Brownian electrorheological fluids as a model for flocculated dispersions

Brownian electrorheological fluids as a model for flocculated dispersionsYvette BaxterDrayton and John F. Brady

Citation: Journal of Rheology (1978-present) 40, 1027 (1996); doi: 10.1122/1.550772 View online: http://dx.doi.org/10.1122/1.550772 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/40/6?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Rheology of dense suspensions of platelike particles J. Rheol. 40, 1211 (1996); 10.1122/1.550798 Colloidal dispersion confined in a planar slit: A density functional approach J. Chem. Phys. 104, 9563 (1996); 10.1063/1.471698 Microstructured fluids: Structure, diffusion and rheology of colloidal dispersions AIP Conf. Proc. 256, 391 (1992); 10.1063/1.42337 Transport properties of concentrated colloidal suspensions AIP Conf. Proc. 256, 359 (1992); 10.1063/1.42331 Flocs: Buildup mechanisms and structures AIP Conf. Proc. 226, 529 (1991); 10.1063/1.40571

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Brownian electrorheological fluids as a modelfor flocculated dispersions

Yvette Baxter-Drayton and John F. Bradya)

Division of Chemistry and Chemical Engineering, California Instituteof Technology, Pasadena, California 91125

(Received 8 December 1995; final revision received 16 August 1996)

Synopsis

The rheological behavior of Brownian electrorheological~ER! fluids is studied as a model forflocculated colloidal dispersions. The ER fluid has the advantages that the interparticle potentialenergy can be varied by simply changing the applied field strength, and the microstructure consistsof essentially linear chains of particles aligned with the field direction. Under simple shear flow, thesuspension has a high-shear-rate Newtonian viscosity and a shear thinning viscosity at lower shearrates. For moderate attractive potential well depths,Umin/kT, the suspension has a low-shearviscosity that scales as exp~Umin/kT!. Furthermore, the low-shear limiting behavior is seen at shearrates that scale as exp~2Umin/kT!. A theory is proposed that makes use of the time scale of diffusionfor aggregated particles out of their mutual potential well,t ; ~a2/D)(kT/Umin!exp~Umin/kT!,much in the spirit of the Eyring theory. Herea is the particle radius andD is the diffusivity of anisolated particle. When the shear rate is nondimensionalized byt, the reduced viscosity data for allfield strengths collapse onto a single universal curve. Although we use a relatively small monolayersuspension, our simulation results compare well to the limited experimental and theoretical work onBrownian ER suspensions. The scaling relationship for the low-shear viscosity has also beenevidenced in other studies of flocculated dispersions. ©1996 Society of Rheology.

I. INTRODUCTION

Aggregating colloidal suspensions exhibit a wide range of rheological behavior fromthick gels to thin pastes. As with Brownian hard-sphere suspensions, the solids concen-tration plays an important role in the suspension dynamics, but in addition, the nature andstrength of the particle interaction forces add a crucial dimension. Aggregation is gener-ally referred to as a nonequilibrium process that converts individually dispersed particlesinto a disordered solid. The disordered solid can be either a flocculated suspension con-sisting of highly porous, weakly aggregated flocs, or a coagulant where the flocs are moreclosely packed@Russel, Saville, and Schowalter~1989!#. Aggregated suspensions are alsodelineated according to the strength of the attractive interaction potential between par-ticles @Buscall, McGowan, and Morton-Jones~1993!#. For example flocculated suspen-sions have an attractive particle potential well depth,Umin, greater than 5–10kT, whilesuspensions withUmin ; 1–5 kT undergo an equilibrium phase separation where thesuspended particles rearrange by diffusion into the structure of lowest energy and are notclassified as flocculated. This is not a clear distinction, however, since given enough timeall suspensions should tend toward equilibrium. In this work, a flocculated or aggregated

a!Corresponding author.

© 1996 by The Society of Rheology, Inc.J. Rheol. 40~6!, November/December 1996 10270148-6055/96/40~6!/1027/30/$10.00


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suspension is taken to be a suspension of particles that form a connected volume span-ning network and includes so-called equilibrium phases as long as the resulting structurespans the suspension volume.

Weakly aggregating suspensions~5kT , Umin , 20kT! exhibit non-Newtonian be-havior; they typically have low- and high-shear-rate viscosities, and shear thin at inter-mediate shear rates@Buscallet al. ~1993!; Melrose and Heyes~1993!#. Suspensions withUmin well above 20kT also shear thin and have a high-shear-rate viscosity, but do notshow a low-shear viscosity, at least over experimentally accessible scales@Buscallet al.~1982!#; they are Bingham plastics with an apparent dynamic yield stress. The rheologyliterature abounds with models for flocculated suspensions. Most are phenomenologicaland offer little insight into the underlying physics of the suspension dynamics@Tanner~1985!#. For example, the Bingham and Casson models often used to fit data for fluidsthat show an apparent yield stress are not predictive and do not apply to all flocculatedsuspensions. A basic microstructural approach incorporating the physics of particle inter-actions and dynamics is needed and is the focus of this work.

A wide variety of ways to induce flocculation are detailed in Russel, Saville, andSchowalter’s~1989! Colloidal Dispersions. Regardless of the flocculation method, how-ever, suspensions with similar particle interaction potentials behave similarly. For thiswork, we use a suspension of polarizable particles in a nonconducting dielectric liquidforming an electrorheological~ER! fluid. Winslow ~1949! first noted that an electric fieldacross a suspension of cornstarch in water thickens the fluid and that the viscosity of thefluid scaled with the square of the applied electric field. Potential applications for thesetunable viscosity fluids range from liquid shocks for magnetically levitated trains@Tech-nology Edge~1992!#, to dampeners for buildings to protect against earthquakes@Ehrgottand Masri~1994!#. The key to the ER effect is the difference in the polarizability of theparticles and the suspending fluid. Applying an electric field across the suspension in-duces particle dipoles, and there is a net attraction between particles whose surfaces areoppositely charged and a net repulsion between particles whose surfaces are similarlycharged, resulting in particle chaining along the field direction.~A difference in electricalconductivity between particles and fluid can have an analogous effect in ac electricfields.! This electrically induced structure is responsible for the increased viscositiesnoted in ER fluids. A completely analogous situation occurs in magnetorheological fluids,where the magnetic permeability mismatch between particles and fluid leads to the samephenomena@Bossiset al. ~1990!#.

In its simplest form, the ER pair potential is the sum of the attractive electrostaticpotential and a hard-sphere repulsion,

u~r,u!

kT5 H` if r , 2a

2lS 2ar D 3 3 cos2 u21

2if r . 2a

,

wherel [ pea3~bE!2/kT. Hereu is the angle between the electric field vector and thevector connecting two particle centers,r 5 2a is the contact between the particles ofradii a, e is the dielectric constant of the fluid,E is the magnitude of the electric field,andb [ ~ep2e!/~ep12e! measures the dielectric mismatch between fluid and particle.For point dipoles aligned along the field directionl gives the potential well depth,Umin/kT.

The ER fluid structure and dynamics are governed by the interplay of polarization,thermal, and viscous forces due separately to the applied field, fluctuations of the solvent

1028 BAXTER-DRAYTON AND BRADY


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particles, and the imposed flow, respectively. The characteristic interparticle interactionforces for an ER fluid are

Force Scale

Thermal kT/aPolarization 12pea2~bE!2

Viscous 6pha2g,

wherek is the Boltzmann constant,T is the temperature,h is the solvent viscosity, andgis the magnitude of the shear rate. The relative importance of these forces are given bythe dimensionless groups

Group Ratio Scale

l polarization/thermal pea3~b E!2/kTPe viscous/thermal 6pha3g/kTMa viscous/polarization hg/2e~bE !2 ,

defining the Peclet, Pe, and Mason, Ma, numbers. Note that the Peclet and Mason num-bers are not independent; their ratio is simply 12l. We shall discuss the suspensionbehavior primarily in terms of Ma andl.

The nature of the static, nonsheared, structure of an ER fluid varies withl, the relativestrengths of the polarization and Brownian forces. Brownian forces tend to disrupt thechaining effect of the electric field and, when dominant, disperse the suspended particles.As Brownian motion becomes less important, asl increases, the attractive electrostaticforces form chains aligned along the electric field direction. Thus, by simply changing theapplied field strength, an ER fluid can be a disordered suspension, a flocculated suspen-sion, or a solid. In short, the relatively simple structure and the ease with which its pairpotential can be manipulated makes the ER fluid an ideal model for studies on flocculatedsuspensions.

The interplay of polarization, viscous, and Brownian forces gives rise to a wide varietyof rheological behavior in ER fluids. At moderate volume fraction and at largel shear,forces gradually degrade the cell spanning network resulting in shear thinning@Marshall,Zukowski, and Goodwin~1989!; Bonnecaze and Brady~1992a!#. At low shear rates thesuspension viscosity scales inversely with shear rate, which is consistent with the exist-ence of a dynamic yield stress. Although Bonnecaze and Brady~1992b! showed that inthe limit of no Brownian forces~1/l [ 0! the suspension has a dynamic yield stress,there has not been a similar study for large but finitel, and it has simply been assumedthat these ER suspensions have a dynamic yield stress. At high shear rates, the particlenetwork is completely destroyed and the suspension viscosity approaches that of the baresuspension with no field applied. Marshallet al. ~1989! found that the Mason numbercollapses the viscosity data for different temperatures, shear rates, and field strengths ontoa single curve, i.e., at high field strengths the suspension rheology is described by onlyMa andf, the suspension volume fraction.

Efforts to make neutrally buoyant ER fluids have turned the focus to smaller particlesand, thus, smallerl, but there is limited rheological data in this regime. Studies for30, l , 2000 @Halsey, Martin, and Adolf~1992!; Melrose ~1992!# indicate no yieldstress; at low shear rates the suspension shear thins with its viscosity scaling asg2a

wherea is less than 1. On the other hand, simulations on a 0.16 volume fraction suspen-sion @Sun and Tao~1995!# showed a low-shear viscosity forl , 90 and a yield stress for

1029FLOCCULATED DISPERSIONS


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l 5 90, their maximuml. Rheological studies forl , 30 were done only at low volumefractions,f 5 0.16 @Sun and Tao~1995!#, where the suspension is not flocculated. Con-trary to the behavior at highl, for finite l the Mason number is not sufficient to collapsethe rheological response onto a single curve as it does not include thermal effects.

There is still no clear picture of the low-shear-rate behavior of ER fluids, particularlywhen flocculated at moderate but finitel. It remains to be seen if they have a dynamicyield stress as for 1/l [ 0 suspensions@Bonnecaze and Brady~1992b!#, or if they havea low-shear viscosity similar to Brownian hard-sphere suspensions,l [ 0 @Phung~1993!#, or dispersed ER suspensions,l ; O~1! @Sun and Tao~1995!#. This is a crucialissue for ER fluids and the answer will have far reaching consequences. The yield stressis often an important design parameter for several potential applications. The ER valve,for example, designed to control flow, is required to sustain a stress. In addition, thisquestion, whether there is a yield stress or a low-shear viscosity, is of generic importanceas it will aid in understanding the flow behavior of flocculated suspensions generally.

Clearly, additional rheological data for smallerl and a wider range of Mason numbersare needed. To this end, Stokesian Dynamics simulations have been used to calculate thesuspension viscosity for a wide range ofl and Ma, with area fractionfA , the ratio of theprojected area of the particles to the area of the cell, of 0.4. Simulations in the absence ofshear show that flocculation occurs atl ' 4. We note that all values ofl defining theboundary between regions of different behavior are approximate; a more accurate deter-mination requires large system sizes. For smallerl, the suspension forms particle chainsthat align along the field direction but do not span the cell; here we find a low-shearviscosity that can be predicted by considering a regular perturbation to the equilibriumstructure. For 4, l , 10 the chains span the cell and aggregate to form thicker clusterswith a hexagonal lattice structure. Here we find a low-shear viscosity that scales expo-nentially with the pair potential well depth. A simple argument based on viscoelastictheory is used to explain this scaling. In addition, the low-shear limiting behavior isobserved wheng , ~D/a2!le2l, whereD 5 kT/6pha is the diffusivity of an isolatedBrownian particle. Abovel ; 10, the suspension consists of a network of flocs made upof strands aligned along the field direction. The flocculated structures formed also have ahexagonal order but there are vacancies within the lattice. Over the shear rate range231025 , Ma , 1023, we find neither a dynamic yield stress nor a low-shear viscosity,instead the viscosity scales approximately as Ma2a, wherea rises from 0.9 to 1. It isargued, however, based on theoretical considerations, that these suspension should alsoshow a low-shear viscosity wheng , ~D/a2)le2l.

Finally, we describe a model for the dynamics of a flocculated suspension based on themicrostructural rearrangements of the particle network, much in the spirit of the Eyringtheory for molecular liquids. By introducing a time scale for diffusion of aggregatedparticles out of their potential well,t ; ~a2/D)(kT/Umin!exp~Umin/kT! @Kramers~1940!; Chandrasekhar~1943!; Hanggiet al. ~1990!; Potaninet al. ~1995!#, we show thatthe low-shear viscosity scales exponentially with the pair potential well depth. Further-more, the ratio oft to the time scale of shear, 1/g, produces a dimensionless shear ratethat collapses our data onto a single universal curve. Thus, the model predicts not onlythe low-shear viscosity, but also the suspension viscosity over a wide range of shear rates,and may apply generally to flocculated dispersions.

II. SIMULATION METHOD

The Stokesian dynamics method with Brownian motion is detailed elsewhere@Bossisand Brady~1987, 1989!; Phung~1993!; Phung, Brady, and Bossis~1996!#, so only the



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important features will be discussed here. ForN rigid, neutrally buoyant particles sus-pended in an incompressible fluid of densityr and viscosityh, the motion of the particlesis governed by the coupledN-body Langevin equation

m•dU

dt5 FH1FB1FP, ~1!

whereU is the particle’s translation/rotational velocity vector,m is its mass/moment ofinertia tensor,FH is the hydrodynamic force,FB is the Brownian force, andFP is theinterparticle force. For colloidal particles the viscous forces dominant the inertial forces~Re5 rga2/h ! 1! such that the fluid dynamics is described by Stokes equation; thus,the hydrodynamic force is given by

FH 5 2RFU•~U2Û&!1RFE :Ê&, ~2!

whereÛ& is the imposed flow at infinity evaluated at the particle centers andÊ& is thesymmetric part of the velocity gradient tensor. The tensorsRFU andRFE are the particleconfiguration-dependent hydrodynamic resistance tensors that relate the hydrodynamicforce and torque on the particles to their motion relative to the fluid and to the imposedshear flow, respectively.

The Brownian force comes from the thermal fluctuations in the fluid and is character-ized by

^FB& 5 0 and ^FB~0!FB~ t !& 5 2kTRFUd~ t !, ~3!

where the angle brackets denote an ensemble average. The amplitude of the correlation ofFB at time 0 and timet comes from the fluctuation–dissipation theorem for theN-bodysystem.

The electrostatic interparticle force is calculated by the method of Bonnecaze andBrady ~1992a!. Starting with the electrostatic energy density,U, for a system of non-charged polarizable particles in an electric field, the force on each particle is given as thederivative of the energy with respect to position

Fa 5 2]U~x!

]xa5 2

1

2 S ]

]xaSE•E D . ~4!

HereU~x! is the electrostatic energy density,xa is the position vector of particlea, SE isthe induced particle dipole, andE is the applied electric field times a unit vector of lengthN, whereN is the number of particles. There is the linear relationship between the chargeand induced dipole, and the potential and electric field:

S qSED 5 C~x!•S F

ED 5 S CqF CqE

CSF CSED •S F

ED , ~5!

which defines the grand capacitance tensorC~x!, that couples the charge and dipole to thepotential and electric field in the same manner that the resistance tensor couples the forceto the velocity for Stokes flow. The subscripts on the tensorsC, relate the chargeq ordipoleSE to the electrostatic potentialF or the electric fieldE . For charge free particlesforming an ER fluid,

SE 5 C~x!•E 5 ~CSF•CqF21•CqE2CSE!•E . ~6!



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The grand capacitance tensor is purely geometric, depending only on the particle con-figuration and the particle-to-fluid dielectric ratio. Combining Eqs.~4! and~5!, the inter-particle force is given by the quadratic form

Fa 5 21

2E•

]C~x!

]xa•E . ~7!

Finally, combining Eqs.~2!–~7!, and the condition that the inertial forces are negli-gible, Eq.~1! is integrated to obtain the evolution equation for the particle positions,

Dx 5 Pe~Û&1RFU21•@RFE :Ê&1Ma21FP# !Dt1¹•RFU

21Dt1X~Dt !, ~8!

whereDx is the change in particle position over a time step ofDt. The last term,X~Dt!,is a random displacement due to Brownian motion that has zero mean and covariancegiven by the inverse of the resistance tensor,

^X& 5 0 and ^X~Dt !X~Dt !& 5 2RFU21Dt. ~9!

Here,x has been made nondimensional bya, t by a2/D, the hydrodynamic forces by6pha2g, and the electrostatic force by 12pea2~bE!2. The parameters Ma and Pe are thedimensionless ratio of the viscous to polarization forces and the viscous to thermal forces,respectively, as discussed in the Introduction. Both can be expressed as ratios of timescales: Ma5 tER/tV and Pe5 tB/tV , wheretER 5 h/~2e~bE!2! gives the time of ag-gregation due to the polarization forces,tV 5 1/g gives the time scale of the shear flow,andtB 5 a2/D, gives the time scale of diffusion of an isolated Brownian particle.

Equation ~8! is the core of the simulation and is used to follow the motion ofNparticles given their initial configuration. The dependence on Pe and Ma in Eq.~8! can bereplaced with either Pe andl, or Ma andl, asl 5 Pe/~12Ma!. We shall generally presentresults in terms of Ma andl, although any two dimensionless groups could be used. Inaddition to Ma, the dynamics depends on the volume fractionf and the relative dielectricconstant ratioep/ef .

For rheology, the suspension bulk stress is needed, and is given by

^(& 5 2^p&I12hÊ&2nkTI1n@^SH&1^SB&1^SP&#. ~10!

Here,^(& is the macroscopic average stress tensor,^p& is a constant denoting the pressurein the incompressible medium, 2hÊ& is the deviatoric stress contribution from the fluid,2nkTI is the isotropic stress associated with the thermal energy of the Brownian par-ticles; I is the isotropic tensor, andn is the number density of particles. The stresses^SH&,^SB&, and ^SP& are associated with hydrodynamics, Brownian motion, and the electro-static interparticle interactions, respectively, and in dimensional form are given by

^SH& 5 2^RSU•RFU21•RFE2RSE&:Ê&, ~11!

^SB& 5 2kT¹•^RSU•RFU21&, ~12!

^SP& 5 2^RSU•RFU211xI !•FP&. ~13!

The hydrodynamic stress is the added mechanical or contact stress due to the resis-tance of the rigid particles to local deformation of the fluid. The Brownian stress is athermodynamic stress that results from the deformation of the equilibrium structure. Theinterparticle stress has two components. The first component,RSU•RFU

21•FP, comes from

the velocity field generated by particle motion arising from the interparticle forces, and ishydrodynamic in origin. The second term,xI•FP, is the direct contribution to the bulkstress from interparticle forces; it is of the same form as occurs in molecular systems.



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The hydrodynamic stress is always positive and scales with the largest length cubed.In contrast, the Brownian and interparticle stresses can be positive or negative dependingon the microstructure. When the polarization forces dominate, particles form chains orclusters along the field direction that tilt, strain, and eventually break under shear. Theimposed shear flow pulls the chained particles apart along the extensional axis, while thepolarization force opposes the fluid and keeps the particles aggregated, resulting in apositive contribution to the stress. The Brownian force, on the other hand, pushes par-ticles apart along the extensional axis resulting in a negative Brownian stress. When theBrownian or viscous force dominates, the interparticle and Brownian stresses are oppo-site in sign to those in the previous case. Shearing clusters particles along the compres-sive axis, which is opposed by Brownian forces resulting in a positive Brownian contri-bution to the stress. In contrast, the polarization force acts with the fluid to push particlestogether resulting in a negative contribution to the bulk stress.

The relative viscosity of the suspension is defined by the ratio of theyx component ofthe bulk stress to theyx component of the bulk rate of strain, with the imposed linearshear flowÛx& 5 gy. In dimensionless form,

hr 5 11hrH

1hrSP

1hrxF

1hrB , ~14!

where the superscriptsH andB indicate the hydrodynamic and Brownian contributions,respectively. The superscriptsSPandxF indicate contributions from the first and secondterms in the interparticle stress@Eq. ~13!#, respectively.

The hydrodynamic resistance and electrostatic capacitance tensors are constructed byincorporating both near-field lubrication and far-field many-body interactions as outlinedin Phung, Brady, and Bossis~1996! and Bonnecaze and Brady~1992a!.

III. RESULTS AND DISCUSSION

The method outlined above was applied to an unbounded monolayer of monodispersedielectric particles with an electric field in they direction and shearing in thex direction,as illustrated in Fig. 1. We used a monolayer because this reduces the simulation time butstill captures the essential physics of an ER suspension. Indeed, Bonnecaze and Brady’s~1992a! Stokesian Dynamics monolayer simulations in the non-Brownian limit comparedquantitatively to experiment. An unbounded suspension is simulated by periodically rep-licated 25 identical spherical particles in a square cell forming a 0.4 area fraction sus-pension. The dielectric constant ratio wasep/e 5 4, giving b 5 1/2. The time step usedvaried from 0.001 to 231025. The equilibrium structure formation for 0, l , ` wasstudied, monitoring the electrostatic energy to assure steady state. The suspension vis-cosity was studied for 0, l < 1012 and 531026 < Ma < `. In this case, steady statewas determined by monitoring the time average viscosity and ensuring that the cellcompleted at least one strain.

A. The suspension structure

The suspension energy,U~x! @see Eq.~4!#, at steady state is plotted as a function oflin Fig. 2. The energy of the initial dispersed configuration generated by Monte Carlomethods is arbitrarily set to 0. The plot shows that the suspension energy approaches anasymptote atl 5 5.8, where the particles have ordered into a single cluster with ahexagonal lattice order as seen in Fig. 3~b!. Forl . 5.8, the suspension energy is slightlylower; in this case, the lattice formed is incomplete with vacancies, and continued simu-lations did not anneal the open structures@Fig. 3~c!#.

To quantify the orientational and positional order of the suspension, we use the orderparameters



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p1,2 51

N(j

N

eib1,2•r j , ~15!

whereb1,2 are unit vectors along and 60° relative to the field direction, respectively, andthe sum is over all particle pairs. A hexagonal lattice hasp1 5 p2 5 1, while a dispersedrandom suspension has vanishing order parameters. Thus,p1 provides a measure ofparticles in contact aligned with the field, andp2 measures the aggregation of chains.

Figure 4 showsp1,2 as functions ofl. Note thatp1 is nonzero forl > 0.8; regardlessof the magnitude, an applied field introduces anisotropy into the suspension structure. Forl 5 0.8, the pair–distribution function at contact parallel and perpendicular to the ap-plied field aregi~2! 5 5.7 andg'~2! 5 1.9, respectively, which should be compared withg~2! 5 3.2 in the absence of the applied field, indicating that there is a preference forparticles in contact to align with the field and a deficit of particles in contact perpendicu-lar to the field. The resulting structure is reminiscent of the weakly anisotropic structurespredicted by Hayter and Pynn~1982! and more recently by Xu and Hass~1993!. Treatingthe particles as point dipoles, Hayter and Pynn used the mean spherical approximation toexpress the equilibrium pair distribution as a series of Legendre polynomials with non-zero even terms. They used only the first two terms to calculate the pair–distributionfunction for smalll. Xu and Hass improved on this method by retaining an additionalterm in the expansion for the pair–distribution function. Both groups found a smallrepulsion at contact perpendicular to the field direction2g'~2! , 12for a comparablevolume fraction of 0.2 for Hayter and Pynn’s work@as detailed in Adriani and Gast~1988!# and 0.25 for Xu and Hass’s work, which is not seen in these simulations. Thisdifference may be a result of the point–dipole approximation used in their theories, whichoverestimates particle repulsion near contact as compared with our multipolar interac-tions.

FIG. 1. Illustration of the monolayer used for the simulations. The electric field is in they direction, while thevelocity is in thex direction with gradient in they direction.



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From Fig. 4 two critical equilibrium transitions can be identified. The first occurs atl ' 1.6 wherep1 increases rapidly, indicating chaining along the field direction. Thesecond transition occurs atl ' 4 wherep2 increases, indicating aggregation of chains.At l 5 5.8, the particles are roughly ordered into a single cluster with a hexagonal latticestructure@Fig. 3~b!#. Note thatp1 and p2 are not identically 1 because the particlescontinue to fluctuate within the lattice structure. Also, note that the periodic boundaryconditions preclude addition of more particles to the clusters so we cannot determine thecharacteristic width of these structures. Indeed, that they form a single cell-spanningcluster is likely to be an artifact of the simulation box size and/or number of particles.Simulations with 50 particles at the same area fraction formed two separate cell-spanningclusters, each two to three particles thick and each with a hexagonal lattice structure asseen in Fig. 5.

For l . 10, the suspension forms a cell-spanning network with a hexagonal order butwith vacancies within the lattice, defining a third, kinetic, transition. This ‘‘kinetically’’frozen state has also been observed experimentally@Klingenberget al. ~1991!# and withsimulations at large to infinitel @Bonnecaze and Brady~1992a!; Melrose~1992!#. Rear-rangement within the structure requires particles to overcome their mutual attractivepotential wells, which occurs with a frequency that scales as~D/a2)le2l @Kramers~1940!; Chandrasekhar~1943!#, such that the open structures may anneal given sufficient

FIG. 2. The electrostatic energy of the suspension at steady state as a function of the ratio of polarization tothermal forces,l. As the suspension forms chains aligned with the field, the energy decreases. Abovel 5 5.8the particles have aggregated into a single structure with a hexagonal lattice order, while forl . 5.8, thehexagonal structure has vacancies in the lattice.



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time. However, the simulations indicated no change in the microstructure or order after adimensionless time of 2, corresponding to approximately 12 s, using typical values fora,T, andh. For most applications the microstructure setup time is a few seconds, and for allpractical purposes the suspension does not achieve its maximum order.

Similar phase behavior has been observed in magnetorheological systems both experi-mentally@Liu et al. ~1993!; Hwang and Wu~1993!# and numerically@Tao ~1993!#. Usingsimulated annealing methods, Tao found that a three-dimensional, 0.15 volume fractionsuspension remained relatively disordered forl < 3. For 3, l , 7 the particles formedrandomly distributed chains and remained separated at steady state. Forl . 7 the par-ticles formed randomly distributed chains that subsequently aggregated into a body-centered-tetragonal lattice cluster. Tao did not find a kinetically arrested structure sincethe Monte Carlo method he used assumes equilibrium. Liuet al. performed experiments

FIG. 3. Instantaneous snapshots of the structure forl 5 0, 5.8, and 17. Atl 5 0.8 few particles aggregatetemporarily and align along the field direction. Atl 5 5.8 all particles have aggregated into a single columnwith a hexagonal lattice order but particles still fluctuate within the lattice. A further increase inl results in thekinetically aggregated open structure.



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with monodisperse ferrofluid emulsion droplets 0.55mm in diameter in an aqueous so-lution forming volume fractions from 0.03 to 0.13.~Liu et al.definedl as the ratio of thedipole energy at contact along the field direction to the thermal energy, which is the samedefinition used here.! Structure formation was probed with optical microscopy and lightscattering. They found that the particles first ordered into chains atl 5 1.5, and, depend-ing on the volume fraction, the chains either remained separated or aggregated to formcolumns upon a further increase inl. For l > 10, the suspension aggregated kineticallyand formed an open structure.

Hwang and Wu~1993! used small angle light scattering to study structure formation inan aqueous suspension of 1mm diam magnetic particles for volume fractions up to 10%subjected to a magnetic field. They identified two transition field strengths,HC1 andHC2. Below HC1 the suspension was a disordered liquid, forHC1 , H , HC2, theparticles assembled into randomly spaced chains along the field direction, and forH . HC2 these chains aggregated into ordered structures. They also found a kineticallyaggregated region forH @ HC2 where the particles formed a gel with no long-rangeorder. Both transition fields were found to be volume fraction dependent;HC1 decayedwith the volume fraction to the21

4 power, whileHC2 decayed as the volume fraction tothe2 1

2 power. At the highest volume fraction studied,f 5 0.1, the single chain regimewas vanishingly small. From Hwang and Wu’s reported field strengths we estimate thatthe suspension formed aggregated columns atl ' 1 and a kinetic gel atl ' 58.

FIG. 4. The order parametersp1 andp2 as defined in Eq.~15! vs l; p1 5 p2 5 1 for a hexagonal lattice andzero for a random dispersion. Forl . 0.8,p1 is greater than 0 indicating a preference for particle pairs to alignalong the field direction. An increase inp2 indicates coalescence of chains. Forl . 10 both order parametersare approaching, but less than, 1 because the structure is arrested in a kinetically controlled open network.



17:28:33

In summary, from simulations we find that flocculation occurs abovel ' 4. Withinthe flocculated region, there is an equilibrium flocculated regime, 4, l , 10, whereparticles order into a hexagonal structure, and a kinetically aggregated regime forl above10 where the suspension forms a hexagonal lattice structure with vacancies within thelattice.

B. The suspension dynamics

Figure 6 shows the simulation results for the relative suspension viscosity as a func-tion of Ma for l from 0.8 to 1012; Bonnecaze and Brady’s~1992! results forl 5 ` areincluded for comparison. The corresponding data are given in Tables I and II. Only thetotal viscosities are reported to save space; the individual contributions@cf. Eq. ~14!# areavailable upon request. At largel the interparticle force contribution dominates, while atlarge Ma the hydrodynamic contribution dominates. The Brownian and interparticle con-tributions oppose each other as discussed in Sec. II. When viscous forces dominate,Ma @ 1, the suspension has a shear viscosity equal to the bare suspension viscosity,regardless ofl. Thus, as Ma→ ` hydrodynamics alone determines the suspension be-havior. As the shear rate or Ma decreases the suspension viscosity increases, as bothpolarization and thermal forces become important; the viscosity now depends onl aswell as Ma.

1. Dispersed regime: l < 4

In the dispersed regime,l , 4, the suspension viscosity shear thins, as one wouldexpect from the analogous behavior of hard-sphere suspensions@van der Werff and deKruif ~1989!; Phung, Brady, and Bossis~1996!#. The low-shear viscosity is found to scaleexponentially with the pair potential well depth and can be estimated by regular pertur-bation theory, similar to that for hard spheres by Brady~1993!, for a square well fluid by

FIG. 5. The steady-state structure forl 5 5.8 with 50 particles. The columns have a hexagonal structure buthave not coalesced into a single column.



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Bergenholtz and Wagner~1994!, or for dilute suspensions of ‘‘sticky’’ spheres@Russel~1984!; Cichocki and Felderhof~1990!#. The method, detailed in the Appendix, modelsthe particle distribution as Boltzmann and neglects all anisotropy in the suspension struc-ture. Also, hydrodynamics are ignored and only the interparticle force contribution to thestress is calculated. A more rigorous analysis can be employed@Brady ~1993!#, but thisapproximate method suffices to show the source of the exponential relationship between

FIG. 6. The suspension viscosity relative to the solvent viscosity as a function of Ma for 0.8< l < `. TheMason number collapses the data only at very high shear rates.

TABLE I. The suspension viscosity for 0.8< l < 5.8. Statistical information is omitted where sampling isinsufficient to produce meaningful standard deviations.

Ma

l

0.8 1.7 2.5 3.3 4.2 5.0 5.8

531026 1870231025 211612.0 3596144 125061361024 20.4 127621.9 250648.8 292617.00.0002 3.7560.1 5.5 15.7613.5 72.169.3 134618.6 155642.40.001 2.87 3.27 5.962.6 18.667.9 30.263.8 37.062.2 45.669.30.005 4.461.1 7.064.30.01 2.4862.3 2.361.5 3.661.5 4.461.7 4.261.6 5.962.1 7.262.10.1 2.460.2 2.660.5 2.660.2 2.560.2 2.560.3 2.560.2 2.560.41.0 2.360.1 2.360.1 2.360.1 2.360.2 2.360.1 2.360.1 2.360.1



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the low-shear viscosity and the pair potential well depth. The low-shear viscosity scalesas the equilibrium pair–distribution function at contact times the perturbation to thestructure caused by flow. The equilibrium pair distribution in turn behaves asg ' gHSe

l,wheregHS is the hard-sphere value; hence, the exponential scaling of the viscosity withl comes directly from the change in the equilibrium pair distribution with the electricfield. A plot of h0 as a function ofl ~Fig. 7! shows that the simple approximate theorypredicts thath0 ; el due tog~2! increasing asel.

2. Equilibrium flocculated regime: 4 < l < 10

In the equilibrium flocculated regime, 4, l , 10, a low-shear viscosity is seen inFig. 6 as Ma→ 0 whose magnitude also scales exponentially withl as shown in Fig. 8.The exponential relationship between the low-shear viscosity and the pair potential isremarkably similar to the Buscallet al. ~1993! observation in concentrated, depletionflocculated, dispersions. The Buscallet al.suspension consisted of a nonaqueous polymerlatex in a low-aromatic white spirit flocculated by adding a soluble nonadsorbing poly-mer, and the apparent pair potential well depth was varied by changing the solublepolymer concentration. They found that the low-shear viscosity increased exponentiallywith the potential well depth for well depths from 2kT to 20kT.

We also see that the shear rate at which shear thinning first occurs scalesinverselywith the exponential ofl as shown in Fig. 9, i.e., the low-shear-rate asymptote is reachedfor shear rates that decrease exponentially ase21.88l. The critical Mason number, Mac ,that marks the transition from shear thinning to the low-shear viscosity plateau for4 , l , 10 was determined from the intercept of the lineshr 5 h0 andhr ; Ma21 inthe shear thinning region. In reality, the transition occurs over a range of Mason numbersand the viscosity was not seen to shear thin as Ma21 for l , 10. Since we were unableto distinguish between the transition and the true shear thinning regions, we assumed ascaling similar to the scaling for very large to infinitel. The ratio of the diffusion timeout of a potential wellt to the shear time 1/g yields a new dimensionless shear rate~ga2/Dl!el. When ~ga2/Dl!el , 1, the suspension dynamics are diffusion limited inthe same manner thatga2/D , 1 marks the diffusion limited dynamics for Brownianhard spheres.

The perturbation method used to estimate the low-shear viscosity forl , 4 will notsuffice for largerl because the suspension structure cannot be described in a simple

TABLE II. The suspension viscosity for 8.3< l < `. Results forl 5 ` are taken from Bonnecaze and Brady~1992a!.

Ma

l

8.3 16.7 83.3 833.3 8.33105 8.331012 `

231025 13106342 14876108 19906123 26206173 25006747 7260624301024 353620.9 3896154 594670.1 582627.6 686692.7 11606122 1250660.80.0002 544629.80.0005 92.60.001 46.664.9 44.9616.7 83.461.2 91.261.1 125631.9 195663.5 120620.10.005 23.166.70.01 10.260.3 14.460.3 18.962.1 20.461.9 23.963.5 22.161.10.05 3.660.50.1 2.660.4 3.060.6 3.460.1 3.560.1 3.860.1 3.4360.10.5 2.360.11.0 2.360.1 2.360.1 2.360.1 2.360.1 2.460.1 2.460.1



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pairwise manner. To illustrate this, the pair–distribution function at contact,g~2!, isplotted as a function ofl in Fig. 10 and clearly shows thatg~2! scales exponentially withl only for l , 4. Abovel ' 4, g~2! approaches a constant asymptote as the suspensionhas formed a cell-spanning network.

To predict the low-shear viscosity in this region ofl, we note from linear viscoelastictheory that for a solidlike structureh0 is given by the product of the shear modulus anda time scale of relaxation,t @Ferry ~1980!#,

h0 5 G0t. ~16!

The shear modulus can be estimated by applying a small strain,g, to the suspension, andis given by the ratio of the stress,(yx , to the strain in the limitg approaches zero:

G0 5 limg → 0

(yx

g. ~17!

Since we are simply interested in the scaling behavior of the stress, we represent thenetwork as a single chain of particles aligned along the field direction~see Fig. 11! towhich a small strain is applied. The major contribution to the stress is from the electro-static restoring force attempting to realign the strained structure with the electric field.

FIG. 7. The low-shear viscosity,h0 5 11hH1hHS1hER, estimated from a regular perturbation analysis of theequilibrium structure detailed in the Appendix, evaluated forf 5 0.27. The polarization and hard-sphereviscosities are given by Eqs.~A11! and ~A12!, respectively, while the hydrodynamic viscosity is the EinsteinresulthH 5

52f. The exponential scaling of the lower-shear viscosity forl , 4 comes from the change in the

equilibrium pair–distribution with the electric field.



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The stress is antisymmetric in ER fluids because of the noncentral polarization forces,and the modulus is properly a tensor. Here we use the restoring force associated with thestraining chain to estimate the modulus@Klingenberg and Zukowski~1990!; Kraynik,Bonnecaze, and Brady~1992!#.

The stress is dominated by electrostatic forces,

S ' 21

2V (a 5 1

N

(b 5 1

N

~xa2xb!FabP , ~18!

whereFabP is the electrostatic force on particlea due to particleb. We approximate the

stress in a pairwise manner from nearest neighbors only, yielding

S 5 2nr12F12P . ~19!

From the interparticle force as the derivative of the pair potential evaluated for twoparticles contracting each other oriented at an angleu relative to the field direction wehave

Syx 5 2kT

a3fl

9

8psinu cosu ~125 cos2 u!. ~20!

For small strainsg 5 0 ! 1 and the modulus is given by

FIG. 8. The low-shear viscosity~Ma→ 0! as a function ofl for l , 10. The magnitude of the low-shearviscosity scales exponentially withl. The line is the fit for 4, l , 6 and is seen to also fitl 5 2.5 and 3.3.



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G0 a3

kT5

9

2pf l. ~21!

To estimate the relaxation time,t, we note that the strained structure relaxes whenparticles diffuse out of their mutual potential wells. This diffusive time is much slowerthan the isolated particle diffusion time,a2/D, as a particle must overcome the highpotential barrier. Kramers~1940! showed@see also Ha¨nggi et al. ~1990!; Potaninet al.~1995!# that two particles in a mutual potential well of depthUmin/kT @ 1 will escape ona time of order~a2/D)(kT/Umin! e

Umin /kT, where we have estimated the second deriva-tive of the potential energy at equilibrium and barrier points as simplyUmin/a

2; it is thistime scale that will determine the low-shear viscosity for ER and other flocculated dis-persions. Thus, withUmin/kT ' l in the prefactor, we have

h0 ; h f eUmin /kT. ~22!

The potential well depthUmin/kT characterizing the escape time differs froml be-cause of two factors. First, the dimensionless ratiol gives the pair potential well depthalong the field direction assuming isolated point dipoles, and is a good approximation fordilute suspensions with weak interactions. However, for the volume fraction and field

FIG. 9. The critical Mason number, Mac , that marks the transition from shear thinning to the low-shearviscosity plateau for 3, l , 10 as a function ofl. Mac was determined from the intercept of the lineshr 5 h0 andhr ; Ma21 in the shear thinning region. In reality, the transition occurs over a range of Masonnumbers and the viscosity was not seen to shear thin as Ma21 for l , 10. Since we were unable to distinguishbetween the transition and the true shear thinning regions, we assumed a scaling similar to the scaling for verylarge to infinitel.



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strengths of interest here, the particles cannot be treated simply as isolated point dipoles.As evidenced by the equilibrium structure studies, the suspension consists of particleaggregates, and since the induced particle dipole is proportional to the local electric fieldrather than the applied external electric field, neighboring particles will affect the pairpotential. The potential well depth between two particles in contact can be estimated from

FIG. 10. The pair–distribution function at contact vs the pair potential well depth. Forl , 4, g(2) scalesexponentially withl indicating that the suspension has not flocculated. Above thisl, however,g(2) saturatesas the suspension becomes flocculated. The dashed line gives thel 5 ` asymptote.

FIG. 11. Illustration of a strained chain. The electrostatic forces attempt to realign the strained chain to itspreferred orientation along the electric field direction.



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the suspension energy,u(r ,u) discussed in the introduction, with the induced particledipoleSE defined in Sec. II replacing the isolated point dipole

Udipole5 SE•T•SE , ~23!

whereT is the dipole–dipole interaction,

T 51

r 3 S d23rr

r 2 D . ~24!

The average potential minimum is obtained by averagingUdipole over all the particlepairs in the suspension, and is depicted as a function ofl in Fig. 12. Note thatUdipole/kTis indeed proportional tol, but is enhanced due to the presence of other particles. Belowthe flocculation point,l gives a good estimate of the potential well depth. When floccu-lated, however, neighboring particles further enhance the pair potential andUdipole/kTscales as 4.5l.

The second factor influencing the potential well depth is the dependence of the inter-particle potential on the relative orientation of the particle pair to the applied electricfield. A particle is more likely to escape from its potential well if it is oriented relative toits neighbor with a significant component perpendicular to the applied field where thepotential is actually repulsive. This angular dependence leads to a decrease in the value of

FIG. 12. The pair potential well depthUdipole/kT as a function ofl. Forl , 4 the point dipole approximation,l, provides a good estimate of the potential well depth. Note the increase in the slope atl ; 4; here theaggregated structure provides enhancement beyond the point dipole limit. The point dipole approximation is oneof two factors that causes the potential well depthUmin/kT that characterizes the escape time to differ froml.



17:28:33

the potential barrier over the enhanced dipole valueUdipole/kT. Further, the networkstructure itself influences the detailed form of the potential well. Thus, whileUmin/kTmust be proportional tol, it is not precisely equal tol, and should depend on the detailednetwork structure formed. Hence, we write

Umin

kT5 b~f!l, ~25!

and find the coefficientb by fitting Eq. ~22! to the experimental data. The data for thelow-shear viscosity yieldsh0/h 5 0.16e1.57l, as shown in Fig. 8. Thus,b ' 1.57, andthe difference between the prefactor, 0.16, and the modulus estimate, shows that thecharacteristic escape time is quantitatively estimated by 0.42e1.57l. The difference be-tween the 1.57 factor in the exponent for the low-shear viscosity and the 1.88 factor forthe transition shear rate for the low-shear thinning behavior is a result of the approximatemethod used to estimate Mac .

Note that the low-shear viscosity scales asel in both the dispersed and equilibriumflocculated regimes, but for different reasons. At smalll the low-shear viscosity increasewith l comes from the pair–distribution function at contact that scales asel; the relax-ation time is still the bare diffusion timea2/D, however. Once the flocculated networkhas formed the pair–distribution function at contact is independent ofl, the relaxationtime is the escape time~a2/Dl!el, and the viscosity is now set by this escape time.

3. Kinetically aggregated regime: l > 10

In the kinetically aggregated regime,l . 10, there is a gradual transition from thehigh-shear-rate Newtonian behavior to shear thinning as Ma decreases. For231025 < Ma < 1023, the suspension viscosity behaves as Ma2a, wherea varies from0.9 forl 5 100, to 1 whenl 5 1012 as shown in Table III. Although forl @ 1 Brown-ian forces are small compared to polarization forces, Ma is not sufficient to collapse thedata over the entire range of shear rate~see Fig. 6!. Instead, the viscosity increases withl at any given Ma below 0.1. At low Ma, neither a low-shear viscosity nor a dynamicyield stress is evident; the low Ma limit has not yet been reached. As discussed above, thelow-shear limiting behavior occurs when~ga2/Dl!eUmin /kT , 1, a very strict require-ment for largel. At present, simulations for smaller Mason numbers are computationallyprohibitive, but these suspensions should show a low-shear viscosity.

4. Comparison to other work

Our simulation results compare well to those of other investigators. Melrose~1992!examined the rheology of an idealized 0.3 volume fraction suspension in the Brownian

TABLE III. Exponents for viscosity dependence for 231025

< Ma < 1023.

l a

` 1.08.331012 1.08.33105 0.9833.3 0.983.3 0.916.7 0.9

0.9



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regime using Brownian dynamics. He neglected all hydrodynamic interactions and mod-eled the interactions between spheres as a summation of electric point dipoles with a softcore repulsive interaction that scales asr236, wherer is the center-to-center separationbetween particles. These simplifications enabled him to model a 3D system. Althoughthis is an advantage since it allows observation of structural variations along all threespatial directions, the method is accurate only in the dilute limit, at low-shear rates, andfor low field strengths. In the range studied, 1023 , Ma , 1021, Melrose found a shearthinning viscosity that scales as Ma2a wherea , 1, and that the Mason number does notcollapse the data onto a single curve. By the two-thirds rule, our 0.4 area fraction isequivalent to a 0.27 volume fraction and allows comparison to his 0.31 volume fractionstudies. Figure 13 shows that the results compare qualitatively well. Quantitatively, how-ever, Melrose’s viscosities are slightly lower in the region of Ma he studied. Note thatMelrose’s simulations do not explore the low Ma limit and, therefore, one cannot con-clude from his work whether or not there is a low-shear limiting viscosity.

Halseyet al. ~1992! studied experimentally a monodisperse 0.1 volume fraction sus-pension of silica spheres 0.75mm in diameter. They estimated their Mason number~ [ 4pMa! by assigning 1 to the shear rate where a suspension with an applied field of4000 V/cm has a suspension viscosity equal to the bare suspension viscosity. Here, weestimate Ma from their reported particle size, shear rates, and field strengths. They did notreport values for the dielectric constants of the solvent methyl–cyclohexanol and silicaspheres, so we use 15 for the solvent$cyclohexanol has a dielectric constant of 15 at

FIG. 13. Comparison of the Stokesian dynamics simulations at an area fraction of 0.4 with the Browniandynamics simulations of Melrose~1992! at a volume fraction 0.31.



17:28:33

25 °C @CRC handbook of Chemistry and Physics~1993!#% and 4 for the silica spheres. Itis estimated that Ma varies from 0.01 for a shear rate of 0.1 s21 to 1.0 for a shear rate of10 s21. Halseyet al. found a power law dependence of the viscosity with the shear rate,h ; g2a, wherea ranged from 0.93 to 0.68 for fields from 103 V/cm to 43103 V/cm,respectively, or in dimensionless terms,l on the order of 102–103. Figure 14 comparestheir l 5 102 results with ourl 5 83 results. Note that the Halseyet al. experimentswere done at Mason numbers larger than those we used to determine the scaling of thelow-shear viscosity, accounting for the higher than expected20.68 slope they reported.Nonetheless, the good comparison between our simulations and the Halseyet al. experi-mental results is encouraging.

Unfortunately, there are no data forl , 30 with which to compare our results. Sunand Tao’s~1995! 0.16 volume fraction simulations do not allow a quantitative compari-son. The transitionl for flocculation is a function of volume fraction and Sun and Tao’ssuspension remained dispersed forl < 30. Qualitatively, however, they also noted alow-shear viscosity when the suspension remained dispersed that scaled exponentiallywith l in agreement with our results forl , 4.

IV. A MODEL FOR THE RHEOLOGY OF FLOCCULATED SUSPENSIONS

The concept of the escape time, the time to diffuse out of the potential minimum, canbe used to formulate a model for the viscosity of aggregating suspensions. The model

FIG. 14. Comparison of simulation viscosities with the experiments of Halsey, Martin, and Adolf~1992!. Thestraight line is from Fig. 2 of their paper and gives their linear fit to the experimental data for an applied fieldof 400 V/cm.



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closely follows the Glasstoneet al. ~1941! viscosity model for molecular liquids anddescribes the suspension rheology in terms of the microscopic rearrangements in thenetwork structure. Similar ideas have been used by Potaninet al. ~1995! to model fractalflocs. At rest, a flocculated suspension consists of a network of particles, each within apotential cage formed by its nearest neighbors as depicted in Fig. 15. Rearrangementrequires breaking a contact pair and forming a new contact, i.e., hopping over a dimen-sionless distanceG relative to the particle size. The activation energy barrier is on theorder of the minimum in the pair potential. For simplicity, the particle potential is repre-sented as being symmetrical about the hopping distance, i.e., the particle’s activated stateis halfway between its initial and final positions. Network rearrangement occurs as par-ticles hop out of their potential well on the time scalet. Under static, nonstressed con-ditions, the initial and final states are equally favorable so there is no net motion of thenetwork; particles hop back and forth with equal probability, and the suspension does notflow or shear.

Under an external stress,s, the suspension acquires an elastic energy densitysg/2e~bE!2, whereg is a macroscopic strain.~The stress is appropriately nondimension-alized by 2e~bE!2 since the induced internal stress will be predominantly electrostatic inorigin and at equilibrium the induced internal stress is equal to the applied stress.! On themicroscopic scale, the acquired energy resides in the pair–interaction potential such thatthe new energy barrier is given by

2Umin

kT6

sa3GV0

lkT, ~26!

FIG. 15. Illustration of the pair potential for a flocculated suspension. The thick lines denote the potential withno external stress, while the thin lines denote the potential with an applied stresss. Rearrangement occurs whenparticle 1 hops a distance ofO~G!.



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whereV0 is the dimensionless volume of the particle pair and depends, as doesG, on thedetails of the network structure and how the macroscopic stress is distributed locally onthe microscale. For example, an affine deformation would have the microscopic strain,G,equal to the macroscopic straing. The dimensionless energy6sa3GV0/lkT is the ad-justment to the work done on a particle as it moves to the top of the potential well with~positive sign! or against~negative sign! the shear direction; the potential well becomesskewed as illustrated in Fig. 15. Unlike the nonstressed case, however, there is now afavored state. The barrier from the strained to relaxed state is smaller than the barrierfrom relaxed to strained state so that on average, more particles will hop to the lowerenergy state resulting in a net shearing motion along the stress direction with shear rateproportional to the net frequency of forward jumps,

g 5D

a2A2

Umin

kTe2Umin /kT FexpSsa3V0G

lkT D2expS2sa3V0G

lkT DG ~27!

52D

a2A2

Umin

kTe2Umin /kT sinhSsa3V0G

lkT D, ~28!

whereA2 is a proportionality constant that depends on the network structure. Specifically,after a pair contact is broken, reformation with another particle requires diffusion not ofan isolated particle@in which caseA2 5 1 in Eq. ~28!# but by diffusion of a large flocstructure. For example, in the case of an isolated strained ER chain, when a pair in thecenter is broken, the top half reconnects with the image of the bottom half. The diffusionis, thus, of a rod, not a single particle, which scales as the cube of the rod length.

Although the relative hopping distanceG will depend on the suspension’s microstruc-ture, e.g., densely packed flocs will result in a smallerG, its variation withl ~once theflocculated structure is formed! is not likely to be large~i.e., greater than order 1!. Thus,we estimate thatV0G 5 F ~f!, a function off, and Eq.~28! gives for the stress

a3

kTs ;

l

F ~f!sinh21SA2 ga2

D

kT

UmineUmin /kTD, ~29!

or a viscosity

h 5 A1~f!1

Masinh21SA2 ga2

D

kT

UmineUmin /kTD, ~30!

whereA1~f! is a proportionality constant dependent onf.With this model, we recover naturally the dimensionless quantity~ga2/D!~kT/Umin!

3 eUmin /kT that gives the ratio of the time scale of diffusion out of a potential well ofdepthUmin/kT to the time scale of shear found in the previous section to give the shearrate below which the low-shear viscosity was reached. From Eq.~30! the low-shearviscosity is achieved when~ga2/D)(kT/Umin! e

Umin /kT ! 1, and Eq.~30! correctly pre-dicts thath → h 0 ; eUmin /kT as g→ 0.

Up to now, only the interparticle force contribution to the viscosity has been discussedsince it is assumed to be the major contributor. Asg→ `, however, the hydrodynamicstress dominates, and the high-shear viscosityh` is purely hydrodynamic. Recalling Eq.~14!, the suspension viscosity can be approximated as

hr2h` 5 11hrB1hr

H1hr

SP1hr

xF2h` ' hr

SP1hr

xF ' hrxF , ~31!



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neglecting the Brownian and interparticle force–flow contributions. Equation~31! as-sumes 11hr

H ; h` , which is true only in the limit of infinite shear rate. For finite Mawhere clusters and chains are formed, the hydrodynamic viscosity is greater thanh` .Nonetheless, as a first approximation we model the difference ofh2h` neglecting thecontributions from Brownian forces and the flow caused by interparticle forces. Thus, wecan write

h2h`

h02h`5

1

A2~ga2/Dl!e1.57l

sinh21SA2 ga2

Dle1.57lD , ~32!

where exp~1.57l! has been substituted for exp~Umin/kT! andl has been substituted forUmin/kT in the prefactor. A plot of the reduced viscosity~h2h`!/~h02h`! as a functionof ~ga2/Dl)e1.57l gives a single universal curve for 4, l , 10 that fits Eq.~32! withA2 5 8.8 as illustrated in Fig. 16.

The model and Eq.~32! should also apply forl . 10. For example, forl 5 17, Eq.~22! is used to estimateh 0, the reduced viscosity is then plotted as a function of~ga2/Dl)e1.57l, and is seen to fit onto the universal curve~Fig. 16!. Similar transforma-tions forl . 17 are possible, but for the shear rates studied the reduced viscosity is toosmall to be plotted. Note that the data forl , 4 also falls on the universal curve; at lowl, h 0 scales ase

l from the change in the equilibrium pair–distribution function, but thecharacteristic time scale for this region isa2/D.

FIG. 16. The reduced viscosity as a function of the dimensionless shear rate~ga2/Dl)e1.57l. When the shearrate is rescaled with the escape time,t, the data collapses onto a single curve.



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V. CONCLUSIONS

In this paper, we presented the results of a study of the dynamics and structure ofdielectrically interacting particles forming an ER fluid. The nature of the static, non-sheared structure is governed byl, the ratio of the polarization to Brownian forces thatequals the pair potential well depth for point dipoles in contact aligned along the field.We found that a 0.4 area fraction suspension remains dispersed whenl , 4. For4 , l , 10, there is an equilibrium flocculated regime where particles order into anhexagonal structure, while forl . 10, the suspension is kinetically flocculated.

The suspension viscosity was described in terms ofl and the Mason number, Ma, thatgives the ratio of the viscous-to-polarization forces. Both the dispersed and weakly floc-culated suspensions,l , 10, have a low-shear viscosity that scales exponentially withlbut for different reasons. In the dispersed regime,l , 4, the scaling can be predictedfrom a regular perturbation analysis of the equilibrium structure. The low-shear viscosityscales with the equilibrium pair–distribution function at contact, which in turn scales asgHSe

l, wheregHS is the hard-sphere value.In the equilibrium flocculated regime, 4, l , 10, a simple model based on linear

viscoelastic theory shows that the low-shear viscosity is given as the product of the shearmodulus and a relaxation time. The major contribution to the stress comes from thepolarization forces and, thus, the shear modulus scales asl. It is the relaxation time scalethat is responsible for the exponential scaling of the low-shear viscosity; since the par-ticles are aggregated, the time scale for escape from their mutual potential wells is thetime scale~a2/Dl) eUmin /kT. The attractive potential well depth,Umin/kT, scales with,but is not precisely,l for two reasons. First, the induced particle dipole has contributionsfrom neighboring particles, an effect that is accounted for in the simulation but is notcaptured with the isolated point dipole model. Second, the electrostatic potential is ori-entation dependent. Particle pairs at an angle to the electric field have a lower potentialwell depth compared to particle pairs aligned along the field, and will escape preferen-tially. Thus, we found that the low-shear viscosity obeys the curveh0/h 5 0.16e1.57l inthe equilibrium flocculated regime.

The ratio of the escape time and the shear time scale gives a new dimensionlessnumber,~ga2/Dl)el, and the low-shear viscosity is seen wheng ! ~D/a2!le21.88l.For l . 10, we did not see a low-shear viscosity, but the shear rates we were able toachieve were not low enough to observe the low-shear limiting behavior.

A microstructural approach to describe the dynamics of flocculated suspensions waspresented. Unlike previous phenomenological models, it incorporates the physics of theparticle interactions and motion to provide a general, widely applicable, and predictivemodel. In this model, it was hypothesized that the network rearrangement process of aflocculated suspension is rate activated. Under stress or shear, the suspension flows whenparticles diffuse out of their mutual potential wells, not on the bare particle diffusion timescalea2/D, but on the escape time scalet ; ~a2/D!~kT/Umin!exp(Umin/kT!. As a result,the relevant dimensionless shear rate is the ratio of the escape time scalet to the timescale of shear, or~ga2/D!~kT/Umin! e

Umin /kT. Our numerical simulations on ER fluidsverified this hypothesis; when the shear rate was scaled witht, a single universal curvewas obtained.

Finally, it is important to note that the distinction between flocculated suspensionswith an apparent yield stress and suspensions with a low-shear viscosity is an artificialone. In theory, all flocculated suspensions, including ER fluids, will flow under stress andshow a low-shear viscosity. One example of this is that although we did not see alow-shear viscosity forl 5 17 over the range of Mason numbers used, the viscosity data



17:28:33

obeys the universal curve described by Eq.~32! when Eq.~22! is used to predicth0.Suspensions with largerl are also expected to have a low-shear viscosity, but at expo-nentially small shear rates that may not be realized in practice.

ACKNOWLEDGMENTS

This work was supported in part by the National Science Foundation through GrantNo. CTS-9420415 and by the Kodak Corporation. The authors also thank Masao Doi foruseful discussions on this topic.

APPENDIX

The equation governing theN-particle distribution is the Smoluckowski equation,

]PN

]t1¹•jN 5 0, ~A1!

where the probability fluxjN is given by

jN 5 UPN1RFU•~FP2kT¹ ln PN!PN , ~A2!

and represents, respectively, contributions from convection, the pair potential, and diffu-sion. Small perturbations to the equilibrium structure are denoted by

PN 5 PN0~12fN!, ~A3!

and the particle flux becomes

jN 5 UPN0 ~12 f N!2PN

0D•¹ f N . ~A4!

In order to proceed analytically, an equation for the pair–distribution functionP2 isobtained by integrating overN-2 particles. Following the method of Brady~1993!, weobtain to the first order inPe,

P20]f2

]t2¹r•^Dr &2

0P20•¹ r f 2 5 2¹ r•Ûr &2

0P20, ~A5!

where the angled brackets denote a conditional average with two particles fixed andf 2 isthe normalized disturbance,^ f 2&2

0 5 Pe f2 . We have also neglected direct couplings to athird particle in writing Eq.~A5!. The quantitiesDr &2

0 and theÛr &20 are the diffusivity

and velocity of the particle pair, respectively. Neglecting all hydrodynamic interactions,and anisotropy, and noting that the scalar functionf 2(r ) 5 2~1/2!f (r )n–Ê&•n, wherenis the dimensionless separation vector between particle centers, Eq.~A5! reduces to,

1

g~r!r2d

dr Sgr2dfdrD26f

r25 2r

d ln g

dr, ~A6!

whereg(r ) is the equilibrium pair–distribution function. For weakly aggregated suspen-sions the pair–distribution function scales as the exponential of the pair potential,

g~r,u! 5 gHS expFlS2ar D3~3 cos2 u21!

2 G, ~A7!



17:28:33

for a dilute suspension. Again, for simplicity, we neglect any anisotropy in the structureand the potential, and use the infinite dilution value of 1 forgHS such that

g~r,0! ' g~r ! ' expFlS2ar D3G. ~A8!

Substituting into Eq.~A6!, the evolution equation for the pair–perturbation function is

d2f

dr212

r

df

dr224l

r 4d f

dr26 f

r 2224l

r 35 0, ~A9!

with boundary conditions

f ; 0 as r → `,

df

dr5 22 at r 5 2.

The first boundary condition ensures no disturbance far away, while the second ensuresno relative flux at the surface of contact of the two particles.

In the absence of hydrodynamic interactions, there is no contribution to the stress fromBrownian motion. The interparticle force, however, is now the sum of a hard-sphererepulsion and the electrostatic force,

FIG. 17. The disturbance functionf ~2! as a function ofl. Note thatf ~2! becomes negative for increasinglyattractive potential wells. This suggests an increase in particle density at angular positions where for hard-sphere suspensions there was a depletion.



17:28:33

FP 5 12 kTnd~r22a!1FER. ~A10!

The hard-sphere contribution to the stress^xF& is

^xF&HS 5 21

2kTn2E nnd~r22a!g~r !dr

5 2kTn2aEr 5 2a

nng~r !dS.

The viscosity is now a sum of the ER viscosity, which from Eq.~71! in Brady ~1993!,becomes

hER 5 227

5lf2E

2

`1

rexpS8l

r3 Df~r!dr, ~A11!

with our expression forg(r ), and the contribution from the hard-sphere repulsion, whichis identical to Eq.~53! from Brady ~1993!,

hHS 5 95 f2el f ~2!, ~A12!

where we have substituted Eq.~A8! evaluated atr 5 2a for g(2) and the infinite dilu-tion diffusivity for the short-time diffusivity.

Equation~A9! was solved numerically using a shooting method. The boundary con-dition for r → ` was satisfied onr 5 100a, and the results forf ~2! are displayed in Fig.17. Note thatf ~2! becomes negative forl . 2, such that the hard-sphere contribution isnegative while the ER contribution is positive.

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Brownian electrorheological fluids as a model for flocculated dispersions

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