Rate Prediction for Homogeneous Nucleation of Methane Hydrate at Moderate Supersaturation Using Transition Interface Sampling

Dissociation line and driving force for nucleation of the nitrogen hydrate from computer simulation. II. Effect of multiple occupancy

In this work, we determine the dissociation line of the nitrogen (N2) hydrate by computer simulation using the TIP4P/Ice model for water and the TraPPE force field for N2. This work is the natural extension of Paper I, in which the dissociation temperature of the N2 hydrate has been obtained at 500, 1000, and 1500 bar [Algaba et al., J. Chem. Phys. 159, 224707 (2023)] using the solubility method and assuming single occupancy. We extend our previous study and determine the dissociation temperature of the N2 hydrate at different pressures, from 500 to 4500 bar, taking into account the single and double occupancy of the N2 molecules in the hydrate structure. We calculate the solubility of N2 in the aqueous solution as a function of temperature when it is in contact with a N2-rich liquid phase and when in contact with the hydrate phase with single and double occupancy via planar interfaces. Both curves intersect at a certain temperature that determines the dissociation temperature at a given pressure. We observe a negligible effect of occupancy on the dissociation temperature. Our findings are in very good agreement with the experimental data taken from the literature. We have also obtained the driving force for the nucleation of the hydrate as a function of temperature and occupancy at several pressures. As in the case of the dissociation line, the effect of occupancy on the driving force for nucleation is negligible. To the best of our knowledge, this is the first time that the effect of the occupancy on the driving force for nucleation of a hydrate that exhibits sII crystallographic structure is studied from computer simulation.

Comparing brute force to transition path sampling for gas hydrate nucleation with a flat interface: comments on time reversal symmetry

Fluid to solid nucleation is often investigated with the rare event method transition path sampling (TPS). I claim that the inherent irreversibility of solid nucleation, even at stationary conditions, calls into question TPS's applicability for determining solid nucleation mechanisms, especially for pre-critical behavior. Even when applied to a phenomenon which displays time reversal asymmetry like solid nucleation, TPS is a good means of exploring phase space and giving trends in post-critical structure, and its ability to facilitate nucleation rate and free energy calculations remains outstanding. Forward-only splitting and ratcheting methods such as forward flux sampling are more attractive for understanding nucleation mechanisms as they do not require time reversal symmetry, but at low driving forces may suffer from the same limitations as brute force: they may never make it to the first ratchet. Here I briefly summarize the TPS method and gas hydrate nucleation simulation literature, focusing on topics within both to facilitate a comparison of brute force hydrate nucleation to transition path sampling of hydrate nucleation. Perhaps anecdotally, the brute force technique results in more crystalline trajectories despite having higher driving forces than TPS. I maintain this difference is because of the inherent irreversibility of hydrate nucleation, meaning its pre-critical behavior cannot accurately be determined by the melting trajectories that comprise approximately half of the configurations in TPS's path ensemble.

LPATH: A Semiautomated Python Tool for Clustering Molecular Pathways

The pathways by which a molecular process transitions to a target state are highly sought-after as direct views of a transition mechanism. While great strides have been made in the physics-based simulation of such pathways, the analysis of these pathways can be a major challenge due to their diversity and variable lengths. Here, we present the LPATH Python tool, which implements a semiautomated method for linguistics-assisted clustering of pathways into distinct classes (or routes). This method involves three steps: 1) discretizing the configurational space into key states, 2) extracting a text-string sequence of key visited states for each pathway, and 3) pairwise matching of pathways based on a text-string similarity score. To circumvent the prohibitive memory requirements of the first step, we have implemented a general two-stage method for clustering conformational states that exploits machine learning. LPATH is primarily designed for use with the WESTPA software for weighted ensemble simulations; however, the tool can also be applied to conventional simulations. As demonstrated for the C7eq to C7ax conformational transition of the alanine dipeptide, LPATH provides physically reasonable classes of pathways and corresponding probabilities.

LPATH: A semi-automated Python tool for clustering molecular pathways

The pathways by which a molecular process transitions to a target state are highly sought-after as direct views of a transition mechanism. While great strides have been made in the physics-based simulation of such pathways, the analysis of these pathways can be a major challenge due to their diversity and variable lengths. Here we present the LPATH Python tool, which implements a semi-automated method for linguistics-assisted clustering of pathways into distinct classes (or routes). This method involves three steps: 1) discretizing the configurational space into key states, 2) extracting a text-string sequence of key visited states for each pathway, and 3) pairwise matching of pathways based on a text-string similarity score. To circumvent the prohibitive memory requirements of the first step, we have implemented a general two-stage method for clustering conformational states that exploits machine learning. LPATH is primarily designed for use with the WESTPA software for weighted ensemble simulations; however, the tool can also be applied to conventional simulations. As demonstrated for the C7eq to C7ax conformational transition of alanine dipeptide, LPATH provides physically reasonable classes of pathways and corresponding probabilities.

Does supercooled water retain its universal nucleation behavior under shear at high pressure?

Understanding the nucleation of homogeneous flow systems at high pressures is vital in protein crystallization and cryopreservation, where high pressure prevents the freezing of biological samples. This study examines the behavior of ice nucleation under shear at various pressures and explores the universal nucleation behavior of the sheared systems applied to supercooled water at higher pressures. In this study, the nucleation rates for the TIP4P/Ice model via a seeding method based on extended classical nucleation theory (CNT) are computed at pressures of 1, 100, 500, 700, and 1000 bar and a constant temperature of 240 K. Using extended CNT with explicitly embodying the shear rate, we analyzed the dependence of pressure on the transport and thermodynamic properties. In line with previous studies, we observed that Δμliq-ice and viscosity decrease while diffusivity increases with an increase in pressure. Furthermore, we showed that the dependence of the nucleation rate on shear at higher pressure is non-monotonic, with the maximum at optimal shear rates between 107 and 108 s-1. Our results demonstrate a non-monotonic pressure dependence of the optimal shear rates, which could originate from a violation of the Stokes-Einstein relation.

Solubility of carbon dioxide in water: Some useful results for hydrate nucleation

In this paper, the solubility of carbon dioxide (CO2) in water along the isobar of 400 bar is determined by computer simulations using the well-known TIP4P/Ice force field for water and the TraPPE model for CO2. In particular, the solubility of CO2 in water when in contact with the CO2 liquid phase and the solubility of CO2 in water when in contact with the hydrate have been determined. The solubility of CO2 in a liquid-liquid system decreases as the temperature increases. The solubility of CO2 in a hydrate-liquid system increases with temperature. The two curves intersect at a certain temperature that determines the dissociation temperature of the hydrate at 400 bar (T3). We compare the predictions with T3 obtained using the direct coexistence technique in a previous work. The results of both methods agree, and we suggest 290(2) K as the value of T3 for this system using the same cutoff distance for dispersive interactions. We also propose a novel and alternative route to evaluate the change in chemical potential for the formation of hydrates along the isobar. The new approach is based on the use of the solubility curve of CO2 when the aqueous solution is in contact with the hydrate phase. It considers rigorously the non-ideality of the aqueous solution of CO2, providing reliable values for the driving force for nucleation of hydrates in good agreement with other thermodynamic routes used. It is shown that the driving force for hydrate nucleation at 400 bar is larger for the methane hydrate than for the carbon dioxide hydrate when compared at the same supercooling. We have also analyzed and discussed the effect of the cutoff distance of dispersive interactions and the occupancy of CO2 on the driving force for nucleation of the hydrate.

Homogeneous nucleation rate of methane hydrate formation under experimental conditions from seeding simulations

In this work, we shall estimate via computer simulations the homogeneous nucleation rate for the methane hydrate at 400 bars for a supercooling of about 35 K. The TIP4P/ICE model and a Lennard-Jones center were used for water and methane, respectively. To estimate the nucleation rate, the seeding technique was employed. Clusters of the methane hydrate of different sizes were inserted into the aqueous phase of a two-phase gas-liquid equilibrium system at 260 K and 400 bars. Using these systems, we determined the size at which the cluster of the hydrate is critical (i.e., it has 50% probability of either growing or melting). Since nucleation rates estimated from the seeding technique are sensitive to the choice of the order parameter used to determine the size of the cluster of the solid, we considered several possibilities. We performed brute force simulations of an aqueous solution of methane in water in which the concentration of methane was several times higher than the equilibrium concentration (i.e., the solution was supersaturated). From brute force runs, we infer the value of the nucleation rate for this system rigorously. Subsequently, seeding runs were carried out for this system, and it was found that only two of the considered order parameters were able to reproduce the value of the nucleation rate obtained from brute force simulations. By using these two order parameters, we estimated the nucleation rate under experimental conditions (400 bars and 260 K) to be of the order of log₁₀ (J/(m³ s)) = -7(5).

Homogeneous nucleation of crystalline methane hydrate in molecular dynamics transition paths sampled under realistic conditions

Methane hydrates are important from a scientific and industrial perspective, and form by nucleation and growth from a supersaturated aqueous solution of methane. Molecular simulation is able to shed light on the process of homogeneous nucleation of hydrates, using straightforward molecular dynamics or rare event enhanced sampling techniques with atomistic and coarse grained force fields. In our previous work [Arjun, T. A. Berendsen, and P. G. Bolhuis, Proc. Natl. Acad. Sci. U. S. A. 116, 19305 (2019)], we performed transition path sampling (TPS) simulations using all atom force fields under moderate driving forces at high pressure, which enabled unbiased atomistic insight into the formation of methane hydrates. The supersaturation in these simulations was influenced by the Laplace pressure induced by the spherical gas reservoir. Here, we investigate the effect of removing this influence. Focusing on the supercooled, supersaturated regime to keep the system size tractable, our TPS simulations indicate that nuclei form amorphous structures below roughly 260 K and crystalline sI structures above 260 K. For these temperatures, the average transition path lengths are significantly longer than in our previous study, pushing the boundaries of what can be achieved with TPS. The temperature to observe a critical nucleus of certain size was roughly 20 K lower compared to a spherical reservoir due to the lower concentration of methane in the solution, yielding a reduced driving force. We analyze the TPS results using a model based on classical nucleation theory. The corresponding free energy barriers are estimated and found to be consistent with previous predictions, thus adding to the overall picture of the hydrate formation process.

Enhanced path sampling using subtrajectory Monte Carlo moves

Path sampling allows the study of rare events, such as chemical reactions, nucleation, and protein folding, via a Monte Carlo (MC) exploration in path space. Instead of configuration points, this method samples short molecular dynamics (MD) trajectories with specific start- and end-conditions. As in configuration MC, its efficiency highly depends on the types of MC moves. Since the last two decades, the central MC move for path sampling has been the so-called shooting move in which a perturbed phase point of the old path is propagated backward and forward in time to generate a new path. Recently, we proposed the subtrajectory moves, stone-skipping (SS) and web-throwing, that are demonstrably more efficient. However, the one-step crossing requirement makes them somewhat more difficult to implement in combination with external MD programs or when the order parameter determination is expensive. In this article, we present strategies to address the issue. The most generic solution is a new member of subtrajectory moves, wire fencing (WF), that is less thrifty than the SS but more versatile. This makes it easier to link path sampling codes with external MD packages and provides a practical solution for cases where the calculation of the order parameter is expensive or not a simple function of geometry. We demonstrate the WF move in a double-well Langevin model, a thin film breaking transition based on classical force fields, and a smaller ruthenium redox reaction at the ab initio level in which the order parameter explicitly depends on the electron density.

Solubility of Methane in Water: Some Useful Results for Hydrate Nucleation

In this paper, the solubility of methane in water along the 400 bar isobar is determined by computer simulations using the TIP4P/Ice force field for water and a simple LJ model for methane. In particular, the solubility of methane in water when in contact with the gas phase and the solubility of methane in water when in contact with the hydrate has been determined. The solubility of methane in a gas-liquid system decreases as temperature increases. The solubility of methane in a hydrate-liquid system increases with temperature. The two curves intersect at a certain temperature that determines the triple point T₃ at a certain pressure. We also determined T₃ by the three-phase direct coexistence method. The results of both methods agree, and we suggest 295(2) K as the value of T₃ for this system. We also analyzed the impact of curvature on the solubility of methane in water. We found that the presence of curvature increases the solubility in both the gas-liquid and hydrate-liquid systems. The change in chemical potential for the formation of hydrate is evaluated along the isobar using two different thermodynamic routes, obtaining good agreement between them. It is shown that the driving force for hydrate nucleation under experimental conditions is higher than that for the formation of pure ice when compared at the same supercooling. We also show that supersaturation (i.e., concentrations above those of the planar interface) increases the driving force for nucleation dramatically. The effect of bubbles can be equivalent to that of an additional supercooling of about 20 K. Having highly supersaturated homogeneous solutions makes possible the spontaneous formation of the hydrate at temperatures as high as 285 K (i.e., 10K below T₃). The crucial role of the concentration of methane for hydrate formation is clearly revealed. Nucleation of the hydrate can be either impossible or easy and fast depending on the concentration of methane which seems to play the leading role in the understanding of the kinetics of hydrate formation.

Nucleation rates from small scale atomistic simulations and transition state theory

The evaluation of nucleation rates from molecular dynamics trajectories is hampered by the slow nucleation time scale and impact of finite size effects. Here, we show that accurate nucleation rates can be obtained in a very general fashion relying only on the free energy barrier, transition state theory, and a simple dynamical correction for diffusive recrossing. In this setup, the time scale problem is overcome by using enhanced sampling methods, in casu metadynamics, whereas the impact of finite size effects can be naturally circumvented by reconstructing the free energy surface from an appropriate ensemble. Approximations from classical nucleation theory are avoided. We demonstrate the accuracy of the approach by calculating macroscopic rates of droplet nucleation from argon vapor, spanning 16 orders of magnitude and in excellent agreement with literature results, all from simulations of very small (512 atom) systems.

An extended autoencoder model for reaction coordinate discovery in rare event molecular dynamics datasets

The reaction coordinate (RC) is the principal collective variable or feature that determines the progress along an activated or reactive process. In a molecular simulation using enhanced sampling, a good description of the RC is crucial for generating sufficient statistics. Moreover, the RC provides invaluable atomistic insight into the process under study. The optimal RC is the committor, which represents the likelihood of a system to evolve toward a given state based on the coordinates of all its particles. As the interpretability of such a high dimensional function is low, a more practical approach is to describe the RC by some low-dimensional molecular collective variables or order parameters. While several methods can perform this dimensionality reduction, they usually require a preselection of these low-dimension collective variables (CVs). Here, we propose to automate this dimensionality reduction using an extended autoencoder, which maps the input (many CVs) onto a lower-dimensional latent space, which is subsequently used for the reconstruction of the input as well as the prediction of the committor function. As a consequence, the latent space is optimized for both reconstruction and committor prediction and is likely to yield the best non-linear low-dimensional representation of the committor. We test our extended autoencoder model on simple but nontrivial toy systems, as well as extensive molecular simulation data of methane hydrate nucleation. The extended autoencoder model can effectively extract the underlying mechanism of a reaction, make reliable predictions about the committor of a given configuration, and potentially even generate new paths representative for a reaction.

The seven deadly sins: When computing crystal nucleation rates, the devil is in the details

The formation of crystals has proven to be one of the most challenging phase transformations to quantitatively model-let alone to actually understand-be it by means of the latest experimental technique or the full arsenal of enhanced sampling approaches at our disposal. One of the most crucial quantities involved with the crystallization process is the nucleation rate, a single elusive number that is supposed to quantify the average probability for a nucleus of critical size to occur within a certain volume and time span. A substantial amount of effort has been devoted to attempt a connection between the crystal nucleation rates computed by means of atomistic simulations and their experimentally measured counterparts. Sadly, this endeavor almost invariably fails to some extent, with the venerable classical nucleation theory typically blamed as the main culprit. Here, we review some of the recent advances in the field, focusing on a number of perhaps more subtle details that are sometimes overlooked when computing nucleation rates. We believe it is important for the community to be aware of the full impact of aspects, such as finite size effects and slow dynamics, that often introduce inconspicuous and yet non-negligible sources of uncertainty into our simulations. In fact, it is key to obtain robust and reproducible trends to be leveraged so as to shed new light on the kinetics of a process, that of crystal nucleation, which is involved into countless practical applications, from the formulation of pharmaceutical drugs to the manufacturing of nano-electronic devices.

Homogenous nucleation rate of CO2 hydrates using transition interface sampling

Carbon dioxide and water can form solid clathrate structures in which water cages encapsulate the gas molecules. Such hydrates have sparked much interest due to their possible application in CO₂ sequestration. How the solid structure forms exactly from the liquid phase via a homogenous nucleation process is still poorly understood. This nucleation event is rare on the molecular timescale even under moderate undercooling or supersaturation conditions because of the large free energy barrier toward crystallization, rendering a brute force simulation of hydrate nucleation unfeasible for moderate undercooling or supersaturation. Here, we perform transition interface sampling simulations to quantify the homogenous nucleation rate for CO₂ hydrate formation using accurate atomistic force fields at 500 bars for three different temperatures between 260 and 273 K. Collecting more than 100 000 pathways comprising roughly two milliseconds of simulation time, we computed a nucleation rate in the amorphous phase of ∼10²¹ nuclei s^-1 cm^-3 for a temperature of 260 K and a rate of ∼10¹² nuclei s^-1 cm^-3 for a temperature of 265 K. For a temperature of 273 K, we find that the hydrate forms an sI crystalline phase with a rate of order of ∼10¹ nuclei s^-1 cm^-3. We compare these rates to classical nucleation theory estimates as well as experiments, and to nucleation rate estimates for methane hydrates and discuss possible causes of the observed differences. Our findings shed light on the kinetics of this important clathrate and should assist in future hydrate formation investigation.

Molecular Understanding of Homogeneous Nucleation of CO2 Hydrates Using Transition Path Sampling

Carbon dioxide hydrate is a solid built from hydrogen-bond stabilized water cages that encapsulate individual CO2 molecules. As potential candidates for reducing greenhouse gases, hydrates have attracted attention from both the industry and scientific community. Under high pressure and low temperature, hydrates are formed spontaneously from a mixture of CO2 and water via nucleation and growth. Yet, for moderate undercooling, i.e., moderate supersaturation, studying hydrate formation with molecular simulations is very challenging due to the high nucleation barriers involved. We investigate the homogeneous nucleation mechanism of CO2 hydrate as a function of temperature using transition path sampling (TPS), which generates ensembles of unbiased dynamical trajectories across the high barrier between the liquid and solid states. The resulting path ensembles reveal that at high driving force (low temperature), amorphous structures are predominantly formed, with 4151062 cages being the most abundant. With increasing temperature, the nucleation mechanism changes, and 51262 becomes the most abundant cage type, giving rise to the crystalline sI structure. Reaction coordinate analysis can reveal the most important collective variable involved in the mechanism. With increasing temperature, we observe a shift from a single feature (size of the nucleus) to a 2-dimensional (size and cage type) variable as the salient ingredient of the reaction coordinate, and then back to only the nucleus size. This finding is in line with the underlying shift from an amorphous to a crystalline nucleation channel. Modeling such complex phase transformations using transition path sampling gives unbiased insight into the molecular mechanisms toward different polymorphs, and how these are determined by thermodynamics and kinetics. This study will be beneficial for researchers aiming to produce such hydrates with different polymorphic forms.

Size dependence of the dissociation process of spherical hydrate particles via microsecond molecular dynamics simulations

The dissociation process of spherical sII mixed methane–propane hydrate particles in liquid hydrocarbon was investigated via microsecond-long molecular dynamics simulations.