Entropy and the Tolman Parameter in Nucleation Theory

Phase separation of a magnetic fluid: Asymptotic states and nonequilibrium kinetics

We study self-assembly in a colloidal suspension of magnetic particles by performing comprehensive molecular dynamics simulations of the Stockmayer (SM) model, which comprises spherical particles decorated by a magnetic moment. The SM potential incorporates dipole-dipole interactions along with the usual Lennard-Jones interaction and exhibits a gas-liquid phase coexistence observed experimentally in magnetic fluids. When this system is quenched from the high-temperature homogeneous phase to the coexistence region, the nonequilibrium evolution to the condensed phase proceeds with the development of spatial as well as magnetic order. We observe density-dependent coarsening mechanisms-a diffusive growth law ℓ(t)∼t^{1/3} in the nucleation regime and hydrodynamics-driven inertial growth law ℓ(t)∼t^{2/3} in the spinodal regimes. [ℓ(t) is the average size of the condensate at time t after the quench.] While the spatial growth is governed by the expected conserved order parameter dynamics, the growth of magnetic order in the spinodal regime exhibits unexpected nonconserved dynamics. The asymptotic morphologies have density-dependent shapes which typically include the isotropic sphere and spherical bubble morphologies in the nucleation region, and the anisotropic cylinder, planar slab, cylindrical bubble morphologies in the spinodal region. The structures are robust and nonvolatile, and exhibit characteristic magnetic properties. For example, the oppositely magnetized hemispheres in the spherical morphology impart the characteristics of a Janus particle to it. The observed structures have versatile applications in catalysis, drug delivery systems, memory devices, and magnetic photonic crystals, to name a few.

Nucleation Rate of N2 and O2 in Cryogenic H2 and He

The homogeneous nucleation of N2 and O2 in cryogenic H2 and He is investigated by using classical molecular dynamics (MD) simulations. The nucleation kinetics of N2 and O2 clusters, including nucleation rate, critical cluster size, and cluster energy, are elucidated in H2 and He carrier gas at thermalization temperatures of 30-80 K and initial gas densities of 5.65 × 1024-2 × 1027 m-3. The energy released from the clusters during nucleation increases the system temperature by 77-138%, consistent with N2 nucleation experiments in supersonic nozzles and the mean-field kinetic nucleation theory. The nucleation rate derived by MD, Jsim, spans across 2.14 × 1029-5.25 × 1036 m-3 s-1 for both N2 and O2 under all conditions. The MD-obtained homogeneous nucleation rate is in agreement with predictions from the self-consistent classical nucleation theory (CNT) at low temperature (<70 K) but is 3-7 orders of magnitude faster than the CNT when temperature exceeds 70 K, consistent with the literature. Increasing temperature and decreasing concentration of the nucleating vapors leads to larger critical cluster sizes. The CNT underpredicts the critical cluster size at cryogenic temperatures below 60 K by 200-700%. The present MD methodology can be used for the direct determination of the nucleation rate and critical cluster size of N2 and O2 under cryogenic conditions, circumventing the assumptions inherent in CNT.

Minimum in the pressure dependence of the interfacial free energy between ice Ih and water

Despite the importance of ice nucleation, this process has been barely explored at negative pressures. Here, we study homogeneous ice nucleation in stretched water by means of molecular dynamics seeding simulations using the TIP4P/Ice model. We observe that the critical nucleus size, interfacial free energy, free energy barrier, and nucleation rate barely change between isobars from -2600 to 500 bars when they are represented as a function of supercooling. This allows us to identify universal empirical expressions for homogeneous ice nucleation in the pressure range from -2600 to 500 bars. We show that this universal behavior arises from the pressure dependence of the interfacial free energy, which we compute by means of the mold integration technique, finding a shallow minimum around -2000 bars. Likewise, we show that the change in the interfacial free energy with pressure is proportional to the excess entropy and the slope of the melting line, exhibiting in the latter a reentrant behavior also at the same negative pressure. Finally, we estimate the excess internal energy and the excess entropy of the ice Ih-water interface.

Kinetics of Precipitation Processes at Non-Zero Input Fluxes of Segregating Particles

We consider the process of formation and growth of clusters of a new phase in segregation processes in solid or liquid solutions in an open system when segregating particles are added continuously to it with a given rate of input fluxes, Φ. As shown here, the value of the input flux significantly affects the number of supercritical clusters formed, their growth kinetics, and, in particular, the coarsening behavior in the late stages of the process. The detailed specification of the respective dependencies is the aim of the present analysis, which combines numerical computations with an analytical treatment of the obtained results. In particular, a treatment of the coarsening kinetics is developed, allowing a description of the development of the number of clusters and their average sizes in the late stages of the segregation processes in open systems, which goes beyond the scope of the classical Lifshitz, Slezov and Wagner theory. As is also shown, in its basic ingredients, this approach supplies us with a general tool for the theoretical description of Ostwald ripening in open systems, or systems where the boundary conditions, like temperature or pressure, vary with time. Having this method at one's disposal supplies us with the possibility that conditions can be theoretically tested, leading to cluster size distributions that are most appropriate for desired applications.

Radial growth rate of near-critical crystal nuclei in poly(l-lactic acid) (PLLA) in Tammann's two-stage development method

The specific features of crystal nucleation widely determine the morphology of the evolving crystalline material. Crystal nucleation is, as a rule, not accessible by direct observation of the nuclei, which develop with time. This limitation is caused by the small size (nanometer scale) of the critical nuclei and the stochastic nature of their formation. We describe an experimental approach to the determination of specific features of the cluster size distribution employing fast scanning calorimetry at scanning rates up to 10 000 K s^-1. The surviving cluster fraction is determined by selectively melting/dissolving clusters smaller than the critical size corresponding to the highest temperature of a short spike positioned between the nucleation and the development stage in Tammann's two-stage method. This approach allows for estimating the time evolution of the radius of the largest detectable clusters in the distribution. Knowing this radius as a function of nucleation time allows for determining a radial growth rate. In the example of poly(l-lactic acid) (PLLA), the order of magnitude estimate of radial growth rates of clusters of about 2-5 nm yields values between 10^-5 and 10^-3 nm s^-1. The radial growth rate of micrometer-sized spherulites is available from optical microscopy. The corresponding values are about three orders of magnitude higher than the values for the nanometer-sized clusters. This difference is explainable by stochastic effects, transient features, and the size dependence of the growth processes on the nanometer scale. The experimental and (order of magnitude) classical nucleation theory estimates agree well.

Effect of Planar Interfaces on Nucleation in Melting and Crystallization

The effect of planar interfaces on nucleation (namely, on the work of critical cluster formation and their shape) is studied both for crystallization and melting. Advancing an approach formulated about 150 years ago by J. W. Gibbs for liquid phase formation at planar liquid−liquid interfaces, we show that nucleation of liquids in the crystal at crystal−vapor planar interfaces proceeds as a rule with a much higher rate compared to nucleation in the bulk of the crystal. Provided the surface tensions crystal−liquid (σcl), liquid−vapor (σlv), and crystal−vapor (σcv) obey the condition σcv=σcl+σlv, the work of critical cluster formation tends to zero; in the range σcv<σcl+σlv, it is less than one half of the work of critical cluster formation for bulk nucleation. The existence of a liquid−vapor planar interface modifies the work of critical cluster formation in crystal nucleation in liquids to a much less significant degree. The work of critical crystal cluster formation is larger than one half of the bulk value of the work of critical cluster formation, reaching this limit at σcv=σcl+σlv. The shape of the critical clusters can be described in both cases by spherical caps with a radius, R, and a width parameter, h. This parameter, h, is the distance from the cutting plane (coinciding with the crystal−vapor and liquid−vapor planar interface, respectively) to the top of the spherical cap. It varies for nucleation of a liquid in a crystal in the range (h/R)≤1 and for crystal nucleation in a liquid in the range 2≥(h/R)≥1. At σcv=σcl+σlv, the ratio (h/R) of the critical cluster for nucleation in melting tends to zero ((h/R)→0). At the same condition, the critical crystallite has the shape of a sphere located tangentially to the liquid−vapor interface inside the liquid ((h/R)≅2). We present experimental data which confirm the results of the theoretical analysis, and potential further developments of the theoretical approach developed here are anticipated.

On the thermodynamics of curved interfaces and the nucleation of hard spheres in a finite system

We determine, for hard spheres, the Helmholtz free energy of a liquid that contains a solid cluster as a function of the size of the solid cluster by means of the formalism of the thermodynamics of curved interfaces. This is done at the constant total number of particles, volume, and temperature. We show that under certain conditions, one may have several local minima in the free energy profile, one for the homogeneous liquid and others for the spherical, cylindrical, and planar solid clusters surrounded by liquid. The variation of the interfacial free energy with the radius of the solid cluster and the distance between equimolar and tension surfaces are inputs from simulation results of nucleation studies. This is possible because stable solid clusters in the canonical ensemble become critical in the isothermal-isobaric ensemble. At each local minimum, we find no difference in chemical potential between the phases. At local maxima, we also find equal chemical potential, albeit in this case the nucleus is unstable. Moreover, the theory allows us to describe the stable solid clusters found in simulations. Therefore, we can use it for any combination of the total number of particles, volume, and global density as long as a minimum in the Helmholtz free energy occurs. We also study under which conditions the absolute minimum in the free energy corresponds to a homogeneous liquid or to a heterogeneous system having either spherical, cylindrical, or planar geometry. This work shows that the thermodynamics of curved interfaces at equilibrium can be used to describe nucleation.

Molecular Dynamics Simulation of Tolman Length and Interfacial Tension of Symmetric Binary Lennard–Jones Liquid

The Tolman length and interfacial tension of partially miscible symmetric binary Lennard–Jones (LJ) fluids (A, B) was revealed by performing a large-scale molecular dynamics (MD) simulation with a sufficient interfacial area and cutting distance. A unique phenomenon was observed in symmetric binary LJ fluids, where two surfaces of tension existed on both sides of an equimolar dividing surface. The range of interaction εAB between the different liquids and the temperature in which the two LJ fluids partially mixed was clarified, and the Tolman length exceeded 3 σ when εAB was strong at higher temperatures. The results show that as the temperature or εAB increases, the Tolman length increases and the interfacial tension decreases. This very long Tolman length indicates that one should be very careful when applying the concept of the liquid–liquid interface in the usual continuum approximation to nanoscale droplets and capillary phase separation in nanopores.

Statistical Approach to Crystal Nucleation in Glass-Forming Liquids

In this work, methods of description of crystal nucleation by using the statistical approach are analyzed. Findings from classical nucleation theory (CNT) for the average time of formation of the first supercritical nucleus are linked with experimental data on nucleation in glass-forming liquids stemming from repetitive cooling protocols both under isothermal and isochronal conditions. It is shown that statistical methods of lifetime analysis, frequently used in medicine, public health, and social and behavioral sciences, are applicable to crystal nucleation problems in glass-forming liquids and are very useful tools for their exploration. Identifying lifetime with the time to nucleate as a random variable in homogeneous and non-homogeneous Poisson processes, solutions for the nucleation rate under steady-state conditions are presented using the hazard rate and related parameters. This approach supplies us with a more detailed description of nucleation going beyond CNT. In particular, we show that cumulative hazard estimation enables one to derive the plotting positions for visually examining distributional model assumptions. As the crystallization of glass-forming melts can involve more than one type of nucleation processes, linear dependencies of the cumulative hazard function are used to facilitate assignment of lifetimes to each nucleation mechanism.

Equivalence between condensation and boiling in a Lennard-Jones fluid

Condensation and boiling are phase transitions highly relevant to industry, geology, and atmospheric science. These phase transitions are initiated by the nucleation of a drop in a supersaturated vapor and of a bubble in an overstretched liquid, respectively. The surface tension between both phases, liquid and vapor, is a key parameter in the development of such nucleation stage. Whereas the surface tension can be readily measured for a flat interface, there are technical and conceptual limitations to obtain it for the curved interface of the nucleus. On the technical side, it is quite difficult to observe a critical nucleus in experiments. From a conceptual point of view, the interfacial free energy depends on the choice of the dividing surface, being the surface of tension the one relevant for nucleation. We bypass the technical limitation by performing simulations of a Lennard-Jones fluid where we equilibrate critical nuclei (both drops and bubbles). Regarding the conceptual hurdle, we find the relevant cluster size by searching the radius that correctly predicts nucleation rates and nucleation free energy barriers when combined with Classical Nucleation Theory. With such definition of the cluster size we find the same value of the surface tension for drops and bubbles of a given radius. Thus, condensation and boiling can be viewed as two sides of the same coin. Finally, we combine the data coming from drops and bubbles to obtain, via two different routes, estimates of the Tolman length, a parameter that allows describing the curvature dependence of the surface tension in a theoretical framework.

Effects of Glass Transition and Structural Relaxation on Crystal Nucleation: Theoretical Description and Model Analysis

In the application of classical nucleation theory (CNT) and all other theoretical models of crystallization of liquids and glasses it is always assumed that nucleation proceeds only after the supercooled liquid or the glass have completed structural relaxation processes towards the metastable equilibrium state. Only employing such an assumption, the thermodynamic driving force of crystallization and the surface tension can be determined in the way it is commonly performed. The present paper is devoted to the theoretical treatment of a different situation, when nucleation proceeds concomitantly with structural relaxation. To treat the nucleation kinetics theoretically for such cases, we need adequate expressions for the thermodynamic driving force and the surface tension accounting for the contributions caused by the deviation of the supercooled liquid from metastable equilibrium. In the present paper, such relations are derived. They are expressed via deviations of structural order parameters from their equilibrium values. Relaxation processes result in changes of the structural order parameters with time. As a consequence, the thermodynamic driving force and surface tension, and basic characteristics of crystal nucleation, such as the work of critical cluster formation and the steady-state nucleation rate, also become time-dependent. We show that this scenario may be realized in the vicinity and below the glass transition temperature, and it may occur only if diffusion (controlling nucleation) and viscosity (controlling the alpha-relaxation process) in the liquid decouple. Analytical estimates are illustrated and confirmed by numerical computations for a model system. The theory is successfully applied to the interpretation of experimental data. Several further consequences of this newly developed theoretical treatment are discussed in detail. In line with our previous investigations, we reconfirm that only when the characteristic times of structural relaxation are of similar order of magnitude or longer than the characteristic times of crystal nucleation, elastic stresses evolving in nucleation may significantly affect this process. Advancing the methods of theoretical analysis of elastic stress effects on nucleation, for the first time expressions are derived for the dependence of the surface tension of critical crystallites on elastic stresses. As the result, a comprehensive theoretical description of crystal nucleation accounting appropriately for the effects of deviations of the liquid from the metastable states and of relaxation on crystal nucleation of glass-forming liquids, including the effect of simultaneous stress evolution and stress relaxation on nucleation, is now available. As one of its applications, this theoretical treatment provides a new tool for the explanation of the low-temperature anomaly in nucleation in silicate and polymer glasses (the so-called "breakdown" of CNT at temperatures below the temperature of the maximum steady-state nucleation rate). We show that this anomaly results from much more complex features of crystal nucleation in glasses caused by deviations from metastable equilibrium (resulting in changes of the thermodynamic driving force, the surface tension, and the work of critical cluster formation, in the necessity to account of structural relaxation and stress effects) than assumed so far. If these effects are properly accounted for, then CNT appropriately describes both the initial, the intermediate, and the final states of crystal nucleation.

Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate

Crystal nucleation can be described by a set of kinetic equations that appropriately account for both the thermodynamic and kinetic factors governing this process. The mathematical analysis of this set of equations allows one to formulate analytical expressions for the basic characteristics of nucleation, i.e., the steady-state nucleation rate and the steady-state cluster-size distribution. These two quantities depend on the work of formation, ΔG(n)=−nΔμ+γn2/3, of crystal clusters of size n and, in particular, on the work of critical cluster formation, ΔG(nc). The first term in the expression for ΔG(n) describes changes in the bulk contributions (expressed by the chemical potential difference, Δμ) to the Gibbs free energy caused by cluster formation, whereas the second one reflects surface contributions (expressed by the surface tension, σ: γ=Ωd02σ, Ω=4π(3/4π)2/3, where d0 is a parameter describing the size of the particles in the liquid undergoing crystallization), n is the number of particles (atoms or molecules) in a crystallite, and n=nc defines the size of the critical crystallite, corresponding to the maximum (in general, a saddle point) of the Gibbs free energy, G. The work of cluster formation is commonly identified with the difference between the Gibbs free energy of a system containing a cluster with n particles and the homogeneous initial state. For the formation of a “cluster” of size n=1, no work is required. However, the commonly used relation for ΔG(n) given above leads to a finite value for n=1. By this reason, for a correct determination of the work of cluster formation, a self-consistency correction should be introduced employing instead of ΔG(n) an expression of the form ΔG˜(n)=ΔG(n)−ΔG(1). Such self-consistency correction is usually omitted assuming that the inequality ΔG(n)≫ΔG(1) holds. In the present paper, we show that: (i) This inequality is frequently not fulfilled in crystal nucleation processes. (ii) The form and the results of the numerical solution of the set of kinetic equations are not affected by self-consistency corrections. However, (iii) the predictions of the analytical relations for the steady-state nucleation rate and the steady-state cluster-size distribution differ considerably in dependence of whether such correction is introduced or not. In particular, neglecting the self-consistency correction overestimates the work of critical cluster formation and leads, consequently, to far too low theoretical values for the steady-state nucleation rates. For the system studied here as a typical example (lithium disilicate, Li2O·2SiO2), the resulting deviations from the correct values may reach 20 orders of magnitude. Consequently, neglecting self-consistency corrections may result in severe errors in the interpretation of experimental data if, as it is usually done, the analytical relations for the steady-state nucleation rate or the steady-state cluster-size distribution are employed for their determination.

Clustering and self-organization in small-scale natural and artificial systems

Physical properties of clusters, i.e. systems composed of a 'small' number of particles, are qualitatively different from those of infinite systems. The general approach to the problem of clustering is suggested. Clusters, as they are seen in the graphs theory, are discussed. Various physical mechanisms of clustering are reviewed. Dimensional properties of clusters are addressed. The dimensionality of clusters governs to a great extent their properties. Weakly and strongly coupled clusters are discussed. Hydrodynamic and capillary interactions giving rise to clusters formation are surveyed. Levitating droplet clusters, turbulent clusters and droplet clusters responsible for the breath-figures self-assembly are considered. Entropy factors influencing clustering are considered. Clustering in biological systems results in non-equilibrium multi-scale assembly, where at each scale, self-driven components come together by consuming energy in order to form the hierarchical structure. This article is part of the theme issue 'Bioinspired materials and surfaces for green science and technology (part 3)'.

Ice-Crystal Nucleation in Water: Thermodynamic Driving Force and Surface Tension. Part I: Theoretical Foundation

A recently developed thermodynamic theory for the determination of the driving force of crystallization and the crystal–melt surface tension is applied to the ice-water system employing the new Thermodynamic Equation of Seawater TEOS-10. The deviations of approximative formulations of the driving force and the surface tension from the exact reference properties are quantified, showing that the proposed simplifications are applicable for low to moderate undercooling and pressure differences to the respective equilibrium state of water. The TEOS-10-based predictions of the ice crystallization rate revealed pressure-induced deceleration of ice nucleation with an increasing pressure, and acceleration of ice nucleation by pressure decrease. This result is in, at least, qualitative agreement with laboratory experiments and computer simulations. Both the temperature and pressure dependencies of the ice-water surface tension were found to be in line with the le Chatelier–Braun principle, in that the surface tension decreases upon increasing degree of metastability of water (by decreasing temperature and pressure), which favors nucleation to move the system back to a stable state. The reason for this behavior is discussed. Finally, the Kauzmann temperature of the ice-water system was found to amount TK=116K, which is far below the temperature of homogeneous freezing. The Kauzmann pressure was found to amount to pK=−212MPa, suggesting favor of homogeneous freezing on exerting a negative pressure on the liquid. In terms of thermodynamic properties entering the theory, the reason for the negative Kauzmann pressure is the higher mass density of water in comparison to ice at the melting point.

Application of the Nucleation Theorem to Crystallization of Liquids: Some General Theoretical Results

Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs’ approach in the thermodynamic description of heterogeneous systems, a general form of the nucleation theorem can be formulated that is valid not only for one-component but generally for multi-component systems. In this analysis, one basic assumption of classical nucleation theory is utilized. In addition, commonly employed in application to crystallization, it is supposed that the bulk properties of the critical clusters are widely identical to the properties of the newly evolving crystal phase. It is shown that the formulation of the nucleation theorem as proposed by Kashchiev [J. Chem. Phys. 76, 5098-5102 (1982)], also relying widely on the standard classical approach in the description of crystal nucleation, holds for multi-component systems as well. The general form of the nucleation theorem derived by us is taken then as the starting point for the derivation of particular forms of this theorem for the cases that the deviation from equilibrium is caused by variations of either composition of the liquid phase, temperature, or pressure. In this procedure, expressions recently developed by us for the curvature dependence of the surface tension, respectively, its dependence on pressure and/or temperature are employed. The basic assumption of classical nucleation theory mentioned above is, however, in general, not true. The bulk and surface properties of the critical crystal clusters may differ considerably from the properties of the evolving macroscopic phases. Such effects can be incorporated into the theoretical description by the application of the generalized Gibbs approach for the specification of the dependence of the properties of critical crystal clusters on the degree of metastability of the liquid phase. Applying this method, it is demonstrated that a similar formulation of the nucleation theorem, as derived based on classical nucleation theory, holds true also in cases when a dependence of the state parameters of the critical clusters on the degree of deviation from equilibrium is appropriately accounted for.

Effects of Microstructure and Sample's Surface to Volume Ratio on Pressure-Induced Nucleation and Transformation to Crystalline and Apparently Amorphous Solids

Heterogeneous nucleation in a polycrystalline solid occurs (i) at the surface of its container, (ii) at its microstructural sites, namely, grain boundaries, grain junctions, and intergrain-strain regions, and (iii) at the defect sites, namely, dislocations, vacancies, and stacking faults (planar defects) in its single crystal grains. We analyze their thermodynamic and kinetic effects in terms of classical nucleation theory by taking into account (a) the increase in Gibbs free energy, G, due to the lattice misfit of the nuclei forming in the parent phase and (b) the decrease in G due to the angle subtended by the nucleus on the external surface, grain boundaries, and grain junctions. Hence we deduce that several combinations of nucleation sites in different materials and also in different polycrystalline samples of the same material may produce the same energy barrier against nucleation and overall growth. The overall nucleation and growth rates are dependent upon the surface to volume ratio, χ, of a sample in a vessel and the vessel's material. Pressurizing a crystalline solid is known to produce either its polymorphic crystal form or a solid that shows no Bragg peaks and appears amorphous. We argue that when self-diffusion rate becomes slower than the pressurizing rate, (dP/dt)_T, a multiplicity of states nucleating at different sites become kinetically frozen on their path to crystal growth. In such a case, the transformed solid would appear amorphous. A solid of high χ would transform to a polymorph when (dP/dt)_T is low and to a state that appears amorphous when (dP/dt)_T is high. Known studies of 0.08-0.1 cm³ volume samples in diamond-anvil high pressure cells provide qualitative evidence of formation of both crystal polymorphs and apparently amorphous solids. Methods are suggested for observing such an occurrence in large polycrystalline samples.

Interfacial free energy of a liquid-solid interface: Its change with curvature

We analyze the changes in the interfacial free energy between a spherical solid cluster and a fluid due to the change of the radius of the solid. Interfacial free energies from nucleation studies using the seeding technique for four different systems, being hard spheres, Lennard-Jones, and two models of water (mW and TIP4P/ICE), were plotted as a function of the inverse of the radius of the solid cluster. In all cases, the interfacial free energy was a linear function of the inverse of the radius of the solid cluster and this is consistent with Tolman's equation. This linear behavior is shown not only in isotherms but also along isobars. The effect of curvature on the interfacial free energy is more pronounced in water, followed by hard spheres, and smaller for Lennard-Jones particles. We show that it is possible to estimate nucleation rates of Lennard-Jones particles at different pressures by using information from simple NpT simulations and taking into account the variation of the interfacial free energy with the radius of the solid cluster. Neglecting the effects of the radius on the interfacial free energy (capillarity approximation) leads to incorrect values of the nucleation rate. For the Lennard-Jones system, the homogeneous nucleation curve is not parallel to the melting curve as was found for water in previous work. This is due to the increase in the interfacial free energy along the coexistence curve as the pressure increases. This work presents a simple and relatively straightforward way to approximately estimate nucleation rates.