Equilibrium Droplet Research Articles

Water on hydroxylated silica surfaces: Work of adhesion, interfacial entropy, and droplet wetting.

In the last few years, much attention has been devoted to the control of the wettability properties of surfaces modified with functional groups. Molecular dynamics (MD) simulation is one of the powerful tools for microscopic analysis providing visual images and mean geometrical shapes of the contact line, e.g., of nanoscale droplets on solid surfaces, while profound understanding of wetting demands quantitative evaluation of the solid-liquid (SL) interfacial tension. In the present work, we examined the wetting of water on neutral and regular hydroxylated silica surfaces with five different area densities of OH groups ρA OH, ranging from a non-hydroxylated surface to a fully hydroxylated one through two theoretical methods: thermodynamic integration (TI) and MD simulations of quasi-two-dimensional equilibrium droplets. For the former, the work of adhesion needed to quasi-statically strip the water film off the solid surface was computed by the phantom wall TI scheme to evaluate the SL interfacial free energy, whereas for the latter, the apparent contact angle θapp was calculated from the droplet density distribution. The theoretical contact angle θYD and the apparent one θapp, both indicating the enhancement of wettability by an increase in ρA OH, presented good quantitative agreement, especially for non-hydroxylated and highly hydroxylated surfaces. On partially hydroxylated surfaces, in which θYD and θapp slightly deviated, the Brownian motion of the droplet was suppressed, possibly due to the pinning of the contact line around the hydroxyl groups. Relations between work of adhesion, interfacial energy, and entropy loss were also analyzed, and their influence on the wettability was discussed.

Quantum self-bound droplets in Bose-Bose mixtures: Effects of higher-order quantum and thermal fluctuations

We systematically study the effects of higher-order quantum and thermal fluctuations on the stabilization of self-bound droplets in Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. We calculate the ground-state energy, the droplet equilibrium density, the depletion and anomalous density of the droplets as well as the critical temperature as a function of the relevant parameters. Our findings are compared with previous analytical predictions and diffusion Monte Carlo simulations. We employ our theory together with the local density approximation for quantum and thermal fluctuations to obtain an extended finite-temperature Gross-Pitaevskii equation. The density profiles and breathing modes of the droplet are deeply examined in terms of the interaction strength and the temperature by numerically solving the developed generalized Gross-Pitaevskii equation.

Myoglobin, expressed in brown adipose tissue of mice, regulates the content and activity of mitochondria and lipid droplets

The identification of novel physiological regulators that stimulate energy expenditure through brown adipose tissue (BAT) activity in substrate catalysis is of utmost importance to understand and treat metabolic diseases. Myoglobin (MB), known to store or transport oxygen in heart and skeletal muscles, has recently been found to bind fatty acids with physiological constants in its oxygenated form (i.e., MBO2). Here, we investigated the in vivo effect of MB expression on BAT activity. In particular, we studied mitochondrial function and lipid metabolism as essential determinants of energy expenditure in this tissue. We show in a MB-null (MBko) mouse model that MB expression in BAT impacts on the activity of brown adipocytes in a twofold manner: i) by elevating mitochondrial density plus maximal respiration capacity, and through that, by stimulating BAT oxidative metabolism along with the organelles` uncoupled respiration; and ii) by influencing the free fatty acids pool towards a palmitate-enriched composition and shifting the lipid droplet (LD) equilibrium towards higher counts of smaller droplets. These metabolic changes were accompanied by the up-regulated expression of thermogenesis markers UCP1, CIDEA, CIDEC, PGC1-α and PPAR-α in the BAT of MB wildtype (MBwt) mice. Along with the emergence of the “browning” BAT morphology, MBwt mice exhibited a leaner phenotype when compared to MBko littermates at 20 weeks of age. Our data shed novel insights into MB's role in linking oxygen and lipid-based thermogenic metabolism. The findings suggest potential new strategies of targeting the MB pathway to treat metabolic disorders related to diminishing energy expenditure.

On the Peculiarities of Realization of Electrostatic Instability of Charged Liquid Surface in Various Geometries

The paper deals with some peculiarities of realization of electrostatic instability of a charged liquid surface on the vertices of the charged and uncharged drops in an external electrostatic field in a cylindrical jet and a flat surface. It was shown that the critical values of the surface density of the electric charge under the mentioned conditions on the threshold of the realization of instability are different in magnitude, despite the phenomenological similarity. Most probably, the reason is the differences (under all mentioned conditions) in the spatial change in the strength of the electrostatic field in the vicinity of the growing (when the charged surface of the liquid is unstable) emission protrusion. Both equilibrium and non-equilibrium forms of droplets, jets, planes and their symmetry were discussed.

Thermodynamic Investigation of Droplet-Droplet and Bubble-Droplet Equilibrium in an Immiscible Medium.

In the absence of external fields, interfacial tensions between different phases dictate the equilibrium morphology of a multiphase system. Depending on the relative magnitudes of these interfacial tensions, a composite system made up of immiscible fluids in contact with one another can exhibit contrasting behavior: the formation of lenses in one case and complete encapsulation in another. Relatively simple concepts such as the spreading coefficient (SC) have been extensively used by many researchers to make predictions. However, these qualitative methods are limited to determining the nature of the equilibrium states and do not provide enough information to calculate the exact equilibrium geometries. Moreover, due to the assumptions made, their validity is questionable at smaller scales where pressure forces due to curvature of the interfaces become significant or in systems where a compressible gas phase is present. Here we investigate equilibrium configurations of two fluid drops suspended in another fluid, which can be seen as a simple building block of more complicated systems. We use Gibbsian composite-system thermodynamics to derive equilibrium conditions and the equation acting as the free energy (thermodynamic potential) for this system. These equations are then numerically solved for an example system consisting of a dodecane drop and an air bubble surrounded by water, and the relative stability of distinct equilibrium shapes is investigated based on free-energy comparisons. Quantitative effects of system parameters such as interfacial tensions, volumes, and the scale of the system on geometry and stability are further explored. Multiphase systems similar to the ones analyzed here have broad applications in microfluidics, atmospheric physics, soft photonics, froth flotation, oil recovery, and some biological phenomena.

Droplets Sit and Slide Anisotropically on Soft, Stretched Substrates.

Anisotropically wetting substrates enable useful control of droplet behavior across a range of applications. Usually, these involve chemically or physically patterning the substrate surface, or applying gradients in properties like temperature or electrical field. Here, we show that a flat, stretched, uniform soft substrate also exhibits asymmetric wetting, both in terms of how droplets slide and in their static shape. Droplet dynamics are strongly affected by stretch: glycerol droplets on silicone substrates with a 23% stretch slide 67% faster in the direction parallel to the applied stretch than in the perpendicular direction. Contrary to classical wetting theory, static droplets in equilibrium appear elongated, oriented parallel to the stretch direction. Both effects arise from droplet-induced deformations of the substrate near the contact line.

The effect of the viscosity ratio on band formation in sheared mixtures of phase-separated polymer solutions with ultralow interface tension

The microscopic structure and dynamics of droplet dispersions formed by phase separation of two Newtonian polymer solutions were observed under simple shear. The phases had equal volumes. Under certain conditions, these dispersions show long, wormlike, extended structures after several minutes of shearing. The development of these structures was recorded in microscopic movies. These were interpreted by calculating two-point grey value correlation functions. The divergence of the correlation in the flow direction was taken as a signature of the onset of the extended structures. It was demonstrated that a viscosity ratio far from the unity of the two phases is necessary for the formation of the long shear-stabilised structures. At a viscosity ratio close to unity, the systems maintain isolated droplets in a dynamic equilibrium of coalescence and break-up. The structure formation is the result of shear-induced coalescence. The need for a difference in viscosity between the phases suggests a relation with shear banding seen in shear-thinning systems.

Optimization and Evaluation of Self-nanoemulsifying Drug Delivery System for Enhanced Bioavailability of Plumbagin.

Plumbagin, a potential bioactive lipophilic molecule, possesses limited solubility and low oral bioavailability. The purpose of the present study was to examine the potential of the self-nanoemulsifying drug delivery system for improving solubility and oral bioavailability of plumbagin. The self-nanoemulsifying drug delivery system was formulated from Capmul MCM (oil), Tween 20 (surfactant), and propylene glycol (cosurfactant). Central composite design was employed as statistical tool to optimize the formulation variables, X1 (oil) and X2 (surfactant: co-surfactant mixture ratio), of the self-nanoemulsifying drug delivery system. The responses studied were droplet size, self-emulsification time, % of drug release in 15 min, and equilibrium solubility. The optimized liquid self-nanoemulsifying drug delivery system was adsorbed on Neusilin US2 and characterized for flow properties, X-ray diffractometry, differential scanning calorimetry, in vitro dissolution, in vivo anti-inflammatory activity, and bioavailability study in Wistar rats, as well as ex vivo permeation study. The droplet size, polydispersity index, self-emulsification time, and equilibrium solubility of the optimized formulation were 58.500 ± 1.170 nm, 0.228 ± 0.012, 17.660 ± 1.520 s, and 34.180 ± 1.380 mg/mL, respectively. Its zeta potential, transmittance value, and cloud point were - 28.200 ± 1.200 mV, 99.200% ± 0.600, and 90 °C, respectively. Drug release was found to be 93.320% ± 1.090. In vivo anti-inflammatory study confirmed more enhanced activity from the self-nanoemulsifying drug delivery system than with pure plumbagin. Pharmacokinetic study in rats revealed that solid self-nanoemulsifying drug delivery system had 4.49-fold higher bioavailability than pure plumbagin. Ex vivo permeation study demonstrated 1.75-fold increased intestinal permeability of the self-nanoemulsifying drug delivery system than pure plumbagin. The developed self-nanoemulsifying drug delivery system is a useful solid platform for improving solubility and oral bioavailability of plumbagin.

Transformative Approach To Investigate the Microphysical Factors Influencing Airborne Transmission of Pathogens.

Emerging outbreaks of airborne pathogenic infections worldwide, such as the current severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic, have raised the need to understand parameters affecting the airborne survival of microbes in order to develop measures for effective infection control. We report a novel experimental strategy, TAMBAS (tandem approach for microphysical and biological assessment of airborne microorganism survival), to explore the synergistic interactions between the physicochemical and biological processes that impact airborne microbe survival in aerosol droplets. This innovative approach provides a unique and detailed understanding of the processes taking place from aerosol droplet generation through to equilibration and viability decay in the local environment, elucidating decay mechanisms not previously described. The impact of evaporation kinetics, solute hygroscopicity and concentration, particle morphology, and equilibrium particle size on airborne survival are reported, using Escherichia coli MRE162 as a benchmark system. For this system, we report that (i) particle crystallization does not directly impact microbe longevity, (ii) bacteria act as crystallization nuclei during droplet drying and equilibration, and (iii) the kinetics of size and compositional change appear to have a larger effect on microbe longevity than the equilibrium solute concentration.IMPORTANCE A transformative approach to identify the physicochemical processes that impact the biological decay rates of bacteria in aerosol droplets is described. It is shown that the evaporation process and changes in the phase and morphology of the aerosol particle during evaporation impact microorganism viability. The equilibrium droplet size was found to affect airborne bacterial viability. Furthermore, the presence of Escherichia coli MRE162 in a droplet does not affect aerosol growth/evaporation but influences the dynamic behavior of the aerosol by processing the culture medium prior to aerosolization, affecting the hygroscopicity of the culture medium; this highlights the importance of the inorganic and organic chemical composition within the aerosolized droplets that impact hygroscopicity. Bacteria also act as crystallization nuclei. The novel approach and data have implications for increased mechanistic understanding of aerosol survival and infectivity in bioaerosol studies spanning the medical, veterinary, farming, and agricultural fields, including the role of microorganisms in atmospheric processing and cloud formation.

Technical note: Equilibrium droplet size distributions in a turbulent cloud chamber with uniform supersaturation

Abstract. In a laboratory cloud chamber that is undergoing Rayleigh–Bénard convection, supersaturation is produced by isobaric mixing. When aerosols (cloud condensation nuclei) are injected into the chamber at a constant rate, and the rate of droplet activation is balanced by the rate of droplet loss, an equilibrium droplet size distribution (DSD) can be achieved. We derived analytic equilibrium DSDs and probability density functions (PDFs) of droplet radius and squared radius for conditions that could occur in such a turbulent cloud chamber when there is uniform supersaturation. We neglected the effects of droplet curvature and solute on the droplet growth rate. The loss rate due to fallout that we used assumes that (1) the droplets are well-mixed by turbulence, (2) when a droplet becomes sufficiently close to the lower boundary, the droplet's terminal velocity determines its probability of fallout per unit time, and (3) a droplet's terminal velocity follows Stokes' law (so it is proportional to its radius squared). Given the chamber height, the analytic PDF is determined by the mean supersaturation alone. From the expression for the PDF of the radius, we obtained analytic expressions for the first five moments of the radius, including moments for truncated DSDs. We used statistics from a set of measured DSDs to check for consistency with the analytic PDF. We found consistency between the theoretical and measured moments, but only when the truncation radius of the measured DSDs was taken into account. This consistency allows us to infer the mean supersaturations that would produce the measured PDFs in the absence of supersaturation fluctuations. We found that accounting for the truncation radius of the measured DSDs is particularly important when comparing the theoretical and measured relative dispersions of the droplet radius. We also included some additional quantities derived from the analytic DSD: droplet sedimentation flux, precipitation flux, and condensation rate.

The Navier–Stokes–Cahn–Hilliard model with a high-order polynomial free energy

In this paper, we present a high-order polynomial free energy for the phase-field model of two-phase incompressible fluids. The model consists of the Navier–Stokes (NS) equation and the Cahn–Hilliard (CH) equation with a high-order polynomial free energy potential. In practice, a quartic polynomial has been used for the bulk free energy in the CH equation. It is well known that the CH equation does not satisfy the maximum principle and the phase-field variable takes shifted values in the bulk phases instead of taking the minimum values of the double-well potential. This phenomenon substantially changes the original volume enclosed by the isosurface of the phase-field function. Furthermore, it requires fine resolution to keep small shapes. To overcome these drawbacks, we propose high-order (higher than fourth order) polynomial free energy potentials. The proposed model is tested for an equilibrium droplet shape in a spherical symmetric configuration and a droplet deformation under a simple shear flow in a fully three-dimensional fluid flow. The computational results demonstrate the superiority of the proposed model with a high-order polynomial potential to the quartic polynomial function in volume conservation property.

Evaluating Stability Conditions for a Rotating Droplet in a Potential Flow

The paper considers issues related to droplet fragmentation in air flow. We present the results of calculating the equilibrium conditions for a rotating droplet. The shape of the droplet is assumed to be ellipsoidal, and the flow is potential. The droplet equilibrium is determined by the balance of pressure due to the surface tension force, the centrifugal force and the aerodynamic effect force of the flow. We obtained an analytic expression linking the equilibrium values of the Weber number, the ellipsoid aspect ratio and the dimensionless swirl parameter. The calculation results showed that droplet swirling has a significant effect on the equilibrium conditions in a narrow range of variation in the ellipsoid aspect ratio. We obtained critical values of the Weber number and the aspect ratios for which the equilibrium state of the droplet becomes unstable.

Dynamics of particle wetting in wet granulation: Micro-scale analysis

Particle wetted surface area affects the probability of particles aggregation in the fluidized bed wet granulation process. Modeling the dynamics of binder spreading over particles and binder evaporation can help to determine the wetted area in time. To this end, a possible chain of events in the particle/droplet level has been proposed to reveal the effect of binder spreading and evaporation on granulation. Then, a second-order model has been introduced based on the key points of spreading, including the initial spreading radius, d∗, and the equilibrium radius. The model is compatible with conventional laws that R(t)∝t0.5 in the inertia-dominant regime and R(t)∝t0.1 in the viscous dominant regime. The binder evaporation has been considered in two situations including the simultaneous evaporation with spreading and evaporation after spreading. In the latter situation, the evaporation model of a sessile droplet on a substrate was employed while in the former situation the evaporation model has been merged with the proposed spreading model. This model has been validated by experimental data and two available theoretical models of spreading. In addition to simplicity and no need for tuning parameter, the present model shows the acceptable accuracy in comparison with the other models. Finally, the effects of impact velocity, equilibrium contact angle, droplet diameter, and temperature on the dynamics of spreading diameter have been investigated using the present model.

Calculation of the Surface Tension of the Vapor–Liquid Interface According to the Gibbs Thermodynamic Definition

A procedure to calculate the surface tension is developed based on a modified lattice gas model (MLGM) for the vapor–liquid system in a strict correspondence to the thermodynamic Gibbs definition for any curvature of the boundary. It is shown that MLGM enables a calculation of molecular distributions in a layered model of the transition region of the interface with regard to the softness of a lattice structure. The state of coexisting phases must satisfy the Yang–Lee theory of condensation, and additional conditions for chemical, thermal, and mechanical equilibria are imposed on the properties of the transition region of an equilibrium droplet with any curvature in each layer. The new calculation procedure for the surface tension is compared with the existing calculation procedures for equilibrium and metastable droplets. A difference in size dependences of the surface tension σ(R) for equilibrium and metastable droplets with a radius R in vicinity of the line σ/σbulk = 1, where σbulk is the bulk surface tension, allows us to formulate the accuracy and reliability criterion for different modeling methods: if the mentioned line exceeds the calculation accuracy σ(R), then it indicates that the method does not correspond to the Gibbs definition.

A coupled immersed interface and grid based particle method for three-dimensional electrohydrodynamic simulations

In the present work, we propose a coupled immersed interface and grid based particle method to solve two-phase electrohydrodynamic problems in three dimensions. The problem considers a leaky dielectric (weakly conducting) droplet immersed in another leaky dielectric fluid under electric field where the non-homogeneous droplet surface charge effect is taken into account. Due to the mismatch of electrical properties between two fluids, the electric potential satisfying Laplace equation with jump conditions across the droplet surface is coupled with the conservation equation for the surface charge density. Consequently, we first develop a three-dimensional augmented immersed interface method (IIM) which incorporates some known jump conditions naturally along the normal direction and check the desired accuracy. Here, the grid based particle method (GBPM) is used to track the interface by the projection of the neighboring Eulerian grid points so no requirement for stitching of parameterizations nor body fitted moving meshes. Within the leaky dielectric framework, the electric stress can be treated as an interfacial force so that both the surface tension and electric force can be formulated in a unified continuum force in the Navier-Stokes equations. A series of numerical tests have been carefully conducted to illustrate the accuracy and applicability of the present method to simulate droplet electrohydrodynamics. In particular, we investigate the droplet equilibrium dynamics under weak and strong electric fields in detail. It is interesting to find out a chaotic tumbling motion with irregular rotating modes which we believe that is the first numerical verification to the recent experiments.

Self-nanoemulsifying ramipril tablets: a novel delivery system for the enhancement of drug dissolution and stability.

Background: Ramipril (RMP) suffers from poor aqueous solubility along with sensitivity to mechanical stress, heat, and moisture. The aim of the current study is to improve RMP solubility and stability by designing solid self-nanoemulsifying drug delivery system (S-SNEDDS) as tablet.Methods: The drug was initially incorporated in different liquid formulations (L-SNEDDS) which were evaluated by equilibrium solubility, droplet size, and zeta potential studies. The optimized formulation was solidified into S-SNEDDS powder by the adsorbent Syloid® and compressed into a self-nanoemulsifying tablet (T-SNEDDS). The optimized tablet was evaluated by drug content uniformity, hardness, friability, disintegration, and dissolution tests. Furthermore, pure RMP, optimized L-SNEDDS, and T-SNEDDS were enrolled in accelerated and long-term stability studies.Results: Among various liquid formulations, F5 L-SNEDDS [capmul MCM/transcutol/HCO-30 (25/25/50%w/w)] showed relatively high drug solubility, nano-scaled droplet size, and high negative zeta potential value. The optimized SNEDDS solidification with Syloid® at ratio (1:1) resulted in a compressible powder with an excellent flowability. The optimized tablet (T-SNEDDS) showed accepted content uniformity, hardness, friability, and disintegration time (<15 minutes). The optimized L-SNEDDS, S-SNEDDS, and T-SNEDDS showed superior enhancement of RMP dissolution compared to the pure drug. Most importantly, T-SNEDDS showed significant (P<0.05) improvement of RMP stability compared to the pure drug and L-SNEDDS in both accelerated and long-term stability studies.Conclusion: RMP-loaded T-SNEDDS offers a potential oral dosage form that provides combined improvement of RMP dissolution and chemical stability.

Thermodynamic Analysis of Adsorption and Line-Tension Contributions to Contact Angles of Small Sessile Droplets

Relations for the contact angles of sessile liquid droplets with axially symmetric or cylindrical shapes and different equilibrium sizes have been considered using a general thermodynamic approach. For a multicomponent system comprising a liquid droplet, a vapor–gas medium, and a solid substrate, influence of dependence of the surface tension on the chemical potentials related to equilibrium droplet size (i.e., the effect of adsorption) and the effect of size-dependent line tension on the contact angles have been discussed. It has been shown that, for an axially symmetric sessile droplet, the effects of adsorption and line tension are, in the general case, comparable and manifest themselves already at the first order with respect to the droplet curvature. For a cylindrical droplet, the effect of adsorption is observed already in the first order with respect to the droplet curvature, while the influence of the line tension manifests itself only in the second order and is governed by the droplet-size dependence of the line tension.

Bridging of liquid drops at chemically structured walls.

Using mesoscopic interfacial models and microscopic density functional theory we study fluid adsorption at a dry wall decorated with three completely wet stripes of width L separated by distances D_{1} and D_{2}. The stripes interact with the fluid with long-range forces inducing a large finite-size contribution to the surface free energy. We show that this nonextensive free-energy contribution scales with lnL and drives different types of bridging transition corresponding to the merging of liquid drops adsorbed at neighboring wetting stripes when the separation between them is molecularly small. We determine the surface phase diagram and show that this exhibits two triple points, where isolated drops, double drops, and triple drops coexist. For the symmetric case, D_{1}=D_{2}≡D, our results also confirm that the equilibrium droplet configuration always has the symmetry of the substrate corresponding to either three isolated drops when D is large or a single triple drop when D is small; however, symmetry-broken configurations do occur in a metastable part of the phase diagram which lies very close to the equilibrium-bridging phase boundary. Implications for phase transitions on other types of patterned surface are considered.

Two regimes of a single n-heptane droplet combustion

This paper is devoted to the burning of an isolated n-heptane droplet in microgravity and analysis of the experimental data presented by FLEX experiment. Developing the classical analytical research methodology, proposed by Forman A. Williams, and using our obtained analytical solutions for the problem of droplet equilibrium evaporation and combustion, the analytical solutions to solve the problem of two regimes of droplet burning were found. To explain the presence of the radiative flame extinction and further droplet burning in cool flame regime, we assumed that two flame surfaces exist, but the first flame surface can be fixed with observation devices, and the second flame surface is an invisible one. According to this hypothesis, we suggested the analytical formulas for droplet and flame histories. The analytical solutions demonstrate a good coincidence with the experimental and numerical results.

Droplet clustering in cyclodextrin-based emulsions mediated by methylcellulose.

Rapid droplet aggregation in cyclodextrin (CD)-stabilized emulsions limits their practical use as material templates. Herein, we formulate mixtures of submicron CD-based emulsion droplets suspended in aqueous solutions of methylcellulose (MC) with various concentrations and molecular weights. We evaluate the effects of MC on the microstructure and stability of the emulsions using different techniques including optical microscopy, laser particle analysis, confocal laser scanning microscopy and multiple light scattering, explore the rheological behavior of the emulsions through large amplitude oscillatory shear experiments, and study the viscoelastic nonlinearities of the emulsions as a function of strain and strain-rate space through nondimensional elastic and viscous Lissajous-Bowditch plots. It is demonstrated that the emulsion droplets are present in the form of small clusters and their size is almost independent of MC concentration and molecular weight. The clustering pattern is also supported by the changes in viscoelastic properties of the emulsions and the intracycle nonlinear behavior of the Lissajous-Bowditch plots. We propose for the first time that glass-like dynamic arrest takes place with the formation of small equilibrium droplet clusters in the situation where the CD-based emulsion droplets are forced by depletion flocculation and kinetic trapping simultaneously exerted by MC.