A dynamical theory of nucleation for colloids and macromolecules

Classical Nucleation Theory for Active Fluid Phase Separation

Classical Nucleation Theory (CNT), linking rare nucleation events to the free-energy landscape of a growing nucleus, is central to understanding phase-change kinetics in passive fluids. Nucleation in nonequilibrium systems is much harder to describe because there is no free energy, but instead a dynamics-dependent quasipotential that typically must be found numerically. Here we extend CNT to a class of active phase-separating systems governed by a minimal field-theoretic model (Active Model B+). In the small noise and supersaturation limits that CNT assumes, we compute analytically the quasipotential, and hence, nucleation barrier, for liquid-vapor phase separation. Crucial to our results, detailed balance, although broken microscopically by activity, is restored along the instanton trajectory, which in CNT involves the nuclear radius as the sole reaction coordinate.

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.

Nucleation precursors compatible with a single energy barrier: catching the nonclassical culprit

In this work we link experimental results of SrSO4 precipitation with a mesoscopic nucleation model (MeNT) to stride towards a cohesive view of the nucleation process integrating both classical and...

Out-of-equilibrium processes in crystallization of organic-inorganic perovskites during spin coating

Complex phenomena are prevalent during the formation of materials, which affect their processing-structure-function relationships. Thin films of methylammonium lead iodide (CH₃NH₃PbI₃, MAPI) are processed by spin coating, antisolvent drop, and annealing of colloidal precursors. The structure and properties of transient and stable phases formed during the process are reported, and the mechanistic insights of the underlying transitions are revealed by combining in situ data from grazing-incidence wide-angle X-ray scattering and photoluminescence spectroscopy. Here, we report the detailed insights on the embryonic stages of organic-inorganic perovskite formation. The physicochemical evolution during the conversion proceeds in four steps: i) An instant nucleation of polydisperse MAPI nanocrystals on antisolvent drop, ii) the instantaneous partial conversion of metastable nanocrystals into orthorhombic solvent-complex by cluster coalescence, iii) the thermal decomposition (dissolution) of the stable solvent-complex into plumboiodide fragments upon evaporation of solvent from the complex and iv) the formation (recrystallization) of cubic MAPI crystals in thin film.

Complex phenomena are prevalent during the formation of materials, and they affect the processing structure-function relationship. Here the authors elucidate the stochastic transformation processes happening during the spin coating of perovskite colloidal precursors by multimodal characterization.

Long-wavelength density fluctuations as nucleation precursors

Recent theories of nucleation that go beyond classical nucleation theory predict that diffusion-limited nucleation of both liquid droplets and of crystals from a low-density vapor (or weak solution) begins with long-wavelength density fluctuations. This means that in the early stages of nucleation, "clusters" can have low density but large spatial extent, which is at odds with the classical picture of arbitrarily small clusters of the condensed phase. We present the results of kinetic Monte Carlo simulations using forward flux sampling to show that these predictions are confirmed, namely, that on average, nucleation begins in the presence of low-amplitude, but spatially extended density fluctuations thus confirming a significant prediction of the nonclassical theory.

Could Microwave Irradiation Cause Misfolding of Peptides?

Microwaves have been experimentally shown to affect the folding dynamics of peptides and proteins. Using molecular dynamics, we performed all-atom simulations of a model β-peptide in aqueous solution where individual degrees of freedom of solvent molecules were decoupled to allow for investigation at non-equilibrium microwave-irradiated conditions. An elevated rotational temperature of the water medium was found to significantly affect the conformation of the peptide due to the weakened hydrogen-bonding interactions with the surrounding solvent molecules. Cluster analysis revealed that microwave irradiation can indeed act as a promoter in the formation of new misfolded peptide structures of the hairpin type, which are generally associated with the onset of several neurodegenerative disorders such as Alzheimer’s, Parkinson’s, Huntington’s, and Creutzfeldt–Jakob diseases as well as certain cancer types such as amyloidosis.

Derivation of an exact, nonequilibrium framework for nucleation: Nucleation is a priori neither diffusive nor Markovian

We discuss the structure of the equation of motion that governs nucleation processes at first order phase transitions. From the underlying microscopic dynamics of a nucleating system, we derive by means of a nonequilibrium projection operator formalism the equation of motion for the size distribution of the nuclei. The equation is exact, i.e., the derivation does not contain approximations. To assess the impact of memory, we express the equation of motion in a form that allows for direct comparison to the Markovian limit. As a numerical test, we have simulated crystal nucleation from a supersaturated melt of particles interacting via a Lennard-Jones potential. The simulation data show effects of non-Markovian dynamics.

Morphology formation in binary mixtures upon gradual destabilisation

Spontaneous liquid-liquid phase separation is commonly understood in terms of phenomenological mean-field theories. These theories correctly predict the structural features of the fluid at sufficiently long time scales and wavelengths. However, these conditions are not met in various examples in biology and materials science where the mixture is slowly destabilised, and phase separation is strongly affected by critical thermal fluctuations. We propose a mechanism of pretransitional structuring of a mixture that approaches the miscibility gap and predict scaling relations that describe how the characteristic feature size of the emerging morphology decreases with an increasing quench rate. These predictions quantitatively agree with our kinetic Monte Carlo and molecular dynamics simulations of a phase-separating binary mixture, as well as with previously reported experimental observations. We discuss how these predictions are affected by non-conserved order parameters (e.g., due to chemical reactions or alignment of liquid-crystalline molecules), hydrodynamics and active transport.

The Generalized Stochastic Smoluchowski Equation

We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy 17, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids.

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.

Instability, Rupture and Fluctuations in Thin Liquid Films: Theory and Computations

Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction, as opposed to the perfectly uncorrelated limit studied by Grün et al. (J Stat Phys 122(6):1261-1291, 2006). We also present a numerical scheme based on a spectral collocation method, which is then utilised to simulate the stochastic thin-film equation. This scheme seems to be very convenient for numerical studies of the stochastic thin-film equation, since it makes it easier to select the frequency modes of the noise (following the spirit of the long-wave approximation). With our numerical scheme we explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we study the effect of the noise intensity on the rupture time, using a large number of sample paths as compared to previous studies.

Breakdown of the Stokes-Einstein relation above the melting temperature in a liquid phase-change material

The dynamic properties of liquid phase-change materials (PCMs), such as viscosity η and the atomic self-diffusion coefficient D, play an essential role in the ultrafast phase switching behavior of novel nonvolatile phase-change memory applications. To connect η to D, the Stokes-Einstein relation (SER) is commonly assumed to be valid at high temperatures near or above the melting temperature T m and is often used for assessing liquid fragility (or crystal growth velocity) of technologically important PCMs. However, using quasi-elastic neutron scattering, we provide experimental evidence for a breakdown of the SER even at temperatures above T m in the high-atomic mobility state of a PCM, Ge1Sb2Te4. This implies that although viscosity may have strongly increased during cooling, diffusivity can remain high owing to early decoupling, being a favorable feature for the fast phase switching behavior of the high-fluidity PCM. We discuss the origin of the observation and propose the possible connection to a metal-semiconductor and fragile-strong transition hidden below T m.

Nucleation of protein crystals - a nanoscopic perspective

Macromolecular phase transitions bear great medical, scientific and industrial relevance, yet the molecular picture of their earliest beginnings is still far from complete. For decades, progress has been hampered by the challenges associated with studying stochastic nucleation phenomena occurring on nanoscopic length scales. In the last 5 years, however, the field has advanced with great strides due to the recent buildout of experimental techniques that allow us to observe details of the nucleation process on the nanoscale. In this review, we present a historical overview and state-of-the-art analysis of protein crystal nucleation from an experimentalist's perspective. After a short introduction of key concepts from classical nucleation theory, we discuss the advancements that have led to the development of alternative models of protein nucleation. We summarize the experimental proof in favour of these various models, but we also focus on some of their shortcomings and experimental blind spots. In our penultimate section we highlight recent works that have provided direct nanoscopic insight into the nucleation of protein crystals. We end with concluding paragraphs discussing outstanding questions and possible strategies to advance the field further in the future.

Classical density functional theory, unconstrained crystallization, and polymorphic behavior

While in principle, classical density functional theory (cDFT) should be a powerful tool for the study of crystallization, in practice this has not so far been the case. Progress has been hampered by technical problems which have plagued the study of the crystalline systems using the most sophisticated fundamental measure theory models. In this paper, the reasons for the difficulties are examined and it is proposed that the tensor functionals currently favored are in fact numerically unstable. By reverting to an older, more heuristic model it is shown that all of the technical difficulties are eliminated. Application to a Lennard-Jones fluid results in a demonstration of power of cDFT to describe crystallization in a highly inhomogeneous system. First, we show that droplets attached to a slightly hydrophobic wall crystallize spontaneously upon being quenched. The resulting crystallites are clearly faceted structures and are predominantly HCP structures. In contrast, droplets in a fully periodic calculational cell remain stable to lower temperatures and eventually show the same spontaneous localization of the density into "atoms" but in an amorphous structure having many of the structural characteristics of a glass. A small change of the protocol leads, at the same temperature, to the formation of crystals, this time with the fcc structure typical of bulk Lennard-Jones solids. The fcc crystals have lower free energy than the amorphous structures which in turn are more stable than the liquid droplets. It is demonstrated that as the temperature is raised, the free energy differences between the structures decrease until the solid clusters become less stable than the liquid droplets and spontaneously melt. The presence of energy barriers separating the various structures is therefore clearly demonstrated.

Dynamical density functional theory with hydrodynamic interactions in confined geometries

We study the dynamics of colloidal fluids in both unconfined geometries and when confined by a hard wall. Under minimal assumptions, we derive a dynamical density functional theory (DDFT) which includes hydrodynamic interactions (HI; bath-mediated forces). By using an efficient numerical scheme based on pseudospectral methods for integro-differential equations, we demonstrate its excellent agreement with the full underlying Langevin equations for systems of hard disks in partial confinement. We further use the derived DDFT formalism to elucidate the crucial effects of HI in confined systems.

Structuring of Fluid Adlayers upon Ongoing Unimolecular Adsorption

Fluids with spatial density variations of single or mixed molecules play a key role in biophysics, soft matter, and materials science. The fluid structures usually form via spinodal decomposition or nucleation following an instantaneous destabilization of the initially disordered fluid. However, in practice, an instantaneous quench is often not viable, and the rate of destabilization may be gradual rather than instantaneous. In this work we show that the commonly used phenomenological descriptions of fluid structuring are inadequate under these conditions. We come to that conclusion in the context of surface catalysis, where we employ kinetic Monte Carlo simulations to describe the unimolecular adsorption of gaseous molecules onto a metal surface. The adsorbates diffuse at the surface and, as a consequence of lateral interactions and due to an ongoing increase of the surface coverage, phase separate into coexisting low- and high-density regions. The typical size of these regions turns out to depend much more strongly on the rate of adsorption than predicted from recently reported phenomenological models. We discuss how this finding contributes to the fundamental understanding of the crossover from liquid-liquid to liquid-solid demixing of solution-cast polymer blends.

Nucleation pathway and kinetics of phase-separating active Brownian particles

Suspensions of purely repulsive but self-propelled Brownian particles might undergo phase separation, a phenomenon that strongly resembles the phase separation of passive particles with attractions. Here we employ computer simulations to study the nucleation kinetics and the microscopic pathway active Brownian disks take in two dimensions when quenched from the homogeneous suspension to propulsion speeds beyond the binodal. We find the same qualitative behavior for the nucleation rate as a function of density as for a passive suspension undergoing liquid-vapor separation, suggesting that the scenario of an effective free energy also extends to the kinetics of phase separation. We study the transition in more detail through a committor analysis and find that transition states are best described by a combination of cluster size and the radial polarization of particles in the cluster.

Mechanism for the stabilization of protein clusters above the solubility curve: the role of non-ideal chemical reactions

Dense protein clusters are known to play an important role in nucleation of protein crystals from dilute solutions. While these have generally been thought to be formed from a metastable phase, the observation of similar, if not identical, clusters above the critical point for the dilute-solution/strong-solution phase transition has thrown this into doubt. Furthermore, the observed clusters are stable for relatively long times. Because protein aggregation plays a central role in some pathologies, understanding the nature of such clusters is an important problem. One mechanism for the stabilization of such structures was proposed by Pan, Vekilov and Lubchenko and was investigated using a dynamical density functional theory model which confirmed the viability of the model. Here, we revisit that model and incorporate additional physics in the form of state-dependent reaction rates. We show by a combination of numerical results and general arguments that the state-dependent rates disrupt the stability mechanism. Finally, we argue that the state-dependent reactions correct unphysical aspects of the model with ideal (state-independent) reactions and that this necessarily leads to the failure of the proposed mechanism.

Formation of calcium sulfate through the aggregation of sub-3 nanometre primary species

The formation pathways of gypsum remain uncertain. Here, using truly in situ and fast time-resolved small-angle X-ray scattering, we quantify the four-stage solution-based nucleation and growth of gypsum (CaSO₄·2H₂O), an important mineral phase on Earth and Mars. The reaction starts through the fast formation of well-defined, primary species of <3 nm in length (stage I), followed in stage II by their arrangement into domains. The variations in volume fractions and electron densities suggest that these fast forming primary species contain Ca–SO₄-cores that self-assemble in stage III into large aggregates. Within the aggregates these well-defined primary species start to grow (stage IV), and fully crystalize into gypsum through a structural rearrangement. Our results allow for a quantitative understanding of how natural calcium sulfate deposits may form on Earth and how a terrestrially unstable phase-like bassanite can persist at low-water activities currently dominating the surface of Mars.

The quantitative understanding of how gypsum nucleates and grows from aqueous solutions is limited. Here, the authors demonstrate how, by using truly in situ and fast time-resolved small-angle X-ray scattering, the four-stage solution-based nucleation and growth of this mineral can be quantified.

Equilibrium fluid-crystal interfacial free energy of bcc-crystallizing aqueous suspensions of polydisperse charged spheres

The interfacial free energy is a central quantity in crystallization from the metastable melt. In suspensions of charged colloidal spheres, nucleation and growth kinetics can be accurately measured from optical experiments. In previous work, from these data effective nonequilibrium values for the interfacial free energy between the emerging bcc nuclei and the adjacent melt in dependence on the chemical potential difference between melt phase and crystal phase were derived using classical nucleation theory (CNT). A strictly linear increase of the interfacial free energy was observed as a function of increased metastability. Here, we further analyze these data for five aqueous suspensions of charged spheres and one binary mixture. We utilize a simple extrapolation scheme and interpret our findings in view of Turnbull's empirical rule. This enables us to present the first systematic experimental estimates for a reduced interfacial free energy, σ(0,bcc), between the bcc-crystal phase and the coexisting equilibrium fluid. Values obtained for σ(0,bcc) are on the order of a few k(B)T. Their values are not correlated to any of the electrostatic interaction parameters but rather show a systematic decrease with increasing size polydispersity and a lower value for the mixture as compared to the pure components. At the same time, σ(0) also shows an approximately linear correlation to the entropy of freezing. The equilibrium interfacial free energy of strictly monodisperse charged spheres may therefore be still greater.

Mechanism for the stabilization of protein clusters above the solubility curve

Pan, Vekilov and Lubchenko [J. Phys. Chem. B, 2010, 114, 7620] have proposed that dense stable protein clusters appearing in weak protein solutions above the solubility curve are composed of protein oligomers. The hypothesis is that a weak solution of oligomer species is unstable with respect to condensation causing the formation of dense, oligomer-rich droplets which are stabilized against growth by the monomer-oligomer reaction. Here, we show that such a combination of processes can be understood using a simple capillary model yielding analytic expressions for the cluster properties which can be used to interpret experimental data. We also construct a microscopic Dynamic Density Functional Theory model and show that it is consistent with the predictions of the capillary model. The viability of the mechanism is thus confirmed and it is shown how the radius of the stable clusters is related to physically interesting quantities such as the monomer-oligomer rate constants.

Unification of classical nucleation theories via a unified Itô-Stratonovich stochastic equation

Classical nucleation theory (CNT) is the most widely used framework to describe the early stage of first-order phase transitions. Unfortunately, the different points of view adopted to derive it yield different kinetic equations for the probability density function, e.g., Zeldovich-Frenkel or Becker-Döring-Tunitskii equations. Starting from a phenomenological stochastic differential equation, a unified equation is obtained in this work. In other words, CNT expressions are recovered by selecting one or another stochastic calculus. Moreover, it is shown that the unified CNT thus obtained produces the same Fokker-Planck equation as that from a recent update of CNT [J. F. Lutsko and M. A. Durán-Olivencia, J. Chem. Phys. 138, 244908 (2013)10.1063/1.4811490] when mass transport is governed by diffusion. Finally, we derive a general induction-time expression along with specific approximations of it to be used under different scenarios, in particular, when the mass-transport mechanism is governed by direct impingement, volume diffusion, surface diffusion, or interface transfer.

A two-parameter extension of classical nucleation theory

A two-variable stochastic model for diffusion-limited nucleation is developed using a formalism derived from fluctuating hydrodynamics. The model is a direct generalization of the standard classical nucleation theory (CNT). The nucleation rate and pathway are calculated in the weak-noise approximation and are shown to be in good agreement with direct numerical simulations for the weak-solution/strong-solution transition in globular proteins. We find that CNT underestimates the time needed for the formation of a critical cluster by two orders of magnitude and that this discrepancy is due to the more complex dynamics of the two variable model and not, as often is assumed, a result of errors in the estimation of the free energy barrier.

Mesoscopic nucleation theory for confined systems: a one-parameter model

Classical nucleation theory has been recently reformulated based on fluctuating hydrodynamics [J. F. Lutsko and M. A. Durán-Olivencia, Classical nucleation theory from a dynamical approach to nucleation, J. Chem. Phys. 138, 244908 (2013). The present work extends this effort to the case of nucleation in confined systems such as small pores and vesicles. The finite available mass imposes a maximal supercritical cluster size and prohibits nucleation altogether if the system is too small. We quantity the effect of system size on the nucleation rate. We also discuss the effect of relaxing the capillary-model assumption of zero interfacial width resulting in significant changes in the nucleation barrier and nucleation rate.

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.

Crystallization kinetics of colloidal model suspensions: recent achievements and new perspectives

Colloidal model systems allow studying crystallization kinetics under fairly ideal conditions, with rather well-characterized pair interactions and minimized external influences. In complementary approaches experiment, analytic theory and simulation have been employed to study colloidal solidification in great detail. These studies were based on advanced optical methods, careful system characterization and sophisticated numerical methods. Over the last decade, both the effects of the type, strength and range of the pair-interaction between the colloidal particles and those of the colloid-specific polydispersity have been addressed in a quantitative way. Key parameters of crystallization have been derived and compared to those of metal systems. These systematic investigations significantly contributed to an enhanced understanding of the crystallization processes in general. Further, new fundamental questions have arisen and (partially) been solved over the last decade: including, for example, a two-step nucleation mechanism in homogeneous nucleation, choice of the crystallization pathway, or the subtle interplay of boundary conditions in heterogeneous nucleation. On the other hand, via the application of both gradients and external fields the competition between different nucleation and growth modes can be controlled and the resulting microstructure be influenced. The present review attempts to cover the interesting developments that have occurred since the turn of the millennium and to identify important novel trends, with particular focus on experimental aspects.

Driving ordering processes in molecular-dynamics simulations

Self-organized criticality describes the emergence of complexity in dynamical nonequilibrium systems. In this paper we introduce a unique approach whereby a driven energy conversion is utilized as a sampling bias for ordered arrangements in molecular dynamics simulations of atomic and molecular fluids. This approach gives rise to dramatically accelerated nucleation rates, by as much as 30 orders of magnitude, without the need of predefined order parameters, which commonly employed rare-event sampling methods rely on. The measured heat fluxes suggest how the approach can be generalized.

Random transitions described by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model

We study random transitions between two metastable states that appear below a critical temperature in a one-dimensional self-gravitating Brownian gas with a modified Poisson equation experiencing a second order phase transition from a homogeneous phase to an inhomogeneous phase [P. H. Chavanis and L. Delfini, Phys. Rev. E 81, 051103 (2010)]. We numerically solve the N-body Langevin equations and the stochastic Smoluchowski-Poisson system, which takes fluctuations (finite N effects) into account. The system switches back and forth between the two metastable states (bistability) and the particles accumulate successively at the center or at the boundary of the domain. We explicitly show that these random transitions exhibit the phenomenology of the ordinary Kramers problem for a Brownian particle in a double-well potential. The distribution of the residence time is Poissonian and the average lifetime of a metastable state is given by the Arrhenius law; i.e., it is proportional to the exponential of the barrier of free energy ΔF divided by the energy of thermal excitation kBT. Since the free energy is proportional to the number of particles N for a system with long-range interactions, the lifetime of metastable states scales as eN and is considerable for N≫1. As a result, in many applications, metastable states of systems with long-range interactions can be considered as stable states. However, for moderate values of N, or close to a critical point, the lifetime of the metastable states is reduced since the barrier of free energy decreases. In that case, the fluctuations become important and the mean field approximation is no more valid. This is the situation considered in this paper. By an appropriate change of notations, our results also apply to bacterial populations experiencing chemotaxis in biology. Their dynamics can be described by a stochastic Keller-Segel model that takes fluctuations into account and goes beyond the usual mean field approximation.

Heterogeneous nucleation of ice on carbon surfaces

Atmospheric aerosols can promote the heterogeneous nucleation of ice, impacting the radiative properties of clouds and Earth's climate. The experimental investigation of heterogeneous freezing of water droplets by carbonaceous particles reveals widespread ice freezing temperatures. It is not known which structural and chemical characteristics of soot account for the variability in ice nucleation efficiency. Here we use molecular dynamics simulations to investigate the nucleation of ice from liquid water in contact with graphitic surfaces. We find that atomically flat carbon surfaces promote heterogeneous nucleation of ice, while molecularly rough surfaces with the same hydrophobicity do not. Graphitic surfaces and other surfaces that promote ice nucleation induce layering in the interfacial water, suggesting that the order imposed by the surface on liquid water may play an important role in the heterogeneous nucleation mechanism. We investigate a large set of graphitic surfaces of various dimensions and radii of curvature and find that variations in nanostructures alone could account for the spread in the freezing temperatures of ice on soot in experiments. We conclude that a characterization of the nanostructure of soot is needed to predict its ice nucleation efficiency.

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.

The microscopic pathway to crystallization in supercooled liquids

Despite its fundamental and technological importance, a microscopic understanding of the crystallization process is still elusive. By computer simulations of the hard-sphere model we reveal the mechanism by which thermal fluctuations drive the transition from the supercooled liquid state to the crystal state. In particular we show that fluctuations in bond orientational order trigger the nucleation process, contrary to the common belief that the transition is initiated by density fluctuations. Moreover, the analysis of bond orientational fluctuations shows that these not only act as seeds of the nucleation process, but also i) determine the particular polymorph which is to be nucleated from them and ii) at high density favour the formation of fivefold structures which can frustrate the formation of crystals. These results can shed new light on our understanding of the relationship between crystallization and vitrification.