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Crystal Polymorph Selection Mechanism of Hard Spheres Hidden in the Fluid

  • Willem Gispen*
    Willem Gispen
    Soft Condensed Matter &and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
    *Email: [email protected]
    More by Willem Gispen
    Orcidhttps://orcid.org/0000-0002-7276-8620
  • , 
  • Gabriele M. Coli
    Gabriele M. Coli
    Soft Condensed Matter &and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
    More by Gabriele M. Coli
    Orcidhttps://orcid.org/0000-0002-5125-8007
  • , 
  • Robin van Damme
    Robin van Damme
    Soft Condensed Matter &and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
    More by Robin van Damme
    Orcidhttps://orcid.org/0000-0003-2871-3726
  • , 
  • C. Patrick Royall
    C. Patrick Royall
    Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, 75005 Paris, France
    H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
    School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom
    More by C. Patrick Royall
  • , and 
  • Marjolein Dijkstra*
    Marjolein Dijkstra
    Soft Condensed Matter &and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands
    *Email: [email protected]
    More by Marjolein Dijkstra
    Orcidhttps://orcid.org/0000-0002-9166-6478
Cite this: ACS Nano 2023, 17, 9, 8807–8814
Publication Date (Web):April 21, 2023
https://doi.org/10.1021/acsnano.3c02182

Copyright © 2023 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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Nucleation plays a critical role in the birth of crystals and is associated with a vast array of phenomena, such as protein crystallization and ice formation in clouds. Despite numerous experimental and theoretical studies, many aspects of the nucleation process, such as the polymorph selection mechanism in the early stages, are far from being understood. Here, we show that the hitherto unexplained excess of particles in a face-centered-cubic (fcc)-like environment, as compared to those in a hexagonal-close-packed (hcp)-like environment, in a crystal nucleus of hard spheres can be explained by the higher order structure in the fluid phase. We show using both simulations and experiments that in the metastable fluid phase, pentagonal bipyramids, clusters with fivefold symmetry known to be inhibitors of crystal nucleation, transform into a different cluster, Siamese dodecahedra. These clusters are closely similar to an fcc subunit, which explains the higher propensity to grow fcc than hcp in hard spheres. We show that our crystallization and polymorph selection mechanism is generic for crystal nucleation from a dense, strongly correlated fluid phase.

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Introduction

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Understanding nucleation is essential in various research fields including determining the molecular structure of proteins through crystallization, drug design in the pharmaceutical industry, and ice crystal formation in clouds. The latter is the largest unknown in the earth’s radiative balance and thus is crucial in the context of climate change and weather forecasts. Nucleation is also important in the crystallization of colloidal and nanoparticle suspensions, which have application perspectives in catalysis, optoelectronics, and plasmonics. (1,2)
However, studying nucleation in molecular systems is extremely challenging, as it is a stochastic and rare process. Additionally, the crystal nuclei sizes are often rather small, and the nuclei grow extremely quickly once they exceed their critical size. Moreover, in most substances, different crystal polymorphs may compete during nucleation, which is particularly important in pharmaceutical sciences and applications. Crystallization of the “undesired” polymorph may lead to neurodegenerative disorders such as Alzheimer’s disease or eye cataract or to reduced solubility/efficacy, and even toxicity of certain drug compounds. (3,4)
Recently, impressive strides have been made in the experimental observation of early stage crystal nucleation. Atomic-resolution in situ electron microscopy has shown the observation of different nucleation pathways leading to different crystal polymorphs of proteins, (5) prenucleation clusters in metal organic frameworks, (6) early stage nucleation pathways of FePt nanocrystals that go beyond classical nucleation theory and nonclassical scenarios, (7) amorphous precursors in protein crystallization, (8) featureless and semiordered clusters of NaCl nanocrystals, (9) and reversible disorder–order transitions of gold crystals. (10) These recent observations challenge current nucleation models and highlight the need for a better theoretical understanding of the crystallization pathways at the earliest stages of nucleation when particles start to order from the metastable fluid phase and select the emerging crystal polymorphs.
Colloidal suspensions are suitable experimental systems for investigating locally heterogeneous phenomena, such as early stage nucleation. The larger sizes and slower time scales of colloidal particles enable direct observation of the nucleation mechanisms. (11,12) However, even for hard spheres, which are arguably the simplest colloidal model system, the polymorph selection mechanism has not yet been revealed. In a hard-sphere system, the hexagonal-close-packed (hcp) crystal is metastable with respect to the face-centered-cubic (fcc) crystal, but the free-energy difference between the two structures is tiny (approximately 10–3kBT per particle). (13,14) Therefore, one might expect to find approximately 50% occurrence of fcc- and hcp-like particles in the crystal nucleus of hard spheres. However, this prediction is not realized in experiments (11,12,15−17) nor in simulations, (18−22) which both show a hitherto unexplained predominance of fcc particles in the final crystal phase.
In this article, we investigate the early stages of nucleation of hard spheres using simulations and particle-resolved experiments to shed light on the selection mechanism of the crystal polymorph. We study the structural transformations in the supersaturated fluid phase that finally lead to crystal nucleation. We find that the crystal embryo exhibits a preference toward fcc-like stacking due to its similarity with the local geometric motifs in the particle arrangements present in the fluid phase.

Results and Discussion

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The Free-Energy Barrier for Nucleation

According to the Stranski–Totomanow conjecture, polymorph selection is governed by the lowest free-energy barrier for nucleation. Sanchez-Burgos et al. (23) argued that fcc is preferred for hard spheres because it has a nucleation barrier marginally lower than that of hcp. They also found that a random-stacked nucleus has a higher nucleation barrier than fcc, even though random stacking commonly occurs in nucleation. This apparent discrepancy begs the crucial question whether the polymorph selection mechanism has a kinetic or thermodynamic origin. To answer this question, we calculate the Gibbs free energy βΔG(nfcc, nhcp) for the formation of a crystal cluster consisting of nfcc fcc-like particles and nhcp hcp-like particles using the umbrella sampling technique; see the Methods section for the technical details.
In Figure 1, we plot βΔG of a crystalline nucleus as a function of the size nfcc + nhcp and composition nfcc/(nfcc + nhcp). The lowest free-energy path on this surface, represented by the dashed line, shows that the crystal nucleus has an excess of fcc-like particles in the early stages of nucleation and that the critical nucleus consists of about 70% fcc-like particles. In the SI, we show that this ratio is not strongly dependent on a specific polymorph detection method, as a variety of methods all find 60%–80% fcc-like particles.

Figure 1

Figure 1. Thermodynamic propensity toward mixed fcc-hcp crystal nuclei in the early stages of crystal nucleation of hard spheres. Gibbs free-energy barrier as a function of the number of fcc and hcp particles nfcc + nhcp and composition nfcc/(nfcc + nhcp). The dots are the composition of crystal nuclei found in our experiments. For both simulations and experiments, the composition of crystal nuclei is determined with the classification scheme described in the Methods Section. The dashed line represents the minimum free-energy path for nucleation, and the solid line represents the critical nucleus size as a function of composition.

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We confirm the same phenomenon in an experimental system consisting of poly(methyl methacrylate) particles. We obtained particle coordinates by tracking them using confocal microscopy. More details on the experiments can be found in the Methods Section. We plot the size and composition of crystal nuclei found in our experiments, showing that the crystal nucleus exhibits a notable excess of fcc-like particles already in an early stage of nucleation.
To provide a rigorous comparison with classical nucleation theory, we also plot the critical nucleus size n* with a solid line. Intriguingly, we observe that the critical nucleus size n* remains nearly constant irrespective of the composition nfcc/(nfcc + nhcp), while the nucleation barrier ΔG* = ΔG(n*) strongly depends on the composition. According to classical nucleation theory, n* and ΔG* should be directly proportional, as Δ G * = 1 2 n * | Δ μ | , where Δμ denotes the supersaturation of the crystal phase with respect to the fluid phase. As the difference in supersaturation between fcc and hcp is very small, the variation of the nucleation barrier with composition cannot be explained by this classical relation. This finding, combined with the observation that fcc is already preferred for very small nucleus sizes, strongly suggests that thermodynamic considerations based solely on surface tension and classical nucleation theory may not provide the full picture and that the metastable fluid deserves further scrutiny.

Topological Structure of the Metastable Fluid

It has been suggested that polymorph selection is hidden in the fluid phase. (21) To study this, we perform molecular dynamics (MD) simulations of a supersaturated fluid of hard spheres. We investigate the structure of the fluid by identifying the geometric motifs of particle assemblies locally present in the fluid phase by using the topological cluster classification (TCC) algorithm. (24) Hereafter, we refer to these local particle assemblies as clusters, as their bond topologies are identical to certain minimum-energy clusters. We first identify rings of three, four, and five particles and then define basic clusters as rings with one or two additional particles, forming pyramids or bipyramids. In total, we distinguish 40 topological clusters composed of these basic clusters. To define a single-particle property, we determine the number of clusters of each type that each particle belongs to. We examine each particle configuration independently and counted the number of clusters. We use the TCC because it offers a more interpretable classification of local structure. Unlike order parameters based on rotational symmetry, (21) which may be difficult to interpret, a topological cluster directly corresponds to a real space arrangement of particles. Furthermore, the TCC does not assume the presence of crystal-like symmetries, providing us with greater resolution on intermediate structures that may not be present in bulk crystal phases.
In Figure 2 we show the impact of these topological clusters on crystallization and polymorph selection in the metastable fluid. First, we study whether the clusters affect crystallization by calculating the probability that particles become crystalline, i.e., become part of either an fcc or hcp cluster. Second, we examine whether the clusters affect polymorph selection by determining the relative probability of particles becoming part of fcc or hcp. Note that the fcc and hcp clusters have a short lifetime in the metastable fluid, as they are smaller than the critical nucleus size. Therefore, we use a small time interval Δ t / β m σ 2 = 1.6 to measure the conversions. To quantify the effect of each individual cluster on these probabilities, we computed the mutual information (MI) between the number of clusters a particle belongs to and the corresponding probabilities. In Figure 2a, we present a scatter plot of the MI on crystallization and polymorph selection. Three clusters, the octahedron (OH), the pentagonal bipyramid (PB), and the Siamese dodecahedron (SD), stand out.

Figure 2

Figure 2. The effect of topological clusters on crystallization and polymorph selection in the metastable fluid. (a) Mutual information (MI) of topological clusters on crystallization and polymorph selection. Each label (e.g., 11F) refers to a different cluster as identified by the topological cluster classification algorithm. (24) Clusters with low MI are not labeled. (b) Typical arrangements of particles in an OH, PB, SD, fcc, and hcp clusters. The ring particles in each cluster are colored light gray. (c) Relative probability to convert to fcc with respect to hcp as a function of the number of OH or PB clusters a particle is part of. (d) Probability to convert to fcc or hcp as a function of the number of PB, OH, or SD clusters it belongs to.

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In Figure 2b we present the geometries of these clusters. The OH is composed of a square ring with two additional particles forming the peaks of the pyramid. Similarly, the PB is a pentagonal ring with two additional particles. The SD is similar to PB, but one of the ring particles is replaced by two particles, one below and one above the ring. Finally, we depict the fcc and hcp clusters consisting of a central particle with its twelve nearest neighbors arranged as in bulk fcc or hcp.
As the OH and PB bipyramids exhibit the highest mutual information with polymorph selection, we show in Figure 2c the relative probability of converting to fcc with respect to hcp as a function of the number of these clusters a particle is part of. We observe that the relative probability to convert to fcc decreases as the number of OH increases. In bulk fcc and hcp, the number of OH is equal, suggesting that OH has an equal preference for fcc and hcp. Therefore, the decrease in the relative probability with the number of OH indicates a drift toward 50%. On the other hand, the relative probability to convert to fcc increases as the number of PB clusters a particle belongs to increases. This finding is peculiar as the fivefold symmetry of PB is incommensurate with the sixfold symmetry of the fcc crystal.
Similarly, we assessed the impact of OH, PB, and SD on crystallization. In Figure 2d, we show the probability of particles to convert to either fcc or hcp as a function of the number of clusters they belong to. We observe that the probability to crystallize decreases with the number of PB clusters a particle belongs to, which is consistent with previous findings that fivefold symmetry suppresses crystallization. (21,25−27) In fact, our observation that PB clusters prefer to convert to fcc suggests that this inhibiting effect of fivefold symmetry on crystallization is due to the reduced conversion to hcp. In contrast, the probability of crystallization increases with the number of SD and OH clusters a particle is part of, with these clusters showing the strongest promotion of crystallization. For instance, the conversion probability of a particle that is part of five OH clusters has a conversion probability of more than twice that of a particle not belonging to any OH cluster.

Topological Structure of Crystal Nuclei

These results suggest that the predominance of fcc is a result of the fivefold structure of the fluid. But how do these clusters actually behave during nucleation? To investigate this, we calculated the number of PB, SD, fcc, and hcp clusters during nucleation. In Figure 3a,b, we present cut-through images of a small crystal nucleus in an experimental sample. We color a particle blue if it belongs to at least three fcc and hcp clusters. This enables us to clearly identify the structure of the surface of the nucleus. We color the particles on the surface and in the surrounding fluid with varying shades of purple and pink depending on the number of SD or PB clusters they belong to. Despite the high density of SD and PB clusters present throughout the fluid, Figure 3a,b shows that the density of these clusters is spatially heterogeneous. Specifically, we observe that the crystal nucleus is surrounded by a high density of SD clusters, whereas the opposite trend is found for the PB clusters, as the PB clusters are depleted near the surface of the crystal nucleus.

Figure 3

Figure 3. The role of PB and SD clusters during crystal nucleation from strongly correlated, dense fluids. (a,b) PMMA spheres imaged with confocal microscopy. (c,d,i) Nearly hard spheres interacting with a WCA potential. (e,f,j) Charged spheres interacting with a Yukawa potential. (g,h,k) Spheres interacting with a Lennard-Jones potential. (a–h) On the left, we show cut-through images of crystal nuclei. The core of the crystal nucleus is colored blue, while the rest of the particles is colored following the scale bar on the left (a,c,e,g) or right (b,d,f,h) depending on the number of PB (a,c,e,g) or SD (b,d,f,h) clusters each particle belongs to. (i–k) On the right, we show the fraction of particles belonging to different clusters as a function of time (solid lines), averaged over multiple different simulated spontaneous nucleation events (transparent lines). The fractions are averaged over 5 (i), 3 (j), and 3 (k) different nucleation events, respectively. Note that a particle can be part of multiple clusters at the same time, and therefore the corresponding fractions add up to a value which is higher than one.

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To demonstrate the universality of this phenomenon, we examined in simulations the crystal nucleation of three different systems: hard spheres, charged spheres, and Lennard-Jones particles. In the charged-sphere system, we used a screened-Coulomb (Yukawa) potential that has a longer and softer repulsive interaction compared with the hard-sphere interaction. The Lennard-Jones system has both attractive and repulsive interactions. We visualize the crystal nuclei in Figure 3c–h, using the same color coding as for the experimental sample in Figure 3a,b. All these systems exhibit a similar heterogeneous structure consisting of high and low density regions of SD and PB clusters in the fluid phase and a crystal nucleus that is surrounded by a high density of SD clusters and a low density of PB clusters. For a quantitative comparison between experiments and simulations, see the SI. Furthermore, our analysis in the SI demonstrates that a high density of SD clusters and a low density of PB clusters are also present near planar solid–fluid interfaces. These findings suggest that the observed behavior is a universal phenomenon that arises from geometric constraints on particle arrangements and is not specific to a particular type of interaction potential.
To understand the role of the topological clusters in the crystallization mechanism, we plot in Figure 3 the fraction of particles belonging to SD (pink line), PB (purple), fcc (blue), and hcp (brown) clusters as a function of time, averaged over multiple spontaneous nucleation trajectories obtained in simulations. Notably, we observe the same behavior in all of these systems. Figure 3i–k reveals that while the fraction of crystalline particles is initially low in the metastable fluid phase, the fraction of fcc clusters is clearly higher than that of hcp clusters at this stage in the fluid phase. The fluid phase already contains 70% fcc and 30% hcp, which agrees well with our umbrella sampling results shown in Figure 1. Furthermore, we observe that the populations of particles in both the SD and PB clusters are already high before crystallization sets in, indicating that the metastable fluid exhibits strong spatial correlations due to packing constraints. At the onset of crystallization, the fraction of PB clusters decreases, reminiscent of the decrease in fivefold symmetry observed at the onset of crystallization. (27) More surprisingly, our analysis reveals that the number of SD clusters increases during the early stages of crystallization, which later decreases to a lower value at the end of the crystallization process, as the fcc structure does not contain SD clusters. Notably, the SD cluster is the only cluster that shows nonmonotonic behavior during nucleation (see Figure S4 of the SI).
To further examine the relationship between SD and PB clusters, we measure the combined fraction of particles belonging to either SD or PB clusters as a function of time (dotted line in Figure 3i–k). Our results indicate that the combined fraction remains constant not only in the metastable fluid phase but also during the early stages of crystallization. We also note that the combined fraction fluctuates less than the individual fractions of the SD and PB clusters. These observations suggest that the increase in SD clusters is a direct consequence of a decrease in PB clusters. Moreover, the constant combined fraction and much smaller fluctuations indicate a reversible conversion between the PB and SD clusters.
In the SI, we quantify the conversions among PB, SD, fcc, and hcp clusters, providing further evidence that the SD cluster acts as an intermediate state between PB clusters and fcc/hcp clusters. Additionally, we show that the findings presented in Figure 3 remain unchanged when we change the dynamics of hard spheres to hydrodynamics or event-driven dynamics. These results lend strong support that the observed crystallization and polymorph selection mechanism is generic for crystal nucleation from a highly correlated, dense fluid phase.

The Nucleation Mechanism

To elucidate how the fivefold structure of the PB and SD clusters becomes integrated into the fcc or hcp crystal lattice, we note that the four particles comprising the pentagonal ring of an SD cluster form a trapezoidal arrangement with two acute and two obtuse angles, as shown in the bottom left of Figure 4c. When these particles arrange in a square configuration, the SD cluster can be identified as a subunit of an fcc or hcp crystal, as illustrated in Figure 4a,b, where the particles are denoted with the same colors to facilitate comparison. These “squared” SD clusters can fully tile both the fcc and hcp lattices without needing any additional particles. However, the “squared” SD cluster fits into the fcc crystal in two ways, parallel (Figure 4a) and perpendicular (Figure 4b) to the hexagonal layers, whereas it fits only in one way in the hcp crystal, namely parallel to the hexagonal layers (Figure 4a). Thus, the SD cluster has greater orientational freedom in forming the fcc lattice, resulting in a 2-to-1 preference for fcc over hcp, which is close to the 70% ratio that we observe in our simulations. We note that this 2-to-1 preference results from the greater symmetry of the fcc crystal, which has been suggested to favor fcc during nucleation. (28)

Figure 4

Figure 4. Transition of SD and PB clusters to fcc subunits. (a) Geometry of the SD cluster (gray and pink particles) parallel to two hexagonal layers of an fcc and hcp crystal. (b) Geometry of the SD cluster perpendicular to the hexagonal layers, which occurs only in fcc. (c) Probability distributions of the internal angles θ of the ring particles of SD and PB clusters before and after the transition to an fcc subunit. The typical arrangements of particles in SD (left) and PB (right) clusters before and after the transition are also shown, with the ring particles colored in gray.

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Given the topological similarity between the SD cluster and the fcc lattice, we speculate that the attachment of fluidlike particles to the solid nucleus proceeds via SD clusters, where the four particles in the pentagonal ring transform from a trapezoidal to a square arrangement. In this way, the SD clusters are fully incorporated into the fcc lattice. To test this hypothesis, we identify the SD clusters that become part of an fcc or hcp cluster during nucleation and measure the distribution of the four angles of the trapezoidal arrangement of the four particles in the pentagonal ring of the SD clusters. As shown in Figure 4c, the distribution is bimodal with peaks at angles smaller and larger than 90°, representing the trapezoidal arrangement, before the transition. However, after the transition, the distribution becomes unimodal with a single peak at around 90°, indicating a square pattern. This transition is further illustrated in Figure 4c, which shows two representative SD clusters before and after the transformation.
To further validate our proposed nucleation mechanism, we measured the distribution of the internal angles of the five particles in the ring of the PB cluster. Before the transition, the distribution is centered around the expected 108° for a fivefold ring. After the transition, we observe a split into three peaks, with the peaks centered around 60°, 90°, and 120°, providing compelling evidence for the transformation of the PB cluster as illustrated in Figure 4c.
This result strongly reinforces our key finding that the fivefold PB clusters, which are abundant in the fluid phase and known to inhibit crystal nucleation, transform into SD clusters and that the SD cluster-mediated attachment of particles to the growing nucleus proceeds via a simple rearrangement of particles into fcc subunits. Hence, the propensity to grow fcc is higher than that for hcp, indicating that the polymorph selection mechanism in hard spheres is already hidden in the higher order structure of the fluid phase.

Conclusions

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In conclusion, we unveil the underlying crystallization and polymorph selection mechanism in a highly correlated, dense fluid phase such as hard spheres during the early stages of nucleation both in simulations and experiments. We demonstrate that the supersaturated fluid is highly dynamic, with reversible transformations between fivefold pentagonal bipyramid and Siamese dodecahedron clusters. Most notably, the Siamese dodecahedra exhibit a close similarity with an fcc subunit, providing an explanation for the as-of-yet-unexplained higher propensity of fcc over hcp in hard spheres. Moreover, we reveal that the polymorph selection mechanism has not only a geometric origin, hidden in the higher-order correlations of the fluid phase, but also a thermodynamic origin, with the lowest free-energy pathway favoring a higher number of fcc-like particles in the early stages of nucleation. Our findings offer valuable insights for controlling nucleation pathways and crystal polymorphs in highly correlated fluid phases and provide a framework for studying nucleation in other systems.

Methods

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Umbrella Sampling Simulations

To investigate the thermodynamic propensity toward fcc-like or hcp-like ordering during the nucleation process, we use umbrella sampling (US) (29) to calculate the nucleation barrier of a system of hard spheres at a pressure of βPσ3 = 17.0. This pressure corresponds to a packing fraction of 0.5352 for the fluid phase. (20) We note that under these conditions, the absolute nucleation rate as predicted by US is in reasonable agreement with experiments. (20) Similar to previous literature, (20,30) we identify the nucleus by using the dot product d6 of local spherical harmonics expansions with l = 6 to define solid-like bonds as those bonds between particle pairs (i, j) for which d6(i, j) > 0.7 and define solid-like particles as those that have at least 7 of such solid-like bonds. Particle neighbors are defined by using a distance cutoff of rc = 1.4σ. The nucleus is then the largest set of solid-like particles that are connected by solid-like bonds. To disentangle fcc-like and hcp-like order, we subsequently classify solid-like particles as fcc-like and hcp-like based on their value of the Steinhardt bond order parameter w4: particles with w4 < 0 are fcc-like and those with w4 ≥ 0 are hcp-like. (31) The number of such particles are nfcc and nhcp, respectively, and we use these to define the US biasing potential:
U b = 1 2 λ fcc ( n fcc − n 0 fcc ) 2 + 1 2 λ hcp ( n hcp − n 0 hcp ) 2
where both coupling constants λfcc and λhcp are set to an equal value of βλfcc = βλhcp = 0.05. This allows us to sample the two-dimensional Gibbs free-energy difference βΔG(nfcc, nhcp) that is the nucleation barrier as a function of the number of fcc-like and hcp-like ordered particles.
We initialize each US window (n0fcc, n0hcp) from a configuration with a nucleus with approximately nfcc ≈ n0fcc and nhcp ≈ n0hcp. For very small nuclei up to n = nfcc + nhcp ∼ 20, we measure the full cluster size distribution instead of only the size of the largest cluster, as the probability of multiple small nuclei appearing simultaneously can be significant. We implement the US scheme by adding additional Monte Carlo bias moves that accept or reject trajectories based on the US bias potential on top of a hard-particle Monte Carlo (HPMC) simulation implemented using HOOMD-blue’s HPMC module. (32,33) Bias moves are performed every MC cycle in order to also sample regions of the free-energy landscape where the gradient is large. Finally, we reconstruct the nucleation barrier by using the weighted histogram analysis method (WHAM), (34) specifically by using the algorithm provided by ref (35).
The minimum free-energy path for nucleation, plotted in Figure 1, is calculated by weighting the composition of nuclei of a given size with the modified Boltzmann factor exp(− 5βΔG(nfcc, nhcp)). The critical nucleus size as a function of its composition, also plotted in Figure 1, is calculated by weighting the size of all nuclei with a certain composition with the inverse modified Boltzmann factor exp(5βΔG(nfcc, nhcp)). We find that these smooth versions of the minimum and maximum functions are more numerically stable against the statistical errors in our US simulations.

Topological Cluster Classification

We require an algorithm that is capable of interpretably quantifying the structure in the metastable fluid. To this end, we use the topological cluster classification (TCC) algorithm. (24) The bonds between particles are detected by using a modified Voronoi construction method. The free parameter fc, controlling the amount of asymmetry that a four-membered ring can show before being identified as two three-membered rings, is set to 0.82.

MD Simulations

In order to generate trajectories in which we observe spontaneous nucleation, we conducted MD simulations in the isothermal–isobaric (NPT) ensemble. For all systems, the temperature T and pressure P are kept constant via the Martyna–Tobias–Klein (MTK) integrator. (36) The simulation box is cubic, and periodic boundary conditions are applied in all directions.
To simulate nearly hard spheres, we use a constant number N = 13500 of particles interacting via a Weeks–Chandler–Andersen (WCA) pair potential, which can straightforwardly be employed in Molecular Dynamics (MD) simulations and which reduces to the hard-sphere potential in the limit that the temperature T → 0. The WCA pair interaction u(rij) is simply a Lennard-Jones potential cut-and-shifted at the minimum of its potential well and reads (37)
u ( r i j ) = { 4 ϵ [ ( σ r i j ) 12 − ( σ r i j ) 6 + 1 4 ] r i j < 2 1 / 6 σ 0 r i j ≥ 2 1 / 6 σ
with rij = | ri – rj| the center-of-mass distance between particle i and j, ri the position of particle i, ϵ the interaction strength, and σ the diameter of each sphere. The steepness of the repulsion between the particles can be tuned by the temperature kBT/ϵ. We set kBT/ϵ = 0.025, which has been used extensively in previous simulation studies to mimic hard spheres. (38−41) The reduced pressure is chosen as βPσeff3 = 17.69, with the thermostat and barostat coupling constants τT = 1.0 τMD and τP = 1.0 τMD, respectively, and τ MD = σ m / ϵ is the MD time unit. The time step is set to Δt = 0.0001τMD, which is small enough to ensure the stability of the simulations. We used the mapping described in refs, (39−41) which results in each particle having an effective diameter σeff ≃ 1.097. With this mapping, the fluid phase has an effective packing fraction of 0.539. The MD simulations for nearly hard spheres are performed using the HOOMD-blue (highly optimized object-oriented many-particle dynamics) software. (32)
To simulate charged spheres, we use a pseudohard-core Yukawa potential. The total interaction is the sum of the pseudohard-core potential (42) and a screened Coulomb (Yukawa) pair potential uY(rij):
β u Y ( r i j ) = β ϵ exp [ − κ σ ( r i j / σ − 1 ) ] r i j / σ
with contact value βϵ and screening length 1/κσ. For this system, we simulate N = 104 particles with contact value βϵ = 81, screening length 1/κσ = 0.125, and pressure βPσ3 = 11.0. Under these conditions, the packing fraction of the fluid is η = πσ3N/6V = 0.203. We used a time step Δt = 0.0025τMD, with the thermostat and barostat coupling constants τT = 100.0Δt and τP = 500.0Δt, respectively. The MD simulations for charged spheres are performed using the LAMMPS molecular dynamics code. (43)
For the Lennard-Jones system, we simulate N = 104 particles at a temperature kBT/ϵ = 1.0 and a pressure Pσ3/ϵ = 11.5. At this temperature and pressure, the liquid has a number density of ρσ3 = 1.04. The interaction potential was truncated and shifted at 2.5σ. We used a time step Δt = 0.005τMD, with the thermostat and barostat coupling constants τT = 100.0Δt and τP = 500.0Δt, respectively. The MD simulations for the Lennard-Jones system are also performed using the LAMMPS molecular dynamics code.
To average multiple nucleation trajectories, such as in Figure 3, we shift the trajectories in time such that the number of solid-like particles reaches 50 at the same time. Additionally, to focus on nucleation, we “zoom in” to a small cubic box containing 200 particles around the center of mass of the nucleus of 50 solid-like particles. Using this procedure, the different nucleation trajectories show qualitatively the same behavior. By plotting the different nucleation trajectories with transparent lines, we aim to give a rough indication of the variance in the data.
The images of topological clusters and crystal nuclei were produced by using the OVITO visualization software. (44)

Experiments

We used poly(methyl methacrylate) (PMMA) particles of diameter 2.00 μm with a polydispersity of 4.0% as determined by static light scattering, which were fluorescently labeled with Rhodamine dye. The particles were dispersed in a density matching mixture of cis decalin and cyclohexyl bormide. Tetrabutyl ammonium bromide salt was used to screen the electrostatic charges. The resulting dispersions were imaged by using a Leica SP5 confocal microscope to obtain real space particle coordinates. Due to the residual electrostatic interactions, the effective hard sphere diameter is 1.02 times that of the physical diameter and thus, we quote experimental values in effective packing fractions. Further details are available in ref (45).

Data Availability

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The code used to generate and analyze the results of this paper is freely available at https://github.com/MarjoleinDijkstraGroupUU/Crystal-Polymorph-Selection-Mechanism-of-Hard-Spheres-Hidden-in-the-Fluid.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnano.3c02182.

  • Gibbs free energy for nucleation in the nfcc–nhcp plane; robustness with respect to polymorph detection method; robustness with respect to dynamics; conversions between PB, SD, fcc, and hcp clusters; behavior of all clusters during nucleation and near planar solid–fluid interfaces; quantitative comparison between experiments and simulations (PDF).

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Author Information

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  • Corresponding Authors
  • Authors
    • Gabriele M. Coli - Soft Condensed Matter &and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands;  Orcidhttps://orcid.org/0000-0002-5125-8007
    • Robin van Damme - Soft Condensed Matter &and Biophysics, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 1, 3584 CC Utrecht, Netherlands;  Orcidhttps://orcid.org/0000-0003-2871-3726
    • C. Patrick Royall - Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, 75005 Paris, France;  H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom;  School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom
  • Author Contributions

    W.G. and G.M.C. contributed equally to this work. G.M.C. and M.D. initiated the project. G.M.C. and W.G. performed the MD simulations and the TCC analysis. US simulations were performed by R.v.D., while experiments were carried out by C.P.R. All authors cowrote the manuscript and discussed the text and interpretation of the results.

  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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The authors are grateful to Roland Roth for his intimate knowledge of complex shapes. M.D. and W.G. acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. ERC-2019-ADG 884902 SoftML). G.M.C. and M.D. acknowledge financial support from the NWO program data-driven science for smart and sustainable energy research (project no.: 16DDS003).

References

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    Sandomirski, K.; Allahyarov, H.; Loewen, E.; Egelhaaf, S. U. Heterogeneous crystallization of hard-sphere colloids near a wall. Soft Matter 2011, 7, 8050,  DOI: 10.1039/c1sm05346a
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    Heterogeneous crystallization of hard-sphere colloids near a wall
    Sandomirski, Kirill; Allahyarov, Elshad; Loewen, Hartmut; Egelhaaf, Stefan U.
    Soft Matter (2011), 7 (18), 8050-8055CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    We investigate the most basic situation of heterogeneous crystn.: crystn. of hard-sphere colloids in the presence of a flat hard wall. Using a combination of confocal microscopy and nonequil. Brownian dynamics simulations, microscopic time-resolved information is obtained on an individual-particle level. Initially, particles near the wall rearrange before an extended regime of crystal growth is found. During growth, we can directly observe a depletion zone in the fluid next to the progressing crystal-fluid interface due to the single-particle information provided by microscopy and simulations. This also allows us to follow the relaxation of the crystal layers and the progression of the crystal-fluid interface. In good agreement between our expts. and simulations, as well as previous studies, the growth rate shows a max. in its dependence on the bulk vol. fraction.
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    Luchnikov, V.; Gervois, A.; Richard, P.; Oger, L.; Troadec, J. Crystallization of dense hard sphere packings: Competition of hcp and fcc close order. J. Mol. Liq. 2002, 96, 185– 194,  DOI: 10.1016/S0167-7322(01)00346-4
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    Crystallization of dense hard sphere packings
    Luchnikov, V.; Gervois, A.; Richard, P.; Oger, L.; Troadec, J. P.
    Journal of Molecular Liquids (2002), 96-97 (), 185-194CODEN: JMLIDT; ISSN:0167-7322. (Elsevier Science S.A.)
    We investigate the kinetics of crystn. in large mol.-dynamics models of dense hard sphere packings (16000 spheres). A sensitive measure for cryst. order is the rotationally invariant Q6 coeff., first introduced by Steinhardt et al. for the neighbors of an atom. The propensity to crystallize depends on the packing fraction and on the way the sample is prepd. The final crystal is of the fcc type. However, the hcp symmetries exist in an intermediate step; practically, mixts. of hcp and fcc clusters may coexist for a long time. The appearance and propagation of the cryst. order is studied by means of the local Q6 parameter, constructed form the 12 atoms the closest to a given atom. For a given packing fraction C, the fcc peak in the distribution function of Q6 increases during the crystn. and keeps the same shape, independently of the state of the crystn. and of the way the sample was initially prepd. The size of fcc atom clusters is studied as a function of time.
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    O’malley, B.; Snook, I. Crystal nucleation in the hard sphere system. Phys. Rev. Lett. 2003, 90, 085702  DOI: 10.1103/PhysRevLett.90.085702
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    Crystal Nucleation in the Hard Sphere System
    O'Malley, Brendan; Snook, Ian
    Physical Review Letters (2003), 90 (8), 085702/1-085702/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    The structure and growth of crystal nuclei that spontaneously form during computer simulations of the simplest nontrivial model of a liq., the hard sphere system, is described. Compact crystal nuclei form at densities within the coexistence region of the phase diagram. The nuclei possess a range of morphologies with a predominance of multiply twinned particles possessing in some cases a significant decahedral character. However the multiply twinned particles do not form from an initial decahedral core but appear to nucleate as blocks of a fcc. crystal partially bounded by stacking faults.
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    Filion, L.; Hermes, M.; Ni, R.; Dijkstra, M. Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques. J. Chem. Phys. 2010, 133, 244115,  DOI: 10.1063/1.3506838
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    Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques
    Filion, L.; Hermes, M.; Ni, R.; Dijkstra, M.
    Journal of Chemical Physics (2010), 133 (24), 244115/1-244115/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Over the last no. of years several simulation methods have been introduced to study rare events such as nucleation. In this paper we examine the crystal nucleation rate of hard spheres using three such numerical techniques: mol. dynamics, forward flux sampling, and a Bennett-Chandler-type theory where the nucleation barrier is detd. using umbrella sampling simulations. The resulting nucleation rates are compared with the exptl. rates of Harland and van Megen. When the rates are examd. in units of the long-time diffusion coeff., we find agreement between all the theor. predicted nucleation rates, however, the exptl. results display a markedly different behavior for low supersatn. Addnl., we examd. the precrit. nuclei arising in the mol. dynamics, forward flux sampling, and umbrella sampling simulations. The structure of the nuclei appears independent of the simulation method, and in all cases, the nuclei contains on av. significantly more face-centered-cubic ordered particles than hexagonal-close-packed ordered particles. (c) 2010 American Institute of Physics.
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    Russo, J.; Tanaka, H. The microscopic pathway to crystallization in supercooled liquids. Sci. Rep. 2012, 2, 1– 8,  DOI: 10.1038/srep00505
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    Leoni, F.; Russo, J. Nonclassical nucleation pathways in stacking-disordered crystals. Physical Review X 2021, 11, 031006  DOI: 10.1103/PhysRevX.11.031006
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    Nonclassical Nucleation Pathways in Stacking-Disordered Crystals
    Leoni, Fabio; Russo, John
    Physical Review X (2021), 11 (3), 031006CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
    The nucleation of crystals from liq. melt is often characterized by a competition between different cryst. structures or polymorphs and can result in nuclei with heterogeneous compns. These mixed-phase nuclei can display nontrivial spatial arrangements, such as layered and onionlike structures, whose compn. varies according to the radial distance, and which so far have been explained on the basis of bulk and surface free-energy differences between the competing phases. Here we extend the generality of these nonclassical nucleation processes, showing that layered and onionlike structures can emerge solely based on structural fluctuations even in the absence of free-energy differences. We consider two examples of competing cryst. structures, hcp and fcc forming in hard spheres relevant for repulsive colloids and dense liqs., and the cubic and hexagonal diamond forming in water relevant also for other group 14 elements such as carbon and silicon. We introduce a novel structural order parameter that combined with a neural-network classification scheme allows us to study the properties of the growing nucleus from the early stages of nucleation. We find that small nuclei have distinct size fluctuations and compns. from the nuclei that emerge from the growth stage. The transition between these two regimes is characterized by the formation of onionlike structures, in which the compn. changes with the distance from the center of the nucleus, similar to what is seen in the two-step nucleation process.
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    Sanchez-Burgos, I.; Sanz, E.; Vega, C.; Espinosa, J. R. Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate. Phys. Chem. Chem. Phys. 2021, 23, 19611– 19626,  DOI: 10.1039/D1CP01784E
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    Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate
    Sanchez-Burgos, Ignacio; Sanz, Eduardo; Vega, Carlos; Espinosa, Jorge R.
    Physical Chemistry Chemical Physics (2021), 23 (35), 19611-19626CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    A detailed computational characterization of the polymorphic nucleation competition between the fcc. and the hcp. hard-sphere crystal phases is provided. By several state-of-the-art simulation techniques, the authors evaluate the melting pressure, chem. p.d., interfacial free energy and nucleation rate of these 2 polymorphs, and of a random stacking mixt. of both crystals. The results highlight that, despite the fact that both polymorphs have very similar stability, the interfacial free energy of the hcp. phase could be marginally higher than that of the fcc. solid, which in consequence, mildly decreases its propensity to nucleate from the liq. compared to the fcc. phase. The abundance of each polymorph in grown crystals was analyzed from different types of inserted nuclei: fcc., hcp. and stacking disordered fcc./hcp. seeds, as well as from those spontaneously emerged from brute force simulations. Post-crit. crystals fundamentally grow maintaining the polymorphic structure of the crit. nucleus, at least until moderately large sizes, since the only crystallog. orientation that allows stacking close-packed disorder is the fcc. (111) plane, or equivalently the hcp. (0001) one. Taken together, the results contribute with 1 more piece to the intricate puzzle of colloidal hard-sphere crystn.
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    Malins, A.; Williams, S. R.; Eggers, J.; Royall, C. P. Identification of Structure in Condensed Matter with the Topological Cluster Classification. J. Chem. Phys. 2013, 139, 234506,  DOI: 10.1063/1.4832897
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    Identification of structure in condensed matter with the topological cluster classification
    Malins, Alex; Williams, Stephen R.; Eggers, Jens; Royall, C. Patrick
    Journal of Chemical Physics (2013), 139 (23), 234506/1-234506/21CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We describe the topol. cluster classification (TCC) algorithm. The TCC detects local structures with bond topologies similar to isolated clusters which minimize the potential energy for a no. of monat. and binary simple liqs. with m ≤ 13 particles. We detail a modified Voronoi bond detection method that optimizes the cluster detection. The method to identify each cluster is outlined, and a test example of Lennard-Jones liq. and crystal phases is considered and critically examd. (c) 2013 American Institute of Physics.
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    Supercooling of liquids
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    Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1952), 215 (), 43-6CODEN: PRLAAZ; ISSN:1364-5021.
    A brief crit. review of recent observations on supercooling of metal liquids. Close-packing arrangements are discussed and nucleation is considered.
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    Taffs, J.; Royall, C. P. The role of fivefold symmetry in suppressing crystallization. Nat. Commun. 2016, 7, 1– 7,  DOI: 10.1038/ncomms13225
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    Karayiannis, N. C.; Malshe, R.; Kröger, M.; de Pablo, J. J.; Laso, M. Evolution of fivefold local symmetry during crystal nucleation and growth in dense hard-sphere packings. Soft Matter 2012, 8, 844– 858,  DOI: 10.1039/C1SM06540H
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    Evolution of fivefold local symmetry during crystal nucleation and growth in dense hard-sphere packings
    Karayiannis, Nikos Ch.; Malshe, Rohit; Kroeger, Martin; de Pablo, Juan J.; Laso, Manuel
    Soft Matter (2012), 8 (3), 844-858CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    Crystal nucleation and growth of monodisperse hard-spheres as a function of packing d. was studied by collision-driven mol. dynamics simulations. Short-range order as 5-fold local symmetry is identified and its dynamical and structural evolution is tracked as the originally amorphous assembly transits to the stable ordered phase. A cluster-based approach shows that hard-sphere configurations having initially a similar av. fraction of 5-fold and ordered sites can crystallize in completely different patterns both in terms of dynamics and morphol. At high vol. fractions crystn. is significantly delayed in assemblies where sites with 5-fold symmetry are abundant. Eventually, once the crystal phase is reached, 5-fold symmetry either diminishes or arranges in specific geometric patterns. Such defects are spatially strongly correlated with twinning planes at cryst. boundaries. A detailed anal. is provided on the structural characteristics of the established crystal morphologies.
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    Martelli, F.; Palmer, J. C. Signatures of sluggish dynamics and local structural ordering during ice nucleation. J. Chem. Phys. 2022, 156, 114502,  DOI: 10.1063/5.0083638
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    Signatures of sluggish dynamics and local structural ordering during ice nucleation
    Martelli, Fausto; Palmer, Jeremy C.
    Journal of Chemical Physics (2022), 156 (11), 114502CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We investigate the microscopic pathway of spontaneous crystn. in the ST2 model of water under deeply supercooled conditions via unbiased classical mol. dynamics simulations. After quenching below the liq.-liq. crit. point, the ST2 model spontaneously separates into low-d. liq. (LDL) and high-d. liq. phases, resp. The LDL phase, which is characterized by lower mol. mobility and enhanced structural order, fosters the formation of a sub-crit. ice nucleus that, after a stabilization time, develops into the crit. nucleus and grows. Polymorphic selection coincides with the development of the sub-crit. nucleus and favors the formation of cubic (Ic) over hexagonal (Ih) ice. We rationalize polymorphic selection in terms of geometric arguments based on differences in the symmetry of second neighbor shells of ice Ic and Ih, which are posited to favor formation of the former. The rapidly growing crit. nucleus absorbs both Ic and Ih crystallites dispersed in the liq. phase, a crystal with stacking faults. Our results are consistent with, and expand upon, recent observations of non-classical nucleation pathways in several systems. (c) 2022 American Institute of Physics.
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    Torrie, G.; Valleau, J. Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. J. Comput. Phys. 1977, 23, 187– 199,  DOI: 10.1016/0021-9991(77)90121-8
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    Auer, S.; Frenkel, D. Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature 2001, 409, 1020– 1023,  DOI: 10.1038/35059035
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    Prediction of absolute crystal-nucleation rate in hard-sphere colloids
    Auer, Stefan; Fenkel, Daan
    Nature (London, United Kingdom) (2001), 409 (6823), 1020-1023CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
    Crystal nucleation is a much-studied phenomenon, yet the rate at which it occurs remains difficult to predict. Small crystal nuclei form spontaneously in supersatd. solns., but unless their size exceeds a crit. value-the so-called crit. nucleus-they will re-dissolve rather than grow. It is this rate-limiting step that proved difficult to probe exptl. The crystal nucleation rate depends on Pcrit, the (very small) probability that a crit. nucleus form spontaneously, and on a kinetic factor (κ) that measures the rate at which crit. nuclei subsequently grow. Given the absence of a priori knowledge of either quantity, classical nucleation expts., with the unconstrained parameters adjusted to fit the observations. This approach yields no ;first principles; prediction of abs. nucleation rate. Here the authors approach the problem from a different angle, simulating the nucleation process in a suspension of hard colloidal spheres, to obtain quant. numerical predictions of the crystal nucleation rate. Large discrepancies between the compute nucleation rates and those deduced from expts. were found.: the orders of magnitude.
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    Accurate determination of crystal structures based on averaged local bond order parameters
    Lechner, Wolfgang; Dellago, Christoph
    Journal of Chemical Physics (2008), 129 (11), 114707/1-114707/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Local bond order parameters based on spherical harmonics, also known as Steinhardt order parameters, are often used to det. crystal structures in mol. simulations. Here we propose a modification of this method in which the complex bond order vectors are averaged over the first neighbor shell of a given particle and the particle itself. As demonstrated using soft particle systems, this averaging procedure considerably improves the accuracy with which different crystal structures can be distinguished. (c) 2008 American Institute of Physics.
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    Anderson, J. A.; Glaser, J.; Glotzer, S. C. HOOMD-blue: A Python package for highperformance molecular dynamics and hard particle Monte Carlo simulations. Comput. Mater. Sci. 2020, 173, 109363,  DOI: 10.1016/j.commatsci.2019.109363
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    HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations
    Anderson, Joshua A.; Glaser, Jens; Glotzer, Sharon C.
    Computational Materials Science (2020), 173 (), 109363CODEN: CMMSEM; ISSN:0927-0256. (Elsevier B.V.)
    HOOMD-blue is a particle simulation engine designed for nano- and colloidal-scale mol. dynamics and hard particle Monte Carlo simulations. It has been actively developed since March 2007 and available open source since August 2008. HOOMD-blue is a Python package with a high performance C++/CUDA backend that we built from the ground up for GPU acceleration. The Python interface allows users to combine HOOMD-blue with other packages in the Python ecosystem to create simulation and anal. workflows. We employ software engineering practices to develop, test, maintain, and expand the code.
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    Anderson, J. A.; Eric Irrgang, M.; Glotzer, S. C. Scalable Metropolis Monte Carlo for simulation of hard shapes. Comput. Phys. Commun. 2016, 204, 21– 30,  DOI: 10.1016/j.cpc.2016.02.024
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    Scalable Metropolis Monte Carlo for simulation of hard shapes
    Anderson, Joshua A.; Eric Irrgang, M.; Glotzer, Sharon C.
    Computer Physics Communications (2016), 204 (), 21-30CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
    We design and implement a scalable hard particle Monte Carlo simulation toolkit (HPMC), and release it open source as part of HOOMD-blue. HPMC runs in parallel on many CPUs and many GPUs using domain decompn. We employ BVH trees instead of cell lists on the CPU for fast performance, esp. with large particle size disparity, and optimize inner loops with SIMD vector intrinsics on the CPU. Our GPU kernel proposes many trial moves in parallel on a checkerboard and uses a block-level queue to redistribute work among threads and avoid divergence. HPMC supports a wide variety of shape classes, including spheres/disks, unions of spheres, convex polygons, convex spheropolygons, concave polygons, ellipsoids/ellipses, convex polyhedra, convex spheropolyhedra, spheres cut by planes, and concave polyhedra. NVT and NPT ensembles can be run in 2D or 3D triclinic boxes. Addnl. integration schemes permit Frenkel-Ladd free energy computations and implicit depletant simulations. In a benchmark system of a fluid of 4096 pentagons, HPMC performs 10 million sweeps in 10 min on 96 CPU cores on XSEDE Comet. The same simulation would take 7.6 h in serial. HPMC also scales to large system sizes, and the same benchmark with 16.8 million particles runs in 1.4 h on 2048 GPUs on OLCF Titan.
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    Journal of Computational Chemistry (1992), 13 (8), 1011-21CODEN: JCCHDD; ISSN:0192-8651.
    The Weighted Histogram Anal. Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomol. Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calcns. This algorithm possesses the following advantages over methods that are currently employed: (1) it provides a built-in est. of sampling errors thereby yielding objective ests. of the optimal location and length of addnl. simulations needed to achieve a desired level of precision; (2) it yields the "best" value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addn. to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better ests. of the free-energy differences. By recasting the Ferrenberg-Swendsen Multiple Histogram equations in a form suitable for mol. mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine.
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    Modularly invariant equations of motion are derived that generate the isothermal-isobaric ensemble as their phase space avs. Isotropic vol. fluctuations and fully flexible simulation cells as well as a hybrid scheme that naturally combines the two motions are considered. The resulting methods are tested on two problems, a particle in a one-dimensional periodic potential and a spherical model of C60 in the solid/fluid phase.
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    Journal of Chemical Physics (1971), 54 (12), 5237-47CODEN: JCPSA6; ISSN:0021-9606.
    The different roles the attractive and repulsive forces play in forming the equil. structure of a Lennard-Jones liq. are discussed. The effects of these forces are most easily sepd. by considering the structure factor (or equivalently, the Fourier transform of the pair-correlation function) rather than the pair-correlation function itself. At intermediate and large wave vectors, the repulsive forces dominate the quant. behavior of the liq. structure factor. The attractions are manifested primarily in the small wave vector part of the structure factor; but, this effect decreases as the d. increases and is almost negligible at reduced ds. >0.65. These conclusions are established by considering the structure factor of a hypothetical ref. system in which the intermol. forces are entirely repulsive and identical to the repulsive forces in a Lennard-Jones fluid. This ref. system structure factor is calcd. with the aid of a simple but accurate approxn. described. The conclusions lead to a very simple prescription for calcg. the radial distribution function of dense liqs. which is more accurate than that obtained by any previously reported theory. The thermodynamic ramifications of the conclusions are presented in the form of calcns. of the free energy, the internal energy (from the energy equation), and the pressure (from the virial equation). The implications of the authors conclusions to perturbation theories for liqs. and to the interpretation of x-ray scattering expts. are discussed.
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    Kawasaki, T.; Tanaka, H. Formation of a Crystal Nucleus from Liquid. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 14036– 14041,  DOI: 10.1073/pnas.1001040107
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    Formation of a crystal nucleus from liquid
    Kawasaki, Takeshi; Tanaka, Hajime
    Proceedings of the National Academy of Sciences of the United States of America (2010), 107 (32), 14036-14041, S14036/1-S14036/6CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
    Crystn. is one of the most fundamental nonequil. phenomena universal to a variety of materials. It has so far been assumed that a supercooled liq. is in a "homogeneous disordered state" before crystn. Contrary to this common belief, we reveal that supercooled colloidal liq. is actually not homogeneous, but has transient medium-range structural order. We find that nucleation preferentially takes place in regions of high structural order via wetting effects, which reduce the crystal-liq. interfacial energy significantly and thus promotes crystal nucleation. This novel scenario provides a clue to solving a long-standing mystery concerning a large discrepancy between the rigorous numerical estn. of the nucleation rate on the basis of the classical nucleation theory and the exptl. obsd. ones. Our finding may shed light not only on the mechanism of crystal nucleation, but also on the fundamental nature of a supercooled liq. state.
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    Filion, L.; Ni, R.; Frenkel, D.; Dijkstra, M. Simulation of Nucleation in Almost Hard- Sphere Colloids: The Discrepancy between Experiment and Simulation Persists. J. Chem. Phys. 2011, 134, 134901,  DOI: 10.1063/1.3572059
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    Simulation of nucleation in almost hard-sphere colloids. The discrepancy between experiment and simulation persists
    Filion, L.; Ni, R.; Frenkel, D.; Dijkstra, M.
    Journal of Chemical Physics (2011), 134 (13), 134901/1-134901/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    In this paper we examine the phase behavior of the Weeks-Chandler-Andersen (WCA) potential with βε = 40. Crystal nucleation in this model system was recently studied by , who argued that the computed nucleation rates agree well with expt., a finding that contradicted earlier simulation results. Here we report an extensive numerical study of crystn. in the WCA model, using 3 totally different techniques (Brownian dynamics, umbrella sampling, and forward flux sampling). All simulations yield essentially the same nucleation rates. However, these rates differ significantly from the values reported by Kawasaki and Tanaka and hence we argue that the huge discrepancy in nucleation rates between simulation and expt. persists. When we map the WCA model onto a hard-sphere system, we find good agreement between the present simulation results and those that had been obtained for hard spheres ; . (c) 2011 American Institute of Physics.
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    Journal of Chemical Physics (2018), 148 (12), 124110/1-124110/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We investigate the kinetics and the free energy landscape of the crystn. of hard spheres from a supersatd. metastable liq. though direct simulations and forward flux sampling. In this first paper, we describe and test two different ways to reconstruct the free energy barriers from the sampled steady state probability distribution of cluster sizes without sampling the equil. distribution. The first method is based on mean first passage times, and the second method is based on splitting probabilities. We verify both methods for a single particle moving in a double-well potential. For the nucleation of hard spheres, these methods allow us to probe a wide range of supersaturations and to reconstruct the kinetics and the free energy landscape from the same simulation. Results are consistent with the scaling predicted by classical nucleation theory although a quant. fit requires a rather large effective interfacial tension. (c) 2018 American Institute of Physics.
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    Combining three numerical methods (forward flux sampling, seeding of droplets, and finite-size droplets), we probe the crystn. of hard spheres over the full range from close to coexistence to the spinodal regime. We show that all three methods allow us to sample different regimes and agree perfectly in the ranges where they overlap. By combining the nucleation work calcd. from forward flux sampling of small droplets and the nucleation theorem, we show how to compute the nucleation work spanning three orders of magnitude. Using a variation of the nucleation theorem, we show how to ext. the pressure difference between the solid droplet and ambient liq. Moreover, combining the nucleation work with the pressure difference allows us to calc. the interfacial tension of small droplets. Our results demonstrate that employing bulk quantities yields inaccurate results for the nucleation rate. (c) 2018 American Institute of Physics.
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    We present a continuous pseudo-hard-sphere potential based on a cut-and-shifted Mie (generalized Lennard-Jones) potential with exponents (50, 49). Using this potential one can mimic the volumetric, structural, and dynamic properties of the discontinuous hard-sphere potential over the whole fluid range. The continuous pseudo potential has the advantage that it may be incorporated directly into off-the-shelf mol.-dynamics code, allowing the user to capitalize on existing hardware and software advances. Simulation results for the compressibility factor of the fluid and solid phases of our pseudo hard spheres are presented and compared both to the Carnahan-Starling equation of state of the fluid and published data, the differences being indistinguishable within simulation uncertainty. The specific form of the potential is employed to simulate flexible chains formed from these pseudo hard spheres at contact (pearl-necklace model) for mc = 4, 5, 7, 8, 16, 20, 100, 201, and 500 monomer segments. The compressibility factor of the chains per unit of monomer, mc, approaches a limiting value at reasonably small values, mc < 50, as predicted by Wertheim's first order thermodn. perturbation theory. Simulation results are also presented for highly asym. mixts. of pseudo hard spheres, with diam. ratios of 3:1, 5:1, 20:1 over the whole compn. range. (c) 2012 American Institute of Physics.
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    Taffs, J.; Williams, S. R.; Tanaka, H.; Royall, C. P. Structure and kinetics in the freezing of nearly hard spheres. Soft Matter 2013, 9, 297– 305,  DOI: 10.1039/C2SM26473K
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    Structure and kinetics in the freezing of nearly hard spheres
    Taffs, Jade; Williams, Stephen R.; Tanaka, Hajime; Royall, C. Patrick
    Soft Matter (2013), 9 (1), 297-305CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
    We consider homogeneous crystn. rates in confocal microscopy expts. on colloidal nearly hard spheres at the single particle level. These we compare with Brownian dynamics simulations by carefully modeling the softness in the colloid interactions with a Yukawa potential, which takes account of the electrostatic charges present in the exptl. system. Both structure and dynamics of the colloidal fluid are very well matched between expt. and simulation, so we have confidence that the system simulated is close to that in the expt. In the regimes we can access, we find reasonable agreement in crystn. rates between expt. and simulations, noting that the larger system size in expts. enables the formation of crit. nuclei and hence crystn. at lower supersaturations than in the simulations. We further examine the metastable fluid with a novel structural anal., the topol. cluster classification. We find that at densities where the hard sphere fluid becomes metastable, the dominant structure is a cluster of m = 10 particles with five-fold symmetry. Analyzing histories of the local environment of single particles, we find fluctuations into cryst. configurations in the metastable fluid, and that the cryst. state a very often preceded by a transition region of frequent hopping between crystal-like environments and other (m ≠ 10) structures.
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  • Abstract

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    Figure 1

    Figure 1. Thermodynamic propensity toward mixed fcc-hcp crystal nuclei in the early stages of crystal nucleation of hard spheres. Gibbs free-energy barrier as a function of the number of fcc and hcp particles nfcc + nhcp and composition nfcc/(nfcc + nhcp). The dots are the composition of crystal nuclei found in our experiments. For both simulations and experiments, the composition of crystal nuclei is determined with the classification scheme described in the Methods Section. The dashed line represents the minimum free-energy path for nucleation, and the solid line represents the critical nucleus size as a function of composition.

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    Figure 2

    Figure 2. The effect of topological clusters on crystallization and polymorph selection in the metastable fluid. (a) Mutual information (MI) of topological clusters on crystallization and polymorph selection. Each label (e.g., 11F) refers to a different cluster as identified by the topological cluster classification algorithm. (24) Clusters with low MI are not labeled. (b) Typical arrangements of particles in an OH, PB, SD, fcc, and hcp clusters. The ring particles in each cluster are colored light gray. (c) Relative probability to convert to fcc with respect to hcp as a function of the number of OH or PB clusters a particle is part of. (d) Probability to convert to fcc or hcp as a function of the number of PB, OH, or SD clusters it belongs to.

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    Figure 3

    Figure 3. The role of PB and SD clusters during crystal nucleation from strongly correlated, dense fluids. (a,b) PMMA spheres imaged with confocal microscopy. (c,d,i) Nearly hard spheres interacting with a WCA potential. (e,f,j) Charged spheres interacting with a Yukawa potential. (g,h,k) Spheres interacting with a Lennard-Jones potential. (a–h) On the left, we show cut-through images of crystal nuclei. The core of the crystal nucleus is colored blue, while the rest of the particles is colored following the scale bar on the left (a,c,e,g) or right (b,d,f,h) depending on the number of PB (a,c,e,g) or SD (b,d,f,h) clusters each particle belongs to. (i–k) On the right, we show the fraction of particles belonging to different clusters as a function of time (solid lines), averaged over multiple different simulated spontaneous nucleation events (transparent lines). The fractions are averaged over 5 (i), 3 (j), and 3 (k) different nucleation events, respectively. Note that a particle can be part of multiple clusters at the same time, and therefore the corresponding fractions add up to a value which is higher than one.

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    Figure 4

    Figure 4. Transition of SD and PB clusters to fcc subunits. (a) Geometry of the SD cluster (gray and pink particles) parallel to two hexagonal layers of an fcc and hcp crystal. (b) Geometry of the SD cluster perpendicular to the hexagonal layers, which occurs only in fcc. (c) Probability distributions of the internal angles θ of the ring particles of SD and PB clusters before and after the transition to an fcc subunit. The typical arrangements of particles in SD (left) and PB (right) clusters before and after the transition are also shown, with the ring particles colored in gray.

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    This article references 45 other publications.

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      Nucleation: theory and applications to protein solutions and colloidal suspensions
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      A review. The status of our understanding of nucleation is described. Both general aspects of nucleation and those specific to protein solns. and colloidal suspensions are considered. We conclude that although we have what we believe is a quite good understanding of homogeneous nucleation in one simple system, hard-sphere colloids, in other systems there are still basic questions that are unanswered. For example, for even the most studied protein, lysozyme, there is an ongoing debate about whether the obsd. nucleation is homogeneous or heterogeneous. We review theor. and simulation work on both homogeneous and heterogeneous nucleation. As heterogeneous nucleation appears to be much more common than homogeneous nucleation, and as earlier reviews have tended to focus more on homogeneous nucleation, we place particular emphasis on heterogeneous nucleation.
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      Palberg, T. Crystallization kinetics of colloidal model suspensions: recent achievements and new perspectives. J. Phys.: Condens. Matter 2014, 26, 333101,  DOI: 10.1088/0953-8984/26/33/333101
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      Crystallization kinetics of colloidal model suspensions: recent achievements and new perspectives
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      Colloidal model systems allow studying crystallization kinetics under fairly ideal conditions, with rather well-characterized pair interactions and minimized external influences. In complementary approaches experiment, analytic theory and simulation have been employed to study colloidal solidification in great detail. These studies were based on advanced optical methods, careful system characterization and sophisticated numerical methods. Over the last decade, both the effects of the type, strength and range of the pair-interaction between the colloidal particles and those of the colloid-specific polydispersity have been addressed in a quantitative way. Key parameters of crystallization have been derived and compared to those of metal systems. These systematic investigations significantly contributed to an enhanced understanding of the crystallization processes in general. Further, new fundamental questions have arisen and (partially) been solved over the last decade: including, for example, a two-step nucleation mechanism in homogeneous nucleation, choice of the crystallization pathway, or the subtle interplay of boundary conditions in heterogeneous nucleation. On the other hand, via the application of both gradients and external fields the competition between different nucleation and growth modes can be controlled and the resulting microstructure be influenced. The present review attempts to cover the interesting developments that have occurred since the turn of the millennium and to identify important novel trends, with particular focus on experimental aspects.
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      Ohm, T.; Kirca, M.; Bohl, J.; Scharnagl, H.; Groβ, W.; März, W. Apolipoprotein E polymorphism influences not only cerebral senile plaque load but also Alzheimer-type neurofibrillary tangle formation. Neuroscience 1995, 66, 583– 587,  DOI: 10.1016/0306-4522(94)00596-W
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      Apolipoprotein E polymorphism influences not only cerebral senile plaque load but also Alzheimer-type neurofibrillary tangle formation
      Ohm, T. G.; Kirca, M.; Bohl, J.; Scharnagl, H.; Gross, W.; Maerz, W.
      Neuroscience (Oxford) (1995), 66 (3), 583-7CODEN: NRSCDN; ISSN:0306-4522. (Elsevier)
      Only recently, evidence was provided that apolipoprotein E allele ε4 located on Chromosome 19 is assocd. with late onset (i.e. senile) sporadic Alzheimer's disease. Histol., Alzheimer's disease is assocd. with intraneuronal neurofibrillary changes and extraneuronal A4/β-amyloid deposition. We set out with a histol. staging system which considers the gradual development of Alzheimer's disease-related histol. changes over time and correlates highly with the cognitive decline ante mortem. Our anal. revealed that both the mean stage for A4/B-amyloid deposits and the mean stage for neurofibrillary tangles get significantly shifted upwards in ε4-carriers. This represents an earlier onset of the histopathol. process of about one decade. The fact that both types of Alzheimer's disease-related changes correlate pos. with the prevalence of the ε4-allele suggests for a causal relationship between the apolipoprotein E polymorphism and the development of Alzheimer's disease.
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      Bauer, J.; Spanton, S.; Henry, R.; Quick, J.; Dziki, W.; Porter, W.; Morris, J. Ritonavir: an extraordinary example of conformational polymorphism. Pharm. Res. 2001, 18, 859– 866,  DOI: 10.1023/A:1011052932607
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      Ritonavir: an extraordinary example of conformational polymorphism
      Bauer, John; Spanton, Stephen; Henry, Rodger; Quick, John; Dziki, Walter; Porter, William; Morris, John
      Pharmaceutical Research (2001), 18 (6), 859-866CODEN: PHREEB; ISSN:0724-8741. (Kluwer Academic/Plenum Publishers)
      Purpose. In the summer of 1998, Norvir semi-solid capsules supplies were threatened as a result of a new much less sol. crystal form of ritonavir. This report provides characterization of the 2 polymorphs and the structures and hydrogen bonding network for each form. Methods. Ritonavir polymorphism was investigated by solid state spectroscopy and microscopy techniques including solid state NMR, near-IR Spectroscopy, powder x-ray Diffraction and single crystal x-ray. A sensitive seed detection test was developed. Results. Ritonavir polymorphs were thoroughly characterized and the structures detd. An unusual conformation was found for form II that results in a strong hydrogen bonding network. A possible mechanism for heterogeneous nucleation of form II was investigated. Conclusions. Ritonavir exhibited conformational polymorphism with two unique crystal lattices having significantly different soly. properties. Although the polymorph (form II) corresponding to the "cis" conformation is a more stable packing arrangement, nucleation, even in the presence of form II seeds, is energetically unfavored except in highly supersatd. solns. The coincidence of a highly supersatd. soln. and a probable heterogeneous nucleation by a degrdn. product resulted in the sudden appearance of the more stable form II polymorph.
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      Van Driessche, A. E.; Van Gerven, N.; Bomans, P. H.; Joosten, R. R.; Friedrich, H.; Gil-Carton, D.; Sommerdijk, N. A.; Sleutel, M. Molecular nucleation mechanisms and control strategies for crystal polymorph selection. Nature 2018, 556, 89– 94,  DOI: 10.1038/nature25971
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      Molecular nucleation mechanisms and control strategies for crystal polymorph selection
      Van Driessche, Alexander E. S.; Van Gerven, Nani; Bomans, Paul H. H.; Joosten, Rick R. M.; Friedrich, Heiner; Gil-Carton, David; Sommerdijk, Nico A. J. M.; Sleutel, Mike
      Nature (London, United Kingdom) (2018), 556 (7699), 89-94CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
      The formation of condensed (compacted) protein phases is assocd. with a wide range of human disorders, such as eye cataracts, amyotrophic lateral sclerosis, sickle cell anemia and Alzheimer's disease. However, condensed protein phases have their uses: as crystals, they are harnessed by structural biologists to elucidate protein structures, or are used as delivery vehicles for pharmaceutical applications. The physiochem. properties of crystals can vary substantially between different forms or structures ('polymorphs') of the same macromol., and dictate their usability in a scientific or industrial context. To gain control over an emerging polymorph, one needs a mol.-level understanding of the pathways that lead to the various macroscopic states and of the mechanisms that govern pathway selection. However, it is still not clear how the embryonic seeds of a macromol. phase are formed, or how these nuclei affect polymorph selection. Here we use time-resolved cryo-transmission electron microscopy to image the nucleation of crystals of the protein glucose isomerase, and to uncover at mol. resoln. the nucleation pathways that lead to two cryst. states and one gelled state. We show that polymorph selection takes place at the earliest stages of structure formation and is based on specific building blocks for each space group. Moreover, we demonstrate control over the system by selectively forming desired polymorphs through site-directed mutagenesis, specifically tuning intermol. bonding or gel seeding. Our results differ from the present picture of protein nucleation, in that we do not identify a metastable dense liq. as the precursor to the cryst. state. Rather, we observe nucleation events that are driven by oriented attachments between subcrit. clusters that already exhibit a degree of crystallinity. These insights suggest ways of controlling macromol. phase transitions, aiding the development of protein-based drug-delivery systems and macromol. crystallog.
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      Xing, J.; Schweighauser, L.; Okada, S.; Harano, K.; Nakamura, E. Atomistic structures and dynamics of prenucleation clusters in MOF-2 and MOF-5 syntheses. Nat. Commun. 2019, 10, 1– 9,  DOI: 10.1038/s41467-019-11564-4
      6
      Author Correction: Mitochondria-specific drug release and reactive oxygen species burst induced by polyprodrug nanoreactors can enhance chemotherapy
      Zhang, Wenjia; Hu, Xianglong; Shen, Qi; Xing, Da
      Nature Communications (2019), 10 (1), 1CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)
      An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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      Zhou, J.; Yang, Y.; Yang, Y.; Kim, D. S.; Yuan, A.; Tian, X.; Ophus, C.; Sun, F.; Schmid, A. K.; Nathanson, M. Observing crystal nucleation in four dimensions using atomic electron tomography. Nature 2019, 570, 500– 503,  DOI: 10.1038/s41586-019-1317-x
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      Observing crystal nucleation in four dimensions using atomic electron tomography
      Zhou, Jihan; Yang, Yongsoo; Yang, Yao; Kim, Dennis S.; Yuan, Andrew; Tian, Xuezeng; Ophus, Colin; Sun, Fan; Schmid, Andreas K.; Nathanson, Michael; Heinz, Hendrik; An, Qi; Zeng, Hao; Ercius, Peter; Miao, Jianwei
      Nature (London, United Kingdom) (2019), 570 (7762), 500-503CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
      Nucleation plays a crit. role in many phys. and biol. phenomena that range from crystn., melting and evapn. to the formation of clouds and the initiation of neurodegenerative diseases1-3. However, nucleation is a challenging process to study exptl., esp. in its early stages, when several atoms or mols. start to form a new phase from a parent phase. A no. of exptl. and computational methods have been used to study nucleation processes4-17, but exptl. detn. of the three-dimensional at. structure and the dynamics of early-stage nuclei has been unachievable. Here the authors use at. electron tomog. to study early-stage nucleation in four dimensions (i.e., including time) at at. resoln. Using FePt nanoparticles as a model system, early-stage nuclei are irregularly shaped, each has a core of one to a few atoms with the max. order parameter, and the order parameter gradient points from the core to the boundary of the nucleus. The authors capture the structure and dynamics of the same nuclei undergoing growth, fluctuation, dissoln., merging and/or division, which are regulated by the order parameter distribution and its gradient. These exptl. observations are corroborated by mol. dynamics simulations of heterogeneous and homogeneous nucleation in liq.-solid phase transitions of Pt. The authors' exptl. and mol. dynamics results indicate that a theory beyond classical nucleation theory1,2,18 is needed to describe early-stage nucleation at the at. scale. The authors anticipate that the reported approach will open the door to the study of many fundamental problems in materials science, nanoscience, condensed matter physics and chem., such as phase transition, at. diffusion, grain boundary dynamics, interface motion, defect dynamics and surface reconstruction with four-dimensional at. resoln.
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      Houben, L.; Weissman, H.; Wolf, S. G.; Rybtchinski, B. A mechanism of ferritin crystallization revealed by cryo-STEM tomography. Nature 2020, 579, 540– 543,  DOI: 10.1038/s41586-020-2104-4
      8
      A mechanism of ferritin crystallization revealed by cryo-STEM tomography
      Houben, Lothar; Weissman, Haim; Wolf, Sharon G.; Rybtchinski, Boris
      Nature (London, United Kingdom) (2020), 579 (7800), 540-543CODEN: NATUAS; ISSN:0028-0836. (Nature Research)
      Abstr.: Protein crystn. is important in structural biol., disease research and pharmaceuticals. It has recently been recognized that nonclassical crystn.-involving initial formation of an amorphous precursor phase-occurs often in protein, org. and inorg. crystn. processes. A two-step nucleation theory has thus been proposed, in which initial low-d., solvated amorphous aggregates subsequently densify, leading to nucleation. This view differs from classical nucleation theory, which implies that cryst. nuclei forming in soln. have the same d. and structure as does the final cryst. state. A protein crystn. mechanism involving this classical pathway has recently been obsd. directly. However, a mol. mechanism of nonclassical protein crystn. has not been established. To det. the nature of the amorphous precursors and whether crystn. takes place within them (and if so, how order develops at the mol. level), three-dimensional (3D) mol.-level imaging of a crystn. process is required. Here we report cryogenic scanning transmission microscopy tomog. of ferritin aggregates at various stages of crystn., followed by 3D reconstruction using simultaneous iterative reconstruction techniques to provide a 3D picture of crystn. with mol. resoln. As cryst. order gradually increased in the studied aggregates, they exhibited an increase in both order and d. from their surface towards their interior. We obsd. no highly ordered small structures typical of a classical nucleation process, and occasionally we obsd. several ordered domains emerging within one amorphous aggregate, a phenomenon not predicted by either classical or two-step nucleation theories. Our mol.-level anal. hints at desolvation as the driver of the continuous order-evolution mechanism, a view that goes beyond current nucleation models, yet is consistent with a broad spectrum of protein crystn. mechanisms.
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    9. 9
      Nakamuro, T.; Sakakibara, M.; Nada, H.; Harano, K.; Nakamura, E. Capturing the Moment of Emergence of Crystal Nucleus from Disorder. J. Am. Chem. Soc. 2021, 143, 1763– 1767,  DOI: 10.1021/jacs.0c12100
      9
      Capturing the Moment of Emergence of Crystal Nucleus from Disorder
      Nakamuro, Takayuki; Sakakibara, Masaya; Nada, Hiroki; Harano, Koji; Nakamura, Eiichi
      Journal of the American Chemical Society (2021), 143 (4), 1763-1767CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      Crystn. is the process of atoms or mols. forming an organized solid via nucleation and growth. Being intrinsically stochastic, the research at an atomistic level was a huge exptl. challenge. In situ detection is reported of a crystal nucleus forming during nucleation/growth of a NaCl nanocrystal, as video recorded in the interior of a vibrating conical C nanotube at 20-40 ms/frame with localization precision of <0.1 nm. NaCl units were seen assembled to form a cluster fluctuating between featureless and semiordered states, which suddenly formed a crystal. Subsequent crystal growth at 298 K and shrinkage at 473 K took place also in a stochastic manner. Productive contributions of the graphitic surface and its mech. vibration were exptl. indicated.
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    10. 10
      Jeon, S.; Heo, T.; Hwang, S.-Y.; Ciston, J.; Bustillo, K. C.; Reed, B. W.; Ham, J.; Kang, S.; Kim, S.; Lim, J. Reversible disorder-order transitions in atomic crystal nucleation. Science 2021, 371, 498– 503,  DOI: 10.1126/science.aaz7555
      10
      Reversible disorder-order transitions in atomic crystal nucleation
      Jeon, Sungho; Heo, Taeyeong; Hwang, Sang-Yeon; Ciston, Jim; Bustillo, Karen C.; Reed, Bryan W.; Ham, Jimin; Kang, Sungsu; Kim, Sungin; Lim, Joowon; Lim, Kitaek; Kim, Ji Soo; Kang, Min-Ho; Bloom, Ruth S.; Hong, Sukjoon; Kim, Kwanpyo; Zettl, Alex; Kim, Woo Youn; Ercius, Peter; Park, Jungwon; Lee, Won Chul
      Science (Washington, DC, United States) (2021), 371 (6528), 498-503CODEN: SCIEAS; ISSN:1095-9203. (American Association for the Advancement of Science)
      Nucleation in at. crystn. remains poorly understood, despite advances in classical nucleation theory. The nucleation process was described to involve a nonclassical mechanism that includes a spontaneous transition from disordered to cryst. states, but a detailed understanding of dynamics requires further study. In situ electron microscopy of heterogeneous nucleation of individual Au nanocrystals with millisecond temporal resoln. shows that the early stage of at. crystn. proceeds through dynamic structural fluctuations between disordered and cryst. states, rather than through a single irreversible transition. The exptl. and theor. analyses support the idea that structural fluctuations originate from size-dependent thermodn. stability of the 2 states in at. clusters. These findings, based on dynamics in a real at. system, reshape and improve the understanding of nucleation mechanisms in at. crystn.
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    11. 11
      Pusey, P.; Van Megen, W.; Bartlett, P.; Ackerson, B.; Rarity, J.; Underwood, S. Structure of crystals of hard colloidal spheres. Phys. Rev. Lett. 1989, 63, 2753,  DOI: 10.1103/PhysRevLett.63.2753
      11
      Structure of crystals of hard colloidal spheres
      Pusey, P. N.; Van Megen, W.; Bartlett, P.; Ackerson, B. J.; Rarity, J. G.; Underwood, S. M.
      Physical Review Letters (1989), 63 (25), 2753-6CODEN: PRLTAO; ISSN:0031-9007.
      The authors report light-scattering measurements of powder diffraction patterns of crystals of essentially hard colloidal spheres. These are consistent with structures formed by stacking close-packed planes of particles in a sequence of permitted lateral positions, A,B,C, which shows a high degree of randomness. Crystals grown slowly, while still contg. many stacking faults, show a tendency towards face-centered-cubic packing; possible explanations for this observation are discussed.
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    12. 12
      Gasser, U.; Weeks, E. R.; Schofield, A.; Pusey, P.; Weitz, D. Real-space imaging of nucleation and growth in colloidal crystallization. Science 2001, 292, 258– 262,  DOI: 10.1126/science.1058457
      12
      Real-space imaging of nucleation and growth in colloidal crystallization
      Gasser, U.; Weeks, Eric R.; Schofield, Andrew; Pusey, P. N.; Weitz, D. A.
      Science (Washington, DC, United States) (2001), 292 (5515), 258-262CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
      Crystn. of concd. colloidal suspensions was studied in real space with laser scanning confocal microscopy. Direct imaging in three dimensions allowed identification and observation of both nucleation and growth of cryst. regions, providing an exptl. measure of properties of the nucleating crystallites. By following their evolution, the authors identified crit. nuclei, detd. nucleation rates, and measured the av. surface tension of the crystal-liq. interface. The structure of the nuclei was the same as the bulk solid phase, random hcp., and their av. shape was rather nonspherical, with rough rather than faceted surfaces.
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    13. 13
      Bolhuis, P. G.; Frenkel, D.; Mau, S.-C.; Huse, D. A. Entropy difference between crystal phases. Nature 1997, 388, 235– 236,  DOI: 10.1038/40779
      13
      Entropy difference between crystal phases
      Bolhuis, P. G.; Frenkel, D.; Mau, Siun-Chuon; Huse, David A.
      Nature (London) (1997), 388 (6639), 235-236CODEN: NATUAS; ISSN:0028-0836. (Macmillan Magazines)
      There is no expanded citation for this reference.
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    14. 14
      Noya, E. G.; Almarza, N. G. Entropy of hard spheres in the close-packing limit. Mol. Phys. 2015, 113, 1061– 1068,  DOI: 10.1080/00268976.2014.982736
      14
      Entropy of hard spheres in the close-packing limit
      Noya, Eva G.; Almarza, Noe G.
      Molecular Physics (2015), 113 (9-10), 1061-1068CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)
      The Helmholtz free energies of the face-centered cubic (FCC) and hcp. (HCP) hard-sphere solids in the close-packing limit have been evaluated using two different approaches based on the Einstein crystal method. Different system sizes and orientations of the crystal with respect to the simulation box have been investigated, both methods giving free energies that are consistent within statistical uncertainty. Our results show that for a given orientation of the crystal and system size, the FCC crystal is always slightly more stable than the HCP, the free-energy difference remaining practically const. with the no. of particles up to the thermodn. limit. In agreement with previous calcns., it is found that the free-energy difference between the HCP and FCC crystals at close packing in the thermodn. limit is 0.001 164(8) NkBT.
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    15. 15
      Dux, C.; Versmold, H. Light diffraction from shear ordered colloidal dispersions. Phys. Rev. Lett. 1997, 78, 1811,  DOI: 10.1103/PhysRevLett.78.1811
      15
      Light diffraction from shear ordered colloidal dispersions
      Dux, Christian; Versmold, Heiner
      Physical Review Letters (1997), 78 (9), 1811-1814CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Light diffraction from shear ordered colloidal dispersions is discussed in terms of the scattering power distribution I(l) along Bragg rods of hexagonal layers. For a charge stabilized dispersion the angle dependence of the light scattering intensity was used to det. I(l), from which conclusions on the mutual registration of the layers, the stacking order, and the kinetics of crystn. can be drawn. For the system under study a structural transition from random close-packed hexagonal layers to faulted twinned fcc. is identified.
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    16. 16
      Cheng, Z.; Zhu, J.; Russel, W. B.; Meyer, W. V.; Chaikin, P. M. Colloidal hard-sphere crystallization kinetics in microgravity and normal gravity. Appl. Opt. 2001, 40, 4146– 4151,  DOI: 10.1364/AO.40.004146
      16
      Colloidal hard-sphere crystallization kinetics in microgravity and normal gravity
      Cheng, Zhengdong; Zhu, Jixiang; Russel, William B.; Meyer, William V.; Chaikin, Paul M.
      Applied Optics (2001), 40 (24), 4146-4151CODEN: APOPAI; ISSN:0003-6935. (Optical Society of America)
      The hard-sphere disorder-order transition serves as the paradigm for crystn. The authors used time-resolved Bragg light scattering from the close-packed planes to measure the kinetics of nucleation and growth of colloidal hard-sphere crystals. The effects of gravity are revealed by comparison of the expts. in microgravity and normal gravity. Crystallites grow faster and larger in microgravity, and the coarsening between crystallites is suppressed by gravity. The fcc. structure was strongly indicated as being the stable structure for hard-sphere crystals. For a sample with a vol. fraction of 0.552, the classic nucleation and growth picture is followed.
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    17. 17
      Sandomirski, K.; Allahyarov, H.; Loewen, E.; Egelhaaf, S. U. Heterogeneous crystallization of hard-sphere colloids near a wall. Soft Matter 2011, 7, 8050,  DOI: 10.1039/c1sm05346a
      17
      Heterogeneous crystallization of hard-sphere colloids near a wall
      Sandomirski, Kirill; Allahyarov, Elshad; Loewen, Hartmut; Egelhaaf, Stefan U.
      Soft Matter (2011), 7 (18), 8050-8055CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      We investigate the most basic situation of heterogeneous crystn.: crystn. of hard-sphere colloids in the presence of a flat hard wall. Using a combination of confocal microscopy and nonequil. Brownian dynamics simulations, microscopic time-resolved information is obtained on an individual-particle level. Initially, particles near the wall rearrange before an extended regime of crystal growth is found. During growth, we can directly observe a depletion zone in the fluid next to the progressing crystal-fluid interface due to the single-particle information provided by microscopy and simulations. This also allows us to follow the relaxation of the crystal layers and the progression of the crystal-fluid interface. In good agreement between our expts. and simulations, as well as previous studies, the growth rate shows a max. in its dependence on the bulk vol. fraction.
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    18. 18
      Luchnikov, V.; Gervois, A.; Richard, P.; Oger, L.; Troadec, J. Crystallization of dense hard sphere packings: Competition of hcp and fcc close order. J. Mol. Liq. 2002, 96, 185– 194,  DOI: 10.1016/S0167-7322(01)00346-4
      18
      Crystallization of dense hard sphere packings
      Luchnikov, V.; Gervois, A.; Richard, P.; Oger, L.; Troadec, J. P.
      Journal of Molecular Liquids (2002), 96-97 (), 185-194CODEN: JMLIDT; ISSN:0167-7322. (Elsevier Science S.A.)
      We investigate the kinetics of crystn. in large mol.-dynamics models of dense hard sphere packings (16000 spheres). A sensitive measure for cryst. order is the rotationally invariant Q6 coeff., first introduced by Steinhardt et al. for the neighbors of an atom. The propensity to crystallize depends on the packing fraction and on the way the sample is prepd. The final crystal is of the fcc type. However, the hcp symmetries exist in an intermediate step; practically, mixts. of hcp and fcc clusters may coexist for a long time. The appearance and propagation of the cryst. order is studied by means of the local Q6 parameter, constructed form the 12 atoms the closest to a given atom. For a given packing fraction C, the fcc peak in the distribution function of Q6 increases during the crystn. and keeps the same shape, independently of the state of the crystn. and of the way the sample was initially prepd. The size of fcc atom clusters is studied as a function of time.
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    19. 19
      O’malley, B.; Snook, I. Crystal nucleation in the hard sphere system. Phys. Rev. Lett. 2003, 90, 085702  DOI: 10.1103/PhysRevLett.90.085702
      19
      Crystal Nucleation in the Hard Sphere System
      O'Malley, Brendan; Snook, Ian
      Physical Review Letters (2003), 90 (8), 085702/1-085702/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The structure and growth of crystal nuclei that spontaneously form during computer simulations of the simplest nontrivial model of a liq., the hard sphere system, is described. Compact crystal nuclei form at densities within the coexistence region of the phase diagram. The nuclei possess a range of morphologies with a predominance of multiply twinned particles possessing in some cases a significant decahedral character. However the multiply twinned particles do not form from an initial decahedral core but appear to nucleate as blocks of a fcc. crystal partially bounded by stacking faults.
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    20. 20
      Filion, L.; Hermes, M.; Ni, R.; Dijkstra, M. Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques. J. Chem. Phys. 2010, 133, 244115,  DOI: 10.1063/1.3506838
      20
      Crystal nucleation of hard spheres using molecular dynamics, umbrella sampling, and forward flux sampling: A comparison of simulation techniques
      Filion, L.; Hermes, M.; Ni, R.; Dijkstra, M.
      Journal of Chemical Physics (2010), 133 (24), 244115/1-244115/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Over the last no. of years several simulation methods have been introduced to study rare events such as nucleation. In this paper we examine the crystal nucleation rate of hard spheres using three such numerical techniques: mol. dynamics, forward flux sampling, and a Bennett-Chandler-type theory where the nucleation barrier is detd. using umbrella sampling simulations. The resulting nucleation rates are compared with the exptl. rates of Harland and van Megen. When the rates are examd. in units of the long-time diffusion coeff., we find agreement between all the theor. predicted nucleation rates, however, the exptl. results display a markedly different behavior for low supersatn. Addnl., we examd. the precrit. nuclei arising in the mol. dynamics, forward flux sampling, and umbrella sampling simulations. The structure of the nuclei appears independent of the simulation method, and in all cases, the nuclei contains on av. significantly more face-centered-cubic ordered particles than hexagonal-close-packed ordered particles. (c) 2010 American Institute of Physics.
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    21. 21
      Russo, J.; Tanaka, H. The microscopic pathway to crystallization in supercooled liquids. Sci. Rep. 2012, 2, 1– 8,  DOI: 10.1038/srep00505
      There is no corresponding record for this reference.
    22. 22
      Leoni, F.; Russo, J. Nonclassical nucleation pathways in stacking-disordered crystals. Physical Review X 2021, 11, 031006  DOI: 10.1103/PhysRevX.11.031006
      22
      Nonclassical Nucleation Pathways in Stacking-Disordered Crystals
      Leoni, Fabio; Russo, John
      Physical Review X (2021), 11 (3), 031006CODEN: PRXHAE; ISSN:2160-3308. (American Physical Society)
      The nucleation of crystals from liq. melt is often characterized by a competition between different cryst. structures or polymorphs and can result in nuclei with heterogeneous compns. These mixed-phase nuclei can display nontrivial spatial arrangements, such as layered and onionlike structures, whose compn. varies according to the radial distance, and which so far have been explained on the basis of bulk and surface free-energy differences between the competing phases. Here we extend the generality of these nonclassical nucleation processes, showing that layered and onionlike structures can emerge solely based on structural fluctuations even in the absence of free-energy differences. We consider two examples of competing cryst. structures, hcp and fcc forming in hard spheres relevant for repulsive colloids and dense liqs., and the cubic and hexagonal diamond forming in water relevant also for other group 14 elements such as carbon and silicon. We introduce a novel structural order parameter that combined with a neural-network classification scheme allows us to study the properties of the growing nucleus from the early stages of nucleation. We find that small nuclei have distinct size fluctuations and compns. from the nuclei that emerge from the growth stage. The transition between these two regimes is characterized by the formation of onionlike structures, in which the compn. changes with the distance from the center of the nucleus, similar to what is seen in the two-step nucleation process.
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    23. 23
      Sanchez-Burgos, I.; Sanz, E.; Vega, C.; Espinosa, J. R. Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate. Phys. Chem. Chem. Phys. 2021, 23, 19611– 19626,  DOI: 10.1039/D1CP01784E
      23
      Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate
      Sanchez-Burgos, Ignacio; Sanz, Eduardo; Vega, Carlos; Espinosa, Jorge R.
      Physical Chemistry Chemical Physics (2021), 23 (35), 19611-19626CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      A detailed computational characterization of the polymorphic nucleation competition between the fcc. and the hcp. hard-sphere crystal phases is provided. By several state-of-the-art simulation techniques, the authors evaluate the melting pressure, chem. p.d., interfacial free energy and nucleation rate of these 2 polymorphs, and of a random stacking mixt. of both crystals. The results highlight that, despite the fact that both polymorphs have very similar stability, the interfacial free energy of the hcp. phase could be marginally higher than that of the fcc. solid, which in consequence, mildly decreases its propensity to nucleate from the liq. compared to the fcc. phase. The abundance of each polymorph in grown crystals was analyzed from different types of inserted nuclei: fcc., hcp. and stacking disordered fcc./hcp. seeds, as well as from those spontaneously emerged from brute force simulations. Post-crit. crystals fundamentally grow maintaining the polymorphic structure of the crit. nucleus, at least until moderately large sizes, since the only crystallog. orientation that allows stacking close-packed disorder is the fcc. (111) plane, or equivalently the hcp. (0001) one. Taken together, the results contribute with 1 more piece to the intricate puzzle of colloidal hard-sphere crystn.
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    24. 24
      Malins, A.; Williams, S. R.; Eggers, J.; Royall, C. P. Identification of Structure in Condensed Matter with the Topological Cluster Classification. J. Chem. Phys. 2013, 139, 234506,  DOI: 10.1063/1.4832897
      24
      Identification of structure in condensed matter with the topological cluster classification
      Malins, Alex; Williams, Stephen R.; Eggers, Jens; Royall, C. Patrick
      Journal of Chemical Physics (2013), 139 (23), 234506/1-234506/21CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We describe the topol. cluster classification (TCC) algorithm. The TCC detects local structures with bond topologies similar to isolated clusters which minimize the potential energy for a no. of monat. and binary simple liqs. with m ≤ 13 particles. We detail a modified Voronoi bond detection method that optimizes the cluster detection. The method to identify each cluster is outlined, and a test example of Lennard-Jones liq. and crystal phases is considered and critically examd. (c) 2013 American Institute of Physics.
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    25. 25
      Frank, F. C. Supercooling of liquids. Proc. R. Soc. Lond. A 1952, 215, 43– 46,  DOI: 10.1098/rspa.1952.0194
      25
      Supercooling of liquids
      Frank, F. C.
      Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences (1952), 215 (), 43-6CODEN: PRLAAZ; ISSN:1364-5021.
      A brief crit. review of recent observations on supercooling of metal liquids. Close-packing arrangements are discussed and nucleation is considered.
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    26. 26
      Taffs, J.; Royall, C. P. The role of fivefold symmetry in suppressing crystallization. Nat. Commun. 2016, 7, 1– 7,  DOI: 10.1038/ncomms13225
      There is no corresponding record for this reference.
    27. 27
      Karayiannis, N. C.; Malshe, R.; Kröger, M.; de Pablo, J. J.; Laso, M. Evolution of fivefold local symmetry during crystal nucleation and growth in dense hard-sphere packings. Soft Matter 2012, 8, 844– 858,  DOI: 10.1039/C1SM06540H
      27
      Evolution of fivefold local symmetry during crystal nucleation and growth in dense hard-sphere packings
      Karayiannis, Nikos Ch.; Malshe, Rohit; Kroeger, Martin; de Pablo, Juan J.; Laso, Manuel
      Soft Matter (2012), 8 (3), 844-858CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      Crystal nucleation and growth of monodisperse hard-spheres as a function of packing d. was studied by collision-driven mol. dynamics simulations. Short-range order as 5-fold local symmetry is identified and its dynamical and structural evolution is tracked as the originally amorphous assembly transits to the stable ordered phase. A cluster-based approach shows that hard-sphere configurations having initially a similar av. fraction of 5-fold and ordered sites can crystallize in completely different patterns both in terms of dynamics and morphol. At high vol. fractions crystn. is significantly delayed in assemblies where sites with 5-fold symmetry are abundant. Eventually, once the crystal phase is reached, 5-fold symmetry either diminishes or arranges in specific geometric patterns. Such defects are spatially strongly correlated with twinning planes at cryst. boundaries. A detailed anal. is provided on the structural characteristics of the established crystal morphologies.
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    28. 28
      Martelli, F.; Palmer, J. C. Signatures of sluggish dynamics and local structural ordering during ice nucleation. J. Chem. Phys. 2022, 156, 114502,  DOI: 10.1063/5.0083638
      28
      Signatures of sluggish dynamics and local structural ordering during ice nucleation
      Martelli, Fausto; Palmer, Jeremy C.
      Journal of Chemical Physics (2022), 156 (11), 114502CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We investigate the microscopic pathway of spontaneous crystn. in the ST2 model of water under deeply supercooled conditions via unbiased classical mol. dynamics simulations. After quenching below the liq.-liq. crit. point, the ST2 model spontaneously separates into low-d. liq. (LDL) and high-d. liq. phases, resp. The LDL phase, which is characterized by lower mol. mobility and enhanced structural order, fosters the formation of a sub-crit. ice nucleus that, after a stabilization time, develops into the crit. nucleus and grows. Polymorphic selection coincides with the development of the sub-crit. nucleus and favors the formation of cubic (Ic) over hexagonal (Ih) ice. We rationalize polymorphic selection in terms of geometric arguments based on differences in the symmetry of second neighbor shells of ice Ic and Ih, which are posited to favor formation of the former. The rapidly growing crit. nucleus absorbs both Ic and Ih crystallites dispersed in the liq. phase, a crystal with stacking faults. Our results are consistent with, and expand upon, recent observations of non-classical nucleation pathways in several systems. (c) 2022 American Institute of Physics.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38Xntlyjs7s%253D&md5=2b0ce09520b715702502808ebd599f91
    29. 29
      Torrie, G.; Valleau, J. Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling. J. Comput. Phys. 1977, 23, 187– 199,  DOI: 10.1016/0021-9991(77)90121-8
      There is no corresponding record for this reference.
    30. 30
      Auer, S.; Frenkel, D. Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature 2001, 409, 1020– 1023,  DOI: 10.1038/35059035
      30
      Prediction of absolute crystal-nucleation rate in hard-sphere colloids
      Auer, Stefan; Fenkel, Daan
      Nature (London, United Kingdom) (2001), 409 (6823), 1020-1023CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
      Crystal nucleation is a much-studied phenomenon, yet the rate at which it occurs remains difficult to predict. Small crystal nuclei form spontaneously in supersatd. solns., but unless their size exceeds a crit. value-the so-called crit. nucleus-they will re-dissolve rather than grow. It is this rate-limiting step that proved difficult to probe exptl. The crystal nucleation rate depends on Pcrit, the (very small) probability that a crit. nucleus form spontaneously, and on a kinetic factor (κ) that measures the rate at which crit. nuclei subsequently grow. Given the absence of a priori knowledge of either quantity, classical nucleation expts., with the unconstrained parameters adjusted to fit the observations. This approach yields no ;first principles; prediction of abs. nucleation rate. Here the authors approach the problem from a different angle, simulating the nucleation process in a suspension of hard colloidal spheres, to obtain quant. numerical predictions of the crystal nucleation rate. Large discrepancies between the compute nucleation rates and those deduced from expts. were found.: the orders of magnitude.
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    31. 31
      Lechner, W.; Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. J. Chem. Phys. 2008, 129, 114707,  DOI: 10.1063/1.2977970
      31
      Accurate determination of crystal structures based on averaged local bond order parameters
      Lechner, Wolfgang; Dellago, Christoph
      Journal of Chemical Physics (2008), 129 (11), 114707/1-114707/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Local bond order parameters based on spherical harmonics, also known as Steinhardt order parameters, are often used to det. crystal structures in mol. simulations. Here we propose a modification of this method in which the complex bond order vectors are averaged over the first neighbor shell of a given particle and the particle itself. As demonstrated using soft particle systems, this averaging procedure considerably improves the accuracy with which different crystal structures can be distinguished. (c) 2008 American Institute of Physics.
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    32. 32
      Anderson, J. A.; Glaser, J.; Glotzer, S. C. HOOMD-blue: A Python package for highperformance molecular dynamics and hard particle Monte Carlo simulations. Comput. Mater. Sci. 2020, 173, 109363,  DOI: 10.1016/j.commatsci.2019.109363
      32
      HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations
      Anderson, Joshua A.; Glaser, Jens; Glotzer, Sharon C.
      Computational Materials Science (2020), 173 (), 109363CODEN: CMMSEM; ISSN:0927-0256. (Elsevier B.V.)
      HOOMD-blue is a particle simulation engine designed for nano- and colloidal-scale mol. dynamics and hard particle Monte Carlo simulations. It has been actively developed since March 2007 and available open source since August 2008. HOOMD-blue is a Python package with a high performance C++/CUDA backend that we built from the ground up for GPU acceleration. The Python interface allows users to combine HOOMD-blue with other packages in the Python ecosystem to create simulation and anal. workflows. We employ software engineering practices to develop, test, maintain, and expand the code.
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    33. 33
      Anderson, J. A.; Eric Irrgang, M.; Glotzer, S. C. Scalable Metropolis Monte Carlo for simulation of hard shapes. Comput. Phys. Commun. 2016, 204, 21– 30,  DOI: 10.1016/j.cpc.2016.02.024
      33
      Scalable Metropolis Monte Carlo for simulation of hard shapes
      Anderson, Joshua A.; Eric Irrgang, M.; Glotzer, Sharon C.
      Computer Physics Communications (2016), 204 (), 21-30CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
      We design and implement a scalable hard particle Monte Carlo simulation toolkit (HPMC), and release it open source as part of HOOMD-blue. HPMC runs in parallel on many CPUs and many GPUs using domain decompn. We employ BVH trees instead of cell lists on the CPU for fast performance, esp. with large particle size disparity, and optimize inner loops with SIMD vector intrinsics on the CPU. Our GPU kernel proposes many trial moves in parallel on a checkerboard and uses a block-level queue to redistribute work among threads and avoid divergence. HPMC supports a wide variety of shape classes, including spheres/disks, unions of spheres, convex polygons, convex spheropolygons, concave polygons, ellipsoids/ellipses, convex polyhedra, convex spheropolyhedra, spheres cut by planes, and concave polyhedra. NVT and NPT ensembles can be run in 2D or 3D triclinic boxes. Addnl. integration schemes permit Frenkel-Ladd free energy computations and implicit depletant simulations. In a benchmark system of a fluid of 4096 pentagons, HPMC performs 10 million sweeps in 10 min on 96 CPU cores on XSEDE Comet. The same simulation would take 7.6 h in serial. HPMC also scales to large system sizes, and the same benchmark with 16.8 million particles runs in 1.4 h on 2048 GPUs on OLCF Titan.
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    34. 34
      Kumar, S.; Rosenberg, J. M.; Bouzida, D.; Swendsen, R. H.; Kollman, P. A. The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J. Comput. Chem. 1992, 13, 1011– 1021,  DOI: 10.1002/jcc.540130812
      34
      The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method
      Kumar, Shankar; Bouzida, Djamal; Swendsen, Robert H.; Kollman, Peter A.; Rosenberg, John M.
      Journal of Computational Chemistry (1992), 13 (8), 1011-21CODEN: JCCHDD; ISSN:0192-8651.
      The Weighted Histogram Anal. Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomol. Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calcns. This algorithm possesses the following advantages over methods that are currently employed: (1) it provides a built-in est. of sampling errors thereby yielding objective ests. of the optimal location and length of addnl. simulations needed to achieve a desired level of precision; (2) it yields the "best" value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addn. to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better ests. of the free-energy differences. By recasting the Ferrenberg-Swendsen Multiple Histogram equations in a form suitable for mol. mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine.
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    35. 35
      Grossfield, A. WHAM: the weighted histogram analysis method. http://membrane.urmc.rochester.edu/wordpress/?page_id=126 (accessed 2021-05-10).
      There is no corresponding record for this reference.
    36. 36
      Martyna, G. J.; Tobias, D. J.; Klein, M. L. Constant Pressure Molecular Dynamics Algorithms. J. Chem. Phys. 1994, 101, 4177– 4189,  DOI: 10.1063/1.467468
      36
      Constant pressure molecular dynamics algorithms
      Martyna, Glenn J.; Tobias, Douglas J.; Klein, Michael L.
      Journal of Chemical Physics (1994), 101 (5), 4177-89CODEN: JCPSA6; ISSN:0021-9606.
      Modularly invariant equations of motion are derived that generate the isothermal-isobaric ensemble as their phase space avs. Isotropic vol. fluctuations and fully flexible simulation cells as well as a hybrid scheme that naturally combines the two motions are considered. The resulting methods are tested on two problems, a particle in a one-dimensional periodic potential and a spherical model of C60 in the solid/fluid phase.
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    37. 37
      Weeks, J. D.; Chandler, D.; Andersen, H. C. Role of Repulsive Forces in Determining the Equilibrium Structure of Simple Liquids. J. Chem. Phys. 1971, 54, 5237– 5247,  DOI: 10.1063/1.1674820
      37
      Role of repulsive forces in determining the equilibrium structure of simple liquids
      Weeks, John D.; Chandler, David; Andersen, Hans C.
      Journal of Chemical Physics (1971), 54 (12), 5237-47CODEN: JCPSA6; ISSN:0021-9606.
      The different roles the attractive and repulsive forces play in forming the equil. structure of a Lennard-Jones liq. are discussed. The effects of these forces are most easily sepd. by considering the structure factor (or equivalently, the Fourier transform of the pair-correlation function) rather than the pair-correlation function itself. At intermediate and large wave vectors, the repulsive forces dominate the quant. behavior of the liq. structure factor. The attractions are manifested primarily in the small wave vector part of the structure factor; but, this effect decreases as the d. increases and is almost negligible at reduced ds. >0.65. These conclusions are established by considering the structure factor of a hypothetical ref. system in which the intermol. forces are entirely repulsive and identical to the repulsive forces in a Lennard-Jones fluid. This ref. system structure factor is calcd. with the aid of a simple but accurate approxn. described. The conclusions lead to a very simple prescription for calcg. the radial distribution function of dense liqs. which is more accurate than that obtained by any previously reported theory. The thermodynamic ramifications of the conclusions are presented in the form of calcns. of the free energy, the internal energy (from the energy equation), and the pressure (from the virial equation). The implications of the authors conclusions to perturbation theories for liqs. and to the interpretation of x-ray scattering expts. are discussed.
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    38. 38
      Kawasaki, T.; Tanaka, H. Formation of a Crystal Nucleus from Liquid. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 14036– 14041,  DOI: 10.1073/pnas.1001040107
      38
      Formation of a crystal nucleus from liquid
      Kawasaki, Takeshi; Tanaka, Hajime
      Proceedings of the National Academy of Sciences of the United States of America (2010), 107 (32), 14036-14041, S14036/1-S14036/6CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      Crystn. is one of the most fundamental nonequil. phenomena universal to a variety of materials. It has so far been assumed that a supercooled liq. is in a "homogeneous disordered state" before crystn. Contrary to this common belief, we reveal that supercooled colloidal liq. is actually not homogeneous, but has transient medium-range structural order. We find that nucleation preferentially takes place in regions of high structural order via wetting effects, which reduce the crystal-liq. interfacial energy significantly and thus promotes crystal nucleation. This novel scenario provides a clue to solving a long-standing mystery concerning a large discrepancy between the rigorous numerical estn. of the nucleation rate on the basis of the classical nucleation theory and the exptl. obsd. ones. Our finding may shed light not only on the mechanism of crystal nucleation, but also on the fundamental nature of a supercooled liq. state.
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    39. 39
      Filion, L.; Ni, R.; Frenkel, D.; Dijkstra, M. Simulation of Nucleation in Almost Hard- Sphere Colloids: The Discrepancy between Experiment and Simulation Persists. J. Chem. Phys. 2011, 134, 134901,  DOI: 10.1063/1.3572059
      39
      Simulation of nucleation in almost hard-sphere colloids. The discrepancy between experiment and simulation persists
      Filion, L.; Ni, R.; Frenkel, D.; Dijkstra, M.
      Journal of Chemical Physics (2011), 134 (13), 134901/1-134901/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      In this paper we examine the phase behavior of the Weeks-Chandler-Andersen (WCA) potential with βε = 40. Crystal nucleation in this model system was recently studied by , who argued that the computed nucleation rates agree well with expt., a finding that contradicted earlier simulation results. Here we report an extensive numerical study of crystn. in the WCA model, using 3 totally different techniques (Brownian dynamics, umbrella sampling, and forward flux sampling). All simulations yield essentially the same nucleation rates. However, these rates differ significantly from the values reported by Kawasaki and Tanaka and hence we argue that the huge discrepancy in nucleation rates between simulation and expt. persists. When we map the WCA model onto a hard-sphere system, we find good agreement between the present simulation results and those that had been obtained for hard spheres ; . (c) 2011 American Institute of Physics.
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    40. 40
      Richard, D.; Speck, T. Crystallization of Hard Spheres Revisited. I. Extracting Kinetics and Free Energy Landscape from Forward Flux Sampling. J. Chem. Phys. 2018, 148, 124110,  DOI: 10.1063/1.5016277
      40
      Crystallization of hard spheres revisited. I. Extracting kinetics and free energy landscape from forward flux sampling
      Richard, David; Speck, Thomas
      Journal of Chemical Physics (2018), 148 (12), 124110/1-124110/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We investigate the kinetics and the free energy landscape of the crystn. of hard spheres from a supersatd. metastable liq. though direct simulations and forward flux sampling. In this first paper, we describe and test two different ways to reconstruct the free energy barriers from the sampled steady state probability distribution of cluster sizes without sampling the equil. distribution. The first method is based on mean first passage times, and the second method is based on splitting probabilities. We verify both methods for a single particle moving in a double-well potential. For the nucleation of hard spheres, these methods allow us to probe a wide range of supersaturations and to reconstruct the kinetics and the free energy landscape from the same simulation. Results are consistent with the scaling predicted by classical nucleation theory although a quant. fit requires a rather large effective interfacial tension. (c) 2018 American Institute of Physics.
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    41. 41
      Richard, D.; Speck, T. Crystallization of Hard Spheres Revisited. II. Thermodynamic Modeling, Nucleation Work, and the Surface of Tension. J. Chem. Phys. 2018, 148, 224102,  DOI: 10.1063/1.5025394
      41
      Crystallization of hard spheres revisited. II. Thermodynamic modeling, nucleation work, and the surface of tension
      Richard, David; Speck, Thomas
      Journal of Chemical Physics (2018), 148 (22), 224102/1-224102/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Combining three numerical methods (forward flux sampling, seeding of droplets, and finite-size droplets), we probe the crystn. of hard spheres over the full range from close to coexistence to the spinodal regime. We show that all three methods allow us to sample different regimes and agree perfectly in the ranges where they overlap. By combining the nucleation work calcd. from forward flux sampling of small droplets and the nucleation theorem, we show how to compute the nucleation work spanning three orders of magnitude. Using a variation of the nucleation theorem, we show how to ext. the pressure difference between the solid droplet and ambient liq. Moreover, combining the nucleation work with the pressure difference allows us to calc. the interfacial tension of small droplets. Our results demonstrate that employing bulk quantities yields inaccurate results for the nucleation rate. (c) 2018 American Institute of Physics.
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    42. 42
      Jover, J.; Haslam, A. J.; Galindo, A.; Jackson, G.; Müller, E. A. Pseudo hard-sphere potential for use in continuous molecular-dynamics simulation of spherical and chain molecules. J. Chem. Phys. 2012, 137, 144505,  DOI: 10.1063/1.4754275
      42
      Pseudo hard-sphere potential for use in continuous molecular-dynamics simulation of spherical and chain molecules
      Jover, J.; Haslam, A. J.; Galindo, A.; Jackson, G.; Mueller, E. A.
      Journal of Chemical Physics (2012), 137 (14), 144505/1-144505/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present a continuous pseudo-hard-sphere potential based on a cut-and-shifted Mie (generalized Lennard-Jones) potential with exponents (50, 49). Using this potential one can mimic the volumetric, structural, and dynamic properties of the discontinuous hard-sphere potential over the whole fluid range. The continuous pseudo potential has the advantage that it may be incorporated directly into off-the-shelf mol.-dynamics code, allowing the user to capitalize on existing hardware and software advances. Simulation results for the compressibility factor of the fluid and solid phases of our pseudo hard spheres are presented and compared both to the Carnahan-Starling equation of state of the fluid and published data, the differences being indistinguishable within simulation uncertainty. The specific form of the potential is employed to simulate flexible chains formed from these pseudo hard spheres at contact (pearl-necklace model) for mc = 4, 5, 7, 8, 16, 20, 100, 201, and 500 monomer segments. The compressibility factor of the chains per unit of monomer, mc, approaches a limiting value at reasonably small values, mc < 50, as predicted by Wertheim's first order thermodn. perturbation theory. Simulation results are also presented for highly asym. mixts. of pseudo hard spheres, with diam. ratios of 3:1, 5:1, 20:1 over the whole compn. range. (c) 2012 American Institute of Physics.
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    43. 43
      Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys. 1995, 117, 1– 19,  DOI: 10.1006/jcph.1995.1039
      43
      Fast parallel algorithms for short-range molecular dynamics
      Plimpton, Steve
      Journal of Computational Physics (1995), 117 (1), 1-19CODEN: JCTPAH; ISSN:0021-9991.
      Three parallel algorithms for classical mol. dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-at. forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for mol. dynamics models which can be difficult to parallelize efficiently - those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a std. Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers - the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C90 processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex mol. dynamics simulations are also discussed.
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    44. 44
      Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool. Modell. Simul. Mater. Sci. Eng. 2010, 18, 015012  DOI: 10.1088/0965-0393/18/1/015012
      There is no corresponding record for this reference.
    45. 45
      Taffs, J.; Williams, S. R.; Tanaka, H.; Royall, C. P. Structure and kinetics in the freezing of nearly hard spheres. Soft Matter 2013, 9, 297– 305,  DOI: 10.1039/C2SM26473K
      45
      Structure and kinetics in the freezing of nearly hard spheres
      Taffs, Jade; Williams, Stephen R.; Tanaka, Hajime; Royall, C. Patrick
      Soft Matter (2013), 9 (1), 297-305CODEN: SMOABF; ISSN:1744-683X. (Royal Society of Chemistry)
      We consider homogeneous crystn. rates in confocal microscopy expts. on colloidal nearly hard spheres at the single particle level. These we compare with Brownian dynamics simulations by carefully modeling the softness in the colloid interactions with a Yukawa potential, which takes account of the electrostatic charges present in the exptl. system. Both structure and dynamics of the colloidal fluid are very well matched between expt. and simulation, so we have confidence that the system simulated is close to that in the expt. In the regimes we can access, we find reasonable agreement in crystn. rates between expt. and simulations, noting that the larger system size in expts. enables the formation of crit. nuclei and hence crystn. at lower supersaturations than in the simulations. We further examine the metastable fluid with a novel structural anal., the topol. cluster classification. We find that at densities where the hard sphere fluid becomes metastable, the dominant structure is a cluster of m = 10 particles with five-fold symmetry. Analyzing histories of the local environment of single particles, we find fluctuations into cryst. configurations in the metastable fluid, and that the cryst. state a very often preceded by a transition region of frequent hopping between crystal-like environments and other (m ≠ 10) structures.
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    • Gibbs free energy for nucleation in the nfcc–nhcp plane; robustness with respect to polymorph detection method; robustness with respect to dynamics; conversions between PB, SD, fcc, and hcp clusters; behavior of all clusters during nucleation and near planar solid–fluid interfaces; quantitative comparison between experiments and simulations (PDF).


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