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
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
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.