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Field induced structure in magneto and electro-rheological fluids
E. Lemaire, Y. Grasselli, G. Bossis
To cite this version:
E. Lemaire, Y. Grasselli, G. Bossis. Field induced structure in magneto and electro-rheological fluids.
Journal de Physique II, EDP Sciences, 1992, 2 (3), pp.359-369. �10.1051/jp2:1992139�. �jpa-00247638�
J. Phys. II France 2 (1992) 359-369 MARCH 1992, PAGE 359
Classification
Physics Abstracts
75.30K 82.70D
Field induced structure in magneto and electro-rheological
fluids
E. Lemaire, Y. Grasselli and G. Bossis
Laboratoire de Physique de la Matibre Condensde (*), Universit6 de Nice-Sophia Antipolis, Parc Valrose, 06034 Nice Cedex, France
(Received 25 September I99I, accepted in final form 4 December J99J)
Abstract. When colloidal suspensions of magnetic particles are submitted to a magnetic field
they undergo a phase separation with the formation of high-density domains whose structure
depends on the initial concentration and on the intensity of the applied field. The shape of these domains and the mechanism which explains their formation is quite different from what happens
in ferrofluids or in ferromagnetic solids where the surface tension associated with the boundaries of the domains plays a major role. We report experimental results conceming the size of the
domains, the critical field associated with this phase separation and some dynamical aspects of the transition. These experimental results are predicted by a simple model based on the estimation of the magnetic energy in a mean field approximation with the structural unit represented by an
ellipsoidal aggregate inside a cylindrical cavity. The similarity with structure formation in
electrorheological fluids is qualitatively demonstrated.
Magnetic colloidal suspensions are usually known as fe«ofluids and are formed of very small fe«omagnetic particles (d)
m 100 h dispersed in various organic or aqueous solvents [Ii. At the other extremity in size, powders of fe«omagnetic materials dispersed in an inert
solvent (for instance a silicone oil) are studied for their ability to change their rheological properties under the application of a magnetic field [2, 3]. Nevertheless, tllese particles are
very polydisperse, irregular in shape and they sediment quickly owing to their high density
and larger size (several microns), so they are less interesting for fundamental studies. We have recently studied [4, 5] the rheological properties of a new kind of magnetic suspension
formed of micronic polystyrene particles synthetized by Rh6ne-Poulenc, which contain tile
same small ferromagnetic particles as those dispersed in a ferrofluid. Since tllese small magnets are randomly oriented inside the polystyrene particles, the resulting magnetic
moment is zero and tllese particles behave as a superparamagnetic material with virtually no
hysteresis. These particles are spherical, their average diameter is 0.8~m and their sedimentation time is long enough to neglect gravitational effects in the structure formation.
When a magnetic field is applied to the suspension each particle will acquire a magnetic
(*) URA 190.
moment m; then, in the dipolar approximation, two particles whose line of centers is
perpendicular to the magnetic field will repel each otller whereas the interaction will be attractive if they are aligned with the direction of the field.
This dipolar attraction in the direction of the field explains the formation of columns of
particles parallel to the magnetic field which have been observed in ferrofluids [6]. A key parameter for the formation of these structures is the ratio of the magnetic energy to the thermal energy : A
=
poH(V/kT where Ho is the vacuum permeability Ho the applied magnetic field and V the volume of the particles. For our particles, at room temperature, with
an average radius of 0.4 ~m we obtain A
= 0.25 H( (Oersted) which gives large values of A for rather small fields (Ho ~10 De). We then expect important changes of structure of these
suspensions for magnetic fields of a few Oersted. This is a significant advantage over ferrofluids where, due to the smaller size of the particles, we need much larger fields (about 1000 Oe) to obtain the same effect. Furthermore the optical observation of the structured
phase is easier due to the larger size of the particles. Actually the more fundamental difference between ferrofluids and magnetic polystyrene spheres is due to the absence of a net
magnetic moment of the latter. This means that a cluster of these magnetic polystyrene spheres has no « surface tension
» (if we neglect very short range interactions) and could not be a stable structure in the absence of a field. Actually this is only true in the absence of an extemal magnetic field, since when the particles have an induced dipole, their energy of
interaction becomes different, depending on their position inside or at the surface of a cluster and on the structure of this cluster. This effect which is due to the difference of local field
between an homogeneous medium and a structured one, gives rise to a surface tension
proportional to the square of the magnetic field [10]. We shall see hereafter that the
corresponding surface energy is low compared to the bulk energy coming from the repulsive
energy between the aggregates. For tile ferrofluid case, the particles have a permanent
moment v, and randomly oriented chains of particles exist in the absence of a field since the ratio of dipolar to thermal energy p~la~ kT is usually between I and 10 at room temperature [7]. It must also be stressed that many studies of drop deforrnation [8] or labyrinthine instability [9], concem ferrofluids drops in an immiscible fluid where surface tension is
inherent to the use of two immiscible fluids.
Having emphasized the difference between suspensions of magnetic polystyrene particles
and ferrofluids we need also to note the similarity of these suspensions with electrorheological (ER) fluids. Upon the application of an electric field ER suspensions also show a fibrillated
structure which confers them a solid aspect with the apparition of a yield stress and an elastic
behaviour. The characterization of this structure and of the field necessary to build it, is at the heart of any potential application of these fluids. In the presence of an electric field each
particle is polarized with an electric dipole m, but this polarization is more difficult to model since it is partly due to the motion of the ionic cloud surrounding each particle. In any event, the interactions developed between electric dipoles are identical to those between magnetic dipoles and we expect to obtain the same kind of structure in electrorheological fluids as in
suspensions of superparamagnetic spheres. This is actually what we observe as illustrated in
figure I. The top view refers to a suspension of hydrated silica particles submitted to an electric field of 10 kV/cm and figure16 corresponds to the suspension of magnetic particles
with a magnetic field H
=
100 Oe. The two structures are indeed very similar and we shall
comment on them in more detail at the end of this paper. The phase separation in ER fluids has been recently predicted by Halsey and Toor [10] who expect a two step process with firstly
a formation of isolated chains of particles and then a slow sticking of theses chains together
until the completion of the phase separation. Another indirect study of this phase transition aspect in ER fluids has been experimentally investigated by Tao et al. [ll]. The structural
N° 3 FIELD INDUCED STRUCTURE IN SUSPENSIONS 361
transition is associated with the formation of a solid network of particles which sticks to the walls of the cell and prevents tile suspension to flow under gravity. Nevertheless this critical field defined through tile appearance of a yield stress is difficult to relate to the characteristics of the structure. In our approach we visualize the structural transformation with the help of a
microscope and image analysis. In that way we have access to the complete information on spatial and temporal evolution of the structure as a function of the applied field.
The magnetic fluid is sealed between two glass plates with parafilm sheets and the thickness of the cell is varied between 100 ~um and 400 ~um by using different spacers. This cell is placed
under a microscope objective and a magnetic field perpendicular to the glass plates is applied through a current generator which is computer controlled. The current and so the magnetic
field is raised from zero to its maximum value with a constant slope and, by image processing,
the average transmitted light recorded by the camera is calculated versus the magnetic field. A
typical curve is shown in figure 2 for a volume fraction of suspension ~fi~ =10 f6. The transmitted light does not change up to a field of a few Oersteds, then we have a fast rise associated witl1the formation of domains, whose concentration in magnetic particles increases with the magnetic field consequently the remaining part of tile suspension becomes less and less concentrated and tile transmitted light increases and reaches a plateau when most of the
particles are incorporated into these domains. Reproducible curves of the transmitted light
versus the magnetic field can only be obtained if tile raising time of tile magnetic field is slow
enough. In figure 2 tile curve A corresponds to an increase of I Oersted per minute and for the same rate of decrease tllere is practically no structural hysteresis. On the contrary,
curve B is obtained out of equilibrium with a faster decrease of the magnetic field (5 Oersted
per minute). In that case the particles do not have enough time to reach tile new equilibrium
state by Brownian motion (the characteristic time for a particle to move over a distance of one
radius r, is r
=
r~/Do= 0.3 s but in the presence of the field these are linear clusters of
particles which rearrange, and their diffusivity is roughly inversly proportional to tlleir
length).
In order to see how tllis phase separation depends on the initial concentration of magnetic particles we have chosen for experimental convenience to follow the field, Hs corresponding
to the inflexion point of the transmission curve. The behavior of this field characterizing a
kind of intermediate state in phase separation is represented in figure 3 versus the
concentration. It clearly appears that the field necessary to produce the structuration of the
suspension increases linearly with the volume fraction. In their experiment on ER fluids Tao et al. find the opposite behavior but their threshold field E~ measures the ability of the solid network of particles to prevent the fluid to flow rattler than its true formation. It is then understandable that, at low volume fraction of particles, the field E~ needs to be increased in order to prevent the fluid to flow since tile solid network which retains the liquid phase
becomes progressively tllinner and thinner as tile volume fraction decreases.
This experimental behavior corresponding to an increase of the structuring field with the
volume fraction and also the evolution of the domain size with the concentration can be
explained by a simple model. Rather than starting from the homogeneous system (here the
suspension at the initial concentration ~fi~) with a study of the growtll of a periodic perturbation of the density near the critical field [12], we examine a situation where
we are
well above the critical field with a well defined phase separation. In practice, what we want to
predict is the average size of the domains and the field we have to apply in order to obtain a
phase separation with a solid network of particles which will be able to give an elastic component to the fluid. These properties are fundamental to guide tile choice of a good
electro- or magneto-rheological fluid. We model a domain by a prolate ellipsoid of revolution with length, I = 2 a, which is the spacing between the plates, and radial semi-axis b (cf.
a)
b)
Fig. I. Structuration of an ER fluid (Fig. la) and a MR fluid (Fig. lb). In both cases the volume fraction of particles is 20 %, The electric field was 10 kV/cm and the magnetic field 100 Oersted
lengthscale of the picture : 250 ~L. Figure lc : structuration of a 2 % magnetic fluid for the same magnetic field note the more regular shape of the aggregates lengthscale of the picture : 500 ~L,
N° 3 FIELD INDUCED STRUCTURE lli SUSPENSIONS 363
C)
Fig, I (continued).
1(a.u.)
D.
MAGNETIC FIELD (OERSTED)
Fig. 2. Average transmitted light through the suspension versus the magnetic field. A) raising time, I Oersted/mn, B) falling time, 5 Oersted/mn.
Fig. 4). This ellipsoid is located in a cylindrical cavity of radius ro. Looking at picture lc which
corresponds to a volume fraction of 2 f6 we see well defined aggregates the black filaments
which can link some aggregates are not visible on the original colored picture the
projection of these aggregates is roughly circular. At higher volume fractions their shapes
Hs (OEBSTED)
o.
o.
VOLUME FRACTION
Fig. 3. Structuration field Hs defined at the point of inflexion of the transmission curve (cf. Fig. 2)
versus the volume fraction of magnetic particles. The continuous line represents a least mean square
line.
d r~
~b
Fig. 4. Schematic representation of a reference ellipsoidal aggregate inside a cylindrical cavity. The outside material is assumed to be homogeneous.
become more complicated (cf. Fig. lb), but are still different from those given by the
labyrinthine instability observed in the ferrofluids they are more branched that elongated.
Even if this model is more doubtful at high volume fraction, we believe that an effective field
theory can continue to capture the dominant terms of the magnetic energy. The choice of an
ellipsoid rather than a cylinder is justified because for the same volume and the same length, I, the ellipsoidal shape gives a lower free energy for a ratio bla
~ 0.3 [14] and experimentally
this ratio is less than 0.I. The magnetic free energy of the sample is given by [I] :
F~
=
M Ho/2 where M is the whole magnetic moment of the sample : M
= N~M~, with
M~ and N~ being respectively the moment of an aggregate (of ellipsoidal shape) and the
number of aggregates. In the absence of the reference aggregate, the field inside the
cylindrical cavity, Hi, is the sum of the extemal field plus the demagnetizing field coming
from the average polarization P
= N~M~/VT that we assume homogeneous in a mean field
N° 3 FIELD INDUCED STRUCTURE IN SUSPENSIONS 365
approach. VT is the total volume of the sample. Then the field H, averaged over a cylindrical
volume of radius b
~ ro will be :
(H,)
= Ho 4 ar(n~) N~ M~/VT (I)
where (n~) corresponds to the demagnetizing factor coming from the outside magnetization
and averaged over the volume of a cylinder of radius b. Considering this average cavity field
as an extemal field, the moment M~ of an ellipsoidal aggregate of intemal relative magnetic permeability p~, is given (in e.m.u.) by [13] :
M~ = (p~ I)Hi 4/3 arb~a (2)
with Hi, the field inside the aggregate, related to tile average cavity field (H,) by :
~
i +
~j~~
i) ~~~
j2bj~j ~ j jl+ejj
~ ~ °~
l e
~
/~ 4 b~
~~
2 e~ ' ~~ ~ fl
or using (I) I
Ho HI ~ ~~~
i + (»~ i)(ne + <nb> ~°m/~P~
n~ is the demagnetizing factor and e the eccentricity of tile ellipsoid: n~-0 for
bli
- 0 (needle shape).
With (I) and (2) the total free magnetic energy per unit volume is :
F~ N
~
M~ Ho 4~i ~P
~j ~~
$ 2 V ~fi ~~~
~ ~
l + (JCp I) n~
+ (n~) fi
I~P
(n~) is given by a series in s
= b/ro, whose first terns are :
~~~~ ~~ ~ /4~~ ~
l '24~~ ~ ~~~)2
~~ ~ ~ ~ i ~
~~~
The radius of the cavity can be approximated by ro
=
d- (b) d being the average distance between aggregates and (b)
=
fit is the equivalent radius of
a cylinder of
same volume as the ellipsoid (cf. Fig. 4). On the other hand the ratio of the semi-axis b of the
ellipsoid to the average distance is fixed by the ratio of the average volume fraction
~fi~ to the volume fraction ~fi~ inside tile aggregate. Assuming a random structure we have :
~
=
j~
fi(7)
d 8
~
Then, the magnetic free energy depends only on one structural parameter, dli, which is obtained by the minimization of F~. It is wo~h noting that we do not need any surface tension to explain the formation of these domains. The two competing mechanisms which are
necessary to obtain the formation of structures whose size is much larger that the average
particle size come from the magnetic energy. This becomes clear if we look at expressions (1)
