PHYSICAL REVIEW E SEPTEMBER 1998VOLUME 58, NUMBER 3

Particle dynamics in magnetorheological suspensions using diffusing-wave spectroscopy

Eric M. Furst and Alice P. Gast*Department of Chemical Engineering, Stanford University, Stanford, California 94305-5025

~Received 12 May 1998!

Magnetorheological suspensions acquire dipole moments in an external magnetic field. Suspensions of thesedipolar particles rapidly aggregate to form long chains. We use diffusing-wave spectroscopy~DWS! anddensity-matched superparamagnetic emulsion droplets to probe the short-wavelength motion of the chains. Themeasured particle displacements are independent of the magnitude of the dipolar interactions at short times, butcharacterized by a constrained, subdiffusive motion at long times that slows as the dipolar interactions areincreased. Our observations show good qualitative agreement with Brownian dynamics simulations of dipolarchains.@S1063-651X~98!08509-2#

PACS number~s!: 83.80.Gv, 83.20.Jp, 82.70.Dd

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I. INTRODUCTION

The unique field-induced rheological response of magtorheological~MR! fluids has made them the focus of cosiderable attention in the past decade@1–6#. MR fluids arecolloidal suspensions of paramagnetic particles in a nonmnetic fluid. These fluids, along with their electrical analogelectrorheological~ER! fluids, have been proposed as cotrollable fluids that are ideal for interfacing electronic cotrols with mechanical systems, such as dampers, clutcand brakes@7,8#. When exposed to an external magnefield, the particles in a MR suspension acquire dipole mments and aggregate to form chains in the field directiThe fluid structure depends on the volume fraction; dilsuspensions form weakly interacting single-particle chawhile when more concentrated, the particle chains cross-laterally into a dense network. The latter structure is respsible for the unique rheological properties of MR suspesions: the quick formation of a network in response toexternal field creates a rapid liquid-to-solid transition. Asifrom their practical applications, MR suspensions are of fdamental interest because they allow us to probe the sture and dynamics of a suspension of particles interactinga ‘‘tunable’’ anisotropic interaction. For instance, Zhan aco-workers recently applied paramagnetic colloids to stuthe effects of collective hydrodynamics on particle sediffusion, taking advantage of the ability to vary and factout direct particle interactions@9#.

Recently, we have studied microscopic properties, sucthe kinetics of chain growth and low-energy structures@10–12#. The suspension dynamics@13,14# are of interest since ihas been proposed that Landau-Peierls thermal fluctuaof dipolar chains could be responsible for long-range attrtions between chains@15#. Rigid dipolar chains producetransverse magnetic field that decreases approximatelexp(2r/a) wherer is the distance from the chain anda isthe dipolar particle radius. Thus, chains separated by atance greater than one diameter should interact weaklycontrast, taking into account thermal fluctuations producemean-squared field that decreases as a power law:A^H2&

*Author to whom correspondence should be addressed.

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Here we report our initial studies of the dynamics of Mfluid suspensions using diffusing-wave spectroscopy~DWS!.Our work utilizes particles that are density matched tosuspending solvent to minimize sedimentation. Previowork examining the fluctuations of dipolar chains used pamagnetic polystyrene beads which were at rest on a gslide @14#. The use of a density-matched system and ligscattering allows us to probe the dynamics of long chafluctuating fully in three dimensions, free of interfacial inteactions or hydrodynamic effects. Additionally, our appliction of DWS to study the dynamics of MR suspensionslows us to probe smaller length and time scales previouinaccessible through single scattering experiments or miscopic observation. Previous work by Ginder used simmethods to examine particle dynamics and structure fortion in ER suspensions@16#. Here the DWS experimentwere used to resolve the oscillatory electrophoretic motionparticles in an applied ac field along with an effective icrease in the diffusivity due to a concentrated network fmation; his aim was not to study the fluctuation spectrumdipolar chains. Also, a thorough understanding of partiinteractions in ER suspensions differs from that in MR spensions due to the effects of surface charge, adsorbed wand field inhomogeneities.

II. EXPERIMENT

We synthesized MR emulsion droplets followingmethod due to Bibette@17#. A ferrofluid ~Rhone-Poulenc!,composed of monodomain iron oxide particles suspendeoctane, is emulsified into water using sodium dodecyl sfate, SDS~Sigma, cmc52.351 g/ml). The rough emulsion ifractionated by seven successive depletion aggregationsSDS micelles. We vary the particle density by manipulatithe amount of octane in the ferrofluid; the particles in thstudy have a density of approximately 1.1 g/ml. We respend the particles in a D2O-SDS solution (r51.10 g/ml) atthe SDS cmc to minimize sedimentation effects in our ligscattering experiments. In this study we use particles ofproximately 240 nm diameter, at a volume fraction,f of

3372 © 1998 The American Physical Society

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PRE 58 3373PARTICLE DYNAMICS IN MAGNETORHEOLOGICAL . . .

0.005 and a magnetic susceptibilityx of 1.2.When placed in an external magnetic field, the partic

interact via an anisotropic dipolar potential

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whereu is the angle the particle centers form with the fiedirection, r is the distance between particle centers, andm5 4

3 pa3m0xH is the induced dipole for a particle of radiusain a field of magnitudeH. We characterize the dipolstrength with@18#

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Our experiments are conducted over a range of dipstrengths:l54, 7, 11, and 16.

The dynamics of our MR fluid are studied using a tranmission geometry@19#. The MR fluid is placed in a spectrophotometer cuvette~Spectrocell! with a path length L510 mm. The cuvette is centered in a uniform magnefield generated by Helmholtz coils. Linearly polarized ligfrom a 35 mW HeNe~Spectraphysics! at 632.8 nm is ex-panded to a plane wave with a 103 laser collimator andilluminates the face of the sample perpendicular to the fidirection. The transmitted diffuse light is collected by twpinholes coupled to a multimode optical fiber. A crospolarizer transmits only depolarized, highly multiply scatered light. The intensity fluctuations are measured usinphotomultiplier ~Thorn EMI! with a built-in amplifier-descriminator. We use a commercial correlator~BrookhavenInstruments, BI-9000AT! to calculate the intensity autocorelation,g(2)(t). The electric field autocorrelationg(1)(t), isfound using the Siegert relation,g(2)(t)5ug(1)(t)u221.

When the magnetic field is applied, the dipolar particrapidly aggregate to form chains. On longer time scales,chains experience lateral interactions and coalesce to fthicker columns. We are interested in the fluctuation-indulateral attraction causing this coalescence. Thus we deoped an experimental protocol to produce conditions whthefluctuations of individual chainsdominate the system dynamics. First, by minimizing the density difference betweour droplets and the suspending fluid, we reducesedimentation-induced aggregation of chains. For ournamic studies, the chains are first grown atl516 for 12 s,and then subjected to the desired field strength. This prova consistent suspension structure as a starting point for msurements at each dipole strength, and avoids the initiallistic motion of particles and small chains occurring whthe field is first applied. Starting at 15 s, we collect autocrelation data for 60 s. To assess the effect of the startime, we repeat the experiments switching the field at 8and starting the autocorrelation at 90 s.

In the diffusion limit of light transport in turbid media,photon executes a random walk with an average step leof l * , characterizing the distance over which its directionrandomized. Since structure and interactions affect the vof l * @19–22#, we measure the diffuse transmittance ofexpanded laser beam to monitor the change in light transproperties during an experiment. Changes inl * are taken

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into account in our analysis. The initial value ofl * is foundfrom by fitting the field-off data. The characteristic absortion length for photons diffusing in a medium having aabsorption coefficienta is l a5Al * /3a @23,16#. This is foundindependently by varying the path length and measuringdiffuse transmittance through a sample without the magnfield.

Light transport properties are also measured by removthe collimator and focusing the incident beam to a posource on the sample cuvette face. The transmitted difflight is collected on a charge-coupled device~CCD! arrayand recorded on videotape at a rate of 30 Hz. Imagescaptured from tape by averaging 20 video frames and pcessed to assess the evolution of anisotropic light transpoour samples after the field is applied and the magnetic chbegin growing.

III. RESULTS AND DISCUSSION

A. Effects of anisotropic light transport

In diffusing wave spectroscopy, the electric-field autocrelation is a weighted sum:

g~1!~r ,t !5E0

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dtP~r ,t!exp@2~c/3l * !k02^Dr 2~ t !&t#,

~3.1!

where ^Dr 2(t)& is the mean-squared displacement of tscatterers andc is the speed of light in the medium. Thprobability P(r ,t) that a diffusing photon will arrive at position r and timet is determined by solving the diffusionequation for the experimental geometry. The MR systemcomplicated by the fact that the suspension structurecomes anisotropic as the particles aggregate to form chaIn this case, the diffusion equation for the light energy desity U is anisotropic:

]U~r ,t!

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21D'¹'2 11/ma!U~r ,t!, ~3.2!

wherema is the characteristic time for photon absorption aD i andD' are the light diffusion constants in the directioparallel and perpendicular to the orientation of the chairespectively. The diffusive probability is found fromP(r ,t)5U(r ,t)/*0

`dtU(r ,t). The anisotropic light diffu-sion problem may be reduced to the familiar isotropic foby rescaling the axes, such that for our geometry we h@24#

F ]

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where D is the average diffusivity andz5AD/D'x. Thisequation is valid as long as the plane-wave conditionsmet experimentally; that is, both of the components ofl *parallel and orthogonal to the orientation of the scatterchains are much less than the width of our expanded beEquation~3.3! is solved using zero net-flux boundary condtions discussed elsewhere@19#.

We note that Eq.~3.1! assumes the dynamic scatterincross section can be approximated with its form for a dil

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3374 PRE 58ERIC M. FURST AND ALICE P. GAST

colloidal suspension@25#. Recently, an anisotropic multipllight scattering formalism has been developed and applielight scattering in nematic liquid crystals@26–29#; however,the nature of light scattering in nematics differs from thatcolloidal systems. For instance, the static structure factonematics diverges at low scattering angleq, and results in avanishing scattering mean free pathl s while maintaining afinite l * @30#.

The magnitude of the light transport anisotropy is estlished through the steady-state transmission of a point-sothrough the sample. Figure 1 shows images of the diffdepolarized intensity collected by our CCD camera forand 150 s after the field is applied atl516. The ratio of themajor and minor axes increases rapidly to 1.6 at 15 s aschains are initially formed. At 45 s, the ratio is 1.8 and icreases slowly to 2.2 at 150 s; this small anisotropy validathe use of Eq.~3.3! with the scaled distancez. The ratios areinsensitive to switching the field strength to lower valueand the images return immediately to the isotropic valwhen the field is turned off.

B. Estimation of l * and l a

The absorption lengthl a is 1.1260.08 mm. Using thisvalue and the calculated particle self-diffusion coefficieDs , we fit the field-off autocorrelation to findl * 5790610mm as shown in Fig. 3. The relatively large value ofl *compared with systems such as polystyrene spheres@20# canbe attributed to the lower contrast of our droplets.

As mentioned previously, changes in the spatial arranment and interactions of scatterers alter the diffuse transtance through the sample. In the case of dense or interacsuspensions,l * is proportional to the weighted average ovthe particle form factorP(q) and structure factorS(q) @19–22#:

l * 21}2k022E

4pq2P~q!S~q!dV. ~3.4!

To assess the effect of the dipolar interactions and ch

FIG. 1. Images showing depolarized diffuse intensity frompoint source transmitted through a 10 mm sample with the fieldand atl516. The intensity is initially isotropic, but becomes aisotropic as chains form. The chains are oriented from top to btom. Each image is 5.0 by 3.7 mm.~a! Field off. ~b! Field on 45 s.~c! Field on 90 s.~d! Field on 150 s.

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structures onthe orientationally averaged value of the tranport mean free path l* , we monitor the diffuse transmittancthrough our sample and measure the normalized increasthe scattering lengthD l * (t)/ l * (0). For L. l a , the diffusetransmittance may be written as@23#

T~L !510l *

3l aexp~2L/ l a!. ~3.5!

Expanding Eq.~3.5! to three terms and dividing byT(0), wefind an expression for the normalized change in the diffutransmittanceDT(t)/T(0) as a function ofD l * (t)/ l * (0)@16#:

DT~ t !

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During the experiments,D l * (t)/ l * (0) increases rapidlyto 0.20 10 s after applying the field, then increases linearly0.24 at 150 s as shown in Fig. 2. Thus,l * varies by less thanapproximately 2% during our autocorrelation measuremeand does not change when the field is stepped to lowerues; it does return immediately tol * (0) when the field isturned off. This suggests that structure dominatesl * andthere is a minimal amount of structural change when the fiis reduced for our dynamic measurements.

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FIG. 2. The normalized change inl * after the magnetic field isapplied. Two data sets are shown: (n) the field is held constant al516. (s) The field is stepped tol57 at 85 s. The inset showthe normalized change in transmittance used to calcuD l * (t)/ l * (0).

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PRE 58 3375PARTICLE DYNAMICS IN MAGNETORHEOLOGICAL . . .

C. DWS results

As shown in Fig. 3, the autocorrelationg(1)(t) for thefield-off data is well described by the isotropic solutionEq. ~3.1! by fitting l * using our measured value forl a andcalculated value forDs . The next three curves showg(1)(t) measured 90 s after applying the field. The autocrelations now show a shift to longer decay times, increaswith higher values ofl. Unlike the isotropic case, the autocorrelations for the dipolar systems cannot be fit withsingle characteristic decay time such as (Dsk0

2)21. Instead,we calculate the mean-squared displacement with t^Dr 2(t)& from the solution of Eq.~3.1!.

Figure 4 showsDr 2(t)& scaled by the particle diameterdfor each dipole strength as a function of scaled timetDsd

22.The straight solid line in each plot represents self-diffusiwhich the field-off data follows very well. When the fieldapplied, the initial mean-squared displacements followsimilar slope, which is less than that of free thredimensional ~3D! diffusion, but consistent with two-dimensional~2D! diffusion ~the dashed, straight lines in Fig4! irrespective of dipole strength. As the time increas^Dr 2(t)& diverges further from the free diffusion limit, exhibiting a constrained, subdiffusive motion. The magnituof deviation from the solid line increases as we increasedipole strength, suggesting that the DWS experiment is ssitive to the small-wavelength motions of the dipolar chaiThe chains should exhibit a spectrum of modes of motiincluding short-time individual ‘‘vibrations’’ of each particle, collective chain motion at intermediate timescales, aeventually the long-time diffusion of the entire chain. Sinthe scattering data for each field strength exhibits the sinitial slope, our experiments appear to measure the Broian motion of the particles comprising the dipolar chainParticle diffusion is hindered by constraint to the cha

FIG. 3. Measured DWS autocorrelation functions for the isotpic MR suspension and three field strengths measured 90 sapplying the initial field. The solid line shows a fit of the solutionEq. ~3.1!. Increasingl shifts the decay to longer times.

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through the dipolar interaction with its neighbors. Movemeperpendicular to the chain direction dominates, indicatedthe initial 2D motion, while the constraint increases asincrease the dipole strength, thereby stiffening the chainis also apparent that the mobility 90 s after applying the fiis reduced further from the freely diffusing limit than thmeasured 15 s after applying the field, possibly due tolateral aggregation of chains with time. This is consistewith the slow increase inl * measured through the diffustransmittance. Lateral aggregation should stiffen the mnetic chains further, and thus decrease chain segmenttion.

D. Brownian dynamics simulations

To test our interpretation of the DWS experiments, wconduct Brownian dynamics simulations of magnetic chaover the time scales observed in our experiments@31,32#.Briefly, the simulations solve the equation of motion of tnth particle, taking into account four forces acting on it:dipolar force between neighboring particlesFi

D,n , a stochas-tic Brownian forceFi

Br,n , Stokes dragFiS,n , and an excluded

volume interactionFiEV,n . Again scaling displacement with

the particle diameterd and time with the characteristic tima free particle diffuses its diameterDsd

22, the dimensionlessequation of motion, neglecting inertia, is written in Einstenotation as

qin5lFi

D,n1FiBr,n1Fi

S,n1cFiEV,n , ~3.7!

where l is the dipole strength. The simulations are coducted forl55, 10, 15, 20, and 26 using ensembles of 5chains for proper averaging. We compute the mean-squdeviation of each particle,Dr n

2(t)& averaged over all particles in the simulation for chains of 20 and 100 particles

-ter

FIG. 4. ^Dr 2(t)& calculated fromg(1)(t) using the solution toEq. ~3.1!. The straight, solid line represents 3D self-diffusion, tdashed line represents 2D self-diffusion, and the five curved slines are simulation results for dipolar chains of 100 particles wl55, 10, 15, 20, and 26, increasing from left to right in the figur~a! DWS data taken 15 s after applying the magnetic field.~b! Datataken 90 s after applying the field.

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3376 PRE 58ERIC M. FURST AND ALICE P. GAST

Since our experiments are most sensitive to lateral fltuations in the chain, we extract the mean-squared displment projected onto a plane orthogonal to the chain dirtion. We compare the 100 particle simulation results toDWS data in Fig. 4. The differences inDr n

2(t)& using 20particles showed no consistent trend and did not exceedexpected accuracy. The simulations capture the qualitaaspects of the experimental results nicely; however, therequantitative discrepancies for both the 15 and 90 s data.instance, the simulations require a higher dipole strengtgenerate similar displacements with time. This could reflthe fact that the simulation neglects the local field creafrom the formation of the dipolar chains as well as highmoment interactions@6#. At this time we should not expecexact quantitative agreement for a variety of reasons incing neglect of hydrodynamic interactions and magnetic inactions between chains.

IV. CONCLUSIONS

As a class of tunable fluids, MR systems are interestdue to their fast rheological response and possible apptions in mechanical systems such as brakes and clutchethis paper, we presented a study of the dynamics of ne

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neutrally buoyant MR suspensions using diffusing-waspectroscopy. By accounting for changes in the light traport properties of the system, we isolate variations inchain dynamics caused by the interactions between theticles. The DWS experiments offer the capability to measthe dynamics of the MR fluid systems on time and lengscales that capture the short-wavelength chain motioBrownian dynamics simulations of dipolar chains suppour experimental results. Both experiments and simulatishow initial particle displacements that are independenfield strength; however, at longer times, we see a constrasub-diffusive movement that increases with the dipolar intactions. We plan to investigate the quantitative agreembetween simulations and experiments by incorporatingdrodynamic interactions and chaining effects into our simlations in the near future.

ACKNOWLEDGMENTS

The authors thank Patrick Doyle for his aid with the simlations, Jerome Bibette for providing the ferrofluid, and MaFermigier for fruitful discussions. Support by NASA~GrantNo. NAG3-1887-1! is gratefully acknowledged.

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