Homogeneous and heterogeneous nucleation of Lennard-Jones liquids

Metastable states assisted homogeneous nucleation in supercooled liquid aluminum alloys: Insights from a phenomenologically coupled atomistic, phase-field, and machine learning investigation

Crystallization due to liquid → solid transformation is observed in many natural and engineering processes. Extant literature indicates that crystallization in supercooled liquids is initiated by precursory metastable phases or states, also called non-classical nucleation. For face-centered cubic (FCC) materials, latest experimental and computational studies suggest that metastable hexagonal-closed packed (HCP) structures facilitate equilibrium FCC formation. However, the underlying nucleation mechanism remains unclear. Here, we examine structural changes and energetic barriers associated with such a non-classical mechanism, by performing molecular dynamics (MD) simulations using pure Al, Al-0.5 at. %Cu, and Al-0.5 at. %Ni (all FCC-formers) and phenomenologically coupling MD results with phase-field (PF) modeling. Such a coupling involved initializing PF simulation domains and constructing Landau polynomials-consistent with MD observations. Unsupervised machine learning was utilized to capture nuclei structures from MD simulations, while neural networks helped in extracting equilibrium interfacial energies from PF modeling. Atomistic simulations showed that precursory nuclei are comprised of collection of metastable-HCP states with medium ranged ordering. The pockets of HCP states later transform to critical nuclei-containing an FCC core and an outer layer of HCP. PF modeling qualitatively replicated the precursory-to-critical nuclei transformation and showed that the energetic barriers between the precursory and critical nuclei are substantially smaller than predictions obtained from classical nucleation theory. Together, these observations permitted us to propose a holistic non-classical mechanism that links triangular motifs within Al-based supercooled liquids to the critical nuclei via in-liquid structural transformations.

Toward an FPGA-based dedicated computer for molecular dynamics simulations

First steps toward a molecular dynamics (MD) implementation in a cluster of field-programmable gate arrays (FPGAs) are presented, reaching a simulation speed of a few microseconds/day. The nodes in this cluster are programmed into a mid-ranged FPGA (Artix 7 XC7A200T), interconnected as a 3D torus by fast optical links. The implemented MD algorithm is highly parallelized and highly pipelined internally. The FPGA cluster is freely scalable in terms of size, i.e., a larger MD system requires more nodes, however, without compromising simulation speed. The performance in terms of energy stability and simulation speed is analyzed. At present, the focus lies on the fast networking, while only minimal MD functionality has been implemented so far, i.e., Lennard-Jones interactions and a thermostat, which were needed to demonstrate the feasibility of the FPGA cluster to run multi-microsecond simulations. To that end, the nucleation of a super-cooled Lennard-Jones liquid is investigated by unbiased MD simulations, which is a difficult MD problem since a high nucleation barrier has to be overcome. Finally, the pathways toward a full MD implementation are outlined. The current implementation will be made available as an open-source development project.

100 Years of the Lennard-Jones Potential

It is now 100 years since Lennard-Jones published his first paper introducing the now famous potential that bears his name. It is therefore timely to reflect on the many achievements, as well as the limitations, of this potential in the theory of atomic and molecular interactions, where applications range from descriptions of intermolecular forces to molecules, clusters, and condensed matter.

Resolving Temperature-Dependent Hydrate Nucleation Pathway: The Role of "Transition Layer"

Understanding the nucleation of natural gas hydrate (NGH) at different conditions has important implications to NGH recovery and other industrial applications, such as gas storage and separation. Herein, vast numbers of hydrate nucleation events are traced via molecular dynamics (MD) simulations at different degrees of supercooling (or driving forces). Specifically, to precisely characterize a hydrate nucleus from an aqueous system during the MD simulation, we develop an evolutionary order parameter (OP) to recognize the nucleus size and shape. Subsequently, the free energy landscapes of hydrate during nucleation are explored by using the newly developed OP. The results suggest that at 270 K (or 0.92 Tm supercooling, where Tm is the melting point), the near-rounded nucleus prevails during the nucleation, as described from the classical nucleation theory. In contrast, at relatively strong driving forces of 0.85 and 0.88 Tm, nonclassical nucleation events arise. Specifically, the pathway toward an elongated nucleus becomes as important as the pathway toward a near-rounded nucleus. To explain the distinct nucleation phenomena at different supercoolings, a notion of a "transition layer" (or liquid-blob-like layer) is proposed. Here, the transition layer is to describe the interfacial region between the nucleus and aqueous solution, and this layer entails two functionalities: (1) it tends to retain CH4 depending on the degrees of supercooling and (2) it facilitates collision among CH4, which thus promote the incorporation of CH4 into nucleus. Our simulation indicates that compared to the near-rounded nucleus, the transition layer surrounding the elongated nucleus is more evident with the higher collision rate among CH4 molecules. As such, the transition layer tends to promote the elongated nucleus pathway, while offsetting the cost of larger surface free energy associated with the elongated nucleus. At 0.92 Tm, however, the transition layer gradually disappears, and classical nucleation events dominate. Overall, the notion of "transition layer" offers deeper insight into the NGH nucleation at different degrees of supercooling and could be extended to describe other types of hydrate nucleation.

Critical comparison of general-purpose collective variables for crystal nucleation

The nucleation of crystals is a prominent phenomenon in science and technology that still lacks a full atomic-scale understanding. Much work has been devoted to identifying order parameters able to track the process, from the inception of early nuclei to their maturing to critical size until growth of an extended crystal. We critically assess and compare two powerful distance-based collective variables, an effective entropy derived from liquid state theory and the path variable based on permutation invariant vectors using the Kob-Andersen binary mixture and a combination of enhanced-sampling techniques. Our findings reveal a comparable ability to drive nucleation when a bias potential is applied, and comparable free-energy barriers and structural features. Yet, we also found an imperfect correlation with the committor probability on the barrier top which was bypassed by changing the order parameter definition.

A high-accuracy machine-learning water model for exploring water nanocluster structures

Water, the most important molecule on the Earth, possesses many essential and unique physical properties that are far from completely understood, partly due to serious difficulties in identifying the precise microscopic structures of water. Hence, identifying the structures of water nanoclusters is a fundamental and challenging issue for studies on the relationship between the macroscopic physical properties of water and its microscopic structures. For large-scale simulations (at the level of nm and ns) of water nanoclusters, a calculation method with simultaneous accuracy at the level of quantum chemistry and efficiency at the level of an empirical potential method is in great demand. Herein, a machine-learning (ML) water model was utilized to explore the microscopic structural features at different length scales for water nanoclusters with a size up to several nm. The ML water model can be employed to efficiently predict the structures of water nanoclusters with a similar accuracy to that of density functional theory and with substantially lower computational resource demands. To validate the low-lying structure search results with experimental spectral results, an ML water model combined with velocity autocorrelation function analysis was used to simulate the vibrational spectra of water nanoclusters with up to thousands of water molecules. By comparing the simulated and experimentally recorded vibrational spectra, the atomic structures determined by a simulation based on the ML water model are all verified. To demonstrate its ability to represent water's structural evolution at large length and time scales, the ML water model was employed to model the structural evolution during the crystal-liquid transition, and the phase transition temperatures of water clusters with different sizes were precisely predicted. The ML water model provides an efficient theoretical calculation tool for exploring the structures and physical properties of water and their relationships, especially for clusters with relatively large sizes and processes with relatively long durations.

The role of the dipole moment orientations in the crystallization tendency of the van der Waals liquids - molecular dynamics simulations

Computer simulations of model systems play a remarkable role in the contemporary studies of structural, dynamic and thermodynamic properties of supercooled liquids. However, the commonly employed model systems, i.e., simple-liquids, do not reflect the internal features of the real molecules, e.g., structural anisotropy and spatial distribution of charges, which might be crucial for the behavior of real materials. In this paper, we use the new model molecules of simple but anisotropic structure, to studies the effect of dipole moment orientation on the crystallization tendency. Our results indicate that proper orientation of the dipole moment could totally change the stability behavior of the system. Consequently, the exchange of a single atom within the molecule causing the change of dipole moment orientation might be crucial for controlling the crystallization tendency. Moreover, employing the classical nucleation theory, we explain the reason for this behavior.

Divining the shape of nascent polymer crystal nuclei

We demonstrate that nascent polymer crystals (i.e., nuclei) are anisotropic entities with neither spherical nor cylindrical geometry, in contrast to previous assumptions. In fact, cylindrical, spherical, and other high symmetry geometries are thermodynamically unfavorable. Moreover, postcritical transitions are necessary to achieve the lamellae that ultimately arise during the crystallization of semicrystalline polymers. We also highlight how inaccurate treatments of polymer nucleation can lead to substantial errors (e.g., orders of magnitude discrepancies in predicted nucleation rates). These insights are based on quantitative analysis of over four million crystal clusters from the crystallization of prototypical entangled polyethylene melts. New comprehensive bottom-up models are needed to capture polymer nucleation.

Reaction coordinates and rate constants for liquid droplet nucleation: Quantifying the interplay between driving force and memory

In this work, we revisit the classic problem of homogeneous nucleation of a liquid droplet in a supersaturated vapor phase. We consider this at different extents of the driving force, or equivalently the supersaturation, and calculate a reaction coordinate (RC) for nucleation as the driving force is varied. The RC is constructed as a linear combination of three order parameters, where one accounts for the number of liquidlike atoms and the other two for local density fluctuations. The RC is calculated from biased and unbiased molecular dynamics (MD) simulations using the spectral gap optimization approach "SGOOP" [P. Tiwary and B. J. Berne, Proc. Natl. Acad. Sci. U. S. A. 113, 2839 (2016)]. Our key finding is that as the supersaturation decreases, the RC ceases to simply be the number of liquidlike atoms, and instead, it becomes important to explicitly consider local density fluctuations that correlate with shape and density variations in the nucleus. All three order parameters are found to have similar barriers in their respective potentials of mean force; however, as the supersaturation decreases, the density fluctuations decorrelate slower and thus carry longer memory. Thus, at lower supersaturations, density fluctuations are non-Markovian and cannot be simply ignored from the RC by virtue of being noise. Finally, we use this optimized RC to calculate nucleation rates in the infrequent metadynamics framework and show that it leads to a more accurate estimate of the nucleation rate with four orders of magnitude acceleration relative to unbiased MD.

Molecular dynamics investigation of the local structure in iron melts and its role in crystal nucleation during rapid solidification

Nucleation process of a bcc crystal after the formation of an MRO cluster.

Thermodynamic and structural anomalies of water nanodroplets

Liquid water nanodroplets are important in earth's climate, and are valuable for studying supercooled water because they resist crystallisation well below the bulk freezing temperature. Bulk liquid water has well-known thermodynamic anomalies, such as a density maximum, and when supercooled is hypothesised to exhibit a liquid-liquid phase transition (LLPT) at elevated pressure. However, it is not known how these bulk anomalies might manifest themselves in nanodroplets. Here we show, using simulations of the TIP4P/2005 water model, that bulk anomalies occur in nanodroplets as small as 360 molecules. We also show that the Laplace pressure inside small droplets reaches 220 MPa at 180 K, conditions close to the LLPT of TIP4P/2005. While the density and pressure inside nanodroplets coincide with bulk values at moderate supercooling, we show that deviations emerge at lower temperature, as well as significant radial density gradients, which arise from and signal the approach to the LLPT.

Lattice induced crystallization of nanodroplets: the role of finite-size effects and substrate properties in controlling polymorphism

Targeting specific technological applications requires the control of nanoparticle properties, especially the crystalline polymorph. Freezing a nanodroplet deposited on a solid substrate leads to the formation of crystalline structures. We study the inherent mechanisms underlying this general phenomenon by means of molecular dynamics simulations. Our work shows that different crystal structures can be selected by finely tuning the solid substrate lattice parameter. Indeed, while for our system, face-centered cubic is usually the most preponderant structure, the growth of two distinct polymorphs, hexagonal centered packing and body-centered cubic, was also observed even when the solid substrate was face-centered cubic. Finally, we also demonstrated that the growth of hexagonal centered packing is conditioned by the appearance of large enough body-centered cubic clusters thus suggesting the presence of a cross-nucleation pathway. Our results provide insights into the impact of nanoscale effects and solid substrate properties towards the growth of polymorphic nanomaterials.

Anisotropic stress inhibits crystallization in Cu-Zr glass-forming liquids

Liquids attain a metastable state without crystallizing by cooling rapidly to a given temperature below the melting point. With increasing supercooling, the nucleation rate would show an increase based on the prediction of the classical nucleation theory. It is generally thought that the nucleation rate will reach the maximum upon approaching the glass transition temperature, T_g, for glass-forming liquids. We report that there exists a supercooled region above T_g in which the crystallization has actually been severely suppressed. Our molecular dynamics simulations show that the growth of embryos in the supercooled Cu₆₀Zr₄₀ melt is subjected to a strong anisotropic stress associated with the dynamic heterogeneity. Its long-range effect drives the embryo to grow into a ramified morphology so that the interface energy dominates over the embryo growth, leading to the suppression of nucleation.

Example of a Fluid-Phase Change Examined with MD Simulation: Evaporative Cooling of a Nanoscale Droplet

We carried out a series of molecular dynamics simulations in order to examine the evaporative cooling of a nanoscale droplet of a Lennard-Jones liquid. After thermally equilibrating a droplet at a temperature T_ini/T_t ≃ 1.2 (T_t is the triple-point temperature), we started the evaporation into vacuum by removing vaporized particles and monitoring the change in droplet size and the temperature inside. As free evaporation proceeds, the droplet reaches a deep supercooled liquid state of T/T_t ≃ 0.7. The temperature was found to be uniform in spite of the fast evaporative cooling on the surface. The time evolution of the evaporating droplet properties was satisfactorily explained with a simple one-dimensional phase-change model. After a sufficiently long run, the supercooled droplet was crystallized into a polycrystalline fcc structure. The crystallization is a stochastic nucleation process. The time and the temperature of inception were evaluated over 42 samples, which indicate the existence of a stability limit.

Thermodynamic formulation of the barrier for heterogeneous pinned nucleation: Implication to the crossover scenarios associated with barrierless and homogeneous nucleation

The effect of contact line pinning on nucleation is reported using continuum thermodynamics. Based on the principle of the free-energy maximization, closed-form expressions in the dimensionless form for the free-energy of the three-phase metastable system and the thermodynamic barrier are formulated with respect to the system geometry and the substrate wettability. The condition of maximality limits the dynamic contact angle within the cluster-phase-phobic regime. The dimensionless nucleation barrier or the potency factor can be divided into two components related to the system geometry and the pinning effect. Depending on the relative value of the equilibrium and the critical dynamic contact angle, the contact line pinning can either have favorable or adverse effects. Associated pinning-depinning transition can also lead to the crossovers related to barrierless and homogeneous nucleation. Contact line tension is found to have a considerable effect during these transitional scenarios. Complete wetting transition associated with barrierless nucleation can take place due to the presence of tensile (negative) line tension. On the other hand, complete drying transition related to homogeneous nucleation can occur when line tension is compressive (positive) in nature. The pinning has a favorable effect only when the substrate wettability is within the cluster-phase-philic regime. There can be favorable, adverse, or no pinning effects when the substrate wettability is within the cluster-phase-phobic regime. Although the contact line is pinned, the minimum value of the potency factor is obtained when equilibrium and dynamic contact angles are equal.

Effect of chain stiffness and temperature on the dynamics and microstructure of crystallizable bead-spring polymer melts

We contrast the dynamics in model unentangled polymer melts of chains of three different stiffnesses: flexible, intermediate, and rodlike. Flexible and rodlike chains, which readily solidify into close-packed crystals (respectively, with randomly oriented and nematically aligned chains), display simple melt dynamics with Arrhenius temperature dependence and a discontinuous change upon solidification. Intermediate-stiffness chains, however, are fragile glass-formers displaying Vogel-Fulcher dynamical arrest, despite the fact that they also possess a nematic-close-packed crystalline ground state. To connect this difference in dynamics to the differing microstructure of the melts, we examine how various measures of structure, including cluster-level metrics recently introduced in studies of colloidal systems, vary with chain stiffness and temperature. No clear static-structural cause of the dynamical arrest is found. However, we find that the intermediate-stiffness chains display qualitatively different dynamical heterogeneity. Specifically, their stringlike motion (cooperative rearrangement) is correlated along chain backbones in a way not found for either flexible or rodlike chains. This activated "crawling" motion is clearly associated with the dynamical arrest observed in these systems, and illustrates one way in which factors controlling the crystallization versus glass formation competition in polymers can depend nonmonotonically on chain stiffness.

Heterogeneous Crystallization on Pairs of Pre-Structured Seeds

Studying the effects of small pre-structured seeds on the crystallization transition in an undercooled monodisperse Lennard-Jones fluid with transition interface path sampling combined with molecular dynamics simulations, we analyze the impact of the simultaneous presence of two seeds with various structures. In the presence of seeds with face- and body-centered cubic structures, we find that decreasing the seed-to-seed distance enhances the probability of the crystalline clusters formed on one of the seeds to grow beyond the critical size, thus, increasing the crystal nucleation rates. In contrast, when seeds have an icosahedral structure, the crystalline clusters form mostly in the bulk. The crystal nucleation rate, however, is also determined by the distance between the seeds with regular structure in which the lattice spacing is equal to the bulk lattice constant, pointing to a heterogeneous crystal nucleation that occurs away from the icosahedrally structured seeds. For slightly squeezed seeds, the effects of the presence of seeds with face- and body-centered cubic structures are reduced in comparison to the regular seeds, and we do not see any effect of the presence of the second seed for seeds with squeezed icosahedral structure.

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.

Crystal Nucleation in Liquids: Open Questions and Future Challenges in Molecular Dynamics Simulations

The nucleation of crystals in liquids is one of nature's most ubiquitous phenomena, playing an important role in areas such as climate change and the production of drugs. As the early stages of nucleation involve exceedingly small time and length scales, atomistic computer simulations can provide unique insights into the microscopic aspects of crystallization. In this review, we take stock of the numerous molecular dynamics simulations that, in the past few decades, have unraveled crucial aspects of crystal nucleation in liquids. We put into context the theoretical framework of classical nucleation theory and the state-of-the-art computational methods by reviewing simulations of such processes as ice nucleation and the crystallization of molecules in solutions. We shall see that molecular dynamics simulations have provided key insights into diverse nucleation scenarios, ranging from colloidal particles to natural gas hydrates, and that, as a result, the general applicability of classical nucleation theory has been repeatedly called into question. We have attempted to identify the most pressing open questions in the field. We believe that, by improving (i) existing interatomic potentials and (ii) currently available enhanced sampling methods, the community can move toward accurate investigations of realistic systems of practical interest, thus bringing simulations a step closer to experiments.

Crystallization of Lennard-Jones nanodroplets: From near melting to deeply supercooled

We carry out molecular dynamics (MD) and Monte Carlo (MC) simulations to characterize nucleation in liquid clusters of 600 Lennard-Jones particles over a broad range of temperatures. We use the formalism of mean first-passage times to determine the rate and find that Classical Nucleation Theory (CNT) predicts the rate quite well, even when employing simple modelling of crystallite shape, chemical potential, surface tension, and particle attachment rate, down to the temperature where the droplet loses metastability and crystallization proceeds through growth-limited nucleation in an unequilibrated liquid. Below this crossover temperature, the nucleation rate is still predicted when MC simulations are used to directly calculate quantities required by CNT. Discrepancy in critical embryo sizes obtained from MD and MC arises when twinned structures with five-fold symmetry provide a competing free energy pathway out of the critical region. We find that crystallization begins with hcp-fcc stacked precritical nuclei and differentiation to various end structures occurs when these embryos become critical. We confirm that using the largest embryo in the system as a reaction coordinate is useful in determining the onset of growth-limited nucleation and show that it gives the same free energy barriers as the full cluster size distribution once the proper reference state is identified. We find that the bulk melting temperature controls the rate, even though the solid-liquid coexistence temperature for the droplet is significantly lower. The value of surface tension that renders close agreement between CNT and direct rate determination is significantly lower than what is expected for the bulk system.

Freezing of Lennard-Jones fluid on a patterned substrate

Using molecular dynamics simulations, we study freezing of Lennard-Jones particles at commensurate substrate with triangular pattern. Throughout the box particles freeze onto the substrate and form close-packed layers. For the moderately attractive substrates, an intermediate hexatic phase between liquid and crystal is detected in the first two layers where the hexatic-solid freezing process is continuous while, counterintuitively, the liquid-hexatic process is of first order. Moreover, we observe that liquid-hexatic and hexatic-solid transitions shift towards higher temperatures with the attraction strength increasing. By contrast, the liquid-hexatic transition shifts faster than the hexatic-solid process, significantly widening the temperature range of the hexatic phase. When this phenomenon appears, freezing in the bulk always proceeds through a first-order transition at the same temperature. In addition, changes in the average structural order (three-dimensional) of the layers indicate that freezing processes in layers near substrates seem to cost the structural order of the bulk particles in their vicinity, and an intermediate prestructural cloud of medium-ordered particles is always observed before the layering freezing.

Heterogeneous nucleation in the low-barrier regime

In simulations of the two-dimensional Ising model, we examine heterogeneous nucleation induced by a small impurity consisting of a line of l fixed spins. As l increases, we identify a limit of stability beyond which the metastable phase is not defined. We evaluate the free energy barrier for nucleation of the stable phase and show that, contrary to expectation, the barrier does not vanish on approach to the limit of stability. We also demonstrate that our values for the height of the barrier yield predictions for the nucleation time (from transition state theory) and the size of the critical cluster (from the nucleation theorem) that are in excellent agreement with direct measurements, even near the limit of stability.

Crystallization on prestructured seeds

The crystallization transition of an undercooled monodisperse Lennard-Jones fluid in the presence of small prestructured seeds is studied with transition path sampling combined with molecular dynamics simulations. Compared to the homogeneous crystallization, clusters of a few particles arranged into a face- and body-centered cubic structure enhance the crystallization, while icosahedrally ordered seeds do not change the reaction rate. We identify two distinct nucleation regimes-close to the seed and in the bulk. Crystallites form close to the face- and body-centered structures and tend to stay away from the icosahedrally ordered seeds.

Binding affinity of a small molecule to an amorphous polymer in a solvent. Part 1: free energy of binding to a binding site

Crystallization is commonly used in a separation and purification process in the production of a wide range of materials in various industries. In industry, crystallization usually starts with heterogeneous nucleation on a foreign surface. The complicated mechanism of heterogeneous nucleation is not well understood; however, we hypothesize that there might be a possible correlation between binding affinity to a surface and enhancement of nucleation. Recent studies show that amorphous polymers can be used to control crystallization, selectively produce pharmaceutical polymorphs, and discover novel pharmaceutical polymorphs. To investigate the possible correlation between the binding affinity of one molecule to key binding sites (local binding) and heterogeneous nucleation activity as well as the possibility of using this binding affinity to help guide the selection of polymers that promote heterogeneous nucleation, we computed the free energy of binding of aspirin to four nonporous cross-linked polymers in an ethanol-water 38 v% mixture. These cross-linked polymers are poly(4-acryloylmorpholine) (PAM), poly(2-carboxyethyl acrylate) (PCEA), poly(4-hydroxylbutyl acrylate) (PHBA), and polystyrene (PS); all of them were cross-linked with divinylbenzene (DVB). These systems were used because their heterogeneous nucleation activities are available in literature, and the ranking is PAM > PCEA > PHBA ≈ PS. We generated three independent surfaces for each polymer and computed the free energy of binding of aspirin to the best binding site that we found on each surface. The average free energies of binding to the best sites of PAM, PCEA, PHBA, and PS are -20.4 ± 1.0, -16.7 ± 1.0, -14.4 ± 1.1, and -13.6 ± 1.1 kcal/mol, respectively. We found that the trend of the magnitudes of the average free energies of binding to the best sites is PAM > PCEA > PHBA ≈ PS. This trend is very similar to that of heterogeneous nucleation activity. Our results suggest the importance of the free energy of binding to key sites (local binding) and the possibility of using this quantity to help guide the selection of polymers that promote heterogeneous nucleation.

Binding affinity of a small molecule to an amorphous polymer in a solvent. Part 2: preferential binding to local sites on a surface

Crystallization, a separation and purification process, is commonly used to produce a wide range of materials in various industries, and it usually begins with heterogeneous nucleation on a foreign surface in industrial practice and most other circumstances. Recent studies show that amorphous polymeric substrates are useful in controlling crystallization and selectively producing pharmaceutical polymorphs. In our previous publication, we investigated the possible correlation of the binding affinity of one molecule to key binding sites (local binding), and the possibility of using this binding affinity to guide the selection of polymers promoting heterogeneous nucleation. The studied systems were aspirin binding to four nonporous cross-linked polymers in ethanol-water 38 v% mixture. Cross-linked with divinylbenzene (DVB), these polymers were poly(4-acryloylmorpholine) (PAM), poly(2-carboxyethyl acrylate) (PCEA), poly(4-hydroxylbutyl acrylate) (PHBA), and polystyrene (PS). We discovered that the trend of the magnitudes of the average free energies of binding to the best sites is very similar to that of heterogeneous nucleation activities. This Article aims to investigate whether or not local binding to key sites is the important variable to describe heterogeneous nucleation as opposed to the overall/average binding affinity of molecules to a surface, and to investigate the possibility of using the overall binding affinity to guide the selection of polymers. We used the polymer surfaces generated from our previous study to calculate the overall binding affinity of aspirin molecules to the surface as measured by the preferential interaction coefficients of aspirin (1 m) to these polymers. We discovered that the trend of the average preferential interaction coefficients does not correlate as well to that of heterogeneous nucleation activities as the free energies of binding to the best sites. We also computed the average numbers of aspirin molecules associated with the areas of the surfaces' best binding sites and found that they correlate better to heterogeneous nucleation activities than the average preferential interaction coefficients. These results further support that local binding is indicative of heterogeneous nucleation. Moreover, we found a weak trend of the distance order parameters of the aspirin molecules to be similar that of heterogeneous nucleation activities. Our results from the two-part study suggest the importance of local binding to heterogeneous nucleation as well as the possibility of using the binding affinity to the local area (the free energy of binding to the best site and the number of nucleating molecules associated with the area of the best binding site) and the distance order parameters to guide the selection of polymers.

Crystallization and aging in hard-sphere glasses

We report new results from our programme of molecular dynamics simulation of hard-sphere systems, focusing on crystallization and glass formation at high concentrations. First we consider a much larger system than hitherto, N = 86 400 equal-sized particles. The results are similar to those obtained with a smaller system, studied previously, showing conventional nucleation and growth of crystals at concentrations near melting and crossing over to a spinodal-like regime at higher concentrations where the free energy barrier to nucleation appears to be negligible. Second, we investigate the dependence on the initial state of the system. We have devised a Monte Carlo 'constrained aging' method to move the particles in such a way that crystallization is discouraged. After a period of such aging, the standard molecular dynamics programme is run. For a system of N = 3200, we find that constrained aging encourages caging of the particles and slows crystallization somewhat. Nevertheless, both aged and unaged systems crystallize at volume fraction φ = 0.61 whereas neither system shows full crystallization in the duration of the simulation at φ = 0.62, a concentration still significantly below that of random close packing.

Polymorphism, crystal nucleation and growth in the phase-field crystal model in 2D and 3D

We apply a simple dynamical density functional theory, the phase-field crystal (PFC) model of overdamped conservative dynamics, to address polymorphism, crystal nucleation, and crystal growth in the diffusion-controlled limit. We refine the phase diagram for 3D, and determine the line free energy in 2D and the height of the nucleation barrier in 2D and 3D for homogeneous and heterogeneous nucleation by solving the respective Euler-Lagrange (EL) equations. We demonstrate that, in the PFC model, the body-centered cubic (bcc), the face-centered cubic (fcc), and the hexagonal close-packed structures (hcp) compete, while the simple cubic structure is unstable, and that phase preference can be tuned by changing the model parameters: close to the critical point the bcc structure is stable, while far from the critical point the fcc prevails, with an hcp stability domain in between. We note that with increasing distance from the critical point the equilibrium shapes vary from the sphere to specific faceted shapes: rhombic dodecahedron (bcc), truncated octahedron (fcc), and hexagonal prism (hcp). Solving the equation of motion of the PFC model supplied with conserved noise, solidification starts with the nucleation of an amorphous precursor phase, into which the stable crystalline phase nucleates. The growth rate is found to be time dependent and anisotropic; this anisotropy depends on the driving force. We show that due to the diffusion-controlled growth mechanism, which is especially relevant for crystal aggregation in colloidal systems, dendritic growth structures evolve in large-scale isothermal single-component PFC simulations. An oscillatory effective pair potential resembling those for model glass formers has been evaluated from structural data of the amorphous phase obtained by instantaneous quenching. Finally, we present results for eutectic solidification in a binary PFC model.

On the existence of a structural instability in sub-critical crystalline fluctuations in a supercooled liquid

The square gradient free energy functional is used to describe the distribution of sub-critical crystalline fluctuations in a supercooled liquid. We show that for any supercooling there exists a cluster radius of 3-4 particle diameters below which the crystalline order is no longer stable. We explore the consequences of this observation.

Glassy dynamics, spinodal fluctuations, and the kinetic limit of nucleation in suspensions of colloidal hard rods

Using simulations we identify three dynamic regimes in supersaturated isotropic fluid states of short hard rods: (i) for moderate supersaturations, we observe nucleation of multilayered crystalline clusters; (ii) at higher supersaturation, we find nucleation of small crystallites which arrange into long-lived locally favored structures that get kinetically arrested; and (iii) at even higher supersaturation, the dynamic arrest is due to the conventional cage-trapping glass transition. For longer rods we find that the formation of the (stable) smectic phase out of a supersaturated isotropic state is strongly suppressed by an isotropic-nematic spinodal instability that causes huge spinodal-like orientation fluctuations with nematic clusters diverging in size. Our results show that glassy dynamics and spinodal instabilities set kinetic limits to nucleation in highly supersaturated hard-rod fluids.

Crossover from nucleation to spinodal decomposition in a condensing vapor

The mechanism controlling the initial step of a phase transition has a tremendous influence on the emerging phase. We study the crossover from a purely nucleation-controlled transition toward spinodal decomposition in a condensing Lennard-Jones vapor using molecular dynamics simulations. We analyze both the kinetics and at the same time the thermodynamics by directly reconstructing the free energy of cluster formation. We estimate the location of the spinodal, which lies at much deeper supersaturations than expected. Moreover, the nucleation barriers we find differ only by a constant from the classical nucleation theory predictions and are in very good agreement with semiempirical scaling relations. In the regime from very small barriers to the spinodal, growth controls the rate of the transition but not its nature because the activation barrier has not yet vanished. Finally, we discuss in detail the influence of the chosen reaction coordinate on the interpretation of such simulation results.

Crystal nucleation in a supercooled liquid with glassy dynamics

In simulations of supercooled, high-density liquid silica we study a range of temperature T in which we find both crystal nucleation as well as the characteristic dynamics of a glass forming liquid, including a breakdown of the Stokes-Einstein relation. We find that the liquid cannot be observed below a homogeneous nucleation limit (HNL) at which the liquid crystallizes faster than it can equilibrate. We show that the HNL would occur at lower T, and perhaps not at all, if the Stokes-Einstein relation were obeyed, and hence that glassy dynamics plays a central role in setting a crystallization limit on the liquid state in this case. We also explore the relation of the HNL to the Kauzmann temperature, and test for spinodal-like effects near the HNL.

Heterogeneous nucleation at a wall near a wetting transition: a Monte Carlo test of the classical theory

While for a slightly supersaturated vapor the free energy barrier ΔF(hom)(*), which needs to be overcome in a homogeneous nucleation event, may be extremely large, nucleation is typically much easier at the walls of the container in which the vapor is located. While no nucleation barrier exists if the walls are wet, for incomplete wetting of the walls, described via a nonzero contact angle Θ, classical theory predicts that nucleation happens through sphere-cap-shaped droplets attracted to the wall, and their formation energy is ΔF(het)(*) = ΔF(hom)(*)f(Θ), with f(Θ) = (1-cosΘ)(2)(2+cosΘ)/4. This prediction is tested through simulations for the simple cubic lattice gas model with nearest-neighbor interactions. The attractive wall is described in terms of a local 'surface field', leading to a critical wetting transition. The variation of the contact angle with the strength of the surface field is determined by using thermodynamic integration methods to obtain the wall free energies which enter Young's equation. Obtaining the chemical potential as a function of the density for a system with periodic boundary conditions (and no walls), the droplet free energy of a spherical droplet in the bulk is obtained for a wide range of droplet radii. Similarly, ΔF(het)(*) is obtained for a system with two parallel walls. We find that the classical theory is fairly accurate if a line tension correction for the contact angle is taken into account.

Theoretical study of hierarchical structures and heredity effect of silicon solidifying on carbon nanotube

The growth of silicon atoms at the heterogeneous surface of carbon nanotubes (CNTs) is studied by means of molecular dynamics simulations. The results indicate that silicon atoms are concentrated to form an “annual ring” structure around CNTs. The structures of CNTs have strong correlation with the stacking sequence of silicon atoms. Heredity effect can be clearly observed during the heterogeneous nucleation. The uniform internal potential field around CNTs results in the formation of annual ring structures.