Thin Film Evolution Equations Research Articles

Reduced-Order Model for Surfactant-Laden Electrified Sessile Droplets.

A comprehensive understanding of the physics of electrowetting of a surfactant-laden droplet is important for applications in rapid healthcare diagnostics. A majority of biological samples examined during point-of-care (POC) diagnostics are biofluids with dissolved surfactants, such as the respiratory droplets containing protein (mucin) and surfactant molecules like dipalmitoylphosphatidylcholine. The presence of these surfactant molecules is anticipated to have a significant impact on the performance of electrowetting-based POC diagnostic devices. A reduced-order model is developed using the weighted residual integral boundary layer theory for the electrowetting of a surfactant-laden sessile droplet in a parallel plate electrode configuration. Thin film evolution equations are obtained for the fluid-fluid interface, the surfactant concentration, the depth-integrated flow rate, and the interfacial charge density. We show that the presence of surfactants opposes and decreases the strength of the electrohydrodynamic flow due to Marangoni stress-driven convection. The droplet then responds to an AC field with a suppressed amplitude of oscillation and the same mean deformation as that under DC forcing. Thus, low-frequency AC forcing with a suitable surfactant can plausibly be employed as a viable alternative to more energy-intensive high-frequency AC forcing.

Surface-tension-driven evolution of a viscoplastic liquid coating the interior of a cylindrical tube

One mechanism for airway closure in the lung is the surface-tension-driven instability of the mucus layer which lines the airway wall. We study the instability of an axisymmetric layer of viscoplastic Bingham liquid coating the interior of a rigid tube, which is a simple model for an airway that takes into account the yield stress of mucus. An evolution equation for the thickness of the liquid layer is derived using long-wave theory, from which we also derive a simpler thin-film evolution equation. In the thin-film case we show that two branches of marginally yielded static solutions of the evolution equation can be used to both predict the size of the initial perturbation required to trigger instability and quantify how increasing the capillary Bingham number (a parameter measuring yield stress relative to surface tension) reduces the final deformation of the layer. Using numerical solutions of the long-wave evolution equation, we quantify how the critical layer thickness required to form a liquid plug in the tube increases as the capillary Bingham number is increased. We discuss the significance of these findings for modelling airway closure in obstructive conditions such as cystic fibrosis, where the mucus layer is often thicker and has a higher yield stress.

Programmable thermocapillary shaping of thin liquid films

We present a method that leverages projected light patterns as a mechanism for freeform deformation of a thin liquid film via the thermocapillary effect. We developed a closed-form solution for the inverse problem of the thin-film evolution equation, allowing us to obtain the projection pattern required in order to achieve a desired topography. We experimentally implement the method using a computer controlled light projector, which illuminates any desired pattern onto the bottom of a fluidic chamber patterned with heat–absorbing metal pads. The resulting heat map induces surface tension gradients in the liquid–air interface, giving rise to thermocapillary flow that deforms the liquid surface. If a polymer is used for the liquid film, it can then be photocured to yield a solid device. Based on the inverse-problem solutions and using this system, we demonstrate the fabrication of several diffractive optical elements, including phase masks for extended depth of field imaging, and for three-dimensional localization microscopy. The entire process, from projection to solidification, is completed in less than five minutes, and yields a sub-nanometric surface quality without any post-processing.

Droplet Sliding on an Inclined Substrate with a Topographical Defect.

Pinning and depinning of droplets on heterogeneous substrates are widely seen in nature and need to be carefully controlled in industrial processes such as substrate cleaning and spray coating. In this work, a two-dimensional droplet sliding on an inclined substrate with a topographical defect is studied with a thin-film evolution equation. Using results from time-dependent finite-difference calculations, we focus our discussion on the dynamic interactions between the sliding droplet and the topographical defect. For a Gaussian defect shape, we find that droplet pinning is primarily determined by the advancing contact line pinning at the defect surface where the topography slope is minimum. We demonstrate that with certain combinations of defect heights and widths, residual droplets can form on the defect as a result of geometric constraints involving the receding droplet meniscus and the defect shape. We show that the delay in sliding caused by the defect is mainly due to the pinning and depinning of the receding contact line, and less affected by the dynamic behavior of the advancing contact line. This topography-induced delay in sliding of an individual droplet may have important implications for controlling the collective sliding behavior of multiple droplets.

Multiscale level-set method for accurate modeling of immiscible two-phase flow with deposited thin films on solid surfaces

We developed a multiscale sharp-interface level-set method for immiscible two-phase flow with a pre-existing thin film on solid surfaces. The lubrication approximation theory is used to model the thin-film equation efficiently. The incompressible Navier–Stokes, level-set, and thin-film evolution equations are coupled sequentially to capture the dynamics occurring at multiple length scales. The Hamilton–Jacobi level-set reinitialization is employed to construct the signed-distance function, which takes into account the deposited thin-film on the solid surface. The proposed multiscale method is validated and shown to match the augmented Young–Laplace equation for a static meniscus in a capillary tube. Viscous bending of the advancing interface over the precursor film is captured by the proposed level-set method and agrees with the Cox–Voinov theory. The advancing bubble surrounded by a wetting film inside a capillary tube is considered, and the predicted film thickness compares well with both theory and experiments. We also demonstrate that the multiscale level-set approach can model immiscible two-phase flow with a capillary number as low as 10−6.

An extended Landau–Levich model for the dragging of a thin liquid film with a propagating surface acoustic wave

In this paper we revisit the Landau and Levich analysis of a coating flow in the case where the flow in the thin liquid film is supported by a Rayleigh surface acoustic wave (SAW), propagating in the solid substrate. Our theoretical analysis reveals that the geometry of the film evolves under the action of the propagating SAW in a manner that is similar to the evolution of films that are being deposited using the dip coating technique. We show that in a steady state the thin-film evolution equation reduces to a generalized Landau–Levich equation with the dragging velocity, imposed by the SAW, depending on the local film thickness. We demonstrate that the generalized Landau–Levich equation has a branch of stable steady state solutions and a branch of unstable solutions. The branches meet at a saddle-node bifurcation point corresponding to the threshold value of the SAW intensity. Below the threshold value no steady states were found and our numerical computations suggest a gradual thinning of the liquid film from its initial geometry.

Note on thin film equations for solutions and suspensions

This note discusses how one may further develop thin film evolution equations for solutions and suspensions. First, we review the time evolution equation of a film or drop of simple liquid under the sole influence of wettability and capillarity and its formulation as a gradient dynamics. Second, we introduce such a gradient dynamics for a film of suspension or solution and show that it is equivalent to thin film equations in the literature. Finally, the new formulation is used to discuss extensions towards solute molecules/particles with net attractive interactions, decomposing solute-solvent systems and a solute-dependent wettability.

The relation of steady evaporating drops fed by an influx and freely evaporating drops

We discuss a thin film evolution equation for a wetting evaporating liquid on a smooth solid substrate. The model is valid for slowly evaporating small sessile droplets when thermal effects are insignificant, while wettability and capillarity play a major role. The model is first employed to study steady evaporating drops that are fed locally through the substrate. An asymptotic analysis focuses on the precursor film and the transition region towards the bulk drop and a numerical continuation of steady drops determines their fully non-linear profiles. Following this, we study the time evolution of freely evaporating drops without influx for several initial drop shapes. As a result we find that drops initially spread if their initial contact angle is larger than the apparent contact angle of large steady evaporating drops with influx. Otherwise they recede right from the beginning.

Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxialgrowth

In the present contribution we review basic mathematical results for three physical systemsinvolving self-organizing solid or liquid films at solid surfaces. The films may undergo astructuring process by dewetting, evaporation/condensation or epitaxial growth,respectively. We highlight similarities and differences of the three systems based on theobservation that in certain limits all of them may be described using models of similarform, i.e. time evolution equations for the film thickness profile. Those equations representgradient dynamics characterized by mobility functions and an underlying energy functional.Two basic steps of mathematical analysis are used to compare the different systems. First,we discuss the linear stability of homogeneous steady states, i.e. flat films, and second thesystematics of non-trivial steady states, i.e. drop/hole states for dewetting films andquantum-dot states in epitaxial growth, respectively. Our aim is to illustrate that theunderlying solution structure might be very complex as in the case of epitaxial growth butcan be better understood when comparing the much simpler results for the dewetting liquidfilm. We furthermore show that the numerical continuation techniques employed can shedsome light on this structure in a more convenient way than time-stepping methods.Finally we discuss that the usage of the employed general formulation does not only relateseemingly unrelated physical systems mathematically, but does allow as well for discussingmodel extensions in a more unified way.

On inertial effects in the Moffatt–Pukhnachov coating-flow problem

The effects are investigated of including inertial terms, in both small- and large-surface-tension limits, in a remodelling of the influential and fundamental problem first formulated by Moffatt and Pukhnachov in 1977: that of viscous thin-film free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field.An analysis of the non-dimensionalizations of previous related literature is made and the precise manner in which different rescalings lead to the asymptotic promotion or demotion of pure-inertial flux terms over gravitational-inertial terms is highlighted. An asymptotic mass-conserving evolution equation for a perturbed-film thickness is derived and solved using two-timescale asymptotics with a strained fast timescale. By using an algebraic manipulator to automate the asymptotics to high orders in the small expansion parameter of the ratio of the film thickness to the cylinder radius, consistent a posteriori truncations are obtained.Via two-timescale and numerical solutions of the evolution equation, new light is shed on diverse effects of inertia in both small- and large-surface-tension limits, in each of which a critical Reynolds number is discovered above which the thin-film evolution equation has no steady-state solution due to the strength of the destabilizing inertial centrifugal force. Extensions of the theory to the treatment of thicker films are discussed.

Thin-film evolution equation for a strained solid film on a deformable substrate: Numerical steady states

We consider the nonlinear behavior of the thin-film evolution equation for a strained solid film on a substrate. The evolution equation describes morphological changes to the film by surface diffusion in response to elastic energy, surface energy, and wetting energy. Due to the thin-film approximation, the elastic response of the film is determined analytically, resulting in a self-contained evolution equation which does not require separate numerical solution of the full three-dimensional elasticity problem. Using a pseudospectral predictor-corrector method we numerically determine the family of steady state solutions to this evolution equation which correspond to quantum dot and quantum ridge morphologies.

Modelling thin-film dewetting on structured substrates and templates: bifurcation analysis and numerical simulations.

We study the dewetting process of a thin liquid film on a chemically patterned solid substrate (template) by means of a thin-film evolution equation incorporating a space-dependent disjoining pressure. Dewetting of a thin film on a homogeneous substrate leads to fluid patterns with a typical length scale, that increases monotonously in time (coarsening). Conditions are identified for the amplitude and periodicity of the heterogeneity that allow to transfer the template pattern onto the liquid structure ("pinning") emerging from the dewetting process. A bifurcation and stability analysis of the possible liquid ridge solutions on a periodically striped substrate reveal parameter ranges where pinning or coarsening ultimately prevail. We obtain an extended parameter range of multistability of the pinning and coarsening morphologies. In this regime, the selected pattern depends sensitively on the initial conditions and potential finite perturbations (noise) in the system as we illustrate with numerical integrations in time. Finally, we discuss the instability to transversal modes leading to a decay of the ridges into rows of drops and show that it may diminish the size of the parameter range where the pinning of the thin film to the template is successful.