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The Yield Stress Equation for the Electrorheological Fluids 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 July 6, 1999. In Final Form: December 6, 1999 From the internal energy and entropy change of an electrorheological (ER) fluid under an external electric field, a general yield stress equation for the particle-type ER fluids is derived. Experiments were designed for testing the validity of our yield stress equation, and it is found that the experimental results and the prediction obtained from our yield stress equation agree very well. The currently observed ER phenomena reported in the literature are also able to be well understood by using our equation, especially the experimental facts which could not be explained well using the previous yield stress equations, for example, the ER behaviors of the BaTiO3/silicone oil system. Our present yield stress equation essentially differs from the previous ones, as it includes not only the dielectric constants of both liquid and solid materials but also a very important parameter related to the dielectric loss. The previous ER mechanism model proposed by us on the basis of experimental results,1 which emphasizes the role of the dielectric loss and particle turning process in the ER response, is thus materialized in this paper. Our results present a powerful tool to precisely estimate the yield stress based on the physical parameters of the suspension materials. They also provide insight into the mechanism of the ER effect and offer the clear implications on how to design high-performance ER fluids.
I. Introduction The electrorheological (ER) fluids are one kind of suspension whose rheological properties can reversibly change several orders of magnitude under a sufficiently strong electric field. The mechanism of the ER effect has been studied for a long time, and many models have been proposed for understanding the observed ER phenomena, accordingly yielding many shear stress equations.2-12 Although these models can work well for some ER systems, their effectiveness is limited in some other cases. For example, one of the most successful models, the conductive model, first proposed by Atten10 and then well developed by Conrad,11 gives very good agreement with the experimental results. However, it cannot give a reasonable explanation for why the dielectric loss of the ER particles was experimentally found also to play an important role in the ER response,13,14 indicating that it has much room to improve. Bonnecaze7 proposed a method to determine the static and dynamic (Bingham) yield stresses in the ER fluids based on a microstructural model, which relates * To whom correspondence may be addressed at the Department of Ceramic and Material Engineering, Rutgers, The State University of New Jersey, 607 Taylor Rd, Piscataway, NJ 08854-8065. Phone: 732-445-7092. Fax: 732-445-3258. E-mail:
[email protected] (1) Hao, T.; Kawai, A.; Ikazaki, F. Langmuir 1998, 14, 1256. (2) Adriani, P. M.; Gast, A. P. Phys. Fluids 1988, 31 (10), 2757. (3) Klingenberg, D. J.; Swol, F. van; Zukoski, C. F. J. Chem. Phys. 1989, 91, 7888. (4) Halsey, T. C.; Toor. W. Phys. Rev. Lett. 1990, 65, 2820. (5) Chen, Y.; Sprecher, A. F.; Conrad, H. J. Appl. Phys. 1991, 70, 6796. (6) Davis, L. C. Appl. Phys. Lett. 1992, 60, 319. (7) Bonnecaze R. T.; Brady, J. F. J. Rheol. 1992, 36 (1), 73. (8) ) Anderson, R. A. Langmuir 1994, 10, 2917. (9) ) Tao, R.; Jiang, Q.; Sim, H. K. Phys. Rev. E 1995, 52, 2727. (10) ) Felici, N.; Foulc, J. N.; Atten, P. In Proceedings of the International Conference on ER Fluids; Tao, R., Roy, G. D., Eds.; World Scientific: Singapore, 1994. (11) ) Wu, C. W.; Conrad, H. Phys. Rev. E 1997, 56, 5789. (12) ) Ma, H.; Wen, W.; Tam, W. Y.; Sheng, P. Phy. Rev. Lett. 1996, 77 (12), 2499. (13) ) Hao, T. Appl. Phys. Lett. 1997, 70 (15), 1956. (14) ) Hao, T.; Xu, Z.; Xu, Y. J.Colloid Interface Sci. 1997, 190, 334.
the yield stress to the electrostatic energy determined from the suspension capacitance matrix in terms of the particle volume fraction and particle-to-medium dielectric constant ratios. This theory is significant indeed as it correlates the yield stress with the microstructure of the ER suspension, and thus it can successfully predict the yield stresses of many systems. However, one of the main derivations of this theory that both the static and dynamic yield stresses will increase with the particle-to-medium dielectric constant ratios is not always valid (a typical exceptional case is the BaTiO3/oil system, see ref 1 herein). Generally speaking, Bonnecaze’s theory is essentially in accordance with the so-called polarization model, which was very popular 5 years ago. However, the disabilities of the polarization model have been realized soon after it was well recognized in that time and have been discussed in the literature. Ma12 also proposed a method to theoretically calculate the frequency-dependent static yield stress of the glass sphere/silicone oil system using the first principles and the Debye relaxation assumption. The calculated static yield stress is in good quantitative agreement with those measured experimentally. However, they neglected a very general fact: For most solid materials, the dipole is almost unable to reorient because the solidification usually fixes the molecule with such rigidity in the lattice that there is little or no orientation of the dipoles even in an extremely strong electric field. Therefore only a few solid materials exhibit the Debye type relaxation behavior, and the limitation of Ma’s yield stress equation is thus obvious. The more accurate and powerful yield stress equation, which would adequately correlate the dielectric properties of particles with the ER properties of the suspension and can well explain the currently observed ER behaviors, has not been obtained yet due to the lack of sufficient deep understanding of the ER mechanism. Recently, we proposed a qualitative model1 on the basis of the experimental results, assuming that the particle turning process and particle polarization process are both important, and the interfacial polarization
10.1021/la990881r CCC: $19.00 © 2000 American Chemical Society Published on Web 02/04/2000
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Langmuir, Vol. 16, No. 7, 2000 3059
would be responsible for the ER effect. A large interfacial polarization (also called the Maxwell-Wagner polarization) would facilitate the particle to attain a large amount of charges on the surface, then leading to the turning of particle along the direction of the applied electric field to form a fibrillation structure; the strength of the fibrillation chains is thus determined by the particle polarization ability, i.e., the particle dielectric constant. To generate sufficiently large interfacial polarization, the particulate material of large dielectric loss was experimentally found to be necessary. An empirical criterion is that the dielectric loss tangent of dispersed particulate material must be larger than 0.1 at 1000 Hz (see ref 1 herein). The non-ER particles were assumed to be unable to turn along the direction of the external electric field to form the fibrillation structure, thanks to the very small dielectric loss. The currently observed ER phenomena can be reasonably understood with our framework. Establishment of the yield stress equation on the basis of our successful model thus is significant and would be also of considerable interest to the electrorheology community. In this paper, we make an attempt to materialize this ER mechanism model into a meaningful theoretical tool to describe the ER behaviors quantitatively. We use a new way (to our knowledge), from the internal energy and entropy change of an ER fluid under a static electric field, to estimate the interparticle force and then the yield stress of whole system and find that the derived equation can reasonably describe the ER phenomena observed so far. We also employ the experimental method to test the validity of our yield stress equation and its derivations. II. Theoretical Treatment 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 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:15
U ) U0(T) +
(
)
∂s E2 s + T 8π ∂T
(1)
where U0(T) represents the internal energy of the ER suspension in the absence of a field and s represents the static dielectric constant of the whole suspension. Thus the internal energy change ∆U due to the applied the electric field is
∆U )
(
)
∂s E2 +T 8π s ∂T
(2)
Accordingly the entropy change ∆S
∆S )
E2 ∂s 8π ∂T
(3)
After an external electric field is applied to the ER suspension, the suspension becomes a body-centered tetragonal (bct) crystalline. The entropy change obviously includes two parts, the one is the particle configuration entropy, which represents the entropy change from the randomly distributed particle state to the bct lattice state; the other is the entropy change from the very weak (15) ) Frohlich, H. The Theory of Dielectrics; Clarendon, Oxford, 1958; Chapter 1.
Figure 1. Schematic illustration of the assumed two-step process during the solidification transition of the ER suspension under an external electric field.
interparticle force state to the exceptionally strong interparticle force state. The former part would contribute to the particle rearrangement, while the latter part would contribute to the ER effect, which is thought to be originally induced by the interfacial polarization (the MaxwellWagner polarization), according to our theory.1 We therefore presumedly thought there are two steps involved in this process: At the first step, the interfacial polarization does not take place, and the dispersed particles are arrayed to the bct lattice, due to the applied electric field; however, the interparticles force in this state is extremely weak and could be negligible. At the second step, the interfacial polarization takes place, making the already well-arranged particles become strongly correlated. These processes are schematically illustrated in Figure 1. According to our previous theories,1,13 the first step would be controlled by the dielectric loss of the dispersed particles, and the second step is controlled by the dielectric constant. To determine the yield stress of the ER suspension, one has to know the internal energy and entropy change of the second step, ∆S2 and ∆U2, respectively. Obviously
∆S2 ) ∆S - ∆S1
(4)
∆U2 ) ∆U - ∆U1
(5)
The above equations mean that one can easily know the second step internal energy and entropy change if the total internal energy and entropy change, as well as the first step internal energy and entropy change, are determined. The parameters, s and ∂s/∂T, involved in eq 1, could be exactly obtained by employing the Maxwell-Wagner model,16 as previous works17-20 clearly show that the Maxwell-Wagner model can well describe the dielectric properties of the ER fluids. Although the MaxwellWagner equation just holds for a diluted suspension (the particle volume fraction is less than 0.1) and the Hanai equation21 is suitable for concentrated suspensions, we still use the Maxwell-Wagner equation in this paper, since these two models are not so different until reaching a (16) ) Wagner, K. W. Arch. Elektrotech. 1914, 2, 371. (17) ) Weiss K. D.; Carlson, J. D. Proceedings of the 3rd International Conference on Electrorheological Fluids; Tao, R., Eds.; World Scientific: Singapore, 1992; p 264. (18) ) Filisko, F. E. Proceedings of the 3rd International Conference on Electrorheological Fluids; Tao, R., Eds.; World Scientific: Singapore, 1992; p 116 (19) ) Hao, T. J.Colloid Interface Sci. 1998, 206, 240. (20) ) Conrad H.; Chen, Y. Progress in Electrorheology; Havelka, K. O., Filisko, F. E., Eds.; Plenum Press: New York, 1995; p 55. (21) ) Hanai, T. In Emulsion Science; Sherman, P., Eds.; Academic Press: 1968; Chapter 5.
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Hao et al.
particle volume fraction larger than 0.4. According to the Maxwell-Wagner equation, the static dielectric constant of whole suspension s can be expressed as
(
s ) sm 1 + 3φ +
)
(6) ∆U2 )
Providing that sm and sp both are the function of T, then
[
ds ) 1 + 3Φ + dT
sp
272smΦ2(sp (2sm
(
]
- 4sm) dsp (7) dT + )3
)
sm
]
sp
sm sp 272smΦ2T(sp
[
[
}
- 4sm) dsp dT + )3
(2sm sp dsm E2 ) U0 + (1 + 3Φ) sm + T + 8π dT
)
27Φ22sm(sp - sm) (2sm + sp)2 54Φ
- 1.5smsp -
sm
sp
sm
Then total internal energy of the step II, U2
∆U2 ) -
[
2 2 E2 27Φ sm(sp - sm) 8π (2 + )2 sm
sp
sp
2sm)
dsm dT
]
(8)
For the first step, we assumed that the interfacial polarization does not take place, which is only physically likely under the condition of sm ) sp. In this case, the static dielectric constant of this assumed system, according to the eq 6, is sm(1 + 3Φ). The influx of electric energy into the assumed system for arraying the disordered particles to the very loose bct lattice thus can be expressed as an analogue to eq 2
[
]
∂sm E2 sm(1 + 3Φ) + (1 + 3Φ)T 8π ∂T
(
) (1 + 3Φ) sm + T and the entropy change
)
dsm E2 dT 8π
]
27Φ22smT(sp - 4sm) dsp (14) dT (2 + )3
27Φ22smT(sp - 4sm) dsp dT (2 + )3
∆U1 )
(13)
where U02 represents the internal energy of the system without any particle interaction. Thus we may say that only ∆U2 contributes to the particle interaction force, i.e., the mechanical strength of the ER suspension. The ∆U2 should be less than zero, as the interparticle force in the ER crystalline lattice is attractive. Thus eq 11 should be expressed
sm
sp
sp
54Φ2smT(2sp - 1.5smsp - 2sm) dsm + dT (2 + )3
3
sm
]
27Φ22smT(sp - 4sm) dsp (12) dT (2 + )3
+
(2sm + sp)
]
2 2 2 E2 54Φ smT(sp - 1.5smsp - sm) dsm 8π dT (2 + )3
sm
smT(2sp
2
sp
U2 ) U02 + ∆U2
54Φ2sm(2sp - 1.5smsp - 2sm) dsm 1 + 3Φ + dT (2 + )3
(
∆S2 )
sm
sp
∂s E2 s + T 8π ∂T 27Φ2sm(sp - sm) E2 ) U0 + m 1 + 3Φ + +T 8π (2 + )2
{[
sp
27Φ22smT(sp - 4sm) dsp (11) dT (2 + )3
where sm and sp are the static dielectric constants of the medium and the particles; Φ is the particle volume fraction. Then
[
54Φ2 smT(2sp - 1.5smsp - 2sm) dsm dT (2 + )3
54Φ2sm(2sp - 1.5smsp - 2sm) dsm dT (2 + )3
U ) U0 +
[
2 2 E2 27Φ sm(sp - sm) + 8π (2sm + sp)2
sm
sm
(10)
Therefore according to the eqs 4 and 5
27φ2sm(sp - sm) (2sm + sp)2
E2 ∂sm ∆S1 ) (1 + 3Φ) 8π ∂T
sp
∆S2 should also be less than zero; as in step II, the entropy obviously decreases substantially, requiring sp > 4sm at dsp/dT > 0 and sp < 4sm at dsp/dT < 0. These are the preconditions of eq 14. The eq 14 gives the internal energy change per unit volume of the whole suspension. For an ER suspension of volume V, particle volume fraction Φ, and particle diameter r, the particles number N in this volume
N)
3VΦ VΦ ) 3 (4/3)πr 4πr3
(15)
then the interparticle energy Uip
Uip )
∆U2V 8πr3∆U2 ) N/2 3Φ
(16)
The free volume per particle Vfp under off-state electric field
(9)
Vfp )
3 V - VΦ 4πr (1 - φ) ) N 3Φ
(17)
It is known that under an electric field the ER particles
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Langmuir, Vol. 16, No. 7, 2000 3061
for dsm/dT and dsp/dT. The Clausius-Mossotti equation23 provides a means to relate the macrocopic dielectric property with its microcopic propertysthe number of molecules per unit volume and the molecule polarizability. The dielectric property changes with temperature, as the number of the molecule per unit volume also changes due to the material thermal expansion. The detailed treatment can be found in ref 23 and further refined in ref 31. According to the derived equation based on the ClausiusMossotti equation
Figure 2. Schematic illustration of the unit cell of the bct lattice formed by the ER particles under an electric field. The radius of the particle is r.
would form the fibrillation chains, of the bct lattice. The unit cell of the bct lattice is schematically illustrated in Figure 2. One center particle would have eight nearest neighbor particles of the interparticle distance 2r, which belongs to the different chain class; besides, one center particle also has another six nearest neighbor center particles also of the interparticle distance 2r, which belong to the same chain class. So the total energy for one particle in the unit cell Uit
112πr3∆U2 Uit ) (8 + 6)Uip ) 3Φ
VfE ) Vfp -
(
)
3
Note that in one unit cell there are two equivalent particles. Assuming that the yield stress τy is the force that can drive one particle to move within the free volume per particle22 thus
|Uit| 112π ) τy ) |∆U2| VfE 4π(1 - Φ) - 18Φ 27Φ22sm(sp - sm) 14E2 ) 4π(1 - Φ) - 18Φ (2sm + sp)2
[
54Φ2 smT(2sp - 1.5smsp - 2sm) dsm + dT (2 + )3 sp
]
27Φ22smT(sp - 4sm) dsp dT (2 + )3 sm
sp
(sm - 1)(sm + 2) dsm )βm dT 3
(18)
4πr (1 - Φ) - 18Φr 2r × x6r × x6r ) 2 3Φ (19)
sm
βp is the particle material linear coefficient of expansion, n is a constant, between 4 and 19.7, depending on the solid crystalline state,23 ps ) (sp - 1)/(sp + 2), and p∞ ) (∞p - 1)/(∞p + 2), p ) [(n - 1)ps - (n + 2)p∞]. The liquid media used in the ER fluids are usually nonpolar material, and according to the Clausius-Mossotti equation,23 too
(22)
where βm is the liquid linear expansion coefficient. Assuming ξ ) sp/sm, eq 20 can be rewritten as
Since the particle could not freely go out of the cell of the lattice, the free volume per particle under an electric field, VfE, would become 3
(sp + 2)2β dsp ) [(n - 1)ps - (n + 2)p∞] ) dT 3 pβp(sp + 2)2 (21) 3
(20)
Equation 20 presents the correlation between the yield stress of the ER suspension and the physical parameters of the used materials. Principally, we can easily get the yield stress value if all the parameters are already known. Equation 20 can be further developed to a more meaningful form. The important thing is to get a general expression (22) ) This is a common way to express the yield stress based on the interparticle force; for example, see Professor M. Doi’s paper: J. Rheol. 1997, 41 (3), 769.
τy )
{
3sm(ξ2 + ξ - 2) 126Φ2E2 + 4π(1 - Φ) - 18Φ (2 + ξ)3 2
2βmT(ξ - 1.5ξ - 1)(2sm + sm - 2)
+ (2 + ξ)3 Tβpp(ξ - 4)(smξ + 2)2 (2 + ξ)3
}
(23)
The physical meaning of eq 23 is discussed in section IV. III. Experimental Section For the purpose of testing the validity of the yield stress equation derived above, eqs 20 and 23, we carried out the experimental works for comparison. We used the poly(dimethylsiloxane) oil as the dispersing medium and the aluminosilicate (zeolite 3A) and BaTiO3 powder as the dispersed particulate materials. The zeolite/silicone oil suspension was selected because it is a typical ER system that has been comprehensively studied. The BaTiO3/silicone oil suspension was selected because it displays amazing ER behavior that could not be easily understood with the currently accepted polarization and conductivity models. The linear expansion coefficient of poly(dimethylsiloxane) oil, βm, is 9.8 × 10-4, provided by the manufacturer. All the materials (polydimethylsilicone oil, Zeolite 3A, and BaTiO3) were purchased and used as received. The zeolite 3A was heated at 500 °C for 1 day to remove the water totally, and the BaTiO3 was heated at 550 °C for 8 h, before they were separately mixed with the silicone oil for making the suspensions. The electrorheological properties of these two suspensions were measured with a rotational viscometer (VT550, Haake), and the dielectric properties were measured with a LCR meter (HP 4284A). As for the dielectric properties of the powder zeolite 3A and BaTiO3, a typical way to determine the dielectric properties of the powder material was employed. The dry powder was compacted into a pellet with 2 cm in diameter and 0.1 cm in thickness, and then both sides of the pellet were coated with Pt by using a coating machine. (23) ) Skanavi, G. I. Dielectric Physics; translated by Y. H. Chen; High Educational: Beijing, 1958.
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Figure 3. Dielectric constant of poly(dimethylsiloxane) oil vs frequency at different temperatures (20-120 °C).
Hao et al.
Figure 4. Static dielectric constant of poly(dimethylsiloxane) oil vs temperature.
IV. Comparison with the Experimental Data Equation 23 obviously indicates that the yield stress of an ER fluid would increase with the square of the applied electric field, the particle volume fraction, and the dielectric constant of the liquid medium, which agree very well with the previous experimental results.24-27 If p is of a positive value, the yield stress thus will increase with ξ increasing, as the numerator increases much faster than the denominator with ξ, which is in consistent with the prediction given by the polarization model28,29 and Bonnecaze’s theory. However, if p is of a negative value, then the yield stress will decrease with ξ increasing, which is unable to be explained by the polarization model but was experimentally observed. The important parameter in eq 23 is p, which features our equation differing from others developed so far. A positive p would obviously benefit the yield stress of the whole suspension; however, a negative one would make the yield stress become small. Furthermore, for some extreme cases, the yield stress would become negative, leading to a negative ER effect (the total shear stress of a suspension would be the shear stress arising from the viscosity of the suspending medium, plus the shear stress from the interparticle potential forces, see ref 30). The parameter p only becomes positive when the dielectric loss tangent of the dispersed solid material is larger than 0.1, as we demonstrated before both experimentally and theoretically.13,14,31 This requirement now is involved in our present yield stress expression. It is known that the particle volume fraction can dramatically influence the yield stress of ER fluid. We first would like to address this issue qualitatively according to our yield stress equation. Although the MaxwellWanger equation only can be applicable for the diluted suspension, the deviation is not so large when applied to the concentrated one.21 We therefore can approximately estimate the yield stress dependence of particle volume fraction using our yield stress equation (20). We consider the silicone oil ER system. Figure 3 shows the dielectric constant of poly(dimethylsiloxane) oil dependence on (24) ) Sprecher, A. F.; Chen, Y.; Choi, Y.; Conrad, H. In Proceedings of the International Conference on ER Fluids; Tao, R., Eds.; World Scientific: Singapore, 1992. (25) ) Marshall, L.; Zukoski, C. F.; Goodwin, J. J. Chem. Soc., Faraday Trans. 1989, 85, 2785. (26) ) Block, H.; Kelley, J. P.; Qin, A.; Watson, T. Langmuir 1990, 6, 6. (27) ) Hao, T.; Chen, Y. H.; Xu, Z. M.; Xu Y. Z.; Huang, Y. Chin. J. Polym. Sci. 1994, 12, 97. (28) ) Klingenberg, D. J.; Zukoski, C. F. Langmuir 1990, 6, 15. (29) ) Davis, L. C. J. Appl. Phys. 1992, 72 (4), 1334. (30) ) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, 1986. (31) ) Hao, T.; Kawai A.; Ikazaki, F. Langmuir 1999, 15, 918.
Figure 5. Normalized yield stress, τy/E2 × 102, vs particle volume fraction at the particle-to-oil dielectric constant ratio ξ ) 10, and dsp/dT ) 0.4. This is computed on the basis of eq 20.
frequency obtained at different temperatures (20-120 °C). The experimental result shows that the dielectric constant decreases very slightly, and no dielectric loss was observed within the measurement frequency range (20-1 × 106 Hz). With temperature increasing, the dielectric constant decreases obviously. Figure 4 shows the dielectric constant at 20 Hz as a function of temperature. For silicone oil it would be reasonable to regard the dielectric constant at low frequency as the static dielectric constant because there is no detectable dielectric loss. Thus according to the Figure 4 sm at room temperature could be valued as 2.69. The slop of the linear regression line in Figure 4 is -2.4 × 10-3 °C-1, which could be regarded as the value of dsm/dT. For a good ER suspension, dsp/dT usually should be larger than zero,31 and here it is assumed to be a value of 0.4. Figure 5 shows the yield stress as the function of the particle volume fraction at the particleto-oil dielectric ratio ξ ) 10, predicted by eq 20. When the particle volume fraction is small, the yield stress slightly increases with the particle volume fraction increasing, which is in a good agreement with the experimental results;25,26 when the particle volume fraction is around 0.35, the yield stress dramatically increases, indicating that a critical volume fraction would exist in the ER suspension. In our previous work,27 we also found that there is a critical volume fraction value in the ER suspension, and once the particle volume fraction is larger than this value, the rheological properties of the ER suspension would dramatically increase. This critical value was experimentally found to be around 0.37 on the basis of rheological investigation27 and was calculated to be around 0.4 in aid of the Flory’s gelation theory and the
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Langmuir, Vol. 16, No. 7, 2000 3063
Figure 6. Dielectric constant of the pure zeolite 3A as a function of frequency obtained at different temperatures (20-120 °C).
Figure 7. Dielectric loss of the pure zeolite 3A as a function of frequency obtained at different temperatures (20-120 °C). Table 1. Static Dielectric Constants of Pure Zeolite 3A
sp
20 °C
40 °C
60 °C
80 °C
100 °C
120 °C
54.08
59.76
63.76
74.55
80
96.15
percolation concept.32 The critical volume fraction value given by the present yield stress equation is quite close to the previous data, indicating that eq 20 gives a pretty good prediction! The influence of particle-to-oil dielectric constant ratio, ξ, on the yield stress of the ER fluids would be an interesting topic, too. Since dsp/dT would vary with ξ, according to eq 21, we would like to address this issue by using eq 23. To do so, one needs to know the important parameter value, βpp, of the solid material. Here we take the zeolite 3A as an example. Figure 6 and Figure 7 show the dielectric constant and dielectric loss of the pure zeolite 3A as a function of frequency obtained at different temperatures (20-120 °C), respectively. Using the ColeCole plot, one can easily determine the static dielectric constant of pure zeolite 3A at different temperature, which are showed in Table 1. According to the data presented in Table 1, dsp/dT could be fixed of the value 0.40. Thus around the room temperature, βpp ) 3.82 × 10-4, according to eq 21. Then on the basis of the eq 23, one can get the numerical relationship between the yield stress and the particle-to-oil dielectric constant ratio of the silicone oil ER system, which is showed in Figure 8. Figure 8 clearly shows at low particle-to-oil dielectric constant ratio, the yield stress would dramatically increase with ξ increasing; however, the yield stress gradually levels off after ξ is more than 60. The sharp increase of the yield stress takes place in the range of ξ < 50, indicating that the materials (32) ) Hao, T.; Xu, Y. Z. J. Colloid Interface Sci. 1996, 181, 581.
Figure 8. Numerical relationship between the normalized yield stress, τy[4π(1 - Φ) - 18Φ]/Φ2E2, and the particle-to-oil dielectric constant ratio, ξ, of the silicone oil ER system, based on eq 23 (at room temperature).
Figure 9. Dielectric constant of pure BaTiO3 as a function of frequency obtained at different temperatures (20-120 °C).
Figure 10. Dielectric loss of pure BaTiO3 as a function of frequency obtained at different temperatures (20-120 °C).
of the static dielectric constant around 150 would potentially display the best ER effect and the materials of the larger static dielectric constant would not always surely display a good ER effect if other properties are not beneficial to the ER effect. It should be noted that the above conclusion is derived under the assumption that the parameter p is of a positive value. If p is negative, the larger particle-to-oil dielectric constant ratio would not generate a larger yield stress. Here we would like to take BaTiO3 as an example, as BaTiO3 usually has a very larger dielectric constant (around 2000, depending on its crystalline state). Figure 9 and Figure 10 show the dielectric constant and dielectric loss of pure BaTiO3 as a function of frequency obtained at different temperatures (20-120 °C), respectively. Also using the Cole-Cole plot, one can determine the static dielectric constant, which is showed
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Figure 11. Static dielectric constant of BaTiO3 as a function of temperature.
Figure 12. Yield stress of BaTiO3/silicone oil suspension vs temperature. The particle volume fraction is 0.20.
in Figure 11 against temperature. As BaTiO3 is a ferroelectric material and its dielectric properties greatly depends on its crystalline state and manufacture method, the static dielectric constant of BaTiO3 displays an unusual temperature dependence: within temperature range 2040 °C, it sharply decreases, while it barely decreases between 60 and 120 °C. At the low-temperature range, dsp/dT is -47.5, and at the high-temperature range, its value is -1.08. On the basis of the static dielectric constant data obtained at different temperatures, one may calculate the yield stress of a BaTiO3/silicone oil suspension according to eq 20. The computed yield stresses at all temperatures (20, 40, 60, 80, 100, and 120 °C) are negative, indicating the BaTiO3/silicone oil suspension would not display the ER effect though the BaTiO3 has a very large dielectric constant, which was already verified experimentally (see ref 1 herein). The predicted negative yield stress means that the interparticles force is not attractive, but instead repulsive. However, whether the BaTiO3/ silicone oil suspension displays the negative ER effect would greatly depend on which interaction, the fieldinduced repulsive force or the off-field original interparticle force, is a main contribution to the apparent mechanical properties of whole suspension. Note that eq 20 is not appropriate to predict the negative ER effect, as it involves the assumption that the particle will form a bct lattice under an electric field. For accuracy reasons, we conducted the experimental measurement of the yield stress of a BaTiO3/silicone oil suspension with the particle volume fraction 0.20. Figure 12 shows the yield stress of a BaTiO3/ silicone oil suspension as a function of temperature, experimentally determined under a zero and 2 kV/mm electric field, separately. The BaTiO3/silicone oil suspension indeed does not show any positive ER effect; in contrast, it gives a slight negative ER effect. As a result,
Hao et al.
Figure 13. Yield stress of the zeolite/silicone oil suspension vs temperature at the electric field E ) 2 kV/mm: solid line, calculated from eq 23; black points, experimental data. φ ) 0.23.
the positive p would be very important for an ER material, which was addressed in detail in our previous paper.31 It would be very interesting and forceful to quantitatively compare our prediction derived from the yield stress equation and the experimental results. We take the zeolite/ silicone oil system of the particle volume fraction 0.23 as an example. Using the static dielectric data given in the Table 1 and the yield stress eq 23, one can easily get the yield stress values at different temperature. The calculated yield stress values vs temperature is showed in Figure 13 as a solid line. For comparison, the experimentally measured values are also shown in Figure 13 as black points. As we can see, the predicted values agree very well with the experimental ones, indicating that eq 23 is actually very powerful. As p is a very important parameter in our present yield stress equation, we therefore would like to concentrate ourselves on parameter p in the following section and to see which physical parameters would greatly influence the p value and then the yield stress. According to the definition
∞p - 1 sp - 1 - (n + 2) p ) (n - 1) sp + 2 ∞p + 2 ∞p - 1 sp - 1 ∞p - 1 -3 ) (n - 1) sp + 2 ∞p + 2 ∞p + 2
(
)
(24)
Assuming sp - ∞p ) A
∞p - 1 A p ) 3(n - 1) -3 (∞p + 2 + A)(∞p + 2) ∞p + 2 (n - ∞p)A - ∞p(∞p + 1) + 2 ) )3 (∞p + 2)(∞p + 2 + A)
[
]
3(n - ∞p)A - 3∞p(∞p + 1) + 6 (∞p + 2)2 + (∞p + 2)A (25) If ∞p is not so large (less than 10 for most small dielectric constant materials), 3(n - ∞p) is always larger than (∞p + 2); thus p would increase with A increasing, i.e., the ER effect would approximately increase with the difference between the dielectric constants below and above the relaxation frequency, which was already experimentally
The Yield Stress Equation
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found by Ikazaki and Kawai.33,34 For a solid material of a leak conductivity σ, an initial conductivity σ0, and the relaxation time related constant θ, it is known that22
sp - ∞p ) 4πθσ0
(26)
and
tgδ )
j 2θ2σ0 σ(1 + ω j 2θ2) + ω
(27)
(∞pω j /4π)(1 + ω j 2θ2) + ω j θσ0
thus
A ) 4πθσ0 )
(
)
∞pω j tgδ -σ 4π ω j 2θ - ω j tgδ
4π(1 + ω j 2θ2)
(28)
where ω j is field frequency and tgδ is the dielectric loss tangent of the solid material. Equation 28 indicates that A would obviously increase with tgδ increasing, provided that ω j θ > tgδ (this always holds, as A should have a positive value). Differentiating eq 28 with respect to σ, one would find there is a maximum value for A at
(
)
∞pω j dtgδ -1 ω j 2θ + ω j tgδ 4π dσ σ) dtgδ ω j dσ
(29)
as the second derivative of A with respect to σ, d2A/dσ2, is negative. Considering that ω j θ > tgδ and (∞pω j /4π)(dtgδ/ j 2θ/4π. For solid materials, one would dσ) . 1, thus σ ≈ ∞pω assume the Debye polarization is unable to occur and the dielectric adsorption phenomenon just stems from the ion j ) (1/θ)(sp/∞p)1/2, the ion polarization,15,22,31 and at ω polarization-induced tgδ would reach a maximum value.22 In this case, the parameter A also reaches at a maximum value, thus σ ≈ sp/4πθ, which indicates that for different dispersed solid material, the yield stress would peak at different particle conductivities. In the poly(acenequinones)/silicone oil system, the yield stress was found to peak at a particle conductivity around 10-5 S/m;26 however, in the oxidized polyacrylonitrile/silicone oil system, the yield stress maximum value was found to occur near 10-7 S/m.14 According to the dielectric data presented in each paper, the optimal conductivity (σ ≈ sp/4πθ) is crudely estimated at 0.22 × 10-5 S/m for the former system and 0.88 × 10-7 S/m for the latter system, agreeing well with the experimental results. The temperature dependence of the yield stress can also be qualitatively analyzed using eq 23. Since σ and tgδ are much sensitive than the dielectric constant to temperature, one would still center on the temperature dependence of the parameter p. For most solid dielectric materials the conductivity would exponentially increase with temperature, thus the conductivity, rather than other parameters, would be a main variable and surely make a big contribution to the yield stress as temperatures vary (see eq 28). The yield stress would also go through a maximum value at the temperature where the conductivity reaches the optimal value. The yield stresses first increasing and then decreasing with temperature were already found experi(33) ) Ikazaki, F.; Kawai, A.; Uchida, K.; Kawakami, T.; Edamura, K.; Sakurai, K; Anzai, H.; Asako, Y. J. Phys. D: Appl. Phys. 1998, 31, 336. (34) ) Kawai, A.; Uchida, K.; Kamiya, K.; Gotoh, A.; Yoda, S.; Urabe, K.; Ikazaki, F. Int. J. Mod. Phys., B 1996, 10, 2849.
mentally.35,36 Accordingly, the yield stress would decrease with temperature if the conductivity of the solid particles is already larger than the optimal value, while the yield stress would increase with temperature if the conductivity is lower than the optimal value, which were also experimentally found before.36 The relaxation time constant θ would also influence the parameter A, finally p substantially. From either eq 26 or eq 28, one would find p will increase with θ increasing; that is, the ER effect would be stronger if the dielectric relaxation is slower. However, a too slow relaxation time (then the slow response time) would make the ER fluids become useless. Generally, the ER response time around 1 ms is favorable, thus requiring the relaxation time be of the same time scale, i.e., a dielectric relaxation frequency around 103 Hz. Block presumedly thought the polarization rate would be important in the ER response process, and too fast or too slow polarization is not beneficial to the ER effect.26 Ikazaki and Kawai experimentally found that the ER fluids whose relaxation frequency is within the range 100-105 Hz would exhibit a large ER effect,33,34 supporting our present derivation. V. Conclusion and Discussion From the internal energy and entropy changes of an ER fluid under an external static electric field, we derive a general yield stress equation for the particle-type ER fluids. We also use the experimental tools to test the validity of the derived yield stress equation and found that the prediction and the experimental results are in very good agreement. In addition, this equation also can give very good predictions in accordance with the previous results obtained so far. For the currently observed ER phenomena, for example, the yield stress increases with the square of the applied electric field, the particle volume fraction, and the dielectric constant of the liquid medium; there is a critical particle volume fraction, 0.37, in the ER suspension; the large dielectric loss tangent value of the dispersed solid particles (the maximum of tgδ larger than 0.1) is found to be necessary for a good ER effect; the larger the difference of the dielectric constant before and after the dielectric dispersion, the larger the ER effect; and the particle conductivity and temperature dependence of the yield stress, etc., can be well understood by using our equation. BaTiO3, with a very large dielectric constant, would not present any ER effect under a dc field according to our yield stress equation, also agreeing with the experimental result. Our present equation is mainly based on two assumptions: the first is that only the interfacial polarization contributes to the ER effect; the second is that the ER particles would form the bct lattice structure under an external electric field. The first assumption gets the dielectric loss tangent of dispersed particulate material involved in the yield stress equation, which is essentially different from the former models. The second assumption lays our equation on the ground of the microstructure of the ER suspension, guaranteeing that our equation can describe the yield stress of the ER suspension properly. This equation involves a very important parameter, p, which gives an appropriate expression for the dielectric loss tangent criterion put forward before, and features our present equation differing from others. When p is positive, the yield stress would increase with the particleto-medium dielectric ratio ξ increasing, as showed by the (35) ) Conrad, H.; Li, Y.; Chen, Y. J. Rheol. 1995, 39 (5), 1041. (36) ) Hao, T.; Yu, H.; Xu, Y. Z. J. Colloid Interface Sci. 1996, 184, 542.
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polarization model and other models. However, the yield stress gradually levels off after ξ > 60. The sharp increase of the yield stress takes place in the range of ξ < 50, indicating that the materials of the static dielectric constant around 150 would potentially display the best ER effect (in silicone oil system) and the materials of the larger static dielectric constant would not always surely display a good ER effect if other properties are not beneficial to the ER effect. When p is negative, the suspension would give a weak or even no ER effect. For the purpose of making the value of p as larger as possible, the very slow relaxation time and suitable particle conductivity are essential. Since our equation is derived from the internal energy change under a static electric field, it is unsuitable to account for the frequency dependence of the yield stress. The internal energy change under an oscillating electric field is too complicated for getting a meaningful result.
Hao et al.
Our equation provides a reasonable relationship between the yield stress and the physical parameters of the ER fluids and thus offers a clear implication on how to design a high-performance ER fluid. To get a good ER fluid, from eq 23, one should use the liquid medium and solid material which are of high dielectric constant (large dielectric constant ratio of the solid material to the liquid). In addition, the solid material must have a large dielectric loss tangent (at least its maximum value larger than 0.1) for keeping the parameter p positive. Our present paper materializes our previously qualitative model to a quantitative mathematical expression, which can well describe the ER behaviors, indicating the correctness of our model, thus shedding light on the mechanism of the ER effect substantially. LA990881R
