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May 2, 2013 - Stability and Metastability of Bromine Clathrate Polymorphs. Andrew H. Nguyen and Valeria Molinero*. Department of Chemistry, The ...
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Stability and Metastability of Bromine Clathrate Polymorphs Andrew H. Nguyen and Valeria Molinero* Department of Chemistry, The University of Utah, 315 South 1400 East, Salt Lake City, Utah 84112-0850, United States ABSTRACT: Clathrate hydrates are crystals in which water forms a network of fully hydrogen-bonded polyhedral cages that contain small guests. Clathrate hydrates occur mostly in two cubic crystal polymorphs, sI and sII. Bromine is one of two guests that yield a hydrate with the tetragonal structure (TS), the topological dual of the Frank−Kasper σ phase. There has been a long-standing disagreement on whether bromine hydrate also forms metastable sI and sII crystals. To date there are no data on the thermodynamic range of stability (e.g., the melting temperatures) of the metastable polymorphs. Here we use molecular dynamics simulations with the coarse-grained model of water mW to (i) investigate the thermodynamic stability of the empty and guest-filled the sI, sII, TS, and HS-I hydrate polymorphs, (ii) develop a coarse-grained model of bromine compatible with mW water, and (iii) evaluate the stability of the bromine hydrate polymorphs. The mW model predicts the same relative energy of the empty clathrate polymorphs and the same phase diagram as a function of water−guest interaction than the fully atomistic TIP4P water model. There is a narrow region in water-guest parameter space for which TS is marginally more stable than sI or sII. We parametrize a coarse-grained model of bromine compatible with mW water and use it to determine the order of stability of the bromine hydrate polymorphs. The melting temperatures of the bromine hydrate polymorphs predicted by the coarse-grained model are 281 ± 1 K for TS, 279 ± 1 K for sII, and 276 ± 1 K for sI. The closeness of the melting temperatures supports the plausibility of formation of metastable sII and sI bromine hydrates. process yields the metastable hexagonal structure I (HS-I).17 The unit cell of sI consists of one 512 and three 51262 cages, while sII contains sixteen 512 and eight 51264 cages per unit cell. The TS and HS-I polymorphs contain all but the largest cages: the unit cell of HS-I combines three 512, two 51262, and two 51263 cages, and TS has five 512, eight 51262, and two 51263 cages per unit cell. Bromine hydrate, first reported by Löwig in 1928,18 was one of the first hydrates produced and played an important role in the development of van der Waals and Platteeuw’s statistical mechanical theory of the thermodynamic stability of clathrates.19 Bromine hydrates synthesized in different laboratory conditions display hydration numbers (ratio of water to guest) that range from about 6 to 12.2,16,18,20−28 Two scenarios have been proposed to explain the unusually large variability in hydration number in bromine hydrates. The first contends that there is a single bromine hydrate polymorph, the tetragonal crystal, and that a wide dispersion in guest occupancy of the water cages accounts for the range of hydration numbers.27 The second scenario involves the existence of multiple clathrate polymorphs, each with a narrow range of hydration numbers.24 Bromine is too large to fill the small 512 cages. If bromine were to fill all large cages, the hydration numbers of the partially filled crystals would be 7.6 for sI, 8.6 for TS, 10 for HS-I, and 17 for sII. Dyadin and Aladko argued the existence of distinct polymorphs from their observation that the hydration numbers

1. INTRODUCTION Clathrate hydrates are crystals in which water forms a network of polyhedral cages that typically contain small non-hydrogen bonding guest molecules.1−5 There are vast reservoirs of methane clathrates in the seafloor, and recent estimations indicate that there is more carbon stored in methane clathrate hydrates than in all the other fossil fuels combined.6,7 Clathrates hydrates are made of combinations of four types of polyhedral water cages: dodecahedra (denoted 512), tetrakaidecahedra (51262), pentakaidecahedra (51263), and hexakaidecahedra (51264). Clathrate hydrates are duals of Frank−Kasper crystals: all water molecules in clathrates are four-coordinated and are located in the position of the voids in the tetrahedrally close-packed Frank−Kasper structures.8,9 In the absence of guest molecules, the clathrate hydrate lattices are less stable than ice, except at very negative pressures.10,11 The interaction between the guests and their host water cages can stabilize clathrate hydrates above the melting point of ice. The size of the guest is the most important factor in determining the selection between different clathrate polymorphs.1,12−14 There are, in principle, an infinite possible number of Frank− Kasper crystals. A total of 27 of these structures have been found in metal alloys.8 Only two are common in clathrate hydrates: cubic structure I (C-sI or sI) and cubic structure II (C-sII or sII).1 Some guests, such as methane, yield the sI and sII hydrate polymorphs depending on the conditions of preparation.15 Very few guests yield clathrate hydrate in polymorphs different from sI and sII: bromine hydrate and dimethyl ether form the rare tetragonal structure (TS),16 and Xe hydrate subjected to a compression and decompression © 2013 American Chemical Society

Received: April 9, 2013 Revised: April 30, 2013 Published: May 2, 2013 6330

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phases of bromine hydrate.36 Using the experimental enthalpy of formation of solid bromine to account for the different ratio of water to guest molecules in the polymorphs, Schofield and Jordan predicted that the energy of formation of bromine clathrates increases in the order pTS < psII < psI, an order that would be consistent with the experimental results of Janda et al.28 More recently, Fleischer and Janda examined the energy of interaction of 47 guests with clathrate cages using molecular mechanics calculations with the Merck Molecular Force Field (MMFF).37 They found that the stabilization of bromine and dimethyl ether in the TS and sII polymorphs are within 0.15 kJ/(mole H2O). The melting temperatures of the TS, sII, and sI polymorphs of bromine hydrate would provide an absolute measure of their stability with respect to the liquid phase and would be key to interpret the accessibility and order of metastable states, but they have not yet been determined by theory or molecular simulations. In this work, we use molecular simulations with the monatomic water model mW to determine the regions of stability of filled and partially filled sI, sII, TS, and HS-I polymorphs as a function of water−guest interaction and to evaluatefor the first timethe melting temperatures of the TS, sI, and sII bromine hydrate polymorphs. The mW model represents each water molecule as a single particle with shortrange anisotropic interactions that mimic hydrogen bonds, and it is ∼180 times computationally more efficient than rigid atomistic models of water with Ewald sums.38 mW has been successfully used to model the structure, anomalies, and phase behavior of liquid, amorphous solid, crystalline, and quasicrystalline water, including the stability, growth, nucleation, and cross-nucleation of clathrate hydrates.10,13,39−57 The phase diagram computed by Matsumoto and Tanaka with TIP4P water and van der Waals Platteeuw theory14 shows the same regions of stability for sI and sII crystals as the one previously computed for mW by Jacobson and Molinero.13 We now include TS and HS-I in the evaluation of the phase diagram and determine the melting temperatures of the four polymorphs for families of guests around the region where TS is stable or has the same stability as the next metastable hydrate. We develop a coarse-grained model of bromine to be used with mW water and use it to evaluate the melting temperatures of the sI, sII, and TS polymorphs. Our simulations indicate that the melting points (i.e., the stability) of bromine hydrates decrease in the order TS > sII > sI. More important, they indicate that the melting temperature of the least stable polymorph, sI, is just 5 K lower than the most stable one, TS. These findings support the possibility of formation of TS, sI, and sII bromine hydrate polymorphs under conditions of mild supercooling, as reported in the most recent experiments.2,28

were not uniformly distributed among all the observed range but were clustered around well-defined values.24 Ripmeester and co-workers,27,29 on the other hand, performed X-ray analysis of 16 crystals of bromine hydrate with hydration numbers ranging from 8.6 to 10.7 and found that all had the tetragonal structure proposed by Allen and Jeffrey.16 More recently, Janda and co-workers used resonance Raman and UV−visible spectroscopy to investigate the evolution of the structure of bromine hydrates with temperature and exposure to water.2,28 Based on their analysis of the spectroscopy of bromine in the 51262, 51263, and 51264 cages,2,28,30−33 they concluded that sII bromine hydrate grew from the TS crystals at temperatures between 256 and 266 K in the presence of excess water vapor and that polycrystalline sI crystals grew from highly supersaturated solutions at high supercooling.2,28 To our knowledge, there is not yet X-ray diffraction of bromine hydrates that conclusively supports the existence of the sI and sII polymorphs. The stable bromine hydrate polymorph, the partially filled TS crystal, has a melting temperature of 278.9 K.2,31 The sII and sI bromine hydrates would be metastable. The possibility of forming metastable phases depends on the difference in stability (i.e., melting temperatures) to the most stable crystal. The melting temperatures of the metastable sII and sII bromine hydrates have not been measured, nor has it been reported whether crystals with different hydration numbers have identical or distinct melting points. A natural question that arises from the limited number of polymorphs of clathrate hydrate crystals, considering the large number of possible Frank−Kasper structures, is what explains the selectivity of clathrate hydrates for so few crystal structures. Matsumoto and Tanaka addressed this question through the calculation of the relative stability of several Frank−Kasper clathrates as a function of the water−guest interaction for a given chemical potential of the guest, using the van der Waals and Platteeuw theory with the free energy of the empty clathrate networks computed with the TIP4P water model.14,34 Matsumoto and Tanaka’s work elegantly demonstrated that, within the approximations of the van der Waals−Platteew theory (i.e., no interaction between guests in different cages, no deformation of the water lattice by presence of hydrates)19 and the TIP4P water model,35 the filled or partially filled sI and sII polymorphs are the stable Frank−Kasper hydrate phases for all guests except for a very narrow range in water−guest parameter space between the regions of stability of partially filled sI (psI) and partially filled sII (psII) for which the stability of TS is indistinguishable from the one of the next stable polymorph, and a region of high water−guest interaction for which their theory predicts that HS-I would be the stable clathrate.14 The range of water−guest interactions that can stabilize the TS crystal is consistent with water−bromine interaction potentials in atomistic force fields.14 The theoretical calculations, which account for the possibility of partial occupancy of the cages, suggest that the partially filled sI and sII polymorphs would be close in stability to TS, supporting the existence of these polymorphs as metastable phases for bromine. Molecular simulations can provide further insight on the stability of bromine hydrates. There are few reports of simulations of bromine hydrates. Kerenskaya et al. used the Merck Molecular Force Field to compute the interaction energy of dihalogen molecules with the four clathrate cages.32 Schofield and Jordan developed a fully atomistic polarizable bromine model and used it along with the polarizable COS/G2 water force field to study the stability of the TS, sI, and sII

2. MODELS AND METHODS Force Fields. Water was modeled with the monatomic water model mW38 which has the functional form of the Stillinger−Weber (SW) potential:58 E=

∑ ∑ ϕ2(rij) + ∑ ∑ ∑ ϕ3(rij , rik , θijk) i

j>i

i

j≠i k>j

⎡ ⎛ ⎞4 ⎤ ⎛ σ⎟ σ ⎞⎟ ⎢ ⎜ ϕ2(rij) = Aε B⎜ ⎟ − 1⎥exp⎜⎜ ⎢ ⎝ rij ⎠ ⎥ ⎝ rij − aσ ⎟⎠ ⎣ ⎦ 6331

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from L were set at 500 atm, as in previous studies.39,42 The energies of empty clathrates were computed at 10 K and 0 atm relative to the hexagonal ice. The melting temperature (Tm) of each crystal was determined using the direct coexistence method in the NpT or NpH ensemble (the latter is more accurate when the system is already very close to the melting point).60−63 Simulation times for the determination of the melting temperatures ranged from 50 to 600 ns per state point.

ϕ3(rij , rik , θijk) = λε[cos θijk − cos θ0]2 ⎛ γσ ⎞ ⎛ γσ ⎞ ⎟⎟exp⎜ × exp⎜⎜ ⎟ ⎝ rij − aσ ⎠ ⎝ rik − aσ ⎠

(1)

where rij is the distance between particles i and j and θijk is the angle subtended by the vectors between the positions of i−j and i−k pairs of particles. The constants are A = 7.049556277, B = 0.6022245584, γ = 1.2, a = 1.8, and θ0 = 109.5°. The tetrahedral parameter of water is λw = 23.15, the characteristic size is σww = 2.3925 Å, and the energy scale is εww = 25.895 kJ/ mol.38 The three-body terms ϕ3 encourage tetrahedrally coordinated “hydrogen-bonded” configurations between water molecules by imposing an energy penalty on water−water− water angles that deviate from 109.5°. Each guest molecule was also modeled as a single particle, butdifferent from waterthe guests do not have anisotropic, hydrogen bonding interactions (i.e., λ is set to zero). The interaction energy has the form of the two-body potential ϕ2 of eq 1 with the same constants (A, B, γ, and a) of the mW model. The guest−guest interactions were characterized by a size σgg = 4.2 Å and a depth of the interaction potential εgg = 1.423 kJ/ mol, as the L guest of ref 39, except for bromine which is parametrized in section C of Results. The guest−water interactions were tuned to yield families of guests with distinct preferences for the TS, sI, and sII clathrate polymorphs. Simulation Settings. Molecular dynamics simulations were performed using LAMMPS.59 The equations of motion were integrated using the velocity-Verlet algorithm with a time step of 10 fs. Simulations were performed in the NpT or NpH ensembles. Temperature and pressure were controlled with the Nose−Hoover algorithm with damping parameters of 5 and 25 ps, respectively. The barostat allowed for independent expansion and contraction of simulation cells in the three directions. Simulations. The guest molecules of this study are too large to occupy the dodecahedral cages and only occupy the large cages (5126n where n > 0). The clathrate partially filled structures (all cages except 512 occupied, indicated with a p before the crystal name) have different hydration numbers; therefore their simulation cells have different numbers of water and guest molecules. The psI system contains 5888 waters and 768 guests, psII 7344 waters and 432 guests, pTS 5504 waters and 640 guests, and pHS-I 10240 waters and 1024 guests. Simulation cells used for the parametrization of the bromine were twice as large as those indicated above. Simulation cells containing each of the clathrate crystals were melted halfway along the (long) x-axis to produce two phases (clathrate and solution). The liquid phase was equilibrated for 10−100 ns before continuing with the simulations. This process allows for phase segregation of a third phase (a guest bubble) if the solubility of the guest in water is lower than the ratio of guest to water in the hydrate. These two-phase or three-phase coexistence simulation cells were used in the simulations of melting. Simulation cells without crystalline phases, containing 10 240 particles consisting of 9216 water and 1024 guest molecules were independently evolved for 100 to 200 ns to determine the solubility of the guests that phase segregate during equilibration in the aqueous liquid phase. The solubility of the guests was determined from the profile of concentration of guest molecules in aqueous phase. Simulations of the empty clathrate and bromine clathrates were performed at 1 atm, while the simulations of the clathrates filled with guests derived

3. RESULTS AND DISCUSSION A. Stability of Guest-Free Clathrates Polymorphs. An extensive analysis of the stability and thermodynamics of empty sI and sII clathrates modeled with mW water has been reported in ref 10 and was shown to be in excellent agreement with available experimental data. In this section we extend the study with the mW model to include the empty TS and HS-I polymorphs and compare the results with those obtained in ref 14 for the energetics of the hydrate polymorphs evaluated with the TIP4P water model. We first determined the relative energies with respect to the most stable clathrate lattice, sII, and the order of stability (through the calculation of the melting temperatures) of guest-free clathrate crystals with structures sI, sII, TS, and HS-I. The crystal structures are shown in Figure 1.

Figure 1. Structures of guest-free sI, sII, TS, and HS-I clathrates. Lines connect neighboring water molecules in clathrate cages. Each clathrate cage is shown in a different color: green represents the 512 cages, blue the 51262 cages, red the 51263 cages, and gray the 51264 cages. The 512 cages exist in all four clathrate hydrates but are hidden in the sI, TS, and HS-I clathrates by the large cages, 5126n (n > 0), with which they share vertices.

The melting temperatures of the four polymorphs and ice and their energies with respect to the most stable polymorph, sII, are shown in Table 1. As the mW simulations were performed at finite temperature and previous work demonstrated that the entropy of melting of ice and empty clathrates are essentially the same,10 we compare the relative energies obtained in the mW simulations to the relative free energy (zero point energy plus free energy of harmonic vibration) reported for the 6332

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the calculation of the melting temperature of the clathrate crystals. Jacobson and Molinero mapped the formation enthalpies of the filled and partially filled (i.e., with only large cages, 5126n n > 0, occupied) sI and sII clathrate polymorphs as a function of the water−guest interaction strength εwg and water−guest characteristic size σwg.13 Approximating that the most stable crystal is the one with the lowest enthalpy of formation, they used the map to predict the region of stability of filled and partially filled sI and II polymorphs. More recently, Matsumoto and Tanaka mapped the relative stability of the sI, sII, TS, and HS-I clathrate crystals as a function of guest size and guest− water interaction using the atomistic TIP4P water model and Lennard−Jones monatomic guests within the statistical mechanical framework of the van der Waals−Platteeuw theory.34 The phase diagram computed for TIP4P water is essentially the same as for the mW model for the filled and partially filled sI and sII crystals. The atomistic study also considered the stability of the TS and HS-I polymorphs and revealed a narrow region of stability for partially filled TS in the region of guest-water parameter space between partially filled sI and partially filled sII, and an even more reduced region of stability for partially filled HS-I in the high εwg zone of the boundary between partially filled TS and partially filled sII.34 We extend the calculation of the stability map with mW water of ref 13 to include the filled and partially filled TS and HS-I crystals. The enthalpies were evaluated at the same conditions of ref 13, p = 100 atm and T = 10 K, and guest− guest potential given by εgg = 1.422 kJ/mol and σgg = 4.08 Å. The enthalpies of formation were computed with respect to that of sII, correcting for the difference in stoichiometry of sII and the filled and partially filled (p) TS and HS-I clathrates with the following equations:

Table 1. Stability of Empty Clathrate Crystals and Ice at 1 atm Tm (K) mW model Ih sII sI TS HS-I a

274 252 245 244 243

± ± ± ± ±

1 2 2 2 2

E − EsII (kJ/mol) mW model

G − GsII (kJ/mol) TIP4P modela

−0.45 ± 0.01 0.0 0.17 ± 0.01 0.20 ± 0.02 0.33 ± 0.01

−0.55 0.0 0.19 0.20 0.37

From ref 14.

atomistic structures in ref 14. There is excellent agreement in the relative energies predicted by the coarse-grained mW water model with previous determinations using the atomistic TIP4P water model. The difference in stabilities between empty sI and sII predicted by mW and TIP4P agrees with the 0.17 kJ/mol deduced by Handa and Tse from the analysis of experimental data.64 The predicted melting temperatures of the water crystals and their relative energies reveal the same order of stability. Ice Ih is the most stable water crystal at the pressures considered in the present study.10 The most stable empty clathrate crystal is sII, followed by sI, TS-I, and finally HS-I, the least stable of the four crystals. The simulations predict that the tetragonal structure is marginally less stable than sI by 0.03 ± 0.03 kJ/mol for mW water and 0.01 kJ/mol (error bar unknown) for the TIP4P model. Empty sI and TS are very close in stability, as confirmed by their indistinguishable melting temperatures: 245 ± 2 K and 244 ± 2 K for sI and TS, respectively. The energy difference between sII and sI is the same for the TIP4P water model, 0.19 kJ/mol compared to 0.17 ± 0.02 kJ/mol for mW model. The atomistic and coarse-grained models predict HS-I to be the least stable of the four empty clathrate polymorphs, with energy 0.37 or 0.33 ± 0.01 kJ/mol, respectively, above that of sII. The atomistic14 and coarse-grained water models show no correlation between the stability of the empty clathrate crystals and the ratio of five- to six-membered water rings. We find, however, a correlation between the energy of the guest-free clathrates and the percentage of water molecules in the network shared by small (512) and large (5126n with n > 0) cages. Each water molecule in clathrate hydrates is a vertex shared by four polyhedral cages. The identity of the polyhedral cages that share a water defines the vertex order parameter.42,43 The potential energy of the empty clathrates can be expressed as a second order polynomial of the percent p of vertices involving at least one small and one large cages, E = 0.0069p2 − 1.262 p + 58.57 (correlation coefficient 0.991), where the energy is measured with respect to that of hexagonal ice and expressed in kJ/mol. It is an open question whether this relation is general and can be used for the prediction of the energies of other empty Frank−Kasper structures made of water. B. Tuning the Stability of the Clathrates by Selection of the Guest−Water Interaction. Having demonstrated that the stability of the water networks is well-described by the mW model, we now determine how the guest−water interaction parameters εwg and σwg modulate the relative stability of clathrate polymorphs. As a first approximation, we estimate the relative stabilities through a map of the enthalpies of formation as a function of εwg and σwg. Using that map as a guide, we then determine the actual thermodynamic stabilities of the sI, sII, TS, and HS-I clathrate polymorphs for selected guests through

sII(1H 2O + 3/17G) = pTS(1H 2O + 5/43G) + 44/731G

(2a)

sII(1H 2O + 3/17G) = TS(1H 2O + 15/86G) + 3/1462G

(2b)

sII(1H 2O + 3/17G) = pHS‐I(1H 2O + 1/10G) + 13/170G

(2c)

sII(1H 2O + 3/17G) = HS‐I(1H 2O + 7/40G) + 1/680G

(2d)

We note that this map, shown in Figure 2, does not account for the effect of entropy (particularly important for the pure guest) in the relative stabilities of the clathrate phases, so it should be considered only as a guide to identify regions of stability of TS and HS-I clathrates. Same as reported for the atomistic model,34 the map of stabilities of the coarse-grained model predicts a narrow region between partially filled sI and partially filled sII for which partially filled TS is the most stable crystal. mW does not predict a region where the HS-I is the most stable crystal for εwg up to 1.7 kJ/mol; this is expected as the HS-I phase becomes stable for TIP4P only for higher water−guest attraction. Figure 2 shows that, on increasing the guest−water characteristic size at moderate water−guest attraction, the stable clathrate phases evolve from sII, to sI, to psI, to pTS, and, finally, psII. The progression mirrors the 6333

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stable clathrate with a melting temperature 327 ± 2 K, psII is metastable with Tm = 318 ± 2 K.39 We tuned the potential around the one corresponding to the L guest by changing the guest size σwg, the interaction strength εwg, and guest concentration to investigate the stability of the tetragonal structure relative to the sI and sII clathrate. We first vary the water−guest characteristic size σwg between 4.1 and 4.4 Å. The melting temperatures (Figure 3) indicate

Figure 2. Parameter map that predicts the stability of hydrate polymorphs as a function of water−guest interaction parameters using the lowest enthalpy of formation. The colored regions indicate which polymorph has the lowest enthalpy of formation: filled sII (blue), filled sI (red), partially filled sI (green), partially filled TS (orange), and partially filled sII (black). pTS is the most stable clathrate polymorph in a thin region between psI and psII. The white squares represent the σwg and εwg derived from the L guest of ref 39 to tune the stability of pTS. A small contribution to the displacement in the stability predicted by enthalpies of formation (colored regions) and those measured by the melting temperatures at each point, arises because the map was determined for methane-like guest−guest interactions given by σgg = 4.08 Å and εgg = 1.42 kJ/mol, while the stability for the guests represented as white squares was determined for σgg = 4.2 Å and εgg = 1.42 kJ/mol. The magenta pentagon indicates the water−guest interaction potential for the coarse-grained bromine, which has σgg = 4.567 Å and εgg = 5.125 kJ/mol.

Figure 3. Melting temperatures of clathrate polymorphs as a function of guest size. The other parameters are those of the L guest, σgg = 4.2 Å, εgg = 1.43 kJ/mol, and εwg = 1.507 kJ/mol. Melting temperatures psI (green), pTS (orange), psII (black), and pHS-I (magenta). The lines are second-order polynomial fits to the data for each polymorph and are shown as a guide to the eye.

increase in average cage size along these structures. All of the clathrate crystals discussed in what follows have guests that are too large to fill the dodecahedral cages; they are partially filled. It should be noted that, in the region of Figure 2 where partially occupied TS was predicted to be the most stable crystal, the difference in the enthalpy of formation of pTS and the next most stable phase, psI, was tiny (0.03 kJ/mol for εwg = 1.5 kJ/mol). This indicates that even for guests that stabilize the tetragonal crystal, such as bromine, its stability would be very close to that of the closest metastable phase. In the region where pTS is the most stable the enthalpy of formation of HS-I is 0.19 kJ/mol higher than the enthalpy of formation of pTS, the most stable clathrate. Overall we find an excellent agreement between the predictions using the mW and TIP4P water models, suggesting that the order of stability of clathrate polymorphs can be well represented with a coarse-grained model without hydrogen atoms or long-range interactions but that accurately mimics the ability of water to produce tetrahedrally coordinated “hydrogen-bonded” structures. We determined the actual stability and order of metastability of the clathrate crystals through the calculation of the melting temperatures of the partially filled sI, sII, TS, and HS-I for the water−guest interaction potentials marked with white squares in Figure 1. Previous determination of the melting point of hydrates of the L guest (σwg = 4.2 Å, εwg = 1.507 kJ/mol, σgg = 4.08 Å, and εgg = 1.42 kJ/mol) indicated that psI is the most

that the tetragonal structure does not emerge as the most stable crystal within this range of guest sizes, although its stability is indistinguishable −within the error bar of the calculations, from the one of psI for σwg between 4.15 and 4.2 Å. For example, the melting temperatures for the L guest, σwg = 4.2 Å, are 325 ± 2 K for pTS and 327 ± 2 K for psI. Small guests, σwg below 4.15 Å, show a clear preference for psI, while psII emerges as the most stable crystal for large guests, σwg above 4.3 Å. The trends with size are as expected from the stability map of Figure 2 but indicate thatfor this family of guestspTS is, at best, as stable as psI. As a second step to identify guests that favor the tetragonal crystal, we tuned the interaction strength between water and guest, εwg, and computed its effect on the stability of the clathrate polymorphs. We started with the L guest for which the difference in melting temperature between pTS and psI is within the error bar, 2 ± 4 K, and determined the melting temperature of psI, psII, and pTS as a function of the interaction strength between water and guest for εwg, from 1.25 to 1.88 kJ/mol (Figure 4). The upward turn of the melting temperatures on decreasing εwg below 1.42 kJ/mol is associated to the decrease in solubility of the guests in water and the formation of a guest-only bubble: the destabilization of the solution results in an increase in the range of stability of the crystal phases. We find that pTS and psI have indistinguishable melting temperatures for εwg below 1.5 kJ/mol and psI is the 6334

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spectroscopy for the formation of psI under conditions of high supersaturation.2,28 In what follows, we develop a coarsegrained model of Br2 as a single particle interacting with other bromine particles and with water through short-range two-body potentials shown in eq 1. We first tuned the guest−guest interaction parameters to εgg = 5.125 kJ/mol and σg = 4.567 Å, to match the experimental enthalpy of vaporization ΔHvap and density of liquid bromine at 298 K and 1 atm (Table 2). We Table 2. Experimental and Predicted Properties of Bromine and Bromine Hydrates experimental results density @ 298 K and 1 atm ΔHvap @ 298 K and 1 atm Tm of TS Tm of sII Tm of sI solubility, xBr2 a

Figure 4. Melting temperature of psI, psII, and pTS as a function of water−guest interaction strength. The other parameters are those of the L guest, σgg = σwg = 4.2 Å and εgg = 1.43 kJ/mol. Melting temperatures of psI (green), pTS (orange), and psII (black). The melting temperatures of sI and TS are indistinguishable below 1.5 kJ/ mol.

3

3.112 g/cm 30.91 kJ/mol 278.9 K2,31 unknowna unknown 0.0038

simulation results 3.12 g/cm3 30.88 kJ/mol 281 ± 1 K 279 ± 1 K 276 ± 1 K 0.0002 ± 0.0001

sII grew from TS when 256 K < T < 266 K.2,31

then tuned the water−guest interaction parameters to εwg = 1.338 kJ/mol and σwg = 4.2 Å, to reproduce the experimental melting temperature of the TS bromine clathrate at 1 atm, 278.9 K.2 We used the new coarse-grained bromine/water potential to evaluate the melting temperatures of partially filled TS, sII and sI polymorphs; the results are shown in Table 2. The abstract figure shows coexistence of TS clathrate, aqueous solution, and bromine liquid at 1 atm and 281 K. To our knowledge this is the first report of melting temperatures of bromine hydrates using molecular simulations. The simulations with the coarse-grained bromine and mW water models indicate that pTS is the most stable crystal (Tm = 281 ± 1 K), closely followed by psII (Tm = 279 ± 2 K), and that sI is the least stable of the three polymorphs (Tm = 276 ± 1 K), the same order suggested by the simulations with fully atomistic polarizable force fields.36 We note that, for bromine hydrate, TS competes with sII, different from what was observed for the family of guests derived from the L guest in section B, for which sI was more stable than sII. The bromine potential derived in this section and the third guest from the left in Figure 4 have the same water−guest interaction parameters; their main difference is on the strength of the guest−guest interaction, which is higher for bromine and has the effect of decreasing its solubility xs in water. The solubility of the coarse-grained bromine measured in the simulations at 298 K and 1 atm is extremely low, only 2− 3 Br2 molecules in the 10 000 water molecules of the simulation cell. That value is one order of magnitude lower than the experimental solubility of bromine in water, 1 Br2 every 259 water molecules at the same conditions (Table 2).26 It is not possible to simultaneously match the experimental solubility and the experimental melting temperature by changing the size and strength of the water−guest interaction. The underestimation of the bromine solubility in water by the coarsegrained model may be related to the absence of highly anisotropic halogen bond interactions in the coarse-grained model. The halogen bond is not active in the hydrate since all of the water oxygen atoms are directly hydrogen bonded and not available for halogen bonding, but it is relevant for the solvation of bromine in liquid water.65 The interaction energy between bromine and the water cages of the clathrates predicted by the coarse-grained model is in

most stable crystal at higher interaction strengths. For this series of guests derived from L, psII remains always metastable, consistent with the approach to the region of stability of pTS from the psI side of the phase diagram shown in Figure 2. In the simulations presented above, the concentration of guest in the aqueous solution is the same as the ratio of guest to water in the crystal, except when limited by guest solubility (i.e., εwg < 1.5 kJ/mol). Each clathrate structure has a different guestto-water ratio; psI is 1:7.66, pTS is 1:8.6, and psII is 1:17. We now investigate whether it is possible to modulate the stability of the tetragonal structure by varying the concentration of the guest molecules in the solution phase. We vary the guest-towater ratios for the L guest (σwg = 4.2 Å, εwg = 1.507 kJ/mol) for which pTS competes with psI, and for its variant with σwg = 4.3 Å, for which the melting temperatures of pTS, psI, and psII are within 2−3 K. We find that the difference in the melting temperature remains unchanged when we evaluate all the melting points with the same concentration. For example, for σwg = 4.2 Å and 1 guest every 7.66 water molecules, Tm of psI is 327 ± 2 K, Tm of pTS is 325 ± 2 K, and Tm of psII is 316 ± 2 K. For with σwg = 4.3 Å and concentration 1:17, Tm of psI is 314 ± 2 K, Tm of pTS is 312 ± 2 K, and Tm of psII is 315 ± 2 K. The melting points of psI and psII are equal for σwg = 4.3 Å. The decrease in concentration shifts the melting temperatures down, though still within the error bar of the calculations. Overall, we find that changing the concentration in each simulation to match the guest-to-water ratios of sI or sII did not result in TS becoming more stable or in any significant changes in the relative stabilities of the crystals. C. Parameterization of a Coarse-Grained Bromine Guest. We have shown that, starting from the L guest, it is possible to find molecules for which the stability of pTS is indistinguishable from the stability of psI. In bromine hydrates, however, the experimental data suggest that pTS competes in stability with psII, not with psI, although there is support from 6335

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metastable sII bromine hydrate grow in the presence of excess water at temperatures between 256 and 266 K, and a polycrystalline sI bromine hydrate film was inferred to form under conditions of very high supersaturation of bromine.2,28,31 We parametrized a coarse-grained model of bromine compatible with mW water which displays the experimental enthalpy of vaporization of liquid bromine and its density at 298 K and 1 atm and the experimental melting temperature of TS and the order of stability of the clathrate polymorphs inferred from the experiments. To our knowledge this is the first coarse-grained model of bromine and the first determination of the melting temperatures of TS, sII and sI bromine polymorphs in simulations. The melting temperatures of the bromine hydrate polymorphs predicted by the coarse-grained model are 281 ± 1 K for TS, 279 ± 1 K for sII, and 276 ± 1 K for sI. The same order of stabilities was inferred from energetic data using fully atomistic polarizable force fields.36 The small difference in the melting point of TS, sII, and sI bromine hydrate predicted by the simulations lend support to the interpretation of the experimental variability of hydration numbers of bromine clathrates as arising from distinct polymorphs with a broad range of stoichiometry. Goldschleger et al. found that exposure of TS bromine hydrate to water vapor between 256 and 266 K resulted in the formation of sII bromine hydrate crystals.2 It is an open question whether this transformation involves the nucleation and growth of sII on a crystal face of TS. We have previously demonstrated that clathrate hydrates grown at high supercooling can cross-nucleate from the most stable to a least stable polymorph, provided that that the metastable crystal is more stable than the solution and that it grows faster than the most stable crystal.42 It is not yet known whether TS clathrates crossnucleate to the polymorph that is closer in stability (sI or sII depending on the guest), the one structurally closer (TS have common layers with sI but not with sII), to the one that grows the fastest (sII is usually faster growing than sI), or do not present cross-nucleation. The mechanisms of growth and crossnucleation of TS hydrates will be addressed in a future publication. Theoretical arguments based on the calculation of the chemical potentials of hydrates using the van der Waals and Platteew theory and the Yarmolyuk and Kripyakevich rule suggest that the stability of all Frank−Kasper structures would be very similar in the boundary between psI and psII.67 The existence of multiple polymorphs with competing order and similar stability is known to favor vitrification.5,68−70 Molecular simulations indicate that clusters of amorphous clathrates, which contain the four polyhedral cages in ratios that do not correspond to the crystals and have no long-range order, are involved in the mechanism of nucleation of crystalline clathrates at a high driving force.39−41,43,71,72 The polymorphism of bromine clathrates makes water/bromine solutions promising candidates for producing bulk amorphous clathrates by hyperquenching in laboratory experiments.

excellent agreement with previous calculations using atomistic force fields. Kerenskaya et al. calculated the association energy of the 51263 cage with a bromine molecule to be −24 kJ/mol using the MMFF force field.32 We determined an interaction energy of −23.98 kJ/mol for the monatomic bromine guest with the 51263 cages of mW water using the master curves of interaction energy as a function of water−guest interaction of ref 39. Similarly, Schofield and Jordan’s simulations with polarizable force fields showed that the bromine molecule is 10.88 kJ/mol more stable in the 51263 cages than the 51262 cages.36 The coarse-grained bromine guest shows the same preference, predicting that bromine is 8.93 kJ/mol more stable in the 51263 cages than in the 51262 cages. Based on the hydration number, morphologies, and spectra of the samples obtained, it has been suggested that sI bromine hydrate could be formed in laboratory experiments.25,27,28 The atomistic and coarse-grained simulations indicate that bromine interacts attractively with the 51262 cages, and the coarse-grained model predicts that the melting point of the sI bromine clathrate is only 5 ± 2 K lower than for the TS crystal, supporting the plausibility of formation of metastable sII and sI bromine clathrate hydrates in laboratory experiments under conditions of medium and high supercooling.

4. CONCLUSIONS We used molecular dynamics simulations with mW water and a series of monatomic guests to investigate the stability of guestfree and guest-filled clathrate polymorphs and to derive a coarse-grained model of bromine that reproduces the experimental density and enthalpy of vaporization of the liquid, the melting point of the TS hydrate and predicts the order of stability of the other polymorphs as deduced from the experiments. The monatomic water model mW predicts the same order of stability and relative energies of the empty clathrates and ice I than the fully atomistic TIP4P water model. These results, along with previous predictions of the stability of sI and sII clathrate hydrates, cubic and hexagonal ice I, and crystalline and quasicrystalline phases of water in confinement, indicate the relative stability of the water networks are accurately represented by the mW model, in spite of its lack of explicit hydrogen atoms or long-range electrostatics interactions. Molecular dynamics simulations with mW water predict the same stability map as a function of the water−guest interaction potentials than van der Waals and Platteeuw theory using TIP4P water.14 On increasing the water−guest characteristic size at moderately high water−guest attraction there is a narrow region of water−guest interaction potentials for which TS is predicted to be the most stable clathrate hydrate. The guest size in that region is too large to fill the dodecahedral cages and derives the highest stability by being in the 51263 water cages. Our results indicate that, even for guests for which TS is predicted to be the most stable hydrate, its melting point is at best slightly higher than the first metastable crystal. The relative order of metastability of the sI and sII polymorphs can be switched by tuning the water−guest and guest−guest interaction potentials. The only known molecules that form TS hydrates in experiments, bromine and dimethyl ether, display competition between TS and sII. In the case of dimethyl ether, sII is the most stable polymorph and TS is the metastable form.66 For bromine, TS is the stable form, with a melting point of 278.9 K,16,27 and it has been reported that single crystals of the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 6336

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation through award CHE-1012651. We thank Masakazu Matsumoto and Kenneth Janda for insightful discussions and the Center for High Performance Computing at the University of Utah for technical support and resources.

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