Coarsening dynamics of dewetting films

Uniform large-area surface patterning achieved by metal dewetting for the top-down fabrication of GaN nanowire ensembles

The dewetting of thin Pt films on different surfaces is investigated as a means to provide the patterning for the top-down fabrication of GaN nanowire ensembles. The transformation from a thin film to an ensemble of nanoislands upon annealing proceeds in good agreement with the void growth model. With increasing annealing duration, the size and shape uniformity of the nanoislands improves. This improvement speeds up for higher annealing temperature. After an optimum annealing duration, the size uniformity deteriorates due to the coalescence of neighboring islands. By changing the Pt film thickness, the nanoisland diameter and density can be quantitatively controlled in a way predicted by a simple thermodynamic model. We demonstrate the uniformity of the nanoisland ensembles for an area larger than 1 cm2. GaN nanowires are fabricated by a sequence of dry and wet etching steps, and these nanowires inherit the diameters and density of the Pt nanoisland ensemble used as a mask. Our study achieves advancements in size uniformity and range of obtainable diameters compared to previous works. This simple, economical, and scalable approach to the top-down fabrication of nanowires is useful for applications requiring large and uniform nanowire ensembles with controllable dimensions.

Coarsening of Quasi Two-Dimensional Emulsions Formed by Islands in Free-Standing Smectic Films

We study the coarsening behavior of assemblies of islands on smectic A freely suspended films in ISS microgravity experiments. The islands can be regarded as liquid inclusions in a two-dimensional fluid in analogy to liquid droplets of the discontinuous phase of an emulsion. The coarsening is effectuated by two processes, predominantly by island coalescence, but to some extend also by Ostwald ripening, whereby large islands grow at the expense of surrounding smaller ones. A peculiarity of this system is that the continuous and the discontinuous phases consist of the same material. We determine the dynamics, analyze the self-similar aging of the island size distribution and discuss characteristic exponents of the mean island growth.

Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for nonlinear wavelength-selection dynamics in (nearly) mass-conserving two-component reaction-diffusion systems independent of the specific mathematical model chosen. Previous work has shown that these systems support an extremely broad band of stable wavelengths, but the mechanism by which a specific wavelength is selected has remained unclear. We show that an interrupted coarsening process selects the wavelength at the threshold to stability. Based on the physical intuition that coarsening is driven by competition for mass and interrupted by weak source terms that break strict mass conservation, we develop a singular perturbation theory for the stability of stationary patterns. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. We find excellent agreement with numerical results throughout the diffusion- and reaction-limited regimes of the dynamics, including the crossover region. Further, we show how, in these limits, the two-component reaction-diffusion systems map to generalized Cahn-Hilliard and conserved Allen-Cahn dynamics, therefore providing a link to these two fundamental scalar field theories. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multicomponent systems with potentially several conservation laws.

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.

Electrokinetic Thin-Film Model for Electrowetting: The Role of Bulk Charges

The electrowetting behavior of a charge-carrying sessile droplet is relevant to applications such as point-of-care diagnostics. Often biomedical assays involve droplets that contain charged molecules such as dissolved ions, proteins, and DNA. In this work, we develop a reduced-order electrokinetic model for electrowetting of such a charge-carrying droplet under a parallel-plate electrode configuration. An inertial-lubrication model based on the weighted residual integral boundary layer (WRIBL) technique is used to obtain evolution equations that describe the spatiotemporal evolution of the fluid-air interface and the depth-integrated flow rate. The solutions to the evolution equations are obtained numerically by using the spectral collocation method. We investigate the role of domain and surface charges, characterized by the Debye length, on droplet wetting. Under low relaxation timescales, both droplet deformation and wetting alteration under an AC field are shown to be equivalent to that under a root-mean-square (RMS) DC field. We show that an electrolytic sessile droplet can exhibit a larger deformation in comparison to the two asymptotic limits of a perfect conductor and a perfect dielectric droplet, corresponding, respectively, to very low and high Debye lengths. The effects of several other parameters such as the inherent equilibrium wettability, permittivity ratio, and electric field strength are also investigated.

Gradient-dynamics model for liquid drops on elastic substrates

The wetting of soft elastic substrates exhibits many features that have no counterpart on rigid surfaces. Modelling the detailed elastocapillary interactions is challenging, and has so far been limited to single contact lines or single drops. Here we propose a reduced long-wave model that captures the main qualitative features of statics and dynamics of soft wetting, but which can be applied to ensembles of droplets. The model has the form of a gradient dynamics on an underlying free energy that reflects capillarity, wettability and compressional elasticity. With the model we first recover the double transition in the equilibrium contact angles that occurs when increasing substrate softness from ideally rigid towards very soft (i.e., liquid). Second, the spreading of single drops of partially and completely wetting liquids is considered showing that known dependencies of the dynamic contact angle on contact line velocity are well reproduced. Finally, we go beyond the single droplet picture and consider the coarsening for a two-drop system as well as for a large ensemble of drops. It is shown that the dominant coarsening mode changes with substrate softness in a nontrivial way.

A hydro-osmotic coarsening theory of biological cavity formation

Fluid-filled biological cavities are ubiquitous, but their collective dynamics has remained largely unexplored from a physical perspective. Based on experimental observations in early embryos, we propose a model where a cavity forms through the coarsening of myriad of pressurized micrometric lumens, that interact by ion and fluid exchanges through the intercellular space. Performing extensive numerical simulations, we find that hydraulic fluxes lead to a self-similar coarsening of lumens in time, characterized by a robust dynamic scaling exponent. The collective dynamics is primarily controlled by hydraulic fluxes, which stem from lumen pressures differences and are dampened by water permeation through the membrane. Passive osmotic heterogeneities play, on the contrary, a minor role on cavity formation but active ion pumping can largely modify the coarsening dynamics: it prevents the lumen network from a collective collapse and gives rise to a novel coalescence-dominated regime exhibiting a distinct scaling law. Interestingly, we prove numerically that spatially biasing ion pumping may be sufficient to position the cavity, suggesting a novel mode of symmetry breaking to control tissue patterning. Providing generic testable predictions, our model forms a comprehensive theoretical basis for hydro-osmotic interaction between biological cavities, that shall find wide applications in embryo and tissue morphogenesis.

Evaporation of Binary-Mixture Liquid Droplets: The Formation of Picoliter Pancakelike Shapes

Small multicomponent droplets are of increasing importance in a plethora of technological applications ranging from the fabrication of self-assembled hierarchical patterns to the design of autonomous fluidic systems. While often far away from equilibrium, involving complex and even chaotic flow fields, it is commonly assumed that in these systems with small drops surface tension keeps the shapes spherical. Here, studying picoliter volatile binary-mixture droplets of isopropanol and 2-butanol, we show that the dominance of surface tension forces at small scales can play a dual role: Minute variations in surface tension along the interface can create Marangoni flows that are strong enough to significantly deform the drop, forming micron-thick pancakelike shapes that are otherwise typical of large puddles. We identify the conditions under which these flattened shapes form and explain why, universally, they relax back to a spherical-cap shape toward the end of drop lifetime. We further show that the formation of pancakelike droplets suppresses the "coffee-ring" effect and leads to uniform deposition of suspended particles. The quantitative agreement between theory and experiment provides a predictive capability to modulate the shape of tiny droplets with implications in a range of technologies from fabrication of miniature optical lenses to coating, printing, and pattern deposition.

Wavelength Selection by Interrupted Coarsening in Reaction-Diffusion Systems

Wavelength selection in reaction-diffusion systems can be understood as a coarsening process that is interrupted by counteracting processes at certain wavelengths. We first show that coarsening in mass-conserving systems is driven by self-amplifying mass transport between neighboring high-density domains. We derive a general coarsening criterion and show that coarsening is generically uninterrupted in two-component systems that conserve mass. The theory is then generalized to study interrupted coarsening and anticoarsening due to weakly broken mass conservation, providing a general path to analyze wavelength selection in pattern formation far from equilibrium.

Molecular simulation of thin liquid films: Thermal fluctuations and instability

The instability of a thin liquid film on a solid surface is studied both by molecular dynamics simulations (MD) and a stochastic thin-film equation (STF), which models thermal fluctuations with white noise. A linear stability analysis of the STF allows us to derive a power spectrum for the surface fluctuations, which is quantitatively validated against the spectrum observed in MD. Thermal fluctuations are shown to be critical to the dynamics of nanoscale films. Compared to the classical instability mechanism, which is driven by disjoining pressure, fluctuations (a) can massively amplify the instability, (b) cause the fluctuation wavelength that is dominant to evolve in time (a single fastest-growing mode does not exist), and (c) decrease the critical wavelength so that classically stable films can be ruptured.

Thin viscoelastic dewetting films of Jeffreys type subjected to gravity and substrate interactions

This work presents a study of the interfacial dynamics of thin viscoelastic films subjected to the gravitational force and substrate interactions induced by the disjoining pressure, in two spatial dimensions. The governing equation is derived as a long-wave approximation of the Navier-Stokes equations for incompressible viscoelastic liquids under the effect of gravity, with the Jeffreys model for viscoelastic stresses. For the particular cases of horizontal or inverted planes, the linear stability analysis is performed to investigate the influence of the physical parameters involved on the growth rate and length scales of instabilities. Numerical simulations of the nonlinear regime of the dewetting process are presented for the particular case of an inverted plane. Both gravity and the disjoining pressure are found to affect not only the length scale of instabilities, but also the final configuration of dewetting, by favoring the formation of satellite droplets, that are suppressed by the slippage with the solid substrate.

Interplay between phase separation and dewetting in PS/PVME thin films: effect of temperature

We studied the effects of temperature on the interplay between dewetting and phase separation at shallow and deep depths at two-phase temperatures in PS/PVME polymer blend thin films. Optical microscopy, AFM measurements, and ellipsometry analysis were performed to investigate the dewetting behavior of the films. At the deep quench depth (phase separation temperature of 115 °C), a two-layer film formed, consisting of a thin PVME layer directly on the surface of a silicon wafer (as the wetting layer) and a bulk layer which was the upper layer. In the bulk layer, the phase separation mechanism was controlled by an apparent nucleation and growth mechanism, which was driven by entropic and anisotropic limitations rather than thermodynamic preferences. After about 106 min of annealing, liquid-liquid dewetting occurred in the interface of the formed layers, triggered by Laplace pressure differences. However, at the shallow quench depth (phase separation temperature of 95 °C), a tri-layered structure formed in the thin films and concentration fluctuations at the interfaces of the formed layers triggered surface fluctuations and instabilities (dewetting phenomenon).

Sliding Drops: Ensemble Statistics from Single Drop Bifurcations

Ensembles of interacting drops that slide down an inclined plate show a dramatically different coarsening behavior as compared to drops on a horizontal plate: As drops of different size slide at different velocities, frequent collisions result in fast coalescence. However, above a certain size individual sliding drops are unstable and break up into smaller drops. Therefore, the long-time dynamics of a large drop ensemble is governed by a balance of merging and splitting. We employ a long-wave film height evolution equation and determine the dynamics of the drop size distribution towards a stationary state from direct numerical simulations on large domains. The main features of the distribution are then related to the bifurcation diagram of individual drops obtained by numerical path continuation. The gained knowledge allows us to develop a Smoluchowski-type statistical model for the ensemble dynamics that well compares to full direct simulations.

Instability, self-organization and pattern formation in thin soft films

The free surface of a thin soft polymer film is often found to become unstable and self-organizes into various meso-scale structures. In this article we classify the instability of a thin polymer film into three broad categories, which are: category 1: instability of an ultra-thin (<100 nm) viscous film engendered by amplification of thermally excited surface capillary waves due to interfacial dispersive van der Waals forces; category 2: instability arising from the attractive inter-surface interactions between the free surface of a soft film exhibiting room temperature elasticity and another rigid surface in its contact proximity; and category 3: instability caused by an externally applied field such as an electric field or a thermal gradient, observed in both viscous and elastic films. We review the salient features of each instability class and highlight how characteristic length scales, feature morphologies, evolution pathways, etc. depend on initial properties such as film thickness, visco-elasticity (rheology), residual stress, and film preparation conditions. We emphasize various possible strategies for aligning and ordering of the otherwise isotropic structures by combining the essential concepts of bottom-up and top-down approaches. A perspective, including a possible future direction of research, novelty and limitations of the methods, particularly in comparison to the existing patterning techniques, is also presented for each setting.

Fully nonlinear dynamics of stochastic thin-film dewetting

The spontaneous formation of droplets via dewetting of a thin fluid film from a solid substrate allows materials nanostructuring. Often, it is crucial to be able to control the evolution, and to produce patterns characterized by regularly spaced droplets. While thermal fluctuations are expected to play a role in the dewetting process, their relevance has remained poorly understood, particularly during the nonlinear stages of evolution that involve droplet formation. Within a stochastic lubrication framework, we show that thermal noise substantially influences the process of droplets formation. Stochastic systems feature a smaller number of droplets with a larger variability in size and space distribution, when compared to their deterministic counterparts. Finally, we discuss the influence of stochasticity on droplet coarsening for asymptotically long times.

Interaction of 3D dewetting nanodroplets on homogeneous and chemically heterogeneous substrates

Long-time interaction of dewetting nanodroplets is investigated using a long-wave approximation method. Although three-dimensional (3D) droplets evolution dynamics exhibits qualitative behavior analogous to two-dimensional (2D) dynamics, there is an extensive quantitative difference between them. 3D dynamics is substantially faster than 2D dynamics. This can be related to the larger curvature and, as a consequence, the larger Laplace pressure difference between the droplets in 3D systems. The influence of various chemical heterogeneities on the behavior of droplets has also been studied. In the case of gradient surfaces, it is shown how the gradient direction could change the dynamics. For a chemical step located between the droplets, the dynamics is enhanced or weakened depending on the initial configuration of the system.

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.

Coarsening modes of clusters of aggregating particles

There are two modes by which clusters of aggregating particles can coalesce: The clusters can merge either (i) by the Ostwald ripening process, in which particles diffuse from one cluster to the other while the cluster centers remain stationary, or (ii) by means of a cluster translation mode, in which the clusters move toward each other and join. To understand in detail the interplay between these different modes, we study a model system of hard particles with an additional attraction between them. The particles diffuse along narrow channels with smooth or periodically corrugated walls, so that the system may be treated as one-dimensional. When the attraction between the particles is strong enough, they aggregate to form clusters. The channel potential influences whether clusters can move easily or not through the system and can prevent cluster motion. We use dynamical density functional theory to study the dynamics of the aggregation process, focusing in particular on the coalescence of two equal-sized clusters. As long as the particle hard-core diameter is nonzero, we find that the coalescence process can be halted by a sufficiently strong corrugation potential. The period of the potential determines the size of the final stable clusters. For the case of smooth channel walls, we demonstrate that there is a crossover in the dominance of the two different coarsening modes, which depends on the strength of the attraction between particles, the cluster sizes, and the separation distance between clusters.

Beyond the Young-Laplace model for cluster growth during dewetting of thin films: effective coarsening exponents and the role of long range dewetting interactions

Long range dewetting forces acting across thin films, such as the fundamental van der Waals interactions, may drive the formation of large clusters (tall multilayer islands) and pits, observed in thin films of diverse materials such as polymers, liquid crystals, and metals. In this study we further develop the methodology of the nonequilibrium statistical mechanics of thin films coarsening within continuum interface dynamics model incorporating long range dewetting interactions. The theoretical test bench model considered here is a generalization of the classical Mullins model for the dynamics of solid film surfaces. By analytic arguments and simulations of the model, we study the coarsening growth laws of clusters formed in thin films due to the dewetting interactions. The ultimate cluster growth scaling laws at long times are strongly universal: Short and long range dewetting interactions yield the same coarsening exponents. However, long range dewetting interactions, such as the van der Waals forces, introduce a distinct long lasting early time scaling behavior characterized by a slow growth of the cluster height/lateral size aspect ratio (i.e., a time-dependent Young angle) and by effective coarsening exponents that depend on cluster size. In this study, we develop a theory capable of analytically calculating these effective size-dependent coarsening exponents characterizing the cluster growth in the early time regime. Such a pronounced early time scaling behavior has been indeed seen in experiments; however, its physical origin has remained elusive to this date. Our theory attributes these observed phenomena to ubiquitous long range dewetting interactions acting across thin solid and liquid films. Our results are also applicable to cluster growth in initially very thin fluid films, formed by depositing a few monolayers or by a submonolayer deposition. Under this condition, the dominant coarsening mechanism is diffusive intercluster mass transport while the cluster coalescence plays a minor role, both in solid and in fluid films.

Coarsening dynamics of nanodroplets on topographically structured substrates

Employing a biharmonic boundary integral method with linear elements, coarsening dynamics of nanodroplets on topographical step heterogeneity is investigated. It is shown that the step height and droplet configuration have an influential effect on the dynamics. Increasing the step height slows down the process while locating the droplets close to the step boosts the coarsening rate. Considering a slip boundary condition enhances the dynamics and reveals a transition in the droplet migration direction. Our results reveal that increasing the surface wettability weakens the dynamics. Various types of the disjoining pressure over the step are also considered and their effects on the coarsening are investigated.

Closed-form expression for the profile of partially wetting two-dimensional droplets under gravity

Analytical solutions for the shape of both hanging and sitting droplets under the effects of gravity and surface tension are presented. The modeling also includes the action of molecular forces arising between the liquid and the substrate, which are responsible for the formation of a stable nanometric film in the region close to the droplet contact line. The shape of the droplet is completely described by an analytical solution that also accounts for the pancake-shaped droplets as a limiting case. We find expressions that relate microscopic and nanoscopic aspects, such as the strengths of the molecular forces and the thickness of the nanometric film, to macroscopic quantities, such as the cross-sectional area and the width of the droplet. We study the effect of gravity on the contact angle and find that for small droplets the contact angle follows a power law with the droplet's size. For sitting droplets we find that the there is an upper limit for the value of the gravity.

Coarsening dynamics of dewetting nanodroplets on chemically patterned substrates

Mesoscopic hydrodynamic equations are solved to investigate coarsening dynamics of two interacting nanodroplets on chemically patterned substrates. The effects of different parameters such as the surface chemical pattern, the slip length, the profile of the disjoining pressure, the size of the droplets, and the contact angles on the coarsening are studied. Our results reveal that the presence of a chemical heterogeneity can enhance or weaken the coarsening dynamics depending on the pattern type and positions of the droplets on the substrate. Also increasing the contact angles to values larger than a critical value may qualitatively change the coarsening process, and the profile of the disjoining pressure and the slippage can appreciably modify the coarsening rate.

Dynamics of nanodroplets on wettability gradient surfaces

A lubrication model is used to study the dynamics of nanoscale droplets on wettability gradient surfaces. The effects of the gradient size, size of the nanodroplets and the slip on the dynamics have been studied. Our results indicate that the position of the center of mass of the droplets can be well described in terms of a third-order polynomial function of the time of the motion for all the cases considered. By increasing the size of the droplets the dynamics increases. It is also shown that the slip can considerably enhance the dynamics. The results have been compared with the results obtained using theoretical models and molecular dynamics simulations.

Coarsening of capillary drops coupled by conduit networks

A system of n spherical-cap drops, coupled by a network of conduits, coarsens due to surface tension forces. The total interfacial energy drives the fluid through the conduits such that, with time, the volume becomes increasingly localized into fewer large drops. The coarsening rate is predicted heuristically for drops coupled by orthogonal networks, a porous medium, and fractal networks of various dimensions. The predicted coarsening law as it depends upon the type and dimension of network, total number of drops, and initial drop volume is compared against numerical simulations of large n . Additionally, distributions of large drop volumes are obtained using a Lifshitz-Slyozov-Wagner (LSW) model. The predicted distributions are independent of network topology; in contrast, simulation results depend weakly on the network dimension. The heuristic coarsening rate laws are recovered using the LSW model for all but a square network topology.

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.

Coarsening in the electrohydrodynamic patterning of thin polymer films

Periodic pillarlike microstructures can be created from initially flat polymer films via the electrohydrodynamic instabilities. Those patterns, however, are metastable. Our experimental observations show that the average pillar size increases slowly after linear growth. Major coarsening events then take place over times several orders of magnitude longer than the linear growth time. For all fill ratios, a logarithmic time dependence of the average pillar size can be identified, i.e., <S> proportional to ln t. Thicker films, however, have faster coarsening rates than thinner films. Linear stability analysis of the pseudosteady states reveals two major coarsening mechanisms, collision and Ostwald ripening, which can also be identified from experimental images. We then reduce the original partial differential equation (PDE) into a pair of ODEs, which govern the interaction between pillars due to the above two coarsening mechanisms. From this, a logarithm scaling law is obtained for both low and high fill ratios and the coarsening rate is slower for lower fill ratios, consistent with experimental observations. We also find that arrays with more uniform sizes tend to start coarsening later, but they coarsen faster than more "disperse" arrays, which could be possibly utilized in experiments for controlling the onset and speed of coarsening. The logarithm scaling in the electrohydrodynamic coarsening phenomenon, which differs from coarsening in spinodal decomposition and dewetting of thin liquid films, is due to the significant nonlinear effect of Maxwell stresses and geometric confinement on the disjoining pressure at both top and bottom electrodes.

Analytical solutions for partially wetting two-dimensional droplets

We present a new analytical solution for the static shape of a two-dimensional droplet in equilibrium with a surrounding thin film on a solid substrate. The modeling includes the effects of capillarity and disjoining-conjoining pressure accounting for intermolecular forces between the solid and the liquid. We derive new analytical solutions for the shape of the droplet, the cross-sectional area, the half-width, and the maximum curvature and inflection points. We study the effects of the size of the droplet on the apparent contact angle. The shape of the droplet in the contact line region is compared with profiles obtained by employing approximations suggested in the literature, and the observed differences are discussed. Finally, we present the time evolution to the steady state to show how the whole profile, including the thin film, evolves to the corresponding stationary configuration.

Nanoparticle assembly via the dewetting of patterned thin metal lines: understanding the instability mechanisms

Nanosecond pulsed laser heating was used to control the assembly of spatially correlated nanoparticles from lithographically patterned pseudo-one-dimensional nickel lines. The evolution of the nickel line instabilities and nanoparticle formation with a correlated size and spacing was observed after a series of laser pulses. To understand the instabilities that direct the nanoparticle assembly, we have carried out nonlinear time-dependent simulations and linear stability analysis based on a simple hydrodynamic model. We find that the simulated time scales and length scales agree well with the experimental results. Interestingly, in both experiments and simulations, the instabilities associated with the line edge, and with the surface perturbation-driven mechanism, are found to result in similar particle sizes and spacings.

Enhanced stability of a dewetting thin liquid film in a single-frequency vibration field

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description is reasonable and the amplitude equation holds. The linear and nonlinear analyses of the amplitude equation and the numerical computations show that such vibration stabilizes the film against dewetting and rupture.

Coarsening of unstable thin films subject to gravity

Thin films of viscous fluids coating hydrophobic substrates are unstable to dewetting instabilities, and long-time evolution leads to the formation of an array of near-equilibrium droplets connected by ultrathin fluid layers. In the absence of gravity, previous use of lubrication theory has shown that coarsening dynamics will ensue-the system will evolve by successively eliminating small drops to yield fewer larger drops. While gravity has only a weak influence on the initial thin film, we show that it has a significant influence on the later stages of the coarsening dynamics, dramatically slowing the rate of coarsening for large drops. Small drops are relatively unaffected, but as coarsening progresses, these aggregate into larger drops whose shape and dynamics are dominated by gravity. The change in the mean drop shape causes a corresponding gradual transition from power-law coarsening to a logarithmic behavior.

Formation and mobility of droplets on composite layered substrates

A mesoscale fluid film placed on a solid support may break up and form droplets. In addition, droplets may exhibit spontaneous translation by modifying the wetting properties of the substrate, resulting in asymmetry in the contact angles. We examine mechanisms for droplet formation and motion on uniform and terraced landscapes, i.e., composite substrates. The fluid film stability, droplet formation and velocity are studied theoretically in the isothermal case using a lubrication approach in one spatial dimension. The droplet properties are found to involve contributions from both the terraced layer thickness and molecular interactions via the disjoining potential.

Perturbation theory for traveling droplets

The motion of chemically driven droplets is analyzed by applying a solvability condition of perturbed hydrodynamic equations affected by the adsorbate concentration. Conditions for traveling bifurcation analogous to a similar transition in activator-inhibitorsystems are obtained. It is shown that interaction of droplets leads to either scattering of mobile droplets or the formation of regular patterns, respectively, at low or high adsorbate diffusivity. The same method is applied to droplets running on growing terrace edges during surface freezing.

Regular surface patterns on Rayleigh-Taylor unstable evaporating films heated from below

We study a thin liquid film with a free surface on the underside of a cooled horizontal substrate. We show that if the fluid is initially in equilibrium with its own vapor in the gas phase below, regular surface patterns in the form of long-wave hexagons having a well-defined lateral length scale are observed. This is in sharp contrast to the case without evaporation where rupture or coarsening to larger and larger patterns is seen in the long time limit. In this way, evaporation could be used for regular structuring of the film surface. Finally, we estimate the finite wave length for the simplified case of an extended Cahn-Hilliard equation.

A nonlocal continuum model for biological aggregation

We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short-range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady-state clumps are reached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions as well, and is demonstrated via numerical simulations in two dimensions.

Particle-surface capillary forces with disjoining pressure

A new, atomic force microscopy (AFM) based experimental setup for the continuous acquisition of friction force data as a function of humidity has been developed. The current model of interactions between wet contacts under the influence of capillary effects, has been amended to include a vertical component due to the disjoining pressure and takes into account the influence of liquid films adsorbed on the surface. This is a 'switching' model, i.e. the contact between nanometer-sized sphere and a flat surface can exist in two distinct states due to capillary bridge formation/destruction as the humidity is varied. The model has been qualitatively verified on samples of differing wettability produced by UV-ozone treatment of polystyrene (PS). Results of AFM analysis of the friction vs. vapor pressure curves collected from the surface are presented. Correlation between important surface properties such as wettability, adsorption, and contact angle and friction force under varying humidity were found and discussed.

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.

Pattern dynamics in a checkerboard map

Differential equations often have solutions in the forms of trains of coherent structures such as pulses and antipulses. For such systems, the methods of singular perturbation theory permit the derivation of pattern maps that predict the sequence of spacings between successive pulses. Here we apply such a procedure to cases where two distinct kinds of pulse (or antipulse) may coexist in the system. In that case, direct application of the method leads to multivalued maps that make for complicated descriptions, especially when the succession of pulse types becomes chaotic. We show how this description may be simplified by using maps arrayed in checkerboard style to provide causal descriptions of both the successions of pulse spacings and the order in which the different kinds of pulse go by.

Spinodal-like dewetting of thermodynamically-stable thin polymer films

Energetic considerations indicate that long-range Van der Waals forces stabilize thin polystyrene (PS) films against height fluctuations on silicon substrates. Nevertheless, we report here on the amplification of capillary waves of specific wavelengths for 15 nm thick PS films on silicon, ultimately leading to dewetting in a "spinodal-like" process. However, the temporal dependence of the wavelength of the growing instability does not agree with the "classical" spinodal dewetting mechanism. Therefore, this phenomenon is ascribed to the existence of "structural" forces resulting either from the restructuring of the films or from density variations within the films during annealing, in accordance with recent theoretical treatments. The process is shown not to be limited to polystyrene films, which indicates the generality of our findings.