Dynamic properties of shear thickening colloidal suspensions

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202 sRMS d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R x 2p=x 2p 0 s2 ddt q ¼ sdffiffi p 1:11sd, could be used 2 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 sc ¼ 0:024 kBT ðjaÞW 2 ð2Þ a 2 lb where l b is the Bjerrum length defined by l b ” e 2 / (4p 0k BT), a is the radios of the particle, and Y s=w se/ k BT is the dimensionless surface potential. The theoretical predictions for s c are found to be in good agreement with the measured values (Table 1). In this table h m is the characteristic separation distance in the incipient hydrocluster state, which is determined directly from j )1 , the Debye length by h m=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 ðc 0x _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 Table 1 Comparison of experimental and theoretical critical shear stress foraqueous silica particle suspensions Volume fraction / ja hm (nm) Theoretical critical shear stress sc (Pa) 0.55 8.4 39 37 20±0.5 0.60 9.4 35 41 46±0.5 Experimental critical shear stress sc (Pa) 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 Fig. 3 Application of Cox-Merz and Delaware-Rutgers rules for 60 vol.% aqueous silica dispersion
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 cc ; ð3Þ gcx where g c is the shear viscosity at the critical point for shear thickening and s c is the critical stress, both 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 D slip 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 cc;app ¼ cc;real þ 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 c c;app ¼ sc g cx 203 2Dslip þ : ð5Þ h In the comparison, we take h 1.25 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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Dynamic properties of shear thickening colloidal suspensions

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