High Resolution Measurements of Viscoelastic Properties of Complex Biological Systems Using Rotating Optical Tweezers

Transcription

1 High Resolution Measurements of Viscoelastic Properties of Complex Biological Systems Using Rotating Optical Tweezers Shu Zhang BSc A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2017 School of Mathematics and Physics

2 ii Abstract The aim of this thesis is to investigate the viscoelastic properties of systems on the microscale at extremely low sample volumes and in highly limited spaces with high temporal resolution. A method based on both passive and active rotational tweezers is presented, which enables measurements of viscoelasticity with high resolution and within a broadband frequency range. Cellular mechanics and rheological properties play an important role in biological processes (e.g., transport of intracellular organelles by molecular motors, cellular processes, endocytic trafficking, axonal retrograde transport in neurons and secretory pathways). All these activities occur in cytoplasm which behaves both as an elastic solid and as a viscous fluid, and is considered viscoelastic. Therefore, the quantitative characterization of viscoelasticity in cells can help us to understand cellular mechanisms. This thesis begins by introducing optical tweezers elucidating transfer of both linear and angular momentum of light and describing methods to apply and measure optical torques on birefringent particles. Microspherical vaterite particles are synthesized for our experiments using a method developed and refined in our lab. These particles are composed of the calcium carbonate mineral vaterite and are birefringent. Next I describe both theoretical and experimental methods of determining viscoelasticity by measuring the angular diffusion of displacement of a trapped vaterite particle for a sufficiently long time and transient angular displacement of this probe flipping between two overlapped optical traps (with a fixed angle φ 0 between two linearly polarised beams). I then show experimental results of different concentrations of viscoelastic fluids and demonstrate a good agreement with conventional rheological technique. This method is applied to the analysis of biologically relevant fluids of tear film coated on a contact lens. The flexibility and sensitivity of rotating optical tweezers in quantitative studies inside biologically relevant environment is shown. Due to the complicated architecture and confined space of cells, measurements of the probe are known to be strongly influenced by boundary conditions. Much knowledge

3 iii has been obtained about the influence of flat walls. However, the effect of non-linear dynamical systems at microscopic scales on rotational motion of micro-objects has not been studied. I describe studies of the drag torque on a vaterite particle rotating near the three-dimensional nano-printed walls with different curvatures. In the final chapter, rotating optical tweezers are used to assess the local viscoelasticity of fluid inside a liposome that represents a simplified model of a biological cell. The measurement of the viscoelasticity of fluid inside a liposome lays the ground work of future studies in cellular environments.

4 iv Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis.

5 v Publications during candidature Lachlan J. Gibson, Shu Zhang, Alexander B. Stilgoe, Timo A. Nieminen, and Halina Rubinsztein-Dunlop, Active rotational and translational microrheology beyond the linear spring regime, Phys. Rev. E 95, (2017). Ann A.M. Bui, Alexander B. Stilgoe, Isaac C.D. Lenton, Lachlan J. Gibson, Anatolii V. Kashchuk, Shu Zhang, Halina Rubinsztein-Dunlop, Timo A. Nieminen, Theory and practice of simulation of optical tweezers, Journal of Quantitative Spectroscopy and Radiative Transfer 195, (2017). A. V. Kashchuk, A. A. M. Bui, S. Zhang, A. Houillot, D. Carberry, A. B. Stilgoe, T. A. Nieminen, and H. Rubinsztein-Dunlop. Optically driven rotating micromachines, chapter 4, pages Elsevier, Shu Zhang, Lachlan J. Gibson, Alexander B. Stilgoe, Itia A. Favre-Bulle, Timo A. Nieminen, and Halina Rubinsztein-Dunlop, Ultrasensitive rotating photonic probes for complex biological systems, Optica 4, (2017) Publications included in this thesis Shu Zhang, Lachlan J. Gibson, Alexander B. Stilgoe, Itia A. Favre-Bulle, Timo A. Nieminen, and Halina Rubinsztein-Dunlop, Ultrasensitive rotating photonic probes for complex biological systems, Optica 4, (2017)

6 vi Contributor Shu Zhang Lachlan J. Gibson Alexander B. Stilgoe Itia A. Favre-bulle Timo A. Nieminen Halina Rubinsztein-Dunlop Statement of contribution Conceived the project (10%) Performed the experiments (80%) Results analysis and interpretation (70%) Wrote the paper (30%) Conceived the project (10%) Theoretical and numerical calculation (80%) Results analysis and interpretation (30%) Wrote the paper (5%) Conceived the project (10%) Wrote the paper (20%) Performed the experiments (20%) Wrote the paper (5%) Conceived the project (10%) Theoretical and numerical calculation (20%) Wrote the paper (20%) Conceived the project (60%) Wrote the paper (20%) Contributions by others to the thesis My supervisors Prof. Halina Rubinsztein-Dunlop, Dr Timo A. Nieminen and Dr Alexander B. Stilgoe have provided support in developing of the initial concept of this project as well as the analysis and interpretation of data, and revision of written material. Chapter 3 describes the results that have been published in Optica. I performed experiments and the analysis of the experimental results. Lachlan J. Gibson developed the theoretical model and analysis of some experimental data. Itia A. Favre-Bulle helped with performing experiments and collecting samples. In Chapter 4, the results in this chapter are in preparation for publication in peer reviewed international journal. I produced samples using Nanoscribe and performed

7 vii experiments. David Carberry helped with building the experimental set up and making samples using Nanoscribe. Lachlan J. Gibson developed some theoretical analysis. We both performed the analysis of the experimental results. Statement of parts of the thesis submitted to qualify for the award of another degree None.

8 viii Acknowledgements To be honest, I have not realized my PhD has come to the end until now. This reminds me of how thankful I am to people who helped and supported me over the past few years. I would like to thank my principle supervisor, Prof. Halina Rubinsztein-Dunlop, for her consistent guidance and advice on my project, and help and encouragement in my life. Every time at our weekly meeting, her physical intuition and outstanding knowledge to approach problems I met lead me on the right path. I am also grateful that she always tries to explain her idea with simple words so that I can understand and listen carefully to her ideas and develop my own. She cares about her students and is the person we can lean. There is a saying that goes, Lighthouses don t go running all over an island looking for boats to save; they just stand there shining. Her enthusiasm and generosity shine like a lighthouse for me to follow and inspire me for the next adventure. I would like to thank Dr Alexander Stilgoe, Dr David Carberry and Dr Timo Nieminen for their assistance and patience in sharing their knowledge and experience of optical tweezers. Alex and David helped me with building experimental set up and Labview programming step by step. Timo has helped me gain a deeper understanding of the theoretical aspects of project. I would like to thank all previous and current members of optical micromanipulation group during these years. Thanks to Prof. Norman Heckenberg, Dr Daryl Preece, Dr Vincent Loke, Itia Favre-bulle, Ann Bui, Anatolii Kashchuk, Lachlan Gibson and Isaac Lenton. I enjoyed my time in the past years in this group. I feel so lucky to be one of the group members. I would like to thank Itia, Swaantje and Ann for sharing experiences and supporting each other for our PhDs. We all did a very good job. Thanks for encouraging me in many ways when I felt frustrated. I will not and never forget those lovely moments of

9 ix dining out, girls talk, conferences, celebrations and crochet and so on with you girls. I would like to thank my new friends I met in Physics Department, Changqiu Yu, Daiqin Su, Beibei Li, Yimin Yu, Xin He and Xiuwen Zhou. Thank you all for organizing after-work activities like hiking, BBQ, fishing and cooking home food. I would also like to thank my BFF, Mia, Vivi and Dongdong. We have known each other since childhood. We ve been through so much, good things, horrible things, and you ve brought so much joy and laughter to my life. Now we live and work in different places and fields, however, no matter what happens we will always keep in touch and support each other. Last but not the least, I would like to thank my family for their unconditional love, especially my parents. I never left home since I was born. Without their support, I could not make up my mind to study in another country. They are my backbone, always are.

10 x Keywords Optical tweezers, Angular momentum, Microrheology, Viscoelasticity, Biomaterials, Hydrodynamics, Wall effects, Liposomes. Australian and New Zealand Standard Research Classifications (ANZSRC) ANZSRC code: , Classical and Physical Optics, 70% ANZSRC code: , Biological Physics, 30% Fields of Research (FoR) Classification FoR code: 0205, Optical Physics, 70% FoR code: 0299, Other Physical Sciences, 30%

11 Contents List of Figures xv List of Tables xix List of Abbreviations xxi 1 Introduction 1 2 Optical tweezers Optical trapping Introduction Basic optical tweezers configuration Rotational Optical tweezers Angular momentum of light Optical torque and measurement of optical torque in optical tweezers Optical trapping of vaterite particles Vaterite particles Applications of birefringent particles in optical tweezers Microrheology with rotational optical tweezers Microrheology of complex fluids xi

12 xii CONTENTS 3.2 Microrheological methods Introduction Passive microrheology Active microrheology Microrheology with optical tweezers High resolution, wide-band microrheology with optical tweezers Experimental set up Tracking laser (HeNe beam) detection Trapping laser (IR beam) detection Theoretical model Results and discussion Celluvisc Tear film Effects of confinement Introduction Early work Wall design and fabrication Wall effects on a rotating vaterite particle Experiments Plane and curved walls Cylindrical and spherical walls Conclusions Biomimetic cell model liposomes Introduction Liposome formation Lipid-film hydration Electroformation Microfluidic device Water in oil droplet transfer

13 CONTENTS xiii 5.3 Measurements inside liposomes Discussion Conclusion Summary Future Outlook References 91

14 xiv CONTENTS

15 List of Figures 2.1 A trap formed from two counter-propagating laser beams The single-beam gradient force optical trap Forces on a particle in the ray optics regime Trap stiffness of optical tweezers Schematic of a basic single optical tweezers system Helical phase structures of LG beam with modes l of 2, 3 and 20 respectively Birefringence and polarisation of light Optical trapping of a microscopic glass cylinder Schematics of experimental approach for measurement of optical torque resulting from the transfer of spin angular momentum The size of vaterite particles varies with the growing time A schematic diagram of the vaterite particle under a linearly polarised beam and a circularly polarised beam Brownian motion of a microscopic particle in a fluid Experimental apparatus for microrheological measurements based on rotating optical tweezers Camera images showing the schematic of the experimental procedure Angle calibration with HeNe signal Trapping laser detection xv

16 xvi LIST OF FIGURES 3.6 Experimental autocorrelation functions Non-linear driving torque Experimentally confirmed nonlinear optical torque for flips in step (II) Minimising the effects of Brownian motion by averaging many flips Normalisation for each flip A comparison between analysis methods in both viscous and viscoelastic fluids A comparison between analysis with different number of flips Typical time recorded of the trajectory and optical torque of a vaterite particle flipping between trap 1 and trap Storage (G ) and loss (G ) moduli vs frequency of a solution of water, 25% and 100% w/w of Celluvisc in water A comparison of the complex shear modulus of tear films of two subjects The variation of complex shear modulus and storage modulus vs. frequency of tear films during the day Experimentally measured drag coefficients as a function of the separation of the the sphere from the surface Wall effects of a sphere rotating near a plane wall Wall effects of concentric spheres Wall effects for a infinitely long circular cylinder The design of the wall for wall effects experiments D laser lithography system of Nanoscribe Regions of one-photon and two-photon polymerisation D preview of structures of the plane wall, curved wall and cylindrical wall assembled on DeScribe Schematic of the 2PP technique of Nanoscribe galvo-mode for fabrication of micron sized walls out of the resin Experimental setup for measuring wall effects Measurements near a plane wall

17 LIST OF FIGURES xvii 4.12 Measurements near curved walls Measurements in the centre of cylindrical walls Theoretical analysis of the wall with a structure like a ring Theoretical simulation of the equator plane of the particle is offset from centre of the ring wall Measurements of wall effects on a particle moving from centre towards the edge of the cylindrical wall Wall effects on a particle rotating inside a liposome Schematics of a cell and a cell model-liposome Representation of the process of producing liposomes based on the lipidfilm hydration Liposmes made by lipid-film hydration method Schematic view of electroformation method Fluorescence images of liposomes made by the electroformation method Schematic illustration of microfluidic assembly line Images of liposomes made by microfluidic method Schematic representation of key steps for producing liposomes with water in oil droplet transfer method Images of liposomes made by water in oil droplet transfer method Active microrheological measurements inside liposomes Errors in particle tracking microrheology experiments

18 xviii LIST OF FIGURES

19 List of Tables 4.1 A comparison of wall effects from Equation (4.2) with exact solutions for a particle rotating at the centre of an infinite cylinder xix

20 xx LIST OF TABLES

21 List of Abbreviations The following list is neither exhaustive nor exclusive, but may be helpful. 2PP Two-photon polymerization 3D Three-dimensional AC Alternating Current AFM Atomic Force Microscope AOM Acousto-optic Modulator CaCl 2 Calcium chloride CAM Camera CMC Carboxymethylcellulose DDAB Dimethyldioctadecylammonium bromide DiIC 18 1,1 -Dioctadecyl-3,3,3,3 -Tetramethylindotricarbocyanine Iodide DiOC 18 3,3 -dioctadecyloxacarbocyanine perchlorate DPPC Dipalmitoylphosphatidylcholine Egg PC L-α-phosphatidylcholine FEM Finite Element Method xxi

22 xxii LIST OF ABBREVIATIONS GWL General Writing Language HeNe Heliumneon IR Infra-Red ITO Indium Tin Oxide K 2 CO 3 Potassium carbonate KOH Potassium hydroxide LED Light-emitting Diode LG Laguerre-Gaussian MgSO 4 Magnesium sulfate MSD Mean Square Displacement NA Numerical Aperture NAPAF Normalised angular position autocorrelation function PBS Polarising Beam Splitter PBS Phosphate Buffered Saline (Chapter 5) PGMEA Propylene Glycol Methyl Ether Acetate PSD Position Sensitive Detector QPD Quadrant Photodiode Detector SLM Spatial Light Modulator UV Ultraviolet W/O Water-in-oil

23 1 Introduction The first optical traps were developed by Arthur Ashkin in 1970 [3, 4]. In these experiments, single-beam traps used the forces of radiation pressure from a laser beam to accelerate freely suspended particles, whereas two-beam traps were based on counter-propagating beams to trap particles. Later, in 1986, Ashkin and co-workers found that using the single-beam gradient force alone is sufficient to create an optical trap which has been labeled optical tweezers [6]. They used a single highly focused laser beam to trap a transparent particle in three dimensions. This simple geometry for optical trapping and optical micromanipulation has found wide application and led the way to further novel research in which exciting and new aspects of light-matter interactions were studied. The ability to use optical tweezers for measurements of pn forces and nm displacements, with high temporal resolution, down to µs is of particular interest to biophysical fields. Optical tweezers rapidly became 1

24 2 INTRODUCTION a key tool in a wide range of biophysical experiments and within microdevices. For example, using optical tweezers enabled trapping and manipulating single cells [7], studies of chromosome movement on the mitotic spindle [11], sorting cells [72] and stretching cells in microfluidics [46]. In addition, the non-invasive nature of optical tweezers force microscopy makes it a powerful tool in microrheology, where the viscous and elastic properties of a medium can be studied through the influence that the liquid has on the motion of an embedded probe particle [16, 138]. Optical tweezers have been successfully used to determine the fluid viscosity with high accuracy. Such measurements require either moving the particle with known velocity through the viscous fluid [122], or analysing the autocorrelation function of the Brownian motion signal of the particle within the optical trap [10, 99]. Apart from simple fluids, the use of optical tweezers have also been extended to studies of the viscoelastic properties of complex fluids [99, 100]. It is based on the observation of the free or driven time-dependent trajectories of tracer particles suspended in the fluid [82, 108, 112]. Viscoelasticity (viscosity and elasticity) is a basic physical property of most biomaterials which plays an essential role in the biological function of cells and tissue including bones, connective tissue and synovial fluids [58]. All materials have rheological properties, which means that their mechanical behaviour exhibits features that can be characterised by both the classical theory of elasticity and by simple Newtonian fluid mechanics [138]. Solids store mechanical energy and are elastic, whereas fluids dissipate mechanical energy and are viscous. In general, most materials such as common foods and the cytoplasm of eukaryotic cells are neither simple liquids nor simple solids; these complex fluids which both store and dissipate energy are viscoelastic, with the relative proportions depending on frequency [61]. Viscoelastic properties of complex fluids can be determined from thermal motion driven by the fluid s molecules. On the other hand, for some materials which are dynamic objects, viscoelastic properties depend on the diffusive-like motion driven by active forces. These diffusive-like movements are distinct from movements arising from thermal diffusions. For example, the cytoplasm of living cells is no longer a static medium due to a variety of

25 3 forces created by molecular motors and other enzymatic activity [48]. Active physical properties of the cytoplasm control key cell functions such as cell adhesion and the movements of cellular organelles. Rheology is the study of the viscoelasticity of materials. Viscoelasticity affects the dynamics of cellular shape changes during locomotion and interacellular transport of vesicles or phagosomes by molecular motors [39]. Such biological materials are normally difficult to obtain in large volumes or they exhibit directional properties with nano-scale structure. Thus, this class of problems require new methods to probe small volumes and extend the probe frequency range. Microrheology is a branch of rheology which has the same principles as conventional bulk rheology, but aims to measure the time-dependent response of the fluid at the micrometre or sub-micrometre level [73]. Microrheological techniques can be classified as either passive or active, depending on whether they monitor the thermal motion or the driven motion, respectively, of probe particles suspended in the fluid. The passive techniques are based on tracking the particles motion such as particle tracking video microscopy [81], atomic force microscopy [76], diffusing wave spectroscopy [93] and optical tweezers [100]. Active microrheology techniques involves driven and active manipulation of small particles, for example, magnetic tweezers [131], optical tweezers [100] or atomic force microscopy [75]. Magnetic tweezers have been used in active microrheology as they can explore higher force levels than optical tweezers [16]. However, there is an intense interest in finding new active microrheology techniques based on optical tweezers due to their versatility and ability to precisely determine position of beads in locations of interest. In addition, the development of special optical devices such as the spatial light modulator has contributed to the expansion of holographic optical tweezers apparatus and applications [111]. This allows optical tweezers to simultaneously manipulate multiple particles so that different interactions between the probes and the surrounding medium can be determined.

26 4 INTRODUCTION With continuous development of optical tweezers, another great application of this technology is rotational microrheology as this opens up exciting new opportunies for locally accessing time-dependent properties of fluids on microscale. A new area which emerged in 1995 is optical rotation. Rubinsztein-Dunlop and co-workers demonstrated that by using spatially modified laser beams angular momentum could be transferred from the light to the particle, which causes the particle to spin [51]. This advance in optical tweezers not only answered fundamental questions about the nature of light but also allowed for the creation of optically driven micromachines. Apart from force, torque is a further parameter of biological relevance, as witnessed by its role in biological process, e.g., DNA replication and transcription [68], bacterial rotational motors [134] and sperm motility [104]. New techniques used for the quantification of optical torques have expanded the potential applications of optical tweezers in biology and have greatly contributed to our biophysical understanding of complex systems. The use of optical tweezers to measure the optical torque applied on the particle was developed by Rubinsztein-Dunlop s group in 2001 and then demonstrated experimentally by them in 2003 [14, 89]. In particular, one technique for rotation makes use of the change in polarisation state of light passing through a birefringent microsphere transfering spin angular momentum of light [41, 95, 123]. For instance, circularly polarised light has spin angular momentum which can be transferred to a birefringent material upon propagation of the beam through the material. When trapped in a viscous fluid with a circularly polarised laser beam, these particles spin at a constant speed as the optical torque is balanced by the hydrodynamic rotational viscous drag. The viscosity can be measured by optically applying torque to the particle and optically measuring both torque and the particle s rotation rate [15, 42, 94]. In these experiments it is assumed that the fluid is purely viscous. In terms of complex fluids, techniques similar to common techniques measuring translational diffusion of the probe microsphere, can be used to determine the viscoelastic properties of the medium by the rotational diffusion [23]. Such a microrheometer, in which rotational

27 5 Brownian motion of a birefringent microsphere within an angular optical trap is observed, was developed by Bennett et al. in 2013 [9]. One advantage of rotational techniques over linear approaches lies in the nature of hydrodynamic interactions between the probe particles and any nearby boundaries. The influence of this coupling is dramatically different for translation and rotation so that the rotational correction is required only at much shorter separations than the translational correction [62]. The extremely short range nature of the rotational case means that, for example, a 10% increase in drag coefficient occurs at a distance of five radii for the translation of a sphere parallel to the surface, but only within one radius of the surface for rotational motion [62]. This allows the viscoelasticity of very localized volumes, down to femtoliter volumes, to be measured. Therefore, applying vaterite particles into a sample allows this approach, based on rotational optical tweezers, to probe the properties of picolitre fluid volumes such as internal biological cell environments, biomedical samples and lab-on-a-chip devices. However, when measuring the viscoelastic properties of complex fluids using diffusion techniques, the results are limited to the material s high end of the frequency response, discarding the essential low frequency response, or supported by other techniques but without showing a clear overlapping region [113]. In this thesis, the work on rotational microrheology is substantially extended. The project aims to combine rotation and active-passive techniques to study the local viscoelsticity over a wide frequency range, which demonstrates great promise for future applications to determine the spatial-temporal properties of hard to obtain and vital biologically relevant and complex fluids. This thesis contains three major parts, which have a common goal of studying microrheology in biological systems such as inside living cells: 1. A microrheometer based on rotational optical tweezers is introduced to access

28 6 INTRODUCTION local viscoelastic properties of complex fluids over a wide frequency range. Measurements of eye drop fluid (Celluvisc) are demonstrated using our method. Further, this method is directly applied to measure the material properties of eye tear film coated on a contact lens. (Chapters 2 and 3) 2. A study of the wall effect on the rotational motion of a probe particle near curved surfaces is carried out. The existing close boundary walls are shown to have an influence on the rotational motion of the particle and thereby influence the microrheological measurements in confined geometries. We also show how the wall effects change with the change in curvature of walls. The effect of cylindrical and spherical walls on the motion of the particle is also investigated. (Chapter 4) 3. We study a bio-mimetic model of living cells. Liposomes, sphere-shaped vesicles consisting of one or more phospholipid bilayers are used, as a model of living cells to study the properties of fluids inside them. The direct measurement of a vaterite particle inside a liposome is demonstrated. (Chapter 5)

29 2 Optical tweezers 2.1 Optical trapping Introduction With the development of the laser, it became possible to use the small momentum of light to exert measurable forces on a small particle. In 1970, Ashkin first demonstrated the trapping of a high refractive index particle in water with counter-propagating laser beams and later stable levitation of 20 µm glass beads in vacuum/air with a single laser beam [3, 4]. The trapping of particles is due to two types of optical forces, referred to as the gradient force, which draws the particle towards the axis of the beam, and the scattering force, which pushes the particle away from the light source. 7

30 8 OPTICAL TWEEZERS In Ashkin s original experiment, as shown in Figure 2.1, the trap relies on both gradient and scattering forces for its operation. When the particle is centred (Figure 2.1 a), the net force is zero as the scattering for both beams is equal. When the particle is moved vertically (Figure 2.1 b), the gradient force provided the vertical confinement. If it is displaced about the the beam axis (Figure 2.1 c), one beam is scattered more than the other beam, and the larger scattering force pushes the particle back to the mid-point. FIGURE 2.1: A trap formed from two counter-propagating laser beams. a, When the particle is centred, light is scattered symmetrically about the beam axis so the net force vanishes. b, If the particle is displaced vertically, the light is deflected by the particle. The particle changes the light s momentum and feels a restoring force. c, If the particle is displaced laterally, one beam is scattered more than another beam. So larger scattering force pushes the particle back to the centre of the trap. Sixteen years later, in 1986, the single-beam gradient force optical trap which is commonly referred to optical tweezers, was first achieved by Ashkin and co-workers [6]. It is the more elegant method since a small object can be trapped and manipulated in three dimensions with the greatest ease and precision. Before, it was thought impossible as the scattering of light on the particle always pushed the particle in the direction of the beam s propagation. The idea was to use a high numerical aperture objective (NA=1.25 in this experiment) to create a tightly focused laser beam so that a gradient force would be larger than the scattering force and other acting forces such as thermal force and gravity (Figure 2.2). The laser beam is tightly focused into a

31 2.1 OPTICAL TRAPPING 9 transparent sphere with a higher refractive index than its surroundings. The balance between scattering and gradient forces makes the particle stably trapped at the equilibrium point which is shifted slightly downstream of the laser focus. For instance, if the laser beam comes from below (as is often the case in optical trapping setups), the equilibrium is usually above the focus point as the scattering force will act in the direction of the laser propagation. If the particle is displaced sideways (Figure 2.2 c), which causes the beam to be deflected, for conservation of momentum a restoring force exerted on the particle makes it travel back to the centre of the trap. When the particle is displaced vertically (Figure 2.2 d), this causes the beam either to be focussed or defocussed. The momentum of the beam thus changes in the axial direction; the restoring force acts to reposition the particle upwards or downwards. These forces are therefore always pointing towards the equilibrium position. FIGURE 2.2: The single-beam gradient force optical trap. a, b, Transfer of linear momentum from the laser beam to the particle forces the particle to the centre of the trap which is slightly above the focus of the laser beam. c, If the particle is displaced to the left or right, it causes the laser beam to be deflected which changes its momentum. For conservation of momentum of light, a restoring force makes the particle move back to the centre of the trap. d, If the particle is displaced vertically, it causes the beam to be either focused or defocused by the particle. Similarly, a restoring force acts on the particle. These are two of the most common types of traps. Since the scattering force from two beams cancels in the centre of the trap for the counter-propagating beams trap as shown in Figure 2.1, the counter-propagating beams trap is able to hold more strongly

32 10 OPTICAL TWEEZERS scattering objects than single-beam traps. But as the two beams are relatively weakly focused compared to the tightly focused single beam, the trap stiffness is very low for this counter-propagating beams trap. Optical tweezers rapidly became a key tool used in many areas of research. As mentioned above, in optical tweezers, forces are generated by the transfer of linear momentum from the beam to the trapped particle. To describe these forces, three approaches are used depending on the ratio of particle size to the wavelength of the trapping beam. For a particle that is much smaller than the wavelength of the trapping beam, the particle can be approximated as a point dipole [34]. The electric field E of the trapping beam induces an oscillating dipole with polarisation p = αe, where α is the particle s polarisability. This, in conjunction with the optical Lorentz force, gives the total optical force acting on the particle. As was mentioned above, the force can be separated neatly into two terms, one proportional to the intensity gradient (the gradient force) and one in the beam propagation direction (the scattering force). In addition, both the gradient force and the scattering force scale linearly with the light intensity. For a spherical particle of diameter much greater than the wavelength of the trapping beam, a ray optics model can be applied to describe optical forces on particles. Within the ray optics approximation, the trapping beam is considered to propagate as rays, so the deflection of light occurs through refraction at interfaces as shown in Figure 2.3. Each photon of the light carries linear momentum p = h/λ, where h is Planck s constant and λ is the wavelength of the laser beam. Reflection and refraction of light by a transparent object causes changes in linear momentum, and due to the conservation of momentum, there is a corresponding restoring force acting on the object. Refraction at the interface in the ray optics model is given by Snell s Law: sinθ 1 / sinθ 2 = n 2 /n 1, where θ 1 is the angle of incidence, θ 2 is the angle of refraction. n 1 is the index of refraction of the incident medium and n 2 is the index of refraction of the refractive medium as shown in Figure 2.3 a. Figure 2.3 b,c shows the path of rays that can generate lateral and axial reaction forces on a spherical particle, respectively. For optical trapping, the interface is at the

33 2.1 OPTICAL TRAPPING 11 surface of the trapped object. Therefore by considering the deflection of each ray of light passing through the particle, the change in momentum of light can be found and the force that opposes it. When the spherical particle has a greater refractive index than its surrounding, the optical force is found to oppose small displacements so that the particle can be trapped at the equilibrium position. FIGURE 2.3: Forces on a particle in the ray optics regime. The total momentum is conserved by an optical force on the particle which opposes the change in linear momentum of light. The particle refractive index is greater than the surroundings which results in a trapping force with an opposite direction of the small displacement of the particle. a, Light at an interface between two medium with refractive index of n 1 and n 2. b, Particle is below the focus and is being moved upward. c, Particle is to the side of the focus and is being pulled back to the focus of the beam. Both rays shown are deflected to the right, so the beam gains momentum to the right and the optical force acts to the left. For a particle with size comparable to the wavelength of laser light, these two methods mentioned above are invalid. In this case if the particle is a homogeneous isotropic sphere, the force can be calculated using the Lorenz Mie scattering theory [71]. In the case of a nonspherical particle and nonparaxial beams the T-matrix method is used, which finds the force based on the relationship between the scattered wave and incident wave [87, 88]. Nieminen and co-workers have developed a software package for performing these calculations [90]. Using an optical tweezers system, we can trap objects ranging from tens of nanometres to tens of micrometres in size with pn range of forces. Moreover, it has been shown that for small displacements from the centre of the optical trap the force is proportional to the displacement [101, 106]. It means the optical force can be treated

34 12 OPTICAL TWEEZERS as a spring force as in Hooke s law, F = κx, where F is the restoring force and κ is the stiffness of the spring (trap), which represents the optical stiffness. The stiffness depends on the wavelength and power of the laser, the size and refractive index of the particle, and the refractive index of the surrounding medium. Figure 2.4 shows theoretically calculated trap stiffnesses using the point dipole approximation, ray optical model and Lorenz Mie theory discussed above [77]. Lorenz Mie scattering methods are the most complicated and are suitable for a wide size ranges of particles. The other two are only valid for very small or very large particles. FIGURE 2.4: Trap stiffness of optical tweezers as a function of the ratio of particle size to the wavelength of the trapping beam using theoretical models of point approximation (blue), ray optics (red) and Mie theory (black) [77] Basic optical tweezers configuration After considering the optical force exerted on the trapped objects, we now show an example of a simple optical tweezers system with a single beam gradient force trap built in experiments (Figure 2.5). With increased development of technology, optical tweezers have been developed for many different kinds of specific applications, but there are some basic features common to most designs. As shown in Figure 2.5, an optical tweezers system basically contains a laser beam focusing near an object. The transmitted laser beam is collected and detected. The illumination is also provided and the image is detected on a CCD camera. The laser beam is collimated and expanded onto the back aperture of a water or oil immersion objective lens with a high

35 2.1 OPTICAL TRAPPING 13 numerical aperture so that the beam can be tightly focused to form an optical trap. In addition, for biological purposes, the trapping laser should typically have a wavelength in the region of nm in order to minimise damage to living cells as in this spectral range the absorption in laser is minimised. Some tweezers systems use a home-made inverted microscope structure so that the scattering force acts against gravity which can decrease the laser power needed. FIGURE 2.5: Schematic of a basic single optical tweezers system. Two types of optical forces are acting in this system: a scattering force, F s, and a gradient force, F g. When the gradient force overcomes the scattering force which happens in the focal area, the particle is trapped. Adding an illumination source to the set-up provides a means for observing the sample. An LED source is recommended since it does not heat the sample as much compared to a halogen lamp. Illumination can be either via transmission or reflection; the transmission approach is illustrated in Figure 2.5. The light is collimated and focused on the sample by a condenser. A dichroic mirror is used to separate the illumination light from the laser beam. A camera is placed at a distance from the tube lens of the objective equal to its focal length. The displacement or position fluctuation of the trapped particle can be measured by a quadrant photodiode detector (QPD), a position sensitive detector (PSD) or a high-speed camera [43, 44, 106]. A QPD measures the intensity difference of the forward-scattered laser light between the left and right (and between the top and bottom) sides of the detection plane, and a PSD gives

36 14 OPTICAL TWEEZERS the deflection of the beam by the particle and measures the position of the integral focus of the incoming light signal. Moreover, the use of optical tweezers for quantitative detection of force requires accurate calibration of the optical trap [18, 19, 86, 106]. With proper calibration techniques for the trap stiffness, the force can be linked to the displacement measured with high efficiency and accuracy by these detection techniques. 2.2 Rotational Optical tweezers Angular momentum of light It is well known that light carries angular momentum as well as linear momentum. The angular momentum of light exists in two forms, orbital angular momentum and spin angular momentum. Optical tweezers have been used as an ideal tool to study the different types of angular momentum since the optical torques involved enable rotation of microscopic objects. Spin angular momentum has been shown to rotate birefringent objects [41] or objects with shape birefringence and orbital angular momentum has been shown to cause rotation of absorbing particles [51, 52] or specifically shaped objects. FIGURE 2.6: Helical phase structures of LG beam with modes l of 02, 03 and 20 respectively. The phase shift ranges from 0 to l when going from 0 to 2π degrees around the beam centre. One example of light carrying orbital momentum is Laguerre Gaussian (LG) modes of light [33]. Laguerre Gaussian beams have a structure where the azimuthal phase

37 2.2 ROTATIONAL OPTICAL TWEEZERS 15 is of the form exp ilφ where l gives the number of 2π cycles in phase around the beam axis and φ is the azimuthal angle [2]. This indicates a helical structure of the beam where the wave front twists around the central beam axis as shown in Figure 2.6. The orbital angular momentum of an order l of LG beam is well defined and carries l h per photon of orbital angular momentum. Strong torques can be applied to the trapped particle by increasing the azimuthal mode index l. In contrast to orbital angular momentum, spin angular momentum due to the polarisation of light offers a maximum momentum of 2 h per photon transferred to the particle [13]. Beams with orbital angular momentum can be generated in many ways such as by the use of plate holograms [53] and spatial light modulators (SLM) [28]. Orbital angular momentum is of great importance for a number of applications both in fields ranging from quantum communication to biophysics; however, in this chapter we mainly discuss spin angular momentum since in our experiments the rotational motion results from transferring spin angular momentum from laser light to the particle. FIGURE 2.7: a, Birefringence and polarisation. Due to the birefringent nature of the materials, particles formed from birefringent materials can act as wave-plates; for example, a birefringent particle about 3 µm thick is a λ/2 plate for 1064 nm laser. b, Beth s experiment, 2 h per photon of angular momentum was transferred to the quartz plate. Light carries ± h spin angular momentum per photon for left or right circularly polarised states respectively, about the direction of propagation. The first experimental observation of spin angular momentum of light was achieved by Beth in 1935 [13]. In this experiment, the torque exerted on a suspended quartz wave plate was observed. This torque was due to the change in polarisation of transmitted light as shown in

38 16 OPTICAL TWEEZERS Figure 2.7 b. Since quartz is a highly birefringent material as shown in Figure 2.7 a, the polarisation state of the incident beam changes resulting in angular momentum transfer from the beam to the object. Birefringent materials have two distinct indices of refraction. A beam will be decomposed into two beams with mutually perpendicular polarisation as they travel at two different speeds. One ray is called the ordinary ray (o) and the other ray is extraordinary ray (e); both obey Snell s law, but each is described by a different refractive index. The relative phase shift D between the ordinary and extraordinary path is determined by the two refractive index values (n o and n e ) and the thickness of the material d, D = 2π(n o n e ) d/λ 0, where λ 0 is the free-space wavelength. FIGURE 2.8: Optical trapping of microscopic glass cylinders. The microcylinder rotated by changing the polarisation of the trapping beam using a half-wave plate, and the optical torque was measured directly by monitoring the change in the degree of right and left circular polarisations in the incident-transmitted light. In 1998, using optical tweezers to rotate microscopic transparent birefringent particles due to transfer of spin angular momentum as originally observed by Beth was reported by Friese et al. The rotation rate of a 1 µm thickness particle was 357 Hz in a laser beam with power of 300 mw [41]. Later based on these experiments, a theoretical method was developed by Nieminen et al. in 2001 which considered measurement of spin angular momentum as well as optical torque generated by transfer of spin angular momentum of light [89]. Changes in the polarisation state of the

39 2.2 ROTATIONAL OPTICAL TWEEZERS 17 scattered light constitutes the basis for a measurement of the optical torque on the particle [14]. In this experiment as shown in Figure 2.8 an elongated glass rod was used. This elongated dielectric particle exhibits shape birefringence, as it has different dielectric polarisabilities along their short and long axes so this creates an effective birefringence. To determine the optical torque applied on the particle, as introduced before, it is important to be able to measure the change in polarisation. Spin angular momentum is commonly associated with circularly polarised light depending on the handedness and it has a magnitude of ± h = ±h/2π per photon. The total angular momentum flux of the beam can be defined by L z = σ z P/ω (2.1) where P is the total power of the beam, ω is the optical frequency and σ z is the coefficient of circular polarisation defined as σ z = P R P L P (2.2) where P R and P L are the powers of the right and left circularly polarised components of the beam and we assume that, total power P = P R + P L. Linearly polarised light can be described as a coherent superposition of two circularly polarised beams with opposite handedness and equal intensity, so there is no net spin angular momentum carried. Elliptically polarised light carries angular momentum depending on the intensity ratio of the right and left circularly polarised light Optical torque and measurement of optical torque in optical tweezers As we have shown, it is a net circular polarised component of light that contributes to the spin angular momentum of the beam. When the beam passes through objects of birefringent material, the polarisation of the light changes, and therefore σ z changes. A reaction torque on the object, which results from the change of angular momentum,

40 18 OPTICAL TWEEZERS is equal to the change in the coefficient of circular polarisation between initial σ zin and emergent polarisation σ zout : τ = (σ zin σ zout )P/ω = σ z P/ω (2.3) where σ z is the change in degree of circular polarisation of the light. FIGURE 2.9: Schematics of experimental approach for measurement of optical torque resulting from the transfer of spin angular momentum. Thus to measure the optical torque, one can measure the change in degree of circular polarisation of the beam σ z and beam power P. If we consider the case where the incoming beam is circularly polarised without a trapped particle (since this case provides us a constant as well as a maximal torque), σ z becomes (1 σ zout ) so that measurements of outgoing polarisation σ zout and beam power gives the torque. As shown in Figure 2.9, the incoming beam is circularly polarised, the outgoing beam passes through a quarter-wave plate and a polarising beam splitter (PBS), and the power is collected at two detectors respectively. These two detectors (D 1 and D 2 ) measure the amplitudes of the right and left circularly polarised components. If there is no particle in the beam, measured powers from the two detectors are the same, D 1 = D 2. Therefore, when the beam passes through birefringent material, the polarisation of the beam changes and the change in degree of circular polarisation could be

41 2.3 OPTICAL TRAPPING OF VATERITE PARTICLES 19 measured by (D 1 D 2 )/(D 1 + D 2 ). Implementation of this method in our experimental setup is shown and described in detail in Chapter Optical trapping of vaterite particles Vaterite particles Rather than relying on natural birefringent materials as they are often irregular in shape, our group developed a chemical technique to grow microspherical birefringent particles of vaterite [95, 123]. Vaterite is a positive uniaxial crystal which has strong birefringence with n e = 1.65, n o = The method described here provides a highly controllable way of producing birefringent vaterites with close to spherical geometry and with varying size. The method of producing vaterite particles represents a further development of previous recipes. 0.1 M aqueous solutions of CaCl 2, KOH, MgSO 4 and K 2 CO 3 were prepared. 20 ml of CaCl 2 solution was pipetted into a 50 ml plastic centrifuge tube. Two drops of KOH and 5 ml MgSO 4 were added to the tube and the solution was gently shaken to mix the solutions well. 2 ml of K 2 CO 3 was then added to the solution. We then have to wait for about three minutes while vaterite particles are being formed with two or three gentle shakes during this period. Finally, 10 drops of Agepon soap were added to stop further growth of vaterite particles. The crystals produced are typically vaterite spheres with diameter of 3 to 4 µm. In practice, we prepared several sets of samples at the same time; the best one of these was used as a seed solution for later refinement. The way we chose the best sample as a seed was determined by looking for a sample with the most particles being spherical and having uniform birefringence. Next, the vaterite particles in the seed solution need to be washed. This is done by first centrifuging the tube for a few minutes until the vaterite particles formed a small clump at the bottom of the centrifuge tube. The solution is then removed and replaced with ethanol, and put in a sonicator to separate the vaterite particles. The procedure was repeated three to four times. Once the seed solution was prepared, then it was used to create

42 20 OPTICAL TWEEZERS further generations of vaterite particles. About 10 drops of the seed solution were added before adding 2 drops of 0.1 M KOH when making a new generation of vaterite particles. To control the size of vaterite particles produced, it is good to observe their growth under a microscope. From Figure 2.10 we can see that the size of the vaterite particle depends on the growing time. In Figure 2.10 a, the size of vaterite particles is about µm in diameter after about one and a half minutes. Between 2 3 minutes, the size reached about 3 4 µm in diameter as shown in Figure 2.10 b,c. The vaterite particles can be as big as 5 7 µm after three and a half minutes (Figure 2.10 d). Therefore the use of a microscope in the production process helps control the timing regime which gives good size and shape of particles. Vaterite particles may dissolve under certain circumstances; the production of silica-coated surface-functionalised particles has been described in Vogel et al. (2009) for biological purposes. Almost perfectly spherical and small vaterite particles were chosen in experiments. One reason for this is that the use of spherical shape simplifies the model used to predict optical torques or Stokes drag acting on the vaterite particle, making it more feasible for use in microrheology. In addition, reducing the size of vaterite particles would be useful for future biological studies, for example, when probing microrheological properties of living cells exposed to a variety of conditions. FIGURE 2.10: Size of vaterite particles varies with the growing time. a-d, The growing time of vaterite particles are about 1.30, 2.30, 3.00 and 3.30 minutes respectively.

43 2.3 OPTICAL TRAPPING OF VATERITE PARTICLES Applications of birefringent particles in optical tweezers Vaterite particles, when trapped in a circularly polarised beam, spin at a constant speed that is proportional to the trapping power and inversely proportional to the viscosity of the surrounding fluid. Under linearly polarised light, such particles align to the polarisation direction as shown in Figure FIGURE 2.11: Schematic diagram of the vaterite particle under a linearly polarised beam (left) and a circularly polarised beam (right). The rotational degree of freedom adds new functionality to optical traps: in addition to allowing fundamental studies of optical angular momentum, it can allow measurements of local viscosity and exert local stresses including shear stresses on cellular systems [33]. One advantage of this rotational technique over linear approaches lies in the nature of the hydrodynamic coupling between the particle and the surrounding boundaries, which is described in more details in Chapter 4. Therefore, measurement of the optical torque (Equation (2.3)) on a spherical particle rotating in a fluid with a constant speed enables the measurement of the local viscosity of the surrounding fluid [15, 94]. It is due to the fact that the optical torque driving the vaterite particle is balanced by viscous drag due to the surrounding medium, τ D = 8πηa 3 Ω. (2.4) Here τ D is the drag torque from the fluid, a is the radius of the sphere, η is the viscosity of a low Reynolds number liquid and Ω is the angular speed of the sphere. The

44 22 OPTICAL TWEEZERS balance of the equations of optical torque and the drag torque gives the expression for the viscosity: η = σp/(8πa 3 Ωω). (2.5) Thus the viscosity can be determined from a few parameters measured in experiments. In addition, more recently, the viscoelasticity of a fluid was measured by observing the Brownian motion of a vaterite particle [9]. Optical tweezers with rotating photonic probes demonstrates great promise for future applications to determine the spatial-temporal properties of fluids. In the next chapter, I will describe in more detail how vaterite particles can be used as a microrheometer to access properties of fluids, and discuss our recent studies of microrheology in both theory and experiment based on rotational optical tweezers.

45 3 Microrheology with rotational optical tweezers 3.1 Microrheology of complex fluids Rheology is the study of the viscoelasticity of materials. It is well known that solids store mechanical energy and are elastic, whereas fluids dissipate mechanical energy and are viscous. Many materials are viscoelastic, which both store and dissipate energy, and with the relative proportions depending on frequency [82]. One approach to characterise rheological properties is to measure the shear modulus as a function of frequency [61]. For conventional rheometers, measurements have been performed on several millilitres of materials by applying a small amplitude oscillatory shear strain, given by 23

46 24 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS γ(t) = γ 0 sin(ωt) (3.1) where γ 0 is the amplitude and ω is the frequency of oscillation. If the strain amplitude γ 0 is small enough (γ 0 1) so that the structure of materials is not disturbed much by the deformation and the material remains in equilibrium, then the affine deformation of the material controls the measured stress. The time-dependent stress is linearly proportional to the strain defined by σ(t) = γ 0 [G (ω) sin(ωt) + G (ω) cos(ωt)] (3.2) G (ω) is called the elastic storage modulus and G (ω) is the viscous loss modulus. In general, the frequency-dependent behaviour of viscoelastic materials is described by the complex shear modulus, which is defined as, G (ω) = G (ω) + ig (ω) (3.3) which consists of a real part, G (ω), the elastic storage modulus, and an imaginary part, G (ω), the viscous loss modulus. Microrheology extends this study to the field where the dynamic behaviour of these materials changes with length scale [108]. In particular, it aims to characterise G on micrometre or sub-micrometre scales. Viscoelastic fluids are prevalent in most biological systems such as biomedical materials and cytoplasm inside living cells. These micro- and nano-structured materials are inhomogeneous depending on both their constituent molecules and structures formed by them; they are also rare and often available only in ultra-small volumes. Rheology measurements have successfully characterised complex fluids such as colloidal suspensions, polymer solutions and gels, emulsions, and surfactant mixtures [40, 61, 74]. However, these conventional techniques are not well-suited for the studies of microrheology since many biological samples are difficult to obtain in large quantities. A rich range of theoretical and experimental analysis is required to study the properites of small volumes of materials more quantitatively [73]. Complex fluids arise frequently in biological

47 3.2 MICRORHEOLOGICAL METHODS 25 systems, typically with length scales at the microscopic cellular level. Furthermore, the microrheology of biological materials (e.g. cytoplasm and tear film) often enables studies of the relevant material properties linked to the development and diagnosis of diseased states [129]. Novel microrheological techniques therefore represent extremely useful tools for our understanding of biophysics and diagnosis of a variety of diseases. 3.2 Microrheological methods Introduction As mentioned before, bulk rheometers require sample volumes of millilitres. In contrast microrheology studies fluids on microscopic scales. Besides this, microrheological approaches offer a series of advantages over bulk techniques [125]. Bulk rheometers provide an average measurement of the bulk response; they do not allow for local measurements of inhomogeneous materials which are sensitive to the length scale. Moreover, a conventional rheometer generally provides the macroscopic response of the material over a restricted range of frequencies (up to tens of hertz) [40, 61]. The microrheological technique is able to access a much wider frequency interval. Microrheology has become a much better established field in both theory and experiment that is able to treat much smaller sample volumes, has an improved frequency sensitivity, provides delicate non-invasive measurements and has an improved sensitivity to intracellular dynamics [25, 31, 108, 125, 133, 138]. The majority of techniques of microrheology is directed towards extracting the rheological properties of a material from the motion of probe particles embedded within it. Based on these techniques, microrheology can be classified as passive microrheology which aims to measure material rheology by thermal fluctuations of embedded probes and active microrheology in which an embedded probe is actively driven by an external force. In Chapter 2, the birefringent vaterite microsphere probe for optical

48 26 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS trapping and rotation was introduced. In this Chapter, I will discuss the details of accessing the viscoelasticity of complex fluids on microscale at very low sample volumes and with a broadband frequency range based on both passive and active rotational motion of a vaterite particle using optical tweezers. We begin by introducing some techniques of passive and active microrheology Passive microrheology Passive microrheological techniques use the unconstrained motions of fluctuating particles suspended in a complex fluid to determine the viscoelasticity of the fluid. For viscous fluids, the unconstrained thermal motion of particles in a purely viscous fluid follows well-known Brownian motion (Figure 3.1). The dynamics of particle motion can be determined from the time dependent position correlation function of individual tracers, known as the mean square displacement (MSD), x 2 (τ) = x(t + τ) x(t) 2 t (3.4) where x is the particle position, τ is the lag time and brackets represent an average over all times t. FIGURE 3.1: a, Brownian motion of a microscopic particle suspended in a viscous fluid medium caused by the liquid molecules. b, Two-dimensional experimentally measured trajectory of a particle suspended in water.

49 3.2 MICRORHEOLOGICAL METHODS 27 The time-average assumes the fluid is always in thermal equilibrium and the material properties do not evolve in time. The one dimensional time dependence of the MSD for isotropic diffusion is given by x 2 (τ) = 2Dτ. (3.5) The viscosity η of the fluid surrounding the microsphere of radius a can be obtained from the MSD using the Stokes Einstein equation: D = k B T/6πηa, where k B is Boltzmann s constant and T is the temperature. In terms of complex fluids, the thermally driven motion of a small particle reflects the complex shear modulus, which is obtained through the generalized Langevin equation [82], m v(t) = f R (t) t 0 ζ(t τ)v(τ)dτ (3.6) where m is the mass of the particle and v(t) is the particle velocity. f R (t) is the random force exerted on the particle, and the integral represents the viscous damping of the fluid with a time dependent memory function ζ(t) to account for the elasticity. Assuming that complex fluids can be treated as isotropic, incompressible continuums around a sphere, the viscoelasticity G(s) can be determined by the unilateral Laplace transform of the MSD using Equation (3.6) [81], G(s) = k B T πas r 2 (s) (3.7) where s is the Laplace frequency. Notably, Equation (3.7) is based on Equation (3.6) which describes the motion of the particle neglecting the sphere s inertia and it is consistent with energy equipartition and the fluctuation-dissipation theorem. Moreover, Equation (3.7) assumes that Stokes drag for viscous fluids (no-slip boundary conditions) can be generalized to the viscoelastic fluid at all s [81]. It has been theoretically shown that using the generalized Stokes Einstein relation to obtain the viscoelasticity of a medium works well within a certain frequency range, ω B < ω < ω [65]. The

50 28 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS lower limit, ω B, is the time scale at which longitudinal or compressional modes become significant compared to the shear modes that are excited. The upper frequency limit, ω, is due to the onset of inertial effects of the material at the length scale of the particle. Therefore it is found that within the frequency range of 10 Hz< ω <100 khz, Equation (3.7) is accurate and valid. Tracking of the thermal motion can be either single-particle tracking or multipleparticle tracking. Experimental approaches have been established with laser-particle tracking [29, 81] and with video-particle tracking [26]. For laser-particle tracking, the thermal motion causes the probe to move off the beam s axis and the deflection of the laser beam can be measured. The power of the laser beam used is quite low so that the optical force is smaller than the thermal force acting on the bead. Detection schemes based on laser light deflection provide measurements with high spatial and temporal resolution as the thermal fluctuation of the particle is constrained by the optical trap. On the other hand, while this gives high-frequency properties of the material, the low frequency statistics are limited. The video-particle tracking technique is based on directly imaging the embedded beads using a camera. In this case, the size of trackable probe particles is limited in standard video microscopy to optical wavelengths ( 0.5µm), since this is the diffraction limited resolution of an optical microscope [124]. The frequency range of particle tracking techniques is usually limited to 50 Hz according to standard video rates, but now can be extended to the khz range using high speed cameras [132]. Passive video particle tracking microrheology provides the low-frequency response, while it does not give strictly local measurements Active microrheology In active microrheological measurements, an external force is applied to drive a probe particle through a material. A microscopic particle suspended in a material is actively driven by a known force or a known displacement and its response is measured so that the viscoelasticity of the material can be determined. Active techniques allow for

51 3.3 MICRORHEOLOGY WITH OPTICAL TWEEZERS 29 applying large stresses to stiff materials in order to obtain measurable strains [108]. Moreover, they can be used to detect non-equilibrium behaviour as the material can be strained beyond the linear regime by large forces [84]. Techniques including atomic force microscopy (AFM) [60, 85], optical and magnetic tweezers [100, 109], plasmonics [55] and magnetic spectroscopy [12] have been used in rheological measurements. Optical tweezers are increasingly finding new active microrheology applications due to their versatility and ability to precisely position beads in locations of interest, and the ability of simultaneously manipulating multiple beads whose surface can be easily functionalised [16]. 3.3 Microrheology with optical tweezers Optical tweezers have been successfully used for the investigation of the rheological properties of complex fluids. In these methods one measures the interaction between optically trapped particles and surrounding mediums [80, 84, 138]. Optical trapping systems enable high resolution measurements of pn forces and nm displacements with temporal resolution down to µs, which offers the advantage of investigating the heterogeneities of the medium without distortions of its structure. In addition to the non-invasive nature of optical tweezers, it becomes one of the most versatile manipulation techniques used in situ such as inside living cells. These factors make optical tweezers well suited for microrheological applications for the study of complex biological systems. Previously, optical tweezers have been employed for both active and passive microrheological measurements. Passive methods typically monitor the thermal fluctuation of a particle suspended in a fluid [98, 124], whilst active techniques require external forces to act on probe particles [15, 16, 94, 100]. However, these methods become difficult to use when examining very small volumes of samples. Furthermore, the ability to orient objects using optical tweezers offers new opportunities for applications in studying fluid properties. More recently, related techniques based on rotational diffusion of particles were demonstrated to extract the complex dynamic modulus from viscoelastic fluids [9, 23]. Rotational microrheology

52 30 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS techniques have greater spatial resolution compared to conventional methods, since the centre of the probe can be tightly constrained and the rotational motion is less sensitive to boundary walls according to Faxén s correction [62]. However, measurements of viscoelastic properties of complex fluids by passive microrheology techniques yield high frequency results with high accuracy but often lack precise low frequency information [113]. It is due to the fact that the presence of a trap constrains the particle to making only small displacements from the equilibrium point, discarding the crucial information related to the long time scale material behaviour. Some of the broadest frequency range microrheology data with optical tweezers has been created by combining passive translational Brownian motion with active microrheology by measuring the transient displacement of a bead moving between two optical traps [100], which is not a strictly local measurement. The next step in the application of optical tweezers instruments for microrheology is trying to combine these hybrid techniques to develop a new method with the highest spatial and temporal resolution. 3.4 High resolution, wide-band microrheology with optical tweezers Here we introduce a microrheometer based on rotational optical tweezers to access local viscoelastic properties of complex fluids over a wide frequency range. The experimental procedure consists of two steps: (I) measuring the angular diffusion of a trapped birefringent vaterite particle for a sufficiently long time; (II) measuring the transient angular displacement of this probe particle flipping between two angular traps (with a fixed angle between two linearly polarised beams) that alternately switch on/off at sufficiently low frequency. These two angular traps are located at the same position in space. The analysis of the first step (I) provides the high frequency viscoelastic properties of the material. The second step (II), by exerting an external torque on the particle, provides information on the low-frequency fluid response. Therefore, by combining both steps, which show a clear overlapping region between

53 3.4 HIGH RESOLUTION, WIDE-BAND MICRORHEOLOGY WITH OPTICAL TWEEZERS 31 the results at low and high frequencies, the full viscoelastic spectrum is obtained Experimental set up FIGURE 3.2: Experimental apparatus for microrheological measurements. In the passive process, the vaterite particle was trapped in a linearly polarised beam for a sufficiently long time, undergoing rotational Brownian motion about beam axis. In the active process, the particle is held in different orientations due to the angle of polarisation between the two linear traps. The angular transit is induced by means of two AOMs alternately switching on/off every few seconds. A camera (CAM) allows imaging of the optical trap. Figure 3.2 demonstrates the experimental apparatus, which enables measurements of viscoelasticity with high resolution and with a broadband frequency range. Measurements of the passive rotational diffusion of a birefringent microsphere trapped within either trap 1 or trap 2 (in step I) provide information about the high-frequency viscoelastic properties of the fluid. To determine the low-frequency fluid response, we measure active transient angular displacements of this probe (step II) flipping between the two overlapped traps. Trap 1 and trap 2, with a fixed angle φ 0 of polarisation, were alternately switched on/off (Figure 3.3). The linearly polarised 1064 nm trapping beam was divided into two equal intensity beams by a polarising beam splitter (PBS). In each path, an acousto-optic modulator (AOM, driven at 27 MHz)

54 32 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS was used to control both the power and on/off state of the beam. A half-wave plate (λ/2) is used to rotate the linear polarisation of the trap 1 beam by π/3. Thus a λ/2 waveplate and two AOMs provide a means to control the power as well as the polarisation of the recombined beam. Then the overlapped two beams with fixed angles of polarisation were focused into the sample through a water immersion objective (60 1.2NA). The scattered light was collected by a condenser and then directed into photodiode detectors (D 3 and D 4, Thorlabs PDA55, Amplified Si Photodetector). A weak circularly polarised, 633 nm HeNe, beam was used to observe (in detectors D 1 and D 2, Thorlabs PDA55, Amplified Si Photodetector) the angular displacement of the particle. These detectors are designed for detection of light signals from DC to 10 MHz. To minimise optical forces caused by the tracking beam, the HeNe beam was adjusted to underfill the objective lens resulting in a relatively large focal spot thus exerting a small gradient force. For a collimated beam in water, the optical force is about 10 nn/w or 10 pn/mw. Since the power of the HeNe beam is less than 40 µw compared to the 20 mw of the trapping beam at the focus, the optical force exerted on the particle by the HeNe beam is negligible. FIGURE 3.3: Camera images with the overlaid schematic of two steps of the experimental procedure: In the passive process (step I), the vaterite particle is trapped in a linearly polarised beam, either trap 1 or trap 2. In the active process (step II), the particle is held in different orientations due to the angle of polarisations between trap 1 and trap 2. The angular transit displacement is created by means of two traps alternately switching on/off Tracking laser (HeNe beam) detection The birefringent vaterite particle allows the transfer of spin angular momentum with the trapping beam, either aligning to a fixed orientation determined by the linear

55 3.4 HIGH RESOLUTION, WIDE-BAND MICRORHEOLOGY WITH OPTICAL TWEEZERS 33 polarisation vector of the trapping beam, or rotating with a constant speed in a circularly polarised beam [15]. In order to measure the angular displacement, a circularly polarised HeNe beam can be used to track the angular position of a vaterite particle. Note that the power of the beam is very low so that the torque exerted on the particle by the tracking beam is negligible compared to the trapping beam. When it passes through the vaterite particle, the birefringence changes the spin angular momentum of the light, and the beam becomes elliptically polarised. The long axis of the polarisation ellipse of the elliptically polarised beam rotates with the optic axis of the particle [89]. By passing the HeNe light through a polarising beam splitter, we detect the power at each detector (D 1 and D 2 in Figure 3.2). The normalised recorded voltage difference varies sinusoidally with twice the rotation rate of the particle, sin 2φ as shown in Figure 3.4. So after each measurement, we introduce a quarter-wave plate into the IR path to make the probe particle spin, which allows us to determine the relationship between V (D 1 and D 2 ) and angular position φ from sinusoidally varying signal. Using this calibration, for both passive and active measurements, the rotational Brownian motion and the transient angular displacement of the particle can be determined. FIGURE 3.4: Signal recorded during rotation of a vaterite particle in water by introducing a quarter-wave plate into the IR path. The voltage difference V between detectors D 1 and D 2 relates sinusoidally to the azimuthal angle of the vaterite particle, V sin 2φ. Through spin of the vaterite particle, the range of V enables an exact relationship between V and φ to be determined.

56 34 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS Trapping laser (IR beam) detection Since the spin angular momentum of the trapping beam is transferred to a birefringent particle, the particle experiences a torque τ given by τ = σ CP P/ω, where σ CP = (P L P R )/P is the change in circular polarisation of the beam with P L and P R being the left and the right circularly polarised components of the beam power, P the beam power, and ω the angular frequency of the light [89]. In practice, we measure the change in the degree of circular polarisation of the trapping beam to determine the optical torque. When a linearly polarised trapping beam passes through a vaterite particle, the polarisation of the beam changes. We use a quarter-wave plate and a polarising beam splitter to separate the transmitted light into two circularly polarised components, and measure the power respectively (D 3 and D 4 in Figure 3.2). The difference between the signals collected by the photodiode detectors gives the change in the degree of circular polarisation of the beam. So when the particle is held in a linearly polarised optical trap, we calibrate the two detectors (D 3 and D 4 ) to make the difference in power equal to zero to match the optical torque exerted on the particle. Furthermore, the optical torque depends on the angle sinusoidally with a period of π. The active rotation is performed by rapidly switching the polarisation of the trapping beam with an AOM. Figure 3.5 shows the experimentally measured data from IR detectors for a vaterite particle flipping in water and 25 percent Celluvisc (a carboxymethylcellulose (CMC) sodium solution, 10 mg/ml.) by weight. As the angle between the optic axis of the particle and polarisation is more than π/4 ( π/3 in experiment), the difference in power (D 3 and D 4 ) increases first. Due to the sine dependence on two times the angle, it reaches the maximum power at the position when the angle becomes π/4, and then decreases down to zero when the particle aligns to the other trap. Thus the relationship between V (D 3 and D 4 ) and the optical torque can be determined.

57 3.4 HIGH RESOLUTION, WIDE-BAND MICRORHEOLOGY WITH OPTICAL TWEEZERS 35 FIGURE 3.5: Typical time recorded of the measured optical torque signal of a 3µm diameter particle flipping between trap 1 and trap 2 (Figure 3.2) repeatedly switching after durations of 4s in two different solutions of water (blue) and 25% w/w Celluvisc in water (black). The optical torque increases along the angular displacement of the particle when one trap switches off and the other trap switches on simultaneously, and then decreases. The maximum torque is achieved as the angular displacement is equal to π/ Theoretical model The first step (I) determines high frequency viscoelasticity by measuring passive Brownian motion, and is performed in the same manner as in Bennett, et al. [9]. As described before by Equation (3.6), the rotational dynamics of the trapped particle fluctuating about equilibrium are modelled by the generalised Langevin equation. Therefore, the evolution of the azimuthal angle φ over time t is governed by I φ = τ(t) t ζ(t t ) φ(t )dt χφ. (3.8) The total torque (I φ) on the probe at time t is the sum of the zero-mean thermal torque (τ(t)) from Brownian motion, the linear optical torque ( χφ, with equilibrium at φ = 0) and the viscoelastic torque ( t ζ(t t ) φ(t )dt with generalised memory function ζ(t)). In the low Reynolds number regime inertial effects are negligible thus the inertial term is dropped:

58 36 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS 0 = τ(t) t ζ(t t ) φ(t )dt χφ. (3.9) The presence of a linear optical restoring torque for angles obtainable by Brownian motion about the equilibrium has previously been validated experimentally [9]. This implies a harmonic potential for the angles sampled permitting the application of the equipartition theorem in calculating the trap stiffness χ via χ = k BT φ 2 (3.10) where k B is the Boltzmann constant and brackets... indicate a measurement averaged over all time. The thermal fluctuations of the trapped vaterite particle are related to the viscoelastic properties of the fluid through analysis of time dependence of the normalised angular position autocorrelation function (NAPAF) defined by Equation (3.11), which, containing a measurement of φ(t), is a function of the time interval t, Φ(t) = φ(t )φ(t + t) φ(t ) 2. (3.11) The complex shear modulus can be obtained directly by transforming the NAPAF with a unilateral Fourier transform defined by Φ(ω) = 0 Φ(t)e iωt dt, (3.12) G (ω) = A tilde represents a transformed variable. χ [ ] iω Φ(ω) 8πa 3. (3.13) 1 iω Φ(ω) Figure 3.6 shows the representative normalized autocorrelation function, Φ, measured by passive Brownian motion of vaterite particles in samples of water (red line in the figure) and 25 percent Celluvisc by weight (blue curve in Figure 3.6).

59 3.4 HIGH RESOLUTION, WIDE-BAND MICRORHEOLOGY WITH OPTICAL TWEEZERS 37 FIGURE 3.6: Experimental autocorrelation functions collected in samples of water (red line) and 25% Celluvisc by weight (blue line). Decaying exponential functions in water (red line) fitted to an exponential function (black line). The raw data presented were collected in a single measurement of five minutes. FIGURE 3.7: Optical torque has a sinusoidal variation with angle. When the angle is less than π/6, the optical torque can be assumed to be linearly dependent on angle. When the particle rotates between two traps over a larger angle, the restoring torque function is sinusoidal because of the waveplate nature of the vaterite probe particle. In a similar way, to resolve low frequency viscoelasticity in step (II), we use a generalised Langevin equation to model the dynamics of the vaterite particle as it rotates between two traps with a fixed angle of polarisations (φ 0 ). This new theory was developed by Gibson et al. [45]. In this active process, the thermal noise τ adds random

60 38 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS error to the measurement. In order to measure the viscoelasticity of the fluid more efficiently, the thermal noise needs to be minimised. One way to reduce the thermal noise is to increase the amplitude of the angle of the particle flipping over, which means to increase the angle φ 0 of polarisations between the two traps. However, as shown in Figure 3.7, if the angular displacement of the vaterite particle is small (less than 30 degrees), the driving torque can be approximated by the Hookean restoring torque χφ. Given the wave-plate properties of vaterite probe particles [89], the optical torque depends on the sine of the angle with a period of π. If the particle rotates between two angular offset linearly polarised traps with a large angle, the restoring torque is beyond this linear regime. Accounting for these conditions, the average dynamics of a probe with radius a becomes 0 = t 0 ζ(t t ) φ(t )dt + χ sin(2φ), (3.14) 2 where the first integral term is the viscoelastic drag torque and the second term is the sinusoidal optical torque with trap stiffness χ. Figure 3.8 a and Figure 3.8 b show the experimental measured angular displacement and torque efficiency of a vaterite particle flipping in water respectively. In Figure 3.8 c, we compare the transient angular displacement from the HeNe detectors with corresponding measured torque efficiency from the IR detectors. They show a good agreement with the sinusoidal optical torque as the particle flips over a large angle. This confirms the theory for non-linear torque in Equation (3.14).

61 3.4 HIGH RESOLUTION, WIDE-BAND MICRORHEOLOGY WITH OPTICAL TWEEZERS 39 FIGURE 3.8: a, Signals from HeNe detectors show transient angular displacements of a particle flipping instantaneously between two traps in water. b, The data for the same particle are collected from IR detectors. This shows that the experimental measured torque efficiency on the particle. The optical torque is small when the particle undergoes rotational Brownian diffusion within one trap. When the initial trap is turned off and the second turned on simultaneously, a sudden torque is applied on the particle. The peaks result when the angle of polarisation between two traps is more than π/4. c, From a single flip in water the angle is tracked via the HeNe laser and compared with the torque efficiency measured by the trapping laser. The measured values (blue dots) show excellent agreement with the least squares fit (solid black line) of the nonlinear optical torque (τ) model τ sin(2(φ 0 φ)), where φ 0 π/3 is the angle of polarisation between two traps. The optical torque increases with the angular displacement of the particle when one trap switches off and the other trap switches on simultaneously, and it then decreases. The maximum torque is achieved when the angular displacement φ is 0.2 from one trap to the other trap and the angle between the optic axis of the particle and the polarisation of the beam φ 0 φ becomes π/4. FIGURE 3.9: Repeatedly flipping the vaterite particle between two traps allows the Brownian noise to be mitigated by averaging over many flips. a, Experimentally measured 220 flips of a vaterite particle. b, By averaging all flips, a noise-reduced signal is obtained. The second way to increase the signal to noise ratio is to measure many flips between traps and then average them together to mitigate the effects of Brownian motion. As

62 40 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS shown in Figure 3.9 a, the data is from an experiment with 220 flips of a vaterite probe in water, and by averaging them, the noise is greatly reduced as shown in Figure 3.9 b. Another source of error has to do with the determination of the initial angle. Since many flips are needed in experiments, in practice small changes in the polarisations of the beam and other changes in the system affect the angle where the particle starts at. Variations in initial angle cause error at the beginning of the flip (Figure 3.10). Therefore, in order to account for changes in the initial position, the model needs to be modified. If we assume that the time it takes for the particle to flip between traps is much longer than the decay time of ζ(t), then equation (3.14) is linearised by making a transformation to a normalised variable, ψ = tan φ tan φ 0. (3.15) Equation (3.14) simplifies to a more linear form 0 = t 0 ζ(t t ) ψ(t )dt + χψ. (3.16) In Equation (3.16), the signal from each flip is normalised. In Figure 3.10 a, we show all the flips. Using the variable transformation on the data with normalisation is shown in Figure 3.10 b. We can see that there is much less error at short times which would give us much better data for high frequencies, and for the rest of it as the particle flips over a large angle the effect of Brownian motion is minimised.

63 3.4 HIGH RESOLUTION, WIDE-BAND MICRORHEOLOGY WITH OPTICAL TWEEZERS 41 FIGURE 3.10: Initial flip angle determination. a, If the optical torque is a non-linear function, then the Langevin equation Equation (3.14) is a non-linear differential equation. This causes error for each repeated measurement in which the initial position of each flip varies. b, By applying the transformation of Equation (3.15) Equation (3.14) is linearised, and there is much less error at the initial position and the effect of Brownian motion is minimised by increasing the flipping angle φ 0. Therefore the complex shear modulus can be obtained via a unilateral Fourier transform, G (ω) = χ [ ] iω ψ(ω) 8πa 3. (3.17) 1 iω ψ(ω) With this new theoretical model, the maximal signal-to-noise ratio is achieved by angularly offsetting the two traps by more than π/4, which is beyond the linear spring regime with theoretical analysis described above. Accordingly, in practice we set φ 0 to be π/3. Results shown in Figure 3.11 demonstrate that the differences in accuracy and precision of different theoretical methods including the assumption of a linear torque and non-linear torque without and with normalisation for flips. We observe the values in a linear relationship at the low frequency regime in Figure 3.11, indicating frequency independent viscosity and elasticity. As can be seen from this figure, the transformation method is more accurate and resolves an additional decade at high frequency. In addition, to reduce the error by 50%, the new theory only requires 4 times the number of flips, as opposed to the old model which requires 8 times. As shown in Figure 3.12, with only 12 flips and 5s for each flip, we could get good results

64 42 M ICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS compared to 10 mins (120 flips) of measurements in Figure 3.12 c. This indicates that this active microrheology theory is less time consuming and, hence, of great benefit to biological studies which can be quite time sensitive. F IGURE 3.11: A comparison between analysis methods in water, a viscous fluid and Celluvisc a strongly viscoelastic fluid. a, Results of averaging 220 flips of a vaterite particle in water. b, Results of averaging 90 flips of a vaterite particle in 100% Celluvisc. In each figure, the blue dashed line is the complex shear modulus calculated using the theory that a linear restoring torque is assumed. The red dashed line is the evaluation using the non-linear torque theory without normalisation for each flip. The black line represents results obtained from the variable transformation analysis by Equation (3.15). All of results are compared with either theoretical values (circles) or conventional macrorheological technique (diamonds). F IGURE 3.12: A comparison between analysis with different number of flips of a vaterite particle in 50% Celluvisc solution, and the time between each flip is 5s. a-c, The averaging number of flips is 1, 12 and 120 respectively. In each figure, the blue dashed line is the loss modulus and the red line represents storage modulus of the fluid. All these values are compared with macrorheological technique (circles).

65 3.5 RESULTS AND DISCUSSION Results and discussion Celluvisc To verify our microrheological technique, we performed experiments in aqueous dilutions of Celluvisc (Allergan), an artificial tear film. Figure 3.13 a shows transient angular displacements of a particle from trap 1 to trap 2 nearly instantaneously while it takes longer to flip in the Celluvisc solution. Figure 3.13 b shows the experimentally measured torque efficiency on the particle from IR detectors. The difference between the viscoelastic natures of the two fluids is clear. We found the viscosity of water to be 0.9 mpa s for both passive and active measurements (Figure 3.14 a), which is in a good agreement with the viscosity of water near room temperature. Experiments were also performed in dilutions of 25 and 100 percent Celluvisc by weight and results (Figure 3.14) were compared with conventional rheometry. All data show a clear overlapping region of agreements between passive and active methods, and also show good quantitative agreements between our method and conventional macrorheological data. FIGURE 3.13: Typical time recorded of the trajectory and optical torque of a 3µm diameter particle flipping between trap 1 and trap 2 repeatedly switching after duration of 4s in water (blue) and 25% Celluvisc solution by weight (black). a,b, Signals from HeNe detectors and IR dectors respectively.

66 44 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS FIGURE 3.14: Storage (G ) and loss (G ) moduli vs. frequency of a solution of water, 25% and 100% w/w of Celluvisc in water. a, The loss modulus G of water is measured using both passive (blue) and active (black) methods. The storage modulus G is excluded as it is zero in water. The complex modulus G (G = G + ig ) is equal to ig in this case. b, The complex shear modulus of a solution of 25% w/w of Celluvisc in water is measured by means of both passive and active microrheology. c, Complex modulus G for 100 percent Celluvisc solution. All data also show excellent agreement with conventional technique (storage (dots) and loss (circles) moduli) Tear film Next, we demonstrate that the technique based on rotational optical tweezers is sensitive and accurate enough to provide useful measurements on biological fluids by measuring the viscoelastic properties of eye tear film coated on a contact lens. This evenly distributed multi-layered fluid structure keeps a proper degree of hydration of the cornea, contributing to overall ocular health and comfort. A collection of problems associated with tear film instability is often referred to as dry eye syndrome [8, 119]. However the structure and function of the tear film are far from being understood. Notably, the dynamics of the tear film on corneal, conjunctival and contact lens surfaces is different as well [56, ]. In general, highly viscous tears would be a detriment to contact lens comfort, while runny tears may not remain on the surface of the contact lens long enough to avoid evaporation and therefore may result in dry eye symptoms. Hence, the properties of tears coated on contact lenses are important factors for successful contact lens use. As the thickness of the tear film is only few microns, and the thickness of the fluid reduces over time, it is difficult to measure with conventional methods. A similar study up to now has not been performed on eye

67 3.5 RESULTS AND DISCUSSION 45 fluid coating on contact lenses [59, 128]. The properties of the fluid film coated on the contact lens of two subjects were examined (Figure ). Daily disposable contacts (DAILIES AquaComfort Plus) were used. After the lens was placed on the eye with a few blinks, it was removed immediately from the eye and placed on the coverslip with a small adhesive spacer. The contact lens was cut to fit the chamber, which also enabled us to assume from the optical point of view that the small piece of the lens was flat. Dried vaterite particles were then transferred to the contact lens and the chamber was enclosed with a coverslip. Experiments were performed at different times of the day (morning, mid-day and afternoon) with two subjects of the same gender and age. Each individual set of data was obtained from a contact lens being worn for a few seconds before commencement of the experiment at room temperature. Figure 3.15 illustrates the mean value of complex shear modulus for each subject at three different times of day. The results presented in Figure 3.15 confirm the complex fluid structure of the tear film and provide distinct rheological properties of this structure for each subject. FIGURE 3.15: A comparison of the complex shear modulus of tear films of two subjects (red and black). a and b show the loss modulus (lines) and storage modulus (dashed lines) vs frequency measured by means of active rotational microrheology and rotational Brownian motion for two subjects, respectively. The individual curves represent averages of three measurements taken at three different times of day. c, The full viscoelastic spectrum is obtained by combining the results shown in a for low frequency and b for high frequency. The data in Figure 3.16 is the average of the data of two measurements to show the change in the viscoelasticity of tear film at different times of the day. Figure 3.16 shows that values of viscosity in the afternoon were smaller than values measured in

68 46 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS the morning across a wide frequency range, and that the elasticity measured in the morning was lower than the values obtained in the afternoon in the low frequency range as shown in Figure 3.16 a. FIGURE 3.16: The variation of complex shear modulus (loss modulus (lines)) and storage modulus (dash lines)) vs frequency of tear films during the day. a, b, The complex shear modulus are obtained using active and passive methods. In both a and b, each line is the average value of results for two subjects. c, The complex shear modulus plot on the same scale as in a and b, showing the low-frequency spectrum agrees well with the high-frequency spectrum. The spectrum of elasticity in the afternoon (green) in the low frequency regime (as shown in a) is higher than that in the morning (blue) and noon (purple) at lower frequencies. Then it crosses the other two spectra and becomes the lowest of the three spectra, which agrees with the high frequency ones as can be seen from b, the elasticity of tear film in the afternoon is smaller than both of those in the morning and noon. The potential of rotational optical tweezers as an accurate rheological technique to access the properties of fluids in highly confined environment has been determined. Although our method of analysis provides a useful quantitative approach, further experimental investigations are necessary to explore the properties of the tear film coated on a contact lens under certain conditions. Apart from dry eye syndrome discussed at the beginning of this section, some other factors are known to affect the properties of the tear film [79, 110], for example, properties of contact lenses (i.e., water content and material), patient-related (e.g., gender, age and ethnicity), and environmental conditions (e.g., temperature and humidity). Furthermore this technique could simply be extended to study other biological fluids. For instance, the synovial fluid, filling the cavity within the synovial joint, is a viscous fluid that has lubrication, metabolic, and regulatory functions within synovial joints [54]. Changes in its rheology have been linked to joint injury and disease. It is reported that the volume of

69 3.5 RESULTS AND DISCUSSION 47 synovial fluid in normal human knee joints is 1 ml, which makes it difficult to conduct rheological measurements with conventional methods. And therefore the active microrheology presents a very attractive alternative. Rotational microrheological techniques have greater spatial resolution compared to conventional methods, since the centre of the probe can be tightly constrained and the rotational motion is less sensitive to boundary wall effects [62]. The viscous drag increased by less than 5% even for a surface as close as one particle radius [21, 69], for rotation about an axis parallel to the surface. For rotation perpendicular to the surface [69], it is difficult to experimentally observe any effect until the probe particle touches the surface [15]. In the next Chapter, we discuss in more detail wall effects on a rotating particle. We have demonstrated that the combined ultrasensitive, broad frequency response rotating photonic probe gives reliable, consistent and very accurate spatially resolved measurements of properties of complex fluids on the microscale.

70 48 MICRORHEOLOGY WITH ROTATIONAL OPTICAL TWEEZERS

71 4 Effects of confinement 4.1 Introduction As we have mentioned in the previous chapter, boundary walls have a strong influence on the drag force on optically trapped objects near any surface. Much knowledge has been gained about the effect of flat surfaces on trapped objects [62, 69]. However, to date the effect of non-linear systems at microscopic scale on rotational motion of micro-objects has not been studied much. In addition, understanding and controlling the behaviour of objects at the micrometre scale such as biomolecule membranes and vesicles are crucial to the study of hydrodynamic process in cell biology and related microfluidic systems. For this reason, recent years have witnessed advances in the development of techniques that extend 49

72 50 EFFECTS OF CONFINEMENT the applied force sensitivity for micron sized objects of interest down to the piconewton range, such as atomic force microscopy, magnetic tweezers and optical tweezers [86]. These technologies have enabled the discovery of the dynamic nature of cell biology including DNA replication and transcription, and biological transport across membranes. Due to the torsional properties of biological objects such as single DNA molecules, bacteria and sperm, torque or force is used as a physical parameter to directly and quantitatively study biological systems. There is growing interest in characterising torques on objects in biologically relevant processes as well as microfluidic devices. The dynamics of a rigid particle suspended in a fluid is typically dependent upon three key factors: the geometry of the probe particle and surrounding boundaries, external forces applied to the particle and the properties of the fluid. Thus the motion of the objects will also give information on the surrounding medium, which allows microrheological study of biological materials such as cytoplasm. In biological systems and processes, such measurements are performed in confined environments, where the influence of boundaries on the motion of the probe becomes non-negligible. Hence, novel detection tools not only require simultaneous measurement of force and torque but are also valuable for calibrations of the hydrodynamic interactions between the probe and the surrounding imposed by near boundary walls. Thanks to recent advancements in 3D printing technology, it is possible to fabricate micro-optical components and microfluidic elements such as filters, micromachines or microfluidic chips [118]. This technology shows great promise for objects of almost any shape or geometry to be synthesized on demand. For example objects suitable for use in optical manipulation with transfer of angular momentum of light can be produced in this way as well as arbitrary shape curved walls. In this chapter, direct measurements of constrained rotational motion of a birefringent sphere trapped in a fluid near micron sized curved surfaces by using rotational optical tweezers are performed. An ultra-high resolution 3D nano-printing technique was used to print walls of the precise curved micron size structures. Experiments inside a spherical vesicle (see more details in Chapter 5), which can be used as a

73 4.2 EARLY WORK 51 biological cell model, are also performed. These experiments are useful for finding out modification of the torque on particles, and thus enable accurate hydrodynamic simulations at the micron-scale. This opens the potential for new sensing approaches under more complex conditions, allowing both dynamic and microrheology studies of biological systems. 4.2 Early work When an object suspended in a fluid is moving close to the boundary wall, the drag force or torque from the fluid acting on this particle is increased relative to that when it is far from the wall, which results in the reduction of the particle s mobility. Studying the hydrodynamic interactions between the particle and the fluid generated by the near wall, known as wall effects, is essential for determining the anisotropic motion of the probe and accurate microrheological properties of the surrounding fluid. The wall effects of an infinite plane on the translation and rotation of a sphere are well known [21, 30, 62, 92]. In terms of studying the behavior of more complex systems, the hindered translational diffusion has been studied extensively for spherical particles moving between two plane walls [38, 49, 66, 70]. In addition, cylindrical geometries were also studied in a few cases, such as macroscopic measurements of the drag coefficient of a millimetre-sized sphere settling along the axis of a closed 3D cylinder [63, 64], and microscopic measurements of diffusion along the axis of long cylindrical microchannels or closed cylindrical cavities [27, 32, 37, 135]. So far, both theoretical and experimental studies beyond simple confining geometries are still lacking. In terms of rotational motion of a spherical particle near a wall, previous experiments have been performed studying wall effects for both translational and rotational motion of a sphere near a flat wall based on optical tweezers [62]. It has been shown that drag enhancement due to a solid flat wall is much more important for translations than rotations, the former decaying as the inverse particle-wall distance and the latter as its inverse third power (see Figure 4.1). In other words, the modification

74 52 EFFECTS OF CONFINEMENT for the drag torque is not necessary when the separation between the particle and the boundary exceeds the particle diameter. This indicates that a rotational probe combined with microrheological techniques is able to access local properties of biological fluids within ultra-small volumes. Therefore, a rotating birefringent particle can be employed as a sensitive microrheometer based on optical tweezers, whereby the advantage is gained by its higher spatial precision compared to translation techniques. FIGURE 4.1: Experimentally measured drag coefficients as a function of the separation of the the sphere from the surface. Results for translation (squares) and rotation (circles). The solid lines are the power series expansions, i.e. Faxén s correction [62]. Results are for the translational drag perpendicular and parallel to a surface, and the rotational drag for spinning about an axis perpendicular and parallel to the surface normal. For wall effects of boundaries on a rotating particle, the infinite plane wall effect has been studied. One approach is described by Faxén s corrections [62]. Since Faxén s correction is a power series approximation, the wall effect is not accurate when the distance between the particle and the boundary is smaller than one radius of the particle. Though the infinite plane wall effect cannot simply be expressed analytically, Chaoui and Feuillebois [21] provide coefficients for a polynomial fit to a series solution derived by Dean and O Neill [30]. This allows the wall effects of an infinite plane on a rotating sphere to be calculated numerically in Matlab as shown in Figure 4.2. The left figure shows the geometry of the configuration that we are considering. We are studying the wall effects when the sphere with radius a approaches the wall and is at

75 4.2 EARLY WORK 53 a distance d from the wall. In addition, for a particle between two plane walls, the viscous torque on a sphere of radius a rotating between two plane boundaries 3a apart would experience a viscous torque only 1.05 times greater than the same sphere rotating in an infinite domain [69]. FIGURE 4.2: Particle near a wall. The left part of the figure shows the geometry of the spherical particle near a surface. The right figure shows wall effects on a rotating sphere as a function of the separation of the sphere from the surface. As illustrated in Figure 4.3, the particle rotating at a constant speed is centred inside the spherical wall, in this special case, wall effects of concentric spheres can be expressed analytically, and is given by: W e = b3 b 3 a 3 (4.1) where b is the radius of the spherical boundary wall and a is the radius of the particle.

76 54 EFFECTS OF CONFINEMENT TABLE 4.1: A comparison of wall effects from Equation (4.2) with exact solutions for a particle rotating at the centre of an infinite cylinder. FIGURE 4.3: Wall effects of concentric spheres. As both radii become equal, b/a 1, the wall effect tends to infinity. As the radius of the outer sphere tends to infinity, b/a, the wall effect tends to one. For a particle that rotates in an infinitely long circular cylinder with radius of R 0, for small (a/r 0 ), the wall effect on the particle is approximately given by [17, 20] W e = [ (a/R 0 ) (a/R 0 ) 10 + O(a/R 0 ) 14 ] 1. (4.2) Table 4.1 shows comparison of approximate and exact results for small a/r 0. Even for a/r 0 as large as 0.9, the error from Equation (4.2) is less than 2%. Moreover, in terms of the wall effect for a/r 0 near unity, the wall effect is obtained by using techniques well-known in lubrication theory expressed as [17]

77 4.3 WALL DESIGN AND FABRICATION 55 W e = [ 1 4(a/R 0 ) 2 ( (a/r0 ) 1 (a/r0 ) 2 tan 1 1 (a/r 0 ) ) 1/2 2(a/R 0 ) ( ) a 2 π] 2 R 0 (4.3) and the expression can be simplified by an approximation where we set a/r 0 =1 in all but the dominant term in Equation (4.3), W e = π 2 3/2 1 (a/r 0 ) = (a/r0 ). (4.4) The results are compared in Figure 4.4. Note that the exact numerical results are valid for sphere-cylinder diameter ratios from 0 to under situations where the moment of inertia is negligible. FIGURE 4.4: Wall effects for infinitely long circular cylinder [20]. W e vs (1 a/r 0 ), the solid line represents the exact values computed from Equation (4.3), the dashed line shows an approximation obtained by Equation (4.4). 4.3 Wall design and fabrication In this chapter, the influence of wall effects on rotational motion in more complex systems will be discussed. To appropriately identify the structure of the walls used in our optical system, we consider an inclined wall (Figure 4.5 b) rather than a vertical wall (Figure 4.5 a) to minimise distortions caused by laser beam interaction with the wall. The numerical aperture (NA) of an optical system is a dimensionless number

78 56 EFFECTS OF CONFINEMENT that characterises the range of angles over which the system can accept or emit light. In particular in microscopy, NA is defined by the maximal half-angle θ of the cone of light that exits the objective, as NA = n sinθ (4.5) in which n is the refractive index of the medium. The NA of the objective used in our experiment is about 1.2; thus the angle of inclination of the wall is set to be 70 so that the trapping laser would not interact with the wall as shown in Figure 4.5. FIGURE 4.5: a, The SEM image of a typical straight wall with height of 40 µm produced using a 2PP technique. b, In order to minimise the laser distortion by the wall, instead of a straight wall (as shown in a), an inclined structure was used in our experiments, and the height of the wall is 4 µm in the middle part where the particle is trapped. In terms of production of the walls, Nanoscribe was used which is a nanoscale 3D direct laser writing systems to print microstructures of walls in 3D. Nanoscribe is a facility that combines a 3D printing technology with user-friendly software and innovative materials as shown in Figure 4.6.

79 4.3 WALL DESIGN AND FABRICATION 57 FIGURE 4.6: Nanoscribe (from Nanoscribe GmbH company in Germany) is the 3D laser lithography system used in our experiments to print micro-sized walls. This high resolution 3D printer enables the rapid fabrication of nano-, micro- and mesostructures with feature sizes starting from hundred nanometres and heights up to several milimetres with layer thickness well below 1 micrometre. The direct laser writing systems based on two-photon polymerisation (2PP) allows for additive manufacturing and maskless lithography with the same device. The structure is made out of a UV photopolymerising resin. During the fabrication process, a pulsed near-ir laser is focused inside a UV photopolymerising material that is photosolidified in only small volumes within the depth of focus [102]. When a femtosecond laser pulse is closely focused onto liquid state monomers, the resulting two photon excitation initiates chemical processes and the formation of features with resolution close to 100nm in size (Figure 4.7). Only a highly localized area around the centre of the focused laser beam can be solidified due to the fact that the absorption probability of two photon excitation is proportional to the square of the energy density of the light, whereas it is proportional to the density itself for single-photon absorption [78]. It can be seen from Figure 4.7 how a photopolymerised volume changes when using 1 photon excitation versus 2 photon excitation. The wall fabrication procedure consists of three steps: Step 1: Model all structures of the wall using the software AutoCAD to draw threedimensional virtual figures using the format STL. There are three structures of walls

80 58 E FFECTS OF CONFINEMENT F IGURE 4.7: Regions of one-photon and two-photon polymerisation. used in our experiments: plane walls, curved walls and cylindrical walls with different diameters. Before printing, we use the software called Describe to generate print jobs in Nanoscribe s proprietary General Writing Language (GWL). Slicing as well as hatching of the 3D design is handled by DeScribe as shown in (Figure 4.8). 3D design can also be programmed manually using DeScribe to control the parameters of the writing process such as laser power, scan speed and sample movement. For the printing mode, there are two different modes, a piezo-mode for arbitrary 3D trajectories and a galvo-mode for ultra-fast structuring in a layer-by-layer process. Galvo-mode is used here to print these walls. F IGURE 4.8: 3D preview of structures of the plane wall, curved wall and cylindrical wall assembled on DeScribe. Step 2: The structure is ready to print, and is printed on a cover slip. The cover

81 4.3 WALL DESIGN AND FABRICATION 59 slip needs to be cleaned carefully. It is washed with acetone solution first and then isopropanol solution. It is recommended to repeat this procedure three times. Then the cover glass is put on the sample holder with a drop of resin on top. The resin used is called IP-Dip, which has a refractive index of 1.52 at the 780 nm wavelength of the laser. After inserting the sample holder into the laser lithography system, the system is controlled by the software NanoWrite. Note that we need to find the interface between the cover slide and the resin manually as the location of the focus of the laser as it starts to write is very important. It determines whether the structure is successfully printed on the cover slide or not. To do this, first the interface is roughly found automatically by the software NanoWrite, and then the position of the objective is manually adjusted to find the interface. To test that the interface is found correctly, we print a simple structure of a square only tens of nanometres in height. Once the objective is in right position, the walls are easily printed automatically by NanoWrite (Figure 4.9). Note that change of laser power can be needed, if it is too high; otherwise it burns the resin. FIGURE 4.9: Schematic of the 2PP technique of Nanoscribe galvo-mode for fabrication of micron sized walls out of the resin. Step 3: When the printing job is finished, the excess resin on the cover slide needs to be removed. The sample is put in a solution of propylene glycol methyl ether acetate (PGMEA) for 8 minutes and then put it into a solution of isopropanol for a few seconds. It is important to make sure that the sample is not left in isopropanol for too long (less than 2 minutes), since it could lead to the structure dissolving.

82 60 EFFECTS OF CONFINEMENT 4.4 Wall effects on a rotating vaterite particle Experiments As I introduced before in chapter 2, the optical torque on a birefringent vaterite particle can be determined by monitoring the change in circular polarisation of the beam [89], and is given by τ = σp/ω (4.6) where σ is the change in circular polarisation, P is the power of an incident beam, and ω is the angular frequency of the optical field. Also, when held in a circularly polarised optical trap, vaterite particles spin at a constant speed. Notably, this indicates that the optical torque, τ, driving the particle is balanced by drag torque τ D from the surrounding medium. In the low Reynolds number limit, the drag torque on a sphere rotating in a fluid far from a boundary is simply written as, τ D = β 0 Ω 0 (4.7) where β 0 is the drag constant and Ω 0 is the angular frequency of the particle in a free fluid. The drag coefficient has the value β 0 = 8πη 0 a 3, η 0 is the viscosity of the surrounding fluid and a is the radius of the probe. Thus, the drag coefficient β 0 = τ/ω 0 (4.8) can be determined as a function of optical torque and rotation rate of the particle. When the particle moves towards the boundary wall, the local drag coefficient β i increases as the motion of the particle is hindered due to the boundary. If measurements are made on the same probe with a constant torque, then the correction for the drag coefficient is the inverse of the rotation frequency of the particle, β i β 0 = Ω 0 Ω i. (4.9)

83 4.4 WALL EFFECTS ON A ROTATING VATERITE PARTICLE 61 We also note that an overestimate viscosity η i η i = Ω 0 Ω i η 0 (4.10) is simply a function of η 0 and rotation rate of the vaterite particle. Experimentally, rotation rates of the trapped particle were determined by monitoring the transmitted light through a linear polariser. The experimental system used here (Figure 4.10) is the same as was described in Chapter 3 but only one optical trap is needed. A λ/4 plate placed in the trapping laser (1064 nm) beam path converts the beam to a circularly polarised beam which allows spinning of vaterite particles. The weak circularly polarised HeNe beam is used in the same way as before to detect the rotation rate of the particle. During the experiment, the vaterite particle was held in the optical trap and brought closer to the boundary wall by moving the x, y, z translation stage (Mad City Labs) with nanometre precision. After passing through the sample, the transmitted trapping and detecting beams were collected by the condenser and recorded by detectors respectively. FIGURE 4.10: Experimental setup for measuring wall effects. Top inset: schematic representation of the sample chamber used in experiments.

84 62 EFFECTS OF CONFINEMENT Plane and curved walls We first validate our method for measurements of wall effects by testing it on a plane wall for which the wall effects are well known. The experimental values of β i /β 0 as a function of d/a (d is the gap width between the particle and the wall) correspond well to those of numerical results for drag coefficients with precision better than even applicable for small gap width down to [21]. We also measured the polarisation shift of the trapping laser in order to monitor the optical torque exerted on the particle and noticed it was stable during the measurement (Figure 4.11). This indicates that the quality of the optical trapping system is not significantly degraded by this structure of the wall. Uncertainties exist in both the size of the individual particle and in the location where it is measured due to the non-spherical shape of vaterite particles and imaging system aberrations (Figure 4.11). FIGURE 4.11: a, Overlapping images of a trapped vaterite particle near a plane wall. The particle is trapped with optical tweezers, and the plane wall is moved by a controlled piezo stage. b, Experimental results (blue) of wall effects of a rotating particle near a plane wall compared with numerical results (red). c, Optical torque efficiency was measured as well during the experiment to make sure the laser beam was not distorted by the wall. When the shape of the wall is not easy to determine, and it becomes curved, there is a dearth of information on the influence of wall curvature on rotational motion of the particle. An interesting question is whether and when the curved wall can be treated as a single flat wall. To address this, experiments with vaterite particles with diameter of 3 4 µm rotating near curved walls with radius R of 2, 2.5 and 3

85 4.4 WALL EFFECTS ON A ROTATING VATERITE PARTICLE 63 µm were performed. Experimental data was compared with the exact solution for a plane wall and are shown in Figure The behaviour of the particle for the case of a 3 µm curved wall is the same as the case near a plane wall. This indicates that the curvature can be neglected for R 2a. We also note that very close to the wall the rotational behaviour is largely dominated by the plane wall. FIGURE 4.12: a, Overlapping images of a trapped particle with radius of 1.5 µm near a curved wall with radius of 3 µm. The curved wall is moved by a controlled piezo stage. b-d, Experiments of a rotating particle with radius of 1.5 µm near curved walls with radii of 2, 2.5 and 3 µm respectively. Experimental results (black) were compared with the numerical results (blue) for a plane wall Cylindrical and spherical walls To further understand the effects of curved walls, we study the impact of cylindrical walls on the rotational motion of the particle as well. We trapped a vaterite particle near the centre of cylindrical walls with five different radii (2.5, 3, 3.5, 4 and 5

86 64 EFFECTS OF CONFINEMENT FIGURE 4.13: a, An image of a trapped vaterite particle at the centre of cylindrical walls with radius of 5 µm. b-e, Experiments of four different particles. Each particle (with radius of a) was trapped at the centre of five cylindrical walls with radii of 2.5, 3, 3.5, 4 and 5 µm. Experimental results of wall effects (black) were compared with both results for a plane wall (red) and an infinite concentric cylindrical wall (blue). µm), and compared the results with both the cases of a single flat wall and exact values for infinite cylinders as mentioned before [20]. As the size of the cylindrical wall becomes larger, the wall effect decreases as shown in Figure It shows experimental results with four different vaterite particles. We note that some of the results are smaller than the theoretical values of infinite cylinder walls. Apart from the error in determining the relative distance between particle and the boundary wall, the particular ring structure of the inclined wall used in my experiment should be also considered as the height of the wall is only about 4 µm, which cannot be treated as an infinite cylinder. To model wall effects of this structure on rotational motion. Gibson et al. in our group has evaluated the wall effects of this structure on a sphere positioned on the z axis at differing heights. While the sphere is rotating on the z axis the flow is axisymmetric, the radial and vertical velocity components are zero and the continuity equation is

87 4.4 WALL EFFECTS ON A ROTATING VATERITE PARTICLE 65 satisfied. The Stokes equation characterises the rotational velocity component by 0 = 2 u θ r r u θ r + 2 u θ z 2 u θ r 2. (4.11) This equation can be solved numerically using Matlab s Partial Differential Equation Toolbox, which utilises a Finite Element Method (FEM). The derivatives of flow velocity at an intermediate boundary between the sphere and cylinder are obtained from the solution which allows the wall effects to be calculated. Figure 4.14 shows the theoretical model for the ring inclined wall. The size of the outer boundary sphere is set to be big so that only the boundaries around the particle influence the motion of the particle. The results as shown in Figure 4.14 were compared with the results of infinite cylinders. The wall effects of the model agree with those of an infinite cylindrical wall when the particle is placed at the centre of the ring. However, if the particle is vertically offset from the centre of the ring wall, the wall effect acting on the particle changes. Figure 4.14 shows one example of the ring with radius of 2.5 µm and the particle with radius of 2 µm. The wall effect decreases when the distance between the centre of the particle and the ring becomes larger, and it drops rapidly as the centre of the particle moves out of the ring. As a next step we calculate the wall effects for a particle with radius of 2 µm at the centre, 0.25 µm below and 0.5 µm below the rings with radii of 2.5, 3, 3.5, 4 and 5 µm (Figure 4.15). The experimental data as shown in Figure 4.13 are compared with the theoretical model for these three conditions. From Figure 4.15 we can see that the wall effect drops quickly once the equatorial plane of the spherical particle is slightly outside the ring structure, and then it does not vary much when the particle moves further away. It is interesting to point out that the equatorial plane of the particle plays an important role in the determining the wall effect. Some of the experimental results did not agree very well with the model of concentric ring structure. This could be due to both inaccuracy in the position detection and the vertical location where the particle is trapped. However, when the radius of the cylindrical wall is three times larger than the radius of the particle, the wall effect on the particle rotating at the

88 66 EFFECTS OF CONFINEMENT FIGURE 4.14: a, Theoretical model for the ring wall. b, Numerical values of wall effects of a rotating particle at the centre of the ring wall (red) were compared with wall effects of a plane wall (purple), concentric spheres (green), concentric infinite cylinder walls (black). Blue dots are exact solutions for a concentric infinite cylinder wall as shown in Table 4.1. c, Numerical results for wall effects (black) of a 2 µm rotating particle at the centre of a ring wall with 2.5 µm in radius and 4 µm in height varies with distance of vertical offset from centre. Infinite cylinder (blue) represents the value of this particle at centre of a infinite cylinder. Cylinder bounds (red) shows the region where the equator plane of the spherical particle is within the ring. centre can be ignored. Care should be taken to ensure that during the experiment the particle is trapped at the centre of the wall. Therefore, before driven close to wall, the particle should be moved up and down to find the location where the rotation rate of the particle is smallest. This is because, as shown in Figure 4.14 c, when the equatorial plane of the particle is almost centred in the ring, the wall effects from the ring wall on the particle reached the maximum value. Next, the vaterite particle was moved from the centre toward the side of the cylindrical wall along the radial direction. In a similar way, the experimentally measured correction to rotational drag coefficients increased as the trapped particle was brought closer to the side as seen in Figure In Figure 4.16, we compare the drag coefficients for four different sizes of cylindrical walls (6, 7, 8 and 10 µm in diameter). Also, we found that when the radius of the wall is larger than 5 µm, the cylindrical wall behaves like a plane wall. We have shown several effects of neighbouring boundaries on the steady flow induced by a rotating particle. Remarkably, the rotational corrections do not become significant until very close to the boundaries, allowing optical application and measurement

89 4.4 WALL EFFECTS ON A ROTATING VATERITE PARTICLE 67 FIGURE 4.15: a, The model of the equator plane of the particle with radius of 2 µm is offset from centre of 0.5 µm. b, Theoretical results for wall effects of a particle at the centre with 0 vertical offset(red), vertical offset from centre of 0.25 µm (blue) and 0.5 µm (black) of rings with radius of 2.5, 3, 3.5, 4 and 5 µm. c, Experimental results from Figure 4.13 b-e are compared with theoretical results in b. FIGURE 4.16: a, Overlapping images of a particle moving from centre towards the edge of the cylindrical wall with radius of 5 µm. b-e, Experimental results of eccentric cylindrical walls with radii of 3, 3.5, 4 and 5 µm (blue) are compared with the results of a particle with radius of 1.5 µm rotating near a plane wall (black), at the centre of cylindrical walls (red) and at the centre of spherical walls (green).

90 68 E FFECTS OF CONFINEMENT of torque to operate under varying biological conditions where the effect becomes more complicated. We aim at using this method for measurements inside a biological cell. As an example of cellular structure, we produced liposomes for our experiments since they resemble the basic compartment structure of all biological cells (see details in chapter 5). These liposomes were formed by transferring lipid-coated water-in-oil microdroplets, which occasionally contained a vaterite particle, through the interface between the oil phase containing lipids to the water phase. Figure 4.17 shows an approximately 20µm liposome with a vaterite probe of diameter 4 µm inside. The rotation rate of the particle was measured at each position as a function of distance from the membrane of the liposome (Figure 4.17). No significant difference, which is about 2% of the maximum angular velocity, existed between the measured value at the centre and near the membrane. F IGURE 4.17: a, Image of a liposome with a vaterite particle inside. b, Experimental measured rotation rate of the particle at different places inside the liposome from centre to the edge. 4.5 Conclusions In this chapter we have studied wall effects on the rotational motion of the particle in more complex systems including curved walls, cylindrical walls and spherical walls.

91 4.5 CONCLUSIONS 69 In terms of curved walls, it was found that the curved wall can be treated as a flat wall for R/a >1.5, and for cylindrical walls this happens at R/a >3. When doing experiments at the centre of the cylindrical wall, we can ignore the wall effect if R/a >3. We did not see distinct wall effects from spherical membranes about cellular sizes. We also have developed a theoretical model to calculate wall effects on rotational motion and in the future the model will be improved to model complex systems such as cells. Therefore, our microrheometer based on rotational optical tweezers allows the measurement of viscoelasticity in more confined systems. It demonstrates great promise for future applications to study biologically relevant fluids such as cytoplasm.

92 70 EFFECTS OF CONFINEMENT

93 5 Biomimetic cell model liposomes 5.1 Introduction So far we have described how rotational optical tweezers can be used to measure local viscoelastic properties of fluids over a wide frequency range. The details of the technique are discussed in Chapter 3. Chapter 4 described the studies of wall effects from near boundaries on the rotational motion of a particle when measurements are taken in confined geometries, which gives us corrections for accurate microrheology. Our ultimate aim is to build an understanding of several important properties of living cells by making measurements inside cells using laser micromanipulation. These measurements could be made using indigenous parts of a cell or by introducing a small particle into the cell and studying effects exerted on the particle that vary with different conditions that the cell is under. These could be measured by monitoring changes 71

94 72 BIOMIMETIC CELL MODEL LIPOSOMES that these conditions introduce on the trapped particle in terms of its position, force and torque. In preparation for these types of measurements we first conduct them in a somewhat simpler-than-real cell model. In our case, we choose liposomes as a cell model. In this chapter, we will discuss the details of using liposomes as cell models to study microrheology inside them. This chapter is concerned with different methods we use to make liposomes containing our vaterite particles. I begin by introducing this biomimetic cell model. Cytoplasm surrounds all the basic organelles in living cells. Deformability of the cytoplasm has a crucial impact on cellular and subcellular processes such as replication and intracellular trafficking [31]. Therefore, viscoelasticity of the cytoplasm is an important physical parameter which helps us to investigate and understand cellular mechanism. Moreover, due to the crowded environment within cytoplasm, the Brownian motion of particles is restricted by proteins and other molecules. In other words, the subdiffusive motion of the particle gives us information on the crowdedness of the cytoplasm at a molecular level [130]. Accordingly, the viscoelasticity of the cytoplasm of living cells is not a static parameter, but is instead dependent on a wide variety of factors [48] such as molecular motors, and varies as the cell undergoes reproduction [103]. FIGURE 5.1: Schematic of a cell (left) and a cell model-liposome (right) that encapsulates a hydrophilic compound (purple) in the interior and intercalates a lipophilic compound (green) within the lipid bilayer. As the cytoplasm is a dynamic and very complex environment, it is a challenging

95 5.1 INTRODUCTION 73 task to quantitatively investigate the mechanical properties of the cytoplasm [36]. Despite some work being done on these systems, e.g., fluorescence correlation spectroscopy [47], trapping cellular organelles [5, 80, 107], video particle tracking microrheology [22], force spectrum microscopy [48] and rotational magnetic spectroscopy [12], it is still extremely difficult to make rapid measurements inside living cells. More studies are still required to discover the full biological implications of mechanical properties of cytoplasm. As a biological cell has a rather complex structure, to gain a detailed understanding of how the rotational microrheology technique works inside living cells, it is essential to have a simple cell model first and measure the viscoelasticity inside it. Figure 5.1 represents the schematic of the liposome cell model. We use liposomes as a cell model in our experiment and in this chapter I will discuss the production of liposomes and some measurements conducted on and inside the liposomes. This bottom-up strategy will provide us a first idea and better understanding of the fluid inside cells. A liposome is a microscopic spherical self-closed structure formed by one or more lipid bilayers with an aqueous phase inside and between the lipid bilayers [117]. It has been widely studied as a simple model of living cells trying to investigate the physical properties of biological membranes and biological processes of living cells, since cell-sized liposomes are similar to natural cell structures in terms of size and membrane composition [50, 67, 91]. For example, many studies have tried to evaluate lateral lipid heterogeneities, membrane budding and fission, activities of reconstituted membrane proteins, and membrane permeabilization caused by added chemical compounds [126]. Moreover, due to their large size, liposomes can be observed by optical microscopy, allowing for real time monitoring and manipulation by optical tweezers. Therefore, in such a microrheological experiment, it is worthwhile to study the viscoelasticity inside a liposome so that we can investigate the effect of the membrane separately. This would provide us with a first idea of viscoelastic properties inside an real cell.

96 74 BIOMIMETIC CELL MODEL LIPOSOMES 5.2 Liposome formation In terms of cell models, we aim to produce cell-size liposomes ( 15 µm in diameter) with vaterite particles inside for microrheology measurements. There are a variety of methods available to produce liposomes [1]. The preparation methods have been classified depending on mean size, polydispersity and lamellarity of liposomes obtained as these are challenging factors with almost all preparation methods [96]. Considering the size and encapsulation of vaterite particles, we tried four different methods as described below Lipid-film hydration FIGURE 5.2: Representation of the process of the formation of liposomes by controlled hydration of a film of bilayer-forming lipids deposited at the base of the flask. a, Lipid film deposited at the base of the flask. b-f, When the PBS buffer was added, the lipid film started swelling with certain time and finally the liposome was formed. The first method we tried is lipid-film hydration (Figure 5.2). This method is also known as spontaneous swelling, natural swelling or the gentle hydration method [120]. A.K. Giddam from the School of Chemistry and Molecular Biosciences at UQ helped

97 5.2 LIPOSOME FORMATION 75 me with this method. In this method, the lipids (Dipalmitoylphosphatidylcholine (DPPC, Avanti polar lipids), dimethyldioctadecylammonium bromide (DDAB, Sigma- Aldrich) and cholesterol (Avanti polar lipids) were taken in the ratio of 5:2:1) were first dissolved in chloroform in a flask, and then the solvent was removed using a rotary evaporator to form a lipid film at the base of the flask. We took 5 mg DPPC, 2 mg DDAB and 1mg cholesterol for each experiment. DDAB here is used to produce electrically charged liposomes. The flask was then connected to a freeze dryer for at least 6 hours to make sure that the chloroform was removed completely. Liposomes were formed when phosphate buffered saline (PBS) buffer at 50 C was added. The buffer solution should be added slowly in increasing quantities about three times to form large vesicles (final volume of the buffer solution is 2 ml). Liposomes with multilamellar structure and different sizes were formed with this method as shown in Figure 5.3 a. FIGURE 5.3: Images of liposomes made by lipid-film hydration method formed by lipid-film hydration composed of 62.5% dipalmitoylphosphatidylcholine(dppc), 25% dimethyldioctadecylammonium bromide (DDAB) and 12.5% cholesterol. In order to put vaterite particles inside liposomes, we tried different ways. First, we tried to add vaterite particles to PBS buffer solution and used this buffer containing particles to produce multilamellar vesicles. Another way is to form the lipid film in the presence of vaterite particles, then add pure PBS buffer solution to form liposomes. However, using these methods, we did not get liposomes with vaterite particles inside. We thought that these difficulties were due to the fact that it is difficult to encapsulate these particles inside liposomes with gentle spontaneous swelling of lipid film since these particles are relatively big in comparison to the size of liposomes. Moreover,

98 76 BIOMIMETIC CELL MODEL LIPOSOMES as both the vaterite particles and lipid film are charged, as shown in Figure 5.3 b, it seems that the liposome was coating around the particle Electroformation Another method for production of liposomes is electroformation. The electroformation method is currently the most widely used method for the preparation of giant vesicles, as it is very reproducible and allows for long term observations and manipulation of individual liposomes [126]. Below we describe our recipe for producing liposomes based on the method described in [57, 127]. The lipids were dissolved in chloroform to form 2 mg/ml lipid solution. For the observation of the vesicles with fluorescent microscopy, the dyes DiIC 18 (Ex, 549 nm; Em, 565 nm) or DiOC 18 (Ex, 484 nm; Em, 501 nm) were added to the lipid solution. Once the lipid solution was prepared, a small drop ( 20 µl) of lipid solution was placed onto an ITO glass slide and spread evenly on the surface using a big plastic pipette tip. Two ITO glass slides with lipid solution were placed in a vacuum desiccator for at least 2 hours to form a lipid film. In this process the evaporation of the organic solvent chloroform is completed. A closed chamber sealed with silicone grease was assembled from these two ITO glass slides with a copper tap for each slide (the sides with conductive coating should face each other) and a 1 mm thick rectangular teflon spacer with two holes as the solution inlet and outlet as shown in Figure 5.4 b. The solution (100 mm sucrose) used to form liposomes was then introduced through the inlet to fill the chamber ( 2 ml). After this, the chamber war connected to an AC field function generator for the electroformation process of liposomes as shown in Figure 5.4 a. The procedure started from the input of an AC voltage with (peak-to-peak) amplitude of 0.5 V and frequency of 10 Hz for 20 min. Then the voltage was increased gradually by 0.5 V per 20 min to 2.0 V. The voltage was then further increased to 2.5 V and kept at this value (2.5 V, 10 Hz) for another 2 hours to further grow liposomes. Then liposomes were detached from the ITO glass substrate by lowering the field frequency to 5 Hz and the voltage was set to 0.5 V for 20 min. Note that if bigger liposomes are

99 5.2 L IPOSOME FORMATION 77 F IGURE 5.4: a, Schematic view of electroformation method. The liposomes are grown on the ITO slide in water by electroformation in AC field. b, The ITO slide, teflon spacer with with holes and silicone grease used to seal the the chamber. required, one can decrease the voltage when liposomes are growing, for example, 2.5 V, 10 Hz for 2 hours could be changed to 0.5 V, 10 Hz for 40 min and 1 V, 10 Hz for 2 hours. The resultant vesicles were gently transferred from the electroformation chamber to a clean vial and stored at 4 C. F IGURE 5.5: Fluorescence images of liposomes made by the electroformation method. Figure 5.5 shows liposomes that were produced by the electroformation method. For observation, we took a small amount of the vesicle solution and added 100 mm glucose solution to it (about times the volume of the vesicle solution). The glucose

100 78 BIOMIMETIC CELL MODEL LIPOSOMES solution helps the liposomes to settle down, so they are easy to observe under the microscope. The size of liposomes we made was about 20 µm. In order to put vaterite particles inside liposomes, similar to the lipid-film hydration method, we either tried to add particles to the growing solution, or dry lipid solution with particles to form lipid films with dried vaterite particles. From Figure 5.5 c we can see both particles and liposomes, but it is still hard to have a liposome as well as the particle in it. We thought it is due to the problem of charge on the particles and that the particle seemed to attach to the membrane. The other problem might have been the size of the particle. However, using this method we could produce plenty of very spherical giant unilamellar vesicles. These can be seen in Figure Microfluidic device In order to make liposomes with particles inside, we also tried another method using a microfluidic device, which is known as an approach to form liposomes with controlled size and higher encapsulation efficiencies than other techniques [121]. The design of the microfluidic channel as shown in Figure 5.6 is based on the design in [83]. As shown in Figure 5.6 a, liposome production using microfluidic channels begins with flow-focusing generation of uniform and controllable lipid-stabilized droplets in oil (dodecane was used here) that contain the cytoplasmic aqueous solution (water in our experiment) inside. The droplets in oil containing lipids (Egg PC) travel to a junction where they merge and co-flow with the extracellular aqueous flow where a lipid-stabilized oil/water interface is formed. Then the triangular guide posts force phase transfer of the droplets from the oil flow through the interface to the extracellular aqueous flow. Finally, vesicles will form and collect through the output channel. The flow rate of the oil must be bigger than the flow rate of water at the T junction to form lipid-stabilized water droplets. To have liposomes with particles inside, we add particles to water trying to form water droplets containing particles at the beginning. The formed liposomes still did not contain vaterite particles. Since the size of the channel is about 60 µm, and particles

101 5.2 LIPOSOME FORMATION 79 tend to clump together within the channel, either decreasing the density of particles or the size of particles would help. However, decreasing the density would decrease the encapsulation efficiency of liposomes. Moreover we did some simple tests on small particles ( 1 µm radius silica beads) with low density to produce water droplets in oil, and we found that to have these beads inside water droplets, the size of droplets needs to be big enough. For example, for 1 µm silica beads the size of droplets is more than 100 µm in diameter as shown in Figure 5.7 b. FIGURE 5.6: Schematic illustration of microfluidic assembly line. a, Oil/lipid flow is introduced at the top left, focusing the cytoplasmic aqueous input to form lipid-stabilized droplets that containing the cytoplasmic aqueous. Liposomes are formed when the droplets pass through triangular posts. The liposomes are collected through the output channel at the bottom right. b, Experimental procedure for forming water droplets in an oil flow: the droplet flow merges with an extracellular aqueous input (blue) to form a lipid-containing oil/water interface adjacent to the droplet flow. FIGURE 5.7: Images of liposomes made by microfluidic method. a, The liposome made without particles; the size is about 20 µm. b, The water droplet containing silica bead in oil.

102 80 BIOMIMETIC CELL MODEL LIPOSOMES Water in oil droplet transfer We also tried a method called the water-in-oil (w/o) emulsion, which allows for high entrapment yields [97, 136, 137]. First, lipids (Egg PC powder) were added to oil prior to ultrasonication for about 60 min at 50 degree and vortex mixing (final lipid concentration in oil was 1 mm). After this step water was introduced at the bottom of another glass tube and covered with the oil containing lipids. This solution was left for about 2 hours at room temperature in order to form a stable oil-water interface. As shown in Figure 5.8, to obtain lipid-stabilized water droplets, we added a small amount water to the oil (the volume of oil is bigger than water) containing lipids and emulsified the mixture by pipetting. Then w/o droplet solution was added to the solution with a oil-water interface; the droplets spontaneously crossed the interface between the oil phase to the water phase. These w/o droplets were converted to liposomes during this transfer. Then we waited for a few hours so that all droplets converted to liposomes completely. FIGURE 5.8: Schematic representation of key steps for producing liposomes with water in oil droplet transfer method. In order to encapsulate vaterite particles inside liposomes, we add particles to water when producing the w/o droplet solution. In addition, we added salt (0.1 M) as well to water which not only helps to neutralize the charge of vaterite particles but also promotes the water droplet migration. We first dissolved lipids in dodecane, as shown in Figure 5.9 a. The liposome we got was more like an oil-in-water droplet since once the particle was trapped with optical tweezers the droplet was trapped as well, and we were not able to move the particle within this structure; the particle remained

103 5.3 MEASUREMENTS INSIDE LIPOSOMES 81 at the centre, and the entire structure moved with the particle. So we tried another oil solution of mineral oil, and this time we obtained liposomes with very thin membranes, and the particle could move freely from the centre toward membrane inside the liposome as shown in Figure 5.9 b. FIGURE 5.9: Images of liposomes made by water in oil droplet transfer method. a, The oil solution used is dodecane. b, The lipids were dissolved in mineral oil. 5.3 Measurements inside liposomes Using liposomes with vaterite particles we measured the properties of fluid inside these liposomes with vaterite particles using our rotational microrheology technique as discussed in chapter 3. First, we tried to move the vaterite particle from the centre to the edge of the liposome to see how this membrane wall affects the rotational motion of the particle. As we have shown before (in chapter 4.4), we did not observe significant effects from the walls of liposomes with diameter of 20 µm. Results shown in Figure 5.10 were from the active microrheology measurements (described in chapter 3). Figure 5.10 shows active microrheological measurements inside liposomes, where the internal solution was comprised of water. The result displayed in Figure 5.10 a is from measurement inside a liposome with radius of 2.5 µm. It agrees with the viscosity of water at room temperature even for this confined environment. However, not all results of the measurements agree very well with the known properties of water. As shown in both Figure 5.10 b-d, we not only detected slightly

104 82 BIOMIMETIC CELL MODEL LIPOSOMES different viscosities from water but also elastic properties of the liquid within the liposome. We thought this is due to crowding inside liposomes since the liposomes could encapsulate particles as well as smaller liposomes. Moreover, for the case of high crowding levels inside a liposome (Figure 5.10 d), there is a big difference between the value measured and the properties of water.

105 5.3 MEASUREMENTS INSIDE LIPOSOMES 83 FIGURE 5.10: Active microrheological measurements inside liposomes with radius of R, and r is the radius of the particle. On the right: results are compared with the viscosity of water at room temperature.