Nucleation versus spinodal decomposition in confined binary solutions

Basic features of spinodal decomposition, on one side, and nucleation, on the other side, and the transition between both mechanisms are analyzed within the framework of a generalized thermodynamic cluster model based on the generalized Gibbs approach. Hereby the clusters, representing the density or composition variations in the system, may change with time both in size and in their intensive state parameters (density and composition, for example). In the first part of the analysis, we consider phase separation processes in dependence on the initial state of the system for the case when changes of the state parameters of the ambient system due to the evolution of the clusters can be neglected as this is the case for cluster formation in an infinite system. As a next step, the effect of changes of the state parameters on cluster evolution is analyzed. Such depletion effects are of importance both for the analysis of phase formation in confined systems and for the understanding of the evolution of ensembles of clusters in large (in the limit infinite) systems. The results of the thermodynamic analysis are employed in both cases to exhibit the effect of thermodynamic constraints on the dynamics of phase separation processes.     

Crossover model for the work of critical cluster formation in nucleation theory

We propose a relation for the work of critical cluster formation in nucleation theory W for the systems with long-range interparticle interactions. The method of bridge functions is used to combine the system behavior at sufficiently small quenches, adequately predicted by the classical nucleation theory, with nonclassical effects at deep quenches in the vicinity of the thermodynamic spinodal, described within the framework of the field theoretical approach with an appropriate Ginzburg-Landau functional. The crossover between the two types of nucleation behavior takes place in the vicinity of the kinetic spinodal where the lifetime of a metastable state is of the order of the relaxation time to local equilibrium. We argue that the kinetic spinodal corresponds to the minimum of the excess number of molecules in the critical cluster. This conjecture leads to the form of W containing no adjustable parameters. The barrier scaling function Gamma = W/W(cl), where W(cl) is the classical nucleation barrier, depends parametrically on temperature through the dimensionless combination of material properties. The results for argon nucleation are presented.     

Spinodal patterns indicating unstable regime of polymer crystallization

It has been considered that crystallization of polymer from melt proceeds via the coexistence of molten matrix and growing crystals that have once overcome a nucleation barrier to a critical size. The nucleation process has often been explained analogously with so-called nucleation and growth (NG) behavior of the phase separation of a binary mixture in metastable conditions, although the crystallization in one-component polymer is not a real component separation but a phase transition. Among the mechanisms of polymer crystallization, the topic is whether a liquid–liquid transition between states of different densities within one-component polymers takes place before the aforementioned nucleation process. The liquid–liquid transition between states, which is probably driven by chain orientation, is also categorized into NG and the controversial spinodal decomposition (SD) type processes depending on the quenching depth. This article provides the optical microscopic observations that favor the occurrence of the SD-like process when a one-component polymer melt is very rapidly quenched below a stability limit, including a drastic morphological change from a spherulitic to a spinodal pattern at the critical (or spinodal) temperature. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1817–1822, 2004     

Cluster kinetics and dynamics during spinodal decomposition

Spinodal decomposition (barrierless phase transition) is a spontaneous phase separation caused by conditions that force the system to become thermodynamically unstable. We consider spinodal decomposition to occur under conditions of large supersaturation S and/or small ratio of interfacial to thermal energies omega, such that the computed number of monomers in a critical nucleus xi*=(omega/ln S)3 is less than unity. The small critical nucleus size is consistent with a negligible energy barrier for initiating condensation. Thus, in contrast to conventional opinion, it is suggested that the spinodal decomposition is related to the homogeneous nucleation of metastable fluids. Population balance equations show how clusters aggregate and rapidly lead to phase separation. Different mass dependences of aggregation rate coefficients are proposed to investigate the fundamental features of spinodal decomposition. When the mass dependency is an integer, the equations are solved by the moment technique to obtain analytical solutions. When the mass dependency is a noninteger, the general cases are solved numerically. All solutions predict the two time regimes observed experimentally: the average length scale of condensed-phase domains increases as a power law with an exponent of 1/3 at early times, followed by a linear increase at longer times.     

A thermodynamically consistent kinetic framework for binary nucleation

The traditional theory for binary homogeneous nucleation follows the classical derivation of the nucleation rate in the supposition of a hypothetical constrained-equilibrium distribution in the calculation of the cluster evaporation rate. This model enables calculation of the nucleation rate, but requires evaluation of the cluster distribution and cluster properties for an unstable equilibrium with supersaturated vapor. An alternate derivation of the classical homomolecular nucleation rate eliminated the need for this nonphysical approximation by calculating the evaporative flux at full thermodynamic equilibrium. The present paper develops that approach for binary nucleation; the framework is readily extended to ternary nucleation. In this analysis, the evaporative flux is evaluated by applying mass balance at full thermodynamic equilibrium of the system under study. This approach eliminates both the need for evaluating cluster properties in an unstable constrained-equilibrium state and ambiguity in the normalization constant required in the nucleation-rate expression. Moreover, it naturally spans the entire composition range between the two pure monomers. The cluster fluxes derived using this new model are similar in form to those of classical derivations, so previously developed methods for evaluation of the net nucleation rate can be applied directly to the new formulation.     

Homogeneous crystal nucleation triggered by spinodal decomposition in polymer solutions

We report dynamic Monte Carlo simulations of polymer crystal nucleation initiated by prior spinodal decomposition in polymer solutions. We observed that the kinetic phase diagrams of homogeneous crystal nucleation appear horizontal in the concentration region below their crossovers with the theoretical liquid-liquid spinodal. When the solution was quenched into the temperature beneath this horizontal boundary, the time evolution of structure factors demonstrated the spinodal decomposition at the early stage of crystal nucleation. In comparison with the case without a prior liquid-liquid demixing, we found that the prior spinodal decomposition can regulate the nanoscale small polymer crystallites toward a larger population, more uniform sizes, and a better spatial homogeneity, whereas chain folding in the crystallites seems little affected.     

The birth of a bubble: a molecular simulation study

We study the nucleation of a bubble in a metastable Lennard-Jones (LJ) fluid, confined to a spherical pore with wetting walls, by a combination of grand canonical, canonical ensemble, and gauge cell Monte Carlo simulation methods complemented by the Voronoi-Delaunay tessellation analysis of statistical geometry of intermolecular cavities. We construct the isotherm of confined fluid in the form of a continuous van der Waals' loop, in which the unstable backward trajectory between the spinodals corresponds to bubble states. We show that as the degree of metastability increases and the fluid becomes progressively stretched, the decrease of fluid density is associated with the evolution of a population of interstitial intermolecular cavities. At the spinodal, the fluid becomes mechanically unstable: Interstitial cavities partly coalesce into a larger cavity located due to the system symmetry around the pore center. This cavity represents a bubble embryo, which grows at the expense of interstitial cavities. The nucleation barrier is calculated by direct thermodynamic integration along the isotherm. We compare our simulation results to the predictions of the classical nucleation theory and experiments on capillary condensation-evaporation of nitrogen in pores of hybrid organic-inorganic mesoporous molecular sieve HMM-3.     

Sensitivity of nucleation phenomena on range of interaction potential

Theoretical and computational investigations of nucleation have been plagued by the sensitivity of the phase diagram to the range of the interaction potential. As the surface tension depends strongly on the range of interaction potential and as the classical nucleation theory (CNT) predicts the free energy barrier to be directly proportional to the cube of the surface tension, one expects a strong sensitivity of nucleation barrier to the range of the potential; however, CNT leaves many aspects unexplored. We find for gas-liquid nucleation in Lennard-Jones system that on increasing the range of interaction the kinetic spinodal (KS) (where the mechanism of nucleation changes from activated to barrierless) shifts deeper into the metastable region. Therefore the system remains metastable for larger value of supersaturation and this allows one to explore the high metastable region without encountering the KS. On increasing the range of interaction, both the critical cluster size and pre-critical minima in the free energy surface of kth largest cluster, at respective kinetic spinodals, shift towards smaller cluster size. In order to separate surface tension contribution to the increase in the barrier from other non-trivial factors, we introduce a new scaling form for surface tension and use it to capture both the temperature and the interaction range dependence of surface tension. Surprisingly, we find only a weak non-trivial contribution from other factors to the free energy barrier of nucleation.     

Smooth transition from metastability to instability in phase separating protein solutions

For insight into the structure and dynamics of phases emerging upon crossing the metastability/instability boundary we monitor with optical microscopy, in real time and in real space, the generation of a dense liquid phase in high-concentration solutions of the protein lysozyme after temperature quenches into thermodynamically defined metastable and unstable regions. We show with this system, which is a poor fit to mean-field assumptions, that the evolution of the structure factor during nucleation is similar to that during spinodal decomposition and reveals no singularity predicted upon crossing the metastability boundary. We introduce two kinetic definitions of the metastability/instability boundary that yield values within approximately 1.5 K, i.e., the boundary appears as an area rather than a line, which is near and above the thermodynamic prediction. Delay times for the appearance of the new phase in the unstable regime are significant, i.e., new-phase growth is hindered by kinetic barriers. While our results agree with predictions of the non-mean-field theories of phase transformations, the experimentally observed behavior is richer than the one envisioned by theory.     

Does conventional nucleation occur during phase separation in polymer blends?

The kinetics of liquid–liquid phase separation in off-critical polymer blends was studied with time-resolved small-angle neutron scattering. Our objective was to study the nature of the nuclei that formed during the initial stages of the phase transition. The blends were composed of model polyolefins—deuterium-labeled poly(methyl butylene) (PMB) and poly(ethyl butylene) (PEB)—with molecular weights of about 200 kg/mol. A direct examination of the initial clustering of molecules before macroscopic phase separation was possible because of the large size of the polymer chains and concomitant entanglement effects. We discovered that the scattering profiles obtained during nucleation merged at a well-defined critical scattering vector. We propose that this is the signature of the critical nucleus and that the size of the critical nucleus is inversely proportional to the magnitude of the critical scattering vector. The kinetic studies were preceded by a thorough characterization of the equilibrium thermodynamic properties of our PMB/PEB blends. The locations of the binodal and spinodal curves of our system are consistent with predictions based on the Flory–Huggins theory. This combination of thermodynamic and kinetic experiments enabled the quantification of the dependence of the size and structure of the critical nuclei on the quench depth. Our results do not agree with any of the previous theories on nucleation. Some aspects of our results are addressed in recent theoretical work by Wang in which the effects of fluctuations on the classical binodal and spinodal curves in polymer blends are incorporated. Both theory and experiment support the notion that the traditional stability limit (spinodal) should be replaced by a metastability limit. Although Wang's theory provides an explanation for some of our observations, many fundamental issues regarding nucleation in polymer blends remain unresolved. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1793–1809, 2004     

Texture formation under phase ordering and phase separation in polymer-liquid crystal mixtures

Computational modeling of texture formation in coupled phase separation-phase ordering processes in polymer/liquid crystal mixtures is performed using a unified model based on the nematic tensor order parameter and gradient orientation elasticity. The computational methods are able to resolve defect nucleation, defect-defect interactions, and defect-particle interactions, as well as global and local morphological features in the concentration and order parameter spatiotemporal behavior. Biphasic structures corresponding to polymer dispersed liquid crystals (PDLCs), crystalline filled nematic (CFNs), and random filled nematics (RFNs) are captured and analyzed using liquid crystal defect physics and structure factors. Under spinodal decomposition due to concentration fluctuations, the PDLC structure emerges, and the nucleation and repulsive interaction of defects within nematic droplets leads to bipolar nematic droplets. Under spinodal decomposition due to ordering fluctuations, the CFNs structure emerges, and the stable polymer droplet crystal is pinned by a lattice of topological defects. For intermediate cases, where the mixture is unstable to both concentration and nematic order fluctuations, the RFN structure emerges, and polymer droplets and fibrils are pinned by a defect network, whose density increases with the curvature of the polymer-liquid crystal interface. The simulations provide an information of the role of topological defects on phase separation-phase ordering processes in polymer-liquid crystal mixtures.     

Neutron scattering and monte carlo determination of the variation of the critical nucleus size with quench depth

We have used a combination of neutron scattering experiments and Monte Carlo simulations to study the initial stages of first-order phase transitions. We focus on quenches wherein the nascent phase is formed by homogeneous nucleation, and we approach the spinodal, i.e., the quench depth at which the original phase becomes unstable. In this regime, we show how critical nuclei sizes are determined from neutron scattering structure factors. Prevailing thought is that the size of the critical nucleus should increase with increasing quench depth and diverge at the spinodal. To the contrary, our experiments and simulations indicate that the critical nucleus size decreases monotonically as quench depth is increased and is finite at the spinodal.     

Prediction of phase separation behaviour from thermodynamic measurements in the one phase region

When a polymer blend is heated to within the unstable region of the temperature - composition diagram spinodal decomposition may be observed using small angle neutron scattering. In the one phase region scattering has been used to obtain the temperature and composition dependence of the second derivative with respect to composition of the Gibbs free energy of mixing. Correlation of these two types of measurement not only tests the current theories of spinodal decomposition, but provides insight into the molecular parameters controlling domain morphology in phase separation blends.     

Crisis of stability of hydration shell of Na<Superscript>+</Superscript> ion in condensing water vapor

The Gibbs energy and equilibrium work of the formation of nuclei of the condensed phase on sodium ions are calculated on the molecular level by a Monte Carlo simulation using a detailed interaction model. The stationary rate of nucleation is estimated based on the data obtained. The presence of ionic impurities only substantially affects the rate of nucleation at strong vapor supersaturation. The nucleus losses its thermodynamic stability with an increase in the size of the nucleus and the barrier is formed depending on the work of formation on the size of the nucleus. An abrupt loss of stability is accompanied by pushing the ion off of the microdroplet surface and the restoration of the network of hydrogen bonds. The effect of pushing an ion to the surface of a cluster greatly depends on many-particle polarization interactions.     

Determining the nucleation rate from the dimer growth probability

A new method is proposed for the determination of the stationary one-component nucleation rate J with the help of data for the growth probability P2 of a dimer which is the smallest cluster of the nucleating phase. The method is based on an exact formula relating J and P2, and is readily applicable to computer simulations of nucleation. Using the method, the dependence of J on the supersaturation s is determined by kinetic Monte Carlo simulations of two-dimensional (2D) nucleation of monolayers on the (100) face of Kossel crystal. The change of J over nearly 11 orders of magnitude is followed and it is found that the classical nucleation theory overestimates the simulation J values by an s-dependent factor. The 2D nucleus size evaluated via the nucleation theorem is described satisfactorily by the classical Gibbs-Thomson equation and its corrected version accounting for the spinodal limit of 2D nucleation.     

Heterogeneous condensation of the Lennard-Jones vapor onto a nanoscale seed particle

The heterogeneous condensation of a Lennard-Jones vapor onto a nanoscale seed particle is studied using molecular dynamics simulations. Measuring the nucleation rate and the height of the free energy barrier using the mean first passage time method shows that the presence of a weakly interacting seed has little effect on the work of forming very small cluster embryos but accelerates the rate by lowering the barrier for larger clusters. We suggest that this results from a competition between the energetic and entropic features of cluster formation in the bulk and at the heterogeneity. As the interaction is increased, the free energy of formation is reduced for all cluster sizes. We also develop a simple phenomenological model of film formation on a small seed that captures the general features of the nucleation process for small heterogeneities. A comparison of our simulation results with the model shows that heterogeneous classical nucleation theory provides a good estimate of the critical size of the film but significantly overestimates the size of the barrier.