On the importance of nucleation solutions for the rupture of thin liquid films

Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling

We investigate a generic two-field Cahn-Hilliard model with variational and nonvariational coupling. It describes, for instance, passive and active ternary mixtures, respectively. Already a linear stability analysis of the homogeneous mixed state shows that activity not only allows for the usual large-scale stationary (Cahn-Hilliard) instability of the well-known passive case but also for small-scale stationary (Turing) and large-scale oscillatory (Hopf) instabilities. In consequence of the Turing instability, activity may completely suppress the usual coarsening dynamics. In a fully nonlinear analysis, we first briefly discuss the passive case before focusing on the active case. Bifurcation diagrams and selected direct time simulations are presented that allow us to establish that nonvariational coupling (i) can partially or completely suppress coarsening and (ii) may lead to the emergence of drifting and oscillatory states. Throughout, we emphasize the relevance of conservation laws and related symmetries for the encountered intricate bifurcation behavior.

Kinetically engendered subspinodal length scales in spontaneous dewetting of thin liquid films

Numerical simulations reveal emergence of subspinodal length scales in spontaneous dewetting of nonslipping unstable thin liquid films on homogeneous substrates if the liquid viscosity decreases with decrease in film thickness.

Final state of a perturbed liquid film inside a container under the effect of solid-liquid molecular forces and gravity

We investigate theoretically the possible final stationary configurations that can be reached by a laterally confined uniform liquid film inside a container. The liquid is under the action of gravity, surface tension, and the molecular interaction with the solid substrate. We study the case when the container is in an upright position as well as when it is turned upside down. The governing parameters of the problem are the initial thickness of the film, the size of the recipient that contains the liquid, and a dimensionless number that quantifies the relative strength of gravity with respect to the molecular interaction. The uniform film is always a possible final state and depending on the value of the parameters, up to three different additional final states may exist, each one consisting in a droplet surrounded by a thin film. We derive analytical expressions for the energy of these possible final configurations and from these we analyze which state is indeed reached. A uniform thin film may show three different behaviors after a perturbation: The system recovers its initial shape after any perturbation, the fluid evolves towards a drop (if more than one is possible, it tends toward that with the thinnest precursor film) for any perturbation, or the system ends as a uniform film or a drop depending on the details of the perturbation.

Stability of thin liquid films and sessile droplets under confinement

The stability of nonvolatile thin liquid films and of sessile droplets is strongly affected by finite size effects. We analyze their stability within the framework of density functional theory using the sharp kink approximation, i.e., on the basis of an effective interface Hamiltonian. We show that finite size effects suppress spinodal dewetting of films because it is driven by a long-wavelength instability. Therefore nonvolatile films are stable if the substrate area is too small. Similarly, nonvolatile droplets connected to a wetting film become unstable if the substrate area is too large. This instability of a nonvolatile sessile droplet turns out to be equivalent to the instability of a volatile drop which can attain chemical equilibrium with its vapor.

Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates

Depinning of two-dimensional liquid ridges and three-dimensional drops on an inclined substrate is studied within the lubrication approximation. The structures are pinned to wetting heterogeneities arising from variations of the strength of the short-range contribution to the disjoining pressure. The case of a periodic array of hydrophobic stripes transverse to the slope is studied in detail using a combination of direct numerical simulation and branch-following techniques. Under appropriate conditions the ridges may either depin and slide downslope as the slope is increased, or first break up into drops via a transverse instability, prior to depinning. The different transition scenarios are examined together with the stability properties of the different possible states of the system.

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

In the present contribution we review basic mathematical results for three physical systems involving self-organizing solid or liquid films at solid surfaces. The films may undergo a structuring process by dewetting, evaporation/condensation or epitaxial growth, respectively. We highlight similarities and differences of the three systems based on the observation that in certain limits all of them may be described using models of similar form, i.e. time evolution equations for the film thickness profile. Those equations represent gradient 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 the systematics of non-trivial steady states, i.e. drop/hole states for dewetting films and quantum-dot states in epitaxial growth, respectively. Our aim is to illustrate that the underlying solution structure might be very complex as in the case of epitaxial growth but can be better understood when comparing the much simpler results for the dewetting liquid film. We furthermore show that the numerical continuation techniques employed can shed some 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 relate seemingly unrelated physical systems mathematically, but does allow as well for discussing model extensions in a more unified way.

Ratchet-driven fluid transport in bounded two-layer films of immiscible liquids

We present a two-layer thin film model allowing us to study the behavior of a general class of 'wettability ratchets' that can be employed to transport a continuous phase. Brownian ratchets, in contrast, are normally used to transport particles or molecules within a continuous carrier fluid without transporting the fluid itself. The wettability ratchet is based on a switchable, spatially asymmetric, periodic interaction of the free surface of the film and the walls. To illustrate the general concept, we focus on an electrical dewetting mechanism based on the effective force exercised by a static electric field on the liquid-liquid interface between two dielectric liquids. In particular, we analyse (i) an on-off ratchet with a constant lateral force resulting in a dewetting-spreading cycle, (ii) a ratchet switching between two shifted potentials that shows a transition between oscillating and sliding drops, and (iii) a flashing external force ratchet. For the three cases, the macroscopic transport is studied in its dependence on spatial and temporal characteristics of the ratchet, and physical properties and volume of the liquids.

Driven drops on heterogeneous substrates: onset of sliding motion

Pinning and depinning of driven drops on heterogeneous substrates is studied as a function of the driving and heterogeneity amplitude. Two types of heterogeneity are considered: a "hydrophobic" defect that blocks the drop in front and a "hydrophilic" one that holds it at the back. Two different types of depinning leading to sliding motion are identified, and the resulting stick-slip motion is studied numerically.

Spinodal instability and pattern formation in thin liquid films confined between two plates

The instability, morphology and pattern formation engendered by the van der Waals force in a thin liquid film of thickness h confined between two closely placed solid surfaces (at distance d > h) are investigated based on nonlinear 3D simulations. The initial and the final stages of dewetting and pattern formation are found to be crucially dependent on the volumetric (thickness) ratio of air and liquid and its deviation from the location of the maximum of the spinodal parameter versus volumetric ratio curve. On a low energy surface, relatively thinner films and wider air gaps favor initial dewetting of the lower plate by the formation of holes, whereas thicker films with thinner air gaps initially evolve by the formation of columns/bridges that join the upper plate. In the later stage of evolution, the initial holes in thinner films evolve into columns/drops, while a rapid coalescence of columns in the thicker films eventually causes formation of holes. Thus, a phase inversion, either from liquid-in-air to air-in-liquid dispersion or vice versa, occurs during the final stages of evolution. A thin film confined between two high-energy solid surfaces forms columns (bridges) only when its mean thickness, h0, is greater than a critical thickness (hc) or the air gap is smaller than a critical distance. The patterns can be aligned by using a topographically patterned confining surface. Conditions on pattern periodicity, amplitude, and the volumetric ratio of air and liquid in the gap are explored for the formation of various types of ordered patterns including annular rings of columns, concentric ripples, parallel channels and a rectangular array of complex features. The results are of significance in soft lithographies such as LISA, soft stamping and capillary force lithography.

Nonlinear stability analysis of a two-layer thin liquid film: Dewetting and autophobic behavior

The nonlinear stability analysis of a liquid film composed of two superposed thin layers of immiscible liquids resting on a solid substrate is performed. It is shown that the coupling of van der Waals interactions in the two layers can lead to an autophobic behavior in the form of spinodal decomposition of two planar liquid layers into a system of localized drops divided by almost planar wetting layers. The results of the weakly nonlinear analysis near the instability threshold are confirmed by the numerical solution of a system of two strongly nonlinear evolution equations for the liquid-liquid and liquid-gas interfaces. The kinetics of the drop coarsening at late stages is studied and is found to be close to that reported for a one-layer film. It is also shown that gravity effects can become significant even for very thin two-layer films.

Self-propelled running droplets on solid substrates driven by chemical reactions

We study chemically driven running droplets on a partially wetting solid substrate by means of coupled evolution equations for the thickness profile of the droplets and the density profile of an adsorbate layer. Two models are introduced corresponding to two qualitatively different types of experiments described in the literature. In both cases an adsorption or desorption reaction underneath the droplets induces a wettability gradient on the substrate and provides the driving force for droplet motion. The difference lies in the behavior of the substrate behind the droplet. In case I the substrate is irreversibly changed whereas in case II it recovers allowing for a periodic droplet movement (as long as the overall system stays far away from equilibrium). Both models allow for a non-saturated and a saturated regime of droplet movement depending on the ratio of the viscous and reactive time scales. In contrast to model I, model II allows for sitting drops at high reaction rate and zero diffusion along the substrate. The transition from running to sitting drops in model II occurs via a super- or subcritical drift-pitchfork bifurcation and may be strongly hysteretic implying a coexistence region of running and sitting drops.

Morphology changes in the evolution of liquid two-layer films

We consider a thin film consisting of two layers of immiscible liquids on a solid horizontal (heated) substrate. Both the free liquid-liquid and the liquid-gas interface of such a bilayer liquid film may be unstable due to effective molecular interactions relevant for ultrathin layers below 100-nm thickness, or due to temperature-gradient-caused Marangoni flows in the heated case. Using a long-wave approximation, we derive coupled evolution equations for the interface profiles for the general nonisothermal situation allowing for slip at the substrate. Linear and nonlinear analyses of the short- and long-time film evolution are performed for isothermal ultrathin layers, taking into account destabilizing long-range and stabilizing short-range molecular interactions. It is shown that the initial instability can be of a varicose, zigzag, or mixed type. However, in the nonlinear stage of the evolution the mode type, and therefore the pattern morphology, can change via switching between two different branches of stationary solutions or via coarsening along a single branch.

Electric field induced instability and pattern formation in thin liquid films

Electrostatic field induced instability, morphology, and patterning of a thin liquid film confined between two electrodes with an air gap are studied on the basis of nonlinear 3D simulations, both for spatially homogeneous and heterogeneous fields. In addition to the spinodal flow resulting from the variation of field because of local thickness changes, a heterogeneous imposed field also moves the liquid from the regions of low field to high field, thus allowing a more precise control of pattern. Hexagonal packing of liquid columns is observed for a spatially homogeneous electric field, which is in accord with the e-field experiments on thin polymer films (Schaffer et al. Nature 2000, 403, 874). For a large liquid volume fraction in the gap, varphi > or = 0.75, the coalescence of columns causes a phase inversion, leading to the formation of air columns or cylindrical holes trapped in the liquid matrix (air-in-liquid dispersion). Locally ordered aligned patterns are formed by imposing a spatial variation of the electrostatic field by using a topographically patterned electrode. For example, multiple rows/lines of liquid columns are formed near the edge of a step-like heterogeneity of the electrode and annular rings of ordered columns or concentric ripples are formed around a heterogeneous circular patch. Simulations predict that the electrode pattern is replicated in the film only when the pattern periodicity, L(p), exceeds the instability length scale on the basis of the minimum interelectrode separation distance, L(p) > or = lambda(m)-d(min). Thus, the formation of secondary structures can be suppressed by employing an electrode with deep grooves and stronger field gradients, which produces almost ideal templating. The number density of the electric field induced patterns can be altered by tuning the mean film thickness (or the volume fraction of liquid in the gap), periodicity and depth (amplitude) of the grooves on the top electrode, and the applied voltage. The implications are in electrostatic lithography, pattern replication in soft materials, and the design and interpretation of thin film experiments involving electric fields.

Pattern formation and dewetting in thin films of liquids showing complete macroscale wetting: from "pancakes"to "swiss cheese"

Based on the complete 3D numerical solutions of the nonlinear thin film equation, we address the problems of surface instability, dynamics, morphological diversity and evolution in unstable thin films of the liquids that display complete macroscale wetting. The twin constraints of complete macroscale wettability and nanoscale instability produce a variety of microscopic morphological phases approximating sharp crystal surfaces with flat tops resembling a mesa or a micro "pancake" or a slice of Swiss cheese. While the maximum thickness of flat regions is found to be independent of the initial film thickness, the precise lateral morphology of microdomains formed depends on the film thickness. As the film thickness is increased, the initial pathway of evolution changes from the formation of small spherical droplets, to long mesas (parapets) and islands, to circular holes, all of which eventually resolve by ripening into a collection of round pancakes at equilibrium. However, beyond a certain transition thickness, a novel metastable honeycombed morphology, resembling a membrane or a slice of Swiss cheese, is uncovered, which is produced by an abrupt "freezing" of the evolution during hole growth. In contrast, the spinodal dewetting in thin films of partially wettable systems always engenders spherical droplets at equilibrium. The equilibrium dewetted area from simulations, as well as from simple mass balance, is shown to decline linearly with the initial film thickness.

Dynamical model for chemically driven running droplets

We propose coupled evolution equations for the thickness of a liquid film and the density of an adsorbate layer on a partially wetting solid substrate. Therein, running droplets are studied assuming a chemical reaction underneath the droplets that induces a wettability gradient on the substrate and provides the driving force for droplet motion. Two different regimes for moving droplets--reaction-limited and saturated regime--are described. They correspond to increasing and decreasing velocities with increasing reaction rates and droplet sizes, respectively. The existence of the two regimes offers a natural explanation of prior experimental observations.

Instabilities in thin-film binary mixtures

Binary mixtures of ultra-thin films are subject to both height and concentration fluctuations, either of which, under the appropriate conditions, can become unstable. We investigate a simple general thermodynamic model which shows that, due to the constraint imposed upon coupling between these fluctuations, a thin-film binary mixture will be less stable than if either height or concentration fluctuations are considered in isolation. We investigate the height dependence of the stability condition, discuss equilibrium conditions when higher-order contributions to the free energy are negligible, and predict the scaling behaviour.

Dynamic domain formation in membranes: thickness-modulation-induced phase separation

A simple model investigates the amplification of fluctuations on membranes constituted of two lipids having different lengths. Van der Waals and electrostatic interactions across the lipid bilayer result in a destabilization favoring thickness variations of the membrane. Close to spontaneous demixing of the two components, the additional gain in free energy due to thickness undulations shifts the stability boundary which promotes phase separation into domains. Interestingly, this effect can be induced by an applied electric field or membrane potential. In biological systems, the dynamic model presented here indicates that electric fields might be important for controlling phase separation and the formation of domains called "rafts".

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.