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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., (© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).)

Nucleation is a key step in the synthesis of a new material from a solution. The well-established lattice-gas models can be used to gain insight into the basic physics of nucleation pathways involving a single nucleus type. In many situations, a solution is supersaturated with respect to more than one precipitating phase. This can generate a population of both stable and metastable nuclei on similar timescales and, hence, complex nucleation pathways involving a competition between the two. In this study, we introduce a lattice-gas model based on two types of interacting dimers representing the particles in a solution. Each type of dimer nucleates to a specific space-filling structure. Our model is tuned such that stable and metastable phases nucleate on a similar timescale. Either structure may nucleate first, with a probability sensitive to the relative rate at which a solute is replenished from their respective reservoirs. We calculate these nucleation rates via forward flux sampling and demonstrate how the resulting data can be used to infer the nucleation outcome and pathway. Possibilities include direct nucleation of the stable phase, domination of long-lived metastable crystallites, and pathways in which the stable phase nucleates only after multiple post-critical nuclei of the metastable phase have appeared.

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.

At atmospheric pressure, hexagonal ice (I h ) is thermodynamically stable relative to cubic ice (I c ), although the magnitude and underlying physical origin of this stability difference are not well defined. Pure I c crystals are not accessible experimentally, and hence computer simulations have often been used to interrogate the relative stabilities of I h and I c ; however, these simulations are dominated by molecular interaction models that ignore the intramolecular flexibility of individual water molecules, do not describe intermolecular hydrogen-bonding with sufficient accuracy, or ignore the role of nuclear quantum effects (NQEs) such as zero-point energy. Here, we show that when comparing the relative stability of I h and I c using a flexible, anharmonic molecular interaction model, while also accurately accounting for NQEs, a new picture emerges: I h is stabilized relative to I c as a result of subtle differences in the intramolecular geometries and intermolecular interactions of water molecules which are modulated by NQEs. Our simulations hence suggest that NQEs are a major contributor to the stabilization of I h under terrestrial conditions and thus contribute to the well-known hexagonal (sixfold) symmetry of ice crystals.

This work reexamines seeded simulation results for NaCl nucleation from a supersaturated aqueous solution at 298.15 K and 1 bar pressure. We present a linear regression approach for analyzing seeded simulation data that provides both nucleation rates and uncertainty estimates. Our results show that rates obtained from seeded simulations rely critically on a precise driving force for the model system. The driving force vs. solute concentration curve need not exactly reproduce that of the real system, but it should accurately describe the thermodynamic properties of the model system. We also show that rate estimates depend strongly on the nucleus size metric. We show that the rate estimates systematically increase as more stringent local order parameters are used to count members of a cluster and provide tentative suggestions for appropriate clustering criteria.

We apply the capillary wave method, based on measurements of fluctuations in a ribbon-like interfacial geometry, to determine the solid-liquid interfacial free energy for both polytypes of ice I and the recently proposed ice 0 within a mono-atomic model of water. We discuss various choices for the molecular order parameter, which distinguishes solid from liquid, and demonstrate the influence of this choice on the interfacial stiffness. We quantify the influence of discretisation error when sampling the interfacial profile and the limits on accuracy imposed by the assumption of quasi one-dimensional geometry. The interfacial free energies of the two ice I polytypes are indistinguishable to within achievable statistical error and the small ambiguity which arises from the choice of order parameter. In the case of ice 0, we find that the large surface unit cell for low index interfaces constrains the width of the interfacial ribbon such that the accuracy of results is reduced. Nevertheless, we establish that the interfacial free energy of ice 0 at its melting temperature is similar to that of ice I under the same conditions. The rationality of a core-shell model for the nucleation of ice I within ice 0 is questioned within the context of our results.

We study a three-species analogue of the Potts lattice gas model of nucleation from solution in a regime where partially disordered solute is a viable thermodynamic phase. Using a multicanonical sampling protocol, we compute phase diagrams for the system, from which we determine a parameter regime where the partially disordered phase is metastable almost everywhere in the temperature-fugacity plane. The resulting model shows non-trivial nucleation and growth behaviour, which we examine via multidimensional free energy calculations. We consider the applicability of the model in capturing the multi-stage nucleation mechanisms of polymorphic biominerals (e.g., CaCO 3 ). We then quantitatively explore the kinetics of nucleation in our model using the increasingly popular "seeding" method. We compare the resulting free energy barrier heights to those obtained via explicit free energy calculations over a wide range of temperatures and fugacities, carefully considering the propagation of statistical error. We find that the ability of the "seeding" method to reproduce accurate free energy barriers is dependent on the degree of supersaturation, and severely limited by the use of a nucleation driving force Δμ computed for bulk phases. We discuss possible reasons for this in terms of underlying kinetic assumptions, and those of classical nucleation theory.

Using ab initio methods, we examine the stability of ice 0, a recently proposed tetragonal form of ice implicated in the homogeneous freezing of water [J. Russo, F. Romano, and H. Tanaka, Nat. Mater. 13, 670 (2014)]. Vibrational frequencies are computed across the complete Brillouin Zone using Density Functional Theory (DFT), to confirm mechanical stability and quantify the free energy of ice 0 relative to ice I(h). The robustness of this result is tested via dispersion corrected semi-local and hybrid DFT, and Quantum Monte-Carlo calculation of lattice energies. Results indicate that popular molecular models only slightly overestimate the stability of ice zero. In addition, we study all possible realisations of proton disorder within the ice zero unit cell, and identify the ground state as ferroelectric. Comparisons are made to other low density metastable forms of ice, suggesting that the ice i structure [C. J. Fennel and J. D. Gezelter, J. Chem. Theory Comput. 1, 662 (2005)] may be equally relevant to ice formation.

A simple Ising-like model for the stacking thermodynamics of ice 1 is constructed for nuclei in supercooled water, and combined with classical nucleation theory. For relative stabilities of cubic and hexagonal ice I within the range of experimental estimates, this predicts critical nuclei are stacking disordered at strong sub-cooling, consistent with recent experiments. At higher temperatures nucleation of pure hexagonal ice is recovered. Lattice-switching Monte-Carlo is applied to accurately compute the relative stability of cubic and hexagonal ice for the popular mW model of water. Results demonstrate that this model fails to adequately capture the relative energetics of the two polytypes, leading to stacking disorder at all temperatures.

Metadynamics is employed to sample the configurations available to calcium carbonate nanoparticles in water, and to map an approximate free energy as a function of crystalline order. These data are used to investigate the validity of bulk and ideal surface energies in predicting structure at the nanoscale. Results indicate that such predictions can determine the structure and morphology of particles as small as 3-4 nm in diameter. Comparisons are made to earlier results on 2 nm particles under constant volume conditions which support nanoconfinement as a mechanism for enhancing the stability of amorphous calcium carbonate. Our results indicate that crystalline calcitelike structure is thermodynamically preferred for nanoparticles as small as 2 nm in the absence of nanoconfinement.

We show that recent developments in the application of metadynamics methods to direct simulations of crystallization make it possible to predict the orientation of crystals grown on self-assembled monolayers. In contrast to previous studies, the method allows for dynamic treatment of the organic component and the inclusion of explicit surface water without the need for computationally intensive interfacial energy calculations or prior knowledge of the interfacial structure. The method is applied to calcite crystallization on carboxylate terminated alkanethiols arrayed on Au (111). We demonstrate that a dynamic treatment of the monolayer is sufficient to reproduce the experimental results without the need to impose epitaxial constraints on the system. We also observe an odd-even effect in the variation of selectivity with organic chain length, reproducing experimentally observed orientations in both cases. Analysis of the ordering process in our simulations suggests a cycle of mutual control in which both the organic and mineral components induce complementary local order across the interface, leading to the formation of a critical crystalline region. The influence of pH, together with some factors that might affect the range of applicability of our method, is discussed.

We introduce a metadynamics based scheme for computing the free energy of nanoparticles as a function of their crystalline order. The method is applied to small nanoparticles of the biomineral calcium carbonate to determine the preferred structure during early stages of crystal growth. For particles 2 nm in diameter, we establish a large energetic preference for amorphous particle morphologies. Particles with partial crystalline order consistent with vaterite are also observed with substantially lower probability. The absence of the stable calcite phase and stability of the amorphous state support recent conjectures that calcite formation starts via the deposition of amorphous calcium carbonate.

The metadynamics method for accelerating rate events in molecular simulations is applied to the problem of ice freezing. We demonstrate homogeneous nucleation and growth of ice at 180 K in the isothermal-isobaric ensemble without the presence of external fields or surfaces. This result represents the first report of continuous and dynamic ice nucleation in a system of freely evolving density. Simulations are conducted using a variety of periodic simulation domains. In all cases the cubic polymorph ice I(c) is grown. The influence of boundary effects on estimates of the nucleation free energy barrier are discussed in relation to differences between this and earlier work.

The advantages of performing Langevin dynamics in extended systems are discussed. A simple Langevin dynamics scheme for producing the canonical ensemble is reviewed, and is then extended to the Hoover ensemble. We show that the resulting equations of motion generate the isobaric-isothermal ensemble. The Parrinello-Rahman ensemble is then discussed and we show that despite the presence of intrinsic probability gradients in this system, a Langevin dynamics approach samples the extended phase space in the correct fashion. The implementation of these methods in the ab initio plane wave density functional theory code CASTEP [M. D. Segall, P. L. D. Lindan, M. J. Probert, C. J. Pickard, P. J. Hasnip, S. J. Clarke, and M. C. Payne, J. Phys.: Condens. Matter 14, 2717 (2003)] is demonstrated., ((c) 2004 American Institute of Physics.)