How to quantify and avoid finite size effects in computational studies of crystal nucleation: The case of heterogeneous ice nucleation

Divergence among Local Structure, Dynamics, and Nucleation Outcome in Heterogeneous Nucleation of Close-Packed Crystals

Heterogeneous crystal nucleation is the dominant mechanism of crystallization in most systems, yet its underlying physics remains an enigma. While emergent interfacial crystalline order precedes heterogeneous nucleation, its importance in the nucleation mechanism is unclear. Here, we use path sampling simulations of two model systems to demonstrate that crystalline order in its traditional sense is not predictive of the outcome of the heterogeneous nucleation of close-packed crystals. Consequently, structure-based collective variables (CVs) that reliably describe homogeneous nucleation can be poor descriptors of heterogeneous nucleation. This divergence between structure and nucleation outcome is accompanied by an intriguing dynamical anomaly, wherein low-coordinated crystalline particles outpace their liquid-like counterparts. We use committor analysis, high-throughput screening, and machine learning to devise CV optimization strategies and present suitable structural heuristics within the metastable fluid for CV prescreening. Employing such optimized CVs is pivotal for properly characterizing the mechanism of heterogeneous nucleation in metallic and colloidal systems.

Ideal conductor/dielectric model (ICDM): A generalized technique to correct for finite-size effects in molecular simulations of hindered ion transport

Molecular simulations serve as indispensable tools for investigating the kinetics and elucidating the mechanism of hindered ion transport across nanoporous membranes. In particular, recent advancements in advanced sampling techniques have made it possible to access translocation timescales spanning several orders of magnitude. In our prior study [Shoemaker et al., J. Chem. Theory Comput. 18, 7142 (2022)], we identified significant finite size artifacts in simulations of pressure-driven hindered ion transport through nanoporous graphitic membranes. We introduced the ideal conductor model, which effectively corrects for such artifacts by assuming the feed to be an ideal conductor. In the present work, we introduce the ideal conductor dielectric model (Icdm), a generalization of our earlier model, which accounts for the dielectric properties of both the membrane and the filtrate. Using the Icdm model substantially enhances the agreement among corrected free energy profiles obtained from systems of varying sizes, with notable improvements observed in regions proximate to the pore exit. Moreover, the model has the capability to consider secondary ion passage events, including the transport of a co-ion subsequent to the traversal of a counter-ion, a feature that is absent in our original model. We also investigate the sensitivity of the new model to various implementation details. The Icdm model offers a universally applicable framework for addressing finite size artifacts in molecular simulations of ion transport. It stands as a significant advancement in our quest to use molecular simulations to comprehensively understand and manipulate ion transport processes through nanoporous membranes.

Interplay of multiple clusters and initial interface positioning for forward flux sampling simulations of crystal nucleation

Forward flux sampling (FFS) is a path sampling technique widely used in computer simulations of crystal nucleation from the melt. In such studies, the order parameter underpinning the progress of the FFS algorithm is often the size of the largest crystalline nucleus. In this work, we investigate the effects of two computational aspects of FFS simulations, using the prototypical Lennard-Jones liquid as our computational test bed. First, we quantify the impact of the positioning of the liquid basin and first interface in the space of the order parameter. In particular, we demonstrate that these choices are key to ensuring the consistency of the FFS results. Second, we focus on the frequently encountered scenario where the population of crystalline nuclei is such that there are multiple clusters of size comparable to the largest one. We demonstrate the contribution of clusters other than the largest cluster to the initial flux; however, we also show that they can be safely ignored for the purposes of converging a full FFS calculation. We also investigate the impact of different clusters merging, a process that appears to be facilitated by substantial spatial correlations-at least at the supercooling considered here. Importantly, all of our results have been obtained as a function of system size, thus contributing to the ongoing discussion on the impact of finite size effects on simulations of crystal nucleation. Overall, this work either provides or justifies several practical guidelines for performing FFS simulations that can also be applied to more complex and/or computationally expensive models.

RSeeds: Rigid Seeding Method for Studying Heterogeneous Crystal Nucleation

Heterogeneous nucleation is the dominant form of liquid-to-solid transition in nature. Although molecular simulations are most uniquely suited to studying nucleation, the waiting time to observe even a single nucleation event can easily exceed the current computational capabilities. Therefore, there exists an imminent need for methods that enable computationally fast and feasible studies of heterogeneous nucleation. Seeding is a technique that has proven to be successful at dramatically expanding the range of computationally accessible nucleation rates in simulation studies of homogeneous crystal nucleation. In this article, we introduce a new seeding method for heterogeneous nucleation called Rigid Seeding (RSeeds). Crystalline seeds are treated as pseudorigid bodies and simulated on a surface with metastable liquid above its melting temperature. This allows the seeds to adapt to the surface and identify favorable seed-surface configurations, which is necessary for reliable predictions of crystal polymorphs that form and the corresponding heterogeneous nucleation rates. We demonstrate and validate RSeeds for heterogeneous ice nucleation on a flexible self-assembled monolayer surface, a mineral surface based on kaolinite, and two model surfaces. RSeeds predicts the correct ice polymorph, exposed crystal plane, and rotation on the surface. RSeeds is semiquantitative and can be used to estimate the critical nucleus size and nucleation rate when combined with classical nucleation theory. We demonstrate that RSeeds can be used to evaluate nucleation rates spanning many orders of magnitude.

Ideal Conductor Model: An Analytical Finite-Size Correction for Nonequilibrium Molecular Dynamics Simulations of Ion Transport through Nanoporous Membranes

Modulating ion transport through nanoporous membranes is critical to many important chemical and biological separation processes. The corresponding transport timescales, however, are often too long to capture accurately using conventional molecular dynamics (MD). Recently, path sampling techniques, such as forward-flux sampling (FFS), have emerged as attractive alternatives for efficiently and accurately estimating arbitrarily long ionic passage times. Here, we use non-equilibrium MD and FFS to explore how the kinetics and mechanisms of pressure-driven chloride transport through a nanoporous graphitic membrane are affected by its lateral dimensions. We not only find ionic passage times and free energy barriers to decrease dramatically upon increasing the membrane surface area but also observe an abrupt and discontinuous change in the locus of the transition state. These strong finite size effects arise due to the cumulative effect of the periodic images of the leading ion entering the pore on the distribution of the induced excess charge at the membrane surface in the feed. By assuming that the feed is an ideal conductor, we analytically derive a finite size correction term that can be computed from the information obtained from a single simulation and successfully use it to obtain corrected free energy profiles with no dependence on the system size. We then estimate ionic passage times in the thermodynamic limit by assuming an Eyring-type dependence of rates on barriers with a size-independent prefactor. This approach constitutes a universal framework for removing finite size artifacts in molecular simulations of ion transport through nanoporous membranes and biological channel proteins.

The role of structural order in heterogeneous ice nucleation

The freezing of water into ice is a key process that is still not fully understood. It generally requires an impurity of some description to initiate the heterogeneous nucleation of the ice crystals. The molecular structure, as well as the extent of structural order within the impurity in question, both play an essential role in determining its effectiveness. However, disentangling these two contributions is a challenge for both experiments and simulations. In this work, we have systematically investigated the ice-nucleating ability of the very same compound, cholesterol, from the crystalline (and thus ordered) form to disordered self-assembled monolayers. Leveraging a combination of experiments and simulations, we identify a “sweet spot” in terms of the surface coverage of the monolayers, whereby cholesterol maximises its ability to nucleate ice (which remains inferior to that of crystalline cholesterol) by enhancing the structural order of the interfacial water molecules. These findings have practical implications for the rational design of synthetic ice-nucleating agents.

The freezing of water into ice is still not fully understood. Here, we investigate the role of structural disorder within the biologically relevant impurities that facilitate this fundamental phase transition.

Toward Automated Sampling of Polymorph Nucleation and Free Energies with the SGOOP and Metadynamics

Understanding the driving forces behind the nucleation of different polymorphs is of great importance for material sciences and the pharmaceutical industry. This includes understanding the reaction coordinate that governs the nucleation process and correctly calculating the relative free energies of different polymorphs. Here, we demonstrate, for the prototypical case of urea nucleation from the melt, how one can learn such a one-dimensional reaction coordinate as a function of prespecified order parameters and use it to perform efficient biased all-atom molecular dynamics simulations. The reaction coordinate is learnt as a function of the generic thermodynamic and structural order parameters using the "spectral gap optimization of order parameters (SGOOP)" approach [Tiwary, P. and Berne, B. J. Proc. Natl. Acad. Sci. U.S.A. (2016)] and is biased using well-tempered metadynamics simulations. The reaction coordinate gives insights into the role played by different structural and thermodynamics order parameters, and the biased simulations obtain accurate relative free energies for different polymorphs. This includes an accurate prediction of the approximate pressure at which urea undergoes a phase transition and one of the metastable polymorphs becomes the most stable conformation. We believe the ideas demonstrated in this work will facilitate efficient sampling of nucleation in complex, generic systems.

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.

Crystallization kinetics of amorphous acetonitrile nanoscale films

We measure the isothermal crystallization kinetics of amorphous acetonitrile films using molecular beam dosing and reflection adsorption infrared spectroscopy techniques. Experiments on a graphene covered Pt(111) substrate revealed that the crystallization rate slows dramatically during long time periods and that the overall kinetics cannot be described by a simple application of the Avrami equation. The crystallization kinetics also have a thickness dependence with the thinner films crystallizing much slower than the thicker ones. Additional experiments showed that decane layers at both the substrate and vacuum interfaces can also affect the crystallization rates. A comparison of the crystallization rates for CH₃CN and CD₃CN films showed only an isotope effect of ∼1.09. When amorphous films were deposited on a crystalline film, the crystalline layer did not act as a template for the formation of a crystalline growth front. These overall results suggest that the crystallization kinetics are complicated, indicating the possibility of multiple nucleation and growth mechanisms.

Role of Nanoscale Interfacial Proximity in Contact Freezing in Water

Contact freezing is a mode of atmospheric ice nucleation in which a collision between a dry ice nucleating particle (INP) and a water droplet results in considerably faster heterogeneous nucleation. The molecular mechanism of such an enhancement is, however, still a mystery. While earlier studies had attributed it to collision-induced transient perturbations, recent experiments point to the pivotal role of nanoscale proximity of the INP and the free interface. By simulating the heterogeneous nucleation of ice within INP-supported nanofilms of two model water-like tetrahedral liquids, we demonstrate that such nanoscale proximity is sufficient for inducing rate increases commensurate with those observed in contact freezing experiments, but only if the free interface has a tendency to enhance homogeneous nucleation. Water is suspected of possessing this latter property, known as surface freezing propensity. Our findings therefore establish a connection between the surface freezing propensity and kinetic enhancement during contact nucleation. We also observe that faster nucleation proceeds through a mechanism markedly distinct from classical heterogeneous nucleation, involving the formation of hourglass-shaped crystalline nuclei that conceive at either interface and that have a lower free energy of formation due to the nanoscale proximity of the interfaces and the modulation of the free interfacial structure by the INP. In addition to providing valuable insights into the physics of contact nucleation, our findings can assist in improving the accuracy of heterogeneous nucleation rate measurements in experiments and in advancing our understanding of ice nucleation on nonuniform surfaces such as organic, polymeric, and biological materials.