Experimental evidence for two-step nucleation in colloidal crystallization

The impact of hydration shell inclusion and chain exclusion in the efficacy of reaction coordinates for homogeneous and heterogeneous ice nucleation

Ice nucleation plays a pivotal role in many natural and industrial processes, and molecular simulations have proven vital in uncovering its kinetics and mechanisms. A fundamental component of such simulations is the choice of an order parameter (OP) that quantifies the progress of nucleation, with the efficacy of an OP typically measured by its ability to predict the committor probabilities. Here, we leverage a machine learning framework introduced in our earlier work [Domingues et al., J. Phys. Chem. Lett. 15, 1279, (2024)] to systematically investigate how key implementation details influence the efficacy of standard Steinhardt OPs in capturing the progress of both homogeneous and heterogeneous ice nucleation. Our analysis identifies distance and q6 cutoffs as the primary determinants of OP performance, regardless of the mode of nucleation. We also examine the impact of two popular refinement strategies, namely chain exclusion and hydration shell inclusion, on OP efficacy. We find neither strategy to exhibit a universally consistent impact. Instead, their efficacy depends strongly on the chosen distance and q6 cutoffs. Chain exclusion enhances OP efficacy when the underlying OP lacks sufficient selectivity, whereas hydration shell inclusion is beneficial for overly selective OPs. Consequently, we demonstrate that selecting optimal combinations of such cutoffs can eliminate the need for these refinement strategies altogether. These findings provide a systematic understanding of how to design and optimize OPs for accurately describing complex nucleation phenomena, offering valuable guidance for improving the predictive power of molecular simulations.

Evolution of a particulate ensemble with fluctuations in particle growth rates

The evolution of an ensemble of spherical crystals in a supersaturated solution is considered with allowance for fluctuations in crystal growth rates and initial crystal-size distribution. Two approaches for constructing the analytical solutions based on the saddle-point method and separation of radial and time functions are developed. Both methods yield similar desupersaturation dynamics, which agree well with the experimental data. The first method gives a crystal-size distribution function decaying with time, consistent with a decrease in solution supersaturation. The second method results in a distribution function with opposite dynamics and works only in the case of exponentially decaying initial crystal-size distribution. Therefore, the first method can be used to determine any characteristics of an ensemble of crystals based on calculating the moments of the distribution function. The use of the second, mathematically simpler method, is suitable only for describing the kinetics of supersaturation removal.

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

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

Growing a single suspended perfect protein crystal in a fully noncontact manner

Nucleation is a fundamental process that determines the structure, morphology, and properties of crystalline materials, and is difficult to control because it is unpredictable. Here, we demonstrate a new method to control the protein crystal nucleation using a magnetic force, where we manipulate the movement and coalescence of nucleation precursors by adding paramagnetic salt into the crystallization solution to constrain the number and position of nucleation. We found that protein nucleation could be significantly affected by the magnetic force that the gradient magnetic fields generate. When the magnetization force is sufficiently enough, nucleation can be confined to the crystallization solution with no interface contact; therefore, only one crystal nucleus appears, which results in noncontact suspension growth of a single crystal in the crystallization solution system. Under these situations, the nucleation rate significantly decreases due to the coalescence of the dense liquid phase, and the crystal growth rate also decreases due to the suppression of convection, which increases the crystal quality. Our findings provide a new method for the noncontact control of crystal nucleation and demonstrate that externally applied physical environments can be used to affect the liquid-liquid phase separation process.

2D capsid formation within an oscillatory energy landscape: orderly self-assembly depends on the interplay between a dynamic potential and intrinsic relaxation times

Multiple dissipative self-assembly protocols designed to create novel structures or to reduce kinetic traps have recently emerged. Specifically, temporal oscillations of particle interactions have been shown effective at both aims, but investigations thus far have focused on systems of simple colloids or their binary mixtures. In this work, we expand our understanding of the effect of temporally oscillating interactions to a two-dimensional coarse-grained viral capsid-like model that undergoes a self-limited assembly. This model includes multiple intrinsic relaxation times due to the internal structure of the capsid subunits and, under certain interaction regimes, proceeds via a two-step nucleation mechanism. We find that oscillations much faster than the local intrinsic relaxation times can be described via a time averaged inter-particle potential across a wide range of interaction strengths, while oscillations much slower than these relaxation times result in structures that adapt to the attraction strength of the current half-cycle. Interestingly, oscillation periods similar to these relaxation times shift the interaction window over which orderly assembly occurs by enabling error correction during the half-cycles with weaker attractions. Our results provide fundamental insights to non-equilibrium self-assembly on temporally variant energy landscapes.

Self-Assembly of Model Three- and Four-Patch Colloidal Particles in Two Dimensions

A coarse-grained effective solvent model of two-patch particles is extended to study the self-assembly of three- and four-patch particles to two-dimensional honeycomb and square lattices, respectively. Employing this model, grand canonical ensemble simulations are done to calculate vapor-liquid equilibria and the critical temperatures for patchy particles of various patch widths. The range of stability of the liquid, although very limited compared to isotropic particles, which interact through a longer-range potential, depends on the patch width and on the number of patches. Biased sampling and unbiased simulations are also done to investigate the mechanism of nucleation and crystal growth for honeycomb and square lattices, self-assembled from three- and four-patch particles, respectively. A two-step mechanism governs the nucleation of both lattices. In the first step, the particles form a dense amorphous network, and in the second step, the particles inside the amorphous network reorient to form crystalline nuclei. Barrier heights for the nucleation of honeycomb and square lattices are 7.8 kBT and 7.4 kBT, which are close to the reported values for the nucleation of the kagome lattice. In agreement with confocal microscopy experiments, the self-assembly in a honeycomb lattice involves the formation of 5- to 7-membered rings. The 5- and 7-membered rings hamper the nucleation of the honeycomb lattice, through defect formation and rotation of the symmetry planes of crystals that form at their surfaces. With the progress of self-assembly, a substantial amount of restructuring of the defects and crystals in their vicinity is needed to heal the defects.

Local Structural Features Elucidate Crystallization of Complex Structures

Complex crystal structures are composed of multiple local environments, and how this type of order emerges spontaneously during crystal growth has yet to be fully understood. We study crystal growth across various structures and along different crystallization pathways, using self-assembly simulations of identical particles that interact via multiwell isotropic pair potentials. We apply an unsupervised machine learning method to features from bond-orientational order metrics to identify different local motifs present during a given structure's crystallization process. In this manner, we distinguish different crystallographic sites in highly complex structures. Tailoring this order parameter to structures of varying complexity and coordination number, we study the emergence of local order along a multistep crystal growth pathway─from a low-density fluid to a high-density, supercooled amorphous liquid droplet and to a bulk crystal. We find a consistent under-coordination of the liquid relative to the average coordination number in the bulk crystal. We use our order parameter to analyze the geometrically frustrated growth of a Frank-Kasper phase and discover how structural defects compete with the formation of crystallographic sites that are more high-coordinated than the liquid environments. The method presented here for classifying order on a particle-by-particle level has broad applicability to future studies of structural self-assembly and crystal growth, and they can aid in the design of building blocks and for targeting pathways of formation of soft-matter structures.

In Situ X-ray Scattering Reveals Coarsening Rates of Superlattices Self-Assembled from Electrostatically Stabilized Metal Nanocrystals Depend Nonmonotonically on Driving Force

Self-assembly of colloidal nanocrystals (NCs) into superlattices (SLs) is an appealing strategy to design hierarchically organized materials with promising functionalities. Mechanistic studies are still needed to uncover the design principles for SL self-assembly, but such studies have been difficult to perform due to the fast time and short length scales of NC systems. To address this challenge, we developed an apparatus to directly measure the evolving phases in situ and in real time of an electrostatically stabilized Au NC solution before, during, and after it is quenched to form SLs using small-angle X-ray scattering. By developing a quantitative model, we fit the time-dependent scattering patterns to obtain the phase diagram of the system and the kinetics of the colloidal and SL phases as a function of varying quench conditions. The extracted phase diagram is consistent with particles whose interactions are short in range relative to their diameter. We find the degree of SL order is primarily determined by fast (subsecond) initial nucleation and growth kinetics, while coarsening at later times depends nonmonotonically on the driving force for self-assembly. We validate these results by direct comparison with simulations and use them to suggest dynamic design principles to optimize the crystallinity within a finite time window. The combination of this measurement methodology, quantitative analysis, and simulation should be generalizable to elucidate and better control the microscopic self-assembly pathways of a wide range of bottom-up assembled systems and architectures.

Multistate Dynamic Pathways for Anisotropic Colloidal Assembly and Reconfiguration

We report the controlled interfacial assembly and reconfiguration of rectangular prism colloidal particles between microstructures of varying positional and orientational order including stable, metastable, and transient states. Structurally diverse states are realized by programming time dependent electric fields that mediate dipolar interactions determining particle position, orientation, compression, and chaining. We identify an order parameter set that defines each state as a combination of the positional and orientational order. These metrics are employed as reaction coordinates to capture the microstructure evolution between initial and final states upon field changes. Assembly trajectory manifolds between states in the low-dimensional reaction coordinate space reveal a dynamic pathway map including information about pathway accessibility, reversibility, and kinetics. By navigating the dynamic pathway map, we demonstrate reconfiguration between states on minute time scales, which is practically useful for particle-based materials processing and device responses. Our findings demonstrate a conceptually general approach to discover dynamic pathways as a basis to control assembly and reconfiguration of self-organizing building blocks that respond to global external stimuli.

Optothermal crystallization of hard spheres in an effective bidimensional geometry

Using colloids effectively confined in two dimensions by a cell with a thickness comparable to the particle size, we investigate the nucleation and growth of crystallites induced by locally heating the solvent with a near-infrared laser beam. The particles, which are "thermophilic," move towards the laser spot solely because of thermophoresis with no convection effects, forming dense clusters whose structure is monitored using two order parameters that gauge the local density and the orientational ordering. We find that ordering takes place when the cluster reaches an average surface density that is still below the upper equilibrium limit for the fluid phase of hard disks, meaning that we do not detect any sign of a proper "two-stage" nucleation from a glass or a polymorphic crystal structure. The crystal obtained at late growth stage displays a remarkable uniformity with a negligible amount of defects, arguably because the incoming particles diffuse, bounce, and displace other particles before settling at the crystal interface. This "fluidization" of the outer crystal edge may resemble the surface enhanced mobility giving rise to ultra-stable glasses by physical vapor deposition.

Controlling polymorph selection during nucleation by tuning the structure of metallic melts

Controlling the polymorphism of crystals is crucial to the design of novel metallic materials with specific properties; however, the atomistic mechanism underlying polymorph selection during crystallization remains unclear. In this work, molecular dynamics simulations combined with well-tempered metadynamics simulations are employed to explore the atomic mechanisms of polymorph selection during the nucleation process of FCC aluminum and copper. Simulation results suggest that the distinct nucleation pathways of both FCC metals originate from different free-energy surfaces of nucleation processes and diverse symmetries of nucleation precursors. The initially forming phase from undercooled melts is most likely to be the one that has the symmetry closest to the precursors. Besides, tiny seeds with diverse crystal symmetries could induce the formation of preordered precursors for nucleation around the seed, leading to the reduction of free-energy barrier and thus the promotion of nucleation. Controlling polymorph selection with tiny seeds is realized by tuning the symmetry of precursors. Our findings not only shed significant light on understanding polymorph selection, but also provide theoretical guidance for better controlling the nucleation pathway in practice.

Controlling crystallization: what liquid structure and dynamics reveal about crystal nucleation mechanisms

Over recent years, molecular simulations have provided invaluable insights into the microscopic processes governing the initial stages of crystal nucleation and growth. A key aspect that has been observed in many different systems is the formation of precursors in the supercooled liquid that precedes the emergence of crystalline nuclei. The structural and dynamical properties of these precursors determine to a large extent the nucleation probability as well as the formation of specific polymorphs. This novel microscopic view on nucleation mechanisms has further implications for our understanding of the nucleating ability and polymorph selectivity of nucleating agents, as these appear to be strongly linked to their ability in modifying structural and dynamical characteristics of the supercooled liquid, namely liquid heterogeneity. In this perspective, we highlight recent progress in exploring the connection between liquid heterogeneity and crystallization, including the effects of templates, and the potential impact for controlling crystallization processes. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Depletion-driven antiferromagnetic, paramagnetic, and ferromagnetic behavior in quasi-two-dimensional buckled colloidal solids

We investigate quasi-two-dimensional buckled colloidal monolayers on a triangular lattice with tunable depletion interactions. Without depletion attraction, the experimental system provides a colloidal analog of the well-known geometrically frustrated Ising antiferromagnet [Y. Han et al., Nature 456, 898-903 (2008)]. In this contribution, we show that the added depletion attraction can influence both the magnitude and sign of an Ising spin coupling constant. As a result, the nearest-neighbor Ising "spin" interactions can be made to vary from antiferromagnetic to para- and ferromagnetic. Using a simple theory, we compute an effective Ising nearest-neighbor coupling constant, and we show how competition between entropic effects permits for the modification of the coupling constant. We then experimentally demonstrate depletion-induced modification of the coupling constant, including its sign, and other behaviors. Depletion interactions are induced by rod-like surfactant micelles that change length with temperature and thus offer means for tuning the depletion attraction in situ. Buckled colloidal suspensions exhibit a crossover from an Ising antiferromagnetic to paramagnetic phase as a function of increasing depletion attraction. Additional dynamical experiments reveal structural arrest in various regimes of the coupling-constant, driven by different mechanisms. In total, this work introduces novel colloidal matter with "magnetic" features and complex dynamics rarely observed in traditional spin systems.

A variational approach to assess reaction coordinates for two-step crystallization

Molecule- and particle-based simulations provide the tools to test, in microscopic detail, the validity of classical nucleation theory. In this endeavor, determining nucleation mechanisms and rates for phase separation requires an appropriately defined reaction coordinate to describe the transformation of an out-of-equilibrium parent phase for which myriad options are available to the simulator. In this article, we describe the application of the variational approach to Markov processes to quantify the suitability of reaction coordinates to study crystallization from supersaturated colloid suspensions. Our analysis indicates that collective variables (CVs) that correlate with the number of particles in the condensed phase, the system potential energy, and approximate configurational entropy often feature as the most appropriate order parameters to quantitatively describe the crystallization process. We apply time-lagged independent component analysis to reduce high-dimensional reaction coordinates constructed from these CVs to build Markov State Models (MSMs), which indicate that two barriers separate a supersaturated fluid phase from crystals in the simulated environment. The MSMs provide consistent estimates for crystal nucleation rates, regardless of the dimensionality of the order parameter space adopted; however, the two-step mechanism is only consistently evident from spectral clustering of the MSMs in higher dimensions. As the method is general and easily transferable, the variational approach we adopt could provide a useful framework to study controls for crystal nucleation.

Direct Imaging of the Kinetic Crystallization Pathway: Simulation and Liquid-Phase Transmission Electron Microscopy Observations

The crystallization of materials from a suspension determines the structure and function of the final product, and numerous pieces of evidence have pointed out that the classical crystallization pathway may not capture the whole picture of the crystallization pathways. However, visualizing the initial nucleation and further growth of a crystal at the nanoscale has been challenging due to the difficulties of imaging individual atoms or nanoparticles during the crystallization process in solution. Recent progress in nanoscale microscopy had tackled this problem by monitoring the dynamic structural evolution of crystallization in a liquid environment. In this review, we summarized several crystallization pathways captured by the liquid-phase transmission electron microscopy technique and compared the observations with computer simulation. Apart from the classical nucleation pathway, we highlight three nonclassical pathways that are both observed in experiments and computer simulations: formation of an amorphous cluster below the critical nucleus size, nucleation of the crystalline phase from an amorphous intermediate, and transition between multiple crystalline structures before achieving the final product. Among these pathways, we also highlight the similarities and differences between the experimental results of the crystallization of single nanocrystals from atoms and the assembly of a colloidal superlattice from a large number of colloidal nanoparticles. By comparing the experimental results with computer simulations, we point out the importance of theory and simulation in developing a mechanistic approach to facilitate the understanding of the crystallization pathway in experimental systems. We also discuss the challenges and future perspectives for investigating the crystallization pathways at the nanoscale with the development of in situ nanoscale imaging techniques and potential applications to the understanding of biomineralization and protein self-assembly.

In Situ Kinetic Observations on Crystal Nucleation and Growth

Nucleation and growth are critical steps in crystallization, which plays an important role in determining crystal structure, size, morphology, and purity. Therefore, understanding the mechanisms of nucleation and growth is crucial to realize the controllable fabrication of crystalline products with desired and reproducible properties. Based on classical models, the initial crystal nucleus is formed by the spontaneous aggregation of ions, atoms, or molecules, and crystal growth is dependent on the monomer's diffusion and the surface reaction. Recently, numerous in situ investigations on crystallization dynamics have uncovered the existence of nonclassical mechanisms. This review provides a summary and highlights the in situ studies of crystal nucleation and growth, with a particular emphasis on the state-of-the-art research progress since the year 2016, and includes technological advances, atomic-scale observations, substrate- and temperature-dependent nucleation and growth, and the progress achieved in the various materials: metals, alloys, metallic compounds, colloids, and proteins. Finally, the forthcoming opportunities and challenges in this fascinating field are discussed.

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

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

Multistep Crystallization of Dynamic Nanoparticle Superlattices in Nonaqueous Solutions

Crystallization is a universal phenomenon underpinning many industrial and natural processes and is fundamental to chemistry and materials science. However, microscopic crystallization pathways of nanoparticle superlattices have been seldom studied mainly owing to the difficulty of real-time observation of individual self-assembling nanoparticles in solution. Here, using in situ electron microscopy, we directly image the full self-assembly pathway from dispersed nanoparticles into ordered superlattices in nonaqueous solution. We show that electron-beam irradiation controls nanoparticle mobility, and the solvent composition largely dictates interparticle interactions and assembly behaviors. We uncover a multistep crystallization pathway consisting of four distinct stages through multi-order-parameter analysis and visualize the formation, migration, and annihilation of multiple types of defects in nanoparticle superlattices. These findings open the door for achieving independent control over imaging conditions and nanoparticle assembly conditions and will enable further study of the microscopic kinetics of assembly and phase transition in nanocolloidal systems.

Revealing the role of liquid preordering in crystallisation of supercooled liquids

The recent discovery of non-classical crystal nucleation pathways has revealed the role of fluctuations in the liquid structural order, not considered in classical nucleation theory. On the other hand, classical crystal growth theory states that crystal growth is independent of interfacial energy, but this is questionable. Here we elucidate the role of liquid structural ordering in crystal nucleation and growth using computer simulations of supercooled liquids. We find that suppressing the crystal-like structural order in the supercooled liquid through a new order-killing strategy can reduce the crystallisation rate by several orders of magnitude. This indicates that crystal-like liquid preordering and the associated interfacial energy reduction play an essential role in nucleation and growth processes, forcing critical modifications of the classical crystal growth theory. Furthermore, we evaluate the importance of this additional factor for different types of liquids. These findings shed new light on the fundamental understanding of crystal growth kinetics.

Influence on crystal nucleation of an order-disorder transition among the subcritical clusters

Studies of nucleation generally focus on the properties of the critical cluster, but the presence of defects within the crystal lattice means that the population of nuclei necessarily evolve through a distribution of precritical clusters with varying degrees of structural disorder on their way to forming a growing stable crystal. To investigate the role precritical clusters play in nucleation, we develop a simple thermodynamic model for crystal nucleation in terms of cluster size and the degree of cluster order that allows us to alter the work of forming the precritical clusters without affecting the properties of the critical cluster. The steady state and transient nucleation behavior of the system are then studied numerically, for different microscopic ordering kinetics. We find that the model exhibits a generic order-disorder transition in the precritical clusters. Independent of the types of ordering kinetics, increasing the accessibility of disordered precritical clusters decreases both the steady state nucleation rate and the nucleation lag time. Furthermore, the interplay between the free-energy surface and the microscopic ordering kinetics leads to three distinct nucleation pathways.

Influence of anisotropy on heterogeneous nucleation of gold nanorod assemblies

In this study, we analysed for the first time heterogeneous nucleation with anisotropic nanoparticles as a model system for non-spherical building units on the nanoscale. Gold nanorods were synthesised and assembled to investigate the phenomenon of heterogeneous nucleation. To determine the influence of the particle shape on heterogeneous nucleation, we utilised gold nanorods with varying aspect ratios, ranging from 3.00 and 2.25 to 1.75, while keeping the surface chemistry constant. First, the nucleation of the gold nanorod assemblies in solution and the process kinetics were analyzed with UV-vis-NIR spectroscopy followed by a microscopic examination of the gold nanorod-based superstructures formed heterogeneously on substrates. Here, positively charged cetyltrimethylammonium bromide (CTAB)-functionalized gold nanorods and negatively charged polystyrene sulfonate (PSS) functionalized substrates ensured the directed heterogeneous nucleation on the substrates. A combination of light microscopy with simultaneous UV-vis-NIR spectroscopy allowed us to observe the gold nanorod-based superstructure formation on the substrates in situ and to determine the nucleation rates of the process. We analysed the resulting data with the classical nucleation theory, which revealed a dominating kinetic term and a negligible thermodynamic term in contrast to ionic systems like calcium carbonate. Our studies consistently exhibit an influence of the aspect ratio on the nucleation behaviour resulting in faster nucleation of superstructures as the aspect ratio decreases. Hence our studies show unprecedented insight into the influence of particle anisotropy on the nucleation and growth of nanorod-based superstructures and reveal significant differences in the nucleation of nanoparticle building units compared to the nucleation of atoms or molecules as building units.

Two-step nucleation in confined geometry: Phase diagram of finite particles on a lattice gas model

We use a degenerated Ising model to describe nucleation and crystallization from solution in a confined two-component system. The free energy is calculated using metadynamics simulation with coordination numbers as the reaction coordinates. We deploy nudged elastic band simulation to determine the minimum energy path and give properties of the crystallization path. In this confined system, depletion effects, which could also be caused by slow material transport in the solution, prevent the post-critical cluster from further growth, and the crystalline state would only be stable at larger cluster sizes. Fluctuation of the higher coupling strength of the crystalline state enables further growth until the crystalline cluster is in equilibrium with the solvent, and this way, a second barrier is crossed. From the parameters and setup, we find necessary conditions for the occurrence of two-step nucleation in our system. These findings can be adapted to real systems as biomineralization, colloidal crystallization, and the solidification of metals.

Exploring Nucleation Pathways in Distinct Physicochemical Environments Unveiling Novel Options to Modulate and Optimize Protein Crystallization

The scientific discussion about classical and nonclassical nucleation theories has lasted for two decades so far. Recently, multiple nucleation pathways and the occurrence and role of metastable intermediates in crystallization processes have attracted increasing attention, following the discovery of functional phase separation, which is now under investigation in different fields of cellular life sciences, providing interesting and novel aspects for conventional crystallization experiments. In this context, more systematic investigations need to be carried out to extend the current knowledge about nucleation processes. In terms of the data we present, a well-studied model protein, glucose isomerase (GI), was employed first to investigate systematically the early stages of the crystallization process, covering condensing and prenucleation ordering of protein molecules in diverse scenarios, including varying ionic and crowding agent conditions, as well as the application of a pulsed electric field (pEF). The main method used to characterize the early events of nucleation was synchronized polarized and depolarized dynamic light scattering (DLS/DDLS), which is capable of collecting the polarized and depolarized component of scattered light from a sample suspension in parallel, thus monitoring the time-resolved evolution of the condensation and geometrical ordering of proteins at the early stages of nucleation. A diffusion interaction parameter, KD, of GI under varying salt conditions was evaluated to discuss how the proportion of specific and non-specific protein–protein interactions affects the nucleation process. The effect of mesoscopic ordered clusters (MOCs) on protein crystallization was explored further by adding different ratios of MOCs induced by a pEF to fresh GI droplets in solution with different PEG concentrations. To emphasize and complement the data and results obtained with GI, a recombinant pyridoxal 5-phosphate (vitamin B6) synthase (Pdx) complex of Staphylococcus aureus assembled from twelve monomers of Pdx1 and twelve monomers of Pdx2 was employed to validate the ability of the pEF influencing the nucleation of complex macromolecules and the effect of MOCs on adjusting the crystallization pathway. In summary, our data revealed multiple nucleation pathways by tuning the proportion of specific and non-specific protein interactions, or by utilizing a pEF which turned out to be efficient to accelerate the nucleation process. Finally, a novel and reproducible experimental strategy, which can adjust and facilitate a crystallization process by pEF-induced MOCs, was summarized and reported for the first time.

Kinetics of Polymer Crystallization with Aggregating Small Crystallites

The isothermal crystallization near the glass transition temperature from the melt state of poly(trimethylene terephthalate) has been studied by wide-angle x-ray diffraction (WAXD), small-angle x-ray scattering (SAXS), and optical microscopy. The SAXS and WAXD results show the crystallization mechanism in which the crystalline nodules cover the entire sample with the formation of aggregation regions. The analysis of the SAXS results using Kolmogorov-Johnson-Mehl-Avrami theory indicates that the formation kinetics of the aggregation regions is of three-dimensional homogeneous nucleation type. The analysis of the SAXS profiles using Sekimoto's theory provides the growth velocity and the nucleation rate of the aggregation region. The temperature dependence of the growth velocity of the aggregation region is a natural extrapolation of that of spherulite to the high supercooling region. The temperature dependence of the nucleation rate of the aggregation region is also represented by the parameters of the spherulitic growth rate. The result of the growth velocities of the aggregation region and the spherulite suggests the existence of precursors at the front of the crystal growth.

Formation and stabilization mechanism of mesoscale clusters in solution

To understand the existence of complex meso-sized solute-rich clusters, which challenge the understanding of phases and phase equilibria, the formation and stabilization mechanisms of clusters in solution during nucleation of crystals and the associated physico-chemical rules are studied in detail. An essential part of the mechanism is the formation of long-lived oligomers between solute molecules. By means of density functional theory simulation and nuclear magnetic resonance experiments, this work showed that the oligomers in solution tend to be π-π stacking dimers. Clusters are formed under the combined effect of diffusion and monomer-dimer reaction. The physically meaningful quantities such as the monomer-dimer reaction rate constants and the diffusion coefficients of both species were obtained by reaction-diffusion kinetics and diffusion-ordered spectroscopy results. The evolution of cluster radius as a function of time, and the qualitative spatial distributions of monomer and dimer densities under steady-state were plotted to better understand the formation process and the nature of the clusters.

Nucleation of glucose isomerase protein crystals in a nonclassical disguise: The role of crystalline precursors

The ability of proteins to self-assemble into complex, hierarchical structures has been the inspiration for the bottom-up design of a class of biomaterials with proteins as their building blocks. The earliest stages of formation often involve the passing of an activation barrier under the form of nucleus formation, a quaternary protein complex that templates incoming molecules to proper registry. For protein crystallization, the consensus has emerged that the fastest route toward a nucleus follows a winding path: first, densification, followed by symmetry formation. In this contribution, we show that this need not be the case for the protein glucose isomerase, which seems to follow the simplest path to a nucleus, making crystalline clusters from the earliest detectable beginnings.

Protein crystallization is an astounding feat of nature. Even though proteins are large, anisotropic molecules with complex, heterogeneous surfaces, they can spontaneously group into two- and three-dimensional arrays with high precision. And yet, the biggest hurdle in this assembly process, the formation of a nucleus, is still poorly understood. In recent years, the two-step nucleation model has emerged as the consensus on the subject, but it still awaits extensive experimental verification. Here, we set out to reconstruct the nucleation pathway of the candidate protein glucose isomerase (GI), for which there have been indications that it may follow a two-step nucleation pathway under certain conditions. We find that the precursor phase present during the early stages of the reaction process is nanoscopic crystallites that have lattice symmetry equivalent to the mature crystals found at the end of a crystallization experiment. Our observations underscore the need for experimental data at a lattice-resolving resolution on other proteins so that a general picture of protein crystal nucleation can be formed.

Protein-polymer mixtures in the colloid limit: Aggregation, sedimentation, and crystallization

While proteins have been treated as particles with a spherically symmetric interaction, of course in reality, the situation is rather more complex. A simple step toward higher complexity is to treat the proteins as non-spherical particles and that is the approach we pursue here. We investigate the phase behavior of the enhanced green fluorescent protein (eGFP) under the addition of a non-adsorbing polymer, polyethylene glycol. From small angle x-ray scattering, we infer that the eGFP undergoes dimerization and we treat the dimers as spherocylinders with aspect ratio L/D - 1 = 1.05. Despite the complex nature of the proteins, we find that the phase behavior is similar to that of hard spherocylinders with an ideal polymer depletant, exhibiting aggregation and, in a small region of the phase diagram, crystallization. By comparing our measurements of the onset of aggregation with predictions for hard colloids and ideal polymers [S. V. Savenko and M. Dijkstra, J. Chem. Phys. 124, 234902 (2006) and Lo Verso et al., Phys. Rev. E 73, 061407 (2006)], we find good agreement, which suggests that the behavior of the eGFP is consistent with that of hard spherocylinders and ideal polymers.

Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate

Hard-sphere crystallization has been widely investigated over the last six decades by means of colloidal suspensions and numerical methods. However, some aspects of its nucleation behaviour are still under debate. Here, we provide a detailed computational characterisation of the polymorphic nucleation competition between the face-centered cubic (fcc) and the hexagonal-close packed (hcp) hard-sphere crystal phases. By means of several state-of-the-art simulation techniques, we evaluate the melting pressure, chemical potential difference, interfacial free energy and nucleation rate of these two polymorphs, as well as of a random stacking mixture of both crystals. Our 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 liquid compared to the fcc phase. Moreover, we analyse the abundance of each polymorph in grown crystals 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. We find that post-critical crystals fundamentally grow maintaining the polymorphic structure of the critical nucleus, at least until moderately large sizes, since the only crystallographic orientation that allows stacking close-packed disorder is the fcc (111) plane, or equivalently the hcp (0001) one. Taken together, our results contribute with one more piece to the intricate puzzle of colloidal hard-sphere crystallization.

Real space analysis of colloidal gels: triumphs, challenges and future directions

Colloidal gels constitute an important class of materials found in many contexts and with a wide range of applications. Yet as matter far from equilibrium, gels exhibit a variety of time-dependent behaviours, which can be perplexing, such as an increase in strength prior to catastrophic failure. Remarkably, such complex phenomena are faithfully captured by an extremely simple model-'sticky spheres'. Here we review progress in our understanding of colloidal gels made through the use of real space analysis and particle resolved studies. We consider the challenges of obtaining a suitable experimental system where the refractive index and density of the colloidal particles is matched to that of the solvent. We review work to obtain a particle-level mechanism for rigidity in gels and the evolution of our understanding of time-dependent behaviour, from early-time aggregation to ageing, before considering the response of colloidal gels to deformation and then move on to more complex systems of anisotropic particles and mixtures. Finally we note some more exotic materials with similar properties.

Crystallisation in a two-dimensional granular system at constant temperature

We study the crystallisation processes occurring in a nonvibrating two-dimensional magnetic granular system at various fixed values of the effective temperature. In this system, the energy loss due to dissipative effects is compensated by the continuous energy input coming into the system from a sinusoidal magnetic field. When this balance leads to high values of the effective temperature, no aggregates are formed, because particles’ kinetic energy prevents them from aggregating. For lower effective temperatures, formation of small aggregates is observed. The smaller the values of the applied field’s amplitude, the larger the number of these disordered aggregates. One also observes that when clusters form at a given effective temperature, the average effective diffusion coefficient decreases as time increases. For medium values of the effective temperature, formation of small crystals is observed. We find that the sixth bond-orientational order parameter and the number of bonds, when considering more than two, are very sensitive for exhibiting the order in the system, even when crystals are still very small.

2D isotropic-nematic transition in colloidal suspensions of ellipsoids

Liquid crystals are important condensed matter systems for technological applications, as well as for fundamental studies. An important unresolved issue is the nature of the phase transition in a two-dimensional (2D) liquid crystal system. In contrast to numerous computational studies reported in the last few decades, there have been no convincing experiments to verify these numerical results. Anisotropic colloids provide an excellent experimental model system to study phase transitions, such as crystallization and glass transition in condensed matter physics with single particle resolution. However, using colloids to probe the two-dimensional liquid crystal transition remains a challenge, since the condensed anisotropic colloids usually become stuck in the metastable glassy state rather than approaching their equilibrium liquid crystal phase. Here we report a method of using an external magnetic field to assist a colloidal system of super-paramagnetic anisotropic particles to overcome the local free energy barriers in the metastable states and approach the equilibrium phase. The experiments demonstrate a 2D isotropic-nematic phase transition with increasing packing density. The effects of the anisotropy of the colloidal particles on the 2D isotropic-nematic transition are explored. Our experimental results are compared with those from previous computational work, and quantitative agreements are reached.

The Ambiguous Functions of the Precursors That Enable Nonclassical Modes of Olanzapine Nucleation and Growth

One of the most consequential assumptions of the classical theories of crystal nucleation and growth is the Szilard postulate, which states that molecules from a supersaturated phase join a nucleus or a growing crystal individually. In the last 20 years, observations in complex biological, geological, and engineered environments have brought to light violations of the Szilard rule, whereby molecules assemble into ordered or disordered precursors that then host and promote nucleation or contribute to fast crystal growth. Nonclassical crystallization has risen to a default mode presumed to operate in the majority of the inspected crystallizing systems. In some cases, the existence of precursors in the growth media is admitted as proof for their role in nucleation and growth. With the example of olanzapine, a marketed drug for schizophrenia and bipolar disorder, we demonstrate that molecular assemblies in the solution selectively participate in crystal nucleation and growth. In aqueous and organic solutions, olanzapine assembles into both mesoscopic solute-rich clusters and dimers. The clusters facilitate nucleation of crystals and crystal form transformations. During growth, however, the clusters land on the crystal surface and transform into defects, but do not support step growth. The dimers are present at low concentrations in the supersaturated solution, yet the crystals grow by the association of dimers, and not of the majority monomers. The observations with olanzapine emphasize that detailed studies of the crystal and solution structures and the dynamics of molecular association may empower classical and nonclassical models that advance the understanding of natural crystallization, and support the design and manufacture of promising functional materials.

Identification of critical nuclei in the rapid solidification via configuration heredity

The identification and characterization of critical nuclei is a long-standing issue in the rapid solidification of metals and alloys. An ambiguous description for their sizes and shapes used to lead to an overestimation or underestimation of homogeneous nucleation ratesITin the framework of classical nucleation theory (CNT). In this paper, a unique method able to distinguish the critical nucleus from numerous embryos is put forward on the basis of configuration heredities of clusters during rapid solidifications. As this technique is applied to analyze the formation and evolution of various fcc-Al single crystal clusters in a large-scale molecular dynamics simulation system, it is found that the sizen_cand geometrical configuration of critical nuclei as well as their liquid-solid interfacial structure can be determined directly. For the present deep super-cooled system with an undercooling ofTm=0.42Tmcal, the average size of critical nuclei is demonstrated to benc̄≈26, but most of which are non-spherical lamellae. Also, their liquid-solid interfaces are revealed to be not an fcc-liquid duplex-phase interface but an fcc/hcp-liquid multi-phase structure. These findings shed some lights on the CNT, and a good agreement with previous simulations and experiments inITindicates this technique can be used to explore the early-stage of nucleation from atomistic levels.

Self-Assembly of Patchy Colloidal Rods into Photonic Crystals Robust to Stacking Faults

Diamond-structured colloidal photonic crystals are much sought-after for their applications in visible light management because of their ability to support a complete photonic band gap (PBG). However, their realization via self-assembly pathways is a long-standing challenge. This challenge is rooted in three fundamental problems: the design of building blocks that assemble into diamond-like structures, the sensitivity of the PBG to stacking faults, and ensuring that the PBG opens at an experimentally attainable refractive index. Here we address these problems simultaneously using a multipronged computational approach. We use reverse engineering to establish the design principles for the rod-connected diamond structure (RCD), the so-called "champion" photonic crystal. We devise two distinct self-assembly routes for designer triblock patchy colloidal rods, both proceeding via tetrahedral clusters to yield a mixed phase of cubic and hexagonal polymorphs closely related to RCD. We use Monte Carlo simulations to show how these routes avoid a metastable amorphous phase. Finally, we show that both the polymorphs support spectrally overlapping PBGs. Importantly, randomly stacked hybrids of these polymorphs also display PBGs, thus circumventing the requirement of polymorph selection in a scalable fabrication method.

Iterative Cup Overlapping: An Efficient Identification Algorithm for Cage Structures of Amorphous Phase Hydrates

Molecular dynamics studies have revealed that the nucleation pathway of clathrate hydrates involves the evolution from amorphous to crystalline hydrates. In this study, complete cages are further classified into the standard edge-saturated cages (SECs) and nonstandard edge-saturated cages (non-SECs). Centered on studying the structure and evolution of non-SECs and SECs, we propose a novel and efficient algorithm, iterative cup overlapping (ICO), to monitor hydrate nucleation and growth in molecular simulations by identifying SECs and discuss possible causes of the instability of non-SECs. Manipulation of topological information makes it possible for ICO to avoid the repeated searches for identified cages and deduce all SECs with low time costs, improving the efficiency of identification significantly. The accuracy and efficiency of ICO were verified by comparing the identification results with other well-proven algorithms. Furthermore, it was found that non-SECs have short lifetimes and eventually decompose or reorganize into more stable structures. Some evidence suggests that the instability of non-SECs is closely related to the hydrogen-bonding configuration of water-ring aggregations that they contain. The spontaneous evolution of the hydrogen-bonding network into the tetrahedral network may be the main factor that causes the conversion of QWRAs and the evolution of non-SECs.

A review of solvent freeze-out technology for protein crystallization

In this review, we summarize important advances in solvent freeze-out (SFO) technology for protein crystallization, including the background of SFO, its fundamental principle, and some crucial conditions and factors for optimizing SFO technology.

Two-step crystallization and solid-solid transitions in binary colloidal mixtures

Crystallization is fundamental to materials science and is central to a variety of applications, ranging from the fabrication of silicon wafers for microelectronics to the determination of protein structures. The basic picture is that a crystal nucleates from a homogeneous fluid by a spontaneous fluctuation that kicks the system over a single free-energy barrier. However, it is becoming apparent that nucleation is often more complicated than this simple picture and, instead, can proceed via multiple transformations of metastable structures along the pathway to the thermodynamic minimum. In this article, we observe, characterize, and model crystallization pathways using DNA-coated colloids. We use optical microscopy to investigate the crystallization of a binary colloidal mixture with single-particle resolution. We observe classical one-step pathways and nonclassical two-step pathways that proceed via a solid-solid transformation of a crystal intermediate. We also use enhanced sampling to compute the free-energy landscapes corresponding to our experiments and show that both one- and two-step pathways are driven by thermodynamics alone. Specifically, the two-step solid-solid transition is governed by a competition between two different crystal phases with free energies that depend on the crystal size. These results extend our understanding of available pathways to crystallization, by showing that size-dependent thermodynamic forces can produce pathways with multiple crystal phases that interconvert without free-energy barriers and could provide approaches to controlling the self-assembly of materials made from colloids.

Slowing down of dynamics and orientational order preceding crystallization in hard-sphere systems

Despite intensive studies in the past decades, the local structure of disordered matter remains widely unknown. We show the results of a coherent x-ray scattering study revealing higher-order correlations in dense colloidal hard-sphere systems in the vicinity of their crystallization and glass transition. With increasing volume fraction, we observe a strong increase in correlations at both medium-range and next-neighbor distances in the supercooled state, both invisible to conventional scattering techniques. Next-neighbor correlations are indicative of ordered precursor clusters preceding crystallization. Furthermore, the increase in such correlations is accompanied by a marked slowing down of the dynamics, proving experimentally a direct relation between orientational order and sample dynamics in a soft matter system. In contrast, correlations continuously increase for nonequilibrated, glassy samples, suggesting that orientational order is reached before the sample slows down to reach (quasi-)equilibrium.

Crystallization in Confinement

Many crystallization processes of great importance, including frost heave, biomineralization, the synthesis of nanomaterials, and scale formation, occur in small volumes rather than bulk solution. Here, the influence of confinement on crystallization processes is described, drawing together information from fields as diverse as bioinspired mineralization, templating, pharmaceuticals, colloidal crystallization, and geochemistry. Experiments are principally conducted within confining systems that offer well-defined environments, varying from droplets in microfluidic devices, to cylindrical pores in filtration membranes, to nanoporous glasses and carbon nanotubes. Dramatic effects are observed, including a stabilization of metastable polymorphs, a depression of freezing points, and the formation of crystals with preferred orientations, modified morphologies, and even structures not seen in bulk. Confinement is also shown to influence crystallization processes over length scales ranging from the atomic to hundreds of micrometers, and to originate from a wide range of mechanisms. The development of an enhanced understanding of the influence of confinement on crystal nucleation and growth will not only provide superior insight into crystallization processes in many real-world environments, but will also enable this phenomenon to be used to control crystallization in applications including nanomaterial synthesis, heavy metal remediation, and the prevention of weathering.

In Situ Monitoring of the Seeding and Growth of Silver Metal-Organic Nanotubes by Liquid-Cell Transmission Electron Microscopy

Metal-organic nanotubes (MONTs) are highly ordered one-dimensional crystalline porous frameworks. Despite being nanomaterials, virtually all studies of MONTs rely on characterization of the bulk crystalline material (micron-sized) by single-crystal X-ray diffraction. For MONTs to achieve their raison d'être as tunable one-dimensional nanomaterials, individual tubes or small finite bundles of tubes must be synthesized and characterized. Therefore, to directly observe their formation under a variety of reaction conditions in solution, we employ liquid-cell transmission electron microscopy (LCTEM), which allows the early stages of MONT assembly to be monitored in real time. Notably, changing the metal-to-ligand ratio alters the local concentrations of reactant monomers, resulting in multiple nucleation and growth pathways and diverse morphologies at the nanoscale. These various initial seeds grow to form the same nanocrystalline needle phase. This approach of employing LCTEM to study these nanomaterials is analogous to monitoring typical homogeneous solution phase reactions by NMR for controlled nanomaterial formation.

Revealing roles of competing local structural orderings in crystallization of polymorphic systems

Most systems have more than two stable crystalline states in the phase diagram, which is known as polymorphism. Crystallization in such a system is often under strong influence of competing orderings linked to those crystals. However, how such competition affects crystal nucleation and ordering toward the final crystalline state is largely unknown. This is primarily because the competition takes place locally and thus is masked by large positional fluctuations. We develop a unique method to correctly identify local symmetries by removing their distortions due to positional fluctuations. This allows us to experimentally access the spatiotemporal fluctuations of local symmetries at a single-particle level in crystallization of a charged colloidal system near the body-centered cubic-face-centered cubic border. Thus, we successfully reveal the crucial roles of competing ordering in the initial selection of polymorphs and the final grain boundary motion toward the most stable state from a microscopic perspective.

Vibrational exciton nanoimaging of phases and domains in porphyrin nanocrystals

Much of the electronic transport, photophysical, or biological functions of molecular materials emerge from intermolecular interactions and associated nanoscale structure and morphology. However, competing phases, defects, and disorder give rise to confinement and many-body localization of the associated wavefunction, disturbing the performance of the material. Here, we employ vibrational excitons as a sensitive local probe of intermolecular coupling in hyperspectral infrared scattering scanning near-field optical microscopy (IR s-SNOM) with complementary small-angle X-ray scattering to map multiscale structure from molecular coupling to long-range order. In the model organic electronic material octaethyl porphyrin ruthenium(II) carbonyl (RuOEP), we observe the evolution of competing ordered and disordered phases, in nucleation, growth, and ripening of porphyrin nanocrystals. From measurement of vibrational exciton delocalization, we identify coexistence of ordered and disordered phases in RuOEP that extend down to the molecular scale. Even when reaching a high degree of macroscopic crystallinity, identify significant local disorder with correlation lengths of only a few nanometers. This minimally invasive approach of vibrational exciton nanospectroscopy and -imaging is generally applicable to provide the molecular-level insight into photoresponse and energy transport in organic photovoltaics, electronics, or proteins.

One-dimensional water nanowires induced by electric fields

Self-aggregation of water vapour molecules under external electric fields is systemically investigated by using molecular dynamics simulations. It is found that small water clusters aggregate into one-dimensional water nanowires along the electric field direction. The electric field strength plays a crucial role in tuning the nanowire structure. Under relatively weak electric fields such as E = 0.1 V Å^-1, square and pentagonal prism-like structures are preferred; when intermediate strength electric fields are applied (E = 1.0 V Å^-1), water nanowires featuring a disordered mixture of four-, five- and six-membered rings are formed; and an open ordered structure which is reminiscent of two-dimensional (2D) ice is observed when the field strength becomes very high (E > 3.0 V Å^-1). Bond parameter analysis based on density-functional theory calculations shows that the electric field affects anisotropically the conformation of water molecules as well as the hydrogen-bond properties. Along the electric field, the H-O bond is stretched and the hydrogen bond shrinks with field strength in contrast to the changes perpendicular to the electric field. As a result, the hydrogen bonding is enhanced along the electric field. Under very high electric fields, the anisotropic hydrogen-bond network opens up via breaking of the bonds perpendicular to the electric field and ultimately relaxes into a loose quasi-2D ordered network.

Voltammetric Determination of the Stochastic Formation Rate and Geometry of Individual H2, N2, and O2 Bubble Nuclei

Herein, we report a general voltammetric method to characterize the electrochemical nucleation rate and nuclei of single nanobubbles. Bubble nucleation is indicated by a sharp peak in the current in the voltammetry of gas-evolving reactions. In contrast to expectations based on the stochastic nature of nucleation events, the peak current signifying a stable nucleus is extremely reproducible over hundreds of cycles (∼3% deviation). By applying classical nucleation theory, this seemingly deterministic behavior can be not only understood but also used to quantify the nucleation rate and size of bubble nuclei. A statistical model is developed whereby properties of single critical nuclei (contact angle, the radius of curvature, activation energy, and Arrhenius pre-exponential factor) can be readily measured from the narrow distribution of peak currents (mean, standard deviation) from hundreds of voltammetric cycles at a nanoelectrode. Single nanobubbles formed from gas-evolving reactions (H₂ from H⁺ reduction, N₂ from N₂H₄ oxidation, O₂ from H₂O₂ oxidation) are analyzed to find that their critical nuclei have contact angles of ∼150, ∼160, and ∼154° for H₂, N₂, and O₂, respectively, corresponding to ∼50, ∼40, and ∼90 gas molecules in each nucleus. The energy barriers for heterogeneous nucleation of H₂, N₂, and O₂ bubbles are, respectively, 2, 0.4, and 0.7% of those required for homogeneous nucleation under the same supersaturation.