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Langmuir 1999, 15, 918-921
Dielectric Criteria for the Electrorheological Effect Tian Hao,* Akiko Kawai, and Fumikazu Ikazaki Reaction Engineering Laboratory, Department of Chemical Systems, National Institute of Materials and Chemical Research, Higashi 1-1, Tsukuba, Ibaraki 305, Japan Received August 20, 1998. In Final Form: December 8, 1998 The suspended particles of an electrorheological (ER) fluid can be easily arrayed as a crystallite of a body-centered tetragonal (bct) lattice structure using an external electric field. Thus the entropy of this process, which is generally recognized as a first-order phase transition, would greatly decrease. From this fact, we theoretically derive that the dielectric loss tangent maximum value of the dispersed particles should be larger than 0.1, which agrees well with the currently observed experimental results; the negative ER effect becomes possible only when dsp/dT < 0, where sp is the static dielectric constant of the dispersed particle material and T is temperature.
Introduction The electrorheological (ER) fluids are a kind of suspension composed of semiconductive solid particles (polymer or inorganic materials) dispersed in an insulating oil (silicone oil, transformer oil, etc.). Under the stimulation of an external electric field, the ER fluids can be solidified, changing from a liquid state to a solid state. This process, schematically illustrated in Figure 1, has been comprehensively studied and is recognized as a first-order phase transition.1,2 At the liquid state (before an electric field is applied), the ER particles are randomly distributed in the medium, and at the solid state (after an electric field is applied), the dispersed particles form a fibrillation structure (columns). Within a column the particles are arrayed as a crystallite of a body-centered tetragonal (bct) lattice, which was determined with a laser diffraction technique.3 However, not all kinds of particles can be arrayed to form a crystallite under an external electric field, so why can ER particles do so? This issue has been targeted for many years; however, no sound models or theories give a reasonable answer. Since the order in ER systems increases dramatically after the application of an electric field, the entropy of these systems would be markedly reduce; that is, the ∆S would be obviously less than zero. From this very basic fact, we would like to show in this Letter what derivations we can obtain. Theoretical Derivation We consider a general ER system. Assuming that the volume of the ER fluid is always kept constant and that the temperature T is the only parameter besides the electric field E considered to be varied, after an electric field is applied, the increase dU of the internal energy U per unit volume of the ER suspension can be expressed as follows, according to the first thermodynamic law and the electromagnetic theory:4
dU ) dQ +
(1) Hao, T.; Chen, Y. H.; Xu, Z. M.; Xu, Y. Z.; Huang, Y. Chin. J. Polym. Sci. 1994, 12, 97. (2) Tao, R.; Woestman, J. T.; Jaggi, N. K. Appl. Phys. Lett. 1989, 55 (18), 1844. (3) Chen, T. J.; Zitter, R. N.; Tao, R. Phys. Rev. Lett. 1992, 68, 2555. (4) Frohlich, H. The Theory of Dielectrics; Clarendon Press: Oxford, U.K., 1958.
(1)
where dQ is the influx of heat per unit volume, D is the electric displacement, and E/4π dD represents the influx of energy into the ER fluid. For the ER fluids, it is already found that, both theoretically and experimentally, there seems to be a critical electric field at which the liquidsolid (disorder-order) transition takes place.2,5,6 The critical electric field is usually less than 200 V/mm, and a typical value is around 45 V/mm for many ER systems.5-7 In the case of the applied electric field slightly larger than the critical field, one may assume that the static dielectric constant of an ER suspension s is independent of E, as the applied electric field is not so strong. Even in a very strong field, the static dielectric constant of many materials would not be substantially changed.8 Then D ) sE; thus,
dD ) d(sE) ) s dE + E ds ) s dE + E
∂s dT (2) ∂T
as s just depends on T. It will be useful to take T and E2 as independent variables; thus, eq 1 can be rewritten as
dU ) dQ +
s E2 ∂s d(E2) + dT ) 8π 4π ∂T ∂U ∂U dT (3) d(E2) + 2 ∂T ∂(E )
One also can think the entropy S is a function of T and E2, as dS ) dQ/T, the second thermodynamic law for a reversible process, so that
dS )
∂S ∂S dT + d(E2) ∂T ∂(E2)
Inserting dQ from eq 3 to eq 4,
dS ) * To whom correspondence should be sent. Phone: 81-298-544667. Fax: 81-298-54-4487. E-mail:
[email protected].
E dD 4π
(
)
(
(4)
)
s 1 ∂U 1 ∂U E2 ∂s dT + d(E2) (5) 2 T ∂T 4π ∂T T ∂(E ) 8π
The condition that dS is a total differential requires that (5) Khusid, B.; Acrivos, A. Phys. Rev. E 1995, 52, 1669. (6) Wen, W.; Men, S. J.; Lu, K. Q. Phys. Rev. E 1997, 55, 3015. (7) Hao, T.; Kawai, A.; Ikazaki, F. Langmuir 1998, 14, 1256. (8) Smyth, C. P. Dielectric Behavior and Structure; McGraw-Hill: New York, Toronto, London, 1955; Chapter 3.
10.1021/la981069b CCC: $18.00 © 1999 American Chemical Society Published on Web 01/30/1999
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Langmuir, Vol. 15, No. 4, 1999 919
Figure 1. Schematic illustration of the structure change of an ER fluid before (a) and after (b) an external electric field is applied. The two parallel dark lines stand for two electrodes.
{(
)} { (
∂ 1 ∂U E2 ∂s ∂(E2) T ∂T 4π ∂T
)
s ∂ 1 ∂U 2 ∂T T ∂(E ) 8π
)}
[
∞ ) sm 1 +
(6)
Carrying out the differentiation, one finds
(
)
(7) t)
Comparing eq 5 and eq 4 and inserting ∂U/∂(E from eq 7 2)
1 ∂s ∂S ) 2 8π ∂T ∂(E )
(8)
integrating
(9)
where S0(T) is the entropy in the absence of a field. Thus
∂s E2 ∂T 8π
(10)
For an ER fluid, ∆S < 0; this obviously requires ∂s/∂T < 0. It is known that the Maxwell-Wagner equation can be used to well describe the dielectric properties of an ER fluid,9-12 and the Maxwell-Wagner polarization is the main source of the ER effect.7,13 If the particle conductivity σp is much larger than that of the medium, a simplified Maxwell-Wagner equation can be expressed14
[
) ∞ 1 +
K (1 + $2t2)
(13)
0(2sm + sp) σ
(14)
where 0, sm, and sp are the static dielectric constants of the vacuum, the medium, and the particles, Φ is the particle volume fraction, and ω is the frequency of the alternating electric field. The static dielectric constant s of an ER fluid is the dielectric constant at ω ) 0; thus, it can be expressed from eq 11 as
s ) ∞(1 + K)
∂s E2 S ) S0(T) + ∂T 8π
∆S )
(12)
9Φsm 2sm + sp
K)
∂s 1 ∂U ) s + T 2 8π ∂T ∂(E )
]
3Φ(sp - sm) 2sm + sp
]
(11)
in this equation (9) Weiss, K. D.; Carlson, J. D. Proceedings of the 3rd International Conference on Electrorheological Fluids; Tao, R., Ed.; World Scientific: Singapore, 1992; p 264. (10) Filisko, F. E. In Proceedings of the International Conference on ER fluids; Tao, R., Ed.; World Scientific: Singapore, 1992; p 116. (11) Hao, T.; Xu, Y. Proceedings of the 5th International Conference on Electrorheological Fluids; Bullough, W., Ed.; World Scientific: Singapore, 1996; p 55. (12) Conrad, H.; Chen, Y. Progress in Electrorheology; Havelka, K. O., Filisko, F. E., Eds.; Plenum Press: New York, 1995; p 55. (13) Hao, T. J. Colloid Interface Sci. 1998, 206, 240. (14) Wagner, K. W. Arch. Elektrotech. 1914, 2, 371.
(15)
Assuming sm and sp both are functions of T,
ds ) dT
[
1 + 3Φ +
]
54Φ2sm(2sp - 1.5smsp - 2sm) dsm dT (2 + )3 sm
sp 272smΦ2(sp
(2sm
- 4sm) dsp (16) dT + )3 sp
For the polar liquid materials, dsm/dT is always less than zero.4 So if sp > 4sm and dsp/dT > 0, ds/dT would be less than zero in any conditions; if sp > 4sm and dsp/dT < 0, the situation becomes complicated, and the ER effect is still theoretically possible if
( )/( )
dsp dsm < dT dT (1 + 3Φ)(2sm + sp)3 + 54Φ2m(2sp - 1.5smsp - 2sm) 27Φ22sm(sp - 4sm)
However, the possibility is not likely, as the entropy decrease is not so large (the particles contribute positively), and the ER solidification process would need a very large entropy decrease to form the bct crystalline structure. So only a weak ER effect or no ER effect would be generated; if the value of (dsp/dT)/(dsm/dT) is larger than the value of
920 Langmuir, Vol. 15, No. 4, 1999
Letters
(1 + 3Φ)(2sm + sp)3 + 54Φ2m(2sp - 1.5smsp - 2sm) 27Φ22sm(sp - 4sm) ds/dT would be larger than zero, a negative ER effect, and the viscosity of the fluid would decrease with increasing applied electric field. If sp < 4sm, physically dsp/dT would unlikely be less than zero, as the small dielectric constant is generally due to the polar group orientation, as well as the electronic and atomic polarization. For polar material with a comparatively small dielectric constant, its dsp/dT, likely of the same order as dsm/dT, would be larger than zero.4 So the ER effect could also possibly be generated under the condition of dsp/dT > 0, but it would still be very weak. Here, we just concentrate on the positive ER situation, that is, dsp/dT > 0. According to the Clausius-Mossotti formula, the static dielectric constant of one material can be expressed as15
-1 4 ) πN(Re + Ra + Ri + Rd) +2 3
(17)
where Re, Ra, Ri, and Rd are the polarizabilities of the electron, the atom, the ion displacement, and the dipole rotation and N is the number of molecules per unit volume. For a solid material, the dipole polarization is negligible, as the solidification usually fixes the molecule with such rigidity in the lattice that there is little or no orientation of the dipole in an electric field.4 Thus for ER particles
sp - 1 4 ) πN(Re + Ra + Ri) sp + 2 3
(
Inserting eq 24 into eq 21 and assuming (sp - 1)/(sp + 2) ) ps and (∞p - 1)/(∞p + 2) ) p∞,
(sp + 2)2β dsp ) [(n - 1)ps - (n + 2)p∞] dT 3
sp - ∞p 1 > ≈ 0.1 sp∞p n
t e-(t -t)/θ dt ) i$σ0E0∫-∞ei$t-(t -t)/θ dt ) ∫-∞t dE dt
di) σ0
)
Since (1/N)dN/dT ) -3β, where β is the material linear coefficient of expansion, eq 20 can be rewritten as
[
]
2 sp - 1 dsp 4πN dRi (sp + 2) ) -3 β+ dT sp + 2 3 dT 3
(21)
n+2 (see ref where Ri ) 2q2/k and k ) (2M(n - 1)q2rn-1 0 )/3r 15), q is the ion charge, k is the elastic bonding coefficient between two ions of opposite sign of charge, M and n are constants, and r0 and r are the equilibrium distance and the distance between the two ions. Assuming Ri and k are functions of temperature due to the fact that r would change with temperature,
Ri dk dRi )dT k dT
(22)
dk 1 dr ) -(n + 2)k dT r dT
(23)
According to definition, (1/r)dr/dT ) β, and from eq 19,
1
1
1
1
i$θσ0 E e-t1/θ e1+i$θ/θ 1 + i$θ 0
(19)
dRi dsp 4 dN 4 3 ) π(Re + Ra + Ri) + πN (20) 2 dT 3 dT 3 dT (sp + 2)
(26)
due to, for most materials, n ≈ 10 (see ref 15). Equation 26 is the dielectric requirement for an ER particle material. The conductivities of most ER particle materials are comparatively low, and one would assume that the dielectric loss results only from the ion displacement polarization. The dielectric loss due to the ion displacement polarization can be calculated from the absorption current. Assuming that the initial conductivity of an ER particle material is σ0 and that the absorption current can be expressed as σ0e-t/θ, with t being the time, θ being a time constant, θ ) (sp + 2)/(∞p + 2)τ, and τ being the relaxation time, under an oscillatory electric field E ) E0eiω˜ t, the induced current density di due to the ion displacement polarization can be expressed as15
(18)
Differentiating eq 18 with respect to T,
(25)
Only when ps/p∞ > (n + 2)/(n - 1), would dsp/dT be larger than zero. This requirement would lead to
At a very high-frequency region, eq 18 would lead to
∞p - 1 4 ) πN(Re + Ra) ∞p + 2 3
)
dRi 3 sp - 1 ∞p - 1 ) Ri(n + 2)β ) (n + 2)β dT 4πN sp + 2 ∞p + 2 (24)
$2θ2σ0
$θσ0 E0ei$t1 + i E0ei$t1 1+$ θ 1 + $2θ2
(27)
2 2
where t1 is a constant. The first term of the above equation has the same phase as the applied electric field and can be called the ohmic component dio, which would contribute to the dielectric loss, while the second term has a (π/2) phase difference with the electric field and can be called the capacitive component dic, which just contributes to the polarization. Because the atomic and electronic polarizations also will contribute to the capacitive current density and the general relation between the induced current density d and the dielectric constant is d ) $E0/ 4π, the current density due to these two kinds of polarizations dae can be expressed as
dae )
$∞pE0 ; 4π
then the total capacitive current density dc can be written as
dc ) dic + dae ) $
(
)
θσ0 ∞p + E 4π 1 + $2θ2 0
(28)
According to definition, the dielectric loss tangent (15) Skanavi, G. I. Dielectric Physics; translated by Chen; Y. H. High Education Press: Beijing, China, 1958.
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Langmuir, Vol. 15, No. 4, 1999 921
dio $θ2σ0 tan δ ) ) dc ∞p (1 + $2θ2) + θσ0 4π
(29)
and the total dielectric constant
)
dc 4πθσ0 4π ) ∞p + $E0 1 + $2θ2
(30)
At a dc field (ω ) 0),
sp ) ∞p + 4πθσ0
(31)
Inserting eq 31 into eq 29,
tan δ )
$θ(sp - ∞p) sp + ∞p$2θ2
(32)
Equation 32 is a general expression for a solid material of a very low conductivity and marked ionic displacement polarization. Differentiating eq 32 with respect to ω, one would find that tan δ has a maximum value
tan δmax )
sp - ∞p 2xsp∞p
(33)
Comparing with eq 26, one would conclude that
tan δmax >
sp - ∞p > 0.1 sp∞p
(34)
as for many ER particle materials, (sp∞p)1/2 is always larger than 2. Equation 34 indicates that the suspended particles can become ordered under an electric field only when the maximum value of the dielectric loss tangent is larger than 0.1. Comparison and Conclusion Hao16,17 experimentally compared the dielectric properties of the ER fluids and the non-ER fluids and empirically proposed the dielectric loss tangent of dispersed solid material around 0.10 at 1000 Hz as a criterion for the ER effect. From our present analysis, the physical basis of this criterion becomes clear. Filisko18 experimentally found that the ions that are able to move freely on the particle surface but cannot move off it would be essential for the water-free zeolite ER systems and further suggested that the ER effect might be associated with the marked dielectric dispersion at low-frequency field.10 Ikazaki19,20 (16) Hao, T. Appl. Phys. Lett. 1997, 70 (15), 1956. (17) Hao, T.; Xu, Z.; Xu, Y. J.Colloid Interface Sci. 1997, 190, 334. (18) Treasurer, U.; Radzilowski, H. L.; Filisko, F. E. J. Rheol. 1991, 35 (6), 105. (19) Ikazaki, F.; Kawai, A.; Uchida, K.; Kawakami, T.; Edamura, K.; Sakurai, K.; Anzai, H.; Asako, Y. J. Phys. D: Appl. Phys. 1998, 31, 336.
and Kawai21 also found that the larger the dielectric constant difference below and above the dielectric relaxation frequency, the stronger the ER effect. All these facts mean the larger dielectric loss of dispersed particles would be important, basically agreeing well with our present theoretical predictions. There are many other theories presented before to predict the ER behaviors (for example, refs 5, 22, and 23). As pointed out in our previous paper,7 they indeed obtained good agreement between the theoretically predicted parameters and the experimentally measured values; however, some discrepancies still exist. The famous conduction model can only be used for the situation that the suspension microstructure has been fully formed.5,7 The presumptions of Khusid’s theory that the dispersed particles and the suspending medium have no intrinsic dielectric dispersion and that the variation of the applied electric field is very slow compared with the polarization rate are not always valid in the ER fluids case.5,7 The previous theories on the negative ER effect24,25 are based on the conduction model. A recent experimental result of the magnesium hydroxide/polymethylphenylsiloxane system26 strongly suggests that the conduction model does not reflect the physical essentials of the ER effect, as this system should have given a positive ER effect according to the conduction model; however, it gives a negative ER effect. Our present analysis shows that the negative ER effect may take place only if dsp/dT < 0; unfortunately, no experimental data are available for further comparison. In conclusion, from the basic fact that the entropy of an ER fluid would greatly reduce after an electric field is applied, we theoretically obtain that the maximum value of the dielectric loss tangent of the dispersed particle should be larger than 0.1; the ER effect might originally result from marked ion displacement polarization in the solid particles, and the negative ER effect only becomes possible if
dsp/dT < 0. Our result may shed light on the mechanism of the ER effect and can be used to understand the currently observed ER phenomena. The findings also have a clear implication on how to select the material to make a highperformance ER fluid, which would be no doubt important in the ER application process. The materials that have a marked dielectric dispersion, that is, the marked ion polarization in the solid case (abundant in mobile ions), are obviously preferred. LA981069B (20) Ikazaki, F.; Kawai, A.; Kawakami, T.; Konishi, M.; Asako, Y. Proceedings of the International Conference on ER fluids; Koyama, K., Nakano, M., Eds.; World Scientific: Singapore, 1998. (21) Kawai, A.; Uchida, K.; Kamiya, K.; Gotoh, A.; Yoda, S.; Urabe, K.; Ikazaki, F. Int. J. Mod. Phys. B 1996, 10, 2849. (22) Atten, P.; Foulc, J.-N.; Felici, N. Int. J. Mod. Phys. B 1994, 8, 2731. (23) Wu, C. W.; Conrad, H. Phys. Rev. E 1997, 56, 5789. (24) Boissy, C.; Atten, P.; Foulc, J.-N. J. Electrostatics 1995, 35, 13. (25) Wu, C. W.; Conrad, H. J. Rheol. 1997, 41 (2), 267. (26) Trlica, J.; Quadrat, O.; Bradna, P.; Pavlinek, V.; Saha, P. J. Rheol. 1996, 40 (5), 943.
