Theory Of Capillarity Research Articles

A Heintze-Karcher inequality with free boundaries and applications to capillarity theory

In this paper we analyze the shape of a droplet inside a smooth container. To characterize their shape in the capillarity regime, we obtain a new form of the Heintze–Karcher inequality for mean convex hypersurfaces with boundary lying on curved substrates.

Solitonic connections in capillarity theory: A review

Solitonic connections in capillarity theory: A review

Single and two-cells shape analysis from energy functionals for three-dimensional vertex models.

Vertex models have been extensively used for simulating the evolution of multicellular systems, and have given rise to important global properties concerning their macroscopic rheology or jamming transitions. These models are based on the definition of an energy functional, which fully determines the cellular response and conclusions. While two-dimensional vertex models have been widely employed, three-dimensional models are far more scarce, mainly due to the large amount of configurations that they may adopt and the complex geometrical transitions they undergo. We here investigate the shape of single and two-cells configurations as a function of the energy terms, and we study the dependence of the final shape on the model parameters: namely the exponent of the term penalising cell-cell adhesion and surface contractility. In single cell analysis, we deduce analytically the radius and limit values of the contractility for linear and quadratic surface energy terms, in 2D and 3D. In two-cells systems, symmetrical and asymmetrical, we deduce the evolution of the aspect ratio and the relative radius. While in functionals with linear surface terms yield the same aspect ratio in 2D and 3D, the configurations when using quadratic surface terms are distinct. We relate our results with well-known solutions from capillarity theory, and verify our analytical findings with a three-dimensional vertex model.

The monotonicity behavior of density profiles at vapor-liquid interfaces of mixtures

In their seminal monograph ’Molecular Theory of Capillarity’, Rowlinson and Widom describe different possible shapes of density profiles at the vapor-liquid interface of mixtures. They postulated that in some instances, density profiles could possibly be non-monotonic, exhibiting either a maximum and/ or a minimum. This contribution revisits this statement in the light of four decades of posterior research. We summarize the distinct morphologies at the vapor-liquid interface suggested in the literature recognizing that the condition of a single minimum in the profile has not yet been reported. Interfacial density profiles with a single maximum as well as fully monotonic density profiles have been observed and reported extensively. The case of a simultaneous maximum and minimum is more controversial, as it has only been predicted using theoretical approaches such as density gradient theory (DGT). This ambiguity is further investigated in this work using the example of the vapor-liquid interface of cyclohexane + butanol. Both DGT in combination with several distinct equations of state and molecular dynamics simulations are used. The results from the two methods are found to be contradictory: while the DGT results predict a maximum/minimum structure, the computer experiment results indicate only a single maximum in the density profiles. This work thereby emphasizes that results from DGT for highly non-ideal mixtures should not be taken for granted.

A relaxation model for the non-isothermal Navier-Stokes-Korteweg equations in confined domains

The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals' theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Secondly, any real gas equation of state is based on a non-convex Helmholtz free energy potential which may cause the eigenvalues of the Jacobian of the first-order fluxes to become imaginary numbers inside the spinodal region.In this work, a thermodynamically consistent relaxation model is presented which is used to approximate the NSK equations. The model is complimented by thermodynamically consistent non-equilibrium boundary conditions which take contact angle effects into account. Due to the relaxation approach, the contribution of the Korteweg tensor in the linear momentum and total energy equations can be reduced to first-order terms which enables a straightforward implementation of contact angle boundary conditions in a numerical scheme. Moreover, the definition of a modified pressure function enables to formulate first-order fluxes which remain strictly hyperbolic in the entire spinodal region. The present work is a generalization of a previously presented parabolic relaxation model for the isothermal NSK equations.A high-order discontinuous Galerkin spectral element method which supports curved elements and hanging nodes is employed to discretize the system. The relaxation model and its corresponding boundary conditions are validated using solutions of the original NSK model and analytical results for one-, two- and three-dimensional test cases. The simulation of a spinodal decomposition in a three-dimensional porous structure underlines the capability of the presented approach.

Mesoscopic approach for nanoscale liquid-vapor interfacial statics and dynamics

Nanoscale liquid-vapor interfacial transport phenomena are of great significance to a variety of applications including evaporation, condescension, boiling and micro/nano-fluidics. In this work, we propose a mesoscopic approach to describe the nanoscale liquid-vapor interfacial statics and dynamics by combining the pseudopotential multiphase lattice Boltzmann method, the theorem of corresponding states and the principle of dynamic similarity. We demonstrate that our mesoscopic predictions of density profile, interfacial thicknesses and surface tensions for planar liquid-vapor interfaces of various real fluids agree very well with the NIST recommended data and molecular theories of capillarity in a very wide temperature range. We quantify the size effects of nano-bubbles and nano-droplets on the surface tension of water under a saturation temperature of 100°C. We show that the surface tensions of nano-bubbles decrease while the surface tensions of nano-droplets increase with increasing size, and the predictions from static cases and dynamic cases are consistent. The mesoscopic approach proposed in this paper could well resolve nanoscale liquid-vapor interfacial phenomena for various real fluids with high resolution, high accuracy and affordable computation cost in a very wide temperature range, which paves the way for quantitative investigations of nanoscale liquid-vapor interfacial transport for real fluids in practical applications.

Intrusion of liquids into liquid-infused surfaces with nanoscale roughness.

We present a theoretical study of the intrusion of an ambient liquid into the pores of a nanocorrugated wall w. The pores are prefilled with a liquid lubricant that adheres to the walls of the pores more strongly than the ambient liquid does. The two liquids are modeled as a binary liquid mixture of two species of particles, A and B. The mixture can decompose into a liquid rich in A particles, representing the ambient liquid, and another one rich in B particles, representing the liquid lubricant. The wall is taken to attract the B particles more strongly than the A particles. The ratio w-A/w-B of these interaction strengths is changed in order to tune the contact angle θ_{AB} formed by the A-rich/B-rich liquid interface between the two fluids and a planar wall, composed of the same material as the one forming the pores. We use classical density functional theory in order to capture the effects of microscopic details on the intrusion transition, which occurs as the concentration of the minority component or the pressure in the bulk of the ambient liquid is varied, moving away from bulk liquid-liquid coexistence within the single-phase domain of the A-rich bulk ambient liquid. These liquid structures have been studied as a function of the contact angle θ_{AB} and for various widths and depths of the pores. We also studied the reverse process in which a pore initially filled with the ambient liquid is refilled with the liquid lubricant. The location of the intrusion transition, with respect to its dependence on the contact angle θ_{AB} and the width of the pore, qualitatively follows the corresponding shift of the capillary-coexistence line away from the bulk liquid-liquid coexistence line, as predicted by a macroscopic capillarity model. Quantitatively, the transition found in the microscopic approach occurs somewhat closer to the bulk liquid-liquid coexistence line than predicted by the macroscopic capillarity model. The quantitative discrepancies become larger for narrower cavities. In cases in which the wall is completely wetted by the lubricant (θ_{AB}=0) and for small contact angles, the reverse transition follows the same path as for intrusion; there is no hysteresis. For larger contact angles, hysteresis is observed. The width of the hysteresis increases with increasing contact angle. A reverse transition is not found inside the domain within which the ambient liquid forms a single phase in the bulk once θ_{AB} exceeds a geometry-dependent threshold value. According to the macroscopic capillarity theory, for the considered geometry, this is the case for θ_{AB}>54.7^{∘}. Our computations show, however, that nanoscale effects shift this threshold value to much higher values. This shift increases strongly if the widths of the pores become smaller (below about ten times the diameter of the A and B particles).

Edge wetting: Steady state of rivulets in wedges

The geometry of rough, textured, fractured, and porous media is topologically complicated. Such media are commonly modeled as bundles of capillary tubes. However, angle-containing geometries can serve as a more realistic portrayal of their internal structure. A basic element inherent to all of these media is an open wedge-like channel. The classical theory of capillarity ignoring intermolecular interactions implies that liquid entering a wedge must propagate indefinitely along its spine when the liquid–gas interface is concave. This is well known as the Concus–Finn condition. In the present paper, we show that steady-state rivulets can be formed in such channels when surface forces are taken into account. We present a simple model based on the disjoining pressure approach and analyze the shape of rivulets in wedges. We also consider the case when the walls of the wedge are soft and can be deformed by the liquid.

How Small Is Too Small for the Capillarity Theory?

We perform molecular dynamics (MD) simulations of nanoscale water capillary bridges (WCBs) expanding between two parallel walls and determine the smallest separation between the walls above which t...

Collapsing and the convex hull property in a soap film capillarity model

Soap films hanging from a wire frame are studied in the framework of capillarity theory. Minimizers in the corresponding variational problem are known to consist of positive volume regions with boundaries of constant mean curvature/pressure, possibly connected by “collapsed” minimal surfaces. We prove here that collapsing only occurs if the mean curvature/pressure of the bulky regions is negative, and that, when this last property holds, the whole soap film lies in the convex hull of its boundary wire frame.

Temperature Dependence of Liquid Contact Angle at a Deformable Solid Surface

To date, it has been reliably shown by direct methods that a sessile droplet on a solid surface causes deformation of the substrate. For solids with rather low elasticity moduli, in particular, elastomers, this phenomenon becomes observable, and the deformation work reaches values sufficiently large to take it into account in the theory when necessary. Within the framework of the current theory of capillarity and the generalized Young equation, it is most convenient to introduce the deformation work (realized along and inside the three-phase contact line) into the line tension. The resultant effective line tension is several orders of magnitude higher than the usual one (for undeformable solids); therefore, it manifests itself even in macroscopic (although small) droplets. While the usual line tension is just a small correction to the classical Young equation, when elastomers are used as substrates, the effective line tension may become the main term and govern surface phenomena. In this communication, the results of work [Budziak, C.J., Varcha-Butler, E.I., and Neumann, A.W., J. Appl. Polym. Sci., 1991, vol. 42, p. 1959], which is devoted to studying the temperature dependence of liquid contact angles at elastomers, are discussed from the aforementioned points of view. The following facts have been explained: the higher the surface tension of a liquid the larger the contact angle at the same substrate; most probably, the contact angle increases with temperature; the temperature dependence of a contact angle cannot be strong because of the mutual compensation of the temperature dependences of the surface tension of a wetting liquid and the elasticity modulus of an elastomeric substrate.

A parabolic relaxation model for the Navier-Stokes-Korteweg equations

The isothermal Navier-Stokes-Korteweg system is a classical diffuse interface model for compressible two-phase flow which grounds in Van Der Waals' theory of capillarity. However, the numerical solution faces two major challenges: due to a third-order dispersion contribution in the momentum equations, extended numerical stencils are required for the flux calculation. Furthermore, the equation of state given by a Van-der-Waals law, exhibits non-monotone behaviour in the two-phase state space leading to imaginary eigenvalues of the Jacobian of the first-order fluxes.In this work a lower-order parabolic relaxation model is used to approximate solutions of the classical NSK equations. Whereas an additional parabolic evolution equation for the relaxation variable has to be solved, the system involves no differential operator of higher as second order. The use of a modified pressure function guarantees that the first-order fluxes remain hyperbolic. Altogether, the relaxation system is directly accessible for standard compressible flow solvers.We use the higher-order Discontinuous Galerkin spectral element method as realized in the open source code FLEXI. The relaxation model is validated against solutions of the original NSK model for a variety of 1D and 2D test cases. Three-dimensional simulations of head-on droplet collisions for a range of different collision Weber numbers underline the capability of the approach.

Energy Stored in Nanoscale Water Capillary Bridges between Patchy Surfaces.

We perform molecular dynamics (MD) simulations of a water capillary bridge (WCB) expanding between two identical chemically heterogeneous surfaces. The model surfaces, based on the structure of silica, are hydrophobic and are decorated by a hydrophilic (hydroxylated silica) patch that is in contact with the WCB. Our MD simulations results, including the WCB profile and forces induced on the walls, are in agreement with capillarity theory even at the smallest wall separations studied, h = 2.5-3 nm. Remarkably, the energy stored in the WCB can be relatively large, with an energy density that is comparable to that harvested by water-responsive materials used in actuators and nanogenerators.

Capillary forces on a small particle at a liquid-vapor interface: Theory and simulation

We study the meniscus on the outside of a small spherical particle with radius $R$ at a liquid-vapor interface. The liquid is confined in a cylindrical container with a finite radius $L$ and has a contact angle $\pi/2$ at the container surface. The center of the particle is placed at various heights along the central axis of the container. By varying $L$, we are able to systematically study the crossover of the meniscus from nanometer to macroscopic scales. The meniscus rise or depression on the particle is found to grow as $\ln (2L/R)$ when $R\ll L\ll \kappa^{-1}$ with $\kappa^{-1}$ being the capillary length and saturate to a value predicted by the Derjaguin-James formula when $R \ll \kappa^{-1} \ll L$. The capillary force on the particle exhibits a linear dependence on the particle's displacement from its equilibrium position at the interface when the displacement is small. The associated spring constant is found to be $2\pi\gamma\ln^{-1} (2L/R)$ for $L\ll \kappa^{-1}$ and saturates to $2\pi\gamma\ln^{-1} (3.7\kappa^{-1}/R)$ for $L\gg \kappa^{-1}$. At nanometer scales, we perform molecular dynamics simulations of the described geometry and the results agree well with the predictions of the macroscopic theory of capillarity. At micrometer to macroscopic scales, comparison to experiments by Anachkov \textit{et al.} [Soft Matter {\bf 12}, 7632 (2016)] shows that the finite span of a liquid-vapor or liquid-liquid interface needs to be considered to interpret experimental data collected with $L \sim \kappa^{-1}$.

The Mechanism of Emulsion Formation during Immiscible Fluid Flow in Geological Materials

On the basis of the Laplasian theory of capillarity, laws of continuous media mechanics, and concepts of physical-chemical mechanics of heterogeneous substances, the mechanism of emulsion formation during immiscible fluid flow in rocks is given. It is shown that this mechanism does not depend on the choice of the porous medium model in certain limits, but is determined by stresses acting on the meniscus and by the presence of slow-moving fluid films on the pore surface. The obtained expression for the minimal radius of capillaries where emulsion can form, combined with relevant experimental results, suggests that setting of the pressure gradient can either trigger or prevent emulsification.

Experimental Confirmation of Predicted Rise Heights in a Surface Tension Configuration Introduced by Laplace

We develop numerical methods and experimental procedures to test theoretical predictions derived independently in Bhatnagar and Finn (The Laplace Parallel Plates Problem in Capillarity Theory. In adjoining issue, 2018) describing the rise height of fluid between two parallel plates immersed in a fluid bath. Our experimental methods use an electrowetting-on-dielectric configuration that enables precise electrical control of contact angle and provides a way to test the theory across a range of contact angles. We present detailed quantitative comparisons of numerical, theoretical, and experimental results, finding agreement among all methods.

Mechanism of Cohesion of Monomolecular Water Film with the β-AgI Crystal Surface under Thermal Fluctuations

The growth of liquid phase nuclei on the surface of some crystals was shown to differ from that described in the classic theory of capillarity. The surface of the base face of a silver iodide crystal is completely covered with a monomolecular water film already in unsaturated vapors, and the wetting conditions of the substrate are determined by the hydrophobic properties of the film surface, but not the crystal surface itself. The mechanism by which the monomolecular film is held on the surface of crystalline silver iodide was studied by the Monte Carlo method at the molecular level. It was found that the adhesion of the film to the surface of the base face of the crystal was by hydrogen bonding with the ions of the second crystallographic layer of the substrate, and the film was thermodynamically stable even in unsaturated water vapor. The film hydrophobicity is due to the deficiency of hydrogen bond donors on its surface. The nanostructural elements on the surface of the aerosol particle can neutralize the hydrophobic properties of the film and thus serve as nucleation centers.

Analytical process model for wire + arc additive manufacturing

An analytical process model for predicting the layer height and wall width from the process parameters was developed for wire + arc additive manufacture of Ti-6Al-4V, which includes inter-pass temperature and material properties. Capillarity theory predicted that cylindrical deposits were produced where the wall width was less than 12 mm (radius <6 mm) due to the large value of the surface tension. Power was predicted with an accuracy of ±20% for a wide range of conditions for pulsed TIG and plasma deposition. Interesting differences in the power requirements were observed where a surface depression was produced with the plasma process due to differences in melting efficiency and/or convection effects. Finally, it was estimated the impact of controlling the workpiece temperature on the accuracy of the deposit geometry.

The tongue as a gripper.

Frogs, chameleons and anteaters are striking examples of animals that can grab food using only their tongue. How does the soft and wet surface of a tongue grip onto objects before they are ingested? Here, we review the diversity of tongue projection methods, tongue roughnesses and tongue coatings, our goal being to highlight conditions for effective grip and mobility. A softer tongue can reach farther: the frog Rana pipiens tongue is 10 times softer than the human tongue and can extend to 130% of its length when propelled in a whip-like motion. Roughness can improve a tongue's grip: the spikes on a penguin Eudyptes chrysolophus tongue can be as large as fingernails, and help the penguin swallow fish. The saliva coating on the tongue, a non-Newtonian biofluid, can either lubricate or adhere to food. Frog saliva is 175 times more viscous than human saliva, adhering the tongue to slippery, furry or feathery food. We pay particular attention to using mathematical models such as the theory of capillarity, elasticity and friction to elucidate the parameters for effective tongue use across a variety of vertebrate species. Finally, we postulate how the use of wet and rough surfaces to simultaneously sense and grip may inspire new strategies in emerging technologies such as soft robots.

Validation of Capillarity Theory at the Nanometer Scale. II: Stability and Rupture of Water Capillary Bridges in Contact with Hydrophobic and Hydrophilic Surfaces

We perform molecular dynamics (MD) simulations of water capillary bridges formed between parallel walls. The underlying structure of the walls corresponds to hydroxilated (crystalline) β-cristobalite, modified to cover a wide range of hydrophobicity/hydrophilicity. The capillary bridges are stretched during the MD simulations, from wall–wall separation h = 5 nm up to h ≈ 7.5 nm, until they become unstable and break. During the stretching process, we calculate the profiles of capillary bridges as well as the force and pressure induced on the walls, among other properties. We find that, for all walls separations and surface hydrophobicity/hydrophilicity considered, the results from MD simulations are in excellent agreement with the predictions from capillarity theory (CT). In addition, we find that CT is able to predict very closely the limit of stability of the capillary bridges, i.e., the value of h at which the bridges break. We also confirm that CT predicts correctly the relationship between the surface...