The magnetic system has been studied for a long time and under a variety of differ-ent aspects. Outstanding among its properties are the long-time stability and the reversible external control of particle interactions. After briefly describing the setup, we will discuss the system’s properties and briefly mention some of the previous studies.
adjustable surface
micrometrical syringe
particle dipoles
B
capillary
Figure 1.1: The colloidal particles sediment to the bottom of a hanging water droplet. The water-air interface can be accurately kept flat with the help of a computer triggered micrometrical syringe. The particles interact with magnetic dipole moments which are induced by an external magnetic field~B.
Varying the strength of this external field allows us to easily and reversibly control the correlation strength in the system.
1.1.1 Setup
The general setup of themagnetic systemhas been described in great detail in the PhD thesis of Klaus Zahn [Zahn 97b]. Here, we would like to focus on three important aspects: the sample cell, the reversible control of correlation strength by an external magnetic field and the observation of the colloidal particles by video-microscopy.
Sample Cell
The most distinctive feature of the sample cell is its ‘hanging droplet’ geometry. A drop of the colloidal suspension is applied to a glass slide which is subsequently turned upside down. The relatively large (diameter σ = 4.7µm) and heavy (mass density 1.7kg/dm3) particles [Zahn 97a] quickly sediment to the bottom of the droplet (∼1min) and are subsequently confined by gravity to the water-air interface. Partic-ular care has been taken to ensure the flatness of this interface. Thus, the glass slide actually consist of two slides which are glued together. In one slide (thickness 1mm), two cylindrical holes have been drilled and they have been connected on the surface of the slide by a capillary. The larger hole (diameter 8mm), now covered on the top side by the second slide, is the probe compartment. Its sharp and perpendicular sides
circumvent any deformations at the edges of the droplet. The curvature at the bottom of the droplet is controlled by adjusting the water volume in the probe compartment via the connected second, smaller volume (diameter 4mm). There, a syringe is dipped into the suspension. The indirect regulation with the second volume had to be cho-sen because the tip of the syringe distorts the surface of the interface. The syringe is driven by a micrometrical motor which is triggered by a computer. By monitoring the conjoint drift of all particles, it is possible to initially adjust the surface to be almost perfectly horizontal. After this initial adjustment, the focus of the optical apparatus is slightly moved away from the interfacial plane. Hence, any vertical displacement of the interface results in a change of the apparent particle size. By triggering the syringe accordingly, any vertical deviation of the interface can be counterbalanced.
This way, the system can be kept stable for a long time because it is possible to com-pensate for evaporation of the solvent. See [Zahn 97b] for an extensive description or [Will 02] for a short overview.
Controlling Correlations
Particle interactions, and thus the correlation strength in the system, can reversibly be tuned by applying an external magnetic field. The polystyrene particles are doped with grains of ferromagnetic iron-oxide. Due to the allocation and the size of the grains (max. 30nm), the colloidal particles do not behave ferromagnetic but rather paramagnetic. Because they remain highly magnetisable with a susceptibility of the order ofχ ≈ 7.6×10−11Am2/T, they are called super-paramagnetic. A description of a precise determination of χ can be found in [Zahn 97a]. An external magnetic field ~Binduces a magnetic dipole moment M~ = χ~B in the particles, which start to repel each other∝B2/r3(see chapter 2.1). For our purposes, the strength of the mag-netic field is varied in the range of 0.1 . . . 1.0mT. As the magnetisation of the parti-cles shows no hysteresis, the interaction strength exclusively depends on the external field~B. One can therefore easily and reversibly switch between different interaction strengths. Measurements at different correlation strengths can be performed with one single system which consists of the very same particles for every set of data.
Observing Particles
The colloidal particles are observed with video-microscopy. Processing the images on the computer, the particles’ positions are extracted. The field of view has a size of 520µm × 440µm containing typically about 103 particles. An example is shown in fig. 1.2. The entire sample contains ∼ 105 particles. The time step between two successive pictures was chosen between 3 and 5s. Given a value D0 ' 0.1µm2/s of the self-diffusion coefficient, a time step corresponds to a mean displacement of
Figure 1.2: Snapshot of the paramagnetic colloidal particles at a typical density. The magnetic field has a medium intensity such that the system is well in the liquid phase.
the particles of ≈ 1µm, which approximately equals the lateral optical resolution.
As already mentioned, the focal plane of the microscope is on purpose slightly off the particle centre’s plane. The direction of any vertical shift of the interface can therefore be identified and compensated by triggering the micrometrical syringe (see the previous paragraph ‘sample cell’ and fig. 1.1).
1.1.2 Properties
Among the important properties of the magnetic system are its two-dimensionality and its phase-behaviour. We will present a short overview in the following.
Two-Dimensionality
Due to their iron oxide doping of approximately 10%, the particle’s density is rather large (mass density 1.7g/cm3) and they sediment relatively fast (∼ 6µm/s) to the bottom of the hanging droplet (compare fig. 1.1). Although the interface itself can be considered perfectly horizontal, there are some additional effects which may influ-ence the two-dimensionality.
Let us first consider the local deformation of the interface. The angleφbetween the horizontal and the solvent interface at particle contact depends on the gravitational forceFgand the forceFγdue to the surface tension. As the ratio ofFg/Fγis only of the
order of 10−6, the deformation angleφis of the order ofµrad and therefore negligible.
Let us now take a look at capillary waves. Given the size of our cell, we can ne-glect inertial waves. The wave vector spectrumζ(~q)is completely determined by the surface tensionγand is given by
h|ζˆ(~q)|2i= kBT
(2π)2γq2. (1.1)
The mean-squared amplitude of the interface is now calculated as hζ2i=
qmin is determined by the system size (qmin = 2π/16mm) andqmaxis limited by the shortest possible wavelength, which we assume to be 1mm. Thus, the mean ampli-tudep
hζ2iis of the order of 1nm, which is entirely negligible on the length scale of colloids.
Let us finally consider the vertical extent of Brownian motion. A vertical displace-ment of∆zincreases the potential energy bymg∆z. With the average thermal energy of 1kBTwe get a displacement of∆z =6nm at room temperature.
This last effect, the Brownian motion, is actually the largest of the three which we discussed above. Compared to the particle diameter σ = 4.7µm it is still small enough to consider this system basically an ideal two-dimensional system. The above estimates are all taken from [Will 02].
Phase Behaviour
With this experimental setup it has been possible to thoroughly study the melting of a two-dimensional crystal which differs substantially from the melting of a three-dimensional crystal. In two dimensions, contrary to the three-three-dimensional case, the density-density correlation function decays at large distances algebraically to zero.
The crystalline order in the systems on hand is thus quasi-long-ranged [Merm 66, Merm 68]. The melting process should therefore differ and it was proposed that there is a continuous transition mediated by the dissociation of dislocation pairs [Kost 73].
It was later shown that the dissociation led to the so-called hexatic phase and a sec-ond transition induced by the formation of disclinations is necessary to reach an isotropic liquid [Halp 78, Youn 79, Nels 79]. This melting process is usually referred to as KTHNY theory (Kosterlitz, Thouless, Halperin, Nelson, Young) and has indeed been observed using not only the decay behaviour of the static correlation functions [Zahn 99] but also the time-dependent correlation functions [Zahn 00]. For the sys-tem on hand, the liquid-hexatic phase transition occurs at Γ = 57 and the hexatic-solid atΓ=60.
The solid phase allows investigations of elastic properties of crystals. The shear modulus has been determined by observing the relaxation of a rotated triangle con-sisting of three neighbouring particles [Will 02]. Using Brownian fluctuations, it has been possible to determine the elastic properties [Zahn 03b], to measure the band structure and to compare it with the harmonic lattice theory [Keim 04] and to confirm in the KTHNY theory the limiting behaviour of the renormalised Young’s modulus when approaching the melting transition [Gr ¨un 04].