Dynamics of capillary rise: Effects of surfactants and capillary geometry

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Dynamics of capillary rise

Effects of capillary geometry and surfactants

Ward Schaefers

July 7, 2018

Report Bachelor Project Physics and Astronomy Studentnumber : 10723579

Size : 15 EC

Faculty and University : Science Park, UvA Institute : WZI

Daily supervisor : Thijs de Goede Supervisor : dr. Noushine Shahidzadeh Examiner : prof. dr. Daniel Bonn

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1 Summaries

1.1 Scientific summary (ENG)

Natural porous media, like bricks, contain an irregular complex network where biosurfactants are present. The imbibition of such a network by a liquid is called capillary rise. Classical laws on capillary rise are based on simple fluids and cylindrical capillaries. In this research we investigate the influence of capillary geometry and the presence of surfactants on the mechanical equilibrium height and dynamics of capillary rise. We compare our results compared to values predicted by theory. We show that capillary geometry has an influence on capillary rise in the form of corner flow depending on the wettability of the liquid. Corner flow can be the explanation of a positive influence on the equilibrium height in square capillaries. Although the presence of surfactants show a positive increase in equilibrium height for both cylindrical and square capillaries results show no significant difference for the dynamics of capillary rise except for one superspreading surfactant.

1.2 Popular scientific summary (NL)

Als je een heel dun buisje in contact brengt met een vloeistof is het mogelijk dat de vloeistof tegen de zwaartekracht in omhoog stijgt. De stijging van de vloeistof wordt ook wel capillar-iteit genoemd. Capillarcapillar-iteit ontstaat door adhesie- en cohesiekrachten. In dit geval is adhesie de aantrekkingskracht tussen de vloeistof en de wand van het buisje. Cohesiekrachten zijn de intermoleculaire krachten tussen de moleculen van het vloeistof. Een wisselwerking tussen deze krachten zorgt voor een stijging van de vloeistof door het buisje.

Op deze manier wordt bijvoorbeeld water getransporteerd door de grond. In de grond zitten namelijk kleine capillairen, kleine buisjes, waardoor het grondwater omhoog wordt getrokken. Ook zou dit kunnen worden waargenomen in oudere stenen die capillaire pori¨en bevatten die wa-ter opnemen, waardoor deze beschadigt wordt.

De meeste wetten over capillariteit worden beschreven met cilindrische capillairen en zijn be-wezen aan de hand van simpele vloeistoffen, zoals water. In het geval van een steen zijn de pori¨en echter bijna nooit cilindrisch en zijn er micro-organismen aanwezig. De organismen kunnen de samenstelling van de vloeistof veranderen waardoor de vloeistof complexere eigenschappen kan krijgen.

In dit onderzoek worden zulke complexe vloeistoffen en verschillende geometrie¨en van capillairen onderzocht. De dynamiek en de uiteindelijke hoogte van de stijging wordt door een high-speed camera vast gelegd. Uit deze data wordt de invloed van geometrie en complexe vloeistoffen op capillaire stijging vergeleken met de huidige theorie.

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List of Figures

1 Close up image of a brick containing different types of pores. Source: N. Shahidzadeh 2018 . . . 5 2 A diagram of forces acting on liquid molecules “Source: Wikipedia.org/surfacetension”

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3 Interfacial tensions working on a droplet on a solid. Where γSL represents the solid-liquid, γSG the solid-gas and γLGthe liquid-gas (in text: γ) surface tension. Source: “Wikipedia.org/wetting” . . . 7 9

5 Representation of surfactant behaviour in a liquid. Source: people.maths.ox.ac.uk 10 6 Dynamical surface tension of (a) CTAB and (b) Tween-80 both at Critical micelle

concentration kindly provided by M. Qazi (2018) . . . 11 7 A schematic of the Du No¨uy ring method. Source: www.kruss-scientific.com/ . . . 12 8 A schematic overview of the setup for the capillary rise experiment . . . 14 9 Schematic representation of CTAB solution on a borosilicate glass plate. The red

circles with a (+) in the middle represent the positively charged represent the hydrophilic part and the zigzag tail represent the hydrophobic part of the molecule. The dashed line on the glass plate represents its negative charge. . . 16 10 Uv-Vis specctrum of pure Methelyne Blue solution (8mg/l) and this solution mixed

with pre-wetted glass beads in water, Tween-80, CTAB and silwet . . . 17 11 (a)-(f) is the capillary rise of Silwet solution through a square capillary over time.

The red lines indicate the height of the fingers . . . 18 12 Liquid fingers in square capillaries: (a) Water , (b) Tween-80 , (c) CTAB, (d) Silwet 19 13 The averages of the finger length in square capillaries . . . 19 14 Average equilibrium height of capillary rise for water and surfactant solutions at t ≈

30 seconds. In orange the cylindrical capillaries, in light blue the square capillaries and the black dashed line represents the theoretical value of Jurin (Equation 5) . . 20 15 Measured position (blue circles) of meniscus for both water (a) and Silwet (b). The

data is fitted with Washburn’s equation (Red line). . . 22 16 Measured Washburn constant of cylindrical (orange) and square (light blue)

cap-illaries for water and surfactant solutions. The dashed black line represents the theoretical Washburn constant calculated with Equation 7 . . . 23

List of Tables

1 Concentration of liquids in molar . . . 12 2 Surface tension and contact angles on borosilicate glass of water, Tween-80 (0.015mM),

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2 Introduction

Capillary rise is the ability of fluids to penetrate porous media. A well-known example of this phenomenon is the penetration of certain fluids in narrow tubes which was already observed by Leonardo da Vinci (1452-1519)(de Gennes, Brochard-Wyart, & Quere, 2004). A few centuries later, James Jurin found that the height of the capillary rise of simple liquids was inversely pro-portional to the radius of the capillary (Jurin, 1719). Most theoretical understanding of capillarity for simple liquids is based on the work of Hagen (Hagen, 1839) , Poiseuille (Poiseuille, 1844), Lucas (Lucas, 1918) and Washburn (Washburn, 1921) which was published about a century ago (Zhmud, Tiberg, & Hallstensson, 2000). Understanding this ubiquitos phenomenon gave insight in different fields, e.g. geological science (Peruzzi, Poli, & Toniolo, 2003; G. Gray & A. Schrefler, 2007; D¨oll, Douville, G¨untner, Schmied, & Wada, 2016), fluid mechanics (Stone, Stroock, & Ajdari, 2004; An-dreotti et al., 2016) but also in the building environment (Flatt, 2002; Topcu, Bilir, & Uyguno˘glu, 2009; Vieira, Alves, De Brito, Correia, & Silva, 2016; Wu, Zhang, Nikolov, & Wasan, 2016).

Figure 1: Close up image of a brick containing different types of pores. Source: N. Shahidzadeh 2018

A simple brick can be described as a porous material containing a complex irregular lattice of interconnected pores (Figure 1) (Valvatne & Blunt, 2004). Wetting liquids can impregnate this network due to capillary rise and may deteriorate the material. For example when salt is present, weathering is caused by the pressure of salt crystallization (Desarnaud, Bonn, & Shahidzadeh, 2016). Most laws on capillary rise depend on the radius of the capillary. In figure 1 we see that real porous material contains pores that are far from cylindrical. Several methods have been cre-ated to estimate an equivalent capillary radius but this still remains a challenge (Masoodi & Pillai, 2012), and for pores containing edges corner flow has to be taken into account (Son, Chen, Kang, Derome, & Carmeliet, 2016; Ponomarenko, Quéré, & Clanet, 2011; Gurumurthy, Rettenmaier, Roisman, Tropea, & Garoff, 2018; Dong & Chatzis, 1995; Bico & Quéré, 2002; Wu et al., 2016). An addition to this problem are the natural liquids impregnating real porous media. These are likely to be more complex than simple liquids since biosurfactants could be present in the form of polysaccharides synthesized by micro-organisms (Desai & Banat, 1997). For this reason is the

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influence of both capillary geometry and the presence of surfactants of paramount importance to liquid transport through any natural complex porous medium.

In this research we will investigate the influence of capillary geometry and the presence of surfac-tants on the mechanical equilibrium height and the dynamics of capillary rise using high-speed imaging. This thesis is structured as followed: first the theoretical background will be reviewed. Secondly, we will describe our method before discussing our results. Finally, the conclusions and outlook will be given on our research.

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3 Theory

The difference between surface energies of wet and dry solid surfaces create a capillary force that is the motivator of capillary rise (Masoodi & Pillai, 2012). To understand the origin of this force, a small introduction of surface tension and wetting is required. In this section a brief introduction will be given on wetting and later on related to the statics of capillary rise and dynamics. Finally, a small theoretical background on surfactants will be given.

3.1 Surface Tension and Wetting

Figure 2: A diagram of forces acting on liquid molecules “Source: Wikipedia.org/surfacetension”

Surface tension is one of the most fundamental properties of liquids and solids. It originates from the molecules at the surface that miss half its interaction with other molecules, making them less energetically favorable and creating a force towards the bulk of the liquid (see Figure2). This intermolecular forces (cohesive forces between molecules) minimize the surface area of a liquid. Therefore, work is required to increase the surface area.The energy needed to increase the surface per unit area is defined as the surface tension (Bormashenko, 2017).

Figure 3: Interfacial tensions working on a droplet on a solid. Where γSL represents the solid-liquid, γSG the solid-gas and γLG the liquid-gas (in text: γ) surface tension. Source: “Wikipedia.org/wetting”

As the surface tension is determined by the intermolecular forces at the interface, the inter-actions between the liquid phase and other phases play an important role. For example, when a

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droplet is placed on a surface the interaction between the solid, liquid and gas phase determines how far this droplet spreads (see Figure 3). The mechanical force balance on the three phase con-tact line is given by Young’s equation (Bonn, Eggers, Indekeu, Meunier, & Rolley, 2009; Young, 1805):

γ · cos(θE) = γSG− γSL (1)

Where γSG, γSL and γ represent the interfacial tensions between solid-gas, solid-liquid and liquid-gas. Respectively, if the γSG> γSL+γ spreading occurs since the system wants to minimize its energy. This difference in energy is referred to as the spreading parameter(de Gennes et al., 2004):

S ≡ γSG− (γSL+ γ) = γ(cos(θE) − 1) (2)

This parameter describes the wettability of liquids on solids. When S = 0 the liquid spreads completely creating a thin film on the solid surface and for S < 0 only partially wetting occurs, which is divided in to mostly wetting (θE < π/2) and mostly non-wetting(θE > π/2). (Figure 3) (Bonn et al., 2009). The spreading of a liquid on a surface can be described by the surface tension of the liquid and the contact angle on the solid. These two parameters give insight on the strength of coherence between the molecules and adherence between the solid and the liquid interfaces.

3.2 Capillary rise

The previous discussed wetting is the origin of the capillary force that drives capillary rise. In this section theoretical background is given on the mechanical equilibrium height and the dynamics of this rise.

3.2.1 Jurin’s Law

If a dry solid has a greater surface energy than a wet solid, the solid would prefer to be wet to minimalize its surface energy. A dry capillary disrupts this equilibrium when in it comes in contact with a wetting liquid and will rise through the capillary. This phenomena can also be described with the imbibition parameter (de Gennes et al., 2004):

I ≡ γSG− γSL (3)

If I > 0 the system would benefit on a energetic level by minimizing its surface energy, so the liquid will adhere to the solid.

In the case of a vertically placed capillary the total energy can be written in terms of height of the liquid in the capillary (h) and the radius of the capillary (r) (de Gennes et al., 2004):

E = −2πrhI + 1 2πr

2_h2_ρg ₍₄₎

The first term in this equation represents the gain in surface energy and the second term represents the gravitational potential with ρ the density of the liquid. By taking the mechanical equilibrium of this equation (E=0) and substituting the imbibition parameter I with equation (1) the following equation for the height arises:

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H = 2 · γ · cos θE

ρ · g · r (5)

Where H is the meniscus height at mechanical equilibrium. This is known as Jurin’s law and describes the equilibrium height of capillary rise. (Jurin, 1719)

3.2.2 Washburn’s Equation

When a liquid rises through a capillary two forces oppose the movement of the liquid: the fluid’s inertia and the viscous friction (de Gennes et al., 2004). When capillaries are small enough the flow of the fluid will be slow enough to neglect its inertia. The equation of motion will be:

Fγ− W = Fη (6)

where Fγ represents the capillary force, W the gravitational potential energy and Fη the viscous force. During the early stages of capillary rise W can also be neglected since Fγ>> W in the short-time limit (Ponomarenko et al., 2011). Simplifying equation 6 leads to the Washburn Equation (Washburn, 1921):

h2(t) =γ · r · cos(θE)

2 · η · t (7)

This equation predicts that the capillary rise is proportional to the square root of time in the short-time limit (h<H) (Zhmud et al., 2000).

3.3 Capillary Geometry and Surfactants

In the previous section, two laws of capillary rise are given. We see that the height is proportional to surface tension and the cosine of the contact angle but inversely proportional to the radius of the capillary (Equation 5). The dynamics were described by Equation 7 showing a proportional dependence on all three parameters. These parameters can be influenced by a different geometry of capillaries or by the presence of surfactants.

3.3.1 Geometry

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Figure 4: (a) schematic of meniscus in cylindrical capillary (Source: Wikipedia.org/meniscus), (b) schematic of corner flow in square capillaries (Source: (Gurumurthy et al., 2018))

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The rise of the bulk in a square capillary is very similar to the rise in a cylindrical capillary since the driving and opposing forces are the same (Gurumurthy et al., 2018). If the capillary is wettable, a concave meniscus is formed by the equilibrium of capillary and gravity forces and can be observed within a cylindrical capillary as shown in figure 4(a) (de Gennes et al., 2004). In the corners of square capillaries the distance between the plates is significantly smaller than the radius of a cylindrical one, creating corner rivulet flow (Figure 4(b)) if the capillary walls are wettable enough (θE < π/2) (Gurumurthy et al., 2018; Bico & Qu´er´e, 2002). According to literature, minor modifications have to be made for the eventual mechanical equilibrium height but a universal modification still has to be found (Ichikawa, Hosokawa, & Maeda, 2004; Gurumurthy et al., 2018; Ouali et al., 2013). However, the dynamics of the bulk in more complex geometries (e.g. square capillaries but also porous media like rocks) can still be described with Washburn’s equation (Ponomarenko et al., 2011), and the dynamics of the corner flow appear to propagate similarly (Dong & Chatzis, 1995).

3.3.2 Surfactants

Figure 5: Representation of surfactant behaviour in a liquid. Source: people.maths.ox.ac.uk

Surfactants are able to change the surface tension and the contact angle when solved in water (Thiele, Snoeijer, Trinschek, & John, 2018). Surfactants are amphiphilic molecules containing hydrophobic tails and hydrophilic heads and adsorb to the liquid-gas interface (Figure 5). Most air molecules and hydrophobic tails are nonpolar leading to a decrease in dissimilarity of the liquid and gas phase therefore lowering the surface tension (Rosen, 1959). When concentration of sur-factants is high, the molecules that can not adsorb aggregate to colloids known as micelles. The concentration of surfactants to completely cover the interface with molecules but no micelles are formed is known as the Critical Micelle Concentration (CMC).

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There exist two main groups of surfactants: non-ionic and ionic, where the latter can either be positively charged (cationic) or negatively charged (anionic) (de Gennes et al., 2004). In this research we will investigate the cationic Cetyltrimethylammonium Bromide (CTAB), the non-ionic Polysorbaat 80 (Tween-80) and the surfactant Silwet L-77 (Silwet). Silwet is sold as a commer-cial product and is promoted as a ’superspreading’ surfactant (Silwet , 2018). Superspreading is still a relatively new phenomenon and is not yet understood to its full extent(Bormashenko, 2017).

Surfactant molecules are large and need time to diffuse to the surface (Figure 5) (Rosen, 1959). For this reason is the surface tension not immediately influenced by the presence of surfactants but decreases over time. In Figure 6 we see that the time-limit of this surface tension is not universal for every surfactants and can differ substantially in time-range. Unfortunately, no literature was found on the dynamic surface tension of Silwet.

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Figure 6: Dynamical surface tension of (a) CTAB and (b) Tween-80 both at Critical micelle concentration kindly provided by M. Qazi (2018)

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4 Experimental setup and method

This research is divided in to two sections. The first section we will determine the liquid char-acteristics and wetting properties of water, Tween-80, CTAB and Silwet. In the second section capillary rise is investigated. In this last section the influence of geometry and surfactants will be researched on the mechanical equilibrium height and the dynamics of capillary rise.

4.1 Liquid and surface characteristics

Table 1: Concentration of liquids in molar

Concentration [mM]

Water

-Tween-80 0.015

CTAB 1

Silwet 0.017

Throughout this research three different surfactant solutions are used and water as a reference. The concentrations of these solutions are given in Table 1. These concentrations were chosen since it is slightly above their critical micelle concentration to make sure that the entire liquid-gas interface is covered with surfactant molecules. Experiments were performed to determine the surface tension, contact angle and adsorption on borosilicate glass, since the capillaries are made of the same material.

4.1.1 Surface Tension

Figure 7: A schematic of the Du No¨uy ring method. Source: www.kruss-scientific.com/

The surface tension was measured with the Du No¨uy ring method (Figure 7). A ring was pulled up from a reservoir (about 20 ml) of the solutions creating a lamella exerting a force on the ring. The maximum force exerted on the ring can be related to the surface tension by the following relation (KRUSS , 2018):

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γ = FM AX L · cos θE

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With FM AX the maximum detected force, L the sum of the inner and outer circumference of the ring and θE the contact angle between the liquid and the ring. Since the ring is made of platinum irridium, a material with a high surface free energy, this angle was considered to be equal to 0. This allowed us to measure the surface tension by measuring FM AX. Using this method, the surface tension of each liquid was measured five times.

4.1.2 Wetting

To measure the contact angles the sessile drop method was used. A droplet was gently placed on a borosilicate substrate to create a situation similar to Figure 3. First, glass plates were thoroughly cleaned with water and ethanol and blown dry with compressed air. The droplets on these glass plates were studied with an optical camera. From these images the contact angle of the liquid on glass was extracted.

An average was taken over 3 to 5 measurements.

4.1.3 Adsorption

To investigate the adsorption of the surfactants on glass Ultraviolet-Visible (Uv-Vis) spectroscopy was used. Uv-Vis spectroscopy is an analytic technique that can determine the concentration of a certain molecule in a solution using the absorbance of (Uv-)light. Methylene blue in water (8mg/liter) is a deep blue positively charged solution with an absorbance peak at 670 nm. When mixed with negatively charged glass beads, the Methylene Blue molecules adsorb to the surface of the glass beads due to the charge difference. The adsorption of Methylene Blue on the glass beads leads to a lower absorbance peak. By using this principle the adsorption on glass of the three surfactant solutions and water can be investigated.

Surfactant adsorption was measured in the following way: For half a hour 250 milligrams of glass beads (diameter of 210 micrometer) were pre-wetted with the solutions in a phial of 3 milliliters. This phial was cautiously shaken to give the liquid the opportunity to reach the entire surface of all glass beads. The liquid and glass beads were seperated by careful use of a pipette after which three ml of Methylene Blue was added to the pre-wetted glass beads. The Methylene Blue and glass beads were left for another half hour. Finally, the liquid was investigated using Uv-Vis spectroscopy to measure the absorbance of Methylene Blue. The final absorbance measurements compared to the absorbance of the ’pure’ Methylene Blue solution gives insight of the adsorbtion of the three surfactant solutions and water on glass beads.

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Figure 8: A schematic overview of the setup for the capillary rise experiment

4.2 Capillary Rise

4.2.1 Setup

To capture the dynamics of capillary rise a high-speed camera setup was prepared (Figure 8). Two batches, consisting of five square (side = 0.5 mm) and five cylindrical capillaries (radius = 0.25 mm), were used to investigate the capillary rise. The cleaning routine of the capillaries was immersion into an alcohol bath, rinsing the capillaries with demineralised water, and emptying the capillary using a dust-free tissue thrice. Afterwards the batch of capillaries was dried for three hours at 80◦C. A droplet with a radius larger than the capillary length (Bonn et al., 2009) was deposited on Parafilm to create a liquid reservoir with a flat top.

For each measurement, a capillary was placed slightly above the reservoir. The reservoir was slowly raised until capillary rise was observed, which was recorderd with a high-speed camera at a frame rate of 1000 frames per second. To enhance the contrast of these images a combination of a LED light and a diffuser sheet was used. When the liquid was in mechanical equilibrium, the height of the meniscus was measured.

Almost the same setup was used to make pictures of the meniscus inside the square capillar-ies. The high-speed camera was replaced by a photo-camera with a microscopic objective (12x Navitar) to create high resolution pictures of the menisci. At least five meniscus images were taken of each liquid.

4.2.2 Dynamic analysis

The high-speed camera produced videos which tracked the dynamics of capillary rise. To deduce the correct height of the meniscus from these images a protocol of image editing was established. First of all, the start image was chosen and cropped to the point that only the capillary was visible. Since it was not always possible to put the capillary exactly perpendicular to the surface of the reservoir, rotational corrections on the video were made. The width of the capillary was

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used as a reference length. At last, all the images were binarised in a way that the liquid rising was separated from the rest of the video’s. For each image, the height of the meniscus was taken and divided by the reference length to give the actual height.

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5 Results and Discussions

As mentioned in the Introduction we investigated the effects of capillary geometry and surfactants on capillary rise. In this section we will discuss our results in the following order: First the surface tension and wetting properties of the Tween-80, CTAB, Silwet and water are discussed since these aspects are linked to capillary rise. Then we discuss our observation of capillary rise and divide this in to three sections: the corner flow in square capillaries, the equilibrium height and at last the dynamics.

5.1 Wetting properties

5.1.1 Surface Tension and Contact Angle

Table 2: Surface tension and contact angles on borosilicate glass of water, Tween-80 (0.015mM), CTAB(1mM), Silwet(0.017mM)

Surface Tension [mN/m] Contact Angle [◦]

Water 72 ± 2 36 ± 5

Tween-80 40 ± 2 25 ± 5

CTAB 38 ± 2 55 ± 5

Silwet 25 ± 2 0 ± 5

The surface tension of the three surfactant solution and water are given in Table 1. Tween-80 and CTAB lower the surface tension by approximately 45 percent compared to water, while Silwet lowers the surface tension even more (65%). From Young’s equation (Equation 1), it follows that a lower surface tension implies a lower contact angle, thus increasing the wettability of the liquid. This was observed for Tween-80 and Silwet. However, a decrease in wettability was found for CTAB in the form of a higher contact angle.

Figure 9: Schematic representation of CTAB solution on a borosilicate glass plate. The red circles with a (+) in the middle represent the positively charged represent the hydrophilic part and the zigzag tail represent the hydrophobic part of the molecule. The dashed line on the glass plate represents its negative charge.

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This contradiction to Young’s law is due to the cationic nature of CTAB. As the glass surface is negatively charged, the molecules not only move to the liquid-gas interface but also to the solid-liquid interface (Figure 9). The hydrophobic tails create a hydrophobic ’brush’ between the solution and the glass plate, decreasing the wettability of the surface and therefore increasing the contact angle. By altering the solid-liquid interface Young’s equation still holds. From our results we can conclude that we have three different types of solutions: a mostly wetting (Tween-80), a mostly non-wetting (CTAB) and a complete wetting (Silwet) solution.

5.1.2 Adsorbtion on glass Methylene Blue Water Tween-80 CTAB Silwet 400 500 600 700 800 0.0 0.5 1.0 1.5 Wavelength(nm) Normalized Intensity

Figure 10: Uv-Vis specctrum of pure Methelyne Blue solution (8mg/l) and this solution mixed with pre-wetted glass beads in water, Tween-80, CTAB and silwet

To confirm the adsorption behaviour of CTAB, the adsorption on glass of all surfactant so-lutions was investigated. In Figure 10 we see that the solution of Methylene Blue mixed with glass beads pre-wetted in water, Tween-80 and Silwet result in a low absorbance of Methylene Blue. The positively charged Methylene Blue molecules adsorb to the glass beads as expected. For CTAB the absorbance is high compared to the other liquids. The only explanation is that the CTAB molecules are adsorbed on the glass beads, making it impossible for Methylene Blue molecules to adsorb on these beads. The adsorption of CTAB on glass leads to the previously discussed decrease in wettability.

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5.2 Capillary rise

This section will be divided in to three subsections. The first section will discuss the influence of geometry on capillary rise. Secondly the results on mechanical equilibrium height will be reviewed. Finally the dynamics of the capillary rise will be given.

5.2.1 Geometry

As stated in section 3.3.1, the main difference between cylindrical and square capillary rise is the corner flow produced within square capillaries. Looking at a typical measurement of Silwet in square capillaries (Figure 11), it is observed that the corner flow produces ’fingers’ during the capillary rise Figure (12). We measured the averaged length of the fingers for each solution, which is plotted in Figure 13.

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

Figure 11: (a)-(f) is the capillary rise of Silwet solution through a square capillary over time. The red lines indicate the height of the fingers

The results indicate that the length of the fingers is directly related to the wettability of the liquid, which correspondents to theory (Gurumurthy et al., 2018; Bico & Qu´er´e, 2002). CTAB’s relative high contact angle forms almost no fingers, resulting in a meniscus shape comparable to the meniscus in cylindrical capillaries (Figure 12(c)). Silwet on the other hand has a high wetta-bility and forms significantly larger finger compared to the other solutions. The exact height of these fingers could not be detected with this setup, but was estimated around 4 cm. This can be seen in Figure 12 where only for Silwet (d) all the corners are completely wetted. This corner flow could be explained by our previous results considering Silwet a complete wetting solution.

In Figure 12 we see that the lengths of the fingers deviate, which is probably caused by two imperfections of our setup. One is that the capillary could not be perfectly placed perpendicular on the reservoir. The other one, is that in order for the capillaries to work properly due to probable

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(c) (d)

Figure 12: Liquid fingers in square capillaries: (a) Water , (b) Tween-80 , (c) CTAB, (d) Silwet

Water CTAB Tween-80 Silwet

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Finger Length (cm )

≈40mm

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surface defects, the tips had to be broken off. The broken tips were skewed, so one side of the came in contact with the reservoir earlier than the other. These two factors might be the reason for the large margins of error in Figure 13.

5.2.2 Mechanical Equilibrium Height

In section 5.1 we found that surfactants influence both the surface tension and contact angle, the two most important parameters of capillary rise. In the previous section it is shown that square capillaries produce corner flow creating fingers. In this section the effects of our previous findings on the mechanical equilibrium height will be discussed.

Cylindrical Capillaries Square Capillaries Jurin's Law

Water Water Tween-80 Tween-80 CTAB CTAB Silwet Silwet 0 1 2 3 4 5 Equilibrium Height (cm )

Figure 14: Average equilibrium height of capillary rise for water and surfactant solutions at t ≈ 30 seconds. In orange the cylindrical capillaries, in light blue the square capillaries and the black dashed line represents the theoretical value of Jurin (Equation 5)

Figure 14 shows the measured equilibrium height of square and cylindrical capillaries. Tween-80 barely lowers the height but for CTAB and Silwet a significant decrease was found compared to water. Our result show that geometry only has a positive influence on the height of capillary rise for water (18%) and Tween-80 (12%), but this was not found for CTAB and Silwet.

The increase in height for square capillaries could be related to the corner flow described in section 5.2.1. Both Tween-80 and water have corner flow during capillary rise while CTAB has none. The corner flow in combination between the intermolecular forces between the molecules of the liquid could be the cause of the increase of height in square capillaries. Water has a larger relative change than Tween-80 and this could possibly be due to their difference in surface tension. However, the identical equilibrium height in cylindrical and square capillaries measured for Silwet

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contradicts this statement. A possible explanation of this is that Silwet seems to prefer to wet the corners instead of rising the bulk. This can be seen in Figure 11 where even after maximum capillary rise height (Figure11(e)) the fingers keep on growing lowering the final meniscus height (Figure 11(f)). Unfortunately this is so far an assumption since we did not compare this to another non-surfactant liquid (e.g. Ethanol) with a similar surface tension.

According to Jurin’s Law (Equation 5), the equilibrium height is proportional to the surface tension and cosine of the contact angle of the fluid. The expected values are represented by the black dashed line in Figure 14. The predicted value of water in a cylindrical capillary agrees with Jurin’s law. On the other hand, the measured equilibrium height of surfactants are significantly higher compared to their theoretical prediction. This could be due to the dynamical surface ten-sion of surfactants as described in section 3.3.2. During capillary rise the surfactant molecules need time to move to the liquid-gas interface when the meniscus rises. Therefore it is possible that the equilibrium surface tension, given in section 5.1 is not reached yet. This would lead to a higher surface tension during the rise and from this follows a higher equilibrium height. There is no literature yet on the dynamical surface tension of Silwet but this could support our assumption.

From our results, we can conclude that capillary geometry plays a role in capillary rise depending on the wetting properties and surface tension of the liquid. Our results suggest that the effect of capillary geometry can be related to the corner flow that pulls the liquids to a higher mechanical equilibrium height. Jurin’s law is not matched for surfactant solutions, which is probably due to their dynamical surface tensions, leading to a higher surface tension during the rise than at their equilibrium. Silwet shows different behaviour than Tween-80 and CTAB showing a preference of creating corner flow in stead of rising the bulk.

5.3 Dynamics

Finally, we investigated the dynamics of capillary rise by tracking the height of meniscus. In Figure 15 two typical datasets are shown where the position is plotted as a function of time. According to the simplified Washburn equation (Equation 7), it is possible to describe capillary rise as h(t) = C ·√t. We define C as the Washburn constant and use this as a measure of the capillary rise ’velocity [m/√t]’ during its rise. We checked if the datasets fitted over√t to obtain a Washburn constant for every solution, which are plotted in Figure 16.

Figure 16 shows that there is no significant influence for the capillary geometry or the presence of Tween-80 and CTAB on capillary rise dynamics. Since Washburn’s equation is proportional to the surface tension and the cosine of the contact angle, this was surprising. The dynamical surface tension could also explain the lack of difference between the three liquids, but this would raise questions for CTAB’s equilibrium height since it is significantly lower compared to water and Tween-80. For Silwet however, the Washburn constant is significantly lower. However, this decrease could be explained by Figure 15(b), where we see that Silwet does not properly fit over the square root of time. For the first 500 ms, Silwet shows a linear dependence on time before slowing down. The superwetting properties of Silwet (θE= 0) might explain this dependence on

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Water cylindrical capillary square root fit

0.0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 30 35 Time (s) Height (mmm ) (a)

Silwet cylindrical capillary Square root fit

0.0 0.1 0.2 0.3 0.4 0 5 10 15 20 Time (s) Height (mm ) (b)

Figure 15: Measured position (blue circles) of meniscus for both water (a) and Silwet (b). The data is fitted with Washburn’s equation (Red line).

time in the early stages of the rise.

All Washburn constants are lower than their theoretical counterpart. We assumed that the cap-illary forces in the beginning of the rise were strong enough to neglect gravity, which is allowed for small capillaries as stated in section 3.2.2.. Apparently, we could not neglect this. However in Figure 15(a) we do see a dependence of a square root of time which was universal for water, Tween-80 and CTAB in square and cylindrical capillaries, so it seem that the gravitational forces only influence the Washburn constant. The velocities that we found are therefore a real indication of the dynamics of capillary rise.

The overall dynamics of capillary rise can not be changed by the use square capillaries. Both results of square and cylindrical are within their margin of error. Only Silwet appears to have a significant impact on the dynamics of the capillary rise.

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Cylindrical Capillaries Square Capillaries Washburn's Equation

Water Water Tween-80 Tween-80 CTAB CTAB Silwet Silwet

0.00 0.02 0.04 0.06 0.08 Washburn Constant [m / s ]

Figure 16: Measured Washburn constant of cylindrical (orange) and square (light blue) capillaries for water and surfactant solutions. The dashed black line represents the theoretical Washburn constant calculated with Equation 7

6 Conclusion and Outlook

In this research we investigated the influence of capillary geometry and presence of surfactants on capillary rise. Using high-speed imaging, we measured the dynamics of water, Tween-80, CTAB and Silwet in cylindrical and square capillaries.

The main difference between capillary rise in cylindrical and square capillaries is the corner flow producing liquid fingers in the latter. Since no fingers were observed for CTAB (a mostly non-wetting solution, microscopic ones for Tween-80 (a mostly non-wetting solution) and macroscopic ones for Silwet (a complete wetting solution) we conclude that corner flow depends on the wettability of the liquid, which is in agreement with literature.

The measured mechanical equilibrium height for water in a cylindrical capillary agrees with Ju-rin’s law indicating that our setup functioned properly. Square Capillaries had a positive influence on the capillary rise for water and Tween-80 but not for CTAB and Silwet. We suspect that this could be related to the corner flow pulling the liquid to a higher equilibrium height but To conclude this more liquids, with different wettability, should be tested. Silwet does not support this statement but showed preferential wetting behaviour, wetting the corners instead of rising the bulk. The presence of surfactants lead to an increase in capillary rise compared to Jurin’s law. This is probably caused by the dynamical surface tension of surfactants leading to a higher surface tension during the rise than when in equilibrium. We looked briefly into a possible relaxation of

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the height of capillary rise for surfactant solutions but only observed minor decrease in height over time (t ≈ 20 min). However, this experiment was not performed under the correct conditions. Doing this properly might give insight of the behaviour of the menisci and fingers over time, e.g. pinning of the meniscus.

From our results we can conclude that the geometry of the capillary and the presence of (most) surfactants do not have a significant influence on the dynamics of capillary rise. The effects of surfactants could again be explained by their dynamical surface tension, since we found this ve-locity in a short time-range. Only Silwet showed a decrease in ’veve-locity’ which could be linked to the preferential wetting of the corners. In addition, Silwet’s position appeared to have a differ-ence dependdiffer-ence on time than the other three liquids, therefore further experiments with complete wetting liquids could be interesting. The dynamics of all liquids could not be compared with Wash-burn’s equation as a result of our assumption for neglecting gravity. Experiments could be done with horizontal capillaries to make this assumption valid or a more complex form of Washburn’s equation has to be analyzed and compared with this data.

7 Acknowledgements

I would like to thank everybody affiliated with the Soft matter group of the University of Am-sterdam for giving me my first real introduction in experimental physics. Especially, Thijs de Goede for his guidance, feedback and humor during this project. Noushine Shahidzadeh for her enthusiast for physics and expertise on soft matter. Daniel Bonn for leading me to new directions within this project encouraging critical thinking about my research. I really appreciated the high level of collegiality of this group.

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Dynamics of capillary rise: Effects of surfactants and capillary geometry