Coronene Molecules in Hexagonal Pores of Tricarboxylic Acids: A

Article pubs.acs.org/JPCC

Coronene Molecules in Hexagonal Pores of Tricarboxylic Acids: A Monte Carlo Study Mantas Šimeṅ as,† Andrius Ibenskas,‡ and Evaldas E. Tornau*,‡ †

Faculty of Physics, Vilnius University, Sauletekio 9, LT-10222 Vilnius, Lithuania Semiconductor Physics Institute, Center for Physical Sciences and Technology, A. Goštauto 11, LT-01108 Vilnius, Lithuania

‡

ABSTRACT: We propose a seven-state (orientation) model with some exclusions to describe the ordering of symmetry-reduced biphenyl-3,4′,5tricarboxylic acid (BHTC) molecules into planar ordered structures on a substrate lattice. The model is based on the three-state Bell−Lavis model, which is used to describe the symmetric triangular molecules. The reduction of molecular symmetry allows us to predict different honeycomb-like structures H-bonded by dimeric and trimeric interactions. The model is supplemented by the guest−host interaction term which enables us to analyze the effect of coronene on the ordering of molecular structures. We find two structures formed by dimeric and trimeric interactions which can contain coronene in their pores and mix in different proportions depending on the coronene concentration. The effect of the host−coronene interaction on the ordering of trimesic (TMA) and benzene−tribenzoic (BTB) acids is also studied using the three-state Bell− Lavis model. We demonstrate that the interaction with coronene might induce the coexistence of a coronene-filled honeycomb phase and a coronene-free superflower phase in the TMA molecular system. We also predict that this interaction might induce the occurrence of a superflower phase in the system of BTB molecules.

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INTRODUCTION Supramolecular chemistry is extensively used for the targeted design of planar self-assembled supramolecular formations on solid surfaces. In recent years many diverse surface structures have been prepared, and quite a large part of them comprise nicely ordered H-bonded molecular networks. The directionality, chemical selectivity, and flexibility of H-bonds allow us to organize open molecular networks with well-defined pores used for the confinement of guest molecules with different functional properties.1−4 Many important prerequisites have to be taken into consideration to organize a planar molecular structure. The electronic5 and symmetry (surface commensurability with molecular network) properties of substrate templates are important. The choice of correct properties and proportions of solute and solvent (ref 6 and refs therein) plays this role when the self-assemblies occur at the solid−liquid interface. Thermodynamic conditions and the concentration of solute (or molecular packing density in vacuum experiments7) are important since by tuning the molecular density the same molecular building block might be used to build different ordered planar molecular structures. Nevertheless, the leading role in building supramolecular entities is played by a detailed knowledge of intermolecular and substrate−molecule interactions.1−4 Under natural conditions a dimeric intermolecular H-bond interaction, featuring a high degree of directionality (with preferred bond angles being 170− 180°) and strong covalency,8 is established. The symmetric cyclic trimeric arrangement is also quite often observed in the ordering of triangular molecules. © 2015 American Chemical Society

Along with experimental (especially STM) measurements, theoretical modeling is necessary to better understand the molecular ordering processes and to explain and predict new surface structures. For such a calculation the experimental findings by the STM are used as a starting material to construct the main model with an appropriate set of chosen interactions. As a result, the possible patterns might be found without performing actual experiments. Moreover, the modeling might be really helpful for the interpretation of superstructure geometry of STM snapshots obtained with submolecular resolution. Theoretical studies of planar H-bonded molecular orderings are performed using ab initio,9−11 molecular mechanics,12,13 and molecular dynamics14,15 calculations or a combination of these studies with Monte Carlo (MC) calculations.14,16−18 Molecular ordering is also modeled using statistical phasetransition models, most frequently lattice models.16−23 In such calculations a molecule is represented by a simple geometrical form with a minimal number of molecular orientations. In addition, only the most characteristic molecular interactions are chosen which have to reveal the main orderings of the model. The models are sometimes supplemented by some exclusion rules.23,24 Here we present a seven-state model and MC calculation for the ordering of triangular molecules of reduced symmetry. This model can be easily reduced to the Bell−Lavis model25−27 Received: July 12, 2015 Revised: August 5, 2015 Published: August 14, 2015 20524

DOI: 10.1021/acs.jpcc.5b06690 J. Phys. Chem. C 2015, 119, 20524−20534

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The Journal of Physical Chemistry C which is used to describe21 the self-assembly of symmetric triangular molecules. This paper is mostly devoted to revealing the role played by guest (coronene) molecules in the ordering of tricarboxylic acid molecules. We consider both symmetryreduced molecules, such as biphenyl-3,4′,5-tricarboxylic acid (BHTC), and symmetric molecules, such as trimesic acid (TMA) and 1,3,5-benzene-tribenzoic acid (BTB) (Figure 1).

BHTC molecule might be nicely incorporated on a substrate with a central phenyl ring located on the site of a substrate lattice in such a way that interactions by carboxyl groups are possible on both short and long bonds of the molecule. The decrease in molecular symmetry from an equilateral to isosceles triangle leads to a very rich phase diagram with a large number of ordered structures. All of these phases are porous honeycomb-like formations; some of them demonstrate the nets of slightly irregular hexagonal pores. The occurrence of structures with trimeric bonds is one of the most interesting problems encountered in studies of the triangular molecules interacting by their vertices. The trimeric bonds are found either at higher molecular density7,29−31 or when foreign molecules are incorporated into pores of a host molecules matrix.28 In the former case they support the formation of flower phases of TMA molecules. In the latter case the guest (coronene molecule) stimulates the occurrence of hexagonal porous formations in the BHTC system which must possess trimeric bonds in order to build a symmetric structure with similar pore sizes. Actually, it is not clear if the increase in density due to guest molecule insertion or the host−guest interaction itself is the main reason for a trimeric arrangement. The ratio of a trimeric to dimeric interaction (per molecule) in DFT calculations for tricarboxylic acids is more or less the same (0.8−0.928,31). As shown by our studies here, such a value of trimeric interaction is sufficient to independently create the phases with trimeric bonds at high molecular densities for the TMA molecular system, while for molecules with lower conformational flexibility (BHTC and BTB) some support of the host−guest interaction is needed. It should be noted that our modeling has some similarity to a well-known “tiling problem” realized experimentally by Blunt et al.32 for a two-dimensional arrangement of p-terphenyl3,5,3″,5″-tetracarboxylic acid (TPTC) on graphite. The authors demonstrated that the unique honeycomb ordering might be achieved by five different assemblies of the TPTC molecule mapped onto a rhombic tile. The STM interpretation becomes cumbersome, kind of a “bird’s eye view”, and the final structure is likely a mixture of possible assemblies or a spin-glass (from the point of view of the phase-transition problem). Here, in our simulations for BHTC molecules, we also find several honeycomb structures demonstrating the same density and symmetry of pores which might be assembled in different ways, with the details of the assembly probably being hardly distinguishable by the STM. This problem is especially clearly manifested while studying experimentally observed28 lowdensity porous coronene-stabilized phase. We found that the pores of the same density might be formed by two similar yet slightly different molecular orderings. Our results imply that an interpretation of the experimental structure as a mixture of the two phases should be more feasible. The paper is organized in the following way. The model for ordering symmetry-reduced molecules, the main interaction parameters, and the methods of calculation are presented in section 2. The ground-state calculations for the BHTC system without coronene are given in section 3.1. The effect of coronene on lower-symmetry (BHTC) and higher-symmetry (TMA and BTB) molecular systems is presented in sections 3.2 and 3.3, respectively.

Figure 1. Schematic representation of (a) TMA, (b) BTB, (c) BHTC, and (d) coronene molecules. Large (small) gray circles denote carbon (hydrogen) atoms, and red circles denote oxygen atoms.

The results are presented here not in a form of rigorous phase diagrams but in a simple form of structure snapshots more appropriate for the prediction of results and the analysis of experimental data. Both TMA and BTB molecules are 3-fold-symmetric hydrogen-bonding units. The TMA molecule has one (central) phenyl ring and three carboxylic groups along each of the three bonds of the molecule. The BTB molecule has four phenyl rings central and three peripheral ones, with the carboxylic group originating from each perypheral phenyl ring. BHTC has two carboxylic groups originating from the central phenyl ring as in a TMA molecule. The third carboxylic group is on a bond elongated by a peripheral phenyl ring as in a BTB molecule. The symmetry of BHTC is lower than its better known counterparts; therefore, the ordering possibilities are more diverse. The BHTC molecules demonstrate very interesting self-assemblies at the solid−liquid interface.28 The dimeric Hbond interactions, analogous to those observed in honeycomb orderings of TMA and BTB molecules, play a crucial role in the formation of ordered porous BHTC structures. Along with the dimeric bonds, a trimeric H-bonding motif similar to that found for the TMA flower structure7,29−31 is experimentally observed in the BHTC system28 when the coronene molecules are added. The modeling of the self-assembly of TMA21,22 and melamine16,24 molecules (both possessing C3 symmetry) was performed by mapping the shape of these molecules onto an equilateral triangle with interacting vertices (TMA) or sides (melamine). The BTB molecule might be mapped onto a larger equilateral triangle with interacting vertices. Following the same logic, the BHTC molecule, which has reduced C2 symmetry, might be mapped onto an isosceles triangle with interacting vertices. The fact that the radius of the BTB molecule is twice that of the TMA molecule can be used as an advantage: the 20525


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The Journal of Physical Chemistry C

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= 1, and if not, then τiji = 0. Thus, the Hamiltonian (per molecule) for BHTC molecules has the form

MODEL AND METHODS OF SIMULATION To describe the ordering of the BHTC molecules, we use the lattice model on a triangular lattice as an underlayer. The lattice is chosen for convenience and simplicity of calculations. The molecule is assumed to be “anchored” to the site of a substrate lattice in such a way that the central phenyl ring, which is bonded to a peripheral ring and two carboxylic groups, has the on-site location. While moving from site to site, the molecule might have any of six orientations on each site with the longer phenyl arm being kind of a “vector” pointing along each of the lattice directions. The length of such a vector is ∼2a, where a is the triangular lattice constant. The orientations of the molecule are shown in Figure 2a. Our model comprises

/BL = −

1 2

∑ ninjeijτiijτjji − i,j

1 3

∑ ninjnkeijkτiijkτjjikτkkji i ,j,k

(1)

where eij and eijk are dimeric and trimeric interactions shown in Figure 2b and coefficients 1/2 and 1/3 are taken to avoid the double and triple counting in the first and second sums, respectively. The dimeric H-bond interactions eij might be of a different type: the long−long interaction, ell, representing the interaction by longer arms of the molecules when the centers of interacting molecules are 4a apart; the short−long interaction esl when the molecules are 3a apart; and the two short−short interactions, ess1 and ess2, when the molecules are 2a apart and their vectors make mutual angles of 180 and 60°, respectively. There are no other distances and angles appropriate to the dimeric H-bond to be established. There is also a trimeric interaction in the model, eijk = et, which is possible if three molecules exist on vertices of an equilateral triangle with a side length 2a(31/2) and their longer arms are oriented toward a mutual center, creating a trimeric Hjik kji ijk bond (Figure 2b), i.e., all variables τijk i = τj = τk = 1 where τi ij ik = τi τi , etc. Note that there might be other trimeric interactions in the model. However, they are not considered here since the structures with such molecular configurations are not experimentally observed and the addition of these interactions would significantly complicate our model. To avoid the overlap of molecules and the occurrence of nonrealistic structures, we also use some exclusion rules which are shown for a molecule in state 1 in the bottom-right drawing of Figure 2b. Here the black dots correspond to sites which cannot be occupied by centers of other molecules, while the crosses show the sites which are forbidden for centers of molecules in states 2 (upper cross) and 6 (lower cross) neighboring state 1 (to avoid the structure with overlapping carboxylic groups on a longer arm of three neighboring molecules). For molecules in other states we use the same kind of exclusions. It should be noted that, most likely, some other exclusions could be taken in the model, but they are important only at high densities of molecules which can hardly be achieved in real experiments. When guest (coronene) molecules are inserted into the BHTC system, we should account for the interaction, ecor, stimulating the occurrence of the ordered BHTC−coronene systems. We assume that this interaction acts between the coronene molecule and BHTC molecule (in any of its six orientations) when the distance between their centers is 2a (Figure 2c). The resulting energy with this interaction taken into account is

Figure 2. States (orientations), interactions, and exclusions of the model: (a) six molecular orientations and (b) main pair and triangular H-bond interactions [LL stands for long−long; SL stands for short− long; SS1 (SS2) stands for short−short interactions of the first (second) type; T stands for trio interactions] of the BHTC molecules. The lower-right drawing depicts the exclusions for the molecule in state 1. The solid black circles show the exclusion for the center of the molecule in any state; the upper and lower crosses show the exclusions for molecules in neighboring states 2 and 6, respectively (the corresponding exclusion rule is used for other molecular states). (c) Two examples of the BHTC−coronene interaction. Both interactions are considered to be equal in magnitude.

six molecular states and a vacancy state which occurs when the site is not occupied by the molecule. Thus, in general, the model is a seven-state model with interacting vertices (carboxylic groups) and some exclusion rules dictated by the size and form of the molecule. Since the length of a longer arm (distance from the center of the central phenyl ring to the carboxylic group) is roughly twice that of a shorter arm, the main H-bond interactions existing in such a system might be conveniently taken into account. Here we employ the Bell−Lavis model25−27 previously used to describe the ordering in a two-dimensional bonded fluid. The lattice-gas variable ni = 1(0) describes the existence (nonexistence) of the center of the BHTC molecule on a site i, while the variable τiji characterizes the orientational ordering: if the H-bond exists between the molecules at sites i and j, then τiji

/BLc = /BL −

∑ nit jecor i,j

(2)

where tj is the lattice-gas variable which represents the existence (tj = 1) or absence (tj = 0) of the center of the coronene molecule on lattice site j. It should be noted that TMA and BTB (Figure 1) molecules, which can be mapped on equilateral triangles, might also be described by the Bell−Lavis model in forms (1) and (2). However, for these high-symmetry molecules the appropriate model is simpler: a molecule in a site of a triangular lattice might have any of the three states: one of two orientations, 20526


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Figure 3. Arrangement of molecules in phases F2 and F4 (c = 3/32), F1 and F3 (c = 2/21), and HG (c = 1/10) and their representation in MC simulations. The contours of occurring hexagonal pores are highlighted in red.

if ΔE < 0 or accepted with the probability ∼exp(−ΔE/kBT) if ΔE > 0. For MC calculations with Kawasaki dynamics we used triangular lattices of sizes L × L commensurate with the expected molecular structures. Mostly, we used L = 72 with periodic boundary conditions. We probed our model for higher values of L, but the obtained results did not differ. Our simulations were performed starting from a higher temperature and using a random initial particle configuration. Then the temperature was lowered below the order−disorder phasetransition point which depends on molecular concentration. Then the temperature was further slightly decreased to reduce thermal fluctuations and obtain a fully ordered structure. The total number of MC steps at the lowest temperature was at least 107. Our calculations of the H-bond dimeric interactions using density functional theory (DFT) and accounting for the van der Waals (vdW) interactions and BSSE correction showed ess1 ≳ esl ≳ ell ≈ ess2, and the magnitudes of the dimeric interactions differ by about 1−1.5% (while the absolute values are in between 9.62 and 9.49 kcal/mol per molecule). Our calculation demonstrates some differences from the former calculations28 in which the vdW contribution and BSSE correction were neglected. However, both calculations were performed for the gas phase and did not account for some very important effects (direct effects of the substrate lattice and indirect effects due to free atoms of the substrate34) which might change the value of interactions. We assume that the difference (∼5%) in values obtained by two DFT calculations for a gas phase might be smaller than that given by accounting for the substrate lattice.9 Since in our MC calculations the molecule−substrate interaction is neglected, we left a larger frame for the variation of dimeric interaction constants. This allows us, by slight tuning of the intermolecular interactions, to predict the possibility of some orderings not seen in experiments. We fixed the values ell = esl as a constant ed to which other constants and temperature are rescaled to a dimensionless form. Note that the ratio of trimeric interaction to the largest dimeric interaction in our DFT calculations was 0.86 (per molecule), in good accordance with previous results.28 We approximately kept this ratio in our MC calculations.

which differ by a 60° rotation of the molecule, or a vacancy state. The dimeric H-bond interaction ed in the three-state Bell−Lavis model can exist (τij = 1 in (1)) at a distance 2a if the ordering of TMA molecules is considered. For BTB molecules this distance is 4a. The trimeric interaction exists if three molecules are arranged on vertices of an equilateral triangle with a side length a(31/2) (for TMA molecules) or 2a(31/2) (for BTB molecules). We performed the MC simulation of the proposed sevenand three-state models using the Metropolis algorithm and Kawasaki and Glauber dynamics.33 When we used the latter dynamics, we considered the energy in the form of / = /BL + μ ∑ ni i

(3)

where μ is the chemical potential of the studied molecules. The Glauber calculation starts when the lattice is randomly filled with molecules in different states corresponding to some fixed value of μ. The interaction energy Ei on a randomly chosen site is calculated. Then the initial state on that site is changed (with equal probability) to one of the six remaining states, and the final energy Ef is calculated. Then the Metropolis procedure is performed: the new state is accepted if ΔE = Ef − Ei < 0,or accepted with the probability ∼exp(−ΔE/kBT) if ΔE > 0. Note that if the forbidden state is chosen then the Metropolis procedure is interrupted and a new molecule is randomly selected. In a process of Glauber dynamics, the molecular concentration is changed while the chemical potential is fixed. For calculations with coronene we used the Kawasaki dynamics and Hamiltonian (eq 2). Employing this method, the diffusion of molecules is simulated for a fixed value of molecular concentration. To decrease the computation time and avoid freezing of the system in metastable states, we employed the version in which the molecule can “jump” to any nonoccupied site at an arbitrary distance. Our simulations start from the disordered random mixture of tricarboxylic acids and coronene. The algorithm is realized by randomly choosing the molecule at initial site (with energy Ei) and the empty site, where the molecule is supposed to jump. One of the six states of the molecule is randomly chosen for the empty site, and the corresponding final energy E f is calculated. Then the Metropolis procedure is performed: the new state is accepted 20527


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RESULTS Ground-State Calculations, ecor = 0. We used MC calculations to determine possible ordered structures occurring in our model for different values of interaction parameters and calculated the ground-state (GS) energies of these structures. For further studies we have chosen the phases which emerged for different BHTC molecular concentrations when the difference in magnitude of dimeric interaction parameters was within ∼20%. The first group of phases consists of four flower-type hexagonal structures shown in Figure 3 which might be observed at lower molecular densities. To describe the density, here we use the concentration, c. The concentration of any particular phase is obtained by calculating the number of lattice sites occupied by molecules and dividing this number by L2. In our calculations we assume that a molecule occupies only one lattice point. For example, a BHTC molecule occupies a site which is under the central phenyl ring. The GS energies of the flower phases are E F1 = −c F1(esl + ess1/2 − μ)

The second group of phases consists of two higher-density phases (Figure 4) with a trio of H-bond interactions. One of

(4)

E F2 = −c F2(esl + ess1/2 − μ)

E F3 = −c F3(esl + ess2 /2 − μ) E F4 = −c F4(ess2 + ell /2 − μ) c F1 = c F3 = 2/21 ≈ 0.095 c F2 = c F4 = 3/32 ≈ 0.094

It is interesting that structures F1 and F2 have the same energy and just slightly different concentrations. The F1 phase features one type of hexagon with two side lengths equal to 2a and four side lengths equal to 3a. The F2 phase demonstrates two types of hexagons: one regular (side length equal to 3a) and the other irregular (three sides have length 2a and three have length 3a). In GS phase diagrams only the F1 phase is emerging due to denser packing. However, in finite-temperature MC calculations both F1 and F2 phases use to occur, most likely due to the higher entropy of the F2 structure. The low-density flower structure F3 has the same geometry and concentration as the F1 phase. The pores of both of these phases consist of four longer sides and two shorter sides of a schematic BHTC triangle, just the shorter sides of the F1 phase are on opposite sides of a hexagonal pore, while in the F3 phase they are close to each other. The most exotic is the F4 phase which consists of a central regular hexagonal pore, created by shorter sides of six BHTC molecules. This pore is surrounded by six irregular hexagons containing three long (4a) and three short (2a) sides (Figure 3). There is one additional structure which has a low molecular density and therefore might be attributed to the flower phases. We call it the hourglass (HG) phase (cHG = 0.1, see Figure 3). The fragments of the HG are experimentally found28 at higher molecular densities in between large domains of a high-density zigzag structure with cZ = 0.143. The relation between the HG and zigzag phases is a separate topic which is planned to be published elsewhere. The range of concentration makes the HG phase a likely competitor of the flower phases in the phase diagram. The GS energy of the HG structure is E HG = −c HG(ell /2 + ess1 − μ)

Figure 4. Arrangement of molecules in phases C1 and C2 (c = 1/8) and their representation in MC simulations. Two hexagons with the six- and four-molecule pores in phase C1 and five- and four-molecule pores in phase C2 are highlighted in red.

these structures, C1 (cC1 = 0.125)), was suggested28 when interpreting the STM images of the BHTC−coronene ordering; in this structure the coronene molecules fill the hexagonal pores formed by the BHTC molecules. The C1 phase is a sequence of two hexagons (both have a side length equal to 2a). The sides of the first hexagon are assembled from six shorter sides of a schematic BHTC triangle. The sides of a second hexagon are made of four longer sides of such a triangle. Furthermore, we call these hexagons “six-molecule” and “fourmolecule” pores. The ratio of the number of six- to fourmolecule pores in the C1 structure is 1:3. Performing MC simulations, we noticed another very similar structure possessing the same molecular density, as well as form and size of hexagonal pores, as C1. We called this phase C2 (cC2 = 0.125). It also demonstrates a sequence of two hexagons (both have a side length equal to 2a): one is formed from four longer sides of a schematic BHTC triangle as a four-molecule pore of C1, and the other hexagon is assembled from five sides of the triangle (two longer sides and then three shorter sides in a row) comprising a five-molecule pore. There are the same numbers of five- and four-molecule pores in the C2 structure. The GS energies of these two phases are

(5) 20528


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The Journal of Physical Chemistry C EC1 = −cC1(ess2 + et /3 − μ)

dimer interactions, ess2/ess1 = 1.0128 and 0.97 (our calculations). Given the similarity of dimeric interactions in DFT calculations, the effect of substrate and free surface atoms might easily shift this ratio in favor of one of the phases. Therefore, the conclusions of the GS analysis certainly do not deny the situation when both phases exist as domains in different regions of the lattice or when they mix. The situation with mixing of these two phases is very similar to that found for a rhombic tiling32 when one type of honeycomb arrangement could be achieved by 5 different assemblies of TPTC molecules on graphite, and the resulting structure is a kind of molecular “spin-glass“. In our case the structure (which might seem unique in STM measurement) might be obtained by two different assemblies, and be either pure phase or their “spin-glass“ mixture. Effect of Coronene on the Occurrence of C1 and C2 Phases. Our GS estimate and following MC calculations demonstrate that the trio interaction has to be at least of the same magnitude (per molecule) as the main dimer interactions in order to form the C1 or C2 phase. The DFT calculations28,31 show that the magnitude of et is lower. This is the most probably why the C1 or C2 phases do not occur in the absence of coronene.28 Meanwhile, it was experimentally shown35 that coronene can be easily incorporated into the honeycomb (six-molecule pore) phase of TMA molecules. It increases the binding energy of the system by 3.9−4.6 kcal/mol (DFT result36). In the BHTC− coronene system the coronene plays the same role of a binding element which effectively strengthens the trio interaction and consequently leads to the formation of the C1 and C2 phases. Here we demonstrate how the coronene−BHTC interaction, ecor, affects the occurrence of these phases when the concentration of coronene, ccor, is varied. It is very interesting that the variation of ccor might lead to switching from one phase to another: for low values of ccor and ess1 ≈ ess2 the C1 phase is preferred due to six molecules surrounding the coronene, while at a high value of ccor which phase would appear depends on the ratio of interaction constants ess1/ess2. Here we perform a Kawasaki-type treatment and therefore neglect the effect of chemical potential. We also introduce the parameter α = ccor/cC1 which is the ratio of coronene coverage to BHTC coverage in any of the C1 or C2 phases (cC1 = cC2). When hexagonal pores of BHTC are fully occupied by the coronene in both these phases (α = 2/3), their GS energy might be written as

(6)

EC2 = −cC2(ess1/3 + 2ess2 /3 + et /3 − μ)

The energies of these two phases are equal when both short− short interactions are equal, ess1 = ess2. Despite the diversity of obtained phases and dimeric interactions, both low-density flower (F) phases (4) including the HG phase (5), i.e., the phases with c ≤ 0.1, and trimerorganizing phases C1 and C2 (further denoted as C) with c = 0.125 can be analyzed using GS phase diagrams. Let us initially neglect a small difference in the magnitude of dimeric bond interactions, ell = esl = ess1 = ess2 = e. Then the GS energies of the F and C phases are EF = −cF(3e/2 − μ) and EC = −cC(e + et/3 − μ), respectively. Here as cF we take cHG as the densest of the first group of phases. The resulting phase diagram is shown in Figure 5, where the line 5et = 3e + 3μ separates the F and C

Figure 5. Ground-state (μ, et) phase diagram. Insets are (ess1, ess2) diagrams at two points on the main diagram: μ/ed = 1.3 and et/ed = 1.0 and 1.8. The cross and dashed line mark the DFT result.28

phases. It should be noted that in this diagram only the interval of 1.2 < μ/e < 1.5 is interesting because its upper bound is the gas-phase limit, while at μ/e < 1.2 the finite temperature MC calculations demonstrate the molecular density which is too large for any of the C phases to be obtained. We have chosen two points in this phase diagram at an optimal value of the chemical potential, μ/e = 1.3, with rather small (et/e = 1) and rather high (et/e = 1.8) trio interactions to compare with the DFT result. Note that the DFT ratio of trimeric to dimeric interaction (0.86 per molecule) is equal to et/e ≈ 1.3 in our MC calculations. At et/e = 1 and 1.8 we perform a further calculation of the GS phase diagrams using formulas 4−6, assuming ell = esl = ed and varying ess1/ed and ess2/ed in small limits (0.9/1.1). The obtained diagrams are shown as insets in Figure 5. The main conclusions are the following: (1) the magnitude of et has to be rather large for the C phases to occur: the C1 phase exists at ess1< ess2 and the C2 phase exists at ess1> ess2; (2) roughly the same inequality separates the F4 and F3 phases on one side and the F1 and HG phases on the other for not too large values of et; (3) the DFT result for et implies that this value is too small to obtain the C1 or C2 phase without additional interactions. Note that structures C1 and C2 are not experimentally observed without coronene.28 Formally, phases C1 and C2 cannot be formed simultaneously; it is the ratio of interaction constants, ess2/ess1 which predetermines which phase would be formed. The gas-phase DFT calculations provide us with very similar magnitudes of

2/3 EC1 = EC1 − 3cC1ecor

(7)

2/3 EC2 = EC2 − 3cC2ecor

It is seen that at full occupancy the term related to interaction ecor brings the same energy contribution to the C1 and C2 phases, despite the difference in coronene surroundings. This difference is important when the parameter α (concentration of coronene ccor) decreases; obviously the coronene is bonded best in six-molecule pores and bonded worse in four-molecule pores. This preference leads to the occurrence of the C1 phase with coronene at the six-molecule centers at α = 1/6 and the C2 phase with coronene chains in five-molecule pores at higher values of α. Taking into account that the decrease in α leads to a gradual reduction of coronene first in four-molecule pores, then in five-molecule-pores, and finally in six-molecule pores, we can write the energy in a more general form: 20529


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Figure 6. (a) Ground-state energies of phases C1 and C2 and their difference ΔE = EαC1 − EαC2 (inset) as a funcion of α = ccor/cC1 at ess1/ed = 1.2 > 1/3 1 ess2/ed = 1.05 (1), ess1/ed = ess2/ed = 1.1 (2), and ess1/ed = 1.05 < ess2/ed = 1.17 (3). Red circles mark the energy of E1/6 C1 and EC2 mixing at α = /4. (See the text.) For convenience, lines 1 are shifted up and lines 3 are shifted down along the E axis by 0.04. (b) Relative number of six-molecule N6 (solid curves) and five-molecule N5 (dashed curves) pores as a function of α obtained in MC calculations. The denotation of curves and values of interaction parameters are the same as in (a). The data is rescaled to dimensionless form with respect to their values in ideal phases C1(N6) and C2 (N5) at α = 2/3. The data points are averages over four to six runs. Lines are guides to the eye. Here et/ed = 1.4 and ecor/ed = 0.6.

Figure 7. Snapshots of MC simulations for different values of α and the ess1/ess2 ratio: (a) α = 2/3 (ideal C2 phase), (b) α = 1/4, (c) α = 1/6 for ess1/ed = 1.2 > ess2/ed = 1.05, (d) α = 2/3 (ideal C1 phase) for ess1/ed = 1.05 < ess2/ed = 1.17 and (e) α = 1/4, (f) α = 1/6 for ess1/ed = ess2/ed = 1.1. For better visibility the coronene molecules are colored red, blue, and gray in six-, five- and any-molecule pores, respectively. Here L = 72, et/ed = 1.4, and ecor/ed = 0.6. α = EC1 − EC1

becomes more preferred at α ⩽ 1/3 (Figure 6). At α = 1/6 the C1 phase completely dominates because the number of coronene molecules and the number of six-molecule pores coincide. It is interesting that the C1 phase is more favorable (for just a slightly decreased interval of α < 1/3) also for ess1 ≳ ess2 (line 1 in Figure 6 and its inset), i.e., for such a ratio of dimeric interactions which favors the C2 phase both at high coronene concentrations (eq 7) and in the absence of coronene molecules (eq 6) but large et. It should be noted that we seldom obtain pure structures in our MC calculations. The analysis of the C1 and C2 phases demonstrates that they can mix very easily due to a common building block of both phases: zigzag chains of four-molecule pores (Figure 4). These chains are the main “sticking” element which easily combines pieces of domains of both phases in any proportion without energy losses due to domain walls. These

⎛1 ⎞ ⎜ + 4α⎟cC1ecor , 1/6 ≤ α ⩽ 2/3 ⎝3 ⎠

⩽ 1/6 = EC1 − 6αcC1ecor , α ⩽ 1/6 EC1 α = EC2 − EC2

⎛1 ⎞ ⎜ + 4α⎟cC1ecor , 1/3 ≤ α ⩽ 2/3 ⎝3 ⎠

⩽ 1/3 = EC2 − 5αcC1ecor , α ⩽ 1/3 EC2

(8)

At 1/3 ≤ α ≤ 2/3, i.e., until all coronene vanishes from the fourmolecule pores of the C2 phase, the energies of both phases are equal if the main dimer interactions in the C1 and C2 phases are equal (ess1 = ess2 and, consequently, EC1 = EC2). With further decreases in α ⩽ 1/3 the coronene leaves four-molecule pores in phase C1 and five-molecule pores in phase C2. As a result, the energy of the C2 phase increases relatively and the C1 phase 20530


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Figure 8. (a) Coronene molecule in a six-molecule pore of the TMA HON phase. Snapshots of the TMA−coronene system for different TMA coverages and different ratios of coronene coverage to TMA coverage: (b) ctma = Ntma/L2 = 1/9 and ccor/ctma = 0.5, (c) ctma = 2/9 and ccor/ctma = 0.05, and (d) ctma = 2/9 and ccor/ctma = 0.17. Here L = 60, et/ed = 1.4, ecor/ed = 0.4 (b, c) and 0.2 (d).

At α < 1/6 there is a lack of coronene to maintain the C1 phase. Therefore, some part of the coronene-filled six-molecule pores are substituted by empty, five-molecule pores. The C1 + C2 structure is maintained until very small values of α are obtained. For very small coronene coverage (α ≈ 0.02) the system cannot maintain the C1 + C2 structure anymore, but still rather large domains of this structure might be found around each coronene molecule. These domains are connected by rather disordered flower-phase and HG hexagons. The fact that so small a number of coronene molecules maintain such large domains of the C1 + C2 phases might also be related to our Kawasaki dynamics, where the jump event occurs between non-neighboring sites. This treatment might create small patches of the C1 + C2 phases over a whole lattice, thus artificially maintaining the overall C1 + C2 phase. On the other hand, it cannot be done if the energy of the C1 + C2 mixture and the corresponding F phase would not be rather similar. At α → 0 the C1 + C2 domains are replaced either by the F phases or a system of disordered hexagons. To facilitate the ordering of our system, we used rather high values of the BHTC−coronene interaction, ecor/ed = 0.6. To find how the ordering depends on this interaction, we performed an extra analysis by varying ecor. It turned out that the results do not change up to ecor/ed = 0.3 for all values of α. For lower values of ecor/ed (0.2 and 0.1) the system maintains the same type of ordering as in Figure 7 at high values of α ≈ 2 /3, but for lower values of α a strong competition of the C1 and C2 domains with the flower-type and HG phases starts to occur. Coronene in Pores of Symmetric Molecules. The importance of the host−coronene interaction in a system of BHTC molecules raises the question of the effect this interaction might have on higher-symmetry molecules TMA and BTB. For these molecules model (2) is much simpler and might be reduced to the three-state Bell−Lavis model.21,27 The honeycomb (HON) phase, which has dimeric bonds only, is experimentally observed in both systems (e.g., TMA,7,29,30 BTB,38,39 and mixed TMA−BTB40). In the TMA system the HON phase is the first member (n = 1) of a homologous series of flower phases. Further members of this series (n = 2,..., ∞) emerge in the TMA molecular system when the coverage of TMA molecules exceeds the coverage of 1 hon the HON phase, chon tma = /6. Increase ctma > ctma is characterized by a decrease in the number of dimeric bonds and an increase in the number of trimeric bonds between the molecules. The last member (n = ∞) of the series is called the superflower (SF) phase and is characterized by trimeric bonds only. Alternatively, one can say that with the increase in molecular

different pieces of domains are six-molecule stars in the C1 phase and zigzag five-molecule chains in the C2 phase. There are several formal reasons for mixed structures to be seen in our numerical experiment. The entropy of a mixed phase at finite temperature can be higher than the entropy of any of the pure phases. Besides, it might also be shown from purely energetic considerations that the mix of the C1 and C2 phases for some values of α can be energetically more favorable than any of the pure phases, C1 or C2. Consider the coronene concentration at α = 1/4 for different values of ess1 and ess2. Note that the most appropriate combination of these two phases at α = 1/4 can be obtained by mixing the C1 phase at stoichiometry αC1 = 1/6 (coronene in six-molecule pores) and the C2 phase at stoichiometry αC2 = 1/3 (zigzag chains of fully coroneneoccupied five-molecule pores) in equal proportions. For ess1 ≲ ess2 the energy of such a mixing is higher than that of the lowerenergy C1 phase; for ess1 = ess2 it is equal to E1/4 C1 , but for ess1 ≳ ess2 the mixture has lower energy than E1/4 C1 . This comparison is shown by red circles at α = 1/4 in Figure 6. Our MC calculations confirm this result: we rather obtain the mixing of coronene-occupied six-molecule pores and coronene-occupied five-molecule zigzag chains than domains of pure phases C1 or C2 at α ≈ 1/4. In Figure 7 we collected the snapshots at α = 2/3, 1/4, and 1/6 reflecting the most characteristic features of our calculations. To improve the visual perception of our snapshots, we marked the coronene molecules in six- and five-molecule pores by different colors. Simultaneously, we calculated the share of each of the phases C1 and C2 in a mixture which might be found by calculating the number of six- and five-molecule pores in each snapshot for different values of 1/6 ≤ α ≤ 2/3 (Figure 6b). The curves were obtained by averaging over four to six runs. In Figure 7 we show two snapshots of ideal C2 (Figure 7a) and C1 (Figure 7d) phases at α = 2/3. They were obtained for ess1 > ess2 and ess1 < ess2, respectively. It should be noted that the pure phases at α = 2/3 occur in about 30% of our simulations for such a choice of interaction parameters as in Figures 6 and 7. We also demonstrate that with the decrease in α the share of six-molecule pores increases both for ess1 > ess2 and ess1 ≤ ess2, and phase mixing is rather prominent at α = 1/4 (Figures 6b and 7b,e). Finally, we demonstrate the result of a competition between values of dimeric interactions (ess1 > ess2) favoring the C2 domain and the value of the coronene concentration (α = 1 /6) favoring the C1 domain in Figure 7c. In Figure 7f we show the structure which is obtained when the former effect is neutral and the latter promotes the C1 domain. The structure is dominated by the six-molecule coronene-filled pores. For ess1 < ess2 and α = 1/6 the C1 domain is almost ideal. 20531


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Figure 9. (a) Three coronene molecules in a six-molecule pore of the BTB HON phase. Snapshots of the BTB−coronene system for different BTB coverages and different ratios of coronene coverage to BTB coverage: (b) cbtb = Nbtb/L2 = 7/180 and ccor/cbtb = 1.5, (c) cbtb = 1/15 and ccor/cbtb = 0.58, and (d) cbtb = 3/40 and ccor/cbtb = 1. Here L = 60, et/ed = 1.2 (b) and 1.4 (c, d) and ecor/ed = 0.2. (e) Coronene molecule in a three-molecule pore of BTB molecules (superflower phase).

higher BTB concentrations because the low conformational flexibility of BTB is opposed to the tendency of dense packing.38 Using our simulations, we tried to determine if the interaction with coronene might give extra stability to the BTB molecular system. We explored two cases: (i) When the coverage of BTB molecules corresponds to the stoichiometric coverage of the HON phase, cbtb ≈ chon btb , but the coronene coverage is even higher. Here we tried to find if the stability of the HON phase with three coronene molecules in a pore is plausible. (ii) When the coverage of BTB molecules exceeds the stoichiometric coverage of the HON phase, cbtb > chon btb . Here we studied if the coronene can induce extra flower-type phases at higher BTB concentrations. As for BHTC molecules, we asked if ecor might support the trio interactions which are not strong enough to independently form the trimeric bonds. The snapshot of our calculations for case (i) is presented in Figure 9b. The prediction15 of three molecules in a pore is also corroborated by our results; we find only one stable structure of coronene molecules in a pore of the BTB HON phase with three coronene molecules arranged symmetrically in an equilateral triangle (Figure 9a). While studying case (ii), we have found that the hexagonal pore organized by three BTB molecules (Figure 9e) has an ideal size to host a coronene molecule. Following the analogy with the TMA orderings, we consider the BTB molecular system of such three-molecule pores to be the SF phase of BTB molecules. Further simulations confirmed our prediction. For the ideal ratio of coronene coverage to BTB coverage we obtain the SF structure with coronene in the center of three-molecule pores (Figure 9d). For a lower value of this ratio we obtain a very interesting two-phase coexistence of the coronene-free HON structure and the coronene-filled SF phase (Figure 9c). It should be noted that the coronene molecule in a sixmolecule pore of the BHTC−coronene or TMA−coronene system should be more stable than that in a three-molecule pore of the BTB−coronene SF system. Actually, the reason for the occurrence of the host−coronene interaction is the weakening of the main dimeric double H-bond since one carboxyl group starts to participate in two H-bonds (host−host dimeric and host−guest, ecor). The negative charge which can be transferred from six H-bonds of the six-molecule pore to coronene is two times less than might be given by three Hbonds in a three-molecule BTB pore. This is seen by comparing the schematic Figures 8a and 9a. We explored very small BTB− coronene interactions, and the coronene-supported BTB SF phase still emerged in our simulations.

density a decrease in the number of larger (honeycomb or sixmolecule) pores and an increase in the number of smaller (triangular or three-molecule and rectangular or four-molecule) pores take place.7,22 The six-molecule pore (Figure 8a) is the same type of pore we encountered when studying the ordering of the C1 phase of BHTC molecules, and this pore can incorporate a coronene molecule. The three- and four-molecule pores in the TMA flower phases are too small to host a coronene molecule. Therefore, in calculations for the TMA−coronene system with ecor > 0 one can expect the coronene in honeycomb (sixmolecule) pores of the HON phase for any concentration of coronene if 2ccor ≤ chon tma , and this is corroborated by the DFT calculations.36 In flower phases the coronene could also be incorporated into the honeycomb pores, but ccor should be rather small since any increase in molecular concentration, either TMA or coronene, might violate the molecular density (and a subtle balance between dimeric and trimeric bonds) at which the flower phases exist. The most interesting results for the TMA−coronene system, obtained by Kawasaki dynamics, are presented in Figure 8. For the optimal ratio of coronene/TMA density equal to 0.5 in a perfect HON phase, the six-molecule pores are fully occupied by the coronene molecules (Figure 8b). For slightly higher TMA concentration (ctma = 2/9) either the flower (n = 2) phase or the HON and n = 3 two-phase coexistence can be found without coronene. In our calculations here we obtain the twophase coexistence. By introducing a rather small amount of coronene interacting with TMA molecules (ecor/ed = 0.4) we obtain very interesting phase separation into three phases: HON, n = 3, and SF (Figure 8c). Further increases in coronene concentration lead to the coexistence of HON and SF phases (Figure 8d). It should be noted that this result agrees with the experimental fact that the SF phase might be found only as high-density fragments in coexistence with other molecules (e.g., alcohols41). The BTB molecules also can form the simplest HON phase.15,38 In contrast to the six-molecule honeycomb pore of the TMA molecules, such a pore of the BTB molecules is too large to appropriately host a coronene molecule since the diameter of the BTB molecule is roughly 2 times larger than that of the TMA (Figure 1). The exact number of coronene guest molecules adsorbed in each pore of the BTB HON phase was not determined by the STM measurements, but it was predicted,15 on the basis of molecular mechanics and molecular dynamics calculations and entropic arguments, that this pore can hold three coronene molecules. It is also known that a pure BTB molecular system does not form any flower phases for 20532


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CONCLUSIONS The coronene molecule is an important element strengthening the ordering of tricarboxylic acid molecules. Owing to the similarity of hexagonal lattices of coronene and graphite, an almost perfect π−π stacking is established between the coronene and substrate. As a result, the TMA HON structure with coronene located in honeycomb pores is almost commensurate with the substrate lattice, while the same structure without coronene demonstrates some incommensuration.37 Thus, it is more likely that the coronene rather than the BHTC molecule is the main condensation center when coronene is added to the BHTC system. In particular, it is known36 that coronene gives extra stability to the hexagonal pore of the TMA HON phase.32 Thus, the occurrence28 of a coronene-filled six-molecule pore could be expected. However, in addition to a dimerically bonded six-molecule pore, further ordering of the BHTC−coronene system requires pores of similar size which also could contain coronene molecules. The trimeric interactions inevitably form such pores since it is impossible to enclose the coronene by BHTC molecules bonded by the dimeric interactions only. In such a way, the trimeric interactions, which are not strong enough to independently organize the C1 structure of BHTC molecules, gain some aid from the neighboring coronene−BHTC interaction. Along with the C1 structure, the MC calculations predict the C2 structure which also creates the hexagonal four-molecule pores with dimeric and trimeric interactions. In addition to those pores, the C2 structure has five-molecule pores and is characterized by alternating zigzag chains of four-molecule and five-molecule pores. The size of a pore is appropriate for holding a coronene molecule. In our calculations the mixing of both C1 and C2 phases is very often observed. The mixing is possible because the zigzag chain of four-molecule pores is a common building block of both C1 and C2 phases, and two short−short dimeric interactions, the only interactions which make the energies of both phases different, have very similar magnitudes. Moreover, here we do not have two-phase coexistence, which is characteristic of two structures of different molecular densities accompanied by the first-order phase transition and a simple mixing of two phases of the same density. Nevertheless, our calculations for ess1 ≈ ess2 demonstrate a certain domination of the C2 phase over the C1 phase at full occupancy of pores by coronene. The question might arise as to if the existence of coronene molecules prompts the mixing of phases C1 and C2. The answer is, most likely, negative: the MC calculation for BHTC molecules for ess1 ≈ ess2 without the addition of coronene but with too large a trio interaction often results in a mixed C1 and C2 structure with the domination of the latter phase. We relate this result to higher entropy of the C2 phase at finite temperature. However, at low coronene concentrations ccor the sixmolecule pores characteristic of the C1 structure are more preferred because a larger number of neighboring BHTC molecules are interacting with the coronene molecule (more dimeric BHTC bonds are being shared to establish the BHTC− coronene interaction). But with increasing ccor the dominating structure might change, and the share of each phase depends on a competition between values of the short−short dimeric interactions, difference in entropy of phases C1 and C2, and coronene concentration.

Since the six-molecule pore in a symmetry-reduced BHTC system is the same as the pore created in a honeycomb phase of symmetric TMA molecules, we studied the effect of corone on ordering in the TMA system. The model was reduced to the three-state Bell−Lavis model used21,27 for TMA molecule ordering. We found that coronene molecules easily fill the pores in a honeycomb phase. For higher TMA densities the coronene still fills the hexagonal pores, but its insertion affects the subtle balance of existing phases. The addition of a small number of coronene molecules leads to a phase coexistence among the honeycomb, third flower, and superflower phases. For slightly higher ccor, this coexistence transforms into a nicely defined coexistence between the coronene-filled honeycomb phase and the coronene-free superflower structure. The pores in a honeycomb phase of the BTB molecular system are much larger than those found in TMA and any of the BHTC phases. Analyzing the sizes of pores which might hold the coronene molecule, we detected the hexagonal threemolecule pore which might be formed in a a structure with trimeric interactions only. These pores of the BTB superflower phase are perfect for holding a coronene, though the host− coronene interaction should be lower than in a TMA−corone six-molecule pore. On the other hand, the BTB system is known as not forming dense phases due to the low conformational flexibility of BTB molecules. Thus, our MC calculations were supposed to demonstrate if the coroneneinduced superflower phase can occur in a BTB molecular system when the BTB−coronene interaction is rather small. Our results predict the formation of the coronene-filled superflower phase. At lower concentrations of coronene they demonstrate the coexistence of a coronene-filled superflower phase with a coronene-free honeycomb structure. At even lower BTB concentration, corresponding to that of the honeycomb phase, and high corone concentration, our calculations demonstrate that three coronene molecules are stable in a hexagonal pore of BTB molecules.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +370 5 2616915. Fax: +370 5 2627123. E-mail: et@et. pfi.lt. Notes

The authors declare no competing financial interest.

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REFERENCES

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