On the Interaction of Capillary Shapes with Solid Surfaces

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On the Interaction

of Capillary Shapes

with Solid Surfaces

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On the Interaction of Capillary Shapes

with Solid Surfaces

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College van Promoties,

in het openbaar te verdedigen op woensdag 4 februari 2015 om 10:00 door

Michiel MUSTERD

scheikundig technoloog geboren te Utrecht

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Prof. dr. ir. M.T. Kreutzer

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. dr. ir. C.R. Kleijn Technische Universiteit Delft, promotor Prof. dr. ir. M.T. Kreutzer Technische Universiteit Delft, promotor

Prof. dr. D. Quéré ESPCI ParisTech

Prof. dr. F. Mugele Universiteit Twente

Prof. dr. ir. J.H. Snoeijer Technische Universiteit Eindhoven Prof. dr. W.R. Rossen Technische Universiteit Delft Dr. ir. V. van Steijn Technische Universiteit Delft

Prof. dr. ir. J.J. Derksen Technische Universiteit Delft, reservelid

Dr. ir. Volkert van Steijn heeft in belangrijke mate bijgedragen aan de totstandkoming van dit proefschrift.

This work was carried out within the framework of the HESTRE (Heterogeneous Catalysis in Structured Reactors) project of ISPT.

Printed in the Netherlands

ISBN 978-94-6186-420-8

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the author.

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Real knowledge is to know the extent of one’s ignorance Confucius

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Summary

On the Interaction of Capillary Shapes with Solid Surfaces

Control over the interaction of droplets with solid surfaces is commonplace in nature. Famous examples are the water-shedding capabilities of the lotus leaf and the water-harvesting skin of certain types of beetles. To date, this type of control remains a challenge in engineering ap-plications. Consider, for example, droplets on the windscreen of a car that need to be wiped off because they unwantedly stick to the glass, or the dripping of droplets from fog harvesting devices, where one actually wants the droplets to stick.

In this thesis, we address the interaction of droplets with solid surfaces that underlies the sticking behavior. In particular, we investigate how droplets stick to chemically and physically heteroge-neous surfaces, and what shape droplets take when squeezed by various confining geometries. The results of these investigations can be used to predict the force needed to set a droplet on a given surface into motion, and to calculate the volume of droplets in confining geometries. An important tool in our studies is the principle of energy minimization, by which we determine the (local) equilibrium shape of a droplet under the influence of surface tension, gravity and solid boundaries. It turns out that the interaction of droplets with solid surfaces can largely be captured in terms of two aspects of the system. The first is the contact line of the droplet, the line where the fluid-fluid interface meets the solid surface. The second is the contact angle distribution around this line, where the contact angle is defined as the angle at which the fluid-fluid interface meets the solid surface.

A core question in our work is how to predict the shape and sticking force of a droplet on a real-istic, thus physically or chemically heterogeneous, inclined surface. The answer to this question constitutes an important step towards a model for the sticking force. We resolve this question by studying the model system of a droplet on an incline, using 2D analytical and 3D numerical en-ergy minimization. An important conclusion from our work is that the maximum sticking force of the droplet is determined by local constraints on repositioning of the contact line, originating from a range of possible contact angles within which the droplet can deform while the contact line remains static. The possible, i.e. accessible, shapes of the droplet during the inclination process are then also determined by constraints on repositioning, which causes the initial shape and deformation history of the droplet to influence the shape of the droplet at roll off and thereby

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the maximum sticking force. Accounting for inaccessible shapes and droplet shape history hence results in accurate predictions of the sticking force, in contrast to currently available models that allow all shapes and are thus bound to yield incorrect predictions.

Based on the insight that local constraints control the global shape of the droplet we developed an approach to analytically predict the deformation of 3D droplets upon inclination and the max-imum sticking force of these droplets. This prediction can be used for a number of engineering applications where the sticking force of droplets is an important factor. Examples include the design and positioning of solar panels and windscreens, as well as more industrial applications such as condensors and chemical reactors with small flow channels.

The analytical model for 3D droplets builds on the findings on the role of local constraints by explicitly modelling the deformation process of the contact line, in contrast to the current state-of-the-art where contact line shapes are either assumed or predicted without regard for the con-straints. To keep the model analytically tractable, we parameterize the deformation of the droplet in terms of the contact line shape and the contact angle distribution around the contact line. In combination with a few physically sound assumptions, based on our earlier findings, this model accurately predicts the deformation and maximum sticking force of a droplet, as compared with experiments and numerical simulations. Unlike currently available models, this model also ac-counts for the history of the droplet in the prediction of the maximum sticking force.

Apart from droplets sticking to surfaces we also use energy minimization to determine the shape of bubbles and droplets in straight microchannels of various cross-sectional geometries. With this, we develop a model to calculate the volume of a droplet on the basis of only the channel geometry and the droplet length as observed from a top view. This determination of volume is for example important to accurately determine the concentration of gaseous species or nutrients in studies of mass transfer rates or growth of cells. The model is based on a physical description of the central part of the droplet, where we calculate the shape of a cross-sectional slice of the droplet based on energy minimization. The volume calculation is completed with two caps on the droplet that smoothly fit onto the central part and mimick the physically realistic shape. We compare our analytical model with 3D numerical energy minimization calculations and find excellent agreement for a large range of droplet lengths and channel geometries, thus validating the model for use in quantitative research.

The findings of this research can, with small adaptations, be applied in related systems, such as pinned gas bubbles or droplets in a shear flow. We discuss the necessary adaptations and what open questions remain in the field of droplet pinning, specifically related to cases where assumptions made in our work do not apply. This results in a series of recommendations for future research on the role of viscous and inertial effects and heterogeneities with sizes on the order of the droplet.

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Samenvatting

Over de Interactie van Capillaire Vormen met Vaste Oppervlakken

In de natuur vinden we vele voorbeelden van fascinerende controle over de interactie tussen druppels en vaste oppervlakken. Bekende voorbeelden zijn het waterafstotende oppervlakte van de lotusplant en het druppelvangende schild van bepaalde kevers. In technische toepassingen is het echter nog altijd een uitdaging om een dergelijke mate van controle te bewerkstelligen. Neem bijvoorbeeld autoruiten, die nog altijd ruitenwissers nodig hebben omdat druppels aan het glas blijven plakken, of apparaten om mist te vangen als drinkwater, waarbij vallende druppels juist ongewenst zijn.

In dit proefschrift beschrijven we de interactie van druppels met vaste oppervlakken die ten grondslag ligt aan het blijven plakken van druppels. We onderzoeken specifiek hoe druppels aan chemisch of fysisch heterogene oppervlakken blijven plakken en wat voor vorm druppels aannemen wanneer ze opgesloten worden door omliggende vaste oppervlakken. De resultaten van ons onderzoek kunnen worden gebruikt om te voorspellen hoeveel kracht er nodig is om een druppel op een gegeven oppervlak in beweging te brengen, en om het volume van druppels in gegeven microkanaalgeometrieën te berekenen.

Een belangrijk instrument in ons onderzoek is het principe van energieminimalisatie. Hiermee bepalen we de (lokale) evenwichtsvorm van druppels op basis van hun oppervlaktespanning, de zwaartekracht en de aanwezigheid van vaste oppervlakken. De interacties in het druppel-oppervlak systeem kunnen voor een groot deel worden beschreven op basis van twee aspecten. Het eerste is de contactlijn van de druppel, de lijn waar vloeistof, gas en vaste stof elkaar raken. Het tweede is de contacthoekverdeling langs deze lijn, waarbij de contacthoek gedefinieerd is als de hoek die het vloeistof-gas oppervlak maakt met het vaste oppervlak.

Een kernvraag in ons werk is hoe we de vorm en kleefkracht van een druppel op een reëel, dus fysisch of chemisch heterogeen, oppervlak kunnen voorspellen. Het antwoord op deze vraag is een belangrijke stap richting een model van de kleefkracht. We beantwoorden deze vraag door het modelsysteem van een druppel op een hellend vlak te bestuderen. Hiervoor maken we gebruik van 2D analytische en 3D numerieke energieminimalisatie. Een belangrijke conclusie van ons werk is dat de maximale kleefkracht van een druppel bepaald wordt door lokale beperkingen in de beweging van de contactlijn. Deze beperkingen komen voort uit een bereik aan contacthoeken

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waarbinnen de druppel kan vervormen terwijl de contactlijn op zijn plek blijft. De mogelijke vormen van de druppel tijdens het kantelproces worden daarom mede bepaald door deze lokale beperkingen, met als gevolg dat de vervormingsgeschiedenis en oorspronkelijke vorm van de druppel van belang zijn voor de maximale kleefkracht van de druppel. Door mee te nemen dat sommige vormen op basis van de geschiedenis van de druppel niet mogelijk zijn, kunnen we nauwkeurige voorspellingen maken van de kleefkracht. Dit in tegenstelling tot modellen die momenteel gebruikt worden, waarin alle druppelvormen toegestaan zijn.

Op basis van ons inzicht dat lokale beperkingen aan de contactlijn de globale vorm van de drup-pel bepalen, hebben we een methode ontwikkeld om analytisch de vervorming en maximale kleefkracht van 3D druppels tijdens kantelen te voorspellen. De voorspelling kan dienst doen in toepassingen waarbij plakkende druppels van belang zijn, zoals het ontwerp en de positionering van zonnepanelen en autoruiten, en het ontwerp van condensors en microreactoren.

Het analytische model voor 3D druppels bouwt voort op onze bevindingen over lokale beper-kingen door expliciet de vervorming van de contactlijn mee te nemen, in tegenstelling tot hui-dige modellen waarin een bepaalde vorm van de contactlijn wordt aangenomen, dan wel wordt aangenomen dat de contactlijn zonder beperkingen kan vervormen. Om het model analytisch oplosbaar te houden, beschrijven we de vervorming van de druppel op basis van de vorm van de contactlijn en de contacthoekverdeling rondom deze lijn. We sluiten het model met fysisch onderbouwde aannames, en valideren het door te vergelijken met experimenten en simulaties. In tegenstelling tot huidige modellen, geeft dit model een correcte beschrijving van het effect van druppelgeschiedenis op de kleefkracht.

Behalve voor het bestuderen van plakkende druppels hebben we energieminimalisatie ook ge-bruikt om de vorm van druppels in microkanalen met verschillend gevormde doorsnedes te be-schrijven. We hebben een model ontwikkeld om het volume van druppels te berekenen op basis van alleen de vorm van het kanaal en de lengte van de druppel gezien van boven. Volumebe-rekeningen zijn bijvoorbeeld van belang voor het nauwkeurig bepalen van de concentratie van voedingsstoffen bij het bestuderen van de groei van cellen. Het model is gebaseerd op een fysisch correcte beschrijving van het middenstuk van de druppel, waarbij we de vorm van een doorsnede van de druppel bepalen op basis van energieminimalisatie. De berekening van het volume maken we vervolgens sluitend door twee kapjes naadloos op het middenstuk van de druppel te laten aansluiten, waarbij we de vorm van de kapjes zo kiezen dat we de realistische vorm van de drup-pel nabootsen. We vergelijken het analytische model met 3D numerieke energieminimalisatie en vinden dat de voorspellingen uitstekend overeenkomen voor een groot bereik aan druppellengtes en kanaalvormen, wat betekent dat het model gebruikt kan worden voor kwantitatief onderzoek. De bevindingen in ons onderzoek kunnen, met wat kleine aanpassingen, toegepast worden op ver-gelijkbare systemen, zoals plakkende gasbellen of druppels onder een schuifspanning. We gaan in op deze aanpassingen en behandelen openstaande vragen in het veld van plakkende druppels. We kijken specifiek naar situaties waar de aannames, gebruikt in dit onderzoek, niet van toepas-sing zijn. Hieruit volgen aanbevelingen voor toekomstig onderzoek aan de rol van viscositeit en traagheid en de rol van heterogeniteiten van vergelijkbare afmetingen als de druppel.

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1. Introduction

1.1 Sketch of the Research Field

1.1.1 Capillary Surfaces

The interface between two immiscible fluids is called a capillary surface. The intermolecular interactions between the two fluids result in an energy per unit surface area, called the surface tension γ. In the absence of other energy contributions, the shape of a capillary surface follows from the minimization of surface energy, which for a uniform surface tension is equivalent to the minimization of surface area. In the presence of fluid flow or body forces such as gravitational, magnetic, or electric forces, the surface energy is not the only contribution to the total energy. The capillary surface then assumes a shape corresponding to a minimum in the total energy of the system. This simple picture explains why the tiny bubbles in champagne are spherical, while rain drops are not.

In this thesis we consider capillary surfaces that are governed by surface tension, gravity and interactions with solid surfaces. We now briefly review the most important theory on this topic.

1.1.2 Droplets on Ideal Solid Surfaces

Droplets sitting on a surface, called sessile droplets, are common examples of capillary surfaces where the shape is determined by surface tension and gravity1. Often occuring in nature, these shapes have been studied for over two centuries. Already in 1805, Young studied the cohesion of fluids and among others addressed the shape of sessile droplets. He found that there is a balance between the three surface tensions associated with the liquid-gas, gas-solid and solid-liquid interfaces,γ, γgsand γsl. This balance results in the specific shape of the droplet near the

solid surface2, where the liquid-air interface meets the solid surface at a fixed angle, the so-called Young contact angle θY, according to γ cos θY = γgs− γsl (Fig. 1.1a). Young described

this relation on the basis of a local argument at the contact line of the droplet, the line where the

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γ γsl γgs (a) (b) θ F_g θ ∆θ (c) θf θb _θ a θr θf θb θa θr θf θf θa θa

Figure 1.1 (a) Schematic of a sessile drop on a (seemingly) smooth surface. (b) Zoom of the contact point

where the droplet touches the microscopic roughness of the surface and a range [θ, θ + ∆θ ] of possible

contact angles exists for which the droplet remains at rest due to the meta-stable states at the kink in the surface roughness. (c) Deformation and motion of the droplet during progressive inclination (top) and the local changes at the contact point (bottom)

gas-liquid interface meets the solid.

Gauss3argued that the same shape can also be derived on the basis of a global energy minimiz-ation argument at the scale of the droplet. For this purpose, Gauss used the principle of virtual work, a technique that is closely related to variational calculus. Interestingly, both approaches result in the same relation between the contact angle and the surface tensions. However, for non-ideal surfaces, which we will address shortly, the local and global approach yield different results. To date, this is the cause of a debate on the approach that should be followed in the study of sessile droplets4–6. For a discussion of the implications and subtleties of the Young equation, the reader is referred to the review by Bonn et al.7, the book by De Gennes et al.8and a recent educational paper by Marchand et al.9.

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Introduction 3

1.1.3 Droplets on Non-Ideal Solid Surfaces

In the investigations by Young and Gauss it is implicitly assumed that the solid surface is per-fectly homogeneous. As it turns out, nearly all real solid surfaces have heterogeneities, making them non-ideal. This can be due to microscopic roughness of the surface (see Fig. 1.1b), or due to chemical variations, or a combination of the two10_{. Heterogeneities have two consequences for}

the droplet-solid interactions: (i) the average contact angle is now a function of the nature of the heterogeneities (size, number density, chemistry, etc) and (ii) at the transition of two ‘patches’ of the heterogeneity the contact angle is not uniquely defined.

The change in average contact angle can be understood as follows. For chemical heterogeneities the contact angle is position dependent due to the spatially varying chemical interaction, thus surface energy. If the heterogeneities are sufficiently small as compared to the droplet size an average contact angle weighted by the surface fraction of each chemical component of the solid can be calculated11. For surface roughness the ‘kinks’ in the surface macroscopically lead to the same effect: spatially varying contact angles. A similar analysis as for the chemical heterogen-eities then results in a multiplication factor for the cosine of the contact angle based on the degree of roughness10,12. A much investigated field is how to tune the surface heterogeneities to create surfaces that exhibit superhydrophobicity. For a thorough review on these tuning approaches, see Bhushan and Jung13_.

The non-unique nature of the contact angle at the transition of different patches only becomes apparent when applying a force to the droplet. Instead of resulting in motion of the droplet, as expected on an ideal surface, the fluid-fluid interface of the droplet deforms within the range of possible contact angles while retaining the same contact line position (Fig. 1.1b). The range of contact angles depends on the exact nature of the heterogeneities, but can macroscopically be captured in two contact angles, called the advancing and receding contact angle, θaand θr.

The advancing contact angle is the angle beyond which the contact line of the droplet advances whereas the receding contact angle is the angle below which the contact line recedes. If the contact angle remains within the range [θr, θa] the contact line remains stuck. The difference

between θaand θris called the contact angle hysteresis. On an ideal surface the advancing and

receding angle are equal to each other and to the Young angle. On any non-ideal surface the advancing angle exceeds the receding angle.

A difference in the contact angle at the front and back of the droplet, and in fact all around its contact line, gives rise to a net force, called the pinning force, that can balance gravity, shear or other external forces. It explains for example why droplets can stick to an inclined surface (Fig. 1.1c). If a force is applied to a stationary droplet that pushes the contact angle locally outside the range of possible contact angles, the contact line of the droplet repositions. For a small applied force the motion of the contact line in this repositioning is slow, so it is safe to assume it to be quasi-static, thus being at a constant contact angle, θa or θr. For contact

lines moving at larger velocities, for example for sliding droplets, droplets during deposition, or vibrated droplets, the contact angle changes due to viscous and inertial forces14.

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Although the droplets with contact angles within the range [θr, θa] are all possible, all except one

shape are meta-stable15. This is a consequence of the surface heterogeneity that results in an energy landscape consisting of several energy barriers that prevent the droplet from relaxing to the global energy minimum found in the research by Gauss. While understood on a conceptual level, the effect of these energy barriers on the global shape of a droplet subject to an external force remains unquantified.

The general aim of this thesis is to quantitatively understand the sticking of droplets to, in par-ticular, non-ideal solid surfaces. In our study of pinning we consider droplets on inclined plates as a model system, but the insights gained in our work can also be applied to droplets under the influence of other forces than gravity, or in other geometries. We now present the open questions that we wish to resolve in this thesis.

1.2 Open Questions

1.2.1 Local Pinning of the Contact Line

The effect of contact angle hysteresis on the sticking of droplets is well known qualitatively. The variation of the contact angle around a droplet results in an unbalanced force that can oppose, for example, gravity. The balance of gravity and surface tension can be non-dimensionalized in the Bond number, Bo = (∆ρgL3)/(γL) = ∆ρgL2_{/γ, where ∆ρ is the density difference of the fluids,}

gthe gravitational acceleration and L a typical length scale of the system. A simple 2D force balance for a droplet on a substrate inclined at the critical angle αc, the angle at which a droplet

rolls off, then gives Bo sin αc∼ (cos θr− cos θa). This balance provides an order of magnitude

estimate for the angle at which a droplet of given volume will roll off from a substrate of known contact angle hysteresis. However, an order of magnitude can be the difference between a 5◦ and a 90◦ tilt angle, rendering the order of magnitude estimate effectively useless. The O(1) multiplication factor that translates the estimate into an exact calculation is known to originate from the shape of the contact line, which in turn depends on the history of the droplet. This motivated us to address the question: “how does local pinning of the contact line translate into hysteresis of the shape of the droplet?”. In answering this question we used energy minimiza-tion and could simultaneously answer the quesminimiza-tion: “Can global energy minimizaminimiza-tion provide meaningful predictions for droplets on hysteretic surfaces?”.

1.2.2 Deformation and Roll-Off of Droplets

With the answers to the questions posed in the previous section, we unraveled the mechanism by which a droplet deforms from its initial shape to the shape it has at the moment it rolls off from a progressively inclined plate. Using this knowledge we could resolve the question “how much force is needed to set a droplet on a hysteretic surface into motion?” by building a model

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Introduction 5

to predict the deformation of a sessile droplet that is tilted from horizontal to the roll-off angle, as a function of the applied gravitational force (tilt angle). With this model we could also address whether a core assumption in the seminal work by Dussan V. and Chow16 on the shape of the contact line is valid, viz. the implicitly made assumption that the critical contact line shape is the shape for vanishing velocity. This leads us to the question “Is the contact line shape of a pinned droplet on the verge of motion the same as that of a droplet slowed down to rest?”

1.2.3 Droplet Shapes in Microchannels

In the preceding sections we have dealt with the fundamentals of droplet-solid interactions by considering the problem of a droplet on an incline. Another example of droplet-solid interaction can be found in droplet microfluidics inside microchannels. Here, the droplets do not interact with a single solid surface, but rather are confined by multiple solid surfaces, the channel walls. The droplet cannot freely deform due to the solid bounds, but, in the absence of flow and body forces, the interface will still meet all solid surfaces at a fixed contact angle arising from the different surface tensions. Looking at the middle of the droplet, away from the caps, it depends on the wetting properties of the walls whether a slice of the droplet completely fills the cross-section of the channel or forms a composite shape of parts that conform to the channel walls and parts that do not17_{. In the full wetting case, which is often used in droplet microfluidic research,}

the contact angle becomes 0 and the droplet is surrounded by thin films separating it from the channel walls. As a result, the only interaction with the walls is due to confinement, not due to chemical interactions. For static droplets in sufficiently small channels (Bo<<1) this process is then dominated by the surface energy between the fluids.

Many researchers use droplet microfluidics as a tool for the study of mass transfer rates18, cell growth19or nanoparticle synthesis20. To obtain quantitative results the volume of droplets should be known, but standard microscopic techniques only provide a two-dimensional top- or bottom-view image of the droplet. This triggered us to answer the question “Given the top-bottom-view image of a droplet in a known channel geometry, what is the volume of the droplet?”. We address this question again with the help of energy minimization to develop a physically sound model for the droplet volume as a function of the channel geometry and the droplet length, that can easily be observed in a topview image.

1.3 Relevance of our Studies

1.3.1 Relevance in Nature

Many living organisms make use of surface heterogeneities to trap or repel water droplets (Fig 1.2). The study of natural systems can not only teach us how these amazing organisms sur-vive, but also help us in developing technological applications based on techniques that nature has

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already perfected. A famous subject of study is the lotus plant, the namesake of the lotus effect. This effect, originating from the combination of surface roughness and hydrophobicity, results in a virtually hysteresis free surface that has self-cleaning properties. It has triggered researchers to study these lotus leafs22and engineer surfaces with similar23or even better properties such as electrically switchable hydrophobicity24or repellency of other liquids than only water25. Al-ternatively, the surface roughness can also be specifically designed to capture droplets, rather than to repel them. Namib desert beetles for example use a combination of wetting and non-wetting patches on their skin to collect dew and pin the water droplets on their back until they drink it26_{(Fig. 1.2(a)). Some spiders also use wetting properties to collect water by specifically}

structuring the silk of their webs27_{. Pitcher plants even have a switchable surface adhesion that}

enables them to catch ants21_{(Fig. 1.2(c)). These natural systems could prove very useful in}

im-proving for example the water harvesting capabilities of solar stills28. However, to apply their principles, we need to understand how contact angle hysteresis and contact line pinning influence their operation.

1.3.2 Relevance in Technology

In consumer products, several applications greatly benefit from understanding the interaction of droplets with solid surfaces. For solar panels, a rain shower will leave an amount of droplets on the solar panel that reflect light away from the panel and leave salts after they evaporate, thus lowering effiency29 (Fig. 1.3(b)). Knowledge of the way to make the droplets roll-off could therefore lead to improved efficiencies. For car windows the same applies. Potentially enabling windscreens that need no wipers, because no droplet sticks to it30. Also in industrial environ-ments, e.g. when condensors are used that collect water on their outside surfaces, it is desirable to remove droplets that reduce the heat transfer rate31 (Fig. 1.3(a)). In displays based on elec-trowetting, where remaining droplets in display pixels can negatively influence viewing quality,

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Dry Wet

Figure 1.2 (a) The namib desert beetle has alternating hydrophilic and hydrophobic patches to catch and direct water to his mouth (Reprinted from http://illustrationstoencourage.files.wordpress.com). (b) The structure of leafs can guide water droplets into specific structures. (c) Pitcher plants can switch their surface adhesion by drying or wetting the structure between the hairs on their leafs and catch ants in this way

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Introduction 7

(a) (b) (c)

Figure 1.3 (a) Droplets condense onto the cooling surface of a condensor and thereby slow the rate of

further condensation if they do not fall off (Image from Miljkovic et al.31_{). (b) Solar panels with low}

surface energy exhibit self-cleaning properties that enhance effiency by keeping the panel free of dirt (Image from c-voltaics.com). (c) Only a limited amount of the packed bed particles gets wetted (by red dye) due to

wetting problems and pinning (Image from Baussaron et al.34).

it is beneficial to control the droplet-solid interactions32_{. For a more extensive overview of}

po-tential applications, see e.g. the section in Berthier and Brakke33_.

1.3.3 Relevance in HESTRE

Another industrial application in which partial wetting can play a significant role is in multiphase catalytic reactors. In particular reactors where the fluid velocities are low or the flow channels are small, such as packed beds and microreactors, are bound to be influenced by the wetting properties of the system. In many droplet microfluidic applications, the surface chemistry of the channel walls is adjusted to ensure full wetting. In a catalytic microreactor or packed bed reactor this is not possible, because the chemistry dictates the nature of the channel walls. This inevitably leads to partial wetting35_{, thus resulting in interaction of the droplet interface with the}

heterogeneous catalyst coated wall. This interaction can have a variety of undesired effects, such as erratic flow in microchannels36, or channeling in packed beds34(Fig. 1.3(c)). Insight in the forces associated with the interface-wall interactions can provide a means to control these flows, even in their partial wetting state.

This is the approach taken in the ISPT project: Heterogeneous Catalysis in Structured Reactors (HESTRE), of which the research in this thesis is a part. The overall goal of HESTRE is to design a structured (micro)reactor with a catalytically active coating on the walls. This thesis focuses on the droplet-wall interaction for quasi-static systems as a fundamentally important step in pre-dicting the flow dynamics of droplet flows in catalyst coated microchannels. The knowledge gained from our detailed studies on the local interaction of the droplet interface with the wall can then be translated into boundary conditions for dynamic flow simulations by Computational Fluid Dynamics (CFD).

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1.4 Outline

The open questions mentioned earlier in this introduction are answered in chapters 3, 4, and 5, respectively. These are self-contained chapters that were published in, or submitted to, peer-reviewed journals. Prior to these chapters, in chapter 2, we present details on our experimental and numerical methods, which are only briefly addressed in the other chapters. We finish the thesis with an epilogue meant to discuss the broader applicability of our findings and future opportunities in this field of research.

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2. Numerical

and

Experimental

Methods

§

2.1 Introduction

In this thesis we use both experiments and numerical simulations to study sessile droplets on inclined surfaces. In this chapter we give a detailed description of both the tilted plate experi-ments1, and the numerical simulations based on energy minimization. For these simulations we use the finite element code SURFACE EVOLVER2,3. Briefly, in SURFACE EVOLVERwe trian-gulate a droplet surface and evolve the droplet shape towards the shape that minimizes the total energy of the system, typically consisting of surface energy and gravitational energy. A more extensive description of our SURFACEEVOLVERapproach is provided in section 2.3.

2.2 Experimental

Experimental data is instrumental in the development and validation of new theoretical models. Resolving small differences in the force that pins droplets to surfaces requires high accuracy experimental data. This section describes how we acquired high accuracy data by reducing com-mon measurements errors.

2.2.1 Measuring Droplet Adhesion

The adhesion of a fluid to a substrate is typically measured by one of two techniques: the Wil-helmy plate technique or the tilting plate technique.

§_{Experimental parts of this chapter were published as: M. Musterd, V. van Steijn, C. R. Kleijn and M. T. Kreutzer.}

Improved Reproducibility of Droplet Pinning Measurements on Microscopically Heterogeneous Surfaces. In: Third European Conference on Microfluidics (muFlu-2012), paper no: 181, Heidelberg, Germany, 3-5 December 2012.

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In the Wilhelmy plate technique the substrate is dipped slowly into a bath of the fluid and then withdrawn4. Measuring the difference in force needed for dipping and withdrawing provides a measure for the adhesion of the fluid to the substrate. A minor disadvantage of this technique is that a relatively large amount of fluid is needed. A more important drawback, from the perspect-ive of droplet adhesion, is the need and difficulty to deduce an advancing and receding contact angle from the measured force. These contact angles are required to account for the shape of the contact line of the droplet in the pinning force calculation.

Conversely, the tilting plate technique provides a direct measurement of the advancing and reced-ing contact angles. A droplet is simply deposited on a horizontal surface and subsequently tilted until gravity overcomes the pinning force and the droplet starts moving with the advancing angle at the front and the receding angle at the back1,5,6. By directly using a droplet on a substrate, effects that specifically arise from the nature of the droplet-substrate interface, such as the shape of the contact line, are also captured. For this reason we opt for the tilting plate technique in studying droplet adhesion.

There are several potential sources of inaccuracies in the tilting plate method that are associ-ated with the preparation of the surface1,7and the deposition of a droplet on the surface8. These inaccuracies compromise the reproducibility of the experiment and cause droplet-to-droplet vari-ations in the measured roll-off angle, and hence, in the pinning force. A common solution is to measure the pinning force several times and use the average1,6, which is potentially biased due to systematic errors. An alternative solution is to minimize the droplet-to-droplet variations. Before addressing how we eliminated uncontrolled droplet-to-droplet variations, we first introduce the measurement principle and setup.

2.2.2 The Tilting Plate Method

Measurement Principle

The measurement principle of the tilting plate is based on a balance of the pinning force, Fp, and

gravity, Fg. At the critical angle αc, at which a droplet of volume V is on the verge of rolling off,

we have

Fp= Fg= ρgV sin αc (2.1)

with ρ the density of the droplet and g the gravitational constant.

The pinning force Fpitself is not only a property of the surface and the liquid, but also of the

initial droplet shape as measured by its base width w after deposition on a horizontal surface. In order to determine the necessary parameters to study the pinning force, we capture the 2D sideview images of the droplet from two directions, one along the rotation axis and one ortho-gonal to that in the plane of the substrate. To study the pinning of droplets of different sizes and different initial shapes, we then extract from these images the base width w, the position of the front and back contact points of the droplet xf and xb, and the contact angles at these positions,

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Methods 13 α = 0ο _α_{= 8}o α = 16o θa 1 mm θr α = αc Substrate Camera Light Light Rotation motor (a) (b) Droplet g g g g x_f x_f 2 Camera1 Camera 1 Axis of rotation α = 0ο Camera 2 w

Figure 2.1 (a) The tilting plate setup with dual cameras and light sources. The pinning force is measured by tilting a droplet of known volume until roll-off. (b) Recorded snapshots from the 2 cameras. Top: camera 2 image to measure the base width w. Bottom: camera 1 images showing the deformation of the droplet and

motion of the (in this case) front contact point (xf) prior to roll-off (compare the position of xf at α = 0◦

and α = 16◦).

θf and θb. At the critical angle we expect θf = θaand θb= θr, but it should be noted that this

can also already occur for tilt angles prior to the critical angle (see Ch. 3). The pinning force can then be written as9,10

Fp= kwγ (cos θr− cos θa) (2.2)

with γ the interfacial tension and k an O(1) parameter that depends on the details of the contact line shape (see Ch. 4).

Before we elaborate on the way to obtain the desired quantities from our recorded tilt sequence, we first discuss the tilt setup.

The Tilting Setup

The setup consists of a custom made tilting plate attached to a microstepper motor (Thorlabs NR360S) with two cameras mounted onto the plate such that they co-rotate with the plate around the same axis (Fig. 2.1). The cameras (The Imaging Source, DMK 21BF04) are equipped with

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a f = 25 mm objective separated from the camera by a 10 mm tube. This allows a field-of-view of approximately 8 × 6 mm2 at a working distance of 10 cm. After placing a substrate on the levelled plate, we manually deposit a droplet of deionized water using a micropipette (Eppendorf, 10 − 100 µL). The plate is rotated in steps of 0.5◦at a rate of 0.25◦/s with a 60 s waiting period after each tilt step (see section 2.2.3 for the reasoning behind this approach). Experiments are performed at room temperature with T = 21.5 ± 1.5◦C. These variations affect the surface tension by less than 0.5 mN/m i.e. < 1%11,12_{. The relative humidity is kept at}

100% by the use of a climate chamber that fits over the tilting plate to prevent evaporation (see section 2.2.3).

Image Processing

We record the tilting process from horizontal to roll-off at a rate of 15 frames per second resulting in images as shown in Fig. 2.1(b). The backlight illumination is adjusted to allow shutter speeds without motion blur and a sharp black-white contrast at the edge of the droplet. Separate frames are processed with an automated Matlab script that follows the steps described below.

1. The image is binarized on the basis of a visually determined black-white threshold for the entire set of image

2. The transition of black to white, the edge, is detected with a built-in edge detection al-gorithm

(a) (b) x_fit=0.08 x_fit=0.02 x_fit=0.25 x_fit=0.08

no endpoint removal

θ=139.9o θ=135.0o θ=135.7o θ=138.3o

Figure 2.2 Test droplet generated with SURFACEEVOLVER. The contact angle is θ = 140◦. (a) A circular

fit of part of the contour fits accurately (orange line), whereas extending that circle to the full contour length clearly illustrates that fitting a full circle will result in inaccuracies (dashed gray line) (b) Zooms of the

contact angle fit for the selected fit fraction, xfit= 0.08, and for longer and shorter fit fractions to illustrate

the misfit and the effect on the calculated contact angle. The last image illustrates a misfit caused by not removing the last point of the contour. The pixels used in fitting are marked with black circles, whereas the fitted curve is orange with the contact point marked with an orange square.

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Methods 15

3. The location of the tilting plate surface, the baseline, is determined.

For the detection of the baseline we make use of the reflection of the droplet in the sub-strate. For a range of assumed baseline positions we calculate the sum of squared residuals between the horizontal position of the 15 pixels above the baseline and the horizontal po-sition of the 15 pixels below the baseline. The correct baseline popo-sition is identified as the one resulting in the lowest sum of squared residuals, indicating that this baseline position best coincides with the mirror plane of the substrate. We repeat this procedure for every frame of a tilt movie and then average the baseline positions to obtain a single baseline height for the entire tilt sequence.

4. The contour of the droplet is determined as the largest, connected part of the edge that is above the baseline

To this end, we remove all elements from the array for the edge that are located below the baseline, as well as any spurious edge elements (e.g. due to shadows) above the baseline that are not connected to the droplet contour.

5. The contact points xf and xbare determined as the two meeting points of the contour and

the baseline

6. The contact angles θf and θbare determined by fitting a circle to 8% of the contour

start-ing at xf and xbrespectively and calculating the contact angles as the inverse tangent of

the slope at the contact points

We use part of a circle as the fit function for part of the contour, because this is exact for a droplet in the absence of gravity and thus correct to leading order if gravity is present. Using only a part of the contour, instead of the entire contour then results in an accurate fit as shown in Fig. 2.2(a). The choice for the part of the contour that should be fitted was made by considering artificially generated test images of droplets with known contact angles and droplet volumes. We extracted these images from SURFACE EVOLVER simu-lations and artificially blurred the droplet edge by 2 pixels to mimick experimental images which are observed to have approximately the same edge blur. We then fitted the contact angle with different lengths of the contour and found that a fit fraction of 8% of the entire contour provides an accurate fit (see Fig. 2.2(b)). We additionally found that it is necessary to remove the point of the contour at the height of the baseline, to avoid errors caused by the nearby reflection (see Fig. 2.2(b) on the right).

Using error propagation we can determine that the relative error in tan θ equals the relative error in the slope of the fitted circle, y(x),

∆(tan θ ) tan θ =

∆(dy/dx)

dy/dx (2.3)

We use a bootstrapping method called ‘residual resampling’ to estimate the relative error in dy/dx. Briefly, in this method the residuals arising from the difference between the fitted func-tion and the data to be fitted are calculated and each datapoint is randomly assigned one of the

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100 110 120 130 140 150 110 120 130 θ0 θfinal V = 25 µL V = 50 µL V = 75 µL Droplet deposition Droplet after deposition

θ0

Droplet after vibration Vibrating droplet

1 mm

θfinal

(a) (b)

Figure 2.3 (a) Procedure to obtain a reproducible initial droplet shape through purposeful vibration. (b)

Effect of vibration in reducing the spread in contact angle from, θ0= 124 ± 22◦to θfinal= 119 ± 6◦. The

effect of vibration is independent of the droplet volume V . Errorbars indicate the difference between left and right contact angles.

residuals. The new dataset is then again fitted to result in a new estimate of the slope. When repeated many times this method results in a normal distribution for dy/dx of which the standard deviation is a good estimate for the error. This procedure resulted in a typical relative error of 2% in the slope. For the typically used contact angles between 70◦and 150◦this gives an estimate of the absolute error of at most ±0.5◦_.

2.2.3 Reducing Measurement Error

Vibration

Mechanical vibration, either purposeful or accidental, can depin the contact line of a droplet by providing sufficient energy to overcome local energy barriers that pin the contact line13–15. During a tilt sequence this effect is undesired, because vibrations can result in roll-off at lower-than-critical tilt angles. Therefore, we mount the tilt table on an optical table (Melles Griot, Product No. 070WL007) to minimize vibrations from the surroundings.

Vibration can also be used to our advantage, by providing a reproducible initial state for a droplet on a hysteretic surface. For this purpose, we use a micro vibration motor (Precision Microdrives Pico Vibe 307-100) attached to the substrate to create 3 − 6 g vibration accelerations to induce capillary-gravity waves that depin the contact line14. Depinning occurs by exciting one of the eigen frequencies of the droplet (typically 30-200 Hz for O(10µL) droplets). We find this ei-gen frequency by scanning through the frequency range of the vibration motor and monitor the response of the droplet (Fig. 2.3(a)). Upon vibration at the eigen frequency, droplets deposited with various initial contact angles θ0 indeed obtain contact angles, θfinal, with little spread as

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Methods 17 0 15 30 45 60 75 90 −2.0 −1.5 −1.0 −0.5 0.0 t (min) ∆ ycm (%) With enclosure Without enclosure

Figure 2.4 Relative shift of the height of the center of mass of a 50 µL water droplet due to evaporation for the case without enclosure (grey triangles) and with enclosure kept at 100% relative humidity (black circles).

shown in Fig. 2.3(b). The vibration thus effectively removes deposition history. Therefore, when studying the impact of initial state on the critical angle (Ch. 3), we do not apply the vibration.

Evaporation

Evaporation of sessile droplets is fast due to the large surface-to-volume ratio. On a hysteretic surface the evaporation mode is initially the constant contact radius (CCR) mode, in which the contact angle changes, while the contact radius remains constant16. The rate of evaporation depends not only on the size of the droplet, but also on its contact angle and the temperature and relative humidity of the surrounding air. Under typical lab conditions (T = 20◦, RH = 60%) complete evaporation of a 50 µL droplet takes a few hours.

To investigate the effect of evaporation we tested how much the height of the center of mass, yCM, of a 50 µL droplet shifts by evaporation under these conditions. Our findings show that a

substantial decrease in the center of mass height occurs within 15 minutes (Fig. 2.4). Evaporation is almost completely absent when using an enclosure with wet tissues to obtain a 100% relative humidity environment. A typical tilt experiment with relaxation periods takes about 30 minutes, thus making the use of the climate chamber a necessity for accurate measurements.

Relaxation

During a tilt sequence, after each tilt step there is a possibility that the contact angle at the front grows bigger than the advancing angle, or that the contact angle at the back shrinks smaller than the receding angle. This leads to local motion of the contact line, which does not necessarily result in global motion of the droplet (i.e. roll-off). Taking the front of the droplet as an example, the force per unit length on the contact line is γ(cos θa− cos θ ). For small deviations from θathis

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0 20 40 60 80 100 120 0 0.05 0.10 0.15 0.20 0.25 0.30 t (s) x_b,f/V1/3 t (s) 450 500 550 600 x_b/V1/3 0.06 0.05 0.04 0.03 0.02 0.07 α=18oα=21o Tilt Wait Tilt Wait Tilt Wait Tilt (a) (b)

Figure 2.5 (a) Motion of front and back contact points, xf (circles) and xb (squares), of two identical

droplets. The plate is tilted at 0.25◦_{/s and rotation is stopped at 18}◦_{and 21}◦_{for the two different droplets}

illustrating the slow, but persistent motion of the contact line. Both droplets eventually roll off. (b) Sample of a tilt-wait sequence that shows how the contact line slowly moves to its new stable position during the waiting stage.

force can be nearly zero, leading to a slow relaxation of the contact line to its new meta-stable position through activated jumps17_.

We investigated the relaxation behaviour by depositing two identical droplets and tilting them at a fixed rate of 0.25◦_{/s to different stopping angles (18}◦_{and 21}◦_{). Monitoring the position of the}

front and back contact points, xf and xb, in time we find that stopping the rotation at 18◦still

leads to roll-off, but at a slower rate than for the droplet tilted to 21◦(Fig. 2.5(a)). This shows that tilting continuously at a fixed rate of 0.25◦/s results in identification of the wrong angle as the roll-off angle.

To accurately identify the roll-off angle, we used a tilt-wait sequence in which the inclined plate is tilted by 0.5◦and then kept at that angle for a specified waiting time. By trial-and-error we established that 60 s of waiting time was sufficient for our experiments as can be seen from a typical tilt-wait sequence in Fig. 2.5(b).

2.2.4 Substrate Preparation - Hydrophobization

Tilting plate measurements can provide a measure of the pinning force on any macroscopically flat surface. For ease of post-processing, in particular automated image analysis, it is conveni-ent to work with hydrophobic surfaces such that the water droplets used in the measuremconveni-ent obtain a clearly visible, thus accurately measurable, contact angle. A common

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hydrophobiza-Methods 19

Native silicon wafer

Si Si Si O OH OH Si OH OH C2H4C6F12 Si Si Si O O Si Si Si OH OH OH Si Si Si O Si Cl Cl OH OH C2H4C6F12 Si Si Si O Si Cl Cl OH OH SiCl 3C 8 F12H 4 C₂H₄C₆F₁₂ SiCl 3C_8F 12H 4

Activated wafer Coated wafer with excess Coated wafer without excess Durable, non-reactive, hydrophobic wafer Step 1 Step 2 Step 3 Step 4 Oxygen plasma CVD Excess removal by evaporation at 220 o_C Passivation with H₂O

Figure 2.6 Fabrication of a durable, non-reactive hydrophobic coating using a recipe with four main steps: 1) activate the substrate, 2) deposit the fluoroalkylsilane, 3) remove excess PFOTCS, and 4) passivate re-active side groups.

tion technique in the field of microfluidics is to coat the substrate with big fluorinated carbon-molecules (C6+F12+)18. In the experiments in chapters 3 and 4 of this thesis we make use of

silicon wafers coated with 1H,1H,2H,2H-perfluorooctyl-trichlorosilane (PFOTCS) as a hydro-phobization agent.

We use chemical vapor deposition (CVD) to prepare the hydrophobized silicon wafers. Prior to this process, we chemically activate the surfaces such that the PFOTCS covalently bonds to the silicon. This is done by exposing the surfaces to oxygen radicals for 2.5 min in a plasma cleaner(P = 35 W). After this pre-treatment step (step 1 in Fig. 2.6) we expose the silicon wafers to the vapor of PFOTCS for 1.5 h by placing them in a vacuum chamber (P ≈ 20 mbar) together with a 2 mL vial of PFOTCS (step 2). We found that it is essential to post-treat the surfaces in two subsequent steps to remove excess PFOTCS (step 3) and passivate unreacted chlorine groups (step 4). Without these steps, the coating reacts with water from the surrounding air or with water from the droplets during the tilt experiments. Excess PFOTCS (Tboil= 192◦C) is

removed by heating the wafer to a temperature of 220◦C for 1.5 h18_{. The wafer is then cooled}

to room temperature and submerged in a deionized-water bath for 1 h (step 4) to passivate the coating by replacing unreacted Cl-groups with OH-groups.

To demonstrate that the coatings made with this recipe are durable and non-reactive, we coated several pieces of bare silicon and used them in tilting plate experiments. We measured the

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di-V = 30 μL V = 50 μL 1 2 3 4 1 2 1 2 0 0.1 0.2 0.3 0.4 0.5

Fresh Aged 2 days

First use Second use

Aged 30 days

Third use (Bo sin α)c

Figure 2.7 The dimensionless roll-off angle measured on silicon wafers coated with PFOTCS according to our recipe. Errorbars indicate the minimum-maximum deviation over three measurements.

mensionless roll-off angle of both 30 and 50 µL water droplets (ρ = 998 kg/m3_{, γ = 72 mN/m).}

This angle should be the same for all samples, irrespective of prior use or age. We confirm this by comparing the dimensionless roll-off angle for two samples that were used immediately after preparation (1, 2 in Fig. 2.7) and for two samples kept in air for two days prior to use (3, 4 in Fig. 2.7). In addition, we compare samples that were used for the first, second, and third time, where samples were kept in air for respectively two and 30 days after their first use. All samples show good agreement in the dimensionless roll-off angle as shown in Fig. 2.7, from which we conclude that the coating is both durable and non-reactive.

2.3 Numerical: S

URFACE

E

VOLVER

2.3.1 Energy Minimization

A freely suspended capillary shape of given volume will become spherical due to energy minim-ization, because a sphere has the minimum surface-to-volume ratio. If the shape sits on a solid substrate of given wetting properties, the spherical shape cannot form due to the solid substrate constraining the capillary shape on one side. Moreover, the wetting properties of the substrate now also contribute to the energy of the droplet-substrate system. Furthermore, in the presence of gravity, the gravitational energy will result in a more flattened droplet with lower gravitational energy, but at the cost of more surface area.

Considering the different energy terms mentioned above and the fact that a capillary shape sitting on a surface can freely deform at the top side, is is clear that the minimization of energy through analytical calculations quickly becomes untractable. The principle of energy minimization,

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how-Methods 21

Initialization

Figure 2.8 Deformation and refinement of an initial ‘droplet’ shape to its minimal energy state.

ever, is straightforward locally, and therefore perfectly suited for numerical optimization. SURFACE EVOLVER is a finite element code that does exactly this: it minimizes the energy of the total system under given constraints on the shape, such as dictated by the presence of a substrate or a channel wall, and of course conservation of mass (volume)2. A given initial shape is triangulated (see Fig. 2.8) and vertices are displaced in the direction of the energy minimum by either a gradient descent or conjugate gradient method, while accounting for the constraints. In this way SURFACE EVOLVER deforms the capillary surface until a (local) energy minimum is found. During the entire calculation the triangulated surface should be continuously refined and re-triangulated to retain a good mesh quality (see Section 2.3.3). In this thesis we have used SURFACE EVOLVERversion 2.5, but new versions of SURFACE EVOLVERare expected to be fully backwards compatible. Therefore, we ran a few tests with the currently available version 2.73, and we indeed found the same results as with version 2.5.

In minimizing the energy of a system, SURFACE EVOLVER can only be used for calculations where a steady-state equilibrium exists, including steady-states in a moving reference frame. Another option is to work under the assumption of quasi-static motion, such that the deformation of a capillary surface is a sequence of static (local) equilibrium shapes. This results in a series of static minimization calculations for slightly changed conditions. In the case of a droplet on a tilting plate, the condition that changes is the direction of the gravity vector, which is changed in a stepwise manner to calculate droplet shapes at subsequent tilt angles. For hysteretic systems, as the ones studied in this thesis, the history of the droplet matters such that the entire sequence of tilt angles from horizontal to the desired tilt angle (typically roll-off) should be simulated. For non-hysteretic systems, the order of the quasi-static calculations does not matter such that the desired situation can directly be calculated.

2.3.2 Pinning Constraint

SURFACE EVOLVER offers a framework in which volume constraints and constraints on the position of the capillary surface can be implemented. For the pinning of the contact line the

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constraint on its position and possible movement depends on the value of the local contact angle, and this type of constraint is not natively supported in SURFACE EVOLVER. Therefore, we have implemented the algorithm developed by Santos et al.19 to include pinning effects (see appendix 2.A for our implementation).

The algorithm mimicks the experimentally observed behaviour that a contact line only advances or recedes when the contact angle is at the advancing or receding angle respectively. To this end, first a standard iteration of SURFACE EVOLVER, not accounting for constraints due to pinning, is performed to obtain a prediction of the next shape and energy state. Then, using this predic-tion, the following steps are taken for each vertex point of the contact line (steps indicated in appendix 2.A)

1. The direction of motion is determined on the basis of the current force on the vertex, F, (i.e. the energy gradient, surprisingly calledv_velocity in SURFACEEVOLVER). Motion that is outward from the droplet center is classified as advancing, inward motion is classified as receding.

2. The magnitude of F per unit length is determined, |F|/l, on the basis of the average length of the two adjacent edges on the contact line.

3. The contact angle, θ , at the vertex is determined from the relation |F|/l = εγ(cos θE−

cos θ ), where ε is +1 for an advancing point and −1 for a receding point

4. If θ is inside the range [θr, θa] then the vertex is constrained and the predicted position is

not adopted, otherwise the vertex is unconstrained.

In case of an unconstrained vertex it turns out that the vertex will then move until the contact angle is (just) inside the range of limiting angles again. For more details on the algorithm the interested reader is referred to Santos et al.19.

2.3.3 Mesh Quality

Mesh quality in SURFACE EVOLVER, as in any numerical code, is important to guarantee the accuracy of the calculations. For triangulated surfaces such as those used in SURFACEEVOLVER

it is hard to give an exact definition of mesh quality, but generally two aspects of the mesh are a good measure for the quality: (i) the ratio of smallest to largest edge length within each triangle, Ti, and (ii) the ratio of the areas of adjacent triangles in the mesh Ai/Ajwhere i and j represent the

current and adjacent triangle. From the perspective of mesh quality an ideal mesh has both ratios close to 1 throughout the mesh. In SURFACE EVOLVER there are two ‘grooming’ options that will adjust the current mesh towards this ideal. Equiangulation, as the word already suggests, results in a flip of the joining edge of two thin triangles to create two triangles with more equally sized edges thus improving Ti (see Fig. 2.9(a)). Vertex averaging is performed by computing

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Methods 23 1 2 1 2 Equiangulate (a) (b) T₁=0.45 T2=0.28 T₁=0.62 T2=0.56 Vertex average

Figure 2.9 Illustration of techniques to improve mesh quality. (a) equiangulation, resulting in a higher ∑ Ti,

(b) vertex averaging that results in an improvement in Ai/Aj.

adjacent triangles, thus effectively averaging triangle areas locally, resulting in an improvement in Ai/Aj(see Fig. 2.9(b)).

Initialization and iteration procedure

In our SURFACEEVOLVERcalculations we use a combination of the above mentioned grooming methods, together with a continuous removal of edges below a threshold length and refinement of edges above a threshold length to keep the mesh quality high. We first initialize the droplet shape on a horizontal surface with a sequence of refinements, iterations, vertex averaging and equiangulation to obtain a starting shape with edges of maximum dimensionless edge length l/V1/3= 0.1. Then we refine the edges of the first four rows of vertices above the contact line one step further to a dimensionless length of 0.05. The initialization is finished with a set of 500 iterations to find the energy minimium.

Then, the inclination angle, α, is stepwise increased by ∆α. At each α, the normal iteration sequence is to perform 10 sets of operations, each consisting of 10 iterations and 1 ‘refinement’ step in which two checks are performed: (1) if any edges have grown larger than a dimensionless length of 0.1 (or 0.05 in the refined region) they are refined. (2) if any edges have become smaller than a dimensionless length of 0.02 (or 0.01 in the refined region), they are removed. The 100 iterations and 10 refinements are then followed by 2 vertex averaging steps if the adaptive maximum vertex displacement per iteration drops below 0.001, a clear indication that there are too large differences in triangle areas (because the maximum displacement adapts to the smallest triangle size). For each tilt angle, this iteration sequence is repeated as many times as needed to obtain a shape for which the maximum change in contact angle and dimensionless contact line position between two iteration sequences is less than 10−4.

On the Interaction of Capillary Shapes with Solid Surfaces

On the Interaction

of Capillary Shapes

with Solid Surfaces

On the Interaction of Capillary Shapes

with Solid Surfaces

Michiel MUSTERD

Contents

Summary

Samenvatting

1. Introduction

1.1

Sketch of the Research Field

1.1.1

Capillary Surfaces

1.1.2

Droplets on Ideal Solid Surfaces

1.1.3

Droplets on Non-Ideal Solid Surfaces

1.2

Open Questions

1.2.1

Local Pinning of the Contact Line

1.2.2

Deformation and Roll-Off of Droplets

1.2.3

Droplet Shapes in Microchannels

1.3

Relevance of our Studies

1.3.1

Relevance in Nature

1.3.2

Relevance in Technology

1.3.3

Relevance in HESTRE

1.4

Outline

Bibliography

2. Numerical

and

Experimental

Methods

§

2.1

Introduction

2.2

Experimental

2.2.1

Measuring Droplet Adhesion

2.2.2

The Tilting Plate Method

2.2.3

Reducing Measurement Error

2.2.4

Substrate Preparation - Hydrophobization

2.3

Numerical: S

URFACE

E

VOLVER

2.3.1

Energy Minimization

2.3.2

Pinning Constraint

2.3.3

Mesh Quality