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Isolated Colloidal Pairs
Figure 2:
Left: Colloidal interactions are measured in a
hermetically sealed glass sample
container, shown in crosssection.
Two spheres are selected with optical tweezers and alternately
trapped and released, their motions being recorded at 1/30 sec
intervals as shown in the sequence of images.
Spheres appear brighter when trapped because of
light backscattered from the optical tweezers.
Right: Interaction potentials [21] for pairs of
polystyrene sulfate spheres in deionized water
at
. Curves are labelled by the spheres' radii.
Solid lines are nonlinear least squares fits to
Eq. (9). Dashed lines are fits
to the SogamiIse theory, Eq. (13).

Since its development, the DLVO theory has profoundly influenced
the study of macroionic systems.
Testing its predictions directly through measurements
on pairs of spheres has become
possible with the recent development of experimental techniques
capable of resolving colloids' delicate interactions without
disturbing them.
These fall into three
categories: (1) measurements based on the equilibrium structure
of low density suspensions
[vondermassen94,kepler94,carbajaltinoco96],
(2) measurements based on the
nonequilibrium trajectories [crocker94,crocker96a,dinsmore96,larsen97]
of spheres positioned and released
by optical tweezers [28]
and (3) measurements based on the dynamics of optically trapped spheres
[20,29].
Methods (1) and (2) take advantage of the Boltzmann relationship

(10) 
between the spheres' pair potential and their equilibrium
pair correlation function, which, for an ergodic sample and
can be evaluated as

(11) 
where
is the mean concentration of spheres in area
and the angle
brackets indicate an average over both time and angles.
The two approaches differ in how they measure
, but agree in their principal result:
isolated pairs of spheres, far from walls and unconfined by
their neighbors, repel each other as predicted by Eq. (9).
Nonequilibrium
optical tweezer measurements, such as the examples in Fig. 2.2
use a pair of optical traps to position two spheres
at reproducible separations in a microscope's
focal plane.
Extinguishing the traps eliminates any perturbing influence
of the intense optical field and frees the particles to move under
random thermal forces and their mutual interaction.
Their motions are captured in individual video fields
at
sec intervals and digitized.
Each pair of consecutive images, such as the examples in
Fig. 2.2, provides one discrete sampling
of the probability
that two spheres initially separated
by will have moved to a time later.
Repeatedly trapping and releasing the spheres over a range of
initial separations enables us to sample
uniformly.
This probability density is the kernel of the master equation
[30] describing how the
nonequilibrium pair distribution function,
,
evolves in time:

(12) 
The domain of integration is conveniently limited in practice by the
core repulsion which suppresses
at small ,
and by the finite range of the interaction which renders
diagonal and
independent of
for large .
Consequently,
Eq. (12) may be discretized and solved as an eigenvalue
problem for the equilibrium distribution
,
whose logarithm is proportional to the pair potential
through Eq. (10).
Data from Ref. [25] for three sizes of anionic
polystyrene sulfate spheres dispersed in deionized water are reproduced
in Fig. 2.2.
Solid curves passing through the data points result from nonlinear
least squares fits to
Eq. (9) for the spheres' effective charges and the
electrolyte's screening length.
As expected [alexander84,lowen92,lowen93,gisler94,belloni98],
the effective charges, ranging from for
up to 22,800 for
,
are one or two orders
of magnitude smaller than the spheres' titratable charges
[2,21].
The screening length of
is comparable to the spheres' diameters and
corresponds to a total ionic strength around
,
a reasonable value for deionized water at
.
These numbers will be useful for comparison with
results in Sec. 5.
The apparent success of the screenedCoulomb functional form
does not validate the DebyeHückel approximation, however,
since the exact theory including all ion correlations has
the same leadingorder behavior [31].
Comparable results were obtained by
Vondermassen et al. [22]
from measurements on optical crosssections
of dilute bulk suspensions at low ionic strength.
Sugimoto et al. [20]
studied pairs of spheres at higher ionic
strength trapped in optical tweezers and were able to measure the van der Waals
contribution.
In all cases, the measured pair potentials agree
at least semiquantitatively with predictions of the DLVO
theory.
The observed pair repulsions at low ionic strength pose a challenge
to theories predicting longranged pairwise attractions.
For example, Sogami and Ise [32,33] proposed
that the colloidal pair potential can develop an attractive tail in
the grand canonical ensemble when
the simple ions' number is allowed to vary:

(13) 
This controversial theory has been quoted widely as a possible
explanation for anomalous colloidal phenomena.
However, Eq. (13) fails to
describe the longrange pair repulsions in Fig. 2.2,
and so cannot be expected to describe manysphere behavior [21]
through superposition.
Next: Metastable Superheated Crystals
Up: Electrostatic Interactions
Previous: The DLVO Theory
David G. Grier
20010116