Crystalline Fluidized Beds
By using a solvent counterflow (i.e., fluidization) to prevent
sedimentation of charged silica spheres, we have established a
steady state polycrystalline colloidal suspension of uniform
concentration. With this technique we varied the steady state
concentration of a single sample by simply changing the flow
rate. Bragg scattering from the fluidized bed allowed for direct
investigation of its crystal structure, stability, uniformity,
and volume fraction, while the inter particle interactions were
directly measured from shear resonances of the bulk. For
strongly repulsive interactions we found that the particles
formed an FCC lattice and sedimented considerably slower than
when in a highly disordered state.
For the published paper see:
Crystalline Fluidized Beds, M.A. Rutgers, J.Z. Xue, E.
Herbolzheimer, W.B. Russel, and P.M. Chaikin, Physical Review E,
51: 4674-4678, 1995.
The experimental setup is as follows:
A fluid reservoir (a) supplies a
pressure drop across a glass capillary (b). The reservoir's height controls the
pressure and thereby the flow rate. The fluidized bed (c) consists of a
rectangular glass tube (1.2 cm X 0.2 cm X 10cm) with an alumina
membrane (d), permeable to fluid only, at the bottom. The liquid flows through
the colloidal suspension (e) and out of the cell (f) where it accumulates in a
long thin tube (g). The flow rate, measured from the rate at which the tube
fills, is typically less than 1 cc every 10 days.
This is a color photo of the colloidal crystals between (e) and (f)
in the above figure. The polycrystal is supported by the drag of
the fluid which is flowing upward at less than microns per second.
The visible iridescence is due to Bragg scattering of visible light
between the crystal planes of the 0.6 micrometer diameter silica
spheres. The volume fraction is about 20 percent. Click on the
picture for a larger jpeg.
The volume fraction dependence of the normalized sedimentation
velocity. This is the first measurement of the flow resistance of a
lattice of spheres. This arrangement if hydrodynamically unstable,
but slight electrostatic repulsions between the spheres stabilize the
lattice. The electrostatic forces are however not strong enough to
support the weight of the crystal. The upward fluid flow does just
that. (broken line) A particular simulation by Ladd for
disordered hard spheres and measurements (diamond, triangle) by Paulin
and Buscall respectively. (solid line) Smooth fit to exact
calculations of Zick and Homsy for an FCC lattice. (X) A measurement
of randomly stacked hard sphere crystals by Paulin . (dot) Our
measurement of an polycrystalline FCC colloidal crystal. (big circle)
The steady state disordered fluidized bed after the addition of
screening ions.
Our development of a colloidal fluidized bed has proven fruitful for
measuring several aspects of the relationship between suspension
microstructure and sedimentation velocity. An important
accomplishment has been the experimental verification of the
calculated settling velocity of an FCC crystal, for volume fractions
down to 0.2. The bed has offered unprecedented control over
suspension volume fraction and solvent properties, both of which were
used to induce changes in the microstructure. We have also shown the
technique useful for growing colloidal crystals in a system which
otherwise cannot support its own weight, and it simultaneously gave an
opportunity to measure the crystal's mechanical properties. The use
of a fluidized bed to study interacting particles has many advantages
that await further exploitation. At lower volume fractions we should
be able to see the unusual phi^(1/3) dependence of the sedimentation
velocity. At higher sedimentation rates we should see hydrodynamic
melting rather than thermal melting. By systematically altering the
salt concentration we can better study the suppression of the velocity
variance due to particle repulsions. Inverting the flow in a much
more strongly interacting system provides an easy way of studying the
osmotic pressure vs. $\phi$. Finally, we have found the technique
useful for purifying a bidisperse particle suspension.
This work was partially supported under NASA NAG3-1158.
Real Hard Spheres in the Laboratory.
Using a high resolution digital X-ray camera we have accurately
measured the density profiles of several sediments of highly
screened polystyrene colloidal spheres suspended in water. From
the integral of the profile with height we directly measured the
osmotic pressure as a function of volume fraction. The results
are in excellent agreement with calculations of the hard sphere
equation of state for both the crystalline and disordered states.
These results demonstrate experimentally that particles with a
hard sphere force law indeed exhibit the liquid solid phase
transition at the predicted volume fractions.
The soon to be published paper is:
Measurement of the Hard-Sphere Equation of State Using Screened
Charged Polystyrene Colloids, M. A. Rutgers, J. H. Dunsmuir,
J.-Z. Xue, W. B. Russel, and P. M. Chaikin, Scheduled for
publication in Phys. Rev. B, March 1996.
Equation of state for 0.720 micrometer core diameter polystyrene
spheres in a 6 milli molar HCl solution. Phi stands for the solid
volume fraction of spheres. (solid line) Calculated equation of state
for true hard spheres. The inset shows a photograph (approx. 1cm
tall) of the sample. (A) supernatant fluid, phi approx 0, (B) liquid
phase, 0 < phi < 0.494, (C) sharp interface, (D) crystalline phase,
0.545 < phi < 0.74.
The X-ray densitometry technique outlined in the publication (see
above) is of considerable value for measuring colloidal interactions.
It is nondestructive, is not sensitive to the optical properties of
the materials, and can have very high spatial resolution. The method
is not only effective for volume fractions up to close packing, but
measures particle interactions for large ranges of concentration in a
single measurement, with a single sample. Our first results using
this technique are in excellent agreement with the calculations for
the hard sphere equation of state and show that tightly screened
polystyrene spheres closely approximate real hard spheres. Moreover
they demonstrate that our particles exhibit the liquid-crystalline
solid transition at the volume fraction predicted theoretically.
This research was supported in part by NASA under
grant NAG3-1158.
As of now, hard spheres have already orbited the earth in the space shuttle and the MIR space station. Soon they will be aboard the ISS.
