Capillarity and Convection-Controlled Assembly in the Spreading of Particulate Suspensions on an Air-Liquid Interface

Capillarity and convection-controlled assembly in the spreading of particulate suspensions on an air-liquid interface

Rajesh Ranjan

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Department of Chemical Engineering and Applied Chemistry University of Toronto

i Capillarity and convection-controlled assembly in the spreading of particulate suspensions on an air-liquid interface Master of Applied Science Rajesh Ranjan Department of Chemical Engineering and Applied Chemistry University of Toronto 2018 Abstract Self-assembly of particles at interfaces has immense potential for printing and coating applications in biological and industrial processes. Several studies on the spreading of pure fluids on an air- liquid interface have been conducted; however, none have examined the spreading characteristics of two-phase fluid materials. In this work, a drop of concentrated suspension of PMMA particles in silicone oil was placed on an aqueous glycerol solution – air interface. Depending on the initial rate of spreading of the suspension, two outcomes were observed: the particles were either swept away by the spreading suspension or organized into an array of two-dimensional networks. The two outcomes were explained by describing the particle motion as being a result of a competition between fluid convection and capillary attraction. This description was confirmed by performing experiments for different particle sizes, volume fractions, the viscosity and salinity of the substrate on the spreading behavior and pattern.

ii Acknowledgement

I would like to express my appreciation to all those people who have contributed to make this work possible through their help and support along the way. My deepest gratitude goes to my supervisor Dr. Arun Ramachandran for giving me the opportunity to work on this interesting project and for his valuable guidance, inspiration, constant encouragement and support through all the phases of the project. I feel indebted to my supervisor for giving abundant freedom to me for pursuing new ideas. I take this opportunity to express my deep sense of gratitude to Dr. Julia A.

Kornfield from California Institute of Technology and Dr. Kathleen J. Stebe from University of

Pennsylvania for thoughtful advice and discussions. I extend my thanks to my summer student,

Srishti Sehgal, for help with the experiments and my lab members for their inputs that helped in the completion of this project.

iii Table of Contents

Acknowledgement ...... iii

Table of Contents ...... iv

List of Figures ...... vi

List of Tables ...... viii

List of Videos ...... ix

1. Motivation ...... 1

2. Literature Review ...... 8

3. Experiment...... 13

3.1 Materials and Methods ...... 13

3.2 Experimental Conditions and Systems Investigated ...... 14

4. Result and Discussions ...... 16

4.1 Effect of Volume Fraction of the PMMA Suspension ...... 32

4.2 Effect of Particle Size ...... 33

4.3 Effect of Viscosity of Suspending Fluid ...... 34

4.4 Effect of Addition of Salt in the Substrate ...... 36

4.5 Effect of Dish Area ...... 37

4.6 Summary of the Behavior of Systems ...... 38

5. Conclusions and Future Work ...... 42

Appendix ...... 44

iv Appendix A: Determination of dependence of radius as a function of time ...... 44

Appendix B: Shape of the interface around individual PMMA particle ...... 46

Appendix C: Precursor film experiments ...... 49

Appendix D: Theoretical prediction of the area coverage ...... 50

Appendix E: Scaling analysis to understand the force balance ...... 53

Appendix F: Dependence of the critical radius (Rc) on particle size and volume of the suspension

introduced during the spreading process ...... 55

References ...... 56

List of Figures Figure 1: Wetting conditions and their relationship to the contact angles ...... 1 Figure 2: Self-cleansing properties of lotus leaf due to surface roughness and hydrophobicity .... 2 Figure 3: Spreading behavior can be categorized based on fluidity of the substrate, the volatility of the drop phase, and the volume fraction of particles in the suspending medium ...... 5 Figure 4: Drop of ethanol mixed with dye defragments into myriads of droplets over a layer of mineral oil ...... 6 Figure 5: Liquid drop spreading over a solid substrate ...... 8 Figure 6: Suspension drop introduced over glycerol b: Subsequent spreading of the suspension drop ...... 13 Figure 7: Image of the PMMA aggregates under 4X magnification b: Image of extended network of aggregate under 4X magnification ...... 16 Figure 8: Image sequence for 40% suspension, 20 μm particles on pure glycerol (initial part of the spreading process) Last frame: after a critical radius, the particle ring/ dam can no longer be sustained, and it fragments ...... 17 Figure 9: Area shows linear dependence on time during the spreading process ...... 18 Figure 10: The spreading front is circular during the initial stage (the spreading front is nearly circular) ...... 18 Figure 11: The spreading front during the later stage of the spreading process (the spreading front starts to deviate from circularity) ...... 19 Figure 12: Image sequence for 40% suspension, 20 μm particles on glycerol (later part of the spreading process once the instability sets in) ...... 20 Figure 13: Schematic of the cross-sectional shape of the spreading drop showing the confinement of the particles and the pure fluid region. The inset shows the shape of interface around PMMA particles sitting at the silicone oil – glycerol interface ...... 20 Figure 14: Comparison of the change in outer radius of the suspension front to inner core radius ...... 21 Figure 15: Regions in the spreading drop phase before the instability ...... 22 Figure 16: Regions in the spreading drop phase after the instability ...... 22 Figure 17: Confocal experiments indicates the presence of silicone oil around the PMMA particles ...... 25

Figure 18: The size of the polygons increases radially outward from the center ...... 26 Figure 19: 1/W varies linearly with radial position (R) ...... 27 Figure 20: Band of polygons at a radial position R growing radially outwards ...... 28 Figure 21: For the same particle size and viscosity of the base liquid lower volume fraction suspensions spread faster (indicated by the lower intensity value at the starting point for subsequent frames) ...... 32 Figure 22: For the same volume fraction of the suspension and base liquid PMMA particles with smaller radius spread out faster (indicated by the lower intensity value at the starting point for subsequent frames) ...... 34 Figure 23: For the same volume fraction of the suspension and particle size systems with lower viscosity of the suspending medium spread out faster (indicated by the lower intensity value at the starting point for subsequent frames) ...... 35 Figure 24: Time required for silicone oil front to reach the wall for varying volume of suspension drop (40 % 20 μm on glycerol) ...... 38

vii

List of Tables Table 1: Physical properties of the aqueous glycerol solution ...... 14 Table 2: Network characteristics (slopes and intercepts) of the systems investigated ...... 30 Table 3: Time required for silicone oil front to reach the wall of the dish for varying volumes of suspension (40 % 20 μm on glycerol) ...... 37 Table 4: Predicted and experimental time for 10 μl of suspension drop (40 % 20 μm on glycerol) to cover the dish area completely ...... 38 Table 5: Summary table of the effects of particle size, volume fraction and viscosity of the substrate liquid without NaCl that does/ does not lead to the formation of a network. (Green: indicates network formation Red: network not formed) ...... 40 Table 6: Summary table of the effects of particle size, volume fraction and viscosity of the substrate liquid (after the addition of NaCl) that does/ does not lead to the formation of a network. (Green: indicates network formation Red: network not formed) ...... 41

viii

List of Videos Video 1: 20 μm 40% PMMA suspension spreading over glycerol

Video 2: Film thinning, particle confinement and monolayer formation (20 um 40% PMMA suspension on glycerol)

Video 3: Precursor film experiment with PMMA particles laden on the surface

1. Motivation The phenomena of wetting and spreading of liquids on solid and liquid surfaces are ubiquitous. Wetting is the ability of a liquid that allows it to maintain a certain contact with a surface. It results from intermolecular interactions when the liquid and the surface come together. A force balance between adhesion and cohesion governs the degree of wettability. Wetting and spreading finds widespread applications in numerous fields, such as inkjet printing, deposition of paints on surfaces or pesticides on leaves and in biology [1] [2] [3]. For example, wetting agents and surface modifiers are crucial elements of paints and coatings. Together they determine the wetting properties and ensure the compatibility between surfaces. The spreading of paint over a surface (see Figure 1) is controlled by the surface tension of the paint and the surface over which it is applied. The wetting state can be understood in terms of the spreading coefficient (S). Whether a phase “o” spreads on another phase “aq” or not depends on the sign of the spreading coefficient [3] [1],

S =aq −  o −  aq/ o (1)

where  aq and  o are the surface tensions of the phases “aq” and “o”, and  aq/ o is their interfacial tension. For S  0 , the drop has a finite contact angle with the surface. For S  0 the drop spreads over the surface while preserving the volume.

Figure 1: Wetting conditions and their relationship to the contact angles

Another classical example is that of a lotus leaf which has self-cleansing properties [4]. The tendency to repel water over the surface of lotus leaf is on account of combination of surface roughness and hydrophobicity as shown in Figure 2. The microrelief of the leaf surface is primarily due to epicuticular wax crystalloids. This also reduces the adhesion of contaminating particles and enhances water repellency.

Superhydrophobic surfaces can be created through biomineralization [5]. Nucleated and grown calcium phosphate leads to an open and porous structure which results in lot of open pores suitable for trapping air. This makes it extremely fitting for a superhydrophobic surface.

Nano-patterned organic thin films find applications in sensors and microelectronics. The pore diameter in the thin film can be adjusted to tens of nm depending on the surfactant patterning agent used as the template and the solvent composition [6]. Precise balance between the surfactant micelles, inorganic precursor, solvent phase and interfaces during evaporative drying and thermal treatment leads to the formation of such a system. Such systems have wide application prospects in fields such as nanotechnology and engineering.

Figure 2: Self-cleansing properties of lotus leaf due to surface roughness and hydrophobicity There have been numerous studies to modify and control surface properties to obtain the desired wetting behavior. Such treatments, e.g. salinization or plasma treatment, transfigure the contact angle and surface energy of liquids with the surface of another solid or liquid. Besides chemical interactions which are short-ranged interactions, van der Waal and electrostatic forces also play an important role in wetting. Isolation of two-dimensional van der Waal layered materials have been used to understand how the wetting behavior of surfaces is affected by van der Waal

2 forces. The intermolecular surface force approach originated with Derjaguin and Landau's and Verwey and Overbeek's (DLVO) theory of colloidal stability [7]. The DLVO theory describes the stability of lyophobic (solvent hating) colloids, considering electrostatic and van der Waals interactions. This approach has been used to not only explain the stability of the wetting-water film but also the value of the contact angle when the wetting film collapses [8]. van der Waals forces exist between all matter and are important component of the surface forces in thin films. The interfacial tension of mercury/water system changes as one applies a voltage to mercury [1]. The interfacial tension depends critically on the ions present in the solution. For the case of a sodium sulfate solution, the Na+ ions attach to mercury and the double layer formed in water is due to the 2 – SO4 ions. The double layer is formed over a screening length and acts as a capacitor. If a positive potential is applied, the Na+ ions are repelled leading to the recovery of surface tension of pure water.

The nature and composition of the fluid (presence of surfactants, particles in the case of suspensions) can alter the long-range and short-range interactions and influence the equilibrium state of liquids on the surface of another liquid or solid. Very few studies have attempted to understand the influence of complex fluids on the spreading dynamics. Experiments using different types of complex fluids like surfactant and polymer solutions have been shown to follow Tanner’s law [9]. For the case of liquids on hydrophobic surfaces, one of the effects of surfactants is to lower the interfacial tension sufficiently to make the spreading coefficient positive. As the drop spreads and new interfacial area is created, surfactants transport themselves to the interface to reduce the interfacial tension. The transport is slower than the rate of increase in area leading to lower dynamic spreading coefficient. This slows down the spreading dynamics. Experimentally it has been shown that the spreading of a shear-thinning fluid (another complex fluid) is slower than that of a Newtonian fluid. The power in the spreading law is slightly smaller than 0.1 and decreases with increasing shear thinning [10].

Presence of impurities on surfaces, roughness of the surface may lead to a more intricate and complicated dynamics during the spreading/ wetting process. Three different contact angles can be defined as the static, receding (minimal) and advancing (maximal) contact angle [11]. For a hydrophilic surface, roughness makes the surface more hydrophilic and for a hydrophobic surface the roughness makes the surface more hydrophobic.

Many theoretical and experimental studies have focused on developing a deeper understanding of the physics of spreading [12] [13] [1] [14] [15] [16], but most of the studies have been restricted to the spreading of single-phase liquids on air-solid and air-liquid interfaces. In this work, we will explore the case of Newtonian suspending fluids containing rigid particles suspended within them, in the absence of surfactants. All such systems can be broadly categorized on the basis of three characteristics:

a) Substrate rigidity: This attributes to the fluidity of the substrate, for example solid substrate, gel like substrate or a liquid substrate. b) Particle volume fraction: Depending on whether the drop phase is single phase, contains particles in dilute concentrations, or contains a high-volume fraction of particles, the spreading behavior can be vastly different. c) Suspending medium volatility: This refers to the volatility of the drop phase or the suspending phase in the case of a single phase and multiphase system respectively.

Based on these properties, existing studies can be classified into the eight regimes shown in Figure 3. For example, the majority of the spreading works have focused on the behavior of a drop spreading on solid surfaces [1] [17]. The spreading of non-volatile single-phase liquids on air-solid interface would lie in the Regime III. Studies of drop spreading on liquid substrates [12] [13] [14] [15] constitute the regime where substrate fluidity becomes important. For example, the spreading of alcohol or another volatile single-phase liquid on liquid-air interface lies in Regime VI. While many of the studies center around the pure drop phase spreading over solid or liquid substrates, the famous ‘coffee ring’ effect [18], ring like pattern left by a particle-laden liquid after it evaporates, constitutes a classical case of system containing dilute suspension spreading over solid substrate where evaporative effects are important (Regime VII). Seminal works on coffee stains have shown that evaporation can induce complex flows by inducing variation in the temperature and composition of the liquids [18].

Marangoni flow, surface tension gradient driven mass transfer along the interface between two fluids, plays a crucial role in the spreading behavior of drop phase which is volatile (e.g. alcohol) on the surface of other liquids. Surface velocity field induced by Marangoni effect for aliphatic alcohols on water or aqueous solution have been studied experimentally [15]. Evaporation causes variation in the temperature. Composition of the liquids is also affected by

Figure 3: Spreading behavior can be categorized based on fluidity of the substrate, the volatility of the drop phase, and the volume fraction of particles in the suspending medium solubilization, adsorption and mass transfer. These variations cause gradient in the surface tensions, thus inducing Marangoni flow. Previous works have studied the effect of introducing alcohol-water mixtures on oil surface [19]. The alcohol-water drop spreads and eventually undergoes a Marangoni outburst, disintegrating the parent drop into hundreds of smaller droplets (See Figure 4). Once the drop has reached the maximum radius it starts to recede and ejects droplets as the rim regions undergoes Rayleigh-Plateau instability. The thinner rim region undergoes more evaporation thus affecting the surface tension value leading to change in sign of the spreading parameter (Figure 4 frame (f) onward). This reverses the direction of spreading and leads to the eventual dewetting (Figure 4 frame (f) through (j)). This gradient in surface tension continues to feed the rim and drive radial flow from the center to the edge of the drop which leads to beading at the edge. Subsequent retraction leads to instability and ejection of droplets. Ejecting droplets

5 causes the central region of the drop to recede, which can undergo further instability resulting in a myriad of droplets.

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

Figure 4: Drop of ethanol mixed with dye defragments into myriads of droplets over a layer of mineral oil Based on the octant in which a system lies in Figure 3, we can see that the dynamics of spreading of a drop on a solid/ liquid substrate can showcase a variety of patterns and behavior. In summary, a non-volatile single-phase liquid form a lens or thin film over the solid/ liquid substrates. In the case of the coffee ring effect, a ring-like deposit is formed along perimeter of the drop. A drop of volatile alcohol deposited on water or oil bath can spread, form a lens or spontaneously fragment into myriad of droplets. Contrary to all such behaviors, we observe the formation of a particle network when a drop of particulate suspension is introduced over a liquid substrate.

In this thesis work, our interest lies in Regime I with high substrate viscosity (water- glycerol solution), moderate to high particle volume fraction (10 % to 40 %) and non-volatile suspending phase (silicone oil). While there have been very few studies of suspensions spreading on a solid surface [20] and over liquid substrates [21], to the best of our knowledge, no study has examined the spreading properties of concentrated (high volume fraction) suspensions on an air- liquid interface. Our experiments aim to at least partially fill this void. The largest particle volume fraction in our experiments is 40 %. We are interested in the spreading behavior of moderately concentrated suspensions, where interparticle interactions are dominated by lubrication interactions. For higher volume fractions (>50%), suspensions act essentially like pastes, and upon shearing, stresses in such suspensions are dominated by particle-particle contacts. While the spreading behavior of pastes is very likely interesting, we avoid this regime due to its complicated physics, and restrict ourselves to volume fractions of 40% or less.

In this study, we show that when a concentrated suspension of particles in a non-volatile suspending medium is introduced on a liquid-air interface, the particles do not spread with the suspending fluid in a homogeneous manner; rather, they can demix spontaneously to either form a two-dimensional network at the interface, or be distributed as fragments (see Video 1).

2. Literature Review A liquid drop when introduced on another surface spreads as the excess capillary potential energy dissipates during its motion. If the surface is rigid the entire dissipation occurs by viscous flows within the liquid. There is a dynamic energy balance between the rate of restoration of capillary potential energy and the viscous dissipation through shear motion within the liquid. If a small drop is placed over a solid surface, the contact angle before equilibrium is attained is greater than the static value. This capillary imbalance leads to spreading force. As the wetting front proceeds the work is dissipated in the form of viscous shear within the fluid. On a more flexible surface such as a soft solid; for example, an elastomer; or a liquid substrate deformation may result near the triple contact line [24]. The deformation at the leading edge and its motion induces additional viscoelastic dissipation for the case of soft solids. For the case of liquid substrate, the sub-phase acts as an additional source of viscous dissipation.

Figure 5: Liquid drop spreading over a solid substrate Consider a liquid on the surface of a solid substrate as shown in Figure 5. The equilibrium contact angle that the drop makes with the surface is given by the Young’s equation.

sv=+  sl cos  eq (2)

Interfacial tension of the solid-gas ( sv ), solid-liquid ( sl ) and liquid-gas ( ) determines the equilibrium state of the drop on the solid surface. The wetting state depends on these surface

8 tension values. If sv+  sl  , a drop will have a finite contact angle with the solid and it a partially wetting state. The contact angle is zero if sv=+  sl  . The drop completely wets the surface in this case and forms a uniform liquid layer over the surface.

Depending on how the drop phase is introduced over the liquid substrate the spreading could occur in different regimes [14].

a) Inertial regime (gravity – inertial): Under this regime the gravitational force balances the acceleration of the drop phase. b) Viscous regime (gravity – viscous): The inertial regime gives way to the viscous regime when the spreading fluid becomes thin. Under this regime, the spreading of the drop phase is balanced by the drag in the viscous boundary layer in the sub-phase. c) Surface tension regime (viscous – surface): In this regime the oil thickness is decreased to an extent that the gravitational forces are no longer important. The spreading process always ends in the surface tension regime. The final area of the spread out drop depends on how the surface tension value changes.

Numerous studies have tried to understand the spreading behavior of a pure liquid drop over a solid and liquid surface. When the liquid drop being introduced contains particles suspended within it the phenomena of self-assembly begins to manifest in addition to the conventional spreading process. Small objects floating on the surface of a fluid are not at rest; they tend to exhibit attractive or repulsive behavior depending on the shape of the interface [25] [2] [26]. Attraction is observed between similar particles that deform the interface in the same direction where a repulsion is seen for two dissimilar particles, for example a light and a heavy particle or a hydrophilic and a hydrophilic particle. Self-assembly refers to aggregation of particles into an organized structure without external assistance. Bubbles on the surface of water autonomously associate with each other to form a bubble raft, which reveals short and long-range order present in the assembly. Depending on the shape of the meniscus the bubbles could also migrate towards or away the wall. These phenomena can be explained based on surface tension. Particles sitting at the surface of fluids tend to distort the interface due to gravitational effects. For particles smaller than 10 μm, interfacial deformation arises due to particle wetting properties. The resulting immersion capillary forces can operate even for very small particles [27]. This leads to an increase in the surface area resulting in the increase in surface energy. In order to decrease the area and

9 hence the surface energy the particles start to move towards one another, when distortions of the same sign from the neighboring particles overlap. Capillary attraction arises spontaneously at fluid interfaces between particles with similar wetting behavior. Similar wetting behavior deforms the interface in the same direction leading to attractive interactions.

In the case of suspensions, besides the role of the physics that govern the spreading behavior of a single phase fluid which is the suspending medium, the particles also undergo assembly because of capillary forces [29] [30] and gravity. Latex aggregates formed by diffusion- limited colloid aggregation (DLCA) and reaction-limited colloid aggregation (RLCA) are known to have fractal dimensions close to 1.8 and 2.1 [21] [31], respectively. Diffusive motion of cluster, which stick on first contact is referred to as diffusion limited cluster aggregation. When the probability of two cluster sticking is low different contacts can be explored before the final adherence (reaction limited cluster aggregation) [32] [33].

Colloidal particles are trapped at liquid surfaces by surface tension and electrostatic forces. Aggregation can be induced in between the colloidal particles by the addition of salt/electrolyte to the liquid substrate, due to the reduction in the repulsive forces, allowing the particles to approach one another such that van der Waal attraction can induce the adhesion [21] [34]. Several forces act on interfacial colloidal particle [31] [33]:

a) Capillarity induced surface dimpling. b) Colloidal particles bear electrostatic charges on the wetted portions. This along with the Debye cloud of counterions constitute electric dipoles when the particles are at the liquid interface (the force is 1/r3 repulsion) [35]. c) There may be steric effects: when two polymer-covered surfaces approach each other the entropy of confining the dangling chains leads to a repulsive entropic force. For the case of overlapping polymer this repulsive force is the steric repulsion. Colloidal particles that aggregate in a solvent can be stabilized by addition of small amount of polymer in the dispersion medium [36]. d) Surface forces: it includes the London–van der Waals force, solvation force, hydrophobic force and steric force.

When a sub-micron sized particle is pinned at an air-water or oil-water interface, the capillary energies retaining the particle at the interface are typically six orders of magnitude larger

10 than the characteristic thermal energy; the pinning of the particle is thus, essentially irreversible. When a collection of particles is adsorbed at an interface, the equilibrium arrangement of the particles depends on the particle shape, electrostatic effects, the surface fraction of the particles, relative density of the particle with respect to the liquid, the viscosity and surface tension of the supporting liquid layer. For example, for weak electrostatic effects, six particle-particle contacts are formed in a stable packing of spheres, driven to assembly on account of gravity by monopolar deformation interactions. A non-spherical micron sized particle produces a quadrupolar deformation of the interface, and self-assembly results from interactions between the quadrupoles when multiple particles are present at the interface. Ellipsoidal particles, for example, form a stable packing of as many as ten particles upon self-assembly on an interface [21] [37].

In the presence of repulsive electrostatic interactions, previous studies have shown that monolayer of spherical particles arrange in a dot paper like pattern. For ellipsoidal particles, they occupy triangular, quadrilateral and other polygonal structures repeatedly. This shows that although the electrostatic forces create a barrier for aggregation, the final structure appears to be determined by the capillary forces [38]. By changing the charge at the air-water interface, introducing oil phase, changing the contact angle and interfacial tension the density of the network like structure formed can be changed. Electrostatic interactions between ions are highly attenuated in water. The electrostatic force between charged particles in solution is inversely proportional to the dielectric constant of the solvent. Charged particles have enhanced electrostatic repulsion at the water-oil interface. The repulsion is a lot weaker on air-water interface. Addition of salt can be used induce aggregation in the monolayers. For anisotropic particles, it has been shown that shape- induce capillary forces can also cause aggregation in the absence of salt [21].

While many studies [21] [39] [38] have investigated the idea of particle self-assembly with quadrupoles with volume fractions between 0.1% and 1%, the network structure that we see for our systems is not prominent with the volume fractions mentioned. When a drop of concentrated suspension is introduced on a liquid-air interface (glycerol solution in this case), the particles do not spread homogeneously. Depending on the particle size, particle volume fraction and the substrate viscosity two outcomes were observed. The system underwent demixing and could either form fragments/ small aggregates of particles that were swept away or it could reorganize itself in the form of a two-dimensional network.

In this work, we have tried to identify the combination of parameters in terms of viscosity of base liquid (liquid layer over which the suspension is introduced), volume fraction of suspension and particle size to form a network. To understand the effect of addition of salt on particle assembly, we prepared 0.5M NaCl glycerol-water solutions (by adding 29.2 g of NaCl per L of the solution) to see its impact on the suspension spreading. Network formation was observed even for the systems where networks did not form initially in the presence of salt. The details of the suspension preparation procedure, the experimental setup and the systems investigated are discussed in the Section 3.

3. Experiment

3.1 Materials and Methods The experimental setup is shown in Figure 6a and 6b. Two retort stands were placed next to each other, such that their bases were facing inwards. The clamps were attached at the same height (ensured using a bubble balance) to hold the acrylic sheet in place. The tripod was set up with the head stretched to maximum height so that the entire petri dish was in the field of view. The camera was mounted so that it was overlooking the sheet from above. The petri dish was placed at the center of the acrylic sheet. The center was marked on the underside of the acrylic sheet. A 150 mm petridish was filled with pure glycerol or glycerol-water mixtures up to a height of 3 mm. This was approximately 55mL of glycerol solution for 150mm petri dish. 10 μl of the suspension of PMMA particles in silicone oil was introduced at the center of the petridish on the air-glycerol solution interface. The video was recorded as soon as the drop was introduced. The drop was then very carefully introduced over the glycerol surface to avoid trapping any air bubble, and to ensure negligible approach velocities of drop so as to avoid inertial effects. The frame rate was set at 25fps (Nikon D5100 DSLR). The video was recorded at 1920 x 1080. The white balance was set to auto, the ISO was set between 100 – 400 (external light source was added to compensate for low ISO values and to eliminate reflections and interference due to ambient light) and f – stop was set to auto. The best exposure for the system was -1.0 at full aperture. A Phantom v711 high speed camera was used for the cases where the rate of spreading was very fast to resolve the process dynamics. The video is recorded until there is no change in the system as it spreads on glycerol solution.

(a) (b)

Figure 6 a: Suspension drop introduced over glycerol b: Subsequent spreading of the suspension drop

3.2 Experimental Conditions and Systems Investigated

3 The volume fraction, , of PMMA particles ( P =1.18 g/cm ) in silicone oil was 10%, 20%, 30% or 40%. Three batches of nearly spherical PMMA particles of diameters 2a = 6 ± 0.3, 10± 0.5, and 20± 1 μm were used. To create PMMA suspensions of the desired volume fraction,

3 the appropriate weights of PMMA particles and silicone oil ( o = 0.97 g/cm , o = 0.5 Pa-s ) were mixed first using a vortex mixer, and sonicated for at least 2 hours to eliminate air bubbles. A thin layer of the resulting suspension was examined under a microscope to ensure that it was visibly bubble free.

The viscosity of the base liquid can be varied by adding DI water. The glycerol-water mixtures used were pure glycerol, a 3:1 mixture, a 2:1 mixture (both by volume) and DI water. The physical properties of the systems are listed in Table 1.

Glycerol: Water Density ( aq ) Viscosity ( aq ) (Pa- (g/cm3) s) Pure Glycerol 1.26 0.906 3:1 1.20 0.041 2:1 1.18 0.020 DI Water 1 0.009 Table 1: Physical properties of the aqueous glycerol solution

The spreading parameter [3] [1] S , defined as aq−−  o  aq/ o , where  aq (64 mN/m) and

 o (20 mN/m) are the surface tensions of glycerol and silicone oil, respectively, and  aq/ o (21.2 mN/m) is the glycerol-silicone oil interfacial tension, is positive (64 – 20 – 21.2 = 22.8 mN/m >0) for the glycerol drop/silicone oil/ air system. Hence, when a drop of silicone oil is introduced over glycerol bath it spreads out almost instantaneously to cover the surface.

In order to understand the effect of the aspect ratio of the final suspending fluid thickness with respect to the particle size on the spreading dynamics we conducted experiments with petri dishes of three different diameters (150, 100 and 70 mm). The height of the glycerol layer was the same for all the three set of experiments and maintained at a constant value of 3 mm. This corresponds to approximately 55, 24 and 9 ml for 150, 100 and 70 mm dish respectively. Drop

14 volumes of 10, 20 and 30 μl were explored for the case of 40%, 20 μm PMMA suspension on pure glycerol for all dish areas to understand the effects of volume on the spreading dynamics.

Addition of salt to the liquid substrate plays an important role in the way the PMMA particles within the suspension drop interact among themselves at the silicone oil – glycerol interface. The behavior of all the systems was also studied after the addition of salt in the sub- phase before the drop is introduced. 0.5 M NaCl solution of pure glycerol, 3:1 glycerol solution, 2:1 glycerol solution and DI waster was prepared by the addition of appropriate amount of salt to the system (29.2 g of NaCl per L of the solution). All the glycerol solution had to be sonicated to eliminate the air bubbles trapped during the dissolution process.

4. Results and Discussion For the case of low volume fraction suspension or low size of PMMA particles or low viscosity of the suspending medium it was found that the particles are swept away. The particle aggregates under these conditions as a result of spreading of the suspension over the liquid substrate is shown in Figure 7a. However, when the volume fraction of the suspension is high or the PMMA particle size is increased or the viscosity of the suspending medium is raised by the addition of glycerol the particles organize themselves in an array of two-dimensional network. The structure of the particles in the form of network is as shown in Figure 7b. For appropriate combinations of particle size, volume fraction and glycerol solution viscosity, the particles can form a percolating network structure that spans the entire surface.

(a) (b)

Figure 7a: Image of the PMMA aggregates under 4X magnification b: Image of extended network of aggregate under 4X magnification The initial phase of the spreading process is the same as that of a single-phase fluid spreading over a liquid substrate. The spreading front is nearly circular as shown in Figure 8, and the area shows a linear dependence on time (discussed in detail later). Images of the top view of the spreading process recorded using a Nikon D5100 DSLR (25 fps at 1920×1080) are shown in Figure 8. After the initial phase, instability sets in and the spreading front deviates from circularity, as depicted by the last two frames in Figure 8.

Figure 8: Image sequence for 40% suspension, 20 μm particles on pure glycerol (initial part of the spreading process) Last frame: after a critical radius, the particle ring/ dam can no longer be sustained, and it fragments

During the initial stages, the suspension spreads in a manner similar to a pure fluid (Figure 8); the suspension appears to be homogeneous and the spreading front is circular in shape. The

22SH radius of the front, R, follows the relationship R=+ R0 c t (see Figure 9), which is expected aq when viscous dissipation occurs over the entire thickness of the substrate [40] [14]. Here H and

aq are the thickness and viscosity of the glycerol layer. The relation is obtained by balancing the spreading force against the viscous drag (Appendix A).

During this time period, the perimeter P of the particle front is the same as 2 R , where

A R is the radius of front obtained using the relation R = . Here, A is area of the circular  region. A plot of P with is linear in nature as revealed by the plot (see Figure 10), suggesting the front is circular.

Figure 9: Area shows linear dependence on time during the spreading process

Figure 10: The spreading front is circular during the initial stage (the spreading front is nearly circular)

After a critical radius the spreading front starts to deviate from circularity. Figure 11 shows the deviation of the spreading front into the non-circular regime. The ratio of P to 2 R has a value greater than 1 confirming the departure from axisymmetric spreading process.

Figure 11: The spreading front during the later stage of the spreading process (the spreading front starts to deviate from circularity) Once the spreading front becomes non-circular the process shows a significantly different behavior than that of a single-phase fluid spreading over a liquid substrate. Figure 12 shows evolution of the spreading process after instability sets in. The initial fragment of PMMA aggregates are swept away by the spreading front. When the spreading slows down the aggregates tend to organize themselves in the form of a self-assembled network. Before we go into the details of the mechanism responsible for the reorganization it would be useful to understand the different regions within the spreading drop phase containing the suspended PMMA particles.

The spreading drop phase can be divided into four regions (Figure 13 shows the side view of the four regions. Refer to Figures 15 and Figure 16 for the top view of the four regions during the initial and final phase of the spreading process respectively):

a) Central region (Region A) which is ≥ 3Dp (PMMA particle diameter)

b) Uniform grey region (Region B) at low magnification which is 2-3 Dp thick

c) Region where the nucleation/ inception of holes takes place (region C) where thickness

~ Dp d) Region where the network formation takes place (the length scale of the lacey structure in this region increases with radial position and with time) or during the initial phase of the spreading process where a bright ring of particles is formed (region D)

Figure 12: Image sequence for 40% suspension, 20 μm particles on glycerol (later part of the spreading process once the instability sets in)

Figure 13: Schematic of the cross-sectional shape of the spreading drop showing the confinement of the particles and the pure fluid region. The inset shows the shape of interface around PMMA particles sitting at the silicone oil – glycerol interface

As spreading continues, an annulus of reduced intensity appears near the suspension front (Figure 8), and its thickness increases with time. The bright interior core changes negligibly in

20 radius with time. Figure 14 compares the change in radius of the outer front and the radius of the inner core. The inner core remains nearly invariant during the initial phase of spreading while the front is circular. Hence, the increase in the annular thickness predominantly due to the advancement of the suspension front. Examination of the annular region in a separate experiment using a microscope fitted with a 10X objective (The four insets on the right in Figure 15 and Figure 16 are the microscopic images of the suspension during the spreading process) shows that the reduced intensity in this region is a consequence of film thinning and particle confinement (see Video 2).

Figure 14: Comparison of the change in outer radius of the suspension front to inner core radius

As the oil suspension spreads, the film thickness and hence the number of particles contained within the film near the spreading edge decreases (Figure 15 and Figure 16, regions A and B). Eventually, the film thickness becomes so small that it can hold only a monolayer of particles adsorbed at the silicone oil – glycerol interface, as confirmed by microscopy images (Figure 15 and Figure 16, region C).

100 μm

Figure 15: Regions in the spreading drop phase before the instability

100 μm

Figure 16: Regions in the spreading drop phase after the instability

3 It becomes essential to understand the wetting behavior of PMMA ( P =1.18 g/cm ) with

3 respect to silicone oil ( o = 0.97 g/cm , o = 0.5 Pa-s  o = 20 mN/m) and glycerol (

3 aq = 1.26 g/cm , aq = 0.906 Pa-s  aq = 64 mN/m) in order to determine the position and placement of the particles at the silicone oil – interface. The contact angle of glycerol over PMMA with silicone oil as the surrounding medium is 115⁰. The PMMA particles tend to wet the silicone oil phase more than glycerol. The bond numbers (Bo) for all the systems are of the order 10 –3 – 10 – 5. Most of the particle volume is in the oil phase.

Confocal experiments done separately indicate that the particles are covered with silicone oil. In these set of experiments, a layer of glycerol was spin coated on a coverslip. Methylene blue dye was dissolved in the silicone oil phase before the preparation of the suspension. The resulting suspension was then introduced over the thin glycerol layer. The region around the PMMA particle and far from the introduced were scanned under cy5 fluorescence. Both areas showed the presence of dye indicating the presence of silicone oil around the PMMA particles. Figure 17a shows the montage of the z-scan as we image from the glycerol layer (completely dark due to the absence of dye) to the particle (black outline surrounded by silicone oil containing the dye) all the way up to the silicone oil layer at the top. The volume view for the same system shown in Figure 17b indicates that the particles sit between the glycerol and silicone oil layer. Perturbation analysis was done to predict the direction in which the interface is deformed. The final shape of the interface and degree of deformation is calculated in Appendix B.

The manifestation of the annular region coincides with the appearance of a bright ring of particles at the suspension front (Figure 15, region D). This is reminiscent of the coffee-ring effect [18], where both particles and fluid arrive at the pinned contact line of a sessile drop, but only fluid leaves the contact line region due to evaporation, ultimately leading to the deposition of particles in the form of a ring. In this case, the relative velocity between the particles and the silicone oil is a result of capillary attraction [41] [2] [42] [38] between the particles adsorbed at the oil-glycerol interface, which slows down the particles. We conducted a separate set of experiments to confirm the presence of pure silicone oil front which is free of particles. The glycerol interface for these experiment is seeded with PMMA particles prior to introducing the suspension drop. A silicone

23 oil front precedes the particle front during the spreading which is responsible for an evaporation like effect leading to particle deposition (see Video 3 and Appendix C). The capillary attraction

Figure 17 a: Montage of z-scan from the glycerol layer to the particle all the way up to the silicone oil layer between the particles creates a particle ring/dam that is manifested as a bright ring (Figure 15, region D). The particle ring is absent when a layer of pure silicone oil of thickness smaller than

24 the particle diameter is created on the surface of glycerol prior to the introduction of the suspension drop.

Figure 17 b: Confocal experiments indicates the presence of silicone oil around the PMMA particles

The area coverage ()a of the monolayer of particles in region C was determined using image analysis of three pictures of the same region during the spreading process. For 20, 10 and 6

μm PMMA particles a = 36.9, 39.3, 42.1 % and the fractal dimensions ()D f = 1.71, 1.73 and 1.80 respectively. The coverage is a weak function of the particle size for the same volume fraction. The fractal dimension can be understood by thinking in terms of a fractal line. Fractal line is too detailed to be an object of one-dimension, but at the same time it is not complicated enough to be two-dimensional. Hence its dimension can be best described using its fractal dimension, which is a number between one and two indicating a more open or closed structure. In our experiments the structure of the network is more open for the larger PMMA particles. The theoretical prediction (see Appendix D) for based on particle and suspending medium volume balances is expected

3 to be  (1 − cos  ) = 42.7%, which agrees reasonably well with the experimental a 4 measurements. As spreading proceeds, the monolayer region C grows in size, and simultaneously it also demixes to form a network, as evident from the increase in the spacing between the particle clusters between region B and region C (see Figure 15). The particle dam also increases in radius as it is pulled out by the radial flow, and maintains its integrity due to a supply of particles from region C. But beyond a critical radius Rc , this supply can no longer be sustained at a rate commensurate with the rate of increase of the perimeter of the dam due to spreading; at this point, the dam fragments (Figure 8). is observed to increase with the suspension volume and volume fraction of the suspension (Refer Appendix F).

Figure 18: The size of the polygons increases radially outward from the center After the dam breaks, the particles continue to spread over in the surface, and the monolayer region C grows in area at the expense of particles drawn from regions A and B. Regions A and B have a higher mobility than region C, and hence the interface between them is susceptible to a miscible viscous fingering instability [43] [44] [45]. As a result, A and B are consumed in an asymmetric, non-circular fashion. All this while, the particles in region C continue to organize themselves in the form of an extended network comprised of polygons. Eventually, regions A and B are consumed completely, and nearly all the particles straddle the oil-glycerol interface in the form of a space spanning network. Figure 18 shows two frames during the spreading process (the

26 size of the polygons increases as we go radially outwards). The central core has completely spread out and the particles are rearranging in the form of a network.

The central core is drawn out over time in a non-axisymmetric fashion (Figure 15 and Figure 16, region A). Particle aggregation after the initial fragmentation/ bursting results in the reorganization of the particle in the form of polygons. The mesh size (See Figure 19) increases radially outwards and the polygons coarsen over time.

Figure 19: 1/W varies linearly with radial position (R)

At any instant of time, the sizes of the polygons in region C increase as we go from the interior to the outer edge of the network (see Figure 18). If the inverse of the average polygon size (W ) is plotted as a function of the radial position R , a linear relationship emerges (Figure 19).

We came up with a mathematical model to explain the linear dependence of inverse of the characteristics length (W) of the polygon on the radial position. This model is based on the conservation of area and perimeter of an annular band of polygons at a given radial position. Consider an annular band of polygons of thickness R at a radial location of R (see Figure 20a and 20b).

Figure 20 a: Band of polygons at a radial position R growing radially outwards The area, ,of this band:

= ()RRR + 22 −  (3)  =(2RRR +  ) 

For RR  ,

Area of the patch ( ) = =2 RR (4)

If W is the azimuthal width of each polygon in the band, then the number of polygons in the band is NRW= 2/ . The total perimeter, P , formed by the particles within the patch is given by:

Perimeter of the polygons ( P ) in the radial band =2 RNR +  (5)

Rearranging the above equations gives:

 =R 2 R (6)

PR− 2 N =  R

Figure 20 b: Band of polygons at a radial position R growing radially outwards

Number of polygons in the patch:

2 R N = (7) W

Eliminating N and R gives the final equation:

1 P    =−    R (8) W     which is the linear relationship observed in Figure 19. As the band moves further away from the center while spreading, it shrinks in the radial direction (R decreases) , and stretches in the azimuthal direction (W increases) , preserving constant values of P and  . A careful examination of the evolution of the polygons reveals that this process is a little more complicated. As the network spreads out radially (Video 1), the sides of smaller polygons at lower radii are broken and assimilated into the larger polygons at higher radii, essentially conserving the perimeter of the polygons. But in spite of the breakup of the polygons, 1/W maintains a linear relationship with R with a fairly constant value of the slope, and therefore the area,  .

Table 2 summarizes the effect of individual factors on the network characteristics (the slopes and the intercepts) of the systems that were investigated. At present we hypothesize that ratio of the area to perimeter is the seeding size of the polygons as the particles start to demix and aggregate. Further investigations are required to confirm this hypothesis and to understand the physical significance of the slopes and the intercepts.

Table 2: Network characteristics (slopes and intercepts) of the systems investigated (values in the bracket for the slope and the intercept indicate the 95 % confidence interval)

We try to understand the particle network formation because of capillary attraction (favors network formation) and spreading forces (opposes network formation). The capillary attraction forces and drag exerted by the velocity field due to spreading determine whether or not a drop of the suspension organizes itself into a network. Depending on the initial rate of spreading, two outcomes are possible: small aggregates formed due to the capillary forces [33] are swept away by the spreading suspension, or the small aggregates organize into a two-dimensional network.

We now attempt to understand the conditions that favor the formation of the particle network. The effect of individual factors can be understood with the help of intensity plots which gives insights into the distribution of the PMMA particles on the surface of the liquid substrate. The intensity as a function of radial distance is plotted from the center in the radially outward direction. The time/frame at which the suspension drops touched the liquid substrate is taken as tref / reference frame. The intensity profile of the reference frame starts with a value very close to 255 indicate a white central region with high particle concentration. The particle concentration over the surface of the underlying liquid decreases with the radial position for a frame and ends up attaining an average value of ~50 corresponding to the average greyness of the background. For the viscous substrates the network evolution process is slow. In these cases, the frame intervals were increased to allow better capture of the dynamics of the process. For this analysis to be consistent it was essential to be able to track the center of the suspension based on which the radial distance was evaluated. The fast spreading dynamics in some of the cases (small particle size, low viscosity of the liquid substrate) led to rapid disintegration of the drop into a network like structure, thus rendering it difficult to ascertain the center of the drop. Such cases did not permit the analysis of all the frames.

4.1 Effect of Volume Fraction of the PMMA Suspension The formation of network is favorable for higher volume fractions of PMMA suspensions. The highest volume fractions that we had in our experiments was 40% and the lowest was 10%. A network was formed for the 40% suspension for all the PMMA particle sizes (20μm, 10μm and 6μm) on pure glycerol as the base liquid. While the tendency to aggregate depends on the

Figure 21: For the same particle size and viscosity of the base liquid lower volume fraction suspensions spread faster (indicated by the lower intensity value at the starting point for subsequent frames)

32 combination of suspension volume, fraction, particle size and the viscosity of the base liquid, it is generally observed that the tendency towards forming an extended network decreases with the decrease in volume fraction of the suspensions. Network formation was not observed for suspension with volume fraction less than 10% even for the largest particle size (20μm) on glycerol.

Intensity values on the Y axis and the rate at which it decays over subsequent time steps indicates the rate of spreading. A wider spread in the intensity values implies a stronger spreading force, whereas a more compact spread of intensity values implies dominant capillary forces. The systems with stronger capillary forces have greater tendency towards network formation. Higher volume fraction results in higher capillary interactions and higher attractive forces. This leads to a greater tendency for a system with high particle volume fraction to form a network.

4.2 Effect of Particle Size Particle size had a direct bearing on the tendency of a suspension forming an organized network at the interface. Capillary forces are more pronounced for the larger particle sizes, for example a 20μm 30% PMMA suspension forms a network on 2:1 glycerol: water solutions but that is not the case for the 10μm and 6μm PMMA beads. Contact of a particle with a fluid layer deforms the shape of the interface resulting in the increase in surface area [41] [2]. A system must expend energy to increase the area of the surface. A larger particle deforms the liquid surface more, and results in stronger capillary attraction [33].

Larger particles have lesser spread in the intensity value indicating dominating capillary forces as compared to smaller particles. This lead to a greater likelihood of a particle network.

Figure 22: For the same volume fraction of the suspension and base liquid PMMA particles with smaller radius spread out faster (indicated by the lower intensity value at the starting point for subsequent frames) 4.3 Effect of Viscosity of Substrate The spreading parameter ( S ) drives the spreading of the suspension on the surface of the liquid. The viscosity of the base liquid affects the rate at which spreading of the suspension takes place. The surface tension of glycerol is 64 mN/m and that of water is 72.8 mN/m at 25 ⁰C. Addition of water to glycerol decreases the viscosity and increases the surface tension of the

Figure 23: For the same volume fraction of the suspension and particle size systems with lower viscosity of the suspending medium spread out faster (indicated by the lower intensity value at the starting point for subsequent frames) glycerol water mixture, thus increasing the value of S . Increasing the spreading force may not allow enough time for the particles to come together to form an extended network. In fact, network formation is not observed even for the case of 40% PMMA suspension of 20μm beads on DI water.

Increasing the proportion of glycerol in the glycerol-water solution, increases the viscosity and decreases the spreading parameter thus slowing down the spreading rate and allowing enough time for the PMMA aggregates to interact and form the network.

Less viscous substrates show a larger spread in the intensity values implying a dominant spreading force. The likelihood of a particle network is more over a more viscous substrate on account of greater interaction time due to slower rate of spreading.

4.4 Effect of Addition of Salt in the Substrate Addition of salt to the liquid substrate plays an important role in the way the PMMA particles within the suspension drop interact among themselves at the silicone oil – glycerol interface. The salt concentration modifies the stability and charge-charge interaction among the PMMA particles in the suspension [46] [47] [21]. The charge-charge interactions are screened out even for a moderate salt concentration. This allows the aggregates to come closer to one another thus resulting in the formation of network even for the cases where it was not possible initially (See Table 5 and Table 6 for the effect on the addition of salt in the suspension).

The addition of NaCl reduces the radial span of the network, the PMMA aggregates extend to a smaller radial extent (for the same time interval) if the sub-phase contains dissolved NaCl with other conditions remaining constant. The range of the repulsive force can be reduced by increasing the electrolyte concentration in the liquid substrate. The Debye screening length decreases and van der Waal attraction would then dominate. Aggregation can be tuned from slow to fast by varying the concentration of salt in the sub-phase. In reaction limited cluster-cluster aggregation the reaction probability is less than 1 (numerous attempts will be necessary to overcome the interaction potential barrier) allowing more contacts to be explored before irreversible aggregation occurs [21] [48] [46].

Another effect of the dissolving salt in the glycerol solution is that it leads to an increase in its viscosity. The viscosity rises with concentration. Salt dissociates to form positive and negative ions, which become surrounded by polar molecules, effectively increasing their size. This produces high inter-particle and electrostatic forces which decreases the mobility of the ions. Bartoli et al. showed that at 20⁰C the viscosity of 0 mole % glycerol is 11.7 P and increases to 17.5 and 21.6 P for 5 and 10 mole % of NaCl respectively [49]. In our experiments we the measured value of the 0.5 M NaCl solution of glycerol to be 10.42±0.13 P as compared to 9.28±0.15 P for

36 pure glycerol at 25⁰C. The increase in viscosity leads to the slowing down of the spreading process, resulting in greater interaction of the aggregates, leading to greater chances of formation of extended network.

4.5 Effect of Dish Area The precursor film experiments showed (Appendix C) that the pure silicone oil front spans up to a finite distance from the center. The limited spread can be observed visually if the surface of the glycerol solution is seeded with PMMA particles prior to the introduction of the suspension drop over it. Constraining the area available for spreading lowers the effect of spreading once the leading edge of the pure fluid front hits the wall. The following table shows the time required for the silicone oil front to reach the wall for small (60 mm), medium (100 mm) and large (150 mm) dish for suspension drops of varying volumes (10, 20 and 30 μl).

Dish diameter (mm) Drop volume (μl) 10 20 30 150 208.73±14.39 s 156.64±3.38 s 130.89±6.94 s 100 97.65±4.32 s 88.10±5.96 s 53.4±4.59 s 60 29.01±3.46 s 25.88±4.73 s 19.88±4.8 s

Table 3: Time required for silicone oil front to reach the wall of the dish for varying volumes of suspension (40 % 20 μm on glycerol) The height of the glycerol layer was the same (~ 3 mm) for all the three set of experiments. This corresponds to 55, 24 and 9 ml for 150, 100 and 60 mm dish respectively. This is to keep the spreading process in the same regime (in this case spreading over thin liquid substrate). As long as the final span/ extent of the silicone oil front is less than the area of the dish, the spreading rate is independent of the dish area. For smaller dish area the silicone oil front reaches the wall of the dish which suppresses the surface tension gradient thus slowing down the spreading process significantly. In the absence of surface-tension-driven flow the reconfiguration of the particle network is only on account of capillary attraction driven by gravity.

The larger drops will spread over a larger area than the small drops. If the area exceeds the dish area the spreading rate decreases dramatically. Once the edge of the pure silicone oil front hits the wall and the subsequent spreading process is gravity driven. When a smaller volume of

37 the suspension is added, or a larger dish is used the rate of spreading is much more faster and the resulting particle network is not affected by the presence of wall.

Figure 24: Time required for silicone oil front to reach the wall for varying volume of suspension drop (40 % 20 μm on glycerol)

We tracked the radius of the leading edge of the silicone oil front for 10 μl of the suspension (40 % 20 μm on glycerol). The area vs time is a linear relation, the slope and the intercept can be used to predict the time at which the area of the spreading silicone oil front covers the entire area of the dish. Table 4 shows the error in the predicted and the experimental values for the system.

Dish Diameter (cm) Area (cm2) Predicted Experimental % Error Time (s) Time (s) 6 28.27 32.97 29.01 13.63 10 78.54 93.39 97.65 4.37 15 176.71 211.40 211.40 1.28 Table 4: Predicted and experimental time for 10 μl of suspension drop (40 % 20 μm on glycerol) to cover the dish area completely 4.6 Summary of the Behavior of Systems A scaling analysis can be used to understand the variables that govern this force balance. As shown in Appendix E, the balance between the azimuthal forces acting on particles due to spreading and the capillary attraction forces is characterized by the dimensionless parameter

oSH  aq/ o   = 2 . Here,  is the separation between the sphere surfaces, aq is R2 −  g a 4 f 2 aq( aq o ) the density of the aqueous phase, and f is a dimensionless factor that depends on the contact angle between the particle, silicone oil and glycerol, and the ratio of the densities of the two liquids. Network formation should be expected for ~ 1 or <<1, i.e. for large particle sizes, large density differences, weak spreading coefficients, large volume fractions (which would lead to small values of ), and thin substrate fluid layers. To verify these predictions, supplementary experiments that varied the particle size, substrate viscosity (by mixing water with glycerol) and particle volume fraction were performed. The results confirm our predictions: large particles, high viscosities and high particle fractions favor the formation of the self-assembled particle network.

A summary of which systems formed networks and the conditions under which they formed them is shown in Table 5 (green indicates the experimental conditions where network is formed and red represents the conditions where network is not formed).

The addition of NaCl facilitates the formation of network by screening the charge effects and increasing the viscosity of the sub-phase which in turn facilitates aggregation. For the same systems mentioned above adding 0.5 M NaCl to the base solution of glycerol and water, the tendency towards network formation is modified as summarized in Table 6 (The blocks shaded in green were cases where network formation was observed; a network was not formed for the cases of the red blocks).

Larger particle size results in greater create greater surface distortion thus leading to higher capillary forces. Large volume fractions lead to a smaller separation between particles leading to greater particle interactions. Increasing the viscosity of the supporting liquid slows down the rate of spreading. Higher surface tension of the surrounding liquid layer leads to faster spreading of the drop phase make it less favorable for network formation. The spreading rate determines if there is sufficient interaction time to create an extended network of aggregates.

The addition of salt leads to screening of charge effects and increase in the viscosity of the sub-phase. In conclusion, large particles, higher particle volume fraction, high substrate viscosity and high salt concentration in the sub-phase favors the formation of particle network.

Table 5: Summary table of the effects of particle size, volume fraction and viscosity of the substrate liquid without NaCl that does/ does not lead to the formation of a network. (Green: indicates network formation Red: network not formed)

Table 6: Summary table of the effects of particle size, volume fraction and viscosity of the substrate liquid (after the addition of NaCl) that does/ does not lead to the formation of a network. (Green: indicates network formation Red: network not formed)

5. Conclusions and Future Work The capillary forces and drag due to the velocity field determines whether a drop of PMMA suspension organizes itself into a network or not. Depending on the initial rate of spreading, two outcomes are possible: the particles form smaller aggregates and are swept away by the spreading suspension, or the smaller aggregates organize into an array of two-dimensional networks. The initial particle aggregates are formed due to the capillary forces [33] pulling the particles together, when the suspension drop is released at the air – glycerol interface. Whether a system does form the extended network depends on the time these aggregates spend near other aggregates. This time is governed by the spreading rate, the viscosity of the liquid substrate over which the suspension is introduced and the size of the particles in the suspension. The particle size determines the degree of deformation of the interface thus enabling the particles to detect other particles in the vicinity. Increasing the particle size enhances the capillary interactions. Volume fraction of the original suspension is found to have a direct effect on the likelihood of an extended network. Lower volume fractions do not form a network even on high viscosity sub-phase whereas a higher volume fraction forms network on a sub-phase with lower viscosity. It is generally found that for the cases of suspensions with higher PMMA volume fraction have larger number of aggregates which can cluster together to form an extended PMMA network. For the cases of low viscosity medium (water, glycerol: water 1:1) the spreading rate is high, and the drag experienced by the aggregates is low which does not allow enough time for the aggregates to form a network like structure. For the higher viscosity mediums (pure glycerol, glycerol: water 3:1), the aggregates encounter slower rates of spreading and higher drag force which allows more time for the interaction of the aggregates [31] [32] [33] resulting in higher chances of an extended network being formed. Addition of salt helps screen charge effects and results in the increase of sub-phase viscosity leading to enhanced chances of a particle network being formed.

In this work, we have demonstrated the self-assembly of particles into an organized network at an interface driven by a combination of flow and capillary attraction. The mesh size of the network can be tuned by adjusting the particle size, the substrate viscosity and the particle volume fraction. At this point we do not understand the complete scaling relation that governs the critical radius which dam of particles accumulated at the advancing suspension front breaks. The physical significance of the slope and intercept for the plot of the inverse of characteristics length

42 scale of the polygon vs the radial distance demands further comprehension. The dimensionless parameter  encapsulates the dependence of the physical parameters as to when the suspension should spread out in the form of a network. Based on  , the radius at which the suspension spreading process slows down to the extent that allows network formation can be calculated. The values for the radii we currently obtain exceeds the dimensions of the petri dish used in the experiments. This could be because of the basis for the calculation of  assumes a system of two particles being attracted due to capillary attraction as opposed to the spreading due to radially outward flow. This demands more accurate multi-particle system analysis. This combined with the scaling criteria for critical radius, network formation would help comprehend the entire physics of the process of self-assembly of the particles in the form of two-dimensional network.

Our findings reveal a better understanding of the spreading behavior of multi-phase fluid/ suspensions on the surface of another fluid layer. The effects of the particle size, wetting behavior of the particles, the physical properties of the suspending and the base liquid on the spreading process was investigated. The comprehension of such systems is highly relevant to a variety of technological applications, including emulsion formulations [50] and floatation technology. The results can be used for improved modelling of the rate of spreading of spilled oil [14] containing suspended solids, since suspensions can be used to mimic oil with suspended sand particles. Further work will improve the fundamental physical understanding of complex fluid spreading on liquid surfaces. We expect this discovery to find applications in the creation of new coating technologies and films embedded with particles.

Appendix

Appendix A: Determination of dependence of radius as a function of time The spreading coefficient for the case of oil on aqueous medium can be defined as:

S =WAOAOW/// −  −  (S1)

The general condition for any liquid to spread on the surface of another liquid is that the spreading coefficient S  0 . Spreading of liquid on the surface of another liquid follows the power law. For the case of partially miscible liquids this relation is given by:

R kt n (S2)

The above equation can be derived through a balance between the spreading force and the viscous drag [28] [14]. If the dissipation occurs in the viscous boundary layer of the underlying liquid (thick layer of base liquid). The thickness of the boundary layer is given by= t . Force balance yields,

V SR R2  (S3) R V t

S 0.5 Rearranging gives, R kt n . n = ¾ and k = . () 0.25

In the case of thin films where the dissipation takes place over the entire bulk, the boundary layer thickness is replaced by H . A similar force balance yields,

V SR R2 H (S4) R V t

0.5 n HS which, leads to the relation R~ kt . Here, n = ½ and k = . In all our experiments, we are  under the thin film conditions. The thickness of the substrate was maintained at 3 mm. For H ~3

44 mm,  reaches H in less than a second, so that the flow is developed across the entire sub-phase in all the experiments.

Appendix B: Shape of the interface around individual PMMA particle

Supplementary Figure S1: Shape of interface around individual PMMA particles sitting at the silicone oil – glycerol interface

Let a PMMA particle of radius be sitting at an interface between silicone oil (density =

o ) and glycerol (density = aq ). The contact angle of the PMMA at the oil-glycerol interface is  and the  is the angle made by the center at the point of contact the interface with respect to the vertical axis of the particle. We performed a regular perturbation to determine the shape of the interface around the PMMA particle to show the wetting behavior of PMMA in a silicone oil- glycerol system.

A force balance on the particle gives:

232 a( paq− p o)(1 − cos 2) + a g (  aq +  o − 2  p ) 2300 (S5)

2 33 +a g(aq −  o)cos  + 2  aq/ o a sin  sin(  −  ) = 0 3

The liquid-liquid interface height is described by z= h( r) , where z is measured from the center of the sphere, and against gravity as shown in the Figure S1. An interfacial normal stress balance [Leal (2007)] gives the governing equation for the function h:

 h  aq/ o  r (S6) paq− p o =( aq − o ) gh − r . 00 rr 2 h 1+  r

The boundary conditions for the function h are dh h= acos , = tan  −  , and h = h . r= asin ( ) r →  dr ra= sin

The last boundary condition at asymptotically large distances from the sphere allows us to write

p− p = − gh . aq00 o( aq o )  (S7)

ˆ ˆ Defining h=− h h , the governing equation for h becomes

ˆ − g 1 h ( aq o ) ˆ (S8) rh= . r r r  aq/ o

h Here we have assumed that 1, i.e., that the height variations are weak, which is valid r

2 (aq− o ) ga in the limit of small Bond numbers. The Bond number Bo is defined as Bo= . In  aq/ o our experiments, the bond numbers are small, on the order of 10-5.

Integrating the above equation and applying the boundary conditions dhˆ =tan( −) and hˆ = 0, we get: dr r→ ra= sin

r K0  l hlˆ = −tan( − ) c , (S9) c asin K1  lc where lc is the capillary length and is given by,

 aq/ o lc = . (S10) (AB− ) g ˆ Applying the boundary condition h=− acos h to the solution above gives an ra= sin expression for the asymptotic height h .

a K0 ( Bo sin ) ha=cos + tan(  −  ) . (S11)  Bo K1 ( Bo sin )

Here, K0 and K1 are modified Bessel’s functions of the second kind, and

 Bo 1 K0 ( Bo sin ) cos+ tan(  − ) ( 1 − cos 2  ) 4 Bo K1 ( Bo sin ) (S12) 

Bo(aq+−  o2  p ) 1 + +Bocos3  + sin  sin(  −  ) = 0. 33(aq− o )

When Bo is zero, i.e. in the absence of gravitational effects,  is equal to . Therefore, can be written as the following regular perturbation expansion, =  +Bo  (1) + ... where Bo (1) is the first effect of gravity.

The order Bo terms yield

+− 2  (1) 1 1 1( aq o p ) 1 3 (S13) =cos ( 1 − cos 2 ) + + cos  sin 4 3− 3 ( aq o )

For a contact angle of  =115 , aq=1.25 g/cc,  o = 0.99 g/cc, and  p = 1.18 g/cc,

 (1) =−0.353 .

K Bo sin (1) 0 ( ) h − acos  lc tan( − Bo ) K1 ( Bo sin )

1 −−ln( Bo sin) (S14) =−a ( Bo (1) ) Bo 12 2 ( Bo sin ) =a − Bo sin  − ln Bo sin  −   0. ( ) ( )

Since ha  cos , the shape of the meniscus is as shown in Figure S1.

Appendix C: Precursor film experiments There is a precursor film of silicone oil which spreads out before the bulk of the suspension drop. Capillary attraction slows down the particles, and the pure fluid moves ahead of the particles. A particle ring is created which is characterized by a bright ring. To confirm this, in a separate set of experiments, the surface of the glycerol solution is laden with PMMA particles. The suspension drop is then introduced over the surface. The experiments confirm the presence of a particle-free, pure oil front that leads the particles (see Video 3).

Supplementary Figure S2. Precursor film preceding the spreading suspension drop (PMMA laden surface)

Appendix D: Theoretical prediction of the area coverage

Area coverage (a ) can be theoretically predicted based on particle and suspending medium volume balances.

Supplementary Figure S3. Schematic of the cross-sectional shape of the spreading drop showing the confinement of the particles and the pure fluid region.

Volume balance of the suspending fluid (Vs and Vpf ) gives:

(S15) dV dV Q =+s pf dt dt (S16) 22 Vs=− () R s R i h s

22 (S17) Vpf=− () R pf R s h pf

Since Ri does not change appreciably with time initially its derivative can be set to zero.

hpf can be taken as a constant. Therefore,

dR 2 d Q= hs + h() R22 − R (S18) sdt pf dt pf s

d Q= h  +  h ()  −  (S19) s pf dt

dR 2 dR2 Where,  s and  = pf dt dt

If the volume fraction of the particles in the suspension is o , particle balance yields:

Q d o =−n () R22 R 4 p s i (S20)  a3 dt 3

Where, np is the number of particles per unit area.

Q d o = n  (S21) 4 p  a3 dt 3

22 a()RR s− i np = 2 2 2 a() Rsi− R

 n = a (S22) p  a2

Simplifying the above equation gives:

Q oa=  4 2  a3  a 3

3 Qa=   (S23) 4 oa

Using S19 we have:

3  h +  h()  −  =  a  4 o s pf a

3 h h  s p  ao= + −1 4 aa (S24)

Also,

has =−(1 cos )

h s =−1 cos (S25) a

h For  =115  ,s = 1.42 a

The precursor film thickness hpf is small in comparison to hs , hence the contribution from the first term dominates. For  = 40 % which is the base case for the experiments,

3  (1 − cos  ) = 42.7%, which agrees reasonably well with the experimental measurements. a 4

Appendix E: Scaling analysis to understand the force balance Since majority of the particle is in the oil phase, the azimuthal stress pulling the particles

v away from each other in the  direction scales as  r a2. o R

Where vr is the radial velocity.

The attractive force between the particles scales as:

 aq/ o 5/2 ga2 g (S26)  af2 aq/ o  aq/ o f is the dimensionless resultant weight of the particle after the subtraction of Archimedes force.

 1 1(aq+−  o2  p ) 1 f =cos( 1 − cos 2 ) + + cos3  and  is the separation between the 4 3− 3 ( aq o ) sphere surfaces.

The ratio of the two is:

vvrr2 oa  o  aq/ o   ==RR. 2 42 (S27)  aq/ o (g) a f 5/2 ga2 g  af2 aq/ o  aq/ o

The spreading force balance gives

v SR~. r R2 aq H (S28)

Using S28 and S29

SH    oR2 aq/ o  SH    ==aq o aq/ o . (S29) 224 2 2 4 2 (g) a f aq R(  g) a f

Network formation should be expected for ~ 1 or <<1. For large particle sizes, large density differences, weak spreading coefficients, large volume fractions (which would lead to small values of  ), and thin substrate fluid layers favor the formation of network.

Appendix F: Dependence of the critical radius (Rc) on particle size and volume of the suspension introduced during the spreading process The critical radius (Rc) is found to increase with the suspension volume. A plot of the Rc vs time is shown in Fig. 2b. The straight line fitted for the points where the deviation from circle can be used to calculate the critical radius at which the particle dam bursts. The following table shows the perimeter (cm) where the dam starts becoming unstable.

Intercept with Parity System Line (cm) 20um 40% Glycerol 30 ul 6.3 20um 40% Glycerol 10 ul 5.4 20um 30% Glycerol 10 ul 4.6

Supplementary Table T3: Radius at which the spreading drop deviates from a circle and instability sets in

The critical radius increases with the increase in volume. For the same volume and particle size the critical radius increases with the volume fraction. Larger volume or higher volume fraction ensures a longer supply of particles at a rate commensurate with the rate of increase of the perimeter of the dam due to spreading. This would sustain the dam for a greater critical radius.

References

[1] P.-G. de Gennes, F. Brochard-Wyart and D. Quere, Capillarity and wetting, vol. 1987, 1987, pp. 369-374.

[2] J. A. Pelesko, Self-Assembly. The Science of Things that put Themselves Together, vol. 1, 2015.

[3] J. Davies and E. Rideal, "Interfacial Phenomena," Interfacial Phenomena, no. Chapter 7, pp. 468-474, 1961.

[4] W. Barthlott and C. Neinhuis, "Purity of the sacred lotus, or escape from contamination in biological surfaces," Planta, vol. 202, no. 1, pp. 1-8, 30 4 1997.

[5] N. Costa and P. M. Maquis, "Biomimetic processing of calcium phosphate coating.," Medical engineering & physics, vol. 20, no. 8, pp. 602-6, 1 11 1998.

[6] A. Fisher, M. Kuemmel, M. Järn, M. Linden, C. Boissière, L. Nicole, C. Sanchez and D. Grosso, "Surface Nanopatterning by Organic/Inorganic Self-Assembly and Selective Local Functionalization," Small, vol. 2, no. 4, pp. 569-574, 1 4 2006.

[7] J. N. Israelachvili, Intermolecular and surface forces, Academic Press, 2011, p. 674.

[8] G. J. Hlrasakl, "Wettability: Fundamentals and Surface Forces".

[9] D. Bonn, J. Eggers, J. Indekeu, J. Meunier and E. Rolley, "Wetting and spreading".

[10] S. Rafai and D. Bonn, "Spreading of non-Newtonian fluids on surfactant solutions on solid surfaces," Physica A, vol. 358, pp. 58-67, 2005.

[11] C. Lam, R. Wu, D. Li, M. Hair and A. Neumann, "Study of the advancing and receding contact angles: liquid sorption as a cause of contact angle hysteresis," Advances in Colloid and Interface Science, vol. 96, no. 1-3, pp. 169-191, 25 2 2002.

[12] D. W. Camp and J. C. Berg, The spreading of oil on water in the surface tension regime, vol. 184, 1987, pp. 445-462.

[13] P. Joos and J. Pintens, "Spreading Kinetics of Liquids on Liquids," Journal of Colloid And Interface Science, vol. 60, no. 3, 1977.

[14] D. P. Hoult, "Oil Spreading on the Sea," 1972.

[15] C. M. Bates, F. Stevens, S. C. Langford and J. T. Dickinson, "Motion and dissolution of drops of sparingly soluble alcohols on water," Langmuir, vol. 24, no. 14, pp. 7193-7199, 2008.

[16] O. Carrier and D. Bonn, "Liquid Spreading," Droplet Wetting and Evaporation, pp. 3-13, 2015.

[17] P.-G. de Gennes, F. Brochard-Wyart and D. Quere, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves.

[18] R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel and T. A. Witten, "Capillary flow as the cause of ring stains from dried liquid drops," Nature, vol. 389, no. 6653, pp. 827- 829, 23 10 1997.

[19] L. Keiser, H. Bense, P. Colinet, J. Bico and E. Reyssat, "Marangoni Bursting: Evaporation- Induced Emulsification of Binary Mixtures on a Liquid Layer," Physical Review Letters, vol. 118, no. 7, pp. 1-5, 2017.

[20] J. Han and C. Kim, "Spreading of a suspension drop on a horizontal surface," Langmuir, vol. 28, no. 5, pp. 2680-2689, 2012.

[21] B. Madivala, J. Fransaer and J. Vermant, "Self-assembly and rheology of ellipsoidal particles at interfaces," Langmuir, vol. 25, no. 5, pp. 2718-2728, 2009.

[22] F. A. L. Dullien, Porous Media : Fluid Transport and Pore Structure., Elsevier Science, 1991, p. 598.

[23] H. A. Barnes, "The yield stress—a review or ‘παντα ρει’—everything flows?," Journal of Non-Newtonian Fluid Mechanics, vol. 81, no. 1-2, pp. 133-178, 1 2 1999.

[24] A. Carré, J.-C. Gastel and M. E. R. Shanahan, "Viscoelastic effects in the spreading of liquids," Nature, vol. 379, no. 6564, pp. 432-434, 1 2 1996.

[25] W. Gifford and L. Scriven, "On the attraction of floating particles," Chemical Engineering Science, vol. 26, no. 3, pp. 287-297, 1 3 1971.

[26] P. A. Kralchevsky and N. D. Denkov, "Capillary forces and structuring in layers of colloid particles," Current Opinion in Colloid and Interface Science, vol. 6, no. 4, pp. 383-401, 2001.

[27] P. A. Kralchevsky and N. D. Denkov, "Capillary forces abd structuring in layers of colloid particles," Current Opinion in Colloid and Interface Science, vol. 6, pp. 383-401, 2001.

[28] M. Santiago-Rosanne, M. Vignes-Adler and M. G. Velarde, "Dissolution of a Drop on a Liquid Surface Leading to Surface Waves and Interfacial Turbulence," Journal of Colloid And Interface Science, vol. 191, pp. 65-80, 1997.

[29] S. Dietrich, "Capillary forces on colloids at fluid interfaces".

[30] V. D. Fakult, "Capillary interactions between colloidal," 2010.

[31] D. J. Robinson and J. C. Earnshaw, "Experimental study of colloidal aggregation in two dimensions. I. Structural aspects," Physical Review A, vol. 46, no. 4, pp. 2045-2054, 1 8 1992.

[32] D. J. Robinson and J. C. Earnshaw, "Experimental study of colloidal aggregation in two dimensions. II. Kinetic aspects," Physical Review A, vol. 46, no. 4, pp. 2055-2064, 1 8 1992.

[33] D. J. Robinson and J. C. Earnshaw, "Experimental study of colloidal aggregation in two dimensions. III. Structural dynamics," Physical Review A, vol. 46, no. 4, pp. 2065-2071, 1 8 1992.

[34] B. Wang, M. Wang, H. Zhang, N. S. Sobal, W. Tong, C. Gao, Y. Wang, M. Giersig, D. Wang and H. Möhwald, "Stepwise interfacial self-assembly of nanoparticles via specific DNA pairing," Physical Chemistry Chemical Physics, vol. 9, no. 48, p. 6313, 2007.

[35] A. J. Hurd, "The electrostatic interaction between interfacial colloidal particles," J. Phys. A: Math. Gen, vol. 18, pp. 1055-1060, 1985.

[36] Y. Liang, N. Hilal, P. Langston and V. Starov, "Interaction forces between colloidal particles in liquid: Theory and experiment," Advances in Colloid and Interface Science, Vols. 134- 135, pp. 151-166, 31 10 2007.

[37] A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato and P. M. Chaikin, "Improving the density of jammed disordered packings using ellipsoids.," Science (New York, N.Y.), vol. 303, no. 5660, pp. 990-3, 13 2 2004.

[38] L. Botto, E. P. Lewandowski, M. Cavallaro and K. J. Stebe, "Capillary interactions between anisotropic particles," Soft Matter, vol. 8, no. 39, p. 9957, 19 9 2012.

[39] P. A. Kralchevsky and K. Nagayama, "Capillary interactions between particles bound to interfaces, liquid films and biomembranes," Advances in Colloid and Interface Science, vol. 85, no. 2, pp. 145-192, 2000.

[40] M. Santiago-Rosanne, M. Vignes-Adler and M. G. Velarde, "On the Spreading of Partially Miscible Liquids," Journal of Colloid And Interface Science, vol. 234, pp. 375-383, 2001.

[41] D. Vella and L. Mahadevan, "The 'Cheerios effect'," vol. 817, no. 2005, 2004.

[42] J. C. Loudet, A. M. Alsayed, J. Zhang and A. G. Yodh, "Capillary interactions between anisotropic colloidal particles," Physical Review Letters, vol. 94, no. 1, pp. 9957-9971, 2005.

[43] A. Ramachandran and D. T. Leighton, "Particle migration in concentrated suspensions undergoing squeeze flow," Journal of Rheology Journal of Rheology, vol. 541, no. 10, 2010.

[44] J. Kim, F. Xu and S. Lee, "Formation and Destabilization of the Particle Band on the Fluid- Fluid Interface," Physical Review Letters, vol. 118, no. 7, pp. 1-6, 2017.

[45] G. M. Homsy, "VISCOUS FINGERING IN POROUS MEDIA," Ann. Rev. Fluid Mech, vol. 19, pp. 271-31, 1987.

[46] S. Promkotra and T. Kangsadan, "Morphological arrangement of two-dimensional aggregated colloid," vol. 020003, p. 020003, 2017.

[47] L. Wang, B. Yuan, J. Lu, S. Tan, F. Liu, L. Yu, Z. He and J. Liu, "Self-Propelled and Long- Time Transport Motion of PVC Particles on a Water Surface," Advanced Materials, vol. 28, no. 21, pp. 4065-4070, 2016.

[48] K. Huang, M. Brinkmann and S. Herminghaus, "Wet granular rafts: aggregation in two dimensions under shear flow," Soft Matter, vol. 8, no. 47, p. 11939, 2012.

[49] F. J. Bartoli, J. N. Birch and G. E. Mcduffie, "Conductivity , Dielectric Relaxation , and Viscosity of NaCl – Glycerol Solutions," vol. 49, no. 4, 1968.

[50] S. U. Pickering, "CXCVI.—Emulsions," J. Chem. Soc., Trans., vol. 91, no. 0, pp. 2001-2021, 1 1 1907.