First published March 22, 1984. Research performed at Bell Labs.

     Abstract: Small metallized spheres (~l mm diameter) are free to move along the bottom plate of a plane-parallel capacitor. Applying high voltage to the plates charges each ball, and the resulting inter-ball repulsive potential leads to a number of striking effects. In particular, we discuss experiments on 2-dimensional crystallization, defect motion, a a vortex tube analog, and spontaneous symmetry breaking. This system is ideally suited for examining the behavior of a small number (1->10,000) of interacting particles, and could lead to a better understanding of liquid crystals, phase transitions, and the cross-over between thermodynamics and statistical mechanics.

     Introduction: We describe a new model system for the interaction of particles in two-dimensions which has the important features of versatility and relevance to many problems of current interest. Such model systems have played an important role in the evolution of physical ideas. For example, Ewing ⁽¹⁾ in 1890 built a square array of gimbled magnets from which he was the first to deduce the origin of hysteresis and magnetic remnance. Thus, models have directly stimulated new insights to old problems. They also give a tangible reality to an otherwise difficult to visualize process. For example, Bragg and Nye's elegant experiments on bubble rafts ⁽²⁾ gave credence to a decade's earlier work on dislocations in solids. Lastly, a carefully designed model can function as an analog computer., which surprisingly may be orders of magnitude faster than the most advanced digital computers. For all these reasons, model systems are an important complement to calculations and actual physical measurements..

     We describe a new type of model which relies on electrostatic, rather than magnetic^(3-5), capillary ⁽⁶⁾ or ionic ⁽⁷⁾ forces to couple the particles. The density, force law, synthetic temperature and boundary conditions all may be varied.  To illustrate the model's versatility, three experiments are discussed. More detailed, quantitative results are reserved for a future paper.

   Experimental Apparatus: The basic model consists of a plane-parallel capacitor (see Fig. 1). The lower plate is a hardened steel disk polished with 0.1 micron grit, while the upper plate is glass coated with a transparent conducting film. Metallized balls, 0.8 +- 0.002 mm in diameter, are the particles. Although nonmagnetic ball bearings are also suitable, we use ceramic balls coated with 1000 angstroms of gold to keep the ball mass low. As a result, they may be floated on a dense liquid or on an air film, rather than resting on the bottom plate. Also, a plausible synthetic temperature can be obtained by vibrating the capacitor cell with a speaker, light particles are more easily driven through a wide range of frequencies.

     A foam rubber wedge couples the speaker to the cell, while a micrometer stage is used for leveling. The speaker is driven by "white noise", or at a single frequency, usually 100 Hz. Variations in friction between the balls and the bottom electrode prevent the balls from moving synchronously with the driving frequency. The entire apparatus is designed to avoid mechanical resonances which would otherwise dominate the motion of the balls.

     High voltage applied to the plates induces an identical charge on each ball. The strength of the inter-ball force depends on the applied voltage and the particle separation, while the form of the force law depends on the ball diameter, plate spacing, and to a lesser extent, the number of balls in the cell. For example, if the plates are separated by a few ball diameters, the induced charge will cause an image charge in the bottom plate, and thus each ball will act as a dipole. For closer spacings, a second image charge in the top plate is relevant. In any case, a simple technique is used to measure the force law. A narrow trench is made in the bottom plate, and a number of balls are placed in the trench. One ball is glued in place some distance from one end of the trench. By slightly tilting the entire cell, the balls form a column with monotonically increasing spacing (see inset of Fig. 2). This spacing can then be compared to the form expected for a variety of force laws. A dipole-dipole force law (i.e. x ^-4) is clearly evident. From the weight of the balls and the angle, the "effective" dipole moment can be calculated; it corresponds to 109 electrons separated by two ball diameters.

     For many applications a repulsive potential is adequate. However, if a "Lennard-Jones" type potential is required, the balls may be floated on a denser liquid. In this case the force law is, for small distances, repulsive, and for large distances, attractive, by capillary action.

     To confine the balls in the cell a detachable metal rim is provided. Charge induced on this ring repels the balls, with a strength and spatial dependence determined by the rim's height and its radius of curvature.

   Experiments: The first experiment is the analog of vortex flux lattices, as seen in rotating superfluid helium ⁽⁸⁾. Instead of actual vortex lines, we use the balls as a model system. Although the force laws are different, and thus the dynamics will not be comparable, the statics should be similar. This is because the spacing between the balls is roughly the same regardless of the pattern, and thus the exact force law can be replaced by a Taylor series expansion around the mean spacing. Indeed, the position of the balls in the 20 patterns of Fig. 3a are extremely close to those calculated for helium vortex lattices ⁽⁹⁾]. Each of these patterns took approximately 30 seconds to anneal, and represent the most frequent "ground state" observed after five independent annealing sequences. The next most frequent states, three examples of which are shown in Fig. 3b, are metastable. As the temperature is raised this state hops to the ground state and, at high enough temperatures, back again. From the temperature and the transition rate the barrier height can be deduced.

     A second experiment is two-dimensional crystallization. Here, 1000 balls are placed in a larger cell and are annealed from above the melting point. With a 100 Hz driving frequency it takes approximatly 4 hours to anneal the last dislocation from the center of the crystal. This crystal can then be remelted either by increasing the driving frequncy (i.e., increasing the effective temperature) or by decreasing the inter-ball coupling (i.e., decreasing the external voltage).

     Qualitatively, one can see dislocations, point defects and grain boundaries move through the crystal. Unlike hard sphere models, there is significant space between the balls and thus the diffusion of particles is an important relaxation process. Quantitatively, the kinetic energy can be related to the driving amplitude, although an accurate temperature measurement will require the more sophisticated apparatus described in the conclusions.

     If two different types of balls are mixed, such as one metallic, and one dielectric, or two of dissimilar sizes, simple crystallization is not observed. Instead, depending on relative concentrations and temperatures, surface segregation, a super-lattice, or phase-separation have been observed. With a better control of the synthetic temperature, it should be possible to map a phase diagram for the two component system.

     In a third experiment a transition induced by boundary conditions is observed. Here, the rim of the cell is a hexagon (see Fig. 4). A perfect hexagonal crystal is expected for 37 or 61 balls which have, respectively, 4 or 5 balls on an edge. For an intermediate number of balls a defect usually appears in the lattice. However, for 48 or 44 balls, a lattice with no voids but reduced symmetry is formed. That is, although the boundary is six-fold symmetric, 48 balls form a three-fold symmetric crystal with a pattern of 4-5-4-5-4-5 balls on the edge, while 44 balls is two-fold symmetric, with a 4-4-5-4-4-5 pattern. This is a clear example of spontaneous symmetry breaking, arising as the simple solution to the complex problem of many interacting particles.

   Conclusions: Even at this early stage of development this model sheds light on a number of interesting phenomena. It can function as an analog computer, provide a foundation for building intuition on complex systems, and through its versatility, suggest new ways to solve old problems. A more advanced version, now under construction, has three important improvements. First, the bottom plate is porous. Air forced through the plate levitates the particles, reducing friction and allowing non-spherical particles to be used. For example, ellipsoidal disks could be used as a model for liquid crystals. Second, the rim is segmented and can be externally modulated in position. The rim then functions as an external heat bath, whose spectrum can be modified from the usual Boltzmann distribution. Lastly, a camera and image processor is used to determine the particle positions, and thus correlation functions and the kinetic and potential energy of the crystal can be determined. In a cell with a small number of particles there are wide fluctuations in the usual thermodynamic state variables, such as temperature or pressure. With this improved system, the small N limits to thermodynamics can be explored. Similarly, statistical mechanics has rarely been systematically tested on a small number of particles with varied interactions, and here again the versatility and speed of this model system should prove valuable.

     Acknowledgements: I thank Len Kopf and Ted Zacios for their valuable help in building the electrostatic cell, and L. Campbell for his comments on the vortex lattice analogy.