Can Guest Occupancy in Binary Clathrate Hydrates Be Tuned through

Sep 8, 2014 - The large guest molecules only fit into the 51264 cages, while the small guest molecules can fit into both types of cages. We find that ...
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Can Guest Occupancy in Binary Clathrate Hydrates Be Tuned through Control of the Growth Temperature? Bin Song, 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 nonstoichiometric compounds comprised of a hydrogen-bonded water network that forms polyhedral cages that can be occupied by small guest molecules. Clathrates are candidate materials for storage and transportation of methane and H2. Promoter molecules, such as THF, reduce the pressure or temperature needed to form clathrates of these gases, resulting in the formation of binary clathrates with the promoter molecule hosted in the large cages of the hydrate. In this work, we study the growth and occupancy of binary clathrates as a function of supercooling of the solution using molecular dynamics simulations with the mW water model and small and large guest molecules of sizes similar to those of H2 and THF, respectively, but that are both highly soluble in water and produce single hydrates with identical melting point. The large guest molecules only fit into the 51264 cages, while the small guest molecules can fit into both types of cages. We find that the large guest act as a kinetic promoter for growth, increasing the rate of uptake of small guests into the clathrate. Our results also indicate that the growth of binary clathrates is limited by the arrangement of guest molecules in the large 51264 cages at the clathrate/solution interface. The occupancy of large cages of binary clathrates can be tuned by varying the growth temperature. The simulations indicate that with increasing supercooling there is an increase in the percentage of 51264 cages occupied by the small guest molecules at the expense of the large guest molecules, while the occupancy of 512 cages remains relatively constant. The results of this work show that the composition of clathrates grown at high driving force does not necessarily reflect the composition of the most stable phase. referred to as double clathrates or binary clathrates. The H2− THF binary clathrates have a cubic structure known as structure II (sII), which is comprised of eight 512 cages and 16 hexakaidecahedral cages (51264) per unit cell. The 51264 cages are larger in size than the 512 cages. H2 is small enough to fit in both cages, while THF can only fit in 51264 cages. As a result of THF occupying the 51264 cages, the loading capacity of H2 in binary clathrates is lowered with respect to pure hydrogen hydrates. Simulation studies have been carried out to explore the possibility of loading multiple H2 into each clathrate cage.22−24 Generally, 512 cages only accommodate one H2, but some studies suggest that it is possible to load two H2 per 512 cage.23,24 The 51264 cages can accommodate up to four hydrogen molecules,22,25 but in the binary clathrates they are mostly, if not fully, occupied by THF or another large promoter. Strategies to increase the load of hydrogen in sII binary clathrates focus on replacing the large promoter by several small H2 molecules in the 51264 cages. An experimental study reported that loading of H2 in H2− THF binary clathrates can be increased to about 4 wt % by growing the hydrates with a concentration of 0.15 mol % of THF, much lower than the stoichiometry of THF hydrate.13 This has been called “the tuning effect” in binary hydrates.9 Several subsequent studies were unable to reproduce the high loads reported in ref 13 and found a maximum load of 1% by

1. INTRODUCTION Clathrate hydrates are nonstoichiometric crystals composed of a hydrogen-bonded water network that encages small guest molecules, such as CO2, CH4, H2, or tetrahydrofuran (THF).1 With the exception of THF, these molecules have very low solubility in liquid water. However, in the solid clathrate hydrates, the concentration of these guests can be hundreds of times higher. Clathrates are usually stable under moderate pressures and temperatures, and transportation and storage of clathrates is not as hazardous as for compressed gases. These qualities of clathrates make them good candidate materials for storage and transport of natural gas and hydrogen.2−4 The maximum loading of H2 in laboratory synthesized clathrate hydrate has reached 5.3 wt %;5−7 however, such synthesis requires extreme pressures as high as 2 kbar,6−8 which poses a high demand on the materials of reactors and high consumption of energy to create and maintain such harsh conditions. One approach to avert this drawback is to use promoter molecules to assist in the formation of clathrate hydrates at lower pressures.9−18 THF is the most commonly used promoter for synthesis of H2 clathrates.10,13,15,19 The formation of clathrate hydrates with other promoters, such as tert-butylamine, cyclobutanone, furan, and methane, has also been investigated.17,18,20,21 Recently, methane has been proposed as a promising alternative to THF, as it can compete with H2 for the small cages, resulting in higher loads of hydrogen.17 The water cages in the clathrate synthesized with THF (or other promoter) and hydrogen contain both types of guests. Clathrates with two species of guest molecules are © 2014 American Chemical Society

Received: May 17, 2014 Revised: July 25, 2014 Published: September 8, 2014 23022

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tuning the concentration of THF.10,14,26−30 A central question addressed in these studies is whether and to which extent H2 can replace THF in large cages of sII clathrates. The formation of methane clathrates with promoter molecules, such as CO2, propane, etc., has also been vigorously studied.31−34 Methane clathrates are a potentially very important energy resource, due to their large natural abundance on the permafrost and ocean floor.1,35 Molecular simulations have shown that addition of 1% propane could shift the most stable structure of methane clathrates from structure I to structure II,34 and experiments have shown that the 2:1 methane to propane ratio efficiently converts ice particles into clathrates.31 These studies have mainly focused on the thermodynamic stability and occupancy of binary clathrates. Molecular simulations can complement experimental methods and offer a detailed picture of molecular processes and transient states that are still beyond the resolution of experimental studies. However, while the growth of single clathrates has been studied to date using molecular simulations,36−56 to our knowledge there have not been studies of the growth of binary clathrates using molecular dynamics simulations, although there has been a noteworthy study of replacement of CH4 by CO2 at the surface of already formed methane hydrate.57 Here we report on the molecular modeling of the growth of binary clathrates of small and large guest molecules that roughly mimic the sizes of H2 and THF and their respective preferences for 512 and 51264 cages. However, unlike H2 and THF, the single clathrates of the small and large guests of this study have the same melting temperature (Tm). The binary clathrates they form are significantly more stable than the two single clathrates. In contrast, H2 clathrates cannot exist above 144 K at 10 kPa,7 while the melting temperature of THF hydrate under ambient pressure is 277.4 K,58 and H2− THF clathrates are only slightly more stable than THF clathrates with Tm = 279.6 K at 50 atm.16 Identical melting temperatures of our model single clathrates also ensure that we do not bias or favor the growth of either single clathrate below their Tm. The small and large guests of this study also have fairly high solubility in water, which enables them to grow clathrate hydrates within time scales accessible to molecular dynamic simulation studies. We compare and contrast the thermodynamics and growth of the binary clathrates with the single clathrates of each of the two guest molecules, and we investigate how the occupancies of the small and large cages in the binary sII clathrates respond to tuning of the growth temperature. We show that the large guest molecules not only add thermodynamic stability to the binary clathrates but also speed up the uptake of the small guests. Our results suggest that the rate-limiting step of the growth of the binary clathrates is the arrangement of the guests in the large 51264 cages at the growing crystal/liquid interface. We find that, for the model guests of this study, the loading of the small guest increases with the driving force for clathrate growth.

E=

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

j>i

i

j≠i k>j

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

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

(1)

where rij is the distance between particles i and j, and θijk is the angle subtended by vectors between the positions i−j and i−k pairs of particles. The parameters, A = 7.049556277, B = 0.6022245584, γ = 1.2, a = 1.8, and θ0 = 109.5°, were adopted from the SW potential, and the tetrahedral parameter of water λ = 23.15, the characteristic size σ = 2.3925 Å, and the energy scale ε = 6.189 kcal/mol were specifically parametrized to reproduce water properties.59 The triplet contribution enforces “hydrogen-bonded” configurations through short-ranged interactions: there are no long-ranged electrostatic interactions in the mW model. The mW model is able to reproduce the density and enthalpy of water, as well as the radial and angular structure and anomalies of liquid water.59 mW water has been used extensively to study hydrophobic interactions of methane in water, the stability, nucleation and growth of clathrates hydrates, and cross nucleation between different hydrate crystals.41,42,52,55,56,61−67 The two guest molecules of this study are the “soluble small” (SS) and “extra large” (XL) united particle solutes. Their interaction potentials have the same functional form as the ϕ2 term of the mW model (eq 1) with different values of parameters εij and σij. The SS and XL guests do not form hydrogen-bonded interactions among themselves or with water, so there are no three-body interactions involving SS and/or XL (i.e., the corresponding λ in eq 1 is 0). The parameters of the XL molecules were determined in ref 62. The small guest SS was parametrized in this work. As the SW potential is very soft, we designed the SS−SS interactions to have a relatively large value of σ to avoid multiple occupancies in small cages. The interacting parameters εSS−XL and σSS−XL are 0.05 kcal/mol and 8 Å, respectively. The weak interaction between SS and XL reduces the mixing of SS and XL to a negligible level, to emulate the low solubility of H2 in THF (which is only 2.70 × 10−4 mol % at 298.15 K and 1 bar68). B. Simulations and Analysis. Molecular dynamics simulations were performed with the massively parallel molecular dynamic simulation code LAMMPS.69 The velocity Verlet algorithm was used to integrate the equations of motions with a time step of 10 fs. All simulations were performed at a pressure of 500 atm. The simulations were evolved in the NpT ensemble keeping the temperature and pressure with the Nosé−Hoover thermostat and barostat with damping constants of 5 and 25 ps, respectively. The simulation cells were periodic in the three Cartesian dimensions. The solubilities of SS and XL in liquid water were computed following the procedures in ref 56. The simulation cells were initially built with two neighboring phases, a water phase and a guest molecules (SS or XL) phase. In both cases the simulation box had a size of about 200 × 75 × 75 Å3. The number of water molecules was 20 807, and the number of guest molecules (SS or XL) was 6856. The system was subsequently equilibrated in separate simulations at 500 atm and T = 304, 320, and 333 K, to allow the dissolution of guest molecules in water to reach

2. MODELS, SIMULATIONS, AND METHODS A. Models. We model water with the short-ranged coarsegrained monatomic water model mW.59 The mW model has the form of the Stillinger−Weber (SW) potential60 and consists of a sum of pair and triplet contributions as shown in eq 1 23023

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cages. Then, at the beginning of the simulations, an energy minimization was performed to avoid the overlap of guest molecules. Finally, 50 ns long simulations were performed at 300 K and 500 atm to obtain average values of the overall potential energy of the unit cell. For the empty sII clathrates the simulations were performed at 240 K, 12 K below its melting temperature.61 The potential energy of a unit cell minus the one of the empty unit cell was then normalized by the number of water molecules (136) to give the average potential energy per water molecule. The growth rates of the three clathrates were computed at two degrees below their respective Tm to assess the intrinsic growth rate, which following ref 41 we define as the growth rate at small thermodynamic driving force, 2 K below the melting point. We calculated the growth rate as the total number of newly grown cages divided by the elapsed growth time and normalized by the area of the clathrate/liquid interface. The XL clathrates and XL−SS clathrates were grown for 50 ns and the SS clathrates for 70 ns. Using the visualization software package VMD,70 we investigated the microscopic steps involved in the growth at the interface of XL−SS clathrates. We studied the effect of the growth temperature on the occupancy of the binary clathrates by growing the XL−SS clathrates at four different temperatures: 304, 312, 320, and 333 K. Five configurations of equilibrated XL−SS clathrate/ solution/guest fluid simulation cells were used as starting points for five independent simulations of growth of the binary clathrates at each temperature. The simulation cells were equilibrated just above their melting temperatures for 50−90 ns keeping the positions of the water molecules in the clathrate phase fixed to avoid melting. We show in section A of the Results and Discussion that the solubilities of the two guests do not display a strong dependence with temperature in the range of this study, hence we use the same starting concentrations of the guest molecules in the liquid phase (the solubility at 333 K in the presence of the two guests, 1:10.7 for SS and 1:17.6 for XL) for the growth of the double clathrates at different temperatures. Growth simulations were run for 50 ns at the three lower temperatures and for 70 ns at 333 K. The occupancies of 512 and 51264 cages of the newly grown regions were computed using the previously mentioned code.

equilibrium. Additional 50 ns long simulations at each of these three temperatures were used to compute the solubility of SS and XL in water from the average of the density profile of water and guest in the aqueous phase. We study single clathrates of SS and XL guests and binary and XL−SS clathrates. The stable crystal phase for all these hydrates is sII. The SS clathrate system consisted of 1989 SS guest molecules and 10 971 water molecules in a box of size of 165 × 50 × 50 Å3. First, a sII clathrate unit cell with a single guest in all 512 and 51264 cages was replicated by 9 × 3 × 3 in the three dimensions. Then, a region from 38 to 165 Å in the x direction was melted, and the molecules in the region from 117 to 165 Å were converted to SS guest molecules. In the end, the simulation cells used for the growth of the SS clathrate contained three phases: the solid clathrate, aqueous solution of SS, and a gas phase of SS. The XL clathrate was created by replicating a unit cell of sII clathrates with a single guest in 51264 cages by 6 × 3 × 3 in the three dimensions. Then, the region from 20 to 102 Å was melted to create a liquid phase. A total number of 7344 water molecules and 432 XL molecules were in a periodic simulation box of size 102 × 51 × 51 Å3. As the concentration of XL in the clathrate is lower than its solubility in liquid water (see section A of results), there was no XL gas phase in the simulation cells used for growth of the XL single clathrates from solution. The simulation cell containing XL−SS clathrate had three phases: a clathrate phase, a liquid phase, and a gas phase composed of SS and XL. The simulation cells used for the growth of the XL−SS clathrate were created in the same fashion as those for the SS clathrate system. The box size of the XL−SS binary clathrate was 179 × 69 × 69 Å3. The number of water, SS, and XL molecules was 17 462, 3030, and 2368, respectively. Note that the SS clathrates and XL−SS clathrates in our simulations can grow in only one direction along the xaxis due to blocking by the gas phase in the other direction. XL clathrates can grow in both directions of the x-axis because of the crystal interfaces the aqueous solution on the two sides. The melting temperatures Tm of the SS and XL−SS hydrates were determined through the direct coexistence method, previously used to compute the Tm of the XL clathrate.62 Simulation cells containing SS clathrates or XL−SS clathrates in equilibrium with the corresponding solution and gas phases were evolved at different temperatures to establish whether the clathrate crystal grew (which means T < Tm) or melted (i.e., T > Tm). The aqueous concentrations of SS in the simulation cells for the growth of single SS and binary XL−SS hydrates were 1:10.4 and 1:10.7, respectively. The aqueous concentration of XL guests in the simulation cells for growth of XL−SS clathrates was 1:17.6. In ref 62, the Tm of XL clathrates was determined in equilibrium with an aqueous solution of XL with concentration 1:17. The occupancy of 512 and 51264 cages of the three clathrates at Tm was determined using a code based on the cage search algorithm of ref 61. The code first identified the four types of cages common in the clathrate hydrates, 512, 51262, 51263, and 51264 cages, and then calculated the center of a cage. If a guest molecule was found within a cutoff distance from the center of a cage, it was counted as being inside the cage. The cutoff distances for the four types of cages 512, 51262, 51263, and 51264 were 3.3, 4.0, 4.1, and 4.2 Å, respectively. The occupancy of a cage is the number of guest molecules inside the cage. We calculated the potential energy of the unit cell of sII clathrates, when empty and with various occupancies of SS and XL molecules. To obtain the unit cell with different occupancies, guest molecules were placed at the center of the

3. RESULTS AND DISCUSSION A. Thermodynamic Stability of Clathrate Hydrates. In this section we characterize thermodynamic properties, such as the melting temperatures and free energy of formation, of the single and binary clathrates of the small SS and large XL guests, as well as the solubility of these molecules in water. The interaction parameters involving the two guest molecules are listed in Table 1. The small guest SS was designed to mainly stabilize the 512 cages and only marginally stabilize the 51264 cages (Figure 1). The XL guests can only fit into 51264 cages; it is prohibitive for them to be incorporated into the small cages Table 1. Guest−Guest (εss, σss) and Guest−Water (εws, σws) Interaction Parameters

a

23024

guest

εss (kcal/mol)

σss (Å)

εws (kcal/mol)

σws (Å)

large (XL)a small (SS)

0.34 0.12

4.50 4.60

0.36 0.183

4.50 3.25

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Table 2, were obtained by growing the clathrates at 2 K below their Tm, conditions for which there are no mass transport limitations (see section B) to clathrate growth. Note that SS and XL single clathrates achieve identical Tm with quite distinct occupancies of the cages: the small 512 cages are mostly empty in the XL hydrate and single occupied SS clathrate, while the large 51264 cages are mostly single occupied in the XL clathrate and half of them doubly occupied in the SS clathrate, with the other half divided among single occupied (∼40%) and triply occupied (∼10%). The 512 cages of the equilibrium XL− SS binary clathrate are exclusively occupied by the small guest. The 51264 cages, however, do not have only large guests: more than 10% of the 51264 cages of the equilibrium binary hydrate are multiply occupied by small guests. To further characterize the difference in thermodynamic stability of the single and binary clathrates, we calculated the potential energy of the sII clathrates with various occupancies of small and large guests, relative to the one of the empty clathrate and normalized by the number of water molecules (Table 3). The stabilization energy of the binary clathrate was

Figure 1. Cage−guest interaction energy E normalized by the strength of the water−solute attraction εws and the number of water molecules in the cage nw, as a function of size of the guest σws, computed in ref 62. The green and orange lines represent the energy of guests in the 512 and 51264 cages, respectively. The dashed lines indicate the sizes of the small guest SS and the large guest XL.

(Figure 1). The most stable crystal for the XL and SS single clathrates is sII.56,62 The solubility of the large guest in water at 320 K is 1:13 in guest:water mole ratio, higher than the concentration of the large guest XL in the single hydrate (1:17) but lower than the full miscibility of THF in water. The small SS guest molecule is similar to H2 in its small effective size; however, the solubility of SS in water at 300 K and 500 bar is 1:10, drastically higher than the solubility of H2 in water. The high solubility of SS enables the growth of clathrates in time scales accessible with molecular dynamic simulations. The solubility of XL and SS in water increases slightly on cooling, which implies that the entropy of hydration of these solutes is negative (as is also the case for hydrogen, methane, and other gases). For SS the solubility is 1:10.7 at 333 K, 1:10 at 320 K, and 1:9 at 304 K and for XL is 1:13.4 at 333 K, 1:13 at 320 K, and 1:13 at 304 K. For solutions containing both guestsas is the case for the growth of double clathratesthe solubility at 333 K is 1:10.7 for SS and 1.17.6. The large guest becomes less soluble in water in the presence of the small guest. The solubility of the large guest in the mixture is almost identical to the 1:17 XL to water ratio in single XL hydrates. By fine-tuning the water−SS interaction strength, we ensure that the melting temperatures Tm of SS and XL single clathrates are identical and equal to 307 K. The XL−SS clathrates have significantly enhanced thermodynamic stability, Tm = 335 K, compared to the two single clathrates (Table 2). The equilibrium occupancies of the clathrates, also shown in

Table 3. Difference in Potential Energies of sII Hydrates with Different Occupancies and the Empty sII Clathrate occupancy of 512 occupancy of 51264 Ep − Epempty (kcal/mol water)

1 × SS 1 × SS −0.24

1 × SS 2 × SS −0.32

empty 1 × XL −0.26

1 × SS 1 × XL −0.63

larger than the sum of the stabilization of a clathrate with only large cages occupied by XL and a clathrate with only the small cages occupied by SS. The additional stabilization may be due to a decrease in low-frequency modes of the water network when all cages are occupied.71 We find that the energies associated with single and double occupancy of the large cages in SS hydrates are comparable, consistent with the actual occupancies shown in Table 2. The unit cell with single occupancy of SS in the 512 cages and single occupancy of XL in 51264 cages had the lowest energy, in agreement with the higher stability of XL−SS clathrates as measured by their melting temperature. However, even though full occupancy of the 51264 cages by the large XL guest would result in the lowest potential energy, there is a sizable fraction of SS molecules in the large cages of the binary hydrate grown under close to equilibrium conditions. The free energy of formation ΔGf(T) of the single and binary clathrates is a measure of the relative stability of these crystals with respect to the solution from which they form. We

Table 2. Melting Temperatures of the Clathrates and Occupancy of the Clathrates Grown Two Degrees under Their Respective Tm empty sII

a

XL single

SS single

XL−SS binary

Tm

252 Ka

307 Kb

307 Kb

335 Kb

occupancy of 51264

100% empty

5% empty 94% single

3% is empty 41% single 47% double 9% triple

87% single XL 9% double SS 4% single SS

occupancy of 512

100% empty

>99% empty

3% empty 95% single 1% double

92% single SS 3% double SS

At 1 atm. bAt 500 atm. 23025

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estimated ΔGf(T) of the three clathrates over a range of temperatures using the melting temperature (at which the corresponding ΔGf(T) = 0) and the entropy of melting ΔSm = −ΔSf, which is the slope of the free energy of formation with temperature. Assuming that the entropy of melting is independent of temperature, the free energy of formation can be written as ΔGf(T) = (T − Tm)ΔSm. The ΔSm of the single and binary clathrates was calculated under two assumptions. First, we assume that water and guests contribute independently to the entropy of melting. The entropy melting per mole of water is 18.5 ± 0.3 J/K for the sII guest-free clathrate at 1 bar,61 and we approximate that it does not change with pressure because (∂S/∂p)T = −(∂V/∂T)P = VβT and both liquid water and the clathrate crystal have similar, and small, molar volumes V and isothermal compressibilities βT. The entropy of melting for the hydrate of a monatomic guest that is slightly smaller than XL (4.3 vs 4.5 Å) and with the same water−guest interaction strength is 21.3 ± 1.7 J/K at 500 bar;41 it has been previously shown that the entropy of melting of clathrates of monatomic guests is quite insensitive to variations of the size and interactions of the guest with water.56 We consider that the guest in the large cage contributes (21.3 ± 1.7−18.5 ± 0.3) J/K = 2.8 ± 2.0 J/K per mole of water to ΔSm. Second, we assume that the entropy of melting contributed by a guest molecule is independent of the type of cage it occupies, the species of the guest molecule, and the coexistence with other guest molecules in the cage. In support of our assumptions, we note that De Leeuw and co-workers computed the entropy of binary sII clathrates with different promoter molecules in 51264 cages using Monte Carlo simulations and found it to be quite insensitive to the species of the promoters in the large cages.21 As the ratio of 512 cages to 51264 cages is 2:1, guest molecules singly occupying the 512 cages will contribute 5.6 ± 4.0 J/K per mole of water. On the basis of occupancies listed in Table 2, the entropy of melting for XL−SS clathrates, SS clathrates, and XL clathrates is estimated to be 26.9 ± 6.1, 28.5 ± 7.1, and 21.1 ± 1.9 J/K per mole of water. We neglect the increase in configurational entropy that arises from a combination of different cage occupancies. The contribution of the entropy of mixing of guests in binary clathrates (ΔSmix = −R/17(xA ln xA + xB ln xB) if the mixing of the guests in the cages were ideal) further stabilizes the binary clathrates, but its contribution is very small (0.25 J/K per mole of water) and can be neglected. With the results obtained under these assumptions, we compute the temperature dependence of the free energy of formation of the three clathrates. Figure 2 shows that the binary clathrate is the most stable phase at all temperatures. This is a robust result because of the large difference in melting temperature between the binary and single clathrates. A hypothetical SS-richer binary clathrate would have a higher free energy than the one with the equilibrium composition of Table 2. Nevertheless, clathrates with a higher fraction of small molecules in the large cages (replacing the large molecules) than the equilibrium binary hydrate could still grow at temperatures for which they are more stable than the mother solution (i.e., when their ΔGf(T) < 0). In summary, the small and large guests of this study have significant solubility in water and form single sII clathrates with equal thermodynamic stability but quite different occupancy of the cages. The equilibrium binary clathrate of these two guests has a 28 K higher melting temperature than the two single clathrates and has mixed occupancy of the large cages: 87% are singly filled with the large molecule, and 13% are multiply filled

Figure 2. Free energy of formation of the clathrates with various occupancies of large (XL) and small (SS) molecules. The free energy is normalized per mole of water molecules. The red, black, and green lines are, respectively, the ΔGf of SS clathrates, XL clathrates, and XL− SS clathrates based on the occupancies in Table 2. The two black circles indicate the Tm of the three clathrates. The blue line represents a hypothetical binary clathrate richer in SS than the binary clathrate of Table 2, with more 51264 cages occupied by SS. The equilibrium binary XL−SS clathrate (green line) does not have all large cages occupied by the large guest but 9 ± 1% of the large cages occupied by two SS and 4 ± 1% by single SS guests, as shown in Table 2. The slopes of the curves are the corresponding entropies of melting of the clathrates. The dashed lines indicate the error bars in the free energy of formation of the clathrates, obtained from the errors in the entropies of melting. Note that even accounting for the uncertainty in the entropies of melting the double clathrate is more stable than the single clathrates in all the temperature ranges of this study.

with the small guest. The binary XL−SS clathrates have higher stability than the single clathrate for all the temperatures of growth to be considered in this study, but metastable SS-rich binary clathrates could also form in the temperature range for which they are more stable than the liquid phase. B. Growth of Single and Binary Clathrates. In the previous section we focused on the thermodynamics of the XL−SS binary clathrates and the two single clathrates. Here we discuss the kinetics and molecular steps involved in the growth of these crystals. First, we grew the three clathrates at 2 K under their respective Tm, the highest temperature at which their growth rate is sufficiently fast to be measured from ∼100 ns simulations. Following ref 41 we call the growth rates obtained under very low and equivalent thermodynamic driving force the intrinsic growth rates of the clathrates. The initial configurations of the three simulation cells are shown in Figure 3. In all cases, the fastest growing face of sII clathrates, the [100] crystal face, is exposed to the liquid solution.50 The time evolution of number of cages per interfacial growing area of the single and binary clathrates grown at 2 K below their Tm is shown in Figure 4. XL clathrate has the highest growth rate among the three. The growth rate of SS clathrate is the lowest, only about 10% of the one of XL clathrate and 20% of the rate of XL−SS clathrate. The very slow growth of the SS clathrate advances through cycles of formation of new cages and partial melting, while both XL and XL−SS clathrates grow at an almost steady rate. We computed the growth rate from the slope of the time evolution of numbers of cages, normalized by the area of the crystal/solution interfaces. The rates at Tm − 2 K are 5.3 cages μs−1 Å−2 for XL clathrate, 2.6 cages μs−1 Å−2 for the XL−SS binary clathrate, and 0.46 23026

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Figure 3. Snapshots of the periodic simulation cells containing the initial configurations of the three clathrates: (a) SS clathrate, (b) XL clathrate, and (c) XL−SS clathrate. On the right of each cell is a solid clathrate phase (green lines represent the water−water bonds in the crystal; guest molecules inside the clathrate are not shown). Water molecules in the liquid phase are shown with cyan balls, SS guests with red balls, and XL guests with orange balls. The arrows above the cells denote the direction in which the clathrates grow. In all cases, the [100] face of sII is exposed to the liquid phase.

Figure 4. Kinetics of clathrate growth, as measured by the difference in the number of cages at time t and at the beginning, Δcages, normalized by the geometric area of the crystal/solution interface. The clathrates are grown at 2 K below their melting temperatures. The growth rates of SS, XL, and XL−SS clathrates are 0.5, 5.6, and 2.6 cages μs−1 Å−2, respectively. The green, blue, and red dash lines are the slopes used for calculation of the growth rates of SS, XL, and XL−SS clathrates. (a) The number of cages in the SS clathrate increases slowly and is subject to strong fluctuations. Some SS clathrate cages dissolve after their formation. (b) The number of cages of XL clathrates rises at an almost constant rate until the growth is complete at ∼30 ns. (c) The growth of SS−XL binary clathrates is also steady in the 50 ns simulation.

cages μs−1 Å−2 for the SS clathrate (the latter is only an estimate, as the growth does not show a steady slope). The very low intrinsic growth rate of the SS clathrate may be attributed to its high fraction of double occupancies in the 51264 cages. As Table 2 shows, nearly half of the large cages have double occupancy. Arranging two small molecules on top of 51264 sites may take a long time because it is entropically disfavored and the direct attraction between two small guests is very small (see Table 1). The crystallization and melting of interfacial cages observed during the growth of SS clathrate (Figure 4) may have the function of opening the 51264 cages with single occupancy to accommodate more guest molecules. Although binary clathrates have a lower concentration of small guests than pure SS clathrate, the difference in growth rates between these crystals is so large that the rate of uptake of the small guest in the binary clathrates is about 5.6 times higher than the rate of uptake in SS clathrates. We conclude that the large guest molecule is a good kinetic promoter for the enclathration of the small guest. Mass transport of guest molecules to the crystal/solution interface and adsorption of guest molecules at that interface influence the kinetics of growth of clathrates.48,49 The growth

rates of the single XL and binary XL−SS clathrates in Figure 4 are relatively constant; they do not level off toward the end of the simulations. This implies that, not unexpectedly, growth of the clathrates close to equilibrium is not limited by mass transport of guest molecules to the crystal/solution interface. Mass transport plays a role, however, in limiting the growth at high supercooling. For example, at 310 K the binary clathrate completes the growth (i.e., crystallizes all the water in the solution) in 19 ns. The average rate of growth of the crystal in the first 9.5 ns was ∼55% higher than in the second half of the growth. Through all the process of growth at 310 K, the concentration of the large guest in the solution was the same or larger than the concentration in the clathrate; the mass transport limitation arose from a decrease in the concentration of SS, which was lower in the solution than in the binary clathrate. The concentration profile of SS in the aqueous solution was flat (diffusion of SS in the solution is fast compared to the rate of growth of the crystal), and there was a 23027

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layer of XL fluid between the solution and the reservoir of SS fluid (Figure 3(c)). This suggests that the slow transport of SS through the XL fluid “barrier” is responsible for the decrease of concentration of SS in the solution and the relative slow down of the growth rate of the binary clathrates at high driving force. The small guest SS shows preferential adsorption to the clathrate/solution interface. The equilibrium density of SS in the 7 Å layer closer to the clathrate surface is 60% higher than the density of SS in the aqueous solution, albeit still lower than the density of the small guest in the single or the binary clathrate. The large guest XL does not adsorb at the clathrate/ solution interface. Nada studied the growth of THF clathrate and suggested that the arrangement of THF molecules is the limiting factor of kinetics of clathrate growth when the concentration of THF in aqueous phase is comparable to its concentration in the clathrate crystal.50 If incorporation of the large guest were the limiting step in the growth of binary clathrates, the intrinsic growth rate of the binary XL−SS the single XL clathrates would be the same. However, the XL single hydrate grows twice as fast as the binary clathrate: the enclathration of the small guest plays a role in limiting the rate of growth of the crystal. To further understand the factors that control the growth rate of XL−SS clathrates, we examined the microscopic steps involved in the growth of the [100] plane of the sII binary hydrate. Figure 5 shows side and front views of a ∼2 nm wide slab of water and guest molecules as it evolves from being just at the surface of the clathrate, to be incorporated into it as the crystal front advances. Only guest molecules within 2 Å of the middle of the slab are shown, to illustrate how the guests accommodate to form a new layer of clathrate. When clathrate cages are just emerging into the slab, the small and large guest molecules are quite randomly distributed in the liquid slab in contact with the clathrate (Figure 5(a)). As growth progresses, large guest molecules begin to arrange into the sites where 51264 cages will grow (these sites are specified by the underlying lattice). However, there are still a large number of SS molecules blocking the large cage sites (Figure 5(b)). When the growth of the water clathrate layer is mostly complete, all XL molecules at the crystal surface are arranged on top of 51264 sites. Some 51264 sites at the surface display single occupancy or double occupancy by the small molecules. Because SS is small enough to fit into both 512 cages and 51264 cages, as growth advances, these small guests arrange into both 512 and 51264 sites. However, XL can only fit into 51264 cages, driving SS away from 51264 sites to 512 sites. Our analysis indicates that, when concentration of guest molecules in the solution is comparable to their concentration in the clathrate and the driving force for growth is small, the rate of growth of binary clathrates is limited by the arrangement of the guest moleculeslarge and small into the large cage sites. Our results agree with the analysis by Nada for the growth of THF sII clathrates in atomistic simulations50 but incorporate the effect of the small guest. The lower intrinsic growth rate of binary clathrates compared to XL clathrates indicates that growth of binary clathrates close to equilibrium is limited by the ordering of both large and small guests in the 51264 cages. The same elementary steps of growth of the binary clathrate are observed in simulations close to equilibrium (333 K) and at high supercooling (310 K). However, next section shows that when the clathrates are grown at high driving forces, water can form the network of cages around guests with a higher tolerance for small guests in the large cage sites, resulting in an

Figure 5. Growth of a new layer of binary clathrate. The panels display side and front views of three configurations along a growth trajectory for the binary clathrate at 25 K below its melting point. The slab is 2 nm wide and contains water (connected by green lines), a small SS guest (red balls), and a large XL guest (gray balls). It should be noted that there are guest molecules throughout the water layer, but only the guest molecules in the middle of the water slab are shown, to make the progression of their arrangement clearer to the viewer. The snapshots are from a simulation at 310 K, and the same molecular steps and qualitative pictures are observed at 333 K, although with a lower number of small guests trapped in the large clathrate cages after the new layer of the crystal completes its growth.

enhancement of the loading of small guests in the binary clathrate. C. Occupancy of Binary Clathrate Hydrates Grown at Different Temperatures. We investigate the effect of growth temperature on the occupancies of the two guest molecules in the small and large cages of sII clathrates, to explore the possibility of tuning the loading of guest molecules through the growth conditions. The distribution of occupancies of newly grown 512 and 51264 cages is shown in Figure 6 for four growth temperatures. As expected from the free energies of formation shown in Figure 2, binary clathrates grow at all temperatures, even below the melting temperature of the single clathrates. The occupancies of 512 cages are relatively insensitive to the change of growth temperatures. More than 90% of the 512 cages have single occupancy of SS across all four temperatures, and a small percentage of the small cages have double SS occupancy while the other ∼5−10% of the cages are empty. In contrast, the large 51264 cages are essentially fully occupied at all temperatures and show a clear trend of change in occupancy with lowering growth temperatures. The most significant change in the occupancies of 51264 cages occurs between 320 and 333 K. 13% of the 51264 cages grown at 333 K are occupied by small molecules, while more than twice that amount, 28%, of the large cages grown at 304 K are occupied by SS molecules. 23028

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scales of growth of the clathrate and the time scale of diffusion of the molecules to the crystal surface and adsorption to the clathrate/solution interface would determine whether an excess concentration of small guest could be sustained along the growth process. In the case of binary clathrates of hydrogen and a soluble promoter, the solubility of the small guest in water would be so small that it may be expected that fast growth would not result in a significant enhancement of H2 loading.

4. CONCLUSIONS This work presents, to our knowledge, the first study of the growth of binary clathrates using molecular simulations. For a binary clathrate to form, it has to be thermodynamically more stable than the two single clathrates. In our model, as well as in the case of H2 with THF and other large promoter molecules, the enhanced stability of the binary clathrates is achieved by stabilization of the large cages and small cages selectively choosing the sizes of guest molecules. The small and large model guests of this study loosely mimic the sizes of H2 and THF, but not their solubility in water or the relative stability of their single hydrates. We have shown that the large guest is able to speed up the growth of clathrate hydrates compared to the rate of growth of clathrates with only the small guest. This results in a significant increase of the rate of enclathration of the small guest molecule. We conclude that the large guest is not only a thermodynamic promoter but also a kinetic promoter for the storage of the small guest in the clathrate crystal. We studied the mechanism of the growth of the binary clathrates and found that, when the concentration of two guest molecules in the solution is comparable to the concentration in the clathrate, the rate-limiting step is the arrangement of the large and small guest molecules at the growing crystal interface. Lower growth rate of the binary clathrates compared to clathrates only with large guest molecules suggests that the small molecules act as obstacles for the arrangement of the large guests in the 51264 cages. Our results indicate that the growth rate of binary clathrates is limited by the ordering of both large and small guests in the 51264 cages. The large molecules only fit into 51264 cages; small guests fit in both cages, but the loading of multiple small guests in the large cages is disfavored by entropy and the very weak attraction between guests. Different from the small guest of our study, the solubility of H2 in water is extremely low, even at high pressures. The growth rate of binary clathrates involving H2 and soluble promoters may be limited by the access of H2 to the growing interface. The simulations indicate that binary clathrates grown at high thermodynamic driving forces, achieved by high supercooling, do not have the composition of the most stable crystal. Clathrates with metastable occupancy, with an excess load of small molecules, can form if they are more stable than the liquid solution and grow faster than the stable clathrates. For the guests of this study, the occupancy of the small guest molecules in the large cages was increased by lowering the temperature, while the occupancy of the small cages was not sensitive to the growth temperature. Therefore, the overall loading of the small guests in the binary clathrates was enhanced by growing the crystals at high supercooling, at the expense of load of large guest molecules. The selection of the sizes of the small and large guest of this study was inspired by H2 and THF. There are, however, two key differences between SS−XL clathrates and H2−THF clathrates. First, the small guest SS is much more soluble

Figure 6. Occupancy of newly grown 512 and 51264 cages by the small SS and large XL guests at four temperatures. For each growth temperature, occupancy of the 512 cages is shown in the left column and occupancy of the 51264 cages in the right column. The growth temperature is labeled below each pair of columns. The colors denote the type of occupancy: orange indicates single XL occupancy, green double occupancy of SS, and blue single occupancy of SS. The percentages presented in the figure are averages over new clathrate cages grown over five independent growth simulations at each temperature; the standard deviation is less than 3% for any temperature, guest, and cage type.

The increase of small guests in the large cages occurs at the expense of the large molecules. Overall, considering both the 512 and 51264 cages, the loading of the small guest in the binary clathrates grown at 304 K is 7% higher than at 333 K. We conclude that, for the pair of guests of this study, the clathrate at low temperatures can increase the load of the small guest. A comparison of the occupancies of Figure 4 with the stabilities of Figure 2 shows that the guest composition of the clathrate grown at high supercooling does not correspond to the most stable crystal. Closing the water cages around small solutes is thermodynamically less favorable than enclosing large solutes, but it still results in a decrease in free energy with respect to the deeply supercooled solution. We note that the enhancement of loading of small guests in Figure 6 is probably a lower bound because analysis of the density profiles of guest and water molecules along the axis of growth of the crystal (i.e., the axis perpendicular to the [100] face of sII) reveals that the concentration of the small guest in the clathrates is higher in the early stages of growth and decreases slowly but steadily as growth proceeds. This decrease in the concentration of small guest in the crystal reflects the overall decrease in concentration of small guest in the aqueous solution as the crystallization advances in the finite simulation cell. The kinetic barrier that the layer of XL imposes to the transport of SS from the fluid guest reservoir to the solution is the main culprit of the depletion of SS in the solution. Guest depletion would not existor be of little importanceif the mass of solution were much larger than the mass of newly grown clathrate. For other combinations of guests, we also expect that clathrates with metastable occupancies, richer in the smaller guest, could grow under high supercooling if the concentration of the small guest in the solution were comparable to the concentration in the clathrate. In those cases, the competition between the time 23029

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(8) Patchkovskii, S.; Tse, J. S. Thermodynamic stability of hydrogen clathrates. Proc. Natl. Acad. Sci. U. S. A. 2003, 100, 14645−50. (9) Sugahara, T.; Haag, J. C.; Prasad, P. S. R.; Warntjes, A. A.; Sloan, E. D.; Sum, A. K.; Koh, C. A. Increasing hydrogen storage capacity using tetrahydrofuran. J. Am. Chem. Soc. 2009, 131, 14616−14617. (10) Strobel, T. A.; Taylor, C. J.; Hester, K. C.; Dec, S. F.; Koh, C. A.; Miller, K. T.; Sloan, E. D., Jr. Molecular hydrogen storage in binary THF-H2 clathrate hydrates. J. Phys. Chem. B 2006, 110, 17121−5. (11) Sabase, Y.; Nagashima, K. Growth mode transition of tetrahydrofuran clathrate hydrates in the guest/host concentration boundary layer. J. Phys. Chem. B 2009, 113, 15304−11. (12) Pefoute, E.; Kemner, E.; Soetens, J. C.; Russina, M.; Desmedt, A. Diffusive motions of molecular hydrogen confined in THF clathrate hydrate. J. Phys. Chem. C 2012, 116, 16823−16829. (13) Lee, H.; Lee, J.-W.; Kim, D. Y.; Park, J.; Seo, Y.-T.; Zeng, H.; Moudrakovski, I. L.; Ratcliffe, C. I.; Ripmeester, J. A. Tuning clathrate hydrates for hydrogen storage. Nature 2005, 434, 743−746. (14) Anderson, R.; Chapoy, A.; Tohidi, B. Phase Relations and Binary Clathrate Hydrate Formation in the System H2−THF−H2O. Langmuir 2007, 23, 3440−3444. (15) Strobel, T. A.; Koh, C. A.; Sloan, E. D. Thermodynamic predictions of various tetrahydrofuran and hydrogen clathrate hydrates. Fluid Phase Equilib. 2009, 280, 61−67. (16) Florusse, L. J.; Peters, C. J.; Schoonman, J.; Hester, K. C.; Koh, C. A.; Dec, S. F.; Marsh, K. N.; Sloan, E. D. Stable low-pressure hydrogen clusters stored in a binary clathrate hydrate. Science 2004, 306, 469−71. (17) Matsumoto, Y.; Grim, R. G.; Khan, N. M.; Sugahara, T.; Ohgaki, K.; Sloan, E. D.; Koh, C. A.; Sum, A. K. Investigating the Thermodynamic Stabilities of Hydrogen and Methane Binary Gas Hydrates. J. Phys. Chem. C 2014, 118, 3783−3788. (18) Veluswamy, H. P.; Chin, W. I.; Linga, P. Clathrate hydrates for hydrogen storage: The impact of tetrahydrofuran, tetra-n-butylammonium bromide and cyclopentane as promoters on the macroscopic kinetics. Int. J. Hydrogen Energy 2014, 39, 16234−16243. (19) Sugahara, T.; Haag, J. C.; Warntjes, A. A.; Prasad, P. S. R.; Sloan, E. D.; Koh, C. A.; Sum, A. K. Large-cage occupancies of hydrogen in binary clathrate hydrates dependent on pressures and guest concentrations. J. Phys. Chem. C 2010, 114, 15218−15222. (20) Prasad, P. S. R.; Sugahara, T.; Sum, A. K.; Sloan, E. D.; Koh, C. A. Hydrogen storage in double clathrates with tert-butylamine. J. Phys. Chem. A 2009, 113, 6540−6543. (21) Atamas, A. A.; Cuppen, H. M.; Koudriachova, M. V.; de Leeuw, S. W. Monte Carlo calculations of the free energy of binary sII hydrogen clathrate hydrates for identifying efficient promoter molecules. J. Phys. Chem. B 2013, 117, 1155−65. (22) Katsumasa, K.; Koga, K.; Tanaka, H. On the thermodynamic stability of hydrogen clathrate hydrates. J. Chem. Phys. 2007, 127, 044509. (23) Amano, S.; Tsuda, T.; Hashimoto, S.; Sugahara, T.; Ohgaki, K. Competitive cage occupancy of hydrogen and argon in structure-II hydrates. Fluid Phase Equilib. 2010, 298, 113−116. (24) Chattaraj, P. K.; Bandaru, S.; Mondal, S. Hydrogen storage in clathrate hydrates. J. Phys. Chem. A 2011, 115, 187−193. (25) Lokshin, K. A.; Zhao, Y.; He, D.; Mao, W. L.; Mao, H.-K.; Hemley, R. J.; Lobanov, M. V.; Greenblatt, M. Structure and Dynamics of Hydrogen Molecules in the Novel Clathrate Hydrate by High Pressure Neutron Diffraction. Phys. Rev. Lett. 2004, 93, 125503. (26) Hashimoto, S.; Sugahara, T.; Sato, H.; Ohgaki, K. Thermodynamic stability of H2 + tetrahydrofuran mixed gas hydrate in nonstoichiometric aqueous solutions. J. Chem. Eng. Data 2007, 52, 517−520. (27) Struzhkin, V. V.; Militzer, B.; Mao, W. L.; Mao, H. K.; Hemley, R. J. Hydrogen storage in molecular clathrates. Chem. Rev. 2007, 107, 4133−4151. (28) Mulder, F. M.; Wagemaker, M.; Van Eijck, L.; Kearley, G. J. Hydrogen in porous tetrahydrofuran clathrate hydrate. ChemPhysChem 2008, 9, 1331−1337.

than H2. Second, SS clathrates and XL clathrates have equal thermodynamic stability, while THF clathrates are more stable than H2 clathrates. By lowering the growth temperature of H2− THF clathrates, the growth of THF clathrates will be more favored compared to H2 clathrates due to the higher stability of the former and lower concentration of H2 in the liquid phase. Rapid uptake of H2 by clathrates would be needed to increase its loading; otherwise there would be sufficient time for the large guest to order properly in the large 51264 cages at the clathrate/solution interface, resulting in the growth of the more stable crystal. If uptake of hydrogen were fast, it would inevitably deplete the already very scarce H2 in the solution phase, resulting in a mass transport limitation that would slow down the incorporation of H2 into the clathrate and the overall rate of crystal growth. While the approach presented here for tuning the occupancy of binary clathrates has the merit of high growth rates compared to methods for tuning occupancy based on lowering the concentration of promoter molecules, it may not be applicable to the important case of binary clathrates of hydrogen. It could be applied, however, to tune the composition of binary clathrates of soluble guests. This study focused on the growth of clathrates of two guests, in which the two of them could be considered thermodynamic promotersbecause the stabilities of the single clathrates are identical−while only one of them, the large guest, is the kinetic promoter of growth. There are still important questions regarding the nucleation of clathrates in the presence of a promoter molecule, such as how the ratio of the promoter to gas molecules in the liquid affect their ratio in the clathrates and how the promoter can facilitate heterogeneous nucleation. These questions will be addressed in future studies.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge support by the National Science Foundation through award CHE-1012651 and technical support and resources from the Center for High performance Computing at the University of Utah.

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REFERENCES

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dx.doi.org/10.1021/jp504852k | J. Phys. Chem. C 2014, 118, 23022−23031