Lorentz Center - Contact Line Instabilities from 4 Jan 2010 through 8 Jan 2010
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    Contact Line Instabilities
    from 4 Jan 2010 through 8 Jan 2010





Vladimir Ajaev

“Modeling of apparent contact lines in evaporating liquid films”


In the studies of evaporating thin films and sessile droplets the contact line is often represented as a transition region between a solution corresponding to the macroscopic shape and an ultra-thin film in which disjoining pressure is important. The standard approach to modeling the latter, based on the theory of London - van der Waals forces, is not adequate for describing many liquids  used in heat transfer applications, most notably water and liquid metals. The focus of the present work is on incorporating realistic two-component models of disjoining pressure into studies of both steady and moving contact lines. An extension of the classical Frumkin-Derjaguin theory to the case of evaporating liquids is proposed and is shown to agree with the contact angle data obtained from numerical simulations of steady contact lines. For moving contact lines,Tanner's law turns out to be followed surprisingly well even for relatively strong evaporation,  although some deviations are observed as the contact line speed is increased. Finally, the development of fingering instability in gravity-driven evaporating liquid films is studied. It has been previously shown that  evaporation can  suppress this instability.  However, increasing the apparent contact angle by adjusting the disjoining pressure parameters tends to have a destabilizing effect and also reduces the wavelength of the fastest growing mode of the instability. (Joint work with T.Gambaryan-Roisman, J.Klentzman, and P.Stephan)






Patrick D. Anderson

“Phase field modeling of drop impact phenomena”


In this talk I discuss a Cahn-Hilliard approach to model the interaction of drops with substrates. The properties of the drop and the substrate can be mechanically and chemically complex complex and phase transitions may occur prior, during and after impact. After validating the modeling by comparison with experiments, a selection of results is presented and discussed.






Bruno Andreotti

“Experimental investigation of the dynamical wetting transition”


We consider experimentally the deposition of a film of viscous liquid on a flat plate being withdrawn from a bath, experimentally and theoretically. For any plate speed U, there is a range of ``thick'' film solutions whose thickness scales like U^(1/2) for small U. These solutions are realized for a partially wetting liquid, while for a perfectly wetting liquid the classical Landau-Levich-Derjaguin (LLD) film is observed, whose thickness scales like U^{2/3}. The thick film is distinguished from the LLD film by a dip in its spatial profile at the transition to the  bath. We calculate the phase diagram for the existence of stationary  film solutions as well as the film profiles, and find excellent agreement  with experiment. We then discuss the conditions under which a film is entrained, when the experiment is started from a static plate plunged in the bath. It is shown that the dynamics of both liquid deposition and long-wavelength perturbations quasi-steadily follow the stationary states. This provides a first experimental access to the entire bifurcation diagram of dynamical wetting.





Martin Bakker

“Inkjet Printing trends and future challenges”


Inkjet printing -especially “thermal inkjet”- is a well known printing technology for the consumer-printing market.

The alternative Piezo-based inkjet printing is now emerging as a technology suited for many new printing applications, which will be shortly illustrated.  Jetting of ever more fluids is presenting a challenge to the designers of print-heads and printing systems. The demand for more nozzles (from 256  to over 30.000) and smaller drops (from 20 pl. to below 1 pl.) requires a good understanding of the behavior of liquids through the print-head and obviously at the spot where the ink drops  are leaving the print-head towards the substrate they are supposed to land on.  The intrinsic reliability - of every ink-droplet leaving the print-head at a consistent angle- should be in the design of the print-head.

M. Bakker, Vice President R&D -Research,  Océ Technologies B.V






Jos P.H. Benschop

"Immersion lithography"


The semiconductor industry has realized an aggressive shrinkage of its smallest dimensions over the last 45 years, commonly referred to as “Moore’s law”. Microlithography is used to define patterns on IC’s. The workhorse for the lithography is optical projection lithography. State-of-the arte water immersion lithography system using 193 nm wavelength and an NA of 1.35 are capable to pattern an 300 mm diameter wafer with lines and spaces below 40 nm in less than 20 seconds. After an introduction into the world of lithography and a brief overview on the history of ASML (from startup in 1984 to market- and technology-leader in a multi-billion € market)  the challenges associated with immersion lithography will be discussed.

Dr.ir. Jos P.H. Benschop, Vice President Research ASML






Lydéric Bocquet

"Contact line problems in the inertial limit"


In this talk I consider several situations where wetting, via the existence of contact lines, has an impact on flows at interfaces in the inertial regime. I will in particular describe the problem of dripping on solid surface and flow separation in the inertial regime. This problem is commonly coined as the "teapot effect". Based on a thorough experimental study of this problem, we show that wettability has a considerable impact on the flow separation. As a paradigm, superhydrophobic surface are shown to suppress dripping. I will discuss a theoretical description of these results, showing that flow separation is fully controlled by the existence of a capillary meniscus connecting the liquid and the solid body. As a further illustration of these results I will present electro-wetting experiments demonstrating the control of dripping using an external voltage.






Tak Shing Chan

“Two-phase hydrodynamic model for air entrainment at the moving contact line”


The moving contact line problems are challenging because they involve multiple length scales. One interesting case arises when an advancing liquid of high viscosity entrains the surrounding phase, such as air. In this presentation, we introduce a hydrodynamic model that generalizes the lubrication theory in order to take into account the velocity fields of the two phases. Assuming that the curvature of the interface is small we derive a differential equation for the interface profile at stationary state. We found that there is a critical capillary number above which no steady meniscus can exist and instability will occur. For example, air  bubbles will be entrained into the liquid at the advancing contact line. However, we found no instability when neglecting the viscosity of the surrounding phase, illustrating the two-phase nature of the problem.






Joël de Coninck

“Influence of solid-liquid interactions on dynamic wetting: evidence from molecular dynamics and experiment”


Some dynamic wetting theory can be extended  to take specific account of solidliquid interactions. By equating the work of adhesion with the surface component of the specific activation free energy of wetting, equations have been derived which show the way in which solidliquid interactions modify both the driving force and the resistance to wetting. For a liquid meniscus advancing across the surface of a solid, these two effects have opposing consequences. Thus, strong interactions increase both the driving force and the resistance, while weak interactions decrease the driving force and the resistance. Because of the form of the relationships, the two effects do not simply cancel out. As a result, the maximum rate at which a liquid can wet a solid may exhibit its own maximum at some intermediate level of interaction. Data taken from both experimental and molecular-dynamics simulations are shown to support these findings, which have significant implications for any process where wetting dynamics are important, such as coating.






Jaap den Toonder, Marco Baragona, Milica Kovacevic Milivojevic, Ralph Maessen, Sergei Shulepov

Microbubbles  and Ultrasound Manipulation”


One of the important defect mechanisms in immersion lithography is the formation air bubbles in the light path underneath the lens. These air bubbles can be formed during wafer scanning due to air entrainment during collisions between water droplets and the meniscus of the liquid film present between the wafer and the lens.

This presentation will discuss the possibility to manipulate these micro-bubbles using ultrasonic waves, so that they can be prevented from reaching the light path during lithography. We will show a combination of numerical and experimental work that we have used to explore this idea, but we will also address some unanswered questions. The particular challenges in this problem are the fact that the bubbles are not free-floating, but sticking to a surface, and the confined geometry.






J. Frits Dijksman and Anke Pierik.

“Fluid dynamical analysis of the distribution of ink jet printed biomolecules in microarray substrates for genotyping applications”


Oligonucleotide microarrays are tools used to analyse samples for the presence of specific DNA sequences. In the system as presented here, specific DNA sequences are first amplified by PCR during which process they are labelled with fluorophores. The amplicons are subsequently hybridized onto an oligonucleotide microarray, which in our case is a porous nylon membrane with microscopic spots.  Each spot on the membrane contains oligonucleotides with a sequence complementary to part of one specific target sequence. The solution containing the amplicons flows by external agitation many times up and down through the porous substrate, thereby reducing the time delaying effect of diffusion. By excitation of the fluorophores the emitted pattern of fluorophores can be detected by a CCD camera. The recorded pattern is a characteristic of the composition of the sample. The oligonucleotide capture probes have been deposited on the substrate by using non-contact piezo ink jet printing, which is the focus of our study. The objective of this study is to understand the mechanisms that determine the distribution of the ink jet printed capture probes inside the membrane. The membrane is a porous medium: the droplets placed on the membrane penetrate in the microstructure of it. The 3D-distribution of the capture probes inside the membrane determines the distribution of the hybridized fluorescent PCR products inside the membrane and thus the emission of light when exposed to the light source. As the 3D-distribution of the capture probes inside the membrane eventually determines the detection efficiency, this parameter can be controlled for optimization of the sensitivity of the assay.

The main issues addressed here are: how are the capture probes distributed inside the membrane and how does this distribution depend on the printing parameters. We will use two model systems to study the influences of different parameters: a single nozzle print head jetting large droplets at a low frequency and a multi-nozzle print head emitting small droplets at a high frequency. In particular, we have investigated the effects when we change from usage of the first system to the second system. Furthermore, we will go into detail how we can obtain smaller spot sizes in order to increase the spot density without having overlapping spots, leading eventually to lower manufacturing costs of microarrays. By controlling the main print parameters influencing the 3D distribution inside the porous medium, the overall batch-to-batch variations can possibly be reduced.

Philips Research Europe, Department Healthcare Devices and Instrumentation.

High Tech Campus 4, 5656 AE Eindhoven, The Netherlands, frits.dijksman@philips.com,  anke.pierik@philips.com






Jens Eggers

“Hydrodynamic theory of the dynamic wetting transition”


A prototypical problem in the study of wetting phenomena is that of a non-wetting solid plunging into or being withdrawn from a liquid bath. In the latter, de-wetting case, a critical speed exists above which a three-phase contact line is no longer sustainable and the solid can no longer remain dry. Instead, the solid is being coated with a liquid film. We explain the nature of this transition by matching the highly curved contact line region to the static profile far away from it.

Affiliation, School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom






Mark Franken (group Jerry Westerweel)

“Towards nanoscale contact line dynamics based on TIRFM”

Click here for pdf abstract






Hyoungsoo Kim

“Experimental Investigation Of Internal Flow Fields Of Immersion Lens Droplet By 3d Ptv

Click here for pdf abstract






Roger Jeurissen

“A more tractable approximation to thin film Marangoni flow”


When a droplet moves over a completely wetting solid substrate, whether by gravity, shear stress due to air flow, or inertia, the droplet  spreads. Typically, the order of magnitude of the spreading speed is coupled to the translational speed. However, when a droplet is moved by surfactant driven Marangoni flow, the spreading speed and the translation speed are completely decoupled. We investigate this unique regime in which the time scale of droplet translation is many orders of magnitude larger or smaller { depending on the droplet size { than the spreading speed. We found that the divergence of the surface velocity in thin film Marangoni flow vanishes. As examples of how these results make flows tractable by analytical methods, the translation of a ridge and the development of a fingering instability are analyzed. The analysis of the fingering instability is corroborated by an experimental observation of the fingering instability.






Laurent Limat

“Corner tip singularities”


When a drop slides down a inclined plane, in partial wetting conditions, a sharp tip, of conical shape develops at its rear front, connecting two straight contact lines inclined with respect to the horizontal.  A similar structure is observed when one pulls a tape or a plate outside of a bath: the contact line exhibits a sawtooth shape involving sometimes several of these "tip singularities". Understanding these objects is important for both fundamental and applied reasons. First it is perhaps a unique case of interaction between a point singularity (the tip) and a line singularity (the contact line), that needs to be understood (see I. Peters, J. Snoeijer , A. Daerr, L.L., PRL 2009). Also, its appearance is ubiquitous in numerous coating processes encountred in Industry. I will review the knowledge that  can be saved from several works on this problem, most of these ones being published these ten last years, and present some still opened questions (works made with T. Podgorski, J.-M. Flesselles, A. Daerr, J. Snoeijer, N. Legrand, E. Rio, in collaboration with H.-A. Stone, B. Andreotti and J. Eggers).

Affiliation: Laboratoire Matière et Systèmes Complexes (MSC), CNRS and University Paris Diderot, UMR 7057, Paris, France.





D.J.C.M. ’t Mannetje, H.T.M. Van den Ende, F. Mugele

“Contact line dynamics in electric fields: a sliding drop”


We studied the hysteresis and dynamics of a droplet sliding between two hydrophobic plates under gravity. In this situation, the driving gravitational force is balanced by contact line friction and viscous damping inside the droplet. Our measurements show clearly that the contact line has the largest effect in the chosen geometry, making it ideal for the study of contact line effects.

When we apply an AC electric field between the two plates, while preventing current from flowing between the two with insulating hydrophobic layers, the droplet moves more quickly. As the driving force remains constant, this shows that the hysteresis is reduced, consistent with measurements on (quasi-)static droplets. This means that the advancing contact angle is reduced, while the receding angle is not. In the field of preventing contact line instabilities, this is useful; it means that the advancing contact line is stabilized while the receding contact line is not de-stabilized. An interesting result is that the effect seems to be dependent on the insulator thickness within the experimentally accessible range (1-6 micrometers).






Len Pismen

“Chemo-capillary instabilities of a contact line”


Equilibrium and motion of a contact line are viewed as analogs of phase equilibrium and motion of an interphase boundary. This point of view makes evident the tendency to minimization of the length of the contact line at equilibrium. The concept of line tension is, however, of limited applicability, in view of a qualitatively different relaxation response of the contact line, compared to a two-dimensional curve. Both the analogy and qualitative distinction extend to a non-equilibrium situation arising due to coupling with reversible substrate modification. Under these conditions, the contact line may suffer a variety of chemo-capillary instabilities (fingering, traveling and oscillatory), similar to those of dissipative structures in nonlinear non-equilibrium systems. The preference order of the various instabilities changes, however, significantly due to a different way the interfacial curvature is relaxed.






Yves Pomeau

“Merging of droplets standing on a solid surface”


A physical problem where dynamical capillarity is involved is the merging of droplets standing on an horizontal solid with finite equilibrium contact angle. Experiments by Beysens and collaborators show a two step dynamics for this process: at short times it happens at non-small capillary number, with a balance between driving by capillary forces and viscous dissipation. At later times, the merging process slows down by orders of magnitude and is ruled by the small mobility of the contact line. I shall focus on the transition between the two regimes.






David Quéré

“Leiden frost in frost Leiden”

In this talk, we discuss the ultra-low friction states generated in a Leidenfrost situation, and focus on the particular case of drops (or solids) evaporating (or sublimating) on hot ratchets. Then, a motion can occur, as first shown by Linke in 2006. We present hypothesis for the origin of the motion and the special law of friction associated to it.

Other contributors to the talk: Marie Le Merrer, Guillaume Lagubeau, Guillaume Dupeux and Christophe Clanet.



Markus Rauscher

“Stability of rivulets on chemical channels”


With the possible application in open microfluidic devices in mind we investigate the stability and dynamics of rivulets and droplets on chemical channels, i.e., on hydrophilic stripes on hydrophobic substrates. Straight rivulets with pinned contact lines are stable if the contact angle is smaller than 90 degrees. Rivulets with larger contact angles are subject to a Rayleigh-Plateau instability and break up into droplets. We investigate the influence of partial pinning, of driving the fluid along the channel, and of the long ranged van-der-Waals forces on this instability.

Partially pinned rivulets turn out to be stable as long as the pressure in the rivulet increases with their cross-sectional area. Driving the fluid in the rivulet can neither stabilize nor destabilize the rivulet, but it leads to the formation of larger droplets. We also observe that driving flow in the chemical channel accelerates the coarse graining of the chain of droplets formed after the break-up of a rivulet.  Van-der-Waals forces do not lead to additional instabilities. In particular a negative line tension of the three phase contact line does not lead to the short wavelength instability expected in a capillary model.






Etienne Rolley

“Wetting and dewetting of liquid drops and liquid films on rough substrates”


We have measured the dynamics of the edge of the van der Waals film close to the prewetting transition, using liquid hydrogen on rough Cesium substrates.

Direct /optical/ observation shows that dewetting occurs through the receding of the film edge. We find that the edge is much more sensitive to the disorder than the contact line which is consistent with the idea that the film edge is a true 1D system while the contact line has a non-local elasticity. In contrast, we could not observe the advancing of the film edge: wetting seems to occur through a progressive thickening of the dry part of the substrate.

Though the geometry of the film edge is very different from the one of the meniscus, the receding dynamics of both systems are very similar: both are thermally activated with a similar activation length which is of the order of the typical length scale of the Cs roughness (10 nm). Indeed, at such a length scale, one cannot forget about the structure of the film edge which is not very different from the one of the meniscus.

So the film edge /looks like/ a 1D system only at large length scale, the cut off being set by the healing length. This explains most of our observations, but does not account for the non-advancing  film edge.






James R. T. Seddon

“Surface Nanobubbles


Immersing a hydrophobic substrate in water leads to the surprising formation of nanoscopic bubbles adhered to the solid.  These nanobubbles are spherical caps, with typical heights and diameters of 10nm and 50nm, respectively.  One of their most interesting features is their stability to diffusion.  Complete bulk diffusion should occur in microseconds, but nanobubbles are stable for at least 10-11 orders of magnitude longer than this.  Moreover, nanobubbles grow during this time, and show no obvious evidence of imminent collapse.

Results are presented of an experimental investigation into the effects of liquid temperature and gas concentration on the formation of surface nanobubbles.  Previously, both effects have always been investigated together, whilst here we independently investigate the role of each.






Sergei Shulepov, Marco Baragona and Marleen van Aartrijk

“Dynamic contact line modeling for immersion lithographic systems: model development and applications”


The development and validation of a VOF model for the description of 3D contact line dynamics is outlined. The model is built on existing finite volume CFD codes in order to easily allow for the description of complex 3D geometries, as encountered in industrial CFD, and in particular, immersion lithography applications.

We show with a few examples that the model is a valuable help in the continuous development of new immersion lithography concepts. We will especially focus on prediction of water loss/ substrate characterization, and on the bubble formation during collisions of two fluid fronts. When possible, results are compared to experimental and/or analytical results.

Future (possible) improvements in understanding of the contact line behavior on microscopic and sub-microscopic scale will be discussed, and  model developments - briefly outlined. 






Todd Squires

Microrheology of fluid interfaces:  viscoelasticity, yielding, visualization and slow recovery of phospholipid monolayers


I will describe a new technique we have developed to measure the interfacial rheology -- the viscous and elastic properties -- of fluid-fluid interfaces, typically laden with some surface-active species (molecular surfactants, copolymers, colloids, etc.).  Using photolithography, we fabricate ferromagnetic, amphiphilic microdisk probes that are ideally suited for active interfacial microrheology.  By applying an oscillatory torque using electromagnets, and measuring the resulting (oscillatory) displacement, we create a small version Couette interfacial rheometer that is exceedingly sensitive to the rheology of the interface.  A novel feature is our ability to directly visualize the interface during the measurement.  In particular, we measure the frequency-dependent linear viscoelastic properties of a molecular monolayer of the phospholipid DPPC in the liquid-condensed phase, and directly visualize the deformation of liquid-crystalline domains under both linear and nonlinear deformations.  We find elastic strain to be stored over a shockingly long relaxation time, particularly for a 2nm-think molecular monolayer.





Arjan van der Bos, Roger Jeurissen, Jos de Jong, Marc van den Berg, Herman Wijshoff, Hans Reinten, Michel Versluis and Detlef Lohse

“Acoustic measurement of bubble size and position in an ink jet printhead


A bubble can be entrained in the ink channel of a piezo-driven ink jet print head leading to malfunction. Here an acoustic sizing method for the volume and position of the bubble is presented. The influence of the bubble on the channel acoustics is detected by the piezo-actuator operating in a sensing mode. The acoustic response is also modeled in a linearized bubble dynamics model and we demonstrate how the size of the bubble and its position within the channel can be inferred from the inverse problem. The validity of the presented method is supported by time-resolved optical observations of the bubble dynamics visualized in a glass connection channel. The combined optical and acoustical recordings are compared to the model, where we find very good agreement.






Sandra van der Graaf, Nina Dziomkina

“Hydrophobic coatings for immersion lithography”


In immersion lithography a water puddle scans over a wafer with a certain scan speed. To enable high scan speeds of the water puddle without water loss, surfaces of the scanner in contact with immersion water should desirebly be hydrophobic. Hydrophobic coatings are used to obtain these hydrophobic surfaces. Such coatings are generally organic coatings with functional groups with low surface energies.

An important requirement for hydrophobic coatings for the use in immersion scanners is that the coating has to be resistant against exposure with Deep Ultraviolet in water and keep its hydrophobic properties for several years. Only a few types of hydrophobic coatings meet these requirements. Results of most promising hydrophobic coatings will be presented.






Kees Venner

“Thin layer flow in Rolling Element Bearings”


At present almost all discussion related to energy is focused on sustainable energy. However, equally important is to reduce energy losses due to friction in the infinitely many situations in our machines and devices where forces are transferred between surfaces in relative motion, i.e. in gears, can-followers, bearings, tyre road contacts, etc. A common and very successful device to reduce friction is a rolling element bearing where friction is reduced due to the rolling motion. At the same time wear is avoided by adding a lubricant which should provide a thin layer separating the roller raceway surfaces in the so-called Elasto-Hydrodynamically Lubricated (EHL) contacts. Bearing service life is determined by two aspects. The first is the fatigue life related to the high stresses and many overrollings. The second aspect is referred to as "grease-life", i.e. the time the lubricant supplied can provide adequate surface separation. The lubricant film in the contact is directly related to the thin layers of oil on the surfaces in the inlet to the contacts, and these supply layers are in turn influenced by many effects such as surface wetting, centrifugal forces, cage effects, and simply the high pressure pushing the lubricant to the side, i.e. the overrolling itself. The aim of research is to develop accurate models to predict supply layer variation and lubricant film thickness. A model will be presented based on thin layer flow assumptions. Model predictions for the effect of centrifugal forces on thin layers on a bearing raceway have been validated experimentally. Also results are presented of accurate prediction of film thickness decay in single EHL contacts in the heavily starved regime which have been validated using optical interferometry.






Olga  I. Vinogradova, Alexei V. Belyaev

“Effective slip in pressure-driven flow past super-hydrophobic stripes”


Super-hydrophobic array of grooves containing trapped gas (stripes), have the potential to greatly reduce drag and enhance mixing phenomena in microfluidic devices. Recent work has focused on idealized cases of stick-perfect slip stripes, with limited  guidance. Here, we analyze the experimentally relevant situation of a pressure-driven flow past striped slip-stick surfaces with arbitrary local slip at the gas sectors. We derive analytical formulas for maximal (longitudinal) and minimal (transverse) directional effective slip lengths that can be used for any surface slip fraction. By representing eigenvalues of the  slip length-tensor, they allow us to obtain the effective slip for any orientation of stripes with respect to the mean flow. Our results imply that flow past stripes is controlled by the ratio of the local slip length to texture size. In case of a large (compared to the texture period) slip at the gas areas, surface anisotropy leads to a tensorial effective slip, by attaining the values predicted earlier for a perfect local slip. Both effective slip lengths and anisotropy of the flow decrease when local slip becomes of the order of texture period. In the case of small slip, we predict simple surface-averaged, isotropic flows (independent of orientation). These results provide a framework for the rational design of super-hydrophobic surfaces and devices.

Faculties of Physics and Chemistry, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia

A. N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119991 Moscow, Russia





Stephen K. Wilson, J. M. Sullivan,  and B. R. Duffy

“A thin rivulet or ridge subject to a prescribed transverse shear stress at its free surface”


We use the lubrication approximation to analyze three closely related problems involving a thin rivulet or ridge of fluid subject to a prescribed uniform transverse shear stress at its free surface, namely a rivulet draining under gravity down a vertical substrate, a rivulet driven by a longitudinal shear stress at its free surface, and a ridge on a horizontal substrate, and find qualitatively similar behavior for all three problems. We show that the free surface profile of an equilibrium rivulet/ridge with pinned contact lines is skewed as the shear stress is increased from zero, and that there is a maximum value of the shear stress beyond which no solution with prescribed semi-width is possible. In practice, one or both of the contact lines will de-pin before the maximum value of the shear stress is reached, and so we consider a rivulet/ridge that de-pins at one or both contact lines. In particular, we find that there is a critical value of the shear stress at which de-pinning first occurs. In the case of de-pinning only at the advancing contact line the rivulet/ridge is flattened and widened as the shear stress is increased from its critical value, and there is a second maximum value of the shear stress beyond which no solution with a prescribed advancing contact angle is possible. In contrast, in the case of de-pinning only at the receding contact line the rivulet/ridge is thickened and narrowed as the shear stress is increased from its critical value and a solution with a prescribed receding contact angle is possible for all values of the shear stress. In general, in the case of de-pinning at both contact lines there is a critical "yield" value of the shear stress beyond which no equilibrium solution is possible and the rivulet/ridge will evolve unsteadily. We also investigate two aspects of the stability of the equilibrium rivulet/ridge solutions, namely the quasi-static stability of a rivulet/ridge to small two-dimensional perturbations, and the linear stability of a rivulet/ridge with pinned contact lines to small three-dimensional perturbations, and find that in both cases the rivulet/ridge is unconditionally stable to perturbations of the kind considered.

Department of Mathematics, University of Strathclyde,

26 Richmond Street, Glasgow G1 1XH, UK






Koen Winkels

"Cornered shapes in contact line dynamics."


When a liquid is deposited onto a solid surface, its dynamics is entirely governed by the contact line separating 'wet' and 'dry' regions. The stability of the contact line is crucial for many practical applications and touches upon a number of fundamental questions. In this talk we present the contact line shapes occurring in a geometry representing an Immersion Lithography system, where the air between the lens and the wafer is replaced by a liquid of a higher refractive index. Above a critical capillary number, the receding contact line is not able to move with the substrate velocity resulting either in a rivulet-like structure or the emission of small (unwanted) drops on the substrate

[1]. Before reaching this limit, the contact line is able to resist the instabilities and breakup by changing its shape. It turns out that the hydrodynamics in immersion lithography is closely related to that of droplets sliding down a window, and can be described by a lubrication model that includes surface tension and viscous stress [2,3].

[1] M. Riepen, F. Evangelista, and S. Donders, Proceedings of 1st

European Conference on Microfluidics, Bologna (2008).

[2] J. H. Snoeijer, E. Rio, N. Le Grand-Piteira and L. Limat, Phys.

Fluids 17, 072101 (2005)

[3] I. Peters, J. H. Snoeijer, A. Daerr, and L. Limat,Phys. Rev.

Lett.103, 114501 (2009)