Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics

Nonisothermal evaporation

Evaporation of a liquid layer on a substrate is examined without the often-used isothermality assumption, i.e., temperature variations are accounted for. Qualitative estimates show that nonisothermality makes the evaporation rate depend on the conditions at which the substrate is maintained. If it is thermally insulated, evaporative cooling dramatically slows evaporation down; the evaporation rate tends to zero with time and cannot be determined by measuring the external parameters only. If, however, the substrate is maintained at a fixed temperature, the heat flux coming from below sustains evaporation at a finite rate, deducible from the fluid's characteristics, relative humidity, and the layer's depth. The qualitative predictions are quantified using the diffuse-interface model applied to a liquid evaporating into its own vapor.

A mathematical model and mesh-free numerical method for contact-line motion in lubrication theory

ABSTRACT

We introduce a mathematical model with a mesh-free numerical method to describe contact-line motion in lubrication theory. We show how the model resolves the singularity at the contact line, and generates smooth profiles for an evolving, spreading droplet. The model describes well the physics of droplet spreading-including Tanner's Law for the evolution of the contact line. The model can be configured to describe complete wetting or partial wetting, and we explore both cases numerically. In the case of partial wetting, the model also admits analytical solutions for the droplet profile, which we present here.

ARTICLE HIGHLIGHTS

We formulate a mathematical model to regularize the contact-line singularity for droplet spreading.The model can be solved using a fast, accurate mesh-free numerical method.Numerical simulations confirm that the model describes the quantitative aspects of droplet spreading well.

Can a liquid drop on a substrate be in equilibrium with saturated vapor?

It is well known that liquid and saturated vapor, separated by a flat interface in an unbounded space, are in equilibrium. One would similarly expect a liquid drop, sitting on a flat substrate, to be in equilibrium with the vapor surrounding it. Yet, it is not: as shown in this work, the drop evaporates. Mathematically, this conclusion is deduced using the diffuse-interface model, but it is also reformulated in terms of the maximum-entropy principle, suggesting model independence. Physically, evaporation of drops is due to the so-called Kelvin effect, which gives rise to a liquid-to-vapor mass flux if the boundary of the liquid phase is convex.

Kinetic model for the phase transition of the van der Waals fluid

This is a continuation of previous works [S. Takata and T. Noguchi, J. Stat. Phys. 172, 880 (2018)JSTPBS0022-471510.1007/s10955-018-2068-z; S. Takata, T. Matsumoto, A. Hirahara, and M. Hattori, Phys. Rev. E 98, 052123 (2018)2470-004510.1103/PhysRevE.98.052123]. The simple model proposed in the previous works is extended to be free from the isothermal assumption. The new model conserves the total mass, momentum, and energy in the periodic domain. A monotone functional is found, assuring the H theorem for the new model. Different approaches are taken to tell apart the stable, the metastable, and the unstable uniform equilibrium state. Numerical simulations are also conducted for spatially one-dimensional cases to demonstrate various features occurring in the time evolution process. A prediction method for the profile at the stationary state is discussed as well.

Nonexistence of two-dimensional sessile drops in the diffuse-interface model

The diffuse-interface model (DIM) is a widely used tool for modeling fluid phenomena involving interfaces, such as sessile drops (liquid drops on a solid substrate, surrounded by saturated vapor) and liquid ridges (two-dimensional sessile drops). In this work, it is proved that, surprisingly, the DIM does not admit solutions describing static liquid ridges. If, however, the vapor-to-liquid density ratio is small-for example, for water at room temperature-the ridges can still be observed as quasistatic states, as their evolution is too slow to be distinguishable from evaporation. Interestingly, the nonexistence theorem cannot be extended to axisymmetric sessile drops and ridges near a vertical wall, which are not ruled out.

Dependence of the surface tension and contact angle on the temperature, as described by the diffuse-interface model

Four results associated with the diffuse-interface model (DIM) for contact lines are reported in this paper. First, a boundary condition is derived, which states that the fluid near a solid wall must have a certain density ρ_{0} depending on the solid's properties. Unlike previous derivations, the one presented here is based on the same physics as the DIM itself and does not require additional assumptions. Second, asymptotic estimates are used to check a conjecture lying at the foundation of the DIM, as well as all other models of contact lines, that liquid-vapor interfaces are nearly isothermal. It turns out that, for water, they are not, although, for a more viscous fluid, they can be. The nonisothermality occurs locally, near the interface, but can still affect the contact-line dynamics. Third, the DIM coupled with a realistic equation of state for water is used to compute the dependence of the surface tension σ on the temperature T, which agrees well with the empirical σ(T). Fourth, the same framework is used to compute the static contact angle of a water-vapor interface. It is shown that, with increasing temperature, the contact angle becomes either 180^{∘} (perfect hydrophobicity) or 0^{∘} (perfect hydrophilicity), depending on whether ρ_{0} matches the density of saturated vapor or liquid, respectively. Such behavior presumably occurs in all fluids, not just water, and for all sufficiently strong variations of parameters, not just that of the temperature, as corroborated by existing observations of drops under variable electric field.

Drop Behavior Influenced by the Correlation Length on Noisy Surfaces

We investigate numerically the role of the correlation length in drop behavior on noisy surfaces. To this aim, a phase field tool has been used. Theoretical results are confirmed by experiments of distilled water drops sitting on stainless steel and silicon surfaces textured by laser-induced periodic self-organized structures: an increase of the noise amplitude results in an amplification of the original behavior (i.e., hydrophobic is getting more hydrophobic, hydrophilic is getting more hydrophilic). Furthermore, computer simulations in two and three spatial dimensions allow for predictions of drop behavior on noisy sloped substrates under a gravitational force, a problem of large interest in controlled motion in micro- and nanofluidics.

Alternating current electrothermal modulated moving contact line dynamics of immiscible binary fluids over patterned surfaces

In this paper, we report the results of our numerical study on incompressible flow of a binary system of two immiscible fluids in a parallel plate capillary using alternating current electrothermal kinetics as the actuation mechanism for flow. The surfaces of the capillary are wetted with two different alternating wettability patches. The dynamic motion of the interface of the two fluids is tracked using a phase-field order parameter-based approach. The results exhibit a stick-slip behavior involving acceleration and deceleration of the interface due to the interplay of electrothermal (Coulomb and dielectric) and surface tension forces. Controlling the interface motion through effective tuning of the chemical characteristics of the surfaces and forcing parameters was explored in detail. Finally, we were able to find a critical value of the dimensionless strength of the alternating current electrothermal force above which the interface "breaks", resulting in the formation of isolated droplets. These results have the potential to improve fundamental understanding and design optimization of various biomedical and physiological systems that involve flow of two or more immiscible fluids over chemically wetted surfaces.

Modeling of nanoscale liquid mixture transport by density functional hydrodynamics

Modeling of multiphase compositional hydrodynamics at nanoscale is performed by means of density functional hydrodynamics (DFH). DFH is the method based on density functional theory and continuum mechanics. This method has been developed by the authors over 20 years and used for modeling in various multiphase hydrodynamic applications. In this paper, DFH was further extended to encompass phenomena inherent in liquids at nanoscale. The new DFH extension is based on the introduction of external potentials for chemical components. These potentials are localized in the vicinity of solid surfaces and take account of the van der Waals forces. A set of numerical examples, including disjoining pressure, film precursors, anomalous rheology, liquid in contact with heterogeneous surface, capillary condensation, and forward and reverse osmosis, is presented to demonstrate modeling capabilities.

A Critical Review of Dynamic Wetting by Complex Fluids: From Newtonian Fluids to Non-Newtonian Fluids and Nanofluids

Dynamic wetting is an important interfacial phenomenon in many industrial applications. There have been many excellent reviews of dynamic wetting, especially on super-hydrophobic surfaces with physical or chemical coatings, porous layers, hybrid micro/nano structures and biomimetic structures. This review summarizes recent research on dynamic wetting from the viewpoint of the fluids rather than the solid surfaces. The reviewed fluids range from simple Newtonian fluids to non-Newtonian fluids and complex nanofluids. The fundamental physical concepts and principles involved in dynamic wetting phenomena are also reviewed. This review focus on recent investigations of dynamic wetting by non-Newtonian fluids, including the latest experimental studies with a thorough review of the best dynamic wetting models for non-Newtonian fluids, to illustrate their successes and limitations. This paper also reports on new results on the still fledgling field of nanofluid wetting kinetics. The challenges of research on nanofluid dynamic wetting is not only due to the lack of nanoscale experimental techniques to probe the complex nanoparticle random motion, but also the lack of multiscale experimental techniques or theories to describe the effects of nanoparticle motion at the nanometer scale (10(-9) m) on the dynamic wetting taking place at the macroscopic scale (10(-3) m). This paper describes the various types of nanofluid dynamic wetting behaviors. Two nanoparticle dissipation modes, the bulk dissipation mode and the local dissipation mode, are proposed to resolve the uncertainties related to the various types of dynamic wetting mechanisms reported in the literature.

Thin Films in Partial Wetting: Internal Selection of Contact-Line Dynamics

When a liquid touches a solid surface, it spreads to minimize the system's energy. The classic thin-film model describes the spreading as an interplay between gravity, capillarity, and viscous forces, but it cannot see an end to this process as it does not account for the nonhydrodynamic liquid-solid interactions. While these interactions are important only close to the contact line, where the liquid, solid, and gas meet, they have macroscopic implications: in the partial-wetting regime, a liquid puddle ultimately stops spreading. We show that by incorporating these intermolecular interactions, the free energy of the system at equilibrium can be cast in a Cahn-Hilliard framework with a height-dependent interfacial tension. Using this free energy, we derive a mesoscopic thin-film model that describes the statics and dynamics of liquid spreading in the partial-wetting regime. The height dependence of the interfacial tension introduces a localized apparent slip in the contact-line region and leads to compactly supported spreading states. In our model, the contact-line dynamics emerge naturally as part of the solution and are therefore nonlocally coupled to the bulk flow. Surprisingly, we find that even in the gravity-dominated regime, the dynamic contact angle follows the Cox-Voinov law.

Can vibrations control drop motion?

We discuss a mechanism for controlled motion of drops with applications for microfluidics and microgravity. The mechanism is the following: a solid plate supporting a liquid droplet is simultaneously subject to lateral and vertical harmonic oscillations. In this way the symmetry of the back-and-forth droplet movement along the substrate under inertial effects is broken and thus will induce a net driven motion of the drop. We study the dependency of the traveled distance on the oscillation parameters (forcing amplitude, frequency, and phase shift between the two perpendicular oscillations) via phase field simulations. The internal flow structure inside the droplet is also investigated. We make predictions on resonance frequencies for drops on a substrate with a varying wettability.

Pulsating electric field modulated contact line dynamics of immiscible binary systems in narrow confinements under an electrical double layer phenomenon

We investigate the interfacial electro-chemical-hydrodynamics of an incompressible immiscible binary fluid system that moves in a narrow fluidic channel under time-periodic electroosmotic effects. We apply an alternating electrical voltage that sets the binary fluids in motion along the channel, whereas the channel walls are lined with chemical patch to alter the wetting characteristics of the surface. We demonstrate that the pulsating nature of the externally applied electric field in conjunction with the wetting characteristics of the surface may lead to some fascinating behavior of the contact line motion; which, in turn, may affect the capillary filling dynamics in an intriguing manner. Our results also unveil the profound influence of two important governing factors actuating the flow, namely, the frequency and amplitude of the time periodic electric field, on the tunability of the capillary filling rate and power requirement for filling the fluids into the channel.

Extending the nonequilibrium square-gradient model with temperature-dependent influence parameters

Nonequilibrium interface phenomena play a key role in crystallization, hydrate formation, pipeline depressurization, and a multitude of other examples. Square gradient theory extended to the nonequilibrium domain is a powerful tool for understanding these processes. The theory gives an accurate prediction of surface tension at equilibrium, only with temperature-dependent influence parameters. We extend in this work the nonequilibrium square gradient model to have temperature-dependent influence parameters. The extension leads to thermodynamic quantities which depend on temperature gradients. Remarkably the Gibbs relation proposed in earlier work is still valid. Also for the extended framework, the "Gibbs surface" described by excess variables is found to be in local equilibrium. The temperature-dependent influence parameters give significantly different interface resistivities (∼9%-50%), due to changed density gradients and additional terms in the enthalpy. The presented framework facilitates a more accurate description of transport across interfaces with square gradient theory.

Thermocapillary-actuated contact-line motion of immiscible binary fluids over substrates with patterned wettability in narrow confinement

We investigate thermocapillary-driven contact-line dynamics of two immiscible fluids in a narrow fluidic confinement comprising substrates with patterned wettability variations. Our study, based on phase field formalism, demonstrates that the velocity of the contact line is a strong function of the combined consequences of the applied thermal gradient and the substrate wetting characteristics. Finally, we evaluate different energy transfer rates and show that the dissipation due to fluid slip over the solid surface plays a dominating role in transferring energy into the contact-line motion. Our analysis, in effect, provides an elegant way of controlling the capillary filling rate in a narrow fluidic confinement by tailoring the applied temperature gradient and the substrate wettability in tandem.

Interfacial dynamics of two immiscible fluids in spatially periodic porous media: the role of substrate wettability

We delineate the contact line dynamics of two immiscible fluids in a medium having spatially periodic porous structures. The flow is driven by an external applied pressure gradient. We bring out the combined consequences of the solid fraction distribution and the substrate wettability on the resulting dynamics of the contact line, by employing phase-field formalism. We capture the sequence of spatiotemporal events leading to formation of liquid bridges by trapping a small amount of displaced phase fluid between two consecutive porous blocks, as dictated by the combinations of substrate wettability and solid fraction. We also demonstrate the existence of a regime of complete interfacial recovery, depending on the parametric space of the governing parameters under concern. Our results essentially demonstrate the intricate mechanisms by virtue of which the wettabilities of the substrates alter the dynamical evolutions of interfaces and the subsequent shapes and sizes of the adsorbed dispersed phases, bearing far-ranging consequences in several practical applications ranging from oil recovery to groundwater flow.

Influence of hydrophobic effects on streaming potential

We study the influence of hydrophobic effects on streaming potential mediated flow through a narrow confinement. In a clear departure from the approach used in prior works, we use a phase-field model to capture the hydrophobicity-induced depletion in the near wall region, and express the variation of viscosity and permittivity across the interfacial layer in terms of the phase-field variable. We then use these in the determination of the flow velocity, and highlight the sensitive interplay between the intrinsic length scale of the electrical double layer and that of the depletion in terms of the variations of an effective normalized viscosity that captures the electroviscous effect. We expect that this work will be an important step forward in the realistic continuum modeling of interfacial physics in the particular context of streaming potential mediated flows.

Electric-field-driven contact-line dynamics of two immiscible fluids over chemically patterned surfaces in narrow confinements

We investigate the contact-line dynamics of two immiscible fluids in a narrow fluidic confinement comprising wettability-gradient surfaces, where the bulk fluid motion is actuated by an externally applied electric field. We assume that the channel walls bear spatially uniform surface potential. Our analysis, based on the diffuse interface formalism, reveals that the contact line undergoes stick-slip motion over the chemical patches and its velocity is a strong function of the interfacial electrokinetics. We also show that the tendency of the contact line of getting pinned to the selected patches can decrease or increase with its progression along the channel, depending on the ratio of the permittivities of the two fluids. Finally, we establish the functional dependency of the time taken by the contact line to move across the patches (capillary filling time) on the combined consequences of interfacial electrochemistry and wettability patterning.

Partial coalescence of sessile drops with different miscible liquids

Using computer simulations in three spatial dimensions, we examine the interaction between two deformable drops consisting of two perfectly miscible liquids sitting on a solid substrate under a given contact angle. Driven by capillarity and assisted by Marangoni effects at the droplet interfaces, several distinct coalescence regimes are achieved after the droplets' collision.

On the moving contact line singularity: asymptotics of a diffuse-interface model

The behaviour of a solid-liquid-gas system near the three-phase contact line is considered using a diffuse-interface model with no-slip at the solid and where the fluid phase is specified by a continuous density field. Relaxation of the classical approach of a sharp liquid-gas interface and careful examination of the asymptotic behaviour as the contact line is approached is shown to resolve the stress and pressure singularities associated with the moving contact line problem. Various features of the model are scrutinised, alongside extensions to incorporate slip, finite-time relaxation of the chemical potential, or a precursor film at the wall.

Thermal singularity and droplet motion in one-component fluids on solid substrates with thermal gradients

Using a continuum model capable of describing the one-component liquid-gas hydrodynamics down to the contact line scale, we carry out numerical simulation and physical analysis for the droplet motion driven by thermal singularity. For liquid droplets in one-component fluids on heated or cooled substrates, the liquid-gas interface is nearly isothermal. Consequently, a thermal singularity occurs at the contact line and the Marangoni effect due to temperature gradient is suppressed. Through evaporation or condensation in the vicinity of the contact line, the thermal singularity makes the contact angle increase with the increasing substrate temperature. This effect on the contact angle can be used to move the droplets on substrates with thermal gradients. Our numerical results for this kind of droplet motion are explained by a simple fluid dynamical model at the droplet length scale. Since the mechanism for droplet motion is based on the change of contact angle, a separation of length scales is exhibited through a comparison between the droplet motion induced by a wettability gradient and that by a thermal gradient. It is shown that the flow field at the droplet length scale is independent of the statics or dynamics at the contact line scale.

Consistent description of electrohydrodynamics in narrow fluidic confinements in the presence of hydrophobic interactions

Electrohydrodynamics in the presence of hydrophobic interactions in narrow confinements is traditionally represented from a continuum viewpoint by a Navier slip-based conceptual paradigm, in which the slip length carries the sole burden of incorporating the effects of substrate wettability on interfacial electromechanics, precluding any explicit dependence of the interfacial potential distribution on the substrate wettability. Here we show that this traditional way of treating electrokinetics-wettability coupling may lead to serious discrepancies in predicting the resultant transport characteristics as manifested through an effective zeta potential. We suggest that an alternative consistent description of the underlying physics through a free-energy-based formalism, in conjunction with considerations of hydrodynamic and electrical property variations consistent with the pertinent phase-field description, may represent the underlying consequences in a more rational manner, as compared to the traditional slip-based model coupled with a two-layer description. Our studies further reveal that the above discrepancies may not occur solely due to the slip-based route of representing the interfacial wettability, but may be additionally attributed to the act of "discretizing" the interfacial phase fraction distribution through an artificial two-layer route.

On the coalescence of sessile drops with miscible liquids

Sessile drops sitting on highly wettable solid substrates fuse in qualitatively different ways after contact, depending on the surface tension gradients between the mixing droplets. In early time evolution the drop coalescence can be fast or delayed (intermittent). In long time evolution a secondary drop formation can occur. We study numerically droplet dynamics during coalescence in two and three spatial dimensions, within a phase field approach. We discuss criteria to distinguish different coalescence regimes. A comparison with recent experiments will be done.

Delayed coalescence of droplets with miscible liquids: Lubrication and phase field theories

Mixing of droplets with a body of different liquids shows an interesting behavior for small contact angles at solid substrate. The droplets interact with each other, a liquid exchange appears between the approaching drops owing to surface tension gradients at the droplets interface. But the drops remain separated for some seconds (up to minutes), until the merging into a single drop occurs (Langmuir 24, 6395 (2008)). We investigate this phenomenon using lubrication approximation and phase field approach. For both methods, 2D quantitative computer simulations for delayed fusion of perfectly miscible thin liquid films/droplets with low contact angles are reported.

Different behaviors of delayed fusion between drops with miscible liquids

Fusion of sessile droplets with body of different liquids is delayed when the approaching drops are sitting on a highly wettable solid substrate. Owing to the surface tension gradients between the mixing drops, a Marangoni driven flow through the connecting channel appears. Experiments of delayed coalescence were recently reported in [Langmuir 24, 6395 (2008)10.1021/la800630w] for millimeters sized drops. For droplets of submillimeter dimensions, capillary forces dominate. The control of interfacial energies becomes an important strategy for manipulating tiny droplets along the solid surfaces. In this paper we present phase field simulations in two spatial dimensions for microdroplets of perfectly miscible liquids. For drops with a given geometry, systematic investigations were performed for different fluid viscosities. Different behaviors are observed from chasing droplets to droplets repelling.

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.

Modeling and simulations for molecular scale hydrodynamics of the moving contact line in immiscible two-phase flows

This paper starts with an introduction to the Onsager principle of minimum energy dissipation which governs the optimal paths of deviation and restoration to equilibrium. Then there is a review of the variational approach to moving contact line hydrodynamics. To demonstrate the validity of our continuum hydrodynamic model, numerical results from model calculations and molecular dynamics simulations are presented for immiscible Couette and Poiseuille flows past homogeneous solid surfaces, with remarkable overall agreement. Our continuum model is also used to study the contact line motion on surfaces patterned with stripes of different contact angles (i.e. surfaces of varying wettability). Continuum calculations predict the stick-slip motion for contact lines moving along these patterned surfaces, in quantitative agreement with molecular dynamics simulation results. This periodic motion is tunable through pattern period (geometry) and contrast in wetting property (chemistry). The consequence of stick-slip contact line motion on energy dissipation is discussed.

Spreading dynamics of power-law fluid droplets

This paper aims at providing a summary of the theoretical models available for non-Newtonian fluid spreading dynamics. Experimental findings and model predictions for a Newtonian fluid spreading test are briefly reviewed. Then how the complete wetting and partial wetting power-law fluids spread over a solid substrate is examined. The possible extension of Newtonian fluid models to power-law fluids is also discussed.

Modelling approaches to the dewetting of evaporating thin films of nanoparticle suspensions

We review recent experiments on dewetting thin films of evaporating colloidal nanoparticle suspensions (nanofluids) and discuss several theoretical approaches to describe the ongoing processes including coupled transport and phase changes. These approaches range from microscopic discrete stochastic theories to mesoscopic continuous deterministic descriptions. In particular, we describe (i) a microscopic kinetic Monte Carlo model, (ii) a dynamical density functional theory and (iii) a hydrodynamic thin film model. Models (i) and (ii) are employed to discuss the formation of polygonal networks, spinodal and branched structures resulting from the dewetting of an ultrathin 'postcursor film' that remains behind a mesoscopic dewetting front. We highlight, in particular, the presence of a transverse instability in the evaporative dewetting front, which results in highly branched fingering structures. The subtle interplay of decomposition in the film and contact line motion is discussed. Finally, we discuss a simple thin film model (iii) of the hydrodynamics on the mesoscale. We employ coupled evolution equations for the film thickness profile and mean particle concentration. The model is used to discuss the self-pinning and depinning of a contact line related to the 'coffee-stain' effect. In the course of the review we discuss the advantages and limitations of the different theories, as well as possible future developments and extensions.

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.

Drops on an arbitrarily wetting substrate: a phase field description

We propose a scheme for studying thin liquid films on a solid substrate using a phase field model. For a van der Waals fluid-far from criticality-the most natural phase field function is the fluid density. The theoretical description is based on the Navier-Stokes equation with extra phase field terms and the continuity equation. In this model free of interface conditions, the contact angle can be controlled through the boundary conditions for the density field at the solid walls [L. M. Pismen and Y. Pomeav, Phys. Rev. E 62, 2480 (2000)]. We investigate the stability of a thin liquid film on a flat homogeneous solid support with variable wettability. For almost hydrophobic surfaces, the liquid film breaks up and transitions from a flat film to drops occur. Finally, we report on two-dimensional numerical simulations for static liquid drops resting on a flat horizontal solid support and for drops sliding down on inclined substrates under gravity effects.

Solvability condition for the moving contact line

We consider the motion of a contact line between a fluid, gas, and solid, as it occurs when a drop advances over a solid surface. This motion is controlled by a microscopic length scale near the contact line, such as a slip length or the precursor thickness. The capillary profile inside the drop is linked to the contact line through an intermediate region which is characterized by an interface slope which varies logarithmically. The intermediate solution contains a single adjustable constant, which can be computed either by matching to the capillary region or to the contact line. We describe a simple method to perform the matching and to compute the required constant. This extends and/or simplifies results known previously. We apply our results to the case of a spreading drop in the presence of an interface potential and derive the equation of motion by combining the inner and outer expansions.

Roughness of moving elastic lines: crack and wetting fronts

We investigate propagating fronts in disordered media that belong to the universality class of wetting contact lines and planar tensile crack fronts. We derive from first principles their nonlinear equations of motion, using the generalized Griffith criterion for crack fronts and three standard mobility laws for contact lines. Then we study their roughness using the self-consistent expansion. When neglecting the irreversibility of fracture and wetting processes, we find a possible dynamic rough phase with a roughness exponent of zeta=1/2 and a dynamic exponent of z=2. When including the irreversibility, we conclude that the front propagation can become history dependent, and thus we consider the value zeta=1/2 as a lower bound for the roughness exponent. Interestingly, for propagating contact line in wetting, where irreversibility is weaker than in fracture, the experimental results are close to 0.5, while for fracture the reported values of 0.55-0.65 are higher.

Phase-field simulations for drops and bubbles

Recently we proposed a phase field model to describe Marangoni convection in a compressible fluid of van der Waals type far from criticality [Eur. Phys. J. B 44, 101 (2005)]. The model previously developed for a two-layer geometry is now extended to drops and bubbles. A randomly distributed initial density evolves towards phase separation and single droplet formation. For a two-component liquid-liquid system we report on numerical simulations for drop Marangoni migration in a vertical thermal gradient.

Phase field theory of heterogeneous crystal nucleation

The phase field approach is used to model heterogeneous crystal nucleation in an undercooled pure liquid in contact with a foreign wall. We discuss various choices for the boundary condition at the wall and determine the properties of critical nuclei, including their free energy of formation and the contact angle as a function of undercooling. For particular choices of boundary conditions, we may realize either an analog of the classical spherical cap model or decidedly nonclassical behavior, where the contact angle decreases from its value taken at the melting point towards complete wetting at a critical undercooling, an analogue of the surface spinodal of liquid-wall interfaces.

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.

Effects of quenching rate and viscosity on spinodal decomposition

Spinodal decomposition of deeply quenched mixtures is studied experimentally, with particular emphasis on the domain growth rate during the late stage of coarsening. We provide some experimental evidence that at high Péclet number, the process is isotropic and the domain growth is linear in time, even at finite quenching rates. In fact, the quenching rate appears to influence the magnitude of the growth rate, but not its scaling law. In the second part of the work we analyze the effect of viscosity on the growth rate. As predicted by the diffuse interface model, we do not find any effect of viscosity on the growth rate of the nucleating drops, although, as expected, the viscosity of the continuous phase does influence the settling speed and thus the total separation time.

Avoided critical behavior in dynamically forced wetting

A solid object can be coated by a nonwetting liquid since a receding contact line cannot exceed a critical speed. In this Letter we study the dynamical wetting transition at which a liquid film gets deposited by withdrawing a vertical plate out of a liquid reservoir. It has recently been predicted that this wetting transition is critical with diverging time scales and coincides with the disappearance of stationary menisci. We demonstrate experimentally and theoretically that the transition is due to the formation of a solitary wave, well below the critical point. As a consequence, relaxation times remain finite at threshold. The structure of the liquid deposited on the plate involves a capillary ridge that does not trivially match the Landau-Levich film.