Permittivity of electrorheological fluids under steady and oscillatory

313

Langmuir 1996,11, 313-317

Permittivity of Electrorheological Fluids under Steady and Oscillatory Shear Douglas Adolf,” Terry Garino, and Brad Hance Materials and Process Sciences Center, Sandia National Laboratories, Albuquerque, New Mexico 87185 Received January 5, 1994. In Final Form: October 19, 1994@ The permittivities of ceramic-based colloidal suspensionsin electric fields were measured under steady and oscillatory shearing flows. Under steady shear flow, a unique relationship defined the dependence of permittivity on viscosity for suspensions of differing volume fractions under various applied fields. Under oscillatory shear flow, the maximum in permittivity coincided with the minimum strain under high fields and low strain amplitudes but coincided with the minimum strain rate under low fields and high strain amplitudes.

Introduction In the preceding article,l we followed the increase in static permittivity as anisotropic structure evolved in a quiescent electrorheological (ER) fluid when a n electric field was a applied. In this article, we present measurements of the static permittivity of the same ER fluids under steady and oscillatory shear a t various applied voltages. Numerous measurements of the shear rate dependent viscosity of ER fluids have been presented previously.2 Most studies have probed fluid response under a high dc field, since applications employing ER fluids will most likely operate under these conditions. These experiments show a shear thinning, power law viscosity, r y-A,where p is the shear rate. The exponent is roughly 1, indicating Bringham plastic behavior u = r j = uy Q L ~where , u is the shear stress and uyis the yield stress. At high shear rates, 17 = +, where 7- equals the off-field viscosity. Marshal et ala3found their data to be well described by the expression

-

+

qlq- = Mn*lMn

+1

(1)

where the Mason number, Mn = t7sy/2e,,us@E)2, is the ratio of hydrodynamic forces to polarization forces, l;ls is the solvent viscosity, co is the permittivity of free space, E is the applied field strength, /3 = ( K ~ K ~ ) + ~ 2 ~ ~K~1 and , K~ are the particle and solvent static permittivities respectively, and Mn* is the Mason number characterizing the approach of r to r-. In a recent study, Martin et aL4 explored the shear rate dependent viscosity for a wellcharacterized, model ER fluid under moderate ac fields. At the highest fields, Bingham plasticlike behavior (A = 1)was observed. At lower applied fields, however, the shear thinning exponent, A, decreased smoothly with field strength to a value of roughly 2/3. The value of 2/3 corresponds to the exponent predicted when the hydrodynamic and polarization torques on a particle aggregate are balanced in steady shearing Using the same model fluid as in the steady shear studies, Martin et aL4systematicallyinvestigated ER fluid

+

Abstract published in Advance ACS Abstracts, December 15, 1994. (1)Adolf, D.; Garino,T. Langmuir 1996, 1 1 , 307. (2) See, for example, the review by: Gast, A. P.; Zukoski, C. F. Adu. Colloid Interface Sci. 1989, 30, 153. (3) Marshall,L.;Zukoski,C. F.;Goodwin,J.W .J . Chem.SOC., Faraday @

Trans. 1 1989.85. 2785. (4)Makin, J. E.; Adolf, D.; Halsey, T. C. J. Colloid Interface Sci. 1994, 167, 437. ~

response in oscillatory flow, for which the strain y(t)= yo sin wt. In these experiments, the stress was sinusoidal as well, but lagged the strain with phase angle 6, u = uo sin (ut 6 ) . No linearviscoelastic behavior was observed in the complex modulus, G*, defined as

+

G*= Gosin(ut + 6 ) = G’ sin(ut)+ G cos ( u t ) with

Go= uJyo, G = Gocos 6 , G = Gosin 6, tan 6 = GIG

-

The modulus was dependent on the applied strain even for low strains with G yo-A, where A was the same field-dependent exponent described above for stedy shearing flows. At constant strain amplitude, the moduli depended on frequency as G G o ~ l - ~These . two results imply that the in-phase component of the complex viscosity, 7’= G l w , depended on the shear rate amplitude, po= w yo, as 7’ yo-*, with obvious similarity to the steady The loss tangent, tan 6, exhibited shear resdults, qno dependence on frequency but decreased dramatically wih increasing field strength, such that G >> G” at fields above 1000Wmm. Therefore, the stress tracked the strain a t high fields, acting as a n elastic solid. However, no true solidlike behavior was ever observed for these model ER fluids, as evinced by our inability to detect a recoverable creep compliance.

- -

--

Experimental Section The fluids used in this study were identical to those fluids studied in the previous paper. The colloidal suspensions 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 steady state viscosities of our suspensionswere measured with a Carri-Med controlled stress rheometer using a parallel annulus geometry, which kept both the electric and shear fields constant. The cell diameter was 60 mm, the annulus width was 3 mm, and the plate separation was 0.15 mm. A sinusoidal 400 Hz voltage signal was applied to the sample using a Wavetek 182A function generator in series with a Trek 10/10 amplifier. Electrical contactwas made through a metal probe just touching a NaCl solution in a resevoir machined close to the shaft of the top rheometer plate. This salt water contact performed as well as a mercury contact while offering less drag on the rheometer. The permittivities of the fluids during steady shear were

0743-746319512411-0313$09.00/00 1995 American Chemical Society

Adolf et al.

314 Langmuir, Vol. 11, No. 1, 1995 1 0'

0.08

.-2

0.06 /

!! , - -

I-

1 o3

/ r

/

I

F

1 o2

F

/ I

0.04 0.02

1 0'

0

0.1

0.2

0.3

0.4

1 oo 10'~

particle volume fraction

Figure 1. Experimental and theoretical viscosities and permittivities of Ti02, SfliO3, and BaTiOa particles in dodecane with no applied electric field. Measured values agree with theoretical predictions at low particle volume fractions but deviate at the highest loadings.

measured using the apparatus described in the previous article with the exception that the sample cell in the present study was rheometer parallel annulus cell. The permittivitiesand complexmoduli during oscillatory flow could not be measured under identical experimental conditions due t o sensitivityproblems. More precisely, the frequencies and strain amplitudes which resulted in the most interesting permittivity data did not yield a robust rheological signal. The resultant complex moduliwere noisy and not reliable. Therefore, only the permittivity data are presented here. We obtained more accurate permittivity data using our Rheometrics RDS-2 controlled strain rheometer rather than the controlled stress CarriMed. The sample cell was similar t o the steady shear cell described above; however, electrical contacts were simply made with carbon brushes.

Results and Discussion Steady Shear. The off-field, high shear rate, steady shear viscosities, qo,ofTiOz, SrTiOs, and BaTiOs particles in dodecane at volume fractions, 4 = 0.1,0.2 and 0.3, are compared in Figure 1predictions for noninteracting hard sphere^.^

(3) Also, shown in Figure 1 are the theoretical and experimental off-fieldpermittivities, K ~of, the same suspensions that were described in the previous article.' Both the experimental premittivities and viscosities agree with predictions at the lowest particle loadings but exhibit stronger dependences on particle volume fraction at higher loadings. This effect may be due to limited off-field aggregation. The steady shear viscosities of these suspensions under an electric field have been discussed previously.6 The specific viscosity, q/qo - 1exhibited a power law dependence on the Mason number, Mn-*, where the exponent A varied from roughly V3 at low applied fields to the Bingham limit, A = 1, at high fields. The apparent viscosities at high shear rates under a n applied field, 7were roughly equal to the off-field viscosities, qo,and the measured viscosities approached 7- at a Mason number of roughly 1. Therefore, the critical shear rate, yc,at which (5)Russell, W. B.;Saville, D. A.; Schowalter, W . R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (6) Garino,T.;Adolf,D.; Hance, B.InProceedings oftheznternational Conference on Electrorheological Fluids: Mechanisms, Properties, Structure, Technology and Applications; Tao, R., Ed.;World Scientific: Singapore, 1992.

10.'

1 0 . ~

l o 2

l o 1

1 00

1vh Figure 2. Specificviscosities of a 20 vol % sflio3 suspension as a function of the Mason number. The specific viscosity exhibits a power-law dependence on the Mason number and vanishes at a Mason number of roughly 1,but exact superposition is destroyed by a dependence of the power-law exponent on the applied field strength. 12 o ZO%,IOOO v/mm b 30%,200 Vimm Z O % , ~ O O vimm

10

.'5

.-..

8

G

6

b

e

E

P

. .. ... 30%,1000

%

v/mm

* *

*

e

. . e

4

2

I 0- '

1 oo

1 0'

shear rate

1 0' (5'

1 o3

')

Figure 3. Permittivities of 20 and 30 vol % suspensions of T i 0 2 particles in dodecane under applied fields of 200,500, and 1000 V/mm at various steady shear rates.

the viscosity departs from q- scales as E-2. A typical set of data for a 20 vol % SrTiOs suspension at applied fields of 200, 500,and 1000 Vlmm is shown in Figure 2. The observed variation of A with field destroys exact data superposition. The permittivities for TiOz, SrTiOa, and BaTiOs suspensions at loadings of 10,20,30vol % under fields of200, 500,and 1000Vlmm as a function ofthe steady shear rate are shown in Figures 3-5. The permittivities increase as volume fraction and field strength increase and as shear rate decreases, as expected. Increasing field strength promotes ordering, while increasing shear rate destroys the anisotropic structure. In the simplest view, the polarization forces are opposed by the hydrodynamic forces, which leads to a dependence of the permittivity on the Mason number, analogous to the behavior ofthe apparent viscosity in eq 1. The reduced permittivities, K I K€or ~ , our S I T ~suspensions O~ are plotted versus Mason number in Figure 6 (Ti02 and BaTiO3 suspensions acted similarly). Clearly, the permittivity attains its off-field value at a Mason number of roughly 1 indicting that the critical shear rate, yc, at which the structure becomes disordered, scales as E-2. Unlike eq 1, however, the permittivity does not exhibit a power-law dependence upon Mn. Rather, the permittivity exhibits a n approximately logarithmic dependence upon M n in the range of shear rates explored.

Permittivity of Electrorheological Fluids 12

o

Langmuir, Vol. 11, No. 1, 1995 315 2.5

1 0 ~ , 2 0 0 VJmm

0 10%,500 Vlmm

0 500 Vlmm (Steady shear) 0 1000 Vlmm (stendy shear)

10

200 V/mm (qulercent) 500 V/mm (qulescent)

20%,1000 V/mm

* .E

2 3096,500 vimm

8

I

0 0

.-

00

e e

E

a

1.5 4

21

1

lo"

1 oo

I

I

1 0'

1 o2

- -.,

1

I 1 o3

-

1 oo

" 1 0'

t

shear rate (i')

Figure 4. Permittivities of 10,20, and 30 vol % suspensions of SrTiOa particles in dodecane under applied fields of 200, 500, and 1000 V/mm at various steady shear rates. 12 -

I

-

10

I

ITp

I

I

I

1 02

1 o3

10 '

or l l M n

Figure 7. Reduced permittivities of 20 vol % SrTiOs suspensions as a function of tlz, (similar to Mn-l and defined in the previous article') for quiescent fluids or as a function of Mn-' for fluids in steady shear.

I

~ O % , I O Ov/mm O

0

.

0

lO%,ZOO Vlmm

o

10%,500 v m m

20%,500 V/mm

**

20%,500 Vlmm

1.6

1.4

i

20%.1000 Vlmm x 30%,200 Vlmm

. . 0

1 . .

0 +

+ 0

*d

+ . a *

+

1.2

I

1 1 1 oo

X X *

=

3

+ .O

i

XD O

.",' I

I

10'

1 02

mo

1

1

o3

Figure 8. Reduced permittivities of 10,20,and 30 vol % SrTiOa suspensions as a function of the reduced viscosities. The data at higher applied fields approximately superpose. 1 .a

I

I

El. 0 10%,200 Vlmm

10%,500 vlmm 0 10%,1000 Vlmm a 20%,200 V/mm 20%,500 vlmm ZO%,IOOOv/mm

1.6 0

+. *, * O $0

1.4 -

+

.

x 30%,200 v/mm + 30%,500 vimm

0

+@ 1.2

-

+BP

o x

1o . ~

-

-

lo"

102

1

oo

Mn

Figure 6. Reduced permittivities of SrTiOs suspensions as a function of the Mason number. The data at higher applied fieldsapproximatelysuperpose,and the permittivity approaches the off-field permittivity at a Mason number of roughly 1. ((KIKJ

- 1)

-

log (Mn>-l

(4)

This relationship cannot be valid a t very low shear rates since the permittivity can only increase to K,, the permittivity of the fully coarsened suspension. The data for applied fields of 500 and 1000 V/mm and all volume fractions superpose fairly well, which suggests that the simple balance of polarization and hydrodynamic forces captures not only the essence of rheological behavior

(Figure 2) but structure under steady shear as well. However, the data for a n applied field of 200 V/mm lie considerably below the data for higher fields. In the previous article describing the evolution of structure in quiescent fluids,' we concluded that fluid behavior at this low applied field was quanlitatively distinct from the behavior at higher fields due to destruction of order by thermal forces during a significant portion of the ac field cycle. We see here and later that this trend continues; the permitivity data for applied fields of 200 V/mm under shearingfiow are also qualitatively different from the data obtained a t higher fields. In Figure 7, we have plotted the reduced permittivities for the 20 vol % SrTiOs suspensions in the quiescent fluid as a function of t/zpand under steady shear as a function of inverse Mason number. (It was shown in the previous article' that the ratio zdt is similar to a Mason number.) The permittivities under even very slow shearing flows are significantly lower that the fully coarsened quiescent permittivity, which is not surprising since we would expect hydrodynamic forces to disrupt easily spanning columns. The structure and rheology can be related by plotting the reduced permittivity versus the reduced viscosity, as in Figure 8, for all SrTiOs suspensions (again, T i 0 2 and BaTiOs suspensions behaved similarly). A n approximately logarithmic dependence of permittivity upon viscosity is apparent. While the data for applied fields of 500 and 1000 V/mm appear to superpose for all particle loadings, the data for fields of 200 V/mm lie significantly

Adolf et al.

316 Langmuir, Vol. 11, No. 1, 1995 7

1

I

I

I

I

- - - - -* a

E

~

,

F,i:;

~

- * - yo=2

4

'

I

-2

-1

I

0

I

1

lo"

I

1 0'

2

(dashed lines) V/mm.

lower, due to the aforementioned effects. The superposition of the data a t higher fields implies that a single physical model relates structure and rheology in this experimental regime of particle loadings, field strength, and shear rate. Oscillatory Flow. Permittivity measurements during oscillatory flow offer a sensitive probe of the relationship between structure and rheology. A sinusoidal strain, yo sin(wt), yields a sinusoidal stress of the same frequency, which leads the strain by a phase angle, 6. Similarly, our measured permittivities oscillated sinusoidally about a mean permittivity, with twice the frequency of the applied strain and with a phase angle, 8.

Figure 10. Permittivity phase angle, 0, for 20 vol % SfliO3 suspensions in an applied field of 500 V/mm as functions of strain amplitude (% strain) or strain frequency (rads). Little

dependence on either parameter is observed. 1 0'

1 00

a C

m

CI

10'

1 o'2 1 0'

+ (A") cos[2(ot - e)]

1 oz

YO strain or frequency (rad/s)

(5)

As discussed later in more detail, we expressed the permittivity phase angle in terms of the cosine so that the phase angles, 6 and 8, vanish in the same limit. The permittivity oscillates a t twice the strain frequency since the sign of the strain is unimportant. We measured the permittivity of 20 vol % SfliO3 suspensions a t strain frequencies of50,100, and 200 r a d s (compared to the ac field frequency of 400 Hz = 2513 rad/ s); strain amplitudes of 0.5, 1, and 2; and applied field amplitudes of 500 and 1000Vlmm. In Figure 9, we display typical data for a strain frequency of 100 rad/s a t all strain and field amplitudes. We will focus on three aspects of the data: the phse angle, 8, the maximum permittivity reached within a shear cycle, K", and the amplitude of the permittivity change within a shear cycle, AK. I t is apparent in Figure 9 that the permittivity phase angle, 8,vanishes as the field becomes large and the strain amplitude becomes small. As discussed above, the complex modulus phase angle, 6, also vanishes in this limit, leading to the cosine formalism in eq 5 such a t 6 and 8vanish simultaneously. In Figures 10 and 11, we present the strain amplitude, yo, and frequency dependences, w , of tan 8 under applied fields of 500 and 1000 Vlmm, respectively. We chose tan 8 instead of 8 itself in analogy with the commonly used rheological loss tangent, tan 6. Little dependence of tan 8 on o is seen a t 500 Vlmm (Figure 10)with a relatively weak dependence on yo. At the higher field (1000 V/mm in Figure ll), still little frequency dependence is noted, but the amplitude dependence is significnt, tan 8 yo1.8. The frequency independence of tan 8 is reminiscent of the behavior of tan 6 measured by

-

1 o3

% strain or frequency (rad/s)

strain

Figure 9. Permittivities of 20 vol % SrTiOs suspensions as a function of oscillatory shear amplitude at a frequency of 100 rads and under applied fields of 500 (solid lines) and 1000

= (K)

1 o2

Figure 11. Permittivity phase angle, 0, for 20 vol % Sr'l'iO~ suspensions in an applied field of 1000 V/mm as functions of

-

strain amplitude (% strain) or strain frequency (rads). Little dependence on frequency is observed, but tan 0 yo1,8.

Martin et al.4, which was also independent of strain frequency. If, in fact, tan 8 = tan 6, then

1-2

(T -

(6)

and the structure would track the stress, not the strain; that is, the most ordered structure would form when the permittivity is greatest which, in turn, would occur when the stress is least. We can attempt to provide a physical interpretation of these result. At low strain amplitudes and high applied fields, the interparticle forces are high and the hydrodynamic forces are low. Therefore, the fluid under oscillatory shear retains the electrode-spanning, columnar structure formed in the quiescent fluid with the columns simply tilting under the applied strain. The column tilt angle, a,with respect to the field equals the shear strain, y = yo sin(wt)x yowt for low strain amplitudes. At zero strain, the columns are perpendicular to the plates, and the permittivity is maximized. Anderson7 has shown that the permittivity of columns tilted a t a n angle a has the form of eq 5 with K , , , ~ = KII, mi,, = K I I- ( K J Y ~ ~ and , 8 = 0, where K I Iand are the suspension permittivities parallel (7) Anderson, R. A. Langmuir 1994,10,2917.

Permittivity of Electrorheological Fluids 7

I

0

(0

Yo

50

0.5

Langmuir, Yol. 1 1 , No. 1, 1995 317

1

E' 500

higher. This curious result implies that the relatively gentle oscillatory shearingjostled the particles from their quiescent structure, which, as discussed in the previous article,l was not fully coarsened due to premature quenching to a more fully ordered state. In fact, we see in Figure 12 that the permittivity under oscillatory shear appears to approach the theoretical, fully coarsened structure, K ~ , K,(=) = K I I , where K,(=) is the quiesent fluid permittivity at long times under high fields and K~~ is Anderson's resdult of eq 7 in the preceding article.' Therefore, the physical interpretation outlined above for the limit of low strain amplitude and high field, in which columns simply tilt with the shear strain, predicted that Km, = K I I and is verified experimentally. Finally, the amplitude of the permittivity change within a shear cycle, AK, exhibited no dependence on shear amplitude or frequency. For E = 500 Vlmm, AK = 0.3 f 0.1 and, for E = 1000 Vlmm, AK = 0.4 f 0.1, which are relatively small changes in permittivity within a shear cycle. From the previous discussions, we concluded that the low strain amplitude, high-field physical model (0 0) of Anderson agreed with experimental measurements a t yo = 0.5 and E = 1kV1mm. Anderson's prediction for AK was AK = ( K I I - ~ ~ ) y ~ However, ~/2. since these calculations for K I I and KL are valid only for conducting particles, while &Sh-"iO~)= 290, we examined the ratio Of fk to Km,, AK/Km, = (1- Ki/Kll)y,,%& to account somewhat for finite particle permittivity. With KII = 13 and K~ = 3.4, A K I K ~= , 0.09 as compared to our experimental result of AK/Km, = 0.06 f 0.01 for yo = 0.5 and E = 1 kV/mm.

-

1

I

1 o.2

1 oo

10''

tlme

1 0'

(5)

Figure 12. Ratio of the maximum permittivity attained in a shear cycle to the off-field permittivity for 20 vol % SfliOa suspensions as a function of time after the field is applied. The permittivity increases with time and, under low strain amplitudes, may even exceed the quiescent fluid permittivity measured at long times. and perpendicular to the field for a columnar system given by eqs 7 and 8 in the previous artic1e.l In the opposite limit of large strain amplitudes and low applied fields, the structure will be greatly distorted from the quiescent structure and will resemble more the structure attained under steady shear. In this limit, then, we would expect the permittivity to reach a maximum a t the lowest strain rate (the maximum strain). Since the permittivity tracks the strain rate, the phase angle, 0, is 90". The experimental results follow these general trends; the phase angle vanishes under low-amplitude strains and high fields and approaches 90" under high-amplitude strains and low fields. We now turn our attention to K", the maximum permittivity reached within one shear cycle. In our experiments, the field was applied to the quiescent fluid for 1min to ensure that the resulting structure coarsened. The oscillatory strain was then applied, and the observed permittivity initially decreased (within 10 ms) and slowly increased (-10 s) as shown in Figure 12. For large strain amplitudes, the increase in permittivity with time was not significant. However, for low strain amplitudes, the permittivity increased by roughly 15%. Examining the data in Figure 12 more closely for the lowest frequency, w = 50 rad/s, and strain amplitude, yo = 0.5, and the largest applied field, E = 1kVImm, we observe that, while the permittivity measured immediately after initiation of the strain was roughly 5% lower than the quiescent permittivity, the ultimate permittivity was roughly 10%

-

Conclusions The data presented here offer a unique insight into the relationship between structure and ER fluid performance. Our fluids were well-characterized and interacted via simple dielectric polarization forces. The collapse of our permittivity data when plotted versus the viscosity implies a single relationship connecting the fluid structure and steady shear rheology. The oscillatory flow results, while more complicated, lay between two physically reasonable limits. Under high fields and low strain amplitudes, the electrode-spanningcolumns simply tilted with the applied strain, resulting in a high permittivity a t low strain. Under low fields and high strain amplitudes the columnar structure was severely disrupted, resulting in a high permittivity a t low strain rate. Acknowledgment. This work was supported by the U S . Department of Energy under Contract No. DEAC-04076DP00789. LA940028M