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A new kinetic Monte Carlo simulation approach for crystal growth from solution is presented. The simulations include an explicit representation of an extended volume of solution above the crystal face and realistically treats voids within the forming crystal, step overhangs and concentration gradients in the mother solution. The effect of impurities on step growth is investigated and it is shown that the results depend on a complex interplay between concentration gradients in solution driven by the consumption of material during step growth, the lifetime of impurities on the crystal surface and the geometry of the steps.

The original derivation of Power Functional Theory, Schmidt and Brader, JCP 138, 214101 (2013), is reworked in some detail with a view to clarifying and simplifying the logic and making explicit the various functional dependencies. We note various issues with the original development and suggest a modification that allows us to avoid them. In the process we also suggest an alternative interpretation of our results that bears surprising similarities to classical Density Functional Theory., Comment: Second author added; completely rewritten

The standard model of classical density-functional theory (cDFT) for pair potentials consists of a hard-sphere functional plus a mean-field term accounting for long ranged attraction. However, most implementations using sophisticated fundamental measure hard-sphere functionals suffer from potential numerical instabilities either due to possible instabilities in the functionals themselves or due to implementations that mix real- and Fourier-space components inconsistently. Here we present a new implementation based on a demonstrably stable hard-sphere functional that is implemented in a completely consistent manner. The present work does not depend on approximate spherical integration schemes and so is much more robust than previous algorithms. The methods are illustrated by calculating phase diagrams for the solid state using the standard Lennard-Jones potential as well as a new class of potentials recently proposed by Wang et al. [Phys. Chem. Chem. Phys. 22, 10624 (2020)PPCPFQ1463-907610.1039/C9CP05445F]. The latter span the range from potentials for small molecules to those appropriate to colloidal systems simply by varying a parameter. We verify that cDFT is able to semiquantitatively reproduce the phase diagram in all cases. We also show that for these problems computationally cheap Gaussian approximations are nearly as good as full minimization based on finite differences., SCOPUS: ar.j, info:eu-repo/semantics/published

The derivation of the state of the art tensorial versions of Fundamental Measure Theory (a form of classical Density Functional Theory for hard spheres) is reexamined in the light of the recently introduced concept of global stability of the density functional based on its boundedness [Lutsko and Lam, Phys. Rev. E 98, 012604 (2018)2470-004510.1103/PhysRevE.98.012604]. It is shown that within the present paradigm, explicit stability of the functional can be achieved only at the cost of giving up accuracy at low densities. It is argued that this is an acceptable trade-off since the main value of DFT lies in the study of dense systems. Explicit calculations for a wide variety of systems show that a proposed explicitly stable functional is competitive in all ways with the popular White Bear models while sharing some of their weaknesses when applied to non-close-packed solids., SCOPUS: ar.j, info:eu-repo/semantics/published

Standard solid-on-solid models of crystal growth represent the appearance and disappearance of molecules on the crystal surface as stochastic events. Here, in order to model more realistically the growth of crystals from solution, we introduce a dynamic model of the fluid in contact with the crystal surface and in this way account explicitly for mass transport in the solution. We present our hybrid model and establish its relation to standard SOS simulations while demonstrating explicit effects of mass transport that are usually ignored in SOS simulations. We then introduce flow in the solution in directions parallel to the crystal surface and observe dramatic effects on crystal growth for flows perpendicular to the step front as well as for flows parallel to the step. In order to understand the latter, quite unexpected feature, we study the effect of flow on isolated islands on a crystal surface and show that flow in the solution can induce spontaneous movement of 2D islands on the crystal surface.

A novel microfluidic device that subjects a solution to a constant shear flow was developed. By taking advantage of the linear velocity profile in a lid driven flow configuration, small volumes (10-5 L) can be subjected to a constant shear profile with a shear rate between 0.1 and 100 s-1 at accurately controlled temperatures between 20 and 50 °C. The tunable shear can be maintained for extensive and fully controlled times. A dedicated microscope setup for visualization enables the on-chip detection of micron-sized crystals, particles, and aggregates. The influence of shear on the crystallization process of the reference protein lysozyme was studied. The results indicate that shear rates between 1 and 10 s-1 decrease solubility and promote nucleation not only in the supersaturated and metastable zones of the phase diagram, but also in the undersaturated zone. A monotonically increasing nucleation rate was observed for shear rates between 1 and 10 s-1. It is anticipated that the presented methodology can shed light on a variety of phase transitions that are influenced by flow., info:eu-repo/semantics/published

Ubiquitous processes in nature and the industry exploit crystallization from multicomponent environments1–5; however, laboratory efforts have focused on the crystallization of pure solutes6,7 and the effects of single growth modifiers8,9. Here we examine the molecular mechanisms employed by pairs of inhibitors in blocking the crystallization of haematin, which is a model organic compound with relevance to the physiology of malaria parasites10,11. We use a combination of scanning probe microscopy and molecular modelling to demonstrate that inhibitor pairs, whose constituents adopt distinct mechanisms of haematin growth inhibition, kink blocking and step pinning12,13, exhibit both synergistic and antagonistic cooperativity depending on the inhibitor combination and applied concentrations. Synergism between two crystal growth modifiers is expected, but the antagonistic cooperativity of haematin inhibitors is not reflected in current crystal growth models. We demonstrate that kink blockers reduce the line tension of step edges, which facilitates both the nucleation of crystal layers and step propagation through the gates created by step pinners. The molecular viewpoint on cooperativity between crystallization modifiers provides guidance on the pairing of modifiers in the synthesis of crystalline materials. The proposed mechanisms indicate strategies to understand and control crystallization in both natural and engineered systems, which occurs in complex multicomponent media1–3,8,9. In a broader context, our results highlight the complexity of crystal–modifier interactions mediated by the structure and dynamics of the crystal interface., SCOPUS: ar.j, info:eu-repo/semantics/published

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., SCOPUS: ar.j, info:eu-repo/semantics/published

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., SCOPUS: ar.j, info:eu-repo/semantics/published

Ubiquitous processes in nature and the industry exploit crystallization from multicomponent environments

The temporal Fokker–Planck equation (Boon et al. in J Stat Phys 3/4: 527, 2003) or propagation–dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. We present two generalizations of the temporal Fokker–Planck equation for the first passage distribution function $$f_j(r,t)$$ of a particle moving on a substrate with time delays $$\tau _j$$ . Both generalizations follow from the first visit recurrence relation. In the first case, the time delays depend on the local concentration, that is the time delay probability $$P_j$$ is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, $$P_j \propto f_j^{\nu - 1}$$ , the generalized Fokker–Planck equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the situation where the time delays are distributed according to a power law, $$P_j \propto \tau _j^{-1-\alpha }$$ (with $$0< \alpha < 2$$ ), in which case we obtain a fractional propagation-dispersion equation which is the temporal analog of the fractional spatial diffusion equation (with space and time interchanged). The analysis shows how certain microscopic mechanisms can lead to non-Gaussian distributions and non-classical scaling exponents.

Solvent-mediated interactions emerge from complex mechanisms that depend on the solute structure, its wetting properties, and the nature of the liquid. While numerous studies have focused on the first two influences, here, we compare the results from water and Lennard-Jones liquid in order to reveal to what extent solvent-mediated interactions are universal with respect to the nature of the liquid. Besides the influence of the liquid, the results were obtained with classical density functional theory and brute-force molecular dynamics simulations which allow us to contrast these two numerical techniques., SCOPUS: ar.j, info:eu-repo/semantics/published

A great deal of experimental evidence suggests that a wide spectrum of phase transitions occur in a multistage manner via the appearance and subsequent transformation of intermediate metastable states. Such multistage mechanisms cannot be explained within the realm of the classical nucleation framework. Hence, there is a strong need to develop new theoretical tools to explain the occurrence and nature of these ubiquitous intermediate phases. Here we outline a unified and self-consistent theoretical framework to describe both classical and nonclassical nucleation. Our framework provides a detailed explanation of the whole multistage nucleation pathway showing in particular that the pathway involves a single energy barrier and it passes through a dense phase, starting from a low-density initial phase, before reaching the final stable state. Moreover, we demonstrate that the kinetics of matter inside subcritical clusters favors the formation of nucleation clusters with an intermediate density, i.e. nucleation precursors. Remarkably, these nucleation precursors are not associated with a local minimum of the thermodynamic potential, as commonly assumed in previous phenomenological approaches. On the contrary, we find that they emerge due to the competition between thermodynamics and kinetics of cluster formation. Thus, the mechanism uncovered for the formation of intermediate phases can be used to explain recently reported experimental findings in crystallization: up to now such phases were assumed a consequence of some complex energy landscape with multiple energy minima. Using fundamental concepts from kinetics and thermodynamics, we provide a satisfactory explanation for the so-called nonclassical nucleation pathways observed in experiments.

The foundation for any discussion of first order phase transitions is classical nucleation theory (CNT). CNT, developed in the first half of the twentieth century, is based on a number of heuristically plausible assumptions and the majority of theoretical work on nucleation is devoted to refining or extending these ideas. Ideally, one would like to derive CNT from a more fundamental description of nucleation so that its extension, development and refinement could be developed systematically. In this paper, such a development is described based on a previously established (Lutsko 2012 J. Chem. Phys. 136 034509) connection between CNT and fluctuating hydrodynamics. Here, this connection is described without the need for artificial assumptions such as spherical symmetry. The results are illustrated by application to CNT with moving clusters (a long-standing problem in the literature) and the construction of CNT for ellipsoidal clusters., SCOPUS: ar.j, info:eu-repo/semantics/published

Targeting specific technological applications requires the control of nanoparticle properties, especially the crystalline polymorph. Freezing a nanodroplet deposited on a solid substrate leads to the formation of crystalline structures. We study the inherent mechanisms underlying this general phenomenon by means of molecular dynamics simulations. Our work shows that different crystal structures can be selected by finely tuning the solid substrate lattice parameter. Indeed, while for our system, face-centered cubic is usually the most preponderant structure, the growth of two distinct polymorphs, hexagonal centered packing and body-centered cubic, was also observed even when the solid substrate was face-centered cubic. Finally, we also demonstrated that the growth of hexagonal centered packing is conditioned by the appearance of large enough body-centered cubic clusters thus suggesting the presence of a cross-nucleation pathway. Our results provide insights into the impact of nanoscale effects and solid substrate properties towards the growth of polymorphic nanomaterials.

The nonlinear theory of anomalous diffusion is based on particle interactions giving an explicit microscopic description of diffusive processes leading to sub-, normal, or super-diffusion as a result competitive effects between attractive and repulsive interactions. We present the explicit analytical solution to the nonlinear diffusion equation which we then use to compute the correlation function which is experimentally measured by correlation spectroscopy. The theoretical results are applicable in particular to the analysis of fluorescence correlation spectroscopy of marked molecules in biological systems. More specifically we consider the case of fluorescently labeled lipids and we find that the nonlinear correlation spectrum reproduces very well the experimental data indicating sub-diffusive molecular motion of lipid molecules in the cell membrane.

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., SCOPUS: ar.j, info:eu-repo/semantics/published

More often than not, minerals formed in nature are grown at low supersaturation and from sources that are impure with respect to the crystals' main building blocks. Quite paradoxically, these conditions are in conflict with the established crystal growth theories that focus on the interplay between the crystal interface and impurities that are present in the growth medium. These theories predict a kinetic dead zone for the cases where low purity is combined with weak driving forces. Hints toward reconciling this apparent disparity have been given by the observation that a specific class of steps, so-called macrosteps, can circumvent the debilitating kinetic effects of impurities in ways that up until now are poorly understood. In this contribution, we examine the mechanism of crystal growth by means of kinetic Monte Carlo simulation at conditions close to impurity-induced kinetic arrest. In agreement with previous reports, we show that as a result of impurity binding to the crystal surface, steps spontaneously group into bunches and later condense into macrosteps. A kinetic analysis demonstrates that these macrosteps are able to evade crystal growth cessation under conditions where single steps are firmly pinned. We identify the mechanism of interstep cooperativity which leads to cessation evasion by macrosteps and demonstrate that it applies to a range of supersaturation and impurity concentration values. On the basis of these findings, we present a model that explains how minerals can grow from mother liquor solutions that would otherwise seem to be nonconducive to crystal growth.

Nanoscopic pores are used in various systems to attract nanoparticles. In general the behaviour is a result of two types of interactions : the material specific affinity and the solvent-mediated influence also called the depletion force. The latter is more universal but also much more complex to understand since it requires modeling both the nanoparticle and the solvent. Here, we employed classical density functional theory to determine the forces acting on a nanoparticle near a nanoscopic pore as a function of its hydrophobicity and its size. A simple capillary model is constructed to predict those depletion forces for various surface properties. For a nanoscopic pore, complexity arises from both the specific geometry and the fact that hydrophobic pores are not necessarily filled with liquid. Taking all of these effects into account and including electrostatic effects, we establish a phase diagram describing the entrance and the rejection of the nanoparticle from the pore.

The growth of crystals from solution is a fundamental process of relevance to such diverse areas as X-ray diffraction structural determination and the role of mineralization in living organisms. A key factor determining the dynamics of crystallization is the effect of impurities on step growth. For over 50 years, all discussions of impurity–step interaction have been framed in the context of the Cabrera–Vermilyea (CV) model for step blocking, which has nevertheless proven difficult to validate experimentally. Here we report on extensive computer simulations which clearly falsify the CV model, suggesting a more complex picture. While reducing to the CV result in certain limits, our approach is more widely applicable, encompassing nontrivial impurity–crystal interactions, mobile impurities, and negative growth, among others.

The last 25 years have seen an explosion of interest in the subject of nucleation driven by new experimental techniques and computer simulation methods. The theoretical community has struggled to keep pace with this onslaught. One of the main reasons is an adherence to the paradigm of classical nucleation theory—a theory more than 80 years old and yet still the dominant conceptual framework. In this chapter, a fundamentally new approach to the theoretical description of nucleation is outlined. It is based on the understanding that nucleation is a nonequilibrium process for which equilibrium approximations are of limited applicability. Nevertheless, the theory reproduces classical nucleation theory (CNT) in appropriate limits while offering the flexibility to fully describe with the new paradigms arising from the experiment and simulation.

A recurring question in nonequilibrium statistical mechanics is what deviation from standard statistical mechanics gives rise to non-Boltzmann behavior and to nonlinear response, which amounts to identifying the emergence of “statistics from dynamics” in systems out of equilibrium. Among several possible analytical developments which have been proposed, the idea of nonextensive statistics introduced by Tsallis about 20 years ago was to develop a statistical mechanical theory for systems out of equilibrium where the Boltzmann distribution no longer holds, and to generalize the Boltzmann entropy by a more general function S q while maintaining the formalism of thermodynamics. From a phenomenological viewpoint, nonextensive statistics appeared to be of interest because maximization of the generalized entropy S q yields the q-exponential distribution which has been successfully used to describe distributions observed in a large class of phenomena, in particular power law distributions for q > 1 . Here we re-examine the validity of the nonextensive formalism for continuous Hamiltonian systems. In particular we consider the q-ideal gas, a model system of quasi-particles where the effect of the interactions are included in the particle properties. On the basis of exact results for the q-ideal gas, we find that the theory is restricted to the range q 1 , which raises the question of its formal validity range for continuous Hamiltonian systems.

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 combinations 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 only appears important in the case of small excluded volumes such as for simple fluids. The DFT calculations are then used to study homogeneous nucleation of the high-density phase from the low-density phase including the nucleation barriers, nucleation pathways and the rate. It is found that the nucleation barriers are typically only a few $k_{B}T$ and that the nucleation rates substantially higher than would be predicted by Classical Nucleation Theory., Comment: To appear in Langmuir

There is increasing evidence that large classes of colloid materials crystallize via a non-standard nucleation mechanism involving intermediate metastable phases. In this paper recent work on the microscopic derivation of the phase diagram and free energy barriers in the nucleation of protein crystals, and on the kinetics of growth of solid particles in the post-nucleation regime is reviewed. The extent to which combined structural and density fluctuations give rise to favourable crystallization pathways involving an intermediate fluid phase is assessed and the connection with experiments in microgravity at ISS (PromISS-2 and NANOSLAB-2) is discussed.

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.

Crystals grow by laying down new layers of material which can either correspond in size to the height of one unit cell (elementary steps) or multiple unit cells (macrosteps). Surprisingly, experiments have shown that macrosteps can grow under conditions of low supersaturation and high impurity density such that elementary step growth is completely arrested. We use atomistic simulations to show that this is due to two effects: the fact that the additional layers bias fluctuations in the position of the bottom layer towards growth and by a transition, as step height increases, from a 2D to a 3D nucleation mechanism.

Pan, Vekilov and Lubchenko[\textit{J. Phys. Chem. B}, 2010, \textbf{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.

Nanoscale self-assembly is naturally subject to impediments at the nanoscale. The recently developed ability to follow processes at the molecular level forces us to resolve older, coarse-grained concepts in terms of their molecular mechanisms. In this Letter, we highlight one such example. We present evidence based on experimental and simulation data that one of the cornerstones of crystal growth theory, the Cabrera-Vermilyea model of step advancement in the presence of impurities, is based on incomplete physics. We demonstrate that the piercing of an impurity fence by elementary steps is not solely determined by the Gibbs-Thomson effect, as assumed by Cabrera-Vermilyea. Our data show that for conditions leading up to growth cessation, step retardation is dominated by the formation of critically sized fluctuations. The growth recovery of steps is counter to what is typically assumed, not instantaneous. Our observations on mesoscopic impurities for lysozyme expose a nucleation-dominated regime of growth that has not been hitherto considered, where the system alternates between zero and near-pure velocity. The time spent by the system in arrest is the nucleation induction time required for the step to amass a supercritical fluctuation that pierces the impurity fence.

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. 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 Classical Nucleation Theory 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., Comment: 25 pages, 12 figures

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-D\"oring-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. Dur\'an-Olivencia, J. Chem. Phys., 2013, 138, 244908] 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., Comment: 1 figure

Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker-Planck equation to account for non-classical diffusion. These nonlinear equations are usually constructed on a phenomenological basis. Here we introduce a nonlinear transformation by defining the $q$-generating function which, when applied to the intermediate scattering function of classical statistical mechanics, yields, in a mathematically systematic derivation, a generalized form of the advection-diffusion equation in Fourier space. Its solutions are discussed and suggest that the $q$-generating function approach should be a useful tool to generalize classical diffusive transport formulations., 5 pages with 3 figures

The expression of the free energy density of a classical crystalline system as a gradient expansion in terms of a set of order parameters is developed using classical density functional theory. The goal here is to extend and complete an earlier derivation by Lowen et al. [Europhys. Lett. 9 (1989) 791]. The limitations of the resulting expressions are also discussed including the boundary conditions needed for finite systems and the fact that the results cannot, at present, be used to take into account elastic relaxation.

The kinetic theory for a fluid of hard spheres which undergo endothermic and/or exothermic reactions with mass transfer is developed. The exact balance equations for concentration, density, velocity and temperature are derived. The Enskog approximation is discussed and used as the basis for the derivation, via the Chapman-Enskog procedure, of the Navier-Stokes-reaction equations under various assumptions about the speed of the chemical reactions. It is shown that the phenomenological description consisting of a reaction-diffusion equation with a convective coupling to the Navier-Stokes equations is of limited applicability., Comment: Submitted to Journal of Chemical Physics

A propagation-dispersion equation is derived for the first passage distribution function of a particle moving on a substrate with time delays. The equation is obtained as the hydrodynamic limit of the first visit equation, an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as time-delayers. The propagation-dispersion equation should be contrasted with the advection-diffusion equation (or the classical Fokker–Planck equation) as it describes a dispersion process in time (instead of diffusion in space) with a drift expressed by a propagation speed with non-zero bounded values. The temporal dispersion coefficient is shown to exhibit a form analogous to Taylor's dispersivity. Physical systems where the propagation-dispersion equation applies are discussed.

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., SCOPUS: ar.j, info:eu-repo/semantics/published

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).JCPSA60021-960610.1063/1.4811490]. 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., SCOPUS: ar.j, info:eu-repo/semantics/published

A model for the pair distribution function of nonequilibrium hard-core fluids is proposed based on a model for the effect of velocity correlations on the structure. Good agreement is found with molecular dynamics simulations of granular fluids and of sheared elastic hard spheres. It is argued that the incorporation of velocity correlations are crucial to correctly modeling atomic scale structure in nonequilibrium fluids., Final corrections after referees' reports. To appear in PRL

The Enskog kinetic equation for hard spheres is the only tractable theory with which the transport properties of a moderately dense gas can be studied. However, relatively little is known about its solutions outside the linear regime. In this paper two approximate nonlinear solutions of the Enskog equation for uniform shear flow are presented: a perturbative solution to second order in the shear rate and to fourth order in velocity moments and a ‘‘nonperturbative’’ moment solution to all orders in the shear rate and to second order in the velocity moments. A comparison to the results of nonequilibrium molecular-dynamics simulations shows that the perturbative results give good estimates of the quadratic corrections to the pressure tensor while the nonperturbative solution gives a semiquantitative description of viscoelastic effects including shear thinning and the normal stresses over a wide range of shear rates. The relevance of these results to the construction of kinetic models of the Enskog equation is also discussed. @S1063-651X~97!08512-7#

The hard-sphere potential has been the most intensively studied interaction model in many areas of statistical mechanics for at least two reasons. The first reason is that hard-spheres exhibit many phenomena observed in real systems, such as the existence of liquid, solid, and metastable phases, and indeed provide a good first approximation for real systems of such properties as liquid structure, transport properties, and both liquid- and solid-phase thermodynamic properties. The second reason is that the hard-sphere interaction is the only one for which a tractable kinetic equation applicable at moderate densities, the Enskog equation [1‐ 3], exists. The Enskog equation was originally proposed on physical grounds as a finite-density generalization of the Boltzmann equation which, although applicable to arbitrary two-body interaction models, is restricted to low densities. The generalization of the Enskog equation by van Beijeren and Ernst [4] is capable of giving a unified description of liquid, solid, and metastable states [5]. However, because of the complexity of the Enskog equation, the only analytic solutions available are perturbative and recently developed numerical techniques [6] have yet to be widely applied so that little is known about its description of systems far from equilibrium. The standard method of analyzing either the Boltzmann or the Enskog equation is the Chapman-Enskog expansion [2,3] which is a perturbative expansion of the one-body distribution function, and the kinetic equation describing it, in terms of the uniformity of the system. For example, in a fluid undergoing uniform shear flow (USF), in which the local macroscopic flow velocity along the x axis varies linearly with position along the y axis, $ us$ rd › ay ˆ x, this amounts to an expansion in powers of the shear rate, a. This expansion has only been performed in the general case to third order in the uniformity parameter and the range of validity of such results is presumably limited to near-equilibrium states; the analytic complexity of the Chapman-Enskog procedure has proven prohibitive of the study of higher-order effects. An alternative method of analysis of the Boltzmann equation is the moment method of Grad [2,7,8] according to which the distribution is expressed as an expansion in terms of velocity about local equilibrium. Keeping all terms in the expansion gives an infinite set of coupled equations for the generally spaceand time-dependent coefficients which, assuming the validity of the expansion, is equivalent to the Boltzmann equation. Approximations are then introduced to truncate or decouple the equations allowing for an approximate solution. The method has found particular use in the study of small-wavelength hydrodynamics near equilibrium [9] where the close connection between the moment method and kinetic models has been exploited. The purpose of this Letter is to show that the method may be used to obtain good approximate solutions to the Enskog equation for systems far from equilibrium. In the following, attention is focused on USF since sheared fluids may be reliably simulated and the connection between theory and simulation is well understood. Indeed, for these reasons, USF has been the subject of numerous investigations over the last 20 years (see, e.g., Ref. [10]), and is often viewed as a prototypical nonequilibrium state free from complicating features such as boundary effects. In addition, it has been known for some time that the hard-sphere system is unstable at high shear rates [11] and theories of the instability depend on knowledge of the one-body distribution [12,13]. Since only perturbative results, which lack important physical effects such as shear thinning, have been available, these theories remain tentative. One of the motivations for the present work has been to allow for a more detailed study of this problem. Both the Boltzmann and Enskog equations may be written in the form

We present a master equation formulation based on a Markovian random walk model that exhibits subdiffusion, classical diffusion, and superdiffusion as a function of a single parameter. The nonclassical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean squared displacement [$\ensuremath{\langle}{r}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{\ensuremath{\gamma}}$ with $0l\ensuremath{\gamma}\ensuremath{\le}1.5$] but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and superdiffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent $\ensuremath{\gamma}$. Simulations based on the master equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.

A dynamical theory of nucleation based on fluctuating hydrodynamics is described. It is developed in detail for the case of diffusion-limited nucleation appropriate to colloids and macro-molecules in solution. By incorporating fluctuations, realistic fluid-transport and realistic free energy models the theory is able to give a unified treatment of both the pre-critical development of fluctuations leading to a critical cluster as well as of post-critical growth. Standard results from classical nucleation theory are shown to follow in the weak noise limit while the generality of the theory allows for many extensions including the description of very high supersaturations (small clusters), multiple order parameters and strong-noise effects to name a few. The theory is applied to homogeneous and heterogeneous nucleation of a model globular protein in a confined volume and it is found that nucleation depends critically on the existence of long-wavelength, small-amplitude density fluctuations.

In a recent paper [Phys. Rev. E 86, 040102(R) (2012)], Santos presented a self-consistency condition that can be used to limit the possible forms of fundamental measure theory. Here, the direct correlation function, resulting from the Santos functional, is derived, and it is found to be very close to the result of the White Bear density functional, except near the origin where it diverges.

The role of metastable liquid phases in vapor-crystal nucleation is studied using Density Functional Theory(DFT). The model gives a semi-quantitatively accurate description of both the vapor-liquid-solid phase diagram for both simple fluids (Lennard-Jones interactions) and of the low-density/high-density/crystal phase diagram for model globular proteins (ten Wolde-Frenkel interaction). The density profile is characterized by two local order parameters, the average density and the crystallinity. The bulk free energy model is supplemented by squared-derivative terms in these order parameters to account for inhomogeneities thus producing a model similar in spirit to phase-field theory. It is shown that for both interaction models, the vapor-crystal part of the phase-diagram can be separated into regions for which metastable liquid phases are more or less stable than the vapor, but always less stable than the solid. The former case allows for the possibility of \emph{double nucleation} whereby liquid droplets nucleate from the vapor followed by a separate nucleation of the solid phase within the liquid droplets. Whether or not this actually occurs depends on the relative free energy barriers for vapor-solid and vapor-liquid nucleation and it is shown that for simple fluids, double nucleation is indeed favored at sufficiently large supersaturation. Finally, by studying the minimum free energy path from the vapor to the solid, the separate possibility of \emph{transient nucleation}, where the vapor-solid transition involves a single nucleation event but the sub-critical clusters tend to be liquid-like while small, is shown to also be possible when the metastable liquid exists, but the supersaturation is too small for double nucleation.