Theoretical evidence for a dense fluid precursor to crystallization

Amorphous precursors of crystallization during spinodal decomposition

A general Landau's free energy functional is used to study the dynamics of crystallization during liquid-solid spinodal decomposition (SD). The strong length scale selectivity imposed during the early stage of SD induces the appearance of small precursors for crystallization with icosahedral order. These precursors grow in densely packed clusters of tetrahedra through the addition of new particles. As the average size of the amorphous nuclei becomes large enough to reduce geometric frustration, crystalline particles with a body-centered cubic symmetry heterogeneously nucleate on the growing clusters. The volume fraction of the crystalline phase is strongly dependent on the depth of quench. At deep quenches, the SD mechanism produces amorphous structures arranged in dense polytetrahedral aggregates.

Transformation properties of entropy production

The transformation properties of entropy production under phase-space partitioning, lumping, or the elimination of intermediate steps and variables in the presence of widely separated time scales are studied. Conditions are derived under which dissipation remains invariant. In systems subjected to external periodic driving the adiabatic, asymptotic, and transient entropy productions are evaluated and the extent to which they can be separately non-negative is determined.

Nucleation

Crystallization starts with nucleation and control of nucleation is crucial for the control of the number, size, perfection, polymorphism and other characteristics of crystalline materials. This is particularly true for crystallization in solution, which is an essential part of processes in the chemical and pharmaceutical industries and a major step in physiological and pathological phenomena. There have been significant recent advances in the understanding of the mechanism of nucleation of crystals in solution. The foremost of these are the two-step mechanism of nucleation and the notion of the solution-crystal spinodal. According to the two-step mechanism, the crystalline nucleus appears inside pre-existing metastable clusters of size several hundred nanometers, which consist of dense liquid and are suspended in the solution. While initially proposed for protein crystals, the applicability of this mechanism has been demonstrated for small molecule organic materials, colloids, polymers, and biominerals. This mechanism helps to explain several long-standing puzzles of crystal nucleation in solution: nucleation rates which are many orders of magnitude lower than theoretical predictions, the significance of the dense protein liquid, and others. At high supersaturations typical of most crystallizing systems, the generation of crystal embryos occurs in the spinodal regime, where the nucleation barrier is negligible. The solution-crystal spinodal helps to understand the role of heterogeneous substrates in nucleation and the selection of crystalline polymorphs. Importantly, these ideas provide powerful tools for control of the nucleation process by varying the solution thermodynamic parameters.

The two-step mechanism of nucleation of crystals in solution

The formation of crystalline nanoparticles starts with nucleation and control of nucleation is crucial for the control of the number, size, perfection, polymorph modification and other characteristics of particles. Recently, there have been significant advances in the understanding of the mechanism of nucleation of crystals in solution. The most significant of these is the two-step mechanism of nucleation, according to which the crystalline nucleus appears inside pre-existing metastable clusters of size several hundred nanometers, which consist of dense liquid and are suspended in the solution. While initially proposed for protein crystals, the applicability of this mechanism has been demonstrated for small-molecule organic and inorganic materials, colloids, and biominerals. This mechanism helps to explain several long-standing puzzles of crystal nucleation in solution: nucleation rates which are many orders of magnitude lower than theoretical predictions, nucleation kinetic dependencies with steady or receding parts at increasing supersaturation, the role of heterogeneous substrates for polymorph selection, the significance of the dense protein liquid, and others. More importantly, this mechanism provides powerful tools for control of the nucleation process by varying the solution thermodynamic parameters so that the volume occupied by the dense liquid shrinks or expands.

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

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

Precursor-mediated crystallization process in suspensions of hard spheres

We report on a large scale computer simulation study of crystal nucleation in hard spheres. Through a combined analysis of real- and reciprocal-space data, a picture of a two-step crystallization process is supported: First, dense, amorphous clusters form which then act as precursors for the nucleation of well-ordered crystallites. This kind of crystallization process has been previously observed in systems that interact via potentials that have an attractive as well as a repulsive part, most prominently in protein solutions. In this context the effect has been attributed to the presence of metastable fluid-fluid demixing. Our simulations, however, show that a purely repulsive system (that has no metastable fluid-fluid coexistence) crystallizes via the same mechanism.

Stochastic resonance in multistable systems: the role of intermediate states

The classical setting of stochastic resonance is extended to account for the presence of an arbitrary number of simultaneously stable steady states. General expressions for the linear response are derived for systems involving one variable. The existence of an optimal value of noise strength and of an optimal number of stable states for which the response is maximized is established.

Low-density/high-density liquid phase transition for model globular proteins

The effect of molecule size (excluded volume) and the range of interaction on the surface tension, phase diagram, and nucleation properties of a model globular protein is investigated using a combination of Monte Carlo simulations and finite temperature classical density functional theory calculations. We use a parametrized potential that can vary smoothly from the standard Lennard-Jones interaction characteristic of simple fluids to the ten Wolde-Frenkel model for the effective interaction of globular proteins in solution. We find that the large excluded volume characteristic of large macromolecules such as proteins is the dominant effect in determining the liquid-vapor surface tension and nucleation properties. The variation of the range of the potential is important only in the case of small excluded volumes such as for simple fluids. The DFT calculations are then used to study the homogeneous nucleation of the high-density phase from the low-density phase including the nucleation barriers, nucleation pathways, and rate. It is found that the nucleation barriers are typically only a few k(B)T and that the nucleation rates are substantially higher than would be predicted by classical nucleation theory.

Ostwald ripening of clusters during protein crystallization

Contrary to classical nucleation theory, protein crystals can nucleate via a two-step process in which the molecular arrangement of the ordered solid phase is preceded by nucleation of a dense amorphous phase. We study the growth of these precrystalline clusters in lysozyme using a combination of dynamic light scattering, optical microscopy, and microfluidics. Clusters display Ostwald ripening growth kinetics but deviate from this trend after nucleation of the crystal phase. This behavior arises from the metastable relationship between clusters and the ordered solid and is explained numerically using a population balance model.

Kinetics of intermediate-mediated self-assembly in nanosized materials: a generic model

We propose in this paper a generic model of a nonstandard aggregation mechanism for self-assembly processes of a class of materials involving the mediation of intermediates consisting of a polydisperse population of nanosized particles. The model accounts for a long induction period in the process. The proposed mechanism also gives insight on future experiments aiming at a more comprehensive picture of the role of self-organization in self-assembly processes.

Diffusion-controlled anisotropic growth of stable and metastable crystal polymorphs in the phase-field crystal model

We use a simple density functional approach on a diffusional time scale, to address freezing to the body-centered cubic (bcc), hexagonal close-packed (hcp), and face-centered cubic (fcc) structures. We observe faceted equilibrium shapes and diffusion-controlled layerwise crystal growth consistent with two-dimensional nucleation. The predicted growth anisotropies are discussed in relation with results from experiment and atomistic simulations. We also demonstrate that varying the lattice constant of a simple cubic substrate, one can tune the epitaxially growing body-centered tetragonal structure between bcc and fcc, and observe a Mullins-Sekerka-Asaro-Tiller-Grinfeld-type instability.

Experimental evidence for two-step nucleation in colloidal crystallization

We investigated the freezing of colloidal spheres in two dimensions with single-particle resolution. Using micron-size, charge-stabilized polystyrene spheres with a temperature-dependent depletion attraction induced by surfactant micelles, we supercooled an initially amorphous (gaslike) system. Particle motions were monitored as crystallization proceeded. At low concentrations, freezing occurred in a single step in a manner consistent with classical nucleation theory. In other samples two-step nucleation was found, in which amorphous clusters grew to approximately 30 particles, then rapidly crystallized. Measured free energies show the role of metastable gas-liquid coexistence, which also enhanced the rate of nucleation following deeper quenches.

Nucleation: what happens at the initial stage?

Crystallizing growth: The initial structure of crystal nuclei is supersaturation-dependent. At low degrees of supersaturation, liquid-like nuclei are formed initially, which undergo a continuous structure transition from liquid-like to crystal-like as the size N increases. This gradual structure evolution substantially lowers the nucleation barrier DeltaG* and facilitates the nucleation relative to the formation of crystal-like clusters from the beginning.

Ripening-dominated crystallization in polydisperse hard-sphere-like colloids

We report on the crystal growth scenario in gravity-matched, polydisperse hard-sphere-like colloids at increasing particle concentration. In the fluid-crystal coexistence region, the crystal size as a function of time shows two separate regimes corresponding to crystal growth and crystal ripening. At higher supersaturation the crystal size grows according to the same power law through the whole experimental window of a few days: crystal growth and ripening merge together. We show that our observations cannot be explained by considering the slowing down of single-particle dynamics due to increasing volume fraction. We suggest that size fractionation occurring at the crystal-fluid interface is the dominant mechanism.

Crystallization and gelation in colloidal systems with short-ranged attractive interactions

We systematically study the relationship between equilibrium and nonequilibrium phase diagrams of a system of short-ranged attractive colloids. Using Monte Carlo and Brownian dynamics simulations we find a window of enhanced crystallization that is limited at high interaction strength by a slowing down of the dynamics and at low interaction strength by the high nucleation barrier. We find that the crystallization is enhanced by the metastable gas-liquid binodal by means of a two-stage crystallization process. First, the formation of a dense liquid is observed and second the crystal nucleates within the dense fluid. In addition, we find at low colloid packing fractions a fluid of clusters, and at higher colloid packing fractions a percolating network due to an arrested gas-liquid phase separation that we identify with gelation. We find that this arrest is due to crystallization at low interaction energy and it is caused by a slowing down of the dynamics at high interaction strength. Likewise, we observe that the clusters which are formed at low colloid packing fractions are crystalline at low interaction energy, but glassy at high interaction energy. The clusters coalesce upon encounter.

Competitive nucleation in reversible probabilistic cellular automata

The problem of competitive nucleation in the framework of probabilistic cellular automata is studied from the dynamical point of view. The dependence of the metastability scenario on the self-interaction is discussed. An intermediate metastable phase, made of two flip-flopping chessboard configurations, shows up depending on the ratio between the magnetic field and the self-interaction. A behavior similar to the one of the stochastic Blume-Capel model with Glauber dynamics is found.

Bio-inspired Synthesis of Mineralized Collagen Fibrils

Mineralized collagen fibrils constitute a basic structural unit of collagenous mineralized tissues such as dentin and bone. Understanding of the mechanisms of collagen mineralization is vital for development of new materials for the hard tissue repair. We carried out bio-inspired mineralization of reconstituted collagen fibrils using poly-l-aspartic acid, as an analog of non-collagenous acidic proteins. Transmission electron microscopy and electron diffraction studies of the reaction products revealed stacks of ribbon-shaped apatitic crystals, deposited within the fibrils with their c-axes co-aligned with the fibril axes. Such structural organization closely resembles mineralized collagen of bone and dentin. Initial mineral deposits formed in the fibrils lacked a long range crystallographic order and transformed into crystals with time. Interestingly, the shape and organization of these amorphous deposits was similar to the crystals found in the mature mineralized fibrils. We demonstrate that the interactions between collagen and poly-l-aspartic acid are essential for the mineralized collagen fibrils formation, while collagen alone does not affect mineral formation and poly-l-aspartic acid inhibits mineralization in a concentration dependant manner. These results provide new insights into basic mechanisms of collagen mineralization and can lead to the development of novel bio-inspired nanostructured materials.

Irreversible thermodynamics of multistep transitions

Conditions under which the evolution equations of a multivariate system can be cast in a variational form are identified. A kinetic potential generating both the deterministic part of the evolution and the probabilistic properties in a suitably defined set of variables is derived and compared to the thermodynamic potentials. The results are illustrated on the kinetics of phase transitions involving intermediate metastable phases and chemical reactions giving rise to two or more intermediate species.

Sickle-cell haemoglobin polymerization: is it the primary pathogenic event of sickle-cell anaemia?

Sickle cell anaemia is associated with a mutant haemoglobin, HbS, which forms polymers in the red blood cells of patients. The primary role of the HbS polymerization for the pathophysiology has been questioned: observations in patients and model organisms contradict deterministic scenarios of sickling crises triggered by polymerization. However, results with knock-out sickle-cell mice, which were cured by delaying HbS polymerization, reconfirm polymerization's primary role. To reconcile the contradictory observations, this article reviews recent findings on two steps in polymerization: homogeneous nucleation of fibres, and their growth. The fibre growth is faster by far than for any other protein ordered structure. This is due to a negligible free-energy barrier for incorporation into a fibre, determined by an entropy gain, stemming from the release of water molecules structured around HbS. The kinetics of fibre nucleation have shown that the formation of the polymer nucleus is preceded by a metastable droplet of a dense liquid. The properties of the dense liquid are sensitive functions of solution composition, including components in micro- and nanomolar amounts. This mechanism allows low-concentration solution components to strongly affect the nucleation kinetics, accounting for the high variability of the disease. These insights can potentially be utilized for control of HbS polymerization and treatment of the disease.

Imaging the Sublimation Dynamics of Colloidal Crystallites

We studied the kinetics of sublimating crystals with single-particle resolution by experiments with colloidal spheres and by computer simulations. A short-range attraction between spheres led to crystallites one to three layers thick. The spheres were tracked with optical microscopy while the attraction was reduced and the crystals sublimated. Large crystallites sublimated by escape of particles from the perimeter. The rate of shrinkage was greatly enhanced, however, when the size decreased to less than 20 to 50 particles, depending on the location in the phase diagram. At this size, the crystallites transformed into a dense amorphous structure, which rapidly vaporized. The enhancement of kinetics by metastable or unstable phases may play a major role in the melting, freezing, and annealing of crystals.

Ginzburg-Landau theory of the liquid-solid interface and nucleation for hard spheres

The Ginzburg-Landau free energy functional for hard spheres is constructed using the fundamental measure theory approach to density functional theory as a starting point. The functional is used to study the liquid-fcc solid planer interface and the properties of small solid clusters nucleating within a liquid. The surface tension for planer interfaces agrees well with simulation and it is found that the properties of the solid clusters are consistent with classical nucleation theory.