Langmuir 1996,11, 307-312
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Time-DependentDielectric Response of Quiescent Electrorheological Fluids Douglas Adolf and Terry Garino Materials and Process Sciences Center, Sandia National Laboratories, A1b uquerq ue, New Mexico 8 7185 Received January 5, 1994. In Final Form: October 19, 1994@ The evolution of structure in a quiescentelectrorheologicalfluid was monitored by followingthe increase in static permittivity after an electric field was applied. The zero-time permittivity agreed well with theoretical predictionsfor a randomly dispersedcolloidal suspension. As time progressed, the permittivity increased as the anisotropic structure evolved, and the rate of increase in permittivity at early times depended on the square of the applied field. For low particle loadings and high fields, the measured permittivity at long times agreed with predictions for dense columnar aggregates. Under lower applied fields or in more concentrated suspensions,however,the intermediatetime coarseningprocess was quenched, and the corresponding final structure was less compact, resulting in a lower permittivity than predicted.
Introduction Under a n applied field E, the particulates in an electrorheological (ER) fluid chain along the field lines, resulting in dramatically increased resistance to flow. Since the viscosity change can occur within milliseconds and is reversible when the field is removed, numerous devices have been patented utilizing these "smart materials"in active control schemes. Examples range from active vibration damping to continuously variable transmissions. However, most devices are not currently feasible due to inadequate field-induced viscosity enhancement. Since the enhanced viscosity results from the anisotropic structure formed by an applied field, attempts to improve ER fluid performance may be aided by a n understanding of how the structure evolves under an electric field. In the present study, we investigated the increase in dielectric constant of a suspension of ceramic particulates after an electric field was applied and related these results to the evolution of structure in the quiescent ER fluid. Colloidal aggregation in an electric field arises from field-induced attractive forces between particles resulting from the mismatch in the frequency-dependent complex dielectric constants, E ( W ) = '6 id', of the particulate E , and the suspending fluid 6f.l For isolated point dipoles, the force between two particles of radius R separated by a distance d is
+
Fp = 24q(-)"."-Ep
+ 2Ef
6
d4 = 24 mfj3
At low-field frequencies (the dc limit), the interparticle force is dominated by the mismatch in fluid and particulate conductivities, uf and a,, where u = O E " ( ~ ) whereas , at high-frequency ac fields, the force is dominated by the mismatch in static permittivites, K = ~ ' ( O ) / Ewhere ~, eo is the permittivity of free space. "he critical field frequency ocseparating these two regimes is approximately2 Abstract published in Advance ACS Abstracts, December 15, 1994. (1)Gast, A.P.; Zukoski, C. F. Adv. Colloid Interface Sci. 1989,30, @
153. (2) Anderson, R. A. In Proceedings of the International Conference
on Electrorheological Fluids: Mechanisms, Properties, Structure, TechnologyandApplicatwns;Tao, R., Ed.; WorldScientific: Singapore, 1992.
In model systems, the formation of structure in a ~ quiescent ER fluid proceeds in a stepwise f a ~ h i o n .The initial stage involves aggregation of isolated particles. An estimate of this time scale, t,, is obtained by balancing the attractive force of eq 1 and the viscous drag, F, = 6n773v,for solvent viscosity vS and particle velocity v = d(d)/dt.
7s s5- (2R)5 " = ( 2 0 ~ ~ j 3 ~ E ' ) ( R5
)
(3)
where the initial particle separation, S, is related to the particle volume fraction, d, by S3 = (4nR3)/(34,).For 4, = 0.10,E = 1kV/mm, 7, = 1.5 cP, K f = 2, and tp* 4 ms. As time progresses, the pairs combine to form chains and columns parallel to the field which span the electrodes, characterized by a time, tcol * E-2. Finally, the columns themselves aggregate and coarsen. Halsey et aL4proposed that the force between columns which drives the coarsening process arises from thermal density fluctuations within the columns that create a fluctuating electric field near the columns. The statistically averaged force between columns, F,, was predicted to be proportional to E6l5. By balancing the attractive force and the viscous drag on correlated sections of the columns, the time scale for coarsening, tc,was found to depend on the field strength as E-o.8. In addition, the distance separating columns increased with time as
(4) The fully coarsened columns have been predicted to crystallize into a body-centered tetragonal s t r u c t ~ r eIn .~ reality, small amounts of particle polydispersity or nonsphericity will most likely disrupt this perfect crystalline state, and the fully coarsened state will resemble a dense amorphous aggregate. While the stepwise aggregation (3)Halsey, T. C.; Toor, W. Phys. Rev.Lett. 1990,65, 2820. (4) Martin, J.E.; Odinek, J.;Halsey, T. C. Phys. Rev.Lett. 1992,69, 1524. ( 5 ) Tao, R.; Sun, J. M. Phys. Rev. Lett. 1991,67, 398.
0743-7463/95/2411-0307~09.~0l0 0 1995 American Chemical Society
308 Langmuir, Vol. 11, No. 1, 1995
Adolf and Garino
scenario described above has been observed by optical microscopy at low particle loadings in model systems,6 evolution of structure at high particle loadings is not so straightforwardly analyzed. The various aggregation stages overlap greatly, a n d optical microscopy6 reveals no obvious sequential evolution of structure. Experimentally, the evolution of structure in quiescent ER fluids has been examined using several different optical techniques. In the studies of Smith a n d Fuller,' the increase in birefringence a n d dichroism were monitored in real time a n d revealed a very weak dependence of the characteristic time on electric field strength (tx 100 m s for E = 1kV/mm). Ginder a n d Elie8monitored the increase in transmitted light intensity (beam parallel to the field) with time (1 It I 300 ms) for an ER fluid with high refractive index contrast between particle a n d fluid (#p = 0.35). In contrast to the birefringence measurements, this characteristic time varied as t E-,, as predicted by e q 2 for the early stages of structure formation. Martin et al.4 investigated the coarsening process ( 1 It I100 s) through measurements ofthe anisotropic two-dimensional scattering pattern (beam orthogonal to the field) formed by a nearly index-matched ER fluid (& < 0.085). The maximum in t h e observed scattering lobes corresponded to the characteristic separation between columns, d,. The experimental results, t c a n d d, (t/tc)0.42, agreed well with the theoretical prediction in eq 4. In the present study, we investigated the evolution of structure by monitoring the increase in dielectric constant as a field is applied offering complimentary information to t h e optical techniques.
-
-
1 00
0
0.2
0.1
0.4
0.3
0.5
volume fraction particles
Figure 1. Permittivity of mixtures of ceramic particulates in poly(ethy1eneglycol) as a function of particulate loading. From the log additivity mixing rule, the particulate permittivities are ~ @ a T i O d= 2000 f 500, Kp(SfliO~) = 270 f 60, and K ~ (TiOz)= 120 f 30.
-
V
Experimental Section The colloidal suspensions prepared for this study contained roughly spherical 1.0 pm diameter (measured by a Horiba Capa 700 centrifugal particle size analyzer) ceramic particulates [titanium oxide (Aldrich Chemical Co.), strontium titanate (Johnson Matthey Chemical Co.), or barium titanate (Transelco)] suspended in dodecane (Fisher Scientific). Polyisobutene succinimide (OLOA 1200, Chevron) at 5 mg/g of particulate was used as a dispersant. The fluids were prepared at solids volume contents of 10, 20, and 30% by sonication. The relative permittivities ofthe T i 0 2 and srTio3 particulates found from handbooks are 1E9and 290,1° respectively. We also measured the relative permittivities of30,40, and50vol%pastes of the particulates in poly(ethy1ene glycol) (Fluka, M w = 4000, Kf = 4.0 measured). Pastes were formed into disks 1.3 cm in diameter and 0.2 cm thick by mixing the dried ceramic powder and the polymer at 150 "C and compadingunder 10 000 psi. The pellet surfaces were coated with a silver paint to form electrodes. The capacitance of each sample was measured at 1 V and 100 kHz. K~ was obtained from the relative permittivity of the paste, K-, using the logarithmic additivity law: log K- = log K~ &log Kf, where 4pand h a r e the volume fractions of particles and polymer (see Figure 1). The particle relative permittivities calculated from this procedure were ~,(BaTi03)= 2000 f 500, 2 )120 f 30, which agree well ~ ~ ( S f i O=3270 ) f60, and ~ ~ ( T i 0 = with the handbook values. Figure 2 shows our apparatus for measuring the dielectric constant of our ER fluids as the field was applied. A sinusoidal 400 Hz voltage signalwas applied to the sample using a Wavetek 182A function generator in series with a Trek 10/10 amplifier. The measured critical field frequency, defined in eq 2, was less
I
I
Figure 2. Schematic of the apparatus used to measure the permittivities ofER fluids. C1 is 48 pF, R1 and R2 are variable from 1to 100 kQ, and the voltage frequency is 400 Hz. The time-dependent suspension permittivity is obtained from measurements of the voltage drop across R1 and R2. than 10 Hz, so these ER fluids operate in the capacitive regime where the interparticle forces (eq 1) are determined by the mismatch in static permittivities of the particle and fluid. Our measured fluid permittivities at 400 Hz, therefore, were almost entirely real (tan 6 < 0.05),and the capacitor (48 pF) was chosen to balance roughly the capacitance of the sample. The timedependent static permittivity ofthe ER fluid was calculated using
+
( 6 )Martin, J. E., private communication. (7) Smith, K. L.; Fuller, G. G. InProceedings of the First International Symposium on Electrorheological Fluids; Conrad, H., Sprecher, A. F., Carlson, J. D., Eds.;NCSU Engineering Publ.: Rayleigh, NC, 1989. (8)Ginder, J. M.; Elie, L. D. In Proceedings of the International Conference on Electrorheological Fluids: Mechanisms, Properties, Tao, R., Ed.: World Scientific: Structure, Technology -. and Applications; .. Singapore, 1992. (9) Kingery, W. D.Introduction to Ceramics;Wiley: New York, 1960. (lO)Megaw, H. D. Ferroelectricity in Crystals; Methuen & Co.: London, 1957.
whereL = 0.3 mm is the plate gap,A = 8.35 cm2is the plate area, and IVil(t)is the time-dependent amplitude of the voltage signal over resistor i. The voltage data were captured by a Macintosh IIci computer equipped with a 16 bit National Instruments A/D board acquiring data at a rate of 4000 Hz. Over 99% of the total voltage drop occurs across the sample and capacitor with the resistors used in this study (1-100 kQ). To ensure that the measured permittivities were in the regime of linear response, we superimposed a 4000 Hz probe signal with one-tenth of the amplitude ofthe 400 Hz structure-forming signal using a second Wavetek function generator in series. The steady-state permittivity calculated from the 4000 Hz signal for a 20 vol % SrTiOa suspension under a 1 kV/mm, 400 Hz signal was no different from the permittivity obtained from the 400 Hz signal directly. Therefore, we used only the 400 Hz signal in the studies described below.
Results and Discussion The time-dependent static permittivities for 10,20, and 30 vol % suspensions of TiOz, SrTi03, a n d BaTi03 in
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Dielectric Response of Electrorheological Fluids
Langmuir, Vol. 11, No. 1, 1995 309 IO%, 67 V/mm lo%, 200 Vlmm lo%, 500 Vlmm v 10%. 1000 Vlmm 1 20%, 67 Vlmm 0 0 A
20%, 200 Vlmm A 20%, 500 Vlmm v 20%, 1000 V/mm 30%, 67 V/mm 30%, 200 Vimm 0
10
A
20
20%, 200 Vimm 20% 500 V/mm 20%, 1000 V/mm B 30%, 67 Vlmm * 30%, 200 Vlmm + 30%, 500 Vlmm x 30%, 1000 V/mm A
30%. 500 Vlmm
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e
10
! I v v
5
1 oo
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.
B
I
0 n
o 0
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10-1
time (5)
time (s)
Figure 3. Increase in permittivitywith time for suspensions of T i 0 2 in dodecane at 20 and 30% particle loadings under applied fields of 67, 200, 500, and 1000 V/mm.
I
lo%, 67 Vlmm lo%, 200 Vlmm A lo%, 500 Vlmm v lo%, 1000 Vlmm 1 20%, 67 Vlmm 20%, 200 Vlmm 0
Figure 5. Increase in permittivity with time for suspensions of BaTiOa in dodecane at 10, 20, and 30% particle loadings under applied fields of 67,200, 500, and 1000 V/mm. 10
0
m 30%,
8
6
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Figure 4. Increase in permittivity with time for suspensions of SrTiOa in dodecane at 10,20,and 30%particle loadings under applied fields of 67, 200, 500, and 1000 V/mm.
dodecane under applied fields of 67,200,500, and 1000 V/mm (peak amplitudes) are shown in Figures 3-5. Several qualitative features are readily apparent from each figure. First, the high-field permittivities increase with time, indicating an ordering of the fluid particulates along the field lines, and the ordering occurs faster under higher applied fields. However, while the permittivities for the lowest applied field (67 V/mm) are larger than the corresponding off-field permittivities, no increase is observed within our experimental time window. Second, higher loadings of particulates result in higher measured permittivities, as expected. Third, at a given particle loading, the long-time permittivity seems to reach a plateau, d-),that depends on the applied field, with higher fields resulting in larger 4-1. Finally, when the results from the three systems are compared, it is obvious that the higher permittivity particulates result in higher measured fluid permittivities, as expected. We will focus our analysis on four aspects of these data: the off-field (t = 0) permittivity, the plateau permittivity K(-), the timeindependent behavior of the permittivity at the lowest applied field, and the field dependence of the ordering.
0.1
0.2
0.3
0.4
volume fraction particles
Figure 6. Measured off-field permittivities as a function of ceramic particle volume fraction. The measured permittivities agree with the point dipole theory at low particle loadings but deviate at higher loadings, indicating limited aggregation.
The measured off-field (E = 1V/mm) permittivities for the suspensions are shown in Figure 6. Also shown is the theoretical prediction for the suspension static permittivity, K, based on point dipole interactions
which has little dependence on the particle permittivity, since p varies only from 0.95 to 0.997 in this study. The data a t low volume fraction of particles agree well with the point dipole theory, since multipole interactions become important only when the particles nearly touch. We observe deviations from eq 6 for &, > 0.1, most probably due to limited off-field particle aggregation at the higher loadings. The long-time plateau permittivities vary with the magnitude of the applied field. We can suggest two possible explanations. In the first, friction of some sort resists the field-induced aggregation, and the structure is frozen in an intermediate state along the path toward the fully coarsened crystal; higher fields result in a more fully coarsened frozen state. This intermediate, quenched state may be less dense and/or less anisotropic than the
Langmuir, Yol. 11, No. 1, 1995 311
Dielectric Response of Electrorheological Fluids particle loadings do not attain the most dense configuration but are quenched prematurely even under an infinite field;that is, the problem ofinternal friction is exacerbated at high particle loadings and results in less compact structures. We do not suggest that the dense columnar aggregates actually change from body-centered tetragonal to body-centered cubic crystalline domains as a result of this quenching. Rather, the columns simply become less dense (most likely amorphous) andor less anisotropic structures at high particle loadings. The limited aggregation in the off-field state at high particle loadings indicated in Figure 6 may lead to this premature quenching at high fields. These results imply that the final structure is determined by competing kinetic processes rather than by equilibrium thermodynamic considerations alone. We return to the observation that the permittivities for the lowest applied field (67 Vlmm) are only marginally larger than the correspondingoff-field permittivities and do not seem to increase within our experimental window. Consider the ratio, A, between the attractive force of eq 1and the disruptive thermal force, kT/R, which equals 1 at an applied field of approximately 5 V/mm in our suspensions.
I = (24nc,P2E2R3IkT)= 1 for E
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600
400
200
'
I
"
800
'
1000
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Figure 10. Ratio of the characteristic length scales associated with Brownian diffusion and field-induced aggregation during portions of the voltage cycle as a function of applied field. For fields less than 200 V/mm, the thermal forces dominate and no large aggregates can form. 2.5 o 10%,500 vimm 1o%,1ooo t 20%,200
5 Vlmm (9)
When the sinusoidal field falls below 5 V/mm in its cycle, Brownian forces tend to destroy the structure formed during the remainder of the cycle by the dipolar attractions. We can define characteristic length scales associated with the Brownian diffusion and the field-induced attraction. The diffusive length LDis simply
'
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vimm
20%,500 V/mm 20K,1000 vimm
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-
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and the dipole migration length LE is given by eq 3 Figure 11. Ratio of the permittivity of SrTiOs suspensions under a n applied field to the off-field permittivity as a function of thP,where tpis the time required to form a pair of particles. Data superpose at early times, but restricted aggregate mobility andor thermal forces destroy the superposition at longer times.
where ZD is the time during which the field is less than 5 Vlmm [5 Vlmm = E sin (otD/2)] and ZE is the time during which the field is greater than 5 Vlmm ( t = ~ nlo - t ~ ) . When LD> L E ,structures formed during t~are destroyed during t ~and , no large-scale anisotropic aggregates can exist. In Figure 10, we have plotted the ratio OfLdLE for our suspensions,which is slightly dependenton the volume fraction of particles. It is clear that thermal agitation dominates at fields below 200 Vlmm, which explains the low magnitude of the 67 Vlmm permittivity. We now investigate the field dependence of the ordering. In Figures 11 and 12, we have scaled the measured permittivity for SrTiO3 and BaTiO3 suspensions by the off-field (t = 0) permittivity and have scaled time by tp(eq 3), the pair formation time (note that t d t is a type of Mason number, Mn =FJF,). The measured permittivities begin to deviate from the off-field permittivity at tlt, zz 1as we would expect. The high-field (E = 500 and 1000 Vlmm) data superpose for titp .c 20, implying that chaining of multiparticle aggregates scales with the applied field in the same E2 fashion. We refer again t o Anderson's calculations'l on a suspension of conducting spheres at 10 vol % to help us understand the structures formed during the ordering process. The off-field permittivity was calculated to be 2.7, the permittivity ofthe fully coarsened structure (given above) was 8.3, and the permittivities for structures
3.5 0
10%,200 Vimm 10%,500 Vimm
fully coarsened
2.5
1
lo"
20%.500 vimm 20%,1000 Vlmm n
00 0
1 00
0
102
1 0'
103
1 04
t IFp
Figure 12. Ratio of the permittivity of BaTiOs suspensions under an applied field to the off-field permittivity as a function of t h p ,where tpis the time required to form a pair of particles. Also shown are the theoretical permittivity ratios" for conducting particles in a fully coarsened system, a system of uncoarsened chains which span the electrodes, and a system of particle pairs.
composed of strictly particle pairs or isolated single chains were 3.3 and 5.1, respectively. Since BaTiOa suspensions were seen to resemble closely suspensions of conducting
Adolf and Garino
312 Langmuir, Vol. 11, No. 1, 1995 spheres, we have indicated Anderson’s calculated permittivities on Figure 12. With these reference points, we conclude that the lack of data superposition a t high fields for t/zp > 20 reflects limited mobility of spanning chains during coarsening, which is exacerbated at high loadings (the quenching phenomena discussed above). In contrast, the aberrant behavior of the data a t lower fields (200 V/mm) reflects cluster disruption by thermal forces, as discussed above. In this case, the particles form large, anisotropic clusters which may not completely span the electrodes.
Conclusions This study offers the first data on the increase in permittivity of an ER fluid with time after an electric field was applied. Moreover, the fluid examined was wellcharacterized, anhydrous, and ordered under an ac field. Therefore, the particleholvent permittivity mismatch determined the polarization force, and we avoided the complications of determining conduction mechanisms in dc fields and wetting or surface tension effects associated with added water.
The data suggested that chains which spanned the electrodes were formed within roughly 20 times the time required to form a particle pair. This early-time chain formation scaled as the square of the applied field. Coarsening of these spanning chains was nonuniversal, and the fluid structure could be quenched prematurely by internal friction. Only at low particle loadings and under high applied fields did we obtain a fully coarsened structure with a permittivity in agreement with theoretical predictions.
Acknowledgment. We thank J. E. Martin for his insights into evolution of structure in ER fluids based on results from his unique light scattering and optical microscopy experiments. We also thank R. A.Anderson for access to his theoretical predictions on permittivities prior to publication and B. G. Hance for invaluable assistance in construction of the apparatus for measuring suspension permittivities. This work was supported by the U.S.Department of Energy under Contract DE-AC-04076DP00789. LA940027U
