Advances in modelling the mechanisms and rheology of electrorheological fluids

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Vol. 11, No. 3, September 1999 pp.169-195

Advances in modelling the mechanisms and rheology of electrorheological fluids

Howard See*

Department of Mechanical & Mechatronic Engineering, The University of Sydney, NSW 2006, Australia

(Received August 2, 1999; final revision received August 19, 1999)

Abstract

An electrorheological fluid (ERF) is typically a suspension of semi-conducting solid particles dispersed in an insulating carrier fluid, and shows a dramatic change in rheological properties when an external electric field is applied. This rapid and reversible change in flow properties has potential application in many electronically controlled mechanical devices, but the development of efficient devices and optimal materials for ERF is still hindered by incomplete understanding of the fundamental physical mechanisms involved. In recent years there have been considerable advances in relating microstructural models to the rheological behaviour, and these will form the basis of this review. Results of the theoretical calculations and simulations will be compared to the key experimental evidence available. An overview of the fundamental physical concepts behind electrorheological fluid behaviour will also be presented.

Key words : electrorheological fluid, ERF, microstructure, yield stress, electric polarisation

1. Introduction

An electrorheological fluid (often abbreviated to ERF) is typically a suspension of solid particles in an insulating carrier fluid, showing a rapid and reversible increase in viscosity under an electric field. This dramatic increase in viscosity is often referred to as the “electrorheological response” (ER response), or sometimes the “Winslow effect”, after Willis Winslow (1949) who first reported the pheno- menon in 1949. An example of the response is shown in Fig. 1, where the shear stress of a suspension of carbonaceous particles in silicone oil evidently increases with electric field strength. As can be seen from Fig 1, the flow behaviour of ERF under shearing deformation can be represented by the Bingham fluid model, where the shear stress τ is given by

τ=τy+ηγ^⋅. (1)

Here γ^⋅is the shear rate, η is the plastic viscosity and τy is the yield stress induced by the electric field. Experimen- tally, it is found that τy depends on the electric field as τy E^α, where α takes values close to 2 for low to moderate field strengths, but often appears to fall below 2 for higher fields. The electrorheological response is usually very rapid: for example upon application of a field of magnitude 1kV/mm, the initial jump in shear stress usually

occurs within several 10's of ms. Accompanying the increased shear resistance under an electric field there is a flow of electric current through the system, and this is illustrated in Fig. 2 for the same carbonaceous particle suspension. Because of these tunable rheological properties, electrorheological fluids have been the focus of much attention in recent years, and extensive review articles have

*Corresponding author: howards@mech.eng.usyd.edu.au

Fig 1. The variation of shear stress τxz with shear rateγ⋅ for an electrorheological fluid consisting of carbonaceous particles dispersed in silicone oil with volume fraction φ∼0.3, under a dc electric field E. The field strength increases from the bottom curve to the top: E=0, 1, 2, 3 kV/mm[Ish- ino and co-workers, 1995].

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170 Korea-Australia Rheology Journal appeared in the literature (Deinega and Vinogradov, 1984;

Block and Kelly, 1988; Gast and Zukoski, 1989; Jordan and shaw, 1989; Zukoski, 1993; Weiss and co-workers, 1993; Parthasarathy and Klingenberg, 1996).

The basic mechanism for this increased flow resistance is thought to be that the external field induces electric polarisation within each particle relative to the carrier fluid (an “electric dipole”), and the resulting electrostatic interaction forces between the particles lead to the formation of aggregates aligned in the direction of the field. The presence of these particle aggregates in the flow field causes the macroscopically measured increase in viscosity, as

shown schematically in Fig 3. In general the polarisation may arise from various mechanisms of charge transport, such as orientation of atomic or molecular dipoles, and interfacial polarisation, with the latter generally thought to be the main contributor to the ER response under the dc or ac fields commonly used. This basic model of particle polarisation leading to aggregation and increased flow resistance, appears to have gained the widest acceptance, and will be the focus of this review paper. It should be pointed out that other models for the field-induced increase in shear stress have been suggested, including field- induced aggregation due to water bridges between particles (Stangroom, 1983; Stangroom, 1991; See and co-workers, 1993; Tamura and co-workers, 1993), electrostatic torque preventing particle rotation in the flow field (Block and Kelly, 1988; Hemp, 1991) and overlap of the diffuse counter ion clouds surrounding neighbouring particles (Klass and Martinek, 1967a; Klass and Martinek, 1967b).

Indeed, it is quite possible that some of these mechanisms operate concomitantly with the polarisation-aggregation process, but for clarity we will focus on only the latter here.

The reader is referred to the original papers and references therein for further information on these mechanisms.

There has been considerable effort put into developing optimal materials for use in ERF. Broadly speaking, the carrier fluid needs to be a liquid with minimal electric conductivity and reasonably low viscosity. The particles are dispersed in the carrier fluid with volume fractions typically 30-40%, and under an electric field the ideal particle material should facilitate the induction of a large polarisation and hence ER response, with minimal drawing of electric current. The particles also need to be durable, non- abrasive, and stable against irreversible flocculation or sed- imentation. Earlier efforts focussed on hydrous particles, such as silica or cellulose particles with moisture content of 5-10% (eg Uejima, 1972), but the fact that water is an active constituent in these systems means that the range of operating temperatures is limited by the freezing and boiling points of the water. Recently, considerable emphasis has been placed on the development of anhydrous materials, which are expected to have a much wider operating temperature range. Advances in our knowledge of the mechanisms of electrorheological behaviour, which will be out- lined in this review, have enabled progress to be made in the selection of optimal materials for ERF. Block and Kelly (1988) and Weiss and co-workers (1993) provide extensive lists of electrorheological materials (particles, carrier fluids, and additives (if any)) that have been reported in the sci- entific and patent literature, covering both hydrous and anhydrous particle systems. In recent years, ERFs com- prising of the anhydrous particle suspensions listed below have received considerable coverage in the literature. It should be noted that this list has been compiled to the best of the author's knowledge, and is not intended as an Fig 2. The variation of electric current density j (current per unit

area of electrode) with dc electric field E, for the same electrorheological fluid as Fig. 1 under a shear rate of 366s^-1 (Ishino and co-workers, 1995).

Fig 3. Schematic of an electrorheological fluid consisting of solid particles dispersed in a carrier fluid, between two electrode plates and under shear flow ·γ. Under the electric field E0

the particles become polarised (indicated by the arrows) due to the mismatch in electrical properties between the particle material and carrier fluid. The electrostatic forces between the particles lead to aggregates aligned in the field direction, causing the increase in shear viscosity. The particle size is exaggerated here: typically particles are several microns and the gap is ~1mm. At low shear rates the microstructure is thought to consist of chains or columns of particles spanning the electrodes as shown here; at higher shear rates smaller aggregates suspended in the flow field are often observed.

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exhaustive survey of currently available ERFs. There is the possibility that some of these ERFs are now commercially available, and for reference the company/corporation names (if known) are indicated in parentheses-this has been done as a service to the reader, but does not imply endorsement or otherwise of any of these materials:

(i) carbonaceous particles dispersed in silicone oil (Bridgestone Corp., Japan) (Ishino and co-workers, 1995; Sakurai and co-workers, 1996; Sakurai and co-workers, 1999);

(ii) particles consisting of a polymer core surrounded by a shell made of 50 nm silica or titanium hydroxide particles, dispersed in silicone oil (Otsubo and Eda- mura, 1994, Otsubo and Edamura, 1995); or a polymer core surrounded by an inner layer of silver and an outer layer of silica (Fujikura Kasei Co., Japan) (Saito and co-workers, 1997);

(iii) sulfonated poly(styrene-co-divinylbenzene) particles in an insulating oil (Nihon Shokubai Corp., Japan) (Kawakami and co-workers, 1997; Ikazaki and co- workers, 1997);

(iv) polyurethane particles dispersed in silicone oil (Bayer AG, Germany) (Placke and co-workers, 1995a; Placke and co-workers, 1995b; Bloodworth and co-workers, 1995);

(v) coated polyaniline particles (Lubrizol Corp. USA) (Havelka and co-workers, 1994);

(vi) complex strontium titanate particles dispersed in silicone oil (Zhang and co-workers, 1998; Ma and co-workers, 1998);

(vii) copolystyrene particles with a polyaniline coating, dispersed in silicone oil (Kuramoto and co-workers, 1995);

(viii) Copolyaniline/sulfonic acid particles dispersed in silicone oil (Cho and co-workers, 1997).

Extensive reviews of the material aspects of ERFs are available (Block and Kelly, 1988; Zukoski, 1993; Weiss and co-workers, 1993) and much work continues to be done from the stand-points of materials science and chemistry to find the optimal combination of material properties. This review paper focuses on the advances made in our understanding of the physical mechanisms behind the electrorheological response, and so selective reference will be made to specific materials and ERFs if they illustrate aspects of a particular physical mechanism.

Concomitant with the investigations into the basic mechanisms (the focus of this paper) and materials of ERFs, there has also been considerable effort put into the development of applications which utilise the tunable rheological properties. For reasons of space, applications of ERF technology will not detailed here, but good reviews can be found in (Hartsock and co-workers, 1991; Coulter and co-workers, 1993; Kohudic, 1994). Recently, there have appeared publications discussing advances in vibration

damping devices (Don and Coulter, 1995; Peel and co- workers, 1996; Brooks, 1997; Marksmeier, 1997, clutches (Whittle and co-workers, 1995a; Papadopoukos, 1997), control valves (Whittle and co-workers, 1995a; Furusho and Sakaguchi, 1997; Choi, 1997) and optical devices (Hunter and co-workers, 1993; Tao, 1997; Fujita and co- workers, 1997; Zhao and co-workers, 1997).

Parallel with the continual strong interest in ERFs, there have been developments of other physical systems which also show dramatic changes in rheological properties upon application of an external field (electric or magnetic).

Although there are some physical similarities with electrorheological behavior, these systems will not be treated in detail in this review. However, for completeness the main types are listed below, together with some references to which the interested reader may wish to refer:

(i) Magnetorheological fluids (MRF). These are typically suspensions of particles that develop magnetic polarisation under a magnetic field, and so are the magnetic counterparts of ERF. Research into MRFs is quite active, and recent publications describe advances in the theoretical modelling (Rosens- weig, 1995; Zhang and Widom,1995; Jolly and co-workers, 1996; Tao and Jiang, 1998; Zhou and co-workers, 1998), experimental investigation of the basic mechanisms (Fermigier and Gast, 1992;

Tan and Jones, 1993; Cutillas and co-workers, 1998;

Furst and Gast, 1998; Nahmad-Molinari and co- workers, 1999), rheological measurements (Bossis and Lemaire, 1991; Lemaire and Bossis, 1991; Weiss and co-workers, 1994; Lemaire and co-workers, 1995; Popplewell and Rosensweig, 1996; Laun and co-workers, 1996), as well as their applications (Carlson and Weiss, 1994; Dyke and Spencer, 1996).

There have also been reports of suspensions of particles which respond under simultaneously applied electric and magnetic fields (Minagawa and co- workers, 1994; Koyama and co-workers, 1995; Wen and co-workers, 1997b).

(ii) Liquid crystal systems under an electric field (Yang and Shine, 1992; Tse and Shine, 1995; Inoue and Maniwa, 1995; Negita, 1996; Lacey and Malins, 1996; Chiang and co-workers, 1997; Yao and Ja- mieson, 1998a; Kimura and co-workers, 1998; Ki- mura and co-workers, 1998b).

(iii) Polymer fluids under an electric field (Tajiri and co- workers, 1997; Tajiri and co-workers, 1998; Orihara and co-workers, 1999; Tajiri and co-workers, 1999).

(iv) Particles dispersed in elastomers (“ER elastomers”) (Sakurai and co-workers, 1999; Sakurai and co- workers, 1998). The viscoelasticity of these compos- ite materials changes dramatically with electric field.

This review will focus on the many advances in our understanding of the mechanisms of electrorheological

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172 Korea-Australia Rheology Journal fluid behaviour, in particular the relationship between the

microstructure, interparticle forces, and the rheological response. Much of the earlier work (up to 1995) on the mechanisms of ERF have been discussed in the extensive review by Parthasarathy and Klingenberg (1996). Thus the main focus of the current article will be on the significant body of work reported since Parthasarathy and Klingen- berg (1996). This article will also present a brief overview of the basic physical concepts behind the electrorheological effect, which is essential background for the detailed discussion of the mechanisms and may also be useful to a newcomer to the field. It is hoped this article will help solidify our knowledge of the basic processes in ERF, and suggest directions for the development of improved materials, devices and predictive models.

The rest of this paper is organised as follows. In Sect 2, the nature of the electric field-induced structures and their relation to the rheological behaviour will be reviewed. In Sect 2.1, the mechanism of particle polarisation under dc and ac electric fields is discussed. The discussion then proceeds in Sect 2.2 to the equation of motion of a particle in an idealised system consisting of uniform spheres suspended in a Newtonian carrier fluid, which is commonly used in particle-level computer simulations. The forces acting on the particle will be described, such as the electrostatic and hydrodynamic interactions, the steric forces preventing particle overlap, as well as the interactions with the electrode walls. After this brief summary of the essential features of ERF modelling, we begin the review of recent advances in our understanding of the mechanisms of ERF behaviour. There are discussions of the initial aggregation process and structures formed in the absence of a macroscopic flow field (Sect 2.3), the behaviour under steady shear flow (Sect 2.4), and the dependence of electric current on shear rate (Sect 2.5). These are followed by discussions of the behaviour under oscillatory shear flow (Sect 2.6), and the response under time-varying electric fields, in particular under ac and step electric fields (Sect 2.7). The significant electrorheological effect observed under squeezing deformations has been the focus of some attention in recent years, and the associated literature is reviewed in Sect 2.8. The models discussed in Sect 2 are based on idealised electrostatic interactions, whereas it is known that under the high electric fields used in elec- trorheology, particularly the very intense fields in the interparticle gaps, the electric characteristics behave in a strongly non-linear fashion. This has major implications for the nature of the electrostatic interactions between the particles and hence the rheological response, and so has been the subject of several investigations recently, which are reviewed in Sect 3. In Sect 4, we consider other depar- tures from the idealised model based on uniform, spherical particles, and discuss advances made with the issues of non-spherical particles (Sect 4.1), particle size effects

(Sect 4.2), blending particles with different conductivities (Sect 4.3) and the use of particles with an internal structure (Sect 4.4). Conclusions and some suggestions for possible future directions in research are presented in Sect 5.

2. Electric field-induced structures and the rheo- logical response

2.1. General features of particle polarisation 2.1.1.An isolated particle under dc field

To introduce the concept of particle polarisation, we consider the simplest case of an isolated sphere of radius a with constant electric conductivity σp and dielectric constant εp, suspended in a carrier fluid of conductivity σc and dielectric constant varepsilon εc. This system is sandwiched between a pair of parallel electrodes with a separation of h0

(a<<h0), with the particle located in the mid-region of the gap away from the electrode surfaces. A constant voltage V0 is applied, inducing an electric field E0= V0/h0 across the electrode gap. For simplicity, we initially treat the direct current case, and the extension of these ideas to alter- nating currents and transient behaviour will be given later.

We use the equations of electrostatics to calculate the polarisation or electric dipole p induced in the particle under the external electric field E0= E0ez which we take to be in the +z direction without loss of generality. We assume that the current flowing through the system has reached steady state. The volume density of free charge will be identically zero since the conductivities and permittivities are uniform in both the particle and the carrier fluid (eg there are no diffuse counterion clouds surrounding each particle). However there will be surface distribution of free charge accumulated at the particle-fluid interface, and this leads to the effective electric polarisation of the particle. To find the electric field E and hence the particle's polarisation, we solve the Laplace equation for the potential φ

∇²φ = 0 (2)

with the following boundary conditions at the interfaces (using an obvious notation)

φp= φc (3)

(4) Eqs (2)-(4) enable the potential φ and hence the electric field E(r) = -∇φ to be found. Solving these equations, we find that the field outside the particle is the same as an electric dipole P = pcondez located at the particle centre with magnitude

pcond= 4πεoεca³βcondE0 (5) where εo is the permittivity of space (εo= 8.85×10^-12C²J^-1m^-1) and

σp

∂φp

∂n --- σc

∂φc

∂n ---

=

(5)

(6) Eq(5) tells us that the polarisation of the particle is proportional to the externally applied electric field and in the same direction, with a magnitude that depends on the degree of mismatch between the particle and carrier fluid conductivities.

2.1.2. Behaviour under ac fields:Maxwell-Wagner polarisation

Consider the same isolated particle as before, but now under a step electric field of magnitude E0 applied from t=0 :

E(t)=0 for t0

=E0 for t0 (7)

Immediately after the application of the field, the mobile charge carriers would have only just begun to move and not travelled any significant distance, and the electrical response of the system will be governed by the dielectric constants of the particle material and carrier fluid (εp, εc) (Davis, 1992a). εp, εc are determined for example by molecular orientations under the electric field, and this response can be assumed to be almost instantaneous (over time scales 10^-6s at the long- est). This initial response is independent of the conductivities of the particle or carrier fluid (σp, σc), and a similar calculation to the above gives for the initial dipole magnitude:

pdielec=4πεoεca³βdielecE0 (8)

where

(9) As time progresses, there is a migration of free charge which builds up at the carrier fluid/particle interface, and has the effect of electrically screening the field in the particle interior (Davis, 1992a). Eventually the electric field reaches a distribution determined by the conductivities of the carrier fluid and particle material, and the dipole is given by eq (5). The characteristic time τMW for charge transport, describing the transition from the dielectric-dominated to the conductivity-dominated behaviours, is given by

(10) This is also sometimes called the “polarisation relaxation time”. It is easy to show that the dipole will change with time as follows (Ginder and co-workers, 1995):

p(t)=4πεoεca³E0 [βcond+(βdielec-βcond)e^-t/τ^MW] (11) The model presented here is generally known as the Maxwell-Wagner model of interfacial polarisation (hence the subscript MW). This is the simplest physical model which takes into account the effects of conductive and

dielectric properties on the polarisation of a particle, and assumes εp, εc, σp, σc are constants. A typical electrorheological fluid designed for use under dc field conditions (as are many proposed devices), such as the carbonaceous particles in silicone oil suspension (Sakurai and co- workers, 1999), would have εp ≈ 5, εc≈ 3, σp ≈ 10^-7mho/m with σc<<σp(at most σc ≈ 10^-10mho/m), which gives us from eq (10) τMW ≈ 10^-3s.

We now calculate the polarisation under an ac electric field E(t)=Re{E0e^iωet}ez of frequency ωe. Given the above time dependent behaviour, we would expect that the polarisation of an isolated particle would vary sinusoidally as e^iωet but with a possible shift in phase, and have a magnitude depending on ωe. An analogous calculation to the previous one yields the complex potential φ^* from which the electric field external to the particle can be found.

As before, it turns out this field is equivalent to an electric dipole P = p(t)ez located at the particle centre with instantaneous magnitude

p(t)=4πεoεca³E0Re{β^*e^iωet} (12) where

(13) Here the complex dielectric constants are given by (14) (15) There is a transition from conductivity-dominated behaviour at low frequencies, to dielectric-dominated behaviour at higher frequencies, with the transition frequency equalling 2π/τMW.

It is the electrostatic interaction between these polarised particles which causes the formation of aggregates aligned with the electric field and hence the increase in flow resistance. The nature and modelling of the forces experienced by the particles in an electrorheological fluid will be discussed in the next section.

2.2. Features of microstructural models and com- puter simulations

2.2.1.The equations of motion

Consider a system of N spherical particles suspended in a Newtonian carrier fluid of viscosity ηc confined between parallel electrodes, as illustrated in Fig 3. For concreteness, we consider the case of shear flow in the x-z plane with shear rateγ(t), caused by holding the bottom electrode sta-^⋅ tionary and sliding the top electrode parallel to itself in the x-direction. The particles occupy a volume fraction φ, and since they are typically a few microns in size Brown- βcond σp–σc

σp+2σc

---

=

βdielec εp–εc

εp+2εc

---

=

τMW

εo(εp+2εc) σp+2σc

---

=

β^* εp*–εc*

εp*+2εc*

---

=

εp*, εc*

εp* εp i σp

ε0ωe

--- –

= εc* εc i σc

ε0ωe

--- –

=

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174 Korea-Australia Rheology Journal ian motion effects can be ignored (these will be briefly

discussed later). The influences of inertia and gravity are also usually not considered in the microstructural models. The effects of Joule heating due to the flow of the electric current are assumed to be negligible. For a particular configuration of particles, the instantaneous force balance equation for the i-th particle can be written as follows:

(16) The terms in this equation will now be explained sepa- rately, with the electrostatic interaction force and the electrode wall interaction force discussed in detail in the following sections.

The term is the drag force experienced by particle i as it moves through the carrier fluid of viscosity ηc, and in general this “hydrodynamic interaction” force depends on the relative positions of other particles in the suspension (labelled j), as well as their velocities and orientation with respect to the externally applied flow field. Al- though some simulations of ERF have accurately incorporated these interactions (eg the Stokesian dynamics technique in Bonnecaze and Brady, 1992a; Bonnecaze and Brady, 1992b; Baxter-Drayton and Brady, 1996), the com- plexity of the equations restricts the simulations to a small number of particles (eg 25). For simplicity, many simulations use the following free-draining Stokes' approximation for the drag force on a translating sphere, which ignores the presence of nearby spheres and hence disregards the hydrodynamic interactions altogether. The Stokes' approximation states that the drag force on sphere i is proportional to the difference between the particle velocity dri/dt and the ambi- ent carrier fluid velocity at the particle center vc(ri, t):

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Here ri= (xi, yi, zi) is the position vector of particle i and vc(ri,t) =γ(t)z^⋅ iex for the case of the shear flow shown in Fig 3.

Despite the apparent crudeness of the Stokes' approximation, its simplicity enables the simulation of a large number of particles, while still capturing the essential physics of the microstructural processes occurring in ERF. Indeed, Parthasarathy and Klingenberg (1999) show that there are only minor qualitative changes if eq (17) is used in place of more accurate formulations of the hydrodynamics. We note that the external flow field γ(t) may take a variety of^⋅ forms: (i) absence of flow (ie γ(t)0, for simulations of^⋅ quiescent aggregation and static structure); (ii) very small shear rates and strains (for yielding calculations); (iii) steady shear γ(t)=^⋅ γ^⋅0 (constant); and (iv) small amplitude oscillatory shear γ(t)= ^⋅ γ0ωsin(ωt).

is the rigid body repulsion force experienced by sphere i due to an adjacent contacting sphere j. This force acts radially between the particle centres, and is a function of the interparticle distance rij (Fig 4) and is often approxi- mated by a soft-core potential, for example a power law (18) where er is the unit vector in the radial direction. The pre- factor A is chosen so that the repulsion and electrostatic attraction forces balance at the appropriate interparticle separation (see eg Parthasarathy and Klingenberg, 1996).

A number of possible values of the exponent n have been reported in the literature-Parthasarathy and Klingenberg (1996) note the range 5<n<37. There are other forms available for the rigid body repulsion force, such as an expo- nential variation with surface-to-surface distance. An important point to note is that small variations in these repulsion models can have significant effects on the nature of the structures formed and even on the rheological response predicted (see the discussion in Parthasarathy and Klin- genberg, 1996). It should also be noted that if an approximation like eq(17) is not used and the hydrodynamic interactions are accurately incorporated in the simulations, there is no need to artificially introduce the repulsion force since the lubrication forces between neighbouring particle surfaces will prevent any overlap.

Employing the free-draining Stokes' approximation eq (17), we can write the governing equation (16) as

(19) This equation is the starting point for most simulations:

commencing with some initial configuration of the particles, eq (19) is integrated numerically in time using Euler's method, enabling the evolution of the particle positions to Fhydro

ij

j i∑≠ Frepuls ij

j i∑≠ Felec

ij j i∑≠ Fwall

+ + + i =0

Fhydro ij j i∑≠

Fwalli

Fhydro ij j i∑≠

Fhydro ij

j i∑≠ =–6πηca dri

---dt

 –vc(ri,t) 

Frepulsij

Frepulsij –A a r_ij

 ----

 ⁿer

=

dri

---dt vc(ri,t) 1 6πηca

--- Frepulsij j i≠

∑

 Felecij j i∑≠ Fwalli



+ +

+

=

Fig 4. Two paticles i and j under the electric field E and the flow field γ⋅(t). The spherical particles have a radius a, conductivity σp, dielectric constant εp, and are dispersed in a carrier fluid assumed to be a Newtonian fluid of viscosity ηc, with conductivity σc, dielectric constant εc.

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be followed. Note that the terms on the right hand side of eq (19) depend only on the positions of the particles.

To model suspensions of very small particles whose motion may be affected by thermal effects (eg sub-micron sized particles), a few simulations have also added a random force to the right hand side of eq (19) to represent Brownian motion effects (eg Melrose and Heyes, 1993). In these simulations, the instantaneous force for each direction is usually calculated from a Gaussian distribution with a zero mean, and a standard deviation depending on the absolute temperature, carrier fluid viscosity and some appropriate time increment. A Monte Carlo simulation approach has also been adopted by some workers (Tao and Sun, 1991a; Tao and Sun, 1991b). An equilibrium statistical mechanics approach was used by Adriani and Gast (1988) who calculated the effect that the polarisation interactions between particles would have on the particle pair distribution function and the bulk shear stress.

2.2.2. Electrostatic force

For a given assembly of particles, it is difficult to exactly calculate the electrostatic interaction force experienced by particle i due to another particle j, since we need to know the electric field distribution and integrate the traction vector of the Maxwell stress tensor over the surface of particle i (δµv is the unit tensor). Here we will introduce the most basic model of the electrostatic interactions. For generality we consider a sinusoidal external electric field E(t) = Re{E0e^iωet}ez applied to the suspension.

Assuming that the particles do not alter each other's distribution of charge (which becomes valid in general if they are widely separated), to lowest approximation the instantaneous force acting on particle i with a polarisation Pi

due to another particle j is given by

(20) which is evaluated at the centre of particle i. Here Ej is the disturbance field created by the particle j. If we use the results for the magnitude (eq(12)) and electric field of an isolated dipole, eq (12) will give after some calculation

(21) where Fs(t) indicates the sinusoidally varying magnitude of the force

(22) Here er, e_θ are unit vectors, and the angle θij describes the relative orientation of the two particles as shown in Fig 4 (our notation follows that of Parthasarathy and Klingen- berg, 1999). Eq(21) is often called the “point dipole approximation” to the electrostatic pair interactions, and is the most commonly used formulation in computer simulations.

Eq(21) greatly underestimates the magnitude of the force

when two particles are in close proximity, but it is useful since it manages to capture the basic anisotropic nature of the electrostatic interactions: particles aligned with the field will experience an attraction force, whereas a particle pair which are offset from this aligned configuration will experience a torque attempting to align the pair with the field, and a repulsion force will act if the particles are placed side-by-side with the line joining their centres perpen- dicular to the field direction.

For high frequency ac fields, the time-averaged (r.m.s.) forceF^s between a pair of particles will determine the aggregation behaviour and hence rheological response, and is given by:

(23) Here and is the modulus of the complex number β^*and a function of the frequency ωe given by

(24) where` =`ωeτMW, and βcond, βdielec, τMW are given by eqs (6), (9), (10). In the limit of very high frequencies (

) we have and we see

again that the interaction force will be dominated by the dielectric constants.

In the case of a dc electric field E(t) = E0ez (constant), we find the following for the force magnitude Fs in eq(21):

(25) where βcond is the conductivity mismatch (eq(6)).

Thus under dc and low frequency ac fields, the interaction forces are governed by the conductivities, but at higher frequencies the dielectric permittivities will determine the interaction magnitude. The frequency at which this transition in behaviour occurs is inversely proportional to the polarisation relaxation time viz 2π/τMW.

The nature of the force in eqs (21)-(25) also agrees with the commonly observed dependence of the yield stress τy, which is expected on physical grounds since τy

would be determined by the strength of the interparticle forces. Further, non-dimensionalising eq(16), a dimensionless quantity called the Mason number Mn drops out. Mn is the ratio of the magnitude of the electrostatic force between a pair of touching particles aligned with the electric field Fs, to the magnitude of the viscous drag force due to the flow field acting to pull the pair apart (proportional to ηca²γ^⋅). The commonly used definition of Mn is (Par- thasarathy and Klingenberg, 1996)

(26) The Mason number Mn indicates the relative importance of the electrostatic and hydrodynamic forces between the particles, and is named after S.G. Mason in recognition for Felecij

Felecij

E_µE_ν–1 2⁄ E²δµν

[ ]

F_elec^ij

Felecij ( )t =pi⋅∇Ej

Felecij ( )t Fs( )t a rij

 ----

 ⁴[(3cos²θij–1)er+sin2θije_θ]

=

F_s( )t =12πεcε0a²E₀²[Re{β^*e^iωet}]²

Fs

〈 〉=12πεcε0a²Erms2 β^{* 2} E_rms=E₀⁄ 2 β^*

β^{* 2} (βdielecωe2+βcond)²+ωe2(βdielec–βcond)² 1+ωe2

( )²

---

= ωe

ωe

β^{* 2}=βdielec2 =[(εp– εc) ε⁄( p+2εc)]²

Fs=12πεcε0a²E02βcond2

E02

Mn ηcγ· 2εcε0β^{* 2}E02

---

=

(8)

176 Korea-Australia Rheology Journal his pioneering studies of the behaviour of particle sus-

pensions under flow and electric fields (eg Arp and Mason, 1977). Good data collapse has been observed for viscosity versus shear rate curves under ac electric fields of different amplitudes, if the curves are replotted using Mn (Marshall and co-workers, 1989). Note that Mn is independent of the particle size a , indicating that the field-induced shear stress is predicted to be independent of particle size, at least at the level of this simplified model.

The point dipole model for the electrostatic interactions presented above is very approximate, and greatly underestimates the force magnitudes. Broadly speaking, the exact interactions are complicated by the two major effects of multipole and multibody effects:

(i) Multipole effects : When two polarised spheres approach, the disturbance field created by one sphere results in electrostatic moments of higher order than the dipole appearing in the other. These higher moments can make a large contribution to the interaction force, but their calculation is very complex and will not be detailed in this review. In the literature there are several papers describing the interaction between two spheres in close contact (Arp and Mason, 1977; Davis, 1964; Bonnecaze and Brady, 1990. Gast and Zukoski (1989) describes a calculation attributed to D.J. Klingenberg. The point dipole equation is recovered when the two spheres are far removed or when the mismatch in electrical properties is very small.

(ii) Multibody interactions : Along with the multipole effect, a further complication is that when more than two spheres are in close proximity the interaction forces are not pairwise additive. For example, the presence of a nearby third sphere can greatly alter the interaction force between a pair of spheres. This effect has been quantified by calculating the force acting between neighbouring particles in a chain or a lattice structure (Anderson, 1994; Clercx and Bossis, 1993). The point dipole approximation assumes that the interaction forces are pairwise additive.

Methods exist for accurately calculating the electrostatic forces taking into account multipole and multibody effects, and these include the electrostatic energy approach devel- oped for non-conducting particles and carrier fluids (Bon- necaze and Brady, 1992a; Bonnecaze and Brady, 1992b;

Baxter-Drayton and Brady, 1996) and multipole expansion techniques (Clercx and Bossis, 1993; see also the book by Jones (1995)). Even with these techniques the calculations are still very complex and have only been used in a few theoretical investigations or simulations (restricted to small numbers of particles eg 25). Many investigators seem sat- isfied in obtaining a qualitative description of the key physical features of ERF behaviour via the point dipole approximation, which because of its simplicity at least

allows simulations of a large number of particles.

2.2.3.Periodic boundary conditions and electrodes Some simulations will not explicitly include the electrodes in the calculation, but will use periodic boundary conditions in the x,y,z directions (see eg Allen and Tildesley, 1991), meaning that the simulation cell is effectively surrounded by identical cells to mimic being in an infinite system (if shear flow is to be modelled, sheared periodic boundary conditions can be imposed in the z direction). If electrodes are to be included in the simulation, the periodic boundary conditions can still be imposed in the x and y directions, and the effect of the electrode walls on the motion of particle i enter through the term in eq(16), which can be written as the sum of two contributions:

(27) Here is the force on the polarised particle i due to the electrode surface (generally assumed to be a perfect con- ductor)- this is an attractive force and is usually modelled by the method of electrostatic images (eg see Klingenberg and co-workers, 1991b). is the repulsion force between the particle surface and the electrode wall, and is often modelled in a similar way to the repulsion force preventing particle overlap (Parthasarathy and Klingenberg, 1996; Klingenberg and co-workers, 1991b). Again, there is no need to introduce this repulsion force if the hydrodynamic effects are accurately accounted for, since the lubrication force between the particle and wall will prevent actual contact.

For simplicity, Fig 3 shows the case of a mono-layer of spheres, where it is assumed that the particle centres all lie in the same plane y = constant. The symmetry of the equations means that the particles will never evolve out of this plane, and this configuration is used very often in simulations for reasons of computational efficiency. In fact, in the case of this monolayer, it is not necessary to explicitly consider the force balance or periodic boundary condition image cells in the y direction. True 3 dimensional simulations have also been performed using the governing equations as above, but with a substantially increased load on computational resources.

2.2.4.Calculation of stress

Many simulations aim to calculate the rheological response, and so require a rule connecting the system microstructure to the macroscopically measured stress. Consider a particular configuration of N particles in a cell of volume V, as in Fig 3. A commonly used approach to determine the stress is to ignore the hydrodynamic interactions, which enables the instantaneous particle-contributed stress tensor τ^p to be written as

(28) Fwalli

F_wallⁱ =F_{imag s}ⁱ _e +Fwall repulsi

Fimagi es

Fwall repulsi

τ^p 1 V----

– riFTOTALi i=1

∑N

=

(9)

Here ri is the position vector of particle i and is the sum of all non-hydrodynamic forces on particle i eg electrostatic, repulsion forces (see eg Doi and Edwards, 1986).

From eq(28) the stress and hence the rheological properties can be determined.

A more exact determination of the stress tensor is possible taking into account the hydrodynamic interactions, although this is computationally rather expensive. For example, in Bonnecaze and Brady (1992a) Bonnecaze and Brady (1992b) and Baxter-Drayton and Brady (1996) the stress tensor for a monolayer of spheres under shear and electric fields was accurately calculated using the general expression derived by Batchelor (1970).

2.3. Initial aggregation process and structure

There is already a substantial body of literature dealing with the kinetics of the particle aggregation process after application of an electric field in a quiescent ERF (ie no externally applied flow field), as well as calculations of the lattice structure formed by the particles within the columns aligned with the electric field (the so-called “ground state”

lattice structure). 2 dimensional simulations (ie a monolayer of spheres) have been used to investigate the aggregation kinetics and initial structure (Klingenberg and co- workers, 1991b; Klingenberg and coworkers, 1989; Jaggi, 1991; See and Doi, 1991; Toor, 1993), and some 3 dimensional simulations have also been reported (Melrose and Heyes, 1993; Tao and Jiang, 1994; Melrose, 1992; Hass, 1993). Experimentally, the types of structure formed in quiescent ERF have been investigated by light scattering techniques (Chen and coworkers, 1992; Smith and Fuller, 1993) and by microscopic observation (Klingenberg and Zukoski, 1990; Klingenberg and co-workers, 1991a; Klin- genberg and co-workers, 1993). Theoretical calculations of the lattice structure formed by the particles within the aggregates have also been presented (Davis, 1992; Fried- berg and Yu, 1992), and some calculations have also included the effects of Brownian motion on the aggregate shape and structure (Tao and Sun, 1991a; Tao and Sun, 1991b; Halsey and Toor, 1990b; Toor and Halsey, 1992;

Martin and co-workers, 1992; Tao, 1993). Much of this earlier work has been reviewed in detail in Parthasarathy and Klingenberg (1996).

Recent work has furthered our knowledge of the process of aggregate formation in a quiescent electrorheological system. Jian and Jiaping (1996) conducted 3 dimensional computer simulations using the point dipole model with Stokes' approximation for the hydrodynamics and found that the ERF will eventually evolve into a body-centred tetragonal lattice. A recent report by Gulley and Tao (1997) supports this, but they also investigated the effects of thermal fluctuations and found that, depending on the relative strengths of the electric field and the thermal fluctuations, the following states can also appear (in order of increasing

thermal effects): nematic liquid crystal-like state (with ordering only in the field direction); polycrystalline state;

and a glass-like state with no appreciable ordering. It should be noted that the extent to which these predictions of microstructure depend on the types of approximation used for the electrostatic and hydrodynamic interactions is unclear. Experimentally, microscopic observations of the aggregation process in glass microspheres dispersed in oil have been reported by Wen and co-workers (1998;

1999), and these support the basic picture of the formation of chains, followed by columns of aggregated particles, with the process accelerated with increasing electric field strength.

The role of particle and carrier fluid conductivity on the field-induced aggregation process in ERF has been investigated from a thermodynamic point of view in an elaborate calculation by Khusid and Acrivos (1995). They derived microstructure-based equations for the free energy of a suspension of conducting particles. One method of calculation used a mean-field approximation and the quasi-static elec- trodynamics equations to find the electric energy up to O (φ²) where φ is the particle volume fraction. From this the average interparticle interaction was obtained, enabling a thermodynamic description of the beginnings of the interaction process. A second method used a statistical approach and the group expansion method to calculate the change in the average electric energy when one particle is removed from the suspension, from which the average interparticle interaction was obtained. Although there was no direct treat- ment of the rheological behaviour, by comparing the average interparticle interaction energy with the thermal energy, it was predicted that conducting particles in a less conducting carrier fluid (ie σp > σc, which is the usual sit- uation) will form aggregates after the application of a dc electric field only if the following equation holds

(29) Here F1 and F2 are non-dimensional functions appearing in the theory, and are usually of O(1). Based on this theory, Khusid and Acrivos recommended that to achieve a quick aggregation of ERF particles, it would be optimal to have the ratio of conductivities and that the permittivity ratio εc/εp should be as large as practicable. Khu- sid and Acrivos found reasonable qualitative agreement between their model and some simulations of the cluster formation process in ERF reported in the literature. It should be noted that this calculation only focussed on the aggregation process, and that to achieve a large interparticle force with minimal electric current, which is the aim of most ERF material development, a larger value of σc/σp

is often optimal (typically in experiments σc/σp~10⁵ or more see eg Sakurai and co-workers (1999). The theory is extended to the case of ac electric fields, but the reader is FTOTALi

6εp

εc

---- 1 F1

---

> 2+(6F2–1)σp

σc

--- σp

σc

---

  ² –

σp⁄σc∼1 10–

(10)

178 Korea-Australia Rheology Journal referred to the original article for details (Khusid and

Acrivos, 1995).

2.4. Steady shear flow

There have been numerous simulations and calculations of the behaviour of particle chains or aggregate structures under shearing deformation, with a particular focus on the relationship between the shear stress and the microstructure. Particle-level simulations of the behaviour under steady shear flow and constant electric field have been reported (Bonnecaze and Brady, 1992a; Bonnecaze and Brady, 1992b; Melrose and Heyes, 1993; Klingenberg and co-workers, 1991b; Klingenberg and co-workers, 1991c;

Takimoto, 1991; Parthasarathy and Klingenberg, 1995a), as well as calculations of the stress when a prescribed microstructure (eg a regular chain or column of particles) is subject to an increasing shear strain, often until the point of chain rupture (essentially a calculation of the yield stress) (Anderson, 1994; Klingenberg and Zukoski, 1990; Davis, 1993; Davis and Ginder, 1994; Gulley and Tao, 1993;

Kraynik and co-workers, 1991; Ginder and Davis, 1993;

Lemaire and co-workers, 1992; Ota and Miyamoto, 1994;

Mokeev and co-workers, 1992; Davis, 1992b). There have also been studies which incorporate in the modelling the effects of Brownian motion of the particles (Melrose and Heyes, 1993; Melrose, 1992; Halsey and Toor, 1990b;

Bailey and co-workers, 1989; Heyes and Melrose, 1990;

Whittle, 1990; Martin and co-workers, 1994a; Halsey and co-workers, 1992; Martin and co-workers, 1994b). Much of this earlier work is summarised in (Parathasarathy and Klingenberg, 1996). It should be noted that for the yield stress τy, a distinction is often made between the “static yield stress” which is the minimum stress that must be applied to a sample for the inception of irreversible flow, and the “dynamic yield stress” which is obtained by extra- polating the curves from steady shearing measurements to zero shear rates (possibly smaller than the static yield stress) (Kraynik and co-workers, 1989).

Recently Baxter-Drayton and Brady have reported computer simulations on non-conducting spheres in a Newto- nian carrier fluid under shearing. They focussed on the low shear rate behaviour of very small ERF particles, and so included Brownian motion effects in their model. The degree of influence of the Brownian motion was character- ised by the dimensionless quantity , which is essentially the ratio of the electrostatic interaction energy to the thermal energy, with k the Boltzmann constant and T the absolute temperature. The Stokesian dynamics method which has been previously used to investigate hard sphere Brownian suspensions via the hydrodynamic resistance tensors (Bossis and Brady, 1989) was used to accurately calculate the hydrodynamic interactions between the particles. The electrostatic interaction forces were obtained from spatial derivatives of the total electrostatic

potential for each instantaneous configuration of particles, calculated via a grand capacitance matrix following the technique used in previous simulations (Bonnecaze and Brady, 1992a; Bonnecaze and Brady, 1992b). In this simulation there were two independent dimensionless groups:

λ and the Mason number Mn (eq(26)), and the interplay between the electrostatic, viscous and Brownian forces was investigated. Note that the commonly used dimensionless quantity the Peclet number Pe, can be written as Pe = 1/12 Mnλ- its magnitude indicates the relative importance of hydrodynamic and Brownian effects. The simulations are carried out on a monolayer of spheres under steady shear flow with sheared periodic boundary conditions. At the low shear rates investigated the majority of chains would be expected to span the electrode gap, and so the rupture and reformation process of these chains under shearing can be reasonably reproduced with the sheared periodic boundary conditions. The projected area fraction of the spheres was kept constant at 0.4.

The simulations were initially carried out on a quiescent system, and it was found that depending on the strength of the Brownian motion (ie the value of λ), different degrees of flocculation occurred : at λ≈ 4 flocculation was first observed; if 4 <λ< 10 the particles reached an equilibrium ordering into a hexagonal lattice; and if λ> 10 there was a hexagonal lattice structure but with vacancies in the lattice. Baxter-Drayton and Brady (1996) then applied a steady shear flow to the system, and found that for the dispersed regime λ<4 there was shear-thinning behaviour, which has also been observed with many hard sphere suspensions. For 4<λ<10, Baxter-Drayton and Brady observed that the low shear viscosity scales expo- nentially with λ. They explained this behaviour by using the following result from linear viscoelastic theory (Ferry, 1980): the zero shear viscosity η0 for a solid-like structure can be written

η0=G0τ (30)

where G0 is the shear modulus and τ is a time scale for stress relaxation. Baxter-Drayton and Brady estimated G0

by considering the restoring force on a single chain of particles under a small shear strain. For τ, they considered the characteristic time required for neighbouring particles to diffuse out of their mutual potential well (which has a depth of Umin/kT>>1), following the idea of Kramers (1940):

(31) Hence eq(30) gives for the zero shear viscosity of the ERF

(32) Umin/kT depends on the interparticle force and the orientation of the two particles with respect to the electric λ=(πεcε0a³βdielec2 E02)⁄kT

τ≈6πηca³⁄Umine^{Umin kT}^⁄

η0≈ηcφe^{Umin kT}^⁄

(11)

field. However, Baxter-Frayton and Brady found that Umin/ kT =βλ was a good fit to the experimental results available, with β taking values from 1.57 to 1.88. For the third case of λ> 10, where the electrostatic interactions begin to domi- nate over the Brownian motion effects, Baxter-Dray- ton and Brady found the shear thinning behaviour often observed in ERFs, with suspension viscosity proportional to Mn^-α where α is between 0.9 and 1 (note that α= 1 is consistent with the Bingham model eq(1) at very low shear rates). Calculations for small Mn become computationally prohibitive, but Baxter-Drayton and Brady predict that in the limit of very small Mn the suspension should approach a finite low-shear viscosity, which would be a departure from the Bingham model.

For comparison with actual ERFs, Baxter-Drayton and Brady (1996) considered the suspension used by Halsey and co-workers (1992), which consisted of a suspension of 0.75 µm silica spheres in methyl-cyclohexanol at 25^oC, placed under a field of 0.1 kV/mm-these values give λ= 83. The shear thinning behaviour found in the simulations for λ≈100 agreed with that observed by Halsey and co-workers (1992). However, there are as yet no experimental results available for comparison for the case where Brownian motion effects are strong (λ< 30). Baxter-Drayton and Brady extend this model of a rate-activated network rearrangement process, to the case of a flocculated suspension with a general interparticle interaction.

Another approach to the microstructural modelling of ERF under shear was the subject of a recent calculation by Martin and Anderson (1996). These workers used the flowing aggregate model of ERF, where, instead of considering chains or columns of particles spanning the electrode gap (or an effectively infinite chain due to the periodic boundary conditions as in the calculation of Baxter-Dray- ton and Brady (1996) just discussed, it is assumed that there are numerous elongated aggregates in the flow field whose presence in the carrier fluid causes the increase in viscosity. In actual fact, the flowing aggregate model is very similar to the spanning chain/column model, since in both formulations the particles are held together by the electrostatic interaction, the strength of which basically determines the degree of flow resistance. Further, as shear rate is gradually increased from zero there is expected to be a smooth transition from the spanning chain/column model to the flowing aggregate picture (Takimoto and co- workers, 1996). The flowing aggregate model has been used by several workers to model ERFs under moderate shear rates (Halsey and Toor, 1990b; Takimoto, 1991; Hal- sey and co-workers, 1992; Martin and co-workers, 1994b).

Martin and Anderson (1996) considered the balance of hydrodynamic and electrostatic forces acting on a single chain of particles, from which they were able to estimate the mechanical stability of chains in a steady shear flow field. They found that the characteristic chain length N

varies with Mason number Mn as . From N

they were able to calculate, using eq(28), the particle-contributed shear stress due to these chains and hence the viscosity. With this model, Martin and Anderson were able to predict the shear thinning behaviour observed in many ERFs with a shear thinning exponent of -1. They extended the calculation to the case of chains confined by electrodes with a finite friction coefficient between the particle surfaces and the electrodes, and found that for certain Mason number regimes the chains adhering to the electrodes would become mechanically unstable. Martin and Anderson proposed that this instability corresponds to the “slip zones”

observed by Klingenberg and Zukoski (1990) in their flow visualisation experiments on ERF. Takimoto (Takimoto, 1998; Takimoto and co-workers, 1996) used the flowing aggregate model to investigate the response of an ERF under shear flow, after the sudden application of an electric field. Takimoto also performed particle-level simulations of aggregates in a flow field (Takimoto, 1998), and found that the scaling laws predicted by the mechanical stability analysis hold over a range of shear rates.

A detailed study of the flowing aggregate model was reported by Martin and co-workers (1998). These workers used light scattering techniques to investigate the changes in chain structure after application of an electric field during shear flow, and they compared the results with predictions from the flowing aggregate model. They used a suspension of 0.7 µm silica spheres suspended in 4 meth- ylcyclo hexanol in a concentric cylinder cell modified for optical experiments. An ac electric field was applied. The aggregation kinetics of a quiescent fluid was examined first, and it appeared that the aggregation process occurred in stages: particles formed thin chains, these chains came together to form columns, which gradually grew thicker with time. The growth of these thicker columns was described by a model based on thermal coarsening, where the thermal force due to the Landau-Peierls instability for 1 dimensional structures was balanced by the friction force experienced by a column translating through the carrier fluid. The next step was to subject the ERF to steady shear flow, and from the light scattering patterns Martin et al deduced the existence of tilted aggregates of particles in the flow field. At the low shear rates investigated (γ=34s^⋅ ^-1) they found that the orientation angle θtilt of the aggregates (measured from the electric field direction) follows a power law dependence on the Mason number Mn : θtilt Mn^-0.326. Martin et al were able to explain this result using the ellipsoidal droplet model, which considers tilted ellipsoidal aggregates of ERF particles in the flow field (Halsey and Toor, 1990b; Martin and co-workers, 1994a; Martin and co-workers, 1994b). The angle of tilt was determined from the balance of the hydrodynamic and electrostatic torques, and the size and aspect ratio of the ellipsoidal aggregates were obtained from an energy minimisation

〈 〉N ∝Mn^–^{1 2}^⁄

(12)

180 Korea-Australia Rheology Journal calculation, where the effect of the depolarisation energy

(favoring long thin columns aligned with the field) com- petes with the effect of electrostatic surface energy (which favours spheroidal aggregates). From these calculations it was found that the tilt angle , in good agree- ment with the light scattering experiments. Martin et al pre- sented another calculation based on the simpler chain model (ie single particle-width chains, as used in Martin and Anderson (1996) and described previously), but this model was not able to predict the θtilt Mn^-0.326 relationship observed experimentally. Martin and co-workers (1998) considered improvements to the single chain model, such as the inclusion of more exact hydrodynamic interactions as well as a local field correction to enable more accurate calculation of the electrostatic interactions between neighbouring particles in a chain. They found that the hydrodynamic approximations have much less impact on the results than the point dipole approximation, which dras- tically underestimates the interaction forces. However, even with these enhancements, Martin et al found that there is no qualitative change in the essential physical results predicted by the single chain model (Martin and co-workers, 1998).

2.5. Electric current during shear flow

The behaviour of electric current through an ERF under steady shearing often reflects changes in the microstructure. Further, from a practical point of view, it is important to be able to predict the behaviour of the electric current drawn by electrorheological devices under operational flowing conditions.

Otsubo and Edamura (1995) performed experiments using a suspension of inorganic shell/organic core com- posite particles in silicone oil under a dc electric field of 1kVmm, and found for shear rates under 10s^-1 the electric current maintained an almost constant value. However, as the shear rate was raised above 10s^-1 the current showed a dramatic drop, which Otsubo and Edamura (1995) attri- buted to breakdown in the chain structure.

A similar behaviour was observed by Chen and Luck- ham (1994), who investigated the variation of current with steady shear rate for a suspension of wateractivated silica particles in corn oil. They found that under a dc field of 1kV/mm, the current began dropping when the shear rateγ exceeded 1s^⋅ ^-1, and at a shear rate of γ^⋅= 1000s^-1 the current's value was less than half that at the low shear rates.

For shear rates above γ^⋅= 1000s^-1 the electric current seemed to remain almost unchanged. The drop in electric current became relatively larger as the particle concentra- tion increased; the highest volume fraction used was φ= 0.2. They hypothesised that the change in current was due to changes in the microstructure with shear rate, and by considering the relationship between the current density j (current per unit area of electrode) and the volume fraction,

found that the data could be fitted quite well by the power law j φ^α, where it was envisaged that α = 1 corresponded to a chained structure and α= 2 to an isotropic network structure. The experiments showed that α changes from 1 to 2 with increasing shear rate.

In a later paper (Chen and co-workers, 1995), Chen and co-workers verified this basic picture by a particle level computer simulation, using the Stokes' approximation and point dipole electrostatic interactions (eqs(17), (21), (25)).

They introduced the idea of the average coordination number of a particle in the ERF, which is the number of neighbouring particles in contact (equivalent to the exponent α in their previous paper (Chen and Luckham, 1994)) and showed this to be an effective way to express the changes in microstructure. Using this to monitor the struc- tural changes with increasing shear rate, Chen et al were able to support their basic premise that microstructural changes could lead to the reduction in electric current with shearing observed experimentally. These workers also si- mulated the change in coordination number as a function of time after the imposition of an electric field, and related the results to the different stages of cluster development ie initial aggregation period, then a slower period of densi- fication. However, the detailed dependency of the current on the coordination number and hence the shear rate, was not explored in this simulation.

See and Saito (1996) calculated the variation of current with shear rate by modifying the layered model of ERF under flow proposed by Klingenberg and Zukoski (1990), and focussed on the role of the conductivities of the particles and carrier fluid. The basic idea of the model was that the increase in shear viscosity of the ERF is due to a layered structure between the electrodes, comprised of the remnants of particle chains adhering to the electrodes by electrostatic image forces, and a freely flowing central liquid layer where all the shear flow is concentrated giving rise to the apparent increase in viscosity. As the shear rate increases the central liquid layer increases in thickness.

Such layered structures have been observed experimentally (Klingenberg and Zukoski, 1990), and this model would appear to be appropriate for reasonably concentrated ERF under moderate to high shear rates. Using this layered model, See and Saito (1996) determined the variation of electric current with shear rate using effective medium theory to estimate the overall conductivity through the liquid layer, assumed to be an instantaneous random network of conducting spheres. The calculation predicted a distinct drop in electric current as shear rate is raised, in qualitative agreement with the experimental observations. The calculations showed there is a minimum value of the shear rate γ^⋅min, above which the electric current will drop with increasing shear rate, until an upper value γ^⋅max is reached.

Above γ^⋅max it was predicted that the current would level θtilt∝Mn^–^{1 3}^⁄