Structure induced in suspensions by a magnetic field

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Structure induced in suspensions by a magnetic field

Y. Grasselli, G. Bossis, E. Lemaire

To cite this version:

Y. Grasselli, G. Bossis, E. Lemaire. Structure induced in suspensions by a magnetic field. Journal de

Physique II, EDP Sciences, 1994, 4 (2), pp.253-263. �10.1051/jp2:1994127�. �jpa-00247959�

Classification Physics Abstracts

75.30 64,75 82.70D 46.60

Structure induced in suspensions by

_a

magnetic field

Y.

Grasselli,

G. Bossis and E. Lemaire

Laboratoire de Physique de la Matikre Condensde (*), Universitd de Nice

Sophia Antipolis,

Parc Valrose, 06108 Nice Cedex 2. France

(Receii>ed 3

February

J993, revised J9 October J993,

accepted

9 November J993)

Abstract. Magnetorheological _as well _as electrorheological fluids exhibit _a phase separation when _a field is

applied

to the

suspension.

We observe the formation of aggregates

elongated

in the direction of the field and whose length L is fixed by ^the thickness of the cell. The average radius

b of these aggregates ^{has been measured for L between 100} ~m and 700 _~m for two volume

fractions 4l 5 % and 4l 10 %. In this range we find for _a magnetorheological ^fluid composed of micronic superparamagnetic panicles an experimental law b _cc a(Lla)~ with _an exponent

n m 0.67 ₌ 0.04. We _{can recover} this experimental law and predict ^the characteristic size of the aggregates ^{from the} knowledge ^{of the}

magnetic permeability

of the

particles

with _a model based on

the hypothesis of close

packed

aggregates. ^The agreement ^{with the}

experiments

is

good especially

at the lower volume fraction.

Actually

on a

larger

_range of thicknesses _our model predicts a

continuous increase of this exponent towards _a final value of _one for large values of

Lla.

1. Introduction.

Magnetorheological suspensions

_are

composed

of

magnetic panicles

whose size, in the micronic range, is _at least _two orders of

magnitude larger

than in the better known

magnetic suspensions,

called ferrofluids. The induced

magnetostatic

force between two

panicles being proportional

to the square of the radii of the

panicles,

it is much

larger

for M. R. fluids than for ferrofluids. As _a counterpart, sedimentation is _no

longer prevented by

^Brownian motion and the micronic range represents an

optimum

size. Under the

application

of _a

magnetic

field

[of

_a few

Oersted]

we observe _a transition from _a

homogeneous liquid suspension

to a solid _one I

].

This transition from _a

liquid

to an apparent ^solid state is due _to the formation of

elongated

aggregates ^of

particles

which _are

aligned parallel

to the

magnetic

field and

join

the

opposite

sides of the cell

containing

the

suspension.

These links _are

responsible

for the apparent ^solid behavior

since,

for instance, if the field is

perpendicular

to the two disks of _a rheometer, _we need _to increase the

imposed

_torque above _a critical value in order _to break these aggregates

(*) CNRS URA 190.

and obtain _a continuous rotation of the upper disk. The _onset of _a

yield

stress

strongly

modifies the

rheological

behavior and opens the

possibility

to realize

hydraulic

devices where the

viscosity

is

electronically

controlled. This

effect,

due _to the

magnetization

of the solid

panicles by

a

magnetic

field _{can, as} well, ^be obtained with dielectric

particles polarized by

the

application

of _an electric field. These

electrorheological

^fluids are the

object

of _a

rising

interest both for their

potential applications

and for the

interesting underlying physics [2-41. Clearly

_we

need _to obtain _a

good knowledge

of the _structure of these

suspensions

the

shape

^of the

aggregates, the distance between them, the internal arrangement ^{of the}

particles

if _we want to be able to

predict

the

change

^{of their}

rheological properties

in the presence of _a field.

For

monodisperse

^dielectric

panicles,

Tao _et al.

[5,

61

Predict

and observe _an internal structure for these aggregates

composed

of chains of

spheres ^(of

^radius

a)

with _a

given

chain surrounded

by

four other _{ones at a} distance _a

/

_with

a shift of _a radius in the direction of the

applied

field and four next-nearest

neighbors

at a distance _a

/,

_{but without shift. Such}

a

configuration

called B.C.T.

(body

centered

tetragonal) gives

the lowest internal energy. In _a

recent paper,

Halsey

^and Toor [71 ^model the aggregate as a

spheroidal droplet

and calculate the

intemal energy on the basis of electrostatics of continuous media.

They

furthermore introduce

a surface energy

U~ proponional

to the square of the

applied

field E

U~

_> ^0.I

p

^~

E~ aS~

^with

p

=

~ l)

S~ is the surface of the aggregate, a the radius of _a

panicle

and _s the ratio of the

permittivity

of the solid

particle

to that of the

suspending

fluid. In this

approximation U~

is

independent

of the

shape

of the aggregate.

Minimizing

the total energy

they obtain,

in the limit where the ratio of the

equatorial

semi-axis to the total

length

b/L tends to zero, a

dependence represented by

b cc a

(Lla

)~'~. In this model the energy of interaction between different aggregates ^is not taken

into _account.

Actually

when the aggregates ^{fill the} gap between the

electrodes,

the existence of

image dipoles

which

impose

the

potential

_on the electrodes is

equivalent

to the introduction of aggregates ^{of infinite}

length.

Then, at this last stage ^of

aggregation,

the

depolarization

field and

consequently

the

repulsive

interaction between the aggregates

dissappears.

We expect the surface energy ^to be

given by

the _same

expression

for

magnetic

and for

dielectric

panicles:

_we

just

have to

replace

_s

by

_p, the

magnetic permeability,

and

E

by

H, the average

magnetic

field inside the

sample.

^{This will} be true as

long

_as the

permeability

does not vary too much with the

magnetic

field. In a

previous

work

[I]

we have

reponed experimental

results about the average size of these aggregates i>ersus the initial volume fraction 4~ of the

suspension

and found _a linear law which _{we were} able _to

explain

with

a model

taking

into _account the balance between the volume energy and the

repulsive

_energy

between aggregates, ^without

introducing

_a surface energy. ^In the

following

section of this paper we propose a model

including

both the

repulsive

_energy between the aggregates ^{and the} surface energy which

predicts

the average size of the aggregates

by

a minimization of the total

magnetic

_energy

analogous

to the _one used for the determination of the deformation of ferrofluids

droplets [8-101.

In section

3,

_{we present some}

experimental

results and compare them with the theoretical

predictions.

We conclude

by

a discussion of the

validity

^{of this model} and of its

application

to

electrorheological suspensions.

2. Theoretical model.

We consider the _case of _a

magnetic suspension

submitted _{to an} extemal

magnetic

field

Ho

^much

higher

than the field

corresponding

to the transition and _we model the structure of the

suspension by

a

hexagonal

_array of

ellipsoids

of transverse semi-axis b and of

length

L

(cf. Fig, I).

In the

dipolar approximation

the ratio A of the

magnetic

_energy to

ptical(

axis

/~

_2b

d

~ ~ ~ _~

ri i

. . . ,

,

~ ~ ~ ~ ~

L _i

~ ~ ~ ~ ~

~ ~ ~ ~ ~

i

Ho

@ @

Fig. I. Schematic

diagram

of the _structure A top ^{view B side view. The} aggregates are assimilated to

prolate spheroids

of semi-axes b and f, _L

=

2 f is the thickness of the cell,

kT is

given by

A

=

(4

_VP

o pi p^~

H~/

₍₄

_kT)

_where

p = I is the

magnetic permeability

of the

liquid phase.

In _our

experimental

_{case we} find A

=

2.63

H~

_{(De ). For the terminal field}

we

have used

(50 Oersted)

_we obtain A

=

6 500.

Owing

to this

high

value of A we can

neglect

the

thermodynamic

forces and suppose ^that the internal structure of the aggregates ^{is the B.C.T.}

equilibrium

structure with _a

density

_~P~

=

2 w/9. This is also

only

under these conditions that

we can use an

equation

similar to

(I)

which _assumes that the surface energy is localized _on the last

layer

^of

particles.

On the other hand the

magnetic

field is still low

enough

so that we can

consider that the

magnetic permeability

_p is almost constant, then the

magnetic

_energy is

given by ii II1

N~ M~ Ho

U~ (2)

with N

~

the number of aggregates ^and

M~

^the

magnetic

moment of _an aggregate. ^If we consider

a

spheroidal (prolate)

aggregate (cf.

Fig.

I) of semi-axes b and

I,

with

bli

< I,

composed

of _an

homogeneous magnetic

medium of average

permeability

_p~, the internal

magnetic

field is

constant inside the aggregate ^{and the}

homogeneous magnetic

_moment is

given by (in

_e-m-u-

~ p~ I

Ho

M~

₌ V~

(3)

4 _{" +}

n~(p~ 1)

where

V~

is the volume of the aggregate ^and n~ the

demagnetizing

factor which for _an

ellipsoid

of

excentricity

_e

= (I

b~li~)~'~

is

given by

n~ =

~~

~~

(In ~

₂

ej ⁽⁴⁾

2e -e

If the ratio of the _{two axes} of the

ellipsoid

_K =2b/L is very small, _we have

n~ 4

(1

+

log

K K ~, and

minimizing

the surface energy

(Eq. (I)) plus

the volume energy

(Eqs. (2)

and

(3))

with respect to b

gives

the behavior b _{cc a}

(Lla

_)~'~

reported

in

[7].

Let _{us see}

more

precisely

the

physical

mechanism.

H~ H~

Fig. 2, The local field, H~, _{on a}

spherical

panicle located _on the surface is lower than the local field, H~, _{on a} panicle in the bulk.

As the

ellipsoid elongates

the

demagnetizing

factor tends _{to zero,} then the

magnetic

moment and the absolute value of the energy increase

continuously

so we could expect ^{that the} average

radius,

b, of the aggregate ^{will tend} to the radius of _a

particle. Actually

_we have to take into

account two other

energies which,

_on the contrary are favorable to the

growth

of

b. One is the

repulsive

_{energy, U~,} between the aggregates ^{and has been discussed in} a

previous

_paper

ill,

and the other,

U~

is _a correction to the volume energy calculated _on the basis of _a continuous medium,

coming

from the finite size of the

particles.

^This contribution

can be understood _{as a} surface _energy which _can be calculated

numerically

and

depends

on the

configuration

of the first _{two or} three

layers

of

panicles

below the surface

[121.

In order _to understand the

origin

of this energy and _to

simply

estimate its value in _{a mean} field

approach,

let _us compare the local

magnetic

field _{on a}

spherical panicle

inside the aggregate

H~

to the local field

H~

_{on a}

particle

located on the surface

(cf. Fig. 2).

For _a

homogeneous ellipsoid

whose main axis is

aligned

with the field in the ₌ direction _we have

H~=Ho+~"~-4wn~P.

The field 4 wP/3 where P

=

M)/V~

_comes from the inner surface formed

by

the

spherical cavity

which represents the exclusion volume of _a

particle (Fig. 2)

and 4

wn~P

^{is the}

demagnetizing

field

coming

from the external surface of the

ellipsoid.

For _a

particle

_on the surface, if _{we assume} that the

polarization

P remains _constant inside the

ellipsoid,

the difference with the former _case will _come from the absence of

polarization

on a half

spherical

surface which will

give

for the surface field

H~mHo+~"~-4wn~P.

The local field _on the

particles

situated _on the surface is then lower

by

_a factor

2 wP/3

relatively

to the bulk local field. Then the real

magnetic

moment of the aggregate,

M~,

^{will be lower than that,}

M),

calculated for _a

homogeneous ellipsoid

M~

₌

M) Ni &mj (5)

where

&mi

_{= a} ^{P2 wP/3 is the difference} between the

dipolar

moment of _a

panicle

inside the aggregate ^{and the} one on the

surface;

aP=

a~p,

with

p

₌

(p~ I)/(p~ +2),

is the

polarizability

of _a

spherical panicle. Ni

is the number of

panicles

_on the surface of the

ellipsoid.

With this

cavity

model

&m,

is

positive

and

corresponds

to _a surface tension since it decreases the total

dipole

of the aggregate ^and so increases the electrostatic energy.

Actually

the value of

&mi

is

quite

^sensitive to the

precise

structure on the surface. With the notation of 11

2]

we have _more

generally &m~

= m

(M~

^{I where} m is the bulk

dipole

^of a

panicle

inside the aggregate and

M~

=

m~/m

the normalized

dipole

of the

layer

_v

(v

₌ I

being

the surface

layer). Using

for the bulk

dipole

_m

=

M)/N~

^where

N~

^{is the number of}

particles

inside the aggregate we can

generalize (5) by writing

M~

₌

M)(I _n~)

with _n~

=

zN~,(ji~ 1). (6)

~P

v

In the

homogeneous cavity

model

only

^the surface

layer

^{is concerned and} we have

(Mi

I

=

(a

^P2 wP/3 )/m, or

using

P

=

N

~

m/V~

and

a P

=

pa~,

_we obtain

(M,

I

= p

~P~/2.

In a lattice model _we have shown in 1121 that the contribution of the surface

layer

was far greater ^{than all} the others, _{so we can} still

keep only

the first

layer.

For _a B-C-C- lattice and

PIP

- oo an exact calculus

gives (M, 1)

=

0.3 for ~P

= 0.65 and

(Mi

-1)

=

0.37 for

~P

=

0.67. We did not calculate this

quantity

for _a B-C-T- lattice but _we do not think that the result would be very different. In any way this result is not far from

p

~P~/2 ^and we shall

keep

this value _{as a}

good

estimate of the relative

change

of

dipole

_{moment on} the surface. We have still to estimate

Ni

the number of

particles

_on the

surface,

_S~

=

2

wb~

_{+ 2}

w

(bile sin~'

e, of the

ellipsoidal

_aggregate. If _we consider _a surface with _a

hexagonal packing

we find N i ⁼ S~/

(2 / _a~).

_This

_packing corresponds

to the denser

planes

of the B. C-T- lattice and _we

shall take it _as the upper realistic limit of

Ni.

With these values of

Ni

and of

(Mi

I _we obtain

S~ w

~~ ~

Va

^~ 3

/

^~~~

If _we introduce the

expression (6)

of

M~

in the total

magnetic

_energy

(cf. Eq. (2)),

the _term with n~ will represent ^the contribution of the surface energy

U~.

On the other

hand,

we also need to consider the

repulsive

interactions between the aggregates. ^If we assume a structure formed

by

a two-dimensional

hexagonal

_array of

ellipsoidal

aggregates, ^each one

bearing

the _same total

moment

M~,

we can find the _moment

M)

of _a

given

aggregate, I, considered as a

homogeneous

medium,

by writing

M~

=

a~

Ho T[

M

(8)

a

I

a

The

quantity T(

is the propagator ^{of the field between} two aggregates, ^and

j,

^of

length

L whose _axes of symmetry are

separated by

a distance

d~~. It _can be obtained

easily

in the

dipolar approximation,

with all the

polarization

located _on the revolution axis of each aggregate. ^{In this} case the field of _one

aggregate

^is

equivalent

to the field H

=

«r/r~

_{of two}

charges

^«~

= ±

M~/L

located _on each

extremity

of the axis of revolution of the

ellipsoid.

Because of the

cylindrical

_{symmetry we can} consider

only

the _z component ^{of the field} in the

sum over all the aggregates. ^{Then the} average

(H~)

=

T[~.M~

over the

length

of the

aggregate of the field

coming

from the aggregate

j

defines the

quantity f~

^{which will} be

given by

rJ ^~ )

~~j /~~~/~ l. (9)

Of _course, if d,~/L ^» I, _{we recover} the

dipolar

propagator ^for

parallel dipoles

in the _same

plane

fj

=

1/d(.

The

polarizability,

_a ^~

corresponds

to the ratio

M)/Ho

for _an isolated aggregate ^{and is}

given by equation (3).

The total

magnetic

_{energy per} unit volume is still

given by equation (2)

but

with _now

M~

^{obtained from the} solution of

equations (6)

and

(8).

We thus obtain

U~ ~PH( (p~

)

(I n~)

I

~P~ 8 w I ₊

n~(p~

I I ₊

n~(I n~)

^~~~~

The ratio ~P/~P~ comes from the

equality N~V~/V=

_~P/~P~. The values of _n= and

n~ are

given by equations (4)

and

(6) respectively

and _n~

= a ~

z f~

represents ^{the effect of the}

J

repulsive

interaction of _one aggregate ^{with all the others. The value} of n~ can be calculated

by summing (9)

on a

triangular

lattice _or

by replacing

the _sum

by

an

integral

_: the _two results _are

within 2 fl _{so we} shall take the

analytical

result which is

(p~-I) jj

^~p

j~lj ^{d( oj j2~P~}

n~ = I + with

do

= b

(l

I

I ₊

ii=(p~

^I )

lPa

L~ L 3 *

The

quantity do

is the radius of _a

cylindrical cavity containing

the reference aggregate ^and

surrounded

by

a

homogeneous

^medium of average

density

~P instead of the

hexagonal

distribution of

figure

I. It is worth

noting

that the _use of

equations (6)

and

(8)

amounts to

introduce in _a self consistent way the surface energy of _an aggregate and the

repulsive

interaction with the other aggregates.

3.

Experimental

results and

comparison

with the model.

The

experiments

have been

performed

with _a

suspension

of

magnetic particles

at two volume fractions ~P

=

lo fl and 4l

=

5 fl. The

panicles

are

spherical

but

polydisperse

with _an average

diameter of 0.8 _~Lm.

They

_are

composed

of the _same

grains

of

magnetite

which

usually

compose a ferrofluid

but,

instead of

being

free in the

liquid phase,

these

grains

are

imprisoned

in _a

polystyrene

matrix. The size of the

grains

is less than 100

I

_and

even if these

grains

do _not rotate in the matrix their

magnetic

moment can

jump

from _one direction of easy

magnetization

to another

by

thermal activation since _we have

KV~

« kT where K is the

anisotropy

constant and

V~

^{the volume of the}

grains

of

magnetite. By measuring

the

magnetization

_{curve on} a dried

powder

we have verified that these

panicles

_were

superparamagnetic

and that _we had _no

hysteresis.

The

magnetic permeability, _p~,

of the

particles

has been deduced from the

measurements of the average

magnetization

of the

suspension

at several volume fractions with

the

help

^of

Bruggeman's theory

11

31.

For _a

magnetic

^field of a few _tens of Oersteds, _we obtain p~ =

37. Note that the

permeability

_was constant in this low field domain. The

suspension

_was

placed

between two transparent ^{disks whose}

distance,

L, is set

by

different spacers from 150 _~Lm to 700 _~Lm. The field is

perpendicular

to the disks and is raised very

slowly

to its terminal value

(50 Oersted)

in order _to get an

equilibrium

structure. The

typical rising

time _was 0. I Oersted _per minute and _we have verified that for

longer

times the results _were the _same.

The terminal value of 50 Oersted

corresponds

to the

completion

^of the

aggregation

_{process as}

observed

by light

transmission _: above this value there _{was no} further increase of the

transmitted

light ill.

The structure is observed with _a

microscope

whose

optical

axis is

parallel

to the field. The

images

_are recorded _{on a} computer ^{and treated with} an

image analysis

in order _to obtain the

position

of the _center of

gravity

of each aggregate. A

typical

view of the

projection

of the aggregates on a

plane perpendicular

to the field is

represented

in

figure

3A for ~P

=

5 fl and in

figure

38 for ~P

= lo fl. At the

highest

volume fraction the

projections

do _not

correspond

to an

ellipsoidal shape

^{for the}

aggregates.

This branched

shape

has _some

similarity

with the Patterns observed in the ferrofluids when the

initially ellipsoidal droplet

bends and distorts in _a

complicated

pattern ^which

depends

on the

intensity

and _on the rate of increase of the

magnetic

field

i141.

The

dependence

of the

shapes

^of the aggregates on the final

amplitude

of the field

yf-~Wj*m&~

, -, o

z,~«

~

_,

M ~j~ I&~

_~^r

* ^~

i

~i

a)

b)

Fig. 3. real stricture observed by

optical

microscopy. The view corresponds _{to a}

projection

on a

plane

perpendicular to the magnetic field (which is along the axis of observation) 3a Volume fraction of 5 %

3b Volume fraction of 10 %.

and _on the time

history

of its

application

has also been observed

recently

on micronic

droplets

of ferrofluids

[lsl.

On the other hand, on another device where the upper disk _can translate

[191,

_{we can}

experimentally

see that all the aggregates connect the _two walls of the cell.

We can measure

experimentally by image analysis

the average

distance, d,

between the

centers of

gravity

of the aggregates. ^{This is}

possible

_as

long

_as these aggregates ^{remain well}

individualized _even if their

projections

_are not

simple

disks (cf.

Fig. 3).

We obtain

d

by taking

the maximum of the

pair

correlation function _g

(r)

of the

separations

between the aggregates

ill.

In order to relate this

experimental

distance _to the semi-axis b of _our idealized

ellipsoidal

_{aggregate we represent} the structured

suspension

_as shown in

figure

I. In this situation _we have

b=dl~~~ ⁽¹²⁾

4w~a

where ~P~ = 0.69 is the volume fraction inside the aggregates.

The

log-log plot

of the

experimental

values of

b,

_versus the thickness of the cell, is

represented

in

figure

4 (solid

circles)

for _a volume fraction of lo fl and in

figure

5 (same

symbols)

for _a volume fraction of 5 fl. These values _can be fitted

by

_a

straight

line with _a

slope

of 0.67 ₌ 0.04 and of 0.68 ₌ 0.04

respectively.

On the _same

figures

_are

plotted

the values

obtained

by

the minimization of the total

magnetic

_energy (cf.

Eq. (lo)

with

respect

to

b for the _same values of L

(empty circles).

We _see that the

predictions

_are 40 fl too low for the volume fraction of lo fl but _are very close to the

experimental

values for 5 fl. The

discrepancy

at ~P

=

lo % is _{not so}

unexpected

ⁱⁿ view of the

non-ellipsoidal shape

of the aggregates. ^We also have to

emphasize

that the intemal

permeability

of the aggregates ^is

directly

obtained from

Bruggeman's theory

_:

(with

_p~

=

37 and

~~

=

0.69 _we obtain p~ =

22)

_{so we} have _no free parameter in this model.

~2

b

I O w

~~o

loo 300 500 700 IOOO

L

Fig. 4.

Log-Log

plot of the _transverse ~emi-axis of the aggregates versus the thickness of the

sample

for _a magnetic suspension of initial volume fraction ~fi =10%.

(o)Theoreticil prediction

from

equation

(10) (Slope ^0.7 ). j.) Experimental results

(Slope

0.67 ₌ 0.04) obtained by image

analysis

with the

help

of

equation

(12) to relate h and d.

~ⁱ

b

~~o

~~~

L

^{~~~ ~~~}

Fig.

5.-Same _as for

figure4

with _a volume fraction: 4l 5§F. (o)Theoretical

prediction

from

equation (10)

(Slope

0.68). (.)

Experimental

results obtained

by

_image

analysis (Slope

0.68 ₌ 0.04).

It appears that this model is

quite

efficient in

predicting

the average size of the aggregates ^if the volume fraction is not too

high.

We have calculated for the same

magnetic suspension

at a

volume fraction of 5 % the size of the aggregates ^for a much

larger

range of cell thicknesses _: lo < L

< lo 000 _~Lm. The results are

presented

in

figure

6

(solid curve)

on a

log-log plot.

We first notice that _we do _not have a

straight

line there is no well defined power law defined for the whole range of distances. For low values of L 10 _~Lm

< L _< 50 _{~Lm we} find _an exponent of

0.35,

for the intermediate range where _we have done _some

experiments,

_we find _an

exponent ^close to

2/3

and in the range lo 000 _~Lm

< L

< 20 000 _{~Lm we} find 0.85.

Ultimately

when the effect of the surface tension becomes

totally negligible

the exponent ^{will tend}

towards

unity

since in this _case all the terms in the energy

only depend

on the ratio

b/L. This is the _case for the dotted _curve in

figure

6 which represents ^the

prediction

of the model without the surface energy

(n~

₌

0).

The other limit where _we

neglect

^the

repulsive

oo

/

i o

b

_, /

, , , , , / / / /

O-1 '

Fig.

6. Theoretical prediction of the _transverse semi-axis, b, of the aggregates versus the thickness L of the sample for _a volume fraction of 59b. Full

theory

equation (10).

Equation

(10) with n~ = 0 _no surface tension.

Equation

(10) with _n~

= 0 _{: no}

repulsion

between aggregates.

energy (n~ ₌ 0

)

and

only

^{take into} account the surface energy is

represented by

the dash-dotted

curve. For

large

thicknesses _we tend _{to an} exponent 2/3 in the range loo

~Lm-700

_{~Lm we} have

a

slope

of 0.57 and _at the

beginning

of the _{curve a}

slope

^{of 0.33.} Note

that,

with the value of the surface tension

given by [71,

the surface effect is dominant _at low thicknesses and in _our

experimental

_range the values _are

only

20-30 §b too low. This would _not appear if _we take the

dipolar approximation

for the surface energy

equation (I)

which

gives

much lower values of b.

Other

experiments

with _a

larger

_range of thicknesses would be useful to

verify

the

predictions

of the model and _{we are}

designing

_{a new} sealed cell which operate ⁱⁿ a

larger

range. Some similar

experiments

have also been carried out

recently by

Liu et al.

[lsl

_{on a}

suspension

of

rigid

ferrotluid

droplets. They

have

explored

the range of thicknesses between lo and 500 _~Lm and found the _same

qualitative

behavior _{as we}

predict

_: there is _no constant

exponent

^{for the variation of b with L and} at low thicknesses _we tend _{to an} exponent ^close to

1/3.

4. Conclusion.

We have shown that it _was

possible

to

predict

the

equilibrium

structure induced

by

a

magnetic

field in _a

magnetic suspension

if _we introduce both the

repulsive

interactions between the aggregates ^{and the surface} energy due to the finite size of the

panicles.

This

study

was

performed

with _a

magnetic suspension mainly

because _we have

precise

measurements of the

magnetic permeability

which furthermore is much easier to obtain than the

permittivity

of _a

suspension (in

the latter _case,

problems

_are met due to the ionic clouds

surrounding

the

particles

and _to the

charge

_on the

electrodes).

We have made _some

preliminary experiments

with _an

electrorheological suspension composed

of silica

panicles

activated

by

water in _a low

viscosity

silicone oil. The cell _was formed of _two

glass

disks coated with transparent electrodes.

Unfortunately

our

particles slightly

aggregate ^{under the effect of} electrostatic forces

they disintegrate

under

mixing

but when the field is turned off, the Brownian motion

itself does not allow to recover a

homogeneous

state. In these conditions _we have _some

irreversible

aggregation

and the model based _{on a} minimization of the electrostatic energy alone will

probably

fail. On the other hand the aggregates were not so well defined and the

uncenainty

_on the distance _was

larger.

Nevertheless _we have tried _some

preliminary experiments

_on this

suspension

and _we have found that in the _same range of thicknesses 100

< L

< 700 _~Lm the size of the aggregates was

increasing

with _an exponent close _to 0.6. As

already

said in the introduction the electrostatic _case is different because the

charges brought by

the electrodes cancel the

depolarizing

field of each aggregate. Then, at first

sight,

the model

should

apply

^if we set the

repulsive

interaction _{to zero} (n~ = 0 in

Eq. (9))

and indeed the exponent ^{of 0.6 is}

quite compatible

with _a surface tension effect

(cf. Fig. 6).

Nevenheless

experiments

with

panicles

which do _not stick

together

or on the electrodes should be made in order to get

equilibrium

structures in the presence ^of an electric field.

A

systematic investigation

with different

particle sizes,

different

permittivities (which

can be varied

by changing

the

frequency

of the

applied field),

should enable _us to decide if this model allows the size of the structures in electro and

magnetorheological

fluids _to be

predicted

or if it is restricted to some set of

parameters.

^The

prediction

of the _structure induced in

quiescent

suspensions by

an external field is the first step ⁱⁿ the

study

of _a broader class of

ordering phenomena,

ⁱⁿ

panicular,

shear induced

ordering

which is found in

complex

fluids like micellar fluids, nematics,

suspensions [16,

171 and

electrorheological suspensions.

In these

last _{ones we} have observed the formation of

stripes

of

particles aligned parallel

or

perpendicular

to the flow

[18, 19]

in the presence of to an electric field and of _a sinusoidal shear.

Actually,

^these E-R- _or M-R- fluids _are very convenient for the

study

^of shear induced

ordering

since _{we can} very

easily modify

the energy of interactions between the solid

particles by changing

the

applied

field. Such systems ^should

help

to the elaboration of _a

theory

able _to

explain

^{shear induced}

ordering.

Acknowledgments.

This work was realized with the financial

suppon

^{of Rh6ne Poulenc. We are also very}

grateful

to A.

Audoly

and B.

Gay-Para

for their technical support.

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