Organization of Paramagnetic and Nonmagnetic Colloidal Particles in Ferrofluid

Organization of Paramagnetic and Nonmagnetic Colloidal Particles in Ferrofluid Von der Universität Bayreuth zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung von Ayan Ray geboren am 27. Nov. 1983 in Kalkutta/Indien 1. Gutachter: Prof. Dr. Th .M. Fischer 2. Gutachter: Prof. Dr. W. Köhler Tag der Einreichung: 17.04.2012 Tag des Kolloquiums: 21.06.2012 Dedicated to my beloved father Prof. K. K. Ray (baba) and mother Mrs. S. Ray (maa) Acknowledgement I would like to thank and express my sincere gratitude to Professor Dr. Thomas Fischer for his kind and caring guidance, and constant support during my doctoral study at the University of Bayreuth, Germany. I would also like to thank him specially for introducing me to the new world of "Dynamics of Soft Matter" in physics. This thesis work would not have been possible without his scientific advice, personal guidance and understanding. Professor Thomas is more like a friend than a supervisor to me. I take this opportunity to thank all my group members Uli Langer, Nebojsa Casic, Tobias Gehring, Saeedeh Aliaskarisohi and Christiane Jungnickel who helped me with their scientific and technical knowledge apart from their personal help whenever required. Getting such a galaxy of friendly colleagues in one place is hard to find now-a-days. Altogether it was a small family with precious sweet memories. I would also like to thank all the members of Experimental Physics V for their constant support whether technical or scientific discussions. At this point I would like to thank specially Mrs. Carmen Kerling for helping me in the IT sector, Mr. Klaus Oetter for his assistance in manufacturing of different machined components at Bayreuth and our group secretary Mrs. Christine Linser helping me with the official, administrative works and her i valuable suggestions towards societal relations. After all, a cup of coffee is an unequalled medium to restore and keep up spirits. Taking this opportunity I would like to thank the Welcome Center of the University of Bayreuth and personally Dr. Cornelia Nicodemus for her help and guidance throughout my stay at Bayreuth. I take this opportunity to express my heartfelt thanks to my contemporary research colleagues and friends - Sonal Di, Swastik Da and boudi (Aditi), Dr. Himadri da and boudi (Dolon), Imran, Somnath, Pratap, Andy, Moritz, Christian and Marrion and many others of the University of Bayreuth for their kind help at required moments and for making my stay at Bayreuth a happy and memorable one. All along this work my beloved friend Sayanti has remained as a source of inspiration, and assurance; I express my heartfelt deep gratitude to Sayanti for her love, patience and mental company at all moments to make this achievement a reality. Finally, I would like to thank my parents, for always being with me throughout my studies and for their endless love and support. Their love and motivation was one of the key to the success of this dissertation. Ayan Ray Contents 1 Introduction 1 1.1 Introduction............................. 1 1.2 Colloidalflower ........................... 5 1.3 Transitionstrength......................... 6 1.4 Colloidalphases........................... 9 2 Materials and Method 13 2.1 Materials .............................. 13 2.1.1 Ferrofluid .......................... 13 2.1.2 MagneticField ....................... 15 2.1.3 Optical Microscopy . . . . . . . . . . . . . . . . . . . . . 15 2.2 Method ............................... 16 2.2.1 Dynamics of self-assembly of flower-shaped magnetic colloidalclusters........................ 16 2.2.2 The transition strength from solid to liquid colloidal dipolar clusters in precessing magnetic fields . . . . . . . 17 2.2.3 Magnetic field controlled composite paramagnetic-diamagnetic colloidalphases....................... 20 iii 3 Colloidal flower 23 4 Transition strength 31 5 Colloidal phases 41 6 Summary 65 List of Figures 1.1 a) schematic representation of colloidal flower-shaped clusters [7] formed in a perpendicular field Hz. The center particle is a paramagnetic particle, which is the core of the flower and the particles around the core are the diamagnetic particles that are referred as petals of the flower. b) represents X, Y and Z are the coordinate axis with Hx, Hyand Hzare the external magnetic fieldrespectively. .......................... 6 1.2 a)Schematic representation of the behaviour of super-paramagnetic and nonmagnetic particles in an applied magnetic field when immersed in a thin film of ferrofluid; Figure 2a indicates that the direction of the applied magnetic field is in the z direction. Figure 2b shows a mixture of nonmagnetic and super-paramagnetic particles immersed in thin film of ferrofluid between two glass cover slips under the influence of the magnetic field. Figure 2c reveals the alignment of dipole moment of ferrofliud and Figure 2d shows the effective magnetic moment of the magnetic and nonmagnetic particles under an external applied magnetic field. It can be noted that Figure 2c and Figure 2d can combine to form Figure 2b. Here represent χthe susceptibility factor. . . . 7 v 1.3 a) schematic representation of diamagnetic cluster formed in a rotating magnetic field H|| in x-y plane. A core to petal size ratio is chosen to form the colloidal cluster. b) represents X, Y and Z are the coordinate axis with Hx, Hyand Hzare the external magnetic field with H|| being the in plane rotating effective magnetic field and being the precession angle. Ωis the external appliedfrequency........................... 9 1.4 Schematic representation of magic angle. . . . . . . . . . . . . . 10 1.5 Schematic presentation of the behaviour of nonmagnetic particles under a rotating magnetic field when immersed in a thin film of ferrofluid. Figure 5a indicates the direction of the applied magnetic field in the x-y plane. Figure 5b shows two different sizes of nonmagnetic particles immersed in thin film of ferrofluid between two glass coverslips under the influence of the rotating magnetic field. Figure 5c reveals the alignment of dipole moment of ferrofliud and Figure 4d shows the effective magnetic moment of nonmagnetic particles under an external applied rotating magnetic field, assuming the effect of ferrofluid to be negligible. It can be observed that Figure 5c and Figure 5d can combine to form figure 5b. . . . . . . . . . . . . . . . . . 11 1.6 Schematic representation of the external magnetic field applied ~ H(t) = ˆ Hcos ϑext~ez+ˆ Hsin ϑext(~exsin Ωt+~eysin 2Ωt). H (t) is the total external magnetic field strength applied to the sample. Where x,y and z are the coordinate axises. . . . . . . . . . . . . 12 2.1 a) schematic representation of sample on top of solenoid and b) Hz being the external static magnetic field in the z direction. Here x, y and z are the coordinate axises. . . . . . . . . . . . . . 17 2.2 a) Schematic representation of arrangements of five sets of solenoid coils and b) the combined rotating magnetic field H|| and the perpendicular field H with being the angular frequency and the precessionangle........................... 19 2.3 a) schematic representation of the arrangement of solenoid coils and b) time dependent magnetic field produced by the five sets of solenoid coils similar to Lissajou curve. . . . . . . . . . . . . . 21 1.1. INTRODUCTION CHAPTER 1. INTRODUCTION situation, all paramagnetic beads have magnetic moments that point into the same direction. We can enrich the structure of the assembly [5] by incorporating diamagnetic particles. Such diamagnets react to an external field with a magnetic moment anti-parallel to the external field. Since diamagnetic susceptibilities of most materials at room temperature are small, we must use a trick to obtain effective diamagnets. This trick consists of immersing nonmagnetic colloids into a ferrofluid. When using a ferrofluid with susceptibility between the zero susceptibility of the non-magnetic colloids and the susceptibility of the paramagnetic colloids the paramagnetic colloids still act as paramagnets while the non-magnetic beads act effectively as diamagnetic particles in the background of the ferrofluid. Such effective diamagnets run under the name magnetic holes. In chapter 5, we expose a mixture of paramagnets and magnetic holes [1] [3] to time dependent external fields to self assemble the mixture into various structures. The question addressed in this chapter is which type of anisotropic structures of the mixed system may be assembled when using various forms of external magnetic field modulations. To answer these questions I have arranged the thesis into the following structure: chapter 1 includes a brief introduction to the thesis with motivation as a subsection. Experimental details have been provided in the subsection titled methodology with the materials parameter of chapter 2. Chapter 3, chapter 4, and chapter 5 are the attached published manuscripts with the results and conclusion. Finally, chapter 6 includes the summary. 4 CHAPTER 1. INTRODUCTION 1.2. COLLOIDAL FLOWER 1.2 Dynamics of self-assembly of flower-shaped magnetic colloidal clusters In chapter 3, we were interested to study the effects of dynamic interactions of paramagnetic and nonmagnetic particles in a 1-dimensional system. We observed single file diffusion present in our system. Single file diffusion refers to the 1-dimensional motion of interacting particles in pores, which are so narrow that the mutual passage of such particles is excluded. Since the sequence of particles in such a situation remains unaffected over time t, leads to deviation from normal diffusion. Such a single file diffusion of colloids in 1-dimensional have already been reported [C. Lutz et al, 2004]. Where the colloidal particles were trapped by a scanning laser beam to a circular optical trap. Our system consists of paramagnetic and nonmagnetic particles immersed in ferrofluid under static magnetic field (magnetic field strength ~ H(t) = ˆ H~ez, z-direction), sandwiched between two glass coverslips. Under such conditions flower shaped magnetic colloids are formed, where the paramagnetic particle is at the center i.e. the core of the flower and the nonmagnetic particles are at the equator, the petals, shown in Figure(1.1). External magnetic field induces magnetic moments in the particles that interacts via the dipole dipole interaction. Due to the presence of static magnetic field in the system of magnetic and nonmagnetic particles immersed in ferrofluid (chapter 3), the effective dipoles i.e. the magnetic dipole minus the ferrofluid background of the two sorts of particles point into opposite directions Figure(1.2). Hence, in presence of static magnetic field the nonmagnetic particles immersed in ferrofluid behaves as diamagnets and the paramagnets behaves still as paramagnets. These diamagnets are attracted towards the core (paramagnet) to 5 1.3. TRANSITION STRENGTH CHAPTER 1. INTRODUCTION Figure 1.1: a) schematic representation of colloidal flower-shaped clusters [7] formed in a perpendicular field Hz. The center particle is a paramagnetic particle, which is the core of the flower and the particles around the core are the diamagnetic particles that are referred as petals of the flower. b) represents X, Y and Z are the coordinate axis with Hx, Hyand Hzare the external magnetic field respectively. form a circular channel. Around the core the diamagnets have a repulsive force between each other and interact by soft-core interactions. The motions of these interacting particles (diamagnets) in the circular array made us motivated to study and characterize the single file diffusion in the self-assembled flower-shaped magnetic colloidal clusters. 1.3 The transition strength from solid to liquid colloidal dipolar cluster in precessing magnetic fields Due to the presence of hard-core and dipolar interactions present in the magnetic colloidal flower system we can study the influence of long range interactions on to the single file diffusion chapter 3. Long-range interactions also 6 CHAPTER 1. INTRODUCTION 1.3. TRANSITION STRENGTH Figure 1.2: a)Schematic representation of the behaviour of super-paramagnetic and nonmagnetic particles in an applied magnetic field when immersed in a thin film of ferrofluid; Figure 2a indicates that the direction of the applied magnetic field is in the z direction. Figure 2b shows a mixture of nonmagnetic and super-paramagnetic particles immersed in thin film of ferrofluid between two glass cover slips under the influence of the magnetic field. Figure 2c reveals the alignment of dipole moment of ferrofliud and Figure 2d shows the effective magnetic moment of the magnetic and nonmagnetic particles under an external applied magnetic field. It can be noted that Figure 2c and Figure 2d can combine to form Figure 2b. Here represent χthe susceptibility factor. play an essential role for phase transitions between differently ordered phases. A first-order phase transitions exhibit a discontinuous change in the order parameter. The change of one phase to other occurs via a coexistence of the two phases. The area of the hysteresis measures the dissipated energy when traversing the coexistence region back and forth. Whereas a second order phase transition is a transition where the order parameter changes continuously at the transition. Second order transitions are associated with critical behavior of response functions as a function of the control parameter while first order transitions exhibit no critical behavior. The formation and rupture of flower-shaped magnetic colloidal clusters can be considered as a finite size phase transitions. It was a question of interest, is the change in the hysteresis could reveal the strength and order of the phase transitions in the system. The formation and rupture of the flower7 1.3. TRANSITION STRENGTH CHAPTER 1. INTRODUCTION shaped magnetic colloidal clusters and diamagnetic clusters takes place with the change in the precession angle, the control parameter. A hysteresis loop is observed when tuned the precession angle, from low to high and vice versa. Whether the study of the width of the hysteresis could reveal the order of the system? Besides, is it possible to define the order and strength by the measuring the response function, the angular velocity of the particles as a function of change in the precession angle? These flower-shaped magnetic colloidal clusters were formed from paramagnetic and nonmagnetic particles immersed in diluted ferrofluid under a static magnetic field in the z-direction and sandwiched between two glass coverslips (Figure (1.1)). Whereas the diamagnetic colloidal clusters were formed from nonmagnetic particles immersed in concentrated ferrofluid under a rotating field Figure(1.3) and sandwiched between two glass coverslips. The magnetic field strength being ~ H(t) = ˆ H(~exsin Ωt+~eycos Ωt), with Ωbeing the angular frequency in x-y plane. The flower-shaped magnetic colloidal were stable at low angles and nearby the magic angle these structures were unstable whereas, the diamagnetic clusters were stable at high angles and their stability decreased reaching towards the magic angle. Here the magic angle ϑmagic is the defined as a unique angle, which is approximately 54.73◦. It is the root of a second-order Legendre polynomial P2(cos θ)=0and interactions depending on this second-order Legendre polynomial vanishes at this angle. Mathematically ϑmagic =θm= arctan √2≈54.73◦,Figure(1.4). As external magnetic field induces magnetic moments in the particles and they interact via dipole dipole interaction. The effective dipoles (diamagnetic cluster formation) i.e. the magnetic dipole minus the ferrofluid background of the two sorts of particles point into same directions (x-y plane), shown in Figure(1.5). Similarly, in 8 CHAPTER 1. INTRODUCTION 1.4. COLLOIDAL PHASES Figure 1.3: a) schematic representation of diamagnetic cluster formed in a rotating magnetic field H|| in x-y plane. A core to petal size ratio is chosen to form the colloidal cluster. b) represents X, Y and Z are the coordinate axis with Hx, Hyand Hzare the external magnetic field with H|| being the in plane rotating effective magnetic field and being the precession angle. Ωis the external applied frequency. both the systems of colloidal flower and diamagnetic cluster due to the presence of external magnetic field the particles interact via dipole dipole interaction. 1.4 Magnetic field controlled composite paramagnetic-diamagnetic colloidal phases Neutralization of opposite charge is one of the major concepts in ordinary matter where two opposite charges cancel each other. The interactions taking place between these opposite charges is isotropic and is independent of direction. This charge neutralization is the key towards the organizations of matter on the atomic and molecular scale leading to self-assembly. It is spontaneous breaking of rotational symmetry [4] and the quantization of angular momentum that produces crystalline structures with forming direct bonds in atoms and molecules. Whereas, neutralization process is different in case of mesoscopic 9 1.4. COLLOIDAL PHASES CHAPTER 1. INTRODUCTION Figure 1.4: Schematic representation of magic angle. sized particles due to the absence of the quantum phenomena and angular momentum being a continuous quantity. In a colloidal system the direct bond formation does not work. Steric interactions are the means to spontaneously break the rotation symmetry to form colloidal crystal for isotropic structures. Direct bond in colloidal system are only possible using intrinsically anisotropic colloidal particles e.g. Janus or ellipsoid particles. One of the other possibilities to use the magnetic or electric dipole moment using an external field. In case for a mixture of paramagnetic and nonmagnetic particles immersed in a magnetic fluid under magnetic field. The effective dipole moment induced due to the same external magnetic field results in pointing the dipoles into opposite direction for paramagnetic and nonmagnetic particles. The induced magnetic moment neutralizes each other similarly like the charge neutralization, forming rich variety of anisotropic self-assembled structures. An attempt has been made to study this charge neutralization of magnetic moments in an external magnetic field resulting in forming different anisotropic structures. 10 CHAPTER 1. INTRODUCTION 1.4. COLLOIDAL PHASES Figure 1.5: Schematic presentation of the behaviour of nonmagnetic particles under a rotating magnetic field when immersed in a thin film of ferrofluid. Figure 5a indicates the direction of the applied magnetic field in the x-y plane. Figure 5b shows two different sizes of nonmagnetic particles immersed in thin film of ferrofluid between two glass coverslips under the influence of the rotating magnetic field. Figure 5c reveals the alignment of dipole moment of ferrofliud and Figure 4d shows the effective magnetic moment of nonmagnetic particles under an external applied rotating magnetic field, assuming the effect of ferrofluid to be negligible. It can be observed that Figure 5c and Figure 5d can combine to form figure 5b. Our system consists of paramagnetic and nonmagnetic particles immersed in ferrofluid under a magnetic field ~ H(t) = ˆ Hcos ϑext~ez+ˆ Hsin ϑext(~exsin Ωt+ ~eysin 2Ωt)as shown in Figure (1.5), sandwiched between two glass coverslips. We use magnetic field with three different frequencies with zero-frequency,Ω and 2 -Ωfrequency along different axes. This magnetic field was applied to the particles such that there is no torque. Dipolar interactions are anisotropic and differ in sign for interactions between similar (paramagnetic or diamagnetic ) particles and opposite (paramagnetic and diamagnetic ) particles. The composite structure of a mixture of diamagnetic s and paramagnets is therefore expected to exhibit a rich variety of structures. These structures will be explored in chapter 5. 11 1.4. COLLOIDAL PHASES CHAPTER 1. INTRODUCTION Figure 1.6: Schematic representation of the external magnetic field applied ~ H(t) = ˆ Hcos ϑext~ez+ˆ Hsin ϑext(~exsin Ωt+~eysin 2Ωt). H (t) is the total external magnetic field strength applied to the sample. Where x,y and z are the coordinate axises. 12 Chapter 2 Materials and Method 2.1 Materials 2.1.1 Ferrofluid Ferrofluid is a complex fluid, which has magnetic properties like solid while being a fluid in its physical state. The ferrofluids contain tiny magnetic materials of the order 10 −12 nm in size in a liquid medium. These nanometersized particles are coated with a stabilizing dispersing agent, which prevents particle agglomeration even under an applied strong magnetic field gradient. Depending on the medium, these ferrofluids can be classified either as (a) oil based or (b) water based. For the current experiments, water based ferrofluids were procured from Ferrotec Ferrosound. Ferrofluid EMG 705 and EMG 707 were two water-based ferrofluids used for the present experiments. The EMG 705 has a saturation magnetization at 22 mT with magnetic susceptibility of 4.04 (SI units) whereas the EMG 707 has 11 mT with susceptibility of 1.51 (SI Units) [Ferrotec Ferrosound USA]. Super-paramagnetic beads Spheri13 2.2. METHOD CHAPTER 2. MATERIALS AND METHOD tured using a Leica high-speed camera (Leica DFC 360 FX). The dynamics of the colloidal flowers and clusters formation were analyzed by using imageprocessing techniques in with the help of a commercially available software package (MATLAB) and open source packages such as ImageJ and Virtual Dub. 2.2.3 Magnetic field controlled composite paramagneticdiamagnetic colloidal phases Sample Preparation: A mixture of paramagnetic particles (diameter 2a = 2.8µm) with nonmagnetic fluorescent (red) polystyrene particles (diameter 2a =1.0µm) immersed in concentrated ferrofluid EMG 707.was prepared in controlled proportions ( paramagnetic 2 : nonmagnetic 4 by volume). This mixture was vigorously shaken to form a homogenous mixture. Using a pipette a small amount 0.5µlof this mixture was placed at the center between two pre-cleaned glass cover slips. External Field and Optical Microscopy: The sample was placed on top of a solenoid, shown in Figure(2.3). A combination of static magnatic field in the z-direction was applied with an in plane time dependent magnetic field . This sample was observed under fluorescence microscope in a reflecting mode. Red fluorescence filter was used to observe the red fluorescence particles whereas the Polarization filter was used to observe the non-fluorescence paramagnetic particles. Observations and recording: Changing the static magnetic field anisotropic structures evolved in 2-dimension and 3-dimension. At high static magnetic field H 26.5 mT colloidal flowers are observed where as decreasing this magnetic field results in forming 3-dimensional anistropic sandwiched structure. Where the paramagnets are at the middle 20 CHAPTER 2. MATERIALS AND METHOD 2.2. METHOD Figure 2.3: a) schematic representation of the arrangement of solenoid coils and b) time dependent magnetic field produced by the five sets of solenoid coils similar to Lissajou curve. layer and the diamagnets are on the either sides of the paramagnets. Movies of these colloidal flowers, sandwiched structures, decorated strings were captured using a Leica camera (Leica DFC 360 FX). 21 2.2. METHOD CHAPTER 2. MATERIALS AND METHOD 22 Chapter 3 Dynamics of self-assembly of flower-shaped magnetic colloidal clusters 23 CHAPTER 3. COLLOIDAL FLOWER Dynamics of self-assembly of flower-shaped magnetic colloidal clusters A. Ray, S. Aliaskarisohi, and T. M. Fischer, Phys. Rev. E 82, 031406 (2010) Copyright by The American Physical Society 2010 DOI: 10.1140/epje/i2008-10421-5 24 Dynamics of self-assembly of flower-shaped magnetic colloidal clusters A. Ray, S. Aliaskarisohi, and T. M. Fischer* Institute of Physics, Universität Bayreuth, Bayreuth 95440, Germany 共Received 11 May 2010; published 24 September 2010兲 In a static magnetic field paramagnetic and nonmagnetic colloids immersed in a ferrofluid self-assemble into fluctuating colloidal flowers. Adsorption and desorption of nonmagnetic petals to larger paramagnetic cores and changes in the petal conformation around the paramagnetic core induce a fluctuating dynamics. We track the motion of colloidal petals on the paramagnetic core. Adsorption and desorption of petals occur on a larger time scale than the rotational diffusion of the petals. Magnetic dipole interactions split the motion of the petals into different modes of rotational diffusion. Modes of rotational diffusion that change the petal conformation are suppressed compared to the conformation invariant rotational diffusion of all petals. The suppression of higher modes of rotational diffusion results in a subdiffusive dynamics of the individual petals. DOI: 10.1103/PhysRevE.82.031406 PACS number共s兲: 82.70.Dd I. INTRODUCTION Colloidal assemblies are mesoscopic systems in thermodynamic equilibrium. Understanding the complex structures of these assemblies, the soft interactions between the individual particles, and the resultant dynamics in real space is of current interest; because colloidal assemblies are being used as models for atomic crystals 关1兴for glasses 关2兴, for van der Waals crystals 关3兴, and as systems for the study of dynamic self-assembly 关4,5兴. The softness of the interactions gives rise to fluctuations around the equilibrium that allows observing directly the transport processes 关6–8兴which lead to the dynamic self-assembly of the system. Diffusion is considered as one of these basic passive means for irreversible transport into equilibrium. It arises from fluctuations of the particle velocity due to stochastic forces. These forces act on the diffusing particles due to collisions with other particles from a reservoir at a certain temperature. In the presence of stochastic and deterministic microscopic forces, macroscopic diffusion can be expressed as the zeroth moment of the particle velocity autocorrelation and/or cross-correlation functions 关9兴. Kubo 关9兴extended a generalized concept of diffusion that allows defining and measuring the diffusion of interacting particles. It has been shown by Erb et al. 关5兴that paramagnetic and nonmagnetic colloidal particles immersed in a ferrofluid can self-assemble into colloidal flowers in a static magnetic field. The colloidal flowers result from the effective dipolar attraction of the paramagnetic colloids in which nonmagnetic particles behave as magnetic holes in the ferrofluidic background. The dipole interaction is a tensorial traceless interaction that depends on the angle between the magnetic moments and the particle separation. For holes sitting at the pole positions above or below the paramagnetic bead the dipole interaction with the paramagnetic bead is repulsive. In the equatorial plane on the other hand it is attractive. The dipole interaction between two magnetic holes on the other hand is repulsive in the plane normal to the magnetic moments and attractive along the direction of the magnetic moments. The planar structure of the colloidal flowers is a result of the complex angular dependency of the dipolar interactions. Here, an attempt has been made to measure the normal modes of diffusion, as well as the adsorption and desorption kinetics of the petals in colloidal flowers using the concept proposed by Kubo 关9兴. Kubo generalized the concept of diffusions for situations where the particle kinetics is a superposition of random motion and directed interactions that force the particles into deterministic directions. The interactions correlate the motion of the particles that would otherwise show a degenerate individual diffusion. The correlations split the individual diffusion into statistically independent normal modes of diffusion. It is demonstrated that the adsorption and desorption kinetics as well as the mode dependence of the normal modes of petal diffusion can be understood by the competition of dipolar forces with the fluctuating forces from the viscous carrier fluid. II. EXPERIMENT We study the superparamagnetic Dynabeads M-270 carboxylic acid, 2.8 ␮ m in diameter 共Cat. No. 143.05 D兲obtained from Invitrogen Dynal 共Oslo, Norway兲, and FluroMax red fluorescent polymer microsphere beads with 1.0 ␮ m diameter 共Cat. No. R0100兲obtained from Duke Scientific 共Palo Alto, CA兲. The particles from Dynal are supplied in concentrations of approximately 2⫻109beads ml−1 共10–30 mg ml−1兲and from Fluro-Max supplied with concentration of approximately 1% volume fraction suspended in water and respective surfactant. Paramagnetic particles are mixed with nonmagnetic particles and diluted ferrofluid EMG 705 FerroTec Ferrosound 共FerroTec GmbH, Germany兲with controlled proportions depending on the experiment. Electric current of 0.43 A was supplied to the water-cooled coils to produce a magnetic field of 10.0 mT, machined at University of Bayreuth. The mixture of the beads with ferrofluids was taken on a precleaned glass slide with a cover slip to reduce the air drift. Static magnetic field from the zdirection was applied to the sample and was observed under the LEICA DM4000B 共Leica Microsystems Wetzlar GmbH, Germany兲fluorescence microscope through 63⫻polarization lens in reflecting mode. Videos were cap- *[email protected] PHYSICAL REVIEW E 82, 031406 共2010兲 1539-3755/2010/82共3兲/031406共6兲©2010 The American Physical Society031406-1 tured using a color charge-coupled device Basler camera 共Basler A311fc兲high frame rate from Basler AG, Germany. III. ADSORPTION AND DESORPTION Nonmagnetic beads of radius a=0.5 ␮ m in a diluted aqueous ferrofluid 共EMG 705 Ferrotec Ferrosound/water =1:4兲adsorb at and desorb from the paramagnetic beads of radius R=1.4 ␮ m. When they adsorb they form a colloidal flower with one paramagnetic bead at the core of the flower surrounded by several nonmagnetic beads forming the petals. A typical colloidal flower is depicted in Fig. 1. The assembly is a dynamic structure and the number of petals N共t兲fluctuates as a function of time because nonmagnetic beads adsorb at and desorb from the paramagnetic core. If we assume a Boltzmann distribution for the number of petals we may extract the potential energy of adsorption of Nbeads U共N兲as U共N兲−U共Nref兲=−kBTln 冉 t共N兲 t共Nref兲 冊 ,共1兲 where t共N兲denotes the total time when one finds the colloidal flower with Npetals, Nref denotes a reference number of petals, and Tis the temperature. In Fig. 2we plot the adsorption potential as a function of the number of petals obtained via Eq. 共1兲by measuring N共t兲over a time duration of 4000 video frames. The adsorption potential shows a pronounced minimum near six petals. Assuming the potential to arise via dipolar attraction of the nonmagnetic beads to the paramagnetic core and due to dipolar repulsion between the equally spaced nonmagnetic petals, we predict a potential of U共N兲=4 ␲ ␮ 0 ␹ F 2H2a3 9共R/a+1兲3N 冋 − 冉 ␹ p ␹ F −1 冊 R3 a3 +1 2兺 j=1 N−1 1 8 sin3共j ␲ /N兲 册 .共2兲 In Eq. 共2兲 ␮ 0denotes the vacuum permeability, ␹ Fand ␹ pare the effective susceptibilities of the ferrofluid and of the paramagnetic particle, and His the external magnetic field. The potential has a minimum for an equilibrium number of particles given approximately by Neq =2 ␲ 冑3冑 ␹ p ␹ F −1R3/2 a3/2.共3兲 The dashed line in Fig. 2shows a fit of the experimental data 共solid line兲obtained from Eq. 共1兲to the theoretical prediction in Eq. 共2兲using ␹ P=0.082 and ␹ F=0.063. Note that the theoretical fit exhibits a minimum around N=7 instead of the value N=6 in the experiment. The 2N-dimensional conformational space of the petals is spanned by the positions 共rj, ␸ j,j=1,...,N兲of the petals. In an N-fold colloidal flower the equilibrium configuration is determined by the conformation rj=R+aand ␸ j=2 ␲ j/N共j=1,...,N兲. A transition to a 共N−1兲-fold flower happens when, for example, the Nth petal separates from the flower 共rN→⬁兲and the remaining N−1 petals rearrange their angular positions ␸ j共j=1,...,N−1兲. We describe the reaction pathway of such a conformational change by the reaction coordinate ⌬r. The position of the Nth petal is rN=R+a+⌬rN, ␸ N=0 and the other beads adapt the positions rj=R+a, ␸ j= ␣ 共⌬rN兲+2关 ␲ − ␣ 共⌬rN兲兴共j−1兲/共N−2兲. The angle 2 ␣ 共⌬rN兲describes the angle between the first and the 共N−1兲th petals that readjust 关from ␣ =2 ␲ /Nto ␣ = ␲ /共N−1兲兴, while the Nth petal leaves the flower 共see top in Fig. 3兲. We compute the reaction pathway such that the remaining petals j=1,...,N−1 adjust their positions to the energy minimum of the dipolar energy of the Npetal system while the Nth petal is fixed at the position rN=R+a+⌬rN. Usually no significant changes in energy are computed when the separation ⌬rNof the leaving petal has exceeded ⌬rN⬎4 ␮ m. Hence, separations larger than 4 ␮ m can be considered as quasi-infinite separations. In Fig. 3we plot the dipolar energy versus the reaction coordinates ⌬rN共N=3,...,11兲for a cascade of transitions from an 11fold colloidal flower toward a flower with two petals. The cascade from the 11-folded flower to the theoretical minimum flower with seven petals is plotted on the left side. The remaining cascade from the minimum sevenfold flower toward a two-petal flower is plotted at the right. The reaction coordinates alternate between the lower 共even N兲and upper FIG. 1. 共Color online兲共a兲Fluorescence microscope image of a six-petaled colloidal flower and 共b兲scheme of a colloidal flower. The paramagnetic core particle is nonfluorescent and hence not visible in the fluorescence image. The nonmagnetic fluorescence petal particles are visualized as bright spots in the fluorescence microscope image. FIG. 2. Adsorption potential of the colloidal petals. The solid line is obtained from the experimental data by using Eq. 共1兲. This potential levels off near 5kBTdue to lack of events. The dashed line is a fit according to Eq. 共2兲. RAY, ALIASKARISOHI, AND FISCHER PHYSICAL REVIEW E 82, 031406 共2010兲 031406-2 共odd N兲axes. Numbers indicate equilibrium flowers of the corresponding number of petals. The potential thus changes from the Npetal flower energy ENto the 共N−1兲petal flower energy EN−1. The potential of a Npetal flower with the Nth petal at a distance ⌬r=5 ␮ m is indistinguishable from the potential energy of a 共N−1兲-petaled flower. This confirms that a petal at a distance ⌬r⬎5 ␮ m can be considered as fully separated from the flower. For the desorption of the seventh petal the energy exhibits a maximum EAalong the reaction pathway. This maximum corresponds to a transition state, i.e., a saddle point in conformational space located at a distance ⌬r7,max⬇0.7 ␮ m from the minimum position of the seventh petal with an activation barrier of the desorption of 共EA−E7兲⬇0.7kBT. The activation energy for the adsorption is 共EA−E6兲⬇0.5kBT. A qualitatively similar transition state is computed between the sevenand eight-petaled flowers. All other transitions in the number of petals show no transition state. Hence, all flowers with N⬍6 and N⬎8 are unstable. The sixand eight-petaled flowers are metastable E6,E8⬎0, and the sevenfold flower is the stable conformation E7=0 for the given parameter set. Assuming an Arrhenius behavior for the rate constant k6→7of the adsorption process of the seventh petal one would expect a rate constant of the order k6→7=kBT 6 ␲ ␩ a共⌬rmax兲−2exp关−共EA−E6兲/kBT兴,共4兲 where ␩ =10−3 Nsm −2 is the ferrofluid viscosity. Inserting the values ⌬rmax⬇0.7 ␮ m and 共EA−E6兲⬇0.5kBTfrom Fig. 3into Eq. 共4兲we obtain k6→7⬇0.3 s−1. In Fig. 4we plot the autocorrelation function of the petal number, 具 ␦ N共t兲 ␦ N共t+ ␶ 兲典,共5兲 where ␦ N共t兲=N共t兲−Neq denotes the petal number fluctuation. The autocorrelation function decays with a typical rate of kex⬇0.3 s−1 in good agreement with the estimate given by Eq. 共4兲. For larger times ␶ ⬎10 s the experimental autocorrelation function becomes statistically unreliable since the number of events 共⬀ ␶ meas- ␶ 兲drops to 1 as the time separation ␶ approaches the time ␶ meas of the measurement. IV. PETAL CONFORMATION AND DYNAMICS Once the petals adsorb to the paramagnetic core there is some freedom of conformation, and one observes flowers with petals equally spaced around the core as well as conformations where the petals are crowded at one side of the core. We define the one-dimensional density of particles as ␳ =N/⌬ ␾ ,共6兲 where ⌬ ␾ denotes the minimum angular range over which the Npetals are distributed and 2 ␲ −⌬ ␾ is the largest gap FIG. 3. 共Color online兲共Top兲Scheme of a N-petaled flower losing the Nth petal along the reaction coordinate ⌬rN, while the angular positions of the remaining petals adjust. 共Bottom兲The potential-energy cascade from a 11-petaled flower via the stable VII petal flower 共left兲toward a two-level flower 共right兲. The flower loses the Nth petal along the reaction coordinate ⌬rN; black curves correspond to the desorption of a N=even petal 共lower abscissa兲, and green 共gray兲curves correspond to the desorption of a N=odd petal 共upper abscissa兲. The energy of a petal separated by ⌬rN=5 ␮ m is indistinguishable from an infinitely separated petal and hence equals to the energy of a 共N−1兲-petaled flower. The numbers labeling the ends of the curves correspond to the number of the petals in the flower. The transition state between sixfold and sevenfold petal flowers 关red 共black兲arrow兴is at a distance of ⌬r=0.7 ␮ m from the equilibrium position of the seventh petal and has an activation energy of EA=0.7kBT. FIG. 4. 共Color online兲The autocorrelation function 具 ␦ N共t兲 ␦ N共t+ ␶ 兲典 versus time as obtained from the experimental data 共solid line兲. The number of petals changes on a time scale of 3 s. The dashed line corresponds to an exponential decay with rate constant 0.3 s−1. The statistical error 共error bars兲of the correlation function increases when the time lag ␶ approaches the time of measurement ␶ meas=70 s. DYNAMICS OF SELF-ASSEMBLY OF FLOWER-SHAPED …PHYSICAL REVIEW E 82, 031406 共2010兲 031406-3 between the petals. We compute the potential energy of a conformation U共 ␳ 兲as U共 ␳ 兲−U共 ␳ ref兲=−kBTln 冉 g共 ␳ ref兲⌬ ␳ reft共 ␳ ,⌬ ␳ 兲 g共 ␳ 兲⌬ ␳ t共 ␳ ref,⌬ ␳ ref兲 冊 ,共7兲 where t共 ␳ ,⌬ ␳ 兲is the total time when the petals in the flower show a density in the interval 关 ␳ , ␳ +⌬ ␳ 兴and where g共 ␳ 兲⬀ 冉 N ␳ −N ␳ hc 冊 N−2 共8兲 is the leading-order approximation for the configurational space density 关10兴available for conformations of density ␳ , whereas ␳ hc=共R/a+1兲/2 is the maximum 共hard-core兲packing density of the petals around the core. Figure 5shows the potential U共 ␳ 兲computed via Eq. 共7兲for flowers consisting of an arbitrary number of petals. The resolution ⌬ ␳ varies with ␳ and is chosen in a way so as to ensure that t共 ␳ ,⌬ ␳ 兲⬎0 for all ␳ . Since the data at higher potential are sparse the resolution 1/⌬ ␳ is best at the minimum and decreases when moving toward higher potential. We find the lowest potential for densities ␳ ⬇1 corresponding to a hexagonal arrangement of the petals with equal spacing of ␲ /3 between the petals. The petal conformation results from the simultaneous minimization of the petal number and the minimization of the dipolar repulsion between the petals. The dipolar repulsion between the petals, however, is weak and allows for significant fluctuations around a conformation. We therefore tracked the angular position ␾ j共t兲关j=1,2,3,...,N共t兲兴 of the adsorbed petals as a function of time. The accuracy of the tracking of ␾ j共t兲was better than 2°. The angular frequency ␻ j共t兲= ␾ ˙j共t兲of each individual petal is a fluctuating function of time. We measure the angular frequency using finite differences of the angular positions of consecutive frames. The frame rate of the camera was 30 frames per second. We define the autocorrelation function of the angular frequency of two petals of a colloidal flower with Npetals as CN共 ␴ , ␶ 兲=具 ␻ j共t兲 ␻ j⫾ ␴ 共t+ ␶ 兲 ␦ „N共t兲−N… ␦ „N共t+ ␶ 兲−N…典. 共9兲 Here, ␴ denotes the neighbor number 共 ␴ =0 is the same particle, ␴ =1 is the nearest neighbor, etc.兲. Both delta functions ␦ (N共t兲−N)and ␦ (N共t+ ␶ 兲−N)discard all times where the petal number deviates from the fixed petal number N from the correlation. In Fig. 6we plot C6共 ␴ , ␶ 兲versus ␶ for ␴ =0,1,2,3. The angular frequencies are correlated for zero time delay 共i.e., ␶ =0兲, showing that part of the petal diffusion can be considered as a Markovian process on the time scale ␶ ⬎0.03 s of the measurement. The most prominent observation is that neighboring petals are not statistically independent. As does the petal autocorrelation function C6共0, ␶ 兲, the petal cross-correlation functions C6共 ␴ ⫽0, ␶ 兲also show the same albeit weaker instantaneous positive correlation. This is a dynamic proof of the deterministic interaction of the petals. Apart from this positive correlation a weak anticorrelated decay is observed for the autocorrelation C6共0, ␶ 兲and the cross correlation C6共 ␴ ⫽0, ␶ 兲for ␶ ⬎0.05 s 共see the inset in Fig. 6兲. It is a measure for the retardation of the interaction. In single file diffusion 关11–13兴, where particles interact only via hard-core repulsion, a strong algebraic anticorrelation significantly alters the diffusion of the particles. Neighboring particles in single file diffusion remain uncorrelated at short times and become anticorrelated only at times typical for the individual diffusion time needed to encounter each other. The retardation of such a hard-core interaction is significant. Single file diffusion becomes most prominent in the thermodynamic limit N→⬁, where the time scale of the simultaneous correlated diffusion of the rigid flower separates from the individual diffusion of the petals. Our system differs from a system exhibiting single file diffusion. It has a small number of petals, and the petals interact instantaneously via the soft dipolar interactions; retardation effects are weak. In no time are the petals allowed to diffuse individually. Hence, the relatively weak delayed anticorrelation follows the instantaneous delta correlation with a relative short delay. The diffusion constant of the petals is given by half the area under the autocorrelation funcFIG. 5. Effective petal potential as a function of the petal density ␳ as obtained from the experimental data via Eq. 共7兲. The dashed line is a linear fit. FIG. 6. 共Color online兲Angular frequency autocorrelation and cross-correlation functions for a colloidal flower with six petals. The black line corresponds to the autocorrelation, while the red, blue, and green lines correspond to cross correlations between nearest 共 ␴ =1兲, second-nearest 共 ␴ =2兲, and third-nearest 共 ␴ =3兲neighbors, respectively. RAY, ALIASKARISOHI, AND FISCHER PHYSICAL REVIEW E 82, 031406 共2010兲 031406-4 tion. While the finite frame rate of the camera broadens the experimental correlation function, the area under the correlation function is not affected by the convolution of the data with the time resolution function of the camera. Hence, the diffusion constants have no significant dependence on the frame rate of recording, DN共 ␴ 兲= 冕 0 ⬁ d ␶ CN共 ␴ , ␶ 兲.共10兲 Equation 共10兲is Kubo’s 关9兴generalization of the concept of diffusion to particles that interact. The interaction of the particles causes the motion of one particle to statistically depend on the motion of another. The statistically dependent motion of the particles can be decomposed into statistically independent normal modes of motion. In Fig. 7we plot the diffusion constant D6共 ␴ 兲versus ␴ . The petals behave like being coupled by soft springs, with petals not diffusing independently, but with neighbors performing a correlated diffusion. The correlation decreases when moving away toward further distant neighbors. We may decompose the correlated motion of the petals into uncorrelated normal modes of diffusion via ␾ 共m,t兲=1 冑N兺 j=1 N e2 ␲ imj/N ␾ j共t兲.共11兲 The corresponding statistically independent diffusion constants of the normal modes, DN共m兲=1 N兺 ␴ =1 N e2 ␲ im ␴ /NDN共 ␴ 兲,共12兲 are plotted in Fig. 8. The mode m=0 has the highest diffusion constant, and the diffusion constant decreases with the mode number m. The mode m=0 corresponds to a rigid rotation of all petals by the same amount. It therefore corresponds to the rotational diffusion of the entire flower that leaves the conformation of the flower unchanged. The higher modes m⬎0 involve relative motion of petals that change the conformation. Such modes are suppressed to diffuse by the dipolar repulsion between the petals. The higher is m, the shorter is the distance 2 ␲ /mbetween petals that are moving in opposite directions. The most likely conformation is an equilibrium conformation such that an m⫽0 mode usually raises the dipolar energy of the system. This explains why the diffusion of higher modes 兩m兩⬎0 is suppressed by the dipole-dipole interaction. Contrary to single file diffusion the diffusion mode of the petals arises from mostly instantaneous response of the flower to conformational changes. In single file diffusion the suppression of higher modes arises from a retarded response to conformational changes that only sets in when one petal diffuses to its neighbor and encounters its hard-core repulsion. In conclusion we have characterized the dynamic fluctuations of magnetic colloidal flowers. These fluctuations can be understood as a result of deterministic forces arising due to dipolar interactions and statistical forces arising from the collisions of the embedding fluid. The soft character of the dipolar interactions places this system between that of a free system and a system interacting via hard-core interactions. The soft confinement of the particles leads to a modedependent diffusion that differs from single file diffusion. The desorption and adsorption of the petals can be understood as activated processes. The colloidal flowers are thus a two-dimensional model system for the dynamics of more complex three-dimensional colloidal assemblies such as Pickering emulsions 关14兴and colloidosomes 关15兴. FIG. 7. Diffusion constant D6共 ␴ 兲versus ␴ . FIG. 8. Normal-mode diffusion constants D6共m兲versus the mode number m. DYNAMICS OF SELF-ASSEMBLY OF FLOWER-SHAPED …PHYSICAL REVIEW E 82, 031406 共2010兲 031406-5 Eur. Phys. J. E (2012) 35:17 Page 3 of 6 Fig. 1. a)-b) Reflection polarization —respectively, fluorescence— microscope image of a colloidal flower consisting of a paramagnetic (non-fluorescent) core of diameter 2a1=2.8µm in an aqueous diluted ferrofluid (EMG707 : H2O = 20 : 80) surrounded by an isotropic ring of diamagnets of diameter a) 2a2=3.1µm and b) 2a2=1.0µm. The images were obtained in a normal field of ˆ H⊥=7mT. c)-e) Fluorescence microscope images of isotropic clusters of diamagnets of diameter c) 2a1=3.1µmand2a2=3.1µm, d) 2a1=3.1µmand2a2=2.0µmande)2a1=9.9µmand 2a2=3.1µm, immersed into an undiluted ferrofluid (EMG 707). The clusters were assembled in an in-plane rotating field of ˆ H=1.62 mT at a precession angular frequency of Ω= 188 s−1. The scale bar in all images corresponds to 3 µm. The movie in the supporting information shows the rotating clusters under the in-plane field of ˆ H=1.62 mT and two different normal fields with a precession angle close and far from the magic angle. fluorescence or reflection microscopy, LEICA DM5000 (Leica Microsystems Wetzlar GmbH, Germany). For colloidal clusters non-magnetic particles with different size diameters were immersed in undiluted ferrofluid EMG 707 sandwiched between two cover slips. The cover was then subjected to a rotating magnetic field where isotropic colloidal clusters are formed. Then a static magnetic field normal to the film was superposed to the rotating in-plane field and the dynamics of the clusters were observed under the fluorescence microscope. The field direction of the magnetic field changes from the air into the ferrofluid film according to ˆ Hferrofluid ⊥= ˆ Hair ⊥/(1 + χF)and ˆ Hferrofluid =ˆ Hair ⊥, and the precession angle in the ferrofluid and in the air are related via tan ϑferrofluid =(1+χF) tan ϑair.χFdenotes the magnetic susceptibility of the ferrofluid. All external fields and external precession angles are given in terms of their values inside the ferrofluid. 3Results Isotropic colloidal flowers were assembled in a static magnetic field normal to the sample consisting of a mixture of paramagnetic and non-magnetic particles dispersed in a Fig. 2. (Colour on-line) Hysteresis loops of the formation and rupture of colloidal flowers and of diamagnetic clusters as a function of the static normal field ˆ H⊥. The colloidal flowers consisted of a paramagnetic (non-fluorescent) core of diameter 2a1=2.8µm in an aqueous diluted ferrofluid (EMG 707 : H2O = 20 : 80) surrounded by an isotropic ring of diamagnets of diameter 2a2=1µm in a rotating field of ˆ H=1.62 mT at a precession angular frequency of Ω= 188 s−1. Blue upward triangles correspond to increasing the normal field and pink downward triangles to decreasing normal field. The diamagnetic clusters consisted of core particles of diameter 2a1= 3.1µm and petals of diameter 2a2=3.1µm immersed in an aqueous undiluted ferrofluid (EMG 707). The rotating in-plane field strength and frequency were the same as for the colloidal flowers. Red circles are measured upon increasing and the green squares upon decreasing the normal field. The inset shows the same hysteresis loops in terms of the precession angle. diluted ferrofluid. It has been shown [11,12] that with the proper dilution the magnetic susceptibility can be tuned to prefer a number of diamagnetic petals absorbing at the magnetic core corresponding to a full monolayer of petals around the core. Such kinds of isotropic colloidal flowers are displayed in fig. 1a)-b). Clusters of a bidisperse (radii a1and a2) mixture of effective diamagnets in a ferrofluid were formed in an in-plane rotating magnetic field. The diamagnetic clusters formed are planar clusters lying in the mid plane of the ferrofluid sample having a rich variety of conformations with different numbers of diamagnets forming one clusters. Amongst this variety we picked out clusters having a core formed by a bead of radius a1surrounded by a complete monolayer of beads with radius a2. Examples of such isotropic diamagnetic clusters are shown in fig. 1c)-e). Both types of clusters were exposed to a precessing magnetic field being a superposition of a rotating magnetic field H(t)= ˆ H[excos Ωt +eysin Ωt] in the plane of the ferrofluid film and a static field H⊥(t)= ˆ H⊥ez. The precession angle is defined by the ratio of this two components of the field via tan ϑ=ˆ H⊥/ˆ H. The external fields reported are those in the ferrofluid film far away from the clusters. In fig. 2 we show the stability of such Page 4 of 6 Eur. Phys. J. E (2012) 35:17 Fig. 3. (Colour on-line) Dependence of the width of the hysteresis loop of the formation and rupture of colloidal flowers (red) and diamagnetic clusters (blue) on the ratio of the core radius and the petal radius. The red and blue lines are fits according to eq. (8). clusters as we sweep the normal component ˆ H⊥of the precessing field. Colloidal flowers are stable for low precession angles (large normal field ˆ H⊥) while clusters of holes are stable at large precession angles (small normal field ˆ H⊥). Decreasing the normal component of the field destabilizes the colloidal flowers and they fall apart at a critical field ˆ H⊥c1. If we start the experiment at ˆ H⊥=0 one observes a mixture of magnetic hole clusters and paramagnetic beads. Colloidal flowers form from this mixture upon surmounting a second threshold ˆ H⊥c2>ˆ H⊥c1.We characterize the width of this hystersis by the difference of the two critical fields ∆ˆ H⊥=ˆ H⊥c2−ˆ H⊥c1. The width of the hystersis ∆ˆ H⊥is a measure of how strongly the transition is of first order. A similar hysteresis is measured when diassembling diamagnetic clusters by increasing the normal component ˆ H⊥of the precessing field and reassembling a cluster of a generically different shape and size when decreasing the field. The strength of the transition both for the colloidal flowers as well as for the magnetic hole clusters depends on the size ratio a1/a2of the colloids of the core and of the petals. In fig. 3 we plot the width of the hystersis ∆ˆ H⊥ versus the size ratio a1/a2. The width of the hystersis increases with the size ratio. The rotating parallel component and the contrast of the imaginary part of the magnetic susceptibility ∆χ′′ of the cluster to the surrounding ferrofluid result in a torque τ=4πµ0∆χ′′Vˆ H2sin2ϑ. Here µ0is the vacuum permeability, Vdenotes the volume of the cluster, and ˆ His the absolute value of the magnetic field. This torque causes the clusters to rotate around their core with an angular frequency ω<Ω. The ratio Fig. 4. The angular velocity of different colloidal flowers and diamagnetic clusters as a function of the precession angle recorded at a constant in-plane rotating field of ˆ H=1.62 mT at a constant precession angular frequency of Ω= 188 s−1. Fig. 5. The angular velocity ratio of different colloidal flowers and diamagnetic clusters near and far from the magic angle as a function of the ratio of the core to the petal radii. ω/ ˆ H2sin2ϑmeasures how efficient the magnetic field rotation is converted into a rotation of the cluster. In fig. 4 we plot the angular frequency ωat fixed in-plane field strength and frequency as a function of the precession angle ϑfor different clusters. Some of the clusters show a speeding up when one approaches the magic angle [13], where the clusters fall apart. Other clusters do not change their angular frequency when changing the normal component of the field. We characterize the cluster speed up by the ratio ωfast/ωslow, where ωfast denotes the angular frequency just before rupture and ωslow is the angular frequency at low (high) precession angle where the colloidal flower (magnetic hole cluster) is stable. Eur. Phys. J. E (2012) 35:17 Page 5 of 6 Fig. 6. The angular velocity ratio of different colloidal flowers and diamagnetic clusters near and far from the magic angle as a function of width of the hysteresis. In fig. 5 we plot the cluster speed up versus the size ratio a1/a2of the colloids of the core and of the petals. The speed up decreases with the size ratio for both the colloidal flowers and for the magnetic hole clusters. Figures 3 and 5 show that both the width of the hystereses and the speed up of the rotation correlate with the ratio of the core-to-the-petal radius acore/apetal.Wemay combine figs. 3 and 5 to measure the speed up as a function of the strength of the first order transition. Hence, in fig. 6 we plot the cluster speed up versus the width of the hysteresis. A large speed up is observed for small hysteresis while no speed up occurs at large hysteresis. 4 Discussion If we consider the core particle to be larger than the particles in the ring it is a good approximation to describe the local magnetic field as that in the absence of the petal particles. Toussaint et al. [14] have shown that image dipoles due to the presence of the ferrofluid glass walls can cause a first-order transition with two stable distances between the diamagnets. Here those effects are neglected since the sample thickness is much larger than the separation of the petals from the core. Neglecting the image dipoles, the field from the core is described by H=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ I+a3 1(χc−χF) 1+χc+2(1+χF) 3rr−r2I r5·Hext,for r>a 1, 3(1+χF) 1+χc+2(1+χF)Hext,for r<a 1, (1) where Idenotes the unit tensor, and χcdenotes the susceptibility of the core particle of radius a1. The effective magnetic moment of the core and petal particles mc=Vc(χc−χF)H(r=0)andmP=VP(χP−χF)H(r= a1+r2) are thus determined by the local field at the particle positions r=0andr=a1+r2, the volumes Vcand VPof the particles and the susceptibility contrasts to the ferrofluid. The dipolar interaction hence reads W=−µ0 4πmc·3rr −r2I r5·mP(2) =−γHext ·3rr −r2I r5+a3 1(χc−χF) 1+χc+ 2(1 + χF) ×3rr −r2I r52·Hext,(3) where γ=µ0 4πVPVc 3(1 + χF) 1+χc+ 2(1 + χF)(χc−χF)(χP−χF) (4) and ris the separation vector between the core and petal particle. The first term in (3) corresponds to the interaction of the petal particle in the unperturbed external field and the second term is the perturbation of the magnetic moment of the petal particle due to the presence of the core particle. For an external field Hext = Hext(sin ϑext[excos Ωt+eysin Ωt]+cos ϑextez)andapetal particle sitting in the equatorial plane at a distance r= a1+r2the time-averaged dipole interaction energy reads W=1 2 γH2 ext (a1+r2)31+ β (1 + r2/a1)3 ×P2(cos ϑext)−4β β+(1+r2/a1)3,(5) where β=(χc−χF) 1+χc+ 2(1 + χF).(6) The first term in (5) corresponds to a renormalized long-range dipole interaction that scales with second Legendre polynomial P2(cos ϑext) of the precession angle ϑext and switches sign when passing the magic angle. This part of the interaction is attractive if (χc−χF)(χP− χF)P2(cos ϑext)<0 and explains the stability of the colloidal flowers (χc−χF>0,χ P−χF<0,P 2(cos ϑext)>0) for small precession angles ϑext <ϑ magic and the stability of the diamagnetic clusters (χc−χF<0,χ P−χF< 0,P 2(cos ϑext)<0) for large precession angles ϑext > ϑmagic. The second term is independent of the precession angle. Its sign does not depend on the sign of the susceptibility contrast sign(χc−χF) of the core particle to the ferrofluid. The second term is repulsive for petal particles that are magnetic holes (χP−χF<0), while for paramagnetic particles it is attractive. The destabilizing correction term is short range. This results in an equilibrium distance of the petal from the core given by r2,min =a13 4β P2(cos ϑext)−β−1,(7) that moves from infinity at the magic angle ϑext =ϑmagic toward the hard-core distance a2as one moves away from Page 6 of 6 Eur. Phys. J. E (2012) 35:17 the magic angle. The picture changes when the dipole interactions between the petals are taken into account as well. Here the different range of both interactions becomes important when summing up the interaction of all petal particles. We expect that in a cluster of Nparticles that the dipole interaction increases with the number of pairs of particles that scales as N2, while the short-range correction increases with the number of nearest neighbor particles that scales like N. This explains the hystereses since once a cluster is formed it can be stabilized by the longrange dipole interactions even when a single pair of particles is not yet stable. The minimum radius of Npetals will hence be different from that of one petal described by eq. (7). We expect the hystereses to roughly scale with the ratio of the long-range to short-range interactions such that ∆H ∝1/(β+(1+a2/a1)3).(8) In fig. 3 we have incorporated curves according to eq. (8) with the prefactor of eq. (8) fitted to the data. The fit agrees well for the colloidal cluster but is less accurate for the colloidal flowers. This is not too suprising since the different susceptibility of the core of the flower adds to the complexity of the phenomenon. The width of the hystereses is a measure for the strength of the first-order transitions. If the transition is weakly first order, some of the second-order critical phenomena are likely to persist. This is what we observe in the critical speeding up. For a second-order transition we would expect the rotation speed of the cluster to diverge. For a weakly first-order transition there is significant increase when approaching the transition, while no significant increase is observed when the transition is strongly first order. The interaction between particles at the magic angle in an isotropic environment vanishes. Some interaction will persist if the larger size of the core renders the environment anisotropic. The self-consistent deviation of the system from isotropic is what stabilizes or destabilizes the particular conformation and renders the transition from second to first order. It is therefore conceivable that the presence of a large core particle is responsible for the strong first-order type of transitions in the clusters with a large core. The corresponding second-order speeding up of the rotation of the cluster is destroyed by the large core and partially persists for smaller core sizes. 5 Conclusions The rotation of colloidal clusters of non-magnetic holes and of mixtures of paramagnetic beads with non-magnetic holes in a ferrofluid in a precessing external magnetic field depends on the precession angle of the external field that serves as a control parameter for the stability of the clusters. Near the magic angle cluster-shape–dependent depolarization fields cause an orientation of the local field deviating from the external field and render cluster transitions weakly or strongly first order. If the transition is weakly first order a critical speeding up of the cluster rotation is observed. No speeding up occurs for strongly firstorder cluster transitions with hysteresis. The strength of the first-order transition is larger the larger the size of the core as compared to the petal particles of the cluster. This work is supported by the German Science Foundation within the cluster of excellence SFB840. Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. References 1. P. Pieranski, Contemp. Phys. 24, 25 (1983). 2. A. vanBlaaderen, R. Ruel, P. Wiltzius, Nature 385, 321 (1997). 3. James E. Martin, Eugene Venturini, Gerald L. Gulley, Jonathan Williamson, Phys. Rev. E 69, 021508 (2004). 4. N. Casic, S. Schreiber, P. Tierno, W. Zimmermann, Th.M. Fischer, EPL 90, 58001 (2010). 5. A.P. Gast, C.F. Zukoski, Adv. Colloid Interface Sci. 30, 153 (1989). 6. N. Osterman, I. Poberaj, J. Dobnikar, D. Frenkel, P. Ziherl, D. Babic, Phys. Rev. Lett. 103, 228301 (2009). 7. G. Helgesen, P.O. Pieranski, A.T. Skjeltorp, Phys. Rev. A 42, 7271 (1990). 8. E.R. Andrew, A. Bradbury, R.G. Eades, Nature 182, 1659 (1958). 9. J. Cernak, G. Helgesen, A.T. Skjeltorp, Phys. Rev. E 70, 031504 (2004). 10. R.M. Erb, H.S. Son, B. Samanta, V.M. Rotello, B.B. Yellen, Nature 457, 999 (2009). 11. K.H. Li, B.B. Yellen, Appl. Phys. Lett. 97, 083105 (2010). 12. A. Ray, S. Aliaskarisohi, Th.M. Fischer, Phys. Rev. E 82, 031406 (2010). 13. P. Tierno, R.M. Muruganathan, Th.M. Fischer, Phys. Rev. Lett. 98, 028301 (2007). 14. R. Toussaint, J. Akselvoll, G. Helgesen, A.T. Skjeltorp, Phys. Rev. E 69, 011407 (2004). CHAPTER 4. TRANSITION STRENGTH 40 Chapter 5 Magnetic field controlled composite paramagnetic-diamagnetic colloidal phases 41 CHAPTER 5. COLLOIDAL PHASES Magnetic field controlled composite paramagnetic-diamagnetic colloidal phases A. Ray, and Th. M. Fischer, Submitted to The Journal of Physical Chemistry B 42 Magnetic field controlled composite paramagnetic-diamagnetic colloidal phases A. Ray, and Th. M. Fischer( Institut für Experimentalphysik, Universität Bayreuth, 95440 Bayreuth, Germany. Abstract We report on differently ordered colloidal phases of a mixture of paramagnetic and diamagnetic colloids subject to a quickly varying time dependent magnetic field. Effectively paramagnetic and effectively diamagnetic colloids are created from paramagnetic and nonmagnetic colloids immersed into a thin film of aqueous ferrofluid. The time averaged dyadic product of the magnetic field with itself serves as a control parameter for a sequence of transitions between differently correlated orientation order between the paramagnetic and diamagnetic colloids. We observe antiand equimagnetic order along directions that are orthogonal to each other. At the magic angle equimagnetic and antimagnetic directions of the colloidal order change via an intervening biaxial ordered phase to a phase were the equimagnetic ordered direction is replaced by an antimagnetic ordering and vice versa. email: [email protected] 1. Introduction Neutralization of opposite charges is one of the driving concepts leading to the organization of matter on the atomic and molecular scale. The interaction between point charges is isotropic and does not depend on direction. It is spontaneous breaking of rotational symmetry and the quantization of angular momentum that nevertheless produces crystalline structures with directed bonds in atoms and molecules. Colloidal particles have been used as a model for atoms on a larger scale1,2. They however are of mesoscopic size, where quantum phenomena are absent, and angular momentum is a continuous quantity. The principles leading to directed bonds in microscopic systems therefore do not work on the colloidal scale. In isotropically interacting colloids steric interactions are the means to spontaneously break rotation symmetry and form a colloidal crystal3. The only possibility of obtaining directed bonds in colloidal systems is by using colloidal particles that are intrinsically anisotropic. For this reason chemists have synthesized Janus particles4,5 and patchy colloids6,7 with surface functionalities that vary as a function of the location on the particle surface. Other possibilities are the use of ellipsoidal particles8-10 the shape of which is different in different directions. A third possibility is to use a magnetic11 or electric12 dipole moment using an external magnetic or electric field. Induced paramagnetic dipoles do not neutralize in an external field but build up an induced magnetization with a macroscopic magnetic moment given by the magnetization of the sample times its volume. The situation changes when considering a mixture of paramagnetic and diamagnetic colloids13. Diamagnets and paramagnets point into opposite directions in the same field. They are able to neutralize each other on a macroscopic scale. In this sense, mixtures of paramagnets and diamagnets in an external magnetic field are a model system for neutralizing, anisotropically interacting particles that posses a variety of mesoscopic arrangements that is richer than that of isotropic colloids and than that of non-neutralizing anisotropic colloids. In the current manuscript, we show a few of the most obvious colloidal phases that form in such a system, when we apply a field varying on time-scales faster than the inter particle dynamics. 2. Experiment Figure 1 a) Scheme of the experimental setup. b) Scheme of the field modulation The field varies in time according to [ ] ttHHt yxextzext Ω+Ω+= 2sinsinsin ˆ cos ˆ )( eeeH ϑϑ with the tip of the magnetic field vector following the black Lissajou figure. This modulation produces the same time averaged dipole interactions as a field precessing at a precession angle of ext ϑ around the z-axis (green cone) but causes no net torque on the colloidal assembly. γ (t) denotes the angle between the particle separation vector rij and the field. Only the projection angles of the field ext ϑ and the bond b ϑ enter in the angular dependence of the time averaged interaction. Whether the interaction between the induced moments mi and mj is attractive or repulsive depends on whether ext ϑ and b ϑ are smaller and larger than the magic angle. The interaction also is proportional to the Ω 2Ω H ϑ ext a b c ϑ b γ ri j x y z mi m j glass ferrofluid Hair H ferrofluid diamagnets while polarization reflection microscopy images visualize the paramagnets. The right figure shows the corresponding angular dependence of the dipole interactions as explained in figure 2. The images in figure a) show the random arrangement in the absence of a magnetic field. Figures b-d are recorded in a magnetic field of ( ) mTHHH yx 82.12/ ˆˆ 22 || =+= and a frequency Ω =120s‐1. The vertical field (precession angle) in the images were b) Hzair=26.5mT ( ° = 10 ext ϑ ) c) Hzair =4.0mT ( °= 49 ext ϑ ) d) Hzair =2.22mT ( °= 64 ext ϑ ) e) Hzair =1.27mT ( ° = 75 ext ϑ ). We applied a field of the form equation 5 with an eccentricity of less than 5%. This ensures that the dyadic product of the magnetic field at two different times is a symmetric tensor ( 0HHHH = ′ − ′)()()()( tttt ), where the bar denotes the time average. As a consequence there is no net time averaged torque onto the colloidal structure14. In what follows we describe the assemblies of paramagnetic and diamagnetic particles as we increase the angle ϑext. Colloidal flowers In a static field = H ˆ21200 A/m, ϑext=0 normal to the ferrofluid film we observe the formation of colloidal flowers. Such flowers form due to the dipolar attraction of diamagnetic particles in the equatorial plane 2/ π ϑ = bof the paramagnets. They have been first discovered by Erb et al.13 They are highly dynamic structures where the petals of the flowers may diffuse15 and they can be easily set into rotation with time dependent magnetic fields having an asymmetric part in the dyadic product16. Figure 3b shows a fluorescence microscope image of such colloidal flowers with 2ap=2.8μm paramagnetic cores and 2ad=1.0μm petals. An ensemble of flowers can be seen via the fluorescent petals of the flower surrounding the non fluorescent paramagnetic cores. The flowers are located in the middle of the sample indicating that gravitation and image dipoles prevent the binding of paramagnetic beads into one dimensional strings with a diamagnetic mantle. Decorated strings Upon increasing the precession angle to ϑext =49° we observe the formation of paramagnetic strings undulating around the middle plane of the film with a period of three to five beads (figure 3c). The entire structure is decorated with a collection of diamagnets that horizontally adsorb to the undulating string at the sides of the string. The bonds between diamagnets and paramagnets in this structure are also in the horizontal plane but perpendicular to the bonds between the paramagnets in the string. A scheme of the decorated strings is shown to the right of figure 3c These strings correspond to the biaxial angular dependence of the dipolar interactions. Sandwiched membranes At precession angles of the order ϑext =64° the paramagnetic beads form membranes instead of strings. These paramagnetic membranes are sandwiched between two layers of diamagnets that adsorbed to the membrane on either side. At the transition angle ϑ=51° the orientation of the membrane normal is in the plane of the ferrofluid making the sandwich structure clearly visible in the fluorescence microscope image. The two diamagnetic adsorption layers appear as brightly fluorescing lines of diamagnetic beads sandwiching the non fluorescent paramagnets. Upon increasing the precession angle the membrane bends (figure 3d) such that part of the membrane normal remains in the horizontal direction while the normal to the lower part of the membrane now aligns with the film normal. Eventually upon further increasing the precession angle the membrane flattens and entirely lies in the film plane (figure 3e), allowing a closer inspection of the diamagnetic order of the absorbed layers. For all systems studied here the paramagnetic membrane is a close-packed two dimensional structure with a hexagonal unit cell with unit vectors having the length of a paramagnetic bead diameter 2ap. The order of the diamagnetic adsorbate on the contrary varies a lot and sensitively depends on the size of the diamagnetic beads, on the concentration ratio of diamagnets versus paramagnets and on the susceptibility of the diluted background ferrofluid. In what follows we describe the order of the diamagnetic adsorbate under various conditions. Paramagnetic crystal enslaved diamagnetic gas phase Figure 3b shows a superposition of a reflection microscopy image of the sample with a fluorescence microscope image of the same sample taken immediately one after another for a tilt angle of ϑext =π/2. The paramagnetic particles order into a series of planar clusters surrounded by regions that are completely depleted of paramagnetic colloids. Within the clusters a crystalline hexagonal arrangement of the paramagnetic beads is observed. The arrangement of the diamagnetic colloids is not completely 0,1 1 10 10-3 10-2 10-1 100 101 beyond unit cell diamagnetic gas diamagnetic enslaved crystal Δr2/A time [sec] within unit cell  Figure 4:Mean square displacement of the diamagnetic beads upon a cluster for a diamagnetic gas Hz=0.8mT, Ω =120s-1, H||=1.82mT, 2ad=1 μ m (black) and for an enslaved crystal Hz=1.01mT, Ω = 120s-1, H||=1.82mT, 2ad=2 μ m (orange). The shaded region corresponds to mean square displacements smaller than the paramagnetic unit cell size. uncorrelated to the paramagnets. Diamagnetic particles from the paramagnetic depleted regions adsorb on top and below the paramagnetic crystalline clusters. As a result the density of diamagnetic particles on top and below the clusters is larger than the density in the paramagnetic depleted regions. The paramagnetic crystal is sandwiched between two layers of diamagnetic gas. The diamagnetic particles perform Brownian motion, and the mean square displacement of the diamagnets increases linearly (figure 4) with a slope defining the gaseous diffusion constant of the diamagnets. The increase of the mean square displacement beyond the area of the unit cell of the paramagnetic crystal shows that the diamagnets remain mobile in this phase. For this reason we call this phase the paramagnetic crystal enslaved diamagnetic gas phase. This does not mean that the diamagnetic gas possesses no order. In figure 5 we plot the radial correlation functions ∑∫−−=Δ Δ+ ji jdid rr r d dd rdr N rrg , 2)( 2 1 )( rr δ and ∑∫−−=Δ Δ+ ji jpip rr r p pp rdr N rrg , 2)( 2 1 )( rr δ of the paramagnets and diamagnets, where Np and Nd are the number of paramagnets and diamagnets in a particular cluster and the ip rand jd rare the positions of the ith paramagnet and the jth diamagnet. While the long range behavior of both correlation functions is goverened by the shape of the cluster, the short range behavior shows that despite of the mobility of the diamagnetic gas, the crystal order of the paramagnet is imprinted upon the gas via the magnetic field modulations from the paramagnet. 110 0,0 0,5 1,0 1,5 paramagnetic sheet diamagnetic enslaved gas r (μm) g[r]  Figure 5 radial correlation function of paramagnetic particles (blue) and diamagnetic particles (red) in a diamagnetic gaseous phase cluster. Although the diamagnetic gas is mobile the crystal structure of the paramagnets is imprinted upon the diamagnets.  The auto-correlation-function of the diamagnets share the peaks occurring in the autocorrelation function of the paramagnets. Since the diameter of the diamagnets is much smaller than that of the paramagnets more than one diamagnet can reside on top and below one paramagnet. We observe a disorder in the occupancy number of the diamagnets of the sites above and below the paramagnetic crystals. A site can be vacant, or have one, two, three or four diamagnets on top of a paramagnet. This disorder is expressed by the substructure in the cross correlation function occurring in the hard core region of the auto correlation function. Paramagnetic crystal enslaved diamagnetic crystal phase Figure 6 top left) Polarization reflection microscope image of an enslaved crystalline phase of the diamagnets recorded atHz=1.01mT, Ω = 120s-1,H||=1.82mT, 2ad=2.0 μ m. The magnetic holes are sitting on top of the paramagnets as sketched in the scheme to the top right. The scheme at the bottom shows a side view with two of the frustrated bonds between the diamagnets shown in yellow. Upon increasing the radii of the diamagnets and upon diluting the ferrofluid we observe a slowing down of the large scale diffusion that eventually stops completely. For a bead diameter ad=2.0μm the diamagnets remain on top and below the paramagnetic particle they reside. In the plot of the meansquare displacement of the diamagnetic beads in figure 5, we observe a much weaker increase of the mean square displacement with time that eventually settles at roughly 1 percent of the area of a paramagnetic unit cell. According to the Lindemann criterion a crystal should melt when the root mean square displacements of its elements amounts for one tenths of the lattice spacing. We would hence expect a diamagnetic crystal to immediately melt under the current conditions. It is, however, not the interactions between the diamagnets but the interaction with the crystal potential of the paramagnets that causes the crystalline order of the diamagnets. The diamagnets are hence enslaved by the paramagnetic crystal and form two crystal layers growing epitaxial with the same unit cell on the paramagnetic crystals. Paramagnetic crystal incommensurate diamagnetic crystal phase For larger densities of the diamagnets and when using concentrated ferrofluids the attraction between the diamagnets overcomes the paramagnetic crystal potential and the diamagnets form close-packed hexagonal crystals on top and below the paramagnetic close-packed hexagonal crystal that has its own unit cell turned by 30 degees with respect to the paramagnetic unit cell. The close-packed cells of the paramagnetic and diamagnetic crystal layers have periodicities defined by the diameters of the paramagnetic and diamagnetic beads that generically are incommensurate. Figure 7 shows such an incommensurate crystal structure. Figure 7: fluorescence- (top left), and polarization reflection microscope image (top middle) of an incommensurate crystalline phase Hz=0mT, Ω =120s‐1,H||=1.82mT, 2ad=2.0 μ m(black). The top right picture shows a scheme of the packing of the paramagnets (red) and diamagnets (green). On the bottom we have a side view scheme of the incommensurate structure, where two paramagnetic diamagnetic bonds that are partially frustrated are shown in yellow. We also observe the formation of disordered structures when neither the interdiamagnetic interaction nor the interaction of the diamagnets with the paramagnets dominates. Under such circumstances diamagnets may form small close packed incommensurate clusters on top of the perfectly ordered paramagnet that follow the periodicity of the paramagnetic lattice on a larger scale. 4 Discussion The structure of the phases observed can all be understood by considering the time averaged dipolar interactions between the constituents (equation 7). Depending on the precession angle of the external field we expect paramagnets to bind to larger structures in bond angle directions that are attractive (violet in figure 2). In this way we obtain an assembly of paramagnets in the attractive bond directions equi b equi b ϑϕ ,that are all pointing with their magnetic moments in the same direction parallel to the external field. The order resembles a ferromagnetic ordering, however, the magnetic moments here are not permanent but are induced by the external field. We hence named the ordering an equimagnetic ordering. The time averaged dipole interaction between diamagnets behaves the same way creating a diamagnetic equimagnetic order with the diamagnetic moments all pointing antiparallel to the magnetic field. Bonds between diamagnets and paramagnets are attractive in bond directions anti b anti b ϑϕ , perpendicular to the equimagnetic bonddirections. In those orthogonal directions (orange bond directions in figure 2) we obtain an antimagnetic order of alternating paraand diamagnets that resembles a ferrimagnet, however, the alternating moments are induced moments not permanent moments. The entire order hence consists of opposite magnetic particles that assemble in an alternating induced antimagnetic sequence in one or two directions while the arrangement is equimagnetic in the remaining directions. Whether the antimagnetic ordering is in plane and the equimagnetic is normal to the film or the other way round is controlled by the precession angle ext ϑ of the external magnetic field. Antimagnetic equatorial ordering magic anti b ϑϑ >and equimagnetic polar magic equi b ϑϑ <ordering is supported by precession angles magicext ϑ ϑ < below the magic angle, while equimagnetic equatorial magic equi b ϑϑ >ordering and antimagnetic polar ordering magic anti b ϑϑ < is supported by angles magicext ϑ ϑ >. It is for this reason colloidal flowers form at magicext ϑ ϑ < while sandwich structures are stable for magicext ϑ ϑ >. When the precession angle of the magnetic field is near magic magicext ϑ ϑ ≈we are in the regime were biaxial ordering prevails with equimagnetic ordering along one equatorial direction magic equi b equi b ϑϑϕ >= ,0 and antimagnetic ordering along magic anti b anti b ϑϑπϕ >= ,2/ the other equatorial direction. At large precession angles we observe the equatorial equimagnetic ordering with crystalline packing of the paramagnets and different types of packing of the diamagnets. The gaseous and different crystalline diamagnetic structures are controlled by the strength of thermal fluctuations and the dipole interactions. Whether the dipole interaction between paramagnets or diamagnets or between diamagnets and paramagnets dominates can be controlled via the susceptibility contrasts that can be changed by diluting the ferrofluid, the size of the particles, and the volume fraction of both types of particles. Small particles are mobile and prefer gaseous phases, large particles are immobile. At low volume fractions of diamagnets d φ in a diluted ferrofluid (1<< F χ ) their interaction with the paramagnets is stronger ( 11 pF χχ ∝) than the interaction between them ( 2 F χ ∝). Each paramagnet binds one diamagnet to its northpole leaving diamagnetic bonds frustrated because the diamagnets are separated more than their close-packed distance. It is for such conditions where we observe the enslaved crystal phase. In concentrated ferrofluid at a high fraction of diamagnets each paramagnet in the membrane can bind more than one diamagnet, the diamagnetic dipole interaction becomes stronger, such that diamagnets form a close packed membrane above the paramagnets as well. As a draw back some of the diamagnets reside at positions with bond angles to the paramagnet that are suboptimal (figure 7 bottom). In this limit paramagnetic diamagnetic bonds are partially frustrated and incommensurate phases are observed. We can estimate the amount of neutralization between the paramagnets and diamagnets by the excess susceptibility [] ddpp ext excess eff H M φχφχχ Δ+Δ==Δ (8) , where p φ , and d φ are the volume fractions of paraand diamagnets. For our samples we had 0>Δ eff χ such that interactions between paramagnets dominate all other dipole interactions. They hence formed structures they also would have formed without the presence of the diamagnets. The diamagnets, however, had to accept the distorted structure of the magnetic field, generated by the paramagnets and arrange themselves accordingly. Presumably when using truly neutralizing mixtures 0≈Δ eff χ , the then more symmetric situation between paraand diamagnets would produce even more interesting superstructures. At present we do not have ferrofluids of sufficient magnetic susceptibility to test such fully dipolar neutralized superstructures. However, even without having explored the full parameter space of possible structures it is clear that the control of the different parameters in the dipole interaction of different particles as well as the control of the volume fraction of particles allows the construction a rich variety of phases in a mixed diamagnetic and paramagnetic system. 5 Conclusions Antimagnetically ordered colloidal phases with alternating arrangements of effectively diamagnetic and paramagnetic particles are formed in mixtures of paramagnetic and diamagnetic colloids immersed into a ferrofluid and subject to a quickly varying time dependent magnetic field. Depending on the mean orientation of the time averaged dyadic product of the external magnetic field the alternating order is observed in the plane of the film in form of colloidal flowers or normal to the film in the form of 2D paramagnetic crystals sandwiched between a diamagnetic gas or crystal. Near the magic angle eccentricity of the modulation creates also biaxial structures. The order of the diamagnetic sandwich layer depends on a subtle balance of parameters entering into the dipole interactions at work between the different particles 6 Acknowledgement We thank Thomas Friedrich for helping with susceptibility measurements of the ferrofluids. This work is supported by the German Science Foundation within the cluster of excellence SFB840. BIBLIOGRAPHY BIBLIOGRAPHY [6] Xia, Y. N. ; Gates, B. ; Li, Z. Y.: Self-assembly approaches to three-dimensional photonic crystals. In: Adv. 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