Eric R. Dufresne, Todd M. Squires
,
Michael P. Brenner
, and David G. Grier
Dept. of Physics, James Franck Institute, and
Institute for Biophysical Dynamics
The University of Chicago, Chicago, IL 60637
Dept. of Physics, Harvard University, Cambridge, MA 02138
Dept. of Mathematics, Massachusetts Institute of Technology,
Cambridge, MA 02139
Date: July 31, 2000
Despite considerable progress over the past two centuries [1] hydrodynamic properties of all but the simplest colloidal systems remain controversial or unexplained. For example, velocity fluctuations in sedimenting colloidal suspensions are predicted to diverge with system size [2]. Experimental observations indicate, on the other hand, that long-wavelength fluctuations are suppressed by an as-yet undiscovered mechanism [3]. One possible explanation is that hydrodynamic coupling to bounding surfaces may influence particles' motions to a greater extent and over a longer range than previously suspected [4]. Such considerations invite a renewed examination of how hydrodynamic coupling to bounding surfaces influences colloidal particles' dynamics.
This Letter describes an experimental and theoretical investigation of two colloidal spheres' diffusion near a flat plate. Related studies have addressed the dynamics of two spheres far from bounding walls [5], and of a single sphere in the presence of one [6] or two walls [7,8]. Confinement by two walls poses particular difficulties since available theoretical predictions apply only for highly symmetric arrangements [9], or else contradict each other [8,10]. The geometry we have chosen avoids some of this complexity while highlighting the range of non-additive hydrodynamic coupling in confined colloidal suspensions.
We combined optical tweezer manipulation [11]
and digital video microscopy [12]
to measure four components of the pair diffusion
tensor for two colloidal spheres
as a function of their center-to-center separation
and of their
height
above a flat glass surface.
Measurements were performed on silica spheres of radius
m (Duke Scientific lot 21024)
dispersed in a layer of water
m thick.
The suspension was
sandwiched between a microscope slide and a #1 coverslip whose surfaces
were stringently cleaned before assembly [13] and
whose edges were hermetically sealed with a
uv cured epoxy (Norland type 88) to prevent evaporation and
suppress bulk fluid flow.
A transparent thin film heater bonded to the microscope slide
and driven by a Lakeshore LM-330
temperature controller maintained the sample volume's temperature
at
C, as measured by a platinum
resistance thermometer.
The addition of 2 mM of NaCl to the solution
minimized electrostatic interactions
among the weakly charged spheres and glass surfaces by reducing the
Debye screening length to 7 nm.
Under these conditions, the individual spheres'
free self-diffusion coefficients are expected to be
,
where
cP is the electrolyte's viscosity [14].
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The spheres' motions were tracked with
an Olympus IMT-2 optical microscope using a
100
NA 1.4 oil immersion objective.
Images acquired with an NEC TI-324A CCD camera were recorded on
a JVC-S822DXU SVHS video deck before being digitized with a
Mutech MV-1350 frame grabber at 1/60 sec intervals.
Field-accurate digitization was assured by interpreting the
vertical interlace time code recorded onto each video field.
The spheres' locations
and
in the image acquired at time
then were measured to within 20 nm
using a computerized centroid tracking algorithm [12].
A pair of spheres was placed reproducibly in a plane parallel
to the glass surfaces using optical tweezers [11].
These optical traps were created
with a solid state laser (Coherent Verdi)
whose beam was brought to a focus within the sample volume
by the microscope's objective.
Resulting optical gradient forces sufficed to
localize a silica sphere at the focal point despite
random thermal forces [11].
Two optical traps were created by alternating the
focused laser spot between two positions
in the focal plane at 200 Hz using a galvanometer-driven mirror
[15].
Diverting the trapping laser onto a beam block every few cycles
freed the spheres to diffuse away from this well defined initial
condition.
Resuming the trap's oscillation between the two trapping points
reset the spheres' positions.
Alternately trapping and releasing the spheres allowed us to
sample their dynamics efficiently in a range of configurations.
Allowing the spheres only msec (5 video fields) of freedom
before retrapping them for 16 msec (less than 1 video field)
ensured that their
out-of-plane diffusion,
m,
caused negligible tracking errors.
Because optical tweezers form in the microscope's focal plane, their
height
relative to the coverslip's surface can be adjusted
from 1 to 30
m with
0.5
m accuracy by adjusting the microscope's focus.
For a given height, we continuously varied
the spheres' initial separation between
2
m and 10
m at 0.025 Hz for a total of 20 minutes.
This procedure yielded 60,000 samples of the spheres' dynamics
in 1/60 sec intervals divided into sequences 5/60 sec long
for each value of
.
These trajectories were decomposed into
cooperative motions
and
relative motions
either
perpendicular or parallel to the initial separation vector, and
binned according to the initial separation,
.
The diffusion coefficients
associated with each
mode of motion
were then obtained from
Fig. 1 shows typical data for one mode of motion
at one height and starting separation.
Diffusion coefficients extracted from
least squares fits to Eq. (1) appear
in Fig. 2
as functions of
for the smallest and largest accessible values of
.
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Particles moving through a fluid excite
large-scale flows through the no-slip boundary condition at
their surfaces.
These flows couple distant particles' motions,
so that each particle's
dynamics depends on the particular configuration of the entire collection.
This dependence is readily calculated using Batchelor's
generalization of Einstein's
classic argument [16]:
The probability to find
particles at equilibrium
in a particular configuration
depends on their interaction
through Boltzmann's distribution,
.
The corresponding force
drives a probability flux
,
where
is the particles' mobility tensor.
The system reaches equilibrium when this interaction-driven flux
is balanced by a diffusive flux
.
It follows that the
-particle
diffusivity is
.
Elements of
parameterize generalized
diffusion relations [17]
To first order in the particle radius, the mobility
tensor for spheres of radius
has the form
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(3) |
If the spheres are well separated, we may approximate
the flow field around a given sphere by a stokeslet,
the flow due a unit point force at the sphere's location.
The Green's function for the flow at
in the
direction
due to a unit force at
in the
direction is
[18]
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(4) |
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(7) |
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(8) |
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(9) |
Eqs. (5) and (6) should suffice for two spheres
far from bounding surfaces.
Similarly, the spheres' motions should decouple when the influence of
a nearby wall dominates; Eqs. (10) and (11)
then should apply.
At intermediate separations, however, neither set
is accurate.
Naively adding the drag coefficients [9]
due to sphere-sphere
and sphere-wall interactions yields
.
Results of this linear superposition approximation appear as
dashed curves in Fig. 2.
While adequate for spheres more than 50 radii from the wall, linear
superposition underestimates the wall's influence
for smaller separations.
A more complete treatment not only resolves these quantitative discrepancies but also reveals an additional influence of the bounding surface on the spheres' dynamics: the highly symmetric and experimentally accessible modes parallel to the wall are no longer independent.
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Each sphere interacts with its own image, its neighbor, and its
neighbor's image.
These influences contribute
to the mobility of sphere
in the
direction.
Eigenvectors of the corresponding diffusivity tensor appear in
Fig. 3.
The independent modes of motion are rotated with respect to the
bounding wall by an amount which depends strongly on both
and
.
Even though the experimentally measured in-plane motions
are not independent, they still
satisfy Eq. (2)
with pair-diffusion coefficients
,
where the positive sign corresponds to collective motion, the negative to
relative motion, and
indicates directions either perpendicular
or parallel to the line connecting the spheres' centers.
Explicitly, we obtain
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(12) |
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(13) |
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To gauge the success of this procedure and to quantify the range over
which the presence of a wall measurably influences colloidal
dynamics, we computed the error-weighted mean-squared
deviation of the predicted
diffusivities from the measured values,
,
where
is the measured diffusivity of mode
at
and
, and
is the statistical
uncertainty in
.
Typical results appear in Fig. 4.
Predictions based on the stokeslet approximation
agree well with measurement
over the entire experimentally accessible range.
Deviations from the linear superposition approximation's predictions,
on the other hand,
are evident out to
m or 30 radii.
The present study demonstrates that a confining surface can influence colloidal dynamics even at large separations, and that this three-surface coupling is accurately described by a leading-order stokeslet approximation. This success suggests that the same formalism can be applied to more general configurations of spheres and bounding surfaces. Our results also reveal that wall-induced hydrodynamic interactions may have influenced nonequilibrium optical tweezer measurements of confined colloidal interactions [20], and could have contributed to the observed attractions between like-charged spheres [21]. This possibility is explored elsewhere [22].
Work at the University of Chicago was supported by the National Science Foundation through grant DMR-9730189, through the MRSEC program of the NSF through grant DMR-9880595, and by the David and Lucile Packard Foundation. Theoretical work was supported by the A.P. Sloan Foundation, the Mathematical Science Division of the National Science Foundation, and a NDSEG Fellowship to TS.