Inertia-Dilatancy Interplay Governs Shear-Thickening Drop Impact

Abstract

Combining high-speed photography with direct force measurements, we investigate the impact dynamics of drops of cornstarch-water mixtures -- a premier example of shear-thickening fluids -- across a wide range of impact conditions. Our study identifies three distinct impact regimes. In addition to the liquid-like and solid-like behaviors generally expected for the impact-induced response of shear-thickening fluids, we uncover a counterintuitive regime in which high-concentration cornstarch-water mixtures display a liquid-like response at the onset of impact when shear rates are high and only transition to a solid-like behavior at later times as shear rates reduce. By integrating the classic drop-impact theory with the Reynolds-Darcy mechanism for dilatancy, we develop a unified model that quantitatively describes the impact dynamics of shear-thickening drops across all regimes. Our work reveals the unexpected response of shear-thickening fluids to ultra-fast deformation and advances fundamental understanding of drop impact for complex fluids.

Figures (5)

• Figure 1: Phase diagram of the impact dynamics of shear-thickening drops. (a–c) Temporal evolution of the impact force, $F(t)$, for drops of cornstarch–water mixtures at: (a) volume fraction $\phi=0.35$, impact velocity $U_0 = 1.2$ m/s; (b) $\phi=0.43$, $U_0 = 3.5$ m/s; and (c) $\phi=0.41$, $U_0 = 5.1$ m/s. Solid lines show averages over five experiments, with shaded regions indicating the standard deviation, while dashed lines correspond to model predictions. Vertical dashed and dotted lines indicate the time of maximum force $t_\text{max}^F$ and the time of maximum spreading $t_{\text{max}}^D$, respectively. $F$ is normalized by $\rho U_0^2 D_0^2$, and time $t$ by $D_0/U_0$. (d) Ratio of the two times, $t_{\text{max}}^F / t_{\text{max}}^D$, plotted as a function of $\phi$ and characteristic shear rate $U_0/D_0$. Three regimes are indicated: ① Inertial, ② Dilatant, and ③ Inertial–dilatant. Circled points correspond to the examples shown in (a–c). The dashed line shows the onset of discontinuous shear thickening, while the dot-dash line indicates the model prediction for the boundary between Regime (2) and (3), corresponding to $\text{ID} = 1$ (Eq. \ref{['eqn:eqn6']}). Symbol shapes denote samples with different $\phi$ (the $y$-axis), and symbol colors indicate the value of $t_{\text{max}}^F / t_{\text{max}}^D$. The gray region marks the shear rates inaccessible to conventional rheometry. A three-dimensional representation of the dataset is provided in Fig. S2 supplemental.
• Figure 2: Liquid-drop impact behavior of shear-thickening fluids. (a) Prefactor $c$ in Eq. \ref{['eqn:eqn1']} as a function of characteristic shear rate $U_0/D_0$. The dashed line shows $c = 3\sqrt{6}/2$. (b) Decay time $\tau$ in Eq. \ref{['eqn:eqn1']} as a function of the time of maximum force $t_{\text{max}}^F$ extracted from experiments. The dashed line indicates $\tau=2t_{\text{max}}^F$. Times are normalized by $D_0/U_0$. Symbol shapes denote $\phi$, while colors indicate $t_{\text{max}}^F / t_{\text{max}}^D$, as in Fig. \ref{['fig:fig1']}(d).
• Figure 3: Early-time impact dynamics. (a) Spreading exponent $n$ as a function of the characteristic shear rate $U_0/D_0$. (b) Angle between the drop sidewalls and the impacted substrate, $\theta$, at the time of maximum force as a function of $U_0/D_0$. The dashed lines indicates the liquid-drop impact behaviors $n=0.5$ (a) and $\theta = 90^\circ$ (b). The inset shows $\theta$ for a $\phi=0.30$ sample. The three regimes are labeled. Symbol shapes denote $\phi$, while colors indicate $t_{\text{max}}^F / t_{\text{max}}^D$, as in Fig. \ref{['fig:fig1']}(d).
• Figure 4: Late-time impact dynamics. Time derivative of the impact force, $\dot{F}$, for the three force curves in Fig. \ref{['fig:fig1']}(a–c). $\dot{F}$ is non-dimensionalized by $\rho U_0^3D_0$ and $t$ by $D_0/U_0$.
• Figure 5: Unified force model. (a) Temporal evolution of impact force in Regime (3) for a volume fraction $\phi=0.41$ mixture at impact velocity $U_0 = 5.1$ m/s. The green solid line shows the average of five experiments, with the shaded region representing the standard deviation. The blue dotted line shows the weighted inertial impact force $(1-w_F)F_i$, the red dotted line shows the weighted dilatant impact force $w_FF_d$, and the black dashed line shows the summation of the two. Force is normalized by $\rho U_0^2D_0^2$ and time by $D_0/U_0$. (b) Ratio of the dilatant momentum $M_d$ to the total momentum $mU_0$ at different $\phi$ and $U_0/D_0$. Lines and symbol shapes are the same as those in Fig. \ref{['fig:fig1']}(d). Symbol colors indicate $M_d/mU_0$.