Magnetorheology in an aging, yield stress matrix fluid - MITweb.mit.edu/doylegroup/pubs/jason_rheoacta_2012.pdf · Rheol Acta (2012) 51:579–593 DOI 10.1007/s00397-012-0632-z ORIGINAL

Rheol Acta (2012) 51:579–593DOI 10.1007/s00397-012-0632-z

ORIGINAL CONTRIBUTION

Magnetorheology in an aging, yield stress matrix fluid

Jason P. Rich · Patrick S. Doyle · Gareth H. McKinley

Received: 7 November 2011 / Revised: 16 April 2012 / Accepted: 17 April 2012 / Published online: 8 May 2012© Springer-Verlag 2012

Abstract Field-induced static and dynamic yieldstresses are explored for magnetorheological (MR) sus-pensions in an aging, yield stress matrix fluid composedof an aqueous dispersion of Laponite® clay. Usinga custom-built magnetorheometry fixture, the MRresponse is studied for magnetic field strengthsup to 1 T and magnetic particle concentrationsup to 30 v%. The yield stress of the matrix fluid,which serves to inhibit sedimentation of dispersedcarbonyl iron magnetic microparticles, is found tohave a negligible effect on the field-induced staticyield stress for sufficient applied fields, and goodagreement is observed between field-induced staticand dynamic yield stresses for all but the lowestfield strengths and particle concentrations. Theseresults, which generally imply a dominance of inter-particle dipolar interactions over the matrix fluid yieldstress, are analyzed by considering a dimensionlessmagnetic yield parameter that quantifies the balanceof stresses on particles. By characterizing the appliedmagnetic field in terms of the average particle

J. P. Rich · P. S. Doyle (B)Department of Chemical Engineering,Massachusetts Institute of Technology,Cambridge, MA, USAe-mail: [email protected]

G. H. McKinley (B)Hatsopoulos Microfluids Laboratory,Department of Mechanical Engineering,Massachusetts Institute of Technology,Cambridge, MA, USAe-mail: [email protected]

magnetization, a rheological master curve is generatedfor the field-induced static yield stress that indicatesa concentration–magnetization superposition. The re-sults presented herein will provide guidance to formula-tors of MR fluids and designers of MR devices who re-quire a field-induced static yield stress and a dispersionthat is essentially indefinitely stable to sedimentation.

Keywords Magnetorheology · Yield stress · Aging ·Clay · Suspensions

Introduction

Magnetorheological (MR) fluids are field-responsivematerials that exhibit fast, dramatic, and reversiblechanges in properties when subjected to a magneticfield. First introduced by Rabinow (1948), MR fluidsare composed of microscopic iron-containing particlessuspended in a matrix fluid. Upon application of amagnetic field, the particles acquire a dipole momentand align to form domain-spanning chains. This field-induced structuring of the suspension leads to sig-nificant changes in rheological properties, includingorder-of-magnitude growth in the steady-shear viscos-ity and the emergence of field-dependent yield stressand viscoelastic behavior (de Vicente et al. 2011a). Thetunability of rheological properties with the appliedmagnetic field provides the basis for a wide varietyof commercial applications of MR fluids, includingautomobile clutches (Rabinow 1948), active dampers(Spencer et al. 1997), seismic vibration control (Dykeet al. 1996), prosthetics (Carlson et al. 2001), precisionpolishing (Kordonski and Golini 1999), and drillingfluids (Zitha 2004). MR fluid research and technology

580 Rheol Acta (2012) 51:579–593

has been reviewed numerous times, with articles focus-ing on rheology and flow properties (de Vicente et al.2011a), models and mechanisms of chain formation(Parthasarathy and Klingenberg 1996; Goncalves et al.2006), MR fluid formulation (Park et al. 2010), andapplications (Klingenberg 2001; Olabi and Grunwald2007).

Matrix fluids in MR suspensions have traditionallybeen aqueous or oil-based Newtonian fluids of mod-erate viscosity. While this type of formulation maxi-mizes the rheological differences between the activatedmaterial and the off-state, particle sedimentation isa major concern in Newtonian matrix fluids due tothe (typically) large density difference between iron-containing particles and the surrounding fluid. To ad-dress this problem, modifications to both the suspendedparticles and the matrix fluid have been proposed. Forexample, according to the Stokes’ drag law, sedimen-tation can be slowed by decreasing the particle size.Experiments have shown, however, that smaller parti-cles generally lead to lower field-induced yield stresses(Lemaire et al. 1995). Additionally, when particle sizesapproach nanometer length scales, Brownian effectslimit the length and strength of the chain structuresthat form under an applied field (Fermigier and Gast1992). Composite particles with lower iron content alsoexhibit slower sedimentation, but the accompanyingdecrease in magnetization again results in diminishedfield-induced rheological properties (Cho and Choi2004). Reasonable success has been achieved throughthe use of stabilizing additives that provide a sterichindrance to particle aggregation. Additives such asferromagnetic nanoparticles (Chin et al. 2001; Lopez-Lopez et al. 2005), fumed silica (Lim et al. 2004), organ-oclays (Lim et al. 2005), and magnetizable nanofibers(Ngatu et al. 2008) have been used for this purpose.Arguably the most robust methods for inhibiting sed-imentation involve modifying the matrix fluid rheolog-ical properties. By employing viscoplastic matrix fluids(Rankin et al. 1999; Park et al. 2011) or thixotropic gel-forming agents such as silica nanoparticles (de Vicenteet al. 2003; Lopez-Lopez et al. 2006), sedimentationcan be prevented essentially indefinitely in quiescentdispersions as long as the yield stress of the matrix fluidexceeds the net stress acting on the particles due togravity and buoyancy (Chhabra 1993).

For the set of experimental conditions consideredby Rankin et al. (1999), results indicate that the ma-trix fluid yield stress has minimal effect on the field-induced dynamic yield stress. The field-induced staticyield stress, however, is also an important property inmany MR fluid applications and is a more direct mea-sure of the “strength” of an MR fluid (Kordonski et al.

2001). The dynamic yield stress is typically measuredby imposing a set of decreasing steady-state shear rates,γ̇ , and extrapolating the resulting shear stresses to γ̇ =0 s−1. In contrast, the static yield stress is defined asthe stress required to induce flow from rest (Nguyenand Boger 1992). For materials that exhibit thixotropyor require a finite time to reform microstructure afterbeing sheared, these two yield stress measures are gen-erally not equal (Møller et al. 2006, 2009; Bonn andDenn 2009). Additionally, in the case of MR fluids, itis reasonable to expect that the effects of a matrix fluidyield stress on field-induced structure and rheology willbe more apparent in static yield stress measurements.The externally applied shear rate in dynamic mea-surements increases the probability that particles willencounter each other and aggregate despite the matrixfluid yield stress, whereas in static measurements mag-netic particles must directly overcome the matrix fluidyield stress in order to form structure and provide anMR response. Because of these complications arisingfrom differences in measuring techniques, and becauseof the practical utility of yield stress matrix fluids ininhibiting sedimentation, the need remains to developa more thorough understanding of the effect of matrixfluid yield stresses on field-induced properties in MRfluids.

The yield stress matrix fluid in the current work iscomposed of an aqueous dispersion of the syntheticclay Laponite®. Often used as a rheological modifierin commercial soft materials, Laponite® clay consistsof nanometric disks that undergo progressive structuralarrest over time when dispersed in water at concen-trations as low as about 1 wt% (Mourchid et al. 1995;Ruzicka et al. 2004, 2006; Jabbari-Farouji et al. 2008).This continual microstructural development, knownas aging, results in complex and time-dependent rhe-ology (Cocard et al. 2000; Joshi and Reddy 2008;Negi and Osuji 2010). Furthermore, the competitionbetween aging and microstructural disruption due toshear (i.e. shear “rejuvenation”) leads to thixotropicbehavior (Abou et al. 2003). Previous work has ad-dressed the bulk rheology and microrheology of aque-ous Laponite® dispersions; for more thorough reviewsof the current understanding of the phase behavior,structure, and rheology of aqueous Laponite® disper-sions, see Ruzicka and Zaccarelli (2011) and Rich et al.(2011b). For the purposes of the current work, thesignificance of the aging behavior of Laponite® disper-sions is that it results in continual growth of the staticyield stress of the matrix fluid. Therefore, differentmatrix fluid yield stresses can be examined simply byallowing the MR composite system to age for differentperiods of time.

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Using a custom-built magnetorheometry fixture(Ocalan 2011), the current work explores thefield-induced static yield stress of MR suspensions in anaging, yield stress matrix fluid composed of an aqueousdispersion of Laponite®. The first section details theformulation of the composite MR fluid, including theaqueous Laponite® matrix fluid, and describes thecustom magnetorheometry fixture and experimentalprotocol. Subsequently, we present results for the field-induced static yield stress as a function of magneticfield strength and age time. Field-induced dynamicand static yield stress measurements are compared atvarious particle concentrations, and observations areexplained by considering the magnitude of appropriatedimensionless groups that characterize the balancebetween the matrix fluid yield stress and inter-particlemagnetic stresses. The final section summarizes the keyfindings and identifies potential areas for future work.

Materials and methods

Magnetic particles and matrix fluid

The magnetic particles providing the MR response inthe present study are CM grade carbonyl iron powder(CIP; BASF, Ludwigshafen, Germany). CM is a “softgrade” consisting of mechanically soft, approximatelyspherical particles with an iron content of about 99.5%by weight. Though the CM grade exhibits a wider sizedistribution than other CIP grades, it is also relatively

economical. Figure 1a shows a scanning electron mi-croscopy image of CIP, providing a sense for the poly-dispersity and irregularity of the particles. The scalebar corresponds to 5 μm. Particle size distributionsof CIP, as measured with a Mastersizer 2000 particlesize analyzer (Malvern Instruments, Worcestershire,UK), are shown in Fig. 1b. Both volume-weightedand number-weighted distributions are shown. Thenumber-weighted distribution gives an average particlediameter of 〈d〉 ≈ 3.7 μm, with a standard deviation ofabout 2 μm. Additionally, from the volume-weighteddistribution it is found that 50% of the powder volumeconsists of particles with diameter d ≤ d50 ≈ 8.6 μm,in quantitative agreement with data provided by themanufacturer.

Although the increased polydispersity and eccentric-ity of CIP particles can complicate analysis, the useof CIP in MR fluids provides functional advantagesover polymer–magnetite composite superparamagneticparticles because of its stronger magnetic properties,which result from the high iron content. Figure 1c showsmagnetization data for CIP, obtained using a vibratingsample magnetometer with a sample of approximately0.02 g of CIP powder. The particles exhibit linear mag-netization up to an applied magnetic field of about B ≈0.1 T. Beyond about B ≈ 0.6 T, the particles exhibita constant saturation magnetization of about Msat ≈190 emu g−1 = 190 A m2 kg−1, which is about ten timesgreater than that of similar-sized polymer–magnetitesuperparamagnetic particles (based on data frommanufacturer, Invitrogen). Though a small remnant

5 µm

a)

1 10 1000

2

4

6

8

10

12

Per

cent

age

num. weightedvol. weighted

db)

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

B [T]

M [e

mu.

g-1]

c)

Fig. 1 The magnetic component of the magnetorheologicalfluid in the present study is CM grade Carbonyl Iron Pow-der (CIP; BASF, Ludwigshafen, Germany). a Scanning elec-tron microscopy (SEM) image of CIP. The powder consists ofapproximately spherical particles exhibiting some polydispersityand irregularity. b Volume-weighted and number-weighted sizedistributions of CIP particles in the present study. Treating CIPparticles as spherical, an average particle diameter of about3.7 μm is extracted from the number-weighted distribution. Addi-tionally, the volume-weighted distribution indicates that 50% of

the powder volume consists of particles with effective diameterd ≤ d50 = 8.6 μm, in agreement with data provided by themanufacturer. c CIP magnetization curve. Solid lines connectdata points and serve to guide the eye. According to the man-ufacturer’s specifications, the particles are ≥99.5% Fe, leading tolarge values of the magnetization M at moderate applied fields.The particles exhibit linear magnetization for small applied fields(B ≤ 0.1 T) and reach a saturation magnetization of about Msat ≈190 emu g−1, denoted by the dashed gray line, above about B ≈0.6 T

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magnetization (∼1 emu/g) has previously been re-ported for similar types of magnetic particles (Phuléet al. 1999), our measurements indicate negligible mag-netic hysteresis for B ≥ 0.05 T.

In the present study, CIP is suspended in an aging,yield stress matrix fluid, and the MR response of thecomposite is explored. The matrix fluid consists of a3.0 wt% aqueous dispersion of the synthetic colloidalclay Laponite® (RD grade, Southern Clay Products,Gonzales, TX). To prepare a sample, dry Laponite®powder is added to an aqueous buffer of pH ≈ 10consisting of 1.8 mM NaOH and 4.1 mM NaHCO3. Thepurpose of the buffer is to avoid the slow dissolutionof Laponite® platelets, which has been observed atlower pH (Thompson and Butterworth 1992), and tofix the solvent ionic strength at I = 5.9 mM. Duringmixing, the dispersion is kept under N2 gas in or-der to prevent the uptake of CO2, which can lowerthe solution pH and contribute to the dissociation ofplatelets (Mourchid et al. 1995; Mourchid and Levitz1998; Martin et al. 2002). After mixing vigorously forat least 1.5 h, the clay dispersion is passed througha 0.8-μm filter, breaking apart most of the remainingaggregates with a strong shear field (Bonn et al. 1999).Petit et al. (2009) demonstrated that about 7% of theinitial Laponite® concentration is lost when dispersionsof about 3 wt% Laponite® are passed through 0.45 μmfilters. However, since the present study uses filters

with larger pores (0.8 μm), it is assumed that filtrationdoes not appreciably change the nominal concentrationof Laponite®. Immediately after filtering, CIP is addedto the desired concentration; and after vortex mixingfor about 30 s, the CIP is dispersed approximatelyhomogeneously. The composite suspension is then de-posited onto the sample plate of the custom MR celldescribed below.

Bulk magnetorheology

The bulk rheology of CIP suspended in aqueousLaponite® dispersions is studied under applied mag-netic fields using the custom-built magnetorheologyfixture designed by Ocalan (2011) that is shown inFig. 2. The magnetic field is generated by passing elec-trical current (up to 5 A) through a coil of coppermagnet wire, which is wrapped around a cylindricalcore of 1018 carbon steel. The fluid sample is placedbetween a non-magnetic aluminum sample plate, whichis fixed directly above the cylindrical core, and a 20 mmdiameter non-magnetic titanium plate geometry thatis attached to the spindle rod of the rheometer. Tominimize wall-slip, the aluminum sample plate is sand-blasted to an RMS roughness of about 3.8 μm, andadhesive-backed 600 grit sandpaper disks (McMaster-Carr, Elmhurst, IL, RMS roughness ≈ 6.0 μm) areattached to the 20 mm top plate. An elastomeric ring

Fig. 2 The custom-built fixture used for magnetorheology ex-periments shown a as a cross-sectional schematic and b mountedon a stress-controlled rheometer [images reproduced with per-mission from Ocalan (2011)]. Dimensions are given in millime-ters. The magnetorheometry fixture consists of copper magnetwire wrapped around a cylindrical core of 1018 carbon steel.The fluid sample fills the space between a non-magnetic alu-minum sample plate and a 20 mm diameter non-magnetic plateof titanium alloy that is attached to the spindle rod of the

rheometer. Silicone oil flows through channels surrounding thecoil, providing temperature control. Two cover plates of 1018carbon steel complete the magnetic circuit, helping to direct thefield uniformly through the sample. A thin slot in the bottom ofthe sample plate allows access for a Gauss probe to measure themagnitude of the applied magnetic field. When a current of about3.5 A passes through the coil, the setup can apply magnetic fieldsup to B ≈ 1 T with high spatial uniformity

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on the outer edge of the top plate helps to preventthe sample from escaping from the gap and climbingthe spindle rod in response to strong magnetic fields.A thin slot in the bottom of the sample plate providesaccess for a Gauss probe to measure the magneticfield directly beneath the sample. The entire fixtureis housed in a casing of 1018 carbon steel, including atop cover that serves to complete the magnetic circuitand direct the magnetic field through the sample, andthen mounted on a stress-controlled rheometer (ARG-2, TA Instruments, New Castle, DE). The casing andcover design prevent the applied magnetic fields frominterfering with the magnetic bearing of the rheometer.Temperature control is achieved by flowing silicone oilthrough channels machined within the casing. Using thefixture, highly uniform magnetic fields up to B ≈ 1 Tcan be applied to the sample. For more detailed infor-mation about the design of the fixture, its capabilities,and analysis of the applied fields, see Ocalan (2011).

After preparing the sample as described above,the fluid is introduced between the rheometer platesand the gap height is set to 0.5 mm. Because of thethixotropic nature of the Laponite® matrix fluid, stepsare taken to ensure consistent initial conditions andpromote reproducibility of results. Initially, the sam-ple is pre-sheared at a rate of γ̇ = 600 s−1 for 10 s,effectively erasing the shear history and resetting theage time, tw, to zero (Fielding et al. 2000; Bonn et al.2002). Though it has been shown that even a strongshear cannot completely ‘rejuvenate’ the aging processin aqueous Laponite® dispersions (Shahin and Joshi2010), this small amount of irreversibility is found tohave a negligible effect on the rheological response ofthe composite to magnetic fields. Subsequent to thepre-shear at γ̇ = 600 s−1, the magnetic component ofthe suspension is structured by applying a relativelyhigh magnetic field of 0.8 T for 30 s, after which thefield is switched off and the material is pre-shearedagain at γ̇ = 250 s−1 for 20 s. Pre-shearing for longerperiods of time had negligible effect on results. Thisprotocol provides consistent magnetic and shear his-tories and imposes reproducible initial conditions formagnetorheology experiments (Deshmukh 2006). Thesuspension is allowed to age at a constant temperatureT = 22.5◦C, and the desired magnetic field is appliedstarting 30 s before performing rheometric tests toprobe the yielding behavior. Initiating the magneticfield at times from about 15 s to 1 min prior to the startof rheometric tests did not significantly affect results.The primary focus of the present work is the static yieldstress, which is measured using continuous ramp tests;starting from a value below the static yield stress, theapplied shear stress is increased continuously until the

dispersion has yielded, allowing the extraction of theflow curve during yielding. The stress is ramped linearlyover a test time of �ttest = 2 min (�ttest = 1 min for10 min age samples), which is small compared to theage of the dispersion tw. Though attempts are generallymade to minimize the rate of stress increase, as longas the initial stress is sufficiently below the static yieldstress and �ttest tw, the exact rate of stress increasehas minimal effect on the results. We note that be-cause the stress is ramped continuously, the measuredflow curves do not necessarily correspond preciselyto steady-state measurements. However, steady-statemeasurements would generally be complicated by theaging behavior of the Laponite® matrix fluid (Fieldinget al. 2000), so that in this case the continuous ramptests provide a consistent and meaningful measure ofthe static yield stress at a particular age time when�ttest/tw 1. To measure the dynamic yield stress,steady-state flow tests are performed in which the shearrate is decreased logarithmically in discrete steps fromγ̇ = 100 s−1 to γ̇ = 0.05 s−1. Starting from higher shearrates has negligible effect on the extracted values ofthe dynamic yield stress. Because the aging behavior ofthe matrix fluid can lead to continually evolving prop-erties, a relatively lenient criteria for reaching steady-state is used (two consecutive 3-second measurementsgiving results within 5% of each other), so that thetime required for a test remains small compared to theage time (about 1 min). The dynamic yield stress isobtained by extrapolating the measured stress values toγ̇ = 0 s−1 (Nguyen and Boger 1992).

Results and discussion

Effects of magnetic field and aging

In Newtonian matrix fluids, chain models for electro-and magnetorheology predict that the yield stress willincrease quadratically with the magnetic field B for lowfield strengths when the magnetic particles are in thelinear magnetization regime (Klingenberg and Zukoski1990). As the magnetic field increases and the parti-cles exhibit nonlinear magnetization, the yield stress ispredicted to scale as B3/2 and eventually become inde-pendent of B as the particle magnetization saturates(Ginder et al. 1996). These scalings have beenconfirmed experimentally for spherical particles (Chinet al. 2001; de Vicente et al. 2010), though somestudies have reported a somewhat weaker dependenceon B (Bossis et al. 2002; Bossis and Lemaire 1991).For MR suspensions stabilized by single-walled carbonnanotubes, Fang et al. (2009) reported a transition in

584 Rheol Acta (2012) 51:579–593

the scaling of the yield stress from B2 to B3/2 withincreasing magnetic field. Similar observations havebeen made in some ER fluid systems (Zhang et al.2010). In matrix fluids composed of viscoplastic grease,Rankin et al. (1999) reported that the field-induceddynamic yield stress scales with Bx, where x decreasesfrom about 1.5 to 0 as the magnetic field increases andthe magnetization of the particles saturates. The valueof x = 1.5 in grease-based matrix fluids for magneticfields below the saturation regime (about 0.05 T to0.35 T) has also been reported by other authors (Parket al. 2001, 2011).

Flow curves from continuous stress ramp tests undermagnetic fields are shown in Fig. 3. The sample consistsof 10 v% CIP (47 wt% CIP) in a 3.0 wt% aqueousLaponite® dispersion at an age time of tw = 10 min.The constant age time ensures consistent matrix fluidproperties for each measurement. For each value of themagnetic field, very small shear rates are observed untila critical shear stress is exceeded, after which the shearrate abruptly increases by several orders of magnitude.This behavior is a definitive characteristic of field-activated yield stress fluids, and indicates a breaking ofthe field-induced microstructure at the critical appliedstress, which corresponds to the static yield stress, τ ys.Larger values of this critical stress are observed as themagnetic field is increased; at B = 1.0 T, the materialcan support stresses about four to five times higher thanat B = 0.2 T without yielding. Beyond about B = 0.6 T,

10-2 10-1 100 101 102 103

103

104

0.20 T0.40 T0.60 T

0.80 T1.00 T

Fig. 3 Typical flow curves from continuous stress ramp testsat various applied magnetic fields. Lines connect data points toguide the eye. The qualitative trends in the data above, whichcorrespond to a 10 v% CIP suspension in a 3.0 wt% aqueousLaponite® dispersion at an age time of 10 min, are representativeof all CIP concentrations and age times. At a given magneticfield, the shear rate is negligibly small up to a critical shearstress that corresponds to the field-induced static yield stress,τ ys. Further increasing the applied stress results in the shear rateincreasing by several orders of magnitude. Note that since thestress is ramped continuously, the data above do not necessarilycorrespond directly to steady-state measurements

the response of the material to applied stress changesminimally with increasing magnetic field, which is a re-sult of the magnetic saturation of the particles at thesehigher field strengths. While the measured shear ratesin these experiments may not necessarily correspondprecisely to steady-state measurements, as describedabove, by maintaining consistent experimental condi-tions it is possible to extract meaningful and repeatablevalues of the field-induced static yield stress, τ ys, whichis determined as the critical stress above which anabrupt increase in the shear rate is observed (Nguyenand Boger 1992).

The field-induced static yield stress for 10 v% CIPis shown as a function of the magnetic field strength,B, in Fig. 4a. The age time for the 3.0 wt% aqueousLaponite® matrix fluid is again kept constant at tw =10 min. Error bars represent the standard deviationbetween measurements on three different samples, pro-viding an indication of the reproducibility of the mea-surements. For applied fields of 0.1 T to 0.5 T, τ ys

grows approximately linearly with B. As the magneticfield is increased further, a plateau is observed so thatfor fields above about B ≈ 0.6 T, τ ys is approximatelyindependent of the magnetic field. This regime is againindicative of the magnetic saturation of CIP particles.Before the onset of magnetic saturation, the magneticfield has a strong effect on the rheology; τ ys increasesby about an order of magnitude from B = 0.1 T to B =1.0 T. These trends are qualitatively similar to thosereported in Newtonian matrix fluids (Ginder et al.1996). Though the regime of quadratic dependence onB predicted for Newtonian matrix fluids has not beenobserved, this is likely because the particles alreadybegin to exhibit nonlinear magnetization effects at B =0.1 T (see Fig. 1c).

A unique aspect of the aqueous Laponite® matrixfluid used in the present study is its aging behavior. Leftquiescent, the rheological properties of the Laponite®dispersion evolve with time as individual clay particlescoordinate and an arrested microstructure developsin the material (Ruzicka and Zaccarelli 2011). Agingresults in growth of both the yield stress and the matrixviscoelasticity (Rich et al. 2011a, b), and generally leadsto a more solid-like material. Figure 4b shows the effectof this aging in the matrix fluid on the magnetorheologyof the 10 v% CIP suspension. Squares represent thestatic yield stress of the 3.0 wt% Laponite® matrix fluidwith no added CIP, which grows steadily with age timeas expected. Without a magnetic field, adding CIP tothe matrix fluid raises the static yield stress by at most60% (at 30 v% CIP), which is small compared to thefield-induced gain in the yield stress. For an appliedfield of B = 0.1 T, a small increase in the field-induced

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0.0 0.2 0.4 0.6 0.8 1.00

2000

4000

6000

8000

10000

12000

B [T]

a)

0 20 40 60 80 100 120

102

103

104

tw [min]

no CIP

0.1 T

0.5 T

0.3 T

b)

Fig. 4 Field-induced static yield stress, τ ys, of a 10 v% CIPsuspension in a 3.0 wt% aqueous Laponite® dispersion. a Field-induced static yield stress as a function of the applied magneticfield, B, at an age time of tw = 10 min (matrix fluid staticyield stress, τ ys,0 ≈ 85 Pa). Error bars represent the standarddeviation of measurements on three different samples. At lowfield strengths, τ ys increases with the applied magnetic field, B.Saturation of the yield stress is observed for B greater than about0.6 T, due primarily to the saturation of the particle magnetiza-tion (see Fig. 1c). τ ys grows by more than an order of magnitudefrom B = 0.1 T to B = 1.0 T. b Static yield stress as a function ofage time, tw , for 3.0 wt% aqueous Laponite® dispersions with noadded CIP and with 10 v% CIP at B = 0.1, 0.3, and 0.5 T. Whilethe matrix fluid yield stress increases with age time, the yieldstress of the composite is essentially independent of age time (i.e.,independent of the matrix fluid yield stress) for all but the lowestmagnetic field. The addition of CIP and magnetic fields as low asB = 0.1 T results in an order of magnitude increase in the staticyield stress over that of the matrix fluid alone

static yield stress is observed between tw = 10 min andtw = 120 min. For larger magnetic fields, however, τ ys

becomes essentially independent of age time, indicatingan insensitivity to matrix fluid properties. Two orders ofmagnitude separate the values of τ ys for the pure matrixfluid and data for a 10 v% dispersion at B = 0.5 T,which again highlights the strong effect of the magneticfield on the rheology.

The independence of τ ys on the matrix fluid rheo-logical properties for moderate to high magnetic fieldstrengths can be understood by considering the relativemagnitude of the different stresses acting on the CIPparticles. Specifically, and in analogy with the approach

of Rankin et al. (1999), the matrix fluid yield stress canbe compared to the inter-particle dipolar stress result-ing from the applied magnetic field. If mutual magneticinduction is neglected so that all the particles are as-sumed to have the same constant dipole moment, theinteraction energy Uij between two spherical dipoleswith centers separated by a distance rij and subject toa uniform external magnetic field is

Uij = m2μ0

4π

(1 − 3 cos2 θ

r3ij

)(1)

where m is the dipole moment, μ0 is the magneticpermeability of the medium (assumed to be equal tothe permeability of free space), and θ is the angle thatthe line connecting the particle centers makes with thedirection of the applied magnetic field. The attractionforce between the particles is maximum when theircenters are aligned with the field (θ = 0). In this case,the magnitude of the force is

Fij∣∣θ=0 =

∣∣∣∣−dUij (θ = 0)

drij

∣∣∣∣ = 3m2μ0

2πr4ij

(2)

This expression can be used to find a characteristicforce by setting rij to the particle diameter, d, which isthe minimum distance between particle centers. A char-acteristic magnetic force between particles is therefore

Fchar = π

24d2μ0 (ρM)2 (3)

Here the dipole moment has been expressed as m =(π/6) d3ρM where ρ is the particle density and M is themagnetization per unit mass. A characteristic magneticstress can be found by dividing Eq. 3 by the surface areaof a spherical particle πd2

τchar = μ0 (ρM)2

24(4)

Comparing the characteristic values obtained fromEq. 4 to the static yield stress of the matrix fluid, τ ys,0,which must be overcome for the particles to move andform a chain-like structure in response to the imposedfield, provides insight into the effect of the matrixfluid on the MR response. This balance of stresses ischaracterized by the following dimensionless group

Y∗M = μ0 (ρM)2

24τys,0(5)

This parameter is similar to the so-called “mag-netic yield parameter” introduced by previous authors(Rankin et al. 1999). Generally, if Y∗

M > 1 chain-likestructures will form under the action of an externalmagnetic field and a bulk MR response will be ob-served, whereas if Y∗

M 1, the yield stress of the matrix

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fluid prevents structure formation. For the magneticCIP particles used in the present experiments, the den-sity of iron is ρ ≈ 7.8 g cm−3, and the magnetization datain Fig. 1c shows that for B ≥ 0.2 T, M ∼ 100 emu g−1 =100 A m2 kg−1. Figure 4b shows that the static yieldstress of the matrix fluid is on the order of 100 Pa. Usingthese numerical values in Eq. 5 gives Y∗

M ≈ 320 � 1.Therefore, the characteristic magnetic stress betweenparticles is much greater than the matrix fluid yieldstress. As a result, it is reasonable to expect that moder-ate changes in the matrix fluid yield stress during agingwill have minimal effect on the magnetorheologicalresponse. For the case of B = 0.1 T, it is helpful toreturn to Eq. 2 and recognize that the average distancebetween particle centers is dependent on the volumefraction of CIP, φCI P. Treating the CIP suspension as ahomogeneous dispersion of monodisperse spheres, thisvolume fraction dependence can be accounted for inan approximate way by replacing rij with dφ

−1/3CI P rather

than simply d. Carrying this change through Eqs. 3–5results in a modification to Y∗

M

Y∗M,φ = μ0 (ρM)2

24τys,0φ

4/3CI P (6)

Applying data from Figs. 1c and 4b for a 10 v% CIPdispersion at B = 0.1 T and a matrix fluid age timeof tw = 120 min results in a value of Y∗

M,φ ≈ 4. Thefact that Y∗

M,φ is close to 1 in this case implies thatthe characteristic magnetic stress on particles exceedsthe matrix fluid yield stress by only a small amount.The magnetorheological response of the composite istherefore expected to reflect a combination of the struc-tures formed by magnetic particles as well as the matrixfluid rheology. This is consistent with the observationthat the matrix fluid and the 10 v% CIP composite atB = 0.1 T exhibit similar rates of growth in the staticyield stress during aging. For reference, in the case ofa 10 v% CIP dispersion and a matrix fluid age time oftw = 120 min, Y∗

M,φ ≈ 15 at B = 0.3 T, so that matrixfluid effects are again expected to be minimal for thishigher magnetic field strength.

An important complication in microstructured fluidsis that rheological properties measured at the bulk scaleoften do not entirely reflect behavior and properties atthe microscopic scale (Waigh 2005; Liu et al. 2006). Inthe present experiment, suspended CIP particles havean average diameter of about 3.7 μm (see Fig. 1b),so the yield stress of the matrix fluid measured viabulk rheology may not be representative of the matrixfluid yield stress at the length scale of the magneticmicroparticles. This effect could result from pores orother microstructures in the matrix fluid that have

similar length scales as the CIP particles. This questionwas addressed in a previous communication (Rich et al.2011a) in which bulk yield stress values were com-pared to nonlinear microrheology magnetic tweezermeasurements in aqueous Laponite® dispersions. Theprobes for microrheology experiments were superpara-magnetic spheres of diameter 4.5 μm, which is similarto the average size of CIP particles in the currentwork. For Laponite® concentrations greater than orequal to about 2.0 wt%, bulk and micro-scale yieldstress measurements were found to agree quantitativelyso long as differences in the flow kinematics for thetwo experiments are correctly taken into account. Thiskinematic correction consists of an order one factor thatapproximately captures the shear contribution to theapplied stress in the micro-scale experiment involvingspherical probe particles. Since the Laponite® concen-tration in the present work is 3.0 wt%, and since theCIP particles are irregular in shape (see Fig. 1a), weneglect this correction factor and consider the matrixfluid yield stress measured via bulk experiments also tobe representative of that on the length scale of the CIPparticles. Further, since the value of the matrix fluidyield stress is used primarily to gain physical insight byevaluating the dimensionless yield parameters definedin Eqs. 5 and 6, an order one correction factor will havenegligible effect on conclusions.

The observation that the field-induced static yieldstress is largely independent of the matrix fluid staticyield stress is consistent with the results of Rankinet al. (1999), who showed similar behavior for thefield-induced dynamic yield stress of CIP suspensionsin viscoplastic greases. Because of the nature of dy-namic yield stress measurements, in which an initial ap-plied shear rate increases the probability that magneticparticles will encounter each other and form chains,and because the measured field-induced dynamic yieldstresses exceeded the matrix fluid yield stresses by 2 to3 orders of magnitude, it is to be expected that the MRdynamic yield stresses measured in the work of Rankinet al. (1999) would be relatively independent of matrixfluid properties. For field-induced static yield stressmeasurements, however, the matrix fluid propertiescan play a more significant role because the magneticparticles must overcome the matrix fluid yield stress inorder to form the chain-like structure and provide anMR response. It is therefore unclear a priori whetherfield-induced static yield stress measurements in yieldstress matrix fluids would exhibit a similar insensitivityto matrix fluid properties as in the case of dynamic mea-surements. From the perspective of formulators of MRfluids, independence of the field-induced yield stresson matrix fluid rheology is likely to be an attractive

Rheol Acta (2012) 51:579–593 587

and advantageous property. As long as the matrix fluidyield stress is sufficient to prevent magnetic particlesedimentation, the exact rheological properties of thematrix fluid have little bearing on the field-responsiverheology. Therefore, the matrix fluid rheology can beoptimized to meet various off-state needs or designedfor other functionalities with little concern for how theactivated material will behave. Care must be taken toaccount for matrix fluid rheological properties, how-ever, at low field strengths and large matrix fluid yieldstresses, as has been discussed. The following section,which examines the role of magnetic particle concen-tration, shows that effects of the matrix fluid yield stressmust also be taken into consideration at low volumefractions of magnetic particles, as implied by Eq. 6.

Effect of magnetic particle concentration

Increasing the concentration of magnetic particles gen-erally enhances the rheological response of MR fluidsto an applied magnetic field (Goncalves et al. 2006).For this reason, volume fractions in commercial appli-cations are often as high as 40 to 50 v% (Jolly et al.1999), despite the fact that increased concentrations ofmagnetic particles also result in an elevated off-stateviscosity. Established models predict that the field-induced yield stress and viscoelastic storage modulus(Ginder et al. 1996), as well as the viscosity (Martin andAnderson 1996), will exhibit a linear dependence onthe volume fraction of magnetic particles (de Vicenteet al. 2011a). While experimental results for the field-induced dynamic yield stress have corroborated thislinear relationship up to surprisingly high concentra-tions in both Newtonian and non-Newtonian matrixfluids, a super-linear increase with volume fraction hasbeen observed above about φCIP = 0.2 (Felt et al.1996; Rankin et al. 1999; Chin et al. 2001; Volkovaet al. 2001). This behavior is thought to result from theformation of thick columnar structures, as opposed tothe single particle-width chains that dominate at lowconcentrations.

In Fig. 5 we show the dependence of the field-induced yield stress on the magnetic field strength,B, for suspensions with different volume fractions ofCIP. The matrix fluid is a 3.0 wt% aqueous Laponite®dispersion at a constant age time of tw = 10 min (τ ys,0 ≈85 Pa). Filled symbols and solid lines represent thestatic yield stress, τ ys, while open symbols and dashedlines represent the dynamic yield stress, τ yd, which hastypically been reported by previous authors for MRcomposites in yielding matrix fluids such as greases(Rankin et al. 1999; Park et al. 2011). For all CIPconcentrations, both yield stress measures grow with

0.0 0.2 0.4 0.6 0.8 1.0

102

103

104

105

B [T]

φφφ

φφφ

Fig. 5 Field-induced yield stress as a function of magnetic fieldfor various CIP volume fractions, φCI P, in a 3.0 wt% aqueousLaponite® dispersion at an age time of tw = 10 min. Both thestatic yield stress, τ ys (f illed symbols and solid lines), and thedynamic yield stress, τ yd (open symbols and dashed lines), areshown. For all CIP concentrations considered in the presentstudy, both measures of the yield stress follow a similar trendwith the applied magnetic field, growing with B and exhibiting aplateau above about 0.6 T. Good agreement is observed betweenthe field-induced static and dynamic yield stresses for all but thelowest CIP concentration at low magnetic fields

the magnetic field up to about B = 0.6 T, beyondwhich a plateau is observed. This trend is generallyconsistent with previous measurements of τ yd in yieldstress matrix fluids (Rankin et al. 1999), though theplateau in Fig. 5 begins at slightly higher field strengths(most likely due to a different size and grade of CIP).For the largest volume fraction examined in the presentwork, field-induced static and dynamic yield stresses upto about 50 kPa are observed. Because the matrix fluidyield stress is significantly larger than the gravitationalstress acting on particles and continually grows as thedispersion ages, sedimentation is prevented essentiallyindefinitely for all CIP concentrations examined. Basedon these results, it is reasonable to anticipate that grav-itationally stable dispersions with higher field-inducedyield stresses could be achieved by further increasingthe CIP concentration to 40 or 50 v%.

With the exception of results for the lowest CIPconcentration at low applied fields, the two measuresτ ys and τ yd are in good agreement, indicating that field-induced thixotropy is negligible. This is despite thethixotropic nature of the aqueous Laponite® matrixfluid; at tw = 10 min, τ ys,0 ≈ 85 Pa while the matrixfluid dynamic yield stress is measured to be τ yd,0 ≈20 Pa. The deviation between τ ys and τ yd at φCI P =0.01 and low applied magnetic fields is most likelydue to the inability of some dispersed magnetic par-ticles to overcome the matrix fluid yield stress and

588 Rheol Acta (2012) 51:579–593

form gap-spanning chains during the static yield stressmeasurement (Rich et al. 2012). In this dilute disper-sion, for which the average distance between particlesis relatively large, inter-particle attractive forces at lowexternal fields may be insufficient to overcome thematrix fluid yield stress. Quantitatively, for τ ys,0 =85 Pa, φCI P = 0.01, and B = 0.2 T (M ≈ 125 emu g−1,see Fig. 1c), inserting parameters into Eq. 6 gives avolume-fraction corrected yield parameter of Y∗

M,φ =1.3. Values close to unity indicate that the matrix fluidyield stress approximately balances attractive forcesbetween particles resulting from dipolar interactions,hindering chain formation. In the dynamic yield stressmeasurement, however, bulk shear and rejuvenationof the matrix fluid disrupt and lower the matrix yieldstress, enabling viscous flow and increasing the likeli-hood that particles will aggregate and form chains, asdiscussed above. Therefore, deviations between τ yd andτ ys are reasonable in this dilute regime at low magneticfield strengths. In particular, the observation that τ yd >

τ ys, in contrast to the case of the pure matrix fluid, isconsistent with the mechanism described above.

Figure 5 shows that the field-induced static anddynamic yield stresses increase substantially with CIPconcentration. This behavior is specifically highlightedin Fig. 6a, where the yield stress results are plotted asa function of φCI P for constant values of the appliedmagnetic field. For clarity, only static yield stress mea-surements for three representative field strengths areshown (B = 0.1, 0.2, and 0.5 T), though the dynamicyield stress data is generally quantitatively similar, aspreviously discussed. At a given field strength, theyield stress increases by more than an order of mag-nitude from φCI P = 0.01 to φCI P = 0.30. A nearlylinear dependence on CIP volume fraction is observed,though a more general power-law relationship is mostappropriate

τys = KφaCI P (7)

The coefficient κ and the power-law exponent α

depend on the field strength. Least-squares fits to Eq. 7are shown by black lines. The ability of the power-lawform to characterize the data in Fig. 6a is representativeof the goodness of fit for other field strengths, andthe power-law fits provide a minimum coefficient ofdetermination of R2 = 0.96. The variation of the fittedparameters in Eq. 7 with the applied magnetic field isshown in Fig. 6b for both the static (filled symbols)and dynamic (open symbols) yield stresses. Squaresspecify the power-law exponent α on the left axis, andcircles give the coefficient κ on the right axis. κ valuesreflect the behavior of the field-induced yield stress,

0.01 0.1102

103

104

105

0.1 T

0.5 T

0.2 T

11

a)

0.0 0.2 0.4 0.6 0.8 1.00.6

0.8

1.0

1.2

1.4

1.6

104

105

B [T]

b)

φ

Fig. 6 a Variation of the field-induced yield stress with CIPvolume fraction in a 3.0 wt% aqueous Laponite® dispersion atan age time of tw = 10 min. For clarity, only static yield stressdata at three representative field strengths are shown here, butboth τ ys and τ yd exhibit a similar power-law dependence onφCI P for all field strengths. The magnitude of τ ys (as well as τ yd)

increases by almost two orders of magnitude from φCI P = 0.01 toφCI P = 0.30. In b, the power-law exponents, α, and coefficients,κ , resulting from least-squares fitting to Eq. 7 (for which theminimum coefficient of determination is R2 = 0.96), are shownfor the field-induced static (f illed symbols) and dynamic (opensymbols) yield stresses. Squares represent the exponent α, givenon the left axis, while circles represent the front factor κ , given onthe right axis. The coefficient κ grows with the magnetic field ina manner that reflects the field dependence of the yield stress, asshown in Fig. 5. Both sets of power-law exponents increase fromα ≈ 0.75 at B = 0.1 T (sub-linear dependence on φCI P) to α ≈ 1.15± 0.06 for B greater than about 0.5 T (super-linear dependenceon φCI P)

increasing by almost two orders of magnitude from B =0.1 T to B = 1 T and exhibiting saturation above aboutB = 0.6 T. Additionally, κ values fitted from static anddynamic yield stress data are in good agreement. Whileall α values are close to unity, signifying a nearly lineardependence of the yield stress on CIP volume fractionas mentioned above, there is a clear trend in whichα increases from α ≈ 0.75 at B = 0.1 T to α ≈ 1.15above about B = 0.5 T. This indicates that the yieldstress increases sub-linearly with φCI P for low fieldstrengths, and super-linearly above about B = 0.5 T.The power-law exponents for τ ys and τ yd are in goodagreement, deviating by less than 15%. The sub-linear

Rheol Acta (2012) 51:579–593 589

volume fraction dependence of the field-induced yieldstress observed here at low field strengths is in contrastto model predictions (Ginder et al. 1996) and previousexperimental results (Felt et al. 1996) for Newtonianmatrix fluids, which show a linear dependence on mag-netic particle concentration for low field strengths anddilute suspensions. Additionally, previous studies ofMR composites in a viscoplastic grease have reported alinear dependence on the volume fraction for B ≈ 0.05–0.2 T and φCI P = 0.02–0.25 (Rankin et al. 1999). Thesediscrepancies are again most likely related to the bal-ance between inter-particle magnetic stresses and thematrix fluid yield stress (Rich et al. 2012). For example,for B = 0.1 T, Y∗

M,φ grows from 0.5 to 50 as φCI P is in-creased from 0.01 to 0.30, implying that measurementswill reflect a relative contribution of the matrix fluidyield stress that diminishes as the CIP concentration isincreased. The fact that τ ys remains close to the matrixfluid yield stress (τ ys,0 = 85 Pa) for B = 0.1 T at low CIPconcentrations is further evidence for the effect of thematrix fluid. As the CIP concentration increases andY∗

M,φ becomes much greater than unity, the magneticresponse is expected to dominate the matrix fluid yieldstress. Fitting Eq. 7 to data spanning this range of Y∗

M,φ

values results in a power-law exponent that averagesthe behavior in these two regimes and indicates a sub-linear dependence of the field-induced yield stress onCIP volume fraction. We note that the volume-fractioncorrected yield parameter has values Y∗

M,φ ≥ 5 underthe conditions examined in the work of Rankin et al.(1999), assuming similar magnetization properties asreported in Fig. 1c.

Figure 5 shows that the field-induced yield stresses ofMR composites with different CIP concentrations ex-hibit similar trends with magnetic field strength, despitedifferences in the magnitude of the yield stress. Thisobservation motivates the question of whether the datacan be shifted to generate one master curve relatingthe field-induced yield stress to the magnetic field fordifferent values of φCI P. Such a master curve is shownin Fig. 7a, where the field-induced static yield stressdata from Fig. 5 is shown as a reduced yield stress,bτ ys, plotted as a function of a reduced magnetic fieldstrength, aB B. By employing the horizontal and verticalshift factors aB and b , respectively, yield stress data fordifferent CIP concentrations has been collapsed ontoa single master curve that increases with the magni-tude of the reduced magnetic field. The data has beenshifted to a reference concentration of φCI P = 0.10. Thefact that such a master curve can be generated sug-gests the systems are dynamically self-similar. Whereasthe behavior of each system is governed by the sameunderlying physical mechanisms, the process is am-

baB

0.01 0.10.1

1

10

a B, b

[-]

b)

0.1 1

103

104

aBB [T]

slope = 1

slope = 2

a)

φφφφφφ

φ

Fig. 7 a Master curve showing the reduced field-induced staticyield stress as a function of the reduced magnetic field strengthin a 3.0 wt% Laponite® matrix fluid at tw = 10 min. Static yieldstress measurements for different CIP concentrations are shiftedto a reference concentration of φCI P = 0.10 by the horizontal andvertical shifting factors aB and b , respectively. As the magnitudeof the reduced magnetic field increases, the logarithmic slope ofthe master curve decreases from 2 to 1 and eventually exhibits aplateau. The only exception is the data at 1 v% CIP, which doesnot appear to follow the same trend as higher CIP concentrations.b Horizontal (aB) and vertical (b) shift factors for the datapresented in (a) as a function of the volume fraction of CIP, φCI P.Both sets of shift factors follow a power-law dependence on φCI P;least-squares fitting results in the relationships aB ≈ 0.56φ−0.26

CI Pand b ≈ 0.05φ−1.33

CI P

plified to a different extent at each concentration. Thelogarithmic slope of the collapsed data is approximately2 for aB B ≤ 0.2 T, decreases to approximately 1 (lineardependence) in the range 0.3 T ≤ aB B ≤ 0.6 T, andsubsequently exhibits a plateau. A notable outlier isthe data at φCI P = 0.01 and the highest field strengths,which does not follow quite the same trend as thedata for all higher CIP concentrations examined in thepresent work. Close inspection of this data shows thatthe field-induced static yield stress for φCI P = 0.01 doesnot exhibit a plateau until about B ≥ 0.8 T (see Fig. 5),as opposed to the plateau observed at about B ≥ 0.6 Tfor higher CIP concentrations that corresponds to satu-ration of the particle magnetization. The reason for thisdelayed plateau is unclear, but our current hypothesisis that this anomaly is likely an additional result of the

590 Rheol Acta (2012) 51:579–593

significant matrix fluid yield stress that hinders chainformation at this dilute CIP concentration for lowermagnetic fields (i.e., Y∗

M,φ ∼ 1), as previously discussed.As the magnetic field is increased, some particles thatare restrained by the matrix fluid at lower magneticfields eventually experience sufficient dipolar forces toovercome the matrix fluid yield stress. This effect maybe sensitive to small changes in the average particlemagnetization near saturation, but is not expected to besignificant for φCI P ≥ 0.02. This behavior could resultin the field-induced yield stress exhibiting a plateau athigher applied magnetic fields at φCI P = 0.01 than forhigher CIP concentrations, leading to the poor collapseobserved in Fig. 7a.

The variation of the shift factors with the volumefraction of CIP is shown in Fig. 7b. While aB variesrelatively little with φCI P, the vertical shift factor b de-creases by about two orders of magnitude from φCI P =0.01 to φCI P = 0.30, reflecting the order of magnitudechanges in the field-induced static yield stress overthis range of concentrations. That is, τ ys values mustbe shifted up at low CIP concentrations and downat high CIP concentrations in order to generate themaster curve. Both shift factors follow a power-lawdependence on φCI P over the range investigated inthe present work. Least-squares fitting to a power-lawform leads to the expressions aB ≈ 0.56φ−0.26

CI P and b ≈0.05φ−1.33

CI P (0.01 ≤ φCI P ≤ 0.30). The master curve inFig. 7, combined with these expressions for the shift fac-tors, can be used to predict the dependence of the field-induced yield stress on the applied magnetic field atCIP concentrations within the range 0.01 ≤ φCI P ≤ 0.30,and to reasonably extrapolate to higher concentrations.

The master curve in Fig. 7 relates the field-inducedyield stress at various CIP concentrations to a macro-scopic, externally-set parameter, the applied magneticfield B. Because different types of magnetic particlesexhibit different magnetization responses to appliedmagnetic fields, the behavior shown in Fig. 7 is expectedto apply strictly for the particular grade of CIP particlesused in the present study. A more general master curvecan be developed, however, by considering the depen-dence of the field-induced yield stress on the averageparticle magnetization, M, which is an internal variablethat characterizes the magnetic response on the particlelevel (Klingenberg et al. 2007). The magnetization canthen be related to the applied field via a magnetizationcurve, as in Fig. 1c. The characteristic inter-particlemagnetic stress, τchar = μ0 (ρM)2 /24, which was intro-duced in Eq. 4, is a physically significant quantity that isset by the average particle magnetization. A correlationor master curve relating the field-induced yield stressand τ char would be applicable for a wide range of mag-

0.01 0.10.1

1

a M [-

]

b)

0 50 100 150 20010-1

100

101

102

a)

φ

φφφφφφ

Fig. 8 a Alternative master curve for field-induced static yieldstress data at various CIP concentrations in a 3.0 wt% Laponite®matrix fluid at tw = 10 min. Here τ ys is plotted as a functionof a reduced characteristic magnetic stress between particles,aMτ char , where aM is a shift factor and τ char is the characteristicinter-particle magnetic stress given in Eq. 4 that is a functionof the average particle magnetization per unit mass, M. Data isagain shifted to a reference concentration of φCI P = 0.10. Sinceτ char accounts for the magnetization properties of the suspendedparticulate phase, the above plot is expected to be more gen-erally applicable for different types of magnetic particles thanthe master curve in Fig. 7a and amounts to a concentration–magnetization superposition. In b, the dependence of the shiftfactor on CIP concentration is shown. A power-law provides areasonable fit (R2 = 0.98), and least-squares fitting results in therelationship aM ≈ 4.9φ0.77

CI P

netic particles because it would be independent of theexact relationship between B and the average particlemagnetization, M.

Figure 8a shows such an alternative master curverelating the field-induced static yield stress to a reducedcharacteristic inter-particle magnetic stress, aMτ char,where aM(φCI P) is the magnetization–volume fractionshift factor. The fact that shifting is only required onone axis to generate this alternative master curve sug-gests that the interactions between particles in field-induced chain structures are effectively scaled and char-acterized by τ char (i.e., by the particle magnetization).As in Fig. 7, yield stress data are again shifted toa reference concentration of φCI P = 0.10. Data fordifferent values of φCI P are successfully collapsed, once

Rheol Acta (2012) 51:579–593 591

again with the sole exception of the high magneticfield results for φCI P = 0.01 as was discussed for theshifting in Fig. 7a. These results effectively amount to aconcentration–magnetization superposition. The field-induced static yield stress, τ ys, increases with the mag-nitude of the reduced characteristic inter-particle stress,exhibiting approximately exponential growth at largevalues of aMτ char (appearing linear on semi-log axes).For aMτ char ≥ 30 kPa the argument of the exponentialis about (53 kPa)−1, as shown by the black dotted line.The shift factor aM increases with the CIP volume frac-tion, and we show in Fig. 8b that the relationship canbe well-approximated as a power-law. Least-squaresfitting results in the expression aM ≈ 4.9φ0.77

CI P (0.01 ≤φCI P ≤ 0.30), where the coefficient of determination isR2 = 0.98. In summary, this alternative master curvefor the field-induced static yield stress τ ys in the rangeaMτ char ≥ 30 kPa can be approximately represented as

τys ≈ A exp(aMτchar

τ ∗)

(aMτchar ≥ 30 kPa) (8)

where A ≈ 1.1 kPa, aM is the shift factor given byaM ≈ 4.9φ0.77

CI P, τ ∗ ≈ 53 kPa, and τ char is the character-istic inter-particle magnetic stress μ0 (ρM)2 /24, whichis related to the average particle magnetization per unitmass M. This master curve provides a compact expres-sion for design applications in which the magnitudeof the yield stress must be predicted for a given fieldstrength, volume fraction, and particle magnetization.The fact that such a master curve can be generatedindicates that the field-induced yield stress in these MRfluids arises from a common physical mechanism thatacts over a range of conditions, and that this mecha-nism depends similarly on particle magnetization andconcentration. Additionally, the superposition demon-strated in Fig. 8 reinforces the suggestion that higherfield-induced yield stresses can be achieved at a givenvolume fraction by employing particles with a highersaturation magnetization.

Conclusions

The dramatic field-responsive rheological behavior ofmagnetorheological (MR) fluids, which results from thefield-induced chaining of iron microparticles suspendedin a matrix fluid, has been successfully employed in thedevelopment of numerous field-activated, “smart” softmaterials. The stability of MR fluids against particlesedimentation remains an important concern, however,especially in applications where re-dispersion after longoff-state times is unfeasible. One proposed solution tothis problem is the use of yield stress matrix fluids, and

previous authors have investigated the field-induceddynamic yield stress of MR composites in viscoplasticmatrix fluids. In the current work, analogous studiesof the field-induced static yield stress have been per-formed in MR suspensions in an aging, yield stressmatrix fluid. MR composites were formulated fromCIP and a matrix fluid consisting of an aqueous dis-persion of Laponite® clay, which is known to exhibita yield stress that grows as the material ages. As aresult, sedimentation of CIP is prevented essentiallyindefinitely. Using a custom-built magnetorheometryfixture, the field-induced static yield stress of this MRcomposite was studied as a function of the applied mag-netic field strength, B, the CIP volume fraction, φCI P,and the age time, tw. Results were used to generatea magnetorheological master curve (Fig. 8) that indi-cates a concentration–magnetization superposition andallows prediction of the field-induced yield stress fordifferent types and volume fractions of magnetic par-ticles under a wide range of conditions. A new dimen-sionless parameter, Y∗

M,φ , was defined (Eq. 6), whichrelates the magnitude of the matrix fluid yield stressto the characteristic inter-particle magnetic attractiveforces at a given particle concentration. For Y∗

M,φ � 1,inter-particle magnetic forces dominate and the field-induced rheology is found to be independent of thematrix fluid yield stress. From a practical perspective ofMR formulations, this behavior implies that as long asY∗

M,φ � 1, the rheology of the yield stress matrix fluidcan be optimized to meet other design demands with-out significantly disrupting the behavior of the field-activated material. Conveniently, the condition Y∗

M,φ �1 is frequently satisfied at the high field strengths andparticle concentrations used in most commercial MRapplications.

While the present study has focused solely on shearmagnetorheology, the need for quantitative under-standing of MR fluids in squeeze flow has recentlybeen highlighted (de Vicente et al. 2011a, b). Becauseyield stress matrix fluids could play a similar rolein preventing particle sedimentation in squeeze flowMR devices, an important question for future workis whether the presence of a matrix fluid yield stresshas significant effects on the field-induced squeeze flowrheology of MR composites. An additional interestingproblem for future studies would be to focus on someof the anomalies documented in the current work at lowvolume fractions of magnetic particles. While this diluteregime has limited appeal for traditional MR fluid ap-plications because of the relatively small field-inducedyield stresses, the data presented here is suggestive ofpotentially interesting new regimes and phenomena,which do not appear to have been explored yet. Novel

592 Rheol Acta (2012) 51:579–593

non-traditional applications could be inspired throughan improved understanding of systems at Y∗

M,φ ∼ 1, forwhich inter-particle attractive forces are approximatelyin balance with the matrix fluid yield stress. Numericalsimulations could aid in elucidating the dynamics andequilibrium microstructures of dipolar particles underthese conditions (Rich et al. 2012).

The results presented here will aid designers of MRdevices and guide formulators of MR suspensions inthe choice of appropriate viscoplastic matrix fluids. Themaster curves, correlations, and scaling relationshipsdescribed in the current study characterize the field-induced static and dynamic yield stress of an MR fluidthat is essentially indefinitely stable to sedimentation.This behavior is especially attractive for applicationssuch as active earthquake dampers or field-responsivedrilling fluids, for which re-suspension of a dense, con-centrated particle phase after long off-state times istypically unfeasible.

Acknowledgements Acknowledgement is made to the Donorsof the American Chemical Society Petroleum Research Fund(ACS-PRF Grant No. 49956-ND9) for financial support of thisresearch. The authors are especially grateful to Dr. Murat Ocalanfor assistance and many helpful discussions regarding the custom-built magnetorheometry fixture. Further acknowledgement isgiven to Ki Wan Bong, Dr. Matthew Helgeson, and Dr. DongHun Kim for help with SEM imaging, particle size characteriza-tion, and Magnetometer measurements, respectively.

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