Equilibrium Droplet Research Articles

Canonical free-energy barrier of particle and polymer cluster formation

A common approach to study nucleation rates is the estimation of free-energy barriers. This usually requires knowledge about the shape of the forming droplet, a task that becomes notoriously difficult in macromolecular setups starting with a proper definition of the cluster boundary. Here we demonstrate a shape-free determination of the free energy for temperature-driven cluster formation in particle as well as polymer systems. Combined with rigorous results on equilibrium droplet formation, this allows for a well-defined finite-size scaling analysis of the effective interfacial free energy at a fixed density. We first verify the theoretical predictions for the formation of a liquid droplet in a supersaturated particle gas by generalized-ensemble Monte Carlo simulations of a Lennard-Jones system. Going one step further, we then generalize this approach to cluster formation in a dilute polymer solution. Our results suggest an analogy with particle condensation, when the macromolecules are interpreted as extended particles.

Influence of the fluid structure on the binding potential: Comparing liquid drop profiles from density functional theory with results from mesoscopic theory.

For a film of liquid on a solid surface, the binding potential g(h) gives the free energy as a function of the film thickness h and also the closely related (structural) disjoining pressure Π=-∂g/∂h. The wetting behaviour of the liquid is encoded in the binding potential and the equilibrium film thickness corresponds to the value at the minimum of g(h). Here, the method we developed in the work of Hughes et al. [J. Chem. Phys. 142, 074702 (2015)], and applied with a simple discrete lattice-gas model, is used with continuum density functional theory (DFT) to calculate the binding potential for a Lennard-Jones fluid and other simple liquids. The DFT used is based on fundamental measure theory and so incorporates the influence of the layered packing of molecules at the surface and the corresponding oscillatory density profile. The binding potential is frequently input in mesoscale models from which liquid drop shapes and even dynamics can be calculated. Here we show that the equilibrium droplet profiles calculated using the mesoscale theory are in good agreement with the profiles calculated directly from the microscopic DFT. For liquids composed of particles where the range of the attraction is much less than the diameter of the particles, we find that at low temperatures g(h) decays in an oscillatory fashion with increasing h, leading to highly structured terraced liquid droplets.

Kinetics of alkaline fading of malachite green and crystal violet in critical solutions

Abstract The reaction kinetics for alkaline fading of malachite green (MG) in the critical binary solution 2-butoxyethanol + water and of crystal violet (CV) in the critical microemulsion water/bis(2-ethylhexyl) sodium sulfosuccinate (AOT)/n-decane with the molar ratio of water to AOT being 30 have been studied by the three-wavelength UV-spectrophotometry at initial reaction stage and various temperatures. It was found that the reaction rates kobs for both above reactions were well described by the Arrhenius equation in the temperature ranges far away from the critical points. The critical slowing down effects were observed in the near-critical regions and the corresponding critical slowing down exponents were determined to be 0.146 ± 0.014 and 0.035 ± 0.006 for the two reactions, respectively. The former exhibited both the dynamic and thermodynamic critical slowing down effects; while the latter showed no thermodynamic singularity. It was attributed to the particular experimental design in this work that made the dimer/monomer droplet equilibrium in the critical microemulsion system no longer affect the reaction kinetics.

Molecular basis for calculating the surface tension of binary droplets

A procedure for calculating the surface tension of droplets consisting of two components in the vapor phase is considered. The calculations are performed using the lattice gas model in the quasi-chemical approximation with allowance for the correlation effects of the nearest interacting molecules. A layered model of the structure of a vapor–liquid interface is used. Ways of calculating the surface tension of droplets with different radii are considered. They are based on different thermodynamic definitions of reference surfaces. Typical dependences of the surface tension of metastable and equilibrium droplets on the droplets’ radii are analyzed for four types of phase diagram. It is found that if the energy of interaction between the components of one type exceeds by 150% the energies of interaction between components of another type and between particles of different types, and if the component with the highest energy of interaction predominates in a droplet, this results in a nonmonotonic profile of the component with the lowest energy of interaction in the region of transition. Mixture components are distributed in the region of transition such that the component with the highest energy of interaction is concentrated on the liquid side and the other component is concentrated on the vapor side. The surface tension of equilibrium droplets is less than that of metastable droplets.

Modelling of a free-surface ferrofluid flow

The Cauchy's stress tensor of a ferrofluid exposed to an external magnetic field is subject to additional magnetic terms. For a linearly magnetizable medium, the terms result in interfacial magnetic force acting on the ferrofluid boundaries. This force changes the characteristics of many free-surface ferrofluid phenomena. The aim of this work is to implement this force into the incompressible Navier-Stokes equations and propose a numerical method to solve them. The interface of ferrofluid is tracked with the use of the characteristic level-set method and additional reinitialization step assures conservation of its volume. Incompressible Navier-Stokes equations are formulated for a divergence-free velocity fields while discrete interfacial forces are treated with continuous surface force model. Velocity-pressure coupling is implemented via the projection method. To predict the magnetic force effect quantitatively, Maxwell's equations for magnetostatics are solved in each time step. Finite element method is utilized for the spatial discretization. At the end of the work, equilibrium droplet shape are compared to known experimental results.

EFFECT OF DROPLETS CONVECTIVE HEATING AND THEIR EVAPORATION ON THE SHIELDING PROPERTIES OF FIRE-FIGHTING WATER CURTAIN

Purpose. This work represents further development of earlier received results of simulation of water curtains used as fire protection walls. Subject of the theoretical analysis is the flat fan sprays which flow from the slot-hole sprinkler. The central objective of the study is theoretical research of processes of droplets convective heating and their evaporation in heated air near the fire seat and their influence on the shielding properties of a fire-fighting water curtain. Approach. In this study the earlier published literary data are applied to fulfill the calculations. The calculated formula allowed to have received numerical results for graphic dependence of droplet equilibrium temperature on air temperature. Other calculated formulae allowed to have executed calculations of the droplet evaporation time and of the time of droplet flight along an axis of a water curtain. According to the executed calculations, for typical parameters of water curtain the droplets heating and their evaporation owing to heat convection with hot air have no significant effect on the shielding properties of fire-fighting water curtain. Findings. Application of this formula to earlier developed mathematical model of thermal shielding allowed expanding its opportunities for practical use. This model can be used for designing of water curtains of fire-prevention appointment, and also for definition of optimum modes of their operation.

A balanced-force control volume finite element method for interfacial flows with surface tension using adaptive anisotropic unstructured meshes

A balanced-force control volume finite element method is presented for three-dimensional interfacial flows with surface tension on adaptive anisotropic unstructured meshes. A new balanced-force algorithm for the continuum surface tension model on unstructured meshes is proposed within an interface capturing framework based on the volume of fluid method, which ensures that the surface tension force and the resulting pressure gradient are exactly balanced. Two approaches are developed for accurate curvature approximation based on the volume fraction on unstructured meshes. The numerical framework also features an anisotropic adaptive mesh algorithm, which can modify unstructured meshes to better represent the underlying physics of interfacial problems and reduce computational effort without sacrificing accuracy. The numerical framework is validated with several benchmark problems for interface advection, surface tension test for equilibrium droplet, and dynamic fluid flow problems (fluid films, bubbles and droplets) in two and three dimensions.

Growth and wetting of water droplet condensed between micron-sized particles and substrate.

We study heterogeneous condensation growth of water droplets on micron-sized particles resting on a level substrate. Through numerical simulations on equilibrium droplet profiles, we find multiple wetting states towards complete wetting of the particle. Specifically, a partially wetting droplet could undergo a spontaneous transition to complete wetting during condensation growth, for contact angles above a threshold minimum. In addition, we find a competitive wetting behavior between the particle and the substrate, and interestingly, a reversal of the wetting dependence on contact angles during late stages of droplet growth. Using quasi-steady assumption, we simulate a growing droplet under a constant condensation flux, and the results are in good agreement with our experimental observations. As a geometric approximation for particle clusters, we propose and validate a pancake model, and with it, show that a particle cluster has greater wetting tendency compared to a single particle. Together, our results indicate a strong interplay between contact angle, capillarity and geometry during condensation growth.

Equilibrium of droplets on a deformable substrate: Influence of disjoining pressure

Equilibrium of liquid droplets on soft deformable substrates is investigated. Disjoining pressure action in the vicinity of the apparent three phase contact line is taken into account. It is shown that the combined disjoining and capillary pressure action determine the substrate deformation. A simplified linear disjoining pressure isotherm and simple Winkler's model to account for the substrate deformation are used which allows deducing analytical solutions for both the liquid profile and the substrate deformation. The apparent equilibrium contact angle of the liquid droplet with the deformable substrate is calculated and its dependency on the system parameters is investigated.

Coalescence Dynamics of PEDOT:PSS Droplets Impacting at Offset on Substrates for Inkjet Printing.

The dynamics of coalescence and consequent spreading of conducting polymer droplets on a solid substrate impacting at an offset are crucial in understanding the stability of inkjet printed patterns, which find application in organic flexible electronic devices. Poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) ( PSS) dispersion in water is a widely used commercial conducting polymer for the fabrication of electron devices. The effects of droplet spacing, impact velocity, substrate hydrophilicity, polymer concentration, and charges on the coalescence of two sessile droplets have been experimentally investigated, and the characteristics of dynamic spreading during the coalescence process are determined through image processing. The equilibrium spreading length of the coalesced droplets decreases with concentration and spacing of the droplets, revealing the necessity of optimum fluid properties (viscosity and surface tension) for the stability of the desired pattern. The droplet's impact energy governs the maximum extent of spreading and receding dynamics, as the velocity gradients developed in polymer droplets during coalescence are a function of the inertia of the fluid elements. Hydrophilicity affects the maximum spreading extent but it has no influence on the equilibrium droplet diameter. The spreading length dynamics of charge-neutralized PSS is found similar to the charged droplets, which show that the charged nature of the polymer does not affect the coalescence behavior. Furthermore, different spreading regimes are identified and the governing forces in each regime are described using a semianalytical formulation derived for the coalescence of two droplets. The model has been found to accurately provide insight into the various mechanisms that play a role during the complex spreading event.

Stochastic Rotation Dynamics simulations of wetting multi-phase flows

Multi-color Stochastic Rotation Dynamics (SRDmc) has been introduced by Inoue et al. as a particle based simulation method to study the flow of emulsion droplets in non-wetting microchannels. In this work, we extend the multi-color method to also account for different wetting conditions. This is achieved by assigning the color information not only to fluid particles but also to virtual wall particles that are required to enforce proper no-slip boundary conditions. To extend the scope of the original SRDmc algorithm to e.g. immiscible two-phase flow with viscosity contrast we implement an angular momentum conserving scheme (SRDmc+). We perform extensive benchmark simulations to show that a mono-phase SRDmc fluid exhibits bulk properties identical to a standard SRD fluid and that SRDmc fluids are applicable to a wide range of immiscible two-phase flows. To quantify the adhesion of a SRDmc+ fluid in contact to the walls we measure the apparent contact angle from sessile droplets in mechanical equilibrium. For a further verification of our wettability implementation we compare the dewetting of a liquid film from a wetting stripe to experimental and numerical studies of interfacial morphologies on chemically structured surfaces.

Instability of a charged droplet in an inhomogeneous electrostatic field of a thin rod

The problem of stability of oscillations of a charged droplet in an inhomogeneous electrostatic field of a thin charged rod is investigated in the nonlinear formulation using the asymptotic expansion in two small parameters, namely, the dimensionless equilibrium droplet strain and the ratio of the droplet oscillation amplitude to the droplet radius. It is shown that, when the droplet charge is less than the Rayleigh critical charge, in the inhomogeneous electrostatic field the droplet instability implementation mechanism remains the same as for the charged droplet in the field of a point charge. As the oscillation mode number increases, the critical field parameter reaches saturation tending to the horizontal asymptotics. The longer the rod, the higher the level of the asymptotics. As the rod length increases, the amplitudes of the related droplet oscillations and the increments of the unstable droplet oscillations in the electrostatic field of the rod decrease.

Dependence of the condensate chemical potential on droplet size in thermodynamics of heterogeneous nucleation within the gradient DFT

A scheme of computation of the condensate chemical potential per molecule as a function of the droplet equimolecular radius for stable and critical droplets on uncharged or charged spherical particle of molecular size at heterogeneous nucleation has been considered. The scheme is based of the gradient density functional theory (DFT) with the van der Waals (vdW) and Carnahan–Starling (СS) models for the hard-sphere contribution to intermolecular interaction in liquid and vapor phases and interfaces. The particle serving as a condensation center in the case of heterogeneous nucleation has been characterized by an attractive short-range molecular potential and the long-range electric Coulomb potential. The dielectric permittivities of the droplet–vapor systems have been taken as known functions of the local condensate density and temperature for polar and nonpolar fluids. Detailed numerical calculations of the density profiles in critical and stable equilibrium droplets at water or argon nucleation in presence of the capillary, electrostatic and molecular forces have been performed. Dependence of the condensate chemical potential in the droplet on the droplet equimolecular radius has been analyzed in the case of homogeneous and heterogeneous nucleation and compared for water and argon within the vdW and СS models for the hard-sphere part of the equation of state.

Shear dynamics of an inverted nematic emulsion.

Here we study theoretically the dynamics of a 2D and a 3D isotropic droplet in a nematic liquid crystal under a shear flow. We find a large repertoire of possible nonequilibrium steady states as a function of the shear rate and of the anchoring of the nematic director field at the droplet surface. We first discuss homeotropic anchoring. For weak anchoring, we recover the typical behaviour of a sheared isotropic droplet in a binary fluid, which rotates, stretches and can be broken by the applied flow. For intermediate anchoring, new possibilities arise due to elastic effects in the nematic fluid. We find that in this regime the 2D droplet can tilt and move in the flow, or tumble incessantly at the centre of the channel. For sufficiently strong anchoring, finally, one or both of the topological defects which form close to the surface of the isotropic droplet in equilibrium detach from it and get dragged deep into the nematic state by the flow. In 3D, instead, the Saturn ring associated with the normal anchoring disclination line can be deformed and shifted downstream by the flow, but remains always localized in the proximity of the droplet, at least for the parameter range we explored. Tangential anchoring in 2D leads to a different dynamic response, as the boojum defects characteristic of this situation can unbind from the droplet under a weaker shear with respect to the normal anchoring case. Our results should stimulate further experiments with inverted liquid crystal emulsions under shear, as most of the predictions can be testable in principle by monitoring the evolution of liquid crystalline orientation patterns or by tracking the position and shape of the droplet over time.

Wetting properties of molecularly rough surfaces.

We employ molecular dynamics simulations to study the wettability of nanoscale rough surfaces in systems governed by Lennard-Jones (LJ) interactions. We consider both smooth and molecularly rough planar surfaces. Solid substrates are modeled as a static collection of LJ particles arranged in a face-centered cubic lattice with the (100) surface exposed to the LJ fluid. Molecularly rough solid surfaces are prepared by removing several strips of LJ atoms from the external layers of the substrate, i.e., forming parallel nanogrooves on the surface. We vary the solid-fluid interactions to investigate strongly and weakly wettable surfaces. We determine the wetting properties by measuring the equilibrium droplet profiles that are in turn used to evaluate the contact angles. Macroscopic arguments, such as those leading to Wenzel's law, suggest that surface roughness always amplifies the wetting properties of a lyophilic surface. However, our results indicate the opposite effect from roughness for microscopically corrugated surfaces, i.e., surface roughness deteriorates the substrate wettability. Adding the roughness to a strongly wettable surface shrinks the surface area wet with the liquid, and it either increases or only marginally affects the contact angle, depending on the degree of liquid adsorption into the nanogrooves. For a weakly wettable surface, the roughness changes the surface character from lyophilic to lyophobic due to a weakening of the solid-fluid interactions by the presence of the nanogrooves and the weaker adsorption of the liquid into the nanogrooves.

Evaporation and condensation of water mist/cloud droplets with thermal radiation

The role of thermal radiation in water droplet evaporation and condensation is investigated theoretically. The primary droplet size regime considered is that between non-continuum gas effects and gas–liquid velocity-slip effects, nominally 10−1–102μm, with mist or clouds (nominally 20-μm diameter) being the primary application of interest. Principles for extending the results to larger droplets are also discussed, even to the radiative–evaporative balance of the oceans. The importance of distributed (volumetric, in-depth) absorption in the droplet, conduction, and advection induced by surface regression are analyzed. It is found that advection within the liquid phase induced by surface regression can be neglected for most conditions and that conduction usually establishes a spatially isothermal droplet. Comparison with experimental results for a laser-irradiated droplet suggests that, due to the isothermal condition, these assumptions are reasonable for predicting evaporation rate. Moreover, it is shown that infrared absorption and emission can be treated as surface phenomena; in-depth effects can be neglected. Extending the analysis to cloud droplets shows that radiation, whether with colder upper layers of the atmosphere, warmer lower layers/ground, or solar radiation, can have a significant influence on cloud droplet growth and stability. For typical ambient conditions the thermal radiation effect near the edge of a cloud can be as strong as the effect of a 0.1% super- or sub-saturation, enough to modify significantly the Kohler curves governing equilibrium droplet size distribution and enough to provide the missing mechanism of warm rain droplet growth between the upper limit of non-radiating, diffusional condensation growth (∼30μm) and the lower limit of turbulent-inertial coalescence growth (∼80μm), the so-called condensation–coalescence “bottleneck”.

Drops on hydrophilic conical fibers: gravity effect and coexistent states.

Controlling the droplet equilibrium location and shape on a conical fiber is essential to industrial applications such as dip-pen nanolithography. In this work, the equilibrium conformations of a drop on a vertical, conical fiber has been investigated by the finite element method, Surface Evolver simulations. Similar to the morphology of a drop on a cylinder, two different types (barrel shape and clam-shell shape) can be obtained. In the absence of gravity, the droplet moves upward (lower curvature) and the total surface energy decays as the drop ascends. Whatever the initial conformation of the drop on a conical fiber, the rising drop exhibits the clam-shell shape eventually and there is no equilibrium location. However, in the presence of gravity, the drop can stop at the equilibrium location stably. For a given contact angle, the clam-shell shape is generally favored for smaller drops but the barrel shape is dominant for larger drops. In a certain range of drop volume, the coexistence of both barrel and clam-shell shapes is observed. For large enough drops, the falling-off state is seen.

Polymeric Droplets on Soft Surfaces: From Neumann’s Triangle to Young’s Law

Shape deformation of polymeric droplets on gel-like surfaces is studied by using a combination of the molecular dynamics simulations and theoretical calculations. On the basis of the results of molecular dynamics simulations, we have developed a theoretical model of droplet shape deformation on elastic surfaces which takes into account surface and elastic energy contributions. Analysis of simulation results in the framework of this model shows that the equilibrium droplet shape is controlled by the dimensionless parameter γSL/GSa, where γSL is the surface tension of the substrate/liquid interface, GS is the shear modulus of the substrate, and a is the contact radius of polymeric droplet. This parameter describes crossover between Neumann’s triangle conditions for liquid droplets on liquid substrates and Young’s law for droplets on rigid substrates. In the limit when the elastocapillary length is much larger than the contact radius, γSL/GS ≫ a, we recover the Neumann’s triangle conditions for liquid droplets floating on liquid substrates. However, in the opposite limit, γSL/GS ≪ a, our model reproduces Young’s law. The model predictions are in a very good agreement with simulation results and experimental data.

Droplet Footprint Control

Controlling droplet shape via surface tension has numerous technological applications, such as droplet lenses and lab-on-a-chip. This motivates a partial differential equation-constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate. We use shape differential calculus to derive an $L^2$ gradient flow approach to compute equilibrium shapes for sessile droplets on substrates. We then develop a gradient-based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint (i.e., the liquid-solid interface has a desired shape). Moreover, we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint. Numerical results are also presented to showcase the method.

Kinetics of the oxidation of iodide ion by persulfate ion in the critical water/bis(2-ethylhexyl) sodium sulfosuccinate/n-decane microemulsions.

In this work, we studied the kinetics of the oxidation of iodide ion by persulfate ion in the critical water/bis(2-ethylhexyl) sodium sulfosuccinate (AOT)/n-decane microemulsions with the molar ratios of water to AOT being 35.0 and 40.8 via the microcalorimetry at various temperatures. It was found that the Arrhenius equation was valid for correlating experimental measurements in the noncritical region, but the slowing down effect existed significantly in the near critical region. We determined the values of the critical slowing down exponent and found it to be 0.187 ± 0.023 and 0.193 ± 0.032, respectively, which agreed well with the theoretical value of 0.207 predicted by the Griffiths-Wheeler rule for the singularity of the dimer/monomer droplet equilibrium in the critical AOT/water/n-decane microemulsions.