Young Law Research Articles

Pressure Anisotropy in Polymer Brushes and Its Effects on Wetting.

Polymer brushes, coatings consisting of densely grafted macromolecules, experience an intrinsic lateral compressive pressure, originating from chain elasticity and excluded volume interactions. This lateral pressure complicates a proper definition of the interface and, thereby, the determination and interpretation of the interfacial tension and its relation to the wetting behavior of brushes. Here, we study the link among grafting-induced compressive lateral pressure in polymer brushes, interfacial tension, and brush wettability using coarse-grained molecular dynamics simulations. We focus on grafting densities and polymer-liquid affinities such that the polymer and liquid do not tend to mix. For these systems, a central result is that the liquid contact angle is independent of the grafting density, which implies that the grafting-induced lateral compressive pressure in the brush does not influence its wettability. Although the definition of brush interfacial tensions is complicated by the grafting-induced pressure, the difference in the interfacial tension between wet and dry brushes is perfectly well-defined. We confirm explicitly from Young's law that this difference offers an accurate description of the brush wettability. We then explore a method to isolate the grafting-induced contribution to the lateral pressure, assuming the interfacial tension is independent of grafting density. This scenario indeed allows disentanglement of interfacial and grafting effects for a broad range of parameters, except close to the mixing point. We separately discuss the latter case in light of autophobic dewetting.

Line tension of sessile droplets: Thermodynamic considerations.

We deduce a thermodynamically consistent diffuse interface model to study the line tension phenomenon of sessile droplets. By extending the standard Cahn-Hilliard model via modifying the free energy functional due to the spatial reflection asymmetry at the substrate, we provide an alternative interpretation for the wall energy. In particular, we find the connection of the line tension effect with the droplet-matrix-substrate triple interactions. This finding reveals that the apparent contact angle deviating from Young's law is contributed by the wall energy reduction as well as the line energy minimization. Besides, the intrinsic negative line tension resulting from the curvature effect is observed in our simulations and shows good accordance with recent experiments [Tan etal. Phys. Rev. Lett. 130, 064003 (2023)0031-900710.1103/PhysRevLett.130.064003]. Moreover, our model sheds light upon the understanding of the wetting edge formation which results from the vying effect of wall energy and line tension.

Asymptotic analysis of multi-phase-field models: A thorough consideration of junctions.

The solutions of multi-phase-field models exhibit boundary layer behavior not only along the binary interfaces but also at the common contacts of three or more phases, i.e., junctions. Hence, to completely determine the asymptotic behavior of a multi-phase-field model, the inner analysis of both types of layers has to be carried out, whereas, traditionally, the junctions part is ignored. This is remedied in the current work for a phase-field model of simple grain growth in two spatial dimensions. Since the junction neighbourhoods are fundamentally different from those of the binary interfaces, pertinent matching conditions had to be derived from scratch, which is also accomplished in a detailed manner. The leading-order matching analysis of the junctions exposed the restrictions present on the interfacial arrangement at the common meeting point, while the next-to-the-leading one uncovered the law governing the instantaneous motion of the latter. In particular, it is predicted for the considered model that the Young's law is always satisfied at a triple point, whether or not it is at rest. Surprisingly, the mobilities and the curvatures of the involving interfaces as well as the driving forces on the them do not affect this result. However, they do play a significant role in determining the instantaneous velocity of the junction point. The study has opened up many new directions for future research.

A phase‐field based model for coupling two‐phase flow with the motion of immersed rigid bodies

AbstractThe interaction of immersed rigid bodies with two‐phase flow is of high interest in many applications. A model for the coupling of a Hohenberg–Halperin type model for two‐phase flow and a fictitious domain method for consideration of rigid bodies is introduced leading to a full multiphase‐field method to address the overall problem. A normalized phase variable is used alongside a method for application of wetting boundary conditions over a diffuse fluid‐solid interface. This enables the representation of capillary effects and different wetting behavior based on Young's law. A number of simulations is conducted in order to validate the model and highlight its ability to handle a variety of setups for two‐phase particulate flow. This includes dynamic wetting situations, the motion of multiple particles within the two‐phase flow and the interaction with arbitrarily shaped solid structures inside the domain.

Droplets on chemically patterned surface: A local free-energy minima analysis

Droplet wetting on solid surfaces is a ubiquitous phenomenon in nature and applications. The wetting behavior of droplets on homogeneous surfaces has been accurately elucidated by the quintessential Young's law. However, on heterogeneous substrates, due to the energy barriers and contact line pinning effect, more than one equilibrated droplet pattern exists, which is more close to reality. Here, we propose a concise mathematical-physical model to delineate the droplet patterns on chemically patterned surfaces: stripe, "chocolate," and "chessboard." The present concept is capable of predicting the number as well as the morphologies of the equilibrated droplets on chemically patterned surfaces. We anticipate that the current work can be applied to fabricate programmable surfaces involving droplet manipulation in integrated circuits, biochips, and smart microelectronics.

Apparent Contact Angles on Lubricant-Impregnated Surfaces/SLIPS: From Superhydrophobicity to Electrowetting.

A fundamental limitation of liquids on many surfaces is their contact line pinning. This limitation can be overcome by infusing a nonvolatile and immiscible liquid or lubricant into the texture or roughness created in or applied onto the solid substrate so that the liquid of interest no longer directly contacts the underlying surface. Such slippery liquid-infused porous surfaces (SLIPS), also known as lubricant-impregnated surfaces, completely remove contact line pinning and contact angle hysteresis. However, although a sessile droplet may rest on such a surface, its contact angle can be only an apparent contact angle because its contact is now with a second liquid and not a solid. Close to the solid, the droplet has a wetting ridge with a force balance of the liquid-liquid and liquid-vapor interfacial tensions described by Neumann's triangle rather than Young's law. Here, we show how, provided the lubricant coating is thin and the wetting ridge is small, a surface free energy approach can be used to obtain an apparent contact angle equation analogous to Young's law using interfacial tensions for the lubricant-vapor and liquid-lubricant and an effective interfacial tension for the combined liquid-lubricant-vapor interfaces. This effective interfacial tension is the sum of the liquid-lubricant and the lubricant-vapor interfacial tensions or the liquid-vapor interfacial tension for a positive and negative spreading power of the lubricant on the liquid, respectively. Using this approach, we then show how Cassie-Baxter, Wenzel, hemiwicking, and other equations for rough, textured or complex geometry surfaces and for electrowetting and dielectrowetting can be used with the Young's law contact angle replaced by the apparent contact angle from the equivalent smooth lubricant-impregnated surface. The resulting equations are consistent with the literature data. These results enable equilibrium contact angle theory for sessile droplets on surfaces to be used widely for surfaces that retain a thin and conformal SLIPS coating.

Sessile Nanodroplets on Elliptical Patches of Enhanced Lyophilicity.

We theoretically investigate the shape of a nanodroplet on a lyophilic elliptical patch in lyophobic surroundings on a flat substrate. To compute the droplet equilibrium shape, we minimize its interfacial free energy using both Surface Evolver and Monte Carlo calculations, finding good agreement between the two methods. We observe different droplet shapes, which are controlled by the droplet volume and the aspect ratio of the ellipse. In particular, we study the behavior of the nanodroplet contact angle along the three-phase contact line, explaining the different droplet shapes. Although the nanodroplet contact angle is constant and fixed by Young’s law inside and outside the elliptical patch, its value varies along the rim of the elliptical patch. We find that because of the pinning of the nanodroplet contact line at the rim of the elliptical patch, which has a nonconstant curvature, there is a regime of aspect ratios of the elliptical patch in which the nanodroplet starts expanding to the lyophobic part of the substrate, although there is still a finite area of the lyophilic patch free to be wetted.

Using immiscible magnetic liquids of different density as a separating medium for magnetic liquid separation

The article presents the results of a new magnetic liquid separation method study. This method differs from known methods by the presence of a bilayer separation medium in the form of immiscible ferromagnetic liquids of different density. When separating in a bilayer separation medium, the particle equilibrium condition on a plane surface, recorded according to Young's law and Neumann's rule, must be supplemented by the linear tension of a curved phase interface. A linear tension is a force and energy barrier that prevents fixing of fine particles of noble metals at the phase interface, and is the reason for their effective extraction into the heavy fraction. The magnetic liquid separation method was tested on concentrates containing platinoids. It was established that during separation in a bilayer medium, the extraction of platinoids into the heavy fraction was 25,89 %, and during separation in a water-based ferromagnetic fluid it was 19,73 %. The quality of the heavy fraction makes it possible to direct it to the hydrometallurgical process, bypassing the copper plant, which increases the extraction of precious metals by 5,0 % abs.

Modeling of static contact angles with curved boundaries using a multiphase lattice Boltzmann method with variable density and viscosity ratios

SummaryNumerical modeling of multiphase flow generally requires a special procedure at the solid wall in order to be consistent with Young's law for static contact angles. The standard approach in the lattice Boltzmann method, which consists of imposing fictive densities at the solid lattice sites, is shown to be deficient for this task. Indeed, fictive mass transfer along the boundary could happen and potentially spoil the numerical results. In particular, when the contact angle is less than 90 degrees, the deficiencies of the standard model are major. Various videos that demonstrate this behavior are provided (Supporting Information). A new approach is proposed and consists of directly imposing the contact angle at the boundaries in much the same way as Dirichlet boundary conditions are generally imposed. The proposed method is able to retrieve analytical solutions for static contact angles in the case of straight and curved boundaries even when variable density and viscosity ratios between the phases are considered. Although the proposed wetting boundary condition is shown to significantly improve the numerical results for one particular class of lattice Boltzmann model, it is believed that other lattice Boltzmann multiphase schemes could also benefit from the underlying ideas of the proposed method. The proposed algorithm is two‐dimensional, and the D2Q9 lattice is used. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical simulation of dripping and jetting in supercritical fluids/liquid micro coflows

In this work, a two dimensional simulation of segmented micro coflows of CO2 and water in microcapillaries (20<T (°C)<50 and 8<p (MPa)<16.5) was carried out using a combination of the one-fluid model and the volume of fluid (VOF) method to describe the two-phase flow and a penalty method to account for the wetting property of the capillary walls. The computational work was validated by comparing numerical and experimental results in both the dripping and jetting regimes. The agreement of the calculated pressure difference across the droplet or jet interface with the Laplace⿿Young's law was assessed as supplementary criteria. The effects of CO2/water interfacial tension (5<Ͽ (mNm⿿1)<35) and wall wettability (contactangle CO2/wall varying from 0 to 180°) on the segmented water⿿supercritical CO2 microflows were specially described. It was shown that switching the wall surface from hydrophilic to hydrophobic by tuning the contact angle allows for changing the droplet curvature so that the continuous water phase eventually undergoes a phase inversion resulting in water droplets/slugs formation in a continuous CO2 phase.

Universal Deformation of Soft Substrates Near a Contact Line and the Direct Measurement of Solid Surface Stresses

Droplets deform soft substrates near their contact lines. Using confocal microscopy, we measure the deformation of silicone gel substrates due to glycerol and fluorinated-oil droplets for a range of droplet radii and substrate thicknesses. For all droplets, the substrate deformation takes a universal shape close to the contact line that depends on liquid composition, but is independent of droplet size and substrate thickness. This shape is determined by a balance of interfacial tensions at the contact line and provides a novel method for direct determination of the surface stresses of soft substrates. Moreover, we measure the change in contact angle with droplet radius and show that Young's law fails for small droplets when their radii approach an elastocapillary length scale. For larger droplets the macroscopic contact angle is constant, consistent with Young's law.

Origin of line tension for a Lennard-Jones nanodroplet

The existence and origin of line tension has remained controversial in literature. To address this issue, we compute the shape of Lennard-Jones nanodrops using molecular dynamics and compare them to density functional theory in the approximation of the sharp kink interface. We show that the deviation from Young’s law is very small and would correspond to a typical line tension length scale (defined as line tension divided by surface tension) similar to the molecular size and decreasing with Young’s angle. We propose an alternative interpretation based on the geometry of the interface at the molecular scale.

Multi-phase field study of the equilibrium state of multi-junctions

Abstract Multiple junctions between different grains of a pure material are investigated using the multi-phase field model in 2D and 3D. According to theoretical considerations of the multi-phase field model equations, there may be quasi-static solutions that depend on the ratio of the interface mobilities. Numerical calculations in 2D and 3D indicate that the system always converges to the static solution as described by Young's law independent of the interface mobilities. No quasi-static solutions are found, which is attributed to the necessity of continuous solutions within the interface region of a phase-field model. The effect of interface mobility on the dynamics of the interface angle is discussed.

All Solids, Including Teflon, Are Hydrophilic (To Some Extent), But Some Have Roughness Induced Hydrophobic Tendencies

It has recently been argued on the basis of four experiments that Teflon can often be regarded as hydrophilic when considering the interaction between liquid water and the solid surface of Teflon [Gao and McCarthy, Langmuir 2008, 24, 9183-9188]. The authors also recommend that more recognition be given to "hydrophilic" and "hydrophobic" as qualitative adjectives and discuss the importance of advancing and receding contact angles. In this work, I use net surface free energy changes for events consisting of (i) a smooth solid wrapping a droplet of water and (ii) a grain attaching to a droplet, to show that all solids with a Young's law contact angle thetae<180 degrees can be considered absolutely hydrophilic. This terminology is true in the sense that attachment of the solid to the liquid is always preferred even though the relative strength decreases as theta(e) increases. However, I also demonstrate that, within a surface free energy model, solids with thetae>90 degrees can be regarded as possessing hydrophobic tendencies with increasing roughness. The effect of the bending rigidity of the substrate is discussed, and a condition for minimum droplet radius for wrapping to occur is given.

Characterization of reservoir systems at elevated pressure

The efficiency in enhanced oil recovery strongly depends on interphase behaviour of the participating fluids and solids. Analysis is carried out using experimental techniques for recording contact angles and interfacial tension at elevated pressure and relating these data by use of known approaches from literature. Even in fairly incompressible liquid–liquid–solid systems the solid–liquid interfacial energies are shown to change at rising pressure. Combination of measured contact angles of a ternary water–n–alkane–dolomite system with those of the respective binary systems enables estimation of surface and interfacial energies of rock surfaces. In the presence of gases, especially methane, the water–n-alkane interfacial tension as well as the contact angle of a (non-wetting) water drop on dolomite in n-alkane is decreased which is only possible if the interfacial energy of n-alkane–dolomite rises based on Young's law. In a water-wet system, wetting by water was also observed to improve under pressure which is attributed to a decreasing water–rock interfacial energy since the water–oil interfacial tension is increased at the same time.

Quantitative phase-field modeling of two-phase growth

A phase-field model that allows for quantitative simulations of low-speed eutectic and peritectic solidification under typical experimental conditions is developed. Its cornerstone is a smooth free-energy functional, specifically designed so that the stable solutions that connect any two phases are completely free of the third phase. For the simplest choice for this functional, the equations of motion for each of the two solid-liquid interfaces can be mapped to the standard phase-field model of single-phase solidification with its quartic double-well potential. By applying the thin-interface asymptotics and by extending the antitrapping current previously developed for this model, all spurious corrections to the dynamics of the solid-liquid interfaces linear in the interface thickness W can be eliminated. This means that, for small enough values of W, simulation results become independent of it. As a consequence, accurate results can be obtained using values of W much larger than the physical interface thickness, which yields a tremendous gain in computational power and makes simulations for realistic experimental parameters feasible. Convergence of the simulation outcome with decreasing W is explicitly demonstrated. Furthermore, the results are compared to a boundary-integral formulation of the corresponding free-boundary problem. Excellent agreement is found, except in the immediate vicinity of bifurcation points, a very sensitive situation where noticeable differences arise. These differences reveal that, in contrast to the standard assumptions of the free-boundary problem, out of equilibrium the diffuse trijunction region of the phase-field model can (i) slightly deviate from Young's law for the contact angles, and (ii) advance in a direction that forms a finite angle with the solid-solid interface at each instant. While the deviation (i) extrapolates to zero in the limit of vanishing interface thickness, the small angle in (ii) remains roughly constant, which indicates that it might be a genuine physical effect, present even for an atomic-scale interface thickness.

Dewetting-induced collapse of hydrophobic particles.

A molecular dynamics study of the depletion of water (drying) around a single and between two hydrophobic nanoscale oblate plates in explicit water as a function of the distance of separation between them, their size, and the strength of the attraction between the plates and the water molecules is presented. A simple macroscopic thermodynamic model based on Young's law successfully predicts drying between the stacked plates and accounts for the free-energy barriers to this drying. However, because drying around a single plate is not macroscopic, a molecular theory is required to describe it. The data are consistent with the rate-determining step in the hydrophobic collapse of the two plates being a large-scale drying fluctuation, characterized by a free-energy barrier that grows with particle size.

Droplet profiles under the influence of van der Waals forces

We study the effect of van der Waals forces on globally energy minimizing profiles for liquid droplets which lie on a solid substrate in a vapour atmosphere and which are assumed to have a uniform cross-section. We prove that for repulsive van der Waals forces as well as for certain short range repulsive-long range attractive forces, there exists a unique globally minimizing profile. Although this profile necessarily contains vertical bounding segments, the height A of the vertical bounding segments can often be demonstrated to be order of magnitude smaller that the overall height B of the droplet. This is the case, in particular, when the droplet is sufficiently large, the Hamaker constant is sufficiently small, and the attractive forces are sufficiently mild. In the presence of repulsive forces only, A is on the order of angströms when B is on the order of millimeters, for realistic parameter values. Moreover, conditions are prescribed under which Young's law is satisfied to leading order despite the appearance of the vertical segments, when the contact angle is measured via an inscribed circle construction at a distance ξ0 from the edge of the droplet, where A [Lt ] ξ0 [Lt ] B.

Line Tension Effect Upon Static Wetting

Adding simply, in the classical capillary model, a constant line density of energy along the contact line leads to ill-posed equilibrium problems. Then, when line tension is present, the equilibrium configuration minimizes a different energy: the relaxedenergy, which explicitly depends on the presence of surface phases (i. e. infinitesimal films) on the boundary of the container. This formulation enables us to describe the modifications of the Young's law and then of equilibrium configurations which are due to line tension.

A singular limit for a system of degenerate Cahn-Hilliard equations

A singular limit is considered for a system of Cahn-Hilliard equations with a degenerate mobility matrix near the deep quench limit. Via formal asymptotics, this singular limit is seen to give rise to geometric motion in which the interfaces between the various pure phases move by motion by minus the surface Laplacian of mean curvature. These interfaces may couple at triple junctions whose evolution is prescribed by Young's law, balance of fluxes, and continuity of the chemical potentials. Short time existence and uniqueness is proven for this limiting geometric motion in the parabolic Hölder space ${ C}_{t, \, p}^{1 + \frac{\alpha}{4}, \, 4 + \alpha}, \, 0 <\alpha < 1$, via parameterization of the interfaces.