Stochastic diffusion interactions and coarsening in a system of droplets growing from a supersaturated gas mixture

In this work we study diffusion interactions among liquid droplets growing in stochastic population by condensation from supersaturated binary gas mixture. During the postnucleation transient regime collective growth of liquid droplets competing for the available water vapor decreases local supersaturation leading to the increase of critical radius and the onset of coarsening process. In coarsening regime the growth of larger droplets is prevailing noticeably broadening the droplet size-distribution function when the condensation process becomes more intensive than the supersaturation yield. Modifications in the kinetic equation are discussed and formulated for a stochastic population of liquid droplets when diffusional interactions among droplets become noteworthy. The kinetic equation for the droplet size-distribution function is solved together with field equations for the mass fraction of disperse liquid phase, mass fraction of water vapor component of moist air, and temperature during diffusion-dominated regime of droplet coarsening. The droplet size and mass distributions are found as functions of the liquid volume fraction, showing considerable broadening of droplet spectra. It is demonstrated that the effect of latent heat of condensation considerably changes coarsening process. The coarsening rate constant, the droplet density (number of droplets per unit volume), the screening length, the mean droplet size, and mass are determined as functions of the temperature, pressure, and liquid volume fraction.     

Postnucleation droplet growth in supersaturated gas with arbitrary vapor concentration

This work concerns the reexamination and extension of the current theory of phase transition dynamics for liquid droplets growing on soluble aerosols from a supersaturated gas mixture for the general case of arbitrary value of vapor concentration. We found that the inconsistency in the common treatment of the vapor diffusion, due to an implicit assumption of the constancy of gas density in the vicinity of a droplet by neglecting its dependency on temperature and vapor concentration, leads to the obvious discrepancy in the Maxwell expression for the growth rate regarding droplets of near critical size. Restoring the correct treatment of the vapor diffusion in terms of the mass concentration of water vapor and taking into the consideration variations of gas density in the vicinity of a droplet in compliance with the equation of state of moist air, we have obtained a new expression for the droplet growth rate valid for an arbitrary value of vapor concentration. The limitations imposed by the molecular kinetic fluxes to postnucleation diffusional growth of small droplets with a large Knudsen number are also reevaluated to include previously neglected physical effects. In particular, the essential contribution of the vapor molecular energy flux into the total kinetic molecular heat flux as well as the temperature variations of mean thermal velocities of air and vapor molecules in the vicinity of the droplet interface have been taken into consideration. Surprisingly significant differences have been found in new expressions derived for the droplet growth rate and droplet temperature, even in the limit of small vapor concentration, if comparing with commonly used results. These findings could help with better interpretation of experimental measurements to infer more reliable data for the mass and thermal accommodations coefficients.     

Self-similar solution to the problem of vapor diffusion toward the droplet nucleated and growing in a vapor-gas medium

The problem of vapor diffusion toward a droplet nucleated and growing in the diffusion regime is exactly solved using the similarity theory. The surface motion of droplets is taken into account in the solution. The constructed nonstationary concentration field of vapor satisfies the diffusion equation, the boundary condition of equilibrium on the surface of growing droplet, and the initial homogeneous condition. According to the found solution, the radius of a droplet is proportional to the square root of the time of its growth. Far from the critical point, at a low ratio between the densities of excess vapor and a liquid droplet, the proportionality coefficient coincides with that resulting from an approximate solution. The balance between the numbers of molecules removed from vapor and those composing a growing droplet exactly corresponds to the obtained solution.     

Study of the volume condensation process in supersaturated vapor by the direct numerical solution of the kinetic equation for the droplet size distribution function

The volume condensation of supersaturated vapor is investigated by the direct numerical solution of the basic kinetic equation for the droplet size distribution function by analogy with the corresponding solution of the Boltzmann kinetic equation. The proposed consideration of the condensation growth of droplets is applicable at any Knudsen number. The method is tested by the example of vapor condensation under the conditions of the rapid development of supersaturation in a vapor-gas mixture as a result of its adiabatic expansion. In a wide range of Knudsen numbers, the results of the modeling are compared with those obtained by the moment method.     

Oil-continuous microemulsions mixed with an amphiphilic graft copolymer or with the parent homopolymer Polymer–droplet interactions as revealed by phase behavior and light scattering

Polymer–droplet interactions have been studied in AOT/water/isooctane oil-continuous microemulsions mixed with an amphiphilic graft copolymer, or with the parent homopolymer (AOT = sodium bis(2-ethylhexyl) sulfosuccinate). The graft copolymer has an oil-soluble poly(dodecyl methacrylate) backbone and water-soluble poly(ethylene glycol) side chains. Pseudo-ternary polymer/droplet/isooctane phase diagrams have been established for both the parent homopolymer and the graft copolymer, and the two types of mixture display entirely different phase behavior. The homopolymer–droplet interaction is repulsive, and a segregative phase separation occurs at high droplet concentrations. By contrast, the graft copolymer–droplet interaction is attractive: the polymer is insoluble in the pure oil, but dissolves in the microemulsion. A comparatively high concentration of droplets is required to solubilize even small amounts of polymer. Static and dynamic light scattering has been performed in order to obtain information on structure and dynamics in the two types of mixture. For optically matched microemulsions, with a vanishing excess polarizability of the droplets, the polymer dominates the intensity of scattered light. The absolute intensity of scattered light increases as phase separation is approached owing to large-scale concentration fluctuations. Dynamic light scattering shows two populations of diffusion coefficients; one population originates from “free” microemulsion droplets and the other from the polymer (for homopolymer mixtures) or from polymer–droplet aggregates (for mixtures with the graft copolymer). The graft copolymer forms large polymer–droplet aggregates with a broad size distribution, which coexist with a significant fraction of free droplets.     

On the closure conjectures for the Gibbsian approximation model of a binary droplet

Within the framework of Gibbsian thermodynamics, a binary droplet is regarded to consist of a uniform interior and dividing surface. The properties of the droplet interior are those of the bulk liquid solution, but the dividing surface is a fictitious phase whose chemical potentials cannot be rigorously determined. The state of the nucleus interior and free energy of nucleus formation can be found without knowing the surface chemical potentials, but the latter are still needed to determine the state of the whole nucleus (including the dividing surface) and develop the kinetics of nucleation. Thus it is necessary to recur to additional conjectures in order to build a complete, thermodynamic, and kinetic theory of nucleation within the framework of the Gibbsian approximation. Here we consider and analyze the problem of closing the Gibbsian approximation droplet model. We identify micro- and Gamma-closure conjectures concerning the surface chemical potentials and excess surface coverages, respectively, for the droplet surface of tension. With these two closure conjectures, the Gibbsian approximation model of a binary droplet becomes complete so that one can determine both the surface and internal characteristics of the whole nucleus and develop the kinetic theory, based on this model. Theoretical results are illustrated by numerical evaluations for binary nucleation in a water-methanol vapor mixture at T=298.15 K. Numerical results show a striking increase in the droplet surface tension with decreasing droplet size at constant overall droplet composition. A comparison of the Gibbsian approximation with density functional calculations for a model surfactant system indicate that the excess surface coverages from the Gibbsian approximation are accurate enough for large droplets and droplets that are not too concentrated with respect to the solute.     

Statistical Approach to the Description of Homogeneous Nucleation in the Vapor-Gas Medium: Effects of the Heat of Phase Transition

The effect of the heat release of phase transition is included in the formalism of the statistical approach in order to allow for the depletion of a substance comprising the metastable phase in the kinetics of the homogeneous nucleation of supersaturated vapor. The diffusion regime of the exchange of molecules between the vapor and the growing droplets is considered on the assumption of instantaneous creation of the initial vapor supersaturation. The time-dependent boundary condition on the surface of a sphere with a fixed radius and a center coinciding with the center of a growing droplet is used in the problem of heat conductivity that allowed us to provide the heat balance of phase transition. The main characteristics of the nucleation stage are calculated for the representative vapor-gas systems. It is shown that the allowance for the effects of the heat release of phase transition resulted in a rather notable change in the kinetics of nucleation even at a sufficiently high concentration of the passive gas.     

附加压力与分散系统的稳定性

较详细地讨论了附加压力与分散系统稳定性间的关系,指出分散相附加压力的降低是分散系统趋向稳定的根本原因。以乳状液为例,附加压力降低不仅减少了液滴间相互碰撞的概率,而且更重要的是,它与液滴表面形成牢固的保护膜密切相关。只有当液滴的附加压力趋近0时,分散系统才达到热力学上稳定的状态,此时乳状液已变成了微乳状液。上述讨论也基本适用于固/液分散系统。     

Growth and motion of heterogeneous water droplets in laminar diffusion chambers: 2. High nanoparticle concentrations

The heterogeneous condensation of water vapor on nanoparticles and the growth of formed droplets are numerically studied under conditions of laminar diffusion chamber (LDC) of a new type with a hot porous wall. The main attention is focused on the growth of heterogeneous droplets in the gas flow at high number densities of nanoparticles exceeding 10⁸ droplet/m³. Under these conditions, there is an interrelation between the growth of droplets and processes of heat and mass transfer in LDC due to vapor depletion and the release of latent heat of phase transition on growing droplets. The efficiency of the coverage of nanoparticles with water film is studied under LDC nonuniform conditions. It is shown that, at initial number densities of nanoparticles (N _d > 10¹¹ droplet/m³), heterogeneous droplets do not grow to optically detected sizes.     

Synthetic Membraneless Droplets for Synaptic-Like Clustering of Lipid Vesicles

In eukaryotic cells, the membraneless organelles (MLOs) formed via liquid-liquid phase separation (LLPS) are found to interact intimately with membranous organelles (MOs). One major mode is the clustering of MOs by MLOs, such as the formation of clusters of synaptic vesicles at nerve terminals mediated by the synapsin-rich MLOs. Aqueous droplets, including complex coacervates and aqueous two-phase systems, have been plausible MLO-mimics to emulate or elucidate biological processes. However, neither of them can cluster lipid vesicles (LVs) like MLOs. In this work, we develop a synthetic droplet assembled from a combination of two different interactions underlying the formation of these two droplets, namely, associative and segregative interactions, which we call segregative-associative (SA) droplets. The SA droplets cluster and disperse LVs recapitulating the key functional features of synapsin condensates, which can be attributed to the weak electrostatic interaction environment provided by SA droplets. This work suggests LLPS with combined segregative and associative interactions as a possible route for synaptic clustering of lipid vesicles and highlights SA droplets as plausible MLO-mimics and models for studying and mimicking related cellular dynamics.     

CFD Simulation of Droplet Formation in a Wide-Type Microfluidic T-Junction

Droplet formation in a wide-type microfluidic T-junction was studied using the computational fluid dynamics (CFD) method. Two distinct regimes of droplet formation were confirmed: dripping and jetting; and, at both regimes, droplet size decreases with an increase in capillary number. CFD simulation demonstrated that droplet formation in the T-junction can be divided into three steps: droplet emergence and growing up; separation with the disperse phase; and detachment from the channel wall. The wettability of the channel wall significantly affects the process of droplet detachment from the channel wall; also, the simulation clearly showed that droplets can be formed only when the continuous phase fluid preferentially wets the channel wall, that is, its contact angle on the wall is smaller than 90°. Finally, the CFD study verified that the disperse phase flow rate can significantly affect the droplet size as well as the mechanism of droplet formation.     

Modeling Injection of Dense Liquid Sprays in Radio Frequency Inductively Coupled Plasmas

A spray model and a droplet collision model are implemented into a radio frequency inductively coupled plasma model. The discrete parcel technique combined with the stochastic Monte Carlo method is used to solve the spray equation and determine the outcomes of droplet collisions in dense sprays. Plasma--spray interactions are considered by adding source terms to the conservation equations of mass, momentum and energy of the plasma phase. Two types of the outcomes of water droplets collisions, coalescence and grazing, are predicted and compared to the experimental and analytical results. The agreement is quite good. The effects of droplet collisions on droplet size distribution of the spray and the spray evaporation are investigated. It is found that the droplet collisions can increase the average droplets size of the spray. For the mono-disperse spray, the collisions can lead to a delay on the spray evaporation. However, for the poly-disperse spray, the effect of droplet collisions on the spray evaporation could not be predicted before the calculation due to the randomness of droplet collisions.     

Kinetics of droplet condensation through a double free-energy barrier

Results are presented for the kinetics of nucleation of liquid droplets from a one-component vapor phase on a planar lyophobic substrate patterned with a large number of easily wettable (lyophilic) circular domains. If the wettability of these lyophilic domains is characterized by a contact angle smaller than pi2, for intermediate values of the supersaturation, the condensation of a droplet on a lyophilic domain occurs through a free-energy barrier with two maxima, that is, through a double barrier. A simple model is proposed for the kinetics of droplet condensation through a double barrier that combines Kramers's [Physica (Utrecht) 7, 284 (1940)] transition rate theory with known results of nucleation theory. In the framework of this model, the solution is derived for the steady-state limit of the nucleation process. The number of lyophilic domains available for droplet condensation reduces with time as domains are occupied by droplets. The problem of droplet condensation through a double barrier is solved taking into account the effect of the time-dependent depletion in the number of available lyophilic domains.     

NMR Study of the Kinetics of Butane and Hexane Adsorption from Vapor Phase by Porous Glasses

The kinetics of butane and hexane sorption from vapor phase by porous glasses is studied by the pulsed NMR technique. The sorption process is revealed to proceed in two stages: monomolecular adsorption and capillary condensation. The rate of adsorption is limited by the rate of adsorbate transfer to the adsorbent surface, with the latter rate being described by the classical diffusion flux. It is shown that ultramicropores are filled simultaneously with the formation of a monolayer. The relative content of molecules in such pores is estimated. At the stage of monomolecular adsorption and at the initial stage of capillary condensation, when the adsorption proceeds from the vapor phase of butane-hexane or butane-deuterated hexane mixtures, butane molecules are predominantly sorbed and followed by their partial displacement by hexane molecules. The rate of the capillary condensation of butane from the mixture is 15–18-fold lower than that from the vapor phase of butane alone which is explained by a decrease in the gradient of chemical potential. It is shown that, when adsorption occurs from a nonequilibrium butane-hexane mixture, anomalous kinetic curves are observed because the driving force of adsorption changes in the course of establishing equilibrium in the liquid phase.     

Modeling of wetting: a study of nanowetting at rough and heterogeneous surfaces

Molecular dynamics simulations were performed to study the behavior of nanoscale water droplets at solid surfaces. Simulations of droplets on heterogeneous patterned surfaces show that the relative sizes of the domains and the droplets play an important role as do the interactions between the solid and the liquid, particularly when the domain width is comparable to the droplet radius. For pillar surfaces, a transition is observed between the Wenzel and the Cassie and Baxter regimes with increasing pillar height. The effects of pillar width and the gap between the pillars were also examined. The simulations show clearly the importance of the detailed topography and composition of the solid surface.     

Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet

We study the effects of Marangoni stresses on the flow in an evaporating sessile droplet, by extending a lubrication analysis and a finite element solution of the flow field in a drying droplet, developed earlier. The temperature distribution within the droplet is obtained from a solution of Laplace's equation, where quasi-steadiness and neglect of convection terms in the heat equation can be justified for small, slowly evaporating droplets. The evaporation flux and temperature profiles along the droplet surface are approximated by simple analytical forms and used as boundary conditions to obtain an axisymmetric analytical flow field from the lubrication theory for relatively flat droplets. A finite element algorithm is also developed to solve simultaneously the vapor concentration, and the thermal and flow fields in the droplet, which shows that the lubrication solution with the Marangoni stress is accurate for contact angles as high as 40 degrees. From our analysis, we find that surfactant contamination, at a surface concentration as small as 300 molecules/microm(2), can almost entirely suppress the Marangoni flow in the evaporating droplet.     

Nucleation and Growth of Amino Acid and Peptide Supramolecular Polymers through Liquid–Liquid Phase Separation

The transition of peptides and proteins from the solution phase into fibrillar structures is a general phenomenon encountered in functional and aberrant biology and is increasingly exploited in soft materials science. However, the fundamental molecular events underpinning the early stages of their assembly and subsequent growth have remained challenging to elucidate. Here, we show that liquid–liquid phase separation into solute‐rich and solute‐poor phases is a fundamental step leading to the nucleation of supramolecular nanofibrils from molecular building blocks, including peptides and even amphiphilic amino acids. The solute‐rich liquid droplets act as nucleation sites, allowing the formation of thermodynamically favorable nanofibrils following Ostwald's step rule. The transition from solution to liquid droplets is entropy driven while the transition from liquid droplets to nanofibrils is mediated by enthalpic interactions and characterized by structural reorganization. These findings shed light on how the nucleation barrier toward the formation of solid phases can be lowered through a kinetic mechanism which proceeds through a metastable liquid phase.     

Calculation of vapor mass flux upon isothermal condensation on spherical droplets in a wide range of Knudsen numbers based on the solution of the Boltzmann kinetic equation

Vapor mass flux density has been calculated on the surface of spherical droplets under almost isothermal conditions. The calculations have been performed at droplet radii corresponding to the free-molecular, intermediate, and continuum flow regimes. The study has been performed by the direct numerical solution of the Boltzmann kinetic equation. The method of characteristics has been adapted for describing transfer processes in spherically symmetric systems. The applicability of different simplified approaches to the calculation of mass flux density on droplet surface has been estimated.