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Visco-elastic studies on an electro-rheological fluid
Colloids Elsevier
and Surfaces, 18 (1986) Science Publishers B.V.,
VISCO-ELASTIC
D. BROOKSa,
STUDIES
J. GOODWINb,
293-312 Amsterdam
293 -
Printed
in The Netherlands
ON AN ELECTRO-RHEOLOGICAL
C!. HJELMb,
L. MARSHALLb
FLUID
and C. ZUKOSKIC
aLaser Engineering, London (United Kingdom) bDepartment of Physical Chemistry, University of Bristol (United Kingdom) ‘Department of Chemical Engineering, University of Illinois, Urbana, IL (U.S.A.) (Received
13 September
1985;
accepted
in final form
11 November
1985)
ABSTRACT The fluid used for these studies consisted of lithium poly(methacrylate) dispersed in a chlorinated hydrocarbon oil. Shear wave propagation was employed to measure the storage and loss moduli at a frequency of 1200 rads-’ . The parameters varied were field strength, volume fraction and water content of the particles. Optical microscopy was used to measure the particle size distribution whilst electrophoresis was used to enable the electro-kinetic potential to be measured. This showed that the particles were positively charged with a potential of circa + 45 mV. Further characterisation of the suspensions was carried out by conductivity measurements and the conductivity of the systems was shown to be due to the counter-ions in the diffuse layer which balances the surface charge. The moduli of the suspensions were found to increase with applied field to a maximum value and then decrease. In addition, the storage moduli were much higher than the loss moduli for all samples at low applied fields, but at high fields both moduli were of similar magnitude. These experimental data will be discussed in terms of the inter-particle forces and the structure in the suspensions.
INTRODUCTION
The pioneering work of Winslow [l] clearly demonstrated that marked changes in the rheological properties of suspensions of particles of high relative permittivity dispersed in an oil of low relative permittivity occurred during the application of a high electric field. Since then many systems have been examinec [2] but much of the research has focussed on the engineering applications and so a quantitative description of the electro-rheological effect has not been available to date. The work presented below goes some way towards providing a detailed analysis of the phenomenon. The fluid used here is a commercial product kindly supplied by Laser Engineering of London and consists of lithium poly(methacrylate) particles in a chlorinated hydrocarbon oil. The principal requirement of an electrorheological or ER fluid is that, on the application of an electric field, the
294
system should change from a low viscosity liquid to a high modulus gel with a minimum of loss in both the electrical and rheological senses. Initially, the suspensions were characterised by microscopy to give the particle size distribution, followed by micro-electrophoresis to give the electrical properties of the particles and finally by conductivity measurements as a function of particle concentration in order to assess the ion distribution around the particles. Two types of rheological measurement were used to follow the changes in the properties with applied field, A static yield stress experiment was used followed by a study of shear wave propagation to give the storage and loss moduli at a moderately high frequency. EXPERIMENTAL
The ER fluid was manufactured by Laser Engineering and was used as supplied. By equilibrating the systems with atmospheres of known humidity, the dispersions had been prepared with well-defined water content in the particles, which consisted of an hygroscopic polyeletrolyte - namely the lithium salt of poly(methacrylic acid). Particle size determination A sample of the fluid was diluted with 60/80 petroleum ether and centrifuged at 4000 rpm for 5 min. The solids were washed five times with petroleum ether, and then a drop of the final suspension was placed on a microscope slide and the solvent allowed to evaporate. Optical micrographs were produced at a magnification of X400. A photograph was taken of a Carl Zeiss stage micrometer to serve as a calibration. The particle size distribution was obtained using a Carl Zeiss TGZ3 Particle Size Analyser, measuring at least 1000 particles for each sample. Similar distributions were found for each sample and the mean number average diameter was found to be 15 pm with a coefficient of variation of 38%. Determination
of electro-kinetic
potential
Electrophoretic mobility measurements were made using a Pen Kern System 3000 automated electrophoresis apparatus. Prior to use the quartz cell was flushed with the chlorinated hydrocarbon continuous phase which was then left in the cell overnight to “condition” it by allowing sufficient time for adsorption equilibrium to be established with any ionic species. A sample of ER fluid was diluted with serum obtained by centrifugation of a sample of fluid with the same water content, and the mobility distribution was determined. The measurements were repeated ten times to ensure that the readings were consistent - a precision of better than + 10% was obtained. A mobility distribution is illustrated in Fig. 1 with a mean
295
-19
-048
Electrophoretic
Fig.
1.
Electrophoretic
6 Mobility
.O% /xlO*m*Vsd
mobi1it.y
distribution
results
from
the
Pen
Kern
system
3000.
electrophoretic mobility of 6.8 x 10-l’ m2 V-’ s-l. Because of the low ionic strength of this non-aqueous colloid, the electro-kinetic potential could be calculated by making use of the Hiickel Equation [3] :
where { is the electro-kinetic potential, u is the electrophoretic mobility, q. (0.075 Pas) is the viscosity and E is the permittivity of the continuous phase with a relative permittivity of 8.5. Hence, at low concentrations the value of c = + 45 mV was calculated.
Suspension
conductivity
The conductivity of the ER fluid was determined as a function of the concentration of particles. Volume fractions in the range 0.1-0.35 were employed and a Wayne-Kerr AC bridge was used for the measurements. A clear increase in relative conductivity was found over this range of concentrations which indicated a particle-dependent contribution. The experimental data are plotted in Fig. 2 and are discussed in detail below.
Rheological
measurements
Two types of rheological experiments were carried out. Firstly, a static shear test was used to give a yield stress. Samples of the ER fluid were placed in an annular slot between cylindrical electrodes, a field was applied, and an increasing torque applied to the outer element until motion was observed between the two elements. The field was increased and the procedure repeated. Figure 3 shows some experimental data, and, in this type of experiment, a steep monotonic increase in yield performance was found over the range of fields used. The second type of experiment performed was the determination of the wave rigidity modulus from the measurement of the shear wave propagation
296
velocity. The apparatus used was a Rank Bros. Shearometer which records the time that a torsional shear wave takes to travel between two parallel plates. These were made of stainless steel and electrical leads were soldered to each plate so that an electrical field could be applied to the samples. Plate separations of up to 2 mm were used with applied potentials of up to 3 kV. In all cases, the currents observed were less than 10 mA and no heating effects were observed. The wave rigidity modulus, E, was calculated from the wave velocity, u, and the suspension density, ps: G = v2ps As the propagation time was measured at different plate separations, the attenuation of the wave was estimated. When this is expressed as the characteristic attenuation distance in wavelengths, b, the storage and loss moduli, G’ and G”, can be calculated from [4] : G’ = G (1 - b2)/(1
+ b2)2
G” = 2G b/(1 + b2 )2 The frequency of the shear wave was approximately 1200 rad s-l . A typical set of data are plotted in Fig. 4. There are two important I
I
1
I
I
I
0004 Volume
Fraction
9
Applied
Fig. 2. Relative conductivity as a function + 40 mV (________________); + 45 mV ( Fig. 3. Static
shear force
as a function
of applied
of volume
fraction:
Voltage
0 mV
) field for fluid at @ = 0.30.
/ kV
(-
-
-
-
-);
I
50
0
I
I
200
I
I
400
Applied
Field
I
1
.
I
_
603 E
/Vmm-’
Applied
Field
E /Vmm-’
Fig. 4. Shear moduli as a function of applied field for a fluid at 4 = 0.223, content of 29.5% and at a temperature of 25.2’C. ?? (0); G” (0); G’ (0).
with a water
Fig. 5. The volume fraction dependence of the storage modulus. $J = 0.223 -); r$= 0.14 (-- - -); @J= 0.09 (__________), f$ = 0.18 (l
(-
);
l
features that were common to all the curves obtained. Firstly, there was a maximum in the moduli which moved to higher fields with decreasing volume fraction (see Fig. 5). The second feature to note is that at low fields G’ > G”, but at high fields this was not the case and then G’ < G”. Although in Fig. 4 the condition G’ = G” occurred at E (the field strength at which the maximum in the modulus was found), this was not always the case and the precision of the determination of the wave attenuation, and hence b, was not sufficiently good to identify this point unambiguously. DISCUSSION
Suspension
conductivity
The ER fluids are suspensions of particles of high conductivity in a continuous phase of much lower conductivity. The conductivity of the continuous phase, co, is related to the concentration of ions in solution by the relationship:
(1)
298
the valence, number concentration where zi, ni and Di are, respectively, and diffusivity of the i-th ionic species, e is the fundamental unit of charge, h is the Boltzmann constant and T is the absolute temperature. The summation is carried out over all the ions in solution. Using Eqn (l), the Debye-Htickel screening parameter, K, can be estimated from the following expression, if all ions are assumed to have the same diffusivity:
where E is the permittivity of the continuous phase, D, is the ion diffusivity in water which has a viscosity r), , and v. is the viscosity of the fluid medium. If the value of D, is taken as 1.3 x 1O-9 m2 s-l (i.e. the. diffusivity of a sodium ion), then an approximate value of KU = 4.8 x 10T3 can be calculated. Hence, using the low surface charge model for the diffuse double layer, a surface charge density of 2.17 x 10d9 Cmm2 corresponds to the measured electro-kinetic potential. This gives an average of 1.35 x 10 lo proton charges per particle. Theories are available to relate the volume fraction dependence of the suspension conductivity to the conductivity of the disperse phase. Chiew and Glandt [5] have recently reviewed the work in this area and derived an expression which extends the region of validity of Maxwell’s mean field calculation for uncharged particles. This reproduces the volume fraction of dispersion conductivity up to the percolation threshold, except when the particles are much more conducting than the continuous phase. In the latter case, the model underestimates the dispersion conductivity but, even under these conditions, the model works reliably up to volume fractions of 0.2. The expression for the relative conductivity can be written: (5
-
=
1 + 2A$l+ (K, - 3A2 )@2 (3) (I-
00
GA)
where u is the conductivity particle conductivity u, , and A=---
of a suspension
with volume
fraction
$ and
a-1 a+2
where cx = u, /a0 and K, is a weak function of volume fraction tabulated in Ref. [ 5 ] . The dashed curve in Fig. 2 was calculated from Eqn (3) assuming that u, /a0 > > 1. Thus even by giving the particles an infinite conductivity, the model underestimates the observed trend. Discrepancies of this kind have recently been observed [6] for the conductivity of concentrated aqueous colloidal systems and the results explained by taking into account the contribution of the counter-ions to the dispersion conductivity. For a suspension of particles with a surface charge density of q C m-‘, the counter-ions contribute
299
-
344
ions to the solvent ionic concentration, where z, is the counter-ion valency. Thus as the volume fraction is increased, the background ionic strength of the suspension rises. By including this increase in ionic concentration (but still neglecting contributions to the suspension conductivity due to charge convection and conduction in the diffuse layer which are also positive but small contributions), it was shown that the relative conductivity of a suspension of charged but non-conducting spheres can be approximated by
[61: u -ZZ
00
[1 + Cl
l-4 i 1 + 0.5@
where the counter-ion c=
I
(4)
contribution
- 32, DC qe@ (1 - qb)ao,hT
This assumes that the electrolyte is symmetrical and each ion has the same diffusivity, D, . Equation 4 was used to calculate the suspension conductivity utilising the surface charge density calculated from the electro-kinetic potential determined at low volume fraction. This gives a much better description of the data than Eqn (3) and therefore it can be concluded that the conductivity of these suspensions over the volume fraction range studied is primarily due to the counter-ions associated with the particles rather than the increasing volume fraction of highly conducting particles. Whilst the lithium poly(methacrylate) particles are highly swollen with water containing a large number of lithium ions, it is not clear that a charge flux or current will be transmitted through these particles when they are suspended in an organic solvent and subjected to an applied field. For this to occur, ionic species would have to be present which would be soluble in both the polar and non-polar phases and also be capable of transferring across the water-oil interface. An analogous situation occurs in a suspension of metal particles in an ionically conducting fluid where an electrochemical reaction must occur for there to be charge transfer at the solid-liquid interface. At present we feel, that under the conditions that these suspensions are used? that there is no significant ionic flux across the interface and, therefore, in terms of suspension conductivity, the wet polyelectrolyte particles behave as non-conductors. This view is entirely consistent with the observation that, at high field strengths, strings of particles are formed between the electrodes with no resultant increase in conductivity such as would be the result of ion transfer between the particles.
300
The high water content and ionic strength are, however, important when modelling the forces between the particles. Because there is little or no charge flux across the particle-solvent interface, the particles can be treated as perfectly polarisable dielectrics and will thus assume a constant surface potential when placed in an electric field. If, as a first approximation, one treats the particle as spheres which have no surface charge then the electrical potential, \k, outside the particles can be found from the solution of Laplace’s Equation:
v2\Ir = 0
(5)
Far from the test particle application of the external of the field is E then: \I, = ErcosO
the potential assumes a linear form due to the field applied in the z-direction. If the magnitude
asr+m
At all particle surfaces the potential will assume a constant tangential components of the applied field are zero: \I, = a constant
on all particle surfaces
(6) value and so the
(7)
The solution to Eqn (5) subject to the boundary conditions (6) and (7) is the problem of highly conducting particles suspended in non-conducting fluids for which solutions are available [7-91 for one or two spheres or for arrays of spheres. Once the electrical potential around the sphere is known, the force on the particles can be calculated from the Maxwell stress at the surface of the particles [lo] and again solutions are available. The solutions for these forces are used below for the development of a model for the rheological properties of the suspensions and the full details of the method of calculation are given in the primary references. The potential around a reference particle depends on the position of all the other particles in the suspension and so the force acting on the reference particle depends on the particle distribution function in addition to the surface potential. In order to achieve some visualisation of the structure of the system, a dilute dispersion of particles was placed between two electrodes in a cell mounted on a microscope stage. When a modest field (50 kV m-i ) was applied between the electrodes the particles were seen to undergo rapid motion for a short period of time (-- 20 s). After this period strings of particles were formed between the electrodes with no free particles being observed. The formation of strings of dielectric particles suspended in both organic and aqueous media under the influence of large AC and DC fields has been reported in the literature [lo]. As the volume fraction is increased individual strings are difficult to pick out, although a stringy structure of particles in close contact can be seen at moderate to low volume fractions.
301 MODELLING
THE VISCO-ELASTICITY
OF ELECTRO-RHEOLOGICAL
FLUIDS
In this section a model is derived which describes the dependence of the high frequency limit of the shear modulus as a function of volume fraction and applied field. A form of the frequency dependent storage and loss moduli are also presented and these are tested against the experimental Models for the visco-elastic behaviour of suspensions rely data. predominantly on descriptions of the pair interaction of particles appropriately averaged to take into account the pair distribution function [15]. A radially symmetric distribution function is normally used, however in this work the model must use an anisotropic pair potential. This should result in an asymmetric pair distribution function being produced at the elevated field strengths used with electro-rheological fluids. Estimation
of the limiting storage modulus,
G(m)
At the high frequency limit, there is only a small loss associated with the response of a suspension to an oscillatory strain. For a linear visco-elastic material, this high frequency limit of the modulus, G(m), is simply related to the energy stored per unit volume, U/V, when the material is subjected to a uniform shear strain of magnitude y as:
u GPh2 -=V
2
(8)
It is assumed that the system contains N particles in the volume V and that the particles are of uniform radius as well as interacting as hard spheres in the absence of an applied field. The spherical particles are composed of a highly conducting core of radius a*surrounded by a strongly anchored steric layer of thickness L. This barrier stabilises the particles against coagulation by van der Waals forces and induced dipolar forces. The compressibility of this layer will be discussed below in more detail but it is the repulsion generated by this layer when the particles are brought together by the applied field which prevents electrical contact between the conducting particles. The repulsive force is a very steep function at small separations [16] so that the minimum separation between adjacent spheres can only change very slowly with electrical field strength. At any field strength, the particles are separated by a centre-to-centre distance of r = 2a, that is for r = 2 (a* + 6), where 6 < L. Whilst ~7* is only a weak function of E, variations in the suspension structure due to changes in 6 will be small. For any particular configuration of the N particles, when the system is strained the energy stored per particle is Ui and is written: 6Xi
302
where _Fi is the force required to displace the i-th particle a distance 63~~. The total energy stored will be the sum of Ui over all the particles in the suspension. Consequently, the shear modulus can be written:
If the strain is sufficiently small that a linear visco-elastic response is maintained, and also if it is assumed that the particle interactions are pairwise additive, the force J’i can be calculated by summation over all inter-particle forces. Expanding the pair interaction forces (_Fij) in a Taylor series about the equilibrium position of the j-th particle, and noting that at equilibrium (i.e. prior to the application of the strain) the total force acting on the i-th particle is zero, then _Fi is written: Fi
C
=-
V_Fij
’
6sij
+
0
(11)
(3cij')
j
where V_Fij is the gradient of the force on the j-th particle due to the i-th particle evaluated prior to the application of the infinitesimal shear strain. The negative sign in Eqn (11) arises due to a change from the externally applied force required to cause a displacement to the force which must be overcome. A linear shear gradient is applied in the x-direction so that 6cij = _yZij, where .Zij is the separation of the two particles in the z-direction. Therefore we can write: G(m)
=
--1:
2V
i=l
c
[V~ij
i=l
lKS
&]zij*
(12)
The factor of l/2 occurs in Eqn (12) in order to prevent each interaction being counted twice. However, the positions of the particles are not on static lattice sites and are only known in a statistical fashion and Eqn (12) must be ensemble averaged to give the most probable value of G(m). This averaging process is simplified by the restriction to pairwise additivity and so: (13) where z is the z-component of the separation between a test particle and a particle at r. The pair distribution function is written as a function of the vectorial separation to emphasise that the structure of the suspension is not necessarily isotropic and is in fact unlikely to be so when under the influence of a high external field. Under conditions where thermal motion contributions are negligible, this expression is a generalised version of that derived by Zwanzig and Mountain [ll] who developed their treatment from the statistical mechanics of condensed phases. In order to reduce
303
Eqn (13) to a more tractable form, approximations for _Fii and g(r) can be made which simplify the mathematical form but retain the essential physics of the problem. The effect
of external
fields on the pair distribution
function
The elastic properties of an electro-rheological fluid are modelled as arising from induced dipole-induced dipole interactions. The force between the particle pairs therefore depends on both the absolute separation and the angle between the line of centres with the direction of the applied field. In the shearometer experiments utilised in this work, the strain was applied perpendicularly to the applied field. Hence, in the analysis given below the electric field is applied in the z-direction whilst the strain occurs in the x-direction. Prior to the application of the external electric field, the particle-pair distribution function is assumed to be radially symmetric and to be adequately described by the pair distribution function for hard spheres. Upon the application of a large external electric field, the particles arrange to form strings such that g(r) = g(r, 0). The form of g(r, 0) can be estimated from the distribution function observed by neutron scattering from ferrofluids under the influence of a uniform magnetic field [12,13]. Particles in these suspensions have a permanent magnetic dipole and the applied magnetic field causes the particles to align into chains. Pynn and Hayter [12,13] have calculated g(r) for these systems using the mean spherical approximation where g@) is written as the sum of Legendre polynomials. These authors found that the distribution function could be modelled well by retaining only the terms up to the second polynomial, P,(cosO) viz. g(r)
=
go(r)
+
g2
(rFJ2
(co@)
As a first approximation in the application of this result rheological fluids, g(_r) is rearranged to the following form: g(c) = A(r) -I B(r) cos20 where B(r) accounts top and bottom of both A(r) and B(r) and the suspension ferro-fluid volume g(_r) vanishes due to The pair interaction
(14)
to electro(15)
for the increased probability of finding particles at the the test sphere (0 = 0 and r, respectively), and in general will depend on both the magnitude of the applied field volume fraction, Pynn and Hayter showed that as their fraction approaches close packing, the asymmetry in particle crowding. force
The electrical interaction force between pairs of particles decays rapidly over a distance comparable with a particle diameter. Consequently, the influence of particles sited at a distance greater than a few particle diameters
304
apart on the value of G(m) will be negligible. Thus in developing an analytical expression for G(m), only nearest neighbour interactions are considered. At the field strengths commonly used with electro-rheological fluids (-5 x 10’ V m-l ) the influence of thermal motion on the positions of the particles is not important, the electrical forces are instead balanced by strong repulsive interactions indicating that where the electrical forces are largest the stabiliser layers are in contact (i.e. the centre-to-centre separation is 2~). This results in the formation of chains of particles. The model developed here for g(_r) is based on this concept and also the maintenance of a constant density of nearest neighbours after the application of the electric field. Let no be the number of nearest neighbours which are distributed in a volume u. prior to the application of the field. When the field is turned on, this shell of nearest neighbours rearranges so that a certain fraction of these particles, X = n2 /no, come into contact with the test sphere and are distributed on average as cos’ 0. The remaining n1 = no - n2 spheres are arranged symmetrically about the test sphere so that the total number concentration of nearest neighbours remains at no/u0 . The number of particles in the first shell can be found from [14] : = 27rN/V j s g(r, 0) sin Or2 drd@
n1 +n,
(16)
0 0 where rm is the location of the first minimum in g(r). As an approximation, we will let A(r) = Ap6(r - rl ) and also let B(F) = B,6(r 2a), where 6(r) is the Dirac delta function. Carrying out the integration in Eqn (16) and equating rz1 and n2 with symmetric and asymmetric contributions, respectively, to g(_r) yields values of A, and B, : n1
v
A,
= -
B,
3n, V = 8nNa2
The the the field
radius of the symmetric shell, rl , is found from the constraint that overall density of particles in this first shell remains constant after field is applied. The volume occupied by the first shell prior to the application can be approximated as:
(17a)
4~rNr ;
(17b)
4n vo = -j--(2a)3tm
(18)
where 4, is the maximum packing fraction approximation corresponds to a pair distribution no V6(r g(r)
=
4nNr
*’
F* )
for the suspension. function given by:
(This
(19)
305
where r* = 244, results in: rl
/@)1’3 .) Application
of the constant
density
constraint
= 2a ( 4,i:“h)1’3
(20)
The electrical force due to an external field acting between a pair of polarisable spheres changes sign as the orientation of the line joining the particle centres varies from 0 to 7r/2 with respect to the field. At 0 = 0 the force is attractive, whilst at. 0 = x/2 the force is repulsive. For angles other than integer multiples of 0 *and n/2, a torque is applied to the particle pair which varies as sin 0 cos 0. The electrical pair interaction force, _FFj, can thus be broken up into three components: _F?j(c) = - fa2E2 [(f~ ( r) cos' 0 + f2(r) sin’ O)i +(f3(r) sin@ cosO)@]
(21)
where fl (r),f2 (r),and f3 (r) are dimensionless functions which depend only on the centre-to-centre separation of the spheres. The calculation of Davis [ 71 and Jeffrey [S] for pairs of highly conducting spheres in a nonconducting fluid, i.e. (up /uO > > l), indicates that fl (r) and its derivatives are much larger than f2(r); f3(r) and their derivatives. Consequently, only fl (r) will be considered in the analysis. The strongly attractive electrical potential which causes the particles to align into chains is opposed by a repulsive potential which arises from the interaction between layers of steric stabiliser at the particle surface. The summation of these potentials will give rise to a minimum in the potential energy curve and ensures that - VFij is positive at the most probable particle separation. The exact functional form of the repulsive potential is not well understood at, present [16]. The scaling theories of deGennes [17,18] and the results of Klein and Pincus [19] suggest that when the surface-tosurface separations are small compared with the stabiliser layer thickness, the repulsive force per unit area between flat plates scales as 6-3, where 6 = (r - 2a) is the pair separation. This result was predicted for polymeric stabiliser layers in both good and poor solvents. However, the force determined experimentally between compressed layers or irreversibly adsorbed polymer on crossed mica cylinders was found to be a repulsion which increased much more rapidly than a_3 [20,21]. Due to the large magnitude of the attractive potential at small 6 in the case being modelled here, we expect any stabiliser layer to be highly compressed and the resulting repulsive force to be a very steep function of pair separation. Therefore, a repulsive force, _F[j, is chosen to reflect these expectations viz. - n* “G=Rf
i (22)
(Iwhere z = L/a, R is a constant
describing
the polymer
properties
but which
306
is independent of electrical field strength and II* is the order of the interaction. It is expected that n * > 3 and indeed it is shown below that the experimental results given here indicate that n * > > 1. Note also that _FFj is radially symmetric. The total pair interaction force is found from the sum of _FFj and _Fij. For our purposes this can be written: &j
,. = EU’ E 2 (/If, - fl (r) cos @)c
where fl = R/ea2 E2 and
(23)
f, = (z/S)” *.
The shear modulus Rewriting Eqn (13), taking into account (15) and_Fij(r) in Eqn (23), results in:
G(m)
=
-a[$)2ea2E2if[[
-
-
X
+
2fl
-
df, dr co9
r
[[A@&(r-r,)
cos2
dfr
0
- 9n,&E2
l(1
- X)4/3
4n(l
6(@/@,
15
+ 6)
-
r
Wfr(j
35
a;
@1
drdOd@
+ 24p -fr (h)
where r = a* i. It should separated by 26. Thus, i2 fl (i) can be written [B] : 6-l
32f 1 0-l)
r^,
63i2
be recalled approaches
+1/3ln6
(In (2) + 2y-ln6)2
1) + Wfr(h
J
(24)
)
15i,
+ 4LQfr(rl ) i5ai
f-1
35
Eqn
cos2 0
2d;
- U2’3
afl(rl ) -- 24 -fdb) a;,
--~ 4 afl h) 35
from
i-B@&(r--2a)cos2
I
yields:
--___4
fl
-
0 sin2 0 cos2 * +
Carrying out the integration
=
of g(r)
cos4 0 sin 0 cos2 @ + sin2 @ sin 0 cos* 0)
rJ‘
G(*)
form
sin3 0 cos2 Cp
+ fl z
i
+A
the
J
1
(25)
that at F = F2 the particles are 2 as 6 -+ 0. For small values of 6,
(26)
307
where r= 0.577216 is Euler’s constant. fi (r) Under the decreases rapidly as 6 increases. described above, 6 will be of the order of l-20 3.5 times a particle radius. Therefore, G(m) is of particles located at r2 and terms evaluated neglected. Similarly for 6 < < 1, ldfi (6)/drl> > > If,(S)/ and Eqn (25) can be simplified:
G(m) = -’
n”‘E2e 357r (1 + 6)
[- dfr/d,]
is singular as 6 -+ 0 but experimental conditions nm while rl will be 2.5dominated by the effects at rl in Eqn (25) can be > Ifi (6)l and Idf,(6)/drl
(27)
where
dfT = dr
-df,
-- 70 df, -
d;
3
di
1
r = (2$26)
For G(m) to take on realistic values, dfr/dr must be negative, while the functional form of fl and f, have been chosen to ensure a positive curvature to the pair interaction energy at the most probable separation, F2. It is not obvious how this curvature will depend on /3 (which it will be recalled is a function of E). In order to determine how the curvature of the potential energy scales on p, - dfr /di Ii, was evaluated for several values of /3, Land rz. For E at lo’-104, it was found that to a high degree of accuracy this function could be written as:
dfT
-
d;
= ;
KP-”
1
where m is a positive number m n
0.79 3
0.52 4
0.41 5
0.25 7
which has values of: 0.19 10
0.07 25
m was also found to be insensitive to the magnitude of fl for @” < 1. Thus, for a sufficiently steep repulsive interaction, G(m) will scale on E*, be insensitive to particle size and vary slowly with volume fraction. (In general, nz is expected to be a weak function of volume fraction.) It was also found that 6/r, which was of the order of 10-3-10-4 (corresponding to 2-20 nm for the experimental system), changed slowly with /3 as would be expected from the steepness of the repulsive barrier. As p decreased however, corresponding to an increase in E, 6 decreased as expected. From this numerical work, it was found that Eqn (27) could be approximated well by: G(m)
= n2qkCE2+m
(28)
where m can be calculated by assuming an order for the repulsive potential, and C is a fitting parameter which will be independent of E and, for
308
sufficiently large n, independent of particle size. However, both C and m are unknown functions of the properties of the steric stabiliser layer. For the systems used here, C and m will have constant values and, once fixed, the changes in the visco-elastic properties of the suspensions with field strength and volume fraction should be predicted by Eqn (28). In deriving this expression, no constraints have been placed on n2, except that its value should be less than ~1~. While there is little doubt that n2 must be a function of volume fraction, the number of touching particles is unlikely to change with field strength once the particle chains have been established. At the field strengths used in these experiments the electrical forces acting on the particles dominates the pair interaction. It can be concluded that, at a fixed volume fraction, the high frequency limit to the elastic modulus will be a (sensitive) function of 6 and be independent of particle size. Comparison
of the model with experiment
Direct comparison of the predictions of Eqn (28) with the data presented in Figs 4 and 5 is hindered as the wave rigidity modulus could only be measured at one frequency. It was possible to extract the storage and loss moduli from the experimental data by utilising the attenuation of the wave [4], but the equivalence of G’ with G(m) is only a good approximation when G’/G” < < 1 and this was rarely the case, and this ratio varied with applied field. However, by utilising the constitutive equation that describes the behaviour of the simplest visco-elastic liquid, i.e. the Maxwell Fluid, a better test of the model can be made. For the purposes of this paper only a simple Maxwell model is used with a single relaxation time, t, . Clearly, a spectral response would be more realistic but the added complexity of using the generalised Maxwell model is not warranted at the current stage of development and the underlying physical arguments are unchanged. The Maxwell model consists of a spring in series with a dashpot and the characteristic relaxation time has a value equal to the ratio of the dashpot viscosity to the spring modulus. The frequency dependence of the dynamic moduli are given by the following relationships:
G'(w) =
(Gm I* Woo) 1 + (wt,
(2W
)I
and as w -+ 00, G’ + G(m) G”(o)
=
(tit, )G(Y 1 + (wt, )*
and with w -+ 0, G” -+ q(O), the zero shear viscosity, t nl
= r?(O)/G(=‘)
(29b) and
309
The origin of the elasticity in an electro-rheological fluid is the forces acting between the particles which result in a network structure, and the viscous loss is associated with the diffusive motion of the structural components. The diffusion of the particles is a function of both the hydrodynamic interactions and the electrostatic interactions between a particle and its neighbours. Equation (28) gives the relationship for G(w) where it can be seen that G(m) scales as @L?X~+~ rz2. However, we do not at present have a similar expression for the zero shear viscosity and so as an initial approximation we will assume the viscous term to be a constant independent of the applied field variation and we can write t, as: trill = K/c+hE2””n2
(30)
Taking the scaling of the viscous loss as the constant K implies firstly that the structural arrangement is independent of field strength and secondly that, as the particles are at the same close separation distance at all fields, the hydrodynamic interactions are constant and dominate the viscous loss. From Fig. 4 it can be seen that the storage and loss moduli are equal at the field strength which gives the maximum value of wave rigidity modulus. Therefore, at this field strength, the relaxation time and the experimental time are equal and we will use a bar above the frequency, time and field strength to indicate this point on the curves, i.e. when E = E, t,
= l/W =I t
at all other fields, t,
f
l/w
Therefore: wt,
=
(31)
Substitution of Eqns (25) and (28) into (26) yields the required for the frequency dependence of the storage and loss moduli:
G’(w,E)
expressions
=
(32a)
(=b)
Figure
6 shows the comparison
of Eqn (32a) with the experimental
values
310
I 0
I
I
I
200
I
400
Applied
Field
Fig. 6. Calculated n2 =4 (. . .); =3(--_--_-), n2
600 E
storage @=O.lS,
I Vmm-’ modulus n2 =4
curves (-
as a function ); 4 = 0.14,
of applied field. nz = 3 (---);
@ = 0.223, $J= 0.09,
of storage modulus for each of the volume fractions investigated. The only fitting parameters are the values of n, and C. At i?, w = W and from Eqn (32a): G:,
=
n2 WS2 2
where Gk is the value of the maximum in the storage modulus. Thus, given a value of C and J!?, Gk was calculated with n2 being taken as a small integer. C was found as an average value from the above expression solved for each of the volume fractions studied. This gave a mean value of .eC = 1.51 x 10e3 * 10%. It can be seen that the same form is predicted, the peak magnitudes are well matched, and the variation of peak height with volume fraction is matched. Table 1 gives the values corresponding to each maximum. Equation 33 was used to estimate the value of K’ when E = z and w = w: E2=-
WK W2
=-
K’
(33) W2
Using the values of 4 and n2 given in Table 1, the values of K’ were calculated. It should be noted that there is only a slight variation of this viscous loss parameter with volume fraction. Figure 7 shows the prediction of Eqn (32b) when compared with the
311
experimental data. This equation is not so successful as it gives a plateau and fails to predict the maximum in the loss modulus with increasing field strength. This is a clear indication that the inter-particle forces should be included in the viscous loss, i.e. there should be a variation in the viscous loss with field strength, however the implication of the maximum is that structural changes many occur which could narrow the calculated peaks. Figure 4 clearly shows that at field strengths below that giving the maximum in G’ the storage modulus is greater than the loss modulus, TABLE
1
Comparison of the experimental calculated from Eqn (32a)
maximal
values
z (kVm_‘)
moduli
with
I
,
those
K’ me2
)
33.6 38.7 38.7 25.4
I
storage
?2 rn-’
400 325 280 200
I
the
Calculated
Experimental
0.09 0.14 0.18 0.23
of
I
I
3 3 4 4
I
x lo-lo
)
32.7 33.6 42.7 27.8
4.32 4.44 4.23 3.68
-
____._.-.-
,/--/
/’
50 /, ‘;v E z 140
/( -
I i
Applied
Field
E IVmm-’
Fig. 7. The applied field dependence of calculated for $ = 0.223, and n2 = 4. G’ (
the
storage modulus and ); G” (- _- _ -).
the
loss
modulus
312
indicating that the period of the shearometer wave is less than the stress relaxation time. However, at higher applied fields the reverse is the case and the relaxation time is less than the wave period. This is in line with a structural rearrangement corresponding to a progressive increase in alignment with the field which could result in a variation in relaxation time. A full spectral description of the relaxation process is currently being investigated. ACKNOWLEDGEMENTS
We wish to thank the Electra-Rheological Research Syndicate Plant Protection Division, for their support during this work.
and ICI,
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
W.M. Winslow, US Patent No. 2, 417, 850, 1947. Yu. F. Deinega and G.V. Vinogradov, Rheol. Acta, 23 (1984) 636. E. Hiickel, Phys. Z., 25 (1924) 204. J.D. Ferry, The Visco-elastic Properties of Polymers, 3rd edn, Wiley, New York, 1981. Y.C. Chiew and E.D. Glandt, J. Colloid Interface Sci., 94 (1983) 90. C.F. Zukoski, Ph.D. Thesis, Princeton University, 1984. M.H. Davis, Q.J. Mech. Appl. Math., 37 (1964) 499. D.J. Jeffrey, J. Inst. Math. Its Appl., 22 (1978) 337. A.S. Sangani and A. Acrivos, Proc. R. Sot. London Ser. A., 386 (1983) 263. T.A. Vorob’eva, I.N. Vlodavets and P.I. Zubov, Colloid J. USSR, 31 (1969) 533. R.W. Zwanzig and R.D. Mountain, J. Chem. Phys., 43 (1965) 4464. J.B. Hayter and R. Pynn, Phys. Rev. Lett., 49 (1982) 1103. R. Pynn and J.B. Hayter, J. Mag. Mag. Mat., 31 (1983) 955. D.A. McQuarrie, Statistical Mechanics, Harper and Row, New York, 1976. J.W. Goodwin, T. Gregory, J.A. Miles and B.C.H. Warren J. Colloid Interface Sci., 97 (1984) 488 D.H. Napper, Polymeric Stabilisation of Colloidal Dispersions, Academic Press, New York, 1983. P.G. deGennes, Macromolecules, 14 (1981) 1637. P.G. deGennes, Macromolecules, 15 (1982) 492. J. Klein and P. Pincus, Macromolecules, 15 (1982) 1129. J. Klein, Nature, 288 (1980) 248. Y. Almog and J. Klein, J. Colloid Interface Sci., 106 (1985) 33.
and Surfaces, 18 (1986) Science Publishers B.V.,
VISCO-ELASTIC
D. BROOKSa,
STUDIES
J. GOODWINb,
293-312 Amsterdam
293 -
Printed
in The Netherlands
ON AN ELECTRO-RHEOLOGICAL
C!. HJELMb,
L. MARSHALLb
FLUID
and C. ZUKOSKIC
aLaser Engineering, London (United Kingdom) bDepartment of Physical Chemistry, University of Bristol (United Kingdom) ‘Department of Chemical Engineering, University of Illinois, Urbana, IL (U.S.A.) (Received
13 September
1985;
accepted
in final form
11 November
1985)
ABSTRACT The fluid used for these studies consisted of lithium poly(methacrylate) dispersed in a chlorinated hydrocarbon oil. Shear wave propagation was employed to measure the storage and loss moduli at a frequency of 1200 rads-’ . The parameters varied were field strength, volume fraction and water content of the particles. Optical microscopy was used to measure the particle size distribution whilst electrophoresis was used to enable the electro-kinetic potential to be measured. This showed that the particles were positively charged with a potential of circa + 45 mV. Further characterisation of the suspensions was carried out by conductivity measurements and the conductivity of the systems was shown to be due to the counter-ions in the diffuse layer which balances the surface charge. The moduli of the suspensions were found to increase with applied field to a maximum value and then decrease. In addition, the storage moduli were much higher than the loss moduli for all samples at low applied fields, but at high fields both moduli were of similar magnitude. These experimental data will be discussed in terms of the inter-particle forces and the structure in the suspensions.
INTRODUCTION
The pioneering work of Winslow [l] clearly demonstrated that marked changes in the rheological properties of suspensions of particles of high relative permittivity dispersed in an oil of low relative permittivity occurred during the application of a high electric field. Since then many systems have been examinec [2] but much of the research has focussed on the engineering applications and so a quantitative description of the electro-rheological effect has not been available to date. The work presented below goes some way towards providing a detailed analysis of the phenomenon. The fluid used here is a commercial product kindly supplied by Laser Engineering of London and consists of lithium poly(methacrylate) particles in a chlorinated hydrocarbon oil. The principal requirement of an electrorheological or ER fluid is that, on the application of an electric field, the
294
system should change from a low viscosity liquid to a high modulus gel with a minimum of loss in both the electrical and rheological senses. Initially, the suspensions were characterised by microscopy to give the particle size distribution, followed by micro-electrophoresis to give the electrical properties of the particles and finally by conductivity measurements as a function of particle concentration in order to assess the ion distribution around the particles. Two types of rheological measurement were used to follow the changes in the properties with applied field, A static yield stress experiment was used followed by a study of shear wave propagation to give the storage and loss moduli at a moderately high frequency. EXPERIMENTAL
The ER fluid was manufactured by Laser Engineering and was used as supplied. By equilibrating the systems with atmospheres of known humidity, the dispersions had been prepared with well-defined water content in the particles, which consisted of an hygroscopic polyeletrolyte - namely the lithium salt of poly(methacrylic acid). Particle size determination A sample of the fluid was diluted with 60/80 petroleum ether and centrifuged at 4000 rpm for 5 min. The solids were washed five times with petroleum ether, and then a drop of the final suspension was placed on a microscope slide and the solvent allowed to evaporate. Optical micrographs were produced at a magnification of X400. A photograph was taken of a Carl Zeiss stage micrometer to serve as a calibration. The particle size distribution was obtained using a Carl Zeiss TGZ3 Particle Size Analyser, measuring at least 1000 particles for each sample. Similar distributions were found for each sample and the mean number average diameter was found to be 15 pm with a coefficient of variation of 38%. Determination
of electro-kinetic
potential
Electrophoretic mobility measurements were made using a Pen Kern System 3000 automated electrophoresis apparatus. Prior to use the quartz cell was flushed with the chlorinated hydrocarbon continuous phase which was then left in the cell overnight to “condition” it by allowing sufficient time for adsorption equilibrium to be established with any ionic species. A sample of ER fluid was diluted with serum obtained by centrifugation of a sample of fluid with the same water content, and the mobility distribution was determined. The measurements were repeated ten times to ensure that the readings were consistent - a precision of better than + 10% was obtained. A mobility distribution is illustrated in Fig. 1 with a mean
295
-19
-048
Electrophoretic
Fig.
1.
Electrophoretic
6 Mobility
.O% /xlO*m*Vsd
mobi1it.y
distribution
results
from
the
Pen
Kern
system
3000.
electrophoretic mobility of 6.8 x 10-l’ m2 V-’ s-l. Because of the low ionic strength of this non-aqueous colloid, the electro-kinetic potential could be calculated by making use of the Hiickel Equation [3] :
where { is the electro-kinetic potential, u is the electrophoretic mobility, q. (0.075 Pas) is the viscosity and E is the permittivity of the continuous phase with a relative permittivity of 8.5. Hence, at low concentrations the value of c = + 45 mV was calculated.
Suspension
conductivity
The conductivity of the ER fluid was determined as a function of the concentration of particles. Volume fractions in the range 0.1-0.35 were employed and a Wayne-Kerr AC bridge was used for the measurements. A clear increase in relative conductivity was found over this range of concentrations which indicated a particle-dependent contribution. The experimental data are plotted in Fig. 2 and are discussed in detail below.
Rheological
measurements
Two types of rheological experiments were carried out. Firstly, a static shear test was used to give a yield stress. Samples of the ER fluid were placed in an annular slot between cylindrical electrodes, a field was applied, and an increasing torque applied to the outer element until motion was observed between the two elements. The field was increased and the procedure repeated. Figure 3 shows some experimental data, and, in this type of experiment, a steep monotonic increase in yield performance was found over the range of fields used. The second type of experiment performed was the determination of the wave rigidity modulus from the measurement of the shear wave propagation
296
velocity. The apparatus used was a Rank Bros. Shearometer which records the time that a torsional shear wave takes to travel between two parallel plates. These were made of stainless steel and electrical leads were soldered to each plate so that an electrical field could be applied to the samples. Plate separations of up to 2 mm were used with applied potentials of up to 3 kV. In all cases, the currents observed were less than 10 mA and no heating effects were observed. The wave rigidity modulus, E, was calculated from the wave velocity, u, and the suspension density, ps: G = v2ps As the propagation time was measured at different plate separations, the attenuation of the wave was estimated. When this is expressed as the characteristic attenuation distance in wavelengths, b, the storage and loss moduli, G’ and G”, can be calculated from [4] : G’ = G (1 - b2)/(1
+ b2)2
G” = 2G b/(1 + b2 )2 The frequency of the shear wave was approximately 1200 rad s-l . A typical set of data are plotted in Fig. 4. There are two important I
I
1
I
I
I
0004 Volume
Fraction
9
Applied
Fig. 2. Relative conductivity as a function + 40 mV (________________); + 45 mV ( Fig. 3. Static
shear force
as a function
of applied
of volume
fraction:
Voltage
0 mV
) field for fluid at @ = 0.30.
/ kV
(-
-
-
-
-);
I
50
0
I
I
200
I
I
400
Applied
Field
I
1
.
I
_
603 E
/Vmm-’
Applied
Field
E /Vmm-’
Fig. 4. Shear moduli as a function of applied field for a fluid at 4 = 0.223, content of 29.5% and at a temperature of 25.2’C. ?? (0); G” (0); G’ (0).
with a water
Fig. 5. The volume fraction dependence of the storage modulus. $J = 0.223 -); r$= 0.14 (-- - -); @J= 0.09 (__________), f$ = 0.18 (l
(-
);
l
features that were common to all the curves obtained. Firstly, there was a maximum in the moduli which moved to higher fields with decreasing volume fraction (see Fig. 5). The second feature to note is that at low fields G’ > G”, but at high fields this was not the case and then G’ < G”. Although in Fig. 4 the condition G’ = G” occurred at E (the field strength at which the maximum in the modulus was found), this was not always the case and the precision of the determination of the wave attenuation, and hence b, was not sufficiently good to identify this point unambiguously. DISCUSSION
Suspension
conductivity
The ER fluids are suspensions of particles of high conductivity in a continuous phase of much lower conductivity. The conductivity of the continuous phase, co, is related to the concentration of ions in solution by the relationship:
(1)
298
the valence, number concentration where zi, ni and Di are, respectively, and diffusivity of the i-th ionic species, e is the fundamental unit of charge, h is the Boltzmann constant and T is the absolute temperature. The summation is carried out over all the ions in solution. Using Eqn (l), the Debye-Htickel screening parameter, K, can be estimated from the following expression, if all ions are assumed to have the same diffusivity:
where E is the permittivity of the continuous phase, D, is the ion diffusivity in water which has a viscosity r), , and v. is the viscosity of the fluid medium. If the value of D, is taken as 1.3 x 1O-9 m2 s-l (i.e. the. diffusivity of a sodium ion), then an approximate value of KU = 4.8 x 10T3 can be calculated. Hence, using the low surface charge model for the diffuse double layer, a surface charge density of 2.17 x 10d9 Cmm2 corresponds to the measured electro-kinetic potential. This gives an average of 1.35 x 10 lo proton charges per particle. Theories are available to relate the volume fraction dependence of the suspension conductivity to the conductivity of the disperse phase. Chiew and Glandt [5] have recently reviewed the work in this area and derived an expression which extends the region of validity of Maxwell’s mean field calculation for uncharged particles. This reproduces the volume fraction of dispersion conductivity up to the percolation threshold, except when the particles are much more conducting than the continuous phase. In the latter case, the model underestimates the dispersion conductivity but, even under these conditions, the model works reliably up to volume fractions of 0.2. The expression for the relative conductivity can be written: (5
-
=
1 + 2A$l+ (K, - 3A2 )@2 (3) (I-
00
GA)
where u is the conductivity particle conductivity u, , and A=---
of a suspension
with volume
fraction
$ and
a-1 a+2
where cx = u, /a0 and K, is a weak function of volume fraction tabulated in Ref. [ 5 ] . The dashed curve in Fig. 2 was calculated from Eqn (3) assuming that u, /a0 > > 1. Thus even by giving the particles an infinite conductivity, the model underestimates the observed trend. Discrepancies of this kind have recently been observed [6] for the conductivity of concentrated aqueous colloidal systems and the results explained by taking into account the contribution of the counter-ions to the dispersion conductivity. For a suspension of particles with a surface charge density of q C m-‘, the counter-ions contribute
299
-
344
ions to the solvent ionic concentration, where z, is the counter-ion valency. Thus as the volume fraction is increased, the background ionic strength of the suspension rises. By including this increase in ionic concentration (but still neglecting contributions to the suspension conductivity due to charge convection and conduction in the diffuse layer which are also positive but small contributions), it was shown that the relative conductivity of a suspension of charged but non-conducting spheres can be approximated by
[61: u -ZZ
00
[1 + Cl
l-4 i 1 + 0.5@
where the counter-ion c=
I
(4)
contribution
- 32, DC qe@ (1 - qb)ao,hT
This assumes that the electrolyte is symmetrical and each ion has the same diffusivity, D, . Equation 4 was used to calculate the suspension conductivity utilising the surface charge density calculated from the electro-kinetic potential determined at low volume fraction. This gives a much better description of the data than Eqn (3) and therefore it can be concluded that the conductivity of these suspensions over the volume fraction range studied is primarily due to the counter-ions associated with the particles rather than the increasing volume fraction of highly conducting particles. Whilst the lithium poly(methacrylate) particles are highly swollen with water containing a large number of lithium ions, it is not clear that a charge flux or current will be transmitted through these particles when they are suspended in an organic solvent and subjected to an applied field. For this to occur, ionic species would have to be present which would be soluble in both the polar and non-polar phases and also be capable of transferring across the water-oil interface. An analogous situation occurs in a suspension of metal particles in an ionically conducting fluid where an electrochemical reaction must occur for there to be charge transfer at the solid-liquid interface. At present we feel, that under the conditions that these suspensions are used? that there is no significant ionic flux across the interface and, therefore, in terms of suspension conductivity, the wet polyelectrolyte particles behave as non-conductors. This view is entirely consistent with the observation that, at high field strengths, strings of particles are formed between the electrodes with no resultant increase in conductivity such as would be the result of ion transfer between the particles.
300
The high water content and ionic strength are, however, important when modelling the forces between the particles. Because there is little or no charge flux across the particle-solvent interface, the particles can be treated as perfectly polarisable dielectrics and will thus assume a constant surface potential when placed in an electric field. If, as a first approximation, one treats the particle as spheres which have no surface charge then the electrical potential, \k, outside the particles can be found from the solution of Laplace’s Equation:
v2\Ir = 0
(5)
Far from the test particle application of the external of the field is E then: \I, = ErcosO
the potential assumes a linear form due to the field applied in the z-direction. If the magnitude
asr+m
At all particle surfaces the potential will assume a constant tangential components of the applied field are zero: \I, = a constant
on all particle surfaces
(6) value and so the
(7)
The solution to Eqn (5) subject to the boundary conditions (6) and (7) is the problem of highly conducting particles suspended in non-conducting fluids for which solutions are available [7-91 for one or two spheres or for arrays of spheres. Once the electrical potential around the sphere is known, the force on the particles can be calculated from the Maxwell stress at the surface of the particles [lo] and again solutions are available. The solutions for these forces are used below for the development of a model for the rheological properties of the suspensions and the full details of the method of calculation are given in the primary references. The potential around a reference particle depends on the position of all the other particles in the suspension and so the force acting on the reference particle depends on the particle distribution function in addition to the surface potential. In order to achieve some visualisation of the structure of the system, a dilute dispersion of particles was placed between two electrodes in a cell mounted on a microscope stage. When a modest field (50 kV m-i ) was applied between the electrodes the particles were seen to undergo rapid motion for a short period of time (-- 20 s). After this period strings of particles were formed between the electrodes with no free particles being observed. The formation of strings of dielectric particles suspended in both organic and aqueous media under the influence of large AC and DC fields has been reported in the literature [lo]. As the volume fraction is increased individual strings are difficult to pick out, although a stringy structure of particles in close contact can be seen at moderate to low volume fractions.
301 MODELLING
THE VISCO-ELASTICITY
OF ELECTRO-RHEOLOGICAL
FLUIDS
In this section a model is derived which describes the dependence of the high frequency limit of the shear modulus as a function of volume fraction and applied field. A form of the frequency dependent storage and loss moduli are also presented and these are tested against the experimental Models for the visco-elastic behaviour of suspensions rely data. predominantly on descriptions of the pair interaction of particles appropriately averaged to take into account the pair distribution function [15]. A radially symmetric distribution function is normally used, however in this work the model must use an anisotropic pair potential. This should result in an asymmetric pair distribution function being produced at the elevated field strengths used with electro-rheological fluids. Estimation
of the limiting storage modulus,
G(m)
At the high frequency limit, there is only a small loss associated with the response of a suspension to an oscillatory strain. For a linear visco-elastic material, this high frequency limit of the modulus, G(m), is simply related to the energy stored per unit volume, U/V, when the material is subjected to a uniform shear strain of magnitude y as:
u GPh2 -=V
2
(8)
It is assumed that the system contains N particles in the volume V and that the particles are of uniform radius as well as interacting as hard spheres in the absence of an applied field. The spherical particles are composed of a highly conducting core of radius a*surrounded by a strongly anchored steric layer of thickness L. This barrier stabilises the particles against coagulation by van der Waals forces and induced dipolar forces. The compressibility of this layer will be discussed below in more detail but it is the repulsion generated by this layer when the particles are brought together by the applied field which prevents electrical contact between the conducting particles. The repulsive force is a very steep function at small separations [16] so that the minimum separation between adjacent spheres can only change very slowly with electrical field strength. At any field strength, the particles are separated by a centre-to-centre distance of r = 2a, that is for r = 2 (a* + 6), where 6 < L. Whilst ~7* is only a weak function of E, variations in the suspension structure due to changes in 6 will be small. For any particular configuration of the N particles, when the system is strained the energy stored per particle is Ui and is written: 6Xi
302
where _Fi is the force required to displace the i-th particle a distance 63~~. The total energy stored will be the sum of Ui over all the particles in the suspension. Consequently, the shear modulus can be written:
If the strain is sufficiently small that a linear visco-elastic response is maintained, and also if it is assumed that the particle interactions are pairwise additive, the force J’i can be calculated by summation over all inter-particle forces. Expanding the pair interaction forces (_Fij) in a Taylor series about the equilibrium position of the j-th particle, and noting that at equilibrium (i.e. prior to the application of the strain) the total force acting on the i-th particle is zero, then _Fi is written: Fi
C
=-
V_Fij
’
6sij
+
0
(11)
(3cij')
j
where V_Fij is the gradient of the force on the j-th particle due to the i-th particle evaluated prior to the application of the infinitesimal shear strain. The negative sign in Eqn (11) arises due to a change from the externally applied force required to cause a displacement to the force which must be overcome. A linear shear gradient is applied in the x-direction so that 6cij = _yZij, where .Zij is the separation of the two particles in the z-direction. Therefore we can write: G(m)
=
--1:
2V
i=l
c
[V~ij
i=l
lKS
&]zij*
(12)
The factor of l/2 occurs in Eqn (12) in order to prevent each interaction being counted twice. However, the positions of the particles are not on static lattice sites and are only known in a statistical fashion and Eqn (12) must be ensemble averaged to give the most probable value of G(m). This averaging process is simplified by the restriction to pairwise additivity and so: (13) where z is the z-component of the separation between a test particle and a particle at r. The pair distribution function is written as a function of the vectorial separation to emphasise that the structure of the suspension is not necessarily isotropic and is in fact unlikely to be so when under the influence of a high external field. Under conditions where thermal motion contributions are negligible, this expression is a generalised version of that derived by Zwanzig and Mountain [ll] who developed their treatment from the statistical mechanics of condensed phases. In order to reduce
303
Eqn (13) to a more tractable form, approximations for _Fii and g(r) can be made which simplify the mathematical form but retain the essential physics of the problem. The effect
of external
fields on the pair distribution
function
The elastic properties of an electro-rheological fluid are modelled as arising from induced dipole-induced dipole interactions. The force between the particle pairs therefore depends on both the absolute separation and the angle between the line of centres with the direction of the applied field. In the shearometer experiments utilised in this work, the strain was applied perpendicularly to the applied field. Hence, in the analysis given below the electric field is applied in the z-direction whilst the strain occurs in the x-direction. Prior to the application of the external electric field, the particle-pair distribution function is assumed to be radially symmetric and to be adequately described by the pair distribution function for hard spheres. Upon the application of a large external electric field, the particles arrange to form strings such that g(r) = g(r, 0). The form of g(r, 0) can be estimated from the distribution function observed by neutron scattering from ferrofluids under the influence of a uniform magnetic field [12,13]. Particles in these suspensions have a permanent magnetic dipole and the applied magnetic field causes the particles to align into chains. Pynn and Hayter [12,13] have calculated g(r) for these systems using the mean spherical approximation where g@) is written as the sum of Legendre polynomials. These authors found that the distribution function could be modelled well by retaining only the terms up to the second polynomial, P,(cosO) viz. g(r)
=
go(r)
+
g2
(rFJ2
(co@)
As a first approximation in the application of this result rheological fluids, g(_r) is rearranged to the following form: g(c) = A(r) -I B(r) cos20 where B(r) accounts top and bottom of both A(r) and B(r) and the suspension ferro-fluid volume g(_r) vanishes due to The pair interaction
(14)
to electro(15)
for the increased probability of finding particles at the the test sphere (0 = 0 and r, respectively), and in general will depend on both the magnitude of the applied field volume fraction, Pynn and Hayter showed that as their fraction approaches close packing, the asymmetry in particle crowding. force
The electrical interaction force between pairs of particles decays rapidly over a distance comparable with a particle diameter. Consequently, the influence of particles sited at a distance greater than a few particle diameters
304
apart on the value of G(m) will be negligible. Thus in developing an analytical expression for G(m), only nearest neighbour interactions are considered. At the field strengths commonly used with electro-rheological fluids (-5 x 10’ V m-l ) the influence of thermal motion on the positions of the particles is not important, the electrical forces are instead balanced by strong repulsive interactions indicating that where the electrical forces are largest the stabiliser layers are in contact (i.e. the centre-to-centre separation is 2~). This results in the formation of chains of particles. The model developed here for g(_r) is based on this concept and also the maintenance of a constant density of nearest neighbours after the application of the electric field. Let no be the number of nearest neighbours which are distributed in a volume u. prior to the application of the field. When the field is turned on, this shell of nearest neighbours rearranges so that a certain fraction of these particles, X = n2 /no, come into contact with the test sphere and are distributed on average as cos’ 0. The remaining n1 = no - n2 spheres are arranged symmetrically about the test sphere so that the total number concentration of nearest neighbours remains at no/u0 . The number of particles in the first shell can be found from [14] : = 27rN/V j s g(r, 0) sin Or2 drd@
n1 +n,
(16)
0 0 where rm is the location of the first minimum in g(r). As an approximation, we will let A(r) = Ap6(r - rl ) and also let B(F) = B,6(r 2a), where 6(r) is the Dirac delta function. Carrying out the integration in Eqn (16) and equating rz1 and n2 with symmetric and asymmetric contributions, respectively, to g(_r) yields values of A, and B, : n1
v
A,
= -
B,
3n, V = 8nNa2
The the the field
radius of the symmetric shell, rl , is found from the constraint that overall density of particles in this first shell remains constant after field is applied. The volume occupied by the first shell prior to the application can be approximated as:
(17a)
4~rNr ;
(17b)
4n vo = -j--(2a)3tm
(18)
where 4, is the maximum packing fraction approximation corresponds to a pair distribution no V6(r g(r)
=
4nNr
*’
F* )
for the suspension. function given by:
(This
(19)
305
where r* = 244, results in: rl
/@)1’3 .) Application
of the constant
density
constraint
= 2a ( 4,i:“h)1’3
(20)
The electrical force due to an external field acting between a pair of polarisable spheres changes sign as the orientation of the line joining the particle centres varies from 0 to 7r/2 with respect to the field. At 0 = 0 the force is attractive, whilst at. 0 = x/2 the force is repulsive. For angles other than integer multiples of 0 *and n/2, a torque is applied to the particle pair which varies as sin 0 cos 0. The electrical pair interaction force, _FFj, can thus be broken up into three components: _F?j(c) = - fa2E2 [(f~ ( r) cos' 0 + f2(r) sin’ O)i +(f3(r) sin@ cosO)@]
(21)
where fl (r),f2 (r),and f3 (r) are dimensionless functions which depend only on the centre-to-centre separation of the spheres. The calculation of Davis [ 71 and Jeffrey [S] for pairs of highly conducting spheres in a nonconducting fluid, i.e. (up /uO > > l), indicates that fl (r) and its derivatives are much larger than f2(r); f3(r) and their derivatives. Consequently, only fl (r) will be considered in the analysis. The strongly attractive electrical potential which causes the particles to align into chains is opposed by a repulsive potential which arises from the interaction between layers of steric stabiliser at the particle surface. The summation of these potentials will give rise to a minimum in the potential energy curve and ensures that - VFij is positive at the most probable particle separation. The exact functional form of the repulsive potential is not well understood at, present [16]. The scaling theories of deGennes [17,18] and the results of Klein and Pincus [19] suggest that when the surface-tosurface separations are small compared with the stabiliser layer thickness, the repulsive force per unit area between flat plates scales as 6-3, where 6 = (r - 2a) is the pair separation. This result was predicted for polymeric stabiliser layers in both good and poor solvents. However, the force determined experimentally between compressed layers or irreversibly adsorbed polymer on crossed mica cylinders was found to be a repulsion which increased much more rapidly than a_3 [20,21]. Due to the large magnitude of the attractive potential at small 6 in the case being modelled here, we expect any stabiliser layer to be highly compressed and the resulting repulsive force to be a very steep function of pair separation. Therefore, a repulsive force, _F[j, is chosen to reflect these expectations viz. - n* “G=Rf
i (22)
(Iwhere z = L/a, R is a constant
describing
the polymer
properties
but which
306
is independent of electrical field strength and II* is the order of the interaction. It is expected that n * > 3 and indeed it is shown below that the experimental results given here indicate that n * > > 1. Note also that _FFj is radially symmetric. The total pair interaction force is found from the sum of _FFj and _Fij. For our purposes this can be written: &j
,. = EU’ E 2 (/If, - fl (r) cos @)c
where fl = R/ea2 E2 and
(23)
f, = (z/S)” *.
The shear modulus Rewriting Eqn (13), taking into account (15) and_Fij(r) in Eqn (23), results in:
G(m)
=
-a[$)2ea2E2if[[
-
-
X
+
2fl
-
df, dr co9
r
[[A@&(r-r,)
cos2
dfr
0
- 9n,&E2
l(1
- X)4/3
4n(l
6(@/@,
15
+ 6)
-
r
Wfr(j
35
a;
@1
drdOd@
+ 24p -fr (h)
where r = a* i. It should separated by 26. Thus, i2 fl (i) can be written [B] : 6-l
32f 1 0-l)
r^,
63i2
be recalled approaches
+1/3ln6
(In (2) + 2y-ln6)2
1) + Wfr(h
J
(24)
)
15i,
+ 4LQfr(rl ) i5ai
f-1
35
Eqn
cos2 0
2d;
- U2’3
afl(rl ) -- 24 -fdb) a;,
--~ 4 afl h) 35
from
i-B@&(r--2a)cos2
I
yields:
--___4
fl
-
0 sin2 0 cos2 * +
Carrying out the integration
=
of g(r)
cos4 0 sin 0 cos2 @ + sin2 @ sin 0 cos* 0)
rJ‘
G(*)
form
sin3 0 cos2 Cp
+ fl z
i
+A
the
J
1
(25)
that at F = F2 the particles are 2 as 6 -+ 0. For small values of 6,
(26)
307
where r= 0.577216 is Euler’s constant. fi (r) Under the decreases rapidly as 6 increases. described above, 6 will be of the order of l-20 3.5 times a particle radius. Therefore, G(m) is of particles located at r2 and terms evaluated neglected. Similarly for 6 < < 1, ldfi (6)/drl> > > If,(S)/ and Eqn (25) can be simplified:
G(m) = -’
n”‘E2e 357r (1 + 6)
[- dfr/d,]
is singular as 6 -+ 0 but experimental conditions nm while rl will be 2.5dominated by the effects at rl in Eqn (25) can be > Ifi (6)l and Idf,(6)/drl
(27)
where
dfT = dr
-df,
-- 70 df, -
d;
3
di
1
r = (2$26)
For G(m) to take on realistic values, dfr/dr must be negative, while the functional form of fl and f, have been chosen to ensure a positive curvature to the pair interaction energy at the most probable separation, F2. It is not obvious how this curvature will depend on /3 (which it will be recalled is a function of E). In order to determine how the curvature of the potential energy scales on p, - dfr /di Ii, was evaluated for several values of /3, Land rz. For E at lo’-104, it was found that to a high degree of accuracy this function could be written as:
dfT
-
d;
= ;
KP-”
1
where m is a positive number m n
0.79 3
0.52 4
0.41 5
0.25 7
which has values of: 0.19 10
0.07 25
m was also found to be insensitive to the magnitude of fl for @” < 1. Thus, for a sufficiently steep repulsive interaction, G(m) will scale on E*, be insensitive to particle size and vary slowly with volume fraction. (In general, nz is expected to be a weak function of volume fraction.) It was also found that 6/r, which was of the order of 10-3-10-4 (corresponding to 2-20 nm for the experimental system), changed slowly with /3 as would be expected from the steepness of the repulsive barrier. As p decreased however, corresponding to an increase in E, 6 decreased as expected. From this numerical work, it was found that Eqn (27) could be approximated well by: G(m)
= n2qkCE2+m
(28)
where m can be calculated by assuming an order for the repulsive potential, and C is a fitting parameter which will be independent of E and, for
308
sufficiently large n, independent of particle size. However, both C and m are unknown functions of the properties of the steric stabiliser layer. For the systems used here, C and m will have constant values and, once fixed, the changes in the visco-elastic properties of the suspensions with field strength and volume fraction should be predicted by Eqn (28). In deriving this expression, no constraints have been placed on n2, except that its value should be less than ~1~. While there is little doubt that n2 must be a function of volume fraction, the number of touching particles is unlikely to change with field strength once the particle chains have been established. At the field strengths used in these experiments the electrical forces acting on the particles dominates the pair interaction. It can be concluded that, at a fixed volume fraction, the high frequency limit to the elastic modulus will be a (sensitive) function of 6 and be independent of particle size. Comparison
of the model with experiment
Direct comparison of the predictions of Eqn (28) with the data presented in Figs 4 and 5 is hindered as the wave rigidity modulus could only be measured at one frequency. It was possible to extract the storage and loss moduli from the experimental data by utilising the attenuation of the wave [4], but the equivalence of G’ with G(m) is only a good approximation when G’/G” < < 1 and this was rarely the case, and this ratio varied with applied field. However, by utilising the constitutive equation that describes the behaviour of the simplest visco-elastic liquid, i.e. the Maxwell Fluid, a better test of the model can be made. For the purposes of this paper only a simple Maxwell model is used with a single relaxation time, t, . Clearly, a spectral response would be more realistic but the added complexity of using the generalised Maxwell model is not warranted at the current stage of development and the underlying physical arguments are unchanged. The Maxwell model consists of a spring in series with a dashpot and the characteristic relaxation time has a value equal to the ratio of the dashpot viscosity to the spring modulus. The frequency dependence of the dynamic moduli are given by the following relationships:
G'(w) =
(Gm I* Woo) 1 + (wt,
(2W
)I
and as w -+ 00, G’ + G(m) G”(o)
=
(tit, )G(Y 1 + (wt, )*
and with w -+ 0, G” -+ q(O), the zero shear viscosity, t nl
= r?(O)/G(=‘)
(29b) and
309
The origin of the elasticity in an electro-rheological fluid is the forces acting between the particles which result in a network structure, and the viscous loss is associated with the diffusive motion of the structural components. The diffusion of the particles is a function of both the hydrodynamic interactions and the electrostatic interactions between a particle and its neighbours. Equation (28) gives the relationship for G(w) where it can be seen that G(m) scales as @L?X~+~ rz2. However, we do not at present have a similar expression for the zero shear viscosity and so as an initial approximation we will assume the viscous term to be a constant independent of the applied field variation and we can write t, as: trill = K/c+hE2””n2
(30)
Taking the scaling of the viscous loss as the constant K implies firstly that the structural arrangement is independent of field strength and secondly that, as the particles are at the same close separation distance at all fields, the hydrodynamic interactions are constant and dominate the viscous loss. From Fig. 4 it can be seen that the storage and loss moduli are equal at the field strength which gives the maximum value of wave rigidity modulus. Therefore, at this field strength, the relaxation time and the experimental time are equal and we will use a bar above the frequency, time and field strength to indicate this point on the curves, i.e. when E = E, t,
= l/W =I t
at all other fields, t,
f
l/w
Therefore: wt,
=
(31)
Substitution of Eqns (25) and (28) into (26) yields the required for the frequency dependence of the storage and loss moduli:
G’(w,E)
expressions
=
(32a)
(=b)
Figure
6 shows the comparison
of Eqn (32a) with the experimental
values
310
I 0
I
I
I
200
I
400
Applied
Field
Fig. 6. Calculated n2 =4 (. . .); =3(--_--_-), n2
600 E
storage @=O.lS,
I Vmm-’ modulus n2 =4
curves (-
as a function ); 4 = 0.14,
of applied field. nz = 3 (---);
@ = 0.223, $J= 0.09,
of storage modulus for each of the volume fractions investigated. The only fitting parameters are the values of n, and C. At i?, w = W and from Eqn (32a): G:,
=
n2 WS2 2
where Gk is the value of the maximum in the storage modulus. Thus, given a value of C and J!?, Gk was calculated with n2 being taken as a small integer. C was found as an average value from the above expression solved for each of the volume fractions studied. This gave a mean value of .eC = 1.51 x 10e3 * 10%. It can be seen that the same form is predicted, the peak magnitudes are well matched, and the variation of peak height with volume fraction is matched. Table 1 gives the values corresponding to each maximum. Equation 33 was used to estimate the value of K’ when E = z and w = w: E2=-
WK W2
=-
K’
(33) W2
Using the values of 4 and n2 given in Table 1, the values of K’ were calculated. It should be noted that there is only a slight variation of this viscous loss parameter with volume fraction. Figure 7 shows the prediction of Eqn (32b) when compared with the
311
experimental data. This equation is not so successful as it gives a plateau and fails to predict the maximum in the loss modulus with increasing field strength. This is a clear indication that the inter-particle forces should be included in the viscous loss, i.e. there should be a variation in the viscous loss with field strength, however the implication of the maximum is that structural changes many occur which could narrow the calculated peaks. Figure 4 clearly shows that at field strengths below that giving the maximum in G’ the storage modulus is greater than the loss modulus, TABLE
1
Comparison of the experimental calculated from Eqn (32a)
maximal
values
z (kVm_‘)
moduli
with
I
,
those
K’ me2
)
33.6 38.7 38.7 25.4
I
storage
?2 rn-’
400 325 280 200
I
the
Calculated
Experimental
0.09 0.14 0.18 0.23
of
I
I
3 3 4 4
I
x lo-lo
)
32.7 33.6 42.7 27.8
4.32 4.44 4.23 3.68
-
____._.-.-
,/--/
/’
50 /, ‘;v E z 140
/( -
I i
Applied
Field
E IVmm-’
Fig. 7. The applied field dependence of calculated for $ = 0.223, and n2 = 4. G’ (
the
storage modulus and ); G” (- _- _ -).
the
loss
modulus
312
indicating that the period of the shearometer wave is less than the stress relaxation time. However, at higher applied fields the reverse is the case and the relaxation time is less than the wave period. This is in line with a structural rearrangement corresponding to a progressive increase in alignment with the field which could result in a variation in relaxation time. A full spectral description of the relaxation process is currently being investigated. ACKNOWLEDGEMENTS
We wish to thank the Electra-Rheological Research Syndicate Plant Protection Division, for their support during this work.
and ICI,
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