Limit of validity of Ostwald's rule of stages in a statistical mechanical model of crystallization

Unraveling the Complex Polymorphic Crystallization Behavior of the Alternating Copolymer DMDS-alt-DVE

A complex crystallization behavior was observed for the alternating copolymer DMDS-alt-DVE synthesized via thiol-ene step-growth polymerization. Understanding the underlying complex crystallization processes of such innovative polythioethers is critical for their application, for example, in polymer coating technologies. These alternating copolymers have polymorphic traits, resulting in different phases that may display distinct crystalline structures. The copolymer DMDS-alt-DVE was studied in an earlier work, where only two crystalline phases were reported: a low melting, L - T_m, and high melting, H - T_m phase. Remarkably, the H - T_m form was only achieved by the previous formation and melting of the L - T_m form. We applied calorimetric techniques encompassing seven orders of magnitude in scanning rates to further explore this complex polymorphic behavior. Most importantly, by rapidly quenching the sample to temperatures well below room temperature, we detected an additional polymorphic form (characterized by a very low melting phase, denoted VL - T_m). Moreover, through tailored thermal protocols, we successfully produced samples containing only one, two, or all three polymorphs, providing insights into their interrelationships. Understanding polymorphism, crystallization, and the resulting morphological differences can have significant implications and potential impact on mechanical resistance and barrier properties.

Mechanism of Nucleation Pathway Selection in Binary Lennard-Jones Solution: A Combined Study of Molecular Dynamics Simulation and Free Energy Analysis

The nucleation process, which is the initial step in particle synthesis, determines the properties of the resultant particles. Although recent studies have observed various nucleation pathways, the physical factors that determine these pathways have not been fully elucidated. Herein, we conducted molecular dynamics simulations in a binary Lennard-Jones system as a model solution and found that the nucleation pathway can be classified into four types depending on microscopic interactions. The key parameters are (1) the strength of the solute-solute interaction and (2) the difference between the strengths of the like-pair and unlike-pair interactions. The increment of the former alters the nucleation mechanism from a two-step to a one-step pathway, whereas that of the latter causes quick assembly of solutes. Moreover, we developed a thermodynamic model based on the formation of core-shell nuclei to calculate the free energy landscapes. Our model successfully described the pathway observed in the simulations and demonstrated that the two parameters, (1) and (2), define the degree of supercooling and supersaturation, respectively. Thus, our model interpreted the microscopic insights from a macroscopic point of view. Because the only inputs required for our model are the interaction parameters, our model can a priori predict the nucleation pathway.

Ritonavir Form III: A New Polymorph After 24 Years

Polymorphism occurs widely in pharmaceutical solids, and must be thoroughly studied during product development. Twenty-four years after ritonavir (RTV) Form II materialized, we report a new polymorph, Form III, discovered via melt crystallization. Form III has a unique PXRD pattern, Raman spectrum, lower melting point and heat of fusion, compared to the known polymorphs, Form I and Form II. It is the least stable form, monotropically, among the three polymorphs. Form III differs from Form I and Form II in molecular conformation and hydrogen bonding motifs in crystal lattice. Nucleation from RTV supercooled liquid is slow, and selected Form III exclusively. The discovery of RTV Form III demonstrates the importance of crystal nucleation studies. Crystallization from supercooled liquids should be incorporated as part of polymorph screening workflow.

Crystal Polymorphism Induced by Surface Tension

We use classical density functional theory (cDFT) to calculate fluid-solid surface tensions for fcc and bcc crystals formed by hard spheres and Lennard-Jones (LJ) particles. For hard spheres, our results show that the recently introduced "explicitly stable" functionals perform as well as the state of the art, and for both interaction potentials, our results compare well to simulation. We use the resulting bulk and interfacial energies for LJ to parametrize a capillary model for the free energy of small solid clusters and thereby determine the relative stability of bcc and fcc LJ clusters. We show a crossover from bcc to fcc stability as cluster size increases, thus providing insight into long-standing tension between simulation results and theoretical expectations. We also confirm that the bcc phase in contact with a vapor is unstable, thus extending earlier zero-temperature results. Our Letter demonstrates the potential of cDFT as an important tool in understanding crystallization and polymorphism.

Kinetic Phase Diagram for Nucleation and Growth of Competing Crystal Polymorphs in Charged Colloids

We determine the kinetic phase diagram for nucleation and growth of crystal phases in a suspension of charged colloids. Exploiting the seeding approach in extensive simulations, we calculate nucleation barrier heights for face-centered cubic (fcc) and body-centered cubic (bcc) phases for varying screening lengths and supersaturations. We determine for the entire metastable fluid region the crystal polymorph with the lowest nucleation barrier, and find a regime close to the triple point where metastable bcc can form due to a lower nucleation barrier, even though fcc is the stable phase. For higher supersaturation, we find that the difference in barrier heights decreases and we observe a mix of hexagonal close-packed, fcc, and bcc structures in the growth of crystalline seeds as well as in spontaneously formed crystals. Our kinetic phase diagram rationalizes the different crystallization mechanisms observed in previous work.

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.

Surface Phase Metastability during Langmuir Evaporation

We have directly imaged the spontaneous formation of metastable surface phase domains on GaAs(001) during Langmuir evaporation. Eventually, these metastable phases transform to the thermodynamically stable parent phase, producing a dynamic phase coexistence with a temperature dependent, time-averaged coverage. Monte Carlo simulations are used to identify the key kinetic processes and investigate the interplay between phase metastability and evolving surface morphology. This is used to explain the measured temperature dependence of the time-averaged coverage.

How crystals form: A theory of nucleation pathways

Recent advances in classical density functional theory are combined with stochastic process theory and rare event techniques to formulate a theoretical description of nucleation, including crystallization, that can predict nonclassical nucleation pathways based on no input other than the interaction potential of the particles making up the system. The theory is formulated directly in terms of the density field, thus forgoing the need to define collective variables. It is illustrated by application to diffusion-limited nucleation of macromolecules in solution for both liquid-liquid separation and crystallization. Both involve nonclassical pathways with crystallization, in particular, proceeding by a two-step mechanism consisting of the formation of a dense-solution droplet followed by ordering originating at the core of the droplet. Furthermore, during the ordering, the free-energy surface shows shallow minima associated with the freezing of liquid into solid shells, which may shed light on the widely observed metastability of nanoscale clusters.

Molecular nucleation mechanisms and control strategies for crystal polymorph selection

The formation of condensed (compacted) protein phases is associated with a wide range of human disorders, such as eye cataracts, amyotrophic lateral sclerosis, sickle cell anaemia 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 physiochemical properties of crystals can vary substantially between different forms or structures ('polymorphs') of the same macromolecule, and dictate their usability in a scientific or industrial context. To gain control over an emerging polymorph, one needs a molecular-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 macromolecular 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 molecular resolution the nucleation pathways that lead to two crystalline 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 intermolecular bonding or gel seeding. Our results differ from the present picture of protein nucleation, in that we do not identify a metastable dense liquid as the precursor to the crystalline state. Rather, we observe nucleation events that are driven by oriented attachments between subcritical clusters that already exhibit a degree of crystallinity. These insights suggest ways of controlling macromolecular phase transitions, aiding the development of protein-based drug-delivery systems and macromolecular crystallography.

Nonclassical nucleation pathways in protein crystallization

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

Black Anatase Formation by Annealing of Amorphous Nanoparticles and the Role of the Ti2O3 Shell in Self-Organized Crystallization by Particle Attachment

We use amorphous titania nanoparticle networks produced by pulsed laser vaporization at room temperature as a model system for understanding the mechanism of formation of black titania. Here, we characterize the transformation of amorphous nanoparticles by annealing in pure Ar at 400 °C, the lowest temperature at which black titania was observed. Atomic resolution electron microscopy methods and electron energy loss spectroscopy show that the onset of crystallization occurs by nucleation of an anatase core that is surrounded by an amorphous Ti₂O₃ shell. The formation of the metastable anatase core before the thermodynamically stable rutile phase occurs according to the Ostwald phase rule. In the second stage the particle size increases by coalescence of already crystallized particles by a self-organized mechanism of crystallization by particle attachment. We show that the Ti₂O₃ shell plays a critical role in both black titania transformation and functionality. At 400 °C, Ti₂O₃ hinders the agglomeration of neighboring particles to maintain a high surface-to-volume ratio that is beneficial for enhanced photocatalytic activity. In agreement with previous results, the thin Ti₂O₃ surface layer acts as a narrow bandgap semiconductor in concert with surface defects to enhance the photocatalytic activity. Our results demonstrate that crystallization by particle attachment can be a highly effective mechanism for optimizing photocatalytic efficiency by controlling the phase, composition, and particle size distribution in a wide range of self-doped defective TiO₂ architectures simply by varying the annealing conditions of amorphous nanoparticles.

Anomalous approach to thermodynamic equilibrium: Structure formation of molecules after vapor deposition

We describe experiments and computer simulations of molecular deposition on a substrate in which the molecules (substituted adenine derivatives) self-assemble into ordered structures. The resulting structures depend strongly on the deposition rate (flux). In particular, there are two competing surface morphologies (α and β), which differ by their topology (interdigitated vs lamellar structure). Experimentally, the α phase dominates at both low and high flux, with the β phase being most important in the intermediate regime. A similar nonmonotonic behavior is observed on varying the substrate temperature. To understand these effects from a theoretical perspective, a lattice model is devised which reproduces qualitatively the topological features of both phases. Via extensive Monte Carlo studies we can, on the one hand, reproduce the experimental results and, on the other hand, obtain a microscopic understanding of the mechanisms behind this anomalous behavior. The results are discussed in terms of an interplay between kinetic trapping and temporal exploration of configuration space.

Nonclassical Nucleation in a Solid-Solid Transition of Confined Hard Spheres

A solid-solid phase transition of colloidal hard spheres confined between two planar hard walls is studied using a combination of molecular dynamics and Monte Carlo simulation. The transition from a solid consisting of five crystalline layers with square symmetry (5□) to a solid consisting of four layers with triangular symmetry (4△) is shown to occur through a nonclassical nucleation mechanism that involves the initial formation of a precritical liquid cluster, within which the cluster of the stable 4△ phase grows. Free-energy calculations show that the transition occurs in one step, crossing a single free-energy barrier, and that the critical nucleus consists of a small 4△ solid cluster wetted by a metastable liquid. In addition, the liquid cluster and the solid cluster are shown to grow at the planar hard walls. We also find that the critical nucleus size increases with supersaturation, which is at odds with classical nucleation theory. The △-solid-like cluster is shown to contain both face-centered-cubic and hexagonal-close-packed ordered particles.

Correlation between thermodynamic anomalies and pathways of ice nucleation in supercooled water

The well-known classical nucleation theory (CNT) for the free energy barrier towards formation of a nucleus of critical size of the new stable phase within the parent metastable phase fails to take into account the influence of other metastable phases having density/order intermediate between the parent metastable phase and the final stable phase. This lacuna can be more serious than capillary approximation or spherical shape assumption made in CNT. This issue is particularly significant in ice nucleation because liquid water shows rich phase diagram consisting of two (high and low density) liquid phases in supercooled state. The explanations of thermodynamic and dynamic anomalies of supercooled water often invoke the possible influence of a liquid-liquid transition between two metastable liquid phases. To investigate both the role of thermodynamic anomalies and presence of distinct metastable liquid phases in supercooled water on ice nucleation, we employ density functional theoretical approach to find nucleation free energy barrier in different regions of phase diagram. The theory makes a number of striking predictions, such as a dramatic lowering of nucleation barrier due to presence of a metastable intermediate phase and crossover in the dependence of free energy barrier on temperature near liquid-liquid critical point. These predictions can be tested by computer simulations as well as by controlled experiments.

The role of bond tangency and bond gap in hard sphere crystallization of chains

We report results from Monte Carlo simulations on dense packings of linear, freely-jointed chains of hard spheres of uniform size. In contrast to our past studies where bonded spheres along the chain backbone were tangent, in the present work a finite tolerance in the bond is allowed. Bond lengths are allowed to fluctuate in the interval [σ, σ + dl], where σ is the sphere diameter. We find that bond tolerance affects the phase behaviour of hard-sphere chains, especially in the close vicinity of the melting transition. First, a critical dl(crit) exists marking the threshold for crystallization, whose value decreases with increasing volume fraction. Second, bond gaps enhance the onset of phase transition by accelerating crystal nucleation and growth. Finally, bond tolerance has an effect on crystal morphologies: in the tangent limit the majority of structures correspond to stack-faulted random hexagonal close packing (rhcp). However, as bond tolerance increases a wealth of diverse structures can be observed: from single fcc (or hcp) crystallites to random hcp/fcc stackings with multiple directions. By extending the simulations over trillions of MC steps (10(12)) we are able to observe crystal-crystal transitions and perfection even for entangled polymer chains in accordance to the Ostwald's rule of stages in crystal polymorphism. Through simple geometric arguments we explain how the presence of rigid or flexible constraints affects crystallization in general atomic and particulate systems. Based on the present results, it can be concluded that proper tuning of bond gaps and of the connectivity network can be a controlling factor for the phase behaviour of model, polymer-based colloidal and granular systems.

Nucleation of metastable aragonite CaCO3 in seawater

Predicting the conditions in which a compound adopts a metastable structure when it crystallizes out of solution is an unsolved and fundamental problem in materials synthesis, and one which, if understood and harnessed, could enable the rational design of synthesis pathways toward or away from metastable structures. Crystallization of metastable phases is particularly accessible via low-temperature solution-based routes, such as chimie douce and hydrothermal synthesis, but although the chemistry of the solution plays a crucial role in governing which polymorph forms, how it does so is poorly understood. Here, we demonstrate an ab initio technique to quantify thermodynamic parameters of surfaces and bulks in equilibrium with an aqueous environment, enabling the calculation of nucleation barriers of competing polymorphs as a function of solution chemistry, thereby predicting the solution conditions governing polymorph selection. We apply this approach to resolve the long-standing "calcite-aragonite problem"--the observation that calcium carbonate precipitates as the metastable aragonite polymorph in marine environments, rather than the stable phase calcite--which is of tremendous relevance to biomineralization, carbon sequestration, paleogeochemistry, and the vulnerability of marine life to ocean acidification. We identify a direct relationship between the calcite surface energy and solution Mg:Ca [corrected] ion concentrations, showing that the calcite nucleation barrier surpasses that of metastable aragonite in solutions with Mg:Ca ratios consistent with modern seawater, allowing aragonite to dominate the kinetics of nucleation. Our ability to quantify how solution parameters distinguish between polymorphs marks an important step toward the ab initio prediction of materials synthesis pathways in solution.

Observing classical nucleation theory at work by monitoring phase transitions with molecular precision

It is widely accepted that many phase transitions do not follow nucleation pathways as envisaged by the classical nucleation theory. Many substances can traverse intermediate states before arriving at the stable phase. The apparent ubiquity of multi-step nucleation has made the inverse question relevant: does multistep nucleation always dominate single-step pathways? Here we provide an explicit example of the classical nucleation mechanism for a system known to exhibit the characteristics of multi-step nucleation. Molecular resolution atomic force microscopy imaging of the two-dimensional nucleation of the protein glucose isomerase demonstrates that the interior of subcritical clusters is in the same state as the crystalline bulk phase. Our data show that despite having all the characteristics typically associated with rich phase behaviour, glucose isomerase 2D crystals are formed classically. These observations illustrate the resurfacing importance of the classical nucleation theory by re-validating some of the key assumptions that have been recently questioned.

Many nanoscale systems can form ordered microphases through non-classical multistep nucleation. Here, the authors report that glucose isomerase, which is known to exhibit the characteristics of multi-step nucleation in 3D, nucleates along the pathway predicted by classical nucleation theory in 2D.

Supercooled water escaping from metastability

The return of supercooled water to a stable equilibrium condition is an irreversible process which, in large enough samples, takes place adiabatically. We investigated this phenomenon in water by fast imaging techniques. As water freezes, large energy and density fluctuations promote the spatial coexistence of solid and liquid phases at different temperatures. Upon synchronously monitoring the time evolution of the local temperature, we observed a sharp dynamic transition between a fast and a slow decay regime at about 266.6 K. We construe the observed phenomenon in terms of the temperature dependence of heat transfers from solid and liquid volumes already at their bulk coexistence temperature towards adjacent still supercooled liquid regions. These findings can be justified by observing that convective motions induced by thermal gradients in a supercooled liquid near coexistence are rapidly suppressed as the nucleated solid fraction overcomes, at low enough temperatures, a characteristic percolation threshold.

Collective translational and rotational Monte Carlo cluster move for general pairwise interaction

Virtual move Monte Carlo is a cluster algorithm which was originally developed for strongly attractive colloidal, molecular, or atomistic systems in order to both approximate the collective dynamics and avoid sampling of unphysical kinetic traps. In this paper, we present the algorithm in the form, which selects the moving cluster through a wider class of virtual states and which is applicable to general pairwise interactions, including hard-core repulsion. The newly proposed way of selecting the cluster increases the acceptance probability by up to several orders of magnitude, especially for rotational moves. The results have their applications in simulations of systems interacting via anisotropic potentials both to enhance the sampling of the phase space and to approximate the dynamics.

Salting out the polar polymorph: analysis by alchemical solvent transformation

We computationally examine how adding NaCl to an aqueous solution with α- and γ-glycine nuclei alters the structure and interfacial energy of the nuclei. The polar γ-glycine nucleus in pure aqueous solution develops a melted layer of amorphous glycine around the nucleus. When NaCl is added, a double layer is formed that stabilizes the polar glycine polymorph and eliminates the surface melted layer. In contrast, the non-polar α-glycine nucleus is largely unaffected by the addition of NaCl. To quantify the stabilizing effect of NaCl on γ-glycine nuclei, we alchemically transform the aqueous glycine solution into a brine solution of glycine. The alchemical transformation is performed both with and without a nucleus in solution and for nuclei of α-glycine and γ-glycine polymorphs. The calculations show that adding 80 mg/ml NaCl reduces the interfacial free energy of a γ-glycine nucleus by 7.7 mJ/m(2) and increases the interfacial free energy of an α-glycine nucleus by 3.1 mJ/m(2). Both results are consistent with experimental reports on nucleation rates which suggest: J(α, brine) < J(γ, brine) < J(α, water). For γ-glycine nuclei, Debye-Hückel theory qualitatively, but not quantitatively, captures the effect of salt addition. Only the alchemical solvent transformation approach can predict the results for both polar and non-polar polymorphs. The results suggest a general "salting out" strategy for obtaining polar polymorphs and also a general approach to computationally estimate the effects of solvent additives on interfacial free energies for nucleation.

Coarse-grained computer simulation of dynamics in thylakoid membranes: methods and opportunities

Coarse-grained simulation is a powerful and well-established suite of computational methods for studying structure and dynamics in nanoscale biophysical systems. As our understanding of the plant photosynthetic apparatus has become increasingly nuanced, opportunities have arisen for coarse-grained simulation to complement experiment by testing hypotheses and making predictions. Here, we give an overview of best practices in coarse-grained simulation, with a focus on techniques and results that are applicable to the plant thylakoid membrane-protein system. We also discuss current research topics for which coarse-grained simulation has the potential to play a key role in advancing the field.

Bubble nucleation in polymer–CO2 mixtures

We combine density-functional theory with the string method to calculate the minimum free energy path of bubble nucleation in two polymer–CO2 mixture systems, poly(methyl methacrylate) (PMMA)–CO2 and polystyrene (PS)–CO2. Nucleation is initiated by saturating the polymer liquid with high pressure CO2 and subsequently reducing the pressure to ambient condition. Below a critical temperature (Tc), we find that there is a discontinuous drop in the nucleation barrier as a function of increased initial CO2 pressure (P0), as a result of an underlying metastable transition from a CO2-rich-vapor phase to a CO2-rich-liquid phase. The nucleation barrier is generally higher for PS–CO2 than for PMMA–CO2 under the same temperature and pressure conditions, and both higher temperature and higher initial pressure are required to lower the nucleation barrier for PS–CO2 to experimentally relevant ranges. Classical nucleation theory completely fails to capture the structural features of the bubble nucleus and severely underestimates the nucleation barrier.

Water in the formation of biogenic minerals: peeling away the hydration layers

Minerals of biogenic origin form and crystallize from aqueous environments at ambient temperatures and pressures. The in vivo environment either intracellular or intercellular, contains many components that modulate both the activity of the ions which associate to form the mineral, as well as the activity and structure of the crowded water. Most of the studies about the mechanism of mineralization, that is, the detailed pathways by which the mineral ions proceed from solution to crystal state, have been carried out in relatively dilute solutions and clean solutions. These studies have considered both thermodynamic and kinetic controls. Most have not considered the water itself. Is the water a passive bystander, or is it intimately a participant in the mineral ion densification reaction? A wide range of experiments show that the mineralization pathways proceed through a series of densification stages with intermediates, such as a "dense liquid" phase and the prenucleation clusters that form within it. This is in contrast to the idea of a single step phase transition, but consistent with the Gibbs concept of discontinuous phase transitions from supersaturated mother liquor to crystal. Further changes in the water structure at every surface and interface during densification guides the free energy trajectory leading to the crystalline state. In vertebrates, mineralization takes place in a hydrated collagen matrix, thus water must be considered as a direct participant. Although different in detail, the crystallization of calcium phosphates, as apatite, and calcium carbonates, as calcite, are mechanistically identical from the viewpoint of water.

Nucleation of a stable solid from melt in the presence of multiple metastable intermediate phases: wetting, Ostwald's step rule, and vanishing polymorphs

In many systems, nucleation of a stable solid may occur in the presence of other (often more than one) metastable phases. These may be polymorphic solids or even liquid phases. Sometimes, the metastable phase might have a lower free energy minimum than the liquid but higher than the stable-solid-phase minimum and have characteristics in between the parent liquid and the globally stable solid phase. In such cases, nucleation of the solid phase from the melt may be facilitated by the metastable phase because the latter can "wet" the interface between the parent and the daughter phases, even though there may be no signature of the existence of metastable phase in the thermodynamic properties of the parent liquid and the stable solid phase. Straightforward application of classical nucleation theory (CNT) is flawed here as it overestimates the nucleation barrier because surface tension is overestimated (by neglecting the metastable phases of intermediate order) while the thermodynamic free energy gap between daughter and parent phases remains unchanged. In this work, we discuss a density functional theory (DFT)-based statistical mechanical approach to explore and quantify such facilitation. We construct a simple order-parameter-dependent free energy surface that we then use in DFT to calculate (i) the order parameter profile, (ii) the overall nucleation free energy barrier, and (iii) the surface tension between the parent liquid and the metastable solid and also parent liquid and stable solid phases. The theory indeed finds that the nucleation free energy barrier can decrease significantly in the presence of wetting. This approach can provide a microscopic explanation of the Ostwald step rule and the well-known phenomenon of "disappearing polymorphs" that depends on temperature and other thermodynamic conditions. Theory reveals a diverse scenario for phase transformation kinetics, some of which may be explored via modern nanoscopic synthetic methods.

Discontinuous Bubble Nucleation Due to a Metastable Condensation Transition in Polymer-CO2 Mixtures

We combine a newly developed density-functional theory with the string method to calculate the minimum free energy path of bubble nucleation in compressible polymer-CO2 mixtures. Nucleation is initiated by saturating the polymer liquid with high pressure CO2 and subsequently reducing the pressure to ambient condition. Below a critical temperature, we find that there is a discontinuous drop in the nucleation barrier with increased initial CO2 pressure, as a result of an underlying metastable transition from a CO2-rich-vapor phase to a CO2-rich-liquid phase. This phenomenon is different from previously proposed nucleation mechanisms involving metastable transitions.

Self-assembly of multicomponent structures in and out of equilibrium

Theories of phase change and self-assembly often invoke the idea of a "quasiequilibrium," a regime in which the nonequilibrium association of building blocks results nonetheless in a structure whose properties are determined solely by an underlying free energy landscape. Here we study a prototypical example of multicomponent self-assembly, a one-dimensional fiber grown from red and blue blocks. We find that if the equilibrium structure possesses compositional correlations different from those characteristic of random mixing, then it cannot be generated without error at any finite growth rate: there is no quasiequilibrium regime. However, by exploiting dynamic scaling, structures characteristic of equilibrium at one point in phase space can be generated, without error, arbitrarily far from equilibrium. Our results, supported by mean-field theory in higher dimensions, thus suggest a "nonperturbative" strategy for multicomponent self-assembly in which the target structure is, by design, not the equilibrium one.

Nucleation of colloids and macromolecules: does the nucleation pathway matter?

A recent description of diffusion-limited nucleation based on fluctuating hydrodynamics that extends classical nucleation theory predicts a very non-classical two-step scenario whereby nucleation is most likely to occur in spatially extended, low-amplitude density fluctuations. In this paper, it is shown how the formalism can be used to determine the maximum probability of observing any proposed nucleation pathway, thus allowing one to address the question as to their relative likelihood, including of the newly proposed pathway compared to classical scenarios. Calculations are presented for the nucleation of high-concentration droplets in a low-concentration solution of globular proteins and it is found that the relative probabilities (new theory compared to classical result) for reaching a critical nucleus containing N(c) molecules scales as e(-N(c)/3) thus indicating that for all but the smallest nuclei, the classical scenario is extremely unlikely.