Two-step vapor-crystal nucleation close below triple point

The impact of colloid-solvent dynamic coupling on the coarsening rate of colloidal phase separation

Phase separation, a fundamental phenomenon in both natural and industrial settings, involves the coarsening of domains over time t to reduce interfacial energy. While well-understood for simple viscous liquid mixtures, the physical laws governing coarsening dynamics in complex fluids, such as colloidal suspensions, remain unclear. Here, we investigate colloidal phase separation through particle-based simulations with and without hydrodynamic interactions (HIs). The former incorporates many-body HIs through momentum conservation, while the latter simplifies their effects into a constant friction coefficient on a particle. In cluster-forming phase separation with HIs, the domain size ℓ grows as ℓ∝t1/3, aligning with the Brownian-coagulation mechanism. Without HIs, ℓ∝t1/5, attributed to an improper calculation of cluster thermal diffusion. For network-forming phase separation, ℓ∝t1/2 with HIs, while ℓ∝t1/3 without HIs. In both cases, network coarsening is governed by the mechanical stress relaxation of the colloid-rich phase, yet with distinct mechanisms: slow solvent permeation through densely packed colloids for the former and free draining for the latter. Our results provide a clear and concise physical picture of colloid-solvent dynamic coupling via momentum conservation, offering valuable insights into the self-organization dynamics of particles like colloids, emulsions, and globular proteins suspended in a fluid.

Kinetic phase diagram for two-step nucleation in colloid-polymer mixtures

Two-step crystallization via a metastable intermediate phase is often regarded as a non-classical process that lies beyond the framework of classical nucleation theory (CNT). In this work, we investigate two-step crystallization in colloid-polymer mixtures via an intermediate liquid phase. Using CNT-based seeding simulations, we construct a kinetic phase diagram that identifies regions of phase space where the critical nucleus is either liquid or crystalline. These predictions are validated using transition path sampling simulations at nine different relevant state points. When the critical nucleus is liquid, crystallization occurs stochastically during the growth phase, whereas for a crystalline critical nucleus, the crystallization process happens pre-critically at a fixed nucleus size. We conclude that CNT-based kinetic phase diagrams are a powerful tool for understanding and predicting "non-classical" crystal nucleation mechanisms.

The bcc coating of Lennard-Jones crystal nuclei vanishes with a change of local structure detection algorithm

Since the influential work of ten Wolde, Ruiz-Montero, and Frenkel [Phys. Rev. Lett. 75, 2714 (1995)], crystal nucleation from a Lennard-Jones fluid has been regarded as a paradigmatic example of metastable crystal ordering at the surface of a critical nucleus. We apply seven commonly used local structure detection algorithms to characterize crystal nuclei obtained from transition path sampling simulations. The polymorph composition of these nuclei varies significantly depending on the algorithm used. Our results indicate that one should be very careful when characterizing the local structure near solid-solid and solid-fluid interfaces. Particles near such interfaces exhibit a local structure distinct from that of bulk fluid or bulk crystal phases. We argue that incorporating outlier detection into the local structure detection method is beneficial, leading to greater confidence in the classification results. Interestingly, the bcc coating nearly disappears when adopting a machine learning method with outlier detection.

From adsorption to crystallization of proteins: Evidence for interface-assisted nucleation

Protein crystallization is among the key processes in biomolecular research, but the underlying mechanisms are still elusive. Here, we address the role of inevitable interfaces for the nucleation process. Quartz crystal microbalance with dissipation monitoring (QCM-D) with simultaneously optical microscopy, confocal microscopy, and grazing-incidence small angle X-rays scattering (GISAXS) were employed to investigate the temporal behavior from the initial stage of protein adsorption to crystallization. Here we studied the crystallization of the Human Serum Albumin (HSA), the most abundant blood protein, in the presence of a charged surface and a trivalent salt. We found evidence for interface-assisted nucleation of crystals. The kinetic stages involved are initial adsorption followed by enhanced adsorption after longer times, subsequent nucleation, and finally crystal growth. The results highlight the importance of interfaces for protein phase behavior and in particular for nucleation.

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.

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.

Interfacial properties of binary azeotropic mixtures of simple fluids: Molecular dynamics simulation and density gradient theory

Interfacial properties of binary azeotropic mixtures of Lennard-Jones truncated and shifted fluids were studied by molecular dynamics (MD) simulation and density gradient theory (DGT) in combination with an equation of state. Three binary mixtures were investigated, which differ in the energetic cross interaction parameter that yields different types of azeotropic behavior. This study covers a wide temperature and composition range. Mixture A exhibits a heteroazeotrope at low temperatures, which changes to a low-boiling azeotrope at high temperatures, mixture B exhibits a low-boiling azeotrope, and mixture C exhibits a high-boiling azeotrope. The phase behavior and fluid interfacial properties as well as their relation were studied. Vapor-liquid, liquid-liquid, and vapor-liquid-liquid equilibria and interfaces were considered. Density profiles, the surface tension, the interfacial thickness, as well as the relative adsorption and enrichment of the components at the interface were studied. The results obtained from the two independent methods (MD and DGT) are overall in good agreement. The results provide insights into the relation of the phase behavior, particularly the azeotropic behavior, of simple fluid mixtures and the corresponding interfacial properties. Strong enrichment was found for the mixture with a heteroazeotrope in the vicinity of the three-phase equilibrium, which is related to a wetting transition.

Extension of the Aggregation-Volume-Bias Monte Carlo Method to the Calculation of Phase Properties of Solid Systems: A Lattice-Based Cluster Approach

The aggregation-volume-bias Monte Carlo method, which has been successful in the calculation of the formation free energies of liquid clusters, is extended to solid systems. This extension is motivated by early studies where disordered clusters are observed when the original method is applied at a temperature even far below the triple point. In order to avoid the formation of disordered aggregates, the insertion of particles is targeted directly toward those crystal lattice sites. Specifically, the insertion volume used to be defined as a spherical volume centered around a given target molecule is now restricted to be around each of the crystal lattice sites near a given target molecule. The free energies obtained for both liquid and solid clusters are then used to extrapolate bulk-phase information such as the chemical potential of the liquid and solid phases at coexistence. Using the temperature and pressure dependencies of the chemical potential information obtained for both liquid and solid phases, the location of the triple point can be determined. For Lennard-Jonesium, the results were found to be in good agreement with previous simulation studies using other approaches.

Multiscale Exploration of Concentration-Dependent Amyloid-β(16-21) Amyloid Nucleation

Atomic descriptions of peptide aggregation nucleation remain lacking due to the difficulty of exploring complex configurational spaces on long time scales. To elucidate this process, we develop a multiscale approach combining a metadynamics-based method with cluster statistical mechanics to derive concentration-dependent free energy surfaces of nucleation at near-atomic resolution. A kinetic transition network of nucleation is then constructed and employed to systematically explore nucleation pathways and kinetics through stochastic simulations. This approach is applied to describe Aβ_16-21 amyloid nucleation, revealing a two-step mechanism involving disordered aggregates at millimolar concentration, and an unexpected mechanism at submillimolar concentrations that exhibits kinetics reminiscent of classical nucleation but atypical pathways involving growing clusters with structured cores wrapped by disordered surface. When this atypical mechanism is operative, critical nucleus size can be reflected by the nucleation reaction order. Collectively, our approach paves the way for a more quantitative and detailed understanding of peptide aggregation nucleation.

Molecular Dynamics Study of Wetting and Adsorption of Binary Mixtures of the Lennard-Jones Truncated and Shifted Fluid on a Planar Wall

The wetting of surfaces is strongly influenced by adsorbate layers. Therefore, in this work, sessile drops and their interaction with adsorbate layers on surfaces were investigated by molecular dynamics simulations. Binary fluid model mixtures were considered. The two components of the fluid mixture have the same pure component parameters, but one component has a stronger and the other a weaker affinity to the surface. Furthermore, the unlike interactions between both components were varied. All interactions were described by the Lennard-Jones truncated and shifted potential with a cutoff radius of 2.5σ. The simulations were carried out at constant temperature for mixtures of different compositions. The parameters were varied systematically and chosen such that cases with partial wetting as well as cases with total wetting were obtained and the relation between the varied molecular parameters and the phenomenological behavior was elucidated. Data on the contact angle as well as on the mole fraction and thickness of the adsorbate layer were obtained, accompanied by information on liquid and gaseous bulk phases and the corresponding phase equilibrium. Also, the influence of the adsorbate layer on the wetting was studied: for a sufficiently thick adsorbate layer, the wall's influence on the wetting vanishes, which is then only determined by the adsorbate layer.

Free energy surface of two-step nucleation

We test the theoretical free energy surface (FES) for two-step nucleation (TSN) proposed by Iwamatsu [J. Chem. Phys. 134, 164508 (2011)] by comparing the predictions of the theory to numerical results for the FES recently reported from Monte Carlo simulations of TSN in a simple lattice system [James et al., J. Chem. Phys. 150, 074501 (2019)]. No adjustable parameters are used to make this comparison. That is, all the parameters of the theory are evaluated directly for the model system, yielding a predicted FES, which we then compare to the FES obtained from simulations. We find that the theoretical FES successfully predicts the numerically evaluated FES over a range of thermodynamic conditions that spans distinct regimes of behavior associated with TSN. All the qualitative features of the FES are captured by the theory, and the quantitative comparison is also very good. Our results demonstrate that Iwamatsu's extension of classical nucleation theory provides an excellent framework for understanding the thermodynamics of TSN.

Molecular Theory of Nucleation from Dilute Phases: Formulation and Application to Lennard-Jones Vapor

In this Letter, we present a molecular theory of nucleation from dilute phases such as vapors or dilute solutions. The theory can model the nonclassical two-step crystal nucleation seen in many systems. When applied to study and analyze the crystal nucleation pathways from Lennard-Jones vapor, we find that prior explanations of the two-step mechanism based on lower barrier height for liquid nuclei is incomplete. The analysis from the molecular theory reveal that a complete explanation would also require consideration of anisotropy in the diffusion constants for growth of liquid droplets vis-á-vis the crystal nuclei.

Nonclassical Nucleation—Role of Metastable Intermediate Phase in Crystal Nucleation: An Editorial Prefix

Classical nucleation theory (CNT), which was established about 90 years ago, represents the most commonly used theory in describing nucleation processes. For a fluid-to-solid phase transition, CNT states that the solutes in a supersaturated solution reversibly form small clusters. Once a cluster reaches its critical size, it becomes thermodynamically stable and is favored for further growth. One of the most important assumptions of CNT is that the nucleation process is described by one reaction coordinate and all order parameters proceed simultaneously. Recent studies in experiments, computer simulations, and theory have revealed nonclassical features in the early stage of nucleation. In particular, the decoupling of order parameters involved during a fluid-to-solid transition leads to the so-called two-step nucleation mechanism, in which a metastable intermediate phase (MIP) exists in parallel to the initial supersaturated solution and the final crystals. These MIPs can be high-density liquid phases, mesoscopic clusters, or preordered states. In this Special Issue, we focus on the role of the various MIPs in the early stage of crystal nucleation of organic materials, metals and alloys, aqueous solutions, minerals, colloids, and proteins, and thus on various scenarios of nonclassical pathways of crystallization.

Interfacial properties of binary mixtures of simple fluids and their relation to the phase diagram

Enrichment at vapour–liquid interfaces can be interpreted as a wetting transition in the vicinity of a three phase equilibrium.

Can the pathway of stepwise nucleation be predicted and controlled?

Predicting the critical nucleus size and the nucleation barrier is of central importance in controlling the dynamics of nucleation. However, as the nucleation of a crystal involves intermediate states, the prediction becomes inaccessible with currently available models. Here, we show that based on single-particle level observations, the properties of crystal nuclei in a microscopic stepwise nucleation (MSN) can be well-quantified by incorporating the size and structure order parameter into the formula of free energy without prior knowledge of interfacial tension. The quantified free energy reveals that the intermediate structures arise from thermodynamics rather than kinetics. Precritical and postcritical nuclei are distinct not only in structure but also in the mechanism of crystalline ordering. The relative stability of intermediate structures and the pathway of nucleation can be well-controlled by supercooling. Our studies offer a successful approach to quantify MSN and shed new light on resolving the long-standing discrepancies between simulations and experiments.

Nanocapillarity and Liquid Bridge-Mediated Force between Colloidal Nanoparticles

In this work, we probe the concept of interface tension for ultrathin adsorbed liquid films on the nanoscale by studying the surface fluctuations of films down to the monolayer. Our results show that the spectrum of film height fluctuations of a liquid-vapor surface may be extended to ultrathin films provided we take into account the interactions of the substrate with the surface. Global fluctuations of the film height are described in terms of disjoining pressure, whereas surface deformations that are proportional to the interface area are accounted for by a film thickness-dependent surface tension. As a proof of concept, we model the capillary forces between colloidal nanoparticles held together by liquid bridges. Our results indicate that the classical equations for capillarity follow very precisely down to the nanoscale provided we account for the film height dependence of the surface tension.

Nonclassical nucleation pathways in protein crystallization

Classical nucleation theory (CNT), which was established about 90 years ago, has been very successful in many research fields, and continues to be the most commonly used theory in describing the nucleation process. For a fluid-to-solid phase transition, CNT states that the solute molecules in a supersaturated solution reversibly form small clusters. Once the cluster size reaches a critical value, it becomes thermodynamically stable and favored for further growth. One of the most important assumptions of CNT is that the nucleation process is described by one reaction coordinate and all order parameters proceed simultaneously. Recent studies in experiments, computer simulations and theory have revealed nonclassical features in the early stage of nucleation. In particular, the decoupling of order parameters involved during a fluid-to-solid transition leads to the so-called two-step nucleation mechanism, in which a metastable intermediate phase (MIP) exists between the initial supersaturated solution and the final crystals. Depending on the exact free energy landscapes, the MIPs can be a high density liquid phase, mesoscopic clusters, or a pre-ordered state. In this review, we focus on the studies of nonclassical pathways in protein crystallization and discuss the applications of the various scenarios of two-step nucleation theory. In particular, we focus on protein solutions in the presence of multivalent salts, which serve as a model protein system to study the nucleation pathways. We wish to point out the unique features of proteins as model systems for further studies.

Nucleation and growth of a core-shell composite nucleus by diffusion

The critical radius of a core-shell-type nucleus grown by diffusion in a phase-separated solution is studied. A kinetic critical radius rather than the thermodynamic critical radius of standard classical nucleation theory can be defined from the diffusional growth equations. It is shown that there exist two kinetic critical radii for the core-shell-type nucleus, for which both the inner-core radius and the outer-shell radius will be stationary. Therefore, these two critical radii correspond to a single critical point of the nucleation path with a single energy barrier even though the nucleation looks like a two-step process. The two radii are given by formulas similar to that of classical nucleation theory if the Ostwald-Freundlich boundary condition is imposed at the surface of the inner nucleus and that of the outer shell. The subsequent growth of a core-shell-type postcritical nucleus follows the classical picture of Ostwald's step rule. Our result is consistent with some of the experimental and numerical results which suggest the core-shell-type critical nucleus.

Super-long ZnO nanofibers and novel nucleation mechanism for a gas-phase environment: spatial linear nucleation

Single-crystal super-long ZnO nanofibers (SZFs) have been prepared based on a newly proposed spatial linear nucleation (SLN) mechanism.

Freezing of supercooledn-decane nanodroplets: from surface driven to frustrated crystallization

Decane condenses to form nanodroplets that freeze from the outside in, until it's so cold that fractal like frustrated crystals form instead.

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.

Contact angle of sessile drops in Lennard-Jones systems

Molecular dynamics simulations are used for studying the contact angle of nanoscale sessile drops on a planar solid wall in a system interacting via the truncated and shifted Lennard-Jones potential. The entire range between total wetting and dewetting is investigated by varying the solid-fluid dispersive interaction energy. The temperature is varied between the triple point and the critical temperature. A correlation is obtained for the contact angle in dependence of the temperature and the dispersive interaction energy. Size effects are studied by varying the number of fluid particles at otherwise constant conditions, using up to 150,000 particles. For particle numbers below 10,000, a decrease of the contact angle is found. This is attributed to a dependence of the solid-liquid surface tension on the droplet size. A convergence to a constant contact angle is observed for larger system sizes. The influence of the wall model is studied by varying the density of the wall. The effective solid-fluid dispersive interaction energy at a contact angle of θ = 90° is found to be independent of temperature and to decrease linearly with the solid density. A correlation is developed that describes the contact angle as a function of the dispersive interaction, the temperature, and the solid density. The density profile of the sessile drop and the surrounding vapor phase is described by a correlation combining a sigmoidal function and an oscillation term.

Computer simulation of epitaxial nucleation of a crystal on a crystalline surface

We present results of computer simulations of crystal nucleation on a crystalline surface, in the Lennard-Jones model. Motivated by the pioneering work of Turnbull and Vonnegut [Ind. Eng. Chem. 44, 1292 (1952)], we investigate the effects of a mismatch between the surface lattice constant and that of the bulk nucleating crystal. We find that the nucleation rate is maximum close to, but not exactly at, zero mismatch. The offset is due to the finite size of the nucleus. In agreement with a number of experiments, we find that even for large mismatches of 10% or more, the formation of the crystal can be epitaxial, meaning that the crystals that nucleate have a fixed orientation with respect to the surface lattice. However, nucleation is not always epitaxial, and loss of epitaxy does affect how the rate varies with mismatch. The surface lattice strongly influences the nucleation rate. We show that the epitaxy observed in our simulations can be predicted using calculations of the potential energy between the surface and the first layer of the nucleating crystal, in the spirit of simple approaches such as that of Hillier and Ward [Phys. Rev. B 54, 14037 (1996)].

Nucleation pathway of core-shell composite nucleus in size and composition space and in component space

The kinetics of nucleation of a core-shell composite nucleus that consists of a core of stable final phase surrounded by a wetting layer of intermediate metastable phase is studied using the kinetic theory of binary nucleation not only in the size and composition space but also in the component space. The steady-state solution of the Fokker-Planck equation is considered. Various formulas for the critical nucleus at the saddle point as well as for the postcritical nucleus are derived. The kinetics of nucleation at the saddle point is more appropriately characterized in the size and composition space, while the kinetics of the postcritical nucleus is more appropriately described in the component space. Although both the free-energy landscape and the reaction rates play decisive role to determine the kinetics of nucleation at the saddle point, the details of the free-energy landscape are irrelevant to the kinetics of the postcritical nucleus.

Crystal Nucleation without Supersaturation

Classical nucleation theory (CNT) has been extensively employed to interpret crystal nucleation phenomena and postulates the formation of an ordered crystalline nucleus directly from vapor or solution. Here, we provide the first experimental demonstration of a two-step mechanism that facilitates deposition of crystals on solid surfaces from vapor. Crucially, this occurs from saturated vapor without the need for supersaturation, conditions that, according to CNT, cannot lead to direct deposition of crystals from vapor. Instead, the process relies on condensation of supercooled liquid in surface cavities below the melting point. Crystals then nucleate in this liquid, leading to rapid deposition of more solid. Such a mechanism has been postulated for atmospheric nucleation of ice on aerosol particles and may have analogies in the crystallization of biominerals via amorphous precursor phases.

Computer simulations of nucleation of nanoparticle superclusters from solution

This paper presents simulation studies of nanoparticle supercluster (NPSC) nucleation from a temperature quenched system. The nanoparticles are represented as 5 nm, spherical gold nanoparticles ligated with alkane thiols. The pair potential accounts for the van der Waals interaction between the metallic cores and ligand-ligand and ligand-solvent interactions. Phenomena well-known for molecular systems are observed including a prenucleation induction period, fluctuating prenucleation clusters that predominately add monomers one at a time, a critical nucleus size, and growth of NPSCs from solution in the presence of an equilibrium supernatant, all consistent with classical nucleation theory. However, only the largest prenucleating clusters are dense, and the cluster size can occasionally range greater than the critical size in the prenucleation regime until a cluster with low enough energy occurs, then nucleation ensues. Late in the nucleation process, the clusters display a crystalline structure that is a random mix of face-centered cubic (fcc) and hexagonal close-packed (hcp) lattices and indistinguishable from a randomized icosahedra structure.

Excess equimolar radius of liquid drops

The curvature dependence of the surface tension is related to the excess equimolar radius of liquid drops, i.e., the deviation of the equimolar radius from the radius defined by the macroscopic capillarity approximation. Based on the Tolman [J. Chem. Phys. 17, 333 (1949)] approach and its interpretation by Nijmeijer et al. [J. Chem. Phys. 96, 565 (1991)], the surface tension of spherical interfaces is analyzed in terms of the pressure difference due to curvature. In the present study, the excess equimolar radius, which can be obtained directly from the density profile, is used instead of the Tolman length. Liquid drops of the truncated and shifted Lennard-Jones fluid are investigated by molecular dynamics simulation in the canonical ensemble, with equimolar radii ranging from 4 to 33 times the Lennard-Jones size parameter σ. In these simulations, the magnitude of the excess equimolar radius is shown to be smaller than σ/2. This suggests that the surface tension of liquid drops at the nanometer length scale is much closer to that of the planar vapor-liquid interface than reported in studies based on the mechanical route.

Contact angle dependence on the fluid-wall dispersive energy

Menisci of the truncated and shifted Lennard-Jones fluid between parallel planar walls are investigated by molecular dynamics simulation. Thereby, the characteristic energy of the unlike dispersive interaction between fluid molecules and wall atoms is systematically varied to determine its influence on the contact angle. The temperature is varied as well, covering most of the range between the triple-point temperature and the critical temperature of the bulk fluid. The transition between obtuse and acute angles is found to occur at a temperature-independent magnitude of the fluid-wall dispersive interaction energy. On the basis of the present simulation results, fluid-wall interaction potentials can be adjusted to contact angle measurements.