Landau, Verwey Overbeek Derjaguin

Edl

Keywords:

Chemical potential of the i-th ionic species ν Coecient of kinematic viscosity ω Angular frequency ω abs Magnitude of vorticity Φ

Résumé

Les écoulements multiphasiques dans les systèmes microuidiques ont récemment trouvé de nombreuses applications dans des domaines tels que le traitement biochimique (par exemple les réacteurs dans des puces, les mélangeurs, l'extraction de l'ADN, et administration de médicaments), ou l'extraction de pétrole à partir de formations rocheuses poreuses. Une des caractéristiques de ces écoulements microuidiques est sa grande surface par rapport au volume, ce qui met en évidence, de façon signicative, le rôle des interfaces multiphasiques dans cette dynamique.

Les interfaces entre les phases peuvent être dénies comme des surfaces continues, perpendiculairement auxquelles il y a une transition nette des propriétés intensives d'une phase à l'autre. Ces interfaces peuvent être classées en grande partie par la nature des phases en interaction tels que le gaz-liquide, le liquide-liquide ou le liquide-solide. La présence de ces interfaces apporte la possibilité de transport de la phase passive par cisaillement à l'interface. Ces mécanismes de transport sont actuellement explorés en détail pour les problèmes de transport aux micro-nano échelles. L'un des principaux dés dans les phénomènes de transport à la microéchelle est d'avoir un actionnement able du débit. Le mécanisme d'actionnement d'écoulement le plus couramment utilisé dans des dispositifs à micro-échelle est de créer un gradient de pression en utilisant une pompe. Ces dispositifs sont volumineux avec une utilisation des pièces en mouvement pour créer un écoulement, et nécessitent donc un entretien fréquent. Au cours de la dernière décennie, l'utilisation de l'électrocinétique grâce à un champ électrique appliqué aux bornes du micro-canal est un mécanisme de génération d'écoulement de plus en plus populaire. Ce type d'écoulement est réalisé grâce aux interactions entre une couche de paroi adhésive chargée électriquement (également connu sous le nom de double couche électrique) et d'un champ électrique appliqué aux bornes du micro-canal, écoulement qualié d'électroosmotique (EO). Ce type de mécanisme a un avantage durable par rapport aux pompes puisqu'ils ne nécessitent pas de composants mobiles. Les écoulements électroosmotiques dans des micro-canaux ayant un potentiel électrique axialement invariant ont typiquement un prol de vitesse quasi-uniforme (écoulement dit de bouchon ), ce qui réduit la possibilité de dispersion de l'espèce transportée et par conséquent, peuvent se révéler ecaces pour le transport d'espèces biologiques dans ces micro-canaux. On considère en général un champ électrique constant, mais il entraîne des réactions électrochimiques aux électrodes (phénomène d'électrolyse non souhaitée pour certaines expériences puisque cela entraîne des uctuations au niveau du débit d'écoulement et peut parfois arrêter l'écoulement tout en en isolant complètement l'électrode). Un autre problème associé aux réactions électrochimv iques au niveau des électrodes est la variation de pH dans les réservoirs où sont posées les électrodes. Un tel phénomène crée un gradient de pH dans le canal affectant l'électrochimie du système et peut éventuellement aecter la mobilité des ions dans le liquide. Des études antérieures ont montré que l'utilisation d'un champ électrique oscillatoire contribue à diminuer les réactions chimiques au niveau des électrodes avec un contrôle spatio-temporel eectif sur l'écoulement de liquide et du mélange. Un tel utilitaire d'écoulement électroosmotique oscillatoire a conduit à de nombreuses études récentes dont, entre autres, l'actionnement de l'écoulement, le stockage de l'énergie, etc. . . Bien que l'écoulement électroosmotique a été prouvé être le mécanisme le plus pratique pour le transport des liquides dans des microcanaux, un des inconvénients pour son utilisation dans les applications habituelles vient du fait que ce liquide transporté doit être électriquement conducteur. Pour le transport d'un liquide non-conducteur on le fait généralement à l'aide d'un liquide immiscible conducteur par cisaillement à l'interface uide-uide. L'étude des systèmes électroosmotiques à deux phases, tels que les systèmes air-eau a mis en évidence le rôle des charges à l'interface sur le prol de vitesse (vitesses diérentes à l'interface et dans le c÷ur du uide), une fonctionnalité qui est inattendu dans un EO classique. Un tel prol de vitesse très dispersif peut aecter le transport des espèces par EO. Les études mentionnées ont été réalisées entre deux couches de uide tout en tenant compte des conditions classiques à l'interface telles que la continuité de la vitesse et de la contrainte de cisaillement à cette interface uide-uide. La présence d'une interface liquide-liquide dans un EO nécessite la caractérisation des paramètres responsables de l'instabilité de tels systèmes, dont il faudra trouver la solution de l'état de base du système non perturbé. La modélisation d'un tel système n'est pas triviale et comporte quelques subtilités attribuées à l'existence des tensions de Maxwell à l'interface qui changent les conditions aux limites de cette dernière.

Ceci est en contraste avec les écoulements dans des canaux entre deux parois rigides pour lesquelles on impose une condition à la limite de vitesse (généralement pas de glissement sur les parois) indépendamment de toute considération sur les tensions de Maxwell. En outre, dans le cas d'un écoulement symétrique entre deux parois rigides, les contraintes hydrodynamique et de Maxwell sont individuellement nuls à la ligne de symétrie. Cependant, pour un écoulement à surface libre, c'est la contrainte totale (hydrodynamique et Maxwell) qui doit être nulle à l'interface. Dans la plupart des micro-dispositifs, l'utilisation du champ électrique pour le transport de uides est limitée par la nature non-conductrice d'un grand nombre de uides (par exemple, les liquides biologiques, les liquides polymériques. . . ). En outre, le comportement d'un tel système lorsqu'il rencontre une discontinuité sous la forme d'une interface est encore sous-explorée. L'un des principaux dés dans la dynamique du lm est de comprendre et de modéliser le seuil de stabilité de l'interface sous l'inuence d'un large éventail de phénomènes physiques inhérents à ce système comme la tension supercielle, la viscosité, les forces de van der Waals et des phénomènes statiques ou dépendants du temps comme la gravité, les gradients de température et les champs électriques. Une analyse détaillée et globale est par conséquent nécessaire, ce qui constitue l'objet de cette thèse. vi Le travail présenté dans cette thèse se concentre sur le rôle des tensions de Maxwell constant et dépendant du temps, des forces capillaires et de la pression de disjonction sur la stabilité de l'interface. An d'étudier l'eet des tensions de Maxwell à l'interface dans des congurations classiques à plaques planes parallèles ainsi que pour des congurations réalistes de canaux rectangulaires, la solution d'un EO à surface libre a été obtenue analytiquement. En prenant en compte une expression complète de la condition limite à la surface libre qui inclut les tensions de Maxwell à cette interface, une analyse paramétrique de l'écoulement en termes de l'épaisseur de la double couche électrique, du rapport de forme du canal et des diérentes valeurs du potentiel zêta uide uide et solide liquide a été entreprise. Grâce au champ de vitesse 2D, il a été montré que les parois ont un eet signicatif sur la distribution des vitesses dans un micro-canal rectangulaire. Les paramètres de contrôle de l'écoulement dans un EO sont, non seulement les propriétés électrochimiques des électrolytes qui contrôlent l'épaisseur de la double couche électrique et les valeurs des potentiels zêta au mur et à l'interface, mais aussi la géométrie du canal à travers son rapport de forme. Une telle étude permet d'identier les paramètres appropriés pour imposer un débit d'écoulement et une distribution des vitesses souhaitée dans un dispositif microuidique. Une analyse de stabilité linéaire basée sur une perturbation à l'interface a été réalisée pour un lm mince d'électrolyte sous des champs électriques continus (constants) et alternatifs (dépendant du temps). Une analyse asymptotique avec une hypothèse de grande longueur d'onde des équations d'Orr-Sommerfeld a été appliquée an de déterminer les seuils de stabilité paramétriques d'un lm mince aqueux. L'accent a été mis sur les eets de la tension de surface, de la pression de disjonction pour l'interaction gaz-liquide-substrat, de l'amplitude et de la fréquence du champ électrique appliqué, ainsi que du potentiel zêta du substrat et de la surface libre. En outre, un dispositif expérimental a été conçu et monté an de caractériser l'écoulement électroosmotique dans un micro-canal rectangulaire. Avec l'aide d'une analyse PTV ( Particle Tracking Velocimetry ), les distributions de vitesse ont été obtenues et comparées aux prédictions théoriques.

Cette comparaison a permis d'estimer le potentiel zêta du PDMS utilisé, valeur conforme à la valeur indiquée dans la littérature. En outre, an d'étudier l'instabilité de l'interface liquide-liquide sous champ électrique périodique en temps, deux cas ont été considérés : le premier cas correspondait à deux uides miscibles et le second à deux uides immiscibles.

Une analyse comparative des prols de vitesse de l'état de base avec et sans contraintes de Maxwell à l'interface, a montré que les gradients de vitesse étaient importants à l'interface liquide-liquide avec les contraintes de Maxwell. De tels gradients sont essentiels à l'instabilité interfaciale sous l'action d'un champ électrique périodique car ils peuvent atténuer ou amplier les ondes à l'interface. L'inuence du c÷ur du uide à partir des oscillations proches paroi est inversement proportionnelle à la fréquence d'excitation. A de basses fréquences, la dispersion du champ de vitesse s'aaiblit et tend vers le champ de vitesse classique dans un écoulement électroosmotique continu. Par la variation de la polarité à l'interface on a constaté que, non vii seulement on peut contrôler la vitesse à l'interface ainsi que le taux de cisaillement, mais également établir une diérence de phase signicative de la vitesse du uide en diérents emplacements transversaux dans le uide. Ceci a pour conséquences d'augmenter les eets de dispersion du champ de vitesse et peut être eectivement utilisé dans le transport contrôlé des espèces dans des dispositifs microuidiques.

(a) Les paramètres avec un eet stabilisateur sur la dynamique du lm sont la tension de surface, la pression de disjonction répulsive (A < 0), la pression osmotique due à la double couche électrique aux interfaces et la dissipation visqueuse. (b) Les phénomènes qui contribuent à l'instabilité du lm sont la pression de disjonction attractive (A > 0), des doubles couches électriques plus minces (nombre de Debye De < < 1), du champ électrique externe conduisant à l'écoulement électroosmotique et des basses fréquences. La stabilité du lm décroît lorsqu'on augmente l'amplitude du champ électrique appliqué et qui est dû à une augmentation des tensions de Maxwell à l'interface. (c) Lorsque la valeur du potentiel zêta du substrat augmente l'écoulement tend à se stabiliser et la composante osmotique de la pression augmente. (d) Dans le cas des uides miscibles, l'objectif était d'étudier la déformation de l'interface liquide-liquide soumise à un EO périodique dans le temps et en présence des transports convectif et diusif. Pour obtenir une interface plane (non perturbée) avec une diusion limitée de ces liquides miscibles, un débit a été imposé au niveau des deux liquides de sorte que ce ux de convection peut atténuer le mélange des deux liquides par diusion interfaciale. Le débit imposé était pris identique dans les deux liquides. Il a été observé que pour une valeur donnée du champ électrique appliqué, l'amplitude des déformations à l'interface diminuait avec l'augmentation du débit imposé, ce qui détermine l'eet stabilisant de la vitesse d'écoulement imposée. En outre, il a également été observé que la réactivité de l'interface (c'est à dire l'amplitude de l'interface perturbée) à la fréquence du champ électrique appliqué, était la plus sensible dans la gamme de 1-5Hz. (e) Dans le cas des uides non miscibles, l'objectif était d'étudier la déformation de l'interface liquide-liquide sous un EO périodique dans le temps et en présence de la force capillaire et du transport convectif. Pour obtenir une interface plane (non-perturbée) contre la force capillaire à l'interface, un débit a été imposé au niveau des liquides pour contrer les instabilités de ces forces de tension de surface à l'origine de la génération des bulles. On a observé qu'il existait une valeur critique du débit d'écoulement au-dessus de laquelle le système reste stable pour la gamme des valeurs du champ électrique appliqué. Cela suggère l'existence d'une compétition dynamique entre l'inertie convective et les contraintes de Maxwell. Pour une valeur donnée de débit d'écoulement, la stabilité du système diminue lors de l'augmentation de l'amplitude du champ électrique. Enn, il a été observé que les hautes fréquences du champ électrique tendent à stabiliser l'écoulement à un débit donné.

Mots-clés: écoulement électroosmotique, microuidique, analyse de stabilité linéaire, analyse en grande longueur d'onde, expériences en uides miscibles/immiscibles. viii journey for me both on professional and personal levels. This journey wouldn't have begun without my acceptance into the PhD project, which was a brainchild of Prof. Sakir Amiroudine and Prof. Suman Chakraborty. I express my sincerest gratitude to Prof. Amiroudine and Dr. Didier Lasseux for believing in my abilities to carry out such a detailed project which contained both theoretical and experimental investigations and was completely new and challenging to all of us. I came to France with two big bags, the rst one full of Indian spices, and the second one full of apprehensions regarding my survival in a completely dierent culture, lifestyle and language. Now, after a time span of three years, when I look back in time, I can not imagine how many amazing people I have met during my journey and how each one of them made my stay equally memorable. I was specially overwhelmed by the warmth and convivial nature of each and every individual that I met in the laboratoire I2M-TREFLE. It was because of the tireless eorts of Muriel, Sylviane, Fred, Marian, Audrey and Alain at ENSAM, and Valerie and Noemie at ENSCBP that I cruised through all the toilsome and non-trivial administrative procedures.

As much as I wanted to learn about the French culture and language, the lesson started right from the very rst day when I met Baptiste as my roommate. It was he who acted as my interpreter and translator during all the necessary administrative visits in the rst few months. Then, I was proudly welcomed in the Francky family (Francky, Sandra, and Flavio) to whom I would like to dedicate all the weekends that I can not to remember anymore. I just have realized that during my entire stay at Bordeaux, I was really lucky to share moments and experiences with so many wonderful people that recounting each and every such moment would surpasse the size of my PhD thesis. Specially, I can not forget all the lunches and motivating discussions together with the Postdocs and PhD students of the TREFLE family, namely, Erik, Vlad, Mario, Nisrine, Essam, Carolina, Andrea, Christophe, Cecile, Elvire, Marie, Alex, Fouzia, Stephanie, Antonio, Mehrez, Tan, Xabi, and Jose.

Moreover, it will be really dicult to forget the Friday evening drinks and dinners at FUFU with Alex, Clementine, Louiza, Antoine, Florent, and Nastaran. I would also like to express my sincerest gratitude to Prof. Annie Colin, Julien, David, Virginie, Pierre, Hughes, Laetitia and the rest of the entire sta of the Laboratory of Future for helping me with the fabrication of my microchips and nding time out of their busy schedules to discuss my experimental issues and making sure that I always go home with a solution. ix My thesis could not have achieved its current state of closure without the support of Julien Monteil from CBMN who accommodated my entire experimental setup in his laboratory and allowed me to use the phase contrast microscope of the lab.

At the very last but not the least, I like to thank all the members of Tir à l'arc de Bégles for welcoming me to their club and helping me learn archery. I would specially like to thank Cédric, Gilles, Yasmina, Laure and Chloé for giving me condence and encouragement time to time, and for all the nice moments that we spent together. Multi-phase ows in micro-scale systems have recently found numerous applications in a wide range of elds like bio-chemical processing such as, lab-on-a-chip reactors [START_REF] Deshmukh | Novel micromixers driven by ow instabilities: Application to post-reactors[END_REF]], mixers [START_REF] Campbell | Microuidic mixers: from microfabricated to self-assembling devices[END_REF]],

DNA extraction [START_REF] Ugaz | Microfabricated electrophoresis systems for DNA sequencing and genotyping applications: current technology and future directions[END_REF]], and drug delivery [START_REF] Squires | Making it stick: convection, reaction and diusion in surface-based biosensors[END_REF]], to oil extraction from porous rock formations [START_REF] Zhang | Wettability alteration and spontaneous imbibition in oil-wet carbonate formations[END_REF]. One of the signicant characteristics of micro-scale ows is high surface to volume ratio, which signicantly highlights the role of multi-phase interfaces in such dynamics. Interfaces between phases can be broadly dened as continuous surfaces, normal to which there is a sharp transition of the intensive properties of one phase to another. Such interfaces can be largely categorized by the nature of the interacting phases such as gas-liquid, liquid-liquid and liquid-solid interfaces. Although within the scope of the present study, the solid interfaces are considered to be rigid and non-compliant.

Within the scope of hydrodynamics, such conning solid surfaces provide boundary conditions on velocity slip and permeability (source or sink). Within the scope of electrodynamics, they either act as constant electric potential surfaces or polarizable dielectrics (see Chapter 2). Gas-liquid and liquid-liquid interfaces, within the purview of hydrodynamics are shear free and deformable interfaces. Electrodynamically, they are considered as charge storing interfaces, showing compliant dynamics under an applied electric eld.

One of the major challenges in microscale transport phenomena is to have a reliable ow actuation. The most commonly used ow actuation mechanism in micro devices is by creating a pressure gradient using a pumping device. Such devices are bulky, use moving parts to create ow, and need frequent maintenance. During the past decade, use of electrokinetics as a ow actuating mechanism is microdevices is becoming more popular. Flow actuation in microchannels due to an externally applied electric eld has found remarkable applications in lab-on-a-chip based microuidics devices and systems [START_REF] Stone | Engineering Flows in Small Devices[END_REF]; [START_REF] Sounart | Lubrication theory for electro-osmotic ow in a non-uniform electrolyte[END_REF]; [START_REF] Xu | Maximum eciency of the electro-osmotic pump[END_REF]; [START_REF] Squires | Microuidics: Fluid physics at the nanoliter scale[END_REF]; [START_REF] Bazant | Diuse-charge dynamics in electrochemical systems[END_REF]; [START_REF] Sheng | Electrorheological Fluids: Mechanisms, Dynamics, and Microuidics Applications[END_REF]; Lee and Li (2006)]. Such ows are eectively realised by interactions between a walladhering charged layer (also known as the electrical double layer) and an externally applied electrical eld, resulting in so-called Electro-Osmotic Flow (EOF) (see Chapter 2), which have an enduring advantage over classical uidic pumps in a sense that these do not require any moving components. Electro-osmotic ows in microchan-nels with axially-invariant interfacial potential typically have plug like (uniform) velocity prole which reduces the possibility of species dispersion and hence, may turn out to be eective for the transport of biological species in micro-devices. The most common form of EOF actuation is by a constant electrical eld, but it has some inherent problems due to electrochemical reactions at the electrodes. Such problems include formation of Hydrogen and Oxygen bubbles due to the hydrolysis of water, which leads to uctuations in the ow rate in microuidic devices and sometimes eventually stops the ow altogether by insulating the electrode [START_REF] Schaeper | Parameters aecting reproducibility in capillary electrophoresis[END_REF]].

Another problem associated with electrochemical reactions at the electrodes is the change of pH at electrode reservoirs [Persat et al. (2009b)]. Such a phenomenon creates a pH gradient in the channel aecting the electro-chemistry of the system and eventually aecting the mobility of ions in the liquid. Some previous studies have shown that the use of time periodic (AC) EOF is instrumental in diminishing the Faradaic reactions at the electrodes and to achieve eective spatio-temporal control over liquid ow and mixing [START_REF] Shin | Mixing enhancement by using electrokinetic instability under time-periodic electric eld[END_REF]; [START_REF] Chakraborty | Generalized model for time periodic electroosmotic ows with overlapping electrical double layers[END_REF]; [START_REF] Chakraborty | Mass ow-rate control through time periodic electroosmotic ows in circular microchannels[END_REF]; [START_REF] Dutta | Analytical Solution of Time Periodic Electroosmotic Flows: Analogies to Stokes' Second Problem[END_REF]]. Such a utility of AC EOF has led to recent increase in the studies exploring various novel applications of AC EOF including ow actuation [START_REF] Ramos | Pumping of liquids with traveling-wave electroosmosis[END_REF]; [START_REF] Green | Fluid ow induced by nonuniform ac electric elds in electrolytes on microelectrodes. I. Experimental measurements[END_REF]; [START_REF] Gonzalez | Fluid ow induced by nonuniform ac electric elds in electrolytes on microelectrodes. II. A linear double-layer analysis[END_REF]; [START_REF] Ramos | AC Electric-Field-Induced Fluid Flow in Microelectrodes[END_REF]; [START_REF] Bose | Enhancement of static incubation time in microuidic cell culture platforms exploiting extended air-liquid interface[END_REF]; [START_REF] Chakraborty | Role of streaming potential on pulsating mass ow rate control in combined electroosmotic and pressure-driven microuidic devices[END_REF]],

energy storage [START_REF] Takami | Laminated Thin Li-Ion Batteries Using a Liquid Electrolyte[END_REF]; [START_REF] Jang | Complex Capacitance Analysis on Leakage Current Appearing in Electric Double-layer Capacitor Carbon Electrode[END_REF]; [START_REF] Kötz | Principles and applications of electrochemical capacitors[END_REF]] etc.

Although EOF has been proven to be the preferred mechanism for liquid transport in microchannels, one of the biggest shortcomings of its usage in mainstream applications comes from its basic requirement of the concerned liquid to be electrically conductive. There have been some attempts towards the transport of non-conductive liquid with the help of an immiscible conductive liquid through shear transfer at the uid-uid interface [Lee and Li (2006); Gao et al. (2005b); [START_REF] Haiwang | Time-dependent model of mixed electroosmotic/pressure-driven three immiscible uids in a rectangular microchannel[END_REF]; Lee et al. (2006); [START_REF] Pascall | Electrokinetics at liquid/liquid interfaces[END_REF]]. The study of two-phase electro-osmotic systems such as air-water systems has highlighted the role of interfacial charges on the velocity prole [Gao et al. (2005a)] such as dierent interfacial and bulk velocities, a feature that is unexpected in a classical EOF. Such a highly dispersive velocity prole can aect the species transport using EOF [START_REF] Griths | Charged species transport, separation, and dispersion in nanoscale channels: autogenous electric eld-ow fractionation[END_REF]]. The mentioned studies have been performed in two-layer EOF while considering classical interface matching conditions such as continuity of the velocity and hydrodynamic shear stress at the uid-uid interface. The presence of a uid-uid interface in an EOF necessitates the characterization of the parameters responsible for instability of such systems, for which one has to ascertain the basic or unperturbed state solution of the system. This is characterized by several intricacies, the modeling of which is not trivial. Those intricacies are attributed to the existence of an interface on which appropriate considerations on Maxwell stress need to be invoked [START_REF] Choi | Is free surface free in micro-scale electrokinetic ows?[END_REF]; [START_REF] Mayur | Free-surface instability in electroosmotic ows of ultrathin liquid lms[END_REF]]. This is in sharp contrast with ows in between rigid boundaries for which one imposes a velocity boundary condition (typically no slip at the walls) irrespective of any consideration on Maxwell stress. Moreover, in case of a symmetric ow between two rigid boundaries, the hydrodynamic stress and the Maxwell stress individually become necessarily zero at the centerline (because of the centerline symmetry). However, for free surface ow, the total stress (hydrodynamic and Maxwell) at a at interface needs to be zero.

In most of the micro-devices, the use of electrical eld for uid transport is limited by the non-conductive nature of a large group of uids (for example, several biological liquids and polymeric liquids). Also, the behavior of such a system when it encounters a material discontinuity in the form of an interface is still under-explored.

One of the major challenges in the lm dynamics is to understand and model their interfacial evolution and stability thresholds under the inuence of a wide range of inherent phenomena like surface tension, viscosity, van der Waals forces and imposed static and time-dependent phenomena like gravity, temperature gradients and electric elds. Although, reviews by [START_REF] Oron | Long-scale evolution of thin liquid lms[END_REF] and [START_REF] Craster | Dynamics and stability of thin liquid lms[END_REF] provide comprehensive details on the contribution of various physical phenomena towards lm dynamics by providing a generalized evolution equation of the lm thickness, owing to a complex interaction of the mentioned phenomena, a detailed and all-inclusive analysis is required, which is not easy and still awaited.

The objective of this thesis is to explore the EOF actuated interfacial dynamics in a two uid system with the help of analytical models and experiemental investigations.

A short overview of the included chapters is presented below, Chapter 4 discusses the stability of a thin electrolytic lm under DC and AC electric elds using linear stability analysis for long wave disturbances while also focusing on the roles of capillary eects and disjoining pressure on lm stability.

Chapter 5 discusses the experimental setup leading to one-uid and two-uid EOF in microchannels while also focusing on the microchannel fabrication using soft lithography techniques and a short discussion on the apparatus used.

Chapter 6 discusses the experimental observations and results for a EOF validation step and interfacial instabilities in miscible and immiscible uids case. The EOF validation step involved estimation of the EOF parameters from a velocity eld which was obtained by the help of Particle Tracking Velocimetry technique.

Chapter 7 discusses the conclusion and the perpectives of this thesis.

Electro-Osmotic Flow and Interfacial

Electro-hydrodynamics ). This process is also called hydration when the solvent is water or solvation for any other polar solvent. Such ions under the inuence of an external electric eld are free to move within the solvent medium and can conduct electric current. Another very interesting aspect of such solvated mobile ions is that they can drag around the surrounding solvent molecules creating a bulk ow within the solvent. The catch, however, is that such solutions are electrically neutral, i.e. the concentrations of cations (positively charges ions) and anions (negatively charged ions) are the same. Hence, on assuming very small dierence in the transport properties of involved cations and anions, the net solvent motion stands canceled. On the other hand, it is observed that when such an ionic solution is brought into contact with a chemically active solid substrate or encounters a material discontinuity in the terms of a uid-uid interface, a highly localized charge separation occurs near that interface which can extend over a scale of tens of nanometers.

This structure has nite charge, which under the inuence of an external electric eld results into a near wall slip velocity, creating a highly non-dispersive plug ow in the bulk of the electrolyte. Such a ow of the bulk solution due to the charge Chapter 2

Electro-Osmotic Flow and Interfacial Electro-hydrodynamics separation near the interfaces is called electro-osmotic ow (EOF). Such a system is being used extensively in microuidic devices owing to its simple, low maintenance and non-dispersive ow actuation. This chapter details the exact mechanism of the mentioned phenomena with the help of thermodynamic and transport equations.

This chapter revisits the interfacial boundary conditions with regards to the classical laws of hydrodynamics and the classical laws of electrodynamics. The boundary conditions developed in this chapter are developed with a general purview, which are used either in their entirety or within various approximations in the following chapters.

Na +

H +¡ O -2 O -2 O -2 O -2 O -2 O -2

Charge transport in bulk Electrolytes

The transport of individual dispersed charges in an electrolytic solution can be simplied by using a mean eld approach, by expressing them as a volume averaged ion concentration. The transport equations for the charged species is called the Nernst-Planck equation [START_REF] Zheng | Poisson-Boltzmann-Nernst-Planck model[END_REF]] and can be written as,

∂c i ∂t = -∇ • J i + r i (2.2.1)
where, c i is the molar concentration of the i th ionic species, J i is the ionic ux which can be written as a combination of three dominating uxes, namely, the diusive ux 2.2 Charge transport in bulk Electrolytes (J D,i ), the electro-migration ux (J E,i ), and the convective ux (J C,i ). The term r i accounts for the ion production due to chemical reactions of neutral species, which act as the source of ions. In this study, we will be focusing on unipolar injection of ions, which means that the ions are generated at the electrodes and there is no other chemical reaction going on in the bulk (r i = 0). The diusive ux is due to the concentration gradient present in the system and is expressed as, J D,i = -D i ∇c i , where D i is the coecient of diusion of the i th ionic species. The electro-migration ux is due to the motion of the free charges dispersed in the solvent under the inuence of an electric eld. It can be expressed as, J E,i = M i F z i c i E, where M i is the molar mobility of the i th ionic species, F is the Faraday's constant which represents 1 mole of the electronic charge, z i is the valence (charge number) of the i th ionic species and E is the electric eld vector. The values of ionic diusivity (D i )

and mobility (M i ) for dierent ionic species is presented in Tab. 2.1. The convective ux is due to the ow eld of the solvent and can be expressed as, J C,i = uc i .

In a dilute solution, the background velocity (u) is easy to dene, which is the mass averaged velocity of the solvent. However, in concentrated solutions, such a velocity becomes more dicult to dene, as the dierence between the ux of an ion relative to the ow of the solvent is not clear. A generalized approach is to treat the motion of all the molecules (ions and solvent) in a coupled manner as in

Stephan-Maxwell equations [START_REF] Bird | Transport Phenomena[END_REF]]. But in this work, dilute electrolytes are considered, where such complexities can be neglected. Hence, the total ionic ux can be represented as,

J i = J C,i + J D,i + J E,i = uc i -D i ∇c i + M i F z i c i E (2.2.2) ions at T = 25°C H + K + N a + Br -Cl -F - I -OH -
∂c i ∂t = -∇ • (uc i -D i ∇c i + M i F z i c i E) (2.2.3)
The molar mobility of ions, M i is related to the ionic diusivity through the Einstein equation as, M i = D i /RT where R is the universal gas constant and T is the ambient temperature. Assuming incompressible ow (∇ • u = 0) for the ionic transport and representing the electric eld as, E = -∇φ, where, φ is the electric potential, the Nernst-Planck equation (see Eq. 2.2.3) can be written as,

∂c i ∂t + (u • ∇) c i = D i ∇ 2 c i + D i F z i RT ∇ • (c i ∇φ) (2.2.4)
The Nernst-Planck equation contains two unknowns namely, the ionic concentration, c i and the electrostatic potential eld, φ. So in order to close the system of equations, one has to use the conservation of charges while considering the electroneutrality, ρ e = i F z i c i = 0, where, ρ e is the charge density of the system. The conservation of charge in the system can be obtained by having a divergence free ux of charges (or current density) as,

∇ • I = 0 (2.2.5)
where, I is the current density which is related to the ionic ux as,

I = i F z i J i = i F z i uc i - i F z i D i ∇c i - i F 2 z 2 i D i RT c i ∇φ = I C + I D + I E (2.2.6)
where, I C is the convective current density, I D is the diusive current density and I E is the electronic (Ohmic) current density. Using the above expression, the conservation of charge expression (see 2.2.5) for incompressible ows (∇ • u = 0) leads to,

∇ • I = i F z i (u•∇) c i - i F z i D i ∇ 2 c i - i D i F 2 z 2 i RT ∇ • (c i ∇φ) = (u•∇) i F z i c i - i F z i D i ∇ 2 c i - i D i F 2 z 2 i RT ∇ • (c i ∇φ) = 0 (2.2.7)
From the electroneutrality condition ( i F z i c i = 0), the conservation of charge expression reduces to, i

F z i D i ∇ 2 c i + i D i F 2 z 2 i RT ∇ • (c i ∇φ) = 0 (2.2.8)
For a binary (i = 2) and symmetric electrolyte (z + = -z -= z), the electroneutrality condition gives,

F z + c + + F z -c -= 0 ⇒ c + = c - (2.2.9)
Taking c + = c -= c, the conservation of charge reduces to,

F z (D + -D -) ∇ 2 c + F 2 z 2 RT (D + + D -) ∇ • (c∇φ) = 0
(2.2.10)

Charge distribution near solid substrates

A solid surface (substrate) which is in contact with an electrolyte develops a surface charge density (which can be associated with a surface potential, φ s ). Some of the most plausible mechanisms causing such a phenomenon can be enlisted as, ionization or dissociation of surface groups and adsorption of ions from the solution [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF]]. In order to maintain the electro-neutrality in the bulk region of the electrolyte, the surface charge of the solid substrate is balanced by a net opposite charge (through a combination of oppositely charged ions (counter-ions)

and like charged ions (co-ions)) in the vicinity of the charged substrate. This ionic distribution physically manifests as a diused cloud of ions screening the substrate potential and is structurally identied to have two distinct regions (see Fig. 2.2). The rst region consists of counter-ions that are strongly attached to the charged substrate owing to strong Coulombic attractions. This region exists as a monolayer of immobile counter-ions and is commonly known as the Stern layer. Beyond the Stern layer, the ions are mobile and the plane separating the immobile and mobile layers of ions is also known as the slipping plane. The electric potential associated with this plane is called the zeta potential (ζ b ), which can be measured experimentally [Kirby and Hasselbrink (2004a)] and hence is commonly used in modeling electro-kinetics.

The second region consists of ions suspended in a structural equilibrium under an attractive Coulombic force due to the charged wall, a repulsive Coulombic force due to the neighboring like charged ions, and the intrinsic thermal motion of the ions.

This two-layered structure is also known as the electric double layer (EDL) and is modeled using the Gouy-Chapman-Stern (GCS) model [START_REF] Lyklema | Fundamentals of Interface and Colloid Science Vol II: Solid-Liquid Interfaces[END_REF]]. The GCS model however has certain assumptions as, 1) Ions are modeled as point charges.

2) The dominant interaction between the charges in the diuse double layer is the Coulombic interaction.

3) The solvent is assumed to be a structureless continuous media with constant dielectric permittivity throughout the double layer.

Although recent advances have been made in proposing more accurate models while considering the steric eects of the ions, ionic solvation etc, the GCS model has been quite eective in modeling the equilibrium distribution of charges near a charged surface. To obtain the equilibrium ionic charge distribution and the related potential eld within the EDL one can assume a steady state with no background ow or external electric eld. The net ux, J i in such a case (u = 0) can be written as,

J i = -M i c i ∇µ i (2.3.1)
where, µ i is the molar chemical potential, which is dened as the change in the free energy of the system upon adding or removing one ion. The expression for the molar chemical potential is,

µ i = µ 0,i + RT ln c i c 0,i + F z i φ sc (2.3.2)
where, µ 0,i is the chemical potential of the initial state, c 0,i is the initial (bulk) ionic concentration and φ sc is the electrostatic potential eld in the uid due to the space-charge distribution. In electro-chemical equilibrium, the ions rearrange rapidly to counter any change in the concentration of the ions keeping the chemical potential constant. Therefore, the electro-chemical equilibrium requires the change in the chemical potential to be zero (

µ i = µ i -µ 0,i = 0) which gives, RT ln c i c 0,i + F z i φ sc = 0 (2.3.3)
which results into a Boltzmann distribution of the charged species as,

c i = c 0,i e -F z i φsc RT (2.3.4)
As from the rst principle of dierentiation, ∇µ i = lim L→0 µ i / L. So, from the Eq.

2.

3.1, we can see that an electro-chemical equilibrium condition ( µ i = 0) within the EDL leads to a zero ionic ux condition too (J i = 0). To complete the set of equations and unknowns (namely c i and φ sc ), we need a closure on the potential distribution, φ sc . From Gauss's law,

ρ e = ∇ • D sc (2.3.5)
where, ρ e is the free charge density and D sc is the electric displacement eld due to the space charge distribution, which for a linear dielectric material can be written as a function of the electric eld in the system,

D sc = ε r ε 0 E sc (2.3.6)
where ε r is the dielectric constant of the medium, ε 0 is the permittivity of the vacuum and E sc is the electric eld distribution in the medium. From electrostatics,

∇ × E sc = 0 E sc = -∇φ sc (2.3.7)
Hence, the resulting equation can be written as,

ρ e = -∇ • (ε r ε 0 ∇φ sc ) (2.3.8)
Upon combining the two equations we can obtain the equation for electric potential as a function of concentration distribution as,

-∇ • (ε r ε 0 ∇φ sc ) = i F z i c i (2.3.9)
Using the concentration expression from Eq. 2.3.4, for a symmetric binary electrolyte (z + = -z -= z) and a solvent with constant permittivity, ε r ε 0 , Eq. 2.3.9 reduces to,

ε r ε 0 ∇ 2 φ sc = - i F z i c i = 2F zc 0 sinh F zφ sc k B T (2.3.10)
where, c 0 is the neutral bulk ionic concentration of the solution. eld can be written as [START_REF] Melcher | Continuum Electromechanics[END_REF]],

Σ M = - ε r ε 0 2 E • E + ε r ε 0 E ⊗ E (2.4.1)
where, E is the total electric eld in the medium. The diuse space charge potential near a charged surface in contact with an electrolyte leads to an electric eld in the electrolyte as, E sc = -∇φ sc . The applied electric eld creates a potential gradient too which can be written as, E app = -∇φ app . So, the net electric eld, E (or potential, φ) in the system is a combination of the two eects as,E = E app + E sc (or φ = φ app + φ sc ).

Total Stress in Electrolytes

Fluids by denition are a phase of matter that ow (deform) indenitely under the inuence of a shearing action. Although, real uids inherently show some resistance to such a ow by virtue of a property called viscosity, the nature of the rate of deformation of uids for a given value of applied shear stress varies for dierent uids and can be used for classication of uids as such. If the rate of the deformation of a uid varies linearly with the shear, it is categorized as a Newtonian uid, if the variation is non-linear, it is categorized as a Non-Newtonian uid. For common uids like water, air and some oils, a linear relationship between the shear and the deformation exists. From Hooke's law for isotropic uids the viscous shear stress tensor (τ ) can be written as,

τ = 2µD + λ (∇ • u) I (2.4.2)
where, λ is the Lamé's rst parameter, µ is the Lamé's second parameter or shear modulus, and I is the identity tensor. The Lamé's parameters are related to the bulk modulus as, K = λ + 2 3 µ. D is the strain rate tensor which is dened as,

D = 1 2 ∇u + ∇u T (2.4.3)
The total state of hydrodynamic stress in the uid can be written as a combination of the hydrostatic component (pressure) and the deviatoric component (shear) as,

Σ H = -pI + τ (2.4.4)
where, p is the hydrostatic pressure. Hence, the net stress tensor (Σ T ) acting on a uid under the inuence of an external electric eld is a combination of the Maxwell stress and the hydrodynamic stress tensor (Σ H = -pI + µ ∇u + ∇u T ) and can be written as, Σ T = Σ H + Σ M . Now with this net stress tensor, the momentum transport equation to describe the ow of a Newtonian uid can be written as,

ρ Du Dt = ∇ • Σ T = ∇ • Σ H + ∇ • Σ M = -∇p + µ∇ 2 u + ∇ • Σ M (2.4.5)
The divergence of the Maxwell stress tensor appears as a body force term in the momentum equations and can be expanded as [START_REF] Melcher | Continuum Electromechanics[END_REF]],

∇ • Σ M = ∇ • - ε r ε 0 2 E • E + ε r ε 0 E ⊗ E = - ε 0 2 E • E∇ε r + ρ e E (2.4.6)
For an isotropic dielectric material (∇ε r = 0), the divergence of the Maxwell stress tensor reduces to the product of the charge distribution in the medium (ρ e ) and the net electric eld. From Eq. 2.3.8, we can re-write the momentum equations as,

ρ Du Dt = -∇p + µ∇ 2 u + ρ e E = -∇p + µ∇ 2 u + ε r ε 0 ∇ 2 φ sc ∇φ (2.4.7)
Upon scaling the velocity as, U = u/u ref , time as, θ = ωt, where, ω is the frequency of the periodic actuation, pressure,

P = p ref h 0 /µu ref , electrostatic potential, Φ = φ/ζ ref , gradient as, ∇ = h 0 ∇
, the momentum transport equation can be written as,

ρωu ref dU dθ + ρu 2 ref h U • ∇ U = - p h 0 ∇P + µu ref h 2 0 ∇2 U + ε r ε 0 ζ 2 ref h 3 0 ∇2 Φ sc ∇Φ ⇒ Wo 2 dU dθ + Re U • ∇ U = -∇P + ∇2 U + γ R E R ∇2 Φ sc ∇Φ (2.4.8)
where, Wo =

ωh 2 0 ν is the Womersley number, Re = u ref h 0 ν is the Reynolds number, γ R = εrε 0 ζ b E 0
µu ref is the electro-viscous ratio, which quanties the relative strength of electrical body forces to viscous forces, E 0 is the strength of the applied electric eld (|E app | = E 0 ), and E R = ζ b E 0 h 0 is the ratio of the strengths of the electric eld in the EDL to the applied electric eld, which can be written as, Φ app = -´Eapp dr = -1 E R ´dR (where Rand r are the dimensionless and dimensional position vectors, respectively). Also, at microscales the viscous eects dominate over inertial eects. So, upon assuming a steady system and neglecting the convective terms, the momentum transport equation can be written as,

-∇P + ∇2 U + γ R E R ∇2 Φ sc ∇Φ = 0
(2.4.9)

Electro-Osmotic Flow in a Parallel Plate System

In order to get an idea of the EOF velocity eld in a parallel plate conguration of charged plates, let us consider two charged plates with a zeta potential as ζ b and separated by a distance 2h 0 in the xz plane. The rst plate is at y = -h 0 and the second one is at y = h 0 (see Fig. Under the inuence of an external electric eld as, E = (E app , 0, 0), no external pressure gradient ( ∇P = 0), and a laminar and fully developed velocity eld (U = (U (Y ), 0, 0)), the momentum transport equation (see Eq. 2.4.9) can be written as,

d 2 U dY 2 - γ R cosh Y De De 2 cosh 1 De = 0
(2.4.12)

U (Y ) = -γ R 1 - cosh Y De cosh 1

De

(2.4.13)

In the limit of thin Debye layers as compared to the characteristic length scales (De 1), the electro-osmotic velocity can be obtained as,

U (Y ) = -γ R (2.4.14)
This dispersion-free plug type velocity prole is observed in many practical microsystems where h 0 λ D ∼ O (0.1 -10nm). In such a limit, the EOF velocity is taken as a slip velocity condition and is called the Helmholtz-Smoluchowski velocity as, u HS = -ε r ε 0 ζ b E app /µ.

Van der Waals Forces

Understanding the role of van der Waals forces in the micro-nanoscale systems is very important as they play a signicant role in a wide range of important phenomena. Some of which are adhesion, surface tension, adsorption, wetting, multiphasic interactions of gases, liquids, and solids, and towards determining the structures of condensed macromolecules such as proteins and polymers. The van der Waals force between atoms and molecules is always present unlike the double-layer interaction.

It is mostly insensitive to variations in electrolyte concentration and pH, and can be considered to be a constant characteristic force for a given set of interacting phases.

Their main features may be summarized as follows [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF]],

1. They are long-range forces and, depending on the situation, can be eective from large distances (> 1nm) down to inter-atomic spacings (about 1).

2. These forces may be repulsive or attractive.

3. Such forces not only bring molecules together but also tend to mutually align or orient them, though this orienting eect is usually weaker than with dipolar interactions.

4. They are not additive; that is the force between two bodies is aected by the presence of other bodies nearby. This is called the nonadditivity of an interaction.

Further, in the presence of electrical double layer interactions, the van der Waals attraction must always exceed the double-layer repulsion at small enough distances since it is a power law interaction. The free energies of interactions per unit area have mostly long-range apolar component only. However, short range repulsion forces are also present and act over distances ≤ 10nm. The short range forces are electrostatic in nature occurring due to the overlap of diuse electric double layers, which form near interfaces involving polar uids [START_REF] Craster | Dynamics and stability of thin liquid lms[END_REF]]. The exact nature of such electric double layer based interactions has been detailed in the previous sections. The sequence of phenomena described above forms the basis of the celebrated DLVO theory of colloidal stability, after Derjaguin, Landau, Verwey

and Overbeek [START_REF] Russel | Colloidal Dispersions[END_REF]]. The free energy per unit volume for the long range interactions can be written as [START_REF] Sharma | Nonlinear Stability, Rupture, and Morphological Phase Separation of Thin Fluid Films on Apolar and Polar Substrates[END_REF]],

φ vdW = a 6πh 3 (2.5.1)
where, a is the Hamaker constant, and h is the distance between the interacting surfaces. The disjoining pressure is dened as, p d = -φ vdW .

Interfacial Boundary Conditions

Continuity of Stress

The uid-uid interfaces are stress free interfaces as they instantly ow under the application of a shearing force. In other words, the net surface traction, which is dened as the surface force per unit area applied by one uid should be completely balanced by the net surface traction by the second uid across the interface. The surface traction (T) is related to the total state of stress (Σ

T = Σ H + Σ M ) at a
point by Cauchy's stress theorem as, T = Σ T • n, where, n is the normal vector to the surface. However, when there is interfacial deformation and variation of the interfacial tension, a nite dierence (jump) in the surface traction across the uiduid interfaces is observed which balances out interfacial forces due to any such deformation or variation. The continuity of stress in such cases require,

Σ T • n = γn (∇•n) -∇ s γ (2.6.1) where, Σ T • n = Σ T 1 • n -Σ T 2
• n, is the dierence in the total stress at the interface between the bounding phases 1 and 2, and ∇ s = ∇-n (n • ∇) is the surface gradient operator and models the variations in interfacial tension such as due to temperature (thermocapillarity), species concentration (solutocapillarity), and electric eld (electrocapillarity). By denition, the surface gradient operator is the gradient operator minus the normal component of the gradient operator. Hence, it is always tangential to the interface and is zero for the systems with constant temperature, surfactant concentration, and the electric eld. The Eq. 2.6.1 can be hence separated into normal and tangential components of the stress. The jump in the normal component of the stress can be attributed to the curvature induced due to the interfacial deformation and can be represented by,

n • Σ T • n = γ∇•n (2.6.2)
Similarly, the jump in the tangential component of the stress can be written as,

t • Σ T • n = -t • ∇ s γ (2.6.3)

Kinematic Condition

In the absence of a net mass ux through the uid-uid interface, i.e. an impermeable interface, the relative velocity component of the uid normal to the interface must be zero. This means that if the interface is stationary, the normal component of the velocity is zero. This idea can be represented by,

(U -V) • n = 0 (2.6.4)
where, U is the uid velocity at the interface and V is the interfacial velocity. If the interface is to be represented by, f (r, t) = 0, the unit normal vector pointing in the region f (r, t) > 0 is given by, n = ∇f |∇f |

. Now, within the reference frame of the interface the material derivative of the interface vanishes ( Df Dt = 0), which gives the interfacial velocity as, V

• ∇f = -∂f ∂t or in a simplied manner, V • n |∇f | = -∂f ∂t .
Hence Eq. 2.6.4 can be rewritten as,

U • ∇f |∇f | = - 1 |∇f | ∂f ∂t (2.6.5) which reduces to, U • ∇f = - ∂f ∂t (2.6.6)

Gauss's law

The Gauss's law relates the electric eld distribution to the electrical charge distribution. It says that the net electrical ux through a closed surface is proportional to the total charge enclosed by the surface. While considering the free charge distribution (i.e. ignoring the polarized charges in a dielectric), one can express it as,

˛C D • dS = ˚V ρ f dV (2.6.7)
where, D is the electric displacement eld and ρ f is the free charge density. From the control volume V (see Fig. 2.4) one can write the above equation as,

ˆD1 •n 1 dx+ ˆ(D 1 + D 2 )•n 2 dy+ ˆD2 •n 3 dx+ ˆ(D 1 + D 2 )•n 4 dy = ρ f x (2.6.8) Using, D 1 =D 1,t i + D 1,n j , D 2 =D 2,t i + D 2,n j, n 1 = -j, n 2 = -i, n 3 = j, and n 4 = i,
Eq. 2.6.8 can be nally expressed as,

D 2,n -D 1,n = ρ f (2.6.9)
where, n is the direction normal to the interface. From the Eq. 2.3.6 and using, E n = -∂φ ∂n , the jump of electrical potential gradient across the interface can be written as,

ε r ε 0 ∂φ ∂n 2 1 = ρ f (2.6.10)

Faraday's law of induction

Faraday's law of induction suggests how a time-varying magnetic eld (B) creates a spatially varying, non-conservative electric eld. In dierential form, it can be written as,

∇ × E = -∂B ∂t (2.6.11) In integral form, it can be written as,

˛E • dl = - ¨∂B ∂t • dS
(2.6.12)

In the absence of the magnetic eld, it shows the conservative nature of the electric eld, that the total work done by an electric eld in a closed loop is zero (see

Fig. 2.5). ˆE1 • t 1 dx + ˆ(E 1 + E 2 ) • t 2 dy + ˆE2 • t 3 dx + ˆ(E 1 + E 2 ) • t 4 dx = 0 (2.6.13) Using, E 1 =E 1,t i + E 1,n j , E 2 =E 2,t i + E 2,n j, t 1 = -i, t 2 = j, t 3 = i, and t 4 = -j
in the Eq. 2.6.13, the jump in the tangential component of electric eld across the interface is obtained as,

E 2,t -E 1,t = 0 (2.6.14)
which, in terms of jump of electric potential can be written as, ∂φ ∂x

Conservation of charge

The conservation equation of charged species across an interface can be obtained by integrating the charge transport equation (see Eq. 2.2.3) across the interface while assuming that there are no chemical reactions taking place.

∂c i ∂t = -∇ • (uc i -D i ∇c i + M i F z i c i E) (2.6.17)
where, for i th ionic species, c i is the molar concentration, u is the convective velocity eld, D i is the ionic diusivity, M i is the ionic mobility, F is the Faraday's constant, z i is the ionic valence and E is the applied electric eld. Upon integrating the above equation across the interface, one can obtain,

∂c i,s ∂t = -∇ • ˆ(uc i ) dS + ˆ∇ • (D i ∇c i ) dS -ˆ∇ • (M i F z i c i E) dS = -u • ∇c i,s + ˆ(D i ∇c i ) • ndl -ˆ(M i F z i c i E) • ndl = -u • (∇ s + n • (n • ∇)) c i,s + [D i ∇c i,s -M i F z i c i,s E] 2 1 • n (2.6.18)
where, c i,s is the surface concentration of the i th species and ∇ s is the surface gradient. In order to obtain the conservation equation of the net surface charge density, q f,s = i F z i c i,s one can multiply the Eq.2.6.18 by F z i and perform a summation over all of the ionic species as,

∂q f,s ∂t = -u • ∇ s q f,s -n • u (∇ • n) q f,s + i D i F z i ∇c i,s -E i F 2 z 2 i M i c i,s 2 1 • n = -u • ∇ s q f,s -u n (∇ • n) q f,s Interfacial Deformation +       i D i F z i ∇c i,s Diffusive Current -Λ m,s E Ohmic Current       2 1 • n (2.6.19)
where, Λ m,s is the molar surface conductivity. Hence, from the above expression, one can see that the contributions to the net change in the surface charge density is by the deformation (extension/shrinkage) of the interface, the diusive surface current and the Ohmic surface current. In the case of a binary (i = 2) and symmetric (z + = z -= z) electrolyte with same diusivity (D + = D -= D), one can re-write the Eq. 2.6.19 as, Further, a contrasting analysis of time periodic (AC) EOF in a thin liquid layer with a at free surface and that of a thin layer of liquid between two rigid boundaries has been carried out. In order to establish a common comparative basis between the two cases, an unperturbed interface for the former case is considered here, which techni- Finally, the thin lm dynamics under DC EOF is explored in a rectangular geometry with various aspect ratios, where the eect of EDL parameters on the uid ow rate is presented. Such a study is important for realistic cases where a gaseous lm lies between the uid and the bounding walls, thus creating a free-surface situation.

∂q f,s ∂t = -u • ∇ s q f,s -n • u (∇ • n) q f,s + [D∇q f,s -Λ m,s E] 2 1 • n (2.
cally
The presence of such gaseous layers is commonly observed near hydrophobic substrates, or sometimes due to the generation of gaseous products as a part of certain electrochemical reactions. Also, it provides a base state ow condition for further studies on instabilities associated with a two-layer EOF in the presence of steady or time periodic electric elds.

Electric potential eld

In this study, the thin lm system is considered to have large lateral extents which results into negligible x-gradients as compared to the y-gradients (hence, ∇ ∼ d/dy).

The space charge distribution within the conductive lm leads to a potential eld, φ sc , where, the solid substrate (y = 0) and the gas-liquid interface (y = h 0 ) are considered to be at constant potential with the associated boundary conditions as,

φ sc (0) = ζ b φ sc (h 0 ) = ζ I (3.2.1)
The potential distribution can be obtained as a solution of the Poisson-Boltzmann equation (PBE) (see Eq. 2.3.10). Upon using the reference scales as, Φ sc = φ sc /ζ b and Y = y/h 0 , the dimensionless PBE (see Eq. 2.3.11) can be written as,

d 2 Φ sc dY 2 = sinh (βΦ sc ) βDe 2 (3.2.2)
where, De = λ D /h 0 is the ratio of the Debye length (λ D ) and the lm thickness (h 0 ), and β = ezζ b k B T is the ionic energy parameter (see Chapter 2). The boundary conditions in the dimensionless form can be written as,

Φ sc (0) = 1 Φ sc (0) = Z R (3.2.3)
where, Z R = ζ I /ζ b is the ratio of the interfacial zeta potential to the substrate zeta potential. The value of the interfacial potential depends upon electrochemistry of the problem and hence, Z R is taken as a variable parameter. Generally, the substrate zeta potential is less than the thermal potential, i.e. β < 1. integrating even when the solution is expected to show smooth characteristics (very little variation) in a region [START_REF] Lambert | Numerical Methods for Ordinary Dierential Systems[END_REF]]. To solve such a highly sti BVP, the Automated Continuation with Deferred Correction (ACDC) method by Cash et al.

( 2001) is used here. This method is based on implicit Runge-Kutta scheme on a Lobatto grid. A Lobatto grid includes boundary points in the integration interval, facilitating the use of an adaptive mesh. Such an adaptive mesh is extremely useful for systems having large spatial gradients like a boundary layer or an EDL. The electric potential due to the externally applied electric eld (Φ app ) can be written in the dimensionless form as,

Φ app (X) = - ˆE0 h 0 ζ b dX = - X E R (3.2.5)
where, Φ app = φ app /ζ b , X = x/h 0 , and E R = ζ b /E 0 h 0 . The net electric potential in the system can be written as a sum of the potential elds due to the space charge distribution (Φ sc ) and the externally applied electric eld (Φ app ). The dimensionless total electric potential of the system can be written as,

Φ (X, Y ) = Φ app (X) + Φ sc (Y ) = - X E R + 1 sinh 1 De Z R sinh Y De + sinh 1 -Y De (3.2.6)
Velocity Field

The hydrodynamics of the at thin electrolytic lm is explored under the inuence of an externally applied electric eld by considering a steady and a laminar velocity eld (U = (U (Y ), 0, 0)) with no externally applied pressure gradient (dP /dX = 0) and a at interface (Y = 1). The base state x-momentum transport equations (see Eq. 2.4.9) in dimensionless form can be written as,

d 2 U b dY 2 -γ R ∂ 2 Φ sc ∂Y 2 = 0 (3.2.7)
and, the base state y-momentum transport equation (see Eq. 2.4.9) can be written as,

- dP b dY + γ R E R ∂Φ sc ∂Y ∂ 2 Φ sc ∂Y 2 = 0 (3.2.8)
At the solid substrate (Y = 0), a no slip boundary condition gives,

U b (0) = 0 (3.2.9)
At the gas-liquid interface (Y = 1), the continuity of shear and normal stresses respectively give,

dU b dY + γ R E R ∂Φ ∂X ∂Φ ∂Y = 0 (3.2.10) P b - γ R E R 2 ∂Φ ∂X 2 - ∂Φ ∂Y 2 = 0 (3.2.
U b (Y ) = -γ R 1 - Z R sinh Y De + sinh 1-Y De sinh 1 De (3.2.12) P b (Y ) = γ R E R 2De 2 sinh 2 1 De Z R cosh Y De -cosh 1 -Y De 2 (3.2.13)
The uid velocity at the free surface is then,

U b (1) = γ R (Z R -1) (3.2.14)
which has a linear dependence upon the zeta-potential at the interface. This dependence, which is by virtue of the polarity of the interfacial potential, is observed to facilitate or delay the free surface stability by either increasing or decreasing the shear stress [START_REF] Choi | Is free surface free in micro-scale electrokinetic ows?[END_REF]].

Electric potential eld

The dimensionless electric potential due to the externally oscillating applied electric eld (E app = E 0 sin (ωt)) can be written in the dimensionless form as,

Φ app (X, Θ) = - ˆX 0 E 0 h 0 ζ b sinΘdX = - X E R sinΘ (3.3.1)
where, Θ = ωt. The net electric potential in the system can be written as a sum of the potential eld due to the space charge distribution (Φ sc ) (see Eq. 3.2.4) and the externally applied electric eld (Φ app ). The dimensionless total electric potential of the system under a time-periodic electric eld can be written as,

Φ (X, Y, Θ) = Φ app (X, Θ) + Φ sc (Y ) (3.3.2)

Velocity eld

Upon non-dimensionalizing the ow variables as, U = u/u ref and without an external pressure gradient the dimensionless momentum conservation Eq. 2.4.8 reduces to,

Wo 2 ∂U ∂Θ = ∂ 2 U ∂Y 2 + γ R E R ∂Φ ∂X ∂ 2 Φ ∂Y 2 = ∂ 2 U ∂Y 2 -γ R sinΘ d 2 Φ sc dY 2 (3.3.3)
where, Wo = ωh 2 0 /ν is the Womersley number expressing the relative strength of temporal inertial force over the viscous dissipation force. The dimensionless ymomentum equation gives,

0 = - dP dY + γ R E R ∂Φ ∂Y ∂ 2 Φ ∂Y 2 = - dP dY + γ R E R dΦ sc dY d 2 Φ sc dY 2 (3.3.4)
The interfacial boundary conditions in dimensionless form (see Eqs. 3.2.10 and

3.2.11 ) are, ∂U ∂Y = γ R sinΘ dΦ sc dY (3.3.5) P (1, Θ) = γ R E R ∂Φ ∂X 2 - ∂Φ ∂Y 2 (3.3.6) Chapter 3
Thin Films in Electrokinetics with no-slip condition at the wall (Y = 0),

U (0, Θ) = 0 (3.3.7)
The solution of the system of equations 3.3.3-3.3.7 can be obtained by decomposing the velocity into time-dependent and space-dependent functions as,

U (Y, Θ) = F (Y )G(Θ) = Im F (Y ) e iΘ (
iWo 2 F = d 2 F dY 2 -γ R d 2 Φ sc dY 2 (3.3.9)
the corresponding boundary conditions are,

F (0) = 0 (3.3.10) dF dY -γ R dΦ sc dY = 0 (3.3.11)
For a generalized potential distribution, the resulting velocity prole can be obtained as,

U (Y, Θ) = γ R Im        Φ sc (Y ) e iΘ - cosh( √ i(1-Y )Wo) cosh( √ iWo) Φ sc (0) e iΘ - cosh( √ i(1-Y )Wo) cosh( √ iWo) e iΘ √ iWo ´1 0 cosh √ iWo (1 -η) Φ sc (η) dη √ iWo 2 e iΘ+ √ iWoY ´Y 0 e - √ iWoη Φ sc (η) dη - √ iWo 2 e iΘ- √ iWoY ´Y 0 e √ iWoη Φ sc (η) dη        (3.3.12)
To obtain the velocity eld for a generalized potential, one needs to perform the involved integrations in Eq. 3.3.12 numerically. In this study, the QUADPACK numerical integration package [START_REF] Favati | Algorithm 691-Improving QUADPACK automatic integration routines[END_REF]] was used which eectively handles the integration of complex functions with the help of an adaptive automatic integration algorithm using Gauss-Kronrod rule. For general cases where the substrate zeta potential lies within the Debye-Hückel limit, upon using a space-charge potential obtained by Debye-Hückel linearization, the velocity expression can be simplied analytically and written as,

U (Y, Θ) = U a (Y ) sin (Θ + ∆ (Y )) (3.3.13)
where, U a (Y ) is the amplitude and ∆ (Y ) is the phase of the velocity. Detailed expressions of the amplitude and the phase are given in the Appendix A1. Similarly with the Debye-Hückel linearization, the pressure distribution in the system can be obtained by solving Eqs. 3.3.4 and 3.3.6 as,

P (Y, Θ) = γ R 4E R cos (2Θ)- γ R 4E R + γ R E R 2De 2 sinh 2 1 De Z R cosh Y De -cosh 1 -Y De 2 (3.3.14)
It is also observed from Eqs. 3.3.3 and 3.3.4 that in the absence of an external pressure gradient, the velocity and pressure distributions are decoupled and are parametrically dependent on the applied electric eld and space-charge potential distribution.

Results and Discussion

It can be seen from Eq. 3. as a solution of the PBE [START_REF] Chakraborty | Generalized model for time periodic electroosmotic ows with overlapping electrical double layers[END_REF]; [START_REF] Dutta | Analytical Solution of Time Periodic Electroosmotic Flows: Analogies to Stokes' Second Problem[END_REF]]. In the second case, the wall boundary condition is a no slip condition along with the previously used symmetry condition at the channel centerline. Similarly, the modeling of a thin free surface ow involves a Stokes equation with no slip condition at the wall and a stress free condition at the free surface. In the absence of various surface phenomena such as surface tension variation, phase change and mass transfer, the stress free condition at the free surface (∂u/∂y = 0) is mathematically the same as a symmetry condition at the channel center (∂u/∂y = 0) in a parallel plate EOF [START_REF] Chakraborty | Generalized model for time periodic electroosmotic ows with overlapping electrical double layers[END_REF]; [START_REF] Dutta | Analytical Solution of Time Periodic Electroosmotic Flows: Analogies to Stokes' Second Problem[END_REF]]. Hence, the commonly studied parallel plate EOF can be put forth as a case without the Maxwell stress, which in the simplied form adds a term involving transverse gradient of electric potential at the free surface (see Eq. 3.3.6). As a consequence, it is seen that, in the presence of very thin EDLs (De = 0.01) and no interfacial potential (Z R = 0), the free surface system behaves as a classical half For cases where the EDL extent is comparable to the characteristic dimensions of the ow, which in the present case is the lm thickness h 0 , the eect of Maxwell stress modifying the shear stress balance criterion at the surface boundary is more pronounced. This leads to changes on the transverse velocity gradients as a function

of the surface potential and potential eld present in the bulk. In the absence of transverse velocity component, the distribution of such a gradient also corresponds to the absolute value of vorticity (ω abs ) eld dened as ω abs = |∇ × U| = ∂U/∂Y , which can account for vorticity induced ow instabilities in the system. To study the eect of Maxwell stress model on vorticity distribution, a comparison between freesurface EOF models with and without Maxwell stress was performed (see Fig. As the ionic energy parameter, β, denes the relative strength of electrostatic energy of ions over their thermal energy, the changes in the velocity distribution due to the relative interplay between the mentioned competing energies is signicant.

To demonstrate the eect of β, four cases have been considered with dierent values of the EDL thickness (De) and β (see Fig. 3.4). It is seen that for lower values of β the dierence in the velocity distribution is signicant for higher EDL thickness (De = 0.1). However, in the case of thin EDL, the dierence is not noticeable. This also explains the reason why the Debye-Hückel linearization of Poisson-Boltzmann equation, which is valid for low values of β, successfully models the thin EDL potential distribution even over a large range of β. The above observation can also be explained by the fact that, when the thermal motion of ions dominates (β < 1), the electric actuation is diused strongly in the EDL and hence the local velocity magnitude is reduced. This diusion of the electrical actuation is enhanced if the EDL is thicker as the presence of a greater amount of space-charge distribution aides in more eective distribution of velocity in the liquid bulk. The electro-viscous ratio, γ R , (see Eq. 3.3.3) is directly proportional to the amplitude of the applied oscillating electric eld and hence, increment of which amplies the ow velocity prole for a given uid. This is clearly highlighted in Fig. 3.5 for the values De = 0.01 and De = 0.1. It can also be seen that the maximum value of the velocity obtained for a thin EDL case (De = 0.01) is greater than in the case with a thicker EDL (De = 0.1), which can be attributed to the fact that in the case of thick EDLs, the electrical energy mobilizes a greater space charge distribution than in the case of a thin EDL. This leads to a lower maximum kinetic energy of the liquid within the EDL. Moreover, γ R also represents a ratio of the classically used Helmholtz-Smoluchowski velocity (u HS ) as a reference velocity scale to the current velocity scale. It is interesting to note that u HS can be used as a reference velocity scale (γ R = 1) when (a) EDL is thin (see Fig. 3.5), (b) forcing frequency is smaller than the viscous relaxation frequency (Wo < 1) (see Fig. 3.6). In remaining cases, u HS overestimates the characteristic velocity sometimes by an order of magnitude.

Also, one can observe from Fig. 3.5 that the velocity oscillations propagate as a damped wave into the bulk. The extent of this bulk penetration of the damped velocity oscillations seem to be unaected by the value of γ R .

Taking a cue from the classical Stokes second problem, the typical penetration depth of momentum diusion of oscillatory ows in a liquid can be estimated by the Stokes penetration depth δ s . Typically, δ s is a function of the forcing frequency as δs h 0 =

ν ωh 2 0 = 1

Wo

, where Wo is the Womersley number representing the relative strength of temporal inertia over viscous force (see Eq. 3.3.3). This inverse relationship between the δ s and Wo, is also observed in Fig. 3.6 which shows the velocity proles as a function of Wo. It is observed that decreasing Wo increases the Stokes penetration depth and in turn enhances the sharp changes in the velocity gradients near the wall. This also results in a stronger diusion of the near wall vorticity in the liquid bulk. However, upon increasing Wo, the viscous time lag of momentum propagation in the liquid also increases which leads to inection points in the velocity proles. Presence of such inection points leads to extrema in the vorticity distribution yielding possible uid mixing and instability [START_REF] Davis | The Stability of Time-Periodic Flows[END_REF]]. It is also observed that the magnitude and extent of vorticity propagation is a strong function of the EDL thickness. This can be explained in terms of electrokinetic energy transfer over the extent of space charge distribution in the liquid. The thinner will be the space charge distribution, the stronger will be the velocity gradients. Another aspect of interfacial potential is the phase lag (see Eq. 3.3.13) between velocities at dierent transverse locations in the ow (see Fig.

Free Surface Flows in Rectangular Geometry under a DC Electric Field

The common approach of 2D EOF models predicts the velocity proles by ignoring the channel 3D connement eects, and are useful when channel aspect ratios are signicantly larger than the EDL thickness. However, due to various microfabrication limitations and practical aspects of microuidic devices, large aspect ratio microchannels are not very common. This requires a 3D analysis of the ow to understand the ow proles especially when the channel dimensions are comparable to the EDL thickness. In this section, electro-osmotic ow of an aqueous electrolyte bounded by an inert gas on the top is studied in a rectangular microchannel and the role of interfacial potential and Maxwell stress generated dynamics are explored.

3.4.1. Mathematical Modeling Figure 3.9.: Schematics of a rectangular channel under free-surface electro-osmotic ow.

The system under study consists of a conductive uid having a planar interface with an inert gas in a microchannel with a rectangular cross-section. A Cartesian co-ordinate system is chosen to describe the channel geometry where the origin is set at the intersection of the bottom wall and the transverse line of symmetry (see Fig. 3.9). The width of the channel is taken as 2w. The height of the conductive uid is taken as h. The conductive uid is considered to be a low concentration symmetric electrolytic solution with constant permittivity (ε f,r ε 0 ). The inert gas is considered to be at constant pressure with a low permittivity (ε g,r ∼ 1) and low viscosity (µ g ∼ 0) . The electrical double layer forms at the interface between the electrolyte and the bounding walls as well as at the liquid-gas interface. All the interfaces to the electrolytic solution develop zeta potentials, the magnitude of which depends upon the electrochemistry of liquid-gas and solid-liquid interactions.

The bottom wall zeta potential is taken as ζ b , the side wall zeta potential is taken as ζ s , and the liquid-gas interfacial zeta potential is taken as ζ I . It is assumed that the external electric eld has no eect on the zeta potentials. The bounding walls are considered to be rigid and a constant electric eld is applied along the channel length (x-direction). Due to the symmetry of the system along z = 0, the governing equations will be solved for 0 ≤ z ≤ w and 0 ≤ y ≤ h.

Electrostatic Potential Distribution

The space charge potential distribution in the electrolyte can be obtained from the classical Poisson-Boltzmann equation (PBE) (see Chapter 2) as,

ε f,r ε 0 ∇ 2 φ sc = 2ez 0 C 0 sinh ez 0 φ sc k B θ (
∂ 2 Φ sc ∂Y 2 + ∂ 2 Φ sc ∂Z 2 = Φ sc De 2 (3.4.2)
where, De = λ D /D h is Debye number, which represents the relative extent of the EDL as compared to the geometric length scale with λ D = ε f,r ε 0 k B T / (2ρ 0 z 2 e 2 ).

The electrostatic potential distribution satises the following boundary conditions,

∂Φ sc ∂Z (Y, 0) = 0 Φ sc (Y, W ) = ζs Φ sc (0, Z) = ζb Φ sc (H, Z) = ζI
Φ sc (Y, Z) = ∞ m=1 2(-1) m-1 ζi sinh 1 De 2 +λ 2 m Y + ζb sinh 1 De 2 +λ 2 m (H-Y ) cos(λmZ) W λmsinh H 1 1 De 2 +λ 2 m + ∞ n=1 2(1+(-1) n-1 ) ζscosh 1 De 2 +λ 2 n Z sin(λnY ) Hλnsinh 1 De 2 +λ 2 n W (3.4.4)
where, λ m = (2m -1) π/ (2W ) and λ n = nπ/H. The electric potential due to the external eld can be calculated as,

Φ app (X) = - ˆEapp D h ζ ref dX = - X E R (3.4.5)
where, X = x/D h , and E R =

ζ ref EappD h
is the ratio of the electric potential due to charge polarization in the electrolyte over the electric potential due to the externally applied electric eld. Hence, the net electric potential eld in the uid is,

Φ (X, Y, Z) = Φ app (X) + Φ sc (Y, Z) (3.4.6)
Velocity Distribution

Since the top uid is considered to be a low permittivity, low viscosity inert gas at a constant pressure, it can be considered at rest. An external electric eld acting on a space charge distribution in the conductive uid generates a Maxwell stress.

This Maxwell stress along with the hydrodynamic stress results into a net stress distribution in the uid (see Chapter 2). Under such an actuation, for a Newtonian uid with incompressible, steady and laminar ow assumption, and in the absence of an external pressure gradient the momentum equation can be written as,

∂ 2 u ∂y 2 + ∂ 2 u ∂z 2 = ε f,r ε 0 ∂φ ∂x ∂ 2 φ ∂y 2 + ∂ 2 φ ∂z 2 (3.4.7)
The boundary condition at the walls (y = 0 and z = ±w) is taken to be a no-slip condition as,

u (0, z) = 0 u (y, w) = 0 (3.4.8)
The continuity of shear stress at the gas-uid interface (y = h),

∂u ∂y (h, z) -ε f,r ε 0 ∂φ sc ∂y (h, z) = 0 (3.4.9)
At the symmetry plane (z = 0), the symmetry condition leads to, ∂u ∂z (y, 0) = 0 (3.4.10) Further, the dimensionless governing equation can be written as,

∂ 2 U ∂Y 2 + ∂ 2 U ∂Z 2 = γ R ∂ 2 Φ sc ∂Y 2 + ∂ 2 Φ sc ∂Z 2
U (0, Z) = 0 U (Y, W ) = 0 ∂U ∂Y (H, Z) -γ R ∂Φ sc ∂Y (H, Z) = 0 ∂U ∂Z (Y, 0) = 0 (3.4.12)
The solution to the system of Eqs. 3.4.11 and 3.4.12 can be obtained by using the method of separation of variables and the resulting velocity can be written as,

U (Y, Z) = ∞ m=1 2(-1) m-1 γ R ζI sinh 1 De 2 +λ 2 m Y + ζb sinh 1 De 2 +λ 2 m (H-Y ) cos(λmZ) W λmsinh H 1 1 De 2 +λ 2 m + ∞ n=1 2(1+(-1) n-1 )γR ζscosh 1 De 2 +λ 2 n Z sin(λnY ) Hλncosh 1 De 2 +λ 2 n W + ∞ m=1 2(-1) m γ R ζb cosh(λm(H-Y ))cos(λmZ) Hλncosh 1 De 2 +λ 2 n W -∞ n=1
2(1+(-1) n-1 )γR ζscosh(λnZ)sin(λnY )

Hλncosh(λnW ) (3.4.13)
The corresponding ow rate in the rectangular cavity can be obtained by,

Q = 2 ˆW 0 ˆH 0 U (Y, Z) dY dZ (3.4.14)

Results and Discussions

Owing to the plug like velocity distribution of EOF in the commonly observed thin EDL limit, the most simplied approach towards modeling an EOF is by solving Stokes equation in the presence of Helmholtz-Smoluchowski slip velocity

(u HS = -ε f,r ε 0 ζ b E app /µ )

Eect of the channel aspect ratio

The channel aspect ratio (H/2W ) determines the eect of channel connement on the ow characteristics. In the contrast with the EDL thickness which is determined by the electrochemistry of the system, the channel geometry is controlled by the limitations of the manufacturing process involved and the physical properties of the material used. From Fig. 3.12, it is seen that the channel aspect ratio controls the eect of side wall generated ow on the overall velocity distribution and in some cases on the centerline velocity prole. If the channel aspect ratio is small, the side wall generated dynamics have a strong eect on the overall velocity prole whereas if the channel has a larger aspect ratio, the dynamics at the top and bottom boundaries dominate the ow prole. In the Fig. 3.12, the Debye number is taken as 0.1 so that the side wall generated EOF is more apparent.

Eect of wall and interfacial zeta potential

The magnitude and polarity of wall and interfacial zeta potential dictate the dominating polarity of charge distribution near the wall or the interface over which the uid actuating electric eld operates. From the Helmholtz-Smoluchowski relation, one can see that for a positive value of the electric eld, the direction of the near wall velocity is opposite to that of the wall zeta potential (see Fig. 3.12a and Fig. 3.13a).

However, the direction of the interfacial velocity follows that of the polarity of the interfacial zeta potential. This interesting observation can be attributed to the basic structure of the EDL where the ions contributing to the zeta potential of the substrate are strongly bonded to the surface. For a rigid and stationary substrate, this layer of immobile ions is followed by a layer of mobile oppositely charged ions. For example, if the surface has a negative zeta potential, the mobile ionic layer will be positively charged and under a positive electric eld will produce a positive velocity.

If the surface is positively charged, the mobile ionic layer will be negatively charged and will produce a negative velocity eld under a positive electric eld. However, at the interface, there is no rigid and immobile layer of ions and hence the velocity direction is the same as the polarity of the interfacial potential under a positive electric eld. This observation is evident in all the velocity plots. From

Flow rate

The net ow rate variation over a range of De manifests various regimes observed in such systems (see Fig. From Fig. 3.14b, it is observed that a reversed polarity of interfacial zeta potential has little eect on the ow rate variation when the Debye layer is thin (De = 0.01).

However, the reversed polarity of side wall zeta potential changes the ow rate signicantly for thin Debye layers (see Fig. 3.14a). The thin Debye layer case is special as it is the most commonly observed phenomena in experimental microuidics. Moreover, recent advances allow specic treatment of channel walls to alter their electrochemical properties, so that one can modulate the substrate zeta potential to overcome the limitations posed on ow rate by channel geometry through micro-fabrication constraints or the magnitude of the actuating eld.

Conclusion

The present work focuses on the role of steady and time-dependent Maxwell stress in free surface electro-osmotic ows. It was highlighted that the Maxwell stress generated dynamics introduces signicant velocity gradients at the free surface as compared to the studies where interfacial Maxwell stress was not taken into account.

Moreover, it was observed that for thick EDLs (or very thin lms) the non-Maxwell stress model signicantly underestimates the velocity and vorticity distributions in the uid lm. Such gradients are instrumental in the interfacial instability under a time-periodic actuation. The bulk penetration of near wall oscillations is inversely proportional to the forcing frequency. At lower frequencies, we observe a signicant suppression of velocity dispersion which asymptotes to the classical dispersion free velocity in a DC electro-osmotic ow. By the variation of interfacial polarity one can not only control the interfacial velocity and the rate of shear transfer but also establish a signicant phase dierence in uid velocity at dierent transverse locations in the uid which enhances the dispersion eects and can be eectively used in controlled species transport in microuidic devices. Further, in order to study the eect of interfacial Maxwell stresses in realistic rectangular channels, the solution of a free surface EOF in a microchannel of rectangular cross-section was achieved analytically. By taking into account a complete expression of the boundary condition at the free surface that includes the Maxwell stress at this interface, a parametric analysis of the ow in terms of the EDL thickness, channel aspect ratio and dierent uid-uid and solid-uid interfacial zeta potential was carried out. With the help of two-dimensional velocity contours, it has been shown that the bounding walls have a signicant eect on the velocity distribution in a rectangular microchannel. The ow controlling parameters in an EOF are not only the electro-chemical properties of the involved uid and electrolytes which control the EDL thickness and wall and interfacial zeta potentials but also the geometry of the channel through the channel aspect ratio. Such a study helps identifying proper control parameters for achieving a desired ow rate and velocity distribution in a microuidic device.

Long Wave Analysis of Thin Films Stability in Electrokinetics

4.1. Overview

The ever growing attention towards understanding the dynamics of thin liquid lms can be attributed to their ubiquitous presence all around us. Starting from biological entities like tear lm in the eyes or mucous linings in the organs, they are also found in man-made objects like bearings, paints, adhesives etc. As widespread is the availability of thin lms in the nature, equally extensive are the properties of the constituent uids and their physico-chemical interactions with their environments.

One of the major challenges in the lm dynamics is to understand and model their interfacial evolution and stability thresholds under the inuence of a wide range of inherent parameters like surface tension, viscosity, van der Waals forces and imposed static and time-dependent phenomena like gravity, temperature gradients and electric elds. Owing to the complex interaction of the mentioned phenomena, a detailed and all-inclusive analysis is required, which is not easy and still awaited. Creating and maintaining thin aqueous lms is a challenging task as one has to understand the roles of various mentioned dynamic phenomena at work that contribute to its rupture.

The role of hydrodynamics in lm stability has been investigated in a great detail in the literature but most of the related studies focus on a homogeneous uid without any other dispersed phase. Sometimes the role of the dispersed phase is limited to modify the surface tension, which is used in the classical dispersion relationships (characteristic stability equations). When the uid in question contains ionic species (either due to self-dissociation of the solvent or dissolved ions), their contribution to lm stability is relatively underexplored in the existing literature. To a large extent, the role of ions is again limited to surface tension modication, which for low concentrations can be safely ignored. However, near the bounding interfaces of thin lms the ionic distribution is not uniform and electro-neutrality is not valid (see Chapter 2). When an electric eld is applied to such a lm, the Coulombic forces acting on this net ionic charge distribution actuates a ow, also called as is negligible as compared to the characteristic length-scale. In such domains, the eect of the EDL on the ow is modeled using a slip velocity, which is obtained by the classical Helmholtz-Smoluchowski formulation and the bulk of the uid remains electro-neutral. This approximation has been substantially helpful to design the ow and mixing process in modern microuidic devices. However, with the technology of miniaturization foraying in the nanometric length scale with some applications like controlled delivery of non-conducting liquids in microuidics devices, patterned conformal coatings, and control of thin lms on electrostatic lm radiators in space applications [START_REF] Kim | The eect of an electrostatic eld on lm ow down an inclined plane[END_REF]], the uid dynamics within the limits of the EDL can not be ignored. It has been established that new modes of instability can be observed when the Debye length is comparable to the lm thickness [START_REF] Qian | Free-surface problems in electrokinetic micro-and nanouidics[END_REF]]. In some of the earlier works [START_REF] Qian | Free-surface problems in electrokinetic micro-and nanouidics[END_REF]; [START_REF] Sadiq | Weakly Nonlinear Stability Analysis of an Electro-Osmotic Thin Film Free Surface Flow[END_REF]; [START_REF] Joo | A new hydrodynamic instability in ultra-thin lm ows induced by electro-osmosis[END_REF]; Choi et al. ( 2010)], the eects of electric eld strength, surface tension and intermolecular van der Waals force on free surface instability, where the length scale is of the order of the Debye length have been discussed. However, the eect of electric eld generated stress (Maxwell stress) at the free surface has been overlooked. This Maxwell stress at the free surface is engendered due to the presence of free charges, the magnitude of which depends upon the uid properties and substrate uid interactions. It has been recently shown that due to the Maxwell stress, the free surface has an active role in electro-osmotic ows and should not be treated as a passive entity [START_REF] Choi | Is free surface free in micro-scale electrokinetic ows?[END_REF]].

In this chapter, the stability of free surface under DC and AC electro-osmotic ow in thin liquid lms is investigated, where the lm thickness can be varied over a range of electrical double layer thickness while considering the relative contribution from the van der Waals forces. The role of interfacial Maxwell stress on thin lm stability is highlighted. This conguration gives some interesting insights into the physics of free surface stability at a scale where various competing forces such as Coulombic force, van der Waals force and surface tension come into play. The eects of the mentioned forces is incorporated in the Navier-Stokes equations and linear stability analysis of the resulting governing equations is performed to obtain the Orr-Sommerfeld equations. Parametric dependence of interfacial stability on surface tension, disjoining pressure, applied electric eld, substrate zeta potential, free surface potential, lm thickness and Debye length is explored. The characteristic stability curve of the system is obtained through an asymptotic analysis of the Orr-Sommerfeld equations in the long wave limit. These equations are solved by the asymptotic long wave expansion method of [START_REF] Yih | Stability of Liquid Flow down an Inclined Plane[END_REF], as for thin viscous lms, the instability is found to occur at long wavelengths [START_REF] Oron | Long-scale evolution of thin liquid lms[END_REF]].

This chapter is divided in three sections. In the rst section, the characteristics of the physical system under investigation is presented. The contributions of various parameters to the governing equations are also worked out, and a generalized set of governing equations is presented. In the second section, linear stability analysis of DC EOF system is performed and the characteristic stability curve is obtained as a solution of the Orr-Sommerfeld equations while the role of various parameters is discussed. Finally, linear stability analysis of AC EOF system is performed followed by an extensive discussion on the parametric dependence of the stability of the system.

Thin lm stability under DC EOF

The system under study consists of a thin electrolyte lm spread over a rigid solid substrate exposed to an inert gaseous atmosphere (see Fig. 3.1). The lm thickness is denoted by h. The dynamics of such a lm is studied under the eect of a longitudinal oscillating electric eld, E app = E 0 i, where E 0 is the magnitude of the applied electric eld. The electrolyte concentration in this study is considered to be small enough in magnitude (∼ 0.1 -10mM ) in order to neglect the liquid property changes due to Joules heating [START_REF] Cetin | Eect of Joule heating on electrokinetic transport[END_REF]

Electric Potential Field

The total electric potential in dimensionless form in the conductive thin lm system can be written as (see Eq. 3.

3.2), Φ(X, Y ) = Φ app (X) + Φ sc (Y ) = - X E R + 1 sinh H De Z R sinh Y De + sinh H -Y De (4.2.1)
The corresponding electric eld can be calculated as E = -∇φ.

Hydrodynamic Equations

Considering a Newtonian uid under incompressible ow in the absence of an externally applied pressure gradient, the conservation of mass and momentum equations can be written as,

∇ • u = 0 (4.2.2) ρ ∂u ∂t + (u • ∇) u = -∇p d + ∇ • Σ T (4.2.3)
where, p d is the disjoining pressure. In situations involving thin lms where the Debye length is of the order of the lm thickness, the eect of intermolecular interactions cannot be ignored. This intermolecular interaction manifests itself in the form of a disjoining pressure term in momentum equations. It is represented as a pressure term in the momentum equations (see Chapter 2) and its dominant molecular component is dened as,

p d = - a 6πh 3 (4.2.4)
where, a is the Hamaker's constant and h is the lm thickness. At the solid substrate (y = 0), no-slip and no-penetration conditions are assumed. At the free surface, which is located at y = h(x, t), with an equilibrium height of h 0 , the jump of shear and normal stresses can be respectively written as, 2.9) where, W o =

t • Σ T • n = 0 (4.2.5) n • Σ T • n = γκ
∂U ∂X + ∂V ∂Y = 0 (4.2.7) Momentum: W o 2 ∂U ∂Θ +ReU ∂U ∂X +ReV ∂U ∂Y = - ∂P ∂X + A H 4 ∂H ∂X + ∂ 2 U ∂X 2 + ∂ 2 U ∂Y 2 +γ R E R ∂Φ ∂X ∂ 2 Φ ∂Y 2 (4.2.8) W o 2 ∂V ∂Θ + ReU ∂V ∂X + ReV ∂V ∂Y = - ∂P ∂Y + ∂ 2 V ∂X 2 + ∂ 2 V ∂Y 2 + γ R E R ∂Φ ∂Y ∂ 2 Φ ∂Y 2 (4.
∂U ∂Y + ∂V ∂X 1 - ∂H ∂X 2 -4 ∂H ∂X ∂U ∂X +γ R E R ∂Φ ∂X ∂Φ ∂Y 1 - ∂H ∂X 2 - ∂H ∂X ∂Φ ∂X 2 - ∂Φ ∂Y 2 = 0 (4.2.11) -P + γ R E R 2 ∂Φ ∂X 2 + ∂Φ ∂Y 2 + 2 1 + ∂H ∂X 2 ∂U ∂X ∂H ∂X 2 -1 - ∂H ∂X ∂U ∂Y + ∂V ∂X + γ R E R 1 + ∂H ∂X 2 ∂Φ ∂X 2 ∂H ∂X 2 + ∂Φ ∂Y 2 - 2γ R E R 1 + ∂H ∂X 2 ∂H ∂X ∂Φ ∂X ∂Φ ∂Y = ∂ 2 H ∂X 2 1 + ∂H ∂X 2 3 2 Ca (4.2.12)
where, Ca =

µu ref γ
is the capillary number. The dimensionless kinematic constraint at the free surface is,

W o 2 Re ∂H ∂Θ + U ∂H ∂X = V (4.2.13)
In the absence of any time periodic actuation, the viscous relaxation time can be chosen as the dominant time scale as, τ ref = h 2 0 /ν. In such a case, the Womersley number, Wo = 1. Accordingly, the velocity reference scale can also be set as the viscous velocity scale as, u ref = ν/h 0 . This leads to a Reynolds number, Re = 1.

One can obtain the equilibrium (base state) velocity prole for the system by assuming a steady, laminar and fully developed ow as (see Chapter 3),

U b (Y ) = -γ R 1 - Z R sinh Y De + sinh 1-Y De sinh 1 De (4.2.14)

Linear Stability Analysis

In order to study the stability behavior of the system under an external disturbance, the ow variables are perturbed by innitesimal disturbances as , where, α = 2πh 0 /λ L is the dimensionless wave number, λ L is the wavelength of the perturbation, and σ is the growth rate. Upon substituting the ow variables with the perturbations mentioned above in the Eqs. 4.2.7-4.2.13, linearizing and eliminating pressure the following Orr-Sommerfeld equation is obtained as,

U (X, Y, Θ) = U b (Y ) + Ũ (X, Y, Θ) (4.2.15) V (X, Y, Θ) = Ṽ (X, Y, Θ) (4.2.16) P (X, Y, Θ) = P b (Y ) + P (X, Y, Θ) (4.2.17) H(X, Θ) = 1 + H(X, Θ)
∂ 2 ∂Y 2 -α 2 2 Ψ(Y )-iαU b ∂ 2 ∂Y 2 -α 2 Ψ(Y )+iα d 2 U b dY 2 Ψ(Y ) = σ ∂ 2 ∂Y 2 -α 2 Ψ(Y ) (4.2.22)
The boundary conditions using the normal mode representation of the perturbation parameters can be written as,

Ψ(0) = ∂ 2 Y Ψ(0) = 0 (4.2.23) ∂ 2 ∂Y 2 + α 2 Ψ(1) + H d 2 U b (1) dY 2 -iαγ R E R H ∂Φ ∂X 2 - ∂Φ ∂Y 2 = 0 (4.2.24) ∂ 2 ∂Y 2 -3α 2 Ψ(1) -σ ∂ Ψ(1) ∂Y -iα U b (1) ∂ Ψ(1) ∂Y -Ψ(1) dU (1) dY = iα H α 2 Ca -A (4.2.25)
In thin lm stability problems, the long wave analysis of the resulting Orr-Sommerfeld equations yields results that capture the essential features of the full numerical solution of the complex set of equations. Hence the stability information of thin lm systems can be recovered without solving the complete set of equations. Here, Yih's method [START_REF] Yih | Stability of Liquid Flow down an Inclined Plane[END_REF]] is used to expand the dependent variables like Ψ and σ in powers of α and solve equations at zeroth and rst orders in α. As a consequence we use the developments, The characteristic stability curves can be graphically presented by plotting the variation of real part of the growth rate, σ R vs α in the range of α ≤ O(1) i.e. in the long wave limit.

Ψ = Ψ0 + α Ψ1 + α 2 Ψ2 . . . (4.2.26) σ = σ 0 + ασ 1 + α 2 σ 2 . . .
The ow actuating mechanism in this study is the externally applied electric eld, which acts as a body force in the uid bulk and manifests itself as the Maxwell stress on the free surface. Its contribution is reected in the two dimensionless numbers namely, γ R and E R . It has to be noted that, by denition, the mentioned dimensionless numbers also depend upon another imposed eld variable -the substrate zeta potential. To study the eect of the imposed electric eld and the substrate zeta potential separately, the following combination of γ R and E R is presented. By varying the ratio (γ R /E R ) one can study the eect of the variation of the electric eld on the stability of the system. While selectively varying the product (γ R E R ) one can study the eect of the variation of the substrate zeta potential on the stability of the system. The other important parameters which have a signicant eect on free surface stability are, the dimensionless Hamaker constant (A), the Debye number (De), the capillary number (C a) and the ratio of the interfacial to substrate zeta potential (Z R ). The real part of the growth rate, σ R as obtained from the solution of the Orr-Sommerfeld equations from asymptotic analysis, can be written as,

σ R = f (Ca)α 4 + g (A, γ R , E R , Z R , De) α 2 (4.2.29)
where,

f (Ca) = - 1 3Ca (4.2.30) g (A, γ R , E R , Z R , De) = A + γ R 2E R -cosech 1 De -Z R coth 1 De 2 γ R E R - 1 De -4De cosech 1 De -5coth 1 De De γ 2 R Z R + 5 48De 4 - 11 24De 2 - 1 2 + 1 De -4De coth 1 De γ 2 R Z 2 R -5cosech 1 De Deγ 2 R Z 2 R + 9De 2 γ 2 R Z 2 R -9De 2 γ 2 R Z R (4.2.31)

Results and Discussion

For xed values of the parameters for a typical water-air system: ρ ∼ 10 3 kg/m 3 , γ ∼ 0.072N/m, a ∼ 10 -19 -10 -20 J , ν ∼ 10 -6 m 2 /s, ε r ∼ 80, ε 0 ∼ 8.85×10 -12 F/m is the permittivity of vacuum. The typical values of controllable parameters are taken as, Σ M from the total stress term, Σ T in the free surface boundary conditions. This connes the contribution of the electric eld on the hydrodynamics as a body force in the momentum equations. The second characteristic stability curve was obtained by keeping the Maxwell stress term in the total stress at the free surface boundary condition. Upon plotting the real part of the growth rate against the wave number for both cases, it was found that ignoring the Maxwell stress on the free surface over-estimates the instability (see Fig. 4.1). This over prediction can signicantly aect the sensitive dependence of the stability of the system which is a function of a large number of the mentioned parameters. The parametric dependence of the stability of the system will be discussed henceforth within the purview of the model that includes the contribution of the Maxwell stress at the free surface. The order of magnitude analysis on Eq. 4. Fig. 4.4b). From these gures it is observed that upon increasing the disjoining pressure, the system becomes unstable. This phenomenon can be explained by the fact that increasing the disjoining pressure pushes the system away from the mechanical equilibrium of the thin lm hence making the system unstable. The eect of the externally applied electric eld, which can be reected through the ratio, γ R /E R , on the stability of the system is clearly observed from Eq. 4.2.31 which shows that the increase of the applied electric eld makes the system more unstable. This is an expected phenomenon, as upon increasing the magnitude of the electric eld, E 0 , the unbalanced interfacial stress components due to the resulting Maxwell stress increase, pushing the system away from equilibrium. This eect is shown in Fig. 4.5 for two values of Ca.

ζ b = 1 -100mV , E 0 = 1 -100kV /cm
2 γ R E R =1 (No interfacial Maxwell stress) γ R E R =100 (No interfacial Maxwell stress) γ R E R =1 (With interfacial Maxwell stress) γ R E R =100 (With interfacial Maxwell stress)
γ R /E R = 0.01 γ R /E R = 0.25 γ R /E R = 1 γ R /E R = 25 γ R /E R = 100 (a) Wave number, α Real part of growth rate, σ R 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 4 γ R /E R = 0.01 γ R /E R = 0.25 γ R /E R = 1 γ R /E R = 25 γ R /E R = 10 (b)
It is also known that, inuenced by the nature of substrate-uid interactions and uid properties, the interfacial zeta potential is determined and can be measured experimentally. Under electro-osmotic ows, the eect of the substrate zeta potential ζ b has a signicant eect on interfacial stability of thin lms. This is shown in Fig. 4.6 for two characteristic values of γ R E R , De, Ca and xed value of A. As can be seen from the expression of the growth factor (Eqs. 4.2.29-4.2.31), the product γ R E R , which is proportional to ζ 2 b has a negative coecient, and leads to a decrease in the growth factor when the substrate zeta potential is increased. Hence, when the substrate zeta potential is increased, the system becomes more stable which is clearly shown in Fig. 4.6a for Z R = -1. However, when the interfacial polarity is same as the substrate, an opposite behaviour is seen (Fig. 4.6b), which is explained below.

Wave number, α

Real part of growth rate, From the basic state velocity distribution (see Eq. 4.2.14), one can see that depend-ing upon the interfacial zeta potential the interface can either reduce the interfacial stress or enhance it. This is also shown in Fig. 4.7 where one can also see that when the interface has an opposite polarity as compared to the substrate, it reduces the interfacial stress and when the free surface polarity is the same as the substrate, it supports the interfacial stress. This relation between interfacial polarity and interfacial stress also manifests itself in the system stability equations as expected. It is observed that the interfacial polarity with respect to the substrate zeta potential, which is represented by the ratio Z R , tends to increase the system stability as it reduces the interfacial stress when Z R is negative i.e. of the opposite polarity as the substrate. The system becomes more unstable when Z R is positive, i.e. of the same polarity as of the substrate as it enhances the interfacial stress. This eect is shown in Fig. 4.8. This can also be attributed to the impact of substrate zeta potential on the distribution of charged species in the bulk of the uid. As the zeta potential of the interface increases, the concentration of counter ion species increases near the substrate, decreasing the ionic concentration in the bulk which screens the magnitude of the net electric potential as felt at the interface. This decreases the contributions of the charged substrate to the interfacial Maxwell stress hence making the system more stable. This mechanism is also clearly highlighted in Fig. 4.6 which shows the growth rate variation for two opposite values of Z R with γ R E R as the varying parameter and all the other parameters being xed. It is observed from this gure that the stability of the system increases upon increasing De. This can be explained by the fact that for a given lm thickness h 0 , an increase in Debye number corresponds to an increasing Debye length, which in turn corresponds to a decrease in ionic concentration as,

σ R 0 0.2 0.4 0.6 0.8 1 -4 -2 0 2 γ R E R = 0.01 γ R E R = 0.25 γ R E R = 1 γ R E R = 100 γ R E R = 2500 (a) Wave number, α Real part of growth rate, σ R 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1.5 2 γ R E R = 0.01 γ R E R = 0.25 γ R E R = 1 γ R E R = 100 γ R E R = 2500 (b)
)Z R = -1, (b)Z R = 1. U b Film height (Y) -2 -1.5 -1 -0.5 0 0 0.2 0.4 0.6 0.8 1 Z R =-1 Z R =-0.5 Z R =0 Z R =0.5 Z R =1
λ D ∝ 1 √ c 0
, where c 0 is the ionic concentration. Hence, a decrease in the ionic concentration drives the system naturally towards stability. The ow actuating mechanism in this problem is the Coulombic force by the external electric eld on to the distribution of ions in the uid. If the ionic concentration c 0 is low, so will be the net forces by the external electric eld on the ionic solution. Hence, in this limit of large De, the ow will be reduced and consequently the interface will stay undisturbed.

Conditions leading to the onset of instability can be complemented by a study of a mode that corresponds to the maximum growth rate of a disturbance. This mode, which is characterized as the most dangerous mode (α critical ), is studied as a function of the set of parameters mentioned above. The most dangerous mode is the wave number for which the characteristic stability curve (Eq. 4.2.29) of a system reaches a maximum. The resulting wave number is obtained as a function of dimensionless parameters through the following equation: It is also observed that the increase in the value of surface tension decreases the value of α critical , which is consistent with the observation that the increase in surface tension increases the stability of the system by decreasing the most dangerous mode.

α critical = -g (A, γ R , E R , Z R , De) 2f (Ca) (4.2.32) with a condition that g (A, γ R , E R , Z R , De) is positive.
It is observed that the most dangerous mode is independent of the Debye number for λ D > h 0 /2 (see Fig. 4.11) and suggests that the most dangerous wavelength is of the order of the lm thickness in this range of De, independently of the other parameters. For a given value of the Debye number, α critical increases with the wall zeta potential (γ R E R ) (see Fig. 4.11). Finally, the most dangerous mode is independent of the polarity of the interface zeta potential (Z R ) and increases with its magnitude (see Fig. 4.12). Also, the increase in the magnitude of the external electric eld (γ R /E R ) increases the value of the most dangerous mode.

Thin lm stability under AC EOF

The system under study consists of a thin electrolyte lm spread over a rigid solid substrate exposed to an inert gaseous atmosphere (see Fig. 3.1). The lm thickness is denoted by h 0 . The dynamics of such a lm is studied under the eect of a longitudinal oscillating electric eld, E app = E 0 sin (ωt) i, where E 0 is the magnitude and ω is the frequency of the applied electric eld. The total electric potential in the conductive thin lm system can be written as (see Eq.

3.3.2), Φ(X, Y, Θ) = Φ app (X, Θ) + Φ sc (Y ) = - X E R sinΘ + 1 sinh H De Z R sinh Y De + sinh H -Y De (4.3.1)
where, Θ = ωt is the dimensionless time.

Hydrodynamic Equations

For the dimensionless conservation equations one can refer to the Eqs. 4.2.7-4.2.13.

Linear Stability Analysis

The perturbations in the ow variables are introduced as, U = U b + Ũ , V = Ṽ , P = P b + P and H = 1 + H. The velocity components are converted into stream function using Ũ = ∂ Ψ/∂Y and Ṽ = -∂ Ψ/∂X. The normal mode solutions to the perturbations are considered with small amplitude and with long wavelength

(λ L h 0 ) as, Ψ(X, Y, Θ) = Ψ(Y, Θ)e iαX (4.3.2) P (X, Y, Θ) = P (Y, Θ)e iαX (4.3.3) H(X, Θ) = H (Θ) e iαX (4.3.4)
where,α = 2πh 0 /λ L is the dimensionless wave number and λ L is the wavelength of the perturbation. Upon substituting the ow variables with the perturbations mentioned above in the Eqs. 4.2.7-4.2.13, linearizing and eliminating pressure the following Orr-Sommerfeld equation is obtained as,

∂ 2 ∂Y 2 -α 2 2 Ψ(Y, Θ) -W o 2 ∂ ∂Θ + iαReU b ∂ 2 ∂Y 2 -α 2 Ψ(Y, Θ) +iαRe ∂ 2 U b ∂Y 2 Ψ(Y, Θ) = 0 (4.3.5)
The boundary conditions using the normal mode representation of the perturbation parameters can be written as,

Ψ(0, Θ) = ∂ 2 Ψ(0, Θ) ∂Y 2 = 0 ∂ 2 ∂Y 2 + α 2 Ψ(1, Θ) + H (Θ) ∂ 2 U b (1, Θ) ∂Y 2 -iαγ R E R H (Θ) ∂Φ ∂X 2 - ∂Φ ∂Y 2 = 0 ∂ 2 ∂Y 2 -3α 2 Ψ(1, Θ) -W o 2 ∂ 2 Ψ(1, Θ) ∂Θ∂Y -iαRe U b (1, Θ) ∂ Ψ(1, Θ) ∂Y -Ψ(1, Θ) ∂U b (1, Θ) ∂Y = iα H (Θ) α 2 Ca -A (4.3.6)
Using Floquet theory for the above eigen value problem, the solution for the perturbation variables can be expressed as, Ψ(Y, Θ) = Ψ(Y, Θ)e σΘ and H(Θ) = Ĥ(Θ)e σΘ where, σ is the Floquet exponent, and Ψ(Y, Θ) and Ĥ(Θ) are time periodic functions.

Upon using asymptotic expansions in small parameter (α 1 ) and expanding the parameters as,

Ψ = Ψ0 + α Ψ1 + α 2 Ψ2 . . . (4.3.7) Ĥ = Ĥ0 + α Ĥ1 + α 2 Ĥ2 . . . (4.3.8) σ = σ 0 + ασ 1 + α 2 σ 2 . . . (4.3.9)
Upon solving the resulting set of equations at dierent orders of α (see Appendix B), the characteristic equation of the system was obtained as,

σ = f (α, γ R , Z R , E R , W o, Re, Ca, De) (4.3.10)

Results and discussions

System Parameters

To estimate the typical values of the dimensionless parameters, an aqueous solution is considered as the working uid where the transport coecients are taken to be of water at the normal temperature and pressure, viz. ρ ∼ 10 3 kg/m 3 , µ ∼ 10 -3 P a • s, ε r ∼ 80, ε 0 ∼ 8.85×10 -12 is the permittivity of vacuum, γ ∼ 0.072N/m is the surface tension between water and air, a ∼ 10 -20 J is the Hamaker constant, the substrate zeta potential is taken as ζ b ∼ 10mV , the applied electric eld, E app ∼ 1kV /cm with a frequency, ω of 1MHz. The characteristic electro-osmotic velocity, u HS can be therefore estimated as 1mm/s. For a lm thickness, h 0 ∼ 100nm, the dimensionless Hamaker constant, A is ∼ 0.1, the electro-viscous ratio, γ R ∼ 1, the ratio E R ∼ 1, Capillary number, Ca ∼ 10 -5 , Reynolds number, Re ∼ 10 -4 . The ow control parameters are varied over a range to illustrate the parametric dependence of the free surface stability of the system.

The ionic concentration in the system is considered to be low (c 0 ∼ 0.1mM ) which gives a Debye length (λ D ) of the order of 30nm. Such a small ionic concentration reduces the non-linear dynamics of ions. The resulting Debye number De varies from 0.01 -0.1.

Instability Mechanism 1. Contribution of Capillary and Disjoining Pressure

The stability of a thin lm under electro-osmotic ow can be attributed to the competing dynamics between the capillary forces through Laplace pressure, van der Waals forces through disjoining pressure and electrostatic forces through the electroosmotic pressure. From the right hand side of the Eq. 4.3.6 we can see that the terms representative of respectively, Laplace pressure (1/C a) and disjoining pressure (A) appear together at the free surface boundary condition in a counterbalancing manner, showing the existence of conicting forces.

When the lm surface is perturbed by a small amplitude disturbance, the induced curvature forces the local Laplace pressure to become greater (or smaller) than the forces in the bulk, creating an outow (or inow) of liquid restoring the equilibrium conguration of the lm. However, the long range nature of the disjoining pressure has a permanent eect on the lm dynamics. A negative disjoining pressure (Π < 0)

between the interfaces leads to attraction between them forcing a lm breakup while a positive disjoining pressure (Π > 0) leads to repulsion between the interfaces, causing a lm build-up (see Fig.

Contribution of EDL

In order to identify that out of the two phenomena, entropic and Coulombic, which one has a dominating contribution towards a lm stability, the following discussion is presented. The diused cloud of counter-ions in an EDL is maintained in an equilibrium through mutual repulsions which forces them away from the oppositely charged substrate (or interface) and hence leads to a congurational entropy. When two such ordered charged clouds (EDLs) are brought closer through a perturbation (e.g. EDL near a free surface is brought closer to an EDL near a solid surface), a repulsive force initiates between the two charged clouds, restoring the equilibrium and stabilizing the lm. From Fig. 4.14, one can observe that a charged free surface (Z R = 0 ) over a charged substrate (γ R E R = 0), is relatively more stable than an uncharged free surface (Z R = 0). The case of a charged free surface over a charged substrate creates two interfaces with diused charge distribution following the stability dynamics mentioned above. Moreover, the symmetry observed in the marginal stability curves (see Fig. 4.14) about Z R = 0 justies the entropic rather than Coulombic dominance on the EDL pressure, where the polarity of the free surface charge cloud does not aect the stability of the lm.

One of the other important parameters associated with the EDL is the extent of the diused charge penetration in the bulk. This extent of the diused charge is characterized by the Debye length (λ D ). The relative extent of the EDL thickness as compared to the lm thickness is represented in this work through the Debye number,

De. For thin EDLs (small De) one can imagine a closer packing of diused ions leading to a higher congurational entropy and hence greater repulsion between the interfaces, leading to a more stable lm. This idea is also observed in the Fig. 4.14

where thinner EDL (De = 0.01) (Fig. 4.14a) is more stable than a thicker EDL (De = 0.1) (Fig. 4.14b).

Contribution of the oscillating Electric eld

An oscillating electric eld acting on a charged interface introduces a time-dependent dispersive eld near the interfaces (see Fig. surface is dependent upon the strength of this vortex which is a function of various parameters like De, which accounts for the diusive extent of the electrical eects in the bulk, the strength of the applied electric eld (γ R /E R ) and the strength of interfacial polarity (γ R E R and Z R ). However, it is also known that any deformation in such an interface is countered by a dissipating viscous stress. The strength of this viscous damping mainly depends upon parameters like the coecient of viscosity and Wo, which accounts for the diusive extent of the viscous eects in the bulk.

The two competing mechanisms mentioned above contribute to the neutral stability characteristics of the system. Upon changing the Reynolds number by keeping all the other parameters xed, which is equivalent to changing the dynamic viscosity, one observes from the marginal stability curves (α c , Wo) that more viscous uids (smaller Re) are more stable as compared to less viscous uids (larger Re) (see Fig. 4.15c). Thin EDLs (smaller De) owing to their smaller spatial extent of charge have high velocity gradients as compared to thicker EDLs (larger De). This is also observed in Fig. 4.15a. Hence, for thin EDLs (smaller De) the lm is expected to be more unstable as compared to lms with thicker EDLs (see Fig. 4.15b). The electroosmotic velocity distribution in the lm is directly proportional to the strength of the applied electric eld. Hence, upon increasing the strength of the applied electric eld, the strength of the free surface vortex is enhanced thus leading to a more unstable system (see Fig.

Conclusion

In this chapter, a detailed analysis of the stability of a free surface of an ultra thin liquid lm under steady and time-dependent electro-osmotic ow conditions was presented. Through long wave asymptotic analysis of the Orr-Sommerfeld equations, parametric stability thresholds of a thin aqueous lm has been explored with a focus on parameters like surface tension, Hamaker's constant for uid-substrate interaction, magnitude of the externally applied electric eld, substrate and free (c) The increase in the value of substrate zeta potential tends to stabilize the ow which in turn enhances the osmotic pressure component.

However, due to a complex interaction of all the above phenomena together, the individual stability thresholds overlap, generating interesting stability trends which are tunable over a wide range of the above mentioned parameters. Such a generalized analysis helps identifying parametric boundaries for sustaining thin lms over a wide range of uid properties and operating conditions.

Part III.

Experimental Studies

Microfabrication and Experimental Setup

Overview

In order to observe and parametrize the interfacial instability between two miscible and immiscible uids, an experimental setup was established to create a microuidic two uid interface, with a possibility of imposing controlled ow rate and electroosmotic ow. To that end, various microuidic chips were fabricated, which were interfaced with syringe pumps for pumping liquids and electric potential generator to actuate the electro-osmotic ow. Another objective of the electro-osmotic experimental setup was to correlate the substrate zeta potential to the electro-osmotic velocity, which by virtue of its non-dispersive nature is easy to measure by velocimetry techniques. Although, this is one of the classical approaches to quantify the substrate zeta potential, it was used as a validation case towards assuring a working electro-osmotic ow setup.

In this chapter an overview of microuidic device fabrication and the details of the experimental apparatus used to create a two uid electro-osmotic ow is provided.

Experimental microuidics has been a widely explored eld which basically starts with fabrication of the microdevices, setting up of the sample injection and control, image acquisition setup and analysis of the results. Although, each of the mentioned steps requires a high degree of precision and control, for each of them, a wide range of choices are available depending upon one's research objective and the resources.

One can either mount a very basic microuidic setup to perform exploratory research or build up an extremely controlled and isolated microuidic environment for critical applications such as in the bio-medical eld.

The rst step towards fabrication of a microuidic device is the choice of the material which embeds the motif of the channel. Grossly, one can categorize the existing materials into soft materials such as Elastomers (e.g. PolyDiMethylSiloxane (PDMS), PolyMethylMethAcrylate (PMMA), NOrland Adhesive 81 (NOA), etc.), Thermoplastics [START_REF] Tsao | Bonding of thermoplastic polymer microuidics[END_REF]], paper [START_REF] Li | A perspective on paper-based microuidics: Current status and future trends[END_REF]] and hard materials like glass, silicon, etc. As each of the mentioned materials have dierent physico-chemical properties, the methodology of microfabrication also varies accordingly. Among the mentioned microfabrication materials, one of the most commonly used materials is PDMS, an elastomer which is well-known for its rapid prototyping, low cost and good optical properties. All the mentioned characteristics of PDMS elastomer make it a good candidate for exploratory research. In this work, the microfabrication was carried out on using PDMS elastomer and hence the methodologies specic to PDMS elastomer will be discussed. Further, the details leading to sample injection in the microchannel are provided where the sample consists of a set of miscible uids namely, aqueous electrolyte and de-ionized water and a set of immiscible uids namely, an aqueous electrolytic solution and an oil. Finally, the details of the image acquisition system is provided which included a basic micro-PIV measurement and a phase-contrast observation of the interface between two uids.

Microfabrication

In order to build microchannels using a PDMS elastomer by the help of soft lithography [START_REF] Xia | Soft lithography[END_REF]], one has to strictly follow a well dened protocol in order to enhance the reproducibility. During the experimental studies, a number of microchip designs were fabricated with variable shapes and sizes and experiments were performed. Although the basic fabrication process of all the microchips remained the same, the fabrication process of a representative chip is discussed further. The representative chip is a cross shaped channel with reservoirs at its ends for inserting the liquids and the electrodes. This kind of chip facilitates the creation of two-uid interface which is exposed to an AC electric eld by the electrodes (see wafer is soaked in a bath of PGMEA for 11 minutes. Further, the wafer is rinsed by Iso-Propanol and blow dried by a jet of Nitrogen gas. If some white traces remain on the wafer, it is soaked again in the bath of the PGMEA and the process is repeated until a clean wafer with the embossed microchannel design is obtained. This henceforth will be referred as the master mold.

Preparation of the PDMS elastomer: For preparing the microchip, around 30grams (g) of SYLGARD 184 PDMS elastomer was mixed with 3g curing agent (10 : 1 w/w ratio and referred to as PDMS-1 later) and degassed using a vacuum chamber until no bubbles were visible anymore. Similarly, around 20g of SYLGARD 184 PDMS elastomer was mixed with 4g of curing agent (5 : 1 w/w ratio and referred to as PDMS-2 later) and degassed. The PDMS-2 preparation is used for sealing o the PDMS-1 microchannel reservoirs (see Fig. 5.4) and coating a glass microscope slide on which the nished PDMS-1 microchip was to be placed. Using a dierent concentration of curing agent-PDMS leads to a better bonding between the layers. The glass slide provides rigidity to the microchip and facilitates its placement on the microscope chipholder. Using a glass slide coated with PDMS-2 also leads to an all-PDMS wall of the microchannel for having a consistent and symmetric zeta-potential prole (see Chapter 6 for details).

Preparation of the glass slide: The glass slide was cleaned with Iso-Propanol and Acetone and blow dried by a jet of Nitrogen gas. It was exposed to plasma for 1min and was then spin-coated by the PDMS-2 solution. The spin-coating was performed at 1750rpm for 30s at an angular acceleration of 300rpm/s.

PDMS curing: The master mold was placed in a Petri dish and the resulting PDMS-1 solution was poured over it. The master mold with the PDMS-1 solution was degassed again to remove any remaining trapped air bubble.

Then it was placed in an oven at a temperature of 65°C for 25mins along with the spin-coated glass slide. The remaining PDMS-2 solution was poured on an empty Petri dish and placed in the oven along with the other solutions.

Preparation of the microchip: The cured PDMS-1 is peeled o from the master mold and the microchannel design is cut-o from it. Now, holes are punched in the microchannel to create the reservoirs and the surface is cleaned using an adhesive tape (Scotch tape). Both of reservoirs attached to the microchannel were 5mm is diameter with a height of ∼ 7mm (see Fig. 5.4). There were two holes punched in the top PDMS layer, one for the liquid carrying tube and another for the Platinum electrode. Both, the PDMS-1 microchannel and the PDMS-2 coated glass chip are exposed to the plasma for 45s and bonded together (see Fig. 5.4). Finally, the cured PDMS-2 layer in the Petridish was cutout in order to cover the reservoirs on the PDMS-1 channel. For each reservoir, two 1mm holes were punched on the PDMS-2 cover, one for the inlet tube and another for the electrode. This PDMS-2 cover was exposed to plasma and placed on the top of the PDMS-1 layer, covering the reservoirs.

This nal assembly of a microchip was placed in an oven at 65°C for 48 hours to enhance bonding between the layers before being used for the experiments. The microscope was interfaced to a computer by a USB camera. The images and the videos made for this study were of the resolution 508 × 384 pixels with a minimum exposure time of 45.28µs (for 10×) and 90.56µs (for 60×) at a frame rate of 26f ps.

Electric Field Generator

The electric eld was applied to the microuidic system by an Agilent 6811B AC power source which had a power rating of 325V A with a RMS voltage of 300V

(see Fig. 5.6b). For the rst part of the experiments, a DC voltage in the range of 50 -400V was applied and for the second part of the experiments, an AC voltage in the range of 50 -300V was applied along with frequencies varying from 0.1 -20Hz.

A platinum wire was used to apply electric eld to the reservoirs attached to the 6. Experimental Investigation of the Electrokinetic Instability 6. 1. Overview In this chapter the experimental investigation of the electro-kinetically induced interfacial instability between two immiscible liquids is studied. To that end, rstly, the EOF setup was tested for a velocity of a combined pressure-driven and electroosmotic ow eld obtained using the Micro-Particle Tracking Velocimetry (µ-PTV)

technique. This helps to identify the liquid-substrate zeta potential, which is one of the most important parameters required to characterize the EOF in a system.

The substrate zeta potential is a function of the various parameters such as, electrolytic concentration, pH of the solution, ionic anity of the substrate, etc. One of the most popular methods used to estimate the substrate zeta potential is by obtaining the velocity eld in an EOF. As, EOF has a plug type velocity prole for thin Debye layers, one can use the Helmholtz-Smoluchowski relationship to obtain the substrate zeta potential after having known the magnitude of the velocity and magnitude of the applied electric eld. Next, an extensive study of electro-kinetic interfacial instability between two immiscible liquids is presented.

Combined Pressure-driven and Electro-Osmotic Flow

One of the most important parameters in order to characterize the EOF in a microchannel, is the substrate zeta potential. One way to obtain this parameter is to measure the EOF velocity in a channel and using Helmholtz-Smoluchowski relationship (see Chapter 2) [START_REF] Saville | Electrokinetic Eects with Small Particles[END_REF]] with known liquid properties such as permittivity (ε r ε 0 ), dynamic viscosity (µ), and the magnitude of the applied electric eld (E app ).

However, obtaining an ideal EOF in an experimental scenario is not easy and multiple factors inuence the measured velocity prole such as, back pressure due to non-similar reservoirs, softness of the substrate material and variation in the liquid properties due to Joules heating etc. Investigating a combined pressure-driven and electro-osmotic ow serves a two-fold purpose, rstly, one can estimate the eect of an induced back-ow on the EOF velocity distribution and secondly, it helps to understand the role of the imposed ow rate during the electro-kinetic interfacial instability of liquids. Hence, an analytical model was developed to explore the uid ow in a rectangular microchannel under a combined eect of an electric eld (E app ) and a applied ow rate (Q P ). The channel under study has a rectangular cross-section (2H × 2W ) and all the walls are assumed to be at a constant zeta potential, ζ b (see Fig.

The electric potential distribution

The electric potential distribution in the microchannel is a combination of the electric potential due to the ionic space-charge distribution and the electric potential due to the applied potential bias. The potential distribution due to the ionic spacecharge distribution in a rectangular channel can be obtained by the 2D form of the Poisson-Boltzmann equation (see 2.3.11), which for low substrate zeta potential (i.e.

with Debye-Hückel approximation) (see 2.3.14) in the non-dimensional form can be written as,

∂ 2 Φ sc ∂Y 2 + ∂ 2 Φ sc ∂Z 2 = Φ sc De 2 (6.2.1)
As the system is symmetric across Y = 0 and Z = 0, the symmetry boundary conditions can be written as,

∂Φ sc ∂Y (0, Z) = ∂Φ sc ∂Z (Y, 0) = 0 (6.2.2)
The channel walls are assumed to be at a constant zeta potential, so the respective boundary conditions are,

Φ sc (H, Z) = Φ sc (Y, W ) = 1 (6.2.3)
6.2 Combined Pressure-driven and Electro-Osmotic Flow Upon solving Eqs.6.2.1-6.2.3, the resulting ionic space charge induced potential distribution can be written as,

Φ sc (Y, Z) = ∞ m=1 2(-1) m-1 cosh 1 De 2 +λ 2 m Z cos(λmY ) Hλmcosh 1 De 2 +λ 2 m L + ∞ n=1 2(-1) n-1 cosh 1 De 2 +λ 2 n Y cos(λnZ) W λncosh 1 De 2 +λ 2 n H (6.2.4)
Now the electric potential eld due to the externally applied electric eld can be written as,

dΦ app dX = - E app D h ζ b = - 1 E R (6.2.5)

The velocity distribution

The momentum conservation equation for a Newtonian electrolytic solution with steady, incompressible, laminar and fully developed dynamics, and under the combined inuence of an externally applied pressure gradient and electric eld can be written as (see Chapter 2),

0 = - dp dx + µ ∂ 2 u ∂y 2 + ∂ 2 u ∂z 2 + ε r ε 0 ∂φ ∂x ∂ 2 φ ∂y 2 + ∂ 2 φ ∂z 2 (6.2.6) ∂ 2 U ∂Y 2 + ∂ 2 U ∂Z 2 = dP dX + γ R ∂ 2 Φ ∂Y 2 + ∂ 2 Φ ∂Z 2 (6.2.7)
where, P = U (H, Z) = U (Y, W ) = 0 (6.2.9)

The above system of equations is linear and hence once can split the velocity eld as a combination of the pressure-driven velocity eld (U P ) and the electro-osmotic veloc- ity eld (U E ). The net velocity eld is then, U (Y, Z) = U P Y, Z, dP dX +U E (Y, Z, γ R ).

Since, a syringe pump is used in the experiments, which imposes a known ow rate rather than a pressure gradient, the pressure gradient term, dP dX can be expressed in the terms of the ow rate by calculating the dimensionless ow rate as,

Q P = 4 ˆW 0 ˆH 0 U P (Y, Z) dY dZ (6.2.10)
The resulting velocity eld for a combined pressure-driven and EOF can be obtained as,

U (Y, Z) =Q P H 2 2 1 -Y 2 H 2 + ∞ m=1 4(-1) n cosh(λmZ)cos(λmY ) λ 3 m H 3 cosh(λmW ) 4H 3 W 3 - ∞ m=1 8sinh(λmW ) λ 5 m Hcosh(λmW ) + γ R ∞ n=1 2(-1) n λ n H      cosh (λ n Z) cosh (λ n W ) - cosh 1 De 2 + λ 2 n Z cosh 1 De 2 + λ 2 n W      cos (λ n Y ) + γ R ∞ q=1 2(-1) q λ q W      cosh (λ q Y ) cosh (λ q H) - cosh 1 De 2 + λ 2 q Y cosh 1 De 2 + λ 2 q H      cos (λ q Z) (6.2.11)
where, λ n|m = (2(n|m)-1)π

2H

and λ q = (2q-1)π

2W

. Here, the rst term of the velocity eld is due to the imposed pressure gradient through an applied ow rate (Q P ) and the second and third terms are the contributions from the EOF (γ R ).

Velocity characterization using µ-PTV

The objective of these set of experiments was to create a basic EOF in a straight microchannel with a rectangular cross-section, and characterize the velocity prole in order to predict the PDMS-aqueous solution zeta potential. To that end a basic EOF setup was established (see Chapter 5 for details) and an EOF was created while seeding the uid by micro-particles. Further, Particle Tracking Velocimetry (PTV) was used to obtain the velocity distribution in the system.

PTV is a velocimetry technique which can be understood as a low particle density Particle Image Velocimetry (PIV) [START_REF] Malik | Particle tracking velocimetry in three-dimensional ows[END_REF]]. This technique measures the Lagrangian velocity of the medium by tracking the displacement of isopycnic (same density) particles over a period of time. The velocity is computed by analyzing two consecutive image frames for the displacement of a particle and dividing it by the time interval between the two frames. Although it is not as accurate as PIV to reproduce velocity elds with high spatial distribution, because the velocity distribution information obtained is quite sparse. It is however quite fast and easy to setup. In order to get the particle images, the liquid seeded with the particles is introduced in the microchannel and is illuminated by a light source. The image sequence is recorded by a camera and then analyzed by PTV Lab [START_REF] Brevis | Integrating cross-correlation and relaxation algorithms for particle tracking velocimetry[END_REF]], a MATLAB based open-source PTV toolbox.

Experimental Procedure

Preparation of the liquid sample

The EOF of an aqueous electrolyte is very sensitive to the electro-chemistry of the system, specically, the salt solubility and ionic conductivity, which in turn are strongly aected by the ambient parameters such as room temperature and pressure.

Moreover, the application of an external electric eld sets o a series of Faradaic reactions, changing the ionic concentration in the reservoirs and hence creating a concentration and pH gradient in the electrolyte. Such properties in turn aect the ow parameters such as substrate zeta potential, Debye length, solvent permittivity and viscosity. So, in order to enhance the reproducibility, one has to ensure constant ionic conductivity, constant pH and constant ambient conditions. To that end, one can prepare the electrolytic solution in bulk and store it in controlled ambient conditions.

The rst step towards preparing an electrically conductive solution for EOF is to prepare a base buer solution. This base buer solution is used to set the pH of the solution and stabilize the electro-chemistry of the solution by maintaining its pH and ionic conductivity during the course of the experiments [Persat et al. (2009a)].

The choice of a buer depends upon the specic application of the system under study. One of the most commonly used buers which mimic a biological system (pH 5.5 -8.6) as well as have minimum experimental side-eects are listed by [START_REF] Good | Hydrogen ion buers for biological research[END_REF] and are commonly known as Goods Buers. During this study a weakly basic HEPES Hemisodium buer (Sigma-Aldrich) was used which has a pH of 7.5 and works within a pH range of 6.8 -8.

Seeding particles

In order to perform the PTV, small particles were introduced in the liquid sample.

As the ideal particles for PTV should have the same density and shear behavior as the liquid medium, it is almost impossible to nd such particles. So, in order to minimize the distinction between the seeding particles and the liquid media, one has to identify various specic forces acting on the particles. One of the most common forces acting on a solid particle in a solid media are buoyancy, drag and Coulombic (Electrophoretic) forces in electrodynamic situations. One can reduce the eect of buoyancy on the particle dynamics by choosing the particles material to have a density matching that of the liquid media. In order to reduce the eect of drag, one can choose spherical particles with very small diameter. Finally, to reduce the electrophoresis one can choose particles with very low surface charge density.

Another aspect regarding the particle sizes is that they have to be large enough to scatter light while small enough to clearly resolve their displacement between two frames. So a balance has to be struck in between drag reduction, light scattering and displacement resolution with respect to particle sizes. Further, the particle distribution in the liquid media has to be chosen such that there is very negligible inuence of velocity eld around. In the present study, Carboxylate modied Polystyrene (latex) beads (Sigma-Aldrich) were used as seeding particles where particle diameter was 0.9 -1.1µm, density was 1.045 -1.055g/l and charge density was ≤ 0.008milliequivalents (mEq). The particles were provided in an aqueous suspension where the solid concentration was 2.5%, which corresponds to about 5 × 10 9 particles/ml. To prepare the liquid sample with seeded particles, 10ml of the liquid sample prepared above was mixed with 20µl of the particle suspension which lead to approximately 10 7 particles/ml of the liquid sample.

Filling and cleaning of the microchannel

The PDMS microchannel that was used in this study was a 5cm long microchannel with a rectangular cross-section of 100µm × 100µm (see Fig. inserting the second electrode after opening the sink tube to the atmosphere. The same process was repeated for each of the subsequent lling process to ensure that both the reservoirs are completely lled with the liquid pumped. In order to clean the microchannel, rstly, the microchannel was washed with de-ionized water for 20mins and then with 0.1M NaOH for 20mins. Finally, the sample solution was introduced in the microchannel and images were recorded for various combinations of applied electric eld and ow rates.

Image acquisition

A phase-contrast microscope (for details see Chapter 5) was used in order to obtain a sequence of images for the PTV. The microscope objective with a magnication of 60× and a numerical aperture of 0.7 was used in these experiments (see Chapter 5). The depth of eld hence calculated lied in the range of 1.2 -1.7µm. The small depth of eld (which approximately corresponds to the particle diameter) is good to image particles in a thin focused plane, nevertheless, out of focus plane traversal of particles was also observed, leading to few blurred particle images. Since particles were imaged against the light source, the maximum amount of light scattered by the particles was in the direction perpendicular to the plane of imaging and hence the images show actually a shadow of the particles. For each reading of the dierent combinations of the ow rate and electric eld, a video of 500 frames was recorded by a CCD camera with a frame rate of 26f ps and exposure time of 90.56µs. Although recording images provide a better spatial resolution, they need special apparatus for pulsed illumination and synchronization of various imaging components. While recording a video, a xed time interval between two frames is used, eliminating the need of a pulsed source. Each image had a resolution of 508×384 in 8-bit grayscale.

Post-processing

The post-processing steps include image frame extraction from the raw video, removal of background noise, correlative analysis of the image frame sequence, calibration of pixel to distance and nally, extraction of the mean velocity distribution.

In order to extract image frames from the raw video, VirtualDub v1.9.11 software was used, where 100 frames were extracted out of 500 recorded frames. Later, the image sequence was imported to ImageJ v1.47 software where the image colors were inverted so that particle shadows appear as bright particles on a dark background (a requirement for using PTV Lab software) (see Fig. 6.2). Next, a background image was chosen to subtract noise from each of the images from the image sequence. Then, a contrast enhancement was carried out and the image sequence was exported. Now, these images were imported into PTV Lab and a region of interest (ROI) encompassing the particles bounded by the channel walls was selected. In order to detect the particles, Gaussian Mask algorithm (a fast algorithm for low density particles) was chosen [START_REF] Brevis | Integrating cross-correlation and relaxation algorithms for particle tracking velocimetry[END_REF]] with the particle correlation threshold set as 0.5, sigma (particle diameter) as 3pixels (px), and intensity threshold as 25/255 on a 8-bit/px grayscale. The cross-correlation parameters for consecutive image frames were xed by setting the interrogation area (mesh size) as 40px, and a minimum image correlation value as 0.1. After setting up the processing parameters, the PTV analysis was allowed to run. After the PTV analysis run was nished, in order to lter out the spurious velocity data, a vector validation step was carried out. By applying a cuto on the obtained velocity magnitudes (such as u > 0 and v ∼ 0)

the spurious velocity data was ltered out. The next step involved the calibration of the image in order to convert pixel information into geometric distance, which for the present system corresponds to ∼ 2.5px/µm. Finally, a mean of the velocity prole is calculated and exported as a .dat le.

Results and Discussions

The reported values of PDMS-aqueous electrolyte surface zeta potential at a pH of 7.5 lies in the range of -20mV to -40mV . In order to compare the experimentally obtained velocity prole with the theoretically obtained velocity prole (see Eq. 6.2.11) one has to estimate the mentioned dimensionless parameters. The The objective of this study was to observe the interfacial behavior of immiscible liquids under time-periodic electric elds and to characterize the critical parameters such as the imposed ow rate, amplitude of the electric eld and its frequency. Since the focus of this study was on electrokinetics, one of the two immiscible liquids was necessarily electrically conductive (an aqueous electrolyte).

It is known that surface tension eects are quite dominant over body forces on microscales where the surface to volume is very high. So in order to observe the eect of surface tension also, two dierent liquid combinations were used, namely, oil and electrolyte, a case of nite surface tension, and de-ionized water and electrolyte, a case of zero surface tension (miscible uids). Further, two dierent uid-uid congurations were tested for interfacial instability, namely, one-interface case and two-interface case (see Fig. 6.4). In the one-interface case, the conductive uid is geometrically bounded by a non-compliant wall which participates in the EOF dynamics by providing a sustained viscous dissipative layer and hence enhancing the stability of the system. The two-interface case presents a case of relatively non-constrained liquid-liquid interface where capillary eects dominate over viscous dissipation. In order to observe the interfacial dynamics in both of the cases, the phase-contrast microscope with a CCD camera was used to record the interfacial dynamics under various combinations of mentioned control parameters. Each of the mentioned setup step is detailed subsequently. In this experimental study of two-phase ow, three distinct liquid samples were created. The rst liquid sample (henceforth referred to as FS-1) was an aqueous electrolyte, which had a HEPES buer base while 0.1M KCl was used to enhance the conductivity of the solution similar to the previous study (see sec. 6. The third liquid sample (henceforth referred to as FS-3) was de-ionized (DI) water

Post-processing

The post-processing steps include processing of the raw video in order to extract amplitude and frequency of the interfacial perturbation. In order to process the raw video, Virtual Dub v1.9.11 software was used, where 200 frames were extracted out of 500 recorded frames. Later, the image sequence was imported to ImageJ v1.47 software and the following procedure was followed. In order to extract the maximum amplitude of the interfacial waves at a x-location, a distribution of pixel intensities was plotted over the 200 frames. Since there was a jump in pixel intensity across the interface, the time taken (number of frames) between two consecutive peaks of pixel corresponded to the frequency of the interfacial vibrations and the dierence between the crest and the trough gave the displacement of the interface at that

x-location. The images were calibrated by using a factor of ∼ 1.25µm/px for the one-interface system and ∼ 1.67µm/px for the two-interface system. Finally, this distribution of interfacial displacement is exported as a .dat le.

Results and Discussions

Miscible liquids (without surface tension)

In this set of experiments, the interfacial behavior between two miscible liquids with a conductivity gradient was studied under three major tunable parameters, namely, the imposed ow rate, the amplitude and the frequency of the AC applied electric eld. In order to have a distinct and planar phase boundary (interface) between the two liquids, it was necessary to minimize the diusion length of Rhodamine B in the transverse direction by imposing a convective ux of the solution. This convective ux was achieved by imposing a ow rate with the help of a syringe pump. In all cases, the direct contact surface between the two uids was L = 10 3 µm in length and H = 100µm in depth. The diusive length of Rhodamine B can be estimated from the Fick's law as, δ = 2 √ Dt, where D is the diusion coecient of Rhodamine B, and t is the time taken by the Rhodamine B to diuse over a length of δ. The time taken by the liquid to traverse the channel length (L) can be estimated as t = LS/Q P , where Q P is the imposed ow rate, S = W × H is the half crosssectional area of the channel with the channel half-width, W = 50µm. So, upon equating the two time spans, (6.4.1) For D = 3.7 × 10 -10 m 2 /s and δ ≤ W , one can estimate a limiting ow rate, Q P ≥ 178nl/min. Another limitation on ow rate is that the magnitude of electro-osmotic velocity (u E,ref ) and the pressure driven velocity (u P,ref ) should be of comparable magnitudes. For a maximum applied electric eld of 80V /cm, one can estimate the corresponding pressure driven ow rate (see Eqs. 6.3.1 and 6.3.2) as ∼ 68nl/min.

Q P = 4DLS δ 2
In the following sets of experiments, we choose to work with a wide range of Q P (25 -1000nl/min) thus going from a atter interface and dominant pressure driven ow, to a relatively diused interface and a dominant electro-osmotic ow.

After doing a series of experiments with various combinations of ow rate, electric eld magnitude and frequencies, it was observed that the maximum amplitude of interfacial oscillation decreases with increasing the ow rate but increases with the magnitude of the electric eld (see Fig. 6.5a). Upon increasing the frequencies, it was observed that the interfacial oscillation follows the applied frequency but at higher frequencies (> 10Hz) the viscous relaxation dominates over the imposed electro-temporal dynamics and hence the interfacial displacement behaves independently of the applied electric eld (see Fig.

Immiscible liquids

In this set of experiments, the interfacial behavior between two immiscible liquids was studied under three major tunable parameters, namely, the imposed ow rate, the amplitude and frequency of the applied AC electric eld. Due to the hydrophobicity of PDMS, the aqueous electrolyte-Hexadecane (ρ Hexa = 0.773g/ml, µ Hexa = 3.34cP at 25 • C) and SPAN-80 solution system acts as a liquid-liquid jet system (see Fig. 6.8). So, in such a case, a surface-tension driven instability is highly likely leading to the breaking of the interface into droplets. It is known that in high Reynolds number ows, minimizing the droplet formation can be avoided by suppressing the Rayleigh-Plateau instability [START_REF] Lasheras | Liquid Jet Instability and Atomization in a Coaxial Gas Stream[END_REF]; [START_REF] Guillot | Stability of a Jet in Conned Pressure-Driven Biphasic Flows at Low Reynolds Numbers[END_REF]].

So, in order to have a planar interface between the two liquids, a ow rate was slowly imposed in both liquids until a at interface was achieved. In all of the studied cases, the direct contact surface between the two uids was L = 10 3 µm long.

These set of studies were performed over a wide range of applied AC electric elds (50 -300V ), frequencies (1 -10Hz) and imposed ow rates (100 -1000nl/min).

The most prominent amplitudes of oscillations of the interface were observed for a frequency of 1 -5Hz. To characterize the stability of the system and identify the critical values of applied electric eld and ow rates leading to interfacial instability, the following approach was used, Firstly, the electric eld was switched o and the applied ow rate was xed at 1000nl/min in each of the uids. Then, the ow rate was reduced slowly (in the steps of 100nl/min) until the liquid-liquid interface started oscillating and broke into droplets. Then, the same step was repeated for dierent values of the magnitudes of the applied electric eld and frequencies. Finally, the critical magnitudes of the ow rates obtained for dierent values of electric eld and frequencies were plotted (see Fig. 6.9).

Further, it was observed that there exists a threshold value of imposed ow rate for the interfacial stability of the immiscible uids. This highlights the competitive dynamics between the convective inertia and the capillary forces. In the presence of a xed value of surface tension, it is observed that the high ow rates (> 500nl/min) are relatively stable. It was also observed that the two-liquid system becomes more stable at higher frequencies (see Fig. 6.9 and Fig. 6.10), which leads to the inference that the time-periodic electric eld induced dynamics dampen the interfacial waves prohibiting their growth.

Conclusions

The study of interfacial instability due to electrokinetics requires an understanding of the complex interactions of various physical phenomena such as capillary forces, convective and temporal inertia, diusion, and Maxwell stresses. In this study, the interfacial instability was explored through two cases, namely, a miscible uids case i.e. in the absence of surface tension, and an immiscible uids case i.e. in the presence of surface tension.

In the miscible uids case, the objective was to study the liquid-liquid interfacial deformation under a time-periodic EOF in the presence of convective and diusive transport. To obtain a non-perturbed (at) interface against a nite diusion of miscible uids, a ow rate was imposed on the liquids so that the ensuing convective ow can hinder the mixing of the two uids by interfacial diusion. The ow rate was kept the same in both the liquids to avoid shear induced (Kelvin-Helmholtz) instability. It was observed that for a given magnitude of applied electric eld, the amplitude of the interfacial deformations decreased with increasing imposed ow rate, which ascertains the stabilizing eect of the imposed ow rate. Further, it was also observed that the responsiveness of the interface (i.e. the amplitude of the perturbed interface) to the frequency of the applied electric eld, was most noticeable in the range of 1 -5Hz.

manufacturer. It was found that the error in voltage output can range from 2% at 10Hz to ∼ 10% at 1Hz.

Electro-chemical

The EOF parameters such as surface zeta potential and Debye length are very sensitive to the electro-chemical reactions, which in turn are sensitive to the ambient temperature and pressure. So, in order to have a reproducible electro-osmotic system, a climate-controlled environment is required. Most of the solutions were prepared in bulk in order to have consistent concentration of the species while performing the experiments. The experiments of which the results are presented in this work, were performed over a period of two months at various times during a day in a non-climate controlled room. The key parameters such as solution pH and conductivity were measured every time before starting the experiments, but, a noticeable variation was observed in those parameters (for e.g. conductivity reduced by a value of 200µS/cm over two days). During the experiments, the Faradaic reactions at the electrodes also create an ionic concentration gradient across the channel which nally results into a pH and conductivity gradients. Such reactions also cause hydrolysis leading to gas formation in the reservoirs thus changing the reservoir pressure dynamically. Also, there are Ohmic losses in the connecting wires and electrodes which aects the actual electric potential reaching the liquids. All these issues could be addressed in future developments by putting a feedback system in the reservoirs changing the reservoir pressure and electrode potential dynamically.

Conclusions and Perspectives

Conclusions

The present work focuses on the role of steady and time-dependent Maxwell stress, capillary force and disjoining pressure on interfacial instability. A comparative base state analysis with the previous studies, which did not account for interfacial Maxwell stress, showed that considering Maxwell stress generated dynamics introduces signicant velocity gradients at the liquid-liquid interface. Such gradients are instrumental in the interfacial instability under a time-periodic actuation as they can either attenuate or amplify the interfacial waves. The bulk penetration of near wall oscillations was found to be inversely proportional to the forcing frequency.

At lower frequencies, a signicant suppression of velocity dispersion was observed which asymptotes to the classical dispersion free velocity in a DC electro-osmotic ow. By the variation of interfacial polarity it was found that, one can not only control the interfacial velocity and the rate of shear transfer, but also establish a signicant phase dierence in uid velocity at dierent transverse locations in the uid which enhances the dispersion eects and can be eectively used in controlled species transport in microuidic devices.

Further, in order to study the eect of interfacial Maxwell stresses in realistic rectangular channels, the solution of a free surface EOF in a microchannel of rectangular cross-section was achieved analytically. By taking into account a complete expression of the boundary condition at the free surface that includes the Maxwell stress at this interface, a parametric analysis of the ow in terms of the EDL thickness, channel aspect ratio and dierent uid-uid and solid-uid interfacial zeta potential was carried out. With the help of two-dimensional velocity contours, it was shown that the bounding walls have a signicant eect on the velocity distribution in a rectangular microchannel. The ow controlling parameters in an EOF were not only the electro-chemical properties of electrolytes which controls the EDL thickness and wall and interfacial zeta potentials, but also the geometry of the channel through the channel aspect ratio. Such a study helps identifying proper control parameters for achieving a desired ow rate and velocity distribution in a microuidic device.

A linear stability analysis of interfacial perturbation was performed for the thin lm of electrolyte under DC and AC electric elds. Through long wave asymptotic analysis of the Orr-Sommerfeld equations, parametric stability thresholds of a thin aqueous lm explored with a focus on surface tension, disjoining pressure for

Perspectives

The analytical modeling of the interfacial stability of a gas-liquid (liquid lm) system provided interesting insights into the marginal stability characteristics of an interface under steady and time-periodic Maxwell stress while including a complex interaction of capillary and van der Waals forces. This subject has however an im- Two-liquid System: Although gas-liquid systems occur in many real life systems as bubbles, foams, biological structures such as tear lm, gas exchange in lung alveoli and blood capillaries, mucous lining on internal organs etc. Modeling a generalized two-liquid system will give a detailed insight into the stability of such interfaces while highlighting the role of density, viscosity and, permittivity gradients across the interface. A gas-liquid system can be considered as a limiting case of a two-liquid system undergoing interfacial dynamics.

Interfacial Dynamics: The linear stability analysis provides the critical value of control parameters leading to neutral stability. Studying the interfacial growth and dynamics provides an interesting insight into lm rupture and subsistence. Also, the use of frozen wave instability of polymer substrates in order to create precisely controlled patterned structures [START_REF] Verma | Electric eld induced instability and pattern formation in thin liquid lms[END_REF], is a very interesting application of electrokinetic instability in dielectric liquids.

The experimental study of the interfacial instability of both miscible uids and immiscible uids provided interesting insights into the critical values of applied parameters such as magnitude and frequency of the electric eld, applied ow rate and electrochemistry (e.g. ionic concentration, pH, conductivity etc.) of the system.on interfacial instability. Considering the wide range of involved parameters which are sensitive to the changes in the environmental conditions, following steps can be performed to reduce the experimental error and explore the interfacial stability in a greater detail, Microchannel Fabrication: Use of hard materials like glass microchannels to reduce the geometric variations in the channel prole. Also, the glass zeta potential has been found to be more consistent and uniform, which helps to validate experimental results with theoretical results much easily.

More precise variation of parameters: Most of the parametric variations such as the magnitude and frequency of the electric eld and ow rate were manually controlled. In order to have a more precise variation of the control parameters, a computer controlled parametric variation can be used.

Feedback Control and Device Calibration: Since all uid and electric connections are exposed to losses some of which are head loss, potential loss etc.

A feedback based parametric control can be established in order to minimize losses in the tubing and electrode connections leading to the microchannel.

  constant . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2

 2 discusses the basic principle of an EOF in an aqueous electrolyte. It also discusses the origin of hydrodynamic and electrodynamic boundary conditions at solid-liquid and liquid-liquid interfaces. Chapter 3 discusses the application of the basic concepts developed in Chapter 2 towards understanding the dynamics of a thin aqueous electrolytic lm under the inuence of DC and AC electric elds. Further, the role of interfacial Maxwell stress on ow control in a free surface ow of an electrolyte in a rectangular microchannel is discussed.

2. 1 .

 1 OverviewDierent states of matter like solid, liquid and gas respond dierently under an externally applied electric eld. Based on their responses, they can be broadly classied into conductors and insulators (dielectrics). The conductors have charges that are free to move under the applied electric eld. In dielectrics, the charges that are bound to the constituting atoms or molecules, can only be displaced (translated or rotated) microscopically around the bounding atom or molecule under the inuence of the external electric eld. This spatial separation of bound charges is also called the polarization of charges. Some of the dielectrics show this polarization under the inuence of an external electric eld while others (for e.g. water), owing to their molecular structure have resident polarization and are called polar molecules. It is because of this property that water can dissolve a large number of ionic compounds and polar non-ionic compounds. Such compounds dissociate into their constituent ions in the aqueous phase and are surrounded by the polar water molecules due to the Coulombic forces (see Fig.2.1

Figure 2 .

 2 Figure 2.1.: Solvation of a Sodium ion (N a + ) by water (H -O -H) molecules.

Figure 2 . 2 .

 22 Figure 2.2.: Structure of an EDL near a positively charged surface with a surface potential (φ s ) showing (a) Electrostatic potential distribution (φ sc ), and (b) distribution of counter-ions (c -(y)) and co-ions (c + (y)).

  2.3). The dimensional scales used are, Y = y/h 0 , U = u/u ref , and Φ = φ/ζ b . The potential distribution in such a system can be obtained by using the PBE with the Debye-Hückel linearization for low zeta potential system (see Eq. 2.3.14). Upon using the boundary conditions as, Φ sc = 1 at the walls and the symmetry condition, dΦ sc /dY = 0 at the channel center (Y = 0), the space-charge potential distribution can be obtained as,

Figure 2 . 3 .

 23 Figure 2.3.: Schematics of a parallel-plate EOF with the representation of velocity and potential distribution

Figure 2 .

 2 Figure 2.4.: Schematics of a control volume (V) and control surface (S) at a charged interface with a surface charge density (ρ f ).

  where, ζ I is the interfacial zeta potential.

Figure 2 .

 2 Figure 2.5.: Schematics of a control surface (S) and the enclosing loop (l) at a uid-uid interface.

  the dynamics of a viscous electrolytic thin lm bounded by a solid surface from below and a low viscosity and low permittivity gas phase from the top. The EOF dynamics of such a lm is studied under steady and time-periodic potential bias, V app (or electric eld, E app = V app /L), where the electrodes are kept L distance apart. The gas-liquid interface is considered to have a constant interfacial potential, ζ I , and the substrate-liquid potential is taken to be ζ b . The height of the thin lm is taken to be h 0 . Firstly, the ow characteristics of the conductive thin lm under a steady (DC) electric eld is explored, where a detailed parametric dependence of the velocity prole is presented.

  represents the basic state of a general thin lm stability problem. In an eort to bring out the interesting interplay between the characteristic length scale of the liquid layer and the characteristic electric double layer (EDL) length scale (Debye length), various orders of the liquid layer thickness to Debye length ratios are addressed. For further generalizations, the possibilities of addressing the eect of high zeta potentials on thin lm dynamics by invoking the non-linear Poisson-Boltzmann equation are considered. The analytical expressions for the electric potential and velocity eld are derived, bringing out important implications of Maxwell stresses at the free surface.

Figure 3 .

 3 Figure 3.1.: Schematics of a 2D thin lm system under EOF

  3.6 that the interfacial boundary condition is no longer a classical hydrodynamic stress free boundary condition. The Maxwell stress, by the virtue of the Coulombic force on the space charge distribution and the free surface potential, contributes to the interfacial dynamics as well. To demonstrate the effects of Maxwell stress on electro-osmotic ows, a comparison of ow characteristics between EOF under a Hele-Shaw conguration (parallel plate ow)[START_REF] Chakraborty | Generalized model for time periodic electroosmotic ows with overlapping electrical double layers[END_REF];[START_REF] Dutta | Analytical Solution of Time Periodic Electroosmotic Flows: Analogies to Stokes' Second Problem[END_REF]] and the free surface EOF has been carried out. Although being two completely dierent physical systems the mentioned problems share a similar physical modeling in terms of governing equation and boundary conditions. The most general modeling of a parallel plate EOF system involves Stokes equations with a Helmholtz-Smoluchowski slip velocity at the walls and a symmetry condition applied on the velocity at the channel centerline. Such a system has been very successful in modeling microscale EOF owing to the thin EDL length scales as compared to the channel length scales. However, for systems where the EDL thickness becomes comparable to the channel length scales, one can add an electrical body force term in the Stokes equations, the involved electric charge distribution being obtained from the electrical potential distribution which results

  channel EOF system where the electrical eects are localized close to the wall and the far eld electric potential is zero. A validation of the present free surface model developed in the previous sections is presented through a comparison with a Hele-Shaw EOF, where the half channel velocity prole was taken from the seminal work of Dutta and Beskok (2001) (see Fig. 3.2). Their usage of Debye length as the characteristic length scale as compared to the lm thickness used in this study leads to dierent dimensionless groups which can be mathematically adjusted without losing any physical details for comparing the results. Their various parameters can be recovered from the present parameters as, κ = Wo × De; η = De; u HS = -u ref (3.3.15) where κ is the dimensionless frequency, η is the dimensionless Debye length and u HS is the Helmholtz-Smoluchowski slip velocity used as the reference velocity by Dutta and Beskok (2001). The excellent agreement between the velocity proles obtained from the present model and the one extracted from Dutta and Beskok (2001) as reported in Fig. 3.2, justies the success of non-Maxwell stress model in thin EDL systems.

Figure 3 .

 3 Figure 3.2.: Velocity proles comparison with Dutta and Beskok (2001), where, κ = Wo×De is the dimensionless frequency used by them. Other xed parameters are De = 0.01, Z R = 0, γ R = 1, Θ = π/2.

  3.3).The vorticity prole of non-Maxwell stress model was taken from[START_REF] Chakraborty | Generalized model for time periodic electroosmotic ows with overlapping electrical double layers[END_REF], whose dimensionless parameters relate to the present parameters as mentioned in Eq.3.3.15. It is seen from the gure that neglecting Maxwell stress can lead to signicant underestimation of the vorticity strengths in the bulk. Also, for thick EDLs the strong presence of velocity gradients renders Helmholtz-Smoluchowski velocity a non-ideal slip condition at the liquid solid interface. In order to strengthen this argument, a further discussion on velocity scales is provided with dierent values of γ R . The resulting velocity gradients lead to dispersion in thick EDL ows over the time and with the help of this study one can identify the regimes of applied frequency and electrolyte characteristics to avoid the dispersion eects in a time periodic EOF.

Figure 3 .

 3 Figure 3.3.: Vorticity prole comparison of Maxwell stress and non-Maxwell stress models [Chakraborty and Srivastava (2007)]. The values of other xed parameters are De = 0.5, α = 0.4278, Z R = 0, γ R = 1.

Figure 3 .

 3 Figure 3.4.: Velocity prole with dierent values of the Debye number (De) and ionic energy parameter (β) for xed values of Womersley number (Wo), electroviscous ratio (γ R ), Z R at time Θ = π/2.

Figure 3

 3 Figure 3.5.: Velocity distribution over the lm thickness with dierent values of Electro-osmotic number, γ R for β = 5, Z R = 0, Wo = 10 at Θ = π/2 with (a) De = 0.01 and (b) De = 0.1.

Figure 3

 3 Figure 3.6.: Velocity distribution over the lm thickness with dierent values of Wo and for xed values of β = 5, Z R = 0 and γ R = 1 at Θ = π/2 with (a) De = 0.01 and (b) De = 0.1

Figure 3 Figure 3

 33 Figure 3.7.: Velocity prole with xed values of De = 0.1, β = 5, γ R = 1 and Wo = 10 with (a) dierent values of interfacial polarity, Z R at Θ = π/2 and (b) at dierent times (Θ) and transverse locations (Y ).

  3.4.1) Upon using the scaling parameters as, Y = y/D h , Z = z/D h , Φ sc = φ sc /ζ ref where D h = 4lh/ (l + h) is the hydraulic diameter of an open rectangular duct and using Debye-Huckel linearization for small wall potentials (ζ b ∼ 25mV ), the governing equation for the electrostatic potential distribution in the EDL can be obtained as,

  = w/D h , H = h/D h , ζs = ζ s /ζ ref , ζb = ζ b /ζ ref and ζI = ζ I /ζ ref . The solution of Eqs. 3.4.2 and 3.4.3 can be obtained by separation of variables as,

  dimensionless boundary conditions are,

Figure 3 .Figure 3 .

 33 Figure 3.10.: Velocity proles for cases with and without interfacial Maxwell stress at the symmetry axis of the channel (Z = 0). The case without Maxwell stress was taken from a study by Gao et al. (2005b). The values of xed parameters are γ R = 1, ζ s = 1, ζ I = 0 and (a) De = 0.01 (b) De = 0.1. The free surface is at Y = H.

Figure 3 .

 3 Figure 3.12.: Velocity contours on the Y-Z plane for dierent aspect ratios where De = 0.1, γ R = 1, ζ s = -1, ζ I = -1 and (a) H/2W = 1, (b) H/2W = 1/3. Solid lines show positive velocity and dashed lines show negative velocity values. The free surface is at Y = H.

Figure 3 .

 3 Figure 3.13.: (a) Velocity contours on the Y-Z plane where De = 0.1, γ R = 1, ζ s = 1, ζ I = -1 and (b) Centerline velocity prole (at Z = 0) where De = 0.1, γ R = 1,ζ s = 1, H/2W = 1/2 for dierent values of ζ I .

Figure 3 .

 3 Figure 3.14.: Dimensionless ow rate variation with (a) Debye number, De for a xed aspect ratio, H/2W = 1 and γ R (b) Electro-viscous number (γ R ), for a xed aspect ratio, H/2W = 1.

  Electro-Osmotic Flow (EOF). Stability of uid interfaces under EOF has been awide eld of research in the past decade[START_REF] Lin | Electrokinetic instability in microchannel ows: A review[END_REF];[START_REF] Zaltzman | Electro-osmotic slip and electroconvective instability[END_REF];[START_REF] Rubinstein | Rupture of thin liquid lms: Generalization of weakly nonlinear theory[END_REF];[START_REF] Sounart | Lubrication theory for electro-osmotic ow in a non-uniform electrolyte[END_REF];[START_REF] Santos | Instability of electro-osmotic channel ow with streamwise conductivity gradients[END_REF]], a variety of stability models have been presented, discussing the eects of ow-actuating parameters, along with substantial experimental evidence to validate the theoretical predictions[START_REF] Oddy | Electrokinetic instability micromixing[END_REF];[START_REF] Posner | Convective instability of electrokinetic ows in a cross-shaped microchannel[END_REF];[START_REF] Oddy | Multiple-species model for electrokinetic instability[END_REF][START_REF] Chen | Convective and absolute electrokinetic instability with conductivity gradients[END_REF]]. Most of the mentioned studies have been concentrated in the domain where the Electrical Double Layer (EDL)

  ], even in the case of an applied electric eld of large amplitude. The low electrolyte concentration also avoids complexities in ow modeling by reducing the nonlinear dependence of electrophoretic mobility of ions on the sparse space-charge distribution[START_REF] Wei | Dielectric relaxation of electrolyte solutions[END_REF];[START_REF] Lorenz | Charge inversion of divalent ionic solutions in silica channels[END_REF];[START_REF] Borukhov | Polyelectrolyte Solutions between Charged Surfaces[END_REF];[START_REF] Yossifon | Nonlinear currentvoltage characteristics of nanochannels[END_REF];[START_REF] Fedorov | Towards understanding the structure and capacitance of electrical double layer in ionic liquids[END_REF];[START_REF] Dufreche | Transport in electrolyte solutions: are ions Brownian particles?[END_REF];[START_REF] Song | Excess charge density and its relationship with surface tension increment at the air-electrolyte solution interface[END_REF]].The solid substrate zeta potential is represented as ζ b , which is a function of the substrate-uid interaction, ionic concentration and pH of the solution[Kirby and Hasselbrink (2004a)]. The liquid surface exposed to a gaseous environment develops a charge which is a function of various parameters like ionic concentration, pH of the solution and valence of the ions involved[START_REF] Manciu | Ions near the air/water interface: I. Compatibility of zeta potential and surface tension experiments[END_REF];[START_REF] Li | Reversal of bubble charge in multivalent inorganic salt solutions-Eect of magnesium[END_REF];[START_REF] Gray-Weale | An explanation for the charge on water's surface[END_REF]]. The associated zeta potential (ζ I ) has been found to vary over a wide range in the reported literature[START_REF] Graciaa | The ζ-Potential of Gas Bubbles[END_REF];[START_REF] Yang | Measurement of the Zeta Potential of Gas Bubbles in Aqueous Solutions by Microelectrophoresis Method[END_REF];[START_REF] Choi | On steady two-uid electroosmotic ow with full interfacial electrostatics[END_REF]].

  is the capillary force with γ being the surface tension and κ the local double mean curvature of the interface. The corresponding dimensionless conservation equations are written using the scaling parameters as, Θ =ω ref t with, ω ref = 1 τ ref ,where, τ ref is the reference time scale,

ω ref h 2 0 ν

 0 is the Womersley number expressing the relative strength of temporal inertial force over the viscous dissipation force, Re = dimensionless boundary conditions at the wall (Y = 0) are no-slip and no-penetration conditions, surface, Y = H(X, Θ), the dimensionless continuity of tangential and normal stress are, respectively,

  tilde correspond to perturbation variables. To reduce the number of dependent variables, the stream function is introduced as, normal mode solutions of the perturbation variables as,Ψ(X, Y, Θ) = Ψ(Y )e iαX+σΘ(4.2.19) P (X, Y, Θ) = P (Y )e iαX+σΘ (4.2.20) H(X, Θ) = He iαX+σΘ (4.2.21)

  , c 0 = 0.01mM with a lm thickness d ranging over 0.1 -10µm. The ow control parameters are varied to illustrate the parametric dependence of the free surface stability of the system. To highlight the role of the interfacial Maxwell stress on the thin lm stability, two characteristic stability curves were worked out. The rst one was obtained by removing the Maxwell stress term,

Figure 4 . 1 .FilmFigure 4 . 2 .

 4142 Figure 4.1.: Comparison of the variation of the real part of growth rate (σ R ) as a function of the wave number(α) for Ca = 1, Z R = 1, De = 0.5, A = 1.

Fig. 4 .

 4 Fig. 4.3 shows the growth rate σ R as a function of the wave number for dierent values of the capillary number, Ca and two values of the Hamaker's constant A.All the other parameters are taken as xed. For both values of A, the system becomes more stable at large values of the wave number as the capillary number, Ca increases. This can also be observed from Eqs. 4.2.29 and 4.2.30. While at the small wave numbers, that is in the domain of long wave disturbances, the eect of surface tension diminishes.It can also be seen that the disjoining pressure represented through the dimensionless constant A, has a signicant eect on the stability of the system in the long wave range. Fig. 4.4 shows the eect of A on the growth rate, σ R for xed values of electrical eld E app (Fig. 4.4a) and substrate zeta potential ζ b (Fig. 4.4b). From

  Figure 4.4.: Growth rate variation with the wave number for dierent values of dimensionless disjoining pressure, A with De = 0.1, Ca = 10, Z R = 0 and (a) γ R /E R = 2500, (b) γ R E R = 0.01.

Figure 4

 4 Figure 4.5.: Growth rate variation with the wave number for dierent values of the applied electric eld with A = 0.1, De = 0.1, Z R = 0 and (a) Ca = 10, (b) Ca = 0.1.

Figure 4

 4 Figure 4.6.: Growth rate variation with the wave number for dierent values of substrate zeta potential with De = 0.5, Ca = 10, A = 0.1 and (a)Z R = -1, (b)Z R = 1.

Figure 4

 4 Figure 4.7.: The base state velocity prole with dierent values of the surface to substrate potential ratio (Z R ). The values of other xed parameters are: γ R = 1, De = 0.1.

  Figure 4.8.: Growth rate variation with the wave number for dierent values of interfacial zeta potential. With xed parameters as, γ R E R = 1, De = 1, A = 0.1, for dierent values of Capillary number as, (a) Ca = 10, (b) Ca = 0.1, and also with γ R /E R = 100, Ca = 10, A = 0.1, for dierent values of Debye number as, (c) De = 0.5, (d) De = 1.

Figure 4 .

 4 Figure 4.10.: Most dangerous wave number (α critical ) variation with the van der Waals parameter (A), for dierent values of the capillary number (C a). Values of other xed parameters are, γ R E R = 1, De = 1, Z R = 0.

Figure 4 .

 4 Figure 4.11.: Most dangerous wave number (α critical ) variation with the Debye number (De) for dierent values of the wall zeta potential (γ R E R ). Values of other xed parameters are, Ca = 1, Z R = 1, A = 0.

Figure 4 .

 4 Figure 4.12.: Most dangerous wave number (α critical ) variation with the zeta po- tential ratio (Z R ) for dierent values of the electric eld parameter (γ R /E R ). Values of other xed parameters are, Ca = 1/10, De = 0.1, A = 1.

  Figure 4.13.: (a) Interfacial displacement under the inuence of disjoining pressure (Π) and Laplace pressure (P L ) under a positive (crest) and negative (trough) perturbation in the interface. Upon xing the value of the parameters as De = 0.1, Z R = 0.01, Re = 10 -4 , γ R = 1, W o = 1, Marginal stability curves showing the critical wave number as a function of the dimensionless Hamaker constant, A in the (b) absence of time-periodic electric eld, and (c) presence of time-periodic electric eld.

  Figure 4.14.: Marginal stability curves showing the critical wave number as a function of the zeta potential ratio (Z R ) with stability trends for dierent values of substrate zeta potential (γ R E R ) in the absence of external electric eld at (a) De = 0.01 (b) De = 0.1 with Ca = 10 -5 and A = 0.1.

  4.15a). It can be seen that maximum magnitude of the vorticity (|ω b | = |∇ × U b | = |∂U b /∂Y |) occurs at both the interfaces (solid-liquid and gas-liquid). It can be understood the deformation of the free

  Figure 4.15.: (a) Base state vorticity (|ω b | = |∇ × U b |) distribution over the lm thickness at γ R = 1, Z R = 1 (b) Marginal stability curves showing the critical wave number as a function of the Debye number (De) with stability trends for dierent values of Womersley number (Wo). Marginal stability curves showing the critical wave number as a function of the Wo with stability trends for dierent values of (c) Reynolds number (d) Electric eld strength (γ R /E R ).

  surface zeta potential. It was observed that ignoring the free surface Maxwell stress under electro-osmotic ow, over-estimates the free surface instability. The presented characteristic stability curve helps to identify the critical values of various parameters presented in this chapter, which can be eective in designing thin lm ow process in microuidic devices. Some of the parameter dependent stability trends are summarized as, (a) The phenomena which are observed to have a stabilizing eect on the lm dynamics are surface tension, repulsive disjoining pressure (A < 0), osmotic pressure due to the EDL at the interfaces and viscous dissipation. (b) The phenomena contributing towards the instability of the lm are attractive disjoining pressure (A > 0), thin EDLs (De 1), external electric eld driving the electro-osmotic ow and low frequencies. The lm stability decreases upon increasing the magnitude of the applied electric eld, which is due to an increase in the tangential Maxwell stresses at the interface.

Fig

  Fig. 5.1).

Figure 5 . 1 .

 51 Figure 5.1.: A representative microchannel design.

Figure 5 . 2 .

 52 Figure 5.2.: Basic steps used for Microfabrication.

Figure 5 . 3 .

 53 Figure 5.3.: A sample photo mask with microchannel motifs

Figure 5

 5 Figure 5.4.: Schematics of the a sample microchip with rectangular cross section and connections

  6.1).

Figure 6 . 1 .

 61 Figure 6.1.: Schematics of a rectangular channel under a combined pressure driven and EOF.

D

  h p ref µu ref is the dimensionless pressure. Upon using the symmetry conditions at the channel centerlines, the boundary conditions at the channel symmetry lines, Y = 0 and Z = 0 can be written as, condition at the channel walls, Y = H and Z = W as,

  5.4). It was prepared by the soft lithography technique as detailed in Chapter 5. During the lling process of the microchannel with the liquids, it was made sure that no visible air bubbles are present in the syringe and the connecting tube. The rst reservoir was allowed to ll until the liquid started coming out from the hole for the rst electrode. Then that hole was closed by inserting the electrode and the second reservoir was allowed to ll keeping the sink tube blocked and second electrode hole open. Similarly, when the liquid started coming out of the hole for the second electrode, it was closed by

  Figure 6.2.: PTV Image pre-processing steps -(a) Image as obtained by the CCD camera, (b) Color inverted image, and (c) Image after background removal.

Figure 6

 6 Figure 6.3.: Estimate of PDMS-aqueous electrolyte zeta potential by matching the EOF velocity proles.

  Figure 6.4.: Schematics of the microchannel used for interfacial instability studies (a) channel schematics for 1-interface conguration (b) channel schematics for 2-interface conguration.

  Figure 6.5.: Interfacial oscillations for miscible liquids case under time periodic electric eld. (a) Amplitude of interfacial oscillations as a function of imposed electrolyte solution ow rate for dierent time-periodic electric elds at a frequency of 1Hz, (b) Interfacial displacement over time at an imposed ow rate of Q elec = 500nl/min, under a time periodic electric eld, E app = 300V /cm and dierent frequencies.

  Figure 6.7.: Interfacial evolution between two miscible liquids under an AC electric eld of 3000V /cm and a frequency of (a)1H z and (b) 2Hz. The dark liquid is a solution of 0.01M Rhodamine B in DI water and the clear liquid is an aqueous electrolyte (see text for details) with a conductivity of 1500µS/cm. Both liquids have an additional imposed ow rate of 500nl/min.

Figure 6

 6 Figure 6.8.: The oil(O)-water(W) interfacial structure in a PDMS-glass microchannel as shown by Guillot and Colin (2005).

  Figure 6.10.: Interfacial evolution under an AC electric eld of 300V /cm at a frequency of (a) 2Hz (b) 5Hz. The central liquid is an aqueous electrolyte (see text for details) with a conductivity of 1500µS/cm and the top and the bottom liquids are a solution of Hexadecane with SPAN 80 at 3%w/w. Both of the liquids have an imposed ow rate of 1000nl/min.

  mense possibility to expand the scope of the present analysis to a more generalized set of by considering the following additions, Space-Charge Potential Distribution: It was obtained as a result of an quasi equilibrium between charge diusion due to localized concentration gradients, Coulombic interactions, and thermal motion of ions. The convective motion was assumed to have negligible inuence on the charge relaxation due to the above mentioned phenomena. This observation is valid for low concentration electrolytes which is common in the naturally occurring biological uids. But for systems like ion exchange membranes, and near electrode surfaces, a complete solution of the charge transport equation (Nernst-Planck equation) might give further insights into ensuing the non-linearity of the system.

  Thin Films in DC Electro-osmosis . . . . . . . . . . . . . . . . . . . . 3.2.1. Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . 3.3. Thin Films in AC Electro-osmosis . . . . . . . . . . . . . . . . . . . . 3.3.1. Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 3.4. Free Surface Flows in Rectangular Geometry under a DC Electric Field 3.4.1. Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . 3.4.2. Results and Discussions . . . . . . . . . . . . . . . . . . . . . 3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Long Wave Analysis of Thin Films Stability in Electrokinetics 4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Thin lm stability under DC EOF . . . . . . . . . . . . . . . . . . . . 4.2.1. Electric Potential Field . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . .

	1. General Introduction
	1.1. Motivation and Overview
	II. Theoretical Studies
	3. Thin Films in Electrokinetics 6.4. Electrokinetically Induced Interfacial Instability . . . . . . . . . . . .
	6.4.1. Experimental Procedure . . . . . . . . . . . . . . . . . . . . .
	xviii

1.1. Motivation and Overview . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Publications from this project . . . . . . . . . . . . . . . . . . . . . . 1.2.1. Peer-reviewed Journals . . . . . . . . . . . . . . . . . . . . . . 1.2.2. Conference Proceedings . . . . . . . . . . . . . . . . . . . . . 2. Electro-Osmotic Flow and Interfacial Electro-hydrodynamics 2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Charge transport in bulk Electrolytes . . . . . . . . . . . . . . . . . . 2.3. Charge distribution near solid substrates . . . . . . . . . . . . . . . . 2.4. Electro-Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Maxwell Stress in Electrolytes . . . . . . . . . . . . . . . . . . 2.4.2. Total Stress in Electrolytes . . . . . . . . . . . . . . . . . . . . 2.4.3. Electro-Osmotic Flow in a Parallel Plate System . . . . . . . . 2.5. Van der Waals Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Interfacial Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 2.6.1. Continuity of Stress . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. Kinematic Condition . . . . . . . . . . . . . . . . . . . . . . . 2.6.3. Gauss's law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4. Faraday's law of induction . . . . . . . . . . . . . . . . . . . . 2.6.5. Conservation of charge . . . . . . . . . . . . . . . . . . . . . . 3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 3.2. 4.2.3. Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . 4.2.4. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 4.3. Thin lm stability under AC EOF . . . . . . . . . . . . . . . . . . . . 4.3.1. Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . 4.3.2. Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . 4.3.3. Results and discussions . . . . . . . . . . . . . . . . . . . . . . 4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Experimental Studies 5. Microfabrication and Experimental Setup 5.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Microfabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Sample injection setup . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Image acquisition system . . . . . . . . . . . . . . . . . . . . . 5.3.3. Electric Field Generator . . . . . . . . . . . . . . . . . . . . . 6. Experimental Investigation of the Electrokinetic Instability 6.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Combined Pressure-driven and Electro-Osmotic Flow . . . . . . . . . 6.2.1. The electric potential distribution . . . . . . . . . . . . . . . . 6.2.2. The velocity distribution . . . . . . . . . . . . . . . . . . . . . 6.3. Velocity characterization using µ-PTV . . . . . . . . . . . . . . . . . 6.3.1. Experimental Procedure . . . . . . . . . . . . . . . . . . . . . 6.3.2. Results and Discussions . . . . . . . . . . . . . . . . . . . . .

  Φ sc = φ sc /ζ ref , ∇ = h 0 ∇, where h 0 is the characteristic length scale D /h 0 is the Debye number with λ D = εrε 0 k B T 2F 2 z 2 c 0 as the Debye length, which represents the extent of the EDL. Hence, the Debye number represents the relative extent of the EDL as compared to the characteristic length scale of the system. For a symmetric (1 : 1) electrolyte at 25°C, the Debye length can be calculated as 0.305 √ c 0 nm. (which can be henceforth taken as the reference potential, ζ ref = ζ b ), lying in the XZ plane at Y = 0 and the electrolyte in the Y > 0 half plane (see Fig. 2.2a).

	and the corresponding dimensionless net charge density can be obtained as, λ Such a non-linear boundary value problem is not very easy to solve analytically and ρe = -De e -Y De 2 (2.3.16)
	requires a numerical solution. However, analytical solutions exist for some simplied
	and practical cases.
	One of the examples to demonstrate potential distribution in an EDL is by consid-ering a charged solid substrate-electrolye system, with a surface zeta potential as 2.4. Electro-Hydrodynamics
	ζ b The potential distribution in the electrolyte can be obtained analytically by solving 2.4.1. Maxwell Stress in Electrolytes
	Eq. 2.3.11, while considering electro-neutrality in the bulk. With the assumption Upon application of an external electric eld on any media with a denite electrical
	that at an innite distance away from the charged surface, the ionic concentration permittivity, a Coulombic force is experienced by the free or bound charges present
	reaches its bulk value (c i (Y → ∞) → c 0,i ) and the electrostatic potential becomes in the media, which react by creating an ionic ux and molecular scale polarization,
	zero (Φ sc (Y → ∞) → 0) which complies with the electro-neutrality condition in respectively. Such a force is a function of the charge distribution within the media
	the bulk, the electric potential distribution in the electrolyte can be written as [Is-which can be represented in a volume averaged stress formalism (Maxwell stress) to
	raelachvili (2011)], facilitate the hydrodynamic coupling of the electrostatically imposed uid ux. The
	Φ sc (Y ) = Maxwell stress tensor (Σ 4 β tanh -1 tanh M ) in a linear dielectric media in the absence of magnetic 4 e (-Y De ) (2.3.12) β
	Scaling the net charge density (ρ e ) by ε r ε 0 ζ b /h 2 0 (see Eq. 2.3.8), the dimensionless
	net charge density ( ρe ) can be written as,
	ρe = -	sinh 4tanh -1 tanh β 4 e (-Y De ) βDe 2	(2.3.13)
	Commonly such electrolytic systems have low substrate zeta potential (ζ b ≤ 25mV ),
	which corresponds to β < 1 for a monovalent symmetric electrolyte at 25 • C. In that
	case, Eq. 2.3.11 can be linearized as (also known as the Debye-Hückel linearization
	[Debye and Huckel (1923)]),
	The above equation is also known as the Poisson-Boltzmann Equation (PBE). Upon non-dimensionalizing ∇2 Φ sc = Φ sc De 2 (2.3.14)
	the PBE using, of the system, Eq. 2.3.10 leads to, which can be solved with the electro-neutrality condition in the bulk (Φ sc (Y → ∞) →
	0) as,
	∇2 Φ sc =	sinh (βΦ sc ) βDe 2	(2.3.11)
	where, β = of the electrostatic energy of ions with respect to the thermal energy of ions, De = k B T is the ionic energy parameter which measures the relative strength ezζ ref Φ sc (Y ) = e -Y De (2.3.15)

  In that limit, the Eq. 3.2.2 can be linearized (seeEq. 2.3.14) and the boundary value problem (Eq.

	3.2.2-3.2.3) can be solved analytically to obtain the following closed form solution,
	Φ sc (Y ) =	1 sinh 1 De	Z R sinh	Y De	+ sinh	1 -Y De	(3.2.4)
	However, for larger wall zeta-potential systems (β ≥ 1), Eqs. 3.2.2 and 3.2.3 have
	to be solved in their non-linear form. This system of equations is a boundary value
	problem (BVP) having two Dirichlet boundary conditions along with a non-linear
	second order ordinary dierential equation, for which no tractable analytical solution
	is available. Hence, to obtain the resulting potential distribution one has to resort
	to numerical tools. It is also observed that for thin EDLs (De ≤ 0.1), 1/βDe 2 is

much larger than 1, making the dierential equation very sti with large variations over short distances (for example, within the EDL). A sti dierential equation is a class of dierential equations where one is forced to use small step size while

  2.8, highlights the relative contribution of the two important phenomena, namely, disjoining pressure and Maxwell stress to the thin lm dynamics. Using Eq. 4.2.8, one can show that the Maxwell stress contribution to the momentum equation represented by γ R E R ∂Φ ∂X ∇ 2 Φ term reduces to the order ∼ O( γ R De 2 ).

	σ R
	Real part of growth rate,	1
		0	0	0.2	Wavenumber, α 0.4 0.6	0.8	1

  2. A 300ml of 10mM HEPES Hemisodium (Molar mass: 249.30g/mol) buer was prepared by mixing 0.7979g of the buer salt into 300ml of de-ionized water. In order to enhance the electrical conductivity of the liquid, a neutral ionic salt was added to this base buer solution and the ionic conductivity of the solution was measured. The ionic salt that was used in the experiments was Potassium Chloride (KCl) (Sigma-Aldrich). A 200ml solution of 0.1M KCl (Molar mass: 74.56g/mol) was prepared by mixing 1.4936g of KCl in 200ml of de-ionized water. Two solutions with dierent electrical conductivities (750µS/cm and 1500µS/cm) were prepared by slowly adding this KCl solution to the HEPES buer.

  3.1). The second liquid sample (henceforth referred to as FS-2) consisted of Hexadecane (C 16 H 34 , Sigma Aldrich) mixed with 3%w/w of SPAN-80 (Sigma Aldrich) surfactant. This leads to a surface tension of 4.05 ± 0.29mN/m[START_REF] Hashimoto | Interfacial instabilities in a microuidic Hele-Shaw cell[END_REF]]. The mentioned concentration of SPAN-80 is its critical micelle concentration (CMC) in Hexadecane and hence the value of the surface tension obtained is the corresponding limiting value. So, in order to prepare the FS-2, 100g of Hexadecane (Molar mass: 226.44g) was mixed with 3g of SPAN-80 (Molar mass: 428.62g). The choice of SPAN-80 as a surfactant to reduce Hexadecane-water surface tension is due to insolubility of SPAN 80 in water and hence it does not interfere with the EOF.

Within an EDL, two important interactions between the ions can be identied, rstly, the repulsive Coulombic interaction between the counter-ions, and secondly, the congurational entropy of the counter-ion distribution, which resists the congurational change due to the Coulombic repulsion[START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF]]. Such a competition between the two phenomena manifests itself in terms of a pressure, which can be termed as the EDL pressure. The EDL pressure distribution in a thin lm can be obtained from the basic state solution of the system (see3.2.13 and 3.3.14).

Acknowledgments

My doctoral journey over the past three years at Bordeaux has been a very enriching

Acknowledgments ix

Nomenclature xi I.

diameter. The Silicon wafer is washed with Acetone and Iso-Propanol respectively and then dried under a jet of Nitrogen gas. The dried wafer is exposed under plasma for 1 minute. Exposing the Silicon wafer to plasma activates the surface and enhances the binding of a photo-sensitive material on its surface.

Spin-coating of the photo-resist: A photo-resist is a light sensitive material which is used to create patterned structures in soft-lithography. There are two kinds of photo-resists -positive and negative. When a positive photoresist is exposed to powerful Ultra-Violet (UV) rays, it becomes soluble in a solvent (also called the developer solvent) while when a negative photo-resist is exposed to UV rays, it polymerizes and becomes insoluble in the developer solvent. This characteristic feature of a photo-resist can be used to create controlled patterns in a microchannel design. A negative photo-resist is spincoated on the activated wafer as mentioned in second step as the channel motif in the photo-mask in rst step is transparent and hence allows the passage of the UV light through it. The spin-coating protocol depends upon the thickness of the photo-resist one wants on the wafer, which is eventually determined by the requirements of the channel height. In this work, the channels were fabricated with a height of 100µm in order to keep the channel aspect ratio 1 : 1 and 1 : 2. So to obtain a height of 100µm, SU8-3050 photo-resist was spin-coated on the Silicon wafer in two steps. During the rst step, the wafer was spin-coated at 500rpm for 10s at the angular acceleration of 100rpm/s and then at 1000rpm for 30s at the angular acceleration of 300rpm/s. Pre-Exposure Bake/Soft Bake: Now, this spin-coated wafer is baked on a hot-plate at the temperature of 95°C for 45mins. This pre-baking step densies the photo-resist by evaporating the coating solvent.

Mask Alignment and UV Exposure: After pre-baking, the spin-coated wafer is exposed under uniform UV illumination using an aligner masked by the photo-negative which contains the microchannel design. Since the photoresist SU8-3050 is a negative photo-resist, which means that the photo-resist is photopolymerized where it is exposed to the UV radiation and hence is rendered insoluble in the photo-resist developer solution. In this work, SUSS MicroTec aligner was used for the UV exposure, where the power of the UV lamp was 36mW/cm 2 Next, spin-coated and baked wafer is covered by the photo mask containing the channel design through the holding mechanism in the aligner and exposed to the UV light for 9s. Post-Exposure Bake/Hard Bake: Next, the UV exposed wafer is placed on the hotplate at a temperature of 95°C for 5mins. Development: In this process, the exposed (and hence polymerized) photoresist on the wafer is put in a solvent (developer) which dissolves the nonpolymerized part of the photo-resist leaving behind the polymerized photoresist in the shape of the microchannel motif as printed on the photo-mask.

The developer solution used here is Propylene Glycol Mono-methyl Ether Ac-sec. 5. 3.3). Details of the various components used are provided below (see Fig. 5.5 and Fig. 5.6 for the schematics and the real setup).

Sample injection setup

For the two immiscible uid system, the uid samples were injected into the microchip using two infusion type syringe pumps with dierent ow rates. The syringe pumps used were of the model Legato100 by KD Scientic with a minimum ow rate of 2.551nl/min and a maximum ow rate of 2.649ml/min. The syringes used were Glass Gas Tight 1mL Fixed Luer Lock Tip syringes by SGE Analytical Science. The syringes were connected to the microchip reservoirs using Teon (PTFE) Tubing with 1/16"OD × 0.5mmID.

Image acquisition system

A phase-contrast microscope was used to distinguish the two phases in the experiments, namely, the tracer particles and the solvent in the rst set of experiments and the oil-phase and the aqueous phase in the second. The microscope used was microchannel. The Platinum wires were soldered to copper wires connected to the AC power source. values of the uid properties were taken as, dielectric constant of the aqueous solution, ε r ∼ 80, permittivity of vacuum, ε 0 ∼ 8.85 × 10 -12 F/m, dynamic viscosity, µ ∼ 1centiP oise (cP ) with the ambient temperature at 293K and pressure at 1atm. With the surface zeta potential varying from 20mV to 40mV and the applied electric eld varying from 10V /cm to 80V /cm , one can estimate the electro-osmotic reference velocity u E,ref as,

where, [x, y] represents the range of the parametric variation. The imposed ow rates used in the following study ranged from 25nl/min to 1000nl/min. Dierent velocity proles were obtained from the PTV for dierent combinations of ow rates and applied electric elds. For a pure EOF, i.e. in the absence of an imposed ow rate ( Qp = 0), the EOF velocity prole as obtained by the PTV analysis can be compared to the theoretical velocity distribution (see Eq. 6.2.11) to estimate the PDMS-electrolyte zeta potential (see Fig. 6.

3). The estimated values

of PDMS-electrolyte zeta potential were found to be within the range of reported values in the existing literature [Kirby and Hasselbrink (2004b)].

Electrokinetically Induced Interfacial Instability

In order to study the impact of electrokinetic eects on the interfacial stability of immiscible liquids, a set of experimental investigations were performed over a varied set of electric eld strengths, frequencies, imposed ow rates and surface tension.

mixed with Rhodamine-B (Sigma-Aldrich) dye. This solution was used along with the FS-1 for a two-uid ow experiment with zero surface tension. Rhodamine-B dye, which has a low diusivity in water (D RB ∼ 3.7 × 10 -6 cm 2 /s), was used to highlight the non-conductive phase i.e. the DI water. A 100ml solution of 0.01M of Rhodamine-B (Molar mass: 479.01g) was made by mixing 0.479g of Rhodamine-B powder in 100ml of DI water.

Filling and cleaning of the microchannel

The PDMS microchannel that was used in this study has 1mm long two-phase interfacial region along with a cross-section of 100µm × 100µm. It was prepared by the soft lithography technique as detailed in the Chapter 5. During the lling process of the microchannel with the liquids, it was made sure that no visible air bubbles are present in the syringe and the connecting tube. Firstly, all the Platinum electrodes were disconnected from the reservoirs and the liquid 1 inlet reservoir was allowed to ll the inlet reservoir until the liquid started coming out from the hole for the rst electrode. Then, the electrode hole of the rst reservoir was closed by inserting the electrode. Then, the outlet reservoir was allowed to ll while keeping the outlet tube blocked and the second electrode hole open. Similarly, when the liquid started coming out of the electrode hole of the outlet reservoir, the hole was plugged by inserting the second electrode and the outlet tube was opened. The same process was repeated for each of the subsequent lling process to ensure that both the reservoirs are completely lled with the liquid pumped. In order to clean the microchannel, rstly, the microchannel was washed with de-ionized water for 20mins

and then with 0.1M NaOH for 20mins. Finally, the FS-1 solution was introduced in the microchannel and then FS-2 (or FS-3) were introduced in the liquid 2 inlet using another syringe pump. The ow rates of the FS-1 and FS-2 (or FS-3) were adjusted in order to have a at and stable interface. Then, the AC electric eld was applied for dierent combinations of electric eld magnitude and frequency and they were varied along with the imposed ow rate until the interface broke or oscillated vigorously.

Image acquisition

The phase-contrast microscope (for details see Chapter 5) was used to record videos of the two-phase interfacial dynamics. The microscope objective with a magnication of 10× and a numerical aperture of 0.7 was used. For each reading of the dierent combination of the ow rate, amplitude and frequency of the electric eld, a video of 500 frames was recorded by a CCD camera with a frame rate of 26f ps and exposure time of 45.28µs. Each video had a resolution of 508 × 384 in 8 -bit grayscale.

In the immiscible uids case, the objective was to study the liquid-liquid interfacial deformation under a time-periodic EOF in the presence of capillary force and convective transport. To obtain a non-perturbed (at) interface against an interface breaking capillary force, a ow rate was imposed on the liquids to counter the surface tension driven instability. It was observed that a critical value of ow rate exists above which the system remains stable for the applied range of electric elds. This suggested the existence of a competing dynamics between the convective inertia and Maxwell stresses. For a given value of ow rate, the stability of the system decreases upon increasing the magnitude of the electric eld. Finally, it was observed that high frequency AC electric elds tend to stabilize the ow at a given ow rate.

6.5. Sources of Errors and Uncertainties

Microchannel

The defects in the microchannel can be attributed to the soft lithography technique used to fabricate the channel. Although the channel dimensions corresponding to the motif design were quite accurate (variations in dimension< 1µm), the height of the channels, which is controlled by the spin coating of the photo-resist and the further pre-exposure processes (see Chapter 5), was found to be varying over 95 -105µm for a target height of 100µm. These irregularities in the channel cross-section can lead to an expanding-contracting channel design thus aecting the ow. Moreover, the use of PDMS chips which is soft in nature and thus compliant to changes in the pressure, can further deform the channel geometry. So, in order to avoid such geometrical uncertainties, glass channels are used commonly, which have a relatively rigid construction but are costly with greater fabrication time. This, however, can be envisaged as a future development of the present experimental setup.

Apparatus

One of the main components of the experimental apparatus is the syringe pump.

The step motor which allows the rotation of the innite screw of the syringe pump produces periodic vibrations. Those vibrations are transmitted to the syringe piston translation and then generate periodic variations of the ow rate. In order to overcome these undesirable pulsations, one can use pressure generators, which do not generate vibrations since pressure is regulated through a pressure controller.

The next important component is the AC power generator. For the current model of the AC power generator (Agilent 6811B) used, it was recommended to have the operating frequency > 45Hz. However, in order to obtain perceivable interfacial oscillations, we had to operate in the frequency range ≤ 10Hz. In that range of operating frequency, the precision on the output voltage was not assured by the gas-uid-substrate interaction, magnitude and frequency of the externally applied electric eld, substrate and free surface zeta potential was highlighted. (c) The increase in the value of substrate zeta potential tends to stabilize the ow which in turn enhances the osmotic pressure component.

A set of experiments were performed in order to characterize the EOF in a rectangular microchannel. With the help of a PTV analysis, velocity distributions were obtained which agreed well to the theoretical values. This was used to estimate PDMS zeta potential, which was found to be within the reported values in the existing literature. Further, in order to study the liquid-liquid interfacial instability under time-periodic electric eld, two sets of cases were considered.The rst set of cases consisted of miscible uids and the second one consisted of immiscible uids.

In the miscible uids case, the objective was to study liquid-liquid interfacial deformation under a time-periodic EOF in the presence of convective and diusive transport. To obtain a non-perturbed (at) interface against a nite diusion of miscible uids, a ow rate was imposed on the liquids so that the ensuing convective ow can hinder the mixing of the two uids by interfacial diusion. The ow rate was kept the same in both the liquids. It was observed that for a given magnitude of applied electric eld, the amplitude of the interfacial deformations decreased with increasing imposed ow rate, which ascertains the stabilizing eect of the imposed ow rate. Further, it was also observed that the responsiveness of the interface (i.e. the amplitude of the perturbed interface) to the frequency of the applied electric eld, was most noticeable in the range of 1 -5Hz.

In the immiscible uids case, the objective was to study liquid-liquid interfacial deformation under a time-periodic EOF in the presence of capillary force and convective transport. To obtain a non-perturbed (at) interface against an interface breaking capillary force, a ow rate was imposed on the liquids to counter the surface tension driven instability. It was observed that a critical value of ow rate exists above which the system remains stable for the applied range of electric elds. This suggested the existence of a competing dynamics between the convective inertia and Maxwell stresses. For a given value of ow rate, the stability of the system decreases upon increasing the magnitude of the electric eld. Finally, it was observed that high frequency ac electric elds tend to stabilize the ow at a given ow rate.