Analysis of the linear tension of two-dimensional droplets on heterogeneous adsorbents

A procedure for analyzing the formation processes of two-dimensional droplets of an adsorbate on a rigid adsorbent support is considered. The molecular theory is based on data on the potential functions between adsorbent atoms and adsorbate molecules. Interactions between nearest neighbors are considered in the quasi-chemical approximation. The internal motions of adsorbent atoms and adsorbate molecules are ignored. Problems of describing the formation of droplets on heterogeneous adsorbents are associated with calculations for binodals (illustrated with the simplest example of two different homogeneous crystal faces) due to the choice of methods for calculating linear tension and the structural model of the region of the liquid–vapor transition. The dependence of the characteristics of droplets in the layered structural model on the method for determining the reference lines of the tension is shown for their metastable and equilibrium states. It is found that for a number of structural parameters, the thermodynamic determination of the line of tensions of metastable droplets can result in nonmonotonic dependences of the linear tension on their radii. The characteristics of two-dimensional liquid–vapor interfaces are compared for two structural models: coordination sphere and layered. It is found that the coordination sphere model allows the exclusion of the structural parameter of the layered model, but both models need refinement at small radii.     

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.     

Direct measurement of critical nucleus size in confined volumes

In crystallization, the critical nucleus size is of pivotal importance. Above this size, it is favorable for the new crystalline phase to form; below this size, the clusters will tend to dissolve rather than grow. To date, there has been no direct method for measuring the critical nucleus size. Instead, the size is typically calculated from the variation of crystallization rates with temperature. This involves using bulk values of the interfacial tension and enthalpy of fusion, which are inappropriate for small critical nucleus sizes. Here, we present a direct method for measuring the size of the critical nucleus, based on observing crystallization temperatures of materials within microemulsions. Using this approach, the number of molecules in the critical nucleus can be found simply by measuring the droplet size. Data on the freezing of water in water-in-oil microemulsions with and without the nucleating agent, heptacosanol, are presented to support our hypothesis. The results show that the critical nucleus contains 90-350 ice molecules for water pool radii of approximately 1.2-1.8 nm for the heptacosanol-doped microemulsions in which heterogeneous nucleation is initiated at the droplet interface. For the microemulsions without heptacosanol, the critical nucleus contains 70-210 ice molecules for water pool radii of approximately 1.2-1.8 nm. The smaller values arise because homogeneous nucleation occurs and therefore the crystallization temperatures are lower. We can also determine how bulk properties are perturbed at the nanoscale, and we find that the ratio of the ice-water interfacial tension to the enthalpy of fusion decreases significantly for water pool radii that are <2 nm.     

Surface Tension of the Vapor–Liquid Interface with Finite Curvature

Based on the division of particles into internal and surface particles, the expression is derived closing the system of equations of classical thermodynamics for curvature-dependent surface tension, equimolar radius, and radius of tension surface. A solution to this system allows one to find the surface tension of new phase nucleus of any size (including minimal) and any sign of surface curvature. The obtained results indicate the weak size dependence of thermodynamic parameters that are the functions of surface tension; it is shown that Tolman's length cannot be determined using experimental determination of these parameters. It is shown that the work of nucleus formation strongly depends on its size and is the function of effective rather than true surface tension. Numerical simulation of clusters by the molecular dynamics method indicates that the pressure inside a fairly small cluster is described by Laplace's formula with the coefficient of surface tension for the plane surface of a liquid that agrees with the proposed theory.     

Jetting liquid marbles: study of the Taylor instability in immersed marbles

The behavior of liquid marbles encapsulated with various powders, immersed in oil, and exposed to a uniform DC field was investigated. At some critical value of the electric field, the Taylor instability of the marble shape took place, accompanied by the appearance of a cone and jetting a small droplet. The squared critical electric field was linear dependent on inverse of the size parameter of the marble. In some cases, the extrapolation of this linear dependence to the zero field gave the finite value of the spherical marble radius corresponding to the Rayleigh limit that meant that the marbles were charged. Lycopodium-coated marbles remained neutral under the action of a DC field, as well as a pure water droplet. Therefore, charging marbles is determined by their powder coverage. The data on effective surface tension at marble–oil interfaces were extracted from the above linear dependence for the uncharged marble. The effective surface tension was measured in parallel by the capillary rise method.     

Multicomponent nucleation: thermodynamically consistent description of the nucleation work

A thermodynamically consistent formula is derived for the nucleation work in multicomponent homogeneous nucleation. The derivation relies on the conservative dividing surface which defines the nucleus as having specific surface energy equal to the specific surface energy sigma0 of the interface between the macroscopically large new and old phases at coexistence. Expressions are given for the radius of the nucleus defined by the conservative dividing surface and by the surface of tension. As a side result, the curvature dependence of the surface tension sigmaT of the nucleus defined by the surface of tension is also determined. The analysis is valid for nuclei of any size, i.e., for nucleation in the whole range of conditions between the binodal and the spinodal of the metastable old phase provided the inequality sigmaT < or = sigma0 is satisfied. It is found that under the conditions of validity of the analysis the nucleation rate is higher than the nucleation rate given by the classical nucleation theory. The general results are applied to nucleation of unary liquids or solids in binary gaseous, liquid or solid mixtures.     

Surface tension of equilibrium spherical drops in the vapor phase

The properties of metastable and equilibrium drops that occur in the vapor phase and differ in the size and position of the dividing surface are compared. Using an equimolecular dividing surface, it was found that the total free energy and the total mass of the substance in drops of the same size over a broad temperature range differ by not more than 0.3%. A similar comparison for a dividing surface chosen from the equality condition of the moments of forces gives rather similar results. Using the surface where the maximum surface tension is attained as the dividing surface is impossible at low temperatures, because under these conditions, the notion of surface tension has no physical meaning. At high temperatures, the difference between the total mass of the substance for metastable and equilibrium drops does not exceed 0.6%. The calculations were carried out over a broad temperature range on the basis of the lattice-gas model in the quasichemical approximation.     

Nucleation of self-associating fluids: free versus activated association

We use the density functional theory of statistical mechanics in a square gradient approximation to analyze the structure, size, and work of formation of critical nuclei in self-associating fluids where association reduces the strength of the interactions between bonded particles. This effect is expected in systems of strongly dipolar particles that associate into chains. In this work we analyze the nucleation behavior of two types of self-associating fluids: a system comprised of particles that can freely associate, and a system in which the association process involves a thermally activated initiation step. For the first case, we explore the properties of critical nuclei in fluids that exhibit a metastable critical point between a vapor phase and a highly associated liquid phase. In fluids where the association dynamics involves an initiation step, we investigate the nucleation behavior in the vicinity of the polymerization transition. In both cases critical nuclei undergo a structural transition that shares many of the features of the coil-globule transition reported in Monte Carlo simulations of strongly dipolar Stockmayer fluids. Our results suggest that the sharp structural transition observed in these simulations is evidence of the existence of a second-order or nearly second-order association transition in these model fluids.     

Dynamic Density Functional Theory for Sheared Polymeric Systems

Summary: Details of the dynamic density functional theory for sheared polymer systems are presented. The method is illustrated on polymer blends with and without compatibilizers. Shearing slows down coarsening of structures at later stages of phase separation, trapping the system in kineticly driven metastable states. Under certain conditions the system is found trapped in a metastable multiple sphere morphology, while under other conditions a sphere‐to‐string transition is observed. The results are sensitive to the (block co)polymer architecture.

The morphology of a symmetric polymer blend in a 32 × 32 box.     

Influence of Chemical Heterogeneity and Nanoscale Roughness on the DLVO Energy Interaction by Spherical Coordinates

In this work, the influence of chemical heterogeneity on the stability of nanocolloidal systems is surveyed with a new method. Zone of influence as a very important parameter for chemical patch surveying is modeled for sphere and flat surface. Surface chemical heterogeneity with specified properties, size, and position are created by spherical coordinate integration method. Rippled sphere model is used to create roughness and the flat surface is created by changing two sphere radius ratio. Using the spherical coordinate system for modeling of surface roughness and chemical heterogeneity is very accurate and fast. Results show that the patches could destabilize the colloidal system at very small sizes. Surface roughness reduces the effect of chemical patches for destabilizing the colloidal system, and with increasing the size of roughness the total DLVO energy interaction increases.     

A limit of stability in supercooled liquid clusters

We examine the metastable liquid phase of a supercooled gold nanocluster by studying the free energy landscape using the largest solidlike embryo as an order parameter. Just below freezing, the free energy exhibits a local minimum at small embryo sizes and a maximum at a larger critical embryo size. At T=660 K the free energy becomes a monotonically decreasing function of the order parameter as the liquid phase becomes unstable, indicating that we have reached a limit of stability. In contrast to the mean-field theory predictions for a spinodal, the size of the critical embryo remains finite as the limit of stability is approached. We also calculate the rate of nucleation, independently from our free energy calculations, and observe a rapid increase in its temperature dependence when the free energy barrier is on the order of kT. We suggest that this supports the idea that freezing becomes a barrierless process at low temperatures.     

Computer simulation of nucleation in a gas-saturated liquid

Molecular dynamics methods have been used to investigate the kinetics of the liquid-gas phase transition in a two-component Lennard-Jones system at negative pressures and elastic stretches of the liquid to values close to spinodal ones. The molecular dynamics system consists of 2048 interacting particles with parameters of the Lennard-Jones potential for argon and neon. Density dependences of pressure and internal energy have been calculated for stable and metastable states of the mixture at a temperature T* approximately 0.7+/-0.01 and three values of the concentration. The location of mechanical and the diffusion spinodals has been determined. It has been established that a gas-saturated mixture retains its stability against finite variations of state variables up to stretches close to the values near the diffusion spinodal. The statistic laws of the process of destruction of the metastable state have been investigated. The lifetimes of the metastable phase have been determined. It is shown that owing to the small height of the potential barrier that separates the microheterogeneous from the homogeneous state a system of finite size has a possibility to make the reverse transition from the microheterogeneous into the homogeneous state. The lifetimes of the system in the microheterogeneous state, as well as the expectation times of the occurrence of a critical nucleus, are described by Poissonian distributions.     

Structure and phase behavior of a confined nanodroplet composed of the flexible chain molecules

A polymer density functional theory has been employed for investigating the structure and phase behaviors of the chain polymer, which is modelled as the tangentially connected sphere chain with an attractive interaction, inside the nanosized pores. The excess free energy of the chain polymer has been approximated as the modified fundamental measure-theory for the hard spheres, the Wertheim's first-order perturbation for the chain connectivity, and the mean-field approximation for the van der Waals contribution. For the value of the chemical potential corresponding to a stable liquid phase in the bulk system and a metastable vapor phase, the flexible chain molecules undergo the liquid-vapor transition as the pore size is reduced; the vapor is the stable phase at small volume, whereas the liquid is the stable phase at large volume. The wide liquid-vapor coexistence curve, which explains the wide range of metastable liquid-vapor states, is observed at low temperature. The increase of temperature and decrease of pore size result in a narrowing of liquid-vapor coexistence curves. The increase of chain length leads to a shift of the liquid-vapor coexistence curve towards lower values of chemical potential. The coexistence curves for the confined phase diagram are contained within the corresponding bulk liquid-vapor coexistence curve. The equilibrium capillary phase transition occurs at a higher chemical potential than in the bulk phase.     

Free-energy landscape of nucleation with an intermediate metastable phase studied using capillarity approximation

Capillarity approximation is used to study the free-energy landscape of nucleation when an intermediate metastable phase exists. The critical nucleus that corresponds to the saddle point of the free-energy landscape as well as the whole free-energy landscape can be studied using this capillarity approximation, and various scenarios of nucleation and growth can be elucidated. In this study, we consider a model in which a stable solid phase nucleates within a metastable vapor phase when an intermediate metastable liquid phase exists. We predict that a composite critical nucleus that consists of a solid core and a liquid wetting layer as well as pure liquid and pure solid critical nuclei can exist depending not only on the supersaturation of the liquid phase relative to that of the vapor phase but also on the wetting behavior of the liquid surrounding the solid. The existence of liquid critical nucleus indicates that the phase transformation from metastable vapor to stable solid occurs via the intermediate metastable liquid phase, which is quite similar to the scenario of nucleation observed in proteins and colloidal systems. By studying the minimum-free-energy path on the free-energy landscape, we can study the evolution of the composition of solid and liquid within nuclei which is not limited to the critical nucleus.     

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.     

Universal Asymptotics of the Thermodynamic Characteristics of the Nucleation on Small Macroscopic Nuclei of Soluble Surfactants

Asymptotic behavior of thermodynamic characteristics of nucleation on small macroscopic nuclei of soluble surfactants at their complete dissolution in a nucleating droplet is studied. It is taken into account that, in the region of small sizes of nuclei and corresponding small sizes of critical nuclei of liquid phase, the chemical potential of condensate and the work of droplet formation are affected by the presence of dense surfactant adsorption monolayer on the droplet surface. It is shown that, as the limiting surface area per surfactant molecule in adsorption monolayer increases, the behavior of thermodynamic characteristics of nucleation in the region of small nucleus sizes is characterized by the transition from asymptotics at the adsorption of almost all substance comprising nucleus in a monolayer to the asymptotics at constant adsorption. The study performed is not limited by the selection of specific adsorption isotherms; therefore, the obtained asymptotic dependences of thermodynamic characteristics on the nucleus size can be considered as universal for the heterogeneous nucleation on the nuclei of soluble surfactants.     

Extended version of the van der Waals capillarity theory

An extended version of the van der Waals capillarity theory describing the liquid-vapor interface in the temperature range from the triple to the critical point is suggested. A model functional of thermodynamic potential for a two-phase Lennard-Jones system taking into account the effect of the highest degree terms of gradient expansion has been constructed. The identity of the thermodynamic and the mechanical definition of Tolman's length has been proved in the framework of the adopted form of functional. The properties of nuclei of the liquid and the vapor phase are described. The paper determines: the work of formation of a nucleus, density profiles, size dependences of the surface tension, and the parameter delta in the Gibbs-Tolman-Koenig-Buff equation.