Colloids are particles or droplets dispersed in a continuous medium, e.g. blood, smoke, milk, or ink. Typical sizes range from100nm−10µm. If dispersed in a uid medium, the particles are subjected to Brownian motion. Colloidal suspensions belong to a class of materials often referred as Soft Matter. Further well known representatives are polymers, micro-emulsions, liquid crystals, or complex bio-materials [33]. The softness of these materials, as emerging from their small shear modulus, originates from their inherent mesoscopic length scales. The shear modulus (unit: energy/length3) of a colloidal crystal in comparison to an atomic crystal can be estimated by using the typical binding energies and the underlying length scales of the constituents [33]:
the binding energies do not dier signicantly (typically 1eV in atomic crystals, kBT ≈ 1/40eV for colloidal particles). However, the lattice constant of a colloidal crystal can be larger by a factor of10000. This results in a shear modulus of a colloidal system being smaller by up to twelve orders of magnitude corresponding quite well to experimentally observed values. Therefore, colloidal systems are extremely soft, and mechanical forces distort them easily.
Colloidal glass formers can be used as model systems to study the local phenomena of the glass formation in 2D and 3D. In experiments, colloidal glasses have the advantage over atomic systems that, besides statistical averaged information about the local structure, single particle resolution is provided by microscopy [11, 24, 34, 35].
A model system of a glass former in 3D is a suspension of colloidal hard spheres [6]:
with increasing packing fraction φ of the spheres, the viscosity is increased until the system becomes dynamically arrested at φ ≈ 58% without crystallization. The glassy dynamics has been widely studied using dynamic light scattering [6, 36, 37].
Additionally, with this model system it was possible to study many local phenomena by confocal microscopy like local structure and local dynamics [11, 35, 38].
In 2D, the binary mixture of colloidal dipoles investigated in this work is a good model system to study the glass transition [24, 26, 40, 39, 41, 42, 43]. It consists of two types of particles with dierent magnetic moments that are conned at a water-air interface by gravity. There, they are subjected to Brownian motion
in two dimensions. Trajectories of particles are obtained by video microscopy on the time and length scales relevant for the glass transition. The system exhibits all typical phenomenological features of a glass former, e.g. drastic increase of relaxation times for increasing interaction strength, no long-range order, and dynamic heterogeneities [24]. The dynamics of the system was compared with MCT and good agreement was found [26]. Partial clustering of small particles was observed [42, 44]
due to the negative nonadditivity of the dipolar binary potential. This leads to a heterogeneous distribution of particle composition which results locally in a coexisting variety of small areas with dierent underlying crystal structures. Thus, heterogenous distribution of small particles suppresses long-range order. Extended stable crystal structures for 2D binary dipoles were found in T = 0 lattice sum calculations [45, 46]
as well as by genetic algorithms [47]. Especially all locally ordered structures discussed in this work were predicted to be stable.
The idea that the disordered structure of this system is made up from particular substructures (triangular structures) was originally discussed in [39, 40]. However, in this work it is proposed that the local order originates from the tendency of the binary mixture towards crystallization.
Crystallization may be geometrically possible provided the relative concentration matches a certain crystal structure, and crystallization is not at odds with a glass transition in a binary system. A decrease in temperature can force a system into a dynamically arrested state due to the strong increase of viscosity before crystallization can establish long-range order. Since the small particles cannot reorganize fast enough, much disorder is 'frozen' in. As a consequence dierent competing crystalline structures appear, while the global structure remains amorphous.
Background
In the section 2.1 of this chapter, the basic experimental system of the 2D colloidal glass former is introduced. The concept of the colloidal monolayer at a water-air interface and the dipolar pair interaction between the colloidal particles are dened, based on the concept rst developed by K. Zahn et al. [48]. This introduction gives sucient background to understand the results of this work. The technical details concerning the experimental setup, sample preparation, and sample quality are explained separately in chapter 3.
A brief description of Mode-Coupling Theory (MCT) is given in section 2.2, one of the leading theoretical concepts to describe the dynamics of the glass transition. It is introduced here, because the dynamics of the system at hand is compared in chapter 4 with a simple glass forming model system of hard discs that was calculated using MCT.
Furthermore, this introduction is used to describe some of the general features of glass formers.
Section 2.3 describes a particularity of the interaction potential in the experimental system: The negative nonadditivity of the binary dipolar pair interaction. It becomes relevant in chapter 5 to explain the observed partial clustering of small particles. This phenomenon is responsible for the heterogeneous local concentration of small particles and therefore for the occurrence of dierent competing crystal structures.
Although the experimental system of binary 2D dipoles is never found in an equilib-rium crystalline state at low system temperatures, stable congurations at T = 0 are theoretically predicted. In fact, there are numerous of them, and they are introduced in section 2.4 to be compared with the local crystalline structures of the experimental system in section 5.2.
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2.1 2D system of binary colloidal dipoles
2.1.1 Pending water drop geometry
The system consists of a suspension of two kinds of spherical and super-paramagnetic colloidal particles A and B with dierent diameters (dA = 4.5µm, dB = 2.8µm) and magnetic susceptibilities (χA ≈ 10 · χB). Due to their high mass density of ρ ≈ 1.5g/cm3, they are conned by gravity to a water-air interface formed by a pending water drop suspended by surface tension in a top sealed cylindrical hole (6mm diameter, 1mm depth) in a glass plate. This basic setup is sketched in Figure 2.1.
A magnetic eld H is applied perpendicularly to the water-air interface inducing a magnetic moment M = χH in each particle leading to a repulsive dipole-dipole pair interaction.
The set of particles is visualized by video microscopy from below the sample and is recorded by an 8-bit CCD camera. The gray scale image of the particles is then analyzed in situ with a computer. The eld of view has a size of ≈1mm2 containing typically 3 × 103 particles, whereas the whole sample contains about up to 105 particles1. Standard image processing is performed to get size, number, and positions of the colloids. A computer controlled syringe, driven by a micro stage, controls the volume of the droplet to reach a completely at surface. To achieve a horizontal interface, the inclination of the whole experimental setup has to be aligned. This inclination is controlled actively by micro-stages with a resolution of ∆α ≈ 1µrad.
After typically several weeks of adjustment and equilibration this provides best equilibrium conditions for long-time stability. During data acquisition the images are analyzed with a frame rate down to 10Hz. Trajectories of all particles in the eld of view can be recorded over several days providing the whole phase space information.
The thermal activated 'out of plane' motion of the particles is expected to be in the range of a few tens of nanometer. Thus, the ensemble is considered as ideally two dimensional.
Information on all relevant time and length scales is available, an advantage com-pared to many other experimental systems. Furthermore, the pair interaction is not only known, but can also be directly controlled over a wide range. For all typical experimental particle distances the dipolar interaction is dominant compared to other interactions between particles like van der Waals forces or surface charges [49].
1These numbers are valid for the binary case. In a one-component sample much higher densities can be reached of up to 10000 particle in the eld of view.
H
air water glass cell in side view
T
x,y
Figure 2.1: Left: Super-paramagnetic colloidal particles conned at a water-air interface due to gravity. The curvature of the interface is actively controlled to be completely at, and the system is considered to be ideally two dimensional. A magnetic eld H perpendicular to the interface induces a magnetic moment Mi in each bead leading to a repulsive dipolar pair interaction. Right: Induced dipole moments Mi tilted with an angle θ with respect to the magnetic eld H. The potential is repulsive for θ > 54.7◦. For lower values of θ it is attractive.
2.1.2 Dipolar pair potential and interaction parameter Γ
The external magnetic eld H induces a dipole moment Mi = χiH in each particle indexed with i where χi is the individual susceptibility of a particle.
The dipolar interaction between two spheres is now regarded as depicted in the right sketch of Figure 2.1. The magnetic spheres are homogeneous, and therefore their magnetic eld induction B1(x) at distance x is that of a point dipole which is
B1(x) = µ0
4π
3(M1·x)x−(x·x)M1
x5 (2.1)
with the permeability of vacuum µ0. This is not only true asymptotically for large x but also close to the sphere as the geometry of a homogeneous magnetized sphere has no higher multipoles [50]. The interaction energyEmagn of two arbitrary oriented point dipoles M1 and M2 with connection vector r is given by
Emagn = −1
2B1·M2 (2.2)
= −µ0 8π
3(r·M1)(r·M2)−r2(M1·M2)
r5 . (2.3)
The factor 12 results from the fact that induced dipole moments are regarded [50].
Both induced dipole moments are parallel to the external eld, which simplies the expression to
Emagn = µ0M1M2 8π
(1−3 cos2θ)
r3 (2.4)
where θ is the angle between magnetic eld and connection vector r as indicated in Figure 2.1. The angular dependence of Emagn shows that the interaction potential is not always repulsive. It vanishes for θ ≈ 54.7◦, and for smaller angles the potential becomes attractive.
Assuming that the magnetic eld is always perpendicular to the connection vector r (particles are conned to a plane perpendicular to the eld), the expression further reduces to
Emagn = µ0
8π · M1M2
r3 . (2.5)
Counterpart of the potential energy Emagn is thermal energy kBT which generates Brownian motion. Thus, a dimensionless interaction parameter Γis introduced by the ratio of potential versus thermal energy: NA big and NB small particles and n is the area density of all particles. In this expression for Γ, the two dierent susceptibilities χA and χB are averaged to an eective susceptibility weighted with the relative concentration ξ. The magnetic energy is thus calculated as if the two particle species were identical with this average susceptibility2.
The denition of Γ is adjusted for reasons of tradition by several factors that are accumulated in the constant γ in equation 2.6: The factor of 12 was omitted andπ3/2 is added. Setting r =n−1/2 implies a square arrangement of dipoles in the plane. Other
2Another denition Γ?= n3/2k(χAH)2
BT for the interaction strength is often used where only the big particles are considered and the contribution of the small particles is neglected [42, 44, 51] (further-more, in this denition the factor of µ4π0 ·π2/3 is not included). For the used experimental parameters of χA/χB ≈0.1 and relative concentrations between 0 < ξ <0.5, the values of the denition used in this work only deviate 0−16% in comparison to the denition Γ? where the small particles are neglected.
crystalline patterns or an amorphous arrangement would lead to another prefactor3. Only the interaction between two neighboring particles is taken into account although the dipolar potential is long-range. Consideration of the interaction of all particles leads to a Madelung constant for a crystalline pattern and a corresponding factor for an amorphous arrangement.
Γ can be interpreted as an inverse temperature Tsys and is the thermodynamic control parameter of the system. The interaction strength is externally controlled only by means of the magnetic eld H. All other parameters in equation 2.6, as the thermal sample temperature T, particle density n, and magnetic susceptibilities χi, are kept constant during the experiment. The system temperature can be changed quasi instantaneously and homogenously over the whole sample due to the direct control via the external magnetic eld.