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Generalized yield stress equation for electrorheological fluids
Journal of Colloid and Interface Science 409 (2013) 259–263
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Short Communication
Generalized yield stress equation for electrorheological fluids Ke Zhang a,b, Ying Dan Liu a, Myung S. Jhon c, Hyoung Jin Choi a,⇑ a
Department of Polymer Science and Engineering, Inha University, Incheon 402-751, Republic of Korea School of Chemical Science and Technology, Harbin Institute Technology, Harbin 150001, Peoples Republic of China c Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA b
a r t i c l e
i n f o
Article history: Received 5 June 2013 Accepted 1 August 2013 Available online 11 August 2013 Keywords: Electrorheological fluid Yield stress Universal equation Critical electric field
a b s t r a c t A new generalized yield stress scaling equation for electrorheological (ER) fluids was developed by introducing the critical electric field (Ec) and material parameter. This equation can be used to describe the dependency of the yield stress on an electric field not only for conventional ER suspensions with a change in slope from 2.0 to 1.5, but also for giant ER fluids with a change in slope from 2.0 to 1.0. The yield stress data obtained from different ER fluid systems with different material parameters was collapsed onto a single curve for the entire range of electric field strengths using the proper scaling method proposed in this study. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction
sy / /K f E20 f ðcÞ
Electrorheological (ER) fluids, which are composed of higher dielectric and/or conducting particles dispersed in an insulating liquid in general, can be converted drastically from a liquid-like to a solid-like phase in the order of milliseconds when either an AC or a DC electric field is applied [1–6]. The rapid response to an external electric field is related to the formation of fibrillar structures aligned along the direction of the electric field due to dielectric or conductive mismatch between the particles and insulating liquid. ER fluids exhibit an increase in shear stress, shear viscosity and dynamic moduli according to the variation in structure. Interestingly, a liquid–solid transition is not only tunable, but also reversible when the electric field is removed. Therefore, ER fluids are also known as ‘‘smart materials’’, which have been used widely in a range of potential engineering applications, such as dampers, actuators, brakes and shock absorbers [2,7,8]. The yield stress of ER fluids as the critical design parameter in an ER device has attracted considerable attention [9]. Several models have been proposed in segmented regions of the electric field strength to study an electric field-dependent yield stress. The polarization model [10], which describes the dielectric response of both liquid media and solid particles arising from the Maxwell–Wagner’s interaction, was introduced to predict the dependence of the yield stress on the applied electric field strength [2]. The yield stress (sy) was found to be proportional to the square of the applied electric field strength (E0), which can be expressed as
where / is the volume fraction of the particles and c = (Kp Kf)/ (Kp + 2Kf) is the dimensionless dielectric mismatch parameter [2]. Kp and Kf are the dielectric permittivities of the particles and fluid, respectively. This polarization model showed excellent agreement with the experimental data for small / and low E0 [9–11]. On the other hand, the conduction model [12,13] showed a power law dependency of the yield stress on the electric field strength, sy / E3=2 0 , under high electric field strengths, which was attributed to the nonlinear conductivity effect induced by the electrical breakdown of ER fluids under high electric field strengths. This change in power law dependency from 2.0 of the polarization model to 1.5 of the conduction model of the ER fluids has been also observed for magnetorheological fluids under an applied magnetic field [14]. On the other hand, neither model can be used to describe the dependence of the shear stress on the entire range of electric field strengths. Previously, Choi et al. [15] proposed a simple scaling yield stress function by introducing the critical electric field strength (Ec) to depict the derivation of the yield stress from polarization, which was confirmed to be suitable for many conventional ER fluids [16,17]. Recently, Seo [18] proposed a new model to describe the electric field dependent yield stress. Compared to conventional ER phenomena, a giant electrorheological (GER) effect from a nanoparticle-based suspension whose static yield stress can reach 130 kPa, exceeding the theoretical upper bound shown by a conventional ER fluid, has been reported [19–21]. In this GER effect, saturation surface polarization in the contact regions of neighboring particles was considered and a almost linear relationship between the static yield stress and applied electric field was observed [21].
⇑ Corresponding author. Fax: +82 32 865 5178. E-mail address: [email protected] (H.J. Choi). 0021-9797/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcis.2013.08.003
ð1Þ
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K. Zhang et al. / Journal of Colloid and Interface Science 409 (2013) 259–263
Fig. 1. Re-plotted dynamic yield stress of both tube-like nano-whiskers [23] and raw material of TiO2 nanoparticles [22] as a function of applied electric field strengths on a log–log scale.
Fig. 3. Re-plotted static yield stress of 15 wt% PANI nanofibers (square point), nanoparticles (circle point) and microparticles (triangle point) [24] as a function of electric field on a log–log scale.
sy ¼ aE20 / E20 ; E0 << Ec
ð3Þ
pffiffiffiffiffi
sy ¼ a Ec E3=2 / E3=2 E0 >> Ec 0 0 ;
ð4Þ
Eq. (3) suggests that sy depends quadratically on the electric field strength at low electric fields, as expected by the polarization 3=2 model [10], and Eq. (4) shows that sy is proportional to E0 at high electric fields, which is in agreement with the results obtained from the nonlinear conduction model [12,13]. Eq. (2) can be scaled as follows:
s^y ðE0 Þ ¼ where
qffiffiffiffi
sy ðE0 Þ ^3=2 tanhð b ¼ 1:313E EÞ sy ðEc Þ b E ¼ E0 =Ec
and
ð5Þ
s^ ¼ sy ðE0 Þ=sy ðEc Þ
with
s^ ¼ aE2c tanhð1Þ ¼ 0:762aE2c .
^ versus b Fig. 2. Plot of s E for both tube-like nano-whiskers and raw material of TiO2 nanoparticles based ER fluids. The solid line is obtained from Eq. (5).
Until now, there is no scaling function that can be used to describe the dependence of the shear stress on an applied electric field for both conventional ER and GER fluids. This communication proposes a generalized scaling function for the normalized yield stress by scaling the applied electric field strengths, which is suitable for describing the correlation between the yield stress and applied electric field for both conventional ER and GER fluids, indicating that the experimental data of the yield stress could be fitted closely to a single line for the entire range of electric fields. To describe the yield stress dependence on the electric field strength for a broad range of electric field strengths, Choi et al. [15] proposed a simple scaling function as follows:
pffiffiffiffiffiffiffiffiffiffiffiffi
sy ðE0 Þ ¼
aE20 tanh E0 =Ec pffiffiffiffiffiffiffiffiffiffiffiffi E0 =Ec
ð2Þ
where the parameter depends on the dielectric property of the fluid and the particle volume fraction. Ec is the critical field strength where the correlation between the yield stress of the ER suspension and electric field strength bridges the polarization model and conduction model. Eq. (2) manages the following asymptotic characteristics at both low and high electric field strengths:
The dynamic yield stress data from two references on the ER effect of TiO2 nanoparticles [22] and titanate nanowhiskers [23], respectively, was used to examine the effect of Eq. (5). The dynamic yield stress of Fig. 6 from Ref. [22] and the dynamic yield stress in Fig. 7 from Ref. [23] were replotted as a function of the applied electric field strengths on a log–log scale, as shown in Fig. 1. Ec was obtained at the crossover of two slopes, i.e. the slope of the polarization model (slope = 2.0) and that of the conduction model (slope = 1.5), which are 0.84 kV/mm for titanate nano-whiskers ^ versus b and 0.93 kV/mm for TiO2 nano-particles. The plot of s E for both titanate nano-whiskers and TiO2 nano-particle-based ER fluids were rearranged and plotted in Fig. 2. Eq. (5), which is represented by a solid line, can normalize the dynamic yield stresses. In addition, polyaniline (PANI) systems with different morphologies were also investigated. Yin et al. [24] synthesized nano-fibrous PANI via a modified oxidative polymerization, which possesses a diameter of hundreds of nanometers and several micrometer lengths. Note that the nano-fibrous PANI based ER fluid exhibits improved suspension stability and larger ER effect when compared to conventional granular PANI [25], PANI nanoparticle [24] and PANI microparticle based ER fluids [24]. To obtain the single scaling function for these two different systems, we first replotted Fig. 7(a) (Ref. [24]) on a log–log scale, and then chose data of 1.35 kV/mm, 1.08 kV/mm, and 0.92 kV/mm as Ec for ER fluid containing PANI nanoparticles, microparticles and nanofibers (as ^ versus b shown in Fig. 3), respectively. The scaling curves (s E) for these systems are plotted as shown in Fig. 4, finding that Eq. (5) fits the experimental data well in the broad range of electric field strengths for different morphology of PANI based ER fluids. This
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K. Zhang et al. / Journal of Colloid and Interface Science 409 (2013) 259–263
As a simple scaling function, Eq. (2) indicates the relationship between the yield stress and applied electric field strength for many conventional ER suspensions using two distinct limiting behaviors along the entire electric field strength range. On the other hand, this model does not describe the yield stress for the GER fluids. The GER fluids expressed a linear relationship with the applied electric field strength. A previous study showed that some GER fluids did not follow this simple linear relationship. Instead, sy was proportional to E2:0 at low E0 and to E1:0 at high E0 0 0 [27]. A novel scaling function was proposed to describe the behavior of yield stress for GER fluids accurately [27]:
sy ðE0 Þ ¼
b for 15 wt% PANI nanofibers (square point), nanoparticles ^ versus E Fig. 4. Plot of s (circle point) and microparticles (triangle point). The solid line is obtained from Eq. (5).
aE20 I1 ðE0 =Ec Þ
ð6Þ
ðE0 =Ec ÞI0 ðE0 =Ec Þ
where I1 and I0 are modified Bessel functions of order 1 and 0, respectively. Two distinct limiting behaviors of Eq. (6) for low and high electric field can be expressed as (i) E0 Ec, I1(E0/Ec) / E0 and I0(E0/Ec) = constant,
sy ðE0 Þ / E20
ð7Þ
pffiffiffiffiffiffiffiffiffi (i) E0 Ec, I1(E0/Ec) and I0 ðE0 =Ec Þ / e = 2px, x
sy ðE0 Þ / E0
ð8Þ
and sy(Ec) is calculated as:
sy ðEc Þ ¼
aE2c I1 ð1Þ I0 ð1Þ
¼ 0:446aE2c
ð9Þ
Eq. (6) can then be scaled as follows:
s^
sy ðE0 Þ 2:242 bEI1 ð bEÞ ^ ¼ E0 with E ¼ sy ðEc Þ Ec EÞ I0 ð b
ð10Þ
Although Eqs. (2) and (6) were proposed to describe the correlation of the yield stress versus electric field strength in a broad range of electric field strengths for conventional ER fluids and GER suspensions, respectively, there is no generalized model that can describe all types of ER fluids including both conventional ER and GER fluids. For this purpose, Eq. (2) was generalized by introducing the material parameter b, Fig. 5. Static yield stress of urea-coated nanoparticles of barium titanate oxalate as function of applied electric field strength from Ref. [19].
sy ðE0 Þ ¼
aE20 tanh
b
b
E0 Ec
ð11Þ
E0 Ec
Eq. (11) can generate two distinct limiting behaviors for low electric field strength and high electric field strength, respectively, using the critical electric field strength, Ec: (i) E0 < < Ec
because tanh
)sy ðE0 Þ ¼
b b 3b b E0 E0 1 E0 E0 ¼ þ ffi 3 Ec Ec Ec Ec
aE20
ð12Þ
b E0 Ec
b E0 Ec
¼ aE20
ð13Þ
(ii) E0 Ec
^ versus b Fig. 6. s E for urea-coated nanoparticles of barium titanate oxalate with different concentrations from Ref. [19]. The solid line is obtained from Eq. (16).
scaling equation has been also found to be good to fit both static and dynamic yield stresses [26] in addition to different volume fraction [15].
E b E b E b b 0 0 0 E0 e Ec e Ec e Ec because tanh ¼ E b E b ffi E b ¼ 1 0 0 0 Ec e Ec þ e Ec e Ec
ð14Þ
aE2 )sy ðE0 Þ ¼ 0b ¼ aEbc E2b / E2b 0 0
ð15Þ
E0 Ec
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K. Zhang et al. / Journal of Colloid and Interface Science 409 (2013) 259–263
Fig. 7. Re-plotted static yield stress of calcium–titanium–oxygen (CTO) precipitate from Ref. [28] with different volume fraction as a function of applied electric field strengths on a log–log scale.
electric field strength on a log–log scale as shown in Fig. 5. Following the change in slope from the polarization model regime (slope = 2.0) to the GER model regime (slope = 1.0), Ec was calculated to be 1.74 and 1.63 kV/mm for a 15% and 30% GER fluid, ^y and b respectively. Both were used to obtain s E and were scaled by Eq. (17). The solid line generated by Eq. (17) was found to collapse the experimental data into a single line as given in Fig. 6. These results suggest that the proposed generalized equation indeed fitted all the data of ER fluids over a broad range of electric field strengths. Furthermore, we also re-plotted static yield stress versus particle volume fraction of calcium–titanium–oxygen (CTO) precipitate GER fluids under different external electric fields (Fig. 1 of Ref. [28]) as shown in Fig. 7. It can be seen that the yield stress increase with volume fraction and it is divided into two parts by critical points according to the slope (One part has the slope of 2.0 at a lower electric filed; the other has the slope of 1.0 under a higher electric field). Ecs are found to be 2.16 kV/mm (15%), 2.19 kV/mm (20%), 2.41 kV/mm (25%), 2.8 kV/mm (30%), 3.23 kV/mm (35%), 3.51 kV/mm (40%), 3.3 kV/mm (45%), and 3.11 kV/mm (50%) for different volume fraction of GER fluids, respectively. A correlation ^Þ and electric field strength ð b between scaled yield stress ðs EÞ is represented in Fig. 8, suggesting that the experimental data of GER fluids correlates very well with the universal scaling Eq. (17). In summary, a new generalized scaling function was proposed for ER fluids to describe the dependency of the yield stress on the electric field strength for the entire range of electric field strengths. This model provides an excellent description of the ER effect not only for conventional ER suspension but also for GER fluids by introducing a critical electric field strength (Ec) and material parameter b. The yield stress–electric field strength plots for both conventional ER fluids and GER suspensions collapsed onto a single curve for the entire range of electric field strengths using Eq. (11). Acknowledgment This study was supported by the National Research Foundation, Korea (NRF-2011-0006592).
^ versus E for calcium–titanium–oxygen (CTO) precipitate with different Fig. 8. s volume fraction from Ref. [28]. The solid line is obtained from Eq. (17).
Eq. (13) shows that at low electric field strength, sy corresponds to E20 . On the other hand, at high electric field strengths, it will be sy / Eb2 0 , as indicated by Eq. (15). Therefore, the dependence of sy on E0 at the high electric field range is up to the parameter, b: 3=2 b = 0.5, sy ðE0 Þ ¼ aE1=2 / E2=3 for the conduction model and c E0 0 b = 1 sy(E0) = aEcE0 / E0 for a GER fluid Eq. (11) can then be scaled as: 2 tanh ðE0 =Ec Þb
2b
s ðE Þ aE0 ðE0 =Ec Þb E0 s^y ðE0 Þ ¼ y 0 ¼ ¼ 1:313 sy ðEc Þ Ec 0:762aE2c
tanh
b E0 Ec
b E 2b tanh ð b EÞ ¼ 1:313 b
ð16Þ
When b = 0.5, Eq. (16) becomes the same as Eq. (5). When b = 1, Eq. (16) can be represented as follows:
s^y ðE0 Þ ¼
sy ðE0 Þ E tanhð b EÞ ¼ 1:313 b sy ðEc Þ
ð17Þ
A yield stress database was selected to confirm the efficiency of Eq. (11). The static yield stress of two GER fluids was re-plotted based on urea-coated nanoparticles of barium titanate oxalate with different particle fractions (15% and 30%), as a function of the
Reference [1] J. Yin, X. Xia, X. Wang, X. Zhao, Soft Matter 7 (2011) 10978. [2] Y.D. Liu, H.J. Choi, Soft Matter 8 (2012) 11961. [3] J. Wu, G. Xu, Y. Cheng, F. Liu, J. Guo, P. Cui, J. Colloid Interface Sci. 378 (2012) 36. [4] S. Guzel, O. Erol, H.I. Unal, J. Appl. Polym. Sci. 124 (2012) 4935. [5] K. Shin, D. Kim, J.C. Cho, H.S. Lim, J.W. Kim, K.D. Suh, J. Colloid Interface Sci. 374 (2012) 18. [6] M. Sedlacik, M. Mrlik, V. Pavlinek, P. Saha, O. Quadrat, Colloid Polym. Sci. 290 (2012) 41. [7] K. Koyanagi, Y. Kakinuma, H. Anzai, K. Sakurai, T. Oshima, J. Intell. Mater. Syst. Struct. 22 (2011) 1685. [8] Y.M. Han, M.S. Seong, S.B. Choi, N.M. Wereley, Proc. Inst. Mech. Eng. Part C 224 (2010) 293. [9] Y. Meheust, K.P.S. Parmar, B. Schjelderupsen, J.O. Fossum, J. Rheol. 55 (2011) 809. [10] X.D. Duan, H. Chen, Y.J. He, W.L. Luo, J. Phys. D – Appl. Phys. 33 (2000) 696. [11] D.J. Klingenberg, C.F. Zukoski, Langmuir 6 (1990) 15. [12] L.C. Davis, Appl. Phys. Lett. 60 (1992) 319. [13] C.W. Wu, H. Conrad, Phys. Rev. E 56 (1997) 5789. [14] J.M. Ginder, L.C. Davis, L.D. Elie, Int. J. Mod. Phys. B 10 (1996) 3293. [15] H.J. Choi, M.S. Cho, J.W. Kim, C.A. Kim, M.S. Jhon, Appl. Phys. Lett. 78 (2001) 3806. [16] F.F. Fang, H.J. Choi, W.S. Ahn, Microporous Mesoporous Mater. 130 (2010) 338. [17] Y.D. Liu, B.J. Park, Y.H. Kim, H.J. Choi, J. Mater. Chem. 21 (2011) 17396. [18] Y. Seo, J. Non-Newtonian Fluid Mech. 166 (2011) 241. [19] W.J. Wen, X.X. Huang, S.H. Yang, K.Q. Lu, P. Sheng, Nat. Mater. 2 (2003) 727. [20] J.B. Yin, X.P. Zhao, Chem. Phys. Lett. 398 (2004) 393. [21] R. Shen, X. Wang, Y. Lu, D. Wang, G. Sun, Z. Cao, K. Lu, Adv. Mater. 21 (2009) 4631. [22] Y. Cheng, X. Liu, J. Guo, F. Liu, Z. Li, G. Xu, P. Cui, Nanotechnology 20 (2009) 055604. [23] J.B. Yin, X.P. Zhao, Nanotechnology 17 (2006) 192.
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Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Short Communication
Generalized yield stress equation for electrorheological fluids Ke Zhang a,b, Ying Dan Liu a, Myung S. Jhon c, Hyoung Jin Choi a,⇑ a
Department of Polymer Science and Engineering, Inha University, Incheon 402-751, Republic of Korea School of Chemical Science and Technology, Harbin Institute Technology, Harbin 150001, Peoples Republic of China c Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA b
a r t i c l e
i n f o
Article history: Received 5 June 2013 Accepted 1 August 2013 Available online 11 August 2013 Keywords: Electrorheological fluid Yield stress Universal equation Critical electric field
a b s t r a c t A new generalized yield stress scaling equation for electrorheological (ER) fluids was developed by introducing the critical electric field (Ec) and material parameter. This equation can be used to describe the dependency of the yield stress on an electric field not only for conventional ER suspensions with a change in slope from 2.0 to 1.5, but also for giant ER fluids with a change in slope from 2.0 to 1.0. The yield stress data obtained from different ER fluid systems with different material parameters was collapsed onto a single curve for the entire range of electric field strengths using the proper scaling method proposed in this study. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction
sy / /K f E20 f ðcÞ
Electrorheological (ER) fluids, which are composed of higher dielectric and/or conducting particles dispersed in an insulating liquid in general, can be converted drastically from a liquid-like to a solid-like phase in the order of milliseconds when either an AC or a DC electric field is applied [1–6]. The rapid response to an external electric field is related to the formation of fibrillar structures aligned along the direction of the electric field due to dielectric or conductive mismatch between the particles and insulating liquid. ER fluids exhibit an increase in shear stress, shear viscosity and dynamic moduli according to the variation in structure. Interestingly, a liquid–solid transition is not only tunable, but also reversible when the electric field is removed. Therefore, ER fluids are also known as ‘‘smart materials’’, which have been used widely in a range of potential engineering applications, such as dampers, actuators, brakes and shock absorbers [2,7,8]. The yield stress of ER fluids as the critical design parameter in an ER device has attracted considerable attention [9]. Several models have been proposed in segmented regions of the electric field strength to study an electric field-dependent yield stress. The polarization model [10], which describes the dielectric response of both liquid media and solid particles arising from the Maxwell–Wagner’s interaction, was introduced to predict the dependence of the yield stress on the applied electric field strength [2]. The yield stress (sy) was found to be proportional to the square of the applied electric field strength (E0), which can be expressed as
where / is the volume fraction of the particles and c = (Kp Kf)/ (Kp + 2Kf) is the dimensionless dielectric mismatch parameter [2]. Kp and Kf are the dielectric permittivities of the particles and fluid, respectively. This polarization model showed excellent agreement with the experimental data for small / and low E0 [9–11]. On the other hand, the conduction model [12,13] showed a power law dependency of the yield stress on the electric field strength, sy / E3=2 0 , under high electric field strengths, which was attributed to the nonlinear conductivity effect induced by the electrical breakdown of ER fluids under high electric field strengths. This change in power law dependency from 2.0 of the polarization model to 1.5 of the conduction model of the ER fluids has been also observed for magnetorheological fluids under an applied magnetic field [14]. On the other hand, neither model can be used to describe the dependence of the shear stress on the entire range of electric field strengths. Previously, Choi et al. [15] proposed a simple scaling yield stress function by introducing the critical electric field strength (Ec) to depict the derivation of the yield stress from polarization, which was confirmed to be suitable for many conventional ER fluids [16,17]. Recently, Seo [18] proposed a new model to describe the electric field dependent yield stress. Compared to conventional ER phenomena, a giant electrorheological (GER) effect from a nanoparticle-based suspension whose static yield stress can reach 130 kPa, exceeding the theoretical upper bound shown by a conventional ER fluid, has been reported [19–21]. In this GER effect, saturation surface polarization in the contact regions of neighboring particles was considered and a almost linear relationship between the static yield stress and applied electric field was observed [21].
⇑ Corresponding author. Fax: +82 32 865 5178. E-mail address: [email protected] (H.J. Choi). 0021-9797/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcis.2013.08.003
ð1Þ
260
K. Zhang et al. / Journal of Colloid and Interface Science 409 (2013) 259–263
Fig. 1. Re-plotted dynamic yield stress of both tube-like nano-whiskers [23] and raw material of TiO2 nanoparticles [22] as a function of applied electric field strengths on a log–log scale.
Fig. 3. Re-plotted static yield stress of 15 wt% PANI nanofibers (square point), nanoparticles (circle point) and microparticles (triangle point) [24] as a function of electric field on a log–log scale.
sy ¼ aE20 / E20 ; E0 << Ec
ð3Þ
pffiffiffiffiffi
sy ¼ a Ec E3=2 / E3=2 E0 >> Ec 0 0 ;
ð4Þ
Eq. (3) suggests that sy depends quadratically on the electric field strength at low electric fields, as expected by the polarization 3=2 model [10], and Eq. (4) shows that sy is proportional to E0 at high electric fields, which is in agreement with the results obtained from the nonlinear conduction model [12,13]. Eq. (2) can be scaled as follows:
s^y ðE0 Þ ¼ where
qffiffiffiffi
sy ðE0 Þ ^3=2 tanhð b ¼ 1:313E EÞ sy ðEc Þ b E ¼ E0 =Ec
and
ð5Þ
s^ ¼ sy ðE0 Þ=sy ðEc Þ
with
s^ ¼ aE2c tanhð1Þ ¼ 0:762aE2c .
^ versus b Fig. 2. Plot of s E for both tube-like nano-whiskers and raw material of TiO2 nanoparticles based ER fluids. The solid line is obtained from Eq. (5).
Until now, there is no scaling function that can be used to describe the dependence of the shear stress on an applied electric field for both conventional ER and GER fluids. This communication proposes a generalized scaling function for the normalized yield stress by scaling the applied electric field strengths, which is suitable for describing the correlation between the yield stress and applied electric field for both conventional ER and GER fluids, indicating that the experimental data of the yield stress could be fitted closely to a single line for the entire range of electric fields. To describe the yield stress dependence on the electric field strength for a broad range of electric field strengths, Choi et al. [15] proposed a simple scaling function as follows:
pffiffiffiffiffiffiffiffiffiffiffiffi
sy ðE0 Þ ¼
aE20 tanh E0 =Ec pffiffiffiffiffiffiffiffiffiffiffiffi E0 =Ec
ð2Þ
where the parameter depends on the dielectric property of the fluid and the particle volume fraction. Ec is the critical field strength where the correlation between the yield stress of the ER suspension and electric field strength bridges the polarization model and conduction model. Eq. (2) manages the following asymptotic characteristics at both low and high electric field strengths:
The dynamic yield stress data from two references on the ER effect of TiO2 nanoparticles [22] and titanate nanowhiskers [23], respectively, was used to examine the effect of Eq. (5). The dynamic yield stress of Fig. 6 from Ref. [22] and the dynamic yield stress in Fig. 7 from Ref. [23] were replotted as a function of the applied electric field strengths on a log–log scale, as shown in Fig. 1. Ec was obtained at the crossover of two slopes, i.e. the slope of the polarization model (slope = 2.0) and that of the conduction model (slope = 1.5), which are 0.84 kV/mm for titanate nano-whiskers ^ versus b and 0.93 kV/mm for TiO2 nano-particles. The plot of s E for both titanate nano-whiskers and TiO2 nano-particle-based ER fluids were rearranged and plotted in Fig. 2. Eq. (5), which is represented by a solid line, can normalize the dynamic yield stresses. In addition, polyaniline (PANI) systems with different morphologies were also investigated. Yin et al. [24] synthesized nano-fibrous PANI via a modified oxidative polymerization, which possesses a diameter of hundreds of nanometers and several micrometer lengths. Note that the nano-fibrous PANI based ER fluid exhibits improved suspension stability and larger ER effect when compared to conventional granular PANI [25], PANI nanoparticle [24] and PANI microparticle based ER fluids [24]. To obtain the single scaling function for these two different systems, we first replotted Fig. 7(a) (Ref. [24]) on a log–log scale, and then chose data of 1.35 kV/mm, 1.08 kV/mm, and 0.92 kV/mm as Ec for ER fluid containing PANI nanoparticles, microparticles and nanofibers (as ^ versus b shown in Fig. 3), respectively. The scaling curves (s E) for these systems are plotted as shown in Fig. 4, finding that Eq. (5) fits the experimental data well in the broad range of electric field strengths for different morphology of PANI based ER fluids. This
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K. Zhang et al. / Journal of Colloid and Interface Science 409 (2013) 259–263
As a simple scaling function, Eq. (2) indicates the relationship between the yield stress and applied electric field strength for many conventional ER suspensions using two distinct limiting behaviors along the entire electric field strength range. On the other hand, this model does not describe the yield stress for the GER fluids. The GER fluids expressed a linear relationship with the applied electric field strength. A previous study showed that some GER fluids did not follow this simple linear relationship. Instead, sy was proportional to E2:0 at low E0 and to E1:0 at high E0 0 0 [27]. A novel scaling function was proposed to describe the behavior of yield stress for GER fluids accurately [27]:
sy ðE0 Þ ¼
b for 15 wt% PANI nanofibers (square point), nanoparticles ^ versus E Fig. 4. Plot of s (circle point) and microparticles (triangle point). The solid line is obtained from Eq. (5).
aE20 I1 ðE0 =Ec Þ
ð6Þ
ðE0 =Ec ÞI0 ðE0 =Ec Þ
where I1 and I0 are modified Bessel functions of order 1 and 0, respectively. Two distinct limiting behaviors of Eq. (6) for low and high electric field can be expressed as (i) E0 Ec, I1(E0/Ec) / E0 and I0(E0/Ec) = constant,
sy ðE0 Þ / E20
ð7Þ
pffiffiffiffiffiffiffiffiffi (i) E0 Ec, I1(E0/Ec) and I0 ðE0 =Ec Þ / e = 2px, x
sy ðE0 Þ / E0
ð8Þ
and sy(Ec) is calculated as:
sy ðEc Þ ¼
aE2c I1 ð1Þ I0 ð1Þ
¼ 0:446aE2c
ð9Þ
Eq. (6) can then be scaled as follows:
s^
sy ðE0 Þ 2:242 bEI1 ð bEÞ ^ ¼ E0 with E ¼ sy ðEc Þ Ec EÞ I0 ð b
ð10Þ
Although Eqs. (2) and (6) were proposed to describe the correlation of the yield stress versus electric field strength in a broad range of electric field strengths for conventional ER fluids and GER suspensions, respectively, there is no generalized model that can describe all types of ER fluids including both conventional ER and GER fluids. For this purpose, Eq. (2) was generalized by introducing the material parameter b, Fig. 5. Static yield stress of urea-coated nanoparticles of barium titanate oxalate as function of applied electric field strength from Ref. [19].
sy ðE0 Þ ¼
aE20 tanh
b
b
E0 Ec
ð11Þ
E0 Ec
Eq. (11) can generate two distinct limiting behaviors for low electric field strength and high electric field strength, respectively, using the critical electric field strength, Ec: (i) E0 < < Ec
because tanh
)sy ðE0 Þ ¼
b b 3b b E0 E0 1 E0 E0 ¼ þ ffi 3 Ec Ec Ec Ec
aE20
ð12Þ
b E0 Ec
b E0 Ec
¼ aE20
ð13Þ
(ii) E0 Ec
^ versus b Fig. 6. s E for urea-coated nanoparticles of barium titanate oxalate with different concentrations from Ref. [19]. The solid line is obtained from Eq. (16).
scaling equation has been also found to be good to fit both static and dynamic yield stresses [26] in addition to different volume fraction [15].
E b E b E b b 0 0 0 E0 e Ec e Ec e Ec because tanh ¼ E b E b ffi E b ¼ 1 0 0 0 Ec e Ec þ e Ec e Ec
ð14Þ
aE2 )sy ðE0 Þ ¼ 0b ¼ aEbc E2b / E2b 0 0
ð15Þ
E0 Ec
262
K. Zhang et al. / Journal of Colloid and Interface Science 409 (2013) 259–263
Fig. 7. Re-plotted static yield stress of calcium–titanium–oxygen (CTO) precipitate from Ref. [28] with different volume fraction as a function of applied electric field strengths on a log–log scale.
electric field strength on a log–log scale as shown in Fig. 5. Following the change in slope from the polarization model regime (slope = 2.0) to the GER model regime (slope = 1.0), Ec was calculated to be 1.74 and 1.63 kV/mm for a 15% and 30% GER fluid, ^y and b respectively. Both were used to obtain s E and were scaled by Eq. (17). The solid line generated by Eq. (17) was found to collapse the experimental data into a single line as given in Fig. 6. These results suggest that the proposed generalized equation indeed fitted all the data of ER fluids over a broad range of electric field strengths. Furthermore, we also re-plotted static yield stress versus particle volume fraction of calcium–titanium–oxygen (CTO) precipitate GER fluids under different external electric fields (Fig. 1 of Ref. [28]) as shown in Fig. 7. It can be seen that the yield stress increase with volume fraction and it is divided into two parts by critical points according to the slope (One part has the slope of 2.0 at a lower electric filed; the other has the slope of 1.0 under a higher electric field). Ecs are found to be 2.16 kV/mm (15%), 2.19 kV/mm (20%), 2.41 kV/mm (25%), 2.8 kV/mm (30%), 3.23 kV/mm (35%), 3.51 kV/mm (40%), 3.3 kV/mm (45%), and 3.11 kV/mm (50%) for different volume fraction of GER fluids, respectively. A correlation ^Þ and electric field strength ð b between scaled yield stress ðs EÞ is represented in Fig. 8, suggesting that the experimental data of GER fluids correlates very well with the universal scaling Eq. (17). In summary, a new generalized scaling function was proposed for ER fluids to describe the dependency of the yield stress on the electric field strength for the entire range of electric field strengths. This model provides an excellent description of the ER effect not only for conventional ER suspension but also for GER fluids by introducing a critical electric field strength (Ec) and material parameter b. The yield stress–electric field strength plots for both conventional ER fluids and GER suspensions collapsed onto a single curve for the entire range of electric field strengths using Eq. (11). Acknowledgment This study was supported by the National Research Foundation, Korea (NRF-2011-0006592).
^ versus E for calcium–titanium–oxygen (CTO) precipitate with different Fig. 8. s volume fraction from Ref. [28]. The solid line is obtained from Eq. (17).
Eq. (13) shows that at low electric field strength, sy corresponds to E20 . On the other hand, at high electric field strengths, it will be sy / Eb2 0 , as indicated by Eq. (15). Therefore, the dependence of sy on E0 at the high electric field range is up to the parameter, b: 3=2 b = 0.5, sy ðE0 Þ ¼ aE1=2 / E2=3 for the conduction model and c E0 0 b = 1 sy(E0) = aEcE0 / E0 for a GER fluid Eq. (11) can then be scaled as: 2 tanh ðE0 =Ec Þb
2b
s ðE Þ aE0 ðE0 =Ec Þb E0 s^y ðE0 Þ ¼ y 0 ¼ ¼ 1:313 sy ðEc Þ Ec 0:762aE2c
tanh
b E0 Ec
b E 2b tanh ð b EÞ ¼ 1:313 b
ð16Þ
When b = 0.5, Eq. (16) becomes the same as Eq. (5). When b = 1, Eq. (16) can be represented as follows:
s^y ðE0 Þ ¼
sy ðE0 Þ E tanhð b EÞ ¼ 1:313 b sy ðEc Þ
ð17Þ
A yield stress database was selected to confirm the efficiency of Eq. (11). The static yield stress of two GER fluids was re-plotted based on urea-coated nanoparticles of barium titanate oxalate with different particle fractions (15% and 30%), as a function of the
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