Cage Occupancies in Nitrogen Clathrate Hydrates from Monte Carlo Simulations

Cage occupancies in nitrogen clathrate hydrates from Monte Carlo simulations

Vincent Ballenegger

Institut UTINAM, Univ. Bourgogne-Franche-ComtÃľ, UMR CNRS 6213, 16, route de Gray, 25030 BesanÃğon Cedex, France

Abstract Their exploitation could be combined with the sequestration of carbon dioxide, in the form of Comparisons of Gibbs ensemble Monte Carlo CO2 hydrate on the seafloor, to help keeping the simulations with experimental data for the cage concentration of this global warming gas under occupancies in N2 clathrate hydrates are per- control.2 Nitrogen molecules were found to act formed to assess the accuracy of such simu- as a promoter in the N2âĂŞassisted CH4 CO2 lations, to refine the effective potentials em- exchange reaction.3 Since flue gas mixtures con- ployed, and to help interpret recently measured tain a mixture of N2 and CO2, a deep under- large cage over small cage occupancy ratios standing of mixed N2 CO2 clathrates, and also [Petuya et al., J. Phys. Chem. C 122, 566 pure N2 and CO2 clathrates as reference sys- (2018)]. Different sets of interaction potentials tems, is important for this technology. Air for N2 N2,N2 H2O and H2O H2O interac- clathrates, containing N2,O2 and other trace tions are considered. Some of them fail to re- gases, are found on Earth below ice caps at a produce the known experimental fact that some depth of 1000 m or more. They are of particu- large cages are doubly occupied at 273 K and lar interest as they contains relics of our ancient high pressures. The best agreement between atmosphere up to 420 kyr back. N2 clathrates simulations and experiments is obtained when may also be present in comets and in some plan- using a new N O interaction potential derived ets or their satellites of our solar system.4,5 in this work by averaging an ab-initio potential There exist 3 clathrate structures at low pres- energy surface for the N2 H2O dimer. sures: two cubic ones (sI, sII) and a hexag- 1 onal one (sH). N2 clathrates crystallize in a Introduction cubic structure: the structure I is kinetically

arXiv:1905.02453v1 [physics.chem-ph] 7 May 2019 favoured, while the sII is the thermodynami- cally stable structure if the pressure is not too A gas clathrate hydrate is a form of crystalline high (a transition to sH is observed at the high ice where the water molecules form a solid net- pressure 0.85 GPa).6–8 The unit cell of the sI work with small and large cavities that encap- crystal contains 46 water molecules forming 8 sulate guests.1 This molecular structure is sta- cages: 2 small and 6 large ones. The unit cell ble at high pressure and/or low temperature. of sII contains 136 water molecules forming 24 Methane (clathrate) hydrates have been found cages: 16 small cages and 8 large cages (these in natural gas pipelines, deep sea sediment and large cages are slightly bigger than in sI). Struc- permafrost regions. They are considered to be ture II is promoted by the ability of the small an enormous possible future energy resource.1

1 17,21 molecule N2 to fill in the numerous small cages using molecular dynamics (MD). van Klav- of this structure. eren et al. have shown also that a N2 hy- The cage occupancies θS and θL of small (S), drate with sII at high pressure remains mechan- respectively large (L), cages in a clathrate is an ically stable even for the case of a full double important quantity for practical applications: occupancy of the large cages.19 In those MD they determine the gas storage capacity (see simulations, the number of guest N2 molecules e.g. Ref.9 for hydrogen storage) and intervene is fixed beforehand and no attempt was made also in the calculation of phase equilibria in- to quantify the thermodynamical stability of volving clathrates.1,10 Occupancies in a nitro- a phase with full or partial double occupa- gen clathrate have been measured, at 273 K and tion. The Monte Carlo (MC) method over- pressures up to 1000 bar using powder neutron comes this difficulty because MC moves with in- diffraction and Rietveld refinement by Chazal- sertion/deletion of guest molecules allow the av- lon and Kuhs11 for both sI and sII. They discov- erage cage occupancy of the simulated clathrate ered that some large cages in sII can be doubly to converge automatically towards its value occupied at pressures above ≈ 300 bar. An ad- at thermodynamical equilibrium. A grand- ditional measurement at 1093 bar and 268.2 K canonical Monte Carlo (GCMC) calculation of in a sI nitrogen clathrate was performed by Qin cage filling in a N2 clathrate has been performed and Kuhs;12 it also shows some double occu- by Klapproth et al.,18 who found an unsatis- pancy of the large cages: θL = 111.9 ± 0.8% factory agreement: simulations strongly over- while θS = 98%. Yet another measurement in a estimated the occupancy when using the sim- sII clathrate at 150 bar and at a lower temper- ple point charge (SPC) water model.1 Patt et ature 258.15 K gave: θL = 121.8 ± 0.6% and al. performed GCMC simulations of N2, CO 13 θS = 96.7 ± 0.3%. Occupancy ratios θL/θS and mixed N2 CO clathrates but only at low have moreover been measured at pressures up temperatures (50, 100 and 150 K) and they did to 200 bar and for temperatures down to 150 K not compare quantitatively with experimental by Petuya et al.6 The latter authors used an- data.22 Their calculations suggest that experi- other technique, Raman spectroscopy, which ments performed on the single-guest N2 and CO does not provide separately the occupancies of clathrates might be sufficient to get information large and small cages, but only their ratio. on the corresponding mixed clathrates. The purpose of the present paper is to per- In the present work, the Gibbs ensemble form Monte Carlo simulations to determine the Monte Carlo (GEMC) technique is employed occupancy of cages in N2 clathrates at thermo- because of its convenience: the experimental dynamical equilibrium for the same conditions pressure and temperature are direct input pa- as in the previous experiments. This will en- rameters for the simulation and the volume of able us to assess the agreement between simu- the simulated clathrate adjusts itself automati- lations and experiments for a wide range of con- cally to the imposed pressure. This can be conditions covering both single and multiple occu- trasted with grand-canonical MC simulations pancy regimes for the cages, and to help inter- where the volume is fixed (and must hence be pret the recently measured occupancy ratios6 in adjusted manually when varying the pressure) terms of separate occupancies θL and θS. No- and where the pressure of the gas is not con- tice that similar comparisons but for the sim- trolled directly but must be deduced from its pler methane clathrate, which forms in struc- fugacity by using an appropriate equation of ture I and which does not show any multiple state. occupancy of cages, have been done by several 1This overestimation was probably due in part to 14–16 authors. their assumption of a rigid water framework. Only few simulations results for a N2 clathrate can be found in the literature.17–22 Horikawa et al., and later van Klaveren et al., have studied the dynamical behavior of encaged N2 molecules

2 Computational details sociated with each N atom. Its parameters (see Table 1) are very close to the model X1 Isobaric-isothermal Gibbs ensemble Monte of Murthy et al.29 that was used in the sim- Carlo simulations were performed with the ulation study by Klapproth:18 the geometric open-source MCCCS Towhee simulation pro- and LJ parameters are almost identical, but gram23 (version 7.2.0) to compute the cage oc- the latter model differs by the fact that it fea- cupancies θS and θL for pressures and temperatures no point charge but a point quadrupole tures where experimental data points are avail- of strength −3.91 × 10−40 C m2 located on the able. In this ensemble two simulation boxes are center of mass (that strength is considered to held at the same constant pressure P and tem- be an adjustable parameter in the X1 model perature T , while the total number of water and and results from a best fit to a wide range of nitrogen molecules across both boxes, is also properties). The Etters model involves 6 in- constant. One box represents the N2 clathrate teraction sites: a repulsion-dispersion interac- crystal and the other the nitrogen gas phase. tion site on each N atom and 4 Coulomb inter- Some additional GCMC simulations were per- action sites along the axis of the molecule at formed with the open-source DL_MONTE-2 the positions ±0.847 and ±1.044 Å, i.e. further simulation code;24 their results were consistent away than the N atoms which are at positions with those of Towhee. ±0.547 Å. Similarly to van Klaveren et al., we have modified slightly the electrostatic descrip- Molecular models tion to reduce the number of interaction sites to 3 and to match the experimentally observed A classical description of the interaction poten- quadrupole moment [−4.7 × 10−40 C m2]. The tials between water and guest molecules has repulsion-dispersion interaction potential in the been used. Short-range and dispersion in- Etters model was obtained from a refitting of ab teractions were computed as a sum of pair- initio calculations to better match experimen- wise Lennard-Jones (LJ) or Buckingham (also tal data for the second virial coefficient and for known as Exp-6) interactions between interac- condensed phases of N2. It is given by a Buck- tion sites located on the H2O and N2 molecules. ingham interaction at short distances, another Coulomb interactions between point partial Buckingham interaction at large distances, and charges were computed using the Ewald sum- a quartic spline fit at intermediate distances mation technique. Different force fields have (see Table 1 and Ref.28 for the parameters). been considered to gauge the sensitivity of This potential is less repulsive than the orig- the predicted cage occupancies on the choice inal ab-initio potential for an isolated pair of of the force field. The water molecules have molecules: this softening was found to be neces- 25 been represented by the TIP4P-Ew and the sary to accurately describe condensed phases.28 26 TIP4P/Ice models. The nitrogen interaction potentials of Potoff et To properly describe double occupancy of al. and of Etters et al. are compared in Fig. 1. cages in a N2 clathrate, a highly accurate model The Etters potential is less repulsive at short for the repulsive interaction between two N2 distances and more attractive near the well’s molecules is required because they are in close contact when occupying the same cage. Two Table 1: Parameters for N N interac- models were used to represent the N molecule: 2 tions the model of Potoff et al.27 which is included in the TraPPE force field, and the model of Model Type σ (Å) /kB (K) γ 28 Etters et al. The Potoff model is a 3-site TraPPE27 LJ 3.31 36.0 - model with a partial charge q = −0.482 a.u. Etters et al.28 Exp-6 3.3779 32.8519 13.1946 (a) on each N atom (which are separated by a dis- Exp-6 3.3269 39.7347 15.0755 (b) tance 1.1 Å) and a partial charge −2q on the (a) if r < 3.0102 Å; (b) if r > 3.4495 Å; see Ref. 28 otherwise center of mass, and with a LJ interaction as-

3 600 −10 tion of the form −15 500 Potoﬀ et al. 12 3 Etters et al. −20 X X X X Si,j,Λ(Ωij) N - EN2−H2O ≈ Ci,j,n,Λ 400 N −25 n V Rij i=N1,N2 j=O,H1,H2 n=2 Λ=0 ) −30 K (

300 (1)

) −35 r ( where Rij are intermolecular site-site distances N -

N 200 −40 V 3.4 3.5 3.6 3.7 3.8 3.9 4 and where the orientation function S intro- r 100 duces anisotropy into the atom-atom interactions. This fit holds when the intermolecular 0 distance R is smaller than 12 Bohr radius (a different fit was proposed for distances larger than −100 3 3.5 4 4.5 5 14 aB). Eq. (1) contains all contributions to the r (Å) interaction energy: exchange-repulsion, dispersion and electrostatic energy, including induc- Figure 1: The N site - N site interaction potential according to two different force fields: Etters tion energy. Since the proposed fit contains too et al.28 (red line) and TraPPE.27 A zoom on the many coefficients (564 in total) to be used in region around the potential minimum is shown in this form in our Monte Carlo simulations, we the inset. have deduced an average interaction potential

VN O(r) by averaging over dimer configurations for a fixed distance r between a N and O atom. minimum. In this calculation, the interaction (1) is first The N2 H2O interaction is particularly im- split into separate interactions with each of the portant in this work because it determines the two sites N1 and N2 of the nitrogen molecule. available cage volume and the potential en- The interactions with the H1 and H2 sites are ergy of the guest molecules inside the cages. added to the N O interaction, i.e. the sum over van Klaveren et al. have shown that the pre- j in eq. (1) is kept as is, to avoid picking up large ferred parallel orientation of two N2 molecules electrostatic contributions. If the molecules in a large cage is dictated by this interaction. were fully rigid and non-polarizable, all elec- In previous simulations on doubly-occupied N2 trostatic contributions would cancel out in this 18–21 clathrates, the N O interaction was de- averaging procedure when the two molecules scribed by a LJ potential, with parameters ob- are not touching each other (because the elec- tained via the Lorentz-Berthelot mixing rules. trostatic potential outside a charged spherical 19 The LJ parameters used in Ref. originate for shell is independent of its radius). The poten- instance from Ref.30 where they were calculated tial VN O(r) arises mainly from the exchange- by combining parameters of the SPC/E water repulsion and dispersion energies, but it does model with those of a N N interaction. The include also contributions from the induction importance of deviations from LB mixing rules energy. We have checked that the same aver- in the modelling of clathrates has been exam- age interaction VN O(r) is obtained, near the ined recently for various gases in the framework potential well minimum and also further away, of the van der Waals-Platteuw model,31 and when subtracting to EN2−H2O the electrostatic also with Monte Carlo simulations in the case interactions computed with the partial charges 32 of argon hydrates. defined in the water and nitrogen molecular To refine the N O interaction, we have con- models. sidered the full potential energy surface of the The potential VN O(r) that results from the dimer N2 H2O determined by Tulegenov et previous averaging procedure is shown in Fig. 2. 33 al. They have calculated 12228 energies for Fitting this potential to a Buckingham interac- different intermolecular configurations with a post-Hartree-Fock method and have proposed a fit of the energy surface to a site-site interac-

4 tion The potentials sets 1, 2 and 3 involve the potentials 1, 2 and 3 for the N O interactions, 6 r γ r −6 Buck −γ r −1 VN−O (r) = e m − , respectively. In the potential set 1, the N2 N2 γ − 6 γ − 6 rm interactions are given by the TraPPE force field, (2) while they are given by the Etters model in the where r = 21/6σ, provides the parameters = m potential sets 2 and 3. No standard mixing rule 66.2 K, σ = 3.32 Å and γ = 14.3. The poten- exist for combining the parameters of the Etters tials obtained by using the Lorentz-Berthelot model for N with those of a water model. In mixing rules (blue and red curves in Fig. 2) 2 the potential sets 2 and 3, the N O repulsion- are quite close to this potential V Buck(r). It N−O dispersion interactions are those listed in Ta- is somewhat more repulsive at short distance, ble 2 independently of the chosen water model. and more attractive at larger distances. The The potential set 2 differs from the potentials depth of the potential well is about 9% more at- used in refs19–21 only by the choice of the wa- tractive than the one in Refs.19,30 (red curve). ter model (SPC/E water was used in those ref- The depth of the potential V Buck(r), deduced N−O erences). The potential set 1, when combined from the ab-initio potential energy surface for a with TIP4P-Ew water, coincides with the inter- dimer, should be accurate, but this potential is action potentials used in Ref.22 likely too repulsive at short distances when applied to a condensed phase, similarly to the case of the N N interaction. We introduce there- Simulation details fore a softer potential (green curve in Fig. 2) The considered clathrate structures I and II by setting γ to 12.2, a value obtained by fitting are formed of 2ÃŮ2ÃŮ2 unit cells taken from 22 the N O potential of Ref. (blue curve) to a Takeuchi et al.;34 they are made up of 368 and Buckingham interaction. This provides the pa- 1088 water molecules, respectively. The total rameters listed in the first column of Table 2. number of cages is thus 64 (for sI) and 192 (for The remaining columns show parameters used sII). The GEMC simulations involved two cubic 22 in previous studies: Ref. for the case of po- boxes. One box was initialized with an empty 19–21,30 tential 1 with TIP4P-Ew water and Refs. hydrate lattice, in a box with initial edge length for the case of potential 2. 24.06 ÃĚ (for sI) or 34.62 ÃĚ (for sII); the other

300 box was initialized with nitrogen molecules: 200 or 400 of them for simulations with the sI or calculated from ab initio potential energy surface Fit to Buckingham interaction sII, respectively. Periodic boundary conditions 200 potential set 3 were applied separately to both boxes and the )

K LB mixing rules

( Ewald method was used to compute Coulomb

potential set 1 (TraPPE + TIP4P-Ew, Ref. 21)

k potential set 1 (TraPPE + TIP4P/Ice) interactions. The LJ and Buckingham interac- / 100 )

r potential set 2 (Murthy + SPC/E, Ref. 29) ( O

- tions were calculated using a cutoﬀ at 10 ÃĚ. N V Within each Monte Carlo cycle, N trial moves 0

Table 2: Parameters for N O interac- −100 3 3.5 4 4.5 5 tions r (Å) Potential 321 19–21,30 Figure 2: The N site - O site interaction potential This work Refs. TraPPE with calculated from the ab-initio potential energy sur- TIP4P-Ew or -Ice face of Tulegenov et al.33 (circles) and its ﬁt to an Type Exp-6 LJ LJ Exp-6 interaction (black line). The Exp-6 and the (K) 66.2 60.74 54.30 / 61.80 Lennard-Jones potentials listed in Table 2 are also σ (Å) 3.32 3.282 3.237 / 3.238 shown (see color key in plot). γ 12.2 - -

5 are attempted, where N equals the total num- riod. ber of molecules (water and nitrogen) present in both boxes. The probabilities for the various MC moves were set as follows: 1% prob- Results and discussion ability to attempt to change the volume of ei- The cage occupancies calculated from our ther the hydrate box or the N gas box (each 2 GEMC simulations are compared with experi- having a 50% probability of being selected); mental data first for conditions where no double 19% probability of attempting to transfer a ni- occupancy of cages has been observed experi- trogen molecule from one box (chosen at ran- mentally (sections 3.1 and 3.2), and then for dom) to the other (no attempt was made to conditions where some large cages were found transfer a water molecule from one box to the to be doubly occupied (section 3.3). The re- other to avoid disrupting the clathrate struc- sults presented in the next two sections pertain ture); 40% probability of attempting to trans- to the simulations of a flexible clathrate with late a randomly selected molecule; and, finally, sII with water-water interactions represented by 40% probability of attempting to rotate a ran- the TIP4P-Ew water model. The results in the domly selected molecule. For the last 2 types case of the TIP4P/Ice model will be discussed of moves, water molecules were chosen more of- in section 3.3. ten than nitrogen molecules: probability 73.1% to select the species H2O versus 26.9% for N2. The maximum displacements (volume, transla- Occupancies at 200 bar tion, rotation) were updated periodically, each The results of our GEMC simulations along the 50 MC cycles, to achieve an acceptance ratio of isobar 200 bar for a (flexible) clathrate with sII 50% for each type of move. are shown in Fig. 3, together with experimental The open-source MCCS Towhee simulation points for the cage occupancy ratio θ /θ de- program was modified to allow to calculate con- L S termined by Raman diffraction6 and with the veniently the individual large and small cage separate occupancies θ = 99.6 ± 0.6% and occupancies. This was done by writing into L θ = 82.2 ± 0.3% at 273 K determined by neu- a specific file, whenever a nitrogen molecule S tron diffraction.11 The simulations predict a would enter or leave the hydrate box, the co- preferential filling of the large cages versus the ordinates of the molecule, the type of move small ones in the considered temperature range, (in or out), the number of MC steps since the in agreement with the experimental data. start of the simulation and the current hydrate The small cages become significantly less oc- box length. An ad-hoc analysis script was then cupied when the temperature approaches the used to deduce from this file and from the ini- dissociation temperature. In that region, the tial positions of all molecules, the number of predicted small cage occupancies depend quite N molecules in each type of cages as function 2 strongly on the choice of the interaction po- of the MC “time”. The simulations were run tentials, making it difficult to obtain quanti- for 50,000 Monte Carlo cycles, giving a total of tatively accurate predictions, especially for θ . 28 400 000 and 74 400 000 MC steps for the sim- S On another hand, this sensitivity makes the ulations with sI and sII, respectively. The num- N hydrate a particularly interesting system for ber of guest molecules inside the clathrate was 2 testing and refining the interactions potentials found to have reached equilibrium after around employed in the simulations. The N N in- 1300 Monte Carlo cycles. The uncertainties 2 2 teractions between molecules in different cages on the calculated cage occupancies are below are almost identical when using the Etters and 1%. Some additional simulations were run by TraPPE force fields. As the cages are here only starting from a fully occupied clathrate, with singly occupied, the sensitivity of the predicted all large cages singly occupied (θ = 100%) or L cage occupancies arises from the N O inter- doubly occupied (θ = 200%). The results were L actions. Cages are found to be more filled on found to be identical after the equilibration pe-

6 1 with Langmuir constant for a cage of type t (t=small or large) 0.9

θ Z Z guest 1 −E (~r,Ω)~ /(kBT ) d 0.8 C (T ) = d~r dΩ~ e t . n Isobar 200 bar t a

kBT Vcage L θ (GEMC, pot. set 1, 2 or 3) θ 0.7 L θS (4) guest θL (exp., Ref. 10) ~ Here f is the fugacity of the gas and Et (~r, Ω) 0.6 θS is the potential energy of a guest molecule with ~ 2 GEMC, pot. set 1, 2 or 3 orientation Ω located at ~r inside a cage of type exp. (Ref. 10) t. A small shift in the binding energy is 1.8 exp. (Ref. 6) N O guest

S demultiplied in the potential energy Et (~r, Ω)

θ 1.6 / L because it affects all interactions of the two N θ 1.4 atoms with the nearby water molecules (small 1.2 and large cages in a sII hydrate are made up 1 of 20 and 28 water molecules, respectively). If 150 170 190 210 230 250 270 the N O interaction energies in a filled cage are T (K) all increased on average by 5 K, the Langmuir Figure 3: Cage occupancies in large and small constant for a small cage will increase at 270 K cages (upper panel), and their ratio (lower panel), by the factor exp(5 · 20 · 2/270) ≈ 2. along isobar 200 bar according to our GEMC sim- The preferential filling of the large cages over ulations with the TIP4P-Ew water model for the 3 the small ones can be understood on the basis of sets of potentials defined in Sct. (color code: blue, eq. (4) for the Langmuir constant. For a small red, green for potential sets 1, 2 and 3 respectively). molecules like N2 (diameter 4.2Å), the potential Experimental points from refs.6,11 are also shown. ~ energy Ecage S(~r, Ω) inside a small cage is lower, i.e. more binding, on average than in a large average when switching from the potential set cage because of the smaller size of the cage (the 1 → 2, despite the latter potential being more small and large cages in sII have average diame- repulsive at short distances (r < 3.5 Å): the ters 7.82 and 9.46 Å, respectively). In the limit “affinity” for filling the cages is increased by of low temperatures, the Langmuir constant for the more attractive N O interactions at larger small cages will therefore be larger than the distances. The increase in affinity is naturally one for large cages (CS > CL), leading to the more pronounced when switching from poten- small cages being more occupied than the large tial set 1 → 3, since the latter potential is even cages. As the temperature is increased, the guest guest more attractive when r > 3.5 Å. factor exp(−(ES − EL )/(kBT )) in the ra- The cage fillings depend, in a non-linear way, tio CS/CL of Langmuir constants becomes less on the N O repulsion at short distance, which favorable to the small cages and the integra- determines the available cage volume, and on tion over the cage volume in eq. (4) favors the large cages. In other words, a small molecule the binding energy N O at the minimum of the potential well, which determines the ener- like N2 will prefer the small cages at low tem- getic attractiveness of a cage. The sensitivity perature because it minimizes the energy U of the system (the entropy plays no role at suffi- to the binding energy N O can be understood by using the celebrated model of van der Waals- ciently low temperature), whereas it will pre- Platteuw,35 which assumes rigid cages. In this fer the large cages at higher temperatures because it increases the entropy S of the system, model, the cage occupancies θS and θL are given by a Langmuir-type formula the equilibrium macroscopic state being the one that minimizes the free energy U − TS. A pref- Ct(T )f erential filling of small cages has indeed been θt = (3) 22 1 + Ct(T )f observed in simulations at low temperatures.

7 36 It is interesting to note that the large cages Langmuir constants for N2 of Ref. The present are almost fully occupied along the isobar prediction of a fully occupied clathrate is not 200 bar according to our simulations. The Ra- sensitive to details of the interaction potentials. man spectroscopy measurements of the ratio Taking into account the rather large error bars θL/θS inform us therefore here directly on the of the experimental points, they are compatible occupancy of the small cages. A partial dou- with our theoretical prediction θL/θS ' 1. ble occupancy of the large cages at low tem- As explained in the previous section, a ten- peratures along this isobar cannot however be dency for the small cages to be more occu- excluded, as discussed in Sct. . pied than the large cages, like in the experi- At 273 K, there is a quite large difference be- mental dataset for P ≤ 100 bar, is expected tween the neutron diffraction data of Ref.11 at low temperatures. From the simulations of 22 (whose uncertainty is of the order of the sym- Ref., the transition from CL(T ) > CS(T ) to bol size) and the Raman spectroscopy data of CL(T ) < CS(T ) is expected to occur at around Ref.6 The data of Ref.11 pertains to a deuter- 80 K. Even with such an “inversion” of the ated clathrate, but it is known that deutera- Langmuir constants, the cages are still expected tion has a negligible impact on the cage oc- to be fully occupied at the considered pres- cupancies, as established in Ref.13 Due to the sures, which are much larger than the disso- sensitivity of the calculated occupancies on the ciation pressure (this can be confirmed by ex- choice of the interaction potentials, this exper- trapolating to higher pressures the simulation imental discrepancy cannot be fully lifted from data at 50 K, 100 K and 150 K of Ref.22). An our GEMC simulations at 200 bar. However, increase of the ratio θL/θS with increasing pres- in view of the occupancies predicted when us- sures could be due to some large cages encaging ing the TIP4P/Ice model and of comparisons two molecules, as further detailed in the next along isotherm 273 K (see Sct. ), and also be- section. cause we expect the potential set 3 to be more 1.4 accurate, the simulations appear to agree bet- Isotherm 150 K ter with the lower value of θ /θ measured in L S 1.2 Ref.11

S θ

Occupancies at 150 K / L θ 0.8 Occupancy ratios θL/θS along the isotherm 150 K have been determined by Raman spec- 0.6 troscopy. Fig. 4 shows a comparison between Exp. (Ref. 6) our GEMC simulations and the corresponding GEMC, pot. set 1, 2 or 3 6 0.4 experimental data. The simulations indicate 20 40 60 80 100 120 140 160 180 200 that all cages are fully (and singly) occupied. P (bar) This can be understood as being due to the quite low temperature and to pressures that Figure 4: Same as for Fig. 3, but for isotherm are signiﬁcantly higher than the dissociation 150 K. pressure Pdiss(150 K) ≈ 3 bar (this value is estimated by interpolating the data of Ref.36). A slight preferential ﬁlling of the large cages Double occupancy can be seen in our simulations for pressures below 20 bar. This agrees with an extrapolation A comparison between simulation and experi- of the isotherm 150 K calculated, for pressures mental results for the cage occupancies along P < 10 bar, by GCMC in Ref.22 It agrees also isotherm 273 K, from 148 bar up to the high with the prediction θL/θS ≈ 1.001 at 150 K pressure 1000 bar, is shown in Fig. 5. The po- and 60 bar obtained by using eq. (3) with the tential sets 1 and 2 underestimate the large and

8 small cage occupancies. They fail moreover to tractive energy contribution (i) remains similar predict that some large cages in a N2 clathrate than for single occupancy, but the repulsive en- are doubly occupied at high pressures, whereas ergy (ii) becomes larger and the new repulsive the experimental evidence for this phenomenon energy (iv) enters into the balance. is strong.11–13 The new potential set 3, intro- The model of a rigid clathrate, which is em- duced in Sct. by averaging an ab-initio poten- ployed in the van der Waals-Platteuw model,35 tial energy surface, does lead to a better qual- neglects the energy cost (ii) and the fluctuations itative agreement with the experimental data: of the cage framework due to the finite tempera- a sizable proportion of doubly occupied large ture. Fig. 7 shows the calculated cage occupan- cages can now be observed in those simulations cies in simulations of a rigid N2 clathrate. In at pressures above ≈ 500 bar. these simulations, only the nitrogen molecules can move, the water molecules are kept fixed in y c

n Isotherm 273 K a perfect crystalline arrangement (no volume

115 a p u c

c change of the clathrate box is furthermore at- o 110 e l

b tempted). The cages are found to be more occu- u ) o d

l % 105 (

a pied in the rigid clathrate approximation, sim- i

t L r a θ 14,37 100 p ilarly to the case of methane hydrates. Re- markably, the simulations with a rigid clathrate 95 Exp. (Ref. 10) GEMC (pot. set 1, 2 or 3) predict, when using our potential set 3, a sizable 100 amount of doubly occupied large cages not only at high pressures along the isotherm 273 K, but 90 also at moderate pressures along the isotherm

) 150 K and at low temperatures along the isobar % (

80 S

θ 200 bar. The simulations for the rigid model indicate thus that the large cages in a N hydrate 70 2 could be doubly occupied under the latter con-

60 ditions. Interestingly, this occurence of double 200 400 600 800 1000 occupancy could explain the increase of the ra- P (bar) 6 tio θL/θS with pressure seen at 150 K in Ref Figure 5: Same as for Fig. 3 but for isotherm 273 K (see Fig. 4). (and without plot of the ratio θL/θS). To check the sensitivity of the results on the model for water-water interactions, we have The occupancy of a cage is sensitive to four performed also simulations with the TIP4P/Ice 26 energetic contributions: (i) the interaction en- water model. As shown in Fig. 8, the occupan- ergy of the guest with the surrounding water cies calculated in a flexible clathrate when using molecules, which determines the potential en- with TIP4P/Ice model are higher than when ergy of a guest inside a cage and the avail- using the TIP4P-Ew model. This could be due able cage volume (ii) the change in water-water to the cage framework being more rigid when interaction energy upon insertion of a guest modelled with TIP4P/Ice because this model molecule (this change is due to deformations features stronger water-water interactions (12% of the water network), (iii) the interaction en- higher partial charges on the interaction sites ergy between guest molecules in different cages, and 30% higher binding energy for the O site and (iv) the interaction energy between a guest âĂŞ O site interaction). Fig. 6 shows the lattice molecule and the other guest(s) within the same constant L along isotherm 273 K in our simula- cage if it is multiply occupied. Predicting cor- tions, for the two considered water models. The rectly multiple occupancy is more difficult than lattice constant is somewhat too small when single occupancy because it is determined by using TIP4P-Ew, whereas it is somewhat too a more delicate energetic balance. When in- large when using TIP4P/Ice. The lattice spac- ing L decreases with increasing pressure almost serting a second N2 molecule in a cage, the at-

9 linearly, as in the experimental data, except The occupancy of the small cages is some- at low pressures where the simulations show a what overestimated (by about 10% at 300 small increase of L with pressure when θS is bar). Interestingly, the simulations with the small. This can be attributed to an expansion TIP4P/Ice model predict, as expected from the of the clathrate caused by an increased filling rigid clathrate simulations (Fig. 7), that some of the cages, which more than compensates the large cages are doubly occupied at 150 K and mechanical compression. 200 bar. As the double-occupancy Raman sig- nature could not be disentangled in the experi- 17.35 6 GEMC (pot. set 1, 2, 3 with TIP4P/Ice) ments of Ref., a verification of this prediction via a neutron diffraction experimental would be 17.3 useful to further confirm the ability of GEMC simulations with the potential set 3 to correctly ) Å ( 17.25 describe the doubly occupancy effect. L Exp. (Ref. 10) Eventually, we note that a further refinement GEMC (TIP4P-Ew) 17.2 of the parameters for the N O interactions can be performed to improve the agreement with the experiments at 273 K. A Buckingham po- 17.15 200 400 600 800 1000 tential with parameters = 66.2 K, γ = 13.2 P (bar) and r0 = 3.24 (i.e. an interaction that is more attractive at short distances and more repul- Figure 6: Cubic lattice constant of a N clathrate 2 sive at large distance than the potential 3) pro- with sII according to our simulations (colored lines, see key in plot) and to the experimental data of vides a slightly better agreement along isotherm Ref.11 (black squares). 273 K (see magenta curve labelled “Pot. set 4” in Fig. 8). It predicts also a higher proportional By opposition to our previous calculations of doubly occupied cages at 200 bar and 150 K. with the TIP4P-Ew model, the cage occupancies are lowered when switching from potential sets 1 → 2 when using the TIP4P/Ice model. Conclusions This is due to the use of the Lorentz-Berthelot We have compared the predictions of Gibbs en- mixing rules in the potential set 1: the bind- semble Monte Carlo simulations with experi- ing energy is higher when associating the N O mental data for the cage occupancies in N2 hy- TraPPE force field with the TIP4P/Ice model drates to assess the accuracy of such simula- than with the TIP4P-Ew model (see Table 2). tions, to refine the effective N O interaction The fact that both θL and θS are underesti- potential beyond the common Lorentz-Berthlot mated along the isotherm 273 K, whereas the mixing rules, and to help interpret cage occu- potentiel set 1 provides a better agreement, pancy ratios θL/θS that have been recently mea- especially for θS, confirms our finding from sured by Raman spectroscopy. Different sets of Sct. that the binding energy = 54.3 K N O interaction potentials for N N ,N H O and associated with the combination of the TraPPE 2 2 2 2 H2O H2O interactions have been considered. force field with the TIP4P-Ew model is too low. The predicted cage fillings are sensitive to the The potential sets 1 and 2 associated with choice of the interaction potentials when the fill- TIP4P/Ice water predict a non-negligible pro- ing differs significantly from 100%. This renders portion of doubly occupied large cages at 273 K accurate predictions difficult, but allows for in- and 1000 bar, but the magnitude of this multi- structive comparisons with experiments to re- ple occupancy is largely underestimated. With fine the effective potentials employed. Among the potential set 3, the occupancy of the the considered sets of interaction potentials, large cages is in much better agreement with some of them are not able to reproduce the the experimental data, and predicts correctly known experimental fact that some large cages that multiple occupancy starts at ≈300 bar.

10 115 Isotherm 273 K ) Isotherm 150 K Isobar 200 bar . 10 Ref p., (Ex 110 θL

105

100

) 95 % (

S θ

, 90

L GEMC, pot. set 1, 2 or 3 θ

θL

Exp. (Ref. 10)

θS

80 0 50 100 150 200 160 180 200 220 240 260 200 400 600 800 1000 P (bar) T (K) P (bar)

Figure 7: Cage occupancies in large (solid lines) and small (dotted lines) cages, along isotherm 150 K (left), isobar 200 bar (center) and isotherm 273 K (right) in a rigid sII N2 hydrate according to our GEMC simulations with the TIP4P-Ew water model for the 3 sets of potentials. Experimental points at 273 K from Ref.11 are also shown.

115 Isotherm 273 K ) Isotherm 150 K Isobar 200 bar . 10 Ref p., (Ex 110 θL

105

100 ) %

( 95

S θ

L 90 θ

GEMC, pot. set 1, 2, 3 or 4

θL

θS (Exp., Ref. 10)

θS

80 0 50 100 150 200 160 180 200 220 240 260 200 400 600 800 1000 P (bar) T (K) P (bar)

Figure 8: Same as for Fig. 7 but for GEMC simulations of a ﬂexible N2 clathrate with the TIP4P/Ice water model. See text for the deﬁnition of the potential 4.

11 are doubly occupied at 273 K and high pres- effective potentials. sures. Acknowledgement This paper falls in the We have found, both from comparisons with frame of the MI2C project funded by the French experimental data and from an analysis for an âĂĲAgence Nationale de la RechercheâĂİ ab-initio N H O potential energy surface, that 2 2 (project ANR-15CE29-0016). The calculations the binding energy at the bottom of the poten- were run on computers of the Institute UTI- tial well for the N O interaction is not attrac- NAM of the University de Franche-ComtÃľ, tive enough when combining the TraPPE force supported by the RÃľgion Franche-ComtÃľ field for N2 with the TIP4P-Ew model for wa- and the Institut des Sciences de lâĂŹUnivers ter. The N O interaction potential 2 (see Ta- (INSU), and also on the supercomputer facili- ble 2) is more binding, but it is too repulsive ties of the MÃľsocentre de calcul de FrancheÂŋ- at short distances for the large cages to become ComtÃľ. doubly occupied. By averaging an ab-initio potential energy surface of N2 H2O, we have in- troduced a new effective Buckingham potential References for N O interactions. That interaction leads to more accurate predictions for the cage oc- (1) Sloan, E. D.; Koh, C. A. Clathrate Hydrates cupancies, especially for the large cages. The of Natural Gases; CRC Press, 3rd ed., 2007. simulations with the TIP4P/Ice water model (2) Ohgaki, K.; Takano, K.; Moritoki, M. Ex- appear to agree somewhat better with experi- ploitation of CH4 hydrates under the Nankai ments than those with the TIP4P-Ew water as Trough in combination with CO2 storage. Ka- far as cage occupancies are concerned. Addi- gaku Kogaku Ronbunshu 1994, 20, 121–123. tional theoretical works on the interactions in- (3) Park, Y.; Kim, D. Y.; Lee, J. W.; Huh, D. G.; volved in N2 hydrates would naturally be useful to further improve the accuracy of GEMC sim- Park, K. P.; Lee, J.; Lee, H. Sequestering car- ulations for this system. bon dioxide into complex structures of natu- Some experimental points for the same con- rally occurring gas hydrates. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 12690–12694. ditions but from different experiments do not agree well with each other (for instance the (4) Thomas, C.; Picaud, S.; Mousis, O.; Balleneg- points of Ref.11 and Ref.6 at 200 bar and T ≈ ger, V. A theoretical investigation into the 270 K, see Fig. 3). The simulations with the trapping of noble gases by clathrates on Ti- TIP4P/Ice water model (and those within the tan. Planetary and Space Science 2008, 56, rigid clathrate model) show that some large 1607–1617. cages can be doubly occupied at 150 K and (5) Mousis, O.; Lunine, J. I.; Picaud, S.; 200 bar. This double occupancy effect could ex- Cordier, D. Volatile inventories in clathrate plain the increase of the ratio θL/θS measured 6 hydrates formed in the primordial nebula. in Ref. (see Figs. 4 and 8). An experimen- Faraday Discuss. 2010, 147, 509–525. tal confirmation with a separate measurement of θL and θS under these conditions would be (6) Petuya, C.; Damay, F.; Chazallon, B.; useful. Experimental values for cage occupan- Bruneel, J.-L.; Desmedt, A. Guest Partition- cies are most instructive in regimes where the ing and Metastability of the Nitrogen Gas Hy- small and/or the large cage occupancies devi- drate. J. Phys. Chem. C 2018, 122, 566–573. ate significantly from 100%, i.e. close to the (7) Loveday, J. S.; Nelmes, R. J.; Klug, D. D.; dissociation pressure (low occupancy) or under Tse, J. S.; Desgreniers, S. Structural system- conditions where some large cages are doubly atics in the clathrate hydrates under pressure. occupied (high pressures or intermediate pres- Can. J. Phys. 2003, 81, 539–544. sure and low temperatures). Such measurements provide stringent tests for the simula- (8) Sasaki, S.; Hori, S.; Kume, T.; Shiumizu, H. tions and enable refinements of the employed Microscopic observation and in situ Raman

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14 Graphical TOC entry

Double Exp. 115% occupancy Sim.

100%

85%

200 600 1000 bar

Doubly occupied large cages in a N2 clathrate hydrate

15 Graphical TOC Entry

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