Solute Precipitate Nucleation: A Review of Theory and Simulation Advances

Stochastic computer experiments of the thermodynamic irreversibility of bulk nanobubbles in supersaturated and weak gas-liquid solutions

Spontaneous gas-bubble nucleation in weak gas-liquid solutions has been a challenging topic in theory, experimentation, and computer simulations. In analogy with recent advances in crystallization and droplet formation studies, the diffusive-shielding stabilization and thermodynamic irreversibility of bulk nanobubble (bNB) mechanisms are revisited and deployed to characterize nucleation processes in a stochastic framework of computer experiments using the large-scale atomic/molecular massively parallel simulator code. Theoretical bases, assumptions, and limitations underlying the irreversibility hypothesis of bNBs, and their computational counterparts, are extensively described and illustrated. In essence, it is established that the irreversibility hypothesis can be numerically investigated by converging the system volume (due to the finiteness of interatomic forces) and the initial dissolved-gas concentration in the solution (due to the single-bNB limitation). Helium nucleation in liquid Pb17Li alloy is selected as a representative case study, where it exhibits typical characteristics of noble-gas/liquid-metal systems. The proposed framework lays down the bases on which the stability of gas-bNBs in weak and supersaturated gas-liquid solutions can be inferred and explained from a novel perspective. In essence, it stochastically marches toward a unique irreversible state along out-of-equilibrium nucleation/growth trajectories. Moreover, it does not attempt to characterize the interface or any interface-related properties, neither theoretically nor computationally. It was concluded that bNBs of a few tens of He-atoms are irreversible when dissolved-He concentrations in the weak gas-liquid solution are at least ∼50 and ∼105 mol m-3 at 600 and 1000 K (and ∼80 MPa), respectively, whereas classical molecular dynamics -estimated solubilities are at least two orders of magnitude smaller.

Machine Learning Nucleation Collective Variables with Graph Neural Networks

The efficient calculation of nucleation collective variables (CVs) is one of the main limitations to the application of enhanced sampling methods to the investigation of nucleation processes in realistic environments. Here we discuss the development of a graph-based model for the approximation of nucleation CVs that enables orders-of-magnitude gains in computational efficiency in the on-the-fly evaluation of nucleation CVs. By performing simulations on a nucleating colloidal system mimicking a multistep nucleation process from solution, we assess the model's efficiency in both postprocessing and on-the-fly biasing of nucleation trajectories with pulling, umbrella sampling, and metadynamics simulations. Moreover, we probe and discuss the transferability of graph-based models of nucleation CVs across systems using the model of a CV based on sixth-order Steinhardt parameters trained on a colloidal system to drive the nucleation of crystalline copper from its melt. Our approach is general and potentially transferable to more complex systems as well as to different CVs.

Nonequilibrium Wet-Dry Cycling Acts as a Catalyst for Chemical Reactions

Recent experimental studies suggest that wet-dry cycles and coexisting phases can each strongly alter chemical processes. The mechanisms of why and to what degree chemical processes are altered when subjected to evaporation and condensation are unclear. To close this gap, we developed a theoretical framework for nondilute chemical reactions subject to nonequilibrium conditions of evaporation and condensation. We find that such conditions can change the half-time of the product's yield by more than an order of magnitude, depending on the substrate-solvent interaction. We show that the cycle frequency strongly affects the chemical turnover when the system is maintained out of equilibrium by wet-dry cycles. There exists a resonance behavior in the cycle frequency where the turnover is maximal. This resonance behavior enables wet-dry cycles to select specific chemical reactions, suggesting a potential mechanism for chemical evolution in prebiotic soups at early Earth.

Engulfment Avalanches and Thermal Hysteresis for Antifreeze Proteins on Supercooled Ice

Antifreeze proteins (AFPs) bind to the ice-water surface and prevent ice growth at temperatures below 0 °C through a Gibbs-Thomson effect. Each adsorbed AFP creates a metastable depression on the surface that locally resists ice growth, until ice engulfs the AFP. We recently predicted the susceptibility to engulfment as a function of AFP size, distance between AFPs, and supercooling [ J. Chem. Phys. 2023, 158, 094501]. For an ensemble of AFPs adsorbed on the ice surface, the most isolated AFPs are the most susceptible, and when an isolated AFP gets engulfed, its former neighbors become more isolated and more susceptible to engulfment. Thus, an initial engulfment event can trigger an avalanche of subsequent engulfment events, leading to a sudden surge of unrestrained ice growth. This work develops a model to predict the supercooling at which the first engulfment event will occur for an ensemble of randomly distributed AFP pinning sites on an ice surface. Specifically, we formulate an inhomogeneous survival probability that accounts for the AFP coverage, the distribution of AFP neighbor distances, the resulting ensemble of engulfment rates, the ice surface area, and the cooling rate. We use the model to predict thermal hysteresis trends and compare with experimental data.

Mechanism of Nucleation Pathway Selection in Binary Lennard-Jones Solution: A Combined Study of Molecular Dynamics Simulation and Free Energy Analysis

The nucleation process, which is the initial step in particle synthesis, determines the properties of the resultant particles. Although recent studies have observed various nucleation pathways, the physical factors that determine these pathways have not been fully elucidated. Herein, we conducted molecular dynamics simulations in a binary Lennard-Jones system as a model solution and found that the nucleation pathway can be classified into four types depending on microscopic interactions. The key parameters are (1) the strength of the solute-solute interaction and (2) the difference between the strengths of the like-pair and unlike-pair interactions. The increment of the former alters the nucleation mechanism from a two-step to a one-step pathway, whereas that of the latter causes quick assembly of solutes. Moreover, we developed a thermodynamic model based on the formation of core-shell nuclei to calculate the free energy landscapes. Our model successfully described the pathway observed in the simulations and demonstrated that the two parameters, (1) and (2), define the degree of supercooling and supersaturation, respectively. Thus, our model interpreted the microscopic insights from a macroscopic point of view. Because the only inputs required for our model are the interaction parameters, our model can a priori predict the nucleation pathway.

Broken bond models, magic-sized clusters, and nucleation theory in nanoparticle synthesis

Magic clusters are metastable faceted nanoparticles that are thought to be important and, sometimes, observable intermediates in the nucleation of certain faceted crystallites. This work develops a broken bond model for spheres with a face-centered-cubic packing that form tetrahedral magic clusters. With just one bond strength parameter, statistical thermodynamics yield a chemical potential driving force, an interfacial free energy, and free energy vs magic cluster size. These properties exactly correspond to those from a previous model by Mule et al. [J. Am. Chem. Soc. 143, 2037 (2021)]. Interestingly, a Tolman length emerges (for both models) when the interfacial area, density, and volume are treated consistently. To describe the kinetic barriers between magic cluster sizes, Mule et al. invoked an energy parameter to penalize the two-dimensional nucleation and growth of new layers in each facet of the tetrahedra. According to the broken bond model, barriers between magic clusters are insignificant without the additional edge energy penalty. We estimate the overall nucleation rate without predicting the rates of formation for intermediate magic clusters by using the Becker-Döring equations. Our results provide a blueprint for constructing free energy models and rate theories for nucleation via magic clusters starting from only atomic-scale interactions and geometric considerations.

Free energy barriers for anti-freeze protein engulfment in ice: Effects of supercooling, footprint size, and spatial separation

Anti-freeze proteins (AFPs) protect organisms at freezing conditions by attaching to the ice surface and arresting its growth. Each adsorbed AFP locally pins the ice surface, resulting in a metastable dimple for which the interfacial forces counteract the driving force for growth. As supercooling increases, these metastable dimples become deeper, until metastability is lost in an engulfment event where the ice irreversibly swallows the AFP. Engulfment resembles nucleation in some respects, and this paper develops a model for the "critical profile" and free energy barrier for the engulfment process. Specifically, we variationally optimize the ice-water interface and estimate the free energy barrier as a function of the supercooling, the AFP footprint size, and the distance to neighboring AFPs on the ice surface. Finally, we use symbolic regression to derive a simple closed-form expression for the free energy barrier as a function of two physically interpretable, dimensionless parameters.

Nucleation of Biomolecular Condensates from Finite-Sized Simulations

The nucleation of protein condensates is a concentration-driven process of assembly. When modeled in the canonical ensemble, condensation is affected by finite-size effects. Here, we present a general and efficient route for obtaining ensemble properties of protein condensates in the macroscopic limit from finite-sized nucleation simulations. The approach is based on a theoretical description of droplet nucleation in the canonical ensemble and enables estimation of thermodynamic and kinetic parameters, such as the macroscopic equilibrium density of the dilute protein phase, the surface tension of the condensates, and nucleation free energy barriers. We apply the method to coarse-grained simulations of NDDX4 and FUS-LC, two phase-separating disordered proteins with different physicochemical characteristics. Our results show that NDDX4 condensate droplets, characterized by lower surface tension, higher solubility, and faster monomer exchange dynamics compared to those of FUS-LC, form with negligible nucleation barriers. In contrast, FUS-LC condensates form via an activated process over a wide range of concentrations.

Equilibrium mechanisms of self-limiting assembly

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (e.g. macromolecules, particles) into ordered multi-unit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits, and the other, a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving non-covalent interactions between identical subunits "measure" and select the size of assembled structures? In this review, we begin with an introduction to the basic statistical mechanical framework for assembly thermodynamics, and use this to highlight the key physical ingredients that ensure equilibrium assembly will terminate at finite dimensions. Then, we introduce examples of self-limiting assembly systems, and classify them within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter - micellization of amphiphiles and shell/tubule formation of tapered subunits - as well as less widely known classes of assemblies, such as short-range attractive/long-range repulsive systems and geometrically-frustrated assemblies. For each of these self-limiting mechanisms, we describe the physical mechanisms that select equilibrium assembly size, as well as potential limitations of finite-size selection. Finally, we discuss alternative mechanisms for finite-size assemblies, and draw contrasts with the size-control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.

Direct evaluation of attachment and detachment rate factors of atoms in crystallizing supercooled liquids

Kinetic rate factors of crystallization have a direct effect on formation and growth of an ordered solid phase in supercooled liquids and glasses. Using the crystallizing Lennard-Jones liquid as an example, in the present work, we perform a direct quantitative estimation of values of the key crystallization kinetic rate factors-the rate g⁺ of particle attachments to a crystalline nucleus and the rate g^- of particle detachments from a nucleus. We propose a numerical approach, according to which a statistical treatment of the results of molecular dynamics simulations was performed without using any model functions and/or fitting parameters. This approach allows one to accurately estimate the critical nucleus size n_c. We find that for the growing nuclei, whose sizes are larger than the critical size n_c, the dependence of these kinetic rate factors on the nucleus size n follows a power law. In the case of the subnucleation regime, when the nuclei are smaller than n_c, the n-dependence of the quantity g⁺ is strongly determined by the inherent microscopic properties of a system, and this dependence cannot be described in the framework of any universal law (for example, a power law). It has been established that the dependence of the growth rate of a crystalline nucleus on its size goes into the stationary regime at the size n > 3n_c particles.

Entropic colloidal crystallization pathways via fluid-fluid transitions and multidimensional prenucleation motifs

Complex crystallization pathways are common in protein crystallization, tetrahedrally coordinated systems, and biomineralization, where single or multiple precursors temporarily appear before the formation of the crystal. The emergence of precursors is often explained by a unique property of the system, such as short-range attraction, directional bonding, or ion association. But, structural characteristics of the prenucleation phases found in multistep crystallization remain unclear, and models are needed for testing and expanding the understanding of fluid-to-solid ordering pathways. Here, we report 3 instances of 2-step crystallization of hard-particle fluids. Crystallization in these systems proceeds via a high-density precursor fluid phase with prenucleation motifs in the form of clusters, fibers and layers, and networks, respectively. The density and diffusivity change across the fluid-fluid phase transition increases with motif dimension. We observe crystal nucleation to be catalyzed by the interface between the 2 fluid phases. The crystals that form are complex, including, notably, a crystal with 432 particles in the cubic unit cell. Our results establish the existence of complex crystallization pathways in entropic systems and reveal prenucleation motifs of various dimensions.

Review-Mathematical Formulations of Electrochemically Gas-Evolving Systems

Electrochemically gas-evolving systems are utilized in alkaline water electrolysis, hydrogen production, and many other applications. To design and optimize these systems, high-fidelity models must account for electron-transfer, chemical reactions, thermodynamics, electrode porosity, and hydrodynamics as well as the interconnectedness of these phenomena. Further complicating these models is the production and presence of bubbles. Bubble nucleation naturally occurs due to the chemical reactions and impacts the reaction rate. Modeling bubble growth requires an accurate accounting of interfacial mass transfer. When the bubble becomes large, detachment occurs and the system is modeled as a two-phase flow where the bubbles can then impact material transport in the bulk. In this paper, we review the governing mathematical models of the physicochemical life cycle of a bubble in an electrolytic medium from a multiscale, multiphysics viewpoint. For each phase of the bubble life cycle, the prevailing mathematical formulations are reviewed and compared with particular attention paid to physicochemical processes and the impact the bubble. Through the review of a broad range of models, we provide a compilation of the current state of bubble modeling in electrochemically gas-evolving systems.

Probing the Occurrence of Soluble Oligomers through Amyloid Aggregation Scaling Laws

Drug discovery frequently relies on the kinetic analysis of physicochemical reactions that are at the origin of the disease state. Amyloid fibril formation has been extensively investigated in relation to prevalent and rare neurodegenerative diseases, but thus far no therapeutic solution has directly arisen from this knowledge. Other aggregation pathways producing smaller, hard-to-detect soluble oligomers are increasingly appointed as the main reason for cell toxicity and cell-to-cell transmissibility. Here we show that amyloid fibrillation kinetics can be used to unveil the protein oligomerization state. This is illustrated for human insulin and ataxin-3, two model proteins for which the amyloidogenic and oligomeric pathways are well characterized. Aggregation curves measured by the standard thioflavin-T (ThT) fluorescence assay are shown to reflect the relative composition of protein monomers and soluble oligomers measured by nuclear magnetic resonance (NMR) for human insulin, and by dynamic light scattering (DLS) for ataxin-3. Unconventional scaling laws of kinetic measurables were explained using a single set of model parameters consisting of two rate constants, and in the case of ataxin-3, an additional order-of-reaction. The same fitted parameters were used in a discretized population balance that adequately describes time-course measurements of fibril size distributions. Our results provide the opportunity to study oligomeric targets using simple, high-throughput compatible, biophysical assays.

Role of Salt, Pressure, and Water Activity on Homogeneous Ice Nucleation

Pure water can be substantially supercooled below the melting temperature without transforming into ice. The achievable supercooling can be enhanced by adding solutes or by applying hydrostatic pressure. Avoiding ice formation is of great importance in the cryopreservation of food or biological samples. In this Letter, we investigate the similarity between the effects of pressure and salt on ice formation using a combination of state-of-the-art simulation techniques. We find that both hinder ice formation by increasing the energetic cost of creating the ice-fluid interface. Moreover, we examine the widely accepted proposal that the ice nucleation rate for different pressures and solute concentrations can be mapped through the activity of water [ Koop , L. ; Tsias , P. Nature , 2000 , 406 , 611 ]. We show that such a proposal is not consistent with the nucleation rates predicted in our simulations because it does not include all parameters affecting ice nucleation. Therefore, even though salt and pressure have a qualitatively similar effect on ice formation, they cannot be quantitatively mapped onto one another.

Diabat Interpolation for Polymorph Free-Energy Differences

Existing methods to compute free-energy differences between polymorphs use harmonic approximations, advanced non-Boltzmann bias sampling techniques, and/or multistage free-energy perturbations. This work demonstrates how Bennett's diabat interpolation method ( J. Comput. Phys. 1976, 22, 245 ) can be combined with energy gaps from lattice-switch Monte Carlo techniques ( Phys. Rev. E 2000, 61, 906 ) to swiftly estimate polymorph free-energy differences. The new method requires only two unbiased molecular dynamics simulations, one for each polymorph. To illustrate the new method, we compute the free-energy difference between face-centered cubic and body-centered cubic polymorphs for a Gaussian core solid. We discuss the justification for parabolic models of the free-energy diabats and similarities to methods that have been used in studies of electron transfer.