Most physical properties of dense colloidal suspensions are either determined or modified by the form of the interaction potential between the individual colloidal spheres. While present models for the pair potential have been used to achieve qualitative agreement between theoretical and observed phase diagrams and rheological properties, detailed measurements are needed to secure quantitative agreement and resolve questions of interpretation surrounding a variety of unexplained experimental phenomena. For example, the dependence of the pairwise interaction on temperature, volume fraction and boundary conditions are still being investigated. In practical applications, pairwise interaction measurements provide detailed information regarding the state of charge and local chemical environment of particles in suspension.

The effective pairwise interaction potential is encoded in the equilibrium distribution of particles in a suspension through the relation

where

is the two-particle correlation function and angle brackets indicate
an average over angles.
Static snapshots in principal can yield measurements of
provided that the suspension's concentration is
low enough to avoid many-body contributions[7].
This restriction was circumvented by Kepler and Fraden [6]
who augmented their imaging measurements of *g*(*r*) with
molecular dynamics simulations to correct for many-body effects.
Direct imaging avoids the problems encountered in efforts to
invert light scattering data to extract pair potentials
[23], since detector noise
contributes little to the error in estimating *g*(*r*).
Geometrically confining colloid to a single layer to
avoid complications from the poor depth resolution of
video microscopy, however, introduces wall-mediated interactions [6].

As with the measurements of diffusivity described above, we use blinking optical tweezers to facilitate measurements of the pair interaction in unconfined colloid. This technique allows us to collect all the necessary data with one pair of spheres located far from the sample container's glass walls and far from other particles. We align a pair of traps in the focal plane along a line parallel to the video lines and separated by no less than ten times the typical distance a free sphere can diffuse in 1/60 sec. Under these conditions, out-of-plane diffusion is sufficiently small that the three-dimensional center-to-center separation can be approximated by its projection into the plane.

Rather than measuring the equilibrium pair distribution *g*(*r*)
directly, we use optical tweezers to create reproducible initial
configurations away from which the pair of particles diffuse.
By considering the dynamics of interacting Brownian particles,
we can extract the equilibrium pair correlation function
from a collection of such trajectories.
Since the inertial damping
time for the system is much smaller than the time scales on which we
measure dynamics, we can approximate
the master equation for the dynamical pair
correlation function *g*(*r*,*t*) by

where is a Markovian propagator [24].
The steady-state solution to eqn. (22) is the equilibrium
pair distribution function, *g*(*r*).
After spatially
discretizing eqn. (22)
the discrete pair correlation function is the
nontrivial eigenvector of
a system of linear equations:

where is the transition probability matrix for a pair of particles initially separated by distance to be separated by distance 1/30 sec later and corresponds to .

In practice, we build up by binning pair trajectory
data according to the initial and final center-to-center separations
in each time step.
Each row of the counting data is normalized independently
to conserve probability in eqn. (23).
A typical example of measured in this manner is
shown in Fig. 7(a).
Details for calculating such matrices are published elsewhere [5].
With a chopper wheel set to turn off the tweezers
for six fields out of every twenty, we typically are able to
videotape 15,000 fields of data in less than half an hour.
To collect data at
a range of different pair separations, we adjust the spacing between the traps
every few minutes by slightly rotating a gimbal-mounted
mirror in the laser projection optics.
We programmed our frame grabber to selectively digitize
only those frames with the traps off by
monitoring each image's background brightness,
which decreases noticeably when the laser is chopped.
Once has been calculated from the trajectory
data, *g*(*r*) can be calculated by
solving eqn. (22).
Its logarithm is then an estimate of the interaction potential
in units of the thermal energy .
The pair potential for the propagator matrix shown in
Fig. 7(a) appears in Fig. 7(b).

The method we have outlined for measuring the effective pair interaction potential rests on very few assumptions regarding the nature of the interaction. We require, for example, non-potential interactions to be negligible to ensure history independence of the propagator. Interactions lacking spherical symmetry such as those between Brownian dipoles or charged ellipsoids also would require a more sophisticated analysis. With these caveats, however, we are free to interpret our results for charged spherical latices within the conventional DLVO theory [25] for colloidal interactions.

The electrostatic part of the DLVO potential resembles a Yukawa interaction

where is the effective charge for spheres of radius ,
is the Debye-Hückel screening length
and is the dielectric coefficient of the suspending fluid.
The screening length depends on the concentration *n* of *z*-valent
counterions and determines the range of the interaction.
The charge renormalization theory of Alexander et al. [26]
suggests that
a sphere's effective charge scales with its radius

where is known
as the Bjerrum length, and *C* is a constant predicted [26] to be
about 10.
The solid line in Fig. 7(b)
is a fit of eqns. (24) and (25)
to our measured interaction potential.
The extracted screening length nm
corresponds to an electrolyte concentration of about M, which
presumably is due to dissolved airborne CO .
The fit value of is consistent with the
values predicted by molecular dynamics simulations [27]
(*C*=7) and
neutron scattering measurements on micelles [28] (*C*=6).

We chose to fit our data to eqn. (24) as a test of the charge renormalization theory of constant surface charge. Fitting to other forms would allow estimation of quantities such as the zeta potential. Consequently, this technique might be considered complementary to electrophoretic surface potential measurements.

Our prototype interaction measurement system reports a reliable interaction curve complete with fits for quantities of interest in roughly 10 hours. Most of this time is spent acquiring and analyzing video frames under software control. Performing more of these operations with specialized yet readily available hardware would reduce the processing time to under 1 hour. This is comparable to the time required for more traditional colloidal characterization measurements. Unlike conventional techniques, moreover, digital video microscopy coupled with blinking optical tweezers can measure the interactions between a particular pair of particles rather than averaging over a sample.

Mon Mar 11 23:01:27 CST 1996