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Rheological Behavior of Electrorheological Fluids E. Lemaire, G. Bossis,' and Y. Grasselli Laboratoire de Physique de la Mati2re Condenske, Parc Valrose, 06108 Nice Cedex 2, France Received July 6, 1992. In Final Form: September 18, 1992 Electrorheological fluids are suspensions of solid particles whose rheological behavior can be strongly modified by the application of an electric field. We analyze the rheograms (stress versus shear rate) of a silica-based fluid with the help of a two spheres model. This model reproduces the hysteresis which is experimentally observed. We emphasize that the previous flow history of the suspension must be known in order to predict ita future behavior. A tentative explanation of these observations is proposed on the basis of changes of structure induced by the flow.
I. Introduction Electrorheological fluids ( E m ) as well as magnetorheological fluids (MRF) are known to present important changes of viscosity under the application of an electric (or a magnetic) field.'I2 This property comes from the transient aggregation of the solid phase due to the attractive forces between the dipolar moments induced on each particle by the external field.3 When the field is switched off, the aggregates disappear under the effect of Brownian motion and the initial viscosity is recovered. This possibility to obtain a rapid modulation of the viscosity with an electronic device has raised interest in these fluids for practical applications in the fields of hydraulics, robotica, and automotive applications (dampers, clutches, valves, etc.). These fluids also give to the theoreticians a good opportunity to study the relation between the average transport properties (viscosity, conductivity, elastic moduli, etc.) of a two phases system and its microstructure. Indeed the application of an external field is a convenient way to modify the structure of suspensions and then to observe, for instance, the change of viscosity which comes from this modification. In previous studies we have analyzed the change of structure in the presence of an electric field4lsand the evolution of the yield stress with the intensity of the magnetic field in a suspension of polystyrene particles containinginclusions of magnetite6 In this paper we present new results, obtained with a silica-based fluid, which are characteristic of the general rheological behavior of magneto- and electrorheologicalfluids. In particular we shall emphasize the role played by the time evolution of the structure and its implication on the different kinds of hysteresis which are observed on the stress versus shear rate curves. This time-dependent aspect is often hidden in usual experimenta where the stress is measur6 for a flow characterized by a given stationary shear rate y. In our experiments we increase the stress at a given rate and measure the instantaneous rate of strain y. This rate of strain can correspond to small strains which test the beginning of the deformationand then the breakdown of the aggregates which link the two cylinders of the cell. The knowledge of the response time of the structure to a given shear stress is fundamental for some applications and we propose a (1) Block, H.;Kelly, J. P. J. Phys. E , Appl. Phys. 1988, 21, 1661. (2) Shulman, Z.P.; Kordomky, V. 1.; Zaltagendler, E. A.; Prokhorow, I. V.; Khusid, B. M.; Demchuk, S. A. Int. J. Multiphase Flow 1986,12, 1935. (3) Jordan, T.C.; Shaw, M. T. IEEE Trans. Electr. Insul. 1989,24, 849. (4) Lemaire, E.;Graeeelli, Y.; Boesis, G.J. Phys. II 1992,2, 359. (5) Boasia, G.;Graaeelli, Y.; Lemaire, E.; Petit, L.; Persello, J. Proceedings of the Conference on E.R. Fluids, 15-16 October 1991, South Illinois University, Carbondale, IL. (6) Lemaire, E.; Boaeis, G. J. Phys. D 1991, 24, 1473.
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simple model, based on two particles hydrodynamicand electrostatic interactions, which is able to explain the experimental observations. In the following section we shall describe the experimental results concerning the stress-strain rate curves and the evolution of the conductivity of the suspension when the stress is increased. In section 111,we shall explain these experimentswith the help of a two particles model. The last part of this paper will be devoted to the discussion of this model and of the role played by shear-induced ordering.
11. Experimental Results The suspension we have used is made of monodispersed (diameter 1 pm) silica particles suspended in a silicon oil of viscosity 500 mPa.s. These particles are blue dyed in order to increase the contrast with silicon oil for optical studies of the structure,5their surface is negatively charged are by SiOH- groups, and the counterions (Na+ and H+) contained in a thin layer (100A) of adsorbed water. This ionic layer is responsiblefor the polarization of the particles in the presence of the external field. The relative permittivity of the suspension has been measured with a variable thickness cell and has been found to be tI = 3.2 for a solid volume fraction 0 = 12% and a frequency of 100 Hz. This is to be compared to el = 2.8 for the silicon oil at room temperature. The internal permittivity, tp,of a particle can be deduced from a mean field theory, like Bruggeman7 or Maxwell-Garnett* theories. From these theories, we get E,, = 7.7. This difference between the permittivity of the particles and that of the suspending fluid is mainly due to the ionic cloud around the particle. The rheological measurements have been carried out on a controlled stress rheometer, Carrimed, in a cylindrical Couette geometry. The external cylinder is connected to the high voltage, and the internal cylinder, which rotates upon the application of a torque, is grounded through an electrode deeping in a mercury reservoir situated on the top of the rotating axis. The respective radii of the inner and outer cylinders are 1.383cm and 1.5cm, and we assume that it is narrow enough to neglect the variation of the shear rate inside the gap. In Figure 1,a typical rheogram obtained with a field E = 1.71 X lo6 V/m at a frequency of 100 Hz is represented. The upper curve corresponds to an increase of the stress with a rate of 10 Pdmin. The discharge curve which corresponds to a decrease of the stress with the same rate does not followthe curve of charge below a stress of 3 Pa. Finally the lower curve gives the rheogram obtained at zero field; in this case there is no hysteresis and the charge and discharge curves are (7) Bruggeman, D. A. G. Ann. Phys. 1962,231,779.
(8) Maxwell-Gamett,J. C. Philos. Trans. R . SOC. London 1904,203, 385.
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Figure 1. Rheogram of a silica-based fluid obtained on a controlled stress rheometer: (-) charge curve for a rate of increase of stress, c = 0.33 Pa/s, and an applied field E = 1.71 X 10sV/m at a frequency of 100 Hz;(-1 discharge curve under the same conditions; (- - -) rheogram with E = 0. The charge and discharge curves are undistinguishable.
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Figure 2. Charge curves obtained for differentrates of increase of the stress, = ct and the same electric field as in Figure 1: (-) c =4 x Pa/s (equilibriumcurve);(- - -) c = 1.2 x 10-2 Pds; (- -) c = 3.3 X 10-' Pa/s. Each new charge is done after homogenization of the suspension in the absence of field.
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indistinguishable. In our experimental conditions the change of viscosity due to the application of the electric field is rather small compared to the one obtained with highly active fluids, but our aim was only the understanding of the mechanisms and we prefer to work with a welldefined suspension than with a highly active one. The charge curve presents three different regions which can be recovered on any ER fluids: In the first zone, which is confounded with the stress axis, the stress increases whereas the shear rate remains practically equal to zero. In this region the applied shear stress is smaller than the static yield stress. The second zone where the shear rate suddendly increases for a stress slightly above the static yield stress corresponds to the breakdown of the aggregates which were rigidly connecting the two cylinders. The third zone is approximately linear and can be approximated by a Bingham law, T = gy + Td, where q is the viscosity of the suspension in the absence of the field and 7d is the dynamic yield stress whose origin is related to the supplementary energy dissipation rate I;B16v in the between two particles polarized presence of the force P1 by the electric field and having a relative velocity 6uSgThis dynamic yield stress can be approximated by an average of the restoring force between two polarized particlealo and so will always be lower than the static yield stress. In this experiment the value of the static yield stress seems to be about 3 Pa, compared to 1Pa for the dynamic one. Actually since we have some hysteresis these values are questionable and we need to look at the evolution of these curves when the rate of increase of the stress is changed. Indeed, as we can see in Figure 2a where we have plotted the charge curves for different rates of increase of the applied stress, we have to be careful if we want to deduce the static yield stress from this experiment;the equilibrium curve is attained at a very low rate (4 X 10-3 Pa/@and the definitive static yield stress is now 1.7 Pa, instead of 3 Pa as we could believe from the inspection of Figure 1. On the contrary the discharge curves are much more similar whatever the rate of decrease of the stress. For all these curves each new charge is preceded by a period of (9) Bonnecaze, R. T.; Brady, J. F. J . Rheol. 1992, 36, 73. (10)Klingenberg, D. J. Ph.D. Thesis, University of Illinois, 1990. (11) Klingenberg, D. J.; Zukoski, C. F. Langmuir 1990, 6, 15. (12)Graeselli, Y.; Bossis, G.; Lemaire, E. Submitted to Phys. Reo. Lett.
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Figure 3. Current versus stress during a cycle of charge and discharge with a rate c = 0.5 Pa/s with the same electricfield as in Figure 1: (-) charge; (- - -) discharge.
homogenization of the suspension with a high shear rate in the absence of electric field, and in that way the curves are well reproducible since we always start with a homogeneous suspension and increase the stress at the same rate. Some insight of what happens in the suspension can be obtained by recording the variation of the electric current passing through the cell of the rheometer when the stress is progressively increased. This variation is shown in Figure 3 for a rate of increase of the stress of 0.33 Pa/s. The principal feature of the upper curve is a sharp decrease of the intensity which corresponds,to a sudden breakdown of the aggregates; on the contrary, for the discharge.curve the intensity increases continuously when the stress decreases, and at zero stress the current has not reached ita initial value. This hysteresis for the current indicates that the connections between the two cylinders rebuilt progressivelyand that the find structure is different from the one which is obtained when the field is switched on from an homogeneous suspension at rest. We shall discuss this point again in section IV, but let us first try to explain, with a simple model, the rheograms obtained in Figure 1 and Figure 2. 111. Two Sphere Model In these experiments we apply a shear stress 8 which increases linearly with time
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Figure 4. Schematic representation of the deformation of a chain of spheres. Note that, in this model, each particle remains at the same height, z, during ita motion.
d = ct
(1) and we measure the rate of strain which results from the motion of the inner cylinder. We shall assume that at rest the structure of the suspension under the action of the electric field consists of chains of spheres which connect the two cylinders. Then, if @ is the volume fraction of the solid particles and a their radius, the applied force per chain will be
F=-2* a2 3 Furthermore we suppose that the chain will deform uniformly under the action of IiB, so that each particle will only move along the direction, y, of the applied force (cf. Figure 4). The difference, 6v, of velocity between any two particles separated by a distance d = a(2 + €1, where t is the normalized gap between the two surfaces is then
The numerical application with a = 2.5 and a field E = 1.71 X lo6 V/m gives 7, = 1.38 Pa. This is slightly lower than the experimental value (T* = 1.7 Pa), but the order of magnitude is respected. Actually, many reasons can explain this difference. Firstly, electric forces are calculated in the case of interfacial polarization coming from the difference of permittivity between the two phases, but this is not well adapted to our suspension where the particles are surrounded by a ionic cloud. Secondly, we have used a model where particles form isolated chains without any interaction between them, while this model is only realistic for very diluted suspensions (@ I0.01). For larger solid volume fractions, the chains group into thicker fibers whose section can contain several hundreds of particles. The dynamics of the deformation can be obtained on the basis of this model by writing that the relative velocity of two spheres is given by the product of their relative mobility by the difference of forces acting on them
6V = [M]*[F"- Fell 1 Mij = -67rpa [G(t)
(6)
eiej + (aiii- eiej)H(D1
the functions G ( t ) and H ( t ) have been tabulated by Batche10r.l~ For small (normalized) separations ( [ / a < 0.1) we can take G ( [ ) = 2t + 1.8t2In 6 - 4t2
(3)
H ( t ) 0.401 - 0.532/h 5 the quantity e is the unit vector along the line of centers of the spheres whose componentsare sin 8 in they direction and cos 8 in the z direction (Figure 4). Then with the help of eqs 2, 5, and 6, we can write for the variation of the distance between two spheres
We can then relate the relative velocities of the first two particles to the forces acting on them. If we neglect the interactions between second neighbor particles, the electrostatic force on particle 2 will cancel by symmetry and it remains for the relative force F1 - F2 between the two particles
In this model there is no motion along the z direction, so we must have the following relation between 0 and t
Fl - F2 = F - F'
(4)
P1is the restoring force on the particle which comes from the electrostatic interaction with the particle 2; this force which tends to align the centers of the particles in the direction of the field has been calculated by Klingenbere and is given by
F'= 127reOt,a2B2E2f ,
(5)
with
In this relation, @ = (a - 1)/(a + 2), where a = ep/c, is the ratio of the permittivity of the particles to the one of the suspension (a = 2.5 at a frequency of 100 Hz). The quantities fll, f r , and fl are equal to 1 in the dipolar approximation but otherwise have been calculatedloas a function of a and of the separation between the two spheres. When the distance, I , between the two particles increases, the electrostatic restoring force P1passes through a maximum which defines the static yield stress
Here t o is the minimum normalized separation between two spheres, when 0 = 0. It corresponds to 2 times the thickness of the ionic double layer whose order of magnitude is a few tens of angstriims, and we have taken 50 = as a reasonable estimation. The numerical solution of eq 7 with the condition given by eq 8 gives the relative displacement versus the time. Then from eq 2 we can relate the applied shear stress at a given time to the rate of strain y = d(tgO)/dtat this sametime. These curves are represented in Figure 5 for the lower ( c = 4 X 10-3 Pa/s) and the higher (c = 0.33 Pa/& rates used in the experiments. Both curves are interrupted when the distance between two particles becomes equal to half the average distance, (d)/2, between the chains (with ( d ) = a ( 2 ~ / 3 @ ) lif/ ~we assume that the chains form a cubic lattice). First we can note than the solid curve which represents the lower rate is a steplike curve which allow the definition of a static yield stress which is the theoretical (13)Batchelor, G.K.J . Fluid Mech. 1976, 74, 1.
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Figure 5. Stress versus shear rate obtained from the numerical solution of eqs 7 and 8: (-) c = 4 X lO-9 Pa/s; (- - -) c = 3.3 x 10-1Pa/s; (- - -) discharge curve, 7 = q.7 + 7d (cf. text). The two-chargecurves are interrupted when the upper particle hae gone over the half distance between two chains ( ( d ) / 2 = (a/2) ( 2 * / 3 w 2 ) .
static yield stress previously obtained is = 1.38 Pa. In this case the rate of increase of the stress is low enough for the particles to have the time to reach their equilibrium position; the restoring force is always almost equal to the applied force and so the rate of strain remains close to zero up to the yield stress. On the contrary when the rate of increase of the applied stress is too high, the particles do not have enough time toreach their equilibrium position and the applied shear stress can be much larger than the maximum restoring force which givesthe staticyield stress, even for separations of the spheres lower than the critical separation. In this second case we obtain the short dashed curve in Figure 5. The comparison with the lower and the higher experimental curves in Figure 2 shows that this model qualitativelyreproducesthe observed behavior of the stress versus the shear rate for the beginning of the charge curve. For a more achieved rheological model we should need to consider what happens after the first breakdown, that is to say we need, for a separation larger than ( d ) / 2 , to introduce the interaction of the test particle with a third one which will model the periodic rebuiling of chains at low shear rate; furthermore at higher shear rates the chains will be broken by the flow. Actually a more elaborated model based on the existence of isolated chains of sphere is very likely unuseful since, in practice, we have thick aggregates? instead of chains. So, we have chosen a simple Bingham law, 7 = qsy + 7d, without any time dependency to represent the remaining of the rheogram. In order to keep the coherence with the previous model, we have taken for i d the experimental value multiplied by the ratio of the predicted static yield stress to the experimental one, which gives T d = 0.8 Pa. This order of magnitude can also be recovered by taking the average value of the restoring along they axis over half the distance between force, P', the two chains
This evaluation of the dynamic yield stress is correct but somewhat misleading since the origin of the dynamicyield stress is not related to an average of the electric force-which would be zero for an isotropicstructure-but to the dissipation associated with this force.9 The calculus of 7d from eqs 9 and 5 gives 7d = 1 Pa, which is close to the experimental value. The Bingham law with this value
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Figure 6. Memory effect on the rheogram: (-) f i t charge and discharge curve with E = 1.71 X 108V/m and c = 10 Pa/&, (- - -) second charge under same conditions as the f i t one, carried out immediately after the fiit discharge; (- -) third charge after a waiting time of 90min in the presence of the field.
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of 7d and the viscosity of the suspension at zero yield is plotted in Figure 5. We have noted that the experimental diecharge curve was more or less followinga Bingham curve without time dependency; this is understandable since the starting situation at high shear rates corresponds to all possible separations between 0 and ( d ) . Then, when the stress decreases below the static yield stress, a large number of particles whose separation is still higher than the critical separation d, = 0.13~(which corresponds to the maximum electric restoring force) will continue to move, the fiial stop corresponding to a stress which is lese than or equal to the dynamical yield stress.
IV. Discussion This simple two spheres model explains the main features of the experimental rheogram obtained with an electrorheologicalfluid. It also predicts the right order of magnitude without any fitting parameter. This model could be useful to determine the minimum separation, 40, between two particles by taking this quantity as a parameter to be determined by a fit. We have already pointed out that these experimenta with different rates of increase of the stress must be preceded by an homogeneization of the suspension at high shear rates and without field. If we do not respect this procedure the curves we obtain are not reproducible. This is demonstrated in Figure 6 where the second charge (long dashed line) is executed immediately after the discharge curve. We see that for this second charge with the same rate of increase (c = 10Pa/mn) as the f i t one, the structure breaks more easily and more progressively. This is very likely due to the formation of a new structure during the discharge which does not have the time to relax toward the one which is stable in the absence of the flow. Actually this hypothesis is supported by the observation of the formation of stripes in an oscillating planar shear flow when the strain amplitudebecomes larger than 1,whatever the frequency used between 1 Hz and 60 H2.I' These stripes, which are parallel to the velocity lines, would be in that case formed at high shear rates and, due to the high viscosity of the suspendingfluid,would persist during the discharge time. Such an anisotropic structure has no reason to exhibit the same yield stress as the initial one (14) Bossis, G.; Lemaire, E.; Petit, L.; Pereello, J. h o g . Colloid Polym.
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Electrorheological Fluids
which was formed of isotropic aggregates oriented along the electric field. Another argument in favor of this explanation comes from the followingobservation: if &r the second discharge we wait a long time (90min) in the presence of the field before to start a new rheogram, we get a new curve (short dashed line) which is closer to the first charge. This "change of structure under the field" can be explained by the slow evolution of this anisotropic stripe structure toward the equilibrium isotropic structure formed in the absence of the flow. We expect such a slow evolution since, on one hand, the thermodynamic force which is proportional to the difference of free energy between the two states seems to be low" and, on the other hand, the hydrodynamicresistance relative to the motion of particles which are nearly at contact increases as the reciprocal of the separation between two particles. Furthermoresome rearrangementscan also involvethe relative motion of clusters of particles, which will still be much slower. We believe that these rearrangementa only slightly distort the structure of the stripes but remain in a metastable state macroscopically very different from the
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initial isotropic one. In fact the charge curve obtained &r this waiting time of 90 min is quite different from the fmt one, indicating that we have still not reach the initial equilibrium structure. In conclusion the rheological behavior of these suspensions in the presence of an electric field is not simple but becomes understandable if we take into account the dynamical aspect through the time evolution of the distance of separation between two particles. When cycles of charge and discharge are done in the presence of the field, we have found a memory effect which can be explained by the formation of a macroscopic order inside the suspension. Further progress in the explanation of this time-dependent rheological behavior should be done with the use of transparent cells in order to observe with a microscope the possible changes in the microstructure.
Acknowledgment. We are very grateful to A. Audoly and B. Gaypara for their technical support. Registry No. Silica, 7631-86-9.
