Sven H. Behrens and David G. Grier
Dept. of Physics, James Franck Institute, and
Institute for Biophysical Dynamics
The University of Chicago, Chicago, IL 60637
Date: May 11, 2001
One of the most prominent open questions in colloid physics concerns the influence of spatial confinement on the interaction between charged colloidal particles. For an isolated pair of similar particles in an electrolyte solution, the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory [1] predicts a short-ranged van der Waals interaction and an electrostatic repulsion, which, for well-separated spherical macroions, takes the familiar screened Coulomb form
Experimentally, the pair interaction energy can be assessed by analyzing the relaxation of two particles released from optical traps [3,4], or by interpreting the liquid structure of an equilibrated dispersion [5,6,7]. Both techniques consistently reveal that polystyrene spheres dispersed in deionized water and closely confined between parallel glass walls experience a long-range pair attraction [6,7,8,9] qualitatively inconsistent with DLVO predictions, and more generally with mean-field [10] or local density theories [11]. Unconfined particles, on the other hand, repel each other as expected [3,5,8],
This Letter reports measurements of colloidal pair interactions in equilibrium near a single wall. The only previous one-wall measurement used optical tweezers to position highly charged polystyrene spheres near a charged glass surface, and then released the spheres to measure their interactions [12]. Although this measurement revealed attractions consistent with those observed between two walls, Squires and Brenner [13] showed that they could have resulted from hydrodynamic flows excited by the spheres' retreat from the wall, a purely kinematic effect. The resulting hydrodynamic interaction would mask any other attraction in non-equilibrium measurements near a single wall. It cannot have influenced the equilibrium long-ranged attractions deduced from the structure of colloidal monolayers confined between two walls [6,7]. Nor does it account for the anomalous attractions measured with optical tweezers between two walls; reanalyzing the data from Ref. [8] reveals no sign of hydrodynamic memory. Moreover, indirect evidence based on the structure and dynamics of metastable crystals [12,14] suggests that a single wall can induce like-charge attractions in suspensions of sufficiently low ionic strength. Our measurements therefore focus on this regime.
Our samples consists of monodisperse silica spheres (
Because of their high specific mass (
),
the particles sediment
to the bottom wall in a matter of minutes,
reaching an equilibrium height
where the force due to gravity
is balanced by the spheres' electrostatic interaction with the
wall's surface charge.
The spheres' unvarying appearance under the microscope
confirms their out-of-plane excursions
to be smaller than
.
Limited out-of-plane motion reduces the possibility of
projection errors, which have been identified [15] as a
concern in earlier studies.
The particles were observed
with an inverted optical microscope (Olympus IMT-2), using a
N.A. 1.4 oil immersion
objective and a
video eyepiece.
These image an area
in the
pixel field of view of an attached charge-coupled device
camera.
The particles' motions were recorded at 30 frames per second
before being digitized with a
(MuTech MV-1350) frame grabber.
Standard image analysis techniques
[4] were used to locate the spheres to within 30 nm.
Following the principle adopted in earlier studies, we measure the spheres'
pair potential by compiling histograms of equilibrium pair separations
into the two-dimensional pair correlation function, .
In the limit of infinite dilution,
is related
to the pair interaction energy through the Boltzmann distribution,
,
where
is the areal density of spheres.
For finite concentration, on the other hand,
the radial distribution function
also reflects neighboring particles' influence, and generally
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(2) |
Although straightforward in principle, imaging measurements
of are subject to subtle sources of error which
can introduce spurious features into
.
These errors arise principally from three sources: truncation
by the limited field of view, statistical undersampling
of suspensions' slow dynamics, and uncorrected many-body
correlations.
Previous studies addressed some, but not all, aspects of these
errors [5,6,7].
Consistency among their results reinforces their
qualitative conclusions, while leaving some doubt regarding
their quantitative accuracy.
For this reason, we outline our methods.
The pair correlation function measures the mean number
of particles per unit area separated from any given sphere by
displacement
.
This average is calculated in practice by
counting the number of
-pairs in a
recorded image and normalizing by the number of particles actually tested
for a partner at separation
,
i.e. by the number of particles in the
``overlap area''
, where
is the set
of points in the field of view and
is the same
set displaced by
.
Summing over angles yields
and hence
.
The field of view in Fig. 1 contains too
few particles to sample accurately.
Increasing the field of view to improve statistics
would degrade particle tracking performance [4].
Consequently, results from independent
images must be averaged
to improve statistics.
Unless the particles are given time to redistribute between
snapshots, however, these additional images will not shed
light on the suspension's equilibrium properties so much
as improve the sampling of a particular transient distribution.
The need for statistically independent samples implies
that the period over which
is averaged must be
chosen with care.
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If the monitored area were so large that edge effects could be
neglected, then a histogram of the particle separations
with bin width
would give
as the average number
of particles separated by distance
from any given particle,
normalized by the corresponding
value
for an ideal system of noninteracting particles. Given
pairs at separation
in a snapshot,
Because the experimental duration increases so rapidly
with dilution and because
controlling concentration, temperature,
and ionic strength over long periods can be difficult, statistical
accuracy strongly favors larger particle concentrations.
On the other hand, extracting the
pair potential from measured correlations
becomes increasingly difficult as increases.
The areal densities between
and 0.1
chosen for this study require no more than 30 minutes sampling and
represent a compromise between statistical and interpretive
accuracy.
Our experimental results for and
at different concentrations
are shown in Fig. 2.
The curves indicate a repulsive core interaction causing particle
depletion from a zone about
wide.
Beyond this is a
preferred nearest-neighbor separation between
and
and the oscillatory correlations typical of
a structured fluid.
The depth of the minimum in
clearly depends
on
and so reflects at least some
many-body contributions.
To ensure that none of the observed correlations result from inhomogeneities in the glass surface's properties, we compared two-dimensional histograms of recorded particle positions with analogous histograms for uniformly distributed random data sets. Differences in these histograms' first two moments vanish with increasing delay time, confirming that each particle's position becomes uncorrelated over time as expected. Thus the substrate potential appears to be featureless on the length- and timescales of our experiment, to within our resolution.
Provided that is free from experimental artifacts,
reliable approximations for
can be obtained from
using the Ornstein-Zernike integral equation with
appropriate closure relations.
Good results for ``soft'' potentials are typically achieved with
the hypernetted chain (HNC) approximation,
whereas the Percus-Yevick (PY) approximation is known to be a
better choice for hard spheres.
The pair interaction potential can be evaluated numerically
in these approximations as
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(4) |
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(5) |
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The absence of minima in the pair potential confirms that
crowding induces the observed oscillations in ,
while the underlying interaction is purely repulsive.
The solid curves in Fig. 3 are fits
to Eq. (1) for
,
, and
an arbitrary additive offset.
The fitting parameters
listed in Table 1
are consistent with surface charging due to silanol dissociation
at an ionic strength around
[18].
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0.051 | 5504 | 6502 | 0.32 | 0.30 |
0.079 | 4312 | 5603 | 0.33 | 0.30 |
0.083 | 4656 | 6039 | 0.32 | 0.29 |
The DLVO theory's success at characterizing our data
might seem surprising given the experimental geometry.
However, the DLVO form has been shown to capture the leading-order
behavior in the Poisson-Boltzmann approximation,
with the bounding wall introducing only an additional dipole
repulsion [19,20] to lowest order.
However, the dipole correction is predicted [20] to be weaker than
our experimental resolution over the experimentally
accessible range of interparticle separations.
However, long-range attractions of the
previously reported strength
[6,7,8,12]
would have been resolved.
A dynamic aspect of the observed particle correlations is illustrated by the van Hove function, plotted in Fig. 4 for
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(6) |
The decay of provides an estimate [16]
of
for the sample with
.
The
measurement period therefore
yields
with a relative error
given our
binning resolution
.
The other samples yield comparably good results.
The absence of a measurable inter-particle attraction in this experiment should be interpreted with care. Our silica spheres carry lower charge densities than the polystyrene particles used in earlier studies, and the sphere-wall separation is considerably smaller. A wall-induced equilibrium attraction weaker than our experimental resolution would have to depend strongly on these parameters. The measured structure of our colloidal monolayers suggests instead that like-charged particles repel each other in pure water, even near a charged wall. Our observations support the electrohydrodynamic explanation [13] for the attraction measured with optical tweezers in Ref. [12], thus distinguishing that attraction from those seen in dispersions confined between two walls, and those implicated in observed anomalous phase behavior in bulk suspensions.
We are grateful to Todd Squires, Michael Brenner and Vladimir Lobaskin for enlightening conversations. This work was supported by the National Science Foundation through Grant Number DMR-9730189 and by the Swiss National Science Foundation.