Two dimensional colloidal Systems

Colloidal systems are part of the field of Soft Condensed Matter. Under this concept several systems are summarized like emulsions, dispersions, fog, dust, but also more complex systems like blood, milk, or biological tissue. Generally it is a mixture of matter on a mesoscopic scale often in different states of aggregation. The adjective 'soft' is applied due to the fact that the interaction between objects on the mesoscopic scale is of the same order of magnitude like that between atoms or molecules in solid sate physics. Since the number of objects per length scale is 104-105 times smaller, the energy densities and therefore the elastic moduli of soft matter (scaling to energy per volume) are 1012-1015 times smaller compared to solid state physics. Therefore a much richer variety of excited states are often found at moderate (biological relevant) temperatures.

For fundamental research it is notably interesting to investigate colloidal suspensions. Microscopic particles of uniformly size and shape are dispersed in a solvent. The particles are small enough to make Brownian motion and have to be described as statistical ensemble but still large enough to be visualized optically. Since the dynamic scales inversely with the size all processes in the ensemble are observable on all relevant time scales. If one has a distance dependent interaction between the particles, different ordering phenomena are observable as function of density and interaction strength. Therefore colloidal systems are excellent model systems for collective phenomena like phase- or glass-transitions.

Phase transitions in 2-d and the hexatic phase (KTHNY-melting)

In our experiment we investigate an ensemble of (super-) paramagnetic colloids (a few micrometer in diameter) which are confined by gravity to a completely flat surface. This surface is made by the lower water/air interface of a hanging droplet which is suspended by surface tension in a cylindrical hole of a glass plate. Two dimensional means that the particles can move in the plane driven by brownian motion but excitations in vertical direction are of the order of 1/1000 of the diameter of the particles and can be neglected. Since the particles are paramagnetic one can introduce a dipolar repulsion between them by applying a magnetic field perpendicular to the 2d layer. If the magnetic field is high enough to overcome thermal motion the particles will arrest in a closed packed crystal which is a hexagonal one in two dimensions (Fig. 1).

Fig. 1: Video microscopy of a hexagonal crystal made by colloids. The magnetic field perpendicular to the 2d layer and therefore the magnetic repulsion between the particles is large enough to dominate thermal motion. The grid is a voronoi construction indicating that almost every particle has six nearest neighbours. The hexagonal crystal has discrete translational and six-fold orientational order.

For low magnetic fields thermal motion dominates the magnetic energy and the colloids are randomly distributed in the plane. The system shows continuous orientational and translational symmetry and appears as an two dimensional fluid. In between the crystalline and the isotropic fluid phase one finds a so called hexatic phase (which is unknown in 3d systems) with continuous translational symmetry but still a discrete orientational symmetry. Therefore it is a fluid phase but with a six-fold director field. Fig. 2 shows the structure factor of the three different phases indicating the different degrees of order in the system. In the hexatic phase the translational order is short range but the orientational one is quasi-long range. This melting sceanrio is driven by the thermal dissociation of two different kinds of topological defects. The theoretical description was given in the 1970ies, by Kosterlitz, Thoules, Halperin, Nelson, and Young and is therefore named KTHNY-theory.

Fig. 2: Structure-factor computed the particle positions a two dimensional colloidal system. from left to right: isotropic fluid, hexatic phase and crystalline phase. The system is visualized by video microscopy and digital image processing provides us with the trajectories of the particles on all relevant length and time scales.

Kibble-Zurek mechanism

At finite cooling rates, a system with continuous phase transition has to fall out of equilibrium: critical fluctuations of the order parameter become progressively slower in the vicinity of the transition temperature. If the time to reach the transition exceedes the critical correlation time, the dynamic is frozen out on long scales. This phenomenon is applies to several system with spntaneously broken symmetry ranging from the Higgs-field in the early universe, quantum fluids and colloidal monolayers. T. W. B. Kibble suggested defect structures (domain walls, strings, and monopoles) to appear during the expansion and cooling of the early universe. The lack of such defects within the visible horizon of the universe mainly motivated inflationary Big Bang theories. W. H. Zurek pointed out that the same principles are relevant within the laboratory when a system obeying a second-order phase transition is cooled at finite rates into the low symmetry phase. Using a colloidal system, the Kibble-Zurek mechanism can be visualized on single particle level on a background of an established microscopic melting formalism.

Fig. 3: The typical size the defects plotted as a function of the cooling rate. Blue lines are power law fits predicted by standard Kibble-Zurek mechanism. Red lines are numerical solutions adapting the Kibble-Zurek mechanism to KTHNY universality class.

2-d Glass transition

Glassy materials such as obsidian have been used by humankind for millions of years. Even so, there is still no theoretical description of glassy materials that is able to explain all of the phenomena observed during the process of vitrification (i.e., the formation of glasses). Even worse, there is no consensus on how to define exactly the glassy state; an open question concerns the essential versus sufficient features of a glass. A supercooled fluid is commonly called a glass if the viscosity exceeds a certain value, irrespective of whether the viscosity gets large but stays finite or diverges at the ideal glass transition. While viscosity refers to liquids, the shear elasticity contains complementary information from the solid state. Here, we demonstrate how an investigation of elastic excitations (phonons) provides further insights into the formation of amorphous solids. We employ microscopy data of a two-dimensional colloidal system containing hundred thousands of (roughly 2300) micron-sized colloidal particles. These colloidal particles are two different sizes, and they are all superparamagnetic such that their interactions can be controlled using an external magnetic field. Our data include video microscopy (obtained at two frames per second monitoring about 2300 particles with an optical resolution of 100 nanometers), and we are able to probe the particles' dynamics after they relax. We measure bulk and shear modulus; these elastic moduli should become discontinuous at the glass transition temperature. We test different system temperatures, and we find, as expected, that the dynamics of the colloidal particles decrease at lower (higher) temperatures. Setting the focus on acoustic spectroscopy we show that shear elasticity develops discontinuously from zero during vitrification.

Fig. 4: Temperature dependent bulk- and shear modulus. The data suggest a jump of moduli at the transition temperature, which is located at Γ = 195. The lower plot zooms in on the ordinate. In the solid, the moduli grow linearly due to increased pressure indicated by dashed lines and we extrapolate this linear behaviour down to the fluid. The inset shows a zoom of ordinate and abscissa: the data are not compatible with a continuous increase of elasticity as e.g. given by the dotted lines..