S

A

YIK

a

ARR2AA

KEYB

1

pfipofltstoaiaddeCs

�afipC

0d

J. Non-Newtonian Fluid Mech. 166 (2011) 241–243

Contents lists available at ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journa l homepage: www.e lsev ier .com/ locate / jnnfm

hort communication

new yield stress scaling function for electrorheological fluids

ongsok Seo ∗

ntellectual Textile Research Center and School of Materials Science and Engineering, College of Engineering, Seoul National University, Shillimdong 56-1,wanakgu, Seoul 151-744, Republic of Korea

r t i c l e i n f o

rticle history:eceived 9 October 2010eceived in revised form

a b s t r a c t

A new scaling function capable of modeling the yield stress behavior of electrorheological (ER) fluidsthrough the full range of electric fields is proposed. In spite of its simple form, a comparison of the modelpredictions with experimental data for both ac and dc fields and the polarization model shows that the

4 November 2010ccepted 25 November 2010vailable online 13 December 2010

eywords:R fluid

proposed model correctly predicts the yield stress behavior both quantitatively as well as qualitatively.© 2010 Elsevier B.V. All rights reserved.

ield stress functioni-power law model

. Introduction

Electrorheological (ER) fluids are suspensions of extremely finearticles in an electrically insulating fluid [1]. When an electriceld is applied to such fluids, attractive interactions among the sus-ended particles cause them to rapidly form a solid-like networkf fibers in the field direction [2]. Thus, the apparent viscosity of ERuids changes reversibly by three orders of magnitude in responseo an electric field [3–5]. A typical ER fluid transforms from a liquidtate to a gel on application of an electric field, and then back tohe liquid state on removal of the field, with response times on therder of milliseconds [6]. The effect is best described in terms ofn electric field dependent shear yield stress [6,7]. The yield stresss one of the critical design parameters in an ER device. A largemount of experimental data has been reported on the electric fieldependence of the yield stress of ER fluids; however, comparison ofata from different studies is made difficult by the use of differentxperimental conditions and devices. To overcome this problem,hoi et al. suggested a scaling function for the normalized yieldtress via scaling of the applied electric field strengths [4,8].

The original model expressed the yield stress (�y) as �y ∝kffˇE2

0 where � is the volume fraction, ˇ = (kp − kf)/(kp + 2kf) is

dimensionless dielectric mismatch parameter, E0 is the electriceld strength, and kp and kf are the dielectric permitivities of thearticle and the fluid, respectively [2,4,9]. However, as noted byhoi et al. [4], experimental yield stress data deviate significantly

∗ Tel.: +822 880 9085; fax: +822 885 9671.E-mail address: ysseo@snu.ac.kr

377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2010.11.010

from this equation and are better represented by the power law;�y ∝ E3/2

0 (m < 2) at high electric field strengths. This is ascribed toelectrical breakdown of ER fluids under high electric field strength[1,3]. Davis and Ginder showed that �y ∝ E3/2

0 for E0 > Ec whereEc is the critical electric field strength [9]. To represent the yieldstress data over a broad electric field strength range, Choi et al.proposed the following hybrid equation:

�y(E0) = ˛E20

(tanh

√E0/Ec√

E0/Ec

)(1)

where ˛ depends on the dielectric constant of the fluid as well asthe particle volume fraction, and Ec represents the critical electricfield originating from the nonlinear conductivity model [4]. Thisexpression also defines the crossover between different power-lawregimes in the �y versus E0 plot (Fig. 1).

The hybrid equation (Eq. (1)) can describe the limiting behaviorsat low and high electric field strengths, respectively.

�y = ˛E20 ∝ E2

0 for E0 << Ec (2a)

�y = ˛√

Ec E3/20 ∝ E3/2

0 for E0 >> Ec (2b)

To fit the experimental data, they normalized Eq. (1) with�yc(Ec) = ˛Ec

2tanh(1) = 0.762˛Ec2, giving rise to the following

dimensionless form:√

� = 1.313E1.5tanh E (3)

where E = E0/Ec and � = �y(E0)/�y(Ec). Normalizing the data with�y(Ec) and Ec, the experimental data could be collapsed into a sin-gle curve (Fig. 1). However, although normalizing the data with

242 Y. Seo / J. Non-Newtonian Fluid M

Fig. 1. Normalized yield stress � versus normalized electric field strength E fora 20 wt% microencapsulated polyaniline-based ER fluid with a melamine-formaldehyde (MF) resin dispersed in silicone oil (weight ratio of polyaniline to MFresin = 10:114 (�), 10: 152 (©) and 10:190 (�)). Data are from Choi et al.[4]. Thelpds

ticpfi

�

�

a

�

wg

Fp

ine is the prediction of the “bi-power law” model (Eq. (7)). The inset shows a com-arison of the original Choi et al’s model and the “bi-power law” model. The largestifference is observed at E = 1 where Choi et al’s model underestimates the yieldtress.

he scaling constant value of tanh(1) affords good fit of the exper-mental data, predictions of Eq. (1) give the largest error in theritical region (inset in Fig. 1) where most experimental data arerovided. To better fit the experimental data, they added onetting-parameter, b, as follows

y(E0) = ˛E20

(tanh (E0/Ec)0.5+b

(E0/Ec)0.5+b

)(4)

This equation can be normalized again as

ˆ = 1.313E1.5+btanh E0.5+b (5)

Then this equation was renormalized once more with ˆ� = � E4b

nd ˆE = E1+2b to give

ˆ = 1.313ˆE1.5

tanh

√ˆE (6)

hich has the same form as Eq. (3). Although the Eq. (6) looksood to fit the experimental data, this arises from the inclusion

ig. 2. Comparison of the predictions of the three models on a linear scale of � in the highlot, Choi et al’s model was corrected to include the normalization constant of tanh (1) =

ech. 166 (2011) 241–243

of the normalization constant, 1/tanh(1) = 1.313, which forces therenormalized equation to satisfy the condition of E = 1 and � = 1,and from using two parameters (Ec and the arbitrary power lawparameter b).

2. New modelling results and discussion

As mentioned above, the behavior of the yield stress versus Ecan be divided at Ec into two regions, encompassing low and highelectric field strengths (Fig. 1). After normalization, these regionsmeet at E = 1 and � = 1. The most appropriate functional form todescribe two yield stress regimes is a combination of two scalingfunctions using a step function:

� = (1 − u(1 − E))E2 + u(1 − E)E1.5 (7)

where u(x) is the unitary step function. We call this “bi-power law”model. For practical reasons, however, a single explicit function thatcan describe the whole behavior would be more desirable. Based onphysical reasoning and the rheological behavior of Bingham fluids[10–12], we introduce the following simple equation to fulfill thenecessary conditions:

�y(E0) = ˛E3/20 (1 − exp(−m′

√E0)) (8)

where m′ is a fitting parameter. This equation has the following twolimiting behaviors at low and high electric field strengths, respec-tively

�y = ˛m′ E20 ∝ E2

0 for E0 << Ec (9a)

�y = ˛E3/20 ∝ E3/2

0 for E0 >> Ec (9b)

Normalizing Eq. (8) with Ec and �y = ˛Ec3/2 gives the following

equation:

� = E3/2(1 − exp(−m√

(E))) (10)

where m = m′√

Ec.Fig. 2 shows a comparison of the new model prediction using Eq.

(7) and those by Choi et al.’s model (Eqs. (3) or (6)) and by bi-powerlaw model (Eq. (7)). The proposed model shows satisfactory over-all agreement with the other formalism over the experimentallyaccessible electric field strength range, although there is a slight

difference at low electric field strength. Fig. 3 shows the varia-tion in yield stress with electric field for the three models, alongwith normalized experimental data. Considering that most of theexperimental data were obtained at high electric field strengths,the proposed model provides very good predictions of the yield

electric field strengths region (A) and low electric field strengths region (B). In this0.762 (Eq. (3) or Eq. (6)).

Y. Seo / J. Non-Newtonian Fluid Mech. 166 (2011) 241–243 243

F s norma om th( easur

sev

atmstcmpEe

3

cflwtqfiEmc

[[

ig. 3. Comparison of the predictions of three models of normalized yield stress � vlinear scale (B). The data are the same as in Fig. 2 except that some data (�) are fr�) are from the ref. [13] for 20 vol% 1.5 �m doubly coated particles in silicone oil m

tress of ER fluids. In addition, compared to Choi et al.’s hyperbolicquation (Eq. (3) or Eq. (6)), Eq. (10) is easier to compute, more con-enient and affords better predictions at high electric field strength.

The differences between the predictions of the proposed modelnd those of the model of Choi et al. become more obvious ifhe yield stress is plotted on a linear scale (Fig. 3(b)). Choi et al’s

odel appears to underestimate the yield stress at low electric fieldtrengths and overestimate it at high electric field strengths dueo the intrinsic shape of the tangent hyperbolic function whereasurrent model shows a very good prediction and an excellent agree-ent with the “bi-power” model. Depending on the value of the

arameter m, the proposed model shows a slight differences atˆ = 1 (� = 0.94 at E = 1), but this difference is much less than Choit al.’s model (� = 0.762 at E = 1).

. Summary

The proposed equation, although not derived from the first prin-iple, provides good predictions of the yield stress behavior of ERuids using a single parameter, m. The model results correlate wellith both ac and dc field data due to the normalization of the data

o collapse them onto a single curve [4]. The proposed model agrees

uite well with the original “bi-power law” model at high electriceld strength. Since magnetorheological fluids behave similarly toR fluids (i.e., the yield stress is proportional to the square of theagnetic field strength (H0) at low H0, but to H0

3/2 at high H0), theurrent model is expected to be also applicable for MR fluids [14,15].

[

[

[[

alized electric field strength E. The data are plotted on both a log-log scale (A) ande reference 12 for 10 wt% polyaniline particle suspension in mineral oil and othersed under a 50 Hz ac field. The value of the fitting parameter m (Eq. (10)) was 2.78.

Acknowledgements

This study was supported by Korea National Research Foun-dation through the Intellectual Textile Research Center at SeoulNational University (Basic Science Research Program # R11-2005-065) and Basic Research Program #RIAM 0427-20100020, andMinistry of Knowledge and Economy (Fundamental R&D Programfor Core Technology of Materials, RIAM 0427-20100043).

References

[1] W. Wen, X. Huang, P. Sheng, Soft Matters 4 (2008) 200.[2] M. Parathasarathy, D.J. Klingenberg, Mat. Sci. Eng. R 17 (1996) 57.[3] W. Wen, X. Huang, S. Yang, K. Lu, P. Sheng, Nat. Mat. 2 (2003) 727.[4] H.J. Choi, M.S. Cho, J.W. Kim, C.A. Kim, C.A. Kim, M.S. Jhon, Appl. Phys. Lett. 78

(2001) 3806.[5] F.F. Fang, J.H. Kim, H.J. Choi, Y. Seo, J. Appl. Polym. Sci. 105 (2007) 1853.[6] S.G. Kim, J.Y. Lim, J.H. Sung, H.J. Choi, Y. Seo, Polymer 48 (2007) 6622.[7] C.H. Hong, H.J. Choi, Y. Seo, Synth. Met. 158 (2008) 72.[8] I.S. Sim, J.W. Kim, H.J. Choi, C.A. Kim, M.S. Jhon, Chem. Mat. 13 (2001)

1243.[9] L.C. Davis, J.M. Ginder, Progress in electrorheology, In: K.O. Mavel Ka, F.E. Filisko

(Eds.), Plenum, N.Y. 1995, p. 107; L.C. Davis, J. Appl. Phys. 81 (1997) 1985.10] T.C. Papanastasiou, J. Rheol. 31 (1987) 385.11] H. Zhu, Y.D. Kim, D. De Kee, J. Non-Newtonian Fluid Mech. 129 (2005) 177.

12] H.J. Lee, B.D. Chin, S.M. Yang, O.O. Park, J. Colloid Interf. Sci. 206 (1998)

424.13] W.Y. Tam, G.H. Yi, W. Wen, H. Ma, M.T. Loy, P. Sheng, Phys. Rev. Lett. 78 (1997)

2987.14] J.M. Ginder, L.C. Davis, L.D. Lee, Int. J. Mod. Phys. B 10 (1996) 3923.15] F.F. Fang, H.J. Choi, M.S. Jhon, Colloids Surf. A. 351 (2009) 46.