Rheol Acta (2003) 42: 199–208 DOI 10.1007/s00397-002-0290-7
Young Sil Lee Norman J. Wagner
Received: 22 June 2002 Accepted: 20 November 2002 Published online: 5 February 2003 Springer-Verlag 2003
Y. S. Lee Æ N. J. Wagner (&) Department of Chemical Engineering and Center for Composite Materials, University of Delaware, Newark, DE19716, USA E-mail:
[email protected]
ORIGINAL CONTRIBUTION
Dynamic properties of shear thickening colloidal suspensions
Abstract The transient shear rheology (i.e., frequency and strain dependence) is compared to the steady rheology for a model colloidal dispersion through the shear thickening transition. Reversible shear thickening is observed and the transition stress compares well to theoretical predictions. Steady and transient shear thickening are observed to occur at the same value of the average stress. The critical strain for shear thickening is found to depend inversely on the frequency at fixed applied stress for low frequencies
Introduction When subjected to increasing shear stress, concentrated colloidal suspensions can exhibit a steep rise in viscosity (Lee and Reder 1972; Hoffman 1974, 1997; Barnes 1989). This shear thickening phenomenon can damage processing equipment and induce dramatic changes in suspension microstructure, such as particle aggregation, which results in poor fluid and coating qualities. On the other hand, this behavior can be exploited in the design of damping and control devices, whereby the fluid can limit the maximum rate of flow through a highly nonlinear response (Laun et al. 1991; Helber et al. 1990). It has been demonstrated that reversible shear thickening in concentrated colloidal suspensions is due to the formation of jamming clusters bound together by hydrodynamic lubrication forces, often denoted by the term ‘‘hydroclusters’’ (Bossis and Brady 1989; Farr et al. 1997; Foss and Brady 2000; Catherall et al. 2000). The microstructure of shearing suspensions has been studied
(high strains), but is limited to an apparent minimum critical strain at higher frequencies. This minimum critical strain is shown to be an artifact of slip. Lissajous plots illustrate the transition in material properties through the shear thickening transition, and the energy dissipated by a shear thickening suspension is analyzed as a function of strain amplitude. Keywords Shear thickening Æ Suspension rheology Æ Colloid Æ Dispersions Æ Dilatancy
by rheo-optical experiments (D’Haene et al. 1993; Bender and Wagner 1995), neutron scattering (Laun et al. 1992; Bender and Wagner 1996; Newstein et al. 1999; Maranzano and Wagner 2001a, 2002) and stress-jump rheological measurements (Kaffashi et al. 1997). The onset of shear thickening in steady shear can now be quantitatively predicted (Maranzano and Wagner 2001a, 2001b) for suspension of hard-spheres and electrostatically stabilized dispersions. Although a significant number of experimental and simulation studies have addressed shear thickening in steady shear flow, only limited work has addressed the viscoelastic properties of a shear thickening fluid (Laun et al. 1991; Boersma et al. 1992; Raghavan and Khan 1997; Mewis and Biebaut 2001). This is partly a consequence of the difficulty of studying shear thickening, which is a highly nonlinear response that is often triggered by relatively high stress levels. Nonlinear oscillatory shear experiments are useful for characterizing the onset of shear thickening as well as for
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determining the time scales required to generate the shear thickening response. Of interest for technological applications is the time scale and the minimum strain required to generate the hydrocluster microstructure underlying the shear thickening response. For example, electrostatically stabilized dispersions have been investigated in both steady and dynamic oscillatory shear flows for possible damping applications. Laun et al. (1991) reported the dynamic strain hardening behavior of a polymer latex dispersion. Their oscillatory test protocol increased the strain amplitude at a given frequency, which leads to a shear thickening rheology. The critical strain for dynamic shear thickening (cc) was observed to decrease with increasing frequency (x), but eventually plateaued at higher frequencies. The low frequency behavior was interpreted in terms of the steady shear behavior, where a critical shear rate (c_ dynamic c_ steady ) must be achieved to thicken. The high c c frequency limiting value suggested that a minimum shear strain (50%) is necessary in each half cycle to cause the dispersion to switch to the high viscosity state. Similar conclusions for low frequencies were reached by Boersma et al. (1992), who investigated monodisperse silica particles suspended in a mixture of glycerol and water. They reported ‘‘flow blockage’’ in oscillatory testing, which was also related to steady shear thickening at low frequencies. Intermediate frequencies yielded a weaker frequency dependence, but no plateau value. Finally, at high frequencies the critical deformation for shear thickening was found to be independent of particle volume fraction, but again scaled inversely with frequency. This latter behavior was attributed to the solid-like response for a sample that is fully in the hydrocluster state. Note that these experiments were also performed on a controlled strain rheometer and the critical strain amplitudes were of order O(10-2) at the highest frequencies. Studies of near hard sphere dispersions (Bender 1995) also confirm the agreement between steady shear thickening and the low frequency dynamic oscillatory response. Raghavan and Khan (1997) observed similar congruence at low frequencies, as well as a high frequency limiting critical strain (ccO(1)) for fumed silica dispersions in poly(propylene glycol). Recently, Mewis and Biebaut (2001) also observed dynamic shear thickening in sterically stabilized colloidal suspensions. They observed that the peak shear stress at the onset of shear thickening in oscillatory flow corresponds to the same steady shear stress measured at the onset of shear thickening, with no evidence of a limiting critical strain down to shear strains of order 0.5. Notably, Mewis and Biebaut (2001) also investigated the shear thickened state by parallel superposition, observing a viscoelastic liquid response in the shear thickened state, not the solid response suggested by Boersma et al. (1992).
In summary, the literature data to date suggests that the onset of strain hardening at low frequencies for concentrated suspensions in oscillatory shear flow can be interpreted in terms of the onset of steady shear thickening. However, there is contradictory evidence as to whether a critical strain is required for oscillatory strain hardening, and as to whether a third, solid-like regime exists at higher frequencies. This issue is relevant for the design of devices based on the shear thickening response (Helber et al. 1990). The goal of this work is to relate the nonlinear viscoelastic properties to the steady shear response for a shear thickening fluid, and to determine if a minimum critical strain is necessary for shear thickening. This is achieved by rheological investigation of a model dispersion. Of particular interest is the critical strain amplitude required for shear thickening in dynamic shear flow and its dependence on frequency. Finally, Lissajous plots are constructed to illustrate the ‘‘switching’’ from liquid to solid observed during deformation, and to determine the energy dissipation’s depend on strain amplitude in a shear thickening fluid.
Experimental Sample preparation and characterization The colloidal silica investigated here was obtained from Nissan Chemicals (MP4540), which is provided as an aqueous suspension (pH=8.5 at 25 C) with a particle concentration of about 40 wt%. The particle size distribution has been characterized with dynamic light scattering and TEM. Figure 1 shows a transmission electron micrograph of the suspension; which is observed to contain a minor fraction of smaller particles. The average particle diameter (z-average) was determined to be 446±8.4 nm by dynamic light scattering, which agrees with the TEM measurements of the large particle fractions. The solution density of the particles has been obtained by measuring the density of the suspension as a function of weight fraction of the particles. The weight fraction of silica was determined gravimetrically after drying the sample at 180 C for 5 h using a convection oven. The density of the silica calculated from this method is 1.78 g/cc. The zeta potential has been determined to be )32 mV from electrophoresis measurements (Brookhaven ZetaPALS) at pH=8.5 and CSALT=0.045 mmol/l. This suspension was concentrated by tabletop centrifugation. The sediment was resuspended using a vortex mixer after adding of small amount of the supernatant liquid. Dilution with the mother liquor provided a series of aqueous silica suspensions. The suspending fluid was also replaced with ethylene glycol (EG) by repeated centrifugation and resuspension with a vortex mixer. This process has been repeated four times to prepare a second series of dispersions of the same particles in ethylene glycol. Rheological measurements The experiments were performed primarily in a stress-controlled rheometer (SR-500, Rheometrics) at 25 C with cone-plate geometry having a cone angle of 0.1 radian and a diameter of 25 mm. Complementary measurements were performed on a Rheometrics ARES controlled strain rheometer. A parallel plate geometry was also used with varying gap size to characterize slip. To prevent adhesive slip between the sample and the rheometer plates, parallel plates of diameter 25 mm were covered with emery cloth (NORTON, E-Z FLEX METALITE K224) using double stick tape. The gaps explored varied between 0.05 mm
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and 1.5 mm. A solvent trap prevented evaporation of the solvent. Three types of dynamic experiments have been performed: stress or strain sweeps at constant frequency, frequency sweeps with a constant imposed stress, and time transient measurement of strain response with fixed stress amplitude and frequency. To remove loading effects, a preshear of 1 s-1 was applied for 60 s prior to further measurement. All measurements presented here were reproducible.
age dynamic shear stress applied to the fluid during oscillatory testing is obtained by integrating the absolute value of the applied stress over one cycle as Z x 2p=x 2s0 ð1Þ sd ¼ jsd jdt ¼ 2p 0 p where sd=s0cosxt and s0 is the maximum stress. Note that the root mean square value, i.e.,
Results and discussion Result of stress and frequency sweeps Figure 2 shows the steady shear viscosity and the complex viscosity as a function of the steady shear stress or average dynamic shear stress, respectively, for the aqueous suspension at volume fractions of u=0.55 (a) and 0.60 (b). Shear thickening is evident in both steady and dynamic measurements. To confirm the reversibility of the shear thickening behavior, the complex viscosity was measured for both ascending and descending stress sweeps, with good agreement. Shear thickening is known to be a stress controlled phenomena; hence one might expect to be able to superimpose steady shear viscosity and complex dynamic viscosity when plotted against applied stress. The aver-
Fig. 1 Transmission electron microscopy of colloidal silica obtained from Nissan Chemicals (MP4540) at a magnification of 40,000
Fig. 2a,b Reversible shear thickening behavior of: a 55 vol.%; b 60 vol.% colloidal silica dispersed in water for both steady and dynamic shearing plotted against the average applied dynamic shear stress
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R x 2p=x 2 sRMS ¼ sd dt ¼ psdffiffi2 1:11sd , could be used d 2p 0 with equal superposition to within the accuracy of our measurements. However, the data do not superimpose when plotted against the maximum applied stress, or the shear rate. At the point of shear thickening, the dynamic viscosity is found to be comparable to the steady shear viscosity measured at the same average stress. Deviations between the complex and steady shear viscosities are evident at both low and high shear stresses, as will be discussed below. Using the model developed by Maranzano and Wagner (2001a, 2001b), the critical shear stress for shear thickening (sc) can be predicted from independent measurements of the particle size, concentration, surface potential, and ionic strength. The equation for the critical shear stress for electrostatically stabilized dispersions is kB T ðjaÞW2 sc ¼ 0:024 ð2Þ a2 lb where lb is the Bjerrum length defined by lb ” e2/ (4p0kBT), a is the radios of the particle, and Ys=wse/ kBT is the dimensionless surface potential. The theoretical predictions for sc are found to be in good agreement with the measured values (Table 1). In this table hm is the characteristic separation distance in the incipient hydrocluster state, which is determined directly from j)1, the Debye length by hm=1.453/j. The extended Cox-Merz rule equates the steady and dynamic viscosities at equivalent shear rate and frequency. As our data is highly nonlinear, this relationship is observed not to hold. For materials with slow relaxing microstructure, a modified Cox-Merx rule has been proposed by Doraiswamy et al. (1991) known also as the ‘‘Delaware-Rutgers’’ rule. The basis of this approach is that slowly relaxing materials respond to the highest applied strain rate applied during the dynamic measurement. Consequently dynamic and steady properties overlay when the highest shear rate experienced during the oscillation is taken as the effective steady shear rate ðc0 x c_ ). Raghavan and Khan (1997) use this rule to correlate the steady shear thickening to strain thickening for their fumed silica suspension. In Fig. 3 we compare data obtained from a frequency sweep experiment with imposed maximum stress of 10 Pa to steady shear data in the sprit of the ‘‘Delaware-Rutgers’’ rule. It is apparent that the ‘‘Delaware-Rutgers’’ rule applies in
the region of shear thinning for the suspension of 60% silica particle, but that the shear thickening transitions do not superimpose. Closer inspection of the data of Raghavan and Khan (Fig. 10 of their work) shows a similar level of disagreement upon shear thickening. Consequently neither extended Cox-Merz nor the Delaware-Rutgers rule correlates dynamic and steady viscosities in the shear thickening regime. This is further confirmed by Mewis and Biebaut (2001), who report qualitative differences in the parallel superposition viscoelastic measurements between shear thinning and shear thickening parts of the flow curve. Thus, although the ‘‘Delaware-Rutgers’’ rule applies to the shear and frequency thinning regions of the flow curve, it fails to correlate the viscosity in the shear thickened state or the onset of the shear thickening transition. This result is consistent with the concept that the shear thickening transition is stress controlled, as demonstrated in Fig. 2 above. As the shear-thickening transition is relatively fast for our materials, and the response is in the nonlinear regime, neither the extended Cox-Merz rule nor the Delaware-Rutgers approach would be expected to hold in the shear thickening regime, as demonstrated. Shown in Fig. 4 is the shear thickening behavior as a function of strain amplitude for the aqueous suspension at 60 vol.% silica. The strain reported here is the measured, peak strain amplitude (this experiment is
Table 1 Comparison of experimental and theoretical critical shear stress foraqueous silica particle suspensions Volume ja fraction / 0.55 0.60
hm Theoretical critical Experimental critical (nm) shear stress sc (Pa) shear stress sc (Pa)
8.4 39 9.4 35
37 41
20±0.5 46±0.5
Fig. 3 Application of Cox-Merz and Delaware-Rutgers rules for 60 vol.% aqueous silica dispersion
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determined from the steady shear data. The lines in Fig. 5 are calculated from the measured steady shear data. The predictions work well for our data (Fig. 5a) and that of Raghavan and Khan (1997) and Mewis and Biebaut (2001) (Fig. 5d) at low frequencies. However, deviations from this prediction become evident at higher frequencies. The data of Boersma et al. (1992) (Fig. 5b) may be limiting toward the predictions, whereas the data of Laun et al. (1991) does not have the corresponding steady shear data for making the prediction. However, note that Laun’s data qualitatively displays the predicted, low frequency behavior. Wall slip has been postulated to influence the high frequency results (Boersma et al. 1992). Figure 6 is a schematic drawing of the postulated deformation with slip between parallel plates, where the slip distance at the wall has been defined as Dslip and h is the gap size. Thus, the measured or apparent strain is the sum of the true or real strain in the sample and the strain due to slip as 2Dslip : ð4Þ h Thus, at the point of shear thickening cc,app(=Dapp/h) is the apparent critical strain for shear thickening and cc,real (=Dreal/h) is the real critical strain for shear thickening. The apparent critical strain has been measured using 25-mm parallel plates where the gap size is varied from 1.5 mm to 0.05 mm. Figure 7 shows the measured apparent critical strain for shear thickening for silica dispersed in both water and ethylene glycol at 25 C. The lines correspond to a fit of Eq. (4) to determine the slip distance. The fits yield slip distances of 0.44 mm for the ethylene glycol based shear thickening fluid and 1.45 mm for the water based shear thickening fluid. However, with the use of roughened plates the critical strain for shear thickening obtained during strain sweep experiments were found to be nearly independent of gap size, as shown in Fig. 7, demonstrating that the slip was substantially reduced. This analysis can explain why the measured critical strain for shear thickening limits at high frequencies. We can model the critical strain shown in Fig. 5 by accounting for a slip distance at the fixture walls as cc;app ¼ cc;real þ
Fig. 4 Strain thickening behavior of 60 vol.% aqueous silica dispersion as a function of dynamic frequency
performed on a stress controlled instrument). From this data, the critical strain for strain thickening was found to depend inversely on the dynamic frequency for low frequencies, as shown in Fig. 5. The frequency dependence of the critical amplitude has been reported for various systems. Laun et al. (1991) and Raghavan and Khan (1997) observed transitional behavior, namely that at low frequencies the critical amplitude decreases inversely with the frequency, while at very high frequencies (x>100 rad/s) the critical amplitude approaches a constant value as shown in Fig. 5. In the present experiments the frequency for the onset of shear thickening is around 30 rad/s and the minimum critical strain was found to be 2. Figure 5 also shows the critical strain for the ethylene glycol suspension of 62 vol.% silica. The overall behavior is similar to that of the aqueous silica suspension, but the minimum critical strain of the ethylene glycol based suspension is lower (0.7). The low frequencies data reported in Fig. 5 can be understood within the context of the model presented by Maranzano and Wagner (2001a, 2001b). As shown, the dynamic oscillatory measurement show shear thickening at the same value of the average applied stress as the steady shear measurements. During the oscillation, shear thickening is observed to occur when s>sc. One can thus estimate the strain required by sc ; ð3Þ cc gc x where gc is the shear viscosity at the critical point for shear thickening and sc is the critical stress, both
cc;app ¼
2Dslip sc þ : gc x h
ð5Þ
In the comparison, we take h1.25 mm, the edge gap, to be a characteristic distance for the cone-plate geometry used for these measurements. In Fig. 8 the lines representing the predictions of Eq. (5) are in good agreement with the observed critical strain as a function of frequency. For comparison, we used our slip measurements and the reported steady shear data to predict the critical strains reported by Raghavan and Khan (1997) for fumed silica in polypropylene glycol. Their data fits well with their steady shear data
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Fig. 5a–d Critical strain for shear thickening plotted as a function of frequency for: a 60 vol.% aqueous and 62 vol.% ethylene glycol based silica dispersions; b silica in glycerol/water by Boersma et al. (1992); c latex in ethylene glycol by Laun et al. (1991); d fumed silica in PPG by Raghavan and Khan (1997) and silica in octanol by Mewis and Biebaut (2001). The lines are predictions of Eq. (3) using the measured steady shear rheology when available
and our slip distance characterized for a silica suspension in ethylene glycol. According to our experiments with roughened plates, the slip can be greatly reduced (i.e., Fig. 7), such that the critical strain for shear thickening no longer exhibits a plateau value for high frequencies (Fig. 8). This shows that the high frequency limiting value of the critical strain observed
in previous experiments with smooth tooling is an artifact of adhesive failure and slip. The experimental analysis of slip demonstrates that the high frequency plateau in the critical strain can be explained as wall slip. The slip distance is expected to depend on the properties of solvent, which to first order might be expected to correlate with the wetting of the tool by the solvent. Indeed, we observe more slip for water than ethylene glycol, which qualitatively follows the expected wetting behavior (Vidal 2002) (contact angles on alumina are reported to be 75.9 for water and 50.4 for EG). Interestingly, this analysis confirms the postulate of Boersma et al. (1992), who surmised that their high frequency behavior, for
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Fig. 6 Schematic illustration of the displacement of a suspension between parallel plates with wall slip
Fig. 8 Measured and predicted critical strain as a function of angular frequency for silica suspension in water (open squares), ethylene glycol (open circles) measured with cone and plate, ethylene glycol (filled circles) measured with emery cloth covered parallel plates and fumed silica suspension in polypropylene glycol (open triangles) from Raghavan and Khan (1997)
Lissajous plots
Fig. 7 Apparent critical strain for shear thickening as a function of gap size between parallel plates. Silica suspension in water (filled squares), ethylene glycol (open circles) with cone and plate and ethylene glycol (filled circles) with emery cloth covered parallel plates
which the critical strain for shear thickening does not plateau (Fig. 5b), was due to a solid-like response without slip.
To understand further the dynamical nature of the shear thickened state, the dynamic oscillatory measurements were analyzed as flows. The energy dissipated by the shear thickening fluid can be obtained by integrating the area contained in a plot of stress vs strain for a dynamic oscillatory test. The flow mechanism can be understood from the shape of the resulting stress-strain curve (Lissajous plot). At 5 Pa of maximum imposed stress and 0.1 rad/s of the frequency, an elliptical hysteresis loop is recorded as shown in Fig. 9. Figures 9, 10, and 11 show the data taken at stress amplitudes of 5 (linear viscoelastic region), 50 Pa (strain thinning nonlinear region), and 500 Pa (strain thickening nonlinear region). For this sample the critical maximum stress (s0) for stress thickening in a dynamic test at x=0.1 rad/s was around s0=80 Pa and the critical stress for shear thickening in steady shear was 46 Pa, which approximately corresponds to sd ¼ 2sp0 . The area enclosed by the ‘‘loop’’ can be interpreted as viscous damping. The angle between the primary axis of the ellipse and the horizontal axis indicates the elastic modulus; a zero degree orientation is Newtonian. As seen, the colloidal dispersion at s\ p2 sd yields a cycle that has both elastic and viscous character, indicating viscoelasticity. In order to explore the difference in mechanical
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Fig. 9 Lissajous plot for aqueous silica dispersion at 60 vol.% and a frequency of 0.1 rad/s and the applied stress amplitude of 5 Pa which is relatively low
Fig. 10 Lissajous plot for aqueous silica dispersion at 60 vol.% and a frequency of 0.1 rad/s and the applied stress amplitude of 50 Pa which is close to the point of shear thickening
properties between shear thinning and thickening region, the angular frequency was held fixed at 0.1 rad/s while the maximum stress was increased stepwise. As seen, the area enclosed increases with increasing stress amplitude, indicating an increase in viscous dissipation. Above maximum imposed stresses of 50 Pa the loops
Fig. 11 Lissajous plot for aqueous silica dispersion at 60 vol.% and a frequency of 0.1 rad/s and the applied stress amplitude of 500 Pa, which is well above the critical stress for shear thickening
deviate strikingly from an elliptical shape. Further increases in stress amplitude above the critical stress converge to a highly non-elliptical shape. This can be interpreted as the superposition of a primarily fluid response for low stresses in the cycle with a primarily elastic response for stresses exceeding the critical stress for shear thickening. Although qualitative, the shape analysis immediately distinguishes this fluid from other, complex behaviors (such as a yielding fluid), and signals the onset of shear thickening in the dispersion. Notice that, on the time scale of the oscillation, the fluid is thickening and ‘‘melting’’, such that the material response time for shear thickening is substantially faster than the experiment’s frequency. The normalized strain (c/cmax) is plotted as a function of the normalized applied stress (s/s0) (sinusoidal frequency x=0.1 rad/s) on a period (Fig. 12). Table 2 gives the normalizing factors for Fig. 12. Below the transition stress for shear thickening (46 Pa) the distortion of stress-strain curve is increasing with stress amplitude. However, above the transition stress for shear thickening, the Lissajous diagrams show the same pattern with increasing imposed maximum stress. Note that the maximum strain limits at high stress (Table 2), which is in agreement with the slip analysis if the sample itself exhibits primarily a solid response. This result is also in good agreement with the observation of Mewis and Biebaut (2001), who observed a unique, viscoelastic master curve for their shear thickening dispersions using parallel superposition. Note, however, that this pattern and the parallel superposition spectrum observed by
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Fig. 13 Energy dissipated per cycle per volume as a function of measured strain at x=0.1 rad/s for 60 vol.% aqueous silica dispersion obtained from integrating the area enclosed in the Lissajous plots
Ud ¼ J cn0 : Fig. 12 Scaled Lissajous plot obtained for 60 vol.% aqueous silica dispersion at x=0.1 rad/s and stress amplitude of 1, 5, 10, 50, 100, 500, and 1000 Pa Table 2 Scaling values for Fig. 12 Angular frequency x (rad/s)
Maximum stress s0 (Pa)
Maximum strain c0
0.1 0.1 0.1 0.1 0.1 0.1 0.1
1 5 10 50 100 500 1000
2.8·10)4 4.8·10)3 1.3·10)2 16 16 17 18
Mewis and Biebaut (2001) must also reflect the large amount of slip observed in the shear thickening state. Analysis of the shear thickening fluid subjected to a dynamic frequency of 0.1 rad/s with stress amplitudes varying from 1 to 1000 Pa shows (Fig. 13) that the energy dissipated is an increasing function of strain amplitude (peak value of measured strain). The energy dissipated during a cycle (Ud) is given by the area enclosed H by the Lissajous plot (Yziquel et al. 1999) (i.e., Ud ¼ sdc). The following phenomenological relation is often used to relate dissipation energy to strain amplitude (Citerne et al. 2001):
ð6Þ
where J and n are material constants. The constant J is referred to as the damping constant and n is called the damping exponent. The energy dissipated per volume per cycle for our silica suspension is proportional to the strain raised to the second power when the fluid is deformed in linear viscoelastic region, as expected. However, above the linear viscoelastic region, the damping exponent is very large of c0 (n15). This abrupt change in behavior signals a change in microstructure and is taken to be a signature of the hydroclustered state.
Conclusions Reversible shear thickening is measured in both oscillatory and steady shear flow for a charge stabilized colloidal suspension in two solvents. The DelawareRutgers rule can correlate suspension rheology prior to shear thickening, but is violated in the shear thickened state. Instead, a superposition of steady and the dynamic viscosities at the point of shear thickening regime is achieved if plotted vs the average stress magnitude. The critical strain required for dynamic shear thickening scales inversely with the frequency for small frequency, while for high frequency, slip leads to an apparent plateau. Consequently, measurements of steady shear thickening and the slip enable predicting both the onset of shear thickening in oscillatory flow, as well as the behavior of the strain amplitude with frequency. The
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damping characteristics of a shear thickening suspension as calculated from the Lissajous plots show a dramatic increase in viscous dissipation upon shear thickening, which is due to the jamming inherent in the hydroclustered state.
Acknowledgments Dr. Eric Wetzel of Army Research Laboratory (MD) is acknowledged for helpful discussions. This work has been supported through the Army Research Laboratory CMR program (Grant No. 33-21-3144-66) through the Center for Composite Materials of the University of Delaware. Funding for equipment from the NSF-MRI (CTS-9977451) is gratefully acknowledged.
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