Particle–particle interactions in electrorheological fluids based on surface
conducting particles
P. Gonon,
a)
J.-N. Foulc, P. Atten, and C. Boissy
Laboratoire d’Electrostatique et de Mate
´
riaux Die
´
lectriques (LEMD), Centre National
de la Recherche Scientifique (CNRS), and Universite
´
Joseph Fourier (UJF) BP 166,
38042 Grenoble Cedex 9, France
͑Received 3 May 1999; accepted for publication 15 September 1999͒
We develop a simple, analytical conduction model for the case of electrorheological fluids based on
surface conducting particles. By modeling two contacting spheres in a dielectric liquid by a
distributed impedances network we derive analytical expressions for the potential and current at the
spheres surface, and for the electric field and the current in the liquid phase. The knowledge of the
electric field in the dielectric liquid allows us to calculate the interparticle interaction force as a
function of the applied voltage. The theoretical interaction force is compared with experimental
results obtained on insulating spheres coated with a thin conducting polyaniline film. We find a good
agreement between the theory and experiment. The materials properties which govern the response
of the system are outlined. In this regard, the product of the liquid conductivity by the sheet
resistance of the surface coating appears as a key parameter. Some applications of this model for the
practical design of electrorheological fluids are given. © 1999 American Institute of Physics.
͓S0021-8979͑99͒06824-3͔
I. INTRODUCTION
Electrorheological ͑ER͒ fluids are smart liquids whose
mechanical properties can be controlled by applying an ex-
ternal electric field. Briefly, an ER fluid consists of a suspen-
sion of micrometer-size particles in a dielectric liquid. When
subjected to an electric field of the order of a few kilovolts
per millimeter, it is observed that particles attract each other
to form a solid network of fibers aligned with the field. This
fibrillation results in a considerable increase in the apparent
viscosity of the fluid which now behaves as a viscoelastic
solid. This apparent ‘‘phase transition,’’ from a liquid state
to a ‘‘gel-like’’ state, takes only a few milliseconds. The
effect is reversible, when the electric field is switched off the
gel-like structure vanishes and the fluid recovers its original
liquid state. This ability to electrically control the apparent
fluid viscosity makes ER fluids potentially very interesting
for numerous electromechanical devices such as valves,
dampers, or clutches for the automotive or robotics industries
for instance.
1
Although ER fluids have been known for some time,
2
the
physical mechanisms at the origin of the ER effect are still
not very clear. Among important points which have to be
clarified is the nature of the particle–particle interactions,
and how these interactions are related to materials properties.
Several recent works have addressed this issue,
3–18
propos-
ing different models for the interparticle interactions. In any
case the main problem is to determine the electric field dis-
tribution in the fluid surrounding the particles ͑usually con-
sidered as spherical͒. Once this is done, the calculation of the
interaction force between spherical particles is performed by
integrating the electrostatic stress over the spheres
surface,
4,5,9,10,13,15–17
or by derivating the electrostatic energy
with respect to the system coordinates.
7,8,11,12,14
Two different approaches have been used to calculate
the electric field distribution in the surrounding dielectric
liquid. In the first approach, the so-called ‘‘polarization
model,’’ particles are treated as polarized bodies. This is a
classical problem of electrostatics where the field distribution
is given by polarized spheres in an insulating medium of
dielectric constant
⑀
L
. The simple calculation treats polar-
ized spheres as dipoles, but calculations including multipolar
terms and many-particle effects have been per-
formed.
3,4,7,8,11,14
In the polarization model the important pa-
rameter is the permittivity ratio
⑀
S
/
⑀
L
, where
⑀
S
and
⑀
L
are
the relative dielectric constants of the solid particles and of
the liquid phase, respectively. When the fluid and the par-
ticles conductivities are considered ͑terms
L
and
S
, re-
spectively͒, a Maxwell–Wagner interfacial polarization term
is taken into account by replacing real dielectric constants
with complex dielectric constants
⑀
L
*
ϭ
⑀
L
Ϫj
L
/
and
⑀
S
*
ϭ
⑀
S
Ϫj
S
/
, where
is the frequency of the applied
field.
6–8
In the second approach, the so-called ‘‘conduction
model,’’ the fluid and the particles are treated as conductive
media. This is the approach used by Conrad and
co-workers,
13,15,16
by Davis,
18
and also by our group.
5,9,10,17
The electric field distribution is calculated by assuming that a
current flows through particles and between their gap in the
liquid phase. The problem is now essentially a problem
of electrodynamics, the important parameter being the ratio
of the conductivities
S
/
L
. For alternating fields it is
convenient to take into account the displacement current
via the complex conductivities
L
*
ϭ
L
ϩj
⑀
L
and
S
*
ϭ
S
ϩj
⑀
S
.
a͒
Electronic mail: gonon@labs.polycnrs-gre.fr
JOURNAL OF APPLIED PHYSICS VOLUME 86, NUMBER 12 15 DECEMBER 1999
71600021-8979/99/86(12)/7160/10/$15.00 © 1999 American Institute of Physics