Local Surface Tension Research Articles

Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach

We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean curvature motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.

Calculation of the surface tension and pressure components from a non-exponential perturbation method of the thermodynamic route

Surface tension is probably the most important interfacial property and a large number of techniques have been devoted to its calculation. Usually, this calculation is carried out using mechanical or thermodynamic definitions. The mechanical route uses an arbitrary choice to affect the contribution of the pairwise force. To overcome this arbitrariness, a thermodynamic route based on the area perturbation (test-area (TA) method) has been developed for the calculation of surface tension. The volume perturbation (VP) method provides an original route to compute the components of the pressure tensor. These two routes are developed from the perturbation theory leading to working expressions using exponential averages of energy. The use of exponential averages makes the calculation strongly dependent on the occurrence of low values of ΔU. Additionally, the decomposition of the energy to obtain local surface tension is nontrivial. From the explicit derivation of the partition function the exponential average is avoided providing an interesting alternative to TA, VP, and mechanical methods. To make a consistent comparison, we study the profiles of the surface tension along the direction normal to the surface for the different definitions and techniques in the cases of liquid-vapor interfaces of acids gases, binary, and apolar systems.

Collagen Unfolding Determines Fluid Transfer in the Interstitial Matrix

Local unfolding of collagen at physiologic temperatures is thought to facilitate interactions with enzymes and scaffold molecules during inflammation, tissue remodeling, and wound healing. Previous data showing high interstitial hydration potential (HP) in human and porcine dermis after collagen thermal unfolding and fibroblast death suggest that it also plays a role in local modulation of interstitial flows. To test this hypothesis, collagen was progressively unfolded in situ, and changes in HP and water influx-rate within the matrix were measured as a function of the extent of unfolding, which was quantified by differential scanning calorimetry in full-thickness dermal samples after timed heat-treatment at 60°C and equilibration at 4°C. HP was determined by osmotic stress techniques, and influx-rates from time-dependent gravimetric changes under 35mmHg osmotic counterpressure. Both increased linearly with the proportion of unfolded collagen: the HP by 1.08 ± 0.16 mmHg, and the influx-rate by 3.19 ± 0.39 μl/min /100g per each 1% of collagen unfolded (R2= 0.93 and 0. 95, respectively). The relative humidity and intensity of T2-weighed magnetic resonance images of the dermis also increased with the extent of collagen unfolding, confirming interfacial energy contributions to the HP - as predicted by the Kelvin relationship - and the expected hydrophobic nature of the newly formed protein/water interfaces, respectively. These results are fully consistent with the hypothesis and point to yet another potentially important function of local collagen unfolding in tissue homeostasis. As a plausible mechanism for HP and influx-rate increases with collagen unfolding, we propose that the surface tension of vapor/water interfaces under exposed hydrophobic clusters is higher than at hydrophilic interfaces; at nanometer scales, these differences generate local surface-tension gradients in the matrix that accelerate water influx and shift the HP.

Local pressure components and surface tension of spherical interfaces. Thermodynamic versus mechanical definitions. I. A mesoscale modeling of droplets

We report mesoscale simulations of spherical drops to investigate the surface tension and mechanical properties. The Monte Carlo simulations are performed with the multibody potential commonly used in the many-body dissipative particle dynamics simulations. We establish here the calculation of the local normal and transverse components of the pressure tensor via the perturbation volume within the thermodynamic route. The different profiles of these components are compared to those calculated using the mechanical approach. To complete the mesoscale modeling of drops, we investigate the curvature dependence of the surface tension in order to calculate the Tolman's length, which is found to be negative.

Temperature distribution inside an evaporating two-dimensional droplet lying on curved or flat substrates

The temperature field inside an evaporating two-dimensional droplet resting on curved or flat isothermal substrates is studied under various evaporation conditions. An analytical solution for the temperature is derived, which can be directly used in the cases of pinned and depinned contact lines as well as in stick-slip evaporation modes. It is found that the temperature drop at the free surface of a droplet on a convex, hydrophobic substrate is far greater than that for flat or concave substrates of the same hydrophobicity. The analytical solution for the temperature field allows the direct estimate of the local temperature gradient and, hence, of the local surface tension gradient that gives rise to Marangoni flow directed from the contact lines to the top of the droplet.

Breakup of an electrified viscous thread with charged surfactants

The dynamics and breakup of electrified viscous jets in the presence of ionic surfactants at the interface are investigated theoretically. Axisymmetric configurations are considered and the jet is surrounded by a concentrically placed cylindrical electrode, which is held at a constant voltage potential. The annular region between the jet and the electrode is taken to be a hydrodynamically passive dielectric medium and an electric field is set up there and drives the flow, along with other physical mechanisms including capillary instability and viscous effects. The jet fluid is taken to be a symmetric electrolyte and proper modeling of the cationic and anionic species is used by considering the Nernst–Planck equations in order to find the volume charge density that influences the electric field in the jet. A positively charged insoluble surfactant is present at the interface, and its evolution, as well as the resulting value of the local surface tension coefficient, is coupled with the voltage potential at the interface. The resulting coupled nonlinear systems are derived and analytical progress is made by carrying out a nonlinear slender jet approximation. The reduced model is described by a number of hydrodynamic, electrical, and electrokinetic parameters, and an extensive computational study is undertaken to elucidate the dynamics along with allied linear properties. It is established that the jet ruptures in finite time provided the outer electrode is sufficiently far away, and numerous examples are given where the dimensionless parameters can be used to control the size of the satellite drops that form beyond the topological transition, as well as the time to break up. It is also shown that pinching solutions follow the self-similar dynamics of clean viscous jets at times close to the breakup time. Finally, a further asymptotic theory is developed for large Debye layers to produce an additional model that incorporates the effects of surface charge diffusion. Numerical solutions establish that the presence of electrostatic and electrokinetic effects increases the sizes of satellites but have a rather weak influence on the time to rupture.

From disjoining pressure to line tension

The thermodynamic line tension of rectangular edge for amorphous or crystalline bodies with dispersion forces is calculated. The dependence of the local surface tension of the face adjacent to the edge on the distance to the edge is calculated preliminarily and it is shown that the linear tension is practically formed at three intermolecular distances to the edge. The order of magnitude of line tension is determined by the product of surface tension and intermolecular (interplanar for crystals) distance.

A multi-component lattice Boltzmann model with non-uniform interfacial tension module for the study of blood flow in the microvasculature

This study aims at analyzing the red blood cell (RBC) deformation and velocity while streaming through venules and through capillaries whose diameters are smaller than the RBC size. The characteristics of the RBC shape change and velocity can potentially help in diagnosing diseases. In this work, the RBC is considered as a surfactant-covered droplet. This is justified by the fact that the cell membrane liquefies under pressure in the capillaries, and this allows the marginalization of its mechanical properties. The RBC membrane is in fact a macro-colloid containing lipid surfactant. When liquefied, it can be considered as a droplet of immiscible hemoglobin covered with lipid surfactant in a plasma surrounding. The local gradient in the surface tension due to non-uniform local interface surfactant distribution is neglected here, and a non-uniform zonal-averaged value of surface tension representative of the surfactant bulk zonal concentration is rather implemented. The interplay between the surface tension geometry and the hydrodynamic conditions determines the droplet shape by affecting a change in its Weber number, and influences its velocity. The Gunstensen lattice Boltzmann model for immiscible fluids is used here since it provides independent adjustment of the local surface tension, and allows the use of fluids with viscosity contrast. The proposed concept was used to investigate the dynamic shape change of the RBC while flowing through the microvasculature, and to explore the Fahraeus and the Fahraeus–Lindqvist effects. Copyright © 2010 John Wiley & Sons, Ltd.

Modelling bubble–particle interaction with dynamic surface tension

Bubble–particle interaction in turbulent flow is a dynamic process by its very nature. Important flotation parameters such as air hold-up, bubble size distribution, and bubble–particle interaction are strongly influenced by the shape and behavior of bubbles in turbulent flow. Bubble dynamics and the behavior of the three-phase interface are determined by the local surface tension and surfactant mass transfer characteristics. For systems with homogeneous distribution of surfactants the Marangoni term vanishes from the balance of bubble surface forces. In flotation however the heterogeneity of spatial and temporal distribution of surfactants requires reformulation of the interfacial stress balance and inclusion of the Marangoni effect. In this paper we propose a method for dynamic modeling of surface tension forces in flotation systems. The understanding of bubble and surfactant dynamics can contribute to greater insight in the physico-chemical behavior of flotation systems.

Thermocapillary actuation by optimized resistor pattern: bubbles and droplets displacing, switching and trapping

We report a novel method for bubble or droplet displacement, capture and switching within a bifurcation channel for applications in digital microfluidics based on the Marangoni effect, i.e. the appearance of thermocapillary tangential interface stresses stemming from local surface tension variations. The specificity of the reported actuation is that heating is provided by an optimized resistor pattern (B. Selva, J. Marchalot and M.-C. Jullien, An optimized resistor pattern for temperature gradient control in microfluidics, J. Micromech. Microeng., 2009, 19, 065002) leading to a constant temperature gradient along a microfluidic cavity. In this context, bubbles or droplets to be actuated entail a surface force originating from the thermal Marangoni effect. This actuator has been characterized (B. Selva, I. Cantat, and M.-C. Jullien, Migration of a bubble towards a higher surface tension under the effect of thermocapillary stress, preprint, 2009) and it was found that the bubble/droplet (called further element) is driven toward a high surface tension region, i.e. toward cold region, and the element velocity increases while decreasing the cavity thickness. Taking advantage of these properties three applications are presented: (1) element displacement, (2) element switching, detailed in a given range of working, in which elements are redirected towards a specific evacuation, (3) a system able to trap, and consequently stop on demand, the elements on an alveolus structure while the continuous phase is still flowing. The strength of this method lies in its simplicity: single layer system, in situ heating leading to a high level of integration, low power consumption (P < 0.4 W), low applied voltage (about 10 V), and finally this system is able to manipulate elements within a flow velocity up to 1 cm s(-1).

Thermodynamic line tension of a molecular solid

For a molecular solid with dispersion forces, a particular expression is derived relating thermodynamic line tension to thermodynamic surface tension and the crystal spacing. Using recent data on the local Irving-Kirkwood stress tensor at the slit edge, local surface tension near the crystal edge and the line tension of the edge are estimated. This is the first estimation of the thermodynamic line tension for a molecular cubic crystal in the literature.

A conservative SPH method for surfactant dynamics

In this paper, a Lagrangian particle method is proposed for the simulation of multiphase flows with surfactant. The model is based on the multiphase smoothed particle hydrodynamics (SPH) framework of Hu and Adams (2006) [1]. Surface-active agents (surfactants) are incorporated into our method by a scalar quantity describing the local concentration of molecules in the bulk phase and on the interface. The surfactant dynamics are written in conservative form, thus global mass of surfactant is conserved exactly. The transport model of the surfactant accounts for advection and diffusion. Within our method, we can simulate insoluble surfactant on an arbitrary interface geometry as well as interfacial transport such as adsorption or desorption. The flow-field dynamics and the surfactant dynamics are coupled through a constitutive equation, which relates the local surfactant concentration to the local surface-tension coefficient. Hence, the surface-tension model includes capillary and Marangoni-forces. The present numerical method is validated by comparison with analytic solutions for diffusion and for surfactant dynamics. More complex simulations of an oscillating bubble, the bubble deformation in a shear flow, and of a Marangoni-force driven bubble show the capabilities of our method to simulate interfacial flows with surfactants.

Steep capillary-gravity waves in oscillatory shear-driven flows

We study steep capillary-gravity waves that form at the interface between two stably stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory nature of the external forcing prevents the waves from overturning, and thus enables the development of steep waves at large forcing. They arise through a supercritical pitchfork bifurcation, characterized by the square root dependence of the height of the wave on the excess vibrational Froude number (W, square root of the ratio of vibrational to gravitational forces). At a critical valueWc, a transition to a linear variation inWis observed. It is accompanied by sharp qualitative changes in the harmonic content of the wave shape, so that trochoidal waves characterize the weakly nonlinear regime, but ‘finger’-like waves form forW≥Wc. In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas forW&lt;Wc, the wavelength exhibits an explicit dependence on the frequency of forcing that is due to the effect of viscosity. Most significantly, the radius of curvature of the wave crests decreases monotonically withWto reach the capillary length forW=Wc, i.e. the lengthscale for which surface tension forces balance gravitational forces. ForW&lt;Wc, gravitational restoring forces dominate, but forW≥Wc, the wave development is increasingly defined by localized surface tension effects.

Metastable contact angles and self-propelled drop movement on chemically heterogeneous surfaces by a mean-field lattice Boltzmann model

Contact angle behaviors on heterogeneous surfaces were studied by the mean-field free-energy lattice Boltzmann method. Our results suggest that the relation for contact angles on heterogeneous surfaces proposed by Israelachvili and Gee [Langmuir, 5, 288 (1989)] is in general not valid for the case examined. Droplet self-propelled movement on chemically patterned surfaces was also modeled by means of a local surface tension gradient.

Surfactant-covered drops between parallel plates

We use a numerical model to study the deformation of surfactant-covered drops in a confined situation. A boundary-integral method with modified Green’s functions, which exactly satisfy the no-slip condition at the wall, is used to solve the flow equations. The resulting velocities are used in the convection–diffusion equation for the surfactant concentration which is handled with a finite volume method. The local surfactant concentration is coupled to the local surface tension via a Langmuir isotherm. The model presented here is three dimensional, but only suited for equi-viscous drops and insoluble surfactants, but can be easily extended.Several examples of the influence of surfactants are shown: for drops in shear flow, surface dilution is of more significance due to the larger deformation of drops in confined situations. For pressure-driven flows, the non-uniform surfactant distribution speeds up the migration towards the center line, while the migration velocity in the velocity direction is lower.

Analysis of an extended pressure finite element space for two-phase incompressible flows

We consider a standard model for incompressible two-phase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in general not aligned with the triangulation that is used in the discretization of the flow problem. This non-alignment causes severe difficulties w.r.t. the discretization of the localized surface tension force and the discretization of the flow variables. In cases with large surface tension forces the pressure has a large jump across the interface. In standard finite element spaces, due to the non-alignment, the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure. In many simulations these effects cause large oscillations of the velocity close to the interface, so-called spurious velocities. In [6] it is shown that an extended finite element space (XFEM) is much better suited for the discretization of the pressure variable. In this paper we derive important properties of the XFEM space. We present (optimal) approximation error bounds and prove that the diagonally scaled mass matrix has a uniformly bounded spectral condition number. Results of numerical experiments are presented that illustrate properties of the XFEM space.

Expressions for local contributions to the surface tension from the virial route

The expression of the surface tension using the virial route has been reinvestigated in order to establish a local version of the surface tension and of its long-range corrections. In fact, giving a local surface tension is very important for the simulation from a methodological viewpoint. It is also of basic interest to associate the profile of the intrinsic part of the surface tension with that of the long-range corrections to make the surface tension calculation consistent between the different approaches that can be used. Working expressions for two-phase systems interacting through dispersion-repulsion (Lennard-Jones) and Coulombic (Ewald summation) interactions are proposed. Different operational expressions of the surface tension are compared in the cases of n -pentane, carbon dioxide, and water liquid-vapor equilibria for which the orders of magnitude between the electrostatic and dispersion forces are different.

The effects of a diffusion controlled surfactant on a viscous drop injected into a viscous medium

The effects of a diffusion controlled surfactant on the evolution of a buoyant viscous drop injected into a viscous fluid are studied numerically for the case of finite bulk convection to resolve neck dynamics and detaching drop volumes. When the drop is formed, its interface initially expands. The surfactant adsorbs and depletes a region around the drop. When the drop is sufficiently elongated, a neck begins to form. The surface contracts rapidly above the neck, driving the surface concentration above its equilibrium value. The surfactant subsequently desorbs into the region adjacent to the interface that had previously been depleted of the surfactant. This creates diffusion fluxes away from the neck that are larger than suggested by an a priori scaling of the governing equations. The rapid flux removes the surfactant effectively from the contracting neck, preventing the occurrence of strong local reductions in the surface tension. Through this mechanism, neck dynamics are altered only weakly for surface coverages x=Γeq∕Γ∞ less than or equal to 0.9. For surface coverages close enough to maximum packing (x=0.92) and for diffusion fluxes that are sufficiently slow, surfactant accumulation can reduce the local surface tension sufficiently to prevent drops from detaching. A phase diagram summarizing neck shapes and regimes where drops fail to detach is presented as a function of Ψ, the ratio of surfactant diffusion rate (between the interface and the bulk) to the rate of surface contraction.

Application of the method of a self-consistent field to the calculation of the local density and surface tension of small drops in a simple fluid

An approach based on the perturbation theory combined with the method of a self-consistent field was used to investigate the dimensional dependence of the specific free surface energy (surface tension) of small drops in a simple fluid. The surface tension decreases with decreasing drop size. The approach proposed also allows one to find the radial density distribution in a small object.

Asymmetric Surface Roughness Formationon Moving Non-isothermal Liquid Coatings

Abstract We present a lubrication model of thermo-capillary flows in moving volatile liquid film coatings. The forced air impingement from a slit nozzle onto the moving coating imposes an axi-symmetric surface temperature distribution, and thus the local surface-tension gradients on the liquid surface. Despite the symmetric temperature profile, local thickness variations became asymmetric and exhibited a particular ridge in downstream and a depression in upstream. The competing feature between the surface-tension-driven and the pressure-driven flows gives a characteristic growth and decay in the surface roughness as the temperature profile travels in the opposite direction to the moving coating. The model prediction showed that the surface roughness was first enhanced and then suppressed with increasing the impinging air velocity, suggesting some directions for achieving more uniform coatings at higher speeds.