On the Eective Magnetic Properties of. Magnetorheological Fluids. November 9, Abstract

On the Eective Magnetic Properties of Magnetorheological Fluids November 9, 998 Tammy M. Simon, F. Reitich, M.R. Jolly, K. Ito, H.T. Banks Abstract Magnetorheological (MR) uids represent a class of smart materials whose rheological properties change in response to the application of a magnetic eld. These uids typically consist of small (m) magnetizable particles dispersed in a non-magnetic carrier uid that generally contains additives such as surfactants and anti-wear agents []. Due to such additives, there is an outer non-magnetic layer on the particles that keeps them from touching. The goal of this paper is to study the eective magnetic behavior of an MR composite as a function of the interparticle distance. To this end, we present and employ a model for the eective magnetic properties of MR uids with periodic microstructure that is based on the theory of homogenization. Finally, we discuss an interpolating formula for the eective permeability of MR uids as an extension of the work of Keller [] and Doyle []. Introduction Magnetorheological (MR) uids are suspensions of micron-sized magnetizable particles (such as iron) in a non-magnetic carrier uid (such as oil or water). The essential characteristic of these materials is that they can be rapidly and reversibly varied from the state of a Newtonian-like uid to that of a sti semi-solid with the application of a moderate magnetic eld. This feature, called the MR eect, is a consequence of the fact that, in the presence of a magnetic eld, the particles magnetize and form chain-like structures that align in the direction of the applied eld as depicted in Figure. This columnar microstructure, in turn, dramatically increases its resistance to an applied shear strain (see Figure ). This feature has inspired the design of new technology and various products such as semi-active dampers, brakes, clutches and numerous other robotic control systems [] (see also http://www.mruid.com). A typical MR uid includes special additives such as surfactants and anti-wear agents that enhance its performance in MR uid based devices []. These additives may form a non-magnetic layer on the inclusions that keeps them from actually touching. This minimum interparticle distance aects the overall magnetic properties of the uids []. Understanding the magnetic response of MR uids and its dependence on microstructure is important to the design of improved Research supported in part by a US Department of Education Graduate Assistance in Areas of National Need (GAANN) Fellowship (TMS), the U.S. Air Force Oce of Scientic Research under grants AFOSR F90-9-- 0 (HTB and KI), AFOSR F90-98--080 (HTB), AFOSR F90-9--0 (TMS), AFOSR F90-9-- 0 (FR), NSF DMS-9 (FR). Center for Research in Scientic Computations, Department of Mathematics, North Carolina State University Raleigh, NC 9 School of Mathematics,University of Minnesota,Minneapolis, MN Advanced Technologies Research Group, Lord Corporation, Thomas Lord Research Center Cary, NC - 900

MR uids. So here we study the eective magnetic properties of these uids as a function of the minimum interparticle distance. Predicting the magnetic properties of these uids, however, is a challenging task due to the highly nonlinear and oscillatory nature of the magnetization of the constituents. As stated, the characteristic size of the particles is several orders of magnitude smaller than the characteristic size of a sample, rendering standard nite element modeling impractical. Thus, in this paper we present a model for the eective magnetic response of MR uids that is based on the theory of homogenization. We obtain the constitutive equations that govern the B-H response of MR uids with periodic microstructure in both the linear and nonlinear regimes. The corresponding numerical results exhibit good agreement with experimental data (from experiments carried out at Lord Corporation). However, it becomes increasingly dicult to obtain robust numerical calculations for very small values of the interparticle distance because the necessary mesh size becomes too small. So we also discuss an interpolating formula for the effective permeability of MR uids for all (positive) values of the interparticle distance. It extends an interpolating relation due to Doyle from a simple cubic array structure to particle-chains. Doyle's work is based on that of Keller who analyzed the asymptotic behavior of the eective property as the interparticle distance approaches zero. This paper is organized as follows. In the next section, we review the mathematical model based on homogenization theory and present analytical formulae for the eective properties of MR uids with periodic microstructure. In x, we present experimental data collected for three MR uids and discuss calculating the permeability of a sample from the experimental B- H curve. Then we discuss the corresponding numerical calculations for the eective properties of MR uids and compare them with experimental data in x. Finally in x, we present an interpolating formula for the eective permeability of MR uids and investigate its applicability by comparing it with the homogenization calculations. NO MAGNETIC FIELD APPLIED MAGNETIC FIELD H APPLIED MAGNETIC FIELD H APPLIED STRAIN γ p p p p γ (a) (b) (c) H H Figure : The MR eect: (a) the particles ow in the absence of magnetic elds; (b) they magnetize and form columns when a magnetic eld is applied. (c) Schematic of the microstructure under an applied shear strain. Shear Stress (kpa) 0 00 80 0 0 0 0 0 0 00 Shear Strain Rate (sec- ) 000 Oersted 000 Oersted 000 Oersted 0 Oersted Figure : Shear strain rates versus shear stresses in the post-yield regime, for various values of the magnetic eld intensity [].

Mathematical model. Periodic microstructure In this section we summarize our model for the eective magnetic properties of MR uids with periodic microstructure that is based on the mathematical theory of homogenization. The results presented here can be found in the literature (see, e.g., [, ]). Let IR n (n= or depending on the geometry that we consider) be a sample of heterogeneous material that, in the presence of a vertical magnetic eld, has periodic microstructure that we now describe. Let Y be the parallelogram dened by Y =?c ; c?cn ; c n for some constants c i and assume that the permeability matrix of the sample is Y-periodic; that is, (x) = (x+c i e i ) for every x IR n and for every i = ; : : :; n. Here the vectors e ; : : :; e n are the unit direction vectors in IR n. Then, as depicted in Figure, Y is the basic periodicity cell of. The constituents of our MR uids are assumed isotropic and therefore, we may write 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 00 00 00 00 00 0000 000 00 Periodicity Cell Y d = - µm Figure : Schematic of the periodic microgeometry assumed here. = ^I where ^ = Now we dene : IR n! M nn by p c in the particles, in the carrier uid. x (x) = ; () where is the ratio of the particle size to the characteristic size of a sample. We note that is Y -periodic in IR n. Then the magnetic eld inside the specimen satises the fundamental equation of magnetostatics that, for periodic structures, can by written as x?r r = 0 () where is the magnetostatics scalar potential. Since the sample is in the presence of a vertical applied eld ~H = H n e n, the boundary conditions that complement () are = 0 at the bottom, = H n at the top () and @ = 0 on the sides of the specimen () @n where ~n is the unit normal vector to the surface @. Since is highly oscillatory, a direct treatment of the governing equations ()-() is impractical. So, in order to nd the macroscopic magnetic properties of MR composites, we employ the theory of homogenization.

. Linear homogenization For low eld intensities, MR uids exhibit linear behavior; that is, is independent of the magnetic eld. So we study the asymptotic behavior of () as! 0. We assume that (x) M nn is Y-periodic, symmetric, uniformly positive denite, and each ij is bounded. Then the theory of homogenization provides the following convergence properties as! 0: where 0 satises the homogenized problem r * r 0 weakly in L (; IR n ) ()? x r * hom r 0 weakly in L (; IR n ) The (constant) homogenized matrix hom is given by ( hom ) ik =?r ( hom r o ) = 0 on : () jy j Z Y 0 @ ik (y) + nx j= ij (y) @w k A dy: () @y j The function w k is the unique solution to the auxiliary problem r ((y)(ek + rw k (y))) = 0 in Y ; w k H # (Y ); (8) where H (Y ) denotes the set of periodic functions in # H (Y ) with zero mean. Here the convergence properties () imply that for suciently small > 0, () provides the eective magnetic permeability of the MR composite (i.e., B ~ = hom H). ~ Calculating the homogenized matrix requires solving the boundary value problems (8) for w k (k = ; ; ) and substituting w k into equation ().. Nonlinear Homogenization Magnetic materials such as iron possess a magnetic saturation M s ; that is, a maximum achievable magnetization that, once reached, does not allow any further magnetization. Thus, for moderate to large valued elds, the permeability is, in fact, a function of the magnetic eld ~ H. Then we study the asymptotic behavior of?r (( x ; r )r ) = 0 (9) where the B-H map (y; ) IR n! (y; ) IR n is assumed to be Y-periodic in y. Furthermore, assuming the map to be Lipschitz-continuous, strictly monotone and coercive in, then one can prove the following convergence properties of (9): where 0 satises the homogenized problem r * r 0 weakly in L (; IR n ) (0)? x ; r r * b hom (r 0 ) weakly in L (; IR n ) The map b hom () is dened for all IR n by b hom () = jy j?r (b hom (r 0 )) = 0: () Z Y (y; + rw (y))( + rw (y))dy; () where w is the unique solution to the auxiliary problem r((y; + rw (y))( + rw (y))) = 0 in Y w H # (Y ): ()

In our calculations, we use an empirical constitutive relation known as the Frohlich-Kennelly relation given by ( H) ~ ( p? ) M s = + () ( p? ) j ~Hj + M s to determine the permeability of the iron particles at a given eld strength H. In comparison to the linear case, calculating the eective constitutive law () requires solving a continuum of the nonlinear equations (). In x we will present numerical results for both equations () and (). data In this section we turn to a discussion of the experimental setup and discuss calculating the permeability of a MR uid sample from the experimental B-H curve. Experiments were conducted (at Lord Corporation) to measure the magnetic properties of three MR uids with particle loadings of 0%, 0%, and 0% by volume. The arrangement used to collect the data (KJS model HG-00 magnetic Hysteresisgraph) is schematically depicted in Figure : a current is passed through an electromagnet and a probe is used to measure the applied eld ~ H where ~ H = H ^k; the magnetic induction ~B = ~B( ~H) can then be evaluated from the cylindrical sample with the aid of a search coil. In Figure we display the resulting intrinsic induction ~B i = ~B? 0 ~H as ELECTROMAGNET H SPECIMEN (MR FLUID) J 0 B(H) Figure : The experimental setup. a function of 0 ~H. Since hysteresis is negligible in each case, in our analysis, we only consider data in the rst quadrant that correspond to the second part of the hysteresis loop. Intrinsic Induction [ T ] 0.8 0. 0. 0. 0 0. 0. 0% 0% 0% 0. 0.8 0. 0 0. µ H [ T ] 0 Figure : magnetic induction curves for three MR uids of dierent iron volume percents.

0 9 8 0% 0% 0% Chosen value. s n r n / 0 0 0 0 0 0 0 0 0 80 90 00 n. 0% 0% 0% Chosen value 0 0 0 00 0 n Figure : Slope and residual mean square of each t using n data points. For the linear permeability of the sample, we are interested in calculating the permeability as measured at very low elds. Since the magnetic induction is approximately linear for small elds, we take the permeability to be the slope of the B-H curve in this region. We calculate this slope by conducting a linear least squares t of the data. While it is important to use enough data so that the results are accurate and robust, the t should be conducted only over the linear regime. To this end, consider the following notation: (H k ; B k ) = kth data point of the B-H curve ~H k = 0 H k s n = slope of the t using the rst n data points b n = y-intercept of the t using n data points B~ n k = s n Hk ~ + b P n r n = n jb n? k= k? B ~ nj k (residual mean square): The slope and residual mean square of each t are shown in Figure. We selected the linear regime as the set of n points whose t yielded the minimum residual mean square and denote this set by the interval (0; H l ). As a result, the permeabilities and H l are summarized Table. e 0 H l (T) 0%.0.0 0%.0.0 0%.9.08 Table. Numerical experiments In this section we present the numerical results for the linear and nonlinear eective constitutive laws based on homogenization theory as stated in x and compare these with the experimental data that was presented in x. The specic geometry that we consider here consists of a n-dimensional (n= or ) static conguration of spherical iron inclusions (such as that in Figure ). We assume the inclusions form single chains in the direction of the applied eld that are periodically dispersed throughout the carrier uid. We denote by a the ratio of the vertical interparticle distance,, to the particle diameter. We take the (xed) period along the direction of the eld to be unity and the period in the other directions to be l (for simplicity, it is assumed that all lengths are dimensionless). Then, the periodic microstructure is characterized by the periodicity cell Y = [0; l] n? [0; ] as is shown in Figure. The physical parameters used here are p = 000; c = and 0 M s = : T (tesla) where 0 = 0? T m=a is the permeability of vacuum.

0000000000 0000000000 α = a r r 0 00 0 0 l Figure : Cross section of the periodicity cell. Effective Permeability 0 9 8 Expermental a=.008 a=.0 a=.0 a=.0 0 0.0 0. 0. 0. 0. 0. 0. Volume Fraction Figure 8: Linear eective permeability computed for the two-dimensional model geometry. First, we present the numerical results based on our two-dimensional model geometry (n = ) for the linear case. The elliptic equation (8) was solved to compute ( hom ) (where ( hom ) represents the eective permeability in the direction of the applied eld) using a standard nite element method based on piecewise-linear elements. In Figure 8, the experimental data are shown along with the results of our homogenization calculations for values of the ratio a = :008; :0; :0; :0. We remark that the numerical calculations for a = :0 agree well with the experimental values for all three uids. In Figures 9-, we depict the eective B-H curves (from linear regime through saturation) based on equations ()-() assuming a = :00; :0. We constructed the eective B-H curves by interpolating the discretely calculated values of b hom () for = H ~ = (0; H ), 0 H (0; 0:8). For each uid, the estimates for a = :0 provides a good t not only in the linear region, but through saturation as well. 0 0 B i Intrinsic Induction [ T ] 0 0 Numerical a=.00 Numerical a=.0 0 0 0 µ 0 H [ T ] 0 0 0 Figure 9. Eective magnetization for the 0% sample (a = :00; :0) compared to experimental data.

Next we discuss the numerical results for the three-dimensional model geometry (n = ) and we only consider the linear case. The linear elliptic equations (8) were solved using a commercial nite-element software package [9]. The results for ( hom ) are shown in Figure for a = :000; :00, :00, :00. We nd the agreement to be reasonable, given the idealized nature of the assumed microstructure. Indeed, some degree of polydispersity was certainly present within our samples and, of course, the true columnar structures were more complex than simple linear chains. In fact, this complexity is accentuated at larger volume fractions. This may explain the slight increase in the deviation as the particle concentration is increased. 0 0 B i Intrinsic Induction [ T ] 0 0 Numerical a=.00 Numerical a=.0 0 0 0 µ 0 H [ T ] 0 0 0 Figure 0: Eective magnetization for the 0% sample (a = :00; :0) compared to experimental data. 0 0 B i Intrinsic Induction [ T ] 0 0 Numerical a=.00 Numerical a=.0 0 0 0 µ H [ T ] 0 0 0 0 Figure : Eective magnetization for the 0% sample (a = :00; :0) compared to experimental data. Effective permeability 8 a=.000 a=.00 a=.00 a=.00 0 0.0 0. 0. 0. 0. 0. 0. 0. Volume Fraction Figure. Linear eective permeability computed for the three-dimensional model geometry. 8

Other approximations for e In this section we discuss an interpolating formula for e of MR composites for n=. We seek an interpolating relation which provides a good approximation for all values of a. To this end, we consider linear composites that contain spherical innitely permeable inclusions (for iron inclusions, p = 000 so this is a reasonable case to consider). It becomes increasingly dicult to obtain accurate numerical estimates as a! 0. However, J.B. Keller showed that, when the particles are uniformly dispersed (on a cubic lattice), the asymptotic behavior of e is logarithmic as a! 0 []. Based on this work, Doyle suggested the interpolating formula e = c [A + B log( c? )] () where is the volume fraction of the inclusions and c = = is the maximum volume fraction attained at a = 0 []. To nd the constants A and B, Doyle then considered the following conditions on e at = 0: e = c () d e d = c : () Equation () was obtained using the well-known asymptotic estimate attributed to Maxwell [0] given by p + c + ( p? c ) e c : (8) p + c? ( p? c ) Maxwell's estimate is valid for small volume fractions and becomes exact as! 0. From ()-(), one can determine A and B and () can be written as : (9) e = c? log? Equation (9), known as Doyle's estimate, is now compared with our homogenization estimates. As is seen in Figure, the agreement is good throughout the entire range of volume fractions. 8 Homogenization Interpolation Effective Permeability 0 0 0. 0. 0. 0. 0. Volume Fraction Figure : Homogenization estimates and equation (9) for the eective permeability. Doyle's estimate is for cubic structures whereas the microstructure for MR uids is columnar. In fact, cubic geometry is a special case of the model geometry depicted in Figure with l =. So it is our intent to seek an extension of Doyle's estimate for l. Our preliminary investigations suggest that the e -component of e is logarithmic as a! 0 or, equivalently, as! c (the 9

details are to be reported in future work). Following the procedure described above, we obtain the following extension for l e = c? c log? c (0) where c = =l is the maximum volume fraction. Of course, (0) coincides with Doyle's estimate for l =. The results for the extreme case c :8% (l = 8) are listed in Table. Also shown is the relative dierence of the interpolation relation from the homogenization estimates. The methods are in reasonable agreement throughout the entire range of possible volume fractions. (%) Homogenization Interpolation Relative Dierence 0.0000.0000 0.0.00.00 : 0? 0.0.00.009 : 0? 0.0.00.0 : 0? 0.0.00.0 :8 0? 0.0.0.0 :9 0? 0.0.09.0 :8 0? 0.0.0.0 : 0? 0.80.099.09 : 0? Table. In Figures - we compare the interpolating relation with estimates obtained from homogenization theory (see Figure ) for a = :00; :00; :00; :000. data are also shown. The interpolating formula is based on ininitely permeable particles, and therefore, yields larger permeabilities than the homogenization model. In comparison to experimental data, the interpolating formula for a = :000 exhibits good agreement with experimental data for all three uids. Interpolation Homogenization a=.00 Interpolation Homogenization a=.00 Effective Permeability Effective Permeability 0 0.0 0. 0. 0. 0. 0. 0. % volume fraction 0 0.0 0. 0. 0. 0. 0. 0. % volume fraction Figure. Homogenization estimates and equation (0) for the eective permeability compared with experimental data for a = :00; :00. 0

Interpolation Homogenization a=.00 8 Interpolation Homogenization a=.000 Effective Permeability Effective Permeability 0 0.0 0. 0. 0. 0. 0. 0. % volume fraction 0 0.0 0. 0. 0. 0. 0. 0. % volume fraction Figure. Homogenization estimates and equation (0) for the eective permeability compared with experimental data for a = :00; :000. Conclusions We presented experimental data that we collected for three MR uids: 0%, 0% and 0% by volume. We discussed calculating the linear permeabilities using least squares and found them to be approximately in the range from?. In order to predict the experimental data, we proposed a model for the eective magnetic properties of MR uids that is based on the theory of homogenization. Analytical formulas for linear and nonlinear media with periodic microstructure were discussed. For the numerical results, we considered microstructures of monodispersed spherical particles arranged periodically in single particle chains. Interestingly, even with this simplied model, the homogenization results predicted the experimental data with acceptable accuracy. However, the method is not restricted to these specic microstructures or particle shapes. We also derived an interpolating formula for the eective permeability of composites that contain innitely permeable inclusions arranged in particle-chains. This relation showed good agreement with experimental data. Hence, it may provide a simple yet reasonably accurate method for computing the eective permeability of MR uids. References [] P. P. Phule and J. M. Ginder, The Materials Science of Field-Responsive Fluids, MRS Bulletin (998), 9{. [] T. M. Simon, F. Reitich, M. R. Jolly, K. Ito, H. T. Banks, Estimation of the Eective Permeability in Magnetorheological Fluids, CRSC Technical Report CRSC-TR98-, N.C. State Univ., October, 998; J. Intell. Material Systems and Structures, submitted. [] J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. (9), 99{99. [] W. T. Doyle, The Clausius-Mossotti problem for cubic arrays of spheres, J. Appl. Phys. 9 (98), 9{9. [] J. D. Carlson, D. M. Catanzarite and K. A. St Clair, Commercial magnetorheological uid devices, Int. J. Mod. Phys. B 0 (99), 8{8. [] M. R. Jolly, J. D. Carlson and B. C. Mu~noz, A model for the behavior magnetorheological materials, J. Smart Mater. & Struct. (99), 0{.

[] A. Defranceschi, An Introduction to Homogenization and G-convergence, Lecture Notes, School on Homogenization, ICTP, Trieste (Sept. -8, 99). [8] R. M. Bozorth, Ferromagnetism, van Nostrand, Toronto (9). [9] Maxwell D Field Simulator, Ansoft Corporation, Pittsburgh, PA. [0] J. C. Maxwell, A Treatise on Electricity and Magnetism, rd Ed., Dover Publications, New York (89).