Auxiliary Material (EPAPS)
Dynamic Self-Assembly of Rings of Charged Metallic Spheres
Bartosz A. Grzybowski,* Jason A. Wiles, and George M. Whitesides*
Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street,
Cambridge, Massachusetts 02138
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Experimental Procedures. Stainless steel spheres (Type 316, 1 mm in diameter,
 = 7.972 g/cm3) were purchased from Small Parts (http://www.smallparts.com). Prior to
use, they were washed in methylene chloride and dried in an oven at 60 C for 2 h. The
spheres (251000) were placed on a flat surface made of a dielectric material. All
surfaces were washed with ethanol and dried in an oven overnight. Before each
experiment, the surfaces were thoroughly discharged using an electrostatic gun. Relative
humidity (RH) was monitored by a digital hygrometer (VWR).
The motions of the spheres depended on their sizes and magnetic properties. In
spheres made of Type 316 stainless steel and having diameters larger than 1.6 mm, the
induced magnetic moments were too weak to make them rotate under the influence of the
rotating external magnet; these spheres remained stationary on the PS surface. In smaller
spheres (0.4 mm in diameter, Type 316 stainless steel) and in spheres made of materials
of higher magnetic susceptibilities (chrome steel, Type 440C steel; sizes 0.41.6 mm),
the induced magnetic moments were large, and interacted with one another. These
spheres tended to aggregate into chain structures that rotated around their centers of mass.
The polarity of the generated charges was established by monitoring the deflection
of the spheres moving on a PS support between two concentric electrodes. These
electrodes were prepared by evaporating a 200-nm-thick layer of gold onto the bottom of
the Petri dish that had a paper ring glued onto it. When the paper ring was removed after
evaporation, the surface had two disjoint regions covered with gold (an inner, circular
patch, and an outer, ring-like patch) that constituted the electrodes.
Calculation of Magnetic Force. Consider a stationary, rectangular permanent
magnet in a Cartesian frame of reference, in which the x axis is along the longest
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dimension L~ 5.6 cm of the magnet, and the z-axis coincides with the axis of rotation of
the magnet; the magnet has the magnetization vector M along the x-axis. To calculate the
magnetic induction produced by the magnet, we use a standard current-sheet method.
The magnet is divided into slices of thickness dx; each slice produces a magnetic field
equivalent to that of a current i=Mcdx flowing in a loop enclosing the slice (c stands for

the velocity of light). Integration over all slices gives the total field B ( x, y, z ) at location
(x,y,z) (Eq. (S1)), where s is the distance between point (x,y,z) and a point on the path of
integration. The integral over the loop can be evaluated analytically, but the integration
along the x-direction has to be carried out numerically. A magnetic point-object located at
(x,y,z) experiences a magnetic force that has vector components proportional to the

B( x, y, z )  M

dl  sˆ
 loop s 2 dx
L / 2
L/2
(S1)
gradient of magnetic induction at this point. We approximated a metallic sphere by such a
point-like object, and calculated the gradient of the magnetic induction in the radial
direction r (pointing from the axis of rotation of the magnet towards the sphere) at the

level of the dielectric surface, Br(x,y,H)/r, where Br is the radial component of B .
Because, as we verified experimentally, the position of a sphere does not change
substantially during one revolution of the magnet, the average radially directed magnetic
force it experiences can be calculated as the time average of Br(x,y,H)/r over one
revolution of the magnet (Eq. (S2)). The dependence of the magnetic force on the radial
position r is shown in Fig. 1(b). The magnetic force is
2
Fm  Br ( x, y, H ) / r 
 B ( x, y, H , ) / r d
r
0
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(S2)
attractive (directed towards the center of the dish) within the annular region circle 15 mm
< r < 25 mm, and repulsive otherwise.
Computer Simulations of Pattern Formation. The executable program “Rings”
is attached. The code and the graphical interface were written in Visual Basic 5.0. The
user-specified input parameters (bold italic) to the program are described below.
The annular surface is represented as a rectangle with open boundary conditions,
so that the spheres leaving at one end of the surface reappear on the other end. This
surface is divided into X  Y square cells onto which M spheres are distributed. The size
of a cell in the graphical window of the program is specified by parameter c (in pixels).
The spheres are randomly distributed over the region of the surface specified by
parameters XL, XU, YL, YU. For example, setting XL = YL = 1, XU = X and YU = Y
gives spheres randomly distributed over the entire specified surface. The constant velocity
of the rolling spheres around the dish is defined by parameter v (usually 24). In addition,
the spheres readjust their positions in response to charges generated on other beads and
on the surface; the maximum value of readjustment in one simulation step is given by
parameter ms (usually 14). The cutoff radius of electrostatic interactions is specified by
R. Parameter B (optional) imposes a confining parabolic magnetic potential in the vertical
direction, so that the spheres experience a force attracting them towards the center (Int(Y1)/2) of the rectangular surface.
The electrostatic constant (dimensionless) is specified by Kel. Kinetics of charge
separation is adjusted by parameters kmag and k_ch. The charging curves can be
visualized (and adjusted) by pressing the Charging Curves button, which opens a
separate window in which charge vs. rate of charge transfer curves can be interactively
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generated. In this window, parameters Qs (surface charge), Qm (maximum charge on the
bead), K (same as kmag) and Kch are specified, and a plot of Q vs dQ is then generated
by pressing the Draw button (scale can be adjusted using the dQ scale parameter).
Once all the parameters are specified, the simulation is initiated in
Simulation/Initiate menu in the main graphics window. The number of simulation steps
can be adjusted interactively, that isthe user can run n1 steps, then another n2 steps and
so on. The two small graphical windows on the left show the charge per bead (upper
window), and the order parameter (lower the window) during the simulation. The order
parameter is defined as the number of beads that have the same vertical position; the
emergence of rings is accompanied by increase in the order parameter.
The charges on the spheres and on the surface are color-coded. Uncharged spheres
are black, and turn red as the spheres become positively-charged. Uncharged surface is
white and becomes dark with increasing negative charge. The results of the simulations
can be saved to a file: a bitmap image storing the image of the surface (File/Save), and a
text file (File/SaveGraphs) storing the values of the order parameter and charge per bead
(from the small, gray windows).
The graphs is Fig. 2(h) were generated using X=125, Y=20. M=200, c=67, v=4,
ms=1, B=0, R=3, k_ch=0.6, kmag=0.15, Kel=1, XL=YL=1, XU=125, YU=20, 1000
steps. Note that the location of the rings (and even their number) could vary between
simulations with the same values of input parameters.
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Comparison with the System of Planetary Rings. Although it might appear that our
system is analogous to the rings of the so-called giant planets (Saturn, Uranus, Jupiter and
Neptune), this analogy is only a geometric one, not one that shares common physical
underpinnings.
In short, the particles in the equatorial planetary rings collide with one
another. In this process, energy is lost, angular momentum redistributed, and the
rings tend to spread out radially and also to flatten (J.A. Burns, D.P. Hamilton,
M.R. Showalter, Sci. Am. February 2002, 64-73). Dense packing of the particles
in the rings gives rise to strong gravitational attractions between these particles –
it is hypothesized that gravitational effects resist the tendency to smear the
particles into perfectly circular bands. Further alterations to the shapes of the
rings are caused by irregularly shaped moons orbiting in close proximity to the
rings. These alterations can lead to discrete distributions of particles within the
rings (“beaded rings” or “libration sites”; P. Goldreich, S. Tremaine, Nature 277,
97-99, 1979, P. Goldreich, S. Tremaine, N. Borderies, Astron. J. 92, 490-494,
1986, C.C. Porco, Science 253, 995-1001, 1991) visually similar to the rings of
metal beads.
Despite visual analogies between the planetary rings and our system, there are
important differences that preclude quantitative (or even qualitative) correspondence:
(a) The particles in the planetary rings interact with one another via
attractive gravitational forces; the charged metallic beads in our system repel one
another by electrostatic forces.
(b) The external potentials acting in the two systems are very different.
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Figure 1 shows the radial profiles of the magnetic potential in our system (left)
and the gravitational potential around a planet (right).
(c) The rolling beads permanently modify the characteristics of the
support (environment) on which they move by laying down a track of charge on
the polymeric surface (Figure 1): When the positively charged beads are removed
from the dish, the negative charges in the polystyrene surface remain. Planetary
dust particles, on the other hand, do not permanently modify the gravitational
field acting on other particles: When dust particles are removed from a planetary
ring (i.e., when the ring is destroyed), there is no residual disturbance of a
planet’s gravity.
Figure 2. Permanent modification of the polymeric surface causes the system to
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evolve in response to its own history.
(d) The radii of our rings vary in experimentally indistinguishable
experiments involving the same numbers of beads, and depend on the
distribution of countercharges imprinted in the polymeric surface; this
distribution is highly sensitive to initial conditions. The radii of planetary rings
are determined by the balance between the central gravitational attraction and
centripetal forces acting on the dust particles. Thus, for given initial momenta,
dust particles always evolve into a ring with approximately the same radius
(“approximately,” since the fine details of the ring structure depend largely on the
interactions between the dust particles).
(e) No “moons” are necessary to evolve our system into rings having beads
equally spaced within each ring; all beads in our system have the same size.
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