university of ljubljana faculty of mathematics and physics department of physics Natan Osterman Study of viscoelastic properties, interparticle potentials and self-ordering in soft matter with magneto-optical tweezers Doctoral thesis ADVISER: assoc. prof. dr. Igor Poberaj COADVISER: assist. prof. dr. Dusan Babic Ljubljana, 2009 univerza v ljubljani fakulteta za matematiko in fiziko oddelek za fiziko Natan Osterman Študij viskoelasticnih lastnosti, meddelcnih potencialov in samourejanja v mehki snovi z magneto-opticno pinceto Doktorska disertacija MENTOR: izr. prof. dr. Igor Poberaj SOMENTOR: doc. dr. Dusan Babic Ljubljana, 2009 Zahvala Iskreno se zahvaljujem mentorju izr. prof. dr. Igorju Poberaju in somentorju doc. dr. Dušanu Babicu za izvrstne ideje, vodenje pri raziskavah, pomoc pri izdelavi doktorskega dela in za cas, ki sta ga v zadnjih petih letih vlozila vame. Dr. Jurij Kotar je izdelal odlično magnetno pinceto in mi pomagal pri njeni uporabi. Raziskave nematskih koloidov so bile uspesne zaradi izjemnega vodstva dr. Mojce Vilfan in sodelovanja prof. dr. Martina Čopiča in dr. Mihe Ravnika. Mnogo koristnih nasvetov mi je dal dr. Miha Škarabot. Pri raziskavah samourejanja koloidnih delcev v magnetnem polju ne bi slo brez doc. dr. Primoza Ziherla in dr. Jureta Dobnikarja. Meritve mikroreologije bakterij so potekale v sodelovanju s skupino izr. prof. dr. Davida Štoparja z Biotehniske fakultete; dr. Tjasa Danevcic in Maja Boric sta vztrajno pripravljali bakterijske vzorce, pri teoreticni obravnavi mi je pomagal dr. Andrej Vilfan. Dr. Pietro Čicuta kindly hosted me in his group at Cavendish Laboratory, Cambridge. Uros Jorgacevski je mojstrsko izdeloval zelene mehanske dele za eksperiment. Blaz Kavcic in mnogi drugi kolegi iz FMF in IJS so mi mnogokrat pomagali na dolgi poti do doktorata. Gospa Jasmin Anzicek je vedno prijazno nasla ugodno resitev birokratskih problemov. Za podporo in spodbudo sem posebno hvalezen druzini. Hvala Gaji, da mi vedno stoji ob strani. Abstract In the PhD thesis we use magneto-optical tweezers to study various aspects of soft matter systems. The first part is focused on interparticle potentials of confined nematic colloids. We precisely measure repulsive liquid crystal mediated force between two colloidal particles with quadrupolar symmetry of the director. The force is found to have 1/x6 dependence on separation for particle surface-to-surface separations smaller than the sample thickness. At larger separations we demonstrate that the interparticle potential decreases exponentially due to the effect of the confinement. In the second part of the thesis we study the relationship between isotropic inter-particle potentials and resulting many body structures. Using magnetic tweezers we induce various interaction potentials in two- and three dimensional systems of superparamagnetic colloidal particles. We demonstrate that a combination of perpendicular external static magnetic field and geometric confinement creates hardcore/soft-shoulder potential which stabilizes many unusual mesophases of colloidal particles. Complex interparticle potentials can be also induced by time modulation of external magnetic field. Precession of the field on a surface of a cone with a "magic" opening angle 54.7° averages out the 1/r3 term of dipolar interparticle interaction. We show that the local field effect gives rise to 1/r6 isotropic attraction between particles and that the potential is pair-wise non-additive. We demonstrate that the interaction stabilizes single colloidal chains and hexagonal close-packed colloidal sheets in 3D. In the last part of the thesis we outline several methods that can be used with optical tweezers to measure the microrheological behavior of materials. We implement a system for passive one-particle and active one- and two-particle microrheology consisting of high-resolution video tracking module and optical tweezers for micromanipulation of probe particles. We test the system on water and then use it to measure the mechanical properties of the growing bacterial population of Vibrio sp. The stiffness of the extracellular matrix is found to be very small, whereas its viscosity increases with age and reaches a maximum of approximately n ~ 3.3 x 10-3 Pas. Keywords: magneto-optical tweezers, complex fluids, colloids, nematic liquid crystal, self-assembly, microrheology PACS: 85.70.Sq, 47.57.-s, 61.30.Jf, 81.16.Dn, 83.85.Cg Povzetek Doktorsko delo obravnava raziskave mehke snovi z magneto-opticno pinceto. V prvem delu se osredotočimo na meddelcne potenciale nematskih koloidov. Natančno izmerimo odbojno strukturno silo med dvema koloidnima delcema s kvadrupolno simetrijo direktorja. Ugotovimo potencno odvisnost sile od meddelcne razdalje, ko je razdalja med povrsinama delcev manjsa od debeline vzorca. Pri vecjih razdaljah meddelcni potencial eksponentno pojema zaradi ograjenosti tekocega kristala. V drugem delu se posvetimo povezavi med meddelcnimi potenciali in nastalimi vecdelcnimi stuktrurami. Z magnetno pinceto induciramo razne tipe meddelcnih potencialov v dvo- in trodimenzionalnih sistemih superparamagnetnih koloidnih delcev v vodi. Pokazamo, da kombinacija pravokotnega staticnega magnetnega polja in geometrijske ograjenosti v dveh dimenzijah ustvari zmehcan izotropni odbojni potencial, ki stabilizira neobicajne mezofaze koloidnih delcev. Kompleksne meddelcne potenciale ustvarimo tudi s casovno modulacijo zunanjega magnetnega polja. Prece-sija polja po povrsini stozca z "magicnim" vrsnim kotom 54.7° izpovpreci 1/r3 clen v interakciji. Z racunom pokazemo, da je zaradi vpliva lokalnih magnetnih polj delcev interakcijski potencial izotropno privlacen z 1/r6 odvisnostjo in parsko neaditiven. Predstavimo stabilne vecdelcne strukture v treh dimenzijah: enojne verige in tesno zlozene heksagonalne "plahte". V zadnjem delu obravnavamo nekaj metod za meritev mikroreoloskih lastnosti z op-ticno pinceto. S kombinacijo opticne pincete, ki sluzi za mikromanipulacijo sond, in modula za videomikroskopijo naredimo sistem za pasivne enodelcne ter aktivne eno-in dvodelcne mikroreoloske meritve. Sistem testiramo v vodi in ga nato uporabimo za meritve mehanskih lastnosti rastoce bakterijske populacije Vibrio sp. Ugotovimo, da je elasticnost ekstracelularne bakterijske matrike zelo majhna, njena viskoznost pa s starostjo narasca do najvecje vrednosti priblizno n ~ 3.3 x 10-3 Pas. Ključne besede: magneto-opticna pinceta, kompleksne tekocine, koloidi, nematski tekoci kristali, samosestava, mikroreologija PACS: 85.70.Sq, 47.57.-s, 61.30.Jf, 81.16.Dn, 83.85.Cg Contents 1 Introduction 7 2 Experimental techniques 13 2.1 Optical tweezers............................................................13 2.1.1 Principle of operation..............................................14 2.1.2 Basic design........................................................15 2.1.3 Force calibration..................................................15 2.1.4 Experimental setup................................................17 2.2 Magnetic tweezers..........................................................19 2.2.1 Superparamagnetic beads ........................................21 2.2.2 Magnetic interaction..............................................21 2.2.3 Calibration of the magnetic force................................23 2.3 Imaging....................................................................24 3 Interparticle potentials of nematic colloids 25 3.1 Introduction ................................................................25 3.1.1 Formal description of interparticle interaction..................27 3.1.2 Effect of confinement..............................................29 3.2 Experimental details......................................................30 3.2.1 Liquid crystal......................................................30 3.2.2 Cell preparation....................................................30 3.2.3 Force calibration..................................................31 3.3 Results and discussion....................................................32 3.3.1 Motion of free beads from different starting separations .... 33 3.3.2 Static measurement of LC-mediated force........................34 3.3.3 Dynamic measurements ..........................................35 3.3.4 Effect of confinement..............................................36 3.3.5 Temperature dependence of interaction..........................38 3.4 Conclusion and outlook....................................................39 4 Engineered potentials 41 4.1 Introduction................................................................41 4.2 Experimental details ......................................................44 4.3 Static magnetic field in 2D system ........................................45 4.3.1 Isotropic repulsion ................................................45 4.3.2 Anisotropic dipolar interaction....................................47 4.4 Static magnetic field in quasi-2D system................................49 4.4.1 Measurement of interparticle potential..........................50 4.4.2 Phase behavior of a system with softened repulsion............51 4.5 Static magnetic field in a 3D system ....................................53 4.5.1 Low filling fractions................................................53 4.5.2 High filling fractions..............................................55 4.6 Rotating magnetic field in 2D system....................................58 4.6.1 Inplane rotation of field ..........................................58 4.6.2 Rotation of tilted magnetic field..................................59 4.6.3 Measurement of interparticle potential..........................63 4.6.4 Many-body effect..................................................64 4.6.5 Assembly of colloidal superstructures............................65 4.7 Residual interaction induced at magic angle rotation ..................66 4.8 Conclusion and outlook....................................................72 5 Viscoelastic properties of bacterial networks 75 5.1 Introduction................................................................75 5.1.1 Passive microrheology ............................................77 5.1.2 Optical tweezers microrheology..................................78 5.1.3 One-particle passive MR..........................................79 5.1.4 One-particle active MR............................................80 5.1.5 Two-particle active MR ..........................................81 5.1.6 Rheology of bacterial networks ..................................83 5.2 Experimental details ......................................................84 5.2.1 Preparation of bacteria ............................................85 5.3 Results and discussion ....................................................85 5.3.1 One-particle passive MR ..........................................85 5.3.2 One-particle active MR ............................................89 5.3.3 Two-particle active MR ..........................................93 5.4 Conclusion and outlook ....................................................96 6 Conclusion 99 A Numerical calculation of interaction energy of N particles 103 B Calculation of pair energy 107 C Particle tracking algorithms 111 Bibliography 113 List of publications related to the work described in this thesis 119 RazSirjeni povzetek v slovenskem jeziku 121 6.1 Uvod....................................121 6.2 Eksperimentalne metode.........................122 6.3 Meddelcni potencial v ograjenem nematiku...............125 6.4 Samourejanje superparamagnetnih koloidov v magnetnem polju . . . 129 6.5 Viskoelasticnost bakterijskih mrez....................138 Chapter Introduction Soft matter [1, 2] are materials with structural and dynamical properties between crystals and fluids comprising a variety of physical states. Examples of such materials range from "simple systems" (liquids, liquid crystals, colloids, polymer suspensions, foams, gels...) to the most complex biological systems of different length scales (proteins, lipids, DNA, RNA molecules, membranes, vesicles, cells, tissues). In spite of various forms of soft matter materials, their properties have common physicochem-ical origins, such as large number of internal degrees of freedom, weak interactions between structural elements (characteristic interaction energy is comparable to room temperature thermal energy), and a delicate balance between entropic and enthalpic contributions to the free energy. Thermal fluctuations, external fields and boundary conditions therefore play a significant role in the structure and properties of these materials, while quantum aspects can be generally neglected. Soft matter is important in a wide range of technological applications and is of fundamental relevance in such diverse fields as chemical, environmental, and food industry. Structural and packaging materials, foams and adhesives, detergents and cosmetics, paints, food additives, lubricants and fuel additives, rubber in tires, liquid crystals for display devices, are classifiable as soft matter. Over the past years, soft matter science has been largely extended in its scope from the traditional areas such as colloids, liquid crystals and polymers to the study of biological systems and the development of novel composites and microfluidic devices. Progress in soft matter research is driven largely by the available experimental techniques. Much of the work is concerned with understanding physics at the microscopic level, especially at the micro- and nanometer length scales. This gives soft matter studies a wide overlap with nanotechnology. Experimental methods can be divided into two broad classes: techniques for observation and techniques for manipulation. Techniques such as static and dynamic light scattering, small-angle x-ray and neutron scattering are used to observe the bulk microscopic structural behavior of investigated samples. Main direct imaging methods include electron microscopy (TEM, SEM), various optical microscopy techniques (bright field, dark field, phase contrast, DIC, fluorescence and laser confocal scanning) and atomic force microscopy. Local micromanipulation techniques are used to locally exert forces either to move structural elements to another position or just to monitor a local response of the material (strain/stress relationship). Typical devices used for micromanipulation of soft materials are micromanipulators, atomic force microscope, magnetic tweezers and optical tweezers. Optical tweezers (also called "laser tweezers") are a versatile tool for manipulation of micron-sized objects such as colloidal particles or living matter (single biomolecules, cell organelles, cells). Tightly focused laser beam is used to optically trap particles with index of refraction higher that of the surrounding medium. Optical tweezers are a non-contact method (in contrast to micromanipulator or AFM), the force is exerted locally on a specific particle, and are therefore perfect tool to manipulate them. When combined with position sensing, tweezers can be used to precisely measure or apply forces in piconewton range, which is the magnitude found in interactions between individual protein molecules. Optical tweezers have also some disadvantages. Sometimes piconewton forces aren't sufficient; nanonewton range would be handier. Laser beam can be partially absorbed by the sample, resulting in local heating which disturbs the measurement. In some soft matter materials, such as liquid crystals, the strong electric field of the focused laser beam can disturb the material and thereby change its properties. A complementary tool to optical tweezers are magnetic tweezers - a device used to exert a force on magnetic particles embedded in a soft matter. Basically, magnetic tweezers consist of an electric current source and a set of coils that generate gradient magnetic field in the sample which results in magnetic force acting on the particles. By appropriate design of coil geometry large magnetic gradients, exerting forces in nanonewton range, can be generated. In contrast with optical tweezers, the force acts on all embedded magnetic particles. The magnitude and the direction of the force strongly depends on a particle position due to inhomogeneous nature of magnetic field gradients. Similar device, homogenous magnetic tweezers1, uses homogenous magnetic field to induce forces between superparamagnetic particles, embedded in an investigated material. In zero magnetic field, the particles don't possess any magnetic dipole moment, but as soon as the external magnetic field is applied magnetic dipole moments in the field direction are induced in the particles, which then interact via standard dipole-dipole interaction. Magnetic tweezers are suitable for precise force measurements since once the magnetic properties of the particles are known, the force between two particles depends only on two known parameters: the field magnitude (which is externally controlled and constant through the sample) and the separation between the particles (which can be directly measured using digital image analysis). A disadvantage of homogenous magnetic tweezers is lack of ability to manipulate single particles. To get "the best of both worlds" we combined optical tweezers with homogeneous magnetic tweezers into magneto-optical tweezers. This combination enables experiments that are not or are hardly feasible by the use of either method alone. We used it to study three open problems in the field of soft matter: (i) interparticle interactions in nematic colloids, (ii) relationship between interparticle potentials and resulting ordered structures in a system of superparamagnetic particles and (iii) the 1 Actually both gradient and homogenous magnetic tweezers are inappropriately called "tweezers" since they don't enable manipulation of only one particle. rheology of complex biological system. In the following paragraphs a brief introduction to the topics of the thesis is given, whereas more comprehensive introductions are at the beginning of each chapter. * Colloidal particles in liquid crystals (LC) are subjected to strong and highly anisotropic long-range forces acting between them. Uniform director field of a LC gets distorted if a particle is introduced into LC. The distortion depends on the size, shape and surface properties of the particle. If the particle has spherical shape and its surface induces planar orientation of the surrounding LC molecules, then the director field exhibits quadrupolar symmetry. If the sphere's surface induces perpendicular orientation of the molecules (homeotropic anchoring), the director field exhibits dipolar or quadrupolar symmetry, depending on the anchoring strength [3, 4]. The particles together with the surrounding perturbation of the director field are often called elastic dipoles or quadrupoles. Interactions between colloidal particles in nematic LCs have been extensively investigated experimentally using combined optical microscopy and laser tweezers. Forces between two elastic dipoles were measured statically by determining the threshold power of optical tweezers, which was required to keep the beads at a given separation [5, 6, 7, 8]. A combination of static and dynamic measurements gave the angular dependence of the force between two elastic quadrupoles [7]. Dynamic measurements, where the trajectory of a released bead was observed, were used to study interactions between elastic quadrupoles and dipoles [9, 10]. Optical tweezers are handy tool to manipulate the colloidal particles but an important aspect has to be considered when used in LC. Strong electric field of the trapping laser significantly alters director field in the vicinity of the laser focus [11] which renders results hard to interpret. The reorientational effects can be reduced by choosing low birefringent LC or lowering the laser power [7, 6] but in both cases optical trap stiffness is reduced as well. An alternative approach that completely avoids these effects is the use of magnetic particles and magnetic field for their manipulation. First report of such interaction measurement was given by Poulin et al. [12]. Static magnetic field was used to analyze forces between homeotropic particles [13]. We implemented a similar idea using combined magneto-optic tweezers, which enable manipulation of single particles using optical tweezers and generation of both attractive and repulsive forces between superparamagnetic particles by using either static or rotating magnetic field. Due to low external magnetic field applied during the experiment (less than 10 mT) the LC director was not distorted. The system was used to study the interaction between planar colloidal particles in nematic LC enclosed in a homeotropic cell. The force between two elastic quadrupoles as a function of their separation was precisely measured for various sample thicknesses to quantify the effect of LC confinement. Temperature dependence of the interaction near the nematic to isotropic phase transition was measured to obtain the critical exponent of the force. One of the fundamental problems of the condensed matter physics is the relationship between the interparticle interactions and the collective structural behavior of many-body systems. The topics is especially interesting from the viewpoint of self-assembly, a process in which a system components (molecules, polymers, colloids or other basic building blocks) spontaneously organize into a larger ordered and/or functional structures [14, 15]. The form of final macroscopic structure is governed by the microscopic interactions between the components. At the microscale self-assembly is present as a concept to build metamaterials (for example, colloidal particles can self-assemble into complex super-structures that could be potentially used as a photonic band-gap materials) and in the field of microfluidics (micromixers, microvalves). The relationship between the interparticle interactions and the resulting structures in atomic and molecular systems has been investigated theoretically and numerically. However, there are no such experimental studies since manipulation of the inter-atomic potentials is impossible. Some 30 years ago colloidal systems [16] had been found to be a suitable model for condensed matter systems. Statistical mechanical concepts known from the theory of liquids can be directly applied to ensemble of colloidal particles to provide a framework for relating the microscopic properties of single particles to the macroscopic properties. Colloidal systems are suitable for observation and in addition their properties can easily be manipulated. A change of particles or solvent properties or the addition of an external field (such as electric or magnetic) alters the macroscopic behavior of the system. Commercially available monodisperse colloidal particles are currently available only in spherical shape. The resulting interparticle potential is isotropic which limits the set of ordered phases; the most common are the hexagonal lattice in two dimensions and the face centered cubic lattice in three dimensions [17, 18]. There are two feasible routes to obtain more complex colloidal structures, either by (i) anisotropic interparticle interactions which can be induced by surface treatment of particles [19], external fields [20] or liquid crystalline solvent [3] or by (ii) isotropic interaction potentials with radial profile more complicated than a simple power law. A simple realization of such interaction is an isotropic potential with radial profile, characterized by two length scales, e.g. a combination of hard-core and softer repulsive part. Numerical simulations predict that such interparticle potential induces a variety of mesophases [21, 22, 23, 24] between the fluid and the close packed crystal. We used magneto-optical tweezers to induce various kinds of isotropic interaction potentials in a system of superparamagnetic colloids. The combination of a static magnetic field and confinement induced complex pair potentials with radial profile that could be tailored by varying the sample cell thickness. At critical thickness the mesophases stabilized by the hard core/soft shoulder interaction potential were observed. Complex isotropic potentials induced by rotating magnetic field were also investigated. Especially interesting interaction potential arises when the magnetic field precesses on a cone with the "magic" opening angle 9p = 54.7°. The 1/r3 term in dipolar interaction energy between the induced magnetic dipoles of the particles vanishes but due to the effects of the local fields the effective pair potential is isotrop-ically attractive. The type of self assembled structures in such potential was found to depend on the filling fraction of particles. We demonstrated that such potential can be used for stabilization colloidal superstructures. ~k ~k ~k The material properties of soft matter originate from their complex structures and dynamics with multiple characteristic length and time scales. An important material property is the shear modulus, which has been traditionally measured using rheometers but in the last two decades a number of microrheological (MR) techniques, used to locally measure viscoelastic parameters, has been developed. Mi-crorheology has several advantages compared to traditional bulk rheometry: small quantities of samples are needed; it is possible to study heterogeneous environments; the upper range of accessible frequencies is higher (up to 105 Hz). This enables new insights into the microscopic foundation of viscoelasticity in soft matter systems. There are two broad classes of MR techniques: active and passive. Passive MR techniques rely on fluctuations of probe particles due to thermal noise. Typically colloidal beads with a diameter between few tens of nanometers to several micrometers are embedded in the material, their free diffusion is observed either with dynamic light scattering [25], laser tracking [26] or videomicrocopy [27] and then a linear shear modulus is calculated. Active MR techniques involve the active manipulation of small probes. The earliest experimental implementation of active MR was based on the manipulation of magnetic beads with an external magnetic field [28] almost a century ago. Nowadays micromechanical tools such as micropipettes [29] and atomic force microscopes [30] are used to directly strain materials, while optical tweezers [31] or magnetic bead mi-crorheometers [32, 33, 34] are used to actively manipulate microparticles embedded in materials. These measurements are similar to conventional mechanical rheological techniques in which an external stress is applied to a sample and the resultant strain is measured to obtain the shear moduli. In this case micron-sized probes locally deform the material and probe the local viscoelastic response. Active measurements allow the possibility of applying sufficiently large stresses to stiff materials to obtain detectable strains. They can also be used to measure non-linear behavior if sufficiently large forces are applied to strain the material beyond the linear regime. MR methods can be further divided according to the number of used probes to one-particle MR, where a motion of a single probe particle is tracked using various experimental techniques to get the viscoelasticity, and to two-particle MR where the correlated motion of two (or more) probes is analyzed to obtain the viscoelastic properties on larger length scales. This allows the characterization of bulk material properties even in the systems that are inhomogeneous on the length scale of the probe particle. Optical tweezers can be employed for one-particle passive MR, where a particle is trapped in a stationary trap and its fluctuations are analyzed, and for one-particle active MR, which is required when thermal fluctuations can't induce measurable level of deformation or when the high-deformation response is studied. In one-particle active MR measurement the probe particle is trapped in an optical trap which position is harmonically oscillated [31, 35]. The complex viscoelastic modulus of the investigated media is calculated from the response of the probe particle. Two-particle active MR involves two optical traps [36]. One particle is trapped in a harmonically oscillating stiff trap and is thus actively deforming the surrounding medium, while a second particle is trapped in a weak stationary trap and is used to monitor the deformation of the medium at a desired location. Using optical tweezers we set up a system for measurements of microrheological properties of soft materials, including complex biological systems. In one experiment we are able to perform various microrheological measurements - one-particle passive and one- and two-particle active MR - to fully characterize local viscoelastic properties of the material. We check the system on rheology of water and then use it to monitor the development of viscoelasticity of bacterial extracellular matrix. r^j The thesis is organized as follows. In the following chapter the basic principles of optical and magnetic tweezers are explained. Our combined magneto-optical tweezers setup used for micromanipulation and imaging of individual particles is described. In Chapter 3 the measurement of interparticle potential of nematic liquid crystal colloids is reported. In Chapter 4 magnetic tweezers are used to engineer potentials between superparamagnetic particles and the formation of macroscopic structures is observed. The measurement of bacterial extracellular matrix rheology using optical tweezers is described in Chapter 5. The final concluding Chapter 6 summarizes the main results and findings presented in the thesis. Chapter ^^_ Experimental techniques In this Chapter we describe the experimental techniques used to manipulate micron-sized particles and to exert and measure piconewton forces. First, a brief theory of operation and a basic design of optical tweezers are explained and then we focus on our implementation of tweezers. It consists of three main components: 1064 nm Nd-YAG CW laser, an inverted microscope with high numerical aperture objective and beam steering system, made of two computer-controlled, orthogonally positioned acousto-optic deflectors that enable quick and precise trap positioning and strength control. Then we describe magnetic tweezers, a device consisting of a computer controlled electrical current source and 3 orthogonally placed coil pairs used to create homogenous magnetic field in a sample. The field induces magnetic dipole moments in superparamagnetic particles which results in dipolar interaction between the particles. The interaction is in general anisotropic, but if the particles are confined to two dimensions it can be made isotropically repulsive or attractive, depending on the orientation of the external magnetic field. We conclude the Chapter with a description of particle imaging and tracking system. A fast CMOS camera is used to record the videos, which are subsequently analyzed off-line with custom particle tracking software. 2.1 Optical tweezers Optical tweezers (also called "laser tweezers") are a micromanipulation technique that uses highly focused laser beam to trap, manipulate and measure forces on micron-sized dielectric particles. The field of optical tweezing was started by Ashkin [37] in early 1970's with the demonstration that micron-sized objects with a refractive index higher than the surrounding medium are drawn towards the center of a weakly focused laser beam. The objects were confined on the optical axis and propelled in the direction of light propagation. Ashkin was able to guide the particles in a liquid or air and even trap them in an inverted geometry where the laser light pressure was balanced by the gravitation force [38]. b a Figure 2.1: Left: Qualitative view of the trapping of a dielectric sphere. The refraction of a typical pair of rays a and b of the trapping beam gives forces Fa and F whose sum is pointing towards focus f. Reprinted from [46]. Right: Micrograph of 2 ^m silica beads trapped in 29 optical traps. The single-beam gradient optical tweezers that enable three dimensional trapping of particles were developed by Ashkin, Chu1, and coworkers in 1986 [39]. Since then there has been continuous development of the technique with many new applications. The technique has matured and today optical tweezers are highly effective research tool used in many fields (for a review see [40] or [41]). They are particularly important in biophysics and biology by providing a noninvasive means to manipulate cells, organelles and particles at sub-micron precision and to measure forces in the range of 0.1-100 pN [42, 43, 44, 45]. 2.1.1 Principle of operation An optical trap is created by focusing a laser beam to a diffraction-limited spot using a high numerical aperture objective. Strong gradient of light intensity near the focus creates a three dimensional potential well for objects with a refractive index higher than that of the surrounding medium. If there is no difference in refractive index between the particle and its surroundings, there is no optical force exerted on the particle. If the refractive index of the particle is lower than that of the medium, the particle is expelled from the beam center. The optical force on a particle in a tightly focused laser beam can be decomposed into two components: a scattering component (radiation pressure) which acts in the direction of the beam propagation and a gradient component (which arises from induced dipole interactions with the electric field gradient) in the direction of highest light intensity. The particle is trapped at the point where the gradient component, the scattering component and the gravitational force balance. The optical force acting on a particle can be analytically calculated in Rayleigh regime where a particle is much smaller or in ray optics regime where a particle is much larger than the wavelength of light used for trapping. However, optical tweezers are almost always used to trap particles with size comparable to the laser xChu was awarded the Nobel Prize in Physics in 1997 for his research in cooling and trapping of atoms with laser light wavelength, i.e. in Mie regime. The calculation of optical force on a trapped particle in this regime is quite complex and has to be performed numerically. For most applications of optical tweezers there is no need for force calculation - the tweezers are experimentally characterized and calibrated before they are used for quantitative measurements. 2.1.2 Basic design A basic version of optical tweezers setup consists of a laser and an objective with high numerical aperture which creates one stationary optical trap. Since this would be quite limiting for experimental work, the setup usually includes a system for trap steering. There are many possibilities for a beam steering, starting from simple manual mirrors piezo-driven mirrors and more complex techniques such as spatial light modulators or acousto-optic deflectors. Most optical tweezers are built around modified commercially available microscopes which are simultaneously used to image the sample. Often inverted microscopes are used because gravitational force on a trapped particle counteracts the scattering force so better trapping is obtain in three dimensions. The laser beam is fed into the objective using a dichroic mirror which reflects the trapping laser and transmits the light used for imaging. Samples are imaged by CMOS or CCD camera. A laser is chosen according to optical tweezers application. Typically, a high power, single mode laser with good pointing stability and low power fluctuations is desired. For biological samples, lasers with infrared wavelengths are used to minimize laser absorption. 2.1.3 Force calibration A spherical particle of refractive index n2 embedded in a medium of refractive index u\ in the proximity of a focused laser beam with wavelength larger than particle size (Rayleigh regime) "feels" a potential u=-^(nM)^- (-) where V is the volume of the particle, c is the speed of light, I0 is the laser intensity and w0 is beam waist diameter at the focus. The optical force on the particle as it moves from the equilibrium position r = 0 is F = -W = ^ (W-r2/W rr. (2.2) cw2 V n2 + 2n{ ) Therefore, for small displacement from the equilibrium position, an optical trap exerts an elastic force Fj = k^ (i = x; y; z), with k being the stiffness of the trap along a given direction. Normal optical traps are produced by a tightly focused Gaussian beams where radial forces are larger than the axial one (kx — ky > kz). In order to measure the force acting on a bead within an optical trap, one has to know the displacement of the bead from the equilibrium position and the stiffness of the trap. There are numerous possibilities to measure the trap stiffness, but perhaps most popular is stiffness calibration from the power spectral density of the trapped particle's fluctuations and by the use of the equipartition theorem. A trapped bead undergoes a confined Brownian motion. If the bead is embedded in a Newtonian fluid2, the displacements from the equilibrium position [x(t),y(t), z(t)] can be described by the Langevin equation3, which for one dimension is di ^c d/^c - .,, . m— = -7- - + F it), (2.3) where k is the trap stiffness, m is the mass and 7 = 6nr/a the viscous drag coefficient of the particle. F(t) represents the random thermal force with average value of zero and a flat power spectrum |F (f )|2 = 47kBT. (2.4) Micron sized particles moving with a typical velocity ^m/s have low Reynolds number, therefore the inertial term in (2.3) can be neglected. If (2.3) is transformed into frequency space, (2.4) can be used to obtain the power spectral density (PSD) of the x(t) fluctuations k T |X(f )|2 = 2( f2 + f2), (2.5) Yn2(fc2 + f 2) where corner frequency fc = (k/2n7) is directly related to trap stiffness k. The flat plateau at low frequencies 47kB T IX (0)|2 k2 (2.6) contains information on the confinement of the particle as well. Figure 2.2: Calibration of optical trap stiffness. Left: Typical power spectral density of a trapped particle in water. Red curve is fit of (2.5) with fc = 4.92 Hz. Right: Trap stiffness k for 1 ^m silica bead trapped in water as a function of laser power. 2Newtonian fluid is a fluid whose stress is linearly proportional to strain rate. The constant of proportionality is the viscosity. Water is an example of Newtonian fluid. 3 The equation holds for sufficiently stiff trap, where the particle fluctuations are always in the harmonic part of the potential well created by the trap. The fastest way to calibrate the trap stiffness k of the trapped particle is by the use of the equipartition theorem 1 ksT = 2 k(x2 >. (2.7) By measuring the average squared displacement (x2> of the fluctuations of the trapped bead, the trap stiffness k = kBT/(x2> is obtained. The trap stiffness is typically linearly dependent on the power of the incident laser beam. This is demonstrated in Fig. 2.2 where we present the stiffness k as a function of laser power for 1 ^m silica beads, trapped in water. 2.1.4 Experimental setup Our laser tweezers are built around a commercial inverted optical microscope (Zeiss, Axiovert 200M) and infrared CW laser. The laser beam is spatially controlled with pair of acousto-optic deflectors (AA Opto-electronic, DTSXY-250-1064-002, frequency range of acoustic waves from 60-90 MHz), driven by a beam steering controller (Aresis, BSC-01) which enables creation of multiple optical traps and their precise positioning. A custom software running on a PC, connected with the beam steering controller and a CMOS camera (Pixelink, PLA-741), is used to manipulate the traps and record experimental videos. The schematics of experimental setup are shown in Fig. 2.4. Figure 2.3: Left: Experimental setup: (a) laser diode driver, (b) laser head, (c) box with acousto-optic deflectors, (d) beam steering controller, (e) inverted microscope, (f) CMOS camera. Right: Acousto-optic deflectors and two mirrors for beam alignment. The laser source is diode pumped ND:YAG laser (Coherent, Compass 10642500) with wavelength 1064 nm and 2.5 W of maximum power. It outputs linearly polarized Gaussian beam in TEM00 mode. The beam is expanded to match the entrance pupil of the acousto-optic deflectors (AODs), which are used to deflect the beam and therefore change its position in the sample. A telescope optics and three mirrors are used to align the beam from the AODs to the entrance pupil of a high numerical aperture water immersion IR-optimized microscope objective (Zeiss, Achroplan 63x/0.9W). Optical elements in the beam path reduce the beam power. Transmission ratio from the laser through AODs, telescope optics and microscope tube lens is 28%, transmission of the objective is 51%, giving the total transmission of 14% from the laser to the sample. In experiments where AOD field flattening was used, the total transmission was additionally reduced to overall 10%. telescope Nd:yag laser 5x Master pc Magtweez pc Figure 2.4: Schematics of the complete magneto-optical tweezers setup. Laser beam is expanded and guided through a pair of orthogonal acousto-optic deflectors to the entrance pupil of the microscope objective. Master PC controls the beam steering controller, a CMOS camera, inverted microscope and has network connection to magnetic tweezers PC that controls the current source for 3 pairs of coils mounted on the microscope. The AOD diffraction angle depends on the frequency of acoustic wave and since the beam steering controller (BSC) creates it by direct digital synthesis (DDS) with relative frequency resolution 10-9, the trap position can be changed in sub-nanometer steps. Trap stiffness is controlled by varying AOD transmission which depends on the acoustic wave amplitude. Multiple traps can be created by time sharing of the laser beam. The beam steering controller supports up to a few thousands individual quasi-simultaneous traps or hundred thousands traps in a sequence. A pair of acoustic waves that define the trap position and stiffness (for x and y AOD) can be switched to different pair with user defined rate between 100 Hz and 100 kHz which is needed for time sharing of the beam for a creation of multiple traps. If the AOD switching rate is high then a particle, trapped in one of N simultaneous traps, "feels" as if it was trapped in a single trap with roughly N times lower stiffness. For example, if switching rate is 100 kHz and there are 10 simultaneous traps, each trap gets a laser pulse, 10-5 s long each 9 ■ 10-5 s. Since the self-diffusion time (t = ~ 0.14 s for particles with radius a = 0.5 ^m) is much longer than that, the particle is in a quasi-stationary trapping potential. On the other side, lower AOD switching rate enable precise control over the movement of individual traps. For example, if the switching rate is 1000 Hz and there are 1000 traps, sequently arranged in a circle, the particle "feels" a single trap that is traveling around a circle with frequency 1 Hz. The efficiency of AODs varies with frequency of acoustic wave which results in trap stiffness variation depending on its position. In order to maintain equal trap stiffness everywhere in a trapping region, AOD field flattening was used. A separate photodiode connected to the beam steering controller was used to scan the laser power across the complete trapping region. A field flattening look-up table was created and subsequently used directly in an acoustic wave generation procedure. 2.2 Magnetic tweezers Magnetic tweezers are an instrument for exerting and measuring forces on magnetic particles using a magnetic field gradient. Forces are typically on the order of pico-to nanonewtons [34]. Due to their simple architecture, magnetic tweezers are one of the most popular and widespread biophysical techniques with typical applications in single-molecule micromanipulation, rheology of soft matter, and studies of forceregulated processes in living cells [47, 48, 49]. Another possibility to exert a magnetic force is to use a combination of a homogenous magnetic field and superparamagnetic particles. The device can't exert a force on a single isolated particle but rather a force acts between induced magnetic dipoles in the particles. By proper modulation of the amplitude and direction of the magnetic field either repulsive or attractive forces can be induced. Homogeneous magnetic field in our setup is generated by three orthogonal sets of coils shown in Fig. 2.5. The magnitude and direction of the field are regulated by a 6-channel computer-controlled electrical current source4. Magnetic field components in x and y direction (xy plane is the imaging plane of the microscope) are produced by two orthogonal pairs of coils, whereas the field component in z direction (direction of optical axis) is produced by a pair of coils that are placed closer together. To obtain a homogenous field the electrical current through two opposing coils should be the same. In left chart of Fig. 2.6 we show the measured magnetic field5 at sample location as a function of a current running through the corresponding pair of coils. The generated field B is proportional to the current I, B = aI; the coefficients for field components in x and y directions are ax = ay = 1.77 mT/A, whereas the coefficient for z direction az = 10.3 mT/A is larger because the distance between z coils is smaller than the distance between x or y coils. The current source can generate arbitrary waveforms on six independent channels using direct digital synthesis which enables arbitrary scaling and rotation of the magnetic field in three dimensions. The maximum DC current is limited to 8A, giving a peak Bz of approximately 80 mT. However, due to high heat dissipation, DC currents up to 4 A that produce static magnetic fields up to 40 mT in z and 4The complete system of magnetic tweezers was designed and built by Dr. Jurij Kotar. 5 The field was measured using high accuracy miniature ratiometric linear Hall-Effect sensor Honeywell SS496A1. The supply voltage 10 V was giving sensitivity 50 mV/mT. Figure 2.5: Left: Magnetic tweezers module mounted on the microscope; a sample cell and the objective are visible in the middle. Pairs of coils for field generation in x and y direction are further apart than coils for z direction. Right: Magnetic field in x direction as a function of position on x axis (a focal point of the microscope objective is at x=0). 7 mT in x and y can be produced and were found to be more than enough for all performed experiments. Often a generation of AC magnetic field is desired. The maximum frequency of AC magnetic field is limited by the inductance of the coils and a maximum output voltage of the current source. To characterize the response of the single coil we observed the shape of generated harmonically oscillating current I = I0 sin(2nf t) at fixed amplitude I0 for different frequencies f. At given amplitude I0 there is a maximum frequency fmax where the shape of the current curve is still sinusoidal, while above fmax the shape is distorted. In Fig. 2.6, where we present the dependence of fmax on coil current, one can see that for current amplitudes up to 1 A the maximum frequency of rotation is well above 1000 Hz, which was important for a generation of isotropically attractive magnetic forces. Figure 2.6: Calibration of magnetic tweezers. Left: Magnetic fields Bz and Bx at sample location as a function of coil current I. Field in z-direction Bz (red squares, red line is linear fit with slope 10 mT/A) is stronger than field in x-direction Bx (black squares, black line is linear fit with slope 1.72 mT/A) due to the layout of six coils. Right: Maximum frequency of non-distorted harmonic current oscillation as a function of current amplitude I. 2.2.1 Superparamagnetic beads A superparamagnetic material is composed of small ferromagnetic clusters (e.g. crystallites) where the clusters are so small that magnetic polarization can spontaneously flip under thermal fluctuations. As a result, the material as a whole is not magnetized except when subjected to externally applied magnetic field. The rate of magnetization decay is governed by the Neel-Arrhenius equation. External magnetic field B aligns the dipole moments of superparamagnetic nanopar-ticles and thus induces magnetic dipole moment m in the particle. The induced dipole moment is proportional to B for small enough magnetic fields: m = XVB. (2.8) Here V is the volume, x is the magnetic susceptibility of the particle and = 4n x 10-7 Vs/Am is the inductance constant. In all magnetic tweezers related experiments we used superparamagnetic beads, composed of Fe2O3 nanoparticles in polymer, produced by Dynal. Different sizes and surface chemistries are available under brand name Dynabeads. We were using the following two types, MyOne Carboxy and M450 Epoxy. Their properties, diameter a, density p, magnetic susceptibility x and magnetic dipole moment in 1mT magnetic field are shown in the following table (data taken from [50]). Type p[g/cm3] X m[Am2]@1mT MyOne Carboxy 1.05 1.7 1.37 2.1 ■ 10-16 M450 4.4 1.6 1.63 1.8 ■ 10-14 2.2.2 Magnetic interaction The interaction of two dipoles When magnetic dipole with magnetic moment m is placed in an external magnetic field B equal but opposite forces arise on each side of the dipole creating a torque M = m x B . (2.9) This torque tries to align the dipole in the direction of magnetic field. The potential energy of such magnetic dipole in the external magnetic field is E = -m ■ B. (2.10) Let the magnetic dipole mi be located at the origin of coordinate system. The magnetic field B1 created by m1 is Bi(r) = ^0 3r(m! ■ r5 - r2m!. (2.11) w 4n r5 v ' If the second magnetic dipole m2 is placed in the magnetic field of the first dipole B 1, the pair interaction energy is, according to (2.10) E = — m2 ■ B1 (2.12) ^o I (mi ■ m2) 3(mi ■ r)(m2 ■ r) 4n \ r3 r5 E = no | ymi ■ ^2) ■ wy1^2 ■ i; | (2 13) The size and sign of interaction depend on the separation and the orientation of the dipoles. In the following sections we focus to particular orientations of the external magnetic field. Static external field in 2D Let us consider the case where two superparamagnetic beads are constrained to 2D plane (enclosed in a thin cell). Let 9 denote the angle between the connecting line of the beads and the direction of the external magnetic field B0. According to (6.3) both induced magnetic dipoles are parallel to the direction of B0 and therefore m1 ■ m2 = m1m2. The pair interaction energy (2.13) simplifies to (1 — 3 cos2 4n r3 In the particles are of the same type and assumption that magnetic dipole moment m is proportional to external magnetic field B0, i.e. in linear regime (6.3), the interaction can be written as E = „ 3 ' (2.14) X2V2B02 (1 — 3 cos 4n^0 r3 E = ^-^- 3 '. (2.15) If the magnetic field is perpendicular to the plane of particles (9 = 90°), the pair interaction energy reduces to E = X^B2. (2.16) If the external magnetic field lies in the plane of particles, the pair interaction depends on 9. If Bo is perpendicular to the connecting line between the particles, the pair interaction is repulsive as in (2.16). If B points in the direction of line between the particles (9 = 0°), the interaction is attractive and twice as strong as the repulsive interaction at the same separation E = — ££ (2,7) The expression for arbitrary configuration of particles in static magnetic field is more complicated. If the first particle is in the center of coordinate system, the second particle reside at (x,y,z) and external magnetic field is described by polar and azimuthal angles B0 = B0(cos 0 sin 9, sin 0 sin 9, cos 9), then the pair interaction energy reads x2vbo a , a , ____tJ2 4n^0r5 With static external magnetic field it is therefore possible to induce isotropic repulsive interaction and anisotropic attraction or repulsion. E = — X-5 (3(x cos 0 sin 9 + y sin 0 sin 9 + z cos 9)2 — (x2 + y2 + z2)) (2.18) 2 Rotating magnetic field in 2D Time variation of the magnetic field enables another important type of interaction - isotropic attraction. If the magnetic field rotates in sample plane, B(t)0 = B0(cos ut, sin ut, 0), the average interaction (2.18) for one revolution of the magnetic field reads E E E 1 f2n I E (x'y X2VB2x2 8n2poX5 X2V 2B2. 8np0x3 0,z = Q, 10-5 J/m2, for weak anchoring values are W < 10-7 J/m2. The long-range pair potential between spherical colloidal particles suspended in a uniform liquid crystal can be analytically calculated using in approximation of multipole expansion [65, 66] (for a review see [3]). It is assumed that the director field around the particle retains the dipolar or quadrupolar structure. The one-constant Frank elastic energy for one particle with anchoring contribution is F = J 1K[(V • n)2 + (V x n)2]d3r - £ 1W(n • v)2dS. (3.4) Anchoring strength W is positive in the case of homeotropic anchoring, while planar anchoring corresponds to negative W. At small anchoring WD/K ^ 1, where D is the diameter of the particle, or sufficiently far away from other particles, the differential equation for the minimum of the Frank free energy can be linearized and solved for director field n of the particle. The superposition principle can be used to obtain superimposed director field of two particles and consequently to calculate the total elastic free energy (3.4). Figure 3.1: Schematic drawing of two colloidal beads with a diameter D immersed in nematic liquid crystal with the director n. The resulting pair interaction potential of two particles with quadrupolar structure [3] can be written as U = C2 (l - 10 cos2 0 + 105/9 cos2 0) (3.5) Here 0 is the angle enclosed by the separation vector r and undisturbed director n (Fig. 3.1) and ci and c2 are quadrupole moments1 of the particles. If the particles 1 The magnitude of quadrupole moment is c a WD4/K. are identical and located in a plane, perpendicular to the director (9 = n/2), the interaction energy reads U = ^, (3.6) x5 where x is the center-to-center interparticle separation. The LC-mediated force between the particles can be obtained by differentiation of (3.6) and reads Flc = -W - (3.7) For the sake of simplicity we introduce the substitutions Cu = c2 and CF = 80Di6c2. The pair interaction energy can be then written as and the interparticle force U = (X/Dy. ' <3'8> 3.1.2 Effect of confinement The energy (3.6) is calculated for interaction between particles in a bulk LC. In actual experiments LC was confined in a cell, whose walls affect the interparticle interaction. The walls limit the modes of the elastic deformation of LC and they can also directly interact with the particles via LC [67]. Therefore the confining walls may have an important effect on the particle interactions and many particle structures formed if the interparticle potential is attractive. The effect of the confinement can be modeled with approach similar to classical electrostatics. The method of charge images has been successfully used in the field of LC to understand the interaction between defect and a flat wall [68, 69]. The interaction between two particles inside the confined cells can be interpreted as the interaction of one particle with an infinite array of the mirror images of the second particle. Particles with tangential anchoring are analogous to electric quadrupoles and homeotropic confining surfaces to parallel conducting plates. The electrostatic potential of a quadrupole between two conducting plates can be written in the form of modified Bessel function K0(kx), which decays exponentially for large x [70]. The coefficient k is determined by the boundary conditions in the direction z, perpendicular to cell walls: the solutions are proportional to sin(kz) and equal to zero at the conducting plates. Using the analogy, assuming infinitely strong homeotropic anchoring on the cell walls and considering the symmetry of the director fields around the immersed particle, the coefficient k = 2n/h is obtained, where h is the distance between the confining walls. 3.2 Experimental details 3.2.1 Liquid crystal The nematic liquid crystal used in all experiments was 4-n-pentyl-4'-cyanobiphenyl, well known under its commercial name 5CB. Its molecular formula is CH3(CH2)4C6H4C6H4CN, molecular weight 249.35, density 1.008g/ml (at 25°C) and nematic-isotropic transition temperature TNI = 35.5°C. It was purchased from Sigma-Aldrich (cat. no. 328510) and used as received without further purification. Dried superparamagnetic spheres Dynal Dynabeads M-450 Epoxy Coated with a diameter of D = 4.4 ^m were suspended in the LC using a shaker and ultrasonic bath. The epoxy coated surface of the spheres induced tangential (planar) anchoring of the LC at the surface, which was confirmed by observation of beads embedded in a planar LC cell under polarizing microscope (Fig. 3.2a). Two point defects (boojums) at the poles of the sphere along the average director orientation n0 are clearly visible. We also observed formation of bead chains at an angle of approximately d ~ 25 — 35° with respect to the average director orientation (Fig. 3.2b). Both are consistent with the quadrupolar nature of the director configuration observed previously [71, 6]. Figure 3.2: Superparamagnetic beads immersed in a thin planar cell with director orientation n0. (a) Polarizing micrograph shows two boojums at the poles of the sphere. (b) Bead aggregate at an angle $ = 25 — 35° with respect to director. 3.2.2 Cell preparation Microscope slide and cover glass have been sonicated in deionized water with 2 wt% detergent for 20 min, rinsed under running water for 20 min, rinsed with deionized water and then blown with dry N2. To induce homeotropic anchoring of LC molecules on the cell walls, the slide and cover glass were dipped in 0.5%wt solution of silane (Aldrich 435708) in water for a few seconds and then rinsed under running water for 10 minutes to remove excess silane. To achieve planar anchoring of liquid crystal on surfaces a Nylon coating and combing protocol was used: 0.25%wt Nylon 6/6 was dissolved in methanol at 50°C. The solution was spin-coated on the glass surface at 3500 rpm. The surface was then rubbed 10 times with a velvet cloth to obtain a planar homogeneous alignment. Samples were made by sandwiching a droplet of the mixture of LC and beads between a slide and a cover slip, which ensured a homogeneous distribution of particles. The cell was glued using a UV-curing glue (Norland Optics). The thickness of the cell was controlled by the volume of the droplet (Vdropiet=area of the cover Figure 3.3: Side view of an experimental cell with thickness h and spheres with a diameter D. Director field configuration around two spheres inducing tangential anchoring in a homeotropic cell. glass x thickness). Experimental cell thicknesses were always less than twice the diameter of the spheres to prevent stacking of the beads on top of each other. Figure 3.4: Cell thickness determination procedure. Left: Side view of a tilted pair of colloidal particles. Right: Top view of two beads in contact. Upper image: No magnetic field, the pair lies in the imaging plane. Lower image: Strong magnetic field perpendicular to the imaging plane reorients the pair. The exact cell thickness h was determined after the cell was assembled. The samples always included some pairs of beads in contact which were then used as a probe to indirectly measure the thickness (Fig. 3.4). Such bead pair tries to align with field direction when the perpendicular magnetic field is applied. The cell thickness can be easily obtained from the projection of spheres' separation a as h = D + VD2 - a2. 3.2.3 Force calibration Although the dependence of the magnetic force on the electric current I and particle separation x was theoretically known (FMR = Aa-, see Sec. 2.2.3) the scaling coefficient A still had to be determined independently because of the slight variations of the magnetic susceptibilities of different beads. This was done by observing the motion of the same pair of beads as used in the actual force measurements. First, the in-plane rotating magnetic field inducing attractive magnetic fields was used to brought the particles almost into contact. The field was then switched off and the particles drifted apart (Fig. 3.6, part A) due to the LC-mediated repulsive force. When Brownian motion prevailed, the magnetic field perpendicular to the plane of beads was switched on, which resulted in a repulsive magnetic force (Fig. 3.6, part B). The latter part of the trajectory can be well described if x(t) dependence is calculated from AI2 x Fmr = A-4- = 7-, (3.10) where the inertial term is neglected. The factor 1/2 comes from the fact that for symmetric interactions the change in the separation x is twice the dislocation of one sphere. The coefficient 7 in Eq. 3.10 is given by the Stokes-Einstein relation Y = kBT/Dc, where kB is the Boltzmann constant, T is the temperature, and Dc is the diffusion coefficient. The coefficient y is obtained separately by analyzing the Brownian motion of a single particle [51]. From the measured value y = 4.2(1 ±0.03) pNs/^m, calculated diffusion coefficient Dc = 9.8 x 10-4 ^m2/s is in a very good agreement with the diffusion coefficient obtained in [51]. From (3.10), the relation (10AI2 Ai/s x(t) = — (t - to)) (3.11) is obtained and fitted to experimental data (Fig. 3.6, part B). The accuracy of calibration factor obtained from the fit is approximately 10%. t[s] Figure 3.5: Interparticle separation as a function of time in force calibration procedure. When two particles are released, the LC-mediated force is pushing them apart (part A). After some time a strong repulsive magnetic field is turned on which drives the particles even further apart (part B). The motion can be well described by (3.11), shown in red. 3.3 Results and discussion To understand the nature of the quadrupolar interparticle interactions in thin LC cells we carried out five different types of experiments: 1. Motion of free beads: beads were brought to various initial separations and then released. Trajectories of the motion were recorded. 2. Static measurements of LC-mediated interaction: magnetic tweezers were used to induce attractive magnetic force that was used to counterbalance repulsive LC-mediated force. Beads were at rest at different separations, depending on the magnitude of the magnetic force. The result was dependence of the LC repulsive force on interparticle separation. 3. Dynamics measurements: beads were brought almost into the contact and then released. The relationship between velocity and separation of beads was obtained. 4. The effect of the confinement: static interparticle interaction was measured in samples of different thicknesses. The long range decay of the interaction was obtained. 5. The temperature dependence of the interaction: static force measurements were performed in a temperature controlled sample. 3.3.1 Motion of free beads from different starting separations Single colloidal particle immersed in LC host undergoes Brownian motion [51]. In the vicinity of other particles the motion of the bead is affected by LC-mediated interparticle interactions, which are repulsive in thin homeotropic cells. The effect of the repulsive interparticle force FLC on particle motion was studied as follows. A suitable pair of beads with good planar surface alignment of LC (checked using crossed polarizers) was found and brought to a separation of about 10 microns with weak optical tweezers; the trap power in the sample was on the order of 10 mW. The optical tweezers were then turned off. The rotating magnetic field inducing attractive magnetic force FMA was applied to move beads to the initial separation x0. When the field was switched off, the beads immediately started to drift apart due to quadrupolar interparticle interaction mediated by the surrounding liquid crystal. We measured particle trajectories x(t) for several different values of x0 ranging from 4.6 ¡m to 5.9 ¡m. As can be seen in Fig. 3.6 particle trajectories were independent of the initial separation x0. The overlapping is extremely good for small separation x, whereas at larger separations (x>6.5 ¡m) the trajectories become strongly influenced by Brownian motion. Figure 3.6: Interparticle separations x as a function of time t for different initial separations x0. The overlapping of the trajectories indicates that the relaxation processes present in the nematic LC during the motion of the bead are fast enough to consider the director configuration around the beads as quasistationary. Such adiabatic behavior is not surprising, as lowest relaxation times of the director field in several microns thick cells are of the order of 10 ms. This is much shorter than any characteristic time scale of the particle motion observed during measurements (for example, the characteristic time for a diffusion of the particle for its radius is approximately 103 s). Another important conclusion is that the parameters of the particle motion do not depend on the velocity. This is true even for very small initial separations (x0 ~4.6 ^m). The motion of the particles is thus determined by the effective drag coefficient Yeff, which includes the LC viscous drag, the impact of the cell walls, and the influence of LC distortions. To obtain the effective drag coefficient jeff = FLC/v as a function of separation x, the interparticle force FLC(x) and the velocity of beads v(x) had to be measured. In the first step, a set of static measurements was performed where the LC-mediated repulsive interparticle force FLC was balanced with the attractive magnetic force FMA. In the second step we did dynamic measurements where the LC repulsive force was balanced by the viscous drag force of beads drifting apart. 3.3.2 Static measurement of LC-mediated force In the static type of measurements the LC-mediated force FLC was balanced by the calibrated magnetic force FMA (Fig. 3.7). There were no net motions of the beads, so all dynamic effects were excluded. Two suitable beads were found and the magnetic force was calibrated as described in Sec. 3.2.3. The beads were brought at a separation of around 10 ^m using optical tweezers which were then turned off for the whole duration of the experiment. Sufficiently strong in-plane rotating magnetic field was applied which induced attractive interaction that brought beads almost2 in contact (wall-to-wall separation was 0.2 ^m). (a) h t (O_c 1 X n0 B (b) © n0 F F F 1 LCV y MA ' MAy — © ' LC ^^ Figure 3.7: Experimental geometry: x is the interparticle distance, B is the in-plane rotating magnetic field, n0 is the director. (a) Side view, h is the cell thickness. (b) Top view with forces acting on the beads. The LC-mediated force FLc is balanced by the attractive magnetic force FMA. 2If the magnetic field was too strong the beads came into the contact and permanently sticked together due to van der Waals interaction. Electric current I through the coils (and thus the magnetic attractive force FMA) was then reduced, so that the equilibrium interparticle distance increased. For each current setting we waited for couple of minutes for equilibration before the video recording of the beads started. For each interparticle separation 2000 frames at speed of 50 fps were captured. Typical chart of interparticle separation as a function of time at a given constant current through the coils is shown in Fig. 3.8. It can be clearly seen that the particles undergo Brownian motion around an average separation, which was in this case 5.71±0.04 ^m. 0 10 20 30 40 time[s] Figure 3.8: Interparticle distance as a function of time at some constant magnetic force. Dashed line indicates average separation. For small magnetic attractive forces, where the bead separation x was as large as 10 ^m, Brownian motion prevailed. At that point, the electrical current through the coils (and therefore the FMA) was increased again, and the whole cycle was repeated several times. The experiment showed that the equilibrium positions of the particles are the same at a given current no matter whether the magnetic field was being increased or decreased. The absence of the hysteresis indicates that with varying particle separation the LC director configuration around the beads changes continuously and reversibly. At given equilibrium bead separation the LC mediated colloidal force FLC is exactly opposite to the magnetic attractive force FMA. From the measured separations and known magnetic forces the dependence of the repulsive LC force FLC on the interparticle separation can be obtained. The force FLC as a function of normalized particle separation x/D is shown in Fig. 3.9. The fit of (3.9) to the experimental data yields CF = 4.69(1 ± 0.18) pN. 3.3.3 Dynamic measurements The goal of dynamic measurements was to obtain the velocity of particles drifting apart as a function of interparticle separation. The drift of beads was induced only by repulsive LC force. The measurements were actually made as a first part of force calibration procedure (Sec. 3.2.3) and then the same pair of beads was used 10 Figure 3.9: Left: Interparticle LC-mediated force as a function of interparticle separation in log-log scale. The solid red line is a fit of the power law dependence 1/(x/D)6 to measured data points. The deviations from the power law at small separations are expected, since (3.9) holds only for particles that are sufficiently separated. The deviations at large separations are due to the confinement effect. Right: Particle drift velocity as a function of normalized interparticle separation x/D in log-log scale. Particles are pushed apart only by repulsive LC-mediated force. The solid line is the best fit of v(x) rc 1/x6. for static measurements. To recapitulate the dynamic measurements procedure, the beads were brought almost in contact using attractive magnetic field. The field was then turned off and the beads were pushed apart only by the LC-mediated repulsive force. The whole process was video recorded at 50 fps. The positions of the particles were obtained offline and used for separation and velocity determination. The LC-mediated repulsive force FLC is balanced by opposing drag force yv: (3.12) If the force FLC is the same as in static experiment (FLC rc 1/x6, Eq. 3.9) then the velocity should be also dependent on separation as v(x) rc 1/x6. (3.13) In Fig. 3.9 we present a typical particle drift velocity v as a function of interparticle distance x as calculated from temporal series of particle positions. Good matching between experimental data and fit with (3.13) indicates that both the interparticle force FLC and the drift velocity follow the same dependence. The effective drag coefficient Ye// = F/v is therefore independent of the separation between the particles for x >4.9 ^m. The obtained value Ye// = 3.8(1 ± 0.2) pNs/^m is in agreement with the drag coefficient obtained from the analysis of single particle Brownian motion. 3.3.4 Effect of confinement To measure the effect of the confining surfaces on the interparticle interaction, cells with different thicknesses were prepared. Here we present results from two typical x flc (x) = Yv(x) = Y o samples: sample A had a thickness of h = 8.0(1 ± 0.05) ^m, corresponding to approximately 1.8 x D and sample B a thickness of h = 6.5(1 ± 0.05) ^m, equivalent to 1.5 x D. The measurements were made in the static mode. The force as a function of separation was then integrated in order to get the interparticle potential and the limiting value was set to zero for large interparticle separations. The obtained potential U(x) is shown in Fig. 3.10 as a log-log plot. The circles and squares are the measured data for sample A and B, respectively. Figure 3.10: Logarithmic plot of the interparticle potential as a function of normalized bead separation x/D for two sample thicknesses, h =1.5 x D (red) and h = 1.8 x D (black). The solid lines are fits of power-law functions at small separations, and dashed of exponential decay for larger separations. In Fig. 3.10 it is clearly seen that the power-law dependence fits the measured data only for a limited range of interparticle separations. The power-law function U(x) = C/(x/D)^ can thus be fitted only for x < 0.9h, yielding the coefficients P = 5.1(1 ± 0.05) and P = 5.4(1 ± 0.10) for samples A and B, respectively, which is consistent with theoretical prediction (3.6). For particle separations comparable to the sample thickness, the influence of the confining walls becomes noticeable. When x > 1.2h, the interparticle potential can be successfully fitted by an exponentially decaying function, U = C2 exp(-x/A). (3.14) The obtained decay length is A = 1.41(1 ± 0.10)^m = 0.18(1 ± 0.15)h for sample A and A = 0.92(1 ±0.15)^m = 0.14(1 ±0.20)h for sample B. The decay length obtained from the the electrostatic analogy (Sec. 3.1.2) for large x decays exponentially with decay length A = 0.16h, which is in excellent agreement with the experiment. The deviation from the power-law dependence of the interparticle potential is due to the confinement: when the separation between two particles becomes comparable to the sample thickness, the strong anchoring on the cell walls relatively reduces the effect of the beads on the director. The long range of the particle-induced deformation is thus suppressed (screened) and decays exponentially with cell thickness as a typical length scale. Since the director deformations cause the interparticle forces, exponential decay is expected also in the force profile. Similar screening effects were extensively studied in the defect annihilation processes [72, 73]. 3.3.5 Temperature dependence of interaction Temperature dependence of the quadrupolar interaction was obtained using modified version of static measurements. We used servo mechanism to control the required magnetic force to hold the beads at the same separation at different temperatures. The measurement started approximately 7 K below the nematic-isotropic phase transition temperature, where two beads were brought to desired separation using optical tweezers which were then turned off. If no magnetic attractive force had been applied, the beads would have drifted apart due to repulsive LC force. Instead, magnetic attraction was controlled using closed feedback loop to keep the beads at the desired separation all the time. The error signal for servo mechanism (the deviation of interparticle distance from the desired separation) was obtained from image analysis of microscope image. After few iterations of the feedback loop the electric current was roughly stabilized and real data acquisition began. The required current needed to keep the beads at desired separation was recorded for few minutes until enough data points were collected. The same process was then repeated at higher temperatures all the way up to the transition temperature TNI. 0.6 7 6 5 4 3 2 1 0 V™ Figure 3.11: LC mediated force FLC between two beads 6 ^m apart as a function of reduced temperature TN1 — T. The solid line is the best fit of Eq. 3.15 to the data. The recorded electric currents at each temperature were used to calculate average LC-mediated force FLC. Typical relation between force FLC and reduced temperature TN1 — T for a 6 ^m interparticle separation (wall-to-wall distance 1.6 ^m) is shown in Fig. 3.11. The interparticle force is decreasing when the temperature is approaching the transition point. The measured data points can be fitted with power function F = a(TN/ - TY (3.15) with the coefficients a = 0.0275 ± 0.005 and P = 0.40 ± 0.01. The power law dependence of FLC can be roughly explained using (3.6). If the constant quadrupole moments are assumed, then the strength of interaction depends solely on Frank elastic constant. For 5CB temperature dependence can be written in form K (T) = a + b(TN1 — Twith pK = 0.50 [74] which is reasonably close to the measured value of P. It has to be emphasized that (3.6) holds for quadrupolar particles in bulk LC and when the fluctuations of the director are small which is clearly not the case in our confined geometry when temperature is close to phase transition. 3.4 Conclusion and outlook We have used magneto-optical tweezers as a tool to precisely measure the forces between microparticles in nematic liquid crystal 5CB. The particles were 4.4 ^m superparamagnetic spheres with planar anchoring of LC in a thin homeotropic cell. Such particles interact as elastic quadrupoles. We performed static and dynamic measurements of the force over four orders of magnitude. In static experiments the repulsive LC-mediated force was balanced by known attractive magnetic force. Image analysis was used to obtain equilibrium separations of the beads and from them the force vs. separation dependence was measured. The measured static LC-mediated force between particles follows 1/x6 dependence for wide range of interparticle separations x, starting from x ~ 4.5 ^m to x ~ 10 ^m. In the dynamic measurements the particles were first brought together with attractive magnetic field. The field was then turned off and the repulsive LC mediated force caused the particles to drift apart. From the trajectories of the particles the dependence of the average particle velocity v on the interparticle separation x was calculated. Combining static and dynamics measurements, the effective drag coefficient jeff = FLC/v was determined and within experimental error found to be independent of the particle separation for x > 4.9 ^m. This is surprising because of significant distortions in the director field around the particles, which could result in a separation dependent drag coefficient. We have also experimentally confirmed that the interaction between colloidal particles in a confined NLC is different from that in a bulk. At particle separations, comparable to sample thickness, there is a crossover of the interparticle potential from power law to exponential decay. The decay length was found to be proportional to the sample thickness, causing a significant reduction of the potential. This occurs because the cell walls induce strong homeotropic alignment and when the separation between particles is comparable or larger than the sample thickness, the LC director field between the spheres is less disturbed and the spheres interact as if they were further apart. Such double-regime behavior and reduction of the long range of the potential has to be considered in designing new colloidal structures, especially in samples, in which the thickness is comparable to the bead size. The knowledge of the interparticle potential is crucial for the optimization of the assembly processes in colloidal crystals. The combined magneto-optical tweezers have proven to be successful tool for colloidal studies in liquid crystals, especially as both repulsive and attractive external force can be precisely generated without any measurable effect of the external fields on the director configuration. Chapter T_ Engineered potentials We study two- and three dimensional systems of superparamagnetic colloidal particles interacting with various potentials that are induced by the external magnetic field. The static field in a 2D system induces a dipolar repulsion or attraction between the particles which results in a formation of open hexagonal lattice of particles in the former and the formation of colloidal chains in the later case. In a quasi-2D system the repulsive potential of the static field is softened which stabilizes many unusual mesophases of particles. More complex interparticle potentials can be induced by time modulation of the magnetic field. If the field precesses on a surface of a cone with the opening angle dp = 54.7°, the 1/r3 term of dipolar interaction vanishes. Local field effects give rise to isotropic 1/r6 attraction between the particles. The pair-wise non-additivity of the interaction stabilizes chains and hexagonal close-packed sheets in 3D. 4.1 Introduction One of the fundamental problems of condensed matter physics is a relationship between interparticle interactions and collective structural behavior of many-body systems. The topic is studied theoretically and computationally using so called "forward" approach of statistical mechanics. The interparticle interactions in the system of interest are described with approximations, then numerical simulations and analytical methods are used to predict the details concerning structural, thermodynamic and kinetic properties of the system. Self-assembly (SA) is the process in which a system components (molecules, polymers, colloids or other basic building blocks) spontaneously organize into a larger ordered and/or functional structures [14, 15]. The form of final macroscopic structure is governed by the microscopic interactions between the components. Examples of SA can be found everywhere, perhaps most spectacular in biology, including the formation of lipid bilayers to produce membranes and the protein folding. Starting at the molecular scale, SA is the fundament for crystallization of organic and inorganic molecules. At the nanoscale it is the foundation for various types of molecular structures such as Langmuir-Blodgett films, self-assembled monolayers of amphiphilic fibres as well as superstructures built from nanoparticles. At the microscale SA is most present as a concept to build metamaterials (for example, colloidal particles can self-assemble into complex super-structures that could be potentially used as a photonic band-gap materials) and in the field of microfluidics (micromixers, microvalves). Complementary insight into the subtle link between microscopic and macroscopic properties can be obtained by the inverse approach [75]. Using inverse statistical mechanical methods interparticle interactions that lead to a desired many-body structure of the system can be found. These interactions should be then somehow induced in colloidal or polymer systems to yield the desired structures at the nanoscopic and microscopic length scales. Optimized potentials would enable self-assembly of particle configurations with novel properties. The relationship between the interparticle interactions and the resulting structures in atomic and molecular systems has been widely theoretically and numerically investigated, due to the small size of the constituents of those systems there are no experimental studies. With modern experimental techniques it is still impossible or very difficult to directly observe an atomic or molecular system and access the typical time scales. Even if direct observation is achievable, manipulation of the inter-atomic potentials is impossible. Some 30 years ago colloidal systems [16] had been found to be suitable model for condensed matter systems. Statistical mechanical concepts known from the theory of liquids can be directly applied to an ensemble of colloidal particles to provide a framework for relating the microscopic properties of single particles to the macroscopic properties of the whole ensemble. The phase behavior of colloidal systems, such as freezing and melting of colloidal crystals, shows striking resemblance to that of atomic or molecular systems. The colloidal systems overcome all difficulties that prevent direct study of atomic systems. Larger size and slower dynamics of colloidal particles (a =1 nm - 1 ^m; t = 1 ms - 1 s) allow the use of readily available experimental techniques, such as optical microscopy, confocal laser scanning microscopy, static and dynamic light scattering, small angle X-ray scattering and neutron scattering, to probe the colloidal suspensions. Typical time scales for processes within colloidal systems are long enough to allow the monitoring of real-time dynamics of the particles using optical microscopy. In addition, properties of colloidal systems can be manipulated: a change of a number density (simple addition of more particles into the system) can be enough to induce a phase transition. Changes of particle or solvent properties or application of an external field (such as electric or magnetic) alters the macroscopic behavior of the system. The limitation of colloidal systems is the spherical shape of most kinds of available monodisperse colloidal particles which results in an isotropic interparticle potential. This is the reason why the most common ordered structures are the hexagonal lattice in two dimensions and the face-centered cubic lattice in three dimensions [17, 18]. There are two routes to more complex colloidal structures. It is clear that anisotropic pair potentials stabilize new kinds of structures that are not known in systems with isotropic repulsion or attraction. Anisotropic interparticle interactions can be induced by surface treatment of particles [19], external fields [20] or liquid crystalline solvent [3]. Another, less obvious path to new types of ordered structures are isotropic interaction potentials with radial profile different from the usual power law dependence of common potentials. A simple realization of such modified interaction potential is an isotropic interaction with radial profile, characterized by two length scales, e.g. a combination of hard-core and softer repulsive part. If the ratio between the ranges of the soft shoulder and the core is around 2, numerical simulations predict that such interparticle potential induces a variety of mesophases [21, 22, 23] between the fluid and the close packed crystal. In two dimensions simulations predict loose and dense hexagonal lattice, liquids of monomers, dimers and trimers, stripe and labyrinthine phases, honeycomb structure etc. Similar behavior has been predicted also in systems of paramagnetic particles limited to 2D plane, interacting with a combination of a dipolar repulsion (induced by perpendicular magnetic field) and a Lennard-Jones interaction [76]. For large shoulder/core ratios, the set of mesophases reduces to micellar, lamellar, and inverted micellar structure [24]. In this Chapter we present a study of a two- and three dimensional systems of micrometer sized superparamagnetic colloidal beads interacting with various types of potentials that were induced by the external magnetic field. The interparticle interaction is a combination of the excluded volume interaction (hard-core potential) and the magnetic interaction between the induced magnetic dipole moments of particles. The contribution of other characteristic colloidal interactions, such as electrostatic and van der Waals interaction, is negligible. We begin with the simplest configuration, a 2D system of particles in an external static magnetic field, which depending on its orientation induces a dipolar repulsion or attraction between the particles. We then concentrate solely on the radially isotropic pair potentials, achieved by the magnetic field perpendicular to the sample plane. In two dimensional system such field induces 1/r3 dipolar repulsion that stabilizes the hexagonal lattice regardless of the surface filling fraction of particles. If a system is quasi two-dimensional (the particles have a limited degree of freedom to move in the vertical direction) the vertical static magnetic field induces isotropic pair potential, whose radial profile can be tailored by varying the sample cell thickness. Depending on the surface filling fraction of particles it stabilizes a lot of mesophases that have been theoretically predicted. In the second part of the Chapter, the focus is on the complex isotropic potentials induced by the rotating magnetic field. Especially interesting potential arises when the magnetic field precesses on a cone with the "magic" opening angle 9p = 54.7°, which causes that the 1/r3 term of the dipolar interaction vanishes. Due to the residual interactions between the particles the interaction potential is isotropically attractive with 1/r6 separation dependance. The Chapter is concluded with a demonstration of stable structures induced by this potential. 4.2 Experimental details Particles used in the experiments were superparamagnetic spheres with diameter a = 1.05 ^m (Dynabeads, MyOne Carboxy) suspended in the original Dynabeads solution. The solution efficiently prevented van der Waals interaction between the particles, so no aggregation was observed. Electrical charge on the beads was heavily screened with estimated screening length of a few nanometers. The interaction between the particles was induced by the external magnetic field, which was generated by three orthogonal pairs of coils as described in Sec. 2.2. Figure 4.1: Side view of the experimental cell of thickness h with 3 colloidal beads of diameter a. The external magnetic field Bo is transverse to the cell walls. The colloidal suspension was introduced into the wedge shaped sample cell formed by 150 ^m thick microscope cover glasses. When the thickness had to be precisely known, the cells were built from two 500 ^m thick glasses in order to minimize bending of the cell walls. These cells were assembled under monochromatic light where the exact thickness map could be determined by the observation of fringes. One of the crucial external parameters was the cell thickness: if it was just slightly more than the bead diameter, the particles were confined to one plane and the system was considered to be two-dimensional. Quasi 2D system was realized in a thicker part of the cell, where particles had also some limited freedom of movement in the transverse direction. The cells, more than two diameters thick, were considered as three dimensional systems. Observation of 3D colloidal structures with a standard optical microscope is quite limited. As soon as the filling fraction is high enough that a layer of particles is formed at the bottom of a sample cell the light wavefront traveling from the focal plane to the microscope objective is disturbed which renders the imaging of the space above the layer difficult or impossible (if there are more than 2 layers). Therefore we were able to observe and accurately characterize the structures formed in cells with thickness less than 3 diameters of colloidal particles. Typically the magnetic field, inducing desired interparticle interaction, was turned on and after relaxation time (a few seconds or minutes) when the system reached (quasi-)equilibrium the snapshots of the resulting mesophases were taken. When we were interested in the dynamics of structure formation, the videos at high frame rates (a few 100 fps) were recorded and then analyzed off-line. Numerical calculation of the interaction energy In order to understand the observed ordered mesophases of superparamagnetic particles in more details, a software for a numerical calculation of the system energy in arbitrary external magnetic field was developed. If the effects of the local fields are taken into account, the energy of the system, in general, can not be expressed analytically (see Sec. 2.2.2) so numerical evaluation was unavoidable. The algorithm is described in Appendix A. 4.3 Static magnetic field in 2D system The analysis of various ordered phases induced by different kinds of interparticle potentials starts with the simplest case: a static magnetic field in two dimensional system. According to the calculations in Sec. 2.2.2 the pair interaction can be either isotropically repulsive in the special case of the perpendicular field or anisotropic in any other case. The behavior of the two dimensional system is governed by two external parameters - the strength of the magnetic interaction and the 2D filling fraction. One possible way of describing the interaction strength is with an interaction constant K defined as the magnetic energy of a pair of beads in the external field perpendicular to the pair. Using (2.16), the energy of such pair can be written as E = ^J^o^ =K/r3 with na6x2B 2 K = na X B . (4.1) 144^0 ( ) Here a is the bead diameter. It is also convenient to define similar dimensionless constant K (inverse "temperature" of the system) as a ratio between the energy of a pair of beads in contact divided by kB T [77] K X = -InTr- (4.2) a3kBT Typical magnetic fields in the experiments ranged from a few mT to a few tens of mT, giving K from a few tens to a few hundreds. At this point it has to be emphasized that our interest was not in the dynamical behavior of the system (crystal growth, melting, meshing...), but rather in the equilibrium or quasi-equilibrium structures formed by different types of interparticle potentials. As the typical K was high, the system reached (quasi)-equilibrium phase in matter of seconds after the interaction was turned on. The surface filling fraction n is defined as a fraction of surface covered with beads n = na2n/4, (4.3) with n being a numerical density of particles (number of particles/area). In a 2D system the maximum n = n/2\f3 ~ 0.907 is reached when the spheres are arranged in the hexagonal close-packed lattice. 4.3.1 Isotropic repulsion If the magnetic field is directed perpendicular to the sample cell, the interaction between a pair of particles is isotropic dipolar repulsion (2.16). We performed two different observations of the phase behavior of the system: at the constant filling fraction as a function of interaction strength and at the constant interaction across the broad density range. The characteristic micrographs of the former observation are presented in Fig. 4.2. Figure 4.2: Micrographs of the typical phases in the 2D colloidal system with constant 2D filling fraction, induced by dipolar repulsion of different strengths. With no magnetic interaction, the system is hard-core liquid (left), which changes to open hexagonal lattice when the interaction is turned on. The inverse temperatures of the system from left to right micrograph are K & 0,10,140, 2250. For low magnetic fields thermal motion dominates the magnetic energy and the colloidal particles are randomly distributed in the plane. The system shows continuous orientational and translational symmetry and appears as two dimensional fluid. As soon as the magnetic repulsion between the particles is large enough to dominate thermal motion some local order arises. At high interaction strengths, the particles arrange into hexagonal crystal which has discrete translational and six-fold orientational order. Typical micrographs of the 2D system with constant interaction strength but increasing surface filling fraction are shown in Fig. 4.3. At low filling fractions, when the average pair interaction energies far below kBT, the system is in the liquid phase. When the filling fraction is increasing the equilibrium structure changes to open hexagonal and then to hexagonal close-packed (HCP) lattice. In the 2D system with the dipole-dipole repulsion the pair interaction energy can be written including filling fraction as E = K/r3 = = K^)3, where the mean separation between the particles is a = \J2^73. If the distances are measured Figure 4.3: Micrographs of the typical phases in 2D colloidal system, induced by dipolar repulsion of constant strength (K & 1000). At low filling fractions (left) the particles are in liquid phase, which changes to open and close-packed hexagonal lattice when filling fraction is increased. in units of the mean separation, the filling fraction merely rescales the temperature and only a single parameter is needed to explore the whole phase space [77, 78]. Differently, when the system is not two dimensional the filling fraction scaling is not present and two parameters are needed to describe it. 4.3.2 Anisotropic dipolar interaction External static magnetic field in the particle plane gives rise to anisotropic interaction potential. The sign and the strength of the interaction depends on the relative position of the particles with respect to the field direction (2.15). In Fig. 4.4 we present the potential energy landscape of a single particle, induced by the magnetic field pointing in y-direction. In order to calculate the energy landscape potential of a desired configuration a probe particle is introduced into the system and is moved across the plane of interest. The fixed particles are depicted with gray color (in this case, there is only one particle) whereas white color represents forbidden space for the centers of other particles due to the excluded volume. Different colors of other points in the image represent the total energy of the system if the probe particle is located in that particular point r. The color scale in this and all other energy landscape images ranges from blue (lowest energy) through green and red towards black (highest energy). Figure 4.4: Left: The xy-plane energy landscape of a particle with external magnetic field Bo in the y-direction. The particle, indicated by gray color, is in the center of the image. White color around it represents the excluded volume (sphere with radius a) for the probe particle. Different colors of other (x, y) points in the image represent reduced energy of the system E/(K/a3) when the center of the probe particle r is at (x,y). The potential is symmetric with respect to the x-axis. Particles attract when 9 < 54.7° and repel when 54.7° <9 < 90°. Right: Two micrographs of the system with in-plane external magnetic field. When the field is turned on first small segments of chains are formed, which then merge into long single chains. At high filling fractions neighboring chains join into wider stripes (not shown). The interparticle force is calculated as a gradient of the energy landscape. In this case, the particles are attracted together when the angle 9 between vector r to the location of the probe particle and the direction of the field is from 9 = 0° to 54.7°, and repelled away from each other when 9 = 54.7° to 90°. Between both regions, when cos2 9 = 1/3, i.e. 9 = 54.7°, the dipolar interaction between the particles vanishes. If the external static magnetic field B0 lies out of the particle plane, it can be divided into two perpendicular components1: the in-plane, which is either repulsive or attractive depending on the relative position of the particles, and the perpendicular component, which induces isotropic repulsion. The interparticle interaction is then a sum of both contributions. -3.0-,—,—T—T—,—T—,—,—,—,—,—x—-—T—,—,— -3.1 -3.2 -3 3 UJ -3.4 -3.5 -3.6 -I-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1- 12 3 4 x/ a), the interaction is pure dipolar repulsion (F > hm), the in-plane force is repulsive for large separations and attractive for small separations. This demonstrates that the system behaves as expected - by simple adjustment of geometrical constraints the interparticle potential can be varied from pure dipolar repulsion to softened repulsion or to a combination of attraction at small separations and weak repulsion at large separations (Fig. 4.6). It is important to realize, that due to the limited but finite degree of freedom of the particles in the vertical direction the pair potential shown in Fig. 6.13 is not preserved in the case of many interacting particles. At low surface filling fractions (n < 0.05) when the average distance between the particles is large compared to rtim] r[Mm] Figure 4.6: Left: Measured in-plane force profiles for 3 different cell thicknesses: r-4 repulsion in thin cell (h w a; red circles), softened repulsion (h w hm; black squares) and over-softened interaction in thick cell (h > hm, green triangles) with the attractive part at small separations. Solid lines are fits of (4.5) for h = a, h = 1.46a, and h = 1.72a, respectively. Right: Schematic drawing of the interaction potentials: dipolar repulsion (h w a; red), softened repulsion (h w hm; black) and over-softened interaction in thick cell (h > hm, green). the cell thickness, the interaction potential is well approximated by simple 1/r3 repulsion. At higher filling fractions (n > 0.2) it is energetically favorable for the particles to reside close to either the upper or the lower wall of the cell, i.e. the distribution of their heights is bimodal. In this situation particle pairs can be found in the following states: up/up, down/down and up/down state. In the first two states the particles "feel" a simple 1/r3 repulsion, while for the up/down state the interaction potential has a shape depicted in Fig. 4.6. 4.4.2 Phase behavior of a system with softened repulsion Phase behavior of the system was studied across a broad density range. The cell was filled with a dense suspension of superparamagnetic spheres, and the measuring site with cell thickness of about hm was located. As neither the cell thickness or the interaction potential couldn't be precisely measured due to the high density of the particles, the following protocol was used. In absence of the magnetic field, weak optical tweezers were used to herd the spheres at a given location into contact such that they formed cluster. Transverse magnetic field was then turned on which caused the cluster to disperse. In the thick part of the cell where the potential was attractive the cluster disintegration was partial, whereas in the thin part with purely repulsive interactions the cluster disintegrated completely. By scanning the cell the site with the desired thickness was identified. In this region of the sample cell we increased the particle density n using the extended array of optical tweezers to locally heat the suspension. The heating induced a hydrodynamic flow which dragged a large number of particles (over 104) toward the trap, thereby increasing the local density almost to close packing. The tweezers were then turned off which stopped the flow, and the magnetic field was turned on. The high density region underwent slow expansion such that the system could be observed at decreasingly smaller colloidal densities. A sizable change of the density took place on a time scale of about 10 s, which is much longer than the typical diffusion time2 of particles (< 10 ms). The expansion was therefore slow enough to ensure quasi-equilibrium at all times, and the hydrodynamic interactions due to expansion ^ 10-3&£T are negligible compared to the magnetic repulsion. The main advantage of this protocol is that it allows one to study the states of a fixed ensemble of spheres at the same location in the cell (and thus at the same cell thickness) across a range of densities. This excludes any deviations in the sample that could affect its behavior in an uncontrolled way and ensures that the phase transitions are induced solely by density variation. FIGURE 4.7: Micrographs of the representative mesophases induced by varying the surface filling fraction n (labeled). In some micrographs, patches of the underlying lattices are emphasized to guide the eye. The micrographs of the most interesting phases observed at various surface filling fractions n are shown in Fig. 4.7. At very high filling fractions (n > 0.5) domains of honeycomb and dense square lattice were observed. At lower density (n = 0.39) dense lattice dispersed into stripe/labyrinthine structure formed predominantly by a single cluster of interlaced strings of touching spheres. Lower filling fraction (n = 0.34) resulted in a chain phase with locally aligned finite-length strings. At intermediate filling fractions (n = 0.31) there was a coexistence of expanded square lattice and chain phase, mainly composed of dimers and trimers. Coexisting expanded hexagonal and square lattice were observed at n = 0.23. When filling fraction was low enough, the system was in expanded hexagonal lattice (n = 0.12) and in liquid phase (n = 0.01). 2 In equilibrium, a thermally excited particle in the potential well formed by its neighbors fluctuates around its mean position by about d ^ (knTahf (n)384K)1/2, where f (n) is a numerical factor which equals 1 at close packing and « 1000 at n ~ 0.1. The typical time needed to traverse this distance is t ^ d2/D, where D = knT/3nva is the diffusion constant; v is the viscosity. In our case, t « 10 ^s at n = 0.5 and t « 10 ms at n = 0.1. The distribution of the heights of the particles can be analyzed to some extent. Close inspection of the micrographs reveals that some beads appear brighter than the others - it is because their position is slightly out of the focal plane of the image. At high filling fractions (n = 0.1 or more for K ~ 250, used in the experiment) the beads are either at z = 0 or z = h. The system therefore resembles an off-lattice two-state spin ensemble with dominant nearest-neighbor anti-ferromagnetic interaction. On the stripe, honeycomb and square lattice with 2, 3, and 4 regularly arranged nearest neighbors a simple ground state with alternating up-down positions of spheres exists, while on the hexagonal lattice the system is frustrated and no ground state with long range order is possible. The experimentally observed mesophases are similar to those found in the numerical simulations. In [76] the structure and phase behavior of a 2D system with a hard-core and purely repulsive core-softened long-range interactions is studied using Monte Carlo simulations. The pair interactions are of the form, E(r) = 4e[(a/r)12 — (a/r)6] + e'(a/r)3, with the energy parameter e' chosen to give a stationary point of inflection in the pair potential. The simulation reveals a variety of interesting states: fluids with chainlike, striped, and 6 — 10 sided polygon structural motifs, low and high-density triangular crystalline phases and defective Kagome lattices. The states, found in simulation, are remarkably close to mesophases found in our experiment although the pair interaction is not the same. Even a much more idealized hard-core-soft-shoulder interaction also gives a similar phase sequence [21, 22]. This suggests that the mechanisms at work as well as the structures they produce are rather robust. 4.5 Static magnetic field in a 3D system The previous Section describes the system of interacting superparamagnetic beads confined in the sample cell with the critical thickness hm ~ 1.447a, where the perpendicular external magnetic field induces softened repulsive radial interaction force between two particles. If the cell thickness is increased, the pair interaction force (4.5) remains repulsive at larger separations but becomes attractive at small separations as demonstrated in Fig. 4.6. In this Section we present the phase diagram of the system as a function of two parameters: filling fraction and the cell thickness. 4.5.1 Low filling fractions When the filling fraction is low enough, stable structures are pairs of particles, which are aligned in the direction of the magnetic field B if the thickness of the sample cell is more than two diameters (h > 2a). Since the interaction between two such pairs is dipolar repulsion, pairs arrange into hexagonal grid as can be seen in Fig. 4.8(c). If the thickness of the cell is smaller but still above the critical thickness (hm < h < 2a), the pairs are tilted as in Fig. 4.8(a), (c). The polar angle 9 between B and the vector £ connecting the centers of the particles depends on the ratio between the cell thickness and the diameter of beads, cos 9 = h/a — 1. The azimuth angle 0 between the x-axis of the system and the projection of £ a be Figure 4.8: Representative micrographs of colloidal mesophases formed at low density at three different cell thicknesses. (a) Thickness is more than the critical thickness; pairs are stable despite the low filling fraction. (b) h ~ 1.5a. (c) The cell is thick enough for pair formation in the field direction. The isotropic repulsion around each colloidal pair results in a hexagonal ordering of pairs. Note: beads in (c) look larger because two beads are stacked on each other. to the xy-plane is random if there are no external forces that would make specific direction more energetically favorable. In the case of a dilute system this is true therefore time evolution of 0(t) is a random walk with diffusion constant determined by the friction of colloidal pairs. However, when the particle density is increased and pairs start to "feel" each other, the energy of a pair is not anymore degenerate in 0. In order to minimize the interaction energy with neighboring pairs, it is favorable for the pair to have the same 0 as its neighbors. Such interaction gives rise to nematic structures, which can be seen in the micrographs (a) and (b) of Fig. 4.8. G^J) O GO ....... — GO GO ♦ Figure 4.9: Nematic arrangement of colloidal pairs. Top left: Side view schematic of a tilted colloidal pair with polar angle 9. Bottom left: Top view schematic of a system of 7 titled pairs in a hexagonal arrangement with lattice constant a = 3a used for numerical calculations. Right: Numerically calculated addition to the energy as a function of azimuth angle 0 of the central pair for different pair polar angles 9 = 50° (green), 30° (red) and 10° (black) in the field B0 = 10 mT. It is evident that the interaction that tries to orient a colloidal pair in the direction of neighbors strongly depends on the pair polar angle. If the cell is thick enough to allow vertical pairs (0 = 0°), this interaction is zero, while on the other hand, when the thickness is a bit more than critical thickness (9m = 63.45°), the interaction is the strongest. We have numerically calculated the addition to the magnetic energy of 7 titled pairs in a hexagonal arrangement with the lattice constant a = 3a in the external magnetic field B0 = 10mT as a function of azimuth angle 0 of the central pair. The results for different pair polar angles, presented in Fig. 4.9 support the observed formation of ordered pairs in Fig. 4.8. 4.5.2 High filling fractions In a high filling fraction regime beads can't remain in isolated clusters, and chains of particles are formed. The chains consist of beads that are either in the contact with the upper or with the lower wall of the sample cell: if a starting bead is on the sample bottom ("down", d for short), then the second bead is pushed to the upper wall ("up", u). Third bead is then again d and so on till the last bead which is either u or d, depending on the number of particles in the chain. Schematic drawing of a tilted pair and a short chain is presented in Fig. 4.10. ▲ u \ , u u Bo h 0 i ' y x d Figure 4.10: Schematic drawing of a tilted pair consisting of u and d particles and a short chain, made from u, d and u particles in a cell with hm < h < 2a. The angle d is the pair tilt angle with respect to the direction of magnetic field Bo The creation of chains is a rapid process. When the magnetic field is turned on, particles join into dimers or trimers in a few tenths of a second, then these clusters merge into longer chains within couple of seconds. Three representative images of the system at high filling fractions are shown in Fig. 4.11. The leftmost is the micrograph of the region where thickness h is just slightly more than critical thickness hm, the polar angle is approximately 9 w 50°. The stable structures under these conditions are long single chains; their interchanging u — d composition can be spotted by close inspection. The chains can form a network with junctions where three chains are connected together. The middle image shows a thicker part of the cell where the polar angle is 9 w 35° and the equilibrium structure are isolated single chains. Their udud... configuration is even more pronounced and due to the optical artifacts they appear thicker. If the polar angle is even lower, chains disintegrate into colloidal pairs. In the right micrograph of Fig. 4.11 the cell is a bit thicker than two bead diameters. The isotropic repulsion around each colloidal pair results in a hexagonal ordering of pairs. In the lower right part of the micrograph two "defects" are emphasized, where three colloidal pairs are bound together by a single particle in the middle. Such formations enable increased filling fractions. To understand the process of chain creation we have numerically calculated the energies of different configurations of basic building blocks as a function of their separation. In Fig. 4.12 we show a comparison of energy per particle as a function Figure 4.11: Representative micrographs of colloidal mesophases formed at high density at three different cell thicknesses. Left: d & 50°, stable structures are long single chains. Precise inspection reveals that every second bead in a chain is slightly out of focus. Middle: d & 35°, stable mesophase is the same type of chains, but with less intersections. Right: d & 0°,h > 2a. The isotropic repulsion around each colloidal pair results in a hexagonal ordering of pairs. Red circle marks two "defects", where three colloidal pairs are bound together by a single bead. of separation for two different configurations which consist of either two ud tilted pairs or ud tilted pair and a single u particle. The calculation was performed for two different cell thicknesses h giving the pair tilt angles 9 = 30° and 9 = 50°. As observed in the experiments, there is a repulsion between tilted pairs if 9 is small. If 9 is larger, there is still repulsion at large separations, but at small separations attraction prevails. Two pairs in the contact are bound in a potential well, whose depth depends on a pair tilt angle 9 - higher 9 induces deeper well. In the case of an ud pair and an isolated u particle there is also a potential barrier, but once the particle is over the barrier, it is strongly bound in udu configuration. This is the mechanism that stabilizes chains In the experiment we never observed chain merging, which indicates interchain repulsion. In order to get the magnitude of this interaction, the energy of two parallel chains composed of up-down interchanging colloidal particles as a function of their separation was numerically calculated. Fig. 4.13 shows the energy per particle Eparticie of such system as a function of the normalized separation d/a between the axes of two chains for two different lengths of chain and for two different polar angles 9. Regardless of the chain length and the polar angle the interchain interaction is always strongly repulsive. The calculated energies can be accurately fitted with power-law Eparticle = A0 + A1(d/a)^, giving exponents ft ranging from —1.37 (100 particles, 9 = 30°) to —1.80 (10 particles, 9 = 50°), which is interesting since the dipolar energy between two particles scales as 1/d3. We have also performed the calculations of the energy of the same system, but with one chain shifted for a half of a basic unit (a sin 9) in the long axis direction. One would expect that two chains in contact, one with configuration of beads udud... and the other with shifted configuration dudu.., have lower energy than in the un-shifted case, but it turns out that the energy difference is negligible. This interchain interaction is different from the interaction between two chains parallel to the field direction (Fig. 4.5), where a shift of one chain changes the interaction from repulsive to attractive. Figure 4.12: Calculated energy per particle for different configurations (side views as insets) at B0 = 10 mT. Left: Energy as a function of surface-to-surface separation x. When a tilted pair of beads in ud configuration is approached by a single u particle, there is a potential barrier of 70kBT for polar angle 9 = 30° (green triangles) or 90kBT for 9 = 50° (inverted blue triangles). If a tilted pair approaches to another tilted pair the potential shape on depends on 9. Low 9 induces repulsive potential (example 9 = 30°, black squares), while at high 9 there is a potential barrier (example 9 = 50°, red circles). Right: Energy as a function of linear density of particles, i.e. the number of particles per particle diameter. Black squares and red circles indicate energy for tilted pairs at polar angles 30° and 50°, respectively. Magenta stars indicate energy of perfect triplets at 9 = 30°. Figure 4.13: Numerically calculated energy per particle for a system of two parallel chains (left image) as a function of their normalized separation D/a. The upper two curves show the energy for chains of 100 beads for two tilt angles, 9 = 30° (red squares) and 9 = 50° (green circles). The lower two curves show energy for short chains, made of 10 beads, also for 9 = 30° (blue triangles) and 9 = 50° (light blue inverted triangles). The fit of power-law function to the data of the lowest curve (dashed) yields fl = -1.65. The results of numerical calculations are in agreement with the experimentally observed mesophases. The calculations explain the formation and behavior of colloidal pairs at low and colloidal chains at high filling fractions. 4.6 Rotating magnetic field in 2D system The rotation of the external magnetic field enables new type of interaction potentials. As explained in Sec. 2.2.2 the rotating field induces attractive effective interaction between dipolar particles and exerts a torque on the particle aggregates. The phenomenon has been studied (in systems of magnetic holes [81], droplets [82], magnetorheological fluids [83]...) and used for self-assembly and rotation of colloidal micropumps [84, 85]. Common feature to the mentioned experiments is the rotation of the field in one direction only which caused the structures to rotate. The stationary rotation frequency is reached when the magnetic torque is balanced by the viscous torque. In contrast, our magnetic tweezers enable bidirectional rotation of the field, therefore the self-assembled structures do not rotate and the viscous forces do not affect their shapes. 4.6.1 Inplane rotation of field If the magnetic field rotates in the particle plane the average dipolar pair interaction is isotropic attraction as calculated in (2.17). Superparamagnetic particles exposed to such magnetic field merge into clusters of close packed hexagonal arrangement. If the field direction is changed, it is energetically favorable for clusters to follow the field. In case of a rotating magnetic field also the clusters rotate. If the frequency of field rotation is low enough (friction torque of the cluster is smaller than the magnetic torque), the clusters can follow the field direction consequently their frequency of rotation is the same as the field frequency. If the field rotation frequency is higher, then in each turn of the field the colloidal cluster rotates just for a part of the full turn (phase-slip regime). The cluster rotation frequency is therefore smaller than the field rotation frequency. In order to prevent the rotation of the colloidal clusters, 4 || |i| ¡¡lip » I Figure 4.14: Micrographs of 2D system with isotropic attraction at three different filling fractions. The resulting structure is always hexagonal close packed lattice with possible defects. the in-plane field was rotated back and forth for 360°. If the effective frequency of field rotation is too small then the clusters "shiver" back and forth, thus as high as possible frequency is preferred. In the experiment the frequency was 400 Hz which was high enough to efficiently suppress the effects of field rotation for fields of a few mT used in experiment. Typical resulting structures of in-plane rotation of the magnetic field for different initial filling fractions of colloidal particles are shown in Fig. 4.14. The images were taken 3 minutes after the field of 4 mT was turned on. As expected, the basic arrangement of beads is a HCP lattice. The 2D crystals, especially in the systems with intermediate filling fractions, have lots of voids (defects) which indicates that they are far from equilibrium. Even if the field is turned on for a long period of time, particles do not arrange into one perfect defect-free HCP cluster due to kinetically arrested particles, i.e. the particles with the too high energy barrier towards more favorable configurations. 4.6.2 Rotation of tilted magnetic field We have already shown that the magnetic field perpendicular to the plane of the particles induces dipolar repulsion and that the rotating in-plane field induces dipolar attraction. Now let us consider a combination of both fields. The combined field Bo = Bo (sin 9p cos 0, sin 9V sin 0, cos 9p) precesses on a cone (Fig.4.15) whose opening angle 9P depends on the relative magnitude of rotating in-plane BXy = \JBx2 + By and vertical Bz components tan 9P = ^BX + By . (4.8) Bz Figure 4.15: The combined external magnetic field Bo = Bz + Bxy field precesses on a cone with opening angle 6p as can be seen in a side-view (left). Top-view (right) reveals the azimuthal angle ^ between the in-plane field component and x-axis. If two particles are in such precessing field, it is natural to expect that the effective interaction in linear magnetization regime is a sum of dipolar repulsion and dipolar attraction. Micrographs of the typical mesophases that grow in a low filling fraction system under different opening angles 0 are shown in Fig. 4.16. If the opening angle of the cone 0P is small, the repulsion prevails and the colloidal beads arrange in a hexagonal lattice. On the other hand, very large opening angles close to 90° make the beads attract each other, so the resulting arrangement is HCP. Interesting effects arise in the intermediate region of the opening angles, where magnitudes of the repulsive and the attractive interactions are of similar magnitude. Similar experiment have been recently performed [86], but their use of the rotating magnetic field resulted in symmetrical rotating clusters, formed by the combined effects of magnetic and viscous forces. When the opening angle 9p is increasing, the repulsive force is decreasing which consequently dissolves the initial hexagonal lattice as can be seen in Fig. 4.16 where 9p = 46°. The interparticle repulsion is still strong enough to prevent beads to come into contact. Surprising effect happens at 9p = 50° - colloidal particles join into chains, which suggests the presence of an attractive force. Theoretically calculated repulsion induced by the vertical component of the field (2.16) is at this 9p larger than the attraction induced by the rotating component of the magnetic field (2.21), therefore particle aggregation is not expected. Experimentally observed stable single chains indicate that the attractive force acts only at chain ends, whereas in the direction perpendicular to the chain there is a repulsion, otherwise 2D crystallites would be formed. The regions around chains are clear of any beads due to this repulsion. Just a slight increase of the cone opening angle to 9p = 52° changes the behavior of the system - colloidal chains interconnect into a network. In each junction exactly three chains are joined together, preferably at an angle of 120°. The angle and empty space between neighboring parallel chains imply the interchain repulsion perpendicular to chain direction. At around 9p = 58° the in-plane component of the magnetic field induces the attractive force that is stronger than other interparticle forces. The remaining single beads are adsorbed to the chains; since there is no repulsion between chains anymore, they start to merge. This is a gradual process, because the attractive force in the chain direction (which also prevents chain bending), is stronger than the attraction between two neighboring chains. Since chains are cross-linked, they can't instantaneously rearrange into energetically more favorable configuration of a HCP lattice. In fact, although there is an attraction among all components of the system, it never reaches the equilibrium configuration because the potential barriers towards energetically more favorable configuration are too high. The colloidal mesophases in a 2D system with higher filling fraction, induced by the precessing magnetic field, are shown in Fig. 4.17. At 9p = 35° the dipolar repulsive force is still strong enough to stabilize expanded hexagonal lattice, whereas at 9p = 39° the onset of chains can be seen. Chains are clearly visible at 9p = 42°, although the majority of beads is still free. Higher opening angle, 9p = 46°, induces cross-linking of the chains. Due to the high filling fraction, the density of chain junctions is higher and average length of chain segments is shorter compared to the system with the lower filling fraction. Almost all particles become part of the network at 9p = 50°. There are only rare occasions of single beads, pushed to the middle of network pockets as a result of repulsive force perpendicular to the chain direction. At 9p = 56° the dipolar attraction prevails, the remaining single particles are drawn into the network and ' m • ~ e £ » » # •••••• • #•* # i • « •. • « • t # • • • . • • * * » • • * * * • •a * • * * * « « • Î!•.•••, • • . « # _ • » • • • • . * • • • • • • • • • • » • • * # • _ ♦ * • • ® ® • • • * * • • m • m • B S • • • — _ • É • * ft • — • - ^ * * • • • • • • • # ..AC • • _ 0 a • O 4 • ^ } r ' ^ AT 1 • • • V V X a • 1 rrrxV * J*—HL. ' /Ov • ° : ] • B S S i*( î.\ » 'Vi 52 m o l • ••• ••»**••* ,*< • •• *o 0 >.v • ••• o 0 . • o • • - • • • • §• # * • CO • c • °o • • Jo ' o° * o° 9 ' ° • • •• • •• • s _ # • * •. ° • • • »•• • • o • • m to •• o © • • • A « « • • * # • A 0 * • o • / " • ^V* 8 * • V / 1 *• • f • • •«>« 1 >••• / ••••• ^r • » / ' a (»„C* '„'»i'' ** ° °# ' •* . • #*„ ® O at O • • o • • O ° 46 • 9 J * * ^ • f f0 J h" \l ¥ "ÄL- ¿"Ii- JT a. X Hk fl. L ';;/" „ - Jm £ \ _/* A _ T) 58 "V j f Figure 4.16: Experimental snapshots of a low filling fraction 2D system with magnetic field precessing on a cone, labeled by the corresponding opening angle of the cone dp. H ^^ZlCjrVL •jrjLj s Figure 4.17: Experimental snapshots of a high filling fraction 2D system with precessing magnetic fields, labeled by the corresponding opening angle dp. the process of the network coarsening begins. Some chains in the micrograph have already torn up and merged with neighboring chains to form 2D crystallites which have lower energy. Qualitatively the same mesophases are found in both systems, either with low or high filling fraction. At small cone opening angle 0p the stable phase is the expanded hexagonal lattice, which dissolves as 0p increases. At some 0p the particles start joining into chains, which attract other particles at ends but repel them sideways. When 0p is high enough, isolated chains interconnect into a network of single chains. Even higher opening angle makes the dipolar attraction the dominant force so the structure collapses into a HCP crystal. There is a quantitative difference between both systems: if micrographs of both systems at 0p = 46° are compared, one can see that in the dilute system, the particles are still well separated, whereas in the dense system, not only the chains have already been formed but they are already interconnected. Higher filling fraction results in smaller average interparticle separation, which means that even the subtle changes of interaction (created by tiny variation of 0p) can induce different equilibrium structure. To explain the observed formation of new structures, we have Figure 4.18: The energy of a pair of particles as a function of surface-to-surface separation x at 9p = 50° (left chart) and 9p = 56° (right chart). Comparison of exact energy calculation (black) with dipole-dipole potential where the effect of the local field is neglected (dashed red). numerically calculated the energies of a few basic configurations. When the magnetic field with 0p that prefers the creation of chains is turned on, isolated beads first merge into shorter chains (typically pairs), which then join into longer chains; consequently we have first analyzed the formation of particle pairs. In Fig. 4.18 we compare the energy E of two particles as a function of their surface-to-surface separation x when the opening angle of field cone is 0p = 50°. At this angle the dipolar interaction between two dipoles of fixed magnitude that follow the direction of the rotating field is repulsive. The exact calculation of energy where the effects of local fields are taken into account reveal an important difference - at large separations the behavior of E(x) is the same, whereas at small separations there is a potential barrier. When a particle is pushed over a potential barrier by thermal fluctuation or by another particle, a stable pair is formed. The attraction (and the resulting formation of pairs) at low 0p, where 1/r3 dipolar interaction should be repulsive, is a clear confirmation of the local field effect, explained in Fig. 4.19. The local magnetic field created by the induced dipole of the first particle reduces the induced dipole of the second particle ("demagnetization") when the direction of the external field is perpendicular to the pair - the pair interaction is less repulsive. When the external field is in the direction of the pair, the local magnetic field created by the first particle increases the induced dipole of the second particle ("magnetization") and makes the pair interaction more attractive. Averaging over one cycle of the magnetic field precession brings an overall attraction that causes the potential well at small interparticle separations. Figure 4.19: Schematics of the local field effect. Bi and B2 are the local magnetic fields of particle 1 and 2, respectively. Left: When the external magnetic field B0 is perpendicular to the pair, the induced dipoles mi and m2 are smaller ("demagnetization"). Right: When the external magnetic field B0 is in the pair direction, the induced dipoles mi and m2 are larger ("magnetization"). 4.6.3 Measurement of interparticle potential The interparticle potential was measured using optical tweezers. We first found a bead that was attached to the surface of the sample cell and another free bead that was used as a force probe. The rest of the measuring procedure was exactly the same as described in Sec. 4.4.1. Two representative force profiles are shown in Fig. 4.20. 0.200.150.10- „ 0.05-Z Q. ^ 0.00 --0.05 --0.10-0.15- Figure 4.20: Left: Measured interparticle force profile for two cone opening angles: 0 = 0, B0=2 mT (black squares) and 0 = 53°, B0=10 mT (red circles). Right: Corresponding calculated pair potentials. Although the measured data points are highly scattered two distinct profile shapes can be clearly seen. Vertical magnetic field (0 = 0, B0 = 2 mT) induces 1/r4 repulsive force between two particles. The force induced by the precessing magnetic field with cone opening angle 9 = 53° (B0 = 10 mT) is attractive at small separations due to the effect of the local fields. We have to stress that the magnitude of the magnetic field, used in the measurement at 9 = 53° had to be 5 times larger to obtain comparable forces at separations of interest because the dipolar repulsion and attraction almost cancel each other at this 9. The force profile measurement demonstrates that the system behaves as expected - by the adjustment of the field cone opening angle 9 the shape of the interparticle potential can be varied from pure dipolar repulsion at 9 = 0 through more complexly shaped at intermediate 9 to pure dipolar attraction at 9 = 90°. 4.6.4 Many-body effect After a pair of beads is formed, the potential around it is not isotropic anymore. Fig. 4.21, where we present the energy landscape of a pair of particles in contact, is a key to understanding the observed formation of chains. Anisotropic nature of the interaction between the pair and a probe particle can be clearly observed on the xy energy landscape. If the probe particle is approached to the pair in the direction of its long axis (x-direction), the total energy of the system is a few kBT lower than the energy when the probe is far away. On the other hand, if the probe approaches the pair from the direction of short axis (y-direction), the total energy is a few kBT higher. Figure 4.21: Calculated energy for 3 particles in xy-plane. Left: The energy landscape around a pair of particles at 6p = 50° and B0 = 3.5 mT. Right: System energy vs. surface-to-surface separation when the probe particle is approaching the pair in two special directions (dashed in the left image): in the direction of the long axis of the pair (E(x), blue) and in the direction of the short axis (E(y), red). Dashed curves represent energies for the same two cases but calculated as a sum of 3 pair energies. The anisotropy of the energy landscape around a pair is altered by the effect of many-body interactions. The interaction between a pair of particles at 9p = 50°, shown in Fig. 4.18, is repulsive at surface-to-surface separations larger than 0.25 ^m and attractive at smaller separations. If the energy of the system of 3 interacting particles was just a sum of three individual pair energies, it would be impossible to obtain the attraction in the direction of a pair. The energy profiles, calculated as a sum of pair energies, are presented in Fig. 4.21 with dashed lines. If another particle is approaching from the direction of short axis, the energy profile E(y), obtained as a sum of 3 pair energies, is similar to the exact many-body energy calculation. This certainly isn't valid if the third particle is coming from the direction of the pair (E(x)). The exact many body calculation reveals attractive potential whereas sum of pair energies results in a repulsive potential. To explain the formation of single chains, the force on a single particle approaching to a pair has to be considered. If the particle is somewhere near the long axis of the pair it is immediately drawn to the pair and a linear structure is formed. On the other hand, if the particle is in the region near the short axis, it is either repelled away or pulled to the either end of the pair, so the resulting structure is again linear. 4.6.5 Assembly of colloidal superstructures When the magnetic field with 9p ~ 50° is turned on in the low filling fraction system the resulting self-assembled structure are single chains of beads (Fig. 4.16), oriented isotropically in all directions. Chain growth is similar to the polymerization, a process where monomer molecules form polymer chains. The growth of colloidal chains decreases both the energy of the system and the filling fraction of beads (monomers). The growing stops when there are no free beads in the vicinity of chain ends. Figure 4.22: Colloidal superstructures assembled using optical tweezers. The attraction between two beads in the direction of chains stabilize 2D structures. The repulsion perpendicular to chains prevents them to collapse into a 2D HCP crystal. (dp = 53°, B0 = 3.5mT) Chains are stable structures at external magnetic field of a few mT. Beads at chain ends are in the potential well of more than 5kBT and are therefore "permanently" bound to the rest of the chain. This enables the formation of more complex structures composed from colloidal chains. We used optical tweezers to manipulate them into desired structures; some of them can be seen in Fig. 4.22. Either be single beads, chains, rings, decorated rings, structures within structures, they are freely fluctuating and undergoing Brownian motion. The simplest structure formed from a chain is a ring, which is obtained when a chain is bent and its ends joined together. The bending costs energy due to the repulsion perpendicular to the chain direction. The bending energy depends on ring diameter: smaller rings have greater bending energy and therefore higher tension. In Fig. 4.23 we present micrographs of the artificially assembled colloidal rings of different sizes. The smallest ring of the left image has ruptured and opened into a straight chain, seen in the right image. Figure 4.23: Colloidal rings repel each other. The rings are in general stable, except rings with small diameters which are less stable due to high bending energy. The smallest ring in the left micrograph has ruptured and transformed into a chain. (0p = 53°, Bq = 3.5mT) The repulsion perpendicular to the chain direction gives rise to an interesting phenomena - repulsion between the structures made of colloidal chains. If two rings are brought together using optical tweezers a weak repulsion can be observed - rings slowly drift away from each other. In an enclosed sample with a high number of rings a crystallization would occur - the rings would arrange into an ordered lattice. If a ring is created within another ring (as in Fig. 4.22), the two rings never touch despite their undulations around equilibrium circular shape. 4.7 Residual interaction induced at magic angle rotation When the opening angle of the field cone equals the magic angle 0m = arccos(1/\/3) « 54.7° the 1/r3 term of dipolar interaction vanishes. The effective interaction between two particles averaged over one cycle of magnetic field rotation is analytically calculated in Appendix A. At magic opening angle 0p = 0m, the lowest non-zero correction to the energy of a pair of induced dipoles in the external magnetic fields reads V3v 3 b 2 1 E = - X V Bq - (4 9) E 16n2^o r6. ( ) Due to fully 3D isotropic nature of the interaction (6.16) which is completely independent of the direction of the major axis of the cone of the external magnetic field, colloidal systems in this section are not divided anymore into 2D, quasi-2D and 3D systems. Instead, the focus is on the interactions and structures induced by the magic angle field rotation in a 3D system. Figure 4.24: Left: Calculated energy landscape around a single particle at magic angle and B0 = 3.5 mT. The potential is spherically symmetric, shown is the xz-plane. Right: Calculated pair energy as a function of the interparticle separation. Red curve is fit of the calculated data with the power-law function with the exponent fl = 6.087. In Fig. 4.24 we present numerically calculated energy of a pair of particles when = 0m. Although the axis of the cone of the field precession is in the z-direction, the energy potential around a single particle is isotropic. The chart in Fig. 4.24 shows the energy of two beads as a function of their center-to-center separation. The calculated data can be fit with power-law dependence3, E(r) = Ar-13, yielding fl = 6.087 ± 0.002, which confirms the analytical calculation of the effective interaction Due to the isotropic 1/r6 energy potential around a bead another bead from the vicinity is attracted and a colloidal pair is formed. Its initial orientation is determined by the positions of the constitutive beads, but later the orientation is random, governed only by thermal fluctuations. In Fig. 4.25 micrographs of self-assembled structures formed in a dilute colloidal system are presented. The colloidal pairs are pointing in all directions, both in xy and xz plane, which is a clear evidence of the isotropic nature of the pair interaction. The energy landscape around a pair of beads, presented in Fig. 4.26, is not isotropic. A third bead is attracted to the pair if approached from the direction of the long axis - the resulting linear configuration of 3 beads has approximately 30 kBT lower energy compared to the case when the third bead is far away from the pair. If the third bead approaches from the side, it reaches a tiny potential barrier (less than kBT). The particle is then drawn to either end of the pair and the resulting structure of 3 particles is linear. If the approaching particle manages to overcome the potential barrier, a triangular configuration of particles is formed. This configuration is stable (it has around 3The slight deviation from fl = 6.00 is due to the higher order terms that are neglected in the analytical calculation. 2.0 2.5 3.0 (6.16). Figure 4.25: Micrographs of the self-assembled colloidal chains formed by magic angle rotating magnetic field B0 = 3.5 mT. Left: Two consecutive snapshots of the low-filling fraction system, the lower one taken 1 s after the upper one. Colloidal pairs (circled) are stable regardless of their orientation. Right: Larger structures, predominantly single chains, which are at some locations joined by additional beads and form double chains. r[Mm] Figure 4.26: Calculated energy for 3 particles at magic angle and B0 = 3.5mT. Left: The energy landscape around a pair. Right: Calculated total energy as a function of surface-to-surface separation between the pair and a third particle. The force is always attractive if the third particle is on long axis (x-direction) of the pair (blue). If it is on short axis (y-direction), the force is attractive at small and slightly repulsive at larger separations (red). 20kBT lower energy than if the third bead is far away), but nonetheless the linear configuration of particles is formed in more than 95% cases due to the potential barrier which directs the third particle to either end of the pair. 1 0 CM 1 ■ 5 8 Figure 4.27: Micrographs of HCP colloidal sheets (dp = dm, Bo = 3.5 mT). Left: Where the local filling fraction of particles is high, chains merge into 2D structures. Right: Micrographs of a self-assembled sheet taken at different microscope focal planes. The number in each image is the height in ^m of the focus with respect to the cell bottom. Black stipes indicate the focused region of the image. When double chains are formed and there are available single beads or free chains in the vicinity, the formation of triple chains begins. Free beads or chains are always adsorbed into the plane of the first two chains. The growth goes on until there are no available building blocks anymore. The resulting structures are 2D HCP sheets as demonstrated in Fig. 4.27. Sheets are stable regardless of their orientation in space. In fact, it is possible to bend them (using optical tweezers) without disrupting their stability. This is demonstrated in Fig. 4.27, where we show 4 micrographs of a sheet, taken at four different focal planes. They confirm that the sheet is bent upwards and stable. Due to sedimentation colloidal sheets settle to the bottom of the sample cell as shown in the left micrograph of Fig. 4.28. Slight decrease of the field cone opening angle eliminates the isotropic nature of the interaction - vertically aligned sheets become more energetically favorable. Smaller clusters reorient in a matter of seconds, while the edge regions of larger sheets start bending upwards. The process finishes when sheets reorient into vertical position and form tube-like structures. To understand the assembly of chains from single beads and the formation of sheets from single chains, we have numerically calculated the energies involved in the process. The model, used for calculations presented in Fig. 4.29, assumes that the equally separated beads join into chain by the reduction of their separation. The results of the per-particle energy calculation as a function of separation are in agreement with the experimental observations - chains are energetically more favorable structure than separated particles. The calculation has been performed for two chain lengths: the shortest possible chain of 2 particles and an "infinitely" long chain, modeled by 200 particles. In the case of the short chain, the resulting Figure 4.28: The effect of decreased field opening angle on colloidal sheets. Left: initial state 0p = 0m, the sheets have settled to the bottom of the cell due to gravity. Middle: After the tiny decrease of 0p the favorable configuration are vertically aligned sheets. Smaller separated clusters of sheets undergo complete reorientation, while larger sheets start bending upwards. Right: The equilibrium formation is the network of vertically aligned sheets (tube-like). Figure 4.29: Calculated energy per particle in the assembly of colloidal structures as a function of separation. Left: Assembly of N single beads into a linear chain. Energy per particle for N = 2 (black) and N = 200 beads (red). Right: Assembly of N colloidal chains of N beads into 2D HCP sheets. Energy for N = 10 (black), N = 20 (red) and N = 30 (green). energy vs. separation dependence is of course the 1/r6 pair potential. Particles in the longer chain are bound approximately 7 times stronger than a pair of particles in contact. The functional form of per-particle energy as a function of separation follows 1/r6'3 dependence. The formation of 2D HCP sheets from single chains is energetically favorable process. Right chart of Fig. 4.29 presents the energy per particle of N chains, each made of N beads, as a function of chain separation. In order to characterize the finite size effects the calculation has been performed for N = 10, 20 and 30. The energy vs. chain separation dependence can be fitted with a power-law with the critical coefficient ft = 4.2. To explain the sheet stability we calculated the energy landscape around an axially symmetric HCP planar sheet composed of 19 particles: one in the center Figure 4.30: Planar sheet of 19 particles, arranged in HCP configuration. dp = dm, B0 = 3.5 mT. Left: The xy energy landscape around a sheet. Middle: The xz energy landscape around the sheet. Right: Calculated addition to the total energy as a function of surface-to-surface separation between the sheet and a probe particle. If the probe particle is in plane of the sheet, the potential is attractive (blue). If the probe is on the plane normal, the potential is repulsive (red). of the coordinate system, 6 around it in the first layer and additional 12 in the second layer (Fig. 4.30). If the probe particle is located in the sheet plane (xy) the interaction is attractive. If it is above or below the sheet in the normal plane (xz) that includes the sheet center, the potential is repulsive. In the right chart we plotted the energy as a function of surface-to-surface separation between the sheet and a probe particle on the two special lines. This calculation explains why the HCP sheets are stable structures - if there are any free particles near the sheet plane, they are drawn into the sheet, while on the contrary particles above or below the sheet are repelled away. A spherically isotropic 1/r6 attractive potential around a single bead promotes the formation of pairs. The pair potential has the cylindrical symmetry - attraction in the direction of long axis and repulsion in the perpendicular direction. Such potential stabilizes chains, 1D structures. The potential around a sheet of particles is attractive in sheet plane and repulsive in perpendicular direction, therefore it stabilizes sheets, 2D structures. A logical question is whether 3D structures are stable as well. To answer this question we have analyzed the energy of a single particle approaching a planar sheet from the transverse direction. Left chart of Fig. 4.31 shows the energy vs. surface-to-surface separation dependence for different sheet sizes. If sheet is made of one bead only, the interaction is familiar pair attraction. If 6 beads are added around the central bead (sheet has diameter of 3a), the interaction changes to repulsive. Adding another layer of beads (sheet diameter is now 5a) makes the interaction even more repulsive. Interesting transition occurs when 3 more layers of beads are added around the sheet: the interaction between the single particle and the sheet is still repulsive, but at very small separations there is a minute potential well - the particle is weakly bound to the sheet. If the sheet diameter is increased, then the potential well becomes even larger. A single particle is therefore stable if it is pushed into a contact with a large sheet of particles. To check the stability of 3D structures we have calculated the energy of 2D HCP sheets merging into a 3D HCP crystal. Right chart of Fig. 4.31 displays per-particle 0 1 2 3 4 5 6 0.6 1.0 1 2 1,4 1.6 1.8 2.0 2.2 wall-to-wall separation [u tn| sheet separation [^.m] Figure 4.31: Left: Calculated energy for a single bead approaching to 2D hexagonal close-packed crystal as a function of separation. Different colors denote different diameters (sizes) of sheets: black - 3a (7 beads), red - 5a (19 beads, inline schematic), green - 11a (92 beads) and blue - 17a (218 beads). Right: Joining of N HCP sheets of NxN beads into 3D crystal. The energy per particle as a function of sheet separation for N = 10 (red) and N = 20 (black). Figure 4.32: Hexagonal close-packed crystallite formed of 57 beads. Left: The crystallite is made of 5 HCP layers consisting of 7, 12, 19 (central), 12 and 7 particles. Middle: The xy energy landscape. Right: The xz energy landscape. energy as a function of sheet separation for two sizes of crystals. Similar to the case, when a single bead approaches to HCP sheet, the energy dependence includes a potential barrier: HCP sheets have to be pushed together, but once a 3D crystal is formed, it remains stable. Our final calculation, the potential landscape around a small crystallite, composed of 57 beads, is presented in Fig. 4.32. As expected, the potential around globular 3D crystallite is isotropically attractive. 4.8 Conclusion and outlook We performed an experimental study of the relationship between the interparticle interactions and the structure formation in a system of micrometer-sized super-paramagnetic colloidal particles. The interaction was induced by the external mag- netic field and controlled by the spatial constraints. The focus was on two types of isotropic interaction potentials: core-softened repulsion in 2D and isotropic 1/r6 attraction in 3D. In a system with the core-softened repulsive interaction several self assembled mesophases were formed depending on the surface filling fraction. We observed the square, hexagonal and honeycomb lattices as well as the labyrinthine structure. This was the first experimental validation of theoretical predictions pertaining to similar pair interactions [21, 22, 76]. The observed two-dimensional structures could have also applicative value as they could be used as templates to promote the growth of colloidal crystals [87, 88]. More complex interparticle potentials were realized by the precession of the magnetic field on a surface of a cone. The variation of the cone opening angle 9p was used to precisely tune the ratio between dipolar repulsion and attraction. At the opening angle, equal to the magic angle 9m ~ 54.7°, the 1/r3 term of dipolar interaction vanishes. The dominant term in the pair potential was found to be 1/r6 isotropic attraction which arises due to the effect of the local magnetic fields. Another striking feature of the system is the many-body effect. Isotropic 1/r6 attraction between the particles and the pair-wise non-additivity of the interaction promote the formation of colloidal pairs from single particles, the growth of longer chains out of particle pairs and larger hexagonal close-packed sheets from small planar clusters and finally the growth of 3D crystals from small crystallites. In principle the interaction can be made even more complex by the addition of other external fields. A simple upgrade of the system would be charged beads and consequently additional 1/r3 electrostatic interparticle repulsion. More complex examples include the use of anisotropic solvent (liquid crystals) or different background potentials, such as patterned substrates or light induced potentials. As demonstrated, the magic angle system could be used to study the interactions between superstructures, manually assembled with optical tweezers or produced in microfluidic circuitry. The crystallization and coalescence of vesicles or more complicated structures could be examined in a system of (decorated) colloidal rings. In conclusion, the magneto-optical tweezers turned out to be the versatile tool to tailor complex interparticle interactions and study the induced condensed phases. Chapter ^J_ Viscoelastic properties of bacterial networks In the last two decades a number of rheological techniques has been developed to measure the material properties of soft matter systems on microscopic scales. These microrheological (MR) techniques can be divided into two broad classes, passive and active, and both classes are further divided according to number of probe particles used to one-particle and two-particle MR. Typically, the MR techniques are demonstrated in homogeneous materials, such as polymer solutions or gels, whereas its use in complex biological materials is far more complex. In this Chapter we present a comparison of 3 MR methods, which were used to measure the viscoelastic modulus of the extracellular matrix of bacteria Vibrio sp. We employed optical tweezers for micromanipulation of 3.22 ^m silica probe particles and high resolution video tracking to perform passive one-particle MR and active one- and two-particle MR. The methods were first tested on water and then used to monitor the temporal evolution of mechanical properties of the growing extracellular matrix. The stiffness of the matrix was found to be very small, less than 2 x 10-6 N/m, whereas its viscosity changes with time and reaches a maximum of approximately n ~ 3.3 x 10-3 Pas. 5.1 Introduction Many material properties of soft matter systems originate from their complex structures and dynamics with multiple characteristic length- and time-scales. An important and consequently frequently studied material property is the shear modulus. Contrary to ordinary solids, the shear modulus of soft materials usually exhibits significant frequency dependence. Such materials are viscoelastic - they exhibit both a viscous and an elastic response to applied stress. Rheology, the experimental and theoretical study of viscoelasticity is of fundamental and practical significance. The measurement of bulk viscoelastic properties has been traditionally performed with mechanical rheometers that probe macroscopic milliliter samples by applying a small amplitude oscillatory shear strain 7(t) = 70 sin(wt) and measuring the resultant shear stress. The upper frequency that can be probed using commercial rheometers is limited to few tens of Hz due to the inertial effects of probe plates. The material structure is not significantly deformed and remains in equilibrium if the shear strain amplitude 70 is small. In this case, the time-dependent stress a(t) is linearly proportional to the strain, Here G'(u) is the response in-phase with the applied strain and is called the elastic or storage modulus, a measure of the storage of elastic energy by the sample. G"(u) is the response out of phase with the applied strain, and in phase with the strain rate, and is called the viscous or loss modulus, a measure of viscous dissipation of energy. The frequency dependent complex shear modulus (or viscoelastic modulus) [89] is defined as In the last two decades a number of techniques has been developed to probe the material properties of soft matter systems ranging from polymer solutions to the interior of living cells on microscopic scales. These techniques are called microrheology (for a review see [90, 91, 92]), as they can be used to locally measure viscoelastic parameters. Microrheology has several advantages compared to traditional bulk rheometry. Only small quantities of samples are needed for microrheologic measurements (typically 1 is enough), which is very convenient for biological systems. Another advantage is the possibility to study inhomogeneous environments, for instance cell interiors. Also the upper range of accessible frequencies is higher, up to 105 Hz. This enables new insights into the microscopic basis of viscoelasticity in these systems. There are two broad classes of microrheological techniques: active and passive. Passive MR techniques rely on fluctuations of probe particles due to thermal noise. Typically chemically inert spherical beads with a diameter between few tens of nanometers to several micrometers are embedded in the material, their free diffusion is observed either with dynamic light scattering [25], laser tracking [26] or videomicrocopy [27] and then a linear shear modulus is calculated. Active MR techniques involve the active manipulation of small probes. The earliest experimental realization of active MR was based on the manipulation of magnetic beads with an external magnetic field [28] almost a century ago. Nowadays micromechanical tools such as micropipettes [29] and atomic force microscopes [30] are used to directly strain materials, while optical tweezers [31] or magnetic bead microrheometers [32, 33, 34] are used to actively manipulate microparticles embedded in materials. These measurements are analogous to conventional mechanical rheological techniques in which an external stress is applied to a sample, and the resultant strain is measured to obtain the shear moduli. In this case, micron-sized probes locally deform the material and probe the local viscoelastic response. Active measurements allow the possibility of applying large stresses to stiff materials in order to obtain detectable strains. They can also be used to measure non-linear behavior if sufficiently large forces are applied to strain the material beyond the linear regime. a(t) = 7o[G'M sin(wt) + G"(u) cos(wt)]. (5.1) G(u) = G'(u) + iG"{u). (5.2) 5.1.1 Passive microrheology In passive MR measurements the rheological properties are obtained from Brownian motion of probe particles that are embedded into a material before the measurement. Particles are tracked either with videomicroscopy (hence also the name "particle-tracking MR") or laser tracking (photodiode detection of laser light scattered from a probe particle). Resolution of the position detection limits the range of materials, because they must be sufficiently soft in order for particles to move detectably with only kBT of available energy. To understand how the thermal motion of embedded micron-sized particles is used to probe the frequency dependent rheology of the surrounding viscoelastic material, it is useful to first consider the motion of spherical tracers in a purely viscous fluid and then generalize to account for elasticity. Micron-sized beads in a purely viscous medium undergo Brownian motion. The dynamics of particle motions is revealed in the position correlation function, also known as the mean square displacement (MSD), defined as: (r2(r)) = (|r(t + t) - r(t)|2) (5.3) where r is the d-dimensional particle position, t is the lag time and the brackets indicate an average over the interval of the measurement. The time-average assumes the fluid is always in thermal equilibrium and the material properties do not evolve in time. The diffusion coefficient D of the Brownian particle in d-dimensional space is obtained from the Einstein-Smoluchowsky diffusion equation, ^2(t ^ = 2dDT. (5.4) The viscosity n of the fluid can be obtained using the Stokes-Einstein relation 67t na ' ( ) once the MSD is known. Here kB is the Boltzmann constant, T is the temperature and a is the radius of beads. On the other hand, motion of the particles in an elastic medium is constrained therefore the MSD reaches an average plateau value (rp) that is set by the elastic modulus of the material. By equating the thermal energy kBT of each bead with its elastic energy 1 km(rp2), the effective spring constant km of the surrounding medium can be obtained. The elastic modulus G' is related to the spring constant by a factor of characteristic length (set by bead radius a) . Using such an energy balance argument, the relation between the elastic modulus and the MSD reads G' - kT. (5.6) (r2)a Maximum passively accessible elastic modulus depends on both the size of the embedded probe and on the ability to resolve small particle displacements of order of rp. The resolution of a particle position detection typically ranges from 1 to 10 nm, thus allowing measurements (with micron-sized probes) of samples with an elastic modulus up to 50-5000 Pa. This range is relatively small compared to that accessible by active MR, but is sufficient to study many interesting complex materials. Viscoelastic properties of soft matter exhibit frequency and length scale dependent response and are therefore characterized by the frequency dependent complex shear modulus G. For such materials, thermally driven motion of embedded probes reflects both the viscous and elastic contributions. Unlike a purely viscous fluid where the MSDs evolve linearly with time (5.4), or elastic material where maximum MSDs are limited, the MSDs of tracers in a complex material scale differently with the lag time t : r2(t)j - ta, (5.7) where a is the diffusive exponent. Depending on the material, the particles may exhibit subdiffusive (01) or even become locally constrained at long times (a = 0). The relationship between ensemble-averaged mean squared displacement of tracer particles in a viscoelastic medium and its complex elastic modulus can be described by generalized Stokes-Einstein equation (GSE), proposed by Mason and Weitz [93]. It is obtained from a combination of Laplace transform of generalized Langevin equation and the equipartition theorem. When thermally driven embedded probe particles are large compared to all structural length scales, the GSE for d-dimensions reads ( ) 3nas(f2(s))', ( ) where s is the Laplace frequency, (r2(s)) and G(s) are the Laplace transforms of (r2(t)) and G(t). The linear frequency dependent viscoelastic response is therefore directly obtained from the MSD of thermal tracers. To compare with bulk rheology measurements, G(s) is transformed into the Fourier domain (s ^ iu) to get complex shear modulus G(u) which is the same quantity as measured with a conventional mechanical rheometer. Passive particle-tracking MR, i.e. the measurement of a viscoelastic modulus from the motion of embedded probe particles has a few drawbacks. Freely diffusing particles often exit from the field of view, most often they diffuse out of a focal plane of the microscope. The results are affected by the size of probe particles [94], since GSER is valid only for particles larger than all characteristic length scales of a material under investigation. To overcome these problems, large particles have to be used, but they tend to sink to the bottom of a sample cell and render the measurement impossible. In heterogenous materials high concentration of probes have to be added to ensure that all regions of interest contain tracers, but might affect the rheological properties of the material. 5.1.2 Optical tweezers microrheology The drawbacks of the passive particle tracking MR can be avoided by the use of optical tweezers. A probe particle is held inside a detection zone by optical trap, therefore only the selected volume of material is investigated (especially convenient for heterogenous samples, such as living matter) and its motion can be tracked for a long time since the probe can't escape from the field of view. Positions can be monitored using laser-particle tracking (quadrant photodiode detection using back-focal-plane interferometry) which covers up to five-decades in frequency range. Even if video-tracking is being used, only a small region of interest around the probe particle has to be recorded, which enables high frame rates (up to 1000 Hz with standard commercially available CMOS cameras). Optical tweezers can be also employed for active MR, which is required when thermal fluctuations can't induce measurable level of deformation or when the highdeformation response is studied. In one-particle active MR measurement, the probe particle is trapped in an optical trap with oscillating position [31, 35]. From the probe's trajectory the amplitude and the phase lag of its response are extracted and used to calculate complex viscoelastic modulus of the investigated media. Two-particle active MR involves two optical traps [36]. One particle is trapped in a harmonically oscillating stiff trap and is thus actively deforming the surrounding medium, while a second particle is trapped in a weak stationary trap and is used to monitor the deformation of the medium at a desired location (Fig. 5.1). 5.1.3 One-particle passive MR Motion of an optically trapped bead with radius a in a purely viscous medium is described by the Langevin equation, kx(t)+ ftX(t) = fR (t), (5.9) where k is the trap stiffness, ft the viscous drag coefficient and fR is the random thermal force acting on the bead. In the Newtonian limit, where viscosity n is time independent for all time scales, the drag coefficient is given by ft = 6nrqr. In complex fluids the viscoelastic response can be time-dependent and is therefore described by the complex shear modulus G(w). In linear regime it is possible to relate the Fourier transform of trapped bead's motion x(w) to the Fourier transform of the thermal force fR(w) [95, 96] by x(w) = a(w) • fR(u), (5.10) where a(w) = a(w) + ia''(w) is the complex compliance. If a trapped bead is in thermal equilibrium with the environment, the fluctuation-dissipation theorem can be used to obtain the imaginary part of the compliance from the experimentally measured power spectrum density of bead motion S(w): *"M=wSS$. (5.11) The real part of the compliance is calculated using the Kramers-Kronig dispersion relation a'(w) = - cos(wt) / a''(w') sin(w't) dw'dt. (5.12) n J 0 J 0 This requires the knowledge of a''(w) over whole frequency range. If laser-particle tracking is used, the frequency range spans to about 105 Hz, but nonetheless care must be taken in interpreting a'(u) because the cosine and sine transforms of finite samples can lead to discontinuities or nonsensical data corresponding to the lowest and highest frequencies studied [97]. Once the compliance is known the complex shear modulus can be expressed using generalized Stokes-Einstein relation G(u) = 6 1 , (5.13) 6na a(u) which holds for an incompressible medium without inertia. Expression is valid only if bead radius a is larger that the mesh size of the network. 5.1.4 One-particle active MR Optical tweezers can be used to drive the probe particle and thereby actively deform the media. The local mechanical properties are calculated from the spatio/temporal response of the embedded probe. The equation of motion for the spherical probe with radius a, neglecting the inertial term and thermal fluctuations, is [k + km]x(t) + 6nnax(t) = kA sin(ut), (5.14) where n is the solvent viscosity and km = 24nG'a(1 — v)/(5 — 6v) is the local stiffness of a material with shear modulus G' and Poisson ratio v, k is the trap stiffness and A and u are the amplitude and angular frequency of the trap movement [31, 98]. The response of the bead to oscillating force is x(t) = D(u)sin[ut — 5(u)], (5.15) with the ratio of amplitudes is d(u) = y k-, (5.16) A ^(k + km)2 + (6nanu)2 V ' and the phase lag t-/ \ 6nanu o(u) = arctan ---—. (5.17) k + km The viscoelastic moduli are calculated as = k \cos d(w) sin 6na 6na k - 1 6na G"(w) = ^n(w) =----V . (5-18) d(w) Moduli in 6.23 are scaled by the trap stiffness k, which depends on the refractive index contrast between the trapped particle and suspending medium. If the refractive index of a suspending medium changes (e.g. some sugar is added into water), the stiffness of a trap is changed, which has to be taken into account when calculating the moduli. In one-particle MR measurements tracers probe viscoelasticity on length scales comparable to their size. In materials heterogeneous on those scales, tracer motion depends on both the local and the bulk rheology in a complex way [99]. Varying probe diameter gives qualitatively different information about network mechanics. When embedded probe particles are larger than all structural length scales of the material, one-particle MR results in a true complex elastic modulus. By contrast, when the embedded particles are approximately equal to or smaller than the structural length scales of the material, particles move within small, mechanically distinct microenvironments and thus their dynamics are no longer directly related to the bulk viscoelastic response [100] but rather to the viscosity of the solvent. Improper size of the probes is not the only possible reason for artifacts in one-particle MR results. Chemical interactions between the probes and a surrounding medium have also strong influence on MR analysis [101], e.g. either probes can adsorb molecules of the medium or a depletion layer is created. 5.1.5 Two-particle active MR These limitations are not present in two-particle microrheology, in which the correlated motion of pairs of particles is analyzed to measure the deformation of a network [102]. Two-particle microrheology is insensitive to local heterogeneities since it probes the response on a length scale defined by the separation between probe particles. It is also less dependent on the coupling between the probe and the medium [101]. In two-particle active MR a complex viscoelastic modulus of the sample is determined from the response of a probe particle to an oscillating force applied to drive particle (Fig. 5.1). Let (x(1),y(1)) denotes the position of the drive particle, (x(2),y(2)) the position of the probe particle and the coordinate system be chosen such that x-axis is parallel (||) and y-axis perpendicular (±) to the line connecting the particle centers. In an isotropic homogeneous medium the full linear response is defined by the complex compliance in the direction of beads a||(w) and in the perpendicular direction «i(u). In Fourier space, the response of the probe particle (2) to arbitrary force F, applied to drive particle (1), is in absence of any optical traps written as [103, 104] In active two-particle optical tweezers MR, both particles are held in optical traps. The trapping potential affects the response of the probe bead, therefore the measured response functions A|| and Ax differ from the true rheological responses a|| and ax of the material. The displacements of the probe particle to the force acting on the drive particle is x(2)M = aiiMFi1^) y(2)M = axMF^M (5.19) x(2)M = AiiMF^M y(2)M = a±mf«m. (5.20) Figure 5.1: Schematic of two-particle MR experiment. A pair of silica spheres with radius a is trapped by a pair of optical traps at a separation R. The same displacements can be also expressed directly in terms of single particle compliance a (5.10), the mutual response functions a\\ and a± (5.19), the trapping forces _k(i) r(i) and the other forces F(i), applied to particles, as x(2)(u) = a(2)F2) - k(2)x(2)] + a||[Fi1' - k(1)x(1)] y (2)(u) = a(2)[Fi2' - k(2)y(2)]+ a±[F(1) - k(1)y(1)]. (5.21) The drive particle is confined in an optical trap with oscillating position x0 (t) = A sin(wt) which results in a net force k(1) [x0(t) — x(1)] acting on the drive particle. The apparent force on the first particle in (5.21) is therefore F(1) = k(1)x0(t). The equations (6.24) and (5.21) can be combined to express the apparent response functions A|| and A± of the second particle as A = all 11 (1 + k(1) a(1) )(1 + k(2) a(2)) — k(1) k(2) a2. A± = (1 + k(1)a(1))(1 + a(2)a(2)) — kWk(2)a2 . (^22) The mutual response functions a|| and a± of an incompressible continuum with shear modulus G(w) are given by the generalized Oseen tensor _ 1 a|1 = 4nRG(w) = snRGR, (5.23) where R is the distance between the particles. Single particle complex compliances (in the case of particles with same radius) are a(1) = a(2) = 1 6naG(w) where a is the particle radius. The expression for viscoelastic modulus, (5.24) k G(u) = G'(u) + iG"(u) = - (5.25) 6na is used in (5.23) and (5.24) which are then inserted into (5.22) to obtain expressions for A|| and A^ in terms of medium viscosity n and stiffness km. In a two-particle active MR experiment the frequency dependence of A|| and A^ is measured then (5.22) is fitted to extract n and km. 5.1.6 Rheology of bacterial networks Bacteria are unicellular microorganisms with typical size of a micrometer, existing in many different shapes, from spheres to rods and spirals. Single cell organisms generally exhibit two different modes of behavior - the first is known as a free floating, where single cells independently float or swim in liquid medium, whereas the second mode is a formation of biofilms where cells are closely packed and firmly attached to each other. Biofilms are produced by excretion of protective and adhesive matrix; their common properties are attachment to surrounding surfaces, structural diversity, complex interactions and extra cellular matrix of polymer substances. The advantages of such environment is enhanced protection of the interior of the community from the unfavorable environment. Due to their role in microbial infections the mechanical properties of biofilms have been widely studied both on macroscale with classical rheometers [105] and on a microscale: with microindentation device [106], AFM, magnetic tweezers, optical tweezers and by observation of Brownian motion of embedded particles. FIGURE 5.2: Bacterial growth curve: logarithm of number of bacteria as a function of time. Arrows indicate at which points samples S1, S2 and S3 were taken for MR measurement. In this Chapter rather than to a formed biofilm of close packed cells we focused to an earlier stage in a development of multi-cellular environment - we were monitoring the rheology of growing population of bacterial cells. When a bacterium is added to nutritious environment, it divides into daughter cells in a process called binary fission. The division repeats again and again and the number of bacteria increases. Its time dependence is described by a growth curve (Fig. 5.2), which can be divided into four phases. In initial, lag phase the growth is slow, because the bacteria need some time to accommodate to the food and nutrients in their new habitat, but as soon as the metabolic machinery of bacteria is adapted the growth comes into log phase where their number is exponentially growing. As more and more bacteria are competing for dwindling nutrients, booming growth stops and the number of bacteria stabilizes; this is called stationary phase. The last is a death phase, when bacteria begin to die due to accumulated toxins and depletion of food. We measured the microrheology of a bacterial solution consisting of a growth medium, bacterial cells and the extracellular matrix (ECM), which is composed of an interlocking mesh of fibrous proteins and long unbranched polysaccharides. Components of the ECM are produced intracellularly by bacterial cells, and secreted into the ECM via exocytosis [107]. Once secreted they aggregate with the existing matrix, so the matrix is getting denser through the time. The bacterial culture was growing in optimal conditions, described below in Sec. 5.2.1. At desired time intervals 2 ml samples were taken from the center of the growth flask and their optical density was measured to determine their position in the growth curve. The samples were then prepared for MR measurements as described in the following Section. 5.2 Experimental details We have used plain silica beads of radius a = 1.61 ^m (Bangs Labs, SS05N) as probe particles for MR. Since their index of refraction (nB = 1.43) is close to that of water (n = 1.33) silica beads can be optically trapped in three dimensions but on the other hand their contrast is low, therefore beads with relatively large diameter had to be used for accurate detection of their positions in bacterial solutions. Typically, 1 ^l of the beads, diluted in water to concentration 10-4/^m3, was added to 500 ^l of bacterial solution. After mixing, approximately 20 ^l of solution was pipetted into a sample chamber, made of two coverslips of different sizes. Silicon grease was holding the coverslips separated at a distance of approximately 100 ^m and was used to completely seal the chamber after filling. The measurements were performed at room temperature, with no special control over the chamber temperature. Figure 5.3: Micrograph of two trapped 3.22 ^m silica beads in the bacterial solution. The scale bar is 10 ^m long. The laser power in all MR experiments was held constant at 300 mW, which resulted in about 30 mW of total power in the sample. Two simultaneous optical traps, each with one trapped particle (Fig. 5.1), were used for all types of MR experiments. A passive and active one- and two-particle MR measurements of one sample consisted of following steps. First, two probe beads were found at the bottom of the sample chamber and trapped in two traps that were 10 ^m apart (Fig. 5.3). Beads were then raised to 30 ^m above the chamber bottom to avoid the influence of confining walls on the measurement. For measurements of passive MR the traps were held at fixed positions. The particle fluctuations were recorded at three different traps stiffnesses: 20%, 50% and 100% of full stiffness. For active MR measurements, the left trap was held at fixed position at 20% of full stiffness and was used only to weakly trap the probe bead. The right trap's stiffness was set to maximum and was used to harmonically oscillate the right bead (drive bead), either in the direction of the line connecting the beads xT(t) = A0 sin(2nft) (5.26) or in the perpendicular direction yT(t) = A0 sin(2nft) with a constant amplitude A0 = 1 ^m. The measurements were performed for different oscillation frequencies f: 0.1, 0.2, 0.5, 1, 2, 5, 10, 20 and 50 Hz. The whole active MR measurement procedure was then repeated with the stiffness of the drive trap set to 50%. 5.2.1 Preparation of bacteria We were measuring the viscoelastic properties of a solution in which bacterial strain Vibrio sp. DSM 14379 [108] was growing. This strain is a bacterium isolate from Skocjanski zatok, Slovenia, belonging to family Vibrionaceae, class Gamma Pro-teobacteria, phylum Proteobacteria. Vibrio is a Gram negative bacteria possessing a curved rod shape and is typically found in marine environments [109]. They do not form spores and are facultative anaerobes. Bacteria were grown aerobically in a minimal medium (M9) overnight in the dark on a rotatory shaker at 200 rpm at 28°C. An inoculum (1% v/v) was transferred to a freshly prepared medium. The growth curve was determined by optical density (OD) at 650 nm (Iskra Photometer MA9510). Samples for MR measurements were taken during the lag (sample 1, 3h old, 0D=0.019), log (sample 2, 9h old, 0D=0.79), and stationary phase (sample 3, 26h old, OD=1.75). Another sample, "mesh-only", was prepared from the sample 3 with removal of bacteria by spinning. 5.3 Results and discussion 5.3.1 One-particle passive MR All MR methods have been tested in water as a control and then used in complex bacterial environment. For one-particle passive MR measurement the trajectories (x(t),y(t)) of optically trapped probe beads were recorded. In Fig. 5.4 we present power spectral density S(w) of probe fluctuations in water; the probe was trapped in an optical trap with stiffness k = 2.1 x 10-6 N/m. Measured data points in S(w) were fitted with (2.5), yielding corner frequency fc = 10.3 Hz and viscous drag coefficient 7 = 3.3 x 10-8 Ns/m. The power spectral density is directly related to the imaginary part of compliance a"(w). In our experimental setup we used video-particle tracking with the acquisition rate of 200 frames per second which limited the highest frequency of the measured f[Hz] Figure 5.4: Power spectral density of fluctuations of a trapped 3.22 ^m silica bead in water. Trap stiffness was k = 2.0 x 10-6N/m. Red line is fit of (2.5) with y = 3.3 x 10-8 Ns/m and fc = 10.3 Hz. power spectrum S(u) and consequently a''(u) to 100 Hz, therefore (5.12) couldn't be evaluated numerically. Since the functional dependence of the spectral density is known (2.5), we used analytical expression for S(u) for the evaluation of the real part of the compliance a' (u). 0.01 0.1 1 10 100 1000 0.01 0,1 1 10 100 1000 © [S1] Q> [s'1] Figure 5.5: Compliance and shear modulus in water, calculated from the analytical expression for S(u) of a trapped particle. Left: Real (black) and imaginary part (dashed red) of the complex compliance a of a weakly trapped probe particle as a function of angular frequency u. The compliance was calculated using (5.11) and (5.12) from PSD in Fig. 5.4. Dashed region indicates the frequency range accessible by the video-particle tracking. Right: Real (black) and imaginary part (dashed red) of the complex shear modulus G(u). In Fig. 5.5 we show the real a'(u) and the imaginary part a''(u) of the calculated compliance of the probe particle in water. The obtained value, a'(0) = 4.7 x 105 m/N, is in a good agreement with the theoretical prediction a' = 1/k = 4.8 x 105 m/N. Using (5.13) a complex viscoelastic modulus G(u) is calculated from the complex compliance a(u). In right chart of Fig. 5.5 the storage (elastic) and loss (viscous) modulus of water are presented. The non-zero storage modulus G'(u) = k/6na = 0.07 Pa is due to the stiffness of the optical trap. The loss modulus of a Newtonian liquid is proportional to the angular frequency, G"(u) = —nu. The measured viscosity of water n = 10.8 x 10-4 Pas is in a good agreement with true viscosity1, which is n = 9.3 x 10-4 Pas at temperature of 23°C. Another possibility to obtain a viscoelastic modulus with a passive one-particle MR is by the use of the generalized Stokes-Einstein equation. The complex shear modulus in the measured range of frequencies can be calculated directly from the MSD of trapped particles using GSER (6.20). t[s] Figure 5.6: Mean square displacement (MSD) of a trapped bead in water as a function of lag time, shown for three traps with different stiffness: weak (k = 0.81 x 10-6N/m, black squares), medium (k = 2.0 x 10-6N/m, red circles) and strong trap (k = 4.0 x 10-6N/m, green triangles). The data points are fitted with Eq. (5.27). To avoid potential numerical artifacts due to the Laplace transforms in (6.20), we used analytical expressions for MSDs of probe particles. Mean square displacement of a trapped particle in viscous media can be described by 100/s) there is an error offset of -0.02 Pa. A possible cause for this error could be a large amplitude of the optical trap oscillation. At high frequencies the trapped bead couldn't follow the trap, its amplitude was approximately 0.4 ^m. The displacement of the bead from the center of the optical trap was in some parts of the oscillation cycle larger than 300 nm (the range of parabolic trapping potential). The equation (5.14) in such case does not hold anymore which explains the observed deviation of G' from zero. The measured loss modulus G'(u) of water is shown in Fig. 5.11. Two data series obtained with optical traps of different stiffnesses collapse onto a linear function G'(u) = nu with n = 9.1 x 10-4 Pas, which is is remarkably close to the real viscosity of water n = 9.3 x 10-4. One-particle active MR was then employed to monitor the viscoelastic properties Figure 5.10: One-particle active MR in water. Left: Apparent storage modulus G' in water as a function of angular frequency. Black stars represent the measurements with a strong trap (k = 4.0 x 10-6 N/m), whereas red squares are obtained with weak trap (k = 2.1 x 10-6 N/m). The constant G' = 0.069 Pa (red line) is consistent with trap stiffness k = G'/(6na) = 2.1 x 10-6 N/m. Right: True storage modulus G'(u) of water. 1 10 100 co [S1] Figure 5.11: Absolute value of the loss modulus G'' of water as a function of angular frequency u, measured with one-particle active MR. Black stars represent the measurements with stiff trap (k = 4.0 x 10-6 N/m), whereas red squares are obtained with weak trap of (k = 2.1 x 10-6 N/m). The data points obtained with weaker trap are fitted with linear function yielding n = 9.1 x 10-4 Pas. of the growing bacterial suspension. Fig. 5.12, where frequency dependent loss modulus of aging suspension is presented, is one of the central results of this Chapter. The measured values of G'' (u) of different samples were fitted with linear function the extract their viscosities. The viscosity of the growth medium nM = 12.4 ± 0.1 x 10-4 Pas was a bit higher than that of water, which is reasonable result, since the medium contains some salts and 0.4% glucose. Viscosities of samples 1, 2 and 3 were found to be nsi = 14.1 ± 0.1 x 10-4 Pas, ns2 = 20.1 ± 0.3 x 10-4 Pas and ns3 = 35.4 ± 1.2 x 10-4 Pas, respectively, whereas viscosity of sample 3 without bacteria was nS3' = 24.5 ± 0.4 x 10-4 Pas. o [s"1] Figure 5.12: Absolute value of the loss modulus G'' of growing bacterial suspension as a function of angular frequency, measured with one-particle active MR. Measured values are fitted with imaginary part of (5.25), yielding viscosities of growth medium (black squares): = 12.4±0.1 x 10-4 Pas, sample 1 (green triangles): nS 1 = 14.1 ±0.1 x 10-4 Pas, sample 2 (blue inverted triangles): ns2 = 20.1 ±0.3 x 10-4 Pas, sample 3 (red circles): ns3 = 35.4± 1.2 x 10-4 Pas, sample 3 without bacteria (red diamonds): ns3' = 24.5 ± 0.4 x 10-4 Pas. The frequency dependence of the true elastic modulus of the growing bacterial suspension measured with one-particle active MR is presented in Fig. 5.13. The modulus G' (u) of the growth medium has same frequency response as that of water - zero stiffness at low frequencies and artifacts at u > 30/s. The measured values of the elastic modulus of bacterial samples are scattered but still some conclusions can be made. G'(u) of the bacterial solution is increasing with frequency from zero at low frequencies to around 0.01 Pa at high frequencies. The differences between measured moduli at high and low frequencies AG' = G'(60/s) — G'(0.6/s) are 0.013 Pa for sample 1, 0.016 Pa for sample 2 and 0.023 Pa for sample 3. The corresponding stiffnesses of the medium km = 6naAG' are then 3.9 x 10-7 N/m, 4.8 x 10-7 N/m and 7.1 x 10-7 N/m. 0.03 0.02 - _ 0.01 -CD CL CD 0.00 --0.01 - 1 10 100 co [S 1] Figure 5.13: Elastic modulus Gf of bacterial suspensions as a function of angular frequency, measured with one-particle active MR. ■ gojisce • v1 ▲ v2 T v3 5.3.3 Two-particle active MR As the last and the most complex, the results of two-particle active MR are presented. In two-particle active MR two optical traps are employed to hold two particles. The first trap is stiff and is used to exert an oscillating force on a drive particle, whereas in the second trap a probe particle that serves as a deformation monitor is weakly trapped. A typical set of trajectories recorded in water at trap oscillation frequency f = 10 Hz is shown in Fig. 5.14. The oscillating optical trap is used as a reference, the drive bead has slightly lower amplitude and lags behind the trap, whereas the probe bead has small amplitude but its phase is in the front of the trap. The frequency dependence of the amplitude and phase of both beads in the experiment where the drive bead was oscillated in the direction of the line connecting the beads are presented in Fig. 5.15. The amplitude and the phase dependence of the second bead were used to calculate the complex response functions using (5.22). The real and the imaginary part of parallel A\\(u) and perpendicular A±(u) response functions are displayed in Fig. 5.16. Rather complicated expressions for the theoretical real and imaginary part of the response functions with two free parameters, the viscosity of the medium n and its stiffness km were fitted to the measured data. The theoretical response functions are in a good agreement with measured points except in the low frequency region where the errors occur due to the low signal-to-noise ratio. Since water is purely viscous medium the low frequency oscillation of the drive trap results in a very small deformation of the medium at the location of the probe bead and therefore the determination of its amplitude and phase has a large relative error. Nevertheless, the measured A|| (u) and A± (u) are best fitted with km = 0 and n = 9.1 x 10-4 Pas and n = 9.2 x 10-4 Pas, respectively. These values are within 5% of true viscosity —i—i—i—i—i—i—i—i—i—i—i— 2,00 2,05 2,10 2,15 2,20 2,25 2,30 t[s] Figure 5.14: Part of the trajectories of the drive trap (black), the drive bead x1 (red squares) and the probe bead x2 (green triangles) in a two-particle active MR measurement in water at the trap oscillation frequency f = 10 Hz and amplitude A = 1 ^m. Figure 5.15: Active two-particle MR in water. Amplitudes D/A (left) and phase lags 5 (right) of the drive bead (black squares) and the probe bead (red circles) as a function of trap oscillation frequency. of water. After its demonstration in water, two-particle active MR was used to determine viscoelasticity of the bacterial network. The real and the imaginary part of the perpendicular two-particle response function of the aging bacterial solution, presented in Fig. 5.17, are the second key result of this Chapter. The measured values have been fitted with (5.22) with three adjustable parameters, viscosity n, stiffness km of the medium and a scaling factor. Although the trap powers were kept constant in different samples, the measured two-particle responses A|| (u) and A^(u) have different magnitudes from sample to sample and the additional linear scaling factor had to be introduced in (5.22) to successfully fit the data. The scaling coefficient does not influence the values of viscosity and stiffness, since the viscosity of a solution is defined by the position of a peak of the real component of the response function Figure 5.16: Real (black squares) and imaginary part (red circles) of apparent two-particle response functions in water. Parallel A|| (left) and perpendicular A± (right) response functions. The fits of Eq. (5.22) to data points of A|| yield n = 9.1 x 10-4 Pas and n = 9.2 x 10-4 Pas for A± (solid curves). (higher viscosity shifts the peak to lower frequencies), while the stiffness is revealed by the low frequency behavior of the response function. Figure 5.17: Real (left) and imaginary (right) part of the apparent two-particle perpendicular response function. Data is shown for the growth medium (black squares), sample 1 (red circles), sample 2 (green triangles), sample 3 (blue inverted diamonds) and sample 3 with bacteria removed (cyan diamonds). Solid lines are the theoretical curves (5.22) with three adjustable parameters: n, km and scaling coefficient. The measured values of the real part of two-particle complex response functions are in a good agreement with the theoretical response given by linearly scaled (5.22), whereas the measured imaginary parts of the response are only in qualitative agreement. The viscosities and stiffnesses of the investigated bacterial samples, obtained from various two-particle response functions, are presented in the following table. Obtained from: Al A'j A|| Aj_ A'w Sample n[10-4 Pas] n[10-4 Pas] n [10-4 Pas] km[10-8 N/m] km[10-8 N/m] G. medium 14.7 14.5 14.9 15 0 Sample 1 18.4 16.7 17.2 4.9 9.4 Sample 2 21.1 19.9 21.8 7.4 9.8 Sample 3 33.1 32.7 34 8.8 10 Mesh-only 27.3 22.2 25 0 7.4 5.4 Conclusion and outlook We used optical tweezers and high resolution video particle tracking to measure the microrheology of the bacterial extracellular matrix, produced by Vibrio sp. The investigated samples were probed with three different MR techniques. The simplest was passive one-particle MR, where thermal fluctuations of 3.22 ^m silica bead, trapped in a weak optical trap, were recorded to calculate the complex viscoelastic modulus of the medium. In active one-particle MR, viscoelastic properties were extracted from the response of a probe particle to a harmonically oscillating optical trap. For two-particle active MR two optical traps were employed. One was used to sinusoidally oscillate one bead, whereas another, weaker trap, was holding the probe bead. The rheological properties of the samples were calculated from the response of the probe bead to the oscillations of the trap. growth medium S1 S2 S3 mesh Figure 5.18: Comparison of measured viscosity n of various samples with one-particle passive MR (MSD, white), one-particle active MR (red) and two-particle active MR (blue - real part, cyan - imaginary part of two-particle response function). All MR techniques were first tested on water, where the measured viscosities ranged from from 8.7 x 10-4 Pas to 10.8 x 10-4 Pas. The true value of the viscosity of water in room temperature region varies from 8.9 — 10.0 x 10-4 Pas, therefore the maximum error was less than 10%. The most accurate results were obtained with active MR - the error of two-particle active MR was less than 5%. After the successful demonstration in water, the techniques were used for monitoring the viscoelastic modulus of the growing bacterial population. Samples of the solution in the lag, log and stationary phase of the growth curve were analyzed using the MR techniques and the summarized results are presented in Fig. 5.18. One-particle passive MR detected the increasing viscosity of the aging bacterial solution, but the values seem to be highly underestimated. The viscosities obtained with the active techniques are in a good agreement - the values in each sample are within ±10% of the average value. The viscosity of the "mesh-only" sample (sample 3 with bacteria removed by spinning) was found to be 25% lower than the viscosity of the sample 3, which is reasonable since spinning removes bacteria and cellular debris and therefore lowers the density of the solution. The conclusion about the storage modulus G' of the extracellular matrix is not completely clear. The storage moduli, measured with one-particle active MR, were on the order of 0.01 Pa, which equivalents to the medium stiffness km ~ 3 x 10-7 N/m. The stiffnesses, found with two-particle MR, had highly scattered values and were on the order of km ~ 5 x 10-8 N/m. The stiffness of bacterial samples was definitely far below the stiffness of the optical traps (k ~ 2 x 10-6 N/m) that were used in the experiments. We demonstrated the use of optical tweezers microrheology to probe the vis-coelastic modulus of a complex biological sample. The best results were obtained with the active MR techniques. The experimental setup will be used to perform more detailed measurements of bacterial cultures and to study the influence of various environmental parameters on the rheological properties of the bacterial network. Chapter Conclusion This thesis presents experimental studies in three subfields of condensed soft matter, namely interparticle potentials in nematic colloids, self-ordering of colloidal particles into complex structures under the influence of external magnetic field and the rheology of a developing biological system. Magneto-optical tweezers, a new micromanipulation tool with combined advantages of optical tweezers and homogenous magnetic tweezers, were used to perform experiments that would be otherwise hardly feasible. * The first part of the thesis deals with interparticle forces between colloidal particles in a confined nematic liquid crystal (LC). The tangential anchoring of LC molecules on the particles induced defects with quadrupolar symmetry which resulted in a repulsive force between the particles. Static experiments, where the LC-mediated repulsion was balanced by attractive magnetic force, enabled precise measurement of the interparticle force over four orders of magnitude. This was the first accurate measurement of the force at particle center-to-center separations as small as 1.1x particle diameter. The 1/r6 force dependence, predicted by theoretical calculations, was confirmed on wide interparticle separation range. In dynamic measurements, where the LC-mediated force was obtained from the velocities of particles drifting apart, we determined the same separation dependence of the force. This confirms that the effective drag coefficient 7 = FLC/v is independent on interparticle separation. The force was found to be temperature dependent, F 1.2h) pa parski potencial eksponentno pojema, U = c2exp(-x/A). Izmerjeni karakteristični dolzini sta A = 0.18(1 ± 0.15)h pri h =1.8 x D in A = 0.14(1 ± 0.20)h pri h =1.5 x D. Elektrostatska potencialna energija dveh kvadrupolov med prevodnima ploscama za velike razdalje eksponentno pojema z karakteristicno dolzino A = 0.16h, kar je izvrstno ujemanje z izmerjenimi dolzinama. Vzrok za eksponentno pojemanje parskega potenciala pri vecjih razdaljah je ogra-jenost TK. Ko postane meddelcna razdalja primerljiva z debelino celice, mocno sidranje TK na njenih stenah relativno zmanjsa vpliv ene kroglic na direktorsko polje na mestu druge kroglice. Interakcija je "zasencena" in z razdaljo pojema eksponentno s karakteristicno dolzino, ki je sorazmerna debelini celice. 6.4 Samourejanje superparamagnetnih koloidov v magnetnem polju Eden izmed pomembnih problemov fizike mehke snovi je povezava med meddelcnim interakcijskim potencialom in vecdelcno strukturo, ki jo tak potencial stabilizira. Tematika je posebno zanimiva s stalisca samosestave (self-assembly), procesa v katerem se gradniki sistema (atomi, molekule, koloidni delci ali drugi osnovni gradniki) spontano organizirajo v vecje urejene in/ali funkcionalne vecdelcne strukture [14, 15]. Obliko koncnih makroskopskih struktur dolocajo meddelcne interakcije na mikroskopski skali. Povezave med meddelcno interakcijo in posledicnimi vecdelcnimi strukturami v atomskih in molekulskih sistemih obicajno studirajo teoreticno in z numericnimi simulacijami, neposrednih eksperimentov pa ni, saj je spreminjanje medatomskih interakcij nemogoce. Priblizno pred tremi desetletji so za koloide ugotovili, da so odlicen model kondenzirane snovi. Koloidi so disperzije trdnih delcev, kapljic ali plinskih mehurckov (znacilne velikosti od nekaj nm do nekaj 10 ^m) v gostiteljskem mediju. Koloidni sistemi so zaradi velikosti primerni za opazovanje, hkrati pa je mozno spreminjati njihove lastnosti, npr. z uporabo koloidnih delcev druge velikosti, spremembo gostiteljskega medija ali vklopom dodatnega zunanjega polja (npr. elektricnega ali magnetnega). Omejitev koloidnih sistemov je krogelna oblika vecine komercialno dostopnih monodisperznih koloidnih delcev, kar se odraza v izotropnem meddelcnem interak-cijskem potencialu. Najpogostejsi urejeni strukturi sta zato heksagonalna mreza v dveh dimenzijah in ploskovno centrirana kubicna mreza v treh dimenzijah [17, 18]. Obstajata dve poti do kompleksnejsih koloidnih struktur: (i) z anizotropnimi med-delcnimi interakcijami, kar se doseze z obdelavo povrsin delcev [19], zunanjimi polji [20], anizotropnim gostiteljskim medijem [3] ali z (ii) izotropnimi meddelcn-imi interakcijami z radialnim profilom, ki je bolj zapleten kot potencni zakon r-^, ki obicajno nastopa v potencialih. Preprost primer take interakcije je izotropni potencial z radialnim profilom z dvema dolžinskima skalama, kombinacijo trdega jedra in mehkejsega odbojnega dela ("hard-core/soft-shoulder"). Numericne simulacije napovedujejo, da taka parska interakcija ustvari mnozico mezofaz [21, 22, 23, 24] med tekocino in tesno zlozeno kristalno mrezo. Z uporabo magneto-opticne pincete smo v sistemu superparamagnetnih koloidnih delcev ustvarili razlicne tipe interakcijskih potencialov in opazovali kaksne urejene faze stabilizirajo. Kombinacija staticnega magnetnega polja in geometrijske ograjenosti ustvari izotropen parski potencial, cigar profil v smeri precno na polje doloca debelina vzorca. Pri kriticni debelini dobimo kombinacijo trdega jedra in zmehcanega odbojnega potenciala, ki stabilizira nove faze. Nato smo se osredotocili na izotropne potenciale, ustvarjene z vrtenjem magnetnega polja. Zanimivo interakcijo dobimo, ce magnetno polje precesira po plascu stozca z "magicnim" kotom 0p = 54.7° ob vrhu osnega preseka. Clen 1/r3 v dipolni interakciji med induciranimi dipolnimi momenti delcev izgine, toda zaradi vplivov lokalnega polja je parski potencial izotropno privlacen. Odvisno od gostote sistema se pri takem potencialu samosestavijo razlicne oblike struktur. Eksperimentalne podrobnosti V poskusih smo uporabili superparamagnetne kroglice s premerom a = 1.05 ^m (Dynabeads, MyOne Carboxy), z magnetno pinceto pa smo ustvarili zunanje magnetno polje, ki je induciralo meddelcne potenciale (slika 6.8). Tipicne velikosti polj v eksperimentih so bile nekaj mT, tako da je bila interakcijska energija dveh dotika-jocih kroglic, postavljenih v smeri zunanjega staticnega magnetnega polja od nekaj 10- do nekaj 100-krat vecja od termicne energije kBT. Osredotoceni smo bili na ravnovesne oz. kvaziravnovesne vecdelcne strukture, ustvarjene z razlicnimi tipi meddelcnih potencialov. Zaradi nizke "temperature" sistema so se strukture vzpostavile v nekaj sekundah po vklopu magnetnega polja. Slika 6.8: Shema eksperimentalne celice debeline h s koloidnimi kroglicami premera a. Zunanje magnetno polje Bo je pravokotno na ravnino vzorca. S staticnim magnetnim poljem je mozno doseci izotropen meddelcni potencial, ce je polje usmerjeno pravokotno na ravnino vzorca. Med delci deluje odbojna dipolna interakcija (6.5), ki povzroci njihovo razporeditev v heksagonalno mrezo. Na sliki 6.9 so prikazane tipicne faze 2D sistema pri konstanti velikosti odbojne interakcije za razlicne povrsinske gostote1 sistema. Pri nizki gostoti je sistem v tekoci fazi, z visanjem gostote pa se delci uredijo v odprto in nato v tesno zlozeno heksagonalno mrezo. 1 Površinska gostota sistema pove deleZ površine, prekrit z delci: n = na2n/4, kjer je n številčna gostota delcev (stevilo delcev/povrsino). Slika 6.9: Slike tipičnih faz 2D sistema superparamagnetnih koloidnih kroglic v magnetnem polju, pravokotno na ravnino vzorca. Pri nizki gostoti je sistem v tekoči fazi (levo). Pri visjih gostotah nastane odprta oz. tesno zlozena heksagonalno mrezo. Statično magnetno polje v kvazi-2D sistemu S statičnim pravokotnim magnetnim poljem in ustrezno geometrijo eksperimentalne celice je moZno doseči zmehčano izotropno odbojno interakcijo. Ce je debelina vzorca malo vecja od premera kroglic, njuni sredisci nista vec strogo omejeni na eno ravnino. Posledicno je pri manjsih meddelcnih razdaljah parska interakcija manj odbojna kot pa v primeru, ce se delca nahajata v isti ravnini. Interakcijska energija dveh delcev se lahko zapise kot r2 - 2z2 E(r, z) = K-..., (6.13) (r2 + z2)5/2 kjer sta r in z horizontalna in vertikalna komponenta razdalje med srediscema obeh kroglic (slika 6.8),K = ^j^f pa interakcijska konstanta. Obliko meddelcnega potenciala neposredno doloca debelina celice. Ce je ta enaka premeru delcev, h = a, ti interagirajo dipolno E = K/r3, pri vecjih debelinah celice je omogocen z > 0 (za dva delca je energetsko najugodneje, da je en pri spodnji steni celice, drug pa pri zgornji), kar zmanjsa meddelcni odboj oz. lahko vodi do privlaka pri majhnih meddelcnih razdaljah. Pri kriticni debelini celice hm = a(1 + 1/V5) « 1.447a (6.14) meddelcna interakcija preide iz odbojne (pri vseh meddelcnih razdaljah) v privlacno (pri malih meddelcnih razdaljah). Z opticno pinceto smo izmerili meddelcno interakcijo v razlicno debelih celicah h. Na sliki 6.10 so prikazani izmerki sile in izracunani parski potenciali, ki so kombinacija trdega jedra, cigar doseg je definiran s premerom kroglic, in zmehcanega odbojnega dela pri h < hm oz. privlacnega dela pri h > hm. V sistemu z zmehcanim odbojnim potencialom (pri kriticni debelini h = hm) smo opazovali urejene mezofaze, ki nastanejo pri razlicnih povrsinskih gostotah sistema. V vzorcu smo poiskali obmocje ustrezne debeline, nato pa s segrevanjem vzorca s pomočjo opticne pincete lokalno nakopicili vec kot 104 delcev. Po vklopu precnega magnetnega polja se je zaradi odbojne interakcije povrsinska gostota sistema zman-jsevala od zacetne n > 0.6 do n ~ 0.01. Na sliki 6.11 so prikazane mezofaze, ki se porajajo pri razlicnih povrsinskih gostotah sistema n. Pri zelo visokih gostotah (n > 0.5) nastanejo domene satovja in goste kvadratne mreze. Z nizanjem povrsinske gostote (n = 0.39) se gosta mreza Slika 6.10: Levo: Izmerjeni profili komponente meddelcne sile v ravnini vzorca za 3 razlicne debeline celice: odboj r-4 v tanki celici (h ^ a; rdeci krogi), zmehcan odboj (h ^ hm; crni kvadrati) in privlacna interakcija v debeli celici (h > hm, zeleni trikotniki). Neprekinjene crte prikazujejo teoreticno odvisnost sile za h = a, h = 1.46a in h = 1.72a. Desno: Parski potenciali: dipolni odboj (h ^ a; rdece), zmehcan odboj (h ^ hm; crno) in privlacen potencial v debeli celici (h > hm, zelena). Slika 6.11: Slike reprezentativnih mezofaz pri razlicnih povrsinskih gostotah n. Na nekaterih slikah so z rdeco poudarjene urejene strukture kroglic. razsiri v labirintno strukturo enojnih verig iz dotikajocih kroglic. Pri nizji gostoti (n = 0.34) so stabilne krajse verige, ki so lokalno poravnane. Pri n = 0.31 opazimo koeksistenco razsirjene kvadratne mreze in kratkih verig, vecinoma dimerov in trimerov. Ti z nizanjem povrsinske gostote (n = 0.23) razpadejo na posamezne kroglice, ki se razporedijo v razsirjeno heksagonalno ali kvadratno mrezo. Ko je gostota dovolj majhna, sistem preide v razsirjeno heksagonalno fazo (n = 0.12) in na koncu v tekoco fazo (n = 0.01). Mezofaze, opazene v eksperimentu, so podobne rezultatom numericnih simulacij [76, 21, 22]. Ceprav so v nekaterih simulacijah za potencial uporabljene mocno idealizirane kombinacije trdega jedra in mehkega odbojnega dela, je ujemanje z eksperimentalnimi fazami presenetljivo dobro, iz cesar sklepamo, da so strukture in mehanizmi nastanka precej robustni. Statično magnetno polje v 3D sistemu Ce je sirina celice vecja od kritične sirine hm & 1.447a, pravokotno magnetno polje povzroči privlacen potencial med delci (slika 6.10). Ce je povrsinska gostota sistema majhna, so stabilni parcki kroglic. Ti se postavijo navpicno (v smeri magnetnega polja), ce le imajo dovolj prostora (h > 2a). Med parcki ddeluje odbojna sila, tako da se razporedijo v heksagonalno mrezo (slika 6.12 (c)). Ce je debelina celice manjsa, a se vedno nad kriticno debelino hm, so parcki nagnjeni ((a), (c)). ^p ■m m w gm w m ~ __ A _ ^ A a b c Slika 6.12: Slike koloidnih mezofaz pri nizki gostoti sistema v treh različnih debelinah celice h (a) h & hm; zaradi rahlega privlaka so parčki stabilni kljub nizki gostoti (b) h & 1.5a. (c) Celica je dovolj debela, da se par postavi pokončno v smer magnetnega polja. Izotropni odboj med pari povzroči razporeditev v heksagonalno mrezo. Pri vecjih gostotah sistema kroglice ne morejo ostati v locenih skupkih, zato se tvorijo enojne verige, ki so sestavljene iz dotikajocih se kroglic. Minimizacija energije povzroci, da se npr. prva kroglica v verigi dotika zgornje stene celice, druga spodnje stene, tretja spet zgornje in tako naprej. Na sliki 6.13 so prikazani trije primeri gostih sistemov pri razlicnih debelinah celic. Pri debelini pod dvema premeroma kroglic (h = 1.6a na levi in h = 1.8a na srednji sliki) so stabilne dolge enojne verige kroglic z lepo vidno izmenjujoco sestavo gor-dol, pri vecji debelini pa se vzpostavijo pari v smeri magnetnega polja (h = 2.1a na desni sliki). Odboj med njimi jih razporedi v heksagonalno mrezo z mrezno razdaljo, ki je odvisna od gostote sistema. Ce je mreza tesno zlozena, se pri povecanju gostote sistema zacnejo vzpostavljati "defekti", kjer se pokoncni parcki povezejo z dodatno skupno kroglico. Urejene strukture smo obravnavali tudi z numericnim izracunom energij. Izracun pojasni stabilnost enojnih verig in pokaze, da se verige med seboj odbijajo, oboje pa je v skladu z opazenimi mezofazami pri nizkih ter visokih gostotah sistema. Vrteče nagnjeno magnetno polje Vrtenje zunanjega magnetnega polja omogoca ustvarjanje se kompleksnejsih inter-akcijskih potencialov. V ravnini vrtece magnetno polje med delci inducira izotropno privlacno interakcijo (6.7) in z navorom deluje na skupke delcev. Pojav je dobro znan [81, 82, 83] in je prakticno uporaben je za samosestavo in vrtenje mikrocr-palk iz koloidnih delcev [84, 85]. Skupna znacilnost omenjenih eksperimentov je Slika 6.13: Tipicne mezofaze za razlicne debeline celice h pri veliki gostoti sistema. Leva: h ^ 1.6a, stabilne strukture so enojne verige. Sredina: h ^ 1.8a, enojne verige z malim stevilom razvejisc Desna: h ^ 2.1a. Navpicni parcki kroglic se zlozijo v heksagonalno mrezo. Rdec krog oznacuje dva "defekta", kjer so tri parcki povezani s skupno kroglico. Slika 6.14: Skupno zunanje magnetno polje Bo = Bz + BXy precesira po obodu stozca z vrsnim kotom 0P. Azimut ^ je kot med ravninsko komponento polja in smerjo osi x. vrtenje polja v eno smer, kar povzroci vrtenje vecdelcnih struktur. Z nasim sistemom lahko magnetno polje izmenicno vrtimo - torej en obrat v smeri urinega kazalca, naslednji v obratni smeri, tako da na strukture efektivno ne deluje magnetni. Pod vplivom nastale izotropno privlacne magnete interakcije delci v dveh dimenzijah tvorijo skupke tesno zlozene heksagonalne mreze. Ce je vklopljena kombinacija navpicnega staticnega polja Bz in v ravnini vrtecega polja Bxy, vektor celotnega magnetnega polja Bo = Bo (sin 0p cos sin 0p sin cos 0p) precesira po obodu stozca z vrsnim kotom 0P (slika 6.14), ki je odvisen od relativne velikosti ravninske BXy = ^BX + By in navpicne komponente polja Bz: tan 0P Bz (6.15) Meddelcno interakcijo je mozno kontrolirati s spreminjanjem kota 0p. Pri navpicnem polju, 0p = 0, je interakcija izotropno odbojna, druga skrajnost, pri 0P = 90°, je izotropno privlacna interakcija med delci v 2D sistemu. V obmocju, ko sta velikosti odboja in privlaka priblizno enaki, pride do zanimivih pojavov. Na sliki 6.15 so predstavljene stabilne faze, ki v sistemu nastanejo pri razlicnih kotih 0p. Ceprav je pri 0p < 54.7° odbojna komponenta dipolne interakcije mocnejsa od privlacne se koloidni delci zdruzijo v stabilno enojno verigo, kar je jasen znak privlaka Slika 6.15: Slike 2D sistema z nizko gostoto delcev v magnetnem polju, ki krozi po obodu stozca pri različnih vrednostih vrsnega kota 9p (označen). Slika 6.16: Energija dveh delcev v krozečem magnetnem polju kot funkcija razdalje med njima pri 9p = 50° (levo) in 9p = 56° (desno). S črno je narisana prava energija, črtkano pa dipolni potencial brez popravkov zaradi lokalnega polja. med njimi. Vzrok za to je vpliv lokalnega magnetnega polja delcev. To se najlazje pojasni na paru dveh delcev: ko krozece magnetno polje kaze v smeri para, se lokalno magnetno polje induciranega dipola prvega delca sesteje z zunanjim magnetnim poljem, kar povzroci vecjo magnetizacijo drugega delca. Ko pa magnetno polje kaze pravokotno na smer para, pa lokalno polje prvega delca zmanjsa magnetno polje na mestu drugega delca, tako da je njegova magnetizacija manjsa. Ta pojav znatno spremeni parski potencial, kar je za dva kota 0p prikazano na sliki 6.16. Poleg vpliva lokalnega polja so v takem sistemu pomembni tudi vecdelcni prispevki Slika 6.17: Izračun energije treh delcev v xy-ravnini Levo: Potencial okoli para delcev pri 6p = 50° in B0 = 3.5 mT. Desno: Energija sistema v odvisnosti od razdalje med parom in testnim delcev za dve posebni smeri (črtkani na levi sliki): v smeri dolge osi (E(x), modra) in kratke osi para (E(y), rdeca). Črtkane krivulje predstavljajo energijo istega sistema, izracunano kot vsoto treh parskih energij. - v sistemu vec delcev energija ni enaka vsoti posameznih parskih energij, kar se lepo ilustrira Ze v sistemu treh delcev. Na sliki 6.17 je z različnimi barvami prikazana energija sistema treh delcev: na sredini je fiksen par delcev (sivo), testni delec pa se premika po xy-ravnini. Barva v vsaki tocki slike simbolizira energijo sistema, ce se sredisce tretjega delca nahaja v tisti tocki. V belo obmocje tretji delec ne more zaradi trdega jedra. Par deluje privlacno v smeri dolge osi, v smeri kratke osi pa odbojno. Taka oblika potenciala povzroci nastanek verig pri nizkih 9p. Pri visjih 9p privlacen potencial postane se mocnejsi, odboj v precni smeri izgine, tako da se verige združijo v mrezo (slika 6.16). Tovrsten potencial je uporaben za stabilizacijo superstruktur iz koloidnih delcev. Nekaj primerov takih struktur, ki smo jih iz posameznih kroglic sestavili s pomocjo opticne pincete, je prikazano na sliki 6.18. Vrteče magnetno polje pri magičnem kotu Ce je vrsni kot stozca po katerem precesira polje enak magicnemu kotu 9m = arccos(1/\/3) ~ 54.7°, v meddelcni dipolni interakciji clen z odvisnostjo 1/r3 izgine. Najnizji nenicelni popravek k energiji para dveh induciranih dipolov v magnetnem polju znasa - x3v 3b0 1 E = . u—. (6.16) 16n2^o r6 v ; Ta interakcija je prostorsko izotropno privlacna, zato se v nadaljevanju osre-dotocimo na 3D sisteme s to interakcijo. Zaradi privlaka pride do zdruzevanja posameznih kroglic v pare, ki so orientirani v vse smeri (slika 6.19). Sila para delcev je privlacna v smeri dolge osi, v precni smeri pa je doseg privlaka priblizno 0.5 ^m, pri vecjih razdaljah pa je rahel odboj. Taka interakcija privede do rasti verig. Z vsakim dodatnim delcem v verigi se precni odboj zmanjsa in tako se pri dolzini okoli 8 delcev ali vec lahko nov delec verigi pridruzi tudi od strani. Tak ravninski Slika 6.18: Stabilne koloidne superstrukture, zgrajene z optično pinceto (0p = 53°, B0 = 3.5 mT). Slika 6.19: Slike samosestavljenih verig iz koloidnih delcev pri rotaciji magnetnega polja Bo = 3.5 mT pri magičnemu kotu. Levo: Dve zaporedni sliki redkega sistema. Zaradi izotropne narave interakcije so pari kroglic stabilni ne glede na njihovo orientacijo. Desno: Po dolgem casu se v redkem sistemu stabilizirajo enojne in dvojne verige delcev. trikotnik delcev (2 iz verige + 1 dodatni) povzroči privlačni potencial okoli njega in predstavlja nukleacijsko jedro za rast 2D struktur. Pri srednji gostoti sistema nastane kombinacija 2D heksagonalno zloZenih skupkov, povezanih z enojnimi verigami delcev (slika 6.20 levo). Pri visoki gostoti sistema se delci zlozijo v tesno zlozeno heksagonalno "plahto" poljubne orientacije, npr. na desni sliki 6.20 je "plahta" zaradi sedimentacije na dnu celice, njen rob pa je zavihan navzgor in seze 8 ^m nad dnom. Nastanek koloidnih "plaht" je energetsko ugoden proces. Na sliki 6.21 je obravnavan ravninski kristal, sestavljen iz 19 tesno zlozenih delcev. V njegovi ravnini je potencial privlacen, tako da pritegne morebitne okoliske proste delce in tako raste. V pravokotni smeri je potencial odbojen kar preprecuje rast v tretjo dimenzijo. 1 0 1 2 1 ■ 5 8 Slika 6.20: Heksagonalni tesno zlozeni 2D kristali (dp = dm, B0 = 3.5 mT). Levo: Na mestih večje lokalne gostote delcev nastanejo 2D kristaliti. Desno: Slike samosestavljene koloidne "plahte" pri različnih gorisčnih ravninah mikroskopa. Stevilka pomeni visino fokusa v ^m. Crni trakovi poudarjajo oster del slik. Slika 6.21: Tesno zlozen 2D kristal iz 19 delcev dp = dm,B0 = 3.5 mT. Levo: Potencial v xy ravnini. Sredina: Potencial v prečni ravnini xz. Desno: Energija dodatnega delca v odvisnosti od razdalje od kristala v smeri x (modro) in v prečni smeri z (rdeče). 6.5 Viskoelastičnost bakterijskih mreZ Ena izmed pomembnih snovnih lastnosti je strizni modul, ki je v kompleksnih materialih ponavadi frekvencno odvisen. Meritev viskoelasticnih lastnosti se tradicionalno opravi z mehanicnimi reometri, ki na merjenec delujejo z oscilirajoco strizno deformacijo y(t) = Yo sin(wt) in merijo napetost merjenca. Ce je amplituda deformacije majhna, je napetost a(t) proporcionalna deformaciji a(t) = 7o[G'M sin(ut) + G'V) cos(^i)]. (6.17) Tu je C(w) odziv v fazi z deformacijo, imenovan elasticni modul, G"(w) pa odziv snovi v fazi s hitrostjo deformacije, imenovan viskozni modul. Frekvencno odvisen strizni modul (imenovan tudi viskoelasticni modul) [89] je definiran kot G(w) = GV) + iG'V). (6.18) V zadnjih dveh desetletjih se je razvilo ogromno tehnik za merjenje materialnih lastnosti na mikroskopski velikostni skali s skupnim imenom mikroreologija (pregled v [90, 91, 92]). V primerjavi s klasicno reologijo ima mikroreologija (MR) vec prednosti: potrebna je samo majhna kolicina vzorca (1 je ze dovolj), mozna je meritev nehomogenih okolij (npr. notranjosti celice), dostopne so visje frekvence (do 105 Hz). MR tehnike se delijo v dva razreda: pasivne in aktivne. Pri pasivnih tehnikah se spremlja termicne fluktuacije sond, ponavadi inertnih kroglic premera od nekaj deset nanometrov do nekaj mikronov, ki so vlozene v merjeni material. Iz njihove difuzije se izracuna linearni kompleksni strizni modul [26, 27]. Aktivna MR je podobna klasicni reologiji, saj se na sodno deluje s silo, pri tem pa se meri deformacijo medija. Za prvi primer aktivne MR se steje Freundlichovo [28] manipulacijo magnetnih kroglic z zunanjim magnetnim poljem pred vec kot 80 leti. Danes se za aktivno MR uporabljajo mikropipete [29], mikroskopi na atomsko silo [30], opticne pincete [31] in mikroreometri z magnetnimi kroglicami [32, 33, 34]. Znotraj obeh razredov obstaja se nadaljnja delitev glede na stevilo uporabljenih sond na enodelcno MR, kjer se viskoelasticnost doloci iz gibanja ene same sonde in na dvodelcno MR, pri kateri se analizira korelirano gibanje dveh (ali vec) sond. Dvodelcna MR omogoca meritev viskoelasticnih lastnosti na vecjih dolzinskih skalah, kar je prikladno v sistemih, ki so nehomogeni na velikostni skali posamezne sonde. Opticna pinceta je uporabno orodje za MR meritve. Pri enodelcni pasivni MR je sonda ujeta v nepremicni opticni pasti, njene fluktuacije pa dajo informacijo o mehanskih lastnostih snovi. Ce je snov pretrda in termicne fluktuacije ne povzrocijo zaznavne deformacije, se lahko uporabi enodelcna aktivna MR, pri kateri je sonda ujeta v sinusno premikajoci opticni pasti [31, 35]. Kompleksni viskoelasticni modul merjene snovi je izracunan iz odziva sonde na gibanje pasti. Pri dvodelcni aktivni MR [36] se uporabljata dve opticni pasti, v vsaki pa je ujeta sonda. S harmonicnim spreminjanjem lege delca v prvi (trdi) pasti se aktivno deformira medij, delec v drugi (sibkejsi) pasti pa je indikator deformacije medija na mestu pasti. V nasih meritvah smo kot sonde uporabili kroglice iz silike s premerom 3.22 ^m. Vse MR metode smo testirali v vodi kot primeru vzorca z znano viskoznostjo (n = 9.3 x 10-4 Pas pri temperaturi 23°C), nato pa smo z njimi analizirali mehanske lastnosti kompleksnega bioloskega vzorca - rastoce bakterijske populacije. Ko se bakterije znajdejo v hranljivem okolju, se zacne proces delitve. Spremembo njihovega stevila skozi cas opisuje rastna krivulja, sestavljena iz 4 faz. V adaptivni fazi se bakterije prilagajajo na novo okolje in ne rastejo. Ko se prilagodijo, preidejo v logaritemsko fazo, kjer je rast eksponentna. Bakterijska kultura se znajde v stacionarni fazi, ko zacne primanjkovati okoliskih hranilnih snovi, nato pa sledi faza odmiranja, ko se njihovo stevilo zmanjsuje zaradi akumuliranih toksinov in pomanjkanja hrane. Spremljali smo reologijo raztopine bakterijskega seva Vibrio sp. DSM 14379 [108]. Vzorci za MR meritev so bili vzeti v adaptivni (vzorec 1), logaritemski (vzorec 2) in stacionarni fazi (vzorec 3). Dodaten vzorec ("samo mreza") smo pripravili iz tretjega vzorca z odstranitvijo bakterij s centrifugo. Merili smo MR celotne bakterijske raztopine, ki sestoji iz gojisca, bakterijskih celic in ekstracelularne matrike (ECM). To sestavlja prepletena mreza vlaknastih beljakovin in dolgih nerazvejanih polisa-haridov. Komponente ECM nastanejo v notranjosti bakterijskih celic, nato pa so z 2.0x10~14 -1.5x10"14 - c3 1 £ Q 1.0x10"14 -5.0x10"15 -0.0 - 0.00 0.02 0.04 0.06 0.08 0.10 t [S] Slika 6.22: Enodelcna pasivna MR. Povprečen kvadrat odmika v odvisnosti od časa za rastočo bakterijsko raztopino. Gojisče (črno), vzorec 1 (rdeče), vzorec 2 (zeleno), vzorec 3 (modra), vzoreč 3 brez bakterij (svetlomodra). eksocitozo izločene iz celice, kjer se združijo z obstoječo matriko. Enodelcna pasivna MR Pri enodelcni pasivni MR se iz zaporednih leg ujetega delca izračuna povprečen kvadrat odmika (angl. mean square displacement) v casu t. Definiran je z (r2(t)) = (|r(t + t) - r(t)|2). (6.19) Tu je r polozaj delca, oklepaji pa pomenijo povprecenje po celem intervalu meritve. (r2) in kompleksni viskoelasticni modul G povezuje posplosena Stokes-Einsteinova enacba [93]. Ce je sonda vecja od vseh strukturnih dolzinskih velikostnih skal, v d-dimenzijah velja ( ) 3nas(r2(s)) ( ) Tu je s frekvenca v Laplaceovem prostoru, (r2(s)) in G(s) Laplaceovi transformaciji (r2(t)) in G(t). Za primerjavo z rezultati klasicne reologije je (G(s) pretvorjen v Fourierev prostor s transformacijo (s ^ iu). Ce predpostavimo, da sta viskoznost n in elasticnost medija km frekvencno neodvisna, velja k G(u) = G'(u) + iG"(u) = - (6.21) 6na V vodi smo z enodelcno pasivno MR merili s tremi razlicno mocnimi pastmi: sibko (k = 0.81 x 10-6 N/m), srednjo (k = 2.0 x 10-6 N/m) in trdo (k = 4.0 x 10-6 N/m) ter za viskoznost dobili vrednosti od 8.7 x 10-4 Pas do n = 9.1 x 10-4 Pas, kar je malo manj od pricakovane vrednosti. Pri meritvi bakterijskih vzorcev metoda se posebno pri večjih močeh optičnih pasti ni delovala dobro, saj so močne pasti vase vlekle bakterije, kar je povzročilo artefakte pri meritvi. Povprečne kvadrate odmika smo zato izmerili v sibki pasti (slika 6.22). Izmerjene bakterijske viskoznosti so bile nM = 10.0 X 10-4 Pas, ns 1 = 12.4 x 10-4 Pas, nS2 = 14.4 x 10-4 Pas, nS3 = 16.0 x 10-4 in ns3' = 13.8 x 10-4 Pas. Enodelcna aktivna MR Pri enodelčni aktivni MR se s krozno frekvenco u in amplitudo A sinusno premika lego pasti z ujeto sondo. Odziv sonde na oscilirajočo silo, x(t) = D(u) sin[ut — S(u)], (6.22) je odvisen od mehanskih lastnosti okoliskega medija. Njegov viskoelastični modul je moč izračunati iz izmerjenega razmerja amplitud d(u) = D(u)/A in faznega zaostanka S(u) za lego pasti kot [31, 98]: G' (u) G" (u) km (u ) k 6na un (u) = — 6na k čos S (u) . d(u) sin S(u) 1 6na d(u) (6.23) S prilagoditvijo linearne funkčije izmerkom G''(u) po enačbi (6.21) smo za viskoznost 350 ® [s"1l Slika 6.23: Absolutna vrednost viskoznega modula G" rastoče bakterijske suspenzije kot funkcija kroZne frekvence, izmerjen z enodelčno aktivno MR. Točkam prilagojena linearna funkcija [imaginarni del enačbe (6.21)] da naslednje viskoznosti: gojisče (črna) nM = 12.4 ± 0.1 x 10-4 Pas, vzoreč 1 (zelena) nS1 = 14.1 ± 0.1 x 10-4 Pas, vzoreč 2 (modra) ns2 = 20.1 ± 0.3 x 10-4 Pas, vzoreč 3 (rdeča) ns3 = 35.4 ± 1.2 x 10-4 Pas, vzoreč 3 brez bakterij (črtkana rdeča) nS3 = 24.5 ± 0.4 x 10-4 Pas. vode dobili n = 9.1 x 10-4 Pas. Prilagoditev linearne funkcije izmerkom G''(u) v bakterijskem mediju (slika 6.23) so dali naslednje rezultate. Viskoznost gojisca nM = 12.4 ± 0.1 x 10-4 Pas je rahlo vecja od viskoznosti vode, kar je smiselno, saj medij vsebuje nekaj soli in 0.4% glukoze. Viskoznosti bakterijskih vzorcev so bile nsi = 14.1 ±0.1 x 10-4 Pas, ns2 = 20.1 ±0.3 x 10-4 Pas in ns3 = 35.4± 1.2 x 10-4 Pas, tretjega vzorca brez bakterij pa nS3' = 24.5 ± 0.4 x 10-4 Pas. Viskoznost raztopine se s casom povecuje, saj stevilo bakterij raste, hkrati pa se gosti ekstracelularna matrika. Frekvencna odvisnost elasticnega modula rastoce bakterijske suspenzije, izmerjena z enodelcno aktivni MR je prikazana na sliki 6.24. G'(u) z vecanjem frekvence narasca od vrednosti nic pri nizkih frekvencah do okoli 0.01 Pa pri visokih. Razlika med vrednostjo pri visoki in nizki frekvenci AG' = G'(60/s) — G'(0.6/s) je 0.013 Pa za vzorec 1, 0.016 Pa za vzorec 2 in 0.023 Pa za vzorec 3. Elasticnosti medija, ki ustrezajo tem elasticnim modulom, so po enacbi km = 6naAG' enake 3.9 x 10-7 N/m, 4.8 x 10-7 N/m in 7.1 x 10-7 N/m. 0.030.02- _ 0.01 -CD CL O 0.00-0.01 - 1 10 100 ® [S1] Slika 6.24: Elastični modul G' bakterijske suspenzije v odvisnosti od krožne frekvence, izmerjen z enodelcno aktivno MR. ■ gojišče • v1 ▲ v2 T v3 t T ■ I Dvodelcna aktivna MR Enodelcna reologija poda pravilne rezultate samo v primeru, ce je sonda vecja od vseh dolzinskih skal medija. Te omejitvi se je moc izogniti z uporabo dvodelcne MR, ki je neobcutljiva na lokalne nehomogenosti, saj meri mehanske lastnosti materiala na dolzinski skali, ki je dolocena z razdaljo med obema delcema. Pri dvodelcni aktivni MR je kompleksni viskoelasticni modul vzorca dolocen iz odziva sonde na oscilirajoco silo, ki deluje na pogonski delec (slika 6.25). Teoreticen opis odziva je tu nekoliko bolj zapleten kot pri enodelcni MR. Slika 6.25: Shema dvodelcne aktive MR meritve. Kroglici z radijem a sta ujeti v dveh opticnih pasteh, sibki in mocni, na razdalji R. Ce sta dva delca ujeta v opticnih pasteh, ki se nahajata na osi x na medsebojni razdalja R, lahko odmike sonde (delca 2) v smereh x in y zaradi delovanja poljubne sile Fi1 oz. Fy(1) na pogonski delec (delec 1) v frekvencnem prostoru zapisemo kot [103, 104] x<2) M = A|| (^)Fi1) M y<2) (w) = A± MFy(1) (w). (6.24) Tu sta A|| in A^ sestavljeni kompleksni dvodelcni odzivni funkciji v vzporedni oz. pravokotni smeri, ki sta zaradi vpliva pasti drugacni od lastnih odzivnih funkcij medija a|| (^) in a^(^). Sestavljeni sta iz lastnega odziva medija, trdot obeh pasti k f1) in k<2) ter odzivov posameznih delcev, a(1) in a<2), na zunanjo silo: All = _a_ 11 (1 + k(1)a(1))(1 + k<2)a<2)) - k(1)k(2)a2' A± = (1 + kWa*1))(1 + a(2)a<2)) - kWk(2)a2 . (6.25) Lastne odzivne funkcije za nestisljivi kontinuum s kompleksnim striznim modulom ) opisuje generaliziran Oseenov tenzor a|| = 1 = snRGv). (6.26) Odzivni funkciji posameznega delca sta v nasem primeru enaki, saj tako za sondo kot tudi pogonski delec uporabljamo enaki kroglici: a*11=a<2' = . <6-27> V dvodelcni meritvi se izmeri sestavljeno kompleksno dvodelcno odzivno funkcijo kot funkcijo frekvence. S prilagoditvijo izraza (6.25) izmerkom se ob znani trdoti obeh Slika 6.26: Realni (levi graf) in imaginarni del (desni graf) sestavljene dvodelcne odzivne funkcije v pravokotni smeri. Tocke različne barv prikazujejo izmerjene vrednosti v gojišču (crno), vzorcu 1 (rdeče), vzorcu 2 (zeleno), vzorcu 3 (temnomodro) in v istem vzorcu z odstranjenimi bakterijami (svetlomodra). Vrednostim je prilagojena enacba (6.25) s tremi prilagodljivimi parametri: n, km in skalirnim koeficientom. pasti dobi viskoznost rq in elastičnost km medija. Pri meritvah, ki smo jih izvedli v vodi, so se izmerjenim A||(w) in A±(u) teoretične oblike odzivov odlično prilegale (pri parametrih: elastičnost km = 0, viskoznosti n = 9.1 x 10-4 Pas za vzporedno in n = 9.2 x 10-4 Pas za pravokotno dvodelčno odzivno funkcijo). Dvodelčno aktivno MR smo nato uporabili za meritev viskoelastičnosti bakterijske mreze. Realni in imaginarni del dvodelčne odzivne funkčije v pravokotni smeri sta predstavljena na sliki 6.26. Izmerkom smo prilagodili teoretični odzivni funkčiji (6.25) s tremi prilagoditvenimi parametri - viskoznostjo n, trdoto medija km in skalirnim faktorjem. Ceprav je bila laserska moč v pasteh v vseh poskusih enaka, imata izmerjeni odzivni funkčiji A\\(u>) in A±_(u>) v različnih vzorčih različno magnitudo, zato je bilo potrebno v (6.25) pri prilagajanju uvesti dodaten skalirni faktor. Ta ne vpliva na vrednosti viskoznosti in elastičnosti medija, saj viskoznost povezana z lego vrha realne komponente odzivne funkčije (pri visji viskoznosti se ta premakne k nizjim frekvenčam), elastičnost pa se v odzivni funkčiji pozna pri nizkih frekvenčah . Izmerjene vrednosti realnega dela kompleksne dvodelčne odzivne funkčije se dobro kvantitativno ujemajo s teoretičnim odzivom (6.25), medtem ko je pri imaginarni del odziva ujemanje s teorijo samo kvalitativno. Viskoznosti in elastičnosti bakterijskih vzorčev, dobljenih iz raznih dvodelčnih odzivnih funkčij, so prikazane v spodnji tabeli. Dobljen iz: Al Al A|| Al A|| Vzorec n[10-4 Pas] n [10-4 Pas] n [10-4 Pas] km[10-8 N/m] km[10-8 N/m] Gojisce 14.7 14.5 14.9 15 0 Vzorec 1 18.4 16.7 17.2 4.9 9.4 Vzorec 2 21.1 19.9 21.8 7.4 9.8 Vzorec 3 33.1 32.7 34 8.8 10 "Samo mreza" 27.3 22.2 25 0 7.4 Sklepi Na sliki 6.27 so povzeti rezultati meritev viskoznosti gojisca in bakterijske suspen-zije v adaptivni (S1), logaritemski (S2) ter stacionarni fazi (S3) z različnimi MR metodami. Enodelcna pasivna MR je sicer zaznala povečevanje viskoznosti, toda vrednosti se v primerjavi z drugimi metodami zdijo močno podcenjene. Rezultati enodelcne aktivne MR so bili podobni rezultatom dvodelcna aktivne MR -izmerki v vseh vzorcih so znotraj ±10% povprecne vrednosti. Viskoznost suspenzije s staranjem narasca do n ~ 3.3 x 10-3 Pas, njena elasticnost pa je velikostnega reda k - 10-7 N/m. 40 i-,-.-,-.-,-.-,-.-,- CZImsd ■ ip ^■2Preal \ \ \— 30- CZl2Pimag 'kili growth medium S1 S2 S3 mesh Slika 6.27: Primerjava izmerjenih viskoznosti n vzorcev z enodelcno pasivno MR (belo), enodelcno aktivno MR (rdece) in dvodelcno aktivno MR (temnomodro - realni del, svet-lomodro - imaginarni del dvodelcne odzivne funkcije). Izjava o avtorstvu Spodaj podpisani Natan Osterman, univ. dipl. fiz., izjavljam, da sem samostojno izdelal pričujočo doktorsko disertacijo in sem njen avtor. V Ljubljani, 25.5.2009 Natan Osterman