Microscopic Contact Angle Research Articles

Transition of Liquid Drops on Microstructured Hygrophobic Surfaces from the Impaled Wenzel State to the "Fakir" Cassie-Baxter State.

Low adhesion of liquids on solid surfaces can be achieved with protrusions that minimize the contact area between the liquid and the solid. The wetting state where an air cushion forms under the drop is known as the Cassie-Baxter state. Surfaces where liquids form macroscopic contact angles above 150° are called superhydrophobic and superhygrophobic, if we refer to water or any liquid, respectively. The Cassie state is desirable for applications, but it is usually unstable compared to the Wenzel state, where the drop is in direct contact with the rough surface. The Cassie-to-Wenzel transition can be triggered by an increase in pressure and vibrations, but the inverse Wenzel-to-Cassie is much more difficult to observe. Here, we examine under what conditions the Wenzel-to-Cassie transition is triggered when the microscopic contact angle changes abruptly. For this, we applied a lubricant of low surface tension around drops that were in the Wenzel state on microstructured surfaces. The increase of the microscopic contact angle lifted the drop from the rough surface, when the pillar height and spacing are large and small, respectively. Numerical calculations for the drop-lubricant interface showed that the surface geometry requirements for the Wenzel-to-Cassie transition are stricter than the ones for the stability of the Cassie state. A surface geometry where the Cassie state is more stable than the Wenzel for a given Laplace pressure of the drop may not always allow the Wenzel-to-Cassie transition to take place. Therefore, the stability of the Cassie state is a necessary but insufficient condition for the inverse transition.

Subgrid moving contact line model for direct numerical simulations of bubble dynamics in pool boiling of pure fluids

This contact line vicinity model is conceived as a subgrid model for the DNS of bubble growth in boiling. The model is based on the hydrodynamic multiscale theory and is suitable for the partial wetting case. On the smallest length scale (distance from the contact line) ∼ 100 nm, the interface slope is controlled by the Voinov angle. It is the static apparent contact angle (ACA) that forms due to evaporation, similarly to previous models neglecting the contact line motion. The calculation of the Voinov angle is performed with the generalized lubrication approximation and includes several nanoscale effects like those of Kelvin and Marangoni, vapor recoil, hydrodynamic slip length and interfacial kinetic resistance. It provides the finite values of the heat flux, pressure and temperature at the contact line. The dynamic ACA is obtained with the Cox-Voinov formula. The microscopic length of the Cox-Voinov formula (Voinov length) is controlled mainly by the hydrodynamic slip. The integral heat flux passing through the contact line vicinity is almost independent of the nanoscale phenomena, with the exception of the interfacial kinetic resistance and is mostly defined by the dynamic ACA. Both the dynamic ACA and the integral heat flux are the main output parameters of the subgrid model, while the local superheating and the microscopic contact angle are the main input parameters. The model is suitable for the grid sizes > 1 µm.

Moving contact line dynamics for capillary-driven microfluidics in wetting transition regime

The dynamics of moving contact lines (MCLs) dominate the behavior of capillary-driven microfluidics, which underlie many applications including microfluidic chips. The capillary displacement dynamics in the quasi-static regime has been extensively studied. However, the behavior of MCLs in the dynamic wetting transition regime remains largely unexplored, and previously established MCL dynamic models may be inadequate. In this study, a novel capillary displacement experiment is introduced, which is achieved by reversely introducing microfluidics with surface tension differences, where the one with low surface tension undergoes the wetting transition. In addition, a generalized Navier boundary condition (GNBC)-based model of capillary displacement dynamics is developed within the framework of diffusive interface theory to investigate the MCL dynamics in the wetting transition regime. The oscillation-relaxation process is experienced for phase interface and microscopic dynamic contact angle θd in the wetting transition regime. Spontaneous filling distance follows dfill*∼t1/2, and reaching quasi-static stage follows dfill*∼t1. The previously neglected mechanism of inertial-viscous competition dominates the early dynamics of such dynamic wetting transition processes. θd∝ucl is observed to be valid solely under conditions where viscosity dominates, but it breaks down in the presence of dominant inertial effects. An escalation in slip substantially diminishes the influence of inertia, with frictional dissipation mediated by slip emerging as the predominant factor in the capillary-driven early dynamics. The origin of uncompensated Young's stress in the GNBC and its correlation with capillary forces is unified, unveiling the underlying physical mechanism governing the dynamics at the MCL. Finally, by decoupling the analysis of viscosity and slip, a new θd-viscous-slip formulation is proposed, in agreement with the model predictions.

Soil conditioning of clay based on interface adhesion mechanism: Microscopic simulation and laboratory experiment

Clogging frequently occurs in the cutter head, excavation chamber or screw conveyor when an earth pressure balance (EPB) shield machine is tunneling in soft or silty clay ground with high clay mineral content. In this paper, montmorillonite, kaolinite, and illite were selected as research objects, and molecular dynamics simulation and laboratory experiment were adopted. At the microscopic scale, dynamic contact behavior and interfacial mechanical behavior of the interface between clay minerals and water/surfactant solution was simulated and the interfacial adhesion and conditioning mechanism between clay minerals and water/surfactant solution was revealed. Thus, sodium dodecyl benzene sulfonate surfactant was selected as the main composition of the soil conditioner. Then, the adhesion stress before and after soil conditioning and the contact angles between clay minerals and water/surfactant solution were tested and analyzed at the macroscopic scale. The result shows that the contact angle between droplet and clay mineral surface is an important parameter to characterize soil adhesion. The simulation rules of the microscopic contact angle are consistent with the experiment results. Furthermore, the adsorption energy between microscopic substances is dominated by electrostatic force, which can reflect the adhesion stress between macroscopic substances. Soil adhesion stress can be effectively decreased by adding the surfactant to the soil conditioner.

Near-wall depletion and layering affect contact line friction of multicomponent liquids

The main causes of energy dissipation in micro- and nanoscale wetting are viscosity and liquid-solid friction localized in the three-phase contact line region. Theoretical models predict the contact line friction coefficient to correlate with the shear viscosity of the wetting fluid. Experiments conducted to investigate such correlation have not singled out a unique scaling law between the two coefficients. We perform molecular dynamics simulations of liquid water-glycerol droplets wetting silicalike surfaces, aimed to demystify the effect of viscosity on contact line friction. The viscosity of the fluid is tuned by changing the relative mass fraction of glycerol in the mixture and it is estimated both via equilibrium and nonequilibrium molecular dynamics simulations. Contact line friction is measured directly by inspecting the velocity of the moving contact line and the microscopic contact angle. It is found that the scaling between contact line friction and viscosity is sublinear, contrary to the prediction of molecular kinetic theory. The disagreement is explained by accounting for the depletion of glycerol in the near-wall region. A correction is proposed, based on multicomponent molecular kinetic theory and the definition of a rescaled interfacial friction coefficient. Published by the American Physical Society 2024

Effect of contact angle on the pressure needed for a liquid to permeate a cylindrical pore

Effective separation of two immiscible liquids with filters requires a difference in the pressures to permeate between the two liquids, and setting the applied pressure between these two pressures. To help design such filters, we present an equation that enables the calculation of the pressure for a liquid phase to permeate through a smooth pore in the shape of a truncated cone as a function of (a) the contact angle of the liquid on the filters and (b) the angle of the pore wall. The equation was derived by considering the interfacial energy required to push a liquid meniscus from the top of the pore to the bottom, and then to exceed the maximum curvature required at the exit. This equation was tested experimentally by adding a hydrostatic head with water on the 3D-printed filters of acrylate polymer while systematically varying the pore radii and contact angle with water. Experimental results showed an increased pressure to permeate with higher contact angles while the equation predicted the opposite. We hypothesized that the reason for the disagreement was the assumption of a smooth pore. For a liquid on a rough pore wall, the curvature of the meniscus is not solely determined from the microscopic contact angle and the pore wall angle, but the liquid would adopt a lower curvature meniscus. Therefore, the developed equation was modified after reflecting the lower curvature, which showed much better agreement with the experimental results. The remaining discrepancy from the theory was attributed to the pressure fluctuation from the fluid flow occurring while adding water.

Structure and migration mechanism of thin liquid film in vicinity of advancing contact line

Thin liquid film near the gas-liquid-solid three phase contact line is the core area of oil and gas production, phase-change heat transfer, and material synthesis systems. Although there are many experimental studies on fluid dynamics in the contact line region, the prediction of contact angle is still difficult, and the bottleneck lies in the special structure of thin liquid film in the contact line region. Because the microstructure of thin liquid film is not well understood, the prediction of dynamic contact angle is always controversial. At present, the main controversial points focus on whether the microscopic contact angle changes with speed, and whether the microscopic contact angle is the same as the macroscopic contact angle. Therefore, it is crucial to monitor the dynamic process of the microscopic contact angle in the thin liquid film region of the contact line. In this work, the wetting system of 50 nm liquid droplets on different solid surfaces is constructed by molecular dynamics simulation, and the structure and migration mechanism of thin liquid film are studied. The structure of the precursor liquid film in the completely wetting droplet advancing contact line region and the nanoscale convex structure in the partially wetting droplet advancing contact line region are obtained. The precursor liquid film is 2–3 molecular layers in thickness, leading the droplet to move forward. However, there is no precursor liquid film in a partially wetting system, and the convex nano-bending larger than 10 nm is formed in the wetting process, resulting in the microscopic contact angle. By comparing the difference between the absolute smooth surface dynamic wetting process and the actual solid surface dynamic wetting process, the dynamic evolution law of the micro contact angle and the macro contact angle with time are obtained for the first time in the simulation. The liquid molecules in the contact line region are tracked and statistically analyzed by means of particle tracer. It is revealed that the liquid molecules in thin liquid film change from sliding mode to rolling mode with speed increasing under the action of solid surface friction, and then the air entrainment at the bottom of the contact line leads to slip and sputtering. The research results are expected to provide theoretical guidance for the following three directions: 1) improving heat transfer efficiency of micro and nano device based on wettability control; 2) improving the imbibition displacement efficiency of shale oil micro-nano matrix based on wettability regulation; 3) constructing universal contact angle prediction model.

On the coupling between direct numerical simulation of nucleate boiling and a micro-region model at the contact line

During the past years, numerical simulations of Nucleate Boiling have attracted increasing attention. If significant advances have already been performed on the conception of accurate numerical schemes accounting for jump conditions at a non-isothermal interface, further studies are required on the modeling of the heat flux in the vicinity of the contact line. Especially, at very low scale (<500 nm) the formation of a micro-region can be assumed, since the observed apparent contact angle can be higher than the microscopic contact angle and depend on the wall superheat. This variation of the apparent contact angle can be related to a temperature discontinuity between the saturation temperature and the local wall temperature at the contact line inducing a micro-region heat flux and an evaporation mass flow rate. Whereas this micro-region acts at very low scale, existing theoretical models suggest a strong influence of the micro-region heat flux on the global heat flux, and so on the overall bubble growth phenomenon. However, the numerical description of this micro-region in the framework of Direct Numerical Simulation (DNS) of nucleate boiling is a major issue due to the large range of spatial scales involved in the whole process. In the present paper, we propose a coupling methodology between an existing micro-region theoretical model and a larger scale DNS solver for Nucleate Boiling in axisymmetric configurations. Several benchmarks are proposed to show the consistency and the accuracy of the proposed method and to highlight the quantitative effect of the micro-region at the bubble scale.

Thin-Film Equations with Singular Potentials: An Alternative Solution to the Contact-Line Paradox

In the regime of lubrication approximation, we look at spreading phenomena under the action of singular potentials of the form P(h)≈h1-m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P(h)\\approx h^{1-m}$$\\end{document} as h→0+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\rightarrow 0^+$$\\end{document} with m>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>1$$\\end{document}, modeling repulsion between the liquid–gas interface and the substrate. We assume zero slippage at the contact line. Based on formal analysis arguments, we report that for any m>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>1$$\\end{document} and any value of the speed (both positive and negative) there exists a three-parameter, hence generic, family of fronts (i.e., traveling-wave solutions with a contact line). A two-parameter family of advancing “linear-log” fronts also exists, having a logarithmically corrected linear behavior in the liquid bulk. All these fronts have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with steady states, fronts have microscopic contact angle equal to π/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi /2$$\\end{document} for all m>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m>1$$\\end{document} and finite energy for all m<3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m<3$$\\end{document}. We also propose a selection criterion for the fronts, based on thermodynamically consistent contact-line conditions modeling friction at the contact line. So as contact-angle conditions do in the case of slippage models, this criterion selects a unique (up to translation) linear-log front for each positive speed. Numerical evidence suggests that, fixed the speed and the frictional coefficient, its shape depends on the spreading coefficient, with steeper fronts in partial wetting and a more prominent precursor region in dry complete wetting.

Construction and characterization of superhydrophobic wood coatings using one-step technique

This study was aiming at enhancing the stain resistance and durability of varnish on wood surfaces, a super hydrophobic solution was created by blending the synthetic ZIF-8/paraffin with Hexadecyltrimethoxysilane (HDTMS) at a specified mass ratio. This superhydrophobic mixture was then directly applied onto the wood varnish coating using a one-step method. The microscopic characterization, transmittance, wear resistance, and water contact angle (WCA) tests of hydrophobic coatings indicated that the mass ratio of ZIF-8 to HDTMS affected the surface microstructure and roughness of wood coatings. When the ratio reached 10:1, 5:1, and 2:1, the nanoparticles were dispersed at the interface of the varnish, and the film forming effect was uniform. The transmittance of the varnish coating was about 90%, and the transmittance of the superhydrophobic coating was about 80%. The water absorption rate of the treated sample was much lower than that of the untreated sample. After 10 cycles of abrasion tests, while the WCA decreased, it remained above 150°. Water droplets on the coating surface can remove pollutants through “rolling washing”, demonstrating good self-cleaning performance. The one-step method enables the rapid creation of superhydrophobic films on the surface of wood varnish coating.

Dynamic contact angle measurement of hydrophilic open microchannels: The role of surface wettability

Wetting dynamics play a major role in many practical applications; however, many fundamental problems remain unresolved, especially in relation to the dynamic contact angle. The present study investigates the movement of a gas–liquid interface in an open microchannel with different materials and wettability. Using a high-speed microscope camera, the shape of the dynamic equilibrium of the gas–liquid interface in a flowing state was recorded, including the microscopic regions near the contact line. The results show that the effects of the surface wetting condition play a crucial role in altering the apparent dynamic contact angle. During the experiment, we observed an interesting phenomenon where the dynamic equilibrium contact angle under flow conditions is larger than the contact angle predicted by the Cox–Voinov law. This may be attributed to the existence of friction between the fluid and the wall surface under flow, which reduces the additional pressure at the gas–liquid interface, and the magnitude of friction is manifested through the hysteresis contact angle. The instability and periodic variation of the microscopic contact angle are caused by the surface heterogeneity near the contact line, namely, the spatial variation of solid–gas and solid–liquid interfacial tensions. Understanding the surface properties can help optimize the interface system's design and improve its efficiency for use.

Effects of bendable P3CT polymers layer on the photovoltaic performance of perovskite solar cells

We report on the formation of bendable and edge-on poly[3-(4-carboxybutyl)thiophene-2,5-diyl] (P3CT) polymers thin layer used as a hole modification layer (HML) in the inverted perovskite solar cell. The aggregations of 2D layer-like P3CT polymers in dimethylformamide (DMF) solution can be formed via aromatic π–π stacking interactions and/or hydrogen-bonding interactions with the different concentration from 0.01 to 0.02 wt%, which highly influences the photovoltaic performance of the inverted perovskite solar cells. The atomic-force microscopic images and water droplet contact angle images show that the P3CT polymers modify the surface properties of the transparent conductive substrate and thereby dominating the formation of perovskite crystalline thin films, which play important roles in the highly efficient and stable perovskite solar cells. It is noted that the V OC (J SC) of the encapsulated solar cells values are maintained to be higher than 1.115 V (22 mA cm−2) after 104 d when an optimized π–π stacked and hydrogen-bonded P3CT polymer is used as the HML. On the other hand, the solar cell showed a high long-term stability by maintaining 85% of the initial power conversion efficiency in the ambient air for 103 d.

Electrowetting lattice Boltzmann method for micro- and nano-droplet manipulations.

Electrowetting has become a widely used tool for manipulating tiny amounts of liquids on surfaces. This paper proposes an electrowetting lattice Boltzmann method for manipulating micro-nano droplets. The hydrodynamics with the nonideal effect is modeled by the chemical-potential multiphase model, in which the phase transition and equilibrium are directly driven by chemical potential. For electrostatics, droplets in the micro-nano scale cannot be considered as equipotential as macroscopic droplets due to the Debye screening effect. Therefore, we linearly discretize the continuous Poisson-Boltzmann equationin a Cartesian coordinate system, and the electric potential distribution is stabilized by iterative computations. The electric potential distribution of droplets at different scales suggests that the electric field can still penetrate micro-nano droplets even with the screening effect. The accuracy of the numerical method is verified by simulating the static equilibrium of the droplet under the applied voltage, and the results show the apparent contact angles agree very well with the Lippmann-Young equation. The microscopic contact angles present some obvious deviations due to the sharp decrease of electric field strength near the three-phase contact point. These are consistent with previously reported experimental and theoretical analyses. Then, the droplet migrations on different electrode structures are simulated, and the results show that droplet speed can be stabilized more quickly due to the more uniform force on the droplet in the closed symmetric electrode structure. Finally, the electrowetting multiphase model is applied to study the lateral rebound of droplets impacting on the electrically heterogeneous surface. The electrostatic force prevents the droplets from contracting on the side which is applied voltage, resulting in the lateral rebound and transport toward the side.

Halochromic Textiles for Real-Time Sensing of Hazardous Chemicals and Personal Protection.

Chemical protective clothing (CPC) has become mandatory when performing various tasks to ensure user protection and prevent chemicals from contacting the skin and causing severe injuries. In addition to protection, there is a need to develop a simple mechanism that can be attached to CPC and be capable of detecting and alerting the user to the presence of harmful chemical agents. In this study, a double-sensor strategy was investigated, using six different pH indicators stamped on cotton and polyester knits to detect acidic and alkaline substances, both liquid and gaseous. Functionalized knits underwent microscopic characterization, air permeability and contact angle evaluation. All samples exhibited hydrophobic behavior (contact angle > 90°) and air permeability values above 2400 L/min/cm2/bar, with the best condition demonstrating a contact angle of 123° and an air permeability of 2412.5 L/min/cm2/bar when the sensor methyl orange and bromocresol purple (MO:BP) was stamped on polyester. The performed tests proved the functionality of the sensors and showed a visible response of all knits when contacting with different chemicals (acids and bases). Polyester functionalized with MO:BP showed the greatest potential, due to its preeminent color change. Herein, the fiber coating process was optimized, enabling the industrial application of the sensors via a stamping method, an alternative to other time- and resource-consuming techniques.

Contact Angle Measurement on Curved Wetting Surfaces in Multiphase Lattice Boltzmann Method.

Contact angle is an essential physical quantity that characterizes the wettability of a substrate. Although it is widely used in the studies of surface wetting, capillary phenomena, and moving contact lines, the contact angle measurements in simulations and experiments are still complicated and time-consuming. In this paper, we present an efficient scheme for the measurement of contact angle on curved wetting surfaces in lattice Boltzmann simulations. The measuring results are in excellent agreement with the theoretical predictions without considering the gravity effect. A series of simulations with various drop sizes and surface curvatures confirm that the present scheme is grid-independent. Then, the scheme is verified in gravitational environments by simulating the deformations of sessile and pendent droplets on the curved wetting surface. The numerical results are highly consistent with experimental observations and support the theoretical analysis that the microscopic contact angle is independent of gravity. Furthermore, the method utilizes only the microscopic geometry of the contact angle and does not depend on the droplet profile; therefore, it can be applied to nonaxisymmetric shapes or moving contact lines. The scheme is applied to capture the dynamic contact angle hysteresis on homogeneous or chemically heterogeneous curved surfaces. Importantly, the accurate contact angle measurement enables the dynamic mechanical analysis of moving contact lines. The present measurement is simple and efficient and can be extended to implementations in various multiphase lattice Boltzmann models.

Improving the photovoltaic performance of inverted perovskite solar cells via manipulating the molecular packing structure of PCBM

In this study, the molecular packing structure of solution-processed phenyl-C61-butyric acid methyl ester (PCBM) thin film was manipulated by varying the volume ratio of chlorobenzene (CB) to bromobenzene (BrB) from 100:0 to 50:50, which largely influences the device performance of the PCBM/perovskite heterojunction solar cells. Absorbance spectra, photoluminescence spectra, atomic force microscopic images and contact angle images were used to investigate the molecular packing structure effects of the PCBM thin films on the device performance of the inverted perovskite solar cells. Our experimental results show that the formation of PCBM aggregates and the contact quality at the PCBM/perovksite interface significantly influence the open-circuit voltage, short-circuit current density and fill factor of the resultant solar cells simultaneously. It is noted that the PCE of the encapsulated inverted CH3NH3PbI3 (MAPbI3) solar cells exhibited a stable and high power conversion efficiency of 18%.

Asymmetry of wetting and de-wetting on high-friction surfaces originates from the same molecular physics

Motion of three-phase contact lines is one of the most relevant research topics of micro- and nano-fluidics. According to many hydrodynamic and molecular models, the dynamics of contact lines is assumed overdamped and dominated by localized liquid–solid friction, entailing the existence of a mobility relation between contact line speed and microscopic contact angle. We present and discuss a set of non-equilibrium atomistic molecular dynamics simulations of water nanodroplets spreading on or confined between silica-like walls, showing the existence of the aforementioned relation and its invariance under wetting modes (“spontaneous” or “forced”). Upon changing the wettability of the walls, it has been noticed that more hydrophilic substrates are easier to wet rather than de-wet; we show how this asymmetry can be automatically captured by a contact line friction model that accounts for the molecular transport between liquid layers. A simple examination of the order and orientation of near-contact-line water molecules corroborates the physical foundation of the model. Furthermore, we present a way to utilize the framework of multicomponent molecular kinetic theory to analyze molecular contributions to the motion of contact lines. Finally, we propose an approach to discriminate between contact line friction models which overcomes the limitations of experimental resolution. This work constitutes a stepping stone toward demystifying wetting dynamics on high-friction hydrophilic substrates and underlines the relevance of contact line friction in modeling the motion of three-phase contact lines.

Spreading Equilibria Under Mildly Singular Potentials: Pancakes Versus Droplets

We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, global minimizers are compactly supported and display a microscopic contact angle of pi /2. Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient for purely repulsive potentials. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.

Preparation and performance evaluation of BiOI photocatalytic film

BiOI photocatalyst powder was prepared using the hydrothermal method. Using BiOI, PAN as the solute and DMF as the solvent, the mixed solution was stirred uniformly, and the photocatalytic film was prepared by an electrospinning machine. The microscopic morphology, pore size, and contact angle were characterised by scanning electron microscopy, N2 adsorption and desorption, and contact angle tests. The photocatalytic performance of the film at different calcination temperatures was explored. The results show that the film has the best photocatalytic performance under the heat treatment conditions of 280℃ and 6 h, and the degradation rate of MB to 10 mg/L is 94.3 %. Meanwhile, different cycles of experiments show that the film photocatalyst has good stability and strong mechanical properties. After four cycles of experiments, the film showed almost no damage and the degradation rate could still reach 93.6 %. The films showed good degradation potential of organic dye wastewater and have good application prospect.

The Cox–Voinov law for traveling waves in the partial wetting regime* *The first author is grateful to Lorenzo Giacomelli and Felix Otto for discussions on work preceding this paper. The second author thanks Floris Roodenburg for discussions. This work is partially based on the second author’s bachelor thesis in applied mathematics prepared under the advice of the first author at

We consider the thin-film equation in {h > 0} with partial-wetting boundary conditions and inhomogeneous mobility of the form m(h) = h 3 + λ 3−n h n , where h ⩾ 0 is the film height, λ > 0 is the slip length, y > 0 denotes the lateral variable, and n ∈ (0, 3) is the mobility exponent parameterizing the nonlinear slip condition. The partial-wetting regime implies the boundary condition ∂ y h = const. > 0 at the triple junction ∂{h > 0} (nonzero microscopic contact angle). Existence and uniqueness of traveling-wave solutions to this problem under the constraint as h → ∞ have been proved in previous work by Chiricotto and Giacomelli (2011 Commun. Appl. Ind. Math. 2 e-388, 16). We are interested in the asymptotics as h ↓ 0 and h → ∞. By reformulating the problem as h ↓ 0 as a dynamical system for the difference between the solution and the microscopic contact angle, values for n are found for which linear as well as nonlinear resonances occur. These resonances lead to a different asymptotic behavior of the solution as h ↓ 0 depending on n. Together with the asymptotics as h → ∞ characterizing the Cox–Voinov law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto (2016 Nonlinearity 29 2497–536), the rigorous asymptotics of traveling-wave solutions to the thin-film equation in partial wetting can be characterized. Furthermore, our approach enables us to analyze the relation between the microscopic and macroscopic contact angle. It is found that the Cox–Voinov law for the macroscopic contact angle depends continuously differentiably on the microscopic contact angle.