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Electronic Theses, Treatises and Dissertations The Graduate School

2019

Detecting Wormlike Micellar Microstructure Using Extensional RheologyRose Omidvar

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FAMU-FSU COLLEGE OF ENGINEERING

DETECTING WORMLIKE MICELLAR MICROSTRUCTURE

USING EXTENSIONAL RHEOLOGY

ROSE OMIDVAR

A Thesis submitted to the Department of Chemical and Biomedical Engineering in partial fulfillment of the requirements for the degree of Master of Science

2019

Rose Omidvar defended this thesis on April 8, 2019. The members of the supervisory committee were:

Hadi Mohammadigoushki Professor Directing Thesis

Subramanin Ramakrishnan Committee Member

Daniel Hallinan Committee Member

The Graduate School has verified and approved the above-named committee members, and certifies that the thesis has been approved in accordance with university requirements.

I dedicate this thesis to my family.

iii ACKNOWLEDGMENTS

I would first like to thank my advisor Dr. Hadi Mohammadigoushki, for all his continuous help and support throughout this study. His hard work and dedication to research has always been motivating me and I am thankful for the opportunity to work with him. I shall extend my special thanks to dear committee members for taking time and reading this manuscript and helping me improve my work with their valuable comments. Last but not least I would like to thank the department of Chemical and Biomedical engineering for their care and support.

iv TABLE OF CONTENTS

List of Tables ...... vi List of Figures ...... vii Abstract ...... ix

1. INTRODUCTION ...... 1 2. LITERATURE REVIEW ...... 13 3. EXPERIMENTAL REULTS ...... 24

APPENDIX A. CABER CALIBRATION ...... 42

References ...... 51

Biographical Sketch ...... 56

v LIST OF TABLES

Table 3.1. List of surfactant solutions studied in this work together with measured and/or calculated rheological parameters ...... 29

Table A.1. List of Newtonian ﬂuids together with their physical and rheological properties ...... 50

vi LIST OF FIGURES

Figure 1.1. Two plate model ...... 2

Figure 1.2. Newtonian and non-Newtonian fluids ...... 3

Figure 1.3. Maxwell model ...... 5

Figure 1.4. Linear and non-linear viscoelastic regime in SAOS experiment ...... 7

Figure 1.5. Uniaxial extension ...... 7

Figure 1.6. Spherical micelle formation ...... 10

Figure 1.7. Formation of viscoelastic wormlike micelles by addition of salt and surfactant ...... 10

Figure 2.1. Falling plate filament stretching rheometer ...... 14

Figure 2.2. Filament stretching rheometer developed by Tirtaatmadja and Sridhar ...... 15

Figure 2.3. Operating space of a ﬁlament stretching rheometer and the known instabilities ...... 16

Figure 2.4. Schematic diagram of capillary break-up extensional rheometer ...... 17

Figure 3.1. Capillary breakup extensional rheometer setup mounted on the optical table ...... 26

Figure 3.2. (a) Steady shear viscosity as a function of imposed shear rate for surfactant solutions. The inset shows the zero shear viscosity as a function of concentration. (b) Storage (G′) and loss (G′′) moduli as a function of angular frequency. The inset shows the shear relaxation time calculated by ﬁtting an n-mode Maxwell model (solid curves) to the data. In (b), ﬁlled symbols correspond to the storage modulus and empty symbols denote the loss modulus ...... 27

Figure 3.3. TEM images of the wormlike micellar solutions for (a) 1.1 wt % and (b) 3 wt % CTAT in de-ionized water ...... 30

Figure 3.4. Filament thinning dynamics for surfactant solutions from the onset of experiments till breakup moment with hi= 1.3 mm and h= 4.2 mm. Images in top row indicate ﬁlament dynamics for a surfactant solution that contains 5 wt% CTAT and the bottom row corresponds to 0.8 wt% CTAT in water ...... 32

Figure 3.5. Filament diameter as a function of time for surfactant solutions along with the best ﬁt to Eq. (3): (a) for dilute surfactant solutions, and (b) for more concentrated solutions...... 33

Figure 3.6. (a) Transient extensional viscosity as a function of Hencky strain for surfactant solutions and (b) Maximum Trouton ratio for surfactant solutions as a function of surfactant concentration ...... 35 vii

Figure 3.7. Extensional relaxation time as a function of surfactant concentration. Inset shows the ratio of the relaxation times as a function of surfactant concentration ...... 38

Figure A.1. CaBER setup. (a) Top view. (b) Side View ...... 42

Figure A.2. Transient evolution of the ﬁlament of a silicon oil as a function of time. tbr refers to the breakup time of the ﬁlament...... 44

Figure A.3. MATLAB code ...... 45

Figure A.4. Example of edge detection code applied on an image ...... 46

Figure A.5. Detection of upper and lower edge location based on pixels ...... 47

Figure A.6. Example of a Diameter Vs. Time plot by MATLAB ...... 48

Figure A.7. Filament diameter as a function of time for three Newtonian ﬂuids. Lines indicate the linear ﬁt to the experimental data obtained by equation (2.4)...... 49

viii ABSTRACT

Viscoelastic wormlike micelles are widely used in variety of industrial processes, and daily life applications such as food, paint, pharmacy, oil-field operations and others. Wormlike micelles usually contain surfactant and salts that are dissolved in water at high concentrations. These systems share many similarities with viscoelastic polymer solutions and follow similar scaling laws. However, unlike polymers, micellar chains have the ability to break and reform and for this reason they are also known as living polymers. Many of the above industrial applications, involve continuous shearing and extensional flows of wormlike micelles. Therefore, a fundamental understanding of micellar dynamics in shear and extensional flows is necessary for optimal design of such processes. Dynamics of viscoelastic wormlike micelles have been extensively studied under shear deformations. However much less is known about the behavior of these systems in predominantly extensional flows. Therefore, a fundamental understanding of micellar dynamics and morphological transitions in extensional flows is needed.

In this project, we studied the nonlinear dynamics of a model wormlike micellar solutions using capillary breakup extensional rheometer (CaBER), shear rheology and Transmission

Electron Microscopy (TEM). Wormlike micellar solutions contain cetyltrimethylammonium tosylate (CTAT) in deionized water over a wide range of surfactant concentrations. Steady shear experiments indicate that the shear relaxation time and the zero-shear rate viscosity increase as a function of surfactant concentration up to a critical threshold, beyond which shear relaxation time drops to smaller values, but zero shear viscosity approaches an asymptotic value. TEM images indicate that as surfactant concentration increases, the micellar length increases and beyond the critical concentration micelles become entangled and shorter in size. Our results indicate that at low surfactant concentrations, where micellar solutions exhibit shear thickening, extensional flows ix lead to extreme elongational thickening possibly due to elongation induced structure (EIS) formation. Within this range of concentration, the extensional relaxation time is fairly constant and as the surfactant concentration increases, the extensional relaxation time increases. More importantly, we have estimated the dimensionless Trouton ratio in extensional flows over a wide range of surfactant concentrations. Trouton ratio is defined as the ratio of transient extensional viscosity over the zero-shear viscosity of the fluid. Our results show that Trouton ratio increases as a function of time in the course of filament thinning dynamics until it asymptotes to a constant value. It turns out that the maximum Trouton ratio decreases as surfactant concentration increases and finally reaches a constant value around Tr ≈ 3 for concentration above the critical concentration. This clearly shows that uniaxial extensional flows and in particular, CaBER is sensitivity to microstructural changes in wormlike micelles.

x CHAPTER 1

INTRODUCTION

Rheology

The concept of rheology which is the science of deformation and flow of matter was initially coined in 1929 by Eugene Cook Bingham [1]. Particularly rheology deals with the behavior of complex viscoelastic materials (e.g. polymer melts and solutions, surfactant micellar solutions, suspensions and glass-forming liquids) that show both solids-like and liquids-like response to force, deformation and time [2]. In the past century, rheological studies have grown exponentially due to increasing use of these complex fluids in daily life as well as industrial applications. Rheology plays an important role in a wide range of industries including, food industry (e.g. production of mayonnaise, creams), paints, household products, consumer products and others. In all these applications the fluid undergoes strong continuous shear or extensional deformations, thus a better understanding of the behavior of these systems in shear and extensional flows under controlled setting provides a framework to achieve their optimal processing conditions

[3].

Shear Rheology in Strong Non-linear Flows

Rheological studies are divided to two categories; linear and non-linear. In the non-linear flows where flow is strong, the flow structure is perturbed and viscosity and stress can be measured. However, in linear flows, the oscillations are so small that the inherent structures remain intact in a sample [4].

Non-Linear Shear Flows

Shear flows which is the most common flow behavior, can be best described with the aid of a two-plate-model (Couette flow). As shown in Figure 1.1 a liquid is located between two parallel plates with the surface area (A). The force (F) is applied to move the upper plate with a constant velocity thereby the layers of fluid sliding over one another with each layer moving faster than the one beneath, while the bottom plate is at rest. The displacement gradient across the sample

(x/h) is defined as the shear strain (γ). This shear strain causes a laminar flow of the velocity (v) which linearly decreases from the moving to the static plate [5].

Figure 1.1. Parallel Plate Model

Unlike solid material, due to the applied stress within a fluid the shear strain will continue to increase, this creates a velocity gradient termed as shear rate which is the rate of change of shear with time;

푑γ 푉 γ̇ = = (1.1) 푑푡 ℎ and shear stress is defined as:

퐹 휎 = (1.2) 퐴

Shear viscosity or dynamic viscosity (η) can be defined as the proportionality constant between the shear stress and shear rate:

휎 퐹ℎ ƞ = = (1.3) 훾̇ 퐴푉

2 Shear viscosity is determined by shearing a sample in rotational rheometers (where the moving surface performs a rotary movement) and measuring its resistance to that shear.

Measuring systems include concentric cylinder, cone and plate, and parallel plates, depending on the test sample [2].

Newtonian and Non-Newtonian Flow Behavior

As shown in Figure 1.2, flow behavior can be characterized as Newtonian and non-

Newtonian. Newtonian material or ideally viscous materials have constant viscosity with applied shear rate and their flow behavior only changes with pressure and temperature. Newtonian behavior is the ideal fluid behavior and a typical example would be water. Non-Newtonians fluids refer to those where the viscosity varies as a function of the applied shear rate or shear stress.

Figure 1.2. Newtonian and non-Newtonian fluids

3 The most common type of non-Newtonian behavior is shear thinning in which fluid becomes less viscous with increasing shear rates. This behavior is typical for polymer melts and polymer solutions. Some materials can also show shear thickening behavior where viscosity increases with increasing shear rate or shear stress. Usually dispersions or particular suspensions with high concentration of solid particles exhibit shear thickening behavior because above critical shear rates the solid cannot readily flow past each other and begin to jam causing a rise in viscosity.

An example of shear thickening fluids is the corn starch solution.

Finally, there are yield stress ﬂuids for which the material ﬂows only when stresses are above a threshold yield stress. Some common examples are ketchup and toothpaste [5].

Weak Linear Shear Flows

When materials are subjected to a shear stress, the rheological properties of materials over a wide range of phase behavior can be expressed in terms of viscous, elastic and viscoelastic responses. In viscous flows, the imposed stress causes the material deformation to occur and to stop, if the stress is removed. Now if the flow behavior of material instantly changes under stress, but when stress is removed it springs back to its original state this material possesses an elastic response. On the other hand, material with viscoelastic properties is considered to cover both viscous and elastic phase behaviors and it becomes time dependent. The simplest rheological model representing viscoelastic liquids is the Maxwell model. As shown in Figure 1.3, in Maxwell model, the elastic properties are represented by the spring (with shear modulus, G0) and the viscous properties are presented by the dashpot (zero shear viscosity ɳ0) which are connected in series.

There are various rheological techniques for probing the viscoelastic behavior of materials, including creep testing, stress relaxation and oscillatory testing. The most effective way to measure

4 viscoelastic properties is oscillatory shear rheometry. These tests provide very useful and convenient rheological characterization of complex fluids or soft materials and help to identify how viscous and how elastic the material is [6,7].

Figure 1.3. Maxwell model

Dynamic Oscillatory Shear Flows

Oscillatory shear tests can be divided into two regimes. One regime deals with the linear viscoelastic response (small amplitude oscillatory shear, SAOS) where oscillations are small and sample remains in its equilibrium position. The other regime is defined by a measurable nonlinear material response (large amplitude oscillatory shear, LAOS). At a deformation-controlled oscillatory measurement, a sinusoidal oscillatory strain (훾(푡) = 훾0sin⁡(휔푡)) is imposed on the

′ sample and the resulting stress response (휎(푡) = G 훾0 sin(휔푡) + G′′훾0 cos(휔푡)) is recorded as a function of time and the viscous and elastic properties of the sample are simultaneously measured.

It is convenient to express the periodically varying functions in terms of the complex modulus, G∗ which is a quantitative measure of material stiffness or resistance to deformation:

퐺∗(휔, 훾) = 퐺′(휔, 훾) + 푖퐺′′(휔, 훾) (1.4)

G′ and G′′ are the real and imaginary parts of the complex modulus G∗. The storage modulus (G′) represents the degree of elastic behavior, i.e. the energy, which is reversibly stored by restoring forces. The loss modulus (G′′) represents the strength of the viscous behavior, i.e. the

5 energy which irreversibly dissipated due to viscous flows. As mentioned earlier the relationship between the complex modulus and the material parameters in the viscoelastic models is best illustrated using the Maxwell model and the behavior of a Maxwell material under harmonic oscillations can be obtained from [7]:

2 ′ 푛 휔 휆푖ƞ푖0 퐺 (휔) = ∑𝑖=1 2 (1.5) 1+(휔휆푖)

′′ 푛 휔ƞ푖0 ⁡⁡⁡⁡⁡⁡⁡⁡⁡퐺 (휔) = ∑𝑖=1 2 (1.6) 1+(휔휆푖)

1 ∗ 2 2 |ƞ (휔)| = ƞ0/(1 + 휔 휆 )2 (1.7)

Where, 휆 denotes the relaxation time of the multi-mode material, n is representative of the number of spring and dashpot pairs, 휔 denotes the angular frequency in rad/s, ƞ∗ is the complex viscosity,

ƞ0 denotes steady zero shear viscosity.

As the applied amplitude (of strain or stress) is increased from small to large at a fixed frequency, a transition between the linear and nonlinear regimes can appear. Figure 1.4, schematically illustrates an oscillatory strain-sweep test in which the frequency is fixed and the applied strain amplitude is varied. In the linear viscoelastic regime (LVER) the strain amplitude is sufficiently small that both viscoelastic moduli are independent of strain amplitude and the oscillatory stress response is sinusoidal. In the LVER, applied stresses are insufficient to cause structural breakdown (yielding) of the structure and hence microstructural properties are being measured. When applied stresses exceed a certain stress, non-linarites appear and measurements can no longer be easily correlated with micro-structural properties. The linear viscoelastic region can be determined from experiment by performing a stress or strain sweep test and observing the point at which the structure begins to yield [6-8].

Figure 1.4. Linear and non-linear viscoelastic regime in SAOS experiment [8].

Extensional Flows

Another phenomenon associated with some non-Newtonian ﬂuids is a dramatic resistance to stretching compared to Newtonian ﬂuids. Several important industrial processes involve a strong elongational flow including, film blowing, fiber spinning, pumping and extrusion. There are three simple types of shear-free elongational flow; 1) Uniaxial elongational flow, 2) Biaxial elongational flow, and 3) Planar elongational flow. Uniaxial extensional flow is the simplest version of these stretching flows.

Figure 1.5. Uniaxial extension

7 In the case of uniaxial extensional flow, consider a cylindrical specimen of initial diameter

Do and length Lo, which is stretched along the axial direction by an applied force F, so that its radius decreases uniformly along the length. For a shear-free extensional flow the axial velocity must be uniform over the cross section of the filament at each axial position. The velocity-gradient tensor is given by:

−1 0 0 1 ∇푣 = 휀̇(푡) ( 0 −1 0) (1.8) 2 0 0 2

If the strain rate⁡휀̇ is constant with time, the ﬂow is steady in both the Eulerian and

Lagrangian frameworks. Then from conservation of volume and integration of the components of

Equation (1.8), the length and diameter of the specimen at any time t vary from their initial values of L0 and D0, respectively, in the following way:

1 퐿(푡) = 퐿 exp(⁡휀̇ 푡) ; ⁡⁡⁡⁡⁡퐷(푡) = 퐷 exp(− ⁡휀̇ 푡) (1.9) 0 0 0 2 0

The total accumulated strain arising from such elongational flows is called Hencky strain which is given by:

퐿(푡) 퐷(푡) 휀 = ∫ 휀0̇ 푡 ⁡푑푡 = ln ( ) = −2 ln ( ) (1.10) 퐿0 퐷0

The extensional viscosity is a fundamental material property of a fluid and it is characterizing the resistance of a material to stretching deformations and is a function of the extension rate and temperature. For Newtonian fluid the extensional viscosity doesn’t change with the strain rate. In fact, the extensional viscosity in Newtonian fluid is three times the shear viscosity of the fluid:

휏푧푧−휏푟푟 2휇휀̇ 0−(−휇휀̇ 0) ƞ퐸 ≡ = = 3휇 (1.11) 휀̇ 0 휀̇ 0

The ratio of the extensional viscosity to the shear viscosity is defined as Trouton ratio; 8 <휏 −휏 > ƞ+ ⁡푇푟 = 푧푧 푟푟 = 퐸, (1.12) ƞ0휀̇ 0 ƞ0

+ where ƞ퐸 is the transient extensional viscosity and ƞ0 is the zero shear rate viscosity of the ﬂuid, respectively [10].

The main focus of this thesis is on surfactant based viscoelastic fluids (or wormlike micelles). In the following, we will describe the general features of wormlike micellar systems.

Wormlike Micellar Solutions as Model Rheological Fluids

Surfactants are amphiphilic molecules that contain both hydrophobic and hydrophilic groups. Surfactants can be categorized as ionized and non-ionized surfactants. When ionized surfactants are dissolved in water they ionize into ions with negative charge, this helps them to bind to positively charged particles such as clay and for this reason they are the best candidates to be used in detergents. Unlike ionized surfactants, non-ionized surfactants can efficiently remove oils, and they generate more foam compare to other type of surfactants. Generally, when surfactants dissolved in water, they diffuse to the water-air interface with the hydrophilic groups facing the water phase, and the hydrophobic groups pointing towards air. The adsorptions of the surfactant molecules at the interface cause the surface tension to decrease [11].

As shown in figure 1.6, above critical micelle concentration (CMC), these surfactant monomers can self-assemble into micelles with the simplest structure being spherical micelles.

Different factors including surfactant charge, number of tails, concentration, temperature, pH, salinity of the solution, ionic strength and shear rate can affect the size and shape of the micellar and generate a transition from spherical micelles to lamellar bilayers and long wormlike micellar structures. [12].

Figure 1.6. Spherical micelle formation

Under certain conditions of surfactant and salt concentration, spherical micelles start to grow to cylindrical micelles that at higher concentration can self-assemble into wormlike micelles which exhibit highly viscoelastic behavior similar to polymer solutions (Figure 1.7) [13].

Figure 1.7. Formation of viscoelastic wormlike micelles by addition of salt and surfactant.

Wormlike micelles and polymer solutions share many similarities. However, unlike polymers, transient network of micelles can break and reform. This have earned them the name of

‘living polymers’. In addition, this specific property has made them applicable in plethora of processes where continuous shearing and elongation is necessary. They are being used as rheological modiﬁers in consumer products such as cosmetics, detergents, paints, and pharmaceuticals also in oil and petrol industry play an important role as drag reducing agents.

10 Many of the above applications involve interaction of wormlike micellar solutions with rotational shear and/or extensional ﬂows [11,14].

Non-linear Shear Rheology of Wormlike Micelles

In the regime of large velocity gradients, wormlike micellar solutions exhibit similar non- linear flow response to that of the observed in polymer solutions. Wormlike micellar systems exhibit a rich rheology, with many distinct phenomena not observed in other macromolecular species. For example, some surfactant solutions show shear thickening behavior at low concentration and at higher concentrations, some wormlike micelle solutions exhibit a complex fluid dynamic behavior known as shear-banding. Sometimes high shear rates disrupt the equilibrium structure of wormlike micelles that yields the typical shear thinning behavior of viscous solution. The amount of salt, pH, temperature or external field may affect their behavior

[15]. Dynamic experiments such as steady-state shear experiments, Oscillatory measurements are often performed to get information on the viscoelastic properties of micellar solutions.

Linear Viscoelastic Rheology of Wormlike Micelles

Wormlike micelles form entanglements in the semi-dilute concentration regime. There are two principal relaxation dynamics in wormlike micelles; Reptation (휆푟푒푝),⁡and breakage⁡(휆푏푟). A competition between reptation and the breaking/reforming time of the micelles gives rise to simple linear rheological behavior, often referred to as ‘Maxwell-like’. In the fast-breaking regime, where the micellar breakage time is much shorter than the reptation time scale, (휆푏푟 << 휆푟푒푝),⁡ the linear viscoelastic response is best described by a single element Maxwell model. The linear behavior is described by two parameters, a plateau modulus G0 related to the entanglement density of the

11 mesh, and a relaxation time λ. This dominant relaxation time is simply the geometric mean of the reptation and breaking/reforming times of the micelles [16]:

1 휆 = (휆푏푟. 휆푟푒푝)2 (1.13)

As mentioned previously dynamic experiments such as steady-state shear experiments and

SAOS are often performed to get information on the viscoelastic properties of a solution. In the regime of small deformations, the dynamic moduli of some viscoelastic wormlike micelle solutions can be accurately modeled by a Maxwell model having just one or two relaxation times.

From oscillatory experiments the relaxation time can be calculated by fitting Maxwell model to experimental data (see chapter 3 for more detail) [13,17].

12 CHAPTER 2

LITERATURE REVIEW

Extensional Rheology

Since the work of Trouton (1906) the thread-forming ability of fluids has attracted the attention of many researchers around the world [18,19,20,21,22]. Although there are a number of commercial rheometers available to measure viscometric properties of fluids in simple shear flows, the situation is far less satisfactory as it relates to extensional rheology. A small number of techniques have been developed for such measurements. The need for extensional techniques arises as dynamic responses of the viscoelastic solutions under predominantly extensional flow, is markedly different than in simple shear [16]. Considering the fact that extensional deformations are responsible for a range of interesting phenomena such as instabilities in bubble rise [23] or sphere sedimentation in wormlike micellar solutions [24,25] sudden failure of the wormlike micellar filament in uniaxial extensional flows [26,27], there is an urgent need for a method of measuring extensional rheological properties. One of the challenges associated with extensional flow measurements is related to measuring the viscosity of low-viscosity solutions. So far varieties of devices have been developed to generate extensional flows for viscoelastic solutions, which can be classified as either flow through systems or stagnation-point devices. Examples of flow through systems are: tubeless siphon (also known as Fano flow), spinning techniques, orifice flows, opposing jets, cross-slot ce1l, and four roll mill. There are some downsides to most of these devices for example, some have an inherent difficulty in the measurement of the velocity gradient to which the liquids are subjected, on the other hand most of these flow geometries are not purely extensional flows and have some shear flow history [9,28,29]. One of the major drawbacks

13 introduced by both flow-through and stagnation-point devices is that the macromolecules undergo quite short residence times. Therefore, more recent studies have utilized filament-stretching rheometers (FiSER) and capillary break up extensional rheometers (CaBER) that can impose uniaxial extensional flows [3,10,30]. As of now, these devices are sufﬁciently well developed that can provide quantitative experimental data for the transient extensional stress growth of viscoelastic ﬂuids.

Filament Stretching Rheometer

The development of a falling cylinder device by Matta & Tytus (1990) has enabled the generation of pure extensional flow and measurements of fluid properties such as elongational viscosity [30]. As shown in figure 2.1, this device consists of two vertically oriented cylinders and a small amount of sample is placed between them. The upper cylinder is held fixed and the lower cylinder, which is initially at rest, is allowed to fall under gravity and stretch the sample. Matta and Tytus were able to generate a nearly pure extensional ﬂow and to calculate the tension in the

ﬂuid ﬁlament during stretching.

Figure 2.1. Falling plate filament stretching rheometer [30]

14 Inspired by Matta & Tytus, in 1997 Tirtaatmadja and Sridhar developed a similar apparatus named filament stretching rheometer which is capable of applying homogenous uniaxial extension on the test fluid because the separation of the two endplates are controllable, and it has enabled the transient extensional flow behavior of mobile fluids to be characterized [31]. In figure 2.2 a representative sketch of a filament stretching rheometer elements is shown.

Figure 2.2. Filament stretching rheometer developed by Tirtaatmadja and Sridhar [31]

Similar to setup built by Matta & Tytus, also in this device, a small sample of fluid is placed between two circular end plates and an initially cylindrical filament of fluid is elongated between the endplates but instead of letting lower plate falling under gravity, the electronic control system imposes a predetermined velocity profile on one or both of the end plates. Also several arrangements of the endplates, motor platens, and sensors are possible. Then according to a carefully controlled separation history L(t),⁡the principal time-resolved measurements of the tensile force in the filament Fz(푡) and the mid-filament Radius Rmid(t) as the endplates move apart can be obtained. The filament diameter can be measured by using devices such as laser micrometers or wire gauges or by analyzing video images of the extending filament. In figure below the typical operating space of a filament stretching device and the known instabilities that

15 limits the operation is plotted. The x-axis indicates the endplate velocity and the y-axis denotes the endplate position [32].

Figure 2.3. Operating space of a ﬁlament stretching rheometer and the known instabilities [32]

An ideal homogeneous uniaxial extensional ﬂow described by equation (2.1) and cylindrical ﬂuid sample is separated exponentially in time and here it is represented as a straight line on this phase diagram, with a slope equal to the imposed strain rate [3].

퐿(푡) = ⁡ 퐿0 exp(휀̇ 푡), (2.1) however, in practice there are limitations due to either the total travel available to the motor platens or by the maximum velocity the motors can provide. Thus, there is a time-dependent effective stretch rate.

Instabilities involved with gravitational sagging, capillarity, and/or elasticity can further limit the accessible operating space and the location of these stability boundaries depends on the material properties of the ﬂuid.

On the other hand, another device based on filament self-thinning and breakup was introduced in 1990 by Bazilevsky et al. (1990) and called “the liquid filament micro-rheometer”

16 (LFR). In LFR, the disks bridged by liquid are rapidly separated and then held at a fixed axial separation, and the subsequent evolution of the mid-filament diameter is monitored during the process of necking and breakup. Based on the Bazilevsky group’s LFR, Cambridge polymer group

(Cambridge, USA) has developed a commercial version of a capillary breakup extensional rheometer (CaBER) [33].

Figure 2.4. Schematic diagram of capillary break-up extensional rheometer

Since its introduction in 1990 the capillary break-up extensional rheometer has become one of the most reliable and accurate method of determining the extensional properties of fluids compare to other rheometers. This technique is the most straight forward way that enables the measurements of the longest extensional relaxation times and extensional stresses of viscoelastic fluids through the monitoring of the capillary thinning and breakup dynamics of a fluid thread. In addition, unlike filament stretching rheometer it has the ability to measure the viscosity of low viscosity fluids [34]. In this project, we performed our experiments using capillary break up extensional rheometer. 17 Capillary Break-up Extensional Rheometry

The CaBER device uses an initially cylindrical volume of fluid, which forms a liquid bridge between circular parallel plates. To achieve a final plate separation a step strain is applied to the sample by lifting the upper plate instantaneously [35]. To minimize gravitational sagging and obtain an approximately cylindrical liquid bridge, the initial separation was chosen to be less than the capillary length defined as:

휎 푙 = , (2.2) 푐푎푝 √휌𝑔 where, σ is the surface tension of the sample, ρ is density of the test fluid, and g = 9.81 m s-2 is the acceleration due to gravity [34]. As a result of the applied strain an unstable fluid bridge forms which gradually thins and finally breaks. The breakage of the fluid filament is driven by surface tension and gravitational forces and resisted against by viscous and/or elastic forces and it is described by a general force balance as:

2 2 dRmid Fz 1 휌𝑔(휋퐿0푅0) 휎 3ƞs { ( )} + 2 + 2 = + (휏푧푧 − 휏푟푟), (2.3) Rmid dt πRmid 2 휋푅푚푖푑 푅푚푖푑

where ƞ푠, 퐹푧, 휏푧푧, 휏푟푟, and σ are the viscosity of the Newtonian solvent, the axial force exerted on sample by the device, the normal stresses in axial (z), in radial (r) directions and the surface tension of the ﬂuid respectively. This equation correlates the temporal evolution of the ﬁlament diameter to the transient extensional properties. It has been shown that for a purely viscous liquid, in absence of gravity and inertia, the mid-plane diameter is found to from the self-similar shape of the decaying viscous filament and it decreases linearly with time according to [35,36]:

휎 퐷(푡) = ⁡ 퐷0 − 푡 (2.4) 7.05ƞ푠

18 On the other hand, for viscoelastic ﬂuids, the liquid bridge initially drains under a predominantly capillary-driven flow that is resisted by the viscous stress arising from the solvent contribution. As time progresses, the strong extensional flow deforms the microstructure of the viscoelastic fluid. Thus, during such intermediate times elastic forces become dominant. Entov and Hinch showed that in absence of gravity and inertia the ﬁlament diameter decays exponentially with time as [37]: ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡

푡 퐷(푡) ∝ exp (− ), (2.5) 3휆퐸 where, 휆퐸 is the extensional relaxation time of the viscoelastic solution. According to Anna and

McKinley, for Boger fluids (dilute polymer solutions with constant viscosity), the measured extensional relaxation time is almost equal to the shear relaxation time 휆퐸≈λ [21].

Finally, at later times, the filament thinning process becomes linear with time and the extensional viscosity levels off around a costant value. Based on a report by Entov and Hinch, at later times, polymer chains reach their finite extensibility and become fully stretched [37]. To account for the late viscous contribution, the filament decay in viscoelastic solutions can be fitted to the following equation:

퐷(푡) = 퐴푒−퐵푡 − 퐶푡 + 퐸, (2.6) where 퐵 = 1/3휆퐸 .We discussed Trouton ratio in previous chapter, since CaBER experiment is

ɳ퐸 essentially shear-free, we define the Trouton ratio as 푇푟 = and η퐸 which represents the transient ɳ0

−휎 extensional viscosity is equal to,⁡ƞ퐸 = . 푑퐷푚푖푑/푑푡

By fitting the Eq. (2.6) to experimental data, the transient extensional properties of the wormlike micellar solutions can be obtained.

19 Extensional Rheology of Wormlike Micelles with CaBER

Dynamics of the wormlike micellar solutions under predominantly shear ﬂows have been extensively investigated over the last few decades [37,38]. However, the extensional rheology of these systems is poorly understood. Recent theory predictions and experiments have shown for

Boger fluids, λE≈λ [39,40] but for wormlike micelle solutions the extensional relaxation time has been found to be quite different from the relaxation time measured in shear [10,42]. The question is, what causes this discrepancy in shear and extensional relaxation time in wormlike micellar solutions. Could it be due to the non-linear effects such as ﬂow induced breakage of the wormlike micelles and/or elongation induced structures (EIS) in extensional flow?

Previous literature shows several investigations on the correlation between the transient extensional properties and the shear rheological properties of the wormlike micellar solutions

[10,40,41,42]. Mainly the focus was on the relation between the extensional relaxation time (휆퐸) and shear relaxation time (λ) for a wide range of wormlike micellar solutions. In a study performed by Rothstein and co-workers transient extensional rheology of a series of wormlike micellar solutions CTAB/NaSal, CPCl/NaSal, C8TAB/NaOA were measured; for equi-molar solutions of

CTAB/NaSal, at low surfactant concentrations the ratio of the extensional relaxation time to the shear relaxation time is always below unity (∼0.1) however, for higher CTAB concentrations, the ratio increases until it reaches a plateau around unity. On the other hand, for equimolar CPCl/NaSal systems, the ratio of these two relaxation times begins from values less than unity (<1) and continues to climb beyond unity at higher concentrations. It has been shown for C8TAB/NaOA wormlike micellar solutions, λE/λ < 0.1 [10]. According to Yesilata et al. (2006) for wormlike micellar solutions based on EHAC/KCl, the extensional relaxation time is three orders of magnitude shorter than the shear relaxation time [41]. More recently, Sachsenheimer et al. [42]

20 proposed a general phase diagram to relate the shear and the extensional relaxation times for a series of wormlike micellar solutions (e.g. CTAB/NaSal and CPCl/NaSal). Their results for steady shear experiments indicate, at low salt concentrations; surfactant solutions exhibit a shear thickening behavior. The shear thickening behavior is due to formation of shear induced structures or SIS. At this range of salt concentration, they found that λE/ λ≫1. This result was related to formation of elongational induced structures (EIS). For higher surfactant concentrations, shear thinning wormlike micelles relax dominantly via reptation mechanism and they suggest λE/λ≪1.

Finally, at higher salt concentrations where breakage and reformation processes are the dominant relaxation mechanisms, they found that λE/λ≈1. We note that the latter results are in disagreement with the results reported by other researchers for surfactant solutions that follow a single-mode

Maxwell model. Therefore, the relationship between these two relaxation times is not settled for the wormlike micellar solutions that are best described by a single-mode Maxwell model.

Structure of the wormlike micellar solutions can be manipulated by varying the ratio of the salt to surfactant concentration at a ﬁxed surfactant concentration. Addition of salt to surfactant solutions increases the rheological properties such as the zero shear viscosity or the relaxation time. This is due to a geometrical transition from spherical micelles to long and entangled micelles.

However, beyond a critical threshold, a further increase of the salt concentration results in reduction of the zero shear viscosity or the relaxation time. Cryo-TEM images suggest two possible mechanisms for reduction of the zero shear viscosity (and the relaxation time) above the critical concentration; (1) transition from linear micelles to branched micellar networks

[45,46,47,48,49,50,51] or (2) shrinkage of the wormlike chain length [52,53,54]. Next question to address would be whether elongational ﬂow is sensitive to any structural transitions such as linear to branched micelles. Chellamutu and Rothstein studied transient extensional rheology of a

21 wormlike micellar solution based on C8TAB/ NaOA and showed that the extensional rheology might be sensitive to the transition from linear micelles to branched micelles [55]. The wormlike micellar solution based on C8TAB/NaOA (30/70) experiences a transition from linear to branched micellar networks around cs ≈4 wt %. Their results showed that as the surfactant concentration increases, the maximum Trouton ratio decreases and for concentrations above cs ≈4 wt%, the maximum Trouton ratio drops to 3 and remains at the same level for concentrations above that.

Trouton ratio is deﬁned as the ratio between transient extensional viscosity and the zero shear viscosity [56]. Chellamutu and Rothstein correlated this behavior to an additional stress relief mechanism due to ghost-like crossing effects of the branched wormlike micelles. Thus, they concluded that the elongational ﬂow is sensitive to formation of branched networks [55]. More recently, Sachsenheimer et al. employed a capillary breakup extensional rheometer, and showed that the transition from a linear to branched wormlike micelles leads to subtler changes in the

ﬁlament life-time [42]. According to Sachsenheimer et al. at a given zero shear viscosity, the

ﬁlament of a branched wormlike micellar solution lasts longer than a linear wormlike micellar solution. Sachsenheimer et al. [42] also explained that the C8TAB/NaOA solutions studied by

Chellamutu and Rothstein, show ﬂow induced structures (SIS) for cs < 4 wt%. Thus, these authors suggested that the dramatic reduction of the Trouton ratio for C8TAB/ NaOA solution studied by

Chellamutu and Rothstein, might be due to a ﬂow induced structure formation rather than the transition from linear to branched structure. Therefore, it is still not clear whether elongational

ﬂow is sensitive to any structural transitions such as linear to branched micelles.

In summary, to the best of our knowledge, it is still an open question whether the extensional rheology is sensitive to microstructural transitions (e.g. linear micelles to branched

22 micelles or shorter linear micelles) in wormlike micellar solutions and also the connection between shear and elongational properties is still not clear.

In this thesis, we study the correlation between the structural transition and elongational properties of a salt free surfactant solution via a combination of shear rheology, TEM imaging and transient extensional rheology. The first goal of this study is to evaluate the structural properties of the wormlike micelles using TEM imaging together with the shear rheology, and to use this knowledge to examine whether transient extensional rheology is sensitive to such structural changes around the critical concentration. The second goal of this study is to investigate the relation between the two relaxation times in CTAT/Water solution, which allows us to examine the general phase diagram suggested by Sachsenheimer et al. Another novelty of this work is that we use a custom-built capillary device which provides an alternative and more affordable way to stretch the fluid filament in a uniaxial fashion.

23 CHAPTER 3

EXPERIMENTAL REULTS

Materials and Methods

A series of wormlike micelle solutions were prepared by dissolving various amount of

Hexadecyltrimethylammonium p-toluenesulfonate (CTAT) in de-ionized water. Surfactant concentration varies from 0.7 %wt to 7 %wt. The samples have been prepared by gently stirring surfactant in water for one day at room temperature T = 24 ± 0.5 °C, followed by at least two days in the oven set in the room temperature, in order to reach the equilibrium state. CTAT is purchased from Sigma-Aldrich and used as received. Using a DuNouy Tensiometer (Fisher Surface

Tensiomat Model 21) the surface tension of all solutions is measured at room temperature. Small amplitude oscillatory shear (SAOS) experiments and steady shear rheology measurements are carried out in a commercial rheometer (model MCR 302, Anton-Paar Co.), using a concentric cylinder geometry with Ri = 13.35 mm and Ro = 14.45 mm. To avoid evaporation, a fresh sample was used for each experiment. TEM technique has been used to directly image the microstructure of the wormlike micellar solutions. Prior to TEM imaging the samples were prepared using a negative-staining protocol. Following this protocol, 3 µL of the surfactant solution is placed on perforated carbon-coated grids. The grids were purchased from Electron Microscopy Sciences.

The sample is allowed to wet the carbon surface for 30 s then using a filter paper the extra solution is carefully blotted from the surface of the micro-grids. Solutions are stained 1–3 times by bobbing the grid in a 2 wt% uranyl acetate solution. The purpose of using uranyl acetate is to provide a high contrast as due to low pH and binding to surface carboxyls, this salt ﬁxes the wormlike chains.

Next, Micro-grids are placed in a petri dish to completely dry out. Dried micro-grids are placed in

24 the holder and transferred to TEM for imaging. Images are observed using a Phillips Transmission

Electron Microscope (model FEI CM120 BioTwin) in low dose mode at 120 kV.

A custom built capillary device was designed to perform transient extensional experiments.

As shown in Figure (3.1) the device employs an electromagnetic ﬁeld which is generated with compact linear electromagnet (model 6873K3 purchased from McMaster-Carr) to generate the uniaxial extensional ﬂow. The electromagnet is connected to a DC power source.

There are two end plates with diameter D = 3 mm and the upper plate is mounted to the electromagnet. By powering on the DC source, the electromagnet quickly lifts the upper plate

(within 40 ms) to the ﬁnal height. This imposes a step strain on the sample. The lower plate is mounted on a vertical micro translation stage (purchased from Thorlabs), and its height is adjustable with high accuracy.

In order to calibrate the device three Newtonian ﬂuids are used and results were compared to literature (see more detail in appendix). Reproducibility was guaranteed by repeating the same experiment three times on the same surfactant solution, each time with a fresh, just loaded sample.

After step strain is applied, the fluid ﬁlament starts to thin until it pinches off. A high-speed camera (Model Phantom Miro-310, Vision Research Inc.) records the thinning process at different capture rates (24 to 8600fps). The combination of the high speed camera and a macro zoom lens

(model VSZ-0745 from VS Technology) provides a 3 mm × 3 mm ﬁeld of view that allows us to measure ﬁlament diameters down to ≈ 20 μm.

Finally, a MATLAB script is developed to quantify the temporal evolution of the mid-

ﬁlament diameter as a function of time (see more detail about this code in appendix).

Figure 3.1. Capillary breakup extensional rheometer setup mounted on the optical table

Results and Discussion

Shear Rheology

Figure 3.2.a shows the steady shear viscosity as a function of shear rate for the micellar solution of CTAT/water at different concentrations. For dilute solutions (CCTAT ≤1 wt%), fluids show a constant viscosity at low shear rates followed by a shear thickening behavior above a critical shear rate. According to the previous literature the shear thickening behavior has been attributed to formation of shear induced structures (SIS) in surfactant solutions [57,58,59]. At higher concentrations (CCTAT ≥1.1 wt%), shear thickening behavior disappears and the surfactant solution exhibits strong viscoelastic behavior with a measurable shear relaxation time. The inset of Figure (3.2.a) shows the zero shear viscosity of the surfactant solutions as a function of surfactant concentration. As surfactant concentration increases, the zero shear viscosity increases and remains fairly constant for concentrations above 2 wt%. In Figure 3.2.b the results of the small amplitude oscillatory shear experiments for surfactant solutions are shown. The experimental results are ﬁtted to an n-mode Maxwell model (n values are given in Table 3.1 to obtain the longest

26 shear relaxation time. By increasing surfactant concentration, the relaxation time increases and

ﬁnally beyond a critical concentration (CCTAT ≈ 2wt%), the shear relaxation time starts to decrease

(cf. Inset of Figure 3.2.a). This reduction is indicating a structural transition for concentrations above 2 wt%.

According to Soltero et al. [60,61] around surfactant concentration of c ≈1.1 wt%,

CTAT/Water system forms long entangled wormlike micelles and a hexagonal phase forms at much higher concentrations around c ≈ 20 wt% However, the exact nature of the micellar microstructural transitions at this concentration range in CTAT/Water system is still unclear.

According to literature there are mainly two hypotheses for microstructural transitions above the critical concentration; (1) transition from linear micelles to branched micellar networks

[45,46,47,48,49,50,51] or (2) shrinkage of the wormlike chain length [52,53,54]. Based on the theoretical study of Lequeux, formation of branched micelles lowers the zero shear viscosity [62].

Since we only report a maximum in the shear relaxation time, so we can surmise from this theory

27 that the CTAT/Water system does not form signiﬁcant branching beyond the critical concentration.

In order to evaluate the type of the microstructural transition around the critical concentration, we invoke two methods; First, shear rheology and then direct microstructural imaging with TEM.

Evaluating the Micellar Microstructure with Shear Rheology

As indicated in Table 3.1, for concentrations beyond 2wt% the viscoelastic response of the surfactant solutions is best described by a single-mode Maxwell model, thus, using Cates’ theory the microstructural properties such as overall length of the wormlike micelles and the entanglement density can be estimated [12,63]. According to Granek and Cates [63]:

퐿̅ 퐺0 푁 = ≅ ′′ , (3.1) 퐿푒 퐺 푚푖푛 where N is the entanglement number, Le is the average micellar chain length between two entanglement points, L is the average micellar length, Go is the plateau modulus and G”min is the

“dip” in loss modulus at high frequencies where elastic modulus shows a plateau. Also the entanglement length can be linked to the plateau modulus as below [64]:

퐾퐵푇 퐺0 ≅ 9/5 6/5, (3.2) 퐿푒 푙푝 where kB and lp are Boltzman constant and the persistence length of the wormlike micellar chains.

The length of the wormlike micelles and the entanglement numbers were calculated using these equations and the average micellar length and the number of entanglements for the range of concentrations studied in this work are shown in table 3.1. It should be noted that based on the data available in the literature, the persistence length of this wormlike micellar solutions are estimated to be around lp ≈11 nm [61,62].

28 Table 3.1. List of surfactant solutions studied in this work together with measured and/or calculated rheological parameters.

CTAT λ(s) n λE(s) G0(Pa) L(µm) D*(mm) N b (wt%)

0.7 - - 0.039 ± 0.01 − 0.05 [58] - - 5.7 ×10

50.8 - - 0.037 ± 0.02 − 0.056 [58] - - 3.5 ×10

0.9 - - 0.022 ± − 0.06 [58] - - 1.2 ×10 0.002

1 0.38 5 0.034 ± − - 47.5 - 5.5 ×10 0.015

1.1 1.31 5 0.031 ± − - 24.5 - 8.6 ×10 0.005

1.4 5.48 3 0.11 ± 0.006 − - 20.3 - 144

1.7 68.4 3 0.164 ± 0.04 6.44 - 11.5 - 14.5

2 114 3 0.38 ± 0.05 10 - 4.38 - 10

3 60.2 1 2.02 ± 0.02 22.8 5.99 0.96 36.4 8.01

4 30.4 1 3.03 ± 0.3 35.29 5.3 0.55 38.3 6.52

5 18.8 1 1.1 ± 0.05 54.9 4.4 0.42 37.8 5.42

6 12.6 1 - 84.8 3.5 0.29 38.5 -

7 8.5 1 - 101.8 2.8 0.18 36.4 -

It is clear that as surfactant concentration increases both wormlike micellar length and the entanglement number increase but beyond the critical concentration (c ≈2 wt%), wormlike micellar length decreases, while the number of entanglements remains constant. It is worth noting that the system of cetylpyridinium chloride/sodium salicylate (CPCl/NaSal) is known to experience linear to branched structural transition above a critical salt to surfactant ratio [49,51]. For CPCl/NaSal system, as the zero shear viscosity approaches the maximum value, both the length of the wormlike

29 micelles and entanglement number increase and beyond the viscosity peak these two structural properties decrease. However, we obtain different results for CTAT/water systems; first, the

CTAT/water solutions only show a peak in shear relaxation time, not in viscosity peak and second, the number of entanglements doesn’t change for concentrations above the critical concentration (2 wt %). Thus, differences between the structural transitions beyond the critical concentrations in these two systems are expected.

TEM Imaging

Now in order to directly examine whether any micellar branches exist above the critical concentration in CTAT/ water solution, we use a Transmission Electron Microscopy (TEM) technique described below.

Figure 3.3. TEM images of the wormlike micellar solutions for (a) 1.1 wt % and (b) 3 wt % CTAT in de-ionized water.

Figure 3.3 shows the TEM images on two surfactant solutions each on different sides of the critical concentration. Figure 3.3.a is the result for the wormlike micellar solution that contains

1.1 wt% surfactant, which has long wormlike micelles along with short and some spherical micelles. By increasing the surfactant concentration beyond the critical concentration, giant wormlike micelles form a strongly entangled, dense network (cf. Figure (3.3.b)). For the solution 30 containing 3 wt% surfactant, we do not observe formation of any micellar branches. Also, prior

Cryo-TEM images reported on CTAT/water system for c = 4.5 wt % and T = 25 °C in (Figure 3 in [65]) show formation of a very dense network with no sign of branched micelles. Therefore, we can conclude that the drop in relaxation time in inset of Figure (3.2) is clearly not due to the wormlike micellar branching. Our hypothesis is that above the critical concentration (2 wt%), wormlike micelles construct dens networks and gradually shrink in length to transform to ordered nematic phases at higher concentrations.

Capillary Breakup Extensional Rheology

Following fluid characterization, we performed transient extensional rheology experiments. As pointed out earlier, we use a custom-built capillary rheometer for this work. The first step to use this device is to calibrate the device. To do so, we used a series of Newtonian

ﬂuids. The detailed procedure for calibrating the device with Newtonian ﬂuids is given in appendix. Our test results show a good match with the viscosities measured by the commercial rheometer (within the error of 5%).

In this section, we are going to discuss the extensional behavior of the surfactant solutions and examine the sensitivity of this ﬂow to micellar shortening in wormlike micellar solutions. The

filament thinning process for two wormlike micellar solutions at different concentrations is shown in Figure 3.4. At the beginning of the experiments, and shortly after the strike time (40 ms), viscous forces are significant. However, at intermediate times elastic forces become dominant and a thin elastic filament starts to forms.

The evolution of the mid-ﬁlament diameter of the surfactant solutions as a function of time for a range of concentrations tested in this work is shown in Figure 3.5. At low concentrations, diameter decays quickly with time and at intermediate times varies exponentially with time (cf.

Figure 3.5). An interesting result is obtained for solutions with c < 1 wt%, even though no viscoelasticity is detected in small amplitude oscillatory shear tests, an elastic filament forms in these solutions that last on average for couple of seconds in transient extensional experiments. We hypothesize that this is due to formation of elongation induced structures (EIS) that tend to create highly stable ﬁlaments.

Figure 3.5. Filament diameter as a function of time for surfactant solutions along with the best ﬁt to Eq. (3): (a) for dilute surfactant solutions, and (b) for more concentrated solutions.

The ratio of the filament life time to the life time of a Newtonian fluid with comparable viscosity is approximately equal to 103. The term EIS was initially hypothesized by Sachsenheimer et al. [42]. According to the report by Sachsenheimer et al. shear thickening surfactant solutions based on CTAB/NaSal exhibit very long filament life time in experiments with CaBER compared to the Newtonian fluids with comparable zero shear viscosity. These authors hypothesized that shear thickening solutions experience elongation induced structures in experiments with CaBER.

However, these authors did not provide further detail of their experiments or any other evidence to corroborate this hypothesis [42].

At higher concentrations, the short and the intermediate response of the ﬁlament is similar to the regime that exhibit EIS. However, during the later stages, solutions with higher concentration show viscous thinning similar to theoretical predictions of Entov and Hinch for polymeric solutions [36]. As surfactant concentration increases, the ﬁlament life time increases up to c = 3 wt% and decreases for concentrations beyond that. This indicates that the transient extensional rheology is affected by the micellar shortening in this wormlike micellar solution. This latter observation is different from the one reported by Sachsenheimer and co-workers, that a

33 branched wormlike micellar solution generally shows a longer ﬁlament life time than a linear wormlike micellar solution. This is not surprising because the nature of microstructural transition in CTAT/water system is different from that of CPCl/NaSal. Included in Figure (3.5), are also the best ﬁts of Eq. (2.6) to the experimental data. As noted earlier, at the beginning of the experiments,

fluid response is Newtonian and therefore, micellar chains have not stretched enough to result in strain hardening. At later times, as the filament diameter decreases, the wormlike micelles exceed the threshold for coil-stretch transition and a strain hardening is observed. According to Bhardwaj et al., the coil-stretch transition occurs for De > 0.5, [10]. Deborah number is defined as De= λ휀̇, where 휀̇⁡represents the strain rate and is given by:

−2 푑퐷 (푡) 휀̇ = 푚푖푑 (3.3) 퐷푚푖푑(푡) 푑푡

Therefore, using Eq. (2.4) in combination with the above criterion on coil-stretch transition, a critical diameter (D*) can be calculated below which the wormlike micelles undergo strain hardening as:

4휎휆 퐷∗ ≈ (3.4) 7.05⁡ƞ0

D* values are noted in Table 3.1. In Figure 3.6, Eq. (2.6) is ﬁtted to the experimental data for diameters below D*. The transient extensional viscosity can be deduced from the ﬁt to the experimental data.

Figure 3.6.a shows the transient extensional viscosity as a function of Hencky strain for surfactant solutions up to c = 5 wt%. For concentrations higher than 5 wt%, extensional experiments are difficult to perform as the surfactant solutions become solid-like and the experimental results are not reproducible.

Therefore, we do not include transient extensional results for concentrations beyond 5 wt%.

퐷푚푖푑(푡) Total accumulated strain (or Hencky strain) is deﬁned as:⁡휀 = ∫ 휀0̇ 푡 ⁡푑푡⁡ = ⁡ −2ln⁡( ). All 퐷0 surfactant solutions show strain hardening and eventually extensional viscosity reaches an asymptotic value. The maximum extensional viscosity is fairly constant at low concentrations 0.7 wt% < c < 1.1 wt%. As surfactant concentration increases, the maximum extensional viscosity increases until c ≈ 3 wt%, and reduces modestly for concentrations higher than that.

Figure 3.6. b shows the maximum Trouton ratio as a function of surfactant concentrations.

At low concentrations, Trouton ratio is extremely high (≈O(105)). As noted before, at low CTAT concentrations, surfactant solutions do not exhibit any viscoelastic response and no shear relaxation time can be measured in the commercial rheometer. Therefore, it is expected that these solutions consist of un-entangled rod-like (i.e. rigid) micelles in the aqueous solution. This hypothesis is supported by Small Angle Neutron Scattering (SANS) experiments performed by

Gamez-Corrales et al. [58]. These researchers have shown that as surfactant concentration increases, the overall micellar length increases approximately from 100–600°A in the 35 concentration rate of CCTAT = 0.1-0.9 wt%. We note that the persistence length of this micellar system is approximately 110°A [60,61]. Therefore, all of CTAT solutions with C = 0.7–0.9 wt% should form rod-like micelles that are not highly extensible. Our TEM images indicate that such short cylindrical micelles are still present in the surfactant solution with C = 1.1 wt% (cf. Figure

3.3). One can also estimate the ﬁnite extensibility parameter for these solutions using the asymptotic analysis of FENE-P model. In this model, steady state extensional viscosity can be related to the ﬁnite extensibility parameter as [10,66,43]:

1 ƞ − ƞ = 2푛 푘 푇휆푏 (1 − + ⋯ ), (3.5) 퐸,∞ 푠 푎 퐵 2휆휀̇

where ηs, na, kB, λ, b and T are solvent viscosity, surfactant concentration per unit volume,

Boltzmann constant, relaxation time, ﬁnite extensibility parameter and temperature, respectively.

For CaBER experiments, the nominal extension rate is presented as ɛ˙= 2/3λ [10]. This leads to⁡ƞ퐸,∞ − ƞ푠 = 2푛푎푘퐵푇휆푏/2. At low concentrations ηE,∞≫ηs, therefore, we can assume that finite extensibility b for these solutions is: b= 2ηE,∞ /na kB Tλ, . We note that for dilute surfactant solutions where a relaxation time cannot be measured with the rheometer, we use λ= ηp/G, where ηp and G are contribution from micellar viscosity and elastic modulus. Elastic modulus is given by G= nakBTλ [67]. Using the above analysis, we can evaluate the extensibility parameter for dilute solutions with C ≤1 wt% (see Table 3.1). It turns out that the finite extensibility parameters for shear thickening solutions are quite high (b > 5.5×103). As noted above, from structural characterization of a similar system with SANS, the overall equilibrium size of the micelles is quite small (∼600°A). Therefore, micelles themselves do not have the necessary length to undergo such big extensions. The only possible way to generate such big finite extensibilities is through aggregation of cylindrical micelles into big wormlike micelles under extensional flows, i.e. elongational induced structures. Such finite extensibility parameters correspond roughly to

36 wormlike micelles with length scale of couple of micrometer (∼1µm) and beyond. This serves as indirect evidence for formation of EIS in dilute surfactant solutions. We note that the above analysis may serve as a good starting point in better understanding of the EIS in dilute surfactant solutions. This observation opens new avenues for further theoretical and experimental analysis of the EIS phenomenon in dilute surfactant solutions. As surfactant concentration increases, the

Trouton ratio keeps decreasing till concentration of 2 wt%. This trend is consistent with previous reports on transient extensional behavior of the polymeric fluids and other wormlike micellar solutions [13,25,68]. Using a similar physical argument used by Rothstein and co-workers we can rationalize this behavior. As surfactant concentration increases, wormlike micellar chains grow in size and form highly entangled systems. As entanglement density increases, the effective molecular weight between each entanglement point decreases. A lower molecular weight between entanglement points leads to a less extensible filament or equivalently weaker strain hardening behavior. As mentioned above, for concentrations beyond 2 wt%, the Trouton ratio remains fairly constant about Tr≈3. If the above physical argument still holds, the extensional data suggest that the entanglement number should remain constant for concentrations higher than 2 wt%. This is in complete agreement with the number of entanglement points estimated from the linear viscoelastic results (see Table 3.1). Therefore, the transient extensional flow data conform well to the microstructural properties inferred from the shear rheology and also the TEM images. We also note that the reduction of the maximum Trouton ratio above c ≈1.7–2 wt%, is not due to flow induced structures, as suggested by Sachsenheimer and coworkers. At this concentration range the

ﬁnite extensibility parameter (see Table 3.1) are low, which indicates that wormlike micelles are simply experiencing strain hardening. Moreover, in agreement with the literature, we report no shear-thickening behavior for concentrations above 1.1 wt% [58], therefor we can conclude that

37 the extensional rheology is sensitive to the micellar shortening in CTAT/water solutions above c

≈2 wt%.

We have also evaluated the extensional relaxation time of the surfactant solutions by using

Eq. (2.6). Figure 3.7 shows the extensional relaxation time as a function of concentration for a wide range of CTAT concentrations in de-ionized water.

Figure 3.7. Extensional relaxation time as a function of surfactant concentration. Inset shows the ratio of the relaxation times as a function of surfactant concentration.

As indicated in Figure (3.7) the extensional relaxation time is fairly constant at low concentrations. This might potentially be due to formation of elongation induced structures (EIS).

As concentration increases, EIS is no longer reported and the extensional relaxation time increases.

The inset of Figure 3.7 shows the ratio of the extensional relaxation time to the shear deformation relaxation time for all surfactant solutions tested in this work. At low concentrations the ratio of these two relaxation times is less than unity, and decreases as surfactant concentration increases.

We know from rheological results that the linear viscoelastic response at low concentrations is best ﬁt to multi-mode Maxwell models. Therefore, this result is consistent with the phase diagram suggested by Sachsenheimer et al. [42]. However, at higher concentrations, where surfactant

38 solutions follow a single-mode Maxwell model (i.e. when the breakage and reformation processes are much faster than reptation), the ratio of these relaxation times still remains below unity. The latter result is consistent with the results of Rothstein [13] and Yesilata [41] and is in disagreement with the results of Sachsenheimer and co-workers [42]. Therefore, the capillary thinning of the

CTAT/water solutions seems to be controlled by non-linear effects. Examples of such non-linear effects could be flow induced breakage of the wormlike micelles and/or elongation induced structures (EIS). We note that in capillary breakup extensional experiments, wormlike micelles are suddenly stretched from equilibrium structure to the final aspect ratio, and this step strain could potentially lead to flow induced breakage of the wormlike micelles and/or EIS.

Summary

In this work we have determined rheological and structural properties of a series of wormlike micellar solutions based on CTAT/water by using a combination of shear rheology, transient extensional rheology and TEM imaging. In steady shear rheology, surfactant solutions exhibit shear thickening behavior at low concentrations, and form long lasting elastic ﬁlaments in transient extensional experiments. This behavior is reflected in very high (∼O(105)) maximum

Trouton ratios. Also, within this range extensional relaxation time was found to be constant

(λE≈0.03 s). Based on the asymptotic analysis of the FENE-P model, we demonstrated that cylindrical micelles should aggregate and grow dramatically in size to achieve Trouton ratios in the order of 105. This serves as a direct support for formation of EIS in dilute surfactant solutions.

As surfactant concentration increases, the shear relaxation time increases and solutions show a stronger viscoelastic behavior. However, beyond a critical concentration, c ≈2 wt%, the shear relaxation time shows a maximum, while the zero shear viscosity reaches an asymptote. TEM

39 images show a mixture of poly-disperse wormlike micelles for concentrations below the critical concentration, and a dense micellar network with no sign of branched structure, above the critical concentrations. Using the shear rheology, we evaluated the microstructural properties of surfactant solutions. As surfactant concentration increases the length of the wormlike micelles, and the entanglement number increase. However, beyond the critical concentration, micellar chains shrink in size and the entanglement number remains constant.

Therefore, we hypothesize that the transition above the relaxation peak is due to shrinkage of the wormlike micellar chains in a highly dense network. More interestingly, as surfactant concentration increases, the maximum Trouton ratio decreases until it reaches an asymptotic value around Tr ≈3 for concentrations beyond 2 wt%. This ﬁnding is in line with the structural properties inferred from the shear rheology. Additionally, we showed that the ratio of the extensional relaxation time to the shear relaxation time is always below unity for solutions that are either described by a single-mode Maxwell model or a multimode Maxwell model. Finally, Chellamuthu and Rothstein [13] also showed that micellar branching in CTAB/NaOA solutions gives a Trouton ratio of 3 in uniaxial extensional ﬂows. Therefore, we can conclude from these works that the extensional rheology is sensitive to structural transitions from linear wormlike micelles to shorter linear micelles or branched micelles.

Future Work In the next study we will examine three important aspects of extensional flows of wormlike micellar solutions. First, we will investigate the capability of extensional tools to differentiate between Linear-Branched microstructural transitions from and linear to shorter linear systems. Up to date differentiating between different types of microstructural transitions are only attainable through TEM imaging which is very expensive, and time consuming. Second, we will probe the 40 potential non-linear effects that may give rise to a short extensional relaxation time in wormlike micelles. In particular, we will investigate the role of flow induced micellar breakage in uniaxial extensions. Third, we will probe the detail form of flow structure in dilute micellar solutions when

EIS forms. This detailed analysis will be performed by a combination of particle tracking velocimetry, small angle neutron and small angle x-ray scattering.

41 APPENDIX A

CABER CALIBRATION

The experimental setup uses a custom-built capillary break up extensional rheometer which is mounted on an optical table. In figure A.1 the part numbers of pieces found in the experimental are as follows:

a b

Figure A.1. CaBER setup. (a) Top view. (b) Side View

1) DC power source

2) CaBER device

3) Power ON/Off Switch

4) Optical table

5) Compact linear electromagnet (model 6873K3 purchased from McMaster-Carr)

6) Upper plate with diameter D = 3 mm

7) Lower plate with diameter D = 3 mm

8) Micro translation stage (Thorlab)

42 Device Calibration

Prior to running experiments on surfactant solutions, the CaBER device was calibrated using a set of three different Newtonian oils. The CaBER device allows us to estimate the shear viscosity of these fluids which subsequently were compared to the measured values in shear rheometry.

Experimental Protocol

First step to run a successful caber experiment is to find the optimum aspect ratio. Aspect ratio is defined as:

ℎ(푡) Ʌ(푡) = , (1) 2푅0

According to the literature the optimum aspect ratio has been reported to be within the range of 0.5 ≤ Ʌ0 ≤ 1.5. If the aspect ratio is not set within the optimum range then the measurements are limited and effects such as ‘reverse squeeze flow’ (at low aspect ratios Ʌ(t) <<

1) or sagging and bulging of the cylindrical sample (at high aspect ratios) might alter the results

[33].

The pin diameter in our setup is 3 mm, and using this number we are able to calculate initial and final height needed for the experiments. For calibration purposes we used initial aspect ratio as 0.5 and final aspect ratio of 1.30 and set the height accordingly.

Next step would be placing the sample between two end plates. We use a pipette to place the fluid sample between two plates such that the sample maintains a cylindrical shape without sagging or bulging. Then by turning on the DC power, the upper plate moves to its final height and stretches the sample. A high-speed camera records the diameter evolution over time. After filament pinch-off moment, we clean the plates using a wipe and inject fresh sample for next round.

To guarantee reproducibility, we repeat each experiment at least three times.

43 Figure A.2 shows a series of snapshots taken at diﬀerent time windows during the ﬁlament thinning process for silicon oil.

Figure A.2. Transient evolution of the ﬁlament of a silicon oil as a function of time. tbr refers to the breakup time of the ﬁlament.

After running the experiments, we use an open source “ImageJ” software to convert the videos to frame sequences.

To quantify the temporal evolution of the mid-ﬁlament diameter as a function of time we developed a MATLAB script (see Figure A.3). This script enables us to detect the temporal evolution of edges of the filament which will consequently be converted to the filament diameter.

Figure A.3. MATLAB code

45 MATLAB Code Break-down

In this section we provide further detail on the MATLAB script. clc; close all; clear all; list = dir('*.tif'); N = length(list); This section imports the images from the folder. for K = 1:N images{K} = imread(list(K).name); BW = edge(images{K}); BWcrop{K}=BW(1:480,280:290); K end

This loop is being used to convert images to black and white and do the edge detection, then crop the images based on the dimensions given (see Figure A.4).

Figure A.4. Example of edge detection code applied on an image

46 for J=1:N tohi{J}=[]; for i=1:10 rose{J}=find(BWcrop{J}(:,i)); L{J}=length(rose{J}); tohi{J}(1,i)=rose{J}(1); tohi{J}(2,i)=rose{J}(L{J}); end upper_edge{J}=max(tohi{J}(1,:)); lower_edge{J}=min(tohi{J}(2,:)); D{J} = abs(upper_edge{J} - lower_edge{J}); I(1,J) = D{J}; J end

Next we define an empty matrix and detect the upper edge and lower edge of the filament within the cropped area (see Figure A.5) and retrieve the numbers in the matrix, by subtracting those numbers diameter of the filament can be obtained.

Upper Edge

Lower Edge

Figure A.5. Detection of upper and lower edge location based on pixels

47 I = (3/380).*I; x = linspace(0,N/3,length(I)); loglog(x,I,'o'); p=1 for w=1:length(I) if I(1,w)<3.5; Y(1,p)=I(1,w); X(1,p)=w./3; p=p+1; end end xlabel('Time[s]') ylabel('Diameter[mm]')

Finally, we need to adjust both time and length scales. First line is scaling the pixels of the pin diameter in the image to the actual pin size. To adjust the time, we use “Linspace” and insert the capture ratio we used to record the video. This allows us to run the code and obtain the diameter vs. time plot in real time and space dimensions.

Figure A.6. Example of a Diameter Vs. Time plot by MATLAB 48 Newtonian Oils results

To calibrate the CaBER device, we chose three Newtonian Oils; Vegetable oil (Oil I), Olive oil (Oil II) and silicon oil (Oil III). We first perform steady shear experiments on these fluids and obtain the shear viscosity (ƞs). See Table A.1.

The filament thinning process in Newtonian fluids is governed by gravity, surface tension, viscous forces and/or inertia. Gravity induces an asymmetry between the top and bottom half of the fluids (sagging) and can be estimated by a Bond number. Bond number is defined as:

휌𝑔푅2 ⁡⁡⁡⁡퐵표 = 0, (2) 휎 where ρ, Ro and σ are ﬂuid density, radius of the ﬁlament and the surface tension.

Figure A.7. Filament diameter as a function of time for three Newtonian ﬂuids. Lines indicate the linear ﬁt to the experimental data obtained by equation (2.4).

49 Gravitational eﬀects become negligible if the Bond number is less than unity. For all Newtonian

fluids tested the Bond number for fluids at rest is smaller than unity. Therefore, the gravitational effects are negligible. This is evident in the snapshots provided in Figure 2. If gravitational effects are negligible the filament thinning process progresses linearly. According to McKinley and

휌𝑔(2푅 )2 Tripathi [35], below a critical Bond number 퐵표 = 0 ≈ 0.1, diameter decays linearly with 휎 time. In our experiments, the exact Bond numbers below which the ﬁlament of the three Newtonian

fluids scales linearly with time are given in table A.1. We note that the listed Bond numbers are less than the suggested one by McKinley and Tripathi [35]. Fitting equation (2.4) to the experimental data gives us the viscosity of the Newtonian fluids (ηc). The estimated viscosities for these three Newtonian fluids are listed in table A.1. It is clear that viscosities obtained with the capillary device are very close to the measured values in the rheometer. Therefore, we can conclude that our device performs correctly.

Table A.1. List of Newtonian ﬂuids together with their physical and rheological properties.

Fluid Ƞs (Ps.s) Ƞc (Ps.s) 흈(mN/m) Bond Number, Bo

Newtonian

Oil I 0.04 0.042 30 1.5×⁡10−2

Oil II 0.072 0.072 35.5 10−2

Oil III 0.975 0.975 23.5 10−2

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55 BIOGRAPHICAL SKETCH

EDUCATION

Master of Science in Chemical Engineering - Complex Fluids Jan 2017 - present Florida State University, Tallahassee, FL, USA

Diploma in Advanced Academic English May 2016 - Sep 2016 Centre for Intensive English Studies, Florida State University, Tallahassee, FL, USA Master of Science in Chemical Engineering - Food Technology Sep 2013 - Sep 2015 Islamic Azad University, Tehran Science and Research branch, Tehran, Iran

Bachelor of Science in Chemical Engineering - Food Technology Sep 2008 - Dec 2012 Department of Chemical Engineering, College of Engineering, Islamic Azad University, Tehran Science and Research branch, Tehran, Iran

PUBLICATIONS

JOURNAL PAPERS

 Rose Omidvar, Alireza Dalili, Ali Mir, and Hadi Mohammadigoushki, “Exploring sensitivity of the extensional flow to wormlike micellar structure”, J. of Non-Newtonian Fluid Mechanics”, Journal of Non-Newtonian Fluid Mechanics. 252 (2018), 48-56.  Rose Omidvar, Shijian Wu, and Hadi Mohammadigoushki. "Detecting wormlike micellar microstructure using extensional rheology." Journal of Rheology 63.1 (2019): 33-44.  Mir Masoud Seyyed Fakhrabadi, Navid Khani, Rose Omidvar, Abbas Rastgoo, “Investigation of elastic and buckling properties of carbon nanocones using molecular mechanics approach”, Computational Materials Science, 2012, 61, 248-256.

CONFERENCE PAPERS

 Rose Omidvar and Hadi Mohammadigoushki, " Extensional Rheology: A Microstructural Probing Technique for Living Polymers", The Society of Rheology, 90th Annual Meeting, October 2018, Houston, Texas, USA.  Rose Omidvar and Hadi Mohammadigoushki,” Filament dynamics in a salt-free viscoelastic surfactant solution”, The Society of Rheology, 89th Annual Meeting, October 2017, Denver, CO/American Physical Society Meeting, March 2018, Los Angeles, CA.