Dipolar interaction

The magnetic momentµis a vector quantity used to measure the tendency of an object to interact with an external magnetic field. The object’s intrinsic magnetic properties are often visualized as emanating from a tiny bar magnet with north and south poles and is therefore also called the magnetic dipole moment. The concept of the magnetic dipole is not restricted to the modeling of atomic-sized particles and can be applied to

1.8 Magnetism and the dipolar interaction

Figure 1.10: Magnetization curves of a ferromagnet (green), an exchange-bias film (red), and an ion-bombarded exchange-bias film (blue)

much larger objects and collections of objects. A compass needle and even the earth itself might be considered giant dipoles and also in our experiment, a rod which consists of several particles can be called dipole. When the field lines of two magnetic moments cross, a dipole-dipole interaction occurs. An alternative and more quantitatively useful definition of the magnetic moment is to model it as arising from a tiny currentI traveling around the edge of a loop of cross sectional area A. The magnetic dipole moment µis a vector defined asµ=IA whose direction is alongA.

The magnetic moment µ will seek to align with an externally applied magnetic field B0. It will experience a torqueτ given by the vector cross product τ =µ×B0. When perfectly aligned parallel to µ↑↑B0, µwill be in its lowest energy state and experience no torque. When pointing opposite to µ↑↓ B₀, µwill be in its highest energy state because extra energy would be required to move and maintain it in this position. For any other direction the energy E of the magnetic momentµ would be given by the vector dot product:E = −µ·B₀. In a viscous fluid subject to an external rotating field a the magnetic torque is balanced by a viscous torque and gives rise to synchronous or asynchronous dynamics of the magnetic moment [70].

When two identically paramagnetic spherical particles immersed in a viscous fluid are

Figure 1.11: Magnetic dipole modelled as a current loop

placed in an external magnetic field, the colloidal particles will experience the interaction between two magnetic dipoles induced by the magnetic field. In Fig. 1.13a), the dipoles are aligned parallel to the field and arranged in an end to end configuration. One pole on one colloid is attracted to the opposite pole on the neighboring colloid. In contrast, when the particles are aligned perpendicular to the magnetic field, as shown in Fig. 1.13b), the particle charges on the neighboring sides of the colloids lead to repulsion. These interactions tend to align the colloidal particles to increase their attraction and reduce their potential energy, thus forming hierarchical arrays of colloidal particles.

Magnetic-field-guided colloidal assembly routes have several common characteristics.

First, the assembly process is driven by magnetic dipole-dipole interactions which are directional in nature and can be either attractive or repulsive, depending on the relative configuration of the particles with respect to the applied field. Second strong magnetic interactions can be effectively and reversibly initiated by application of an external field, providing enough driving force for the rapid assembly of colloidal particles. There are generally two types of magnetic interactions experienced by colloidal particles in external magnetic fields, orienting from their permanent or induced dipole moments. The inter-particle dipole-dipole force describes the interaction of a dipole with the magnetic field induced by another dipole the packing force results from the gradient of the external magnetic field. When the inter particle dipole-dipole force is strong enough to overcome thermal fluctuations, the alignment of the dipolar particles along the direction of their magnetic moments is the direct result of the directional dipole-dipole force.

1.8 Magnetism and the dipolar interaction

Figure 1.12: (A) For any random direction, the energy (E) of the magnetic moment (µ) would beE =−µ·B₀ (B)B₀ aligned opposite toµ↑↓B₀ and it has the highest

energy state. (C) B₀ aligned parallel to µ↑↑B₀ and it has the lowest energy state.

Figure 1.13: (The magnetic moment of a paramagnetic particle with a dipole moment in the same direction as the external magnetic field causes a) attraction for particles separated along the direction of the field and b) repulsion for particles separated in a direction perpendicular to the magnetic field.

Chapter 2

Materials and Methods

A good experiment must be simple. It should be as straightforward as possible. However, the experiment is not theory and nature is not as kind as I would wish. It puts certain difficulties in the way of the experiment that one must eliminate prior to having success.

I have organized this chapter by first introducing the setup used for the experiment and then talking about the obstacles nature put in the way of my experiments and how I succeeded in eliminating those problems. Finally, I talk about obstacles that did not get out of my way but developed into the major topic of this thesis.

2.1 Setup

Figure 2.1: a) Picture of the experimental setup. The polarization microscope is equipped with a CCD camera on top and a set of coils on the slide table. b) Close up on the arrangement of coils. There is one coil for each main directionx-,y-, and z-direction.

The magnetic pattern is placed on top of the z coil

The experimental setup is shown in Fig. 2.1. The most important part is the polarization microscope DM2500P from Leica. We use this microscope to visualize our colloidal particles and the magnetic structures. We put three coils on top of the slide table to generate the external magnetic fields (2.1). We have a special arrangement of the coils

which I have shown in Fig. 2.1b to superpose time-dependent, homogeneous external magnetic fields to the heterogeneous field of the magnetic structures. One of three coils each generates the x-, the y-, and the z-component of H_ext(t). Due to the macroscopic dimension of the coils, we can in good approximation assume that the magnetic fields are homogeneous on the observed mesoscopic length scale, however, we need to place the pattern exactly in centre of the three middle axes of the x-, y-, andz-coils. To generate the time-dependent fields the coils are connected to three bi-polar amplifiers (Kepco BOP 20-50GL) (Fig. 2.2.b) that are fed by three channels of a wave generator (Aim-TTi TGA 1244) 2.2.a) with four channels.

Figure 2.2: a) The four channel wave generator (Aim-TTi TGA 1244) b) the bipolar amplifier (Kepco BOP 20-50GL)

The wave generator is capable of playing arbitrary waveforms that were beforehand created with a Matlab program. To convert the applied voltage into a defined field strength the coils are calibrated with a Gauss meter (LakeShore 410). Like this, it is possible to apply any desired, time-dependent modulation of the external magnetic field.

The magnetic patterns were placed directly on top of the coil. Then we put the colloids on top of our magnetic pattern with a pipette.

A CCD-camera (Leica DFC360 FX) was attached on top of the microscope (Fig. 2.1).

Together with the commercial software StreamPix, the dynamics could be recorded with a resolution of 1392×1040 at 20 frames per second. The particle trajectories were extracted from the videos using manually tracking options in ImageJ.