Heterogeneity and the Role of Normal Stresses during the Extensional Thinning of Non-Brownian Shear-Thickening Fluids

We contrast the extensional and shear dynamics of non-Brownian suspensions as a function of particle concentration. We show that the thinning rate selected during the viscoelastic pinch-off of a liquid bridge is related to the shear rate at which normal stresses become positive, which differs from the shear rate at the onset of shear thickening. By tracking particles, we demonstrate that the extensional flow is heterogeneous, with local variations of the volume fraction consistent with self-dilution. This nonuniform structure is the cause of the buckling of the threads formed after breakup.

Figure 1

(a) Shear rheology of the suspensions, measured with a gap width $w=1\mathrm{mm}$. The viscosity of a 10-cSt silicone oil is shown as a calibration for the measurements of low viscosities. (b) Thinning dynamics $2h_{\min }$ versus ($t_0-t$) for three different suspensions, where $t_0$ is the pinch-off time. For clarity, time is shifted by 4 ms for $[CS]=30\mathrm{wt}\%$ and by 5 ms for $[CS]=34\mathrm{wt}\%$. Continuous lines: fits to the experimental data; $h_{\min }\propto (t_0-t{)}^{2/3}$ for $[CS]=26\mathrm{wt}\%$, $h_{\min }\propto (t_0-t)$ for $[CS]=30\mathrm{wt}\%$ and $h_{\min }\propto \exp \mathbf{(}\alpha (t_0-t)\mathbf{)}$ for $[CS]=34\mathrm{wt}\%$. Inset: semilog plot of $h_{\min }$ versus ($t_0-t$) in the viscoelastic regime. (c) Pinch-off of the suspensions. Image 1: inviscidlike; image 2: viscouslike; image 3: viscoelasticlike. Scale bar: 2 mm.

Figure 2

(a) Comparison between the relaxation rate $\omega$ (□) obtained from the thinning dynamics and the shear rate ${\dot{\gamma }}_N$ (■) as a function of [$CS$]. The gap width is $w=h_{\min ,c}$. Inset: Apparent shear viscosity ${\eta }_s$ (▴) and first normal stress difference $N_1$ (▵) as a function of $\dot{\gamma }$ for $[CS]=36\mathrm{wt}\%$ [see Fig. (b) in Supplemental Material [16]]. (b) Comparison between the extensional viscosity ${\eta }_e$ (□) and the shear viscosity ${\eta }_s$ (■) as a function of confinement ($2h_{\min }$ and gap width $w$) for $[CS]=36\mathrm{wt}\%$. Inset: Evolution of ${\eta }_e/{\eta }_s$ as a function of the confinement for $[CS]=36\mathrm{wt}\%$ (●) and 38 wt % (○). In both graphs, the shaded area corresponds to the transition leading to breakup.

Figure 3

(a) Image of the starch grains in the bridge close to breakup. (b) Strain $ϵ$ versus ($t_0-t$) for $[CS]=37\mathrm{wt}\%$ in a jammed volume (■) and in a flowing volume (□). (c) Thinning dynamics at breakup location for $[CS]=37\mathrm{wt}\%$ (△) and 39 wt % (□). Continuous lines: linear fits to the data. The slope is $v_t=114\mathrm{mm}{\mathrm{s}}^{-1}$ for $[CS]=37\mathrm{wt}\%$ and $v_t=133\mathrm{mm}{\mathrm{s}}^{-1}$ for $[CS]=39\mathrm{wt}\%$. Time is shifted by 1 ms for $[CS]=37\mathrm{wt}\%$.

Figure 4

(a) Sketch of the state of the suspension along the bridge close to breakup. (b) Images of a thread after breakup showing the bending of the thread around its fluid parts, $[CS]=37\mathrm{wt}\%$ and $t_r={\omega }^{-1}\simeq 10\mathrm{ms}$. Scale bar: $500\mu \mathrm{m}$.