Hydrate Nucleation and Inhibition: New Methods for Deeper Understanding

P. Mark Rodger*

Department of Chemistry, and Centre for Scientific ComputingUniversity of Warwick

Coventry, CV4 7ALUNITED KINGDOM

ABSTRACTWe present results of a series of molecular dynamics simulations of both the nucleation of methane hydrate and of the effect of various chemical additives on the hydrate formation and growth. It is shown that two different mechanisms for inhibition are observed: one is to destabilize small clusters and thereby increase the critical nucleus size needed for sustained

crystal growth and surface involves no adsorption to the hydrate surface; the second involves surface adsorption but actually works by blocking growth at the sub-micron scale rather than by inhibiting nucleation. Some developments to underpin routine use of molecular modeling to characterize a wide range of additives are discussed.

Keywords: gas hydrates, kinetic inhibitors, molecular dynamics, metadynamics

* Corresponding author: p.m.rodger@warwick.ac.uk

NOMENCLATUREfi Ageneric symbol for order parameters

(measures of local, molecular order)f3 A measure of tetrahedral order within a

hydrogen bonding network

f4 A measure of the torsion angles characterizing a hydrogen bond network

f4t Another, independent, measure related to the torsion angles characterizing a hydrogen bond network

P(m |n) The cluster size transition probability,

which measures the probability and magnitude of cluster growth or shrinkage over a fixed time.

INTRODUCTIONA major target for controlling hydrate formation

remains the nucleation stage. There is a demand for ever more potent but environmentally-friendly additives that can interfere with either nucleation or the early stages of growth to deliver reliable prevention of hydrate formation under ever moreextreme production conditions. At the same time,

recent advances in our understanding of crystal

formation in areas such as biomineralisation have shown how control of the nucleation and early growth mechanisms can generate exquisite control over the physical properties of the resulting solids; if similar control can be effected with hydrates it

wo uld bring considerable advantages for those emerging technologies that seek to exploit hydrate formation.The nucleation and initial growth of hydrates occurs on the nanoscale, and as such is beyond the

resolution of many common experimental techniques used to study hydrates. It is, however, the lengthscale for which molecular simulation is ideal, and so simulations of hydrate nucleation are potentially a very powerful tool for developing additives to control nucleation. Unfortunately, the

timescale accessible to conventional simulations is much shorter than that on which crystal nucleation typically occurs, and so while there have been a few studies over the last decade that have simulated nucleation[1-3] and low dosage inhibition,[4-6] implementing standard molecular

dynamics simulations of methane hydrate formation is still very challenging.

Proceedings of the 7th International Conference on Gas Hydrates (ICGH 2011),Edinburgh, Scotland, United Kingdom, July 17-21, 2011.

Over the last few years, new molecular dynamics methods have been developed that are very well suited to simulating rare events. In particular, metadynamics can be used to simulate crystal nucleation[7] from solution[8] repeatably, and can calculate the free energy landscape associated with

the crystal formation.[9] The method is very well suited to polymorphism, and also has been shown to provide rigorous thermodynamic information on the way chemical additives affect the thermodynamics of the crystal nucleation.

In this paper we will present our work on simulations of methane hydrate formation, with and without low dosage inhibitors present. We will discuss the mechanistic insights that can be provided by conventional molecular dynamics and consider how technique developments such as

metadynamics might provide the next large step towards deepening this insight.

METHODSMolecular Dynamics simulationsThe procedures used to simulate that the water /

methane interface have been described elsewhere[2 -

3] and only summarised here. The simulations were of a water film containing dissolved methane and surrounded (above and below) by a methane fluid. This was created by constructing a 3×3×4 array of

methane hydrate unit cells, creating a surface by expanding the z direction of the simulation supercell to include a 60 Å vacuum gap, and then performing MD simulations for varying lengths of time at 300–330 K. Six different configurations have been prepared in this way, differing in the

length of time for which the melting proceeded. The shortest melt yielded a water film in which no residual hydrate structure was observed and which contained a methane concentration 50% of the fully occupied hydrate (the rest of the methane having entered the “vacuum” to give a methane

gas/fluid phase); we denote this as system A; the longer melts yielded successively less methane in the water film: 25% of the fully occupied hydrate (B), 10% (C), 5% (D), 2% (E) and 1% (F). About 800 additional methane molecules were then

added to the gas phase to give a pressure of 30 MPa at 250 K. The final dimensions of the system

were 3636108Å, and the water film was about 45 Å thick. Nucleation simulations were then conducted in the NVT ensemble, with trajectories followed for 25–50 ns.The simulations were performed using

DLPOLY_2[10] with a 1 fs time step. The smooth

particle mesh Ewald (SPME) method[11] was used for electrostatic interactions with all real space force calculations truncated at 12 Å. The SPC model was used for water and a single-site Lennard-Jones potential was used for methane; these potentials have been found to give a good

representation of methane hydrate in previous work.[12] Bond length and angle constraints were implemented using SHAKE.[13]

Analysis of hydrate structureA key element to probing the onset of hydrate

formation, and hence quantifying the effect of various inhibitors, is to be able to identify hydrate structure at a molecular level. A number of studies have done this by identifying immobilized water molecules, [14-15] as hydrate-like clusters tend to diffuse together, and hence diffuse as

supermolecule with much larger mass. This is certainly a sufficient criterion to know that solid-like behaviour has emerged, but it does introduce a lower bound on the ability to resolve transient structure, and such transient structure may be crucial to the activity of inhibitors. In our work we

have therefore adopted a more structure-based definition that recognizes the inherent variability of local structure within the stable bulk phases (for example, due to the presence of vibrational motion within any crystalline phase). Our method

proceeds by defining an appropriate set of parameters that quantify the structural environment of individual water molecules—we call these order parameters and refer to the set as {fi}. The characteristic distribution of each order parameter is then measured in simulations of the

stable bulk phase. It is then possible to calculate the {fi} about a molecule in any environment and determine whether it is consistent with a unique bulk phase. Since the distribution of order parameters is found to be essentially Gaussian, it is only ever possible to ensure consistency within a

specified statistical significance; in our work we have adopted a 95% significance level, and therefore associate a molecule with a particular phase if each element of its set of order parameters is within two standard deviations of the bulk phase

average value (where the standard deviations are also those of the bulk phase). In this way it is possible to identify the hydrate (or any other possible phases) component from any instantaneous configuration, and hence to identify any rapid fluctuations of within the network of

“hydrate” water. In applications to hydrate formation we have found that a set of three order

parameters to be particularly useful in distinguishing bet ween ice, hydrate and liquid water:[16] a measure of tetrahedrality (f3) and of the

torsion angle (f4) and triple product (f4t) associated with three adjacent links in the hydrogen bonded network between water

molecules. We note, however, that these order parameters will not distinguish between different hydrate phases, and so would need to be extended in order to elucidate polymorph selection from the amorphous hydrate phase that has now been found in many simulations of hydrate formation.[2, 17-18]

A major advantage of this approach is that it enables us to the kinetics of the H-bonded network with very good time resolution. In particular, it has allowed us to quantify the propensity for any given sized hydrate cluster to either grow or shrink over periods as short as 1 ps. It is therefore possible to

begin to get detailed information about how a given inhibitor affects the evolution of hydrate within the system. To quantify this we have calculated conditional transition probabilities, P(m |n) which determine the probability that a

water molecule that is currently part of a hydrate cluster containing m water molecules will be part of a cluster containing n water molecules at a specified time later (taken to be 1 ps in our work). These probabilities have been calculated by noting all occurrences of a cluster of size m in our

simulations, and then constructing a histogram of the cluster sizes that the constituent water molecules are involved in 1 ps later.

METHANE HYDRATE NUCLEATION AND EARLY GROWTHThe growth of hydrate within simulations A, B and C has been reported previously.[3] The data is summarized in Figure 1.It is seen that at the highest aqueous methane concentration, the onset of hydrate growth is immediate, though not entirely monotonic; at the lower concentrations

(10-25% of the fully occupied methane hydrate) there is an induction time of about 10 ns before the onset of growth. For the lowest concentrations (simulations D–F), there was no observable onset of hydrate growth over 50 ns of simulation,

although there was evidence of a net flux of methane from the methane fluid phase into the aqueous phase. It should also be noted that the progression of growth was observed to be stochastic, as would be expected for the very early stages of nucleation and growth. Figure 2 shows a

series of simulations started from a configuration

taken after about 17 ns of simulation A; the simulations differ only in the velocities allocated to each particle at the beginning of the simulation, but the subsequent hydrate growth differs significantly in each case.

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 10 20 30 40 50

t / ns

hyd

rate

fra

ctio

n

A

B

C

Figure 1: hydrate fraction of water for simulations

A, B and C

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2t / ns

hyd

rate

fra

cti

on

A1

A2

A3

Figure 2: hydrate fraction of water for three simulations started from a configuration taken about 17 ns into simulation A. The simulations

differ in the allocation of velocities to the atoms.

Contour maps of the hydrate growth transition probability, P(m |n), for the initial growth of simulation a is depicted in Figure 3. The line m=n(shown in white on the plots) represents the stable

cluster: neither growing nor shrinking. Contours above and to the left of this line indicate the probability of the hydrate cluster shrinking, while contours below and to the right represent growth. Contours far from this line indicate the probability

that a large number of hydrate water molecules left (joined) the cluster in just 1 ps. When averaged over longer periods of simulation, the contour plots appear remarkably well balanced bet ween growth and shrinkage, and even though rapid growth of hydrate is observed through this

simulation, the hydrate particles are clearly undergoing rapid oscillations in size. It is also interesting to note that many of these exchanges involve significant numbers of water molecules, suggesting that small clusters (and not just individual water molecules) are consistently

joining and leaving the larger clusters. In fact, visual inspection of the hydrate networks indicates that this relates to flickering dendrite-like networks of hydrate-water around the surface of the cluster forming and disintegrating on short

timescales.Figure 3 actually shows P(m |n) calculated over three different time intervals (0–1.5, 0–4.2 and 0–6.9 ns). It is important to realize that the transition probabilities for a particle of size m cannot be calculated until a particle of that size has been

observed; hence, given that simulation A displays consistent hydrate growth, the transition probability extends to larger cluster sizes as the length of simulation increases. What is most interesting about these different time-averages is that the leading edge (the region of the transition

probabilities relating to the largest clusters observed up to that time) always shows a strong bias to shrinkage. This was also seen in simulations B and C once growth started. It would appear that even when there is a substantial

driving force to cluster growth, many false trial additions are still necessary before stable additions are found, and so suggests that the nucleation process is nonclassical.

LOW DOSAGE INHIBITORSHaving been able to observe hydrate nucleation and growth consistently in simulations, it now becomes possible to incorporate chemical additives and observer their effect on subsequent growth. Care must be taken to ensure that the insertion of the additive does not automatically

disrupt the emerging hydrate structure, and so we have adopted a protocol in which the additive is inserted into the methane phase, the water is constrained while any stress due to the introduction of the additive is allowed to relax, and

then conventional molecular dynamics simulations are continued on the extended system. We have conducted studies on a number of oligomeric potential inhibitors, included PVP, PVCap, PDMAEMA, and copolymers using these and various related monomers. In general, two

architypes for the action of the inhibitor have been found. For the first, characterized by PVP, there is

no surface binding of the inhibitor to the hydrate

cluster, but small-to-moderate size clusters are considerably destabilized by the presence of the inhibitor in its vicinity. These represent true kinetic inhibitors, where the presence of the inhibitor increases the critical cluster size for

Figure 3 conditional cluster transition

probabilities, P(m |n) for hydrate cluster growth and decay, calculated from the first

segment of simulation A.

4.2 ns

1.5 ns

6.9 ns

nucleation, and so delays the onset of nucleation. With the second archetype, characterized by PVCAP and PDMAEMA, the inhibitor is found to bind to the surface of the hydrate particle, and it is actually found to increase the rate of hydrate growth within the simulations. At first thought this

wo uld appear to contradict experimental findings, but it is important to realize that the length scales for experiments and simulations are very different: experiments often note nucleation only when micrometer sized crystals appear, whereas the sale

is nanometers in simulations. Formally, the critical nucleus is the size of cluster beyond which particle growth becomes energetically favourable, and most of the estimates of this critical size suggest it is on the nano-scale.[19-20] Thus there is a substantial growth period separating the formal

nucleation process and its experimental observation. Strong adsorption to a surface normally requires effective pattern matching between the adsorbent and adsorbate, and where such pattern matching occurs it should not be surprising that the additive would stabilise—

perhaps even induce—the presence of embryonic hydrate structure. However, the presence of the bound additive will hinder subsequent growth of the nanocrystal, since it will block access of water and methane to that surface. Thus promoting

nucleation and growth at the nanoscale can be consistent with delaying the appearance of hydrates at the micro-scale.The transition probability for methane hydrate in the presence of one of the second architypes (PDMAEMA) is shown in Figure 4. Both the

inhibited and uninhibited systems share two main features: (1) a continuous range of cluster sizes between 150 and 400 that are more likely to grow at the small end and more likely to shrink at the large end; and (2) a population of small but substantial clusters (involving up to 80 water

molecules) that are more likely to decay than grow—unless they become attached to an existing large cluster (shown by the contours for clusters where the “old” size is about 50 and the “new” size is above 200). The differences between the

two simulations is largely one of scale, but they are significant. Most intriguingly, in the presence of the PDMAEMA, the population of small clusters extends to larger sizes (more like 80 than the 50 seen without additive present), these tend to add to larger clusters (above, rather than below,

300 water molecules) and there is much greater probability that clusters of around 200 water

molecules will grow by additions of just a few water molecules at a time (seen in the green contours just below the m=n line for the simulation with PDMAEMA). We conclude that the oligomeric additive tends to stabilize hydrate clusters, and in particular provides a larger

population of significant clusters (> 50 water molecules) to aggregate with the dominant cluster and so promotes growth on the nanoscale.

Figure 4 transicluster size transition probabilities for simulations without (top) and with (bottom)

inhibitor (PDMAEMA). Simulations were started

from the same water configuration, which was taken from about 17 ns into simulation A

CONCLUSIONSThe work presented in this paper shows how considerable insight can be gained into the activity of low dosage hydrate inhibitors once it is possible to simulate the nucleation process repeatably. In

particular, we have identified t wo classes of inhibitor activity, one of which is consistent with conventional ideas of surface adsorption and relies on structural matches between the inhibitor and the hydrate, whereas the other is more of a dynamic effect and actually relies on structural

mismatches.[21]

In our work we have shown that supersaturation of methane in a thin water film can be used to effect nucleation consistently on timescales as short as tens of nanoseconds; but we note that similar

strategies have not proved successful propane because the dissolved propane was found to aggregate too quickly. The work of Wu et al. has shown that much more realistic methane concentrations can be used if simulation times can be pushed into the microsecond regime,{[1] but

with current techniques and technology thisremains computationally too expensive to use in comprehensive studies of inhibitors. Considerable progress has also been made by imposing thermodynamic gradients on the simulations[15, 22-

23] and with coarse grained water models.[18, 24-25]

However, both these techniques still have problems to resolve before they can be applied to polymeric additives. In the case of coarse-grained molecular dynamics, it will be necessary to develop coarse-grained potentials for the additives,

and such potentials do not readily reproduce the secondary structure of macromolecules—something that has been shown to be significant for hydrate inhibitor activity.[5] Similarly, it will be necessary to characterize the effect of any imposed gradients on the conformation of polymeric

inhibitors before these techniques can be extended to inhibitor studies.Recent work has shown that smarter molecular dynamics methods, such as metadynamics, can be used to sample infrequent events like crystal nucleation within timescales readily accessible to

molecular simulation. Metadynamics uses the history of the simulation to modify the intermolecular potentials in a way that encourages the system to explore new regions of the state space; at the same time, it does this in a way that

enables the free energy landscape characterizing the different states to be recovered. The method has been used to simulate phases transitions between different states of CaCO3

[26] and even to determine how proteins can change the thermodynamic landscape accessible to CaCO3

nanoparticles.[9] Most significantly for the hydrate applications, it has been used to simulate the

Figure 5 snapshots taken from the beginning (top), middle and end (bottom) of a 20 ns NPTmetadynamics simulation of TIP4P water. The emergence of ice is clearly seen in the top and

bottom right of the middle configuration.

spontaneous formation of ice from liquid water under NPT conditions (i.e. so that the expansion on freezing is recovered as a natural observable of the simulation) and has been show to produce ice nucleation repeatedly on a 20 ns timescale. Preliminary results on the application of

metadynamics to hydrate formation will be presented at the ICGH7 conference.

ACKNOWLEDGEMENTSThe author is grateful to the contributions of many

coworkers, but particularly to R.W. Hawtin and D. Quigley in the context of this paper.

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