May 6, 2002 - Karl von Pfeil, Michael D. Graham, and Daniel J. Klingenberg*. Department of Chemical ..... [2] W. M. Winslow, J. Appl. Phys. 20, 1137 (1949).
VOLUME 88, NUMBER 18
6 MAY 2002
PHYSICAL REVIEW LETTERS
Pattern Formation in Flowing Electrorheological Fluids Karl von Pfeil, Michael D. Graham, and Daniel J. Klingenberg* Department of Chemical Engineering, University of Wisconsin, 1415 Engineering Drive, Madison, Wisconsin 53706
Jeffrey F. Morris School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100 (Received 27 August 2001; published 18 April 2002) A two-fluid continuum model is developed to describe mass transport in electro- and magnetorheological suspensions. The particle flux is related to the field-induced stresses. Solutions of the resulting mass balance show column formation in the absence of flow, and stripe formation when a suspension is subjected simultaneously to an applied electric field and shear flow. DOI: 10.1103/PhysRevLett.88.188301
Electro- and magnetorheological (ER and MR) fluids are particulate suspensions whose rheological properties are dramatically altered by electric and magnetic fields, respectively [1–4]. In shear flow, applied fields can increase the apparent viscosity by several orders of magnitude. This phenomenon is now being exploited in commercial applications [5,6]. Along with the rheological phenomena, the applied fields also induce pattern formation: in quiescent suspensions, the applied field causes the formation of particulate columns oriented in the direction of the field [2,4], while, in flowing suspensions, the field causes the formation of particulate stripes oriented in the flow direction [7–11]. These patterns are intimately connected to the rheological phenomena. Particle-level models and simulations [4,12] have been valuable for understanding the relationships between particle properties, interactions, and macroscopic behavior. Unfortunately, these techniques are limited to small numbers of particles and hence are not well suited for describing phenomena involving large numbers of particles, or predicting the behavior of a suspension throughout an entire device. These issues are particularly important in situations where the field, temperature, or particle concentration are not uniform. In this Letter we develop and study a continuum description of ER fluids. This is accomplished by solving a conservation equation for the particle concentration, using an expression for the particle flux obtained from a momentum balance. Using this two-fluid approach, one obtains patterns observed experimentally. For electric fields applied to quiescent suspensions, this approach captures the formation of particulate columns; for electric fields applied to flowing suspensions, it captures the formation of particulate stripes oriented in the flow direction. The analysis is illustrated below for ER fluids. A similar analysis for MR fluids would follow directly. Following Morris and Boulay [13] and related work [14,15], we begin with the general mass conservation equation. Averaging over the particulate volume and dividing by the particle density yields the particle-phase conserva188301-1
0031-9007兾02兾 88(18)兾188301(4)$20.00
PACS numbers: 83.80.Gv, 47.15.Pn, 77.22.Ch, 77.65.Bn
tion equation ≠f 1 具u典 ? =f 苷 2= ? j , (1) ≠t where f is the particle volume fraction, 具u典 is the suspension average velocity, j ⬅ f共U 2 具u典兲 is the particle flux relative to the mean suspension motion, and U is the local average velocity of the particulate phase. Averaging the general momentum balance over the particle phase and restricting attention to low Reynolds numbers and neutrally buoyant, monodisperse spheres yields the relation for the particle flux j苷
2a2 f共f兲= ? s 共 p兲 , 9hc
(2)
where a is the particle radius, hc is the viscosity of the suspending fluid, f共f兲 苷 共1 2 f兲4 is a hindered mobility (also termed a sedimentation function), and s 共 p兲 is the particle contribution to the stress. Below we obtain the electrostatic part of s 共 p兲 . We consider the suspension to be composed of dielectric spheres immersed in a dielectric suspending fluid (conducting spheres in a weakly conducting fluid are treated analogously). The suspension structure is assumed to be isotropic and amorphous. Such a structure is reasonable for dilute quiescent or flowing suspensions prior to the application of an electric field, as well as for the initial stages of structure formation following the application of an electric field. For such a suspension, the electrostatic stress is ∑µ ∂ ∏ 1 1 a1 EE 2 共e 1 a2 兲E 2 d , (3) s 共E兲 苷 e0 e 2 2 2 212 where e0 苷 8.8542 3 10 F兾m is the permittivity of free space, and e, a1 , and a2 are described as follows. A self-consistent mean-field analysis in the point-dipole limit yields a suspension dielectric constant [16] e共f兲 苷 ec
1 1 2bf , 1 2 bf
(4)
where ec is the dielectric constant of the suspending fluid, b 苷 共ep 2 ec 兲兾共ep 1 2ec 兲, and ep is the dielectric © 2002 The American Physical Society
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constant of the particulate material. Assuming that the disperse and continuous phases themselves are not electrostrictive, the electrostriction coefficients of the suspension, a1 and a2 , are [16] a1 苷 0 and a2 共f兲 苷 2
共 e共f兲 2 ec 兲 共 e共f兲 1 2ec 兲 . 3ec
(5)
Taking the divergence of Eq. (3) and utilizing Maxwell’s equations = ? D 苷 r 共e兲 (r 共e兲 is the free charge density) and = 3 E 苷 0 gives = ? s 共E兲 苷 r 共e兲 E 2 2 e0 E 2 =e 2 2 e0 =共a2 E 2 兲 . 1
1
(6)
The terms on the right side are body forces. For the present analysis, r 共e兲 苷 0, and the remaining terms are body forces exerted on the particles; thus = ? s 共E兲 contains only the influence of particle contributions to the stress (i.e., = ? s 共E兲 苷 = ? s 共 p,E兲 ). The particle stress will also contain a hydrodynamic contribution [13]. This contribution is neglected here in the interest of space and to illustrate the effects of the electrostatic driving force. Hydrodynamic contributions will be addressed in a future publication. Combining Eqs. (1), (2), and (6) yields a conservation equation for the evolution of the volume fraction when a field is applied to a flowing suspension. Here we consider ᠨ 0, 0兲] and limit attenonly simple shear flow [具u典 苷 共gz, tion to small fluctuations in concentration. For an applied field E 苷 共0, 0, E0 兲, the conservation equation becomes µ 2 0 ∂ ≠f 0 ≠ f ≠f0 ≠2 f 0 ≠2 f 0 ᠨ 1 gz 苷 2M , 1 2k ≠t ≠x ≠x 2 ≠y 2 ≠z 2 (7) where f 0 共x, t兲 ⬅ f共x, t兲 2 f0 is the deviation of the volume fraction from the average value f0 , µ ∂ da2 a2 e0 E02 f共f0 兲 de 1 , (8) M 苷2 9hc df df f0 苷
2a2 e0 ec b 2 E02 f共f0 兲f0 , 3hc 共1 2 bf0 兲3
(9)
and k 苷 2共1 2 bf0 兲兾共1 1 2bf0 兲 . 0. Equation (7) is similar to a convection-diffusion equation, with anisotropic diffusion and a negative diffusivity in the x and y directions. However, the particle phase motion has nothing to do with Brownian diffusion, but rather is caused by electrostatic forces. In fact, for typical ER suspensions, Brownian motion is negligible: for a 苷 1 3 1025 m, T 苷 298 K, and hc 苷 0.1 Pa s, the Brownian diffusivity is D0 苷 kT兾6phc a 苷 2.2 3 10216 m2 兾s, whereas M 苷 1.1 3 1029 m2 兾s for ec 苷 2, b 苷 1, E0 苷 1 3 106 V兾m, and f0 苷 0.1. The negative apparent diffusivity is reminiscent of spinodal decomposition in phase separating systems [17]. Indeed, a uniform suspension in an electrostatic field is 188301-2
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PHYSICAL REVIEW LETTERS
thermodynamically unstable. Consider the electrostatic contribution to the free energy, F E 苷 2e0 e共f兲E 2 兾2 [18]. Since ≠2 e兾≠f 2 . 0 [e.g., Eq. (4)], ≠2 F E 兾≠f 2 , 0, which implies that the free energy can be reduced by separating the system into dilute and concentrated phases. Below we show that Eq. (7) predicts that the application of an electric field to a uniform suspension indeed causes the formation of more concentrated “phases.” We show that for a quiescent suspension the field induces the formation of particle-rich columns oriented in the direction of the applied field; for sheared suspensions, the field induces the formation of particle-rich stripes oriented in the flow direction. These structures are consistent with those commonly observed experimentally. Consider first the application of an electric field to a quiescent suspension. The electrodes are located at z 苷 0 and L. We seek a solution of Eq. (7) of the form f0 共x, t兲 苷 f共z兲ei共kx x1ky y兲 est ,
(10)
which represents fluctuations with sinusoidal variation in the x and y directions. The z dependence is yet to be determined. The growth rate of the fluctuations is represented by s. The full solution to Eq. (7) would be composed of an infinite series of many terms of the form of Eq. (10). Substitution of Eq. (10) into Eq. (7) yields kf 00 共z兲 1 共kx2 1 ky2 2 s兾M兲f共z兲 苷 0 .
(11)
Using the no-flux boundary conditions f 共z兲 苷 0 at z 苷 0 and z 苷 L yields 0
f共z兲 苷 cosnpz兾L
n 苷 0, 1, . . . ,
(12)
sL2 苷 共kx L兲2 1 共ky L兲2 2 kn2 p 2 . M
(13)
and
Fluctuations will grow for all s . 0; the fastest growing fluctuations (for any given kx and ky ) will occur for n 苷 0 and grow with rate smax 兾M 苷 kx2 1 ky2 . 0. This result has several implications for the resulting suspension structures. First, since the fastest growing fluctuations occur for n 苷 0, the resulting concentration profile should not depend on z. Second, the maximum growth rate increases with kx and ky ; in fact, the s ! ` as kx and ky ! `. This implies that the fastest growing fluctuations will be thin structures. Third, the growth rate is symmetric with respect to kx and ky , suggesting that if the initial fluctuations are random and isotropic, the resulting structures will be cylindrical columns; i.e., columns of particles oriented in the z direction, as commonly observed (a nonlinear analysis is required to determine the structures resulting from arbitrary initial conditions [19]). This continuum analysis will break down at small length scales. The fastest growing wave numbers are certainly limited to at most roughly the inverse of the particle diameter. We also expect nonlocal polarization (i.e., “surface 188301-2
VOLUME 88, NUMBER 18
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PHYSICAL REVIEW LETTERS
tension” [20]) to affect high jkj growth rates and thus the thickness of the resulting structures. However, these features will not alter the conclusion that columnar structures are produced. A finite difference numerical solution of Eq. (7) is illustrated in Fig. 1. No-flux boundary conditions were applied at the electrodes; periodic boundary conditions were applied at x, y 苷 0, L to represent a system of infinite extent in the x and y directions. The initial concentration consisted of small, random fluctuations about an average volume fraction f0 苷 0.1. The evolution of the concentration fluctuations is illustrated in Fig. 1, where positive fluctuations are indicated by dark cubes, and negative fluctuations are indicated by light cubes. The initial distribution and a distribution at a later time (tM兾L2 苷 0.05) are shown. As time progresses, the concentration fluctuations form columnar structures, which are similar to the fibrous aggregates commonly observed in ER and MR suspensions. The mechanism of column formation is revealed by considering the change in electrostatic free energy associatedR with concentration fluctuations, given by dF E 苷 2e0 V de共f兲E ? E0 dV兾2 [18], where de is the fluctuation in dielectric constant caused by the concentration fluctuation, E is the electric field, and E0 is the electric field prior to the fluctuation. Consider first a parallel concentration fluctuation of the form f共x兲 苷 f0 1 Ak cosbk x with Ak ø 1. In this case the field is unaltered, and the change in the electrostatic free energy is dF E 苷 2e0 E02 共d 2 e兾df 2 兲Ak2 V 兾8 , 0, implying that such fluctuations are unstable and will continue to grow. Next consider perpendicular fluctuations of the form f共z兲 苷 f0 1 A⬜ cosb⬜ z. In this case the field is altered by the fluctuation, and the resulting change in free energy is dF E 苷 2 V 兾4 . 0, im1e0 E02 关共de兾df兲2兾e 2 共d 2 e兾df 2 兲兾2兴A⬜ plying that such fluctuations are stable and will decay. Columnar fluctuations parallel to the applied field, which reduce the system free energy, will thus be favored. Now we consider the evolution of the concentration profile when an electric field is applied to a uniform suspension under shear flow. Again we seek a solution of the form of Eq. (10), which upon substitution into Eq. (7) yields the
ordinary differential equation for f共z兲, ᠨ x z兾M兲f 苷 0 . (14) kf 00 共z兲 1 共kx2 1 ky2 2 s兾M 2 i gk Solution of this differential equation with boundary conditions f 0 共z兲 苷 0 at z 苷 0, L provides the eigenvalue s as a function of the wave numbers kx and ky . Again, positive values of s共kx , ky 兲 indicate unstable or growing fluctuations, while negative values of s共kx , ky 兲 indicate stable or decaying fluctuations. Equation (14) was solved numerically to determine ᠨ 2 兾M 苷 103 and 104 are s共kx , ky 兲. The results for gL illustrated in Fig. 2, where contour plots of sL2 兾M as a function of kx and ky are presented. The s 苷 0 contours in Fig. 2(a) and 2(b) represent the stability boundaries. The s 苷 0 contours are roughly semicircles, which grow with increasing shear rate. Thus certain fluctuations tend to be stabilized by shear flow. As discussed above the present model breaks down at large wave numbers. This arises from neglecting the physical dimensions of the particles, as well as neglecting short range interactions between particles. Including these features will tend to stabilize large jkj fluctuations. Combining this information with that presented in Fig. 2 implies that, at sufficiently large shear rates, the only unstable fluctuations will be those with small kx and all ky below the large jkj cutoff (i.e., the region between the s 苷 0 contour and the ky axis). Within this region, the most unstable fluctuations will be those with kx 苷 0 and nonzero ky , i.e., the dominant structure formed by applying an electric field to a flowing suspension is predicted to be stripes of particlerich regions oriented in the flow direction, as observed experimentally [7 –11]. Equation (7) has also been solved by the finite difference method to illustrate the stripe formation process. No flux boundary conditions were applied at the electrodes; periodic boundary conditions were applied at x, y 苷 0, L. The evolution of concentration fluctuations is illustrated ᠨ 2 兾M 苷 104 and f0 苷 0.1. The initially in Fig. 3 for gL random fluctuations eventually redistribute into stripes oriented in the flow direction, as described above. For the
(a)
(b)
L
E
z 0 0
L x
L 0
y
z
0
10
0 0
L
x
L 0
y
FIG. 1. (a) Initial structure and (b) structure after the application of an electric field to a quiescent suspension (tM兾L2 苷 0.05). Dark: positive concentration fluctuation; light: negative concentration fluctuation.
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2000 4000
20
U
30
ky L
L
30
ky L
(b)
U
401000
40
(a)
S
1000
-1000
20 0
0
10
S
0
0 0
20
40
kx L
60
0
20
40
60
kx L
FIG. 2. Contour plots of sL2 兾M as a function of kx L and ᠨ 2 兾M 苷 103 and (b) gL ᠨ 2 兾M 苷 104 . Stable and ky L for (a) gL unstable regions are denoted by S and U, respectively.
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PHYSICAL REVIEW LETTERS
(a)
(b)
L
E
z 0 0
L
x
L 0
y
z
L
z
x 0 0
x
L L0
y
FIG. 3. (a) Initial structure and (b) structure after the applicaᠨ 2 兾M 苷 104 , tion of an electric field to a flowing suspension (gL tM兾L2 苷 0.05). Dark: positive concentration fluctuation; light: negative concentration fluctuation.
parameter values listed following Eq. (9) along with L 苷 ᠨ 2 兾M 苷 104 corresponds to a shear rate of 1023 m, gL 21 gᠨ 苷 10 s . The mechanism of stripe formation in shear flow is related to the mechanism of column formation in quiescent suspensions. Fluctuations grow when parallel to the field and decay when perpendicular to the field. In shear flow, fluctuations parallel to the applied field will be rotated (and stretched) toward the flow direction and thus will become stabilized, as illustrated in Fig. 4. The result is a uniform concentration in the plane of shear. However, since fluctuations in the vorticity direction [i.e., k 苷 共0, ky , 0兲] are unaffected by shear and will continue to grow, the resulting structure will be sheets of higher concentration in the plane of shear (i.e., stripes oriented in the flow direction). To summarize, a two-fluid continuum model for mass transport in ER suspensions has been presented. The particle flux is related to the divergence of the particle contribution to the stress, which in turn is related to the suspension dielectric and electrostrictive properties. Solutions of the resulting particle conservation equation capture common observations: column formation in quiescent suspensions and stripe formation in sheared suspensions. Column formation arises because only these structures produce a decrease in the free energy of polarization. Stripe formation in shear flow arises because the flow stabilizes fluctuations in the plane of shear. Future work will involve refining and improving this continuum approach. Hydrodynamic interactions must be included to probe the behavior when the concentration within the stripes gets large. Nonlocal polarization contributions (i.e., surface tension [20]) to the stress and flux should be included to correctly capture the large jkj behavior. Effects of particle interactions and structural anisotropy should be included to completely capture the structure evolution process in ER and MR suspensions.
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Rotation by shear flow
φ’(x,t)
6 MAY 2002
Stable (perp. fluctuation) Unstable (par. fluctuation)
FIG. 4. Illustration of the stabilization of parallel fluctuations by simple shear flow.
This research was supported in part by the NSF-funded University of Wisconsin Materials Research Science and Engineering Center on Nanostructured Materials and Interfaces, Grant No. DMR-0079983, and by the German Academic Exchange Service (DAAD).
*Corresponding author. [1] J. Rabinow, AIEE Trans. 67, 1308 (1948). [2] W. M. Winslow, J. Appl. Phys. 20, 1137 (1949). [3] J. M. Ginder, in Encyclopedia of Applied Physics, edited by G. L. Trigg (VCH, Weinheim, 1996), Vol. 16, p. 487. [4] M. Parthasarathy and D. J. Klingenberg, Mater. Sci. Eng. Rep. R17, 57 (1996). [5] J. D. Carlson and J. L. Sproston, in Actuator 2000 — Proceedings of the 7th International Conference on New Actuators, edited by H. Borgmann (Messe Bremen GmbH, Bremen, 2000), p. 126. [6] D. J. Klingenberg, AIChE J. 47, 246 (2001). [7] G. Bossis, E. Lemaire, J. Persello, and L. Petit, Prog. Colloid Polym. Sci. 89, 1 (1992). [8] G. Bossis, Y. Grasselli, E. Lemaire, J. Persello, and L. Petit, Europhys. Lett. 25, 335 (1994). [9] S. Henley and F. E. Filisko, J. Rheol. 43, 1323 (1999). [10] O. Volkova, S. Cutillas, and G. Bossis, Phys. Rev. Lett. 82, 233 (1999). [11] S. Vieira, L. B. Pompeo Neto, and A. C. F. Arruda, J. Rheol. 44, 1139 (2000). [12] Y. Baxter-Drayton and J. F. Brady, J. Rheol. 40, 1027–1056 (1996). [13] J. F. Morris and F. Boulay, J. Rheol. 43, 1213 (1999). [14] P. R. Nott and J. F. Brady, J. Fluid Mech. 275, 157 (1994). [15] P. Mills and P. Snabre, J. Phys. II (France) 5, 1597 (1995). [16] Y. M. Shkel and D. J. Klingenberg, J. Appl. Phys. 83, 7834 (1998). [17] J. W. Cahn, J. Chem. Phys. 42, 93 (1965). [18] J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). [19] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993). [20] G. Bossis, H. Clercx, Y. Grasselli, and E. Lemaire, Int. J. Mod. Phys. B 8, 2747 (1994).
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