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1 This article was downloaded by: [Universitetsbiblioteket i Bergen] On: 20 September 2014, At: 03:27 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Molecular Physics: An International Journal at the Interface Between Chemistry and Physics Publication details, including instructions for authors and subscription information: Molecular dynamics simulations of PVP kinetic inhibitor in liquid water and hydrate/liquid water systems By BJORN KVAMME, GEIR HUSEBY & OLE KRISTIAN FORRISDAHL Published online: 03 Dec To cite this article: By BJORN KVAMME, GEIR HUSEBY & OLE KRISTIAN FORRISDAHL (1997) Molecular dynamics simulations of PVP kinetic inhibitor in liquid water and hydrate/liquid water systems, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 90:6, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 MOLECULAR PHYSICS, 1997, VOL. 90, NO. 6, 979 ± 991 Molecular dynamics simulations of PVP kinetic inhibitor in liquid water and hydrate/liquid water systems By BJé RN KVAMME, GEIR HUSEBY and OLE KRISTIAN Fé RRISDAHL Telemark Institute of Technology, HiT/TF, Kjù lnes Ring 56, N-3914 Porsgrunn, Norway (Received 29 February 1996; revised version accepted 17 November 1996) The possible e ects of PVP (poly(n-vinylpyrrolidone)) on the properties of liquid and water in clathrate hydrate has been investigated using NV T molecular dynamics simulations. A model for a monomer of the PVP polymer is immersed in three systems, liquid water, a unit cell of a hydrate in liquid water with a hydrate former and a third system where some of the liquid water molecules of this last system are replaced by a PVP monomer. Both molecular dynamics simulation and integral equation theory predict hydrogen bonding between the double bonded oxygen in the PVP ring and hydrogens in water. For the composite system, the PVP monomer has a preference for hydrogen bonding to hydrogens from the water molecules at the surface of the hydrate lattice. The simulations indicate that the PVP monomer tends to orient perpendicular to the hydrate surface. For the model systems in this study PVP may form hydrogen bonds with liquid water through the double bonded oxygen in the ring. When a hydrate crystal is immersed in the liquid water phase this hydrogen bonding is shifted towards the hydrate due to a more favourable Coulomb interaction involving hydrogens from more than one water molecule at the hydrate surface. The PVP monomer has a preference for perpendicular orientation with respect to the hydrate surface. A scheme is suggested for the characterization of kinetic hydrate inhibitors based on molecular dynamics simulations and on three basic properties. In addition to the energy between the active groups of the inhibitor and hydrate water another point of focus is the free energy changes in the interactions between the inhibitor and water as the charges are changed from zero to the original model charges. In particular the di erence between this integral for the (hydrate water)± (PVP monomer) interaction and the (liquid water)± (PVP inhibitor) interaction should re ect the driving forces in freezing the inhibitor out from the liquid water phase and onto the hydrate surface. The third property in the characterization scheme is the di usivities of groups connecting to the hydrate crystal, relative to the di usivities of the hydrate crystal. Results are presented from simulations where a small cavity with a methane model as a guest is immersed in liquid water with free methane molecules at a temperature of 150 K. Changes in structure, di usivities and energy indicate a tendency towards a more solid-like structurde around the cavity. 1. Introduction The formation of natural gas hydrates in pipelines and production equipment is a potential problem. Classically the inhibition of hydrate formation has been to shift the conditions for equilibrium between water in the hydrate and liquid water by adding alcohol or other substances to the system. The extra cost of chemicals and processing has turned the focus towards other types of chemical that will shift the kinetics for hydrate formation. Also, chemicals that change the behaviour of hydrate crystals towards small particles with minor tendencies to agglomerate are being tested in academic and industrial laboratories around the world. Common to these two types of kinetic inhibitors is the ppm concentration range of the inhibitor added. One of these kinetic inhibitors is PVP (poly(n-vinylpyrrolidone)). This component has been tested extensively by macroscopic kinetic experiments in several laboratories around the world, see, e.g., Anselme et al. [1]. A model for the monomers of this polymer is shown in gure 1. The polymer is constructed by connecting these monomers by the methyl groups outside the ring. An important property of this component is that it is hygroscopic and that it can adsorb signi cant amounts of water before it actually feels wet. IR spectroscopic studies also indicate a high degree of hydrogen bonding between the PVP and water [2, 3]. A number of other potential kinetic inhibitors are being tested today (see, e.g., Urdahl et al. [4] for some examples of other components that have been subject to di erent types of macroscopic kinetic experiments). In order to construct e ective kinetic inhibitors it is convenient to know more about the microscopic behaviour of these inhibitors in liquid water and on the surface of established hydrate 0026± 8976/97 $ Ñ 1997 Taylor & Francis Ltd.

3 980 B. Kvamme et al. nuclei. As a tool for establishing some of this knowledge we use molecular dynamics simulations. The aim of this work is twofold. First we want to know more about the possible behaviour of kinetic inhibitors in liquid water and in liquid water in contact with the hydrate. In addition we want to compose a scheme of molecular dynamics simulations and caluclations that can be used for testing potential inhibitors prior to expensive testing in the laboratory or pilot scale equipment. Such a scheme may also be used to examine the e ects of possible molecular modi cations that could lead to better kinetic inhibitors. If such a scheme is to have the desired value the results from this study should contain properties that are related to mass transport and free energy di erences across the di erent phase contact surfaces for the systems under consideration. In this rst part of our work in this direction we will limit ourselves to one component and we start with a model for PVP monomer to represent the kinetic inhibitors. Since PVP is hygroscopic and is always added to the liquid before any hydrate is present, a realistic simulation study has to imply PVP in the presence liquid water as well as hydrate. In this study we are not very concerned about the macroscopic surface of the hydrate since we expect it to be in uenced by the presence of PVP [3]and is correspondingly expected to be fairly non-ideal when compared with ideal surfaces that can be constructed from the pentagonal and hexagonal faces of the hydrate structures. As a consequence we apply a unit crystal of structure II constructed from experimental data for the hydrate phase, since this would give us the correct balance between large and small cavities. We will use a model intended to imitate Freon 11 for the hydrate forming molecules. In section 2 we present some details on the simulation procedures and simulation systems. In section 3 we present results from molecular dynamics simulations of a PVP monomer in liquid water. For comparison we also present some results from integral equation theory. In section 4 we discuss results from two composite simulation systems, both containing a unit cell of structure II hydrate. The large cavities of this unit cell are lled with a model for Freon 11. In the rst of these simulation systems we immerse this crystal in a liquid consisting of water and Freon 11 model molecules. This is the reference system (without inhibitor) for comparisons with another system where some of the liquid water molecules are replaced by a PVP monomer. The aim of these simulations is twofold. First we want to examine if these systems will grow to larger crystals. If they do it is of interest to examine possible di erences in growth rates and patterns of the growth process. For the PVP system we are interested in the speci c interactions between the PVP monomer and water in the liquid and hydrate phases. In section 5 we discuss results from another simulation set up where we start with a small cavity immersed in liquid water and methane. The cavity is lled with methane. These simulations are part of a prestudy. The main goal is to investigate whether a small cavity can support the growth to a larger clathrate. And if the indications of such a growth process is strong enough we will continue these studies using larger systems. 2. Computational details For the molecular dynamics (MD) simulations we have modi ed and extended the McMoldyn package, originally written by Laaksonen [5]. We simulate four di erent systems. In the rst of these, later denoted as system I, we simulate a system of 256 molecules consisting of one PVP monomer and 255 water molecules according to the model in tables 1 and 2. The simulation box is cubic and the volumne is kept constant during the simulation, with periodic boundary conditions in three dimensions. Long range Coulomb forces are treated by the Ewald summation method [6]. The temperature is controlled according to the thermostat procedure of Hoover [7, 8] using separate thermostats for the rotational and translational degrees of freedom. The control parameters for the two thermostats are xed according to an analysis of the frequencies for pure liquid model water at 273 K according to the procedures originally suggested by Nose [9] and discussed in more detail by Di Tolla and Ronchetti [10]. Simulations are started from an fcc lattice. A few thousand time steps (step size s) of temperature scaling are used in order to obtain a stable start con guration for the NV T simulations. The RISM results are obtained from the dielectrically corrected version of the RISM equation [11]. Computational details are given elsewhere [12, 13]. Simulation system II is a composite system where a unit cell of hydrate structure II [14± 20] ( A Ê cell constant) is immersed in a liquid mixture of 273 liquid water molecules and 41 apolar molecules intended to imitate Freon 11. In these simulations Freon 11 is represented as a spherical Lennard-Jones molecule with Lennard-Jones parameters s = A Ê and e /k B = K. These parameters are estimated using the corresponding states relations of Stiel and Thodos [21]. All the large cavities of the unit cell are lled with these Freon 11 model molecules. Small cavities are empty. TIP4P [22] is used as a model for water. The system is initially arranged in a cubic box where 81 liquid water molecules are placed in the top layer, followed by a layer of 64 water and 16 Freon 11. Then follows the unit crystal. A layer of 25 Freon 11

4 MD of PV P in water 981 Table 1. Interaction parameters and intramolecuoar structure for the PVP model. Site s /AÊ (e /k B ) /K q /C X/A Ê Y /AÊ Z /AÊ 1 C (ring) a CH 2 (ring) a CH 2 (ring) a CH 2 (ring) a O b N a CH 3 (chain) c CH (chain) a CH 2 (chain) c CH 3 (chain) c a From Maliniak et al. [33]. b From English and Venables [35]. c From Toxvñ rd [34]. Table 2. Water parameters (HOH angle is ë ; OH bond length is AÊ ). s /AÊ (e /k B ) /K q/c molecules is placed below the crystal. The rest of the liquid water molecules are placed on the sides of the hydrate crystal. Simulation system III is obtained from system II by replacing 41 water molecules in the central part of the top layer of the box with a model for a PVP monomer. Simulation system IV consists of 6 methane and 160 water (TIP4P) molecules, where 20 of the water molecules constitute a small cavity surrounding one of the methane molecules. The methane molecules are modelled according to the OPLS potential with s = A Ê and e /k B = A Ê. The small cavity is located initially at the centre of the simulation box and the 5 free methanes are lined up along one of the edges of the simulation box. The composite systems (II, III and IV) probably will not be in internal mechanical equilibrium at the start and it is uncertain whether they will reach internal mechanical equilibrium during the length of the performed simulations. As such the uncertainties in the calculated average pressure for the total box may be too large to enable the use of sophisticated pressure control algoithms [23]. It is, however, important to allow for some adjustments in volume in order to obtain a reasonable (with reference to average system pressure) density at the start and to allow the system to expand as a consequence of possible growth. We have therefore used a procedure where we adjust the O H volume slightly up or down if the pressure is too high or too low. This volume adjustment is applied every 5000 steps on the basis of the average pressure of the 5000 previous pressures calculated. If the average pressure during this period of time is outside a given range (see sections 4 and 5) the volume is adjusted by multiplying the box length with a factor of or , depending on the sign of the pressure deviation time steps is of course a short time in terms of retention times and equilibrium considerations for these systems, but by experience we nd that this period is su cient to reduce the oscillations in energy to a minimum. The work exchange related to these volume changes is small. 3. PVP in liquid water The PVP monomer is schematically illustrated in gure 1. The letter C denotes C, CH, CH2 and CH3 respectively. The corresponding model parameters and intramolecular structure are listed in tables 1 and 2 respectively. The model parameters are as indicated in the caption, collected from di erent sources, and the constructed model is not veri ed directly for speci c properties of PVP. Some of the parameters are generalized group parameters that have been used in other studies for a number of di erent components. The signi cant dipole moments in the ring are calculated from dipole moments of other components containing similar monomers of groups, and are expected to be fairly reasonable except for the nitrogen charges, which are slightly higher than expected. In the continuation of the work presented in this manuscript we have modi ed the potential and we also intend to investigate a exible model for the PVP monomer. The simulations presented are intended primarily to explore the qualitative trends. Details of the simulation set up (system I) are given in

5 982 B. Kvamme et al. Figure 1. Schematic picture of the PVCP monomer. C denotes either C, CH, CH 2 or CH 3. The oxygen is double bonded to C. Intramolecular structure is given in table 2. Figure 2. Oxygen± oxygen correlation functions at K and a density of kg m - 3. The solid line is the result from integral equation theory and the dashed line is the the previous section. The simulated structure for this system is compared with integral theory results for a PVP monomer at a mole fraction PVP monomer corresponding to 1/256 using the hypernetted chain closure (HNC). For this particular part of the study we have applied a water potential that is described in table 2. The reason for using a di erent water potential for this part of the study is that it is very hard to obtain a converged solution of the TIP4P [22]potential by RISM theory due to the short intramolecular distance between the centre of mass of oxygen and the location of the negative charge on the oxygen. The estimated structure from the applied potential is compared with experimental neutron data for pure water from Soper [24] as well as MD simulations in gures 2, 3 and 4. RISM-type of theory will not give the correct positions and heights of the main maxima and minima [12] for potentials like SPC and TIPS when compared with experimental neutron data. These gures are thus included as a veri - cation that the integral results as well as MD results are reasonable with respect to measured structure of real water for the proposed potential that is applied in this section. The calculated energies from this model are slightly lower than the experimental value [25] of kj mol - 1 at K. The corresponding RISM value is kj mol - 1 and the value from molecular dynamics simulation is kj mol - 1. Both of the latter values are calculated using the experimentally measured liquid density [25]. There are a total of 78 distinct correlation functions in this system. For the particular aspects discussed in this paper we will only present a few of the most relevant. The most interesting correlation function in this system are the correlations between water and the oxygen in PVP. Correlations between hydrogen in liquid water corresponding result from NV T molecular dynamics simulations, both for the water model described in the text. The dotted lines are neutron data for real water by Soper [24]. Figure 3. Hydrogen± hydrogen correlation functions at K and a density of kg m - 3 Notations as for gure 2. and the double-bonded oxygen in the PVP ring is presented in gure 5. Results from integral theory and MD both predict a hydrogen bonded structure around the PVP oxygen. There is no indication of hydrogen bonding to the nitrogen in the PVP ring. The results presented in gure 5 indicate that the PVP kinetic inhibitor is active also at the stage where no hydrate cores are present. The hydrogen bonding to the oxygen in the PVP ring will make the ring partially or completely soluble in water. From the MD simula-

6 MD of PV P in water 983 Figure 4. Hydrogen± oxygen correlation functions at K and a density of kg m - 3 Notations as for gure 2. Figure 5. Correlation functions for hydrogen in liquid water and the double bonded oxygen in PVP. The solid line is the result from integral equation theory and the dashed line is the result from MD simulations. Temperature is K, density is kg m - 3, and mole fraction PVP monomer in the water is tion the Coulomb energy between the PVP monomer and water is on average kj mol - 1 while the corresponding short-range contributions to the energy is kj mol - 1, i.e., a total energy between water and PVP monomer of kj mol - 1. The corresponding interaction energies between the PVP polymer and the surface of water, later denoted as the adsorption energy, will exceed that of the most common hydrate formers in natural gas mixtures (methane, ethane and other hydrocarbons up to C 4 plus some inorganic gases). Even small concentrations, in the ppm range relative to the amount of water in bulk phase, may e ciently reduce the free space available to hydrate formers on the water surface. Thus, in the development of a theoretical characterization scheme for kinetic hydrate inhibitors it is also important to consider the stage before stable hydrate cores are present. For this purpose RISM theory may represent a convenient and feasible alternative to MD if we make some approximations. The PVP backbone will be almost insoluble in water. First we arti cially `release the water soluble components from the backbone and dissolve them in water at a concentration corresponding to the average concentration of such components in a real polymer monomer with these soluble parts completely in the water phase. For the PVP polymer this means that we dissolve a number of rings (see gure 1) in water. The average structure and corresponding energy for the PVP/water interactions give a route to an estimate for the average adsorption energy for the soluble part of the PVP polymer. The justi cation for this procedure is that although the soluted rings are not in a true sense uniformly distributed we expect at least some regularity in the arrangement of rings in the water phase and in an approximate sense the procedure will represent a simple and feasible approach for an estimation of the energy between the soluble part of the inhibitor and liquid water. For the backbone we may assume that it is completely at the water surface. With the assumption that the water surface is at or alternatively that the backbone penetrates the surface according to a pure static force balance where the weight alone of the polymer is approximately assumed to cause the penetration. In either case it is possible to make a rough estimate of the additional adsorption energy due to the backbone. In summary, it appears possible to obtain qualitative estimates for the adsorption energies of PVP and other similar kinetic inhibitors onto the water surface. This could then serve as a rst indication of the relative adsorption abilities of di erent inhibitors onto the liquid water surface/interface. These energies may then be applied also in adsorption theory through a grand canonical partition function for the rst adsorbed layer on the water surface in order to estimate relative surface fractions of inhibitor and the di erent hydrate formers in the hydrate forming phase. Within this approach it is also possible to examine the e ects of di erent chain length distributions in the polymer. According to these procedures the adsorption energy will be directly proportional to the number of segments in the polymer, where one segment is the smallest repeating unit in the polymer. This information coupled to the information on the excluded surface area of polymers as a function of chain length can be used to

7 984 B. Kvamme et al. estimate the average structure of the surface adsorption. The range in chain length distribution is expected to in uence the inhibition e ect since a narrow distribution will have a lower degree of surface lling than a broader one for the same type of polymer (for the particular model of PVP in gure 1). 4. PVP monomer in composite systems System II is constructed to be a reference system for system III which contains the inhibitor. If the hydrate grows in the two systems it would be interesting to compare the growth rates and details of how the inhibitor monomer might interfere with the growth process. The properties of this model molecule have not been tested against any mesured properties of real Freon 11. As such it is more correct to interpret the simulation results as results for a speci c model system. Water molecules are represented by the TIP4P [22] potential. A certain degree of subcooling is necessary in experiments as well as in simulations. A degree of subcooling (relative to the temperature of hydrate equilibrium) of 5± 20 K is frequently used in macroscopic experiments on kinetics of hydrate formation. Since the freezing point of TIP4P may be 240 K [24] or lower we have chosen 180 K in simulations on these two systems. System II is started from an average density of g cm - 3, which corresponds to a cubic simulation box of length A Ê. At the end of the simulations (after 175 ps) the density is g cm - 3 and the corresponding simulation box has a length of AÊ. System III is started at the same initial box length and the corresponding values for densities are g cm - 3 and g cm - 3 for start and end (after 175 ps), respectively. The box length at the end of simulations in this system is AÊ. PVP monomer is initially in the liquid phase and from gure 6 it is seen that the double bonded oxygen is hydrogen bonded to liquid water. During the progress of the simulation the hydrogen bond structure is shifted from the liquid water towards hydrogens on the surface of the hydrate, as illustrated in gure 7. The three rst peaks are sharp and the locations of maxima correspond well with the locations of water neighbours in the surroundings to the directly hydrogen bonded water in the lattice. A qualitative picture of the most likely orientations of the inhibitor relative to the surface of the crystal can be obtained by considering the correlation function between the oxygen in the hydrate and the di erent groups in the PVP monomer. In gure 8 we plot correlation function between oxygen in hydrate and the methyl groups in the backbone. From gures 7 and 8 it is seen that the inhibitor has a tendency to orient perpendicular to the surface of the Figure 6. Correlation functions for hydrogen in liquid water and the double bonded oxygen in the PVP ring for the composite system. The di erent lines indicate simulation time: 15 ps (dotted line); 55 ps (dash± dot); 115 ps (dashed); and 175 ps (solid). Figure 7. Correlation functions for hydrogen in hydrate water and the double bonded oxygen in the PVP ring for the composite system. Notation as for gure 6. hydrate. This obviously is an e ect of the liquid water environment since a PVP monomer free of entropy constraints and competition from liquid water molecules would prefer to lie at on the water surface in order to increase the interaction energy between the PVP monomer and the hydrate surface. The particular orientation could have been illustrated more clearly if we had sampled (as we will do in future work along these lines) the correlations between the centre of the hydrate crystal and the sites of the PVP monomer. An analysis of the distances between the three rst peaks in the correlation

8 MD of PV P in water 985 Figure 8. Correlation function between oxygen in the hydrate and the methyl groups outside of the ring after 175 ps of simulations in system III. The dotted line is the methyl groups that are connected at the opposite side of the ring with reference to the double bonded oxygen (see gure 1). The dashed and solid lines are for the CH 2 groups outside of the ring, the solid line representing the group closest to the double bonded oxygen in the ring. functions for hydrogen in hydrate water and oxygen inhibitor ( gure 7) shows that the peak locations correspond perfectly to locations of simultaneous favourable contact with a central hydrate water and its neighbouring water molecules in the hydrate. These results suggest that the inhibitor may stick to the surface of the established nuclei and slow down the growth due to steric hindrance. If we extend this reasoning to a more complete polymer we may expect di erent PVP monomers in the polymer to stick to one or several di erent nuclei. The distances between the di erent rings in the polymer monomer will also make it di cult to obtain favourable structuring of hydrate between neighbouring rings. Interconnections between hydrate nuclei in the water part of the interface are thus likely to occur below the rings of the PVP polymer. Depending on the supersaturation of water at the hydrate former side of the interface there is also a possibility of independent structuring towards hydrate at the top of the apolar part of the PVP polymer. In either case the di usion rates of water and the hydrate former across the polymer covered surface and the established nuclei are likely to be slow compared with the situation without PVP. There is of course an inherent limitation in these present simulations since the PVP monomer is modelled as a rigid and fairly small molecule. In order to get a better picture of the e ects of PVP it is important to simulate a larger PVP segment, at least two or three monomers in a chain. Figure 9. Di usivities for the PVP monomer (solid), liquid water (dashed) and liquid Freon 11 (dash± dot) as functions of simulation time in picoseconds for system III. In gure 9 we plot the di usivities for the liquid phase molecules in system III. The water result will be an average over water molecules that are a ected by the apolar molecules (Freon 11 and apolar groups in PVP monomer), `free liquid water molecules, and water molecules that are a ected by the water molecules at the crystal surface. For water and for Freon 11 these di usivities re ect transport since these molecules are spherical or approximately spherical with respect to mass distribution. For the PVP monomer, on the other hand, oscillations from a preferential perpendicular orientation ( gure 8) with respect to the hydrate surface will move the centre of mass and thus contribute to the calculated di usivity. In a characterization scheme we should sample displacements of the oxygen group rather than the centre of mass for the total inhibitor monomer. There is also the possibiity of simultaneous movement of the hydrate crystal and the inhibitor which also may contribute to the sampled di usion of PVP monomer. In gure 10 we plot the system energies for system III as a function of simulation time. The average trend is that the changes in energy are gradually reducing. Although the gradient in the energy change is getting smaller we cannot state that the system is in equilibrium. The equilibrium state is unknown. It is known that TIP4P freezes at 240 K or lower (Karim and Haymet [26]), although the freezing point has not yet been determined rigorously. Whether TIP4P is actually able to form hydrate from liquid water and hydrate former is not veri ed in any simulations so far. What we do know is that TIP4P as well as SPC may form a number of

9 986 B. Kvamme et al. Figure 10. Average system energy of system III as a function of the simulation time in picoseconds. Dots indicate standard deviation. di erent ice structures that are not known experimentally (Clancy [27]). In summary we cannot expect that a system consisting of a pre xed hydrate unit cell and a liquid phase should equilibrate at 180 K within reasonable simulation times. As indicated above, we do not know what phase(s) the system would end up in. For the purpose of establishing a characterization procedure to distinguish between the di erent kinetic inhibitors this may not be necessary. The important thing is that all systems are evaluated by the same procedure (same temperature, same simulation time, same type of guest molecule, same starting con guration, etc.). Since the system is in a state of slow variations in energy the simulation may be long enough for the purpose. In gure 11 we show the changes in average energy between the PVP monomer and liquid water molecules. The average values are kj mol - 1 and kj mol - 1 for Coulombic and Lennard-Jones energies between PVP monomer and water, respectively. The corresponding average energies between PVP monomer and hydrate water molecules are plotted in gure 12. The average values for the interactions between PVP and hydrate water are kj mol - 1 and kj mol - 1. The Coulombic contribution between the PVP monomer and hydrate water is signi - cantly larger than the corresponding contribution between PVP monomer and liquid water. Furthermore, since the PVP oxygen tends to stick to the hydrate water ( gure 7) it is likely that the Coulombic contribution between liquid water and PVP monomer is due mainly to the interactions with the nitrogen group in the ring. One of the roles of the nitrogen group thus may be that it increases the water solubility of the PVP ring. Another Figure 11. Average energies between PVP monomer and liquid water as a function of the simulation time for system III. The Coulombic contributions are indicated by the dash± dot line, the dashed line is the Lennard- Jones contribution and the solid line is the total energy. Figure 12. Average energies between PVP monomer and hydrate water as a function of the simulation time for system III. The Coulombic contributions are indicated by the dash± dot line, the dashed line is the Lennard- Jones contribution and the solid line is the total energy. interesting aspect from gure 12 is that the Lennard- Jones contributions are certainly signi cant, and the `attachment energy (Carver et al. [28]) based on only the Coulombic contribution may be insu cient for the purpose of classifying hydrate inhibitors. 5. A scheme for characterization of kinetic hydrate inhibitors In the previous sections we have described some characteristic features of composite systems containing

10 MD of PV P in water 987 kinetic hydrate inhibitors or, rather, monomers of these. The energy between the kinetic inhibitor and hydrate water is one quantity that can describe the energies that holds the kinetic inhibitor at the hydrate surface. Another important quantity is the di usion constant which describes the average transport of the PVP monomer through the system. As indicated in the previous section we would prefer to sample the di usion of the functional atoms or groups in future simulations. In addition we need a quantity that is proportional to the surface tension between liquid water and hydrate water. Since we intend to simulate the systems at the same temperatures we might anticipate that the chemical potential of hydrate water will be altered only slightly due to the inclusion of di erent types of kinetic inhibitor. This is due to the fact that the number of water molecules in the hydrate is large compared with the single kinetic inhibitor monomer. The same goes for the liquid water molecules. Furthermore, if the kinetic inhibitor is free from partial charges it will behave similarly to any other apolar molecule, and it will be almost insoluble in water. If any inhibition e ect is seen from these apolar chains it will almost entirely be an e ect due to steric hindrance of hydrate former transport towards water. If we integrate the changes in energy between the kinetic inhibitor and water in liquid and hydrate phases we can sample the change in chemical potential of the kinetic inhibitor and the corresponding contacts (i.e., liquid water and hydrate water). This is a quantity that will be proportional to the change in surface tension from the apolar chain to the partially charged chain. In order to accomplish this we scale the charges on all sites in the PVP monomer by a scaling factor that varies between 0 and 1 and integrate the corresponding energies calculated from MD simulations. The integral over the changes in energy between the PVP monomer and water in the hydrate should be as low as possible. The corresponding number for the liquid water and PVP monomer interaction should be as low as possible to ensure good solubility but small enough to ensure that the backbone essentially will still be on the surface of water. In addition the di erence between the integral of the changes in the energies between water in hydrate and liquid, respectively, and PVP monomer should be as low as possible. This di erence will re ect the driving force in the process of `freezing the hydrate out from the liquid water phase and connecting it to the hydrate surface. Relative to a practical situation where the PVP is present in liquid water before the onset of hydrate formation, a decreasing value for this number will re ect an increasing tendency for the PVP to connect to hydrates that are in the formation process. In view of this it is likely that the hydrate surface that is formed in the presence of kinetic inhibitors will be fairly non-ideal when compared with ideal surfaces that can be constructed from the geometry of the hydrate cavities, as in the study by Carver et al. [28]. Figures 13 and 14 shows the energies between water in the hydrate and the liquid, respectively, and the PVP monomer. If these results were to be used in a comparison between di erent inhibitors, as we will demonstrate in the continuation of the work presented here, we would prefer to use a ner grid for the nal part of the integration. This Figure 13. Coulombic (dashed) and Lennard-Jones (dash± dot) contributions to the average interaction energy (solid) between water in hydrate and PVP monomer as a function of the scaling of charges in the PVP monomer. Figure 14. Coulombic (dashed) and Lennard-Jones (dash± dot) contributions to the average interaction energy (solid) between liquid water and PVP monomer as a function of the scaling of charges in the PVP monomer.

11 988 B. Kvamme et al. Table 3. Calculated properties of the composite system. Energy integral denote the value of the integral over the energy as a function of the scaled charges on the PVP monomer. Coul. denotes the coulombic contributions and L-J denotes the short-range contributions to the total energy. Property Total Coul. L-J Total potential energy/kj mol Hydrate water± PVP monomer/kj mol Liquid water± PVP monomer/kj mol Energy integral hydrate water± PVP monomer/kj mol Energy integral liquid water± PVP monomer/kj mol Di erence in free energy change/kj mol Di usion of hydrate water/m 2 s E-09 Di usion of liquid water/m 2 s E-09 Di usion of freon hydrate/m 2 s E-09 Di usion of freon liquid/m 2 s E-08 Di usion of PVP monomer/m 2 s E-09 integration, where only energies are needed, could be performed more e ciently by a Monte Carlo procedure. Since the main purpose of this paper is to present a procedure for the use of molecular dynamics simulations as a tool in the characterization of hydrate kinetic inhibitors, we present the integrated numbers obtained from the values in gures 13 and 14 and consider these values as qualitative indications. In table 3 we list nal values for some key properties that are the basis for the characterization procedure proposed here. As indicated above, the scaling of the charges will re ect only the changes in chemical potential from that of a completely apolar polymer. This means that a comparison between the di erent kinetic inhibitors for the speci c e ect of the rate reduction that is due to the blockage of the surface of the apolar reference systems has to be considered separately. For a number of these kinetic inhibitors the backbone is an alkane chain. With proper simpli cations of the attached groups it might be possible to use simpli ed theories for polymer packing properties for this part of the comparison. RISM theory might be one option for this purpose. When considering the di usivities in table 3 we note that Freon 11 in the hydrate has limited space for movement in the cavities and a signi cant portion of the di usivity of freon in the hydrate is likely to be the collective movement of the crystal in the simulation box. It is therefore reasonable that this value is close to the di usion of hydrate water. As indicated previously the di usivity of the PVP monomer re ects the movement of the centre of mass through the box as well as oscillations around a perpendicular orientation relative to an attached oxygen and posibly also other rotational degrees of freedom. Some of these contributions might also be a result of the initial stages where PVP is in the liquid phase. In future studies we will sample di usivities of all charged groups in the inhibitors in addition to the di usivity of the hydrate crystal. Unfortunately very few fundamental studies have been made regarding PVP, at least within the open published literature. Since we actually are studying a monomer of the PVP we cannot compare our results with any experimental data of real substance at this stage. Some of the properties suggested in this scheme, like for instance the surface tension between PVP and liquid water, probably are measurable and might be used for the purpose of model veri cation. However, as discussed previously the proposed scheme re ects a set of values that will basically have to be used for comparison between di erent inhibitors. As such the largest value of any simulation scheme for this purpose will be in connection with experimental data. In particular it would be fruitful to compare simulated results for some of the components that have been studied extensively in the laboratory in order to draw lines between experimental ndings and key results from the simulations for the same type of components. This is the direction of our future work in this area. 6. Small cavity in liquid water and methane At least one of the hypotheses [29] concerning the initial formation of hydrate nuclei postulates that the formation of small cavities is the rst step in the formation of hydrate nuclei. In a recent study we simulated hydrate structure I with SPC for water and OPLS for methane [30] during melting and refreezing. It was not possible within the speci c procedures applied in that study to nd any traces of particles or cavities that were more stable than the rest of the system. It was not possible within the speci c procedure to obain any hydrate by refreezing the melted system. System IV (see section 2) therefore is designed as a system for a

12 MD of PV P in water 989 prestudy of whether a small cavity is able to support the formation of hydrate or not. The time scales in molecular dynamics will probably limit the possibility of observing extensive growth but even indications of regrouping and structuring towards possible growth could justify more extensive work in this direction. Other types of simulation will be necessary [31] in order to comment on the probability of the small cavity formation being the rst step in the hydrate formation process as opposed to the possibility of a more complex and collective process. The OPLS methane parameters are fairly close to the parameters for methane (s = AÊ and e /k B = K) applied in a recent study on hydrate equilibrium [32] using TIP4P for water. The calculated equilibrium pressures were found to be in fair agreement with experimental data for real methane hydrate in equilibrium with ice. This system has been simulated at 150 K. The system is started from an initial con guration where the hydrate cavity with one of the methane molecules is located in the centre of a cubic box of length A Ê. The 5 free methane molecules are located along one of the top edges. The volume is adjusted slightly according to the procedure described and within `acceptable pressure limits of 100± 300 bar. The nal density of this system is g cm - 3 after 500 ps. Any possible growth of the hydrate on the single cavity should result in signi cant changes in the correlations between methane inside the cavity and the methanes in the liquid phase outside. Solid-like structuring around the cavity should also result in more structured correlations between the oxygens in the cavity and the oxygens outside the cavity. As a third indication we also may investigate the changes in energy of the system. Since the volume changes are fairly small the direct changes in energy associated with the limited volume changes of the box are fairly small. Changes in energy due to a transition towards larger portions of the solid-like phase will, on the other hand, result in signi cant changes in the system energy, and may be compared with corresponding experimental values for a transition to hydrate. There will of course be some di erences between the model system and real water, but in a qualitative sense the experimental values should give an indication. In gure 15 we have plotted the correlations between the liquid methane and methane in the cavity for the system at 150 K. The rst region of peaks after 500 ps is representative for distances corresponding to situations where the free methane is located at the centre of the pentagonal water sides of the cavity. Distances corresponding to the next maxima corespond to distances that can also be typical for guest± guest distances Figure 15. Correlation between methane in the small cavity and free methane as a function of simulation times: 20 ps (dotted); 140 ps (dash± dot); 340 ps (dash); and 500 ps (solid). Figure 16. Correlations between the oxygen in hydrate and the free methane. Simulation time notations as for gure 13. in the hydrate. The corresponding correlations between oxygen in the cavity and methanes in the liquid phase are plotted in gure 16. These correlation functions and a closer analysis of the molecular positions in the ensemble after 500 ps shows that the methanes are located in the surface or close to the surface of the cavity. This is also veri ed by gure 17, which shows the correlations for free methane. There are still some direct contacts between methane molecules but there is a clear shift towards a spreading of the molecules. Since these correlation functions are the average of all

13 990 B. Kvamme et al. Figure 17. Correlations between methane in the cavity and free methane. Notations as for gure 13. Figure 18. Correlations between oxygen in the cavity and the water molecules constituting the cavity. Notations as for gure 13. previous con gurations we may expect that the instant distribution is more pronounced in this direction. Correlation functions for methane in the cavity and oxygens constituting the cavity wall show only minor changes in structure during the history of the simulation, as is seen from gure 18. The calculated selfdi usion coe cients are plotted in gure 19. Coe cients for water and methane in the cavity will re ect both collective movement of the cavity and librational movements. The di usion of methane and water is more illustrative and the trend indicates a process towards freezing. The signi cant reduction in energy with time also veri es this observation, see gure 20. Figure 19. Di usion coe cients for water in the hyderate (solid), in liquid water (dashed), methane in the hydrate (dash± dot), and free methane (dotted). Figure 20. Average system enthalpy as a function of the simulation time. In summary, the observations above has led to the establishment of a new study along these lines but using a larger simulation system [31]. 7. Conclusion For the model systems in this study we nd that PVP may form hydrogen bonds with liquid water through the double bonded oxygen in the ring. When a hydrate crystal is immersed in the liquid water phase this hydrogen bonding is shifted towards the hydrate due to a more favourable Coulomb interaction involving hydrogens from more than one water molecule at the hydrate surface. The PVP monomer has preference for

14 MD of PV P in water 991 perpendicular orientation with respect to the hydrate surface. We have suggested a scheme for the characterization of kinetic hydrate inhibitors based on molecular dynamics simulations for three basic properties. In addition to the energy between the active groups of the inhibitor and hydrate water, we propose the calculation of free energy changes in the interactions between inhibitor and water as the charges are changed from zero to the original model charges. In particular, the di erence between this integral for the hydrate water± PVP monomer interaction and the liquid water± PVP inhibitor should re ect the driving forces in freezing the inhibitor out from the liquid water phase and onto the hydrate surface. The third property in the characterization scheme is the di usivities of groups connecting to the hydrate crystal, relative to the di usivities of the hydrate crystal. The possible growth from a small cavity as seed in a system of liquid water and methane also has been investigated. The simulations show changes in structure, energy and di usivities that may be associated with a phase transition towards a more solid phase. References [1] Anselme, M. J., Reijnhout, M. J., and Muijs, H. M., 1993, Netherlands Patent No [2] Molyneux, P., 1975, W ater: A Comprehensive Treatise, Vol. 4, edited by F. Franks (New York: Plenum Press), p [3] Beerbower, A., Kaye, L. A., and Pattison, D. A., 1967, Chem. Eng., 74, 118. [4] Urdahl, O., Lund, A., Mørk, P., and Nilsen, T., 1995, Chem. Eng. Sci., 50, 863. [5] Laaksonen, A., 1986, Comp. Phys., 42, 271. [6] Allen, M. P., and Tildesley, D. J., 1987, Computer Simulation of L iquids (Oxford: Clarendon Press). [7] Hoover, W. G., 1985, Phys. Rev. L ett., 31, [8] Toxvaerd, SA., 1990, Molec. Phys., 72, 159. [9] Nose, S., 1984, Molec. Phys., 52, 255. [10] Di Tolla, F. D., and Ronchetti, M., 1993, Phys. Rev. E, 48, [11] Perkyns, J. S., and Pettitt, B. M., 1992, J. chem. Phys., 97, [12] Kvamme, B., 1994, Fluid Phase Equilibria, 100, 157. [13] Kvamme, B., 1995, Int. J. Thermophys., 16, 743. [14] Stackelberg, M. von, 1949, Naturwiss., 36, 327. [15] Stackelberg, M. von, 1954, Z. Elektrochem., 58, 104. [16] Stackelberg, M. von, 1956, Rec. Trav. chim., 75, 902. [17] Stackelberg, M. von, and Fru hbuss, H., 1954, Z. Elektrochem., 58, 99. [18] Stackelberg, M. von, and Jans, W., 1954, Z. Elektrochem., 58, 162. [19] Stackelberg, M. von, and Meinhold, W., 1954, Z. Elektrochem., 58, 40. [20] Stackelberg, M. von, and Muller, H. R., 1954, Z. Elektrochem., 58, 25. [21] Stiel, L. I., and Thodos, G., 1962, J. chem. Eng Data, 7, 234. [22] Jorgensen, W. L., Chandrasekjhar, J., Madura, J. D., Impey, R. W., and Klein, M. L., 1983, J. chem. Phys., 79, 2. [23] Andersen, H. C., 1980, J. chem. Phys., 72, [24] Soper, A., unpublished. [25] Weast, R. C., 1977, Handbook of Chemistry and Physics, 58th Edn (Baton Rouge, FL: CRC Press). [26] Karim, O. A., and Haymet, A. D. J., 1987, Chem. Phys. L ett., 138, 531. [27] Clancy, P., 1995, AIChE annual meeting, Miami, 2± 6 Nov. [28] Carver, T. J., Drew, M. G., and Rodger, P. M., 1995, J. chem. Soc. Faraday Trans, 91, [29] Sloan, E. D., 1990, Clathrate Hydrates of Natural Gases (New York: Dekker). [30] Førrisdashl, O. K., Kvamme, B., and Haymet, A. D. J., 1997, Molec. Phys., in press. [31] Førrisdashl, O. K., and Kvamme, B., unpublished. [32] Kvamme, B., and Tanaka, H., 1995, J. phys. Chem., 99, [33] Maliniak, A., Laaksonen, A., and Korppi-Tommola, J., 1990, J. Amer. chem. Soc., 112, 89. [34] Toxvaerd, S., 1990, Molec. Phys., 72, 159. [35] English, C. A., and Vena bles, J. A., 1974, Proc. R. Soc. L ond. A, 340, 57.