressed) to 15.9 kcal/mol (compressed by p = 4%) (see Table 2). In the DFT optimization, the hexagonal cage molecules were immobilized by freezing their benzene rings, while their other parts were

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Table 2: DFT-calculated energies (kcal/mol) for the TMA-TMA dimer and empty/filled TMA hexagon for different TMA lattice compressions, p. Compression (dTMA , nm) 0% (1.654)

2% (1.621)

3% (1.600)

4% (1.588)

Dimer, E2 (+ BSSE) -23.3 (+4.0) E2 (0%) − E2 (p%) Empty hexagon, E6 -139.7 Filled hexagon: NC position, E7nc -157.2 C position, E7c -151.6 E7nc − E7c -5.6 E7nc − E6 -17.5 E7c − E6 -11.9

-21.7 (+4.4) -1.6 -130.4

-18.7 (+4.6) -4.6 -112.1

-15.9 (+4.8) -7.4 -95.2

-147.6 -144.9 -2.7 -17.2 -14.5

-128.9 -128.3 -0.6 -16.8 -16.2

-111.7 -112.3 +0.6 -16.5 -17.1

relaxed in the plane without any additional constraints. The optimized NC and C versions of the hexagon compressed by 3% are shown in Figure 3 and compared with the uncompressed situation. Their energies differ by -0.6 kcal/mol, therefore the NC arrangement at p = 3% still has a very small advantage which vanishes at approximately 3.5%. At 4% compression, the C position becomes more favorable (see Table 2). The comparison of the compressed and uncompressed lattices in Figure 3 for both NC and C positions of the inner molecule demonstrates how the bonding environment of the inner molecule is affected by the compression. In the NC case, the geometry of the 4-7 and 5-7 interactions almost does not change due to compression (compare, for example, shaded and main arrangements in Figure 3A). As for the C position (Figure 3B), the interactions of the central molecule with six cage molecules are long-ranged and therefore very weak in an uncompressed case. Thus, very small decrease of the bonding distance due to compression does not significantly change the situation. There are two effects in our simulations which could be indirectly attributed to the molecule-substrate interactions: the compression of the hexagonal cage and the tethering of the inner molecule at the C position. The decrease of E2 and E7nc − E7c allows us to roughly estimate this interaction. Studying the energy penalty occurring due to the compression and

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B

A

1

1

2

6

7

7

3

5

2

6

3

5

4

4

Figure 3: Superposition of the optimized NC (A) and C (B) guest arrangements in the uncompressed (shaded, lower level) and compressed at 3% (upper level) TMA hexagon with inner molecule. In A the inner molecule and in B the center of the hexagon are superposed. In A pink and black circles mark uncompressed and compressed hexagon center, respectively. The simulations were performed with fixed benzene rings of all six hexamer molecules 1-6, but inner molecule 7 was unfixed. tethering, we can see that the C position is energetically favorable at approximate compression of 3.5% (reduction of E2 by ∼ 6 kcal/mol). At the same time, for the uncompressed case the energy loss due to tethering at the center is the largest (-5.6 kcal/mol). This can give us a rough estimate of the variation range for the molecule-substrate interaction (0 - 6 kcal/mol) which we introduce in the MC simulations of the next section.

MC Simulations: Model and Results The model of the TMA ordering is constructed on a triangular lattice with a lattice constant a. The TMA molecule is allowed to move over the sites of this lattice with the central benzene ring assumed to be tethered to the site. Two TMA molecules form the main dimeric bond when they are 6a apart and their carboxyl groups are stretched towards each other facilitating the H-bond formation (the length is calculated between the centers of both molecules). In such an intermolecular position the main dimeric interaction e0 (= E2 ) is established. We

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also assume that the molecules prefer to lie 6a apart, because there are centers of attraction (CA) for molecules on some chosen sites of the substrate lattice, and these CAs are 6a apart. Due to the lack of appropriate data, here we do not specify what these CAs could correspond to on a real lattice (HOPG and noble metal surfaces). We just state that they are the sites, where the molecule feels the largest molecule-substrate interaction, es , compared with other sites of the underlying lattice, i.e. relatively largest diffusion barrier. They could be on-top, hollow, bridge sites, or locations which are occupied due to the balance between different forces. Thus, when the perfect honeycomb structure is assembled, 2/3 of these CA sites are occupied by the TMA molecules and 1/3 correspond to the unoccupied centers of the HON cages (shown as black spots in Figure 4). Apart from the main intermolecular interaction e0 and the molecule-substrate interaction es , the following intermolecular interactions of the TMA molecules are also taken into √ nc account: two TMA-TMA interactions, enc 1 and e2 , when the molecules are a 21 apart, and two TMA-TMA interactions, ec1 and ec2 , when the molecules are 6a apart (as for e0 ), but in contrast to e0 their mutual orientations are unfavorable to organize the dimeric bond (see Figure 4). The former pair of interactions were taken in accordance with our DFT calculations (see the importance of the 4-7 and 5-7 interactions in Table 1 for NC position of molecule 7 in a hexagonal cage). The latter pair of interactions might be important when the guest molecule occupies the C position in a cage. Note, that the C position is additionally strengthened by the molecule-substrate interaction. In our calculations, to avoid the overlap √ of molecules, we exclude the intermolecular distances which are below a 21. The total energy of the TMA molecules is described by the lattice-gas Hamiltonian which √ comprises a contribution of the TMA-TMA interactions at 6a and a 21 and the substrateTMA interactions at the CA sites X X 1 X 1 H = − e6a ni nj − ea√21 ni nj − e s ni , 2 2 6a i,j i,j

(1)

where ni stands for occupancy of the lattice site i and is equal to 1 or 0 if the site is filled by 13



Figure 4: Interactions and exclusions of the proposed model in two hexagonal cages with NC and C position of the guest molecule. Gray dots in the background denote the sites of the underlying lattice, while the black sites correspond to the CA sites. The circle denote the exclusion radius valid between centers of two molecules. the center of the molecule or empty, respectively. It should be emphasized that this variable is valid only if the molecules are in a corresponding mutual orientations towards each other. All these orientations are shown in Figure 4. Here e6a corresponds to either e0 , ec1 and ec2 , nc while ea√21 - to enc 1 or e2 . Interactions at all other distances are either zero or excluded. The

denotation 6a in the sum of the last term demonstrates that molecule-substrate interaction is valid only at sites which are 6a apart, but for other sites ni is zero. In our MC procedure a site might be unoccupied or TMA molecule can have any of two states (orientations) on each site. These states differ in 60 degrees rotation of the molecule. They are presented in Figure 4, where the interactions of two molecules in different states, c nc e0 , ec1 and enc 1 , and two molecules of the same state, e2 and e2 , are demonstrated. The choice

of appropriate orientation and corresponding interaction is not specified in the Hamiltonian and is selected during the MC procedure. The values of these interactions are chosen with respect to the main interaction constant nc nc nc e0 . The NC interactions are taken as ǫnc 1 = e1 /e0 = 0.5 and ǫ2 = e2 /e0 = 0.25 roughly

in accordance with our DFT calculations. The molecule-substrate interaction is fixed to be ǫs = es /e0 = 0.25. This interaction is evaluated from our DFT calculations assuming that 14


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the total energy difference between the optimal configuration and the configuration with the central molecule tethered to the center (5.6 kcal/mol) is attributed entirely to es . The values of the C interactions, ec1 and ec2 , are the variable parameters of our model. They are very small without the HON compression (see the energy difference E7c −E6 = −3ec1 −3ec2 in Table 2 at p = 0), due to a large intermolecular separation, as well as due to unfavorable mutual orientations, and therefore we assume ec1 = ec2 . With increasing p, their values notably increase (although they remain much smaller than e0 ), making the C arrangement energetically favorable. We vary ǫc1 = ec1 /e0 between 0.05 and 0.15. In the ground state (T = 0), the interaction energy of a molecule entrapped in a cage can be easily calculated. It is equal to Ec = −3(ǫc1 + ǫc2 ) − ǫs = −6ǫc1 − ǫs for the C position nc of the guest molecule and Enc = −ǫnc 1 − ǫ2 - for the NC one. Thus, the equation Ec = Enc c c∗ gives us the critical value of ǫc∗ 1 = 0.083. For ǫ1 < ǫ1 , the NC position of the guest molecule

is favored, while the C configuration is preferred for ǫc1 > ǫc∗ 1 at T = 0. The entrapment of molecules into the honeycomb cages is observed below the disorderedto-HON phase transition temperature Tc when the cages start to develop. The phase transition is determined as the peak in temperature dependence of the specific heat. At this temperature the hexagons are still not perfect, and active substitution between inner and cage molecules is observed even at low density of the TMA molecules. With decrease of temperature the system develops into a perfect structure of HON cages, and the inner molecules remain in the cages when the density of molecules exceeds the stoichiometric density of the honeycomb phase. In our modeling, we considered several values of interaction parameters ǫc1 and calculated temperature dependences of the order parameters for C and NC positions of guest molecules, ηc = Nc /N0 and ηnc = 1 − ηc = Nnc /N0 , respectively. Here Nc (Nnc ) is the number of entrapped molecules in the C (NC) position and N0 = N − Nst is the number of molecules located inside the cages, while N and Nst stand for the total number of molecules and the number of molecules in the stoichiometric (perfect) HON phase.

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Figure 5: (A) Temperature dependence of the order parameter ηc = 1 − ηnc for ǫc1 = 0.05, 0.07, 0.083, 0.1, 0.12, and 0.15. Snapshots of the system taken at various temperatures for nc (B) ǫc1 = 0.1 and (C) ǫc1 = 0.07. Other parameters are ǫnc 1 = 0.5, ǫ2 = 0.25, and ǫs = 0.25. In Figure 5A we demonstrate a temperature dependence of the order parameter ηc for several values of interaction ǫc1 . The data is obtained for a temperature range when the hexagonal cages are firmly built. At low temperature the results confirm our ground state predictions (C position for ǫc1 > 0.083 and NC for ǫc1 < 0.083), but for the intermediate temperatures the NC position is preferred in both cases, except for very large values of ǫc1 > 0.15. Clearly, such a result is caused by the higher entropy of the NC position: there are only two C and twelve NC positions in the cage. This can be easily seen from the value (≈ 0.14) of the order parameter ηc = (1 − ηnc ) at intermediate temperatures and ǫc∗ 1 ≈ 0.083. The latter observation is supported by the snapshots of the system in Figure 5B,C, where temperature dependences of the guest positions are presented for both low-temperature C and NC arrangements. In a former case (ǫc1 = 0.1), at relatively higher temperature (kB T /e0 = 0.25) the guest molecules are mostly observed in the NC position. Even at sufficiently low temperature (0.05), half of them stay in this position, and only at very low temperature the C arrangement completely prevails. Certainly, with increase of ǫc1 or ǫs the tendency for the C position would be strong even at intermediate temperatures (see, for example, the curve for ǫc1 = 0.15 in Figure 5A). For low-temperature NC arrangement

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(ǫc1 = 0.07, Figure 5C) only a few molecules are seen in the C position at relatively higher temperature and almost none already at 0.05 (not shown here).

Conclusions We have demonstrated by the DFT calculations that the NC position of the guest TMA molecule in a hexagonal cage, assembled of six TMA molecules, is more favorable than the C one. The NC arrangement was found to be mainly caused by the intermolecular interactions between the guest molecule and two cage molecules. Compression of the cage, i.e. reduction of six intermolecular distances between the cage molecules, leads to decrease of the energy difference between the NC and C positions of the guest and the vanishing of this difference at approximately 3.5% of compression. Such a compression level corresponds to a lattice constant of the HON structure dTMA = 1.60 nm found on some of densely-packed metallic surfaces. 31,35 Based on our DFT calculations, we proposed a model for MC simulation of a molecular ensemble of TMA molecules on the triangular lattice. In our model, the formed HON structure suggests a multitude of positioning possibilities for a guest TMA molecule in a hexagonal cage. In the model we relate the compression used in the DFT calculations to the enhancement of interactions maintaining the central position of the guest molecule. Apart from these, we introduce three other types of interactions: (i) maintaining the cage, (ii) causing the NC position of the guest molecule, and (iii) molecule-substrate interaction at the special sites of the lattice. The results reveal a simple relation between all types of interactions at very low temperature. It is shown that the NC position is preferred if the molecule-surface interaction at the center of the cage and the interactions maintaining the C position are not too large. This position is found to be favorable also at intermediate temperatures due to an entropic factor: there are twelve NC and only two C positions for a TMA molecule in a hexagonal cage.

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Acknowledgement A.I. and E.E.T. acknowledge the financial support of the Research Council of Lithuania (long-term program “Nanostructured materials and electronics”).

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Trimesic Acid Molecule in a Hexagonal Pore: Central versus

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