Calculating the Phase Behavior of Gas-Hydrate-Forming Systems from

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Calculating the Phase Behavior of Gas-Hydrate-Forming Systems from Molecular Models S. J. Wierzchowski and P. A. Monson* Department of Chemical Engineering, UniVersity of Massachusetts, Amherst, Massachusetts 01003

We describe a calculation of the phase behavior of the methane-water system, including the structure-I hydrate phase, starting from a model of the intermolecular forces in the system and using Monte Carlo simulations and theory. The approach we use differs from previous calculations of methane hydrate phase behavior in that it does not treat the water molecules in the hydrate as a static or harmonic lattice, i.e., thermal fluctuations are fully incorporated. Our approach is quite general, but we illustrate it here for perhaps the simplest model capable of forming stable hydrate-like structures. This is a mixture of network-forming associating hard spheres and nonassociating hard spheres. With this model as a reference system, we add dispersion forces and dipole-dipole interactions as perturbations. Monte Carlo simulations were used to determine the solid-phase properties of this model, and the fluid phases were treated using an accurate thermodynamic perturbation theory. Our results show that this simple molecular model is able to describe the phase diagram in qualitative agreement with the experiment. In particular, it correctly describes the region of stability of a single-phase methane structure-I hydrate and its dependence on temperature, pressure, and composition. I. Introduction Understanding the thermodynamics and stability of gas hydrates is important both from industrial and environmental perspectives.1 The formation of gas hydrates in oil and gas pipelines is an important area of application. In this context, the main task is maintaining flow assurance through manipulation of flow compositions, specifically by addition of inhibitors, or by controlling the state conditions and operating where gas hydrates are unstable thermodynamically. The existence of massive formations of gas hydrates at the bottom of deep oceans and in the permafrost represents an enormous potential resource of energy. On the other hand, uncontrolled release of the gases from these hydrate formations could contribute significantly to global warming. The phase behavior of gas hydrates has been studied experimentally for several decades. Kobayashi and Katz2 provided an instructive paper as early as 1949 on the temperature-composition phase behavior of methane hydrates by utilizing several previous experimental studies. Similarly, Harmens and Sloan3 highlighted important features of the temperature-compositon phase behavior of propane hydrates. A large number of the experimental pressure-temperature studies have been compiled by Sloan,1 providing an excellent reference for the phase behavior of gas hydrates. Recently, the phase diagram of Kobayashi and Katz has been modified by Sloan and coworkers4-6 to account for a variation in the composition of the methane hydrate. The statistical mechanical theory of van der Waals and Platteeuw (VDWP)7 is the most widely used methodology to calculate gas hydrate phase equilibrium and has been extensively applied, including to the case of mixed gas hydrates.1 In more recent work, various groups8-23 have extended the VDWP formulation by making more rigorous calculations of the cell partition functions that appear in the theory at the cost of some additional complexity in the calculations. The success of the * To whom correspondence should be addressed. E-mail: [email protected].

VDWP approach and its extensions has led Sloan to state that “currently, the calculation of hydrate equilibria is the best exemplar of the industrial use of statistical thermodynamics on a routine basis”.1 Nevertheless, a central feature of this kind of approach is the water molecules creating a static external field for the encaged molecules, and this does not address the role of the water-water interactions on hydrate stability. The properties of gas hydrates have also been determined by lattice dynamics techniques.24-26 These techniques go beyond the VDWP approach in that they take into account the vibrational and librational motion of the water molecules. However, such calculations are limited to the range of temperatures where the harmonic approximation is correct, although anharmonicities can be included. Computer simulations of hydrates using molecular dynamics and Monte Carlo simulations offer the possibility of an exact calculation of the properties for a given molecular model without the assumptions in the VDWP or lattice dynamics approaches. Many such calculations have been published addressing various issues in hydrate behavior, including assessments of the approximations built into the VDWP theory27,28 and calculations of the vibrational spectra of the hydrate solid.29-35 The calculation of the entire phase diagram for an intermolecular potential model of a methane-water-hydrate system via statistical mechanics is a major undertaking and, to our knowledge, has not been attempted in the past. In this paper, we present such a calculation. Our focus is on what is, perhaps, the simplest model of intermolecular forces that could describe hydrate formation. This is a binary mixture of associating hard spheres (with directional association interactions modeling hydrogen bonds) with hard spheres that do not associate.36,37 This “primitive” model exhibits an essential feature of the solution thermodynamics of water/alkane mixturessthe tendency of one component (water) to form networks while the other (alkane) does not. It is this feature that is central to the phenomenon of hydrophobicity.38,39 To this model the effect of dispersion forces and dipole-dipole interactions can be added perturbatively. Moreover, quite accurate descriptions of the fluid phase behavior are available from thermodynamic perturbation

10.1021/ie050875s CCC: $33.50 © 2006 American Chemical Society Published on Web 12/08/2005

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theory.36,37,40-43 This allows us to focus our computational efforts on the solid phases. The associating hard sphere model exhibits solid-fluid phase behavior similar to that of water, and this has already been studied in recent work;44 the solid-fluid phase behavior of the other component, hard spheres, is, of course, well-known.45 We have developed a technique for determining the methane and water chemical potentials from Monte Carlo simulations of gas hydrates.46 This method uses an isobaric semigrand ensemble Monte Carlo method to determine the methane and water chemical potentials versus the hydrate occupancy at a fixed temperature and pressure. The free energies of the fully occupied and empty hydrate solids can be calculated by an adaptation of a method due to Frenkel and Ladd,47 which uses a reversible path between the solid state of interest and a classical Einstein crystal. Our results show that this rather simplified model can describe many of the important features of the methane-water-hydrate phase diagram. In particular, the temperature versus composition behavior at a fixed pressure of the model is similar to that presented by Kobayashi and Katz2 and exhibits the stability of the hydrate over a range of compositions shown by Sloan.4 The outline of our paper is as follows. In the next section we describe the molecular model, in Section III we describe our computational methodologies, and Section IV gives a presentation of our results. Section V gives a summary of our results and conclusions. II. Methodology A. Molecular Model. As stated in the Introduction, the model we use here starts with the “primitive” model of water/alkane systems developed by Nezbeda and co-workers.36,37 This consists of a binary mixture of associating hard spheres (the primitive model of water of Nezbeda and co-workers,36,37 or PW) and hard spheres (HS). The PW/HS mixture model exhibits many of the physical properties necessary to model methane hydrate phase behavior. A tetrahedrally bonded network for both the low-density ice and gas hydrate structures is formed, and the solids are mechanically stable. Moreover, previous studies show the utility of the molecular model in calculating thermodynamic properties of the pure component PW model44,48 and the PW/HS binary mixture.36,37,49 The PW model consists of a hard sphere of diameter σw with tetrahedrally coordinated attractive square-well sites with well width λw. These sites give rise to low coordination number association interactions which model hydrogen bonding in real water. The four attractive sites are divided into two H sites and two O sites. Only O-H interactions between different molecules contribute to the intermolecular potential. The H sites are centered on the surface of the PW hard sphere, while the O sites lie slightly inside the surface of the hard sphere, a distance of 0.45σw from the center of the sphere. The square-well interaction strength is denoted by w. In the PW/HS mixture, we use a value of the HS diameter, σm, of 1.25σw. This is based on the ratio of Lennard-Jones 12-6 collision diameters used in the models of methane50 and water.51 The PW/HS system is the base model for this study. To include the effects of dispersion interactions in the system, we add a van der Waals mean field to the free energy of the system. The PW/HS model with dispersion forces will be referred to as model I. Following Nezbeda and Pavlicek52 and also Muller and Gubbins,53 we can add a dipole-dipole interaction to the PW model to describe the effects of longer-ranged electrostatic interactions. The dipole vector is oriented so that it bisects the vectors from the molecule center to the H sites. The long-range

interactions of the dipole are accounted for through an Ewald summation technique.54,55 The PW/HS model with dispersion forces and added dipole will be referred to as model II. B. Solid-Phase Thermodynamics. Our procedure for calculating the free energies and chemical potentials of hydrate solids is presented in detail elsewhere.46 Our method uses two thermodynamic integrations, one where we calculate the chemical potentials of water and methane in the hydrate as functions of the hydrate composition using the empty hydrate as the reference state for the water chemical potential. We have (for fixed temperature, pressure, and amount of water in the hydrate) that

Nwµw ) Nwµw0 -

∫-∞µ Nm dµ′m m

(1)

where Nw and Nm denote the number of molecules of methane and water, respectively, and µi denotes the chemical potential of component i with the superscript 0 denoting the empty hydrate state for the water chemical potential. We can calculate Nm versus µm in an isobaric semigrand ensemble Monte Carlo simulation of the hydrate in which the variables fixed are Nw, µm, P, and T. We calculate the free energies of the fully occupied hydrate and empty hydrate states using the Frenkel-Ladd method.47 In this method, the free energy is obtained by another thermodynamic integration along a reversible path connecting the solid of interest with a classical Einstein crystal.47 This method has been used to calculate the free energies of a wide range of solids.44,47,56-64 The calculation of the free energies for both the empty and filled states of the hydrate allows us to check for thermodynamic consistency of the results from eq 1. In this work, the above methodology is applied to our base model, the PW/HS system. Dispersion forces are added via a van der Waals mean field (model I), and dipole-dipole interactions can also be added (model II). The dipole moment contribution to the solid free energy is treated as a perturbation and is represented here by a sum over static dipole lattice energies.55,65,66 We have tested this approximation against Monte Carlo simulations of the PW/HS system with added dipolar interactions between the PW molecules.46 C. Fluid-Phase Thermodynamics. Nezbeda and co-workers36,37 have applied Wertheim’s40-43 thermodynamic perturbation theory (TPT) to the PW/HS system. We have made use of their formulation in this work. In TPT for the PW/HS system, the free energy is written as

APW ) AHS,mix + AHB

(2)

where AHS,mix is the free energy of a hard sphere mixture and AHB is the contribution to the free energy from the hydrogen bonding created by the square-well association sites in the PW model. We refer the reader to the papers by Nezbeda and coworkers where the application of TPT in this context is presented in more detail.36,37 Recently, the theory has been tested by comparison with Monte Carlo simulations of the PW/HS system and shown to be quite accurate in predicting thermodynamic properties, including the composition dependence of the chemical potential.49 For the fluid phase, the dipolar contributions to the free energy are determined using a perturbation theory developed by Rushbrooke et al.67 We have

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2π µ4 FI AD 9 T 2 ) N 5π µ2 I3 F 1+ 36 T I2

(3)

where T is the absolute temperature, F is the density, and I2 and I3 are given by

I2 )

1 - 0.3618F - 0.3205F2 + 0.1078F3 (1 - 0.5236F)2

(4)

1 + 0.62378F - 0.11658F2 1 - 0.59056F + 0.20059F2

(5)

I3 )

The dispersion interactions are introduced through a mean field approximation of the van der Waals type. The contribution to the free energy has the familar form

AvdW ) F[(1 - xm)2aw + 2(1 - xm)xm(amaw)1/2 + xm2am] N (6) where xm is the mole fraction of methane. The parameters aw and am for water and methane, respectively, determine the magnitude of the dispersion interactions. As mentioned above, we have incorporated the different kinds of interactions considered here into two models of the methanewater system. Model I consists of the PW/HS model plus dispersion interactions, and model II adds dipolar interactions to model I. In model I, we set aw/(wσ3) ) 0.6 and am/(wσ3) ) 1.2. The parameters of model 2 are aw/(wσ3) ) 0.8, am/w ) 0.8, and µ2/(wσ3) ) 0.04. D. Phase Equilibrium Conditions. The equilibrium conditions for coexistence of two phases are found using expressions for the Gibbs free energy of each phase. The equality of the chemical potentials of the components between phases in equilibrium can be expressed in the form36 R

[

R

G (T,P,xw) - xw

]

∂GR(T,P,xw) ∂xw

)

[

T,P

Gβ(T,P,xw) - xwβ

[

] [

∂GR(T,P,xw) ∂xw

)

T,P

]

∂Gβ(T,P,xw) ∂xw

]


(7)

T,P

(8)

T,P

where the superscripts R and β denote different phases and xw is the mole fraction of the water. These expressions are used to determine the fluid phase equilibrium for our model and also the equilibrium between the structure-1 hydrate and fluid mixtures. They are derived from the tangent intercept equations for the chemical potentials68 and are notationally convenient to use in this work where we determine the Gibbs free energy and then calculate its composition derivative by finite difference methods. For equilibrium involving one pure solid and a mixture, we use the equilibrium in criteria in the form

GRm(T,P) ) Gβ(T,P,xw) - xβw for pure solid methane and

[

]


T,P

(9)

GRw(T,P) ) Gβ(T,P,xw) + (1 - xβw)

[

]


(10)

T,P

for ice. We use these expressions for the equilibria of ice + fluid, solid methane + fluid, ice + hydrate, and solid methane + hydrate. By applying the phase equilibrium criteria, T-x diagrams are calculated at fixed P. Three-phase equilibria are identified by crossings of two-phase lines. III. Results and Discussion A major motivation of this work is to see if it is possible to predict the qualitative features of methane-water phase behavior via a simple molecular model. Accordingly, the methods explained in the previous sections are implemented to produce the T-P-x phase behavior of the methane-water binary system including solid, liquid, and vapor phases. It is worthwhile first to examine the pure-component phase behavior of the model. Figure 1 shows the water T-F diagram for model I, and three phases are shown, ice (I), liquid (L), and vapor (V). Notice that the model correctly describes the fact that water expands on freezing at low pressure. We have not included the possibility of the other ice phases in this work, since they do not feature in the methane-water phase diagram at ambient pressures. The water model used here is one of the simplest to date to reveal the presence of solid-fluid and fluid-fluid equilibria of water. Vega and Monson44 have presented a more complete study of the phase behavior of the PW model without added dispersion forces. The critical point of water in model I is at T/c ) kTc/ ) 0.15167, P/c ) Pcσ3/ ) 0.016581, and F/c ) Fcσ3 ) 0.29515. The ratio of critical properties between methane and water in w m w the model are Tm c /Tc ) 0.729 and Pc /Pc ) 0.305. From the m w w experiment, these ratios are Tc /Tc ) 0.295 and Pm c /Pc ) 0.207. Another point of comparison with the experiment is the ratio of the properties at the triple point to those at the critical point. For water, from the experiment we have Twt /Twc ) 0.422 and Pwt /Pwc ) 2.75 × 10-5, while for model I, we have Twt /Twc ) 0.837 and Pwt /Pwc ) 0.111. The T-F diagram for water in model II is given in Figure 2, with T/c ) 0.17253, P/c ) 0.017718, and F/c ) 0.28225. It is seen from Figures 1 and 2 that Twt /Twc is improved by the inclusion of electrostatic interactions in model II with Twt /Twc ) 0.725 and Pwt /Pwc ) 0.019. w m w Also, the ratios Tm c /Tc ) 0.427 and Pc /Pc ) 0.191 are more in accord with the experimental values. For methane, the phase diagrams of models I and II are identical. The mean field assumption for the effect of attractive forces means that the solid-fluid transition is derived primarily from the underlying transition in the hard sphere system. In this case, the model m m m predicts Tm t /Tc ) 0.466 and Pt /Pc ) 0.003. This compares m m m with experimental values of Tm t /Tc ) 0.476 and Pt /Pc ) 0.0025. Thus, both models I and II are more accurate in their treatment of methane than of water, as might be expected. The essential point here is that we have a model that correctly describes the qualitative features of the pure-component phase behavior, even if the quantitative predictions are not entirely satisfactory (e.g., both our models appear to overestimate the stability of hydrogen-bonded solid phases). We now present the phase behavior from model I through several T-x diagrams at various pressures, similar to the discussion by Kobayashi and Katz2 of the experimental behavior. In Figure 3, a phase diagram is given for a pressure (P/Pwc ) 0.305) slightly above the critical pressure of methane. In labeling phases on the diagram, the same notation is used here as was


Figure 1. Model I water T/Tc vs F/Fc phase diagram. The diagram contains three two-phase regions along with vapor (V), liquid (L), and solid (S) pure-phase regions.

Figure 2. Model II water T/Tc vs F/Fc phase diagram. Comparison with Figure 1 reveals the improved Tc/Tt ratio.

Figure 3. Model I methane/water T-x phase behavior at P/Pwc ) 0.305. Phases present are methane hydrate (H), vapor (V), water-rich liquid (L1), methane-rich liquid (L2), solid methane (M), and ice (I). The pressure is set slightly above the critical point of methane to highlight the presence of a V + L2 region and for comparison with the results of Kobayashi and Katz2 and Sloan.4

used by Kobayashi and Katz2 with I, V, L1, L2, M, and H representing ice, vapor, water-rich liquid, methane-rich liquid, solid methane, and methane hydrate, respectively. Excluding single-component phases, Figure 3 identifies four single-phase regions, L1, L2, H, and V. The experimental phase diagram in this pressure region has been discussed recently by Sloan and co-workers,4-6 suggesting the existence of a variable-composi-

Figure 4. Model I methane/water T-x phase behavior at P/Pwc ) 0.305 expanding the methane-rich portion from Figure 3 to show the vaporliquid coexistence.

Figure 5. Model I methane/water T-x phase behavior at P/Pwc ) 0.305 expanding the methane-rich portion from Figure 3 to show the vaporliquid coexistence.

tion H region, rather than a fixed composition corresponding to a structure-I hydrate with all cavities occupied by methane molecules, as suggested by Kobayashi and Katz.2 We return to this feature later. Our phase diagram includes all the equilibria seen experimentally. At the methane-rich area of the phase diagram, we have phase behavior involving vapor, liquid, solid methane, and hydrate phases. The behavior is seen more clearly in Figures 4 and 5, which expand the composition axis at high methane mole fractions. Figure 4 shows the V + L2, region which terminates at a critical point since the pressure is slightly above Pm c . As the L2 region is followed into lower temperatures, a eutectic develops where the M, H, and L2 phases intersect (Figure 5). This latter feature is subtle and difficult to detect numerically. A portion of the water-rich area of Figure 3 is expanded in Figure 6, showing a eutectic involving the L1, H, and I phases. The phase behavior seen in Figure 6 is central to showing the stability of methane hydrates. As seen experimentally, the methane hydrate melting temperature is slightly above that of ice. The L1 phase in model I is stable over a wider range of composition than seen in the experiment, with a maximum value of xm ) 0.025 as shown in Figure 6, while the experimental methane solubility is xm < 0.001. This higher solubility of methane in water in the model reflects the large values of the methane-water dispersion interaction. Figure 7 expands the composition scale in the region of the hydrate compositions and highlights another interesting feature, the variable-composition methane hydrate region. Notice that all compositions of the

428


Figure 6. Model I methane/water T-x phase behavior at P/Pwc ) 0.305 expanding the water-rich portion from Figure 3.

Figure 7. Model I methane/water T-x phase behavior at P/Pwc ) 0.305 expanding the hydrate single-phase portion from Figure 3.

hydrate here correspond to incomplete filling of the cavities by methane molecules. Complete filling (8 CH4/46 H2O) is only approached at low temperatures. The hydrate reaches its limit of thermodynamic stability at a single composition, 7.8 CH4/ 46 H2O. The hydrate region at first broadens in composition as the temperature is decreased from this temperature and then narrows, with the composition trending toward that of fully occupied hydrate 8 CH4/46 H2O. This is slightly different from the schematic behavior indicated by Sloan,4 which shows that the methane content of the hydrate in equilibrium with ice decreases as the temperature is lowered. The behavior shown here seems more plausible and is a feature of the phase diagrams for both models I and II. Figures 8 and 9 show the effect of increasing the pressure on the results from model I. In Figure 8, we set a pressure between the critical points of water and methane (P/Pwc ) 0.723). The primary changes in the phase diagram are an increase in the solubility of methane in water at the higher pressure, which expands the L1 region, and the disappearance of the V + L2 region, since the pressure is now substantially higher than the critical pressure of methane. The phase diagram is otherwise qualitatively similar to that in Figure 3, with only a change in the magnitude of each region. Figure 9 shows the results for P/Pwc ) 1.024. We see that the V + L1 region no longer extends to the pure water limit, since the pressure is now above the critical pressure of water in the model. In Figures 10-13, the phase behavior of model II is illustrated. Figures 10-12 show behavior comparable to that

Figure 8. Model I methane/water T-x phase behavior at P/Pwc ) 0.723. The pressure is set between the critical point of methane and water showing a disappearance of the H + V equilibrium shown in Figure 3.

Figure 9. Model I methane/water T-x phase behavior at P/Pwc ) 1.024. The pressure is set above the critical point of water to highlight the critical point composition of the fluid that is now not pure water, as seen in Figures 3 and 8. We see that the V + L1 region no longer extends to the pure water limit since the pressure is now above the critical pressure of water in the model.

Figure 10. Model II methane/water T-x phase behavior at P/Pwc ) 0.191. The pressure is set slightly above the critical point of methane. We note the decrease in the methane solubility in L1 relative to that seen in Figure 3.

in Figures 3, 8, and 9. A key difference in the quantitative behavior between models I and II is the much lower solubility of methane in water for model II (xm < 0.0006) at lower pressure. This is a consequence of the inclusion of the dipole-


(P/Pwc ) 0.959) and Figure 12 (P/Pwc ) 1.500) are calculated at pressures similar to those used for Figures 8 and 9. It is again observed how model II accounts more realistically for the change in the solubility of methane in water with pressure. The L1 + H region is seen to increase with increased pressure, a feature not as apparent for model I. Figure 13 depicts the results for a pressure below the critical point of methane (P/Pwc ) 0.112). The state conditions reveal an I + V region not seen in the previous figures. It is interesting that the hydrate remains in equilibrium with the fluid phase even at low pressures, though no L1 + H region exists. IV. Summary and Conclusions

Figure 11. Model II methane/water T-x phase behavior at P/Pwc ) 0.959. The pressure is set between the critical point of methane and water showing a disappearance of the H + V equilibrium shown in Figure 10. These results can be compared with those for model I in Figure 8.

Figure 12. Model II methane/water T-x phase behavior at P/Pwc ) 1.500. The pressure is set above the critical point of water. These results can be compared with those for model I in Figure 9.

In this work, we have presented calculations of the methanewater phase diagram including hydrate and other solid phases for two molecular models. Our goal here has been two-fold. In the first instance, we have sought to establish the feasibility of calculating the phase behavior of these systems without the kinds of assumptions used in previous modeling work based on the VDWP theory of hydrate phases and its extensions. We have also investigated whether simple models of the intermolecular forces in a methane-water system can yield a qualitatively correct picture of the phase behavior. Our model I is essentially a mixture of a network-forming component (modeling water) and a component that does not form networks (modeling methane), with dispersion forces added perturbatively. The role of the dispersion forces in this model are to stabilize the structure-I hydrate relative to ice and to improve the prediction of vapor-liquid equilibrium. It is remarkable that this simple model yields a phase diagram in qualitative agreement with the experiment.2,4 We note especially the prediction of a single-phase region of hydrate stability over a range of compositions.4-6 The addition of dipole-dipole interactions between the water molecules in model II makes the predicted phase behavior more realistic. In particular, model II gives a better description of the solubility of methane in water at lower pressures, which is unrealistically large in model I. Interestingly, this model also exhibits a region of coexistence between ice and a methane-rich vapor at lower temperatures and pressures that has not been reported in experiments on these systems. The present study addresses the phase behavior at low pressure for the simplest hydrate-forming system, a methanewater system. There are significant opportunities for expanding the scope of the present work. In the first instance, we plan to consider the case of other alkanes, such as ethane and propane, by changing the size of the alkane component in the present models and including other structures such as structure-II and structure-H as well as recently discovered high-pressure hydrate structures.69-71 Acknowledgment

Figure 13. Model II methane/water T-x phase behavior at P/Pwc ) 0.112. The pressure is set below the critical point of methane to show the presence of a V + I region not present in the other figures.

This work was supported by a grant from the U.S. Department of Energy (contract no. DE-FG02-90ER14150). Literature Cited

dipole interaction between the water molecules in model II and the accompanying reduction in the dispersion interaction w parameters. Model II also yields more realistic values of Tm c /Tc m w and Pc /Pc , as mentioned earlier. Another important difference in the phase behaviors of the two models is in the temperatures at which three-phase equilibria occurs and is exemplified when Figure 10 is compared to Figure 3. The results in Figure 11

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ReceiVed for reView July 26, 2005 ReVised manuscript receiVed October 16, 2005 Accepted November 7, 2005 IE050875S

Calculating the Phase Behavior of Gas-Hydrate-Forming Systems from

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