Reaction coordinates for the crystal nucleation of colloidal suspensions extracted from the reweighted path ensemble

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.

"Sounding" out crystal nuclei-A mathematical-physical and experimental investigation

We outline techniques for the control and measurement of the nucleation of crystalline materials. Small angle x-ray scattering/wide angle x-ray scattering x-ray diffraction measurements are presented that demonstrate the impact of low power, continuous, non-cavitational ultrasound on the nucleation and crystallization of a wax-n-eicosane dissolved in a heptane/toluene solvent. A mathematical-physical approach based on the rectification of heat and mass transport by such a low power oscillating pressure field is outlined, and it is suggested that this approach be combined with dissipative particle dynamics computational modeling to develop a predictive method capable of modeling the impact of low power oscillating pressure fields (acoustics and ultrasonics) on a wide range of nucleating systems. Combining the ultrasound pitch and catch speed of sound measurements with low power harmonically oscillating pressure fields to monitor and control nucleation presents the prospect of entirely new industrially significant methods of process control in crystallization. It also offers new insights into nucleation processes in general. However, for the acoustic control technique to be widely applied , further theoretical and modeling work will be necessary since, at present, we are unable to predict the precise effect of low power ultrasound in any given situation.

Bond Orientational Order Parameters for Classifying Solid-like Clusters in a Lennard-Jones System near Liquid-Solid Transition and at Solid States

In this paper, we introduced an order parameter, named the local structure similarity (LSS), to measure the resemblance of a cluster structure in a liquid with respect to a perfect crystal. The LSS is based on a dot product of two bond orientational order complex vectors, with one vector associated with a particle in a liquid and the other vector with a particle in a crystal. The calculation of the LSS should scan the entire space of the Euler angles determined by the two coordinate frames describing individually the liquid and the crystal. The effectiveness of the LSS was examined by solid-like clusters in a Lennard-Jones (LJ) system near its liquid-solid phase transition and at solid states below its melting point, where the thermodynamic states of the LJ system were obtained by simulation annealing. The LSS measure was utilized to scrutinize the fcc-like, hcp-like, and bcc-like clusters classified by criteria based on W₄ and W₆ order parameters. As indicated by our results, the two ways of classification are consistent for fcc-like and hcp-like clusters, which are in a close resemblance to their crystalline counterparts. However, the classification with positive W₆ for bcc-like clusters is inconsistent with the results of the LSS measure, which was confirmed by clusters in a LJ system confined between two parallel slabs of particles in the bcc structure arrangement.

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.

Derivation of an exact, nonequilibrium framework for nucleation: Nucleation is a priori neither diffusive nor Markovian

We discuss the structure of the equation of motion that governs nucleation processes at first order phase transitions. From the underlying microscopic dynamics of a nucleating system, we derive by means of a nonequilibrium projection operator formalism the equation of motion for the size distribution of the nuclei. The equation is exact, i.e., the derivation does not contain approximations. To assess the impact of memory, we express the equation of motion in a form that allows for direct comparison to the Markovian limit. As a numerical test, we have simulated crystal nucleation from a supersaturated melt of particles interacting via a Lennard-Jones potential. The simulation data show effects of non-Markovian dynamics.

A generalized deep learning approach for local structure identification in molecular simulations

Identifying local structure in molecular simulations is of utmost importance. The most common existing approach to identify local structure is to calculate some geometrical quantity referred to as an order parameter. In simple cases order parameters are physically intuitive and trivial to develop (e.g., ion-pair distance), however in most cases, order parameter development becomes a much more difficult endeavor (e.g., crystal structure identification). Using ideas from computer vision, we adapt a specific type of neural network called a PointNet to identify local structural environments in molecular simulations. A primary challenge in applying machine learning techniques to simulation is selecting the appropriate input features. This challenge is system-specific and requires significant human input and intuition. In contrast, our approach is a generic framework that requires no system-specific feature engineering and operates on the raw output of the simulations, i.e., atomic positions. We demonstrate the method on crystal structure identification in Lennard-Jones (four different phases), water (eight different phases), and mesophase (six different phases) systems. The method achieves as high as 99.5% accuracy in crystal structure identification. The method is applicable to heterogeneous nucleation and it can even predict the crystal phases of atoms near external interfaces. We demonstrate the versatility of our approach by using our method to identify surface hydrophobicity based solely upon positions and orientations of surrounding water molecules. Our results suggest the approach will be broadly applicable to many types of local structure in simulations.

Free energy landscape and molecular pathways of gas hydrate nucleation

Despite the significance of gas hydrates in diverse areas, a quantitative knowledge of hydrate formation at a molecular level is missing. The impediment to acquiring this understanding is primarily attributed to the stochastic nature and ultra-fine scales of nucleation events, posing a great challenge for both experiment and simulation to explore hydrate nucleation. Here we employ advanced molecular simulation methods, including forward flux sampling (FFS), p_B histogram analysis, and backward flux sampling, to overcome the limit of direct molecular simulation for exploring both the free energy landscape and molecular pathways of hydrate nucleation. First we test the half-cage order parameter (H-COP) which we developed for driving FFS, through conducting the p_B histogram analysis. Our results indeed show that H-COP describes well the reaction coordinates of hydrate nucleation. Through the verified order parameter, we then directly compute the free energy landscape for hydrate nucleation by combining both forward and backward flux sampling. The calculated stationary distribution density, which is obtained independently of nucleation theory, is found to fit well against the classical nucleation theory (CNT). Subsequent analysis of the obtained large ensemble of hydrate nucleation trajectories show that although on average, hydrate formation is facilitated by a two-step like mechanism involving a gradual transition from an amorphous to a crystalline structure, there also exist nucleation pathways where hydrate crystallizes directly, without going through the amorphous stage. The CNT-like free energy profile and the structural diversity suggest the existence of multiple active transition pathways for hydrate nucleation, and possibly also imply the near degeneracy in their free energy profiles among different pathways. Our results thus bring a new perspective to the long standing question of how hydrates crystallize.

Local bond order parameters for accurate determination of crystal structures in two and three dimensions

Local order parameters for the characterization of liquid and different two- and three-dimensional crystalline structures are presented.

Pathways to self-organization: Crystallization via nucleation and growth

Crystallization, a prototypical self-organization process during which a disordered state spontaneously transforms into a crystal characterized by a regular arrangement of its building blocks, usually proceeds by nucleation and growth. In the initial stages of the transformation, a localized nucleus of the new phase forms in the old one due to a random fluctuation. Most of these nuclei disappear after a short time, but rarely a crystalline embryo may reach a critical size after which further growth becomes thermodynamically favorable and the entire system is converted into the new phase. In this article, we will discuss several theoretical concepts and computational methods to study crystallization. More specifically, we will address the rare event problem arising in the simulation of nucleation processes and explain how to calculate nucleation rates accurately. Particular attention is directed towards discussing statistical tools to analyze crystallization trajectories and identify the transition mechanism.

Pseudo-one-dimensional nucleation in dilute polymer solutions

Pathogenic protein fibrils have been shown in vitro to have nucleation-dependent kinetics despite the fact that one-dimensional structures do not have the size-dependent surface energy responsible for the lag time in classical theory. We present a theory showing that the conformational entropy of the peptide chains creates a free-energy barrier that is analogous to the translational entropy barrier in higher dimensions. We find that the dynamics of polymer rearrangement make it very unlikely for nucleation to succeed along the lowest free-energy trajectory, meaning that most of the nucleation flux avoids the free-energy saddle point. We use these results to construct a three-dimensional model for amyloid nucleation that accounts for conformational entropy, backbone H bonds, and side-chain interactions to compute nucleation rates as a function of concentration.

A string reaction coordinate for the folding of a polymer chain

We investigate the crystallization mechanism of a single, flexible homopolymer chain with short range attractions. For a sufficiently narrow attractive well, the system undergoes a first-order like freezing transition from an expanded disordered coil to a compact crystalline state. Based on a maximum likelihood analysis of committor values computed for configurations obtained by Wang-Landau sampling, we construct a non-linear string reaction coordinate for the coil-to-crystal transition. In contrast to a linear reaction coordinate, the string reaction coordinate captures the effect of different degrees of freedom controlling different stages of the transition. Our analysis indicates that a combination of the energy and the global crystallinity parameter Q6 provide the most accurate measure for the progress of the transition. While the crystallinity paramter Q6 is most relevant in the initial stages of the crystallization, the later stages are dominated by a decrease in the potential energy.

Reaction coordinate of incipient methane clathrate hydrate nucleation

Nucleation from solution is a ubiquitous phenomenon with relevance to myriad scientific disciplines, including pharmaceuticals, biomineralization, and disease. One prominent example is the nucleation of clathrate hydrates, multicomponent crystalline inclusion compounds relevant to the energy industry where they block pipelines and also constitute a potential vast energy resource. Despite their importance, the molecular mechanism of incipient hydrate formation remains unknown. Herein, we employ advanced molecular simulation tools (pB histogram, equilibrium path sampling) to provide a statistical-mechanical basis for extracting physical insight into the molecular steps by which clathrates form. Through testing the Mutually Coordinated Guest (MCG) order parameter, we demonstrate that both guest (methane) and host (water) structuring are crucial to accurately describe the nucleation of hydrates and determine a critical nucleus size of MCG-1 = 16 at 255 K and 500 bar. Equipped with a validated (and novel) reaction coordinate, subsequent equilibrium path sampling simulations yield the free energy barrier and nucleation rate. The resulting quantitative nucleation process is described by the MCG clustering mechanism. This constitutes a significant advance in the field of hydrates research, as the fitness of a molecular descriptor has never been statistically verified. More broadly, this work has significance to a wide range of multicomponent nucleation contexts wherein the formation mechanism depends on contributions from both solute and solvent.

Evidence of multi-step nucleation leading to various crystallization pathways from an Fe-O-Al melt

The crystallization process from a solution begins with nucleation, which determines the structure and size of the resulting crystals. Further understanding of multi-pathway crystallizations from solution through two-step nucleation mechanisms is needed. This study uses density functional theory to probe the thermodynamic properties of alumina clusters at high temperature and reveals the thermodynamic relationship between these clusters and the saturation levels of dissolved oxygen and aluminum in an Fe–O–Al melt. Based on the thermodynamics of cluster formation and the experimental evidence for both excess oxygen in the Fe-O-Al melt and for alumina with a polycrystalline structure in solidified iron, we demonstrate that the appearance of various types of clusters that depends on the saturation ratio determines the nucleation steps that lead to the various crystallization pathways. Such mechanisms may also be important in nucleation and crystallization from solution.

Driving ordering processes in molecular-dynamics simulations

Self-organized criticality describes the emergence of complexity in dynamical nonequilibrium systems. In this paper we introduce a unique approach whereby a driven energy conversion is utilized as a sampling bias for ordered arrangements in molecular dynamics simulations of atomic and molecular fluids. This approach gives rise to dramatically accelerated nucleation rates, by as much as 30 orders of magnitude, without the need of predefined order parameters, which commonly employed rare-event sampling methods rely on. The measured heat fluxes suggest how the approach can be generalized.

Recent developments in methods for identifying reaction coordinates

In the study of rare events in complex systems with many degrees of freedom, a key element is to identify the reaction coordinates of a given process. Over recent years, a number of methods and protocols have been developed to extract the reaction coordinates based on limited information from molecular dynamics simulations. In this review, we provide a brief survey over a number of major methods developed in the past decade, some of which are discussed in greater detail, to provide an overview of the problems that are partially solved and challenges that still remain. A particular emphasis has been placed on methods for identifying reaction coordinates that are related to the committor.

Computational study of peptide permeation through membrane: Searching for hidden slow variables

Atomically detailed molecular dynamics trajectories in conjunction with Milestoning are used to analyze the different contributions of coarse variables to the permeation process of a small peptide (N-acetyl-L-tryptophanamide, NATA) through a 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) membrane. The peptide reverses its overall orientation as it permeates through the biological bilayer. The large change in orientation is investigated explicitly but is shown to impact the free energy landscape and permeation time only moderately. Nevertheless, a significant difference in permeation properties of the two halves of the membrane suggests the presence of other hidden slow variables. We speculate, based on calculation of the potential of mean force, that a conformational transition of NATA makes significant contribution to these differences. Other candidates for hidden slow variables may include water permeation and collective motions of phospholipids.

Experiments and comprehensive simulations of the formation of a helical turn

We investigate the kinetics and thermodynamics of a helical turn formation in the peptide Ac-WAAAH-NH(2). NMR measurements indicate that this peptide has significant tendency to form a structure of a helical turn, while temperature dependent CD establishes the helix fraction at different temperatures. Molecular dynamics and milestoning simulations agree with experimental observables and suggest an atomically detailed picture for the turn formation. Using a network representation, two alternative mechanisms of folding are identified: (i) a direct co-operative mechanism from the unfolded to the folded state without intermediate formation of hydrogen bonds and (ii) an indirect mechanism with structural intermediates with two residues in a helical conformation. This picture is consistent with kinetic measurements that reveal two experimental time scales of sub-nanosecond and several nanoseconds.