High shear rate viscometry

Abstract

We investigate the use of two distinct and complementary approaches in measuring the viscometric properties of low viscosity complex fluids at high shear rates up to 80,000 s⁻¹. Firstly, we adapt commercial controlled-stress and controlled-rate rheometers to access elevated shear rates by using parallel-plate fixtures with very small gap settings (down to 30 μm). The resulting apparent viscosities are gap dependent and systematically in error, but the data can be corrected—at least for Newtonian fluids—via a simple linear gap correction originally presented by Connelly and Greener, J. Rheol, 29(2):209–226, 1985). Secondly, we use a microfabricated rheometer-on-a-chip to measure the steady flow curve in rectangular microchannels. The Weissenberg–Rabinowitsch–Mooney analysis is used to convert measurements of the pressure-drop/flow-rate relationship into the true wall-shear rate and the corresponding rate-dependent viscosity. Microchannel measurements are presented for a range of Newtonian calibration oils, a weakly shear-thinning dilute solution of poly(ethylene oxide), a strongly shear-thinning concentrated solution of xanthan gum, and a wormlike micelle solution that exhibits shear banding at a critical stress. Excellent agreement between the two approaches is obtained for the Newtonian calibration oils, and the relative benefits of each technique are compared and contrasted by considering the physical processes and instrumental limitations that bound the operating spaces for each device.

Appendix

In this appendix, we outline the difficulties in adapting the expression for the rate-dependent viscosity measured using a parallel plate device (Eq. 7):

$$ \eta_{\rm true}(\dot{\gamma}_{\rm true}) = \frac{\mathcal{T}/2 \pi R^3}{\dot{\gamma}_{\rm true}}\left[3+\frac{{\rm d} (\ln \mathcal{T}/2 \pi R^3)}{{\rm d}(\ln\dot{\gamma}_{\rm true})}\right], $$

(28)

to include the linear gap error analysis of Connelly and Greener. We start by noting that the analysis can follow two distinct approaches: one, to find the gap error ε where the apparent shear rate $\dot{\gamma}_{\rm a}$ is held constant and the gap height H is altered and, the other, to find the true rate-dependent viscosity where the gap height H is held constant and the shear rate or rotation rate is altered.

Where possible the analysis to find the gap height error ε should be performed in the zero shear rate region so that the fluid viscosity is not a function of applied shear rate. However, for fluids that start shear thinning at very low shear rates, this can result in shear stresses below the resolution of the rheometer torque transducer, and it is not possible to access this region. In this case, we expect the apparent viscosity to be a function of both the true shear rate and the gap setting, and we want to find the gap error ε as a function of the apparent shear rate $\dot{\gamma}_{\rm a} = {\Omega R}/{H}$, which can be commanded in the software. We start from the expression for the true shear rate $\dot{\gamma}_{\rm true} = {\Omega R}/(H + \epsilon)$:

$$ {\dot{\gamma}_{\rm true}} = {\dot{\gamma}_{\rm a}}\left(\frac{H}{H + \epsilon}\right), $$

(29)

and, differentiating using the chain rule, we find:

$$ \left.\frac{{\rm{d}}\dot{\gamma}_{\rm true}}{{\rm{d}}\dot{\gamma}_{\rm a}}\right|_{\Omega} = \left(\frac{\partial\dot{\gamma}_{\rm true}}{\partial H}\right)_{\Omega}\frac{{\rm d}H}{{\rm d}\dot{\gamma}_{\rm a}}=\left(\frac{H}{H + \epsilon}\right)^2. $$

(30)

Thus, we can rewrite Eq. 28 in terms of the apparent shear rate:

$$ \begin{array}{*{20}l} \eta_{\rm true} &=& \frac{\mathcal{T}/2 \pi R^3}{\dot{\gamma}_{\rm a}} \left(\!\frac{H + \epsilon}{H}\!\right)\times\\ &&\times\left[3\!+\!\left(\!\frac{H}{H+\epsilon}\!\right)\left.\frac{{\rm d} (\ln \mathcal{T}/2 \pi R^3)}{{\rm d}(\ln\dot{\gamma}_{\rm a})}\right|_{\Omega}\right]. \end{array}$$

(31)

To solve for the gap error ε, we cast Eq. 31 in terms of the apparent viscosity η _a (c.f. Eq. 6):

$$ \begin{array}{*{20}l} \frac{1}{\eta_{\rm a}} &=& \frac{1}{\eta_{\rm true}} \left(\!1+\frac{\epsilon}{H}\!\right)\frac{1}{4}\times \\ &&\times\left[3\!+\!\left(\!\frac{1}{1+\epsilon/H}\!\right)\left.\frac{{\rm d} (\ln \mathcal{T}/2 \pi R^3)}{{\rm d}(\ln\dot{\gamma}_{\rm a})}\right|_{\Omega}\right], \end{array}$$

(32)

and we note that, for a complex liquid outside the zero shear-viscosity regime, 1/η _a is a nonlinear function of 1/H. For a Newtonian viscous response in which $\tau\sim\dot{\gamma}_{\rm a}$, we recover Eq. 6, and 1/η _a varies linearly with 1/H.

We now turn to the case where we wish to determine the rate-dependent viscosity as a function of the apparent shear rate for a fixed gap height and known gap error. In this instance:

$$ \left.\frac{{\rm{d}}\dot{\gamma}_{\rm true}}{{\rm{d}}\dot{\gamma}_{\rm a}}\right|_{H} = \left(\frac{\partial\dot{\gamma}_{\rm true}}{{\partial}\Omega}\right)_{H}\frac{{\rm d}\Omega}{{\rm{d}}\dot{\gamma}_{\rm a}}=\left(\frac{H}{H + \epsilon}\right), $$

(33)

and, for measurements performed over a range of rotation rates at a fixed gap height, Eq. 28 becomes (cf. Eq. 7):

$$ \eta_{\rm true}(\dot{\gamma}_{\rm true}) = \eta_{\rm a}\left(1+\frac{\epsilon}{H}\right)\frac{1}{4}\left[3+\left.\frac{{\rm d} (\ln \mathcal{T}/2 \pi R^3)}{{\rm d}(\ln\dot{\gamma}_{\rm a})}\right|_{H}\right]. $$

(34)

It is therefore clear that, for shear-thinning fluids, slightly different results can be obtained depending on whether measurements are performed at constant gap (varying rotation rate) or constant rotation rate (and varying the shear rate by changing the gap). It is instructive to compare the magnitude of the leading order correction due to (a) the error in gap height ε and (b) the derivative of the logarithmic term due to a rate-dependent viscosity. At small gaps, $H\sim50~\upmu{\rm m}$, $\epsilon\sim30~\upmu{\rm m}$, and the correction provided by the gap correction term in Eq. 34 can be O(100%). On the other hand, the corrections associated with the shear-thinning term in square brackets are much smaller. At low apparent shear rates, a ratio of Eq. 32 or 34 with the linear form of Eq. 6 in the text shows that the results in Eqs. 32 and 34 differ by at most [3+(1+ε/H)⁻¹]/4. In the case of an extremely shear-thinning fluid in which

$$ \frac{{\rm d} (\ln \mathcal{T}/2 \pi R^3)}{{\rm d}(\ln\dot{\gamma}_{\rm a})}\rightarrow0, $$

both Eqs. 32 and 34 approach the same limit and the additional correction to the viscosity is only 25%. Given that the majority of fluids do not shear thin this strongly, the additional error due to a shear-rate-dependent viscosity in many cases will be less than 10% of that due to gap error. Thus, we conclude that, in the majority of experiments, applying the gap error analysis alone will provide a good estimate of the shear-rate-dependent viscosity.