Hydrodynamic coupling in polygonal arrays of colloids: Experimental and analytical results
Giovanni M. Cicuta,
1
Jurij Kotar,
2
Aidan T. Brown,
2
Ji-Ho Noh,
2
and Pietro Cicuta
2
1
Dip. Fisica, Università di Parma and INFN, Sez. Milano-Bicocca, Gruppo di Parma, Italy
2
Cavendish Laboratory and Nanoscience Centre, University of Cambridge, Cambridge CB3 0HE, United Kingdom
Received 13 April 2010; published 18 May 2010
Colloidal particles are trapped harmonically on the vertices of planar regular polygons, using optical twee-
zers. The particles interact with each other via hydrodynamic coupling, which can be described adequately by
Oseen’s tensor. Because of the interaction, the dynamics of any individual sphere is complex. Thermal motion
results in a spectrum of relaxation times. The configuration of a system of N particles can be decomposed into
2N normal modes. In this work it is shown how to calculate these modes and their relaxation time scale
analytically. The mathematical structure of the matrix of interaction leads to general properties for the sym-
metry of the normal modes and their dynamics, differing between the cases of even and odd N. The theory is
compared to experiments performed on a range of rings with 3 N 10, varying also the trap stiffness and the
distance between particles.
DOI: 10.1103/PhysRevE.81.051403 PACS numbers: 82.70.Dd, 87.80.Cc, 05.60.Cd
I. INTRODUCTION
Colloidal particles are subject to a variety of direct inter-
actions, which can be tuned and controlled to make excellent
model systems or industrial products. Colloidal particles are
also coupled to each other hydrodynamically. This is a quali-
tatively different interaction, depending on the interparticle
velocity as well as the spatial configuration. In contrast to
direct interactions which can be effectively switched off, hy-
drodynamic coupling can be tuned but never completely
screened. The motion of one particle always causes flow
which influences other neighboring particles. Hydrodynamic
coupling is an important feature in biological flows 1 and is
thought, for example, to be at the root of synchronization of
cilia beats, leading to metachronal waves 2,3. It appears in
natural and artificial microfluidic conditions 4,5 and can be
put to use in swimming or pumping strategies at low Rey-
nolds number Re6–9. For the case of spherical particles,
Oseen calculated the form of interaction in the far-field limit
that is when the distance between the spheres is much larger
than their diameter 10. As a further consequence of hydro-
dynamic interaction, there are correlations in the Brownian
fluctuations of different particles. This is well known, and
indeed exploited, in two-particle passive microrheology,
where the statistical correlation in the motion of two tracer
particles enables one to extract the solvent viscosity and
even the viscoelasticity in complex fluids11,12. To main-
tain an average distance and acquire good statistics, the
tracer spheres can be trapped by focused laser beams, effec-
tively confining each bead in a harmonic well. In this sce-
nario the average position of the spheres is well defined, but
the beads are constantly subjected to thermal fluctuations
which displace them from the minimum of the trap potential.
In liquids, optically trapping spheres of colloidal sizes, the
relaxation dynamics is always overdamped; an investigation
of underdamped hydrodynamic interaction was carried out
on water droplets suspended in air 13. Even considering
just simple Newtonian liquids, the systems of this type that
can be treated exactly in the Oseen limit have until now been
limited to either a two particles 14 or b an infinite linear
array of particles 15. Clearly only the first case is experi-
mentally accessible. There have been two experimental stud-
ies going beyond the two-sphere situation: the modes of mo-
tion in linear arrays of ten spheres have been studied in 15;
Di Leonardo et al. considered an arrangement of eight
spheres at the vertices of a regular octagon 16. In both
cases, the eigenmodes of system and their dynamics were
calculated approximately and compared to the observed
cross-correlated motions with good agreement. In this pa-
per we show that the dynamics of systems where spheres are
positioned on the vertices of arbitrary planar regular poly-
gons can be solved exactly within Oseen’s description of
hydrodynamics. We use this powerful theoretical result to
analyze the dynamical modes of systems from three to ten
spheres, regularly positioned on a ring to form polygonal
structures, and compare these predictions to experimental re-
sults.
II. EXPERIMENTAL SYSTEM
Optical traps are used to confine colloidal beads within
harmonic potentials; the system hardware is described in
greater detail in 7. In this work, a varying number of silica
beads of radius a = 1.75 m Bangslabs are trapped by a
time-shared laser beam. A pair of acousto-optical deflectors
AODs allows the positioning of the laser beam in the x , y
plane with subnanometer precision and at a rate high enough
that a bead does not diffuse significantly in the time that the
beam cycles through the other beads.
The solvent in which the beads are suspended is a glyc-
erol Fisher, Analysis Grade water Ultrapure grade, ELGA
solution 50% w/w, giving a nominal viscosity of
=6.00 mPa s at 20 °C 17. Experiments are performed in a
temperature controlled laboratory, T =21 °C. The trapping
plane is positioned 20 1 m above the flat bottom of the
sample, in a sample volume that is around 100 m thick.
A trapped colloidal particle undergoes overdamped sto-
chastic motion, driven by thermal forces which include the
indirect effect of fluctuations of all other particles 14. If
only one bead is present in the system, its motion in the
PHYSICAL REVIEW E 81, 051403 2010
1539-3755/2010/815/0514038 ©2010 The American Physical Society 051403-1