Collective stiffening of soft hair assemblies

Many living systems use assemblies of soft and slender structures whose deflections allow them to mechanically probe their immediate environment. In this work, we study the collective response of artificial soft hair assemblies to a shear flow by imaging their deflections. At all hair densities, the deflection is found to be proportional to the local shear stress with a proportionality factor that decreases with density. The measured collective stiffening of hairs is modeled both with a microscopic elastohydrodynamic model that takes into account long-range hydrodynamic hair-hair interactions and a phenomenological model that treats the hair assemblies as an effective porous medium. While the microscopic model is in reasonable agreement with the experiments at low hair density, the phenomenological model is found to be predictive across the entire density range.

Figure 1

(a) Sketch of the experimental setup (side view along a diameter). (b) Composite image of a PDMS substrate with a density of pillars $n\approx 4{\mathrm{mm}}^{-2}$ obtained with fluorescence imaging. The green color indicates the presence of green fluorescent particles on the pillar's tips. (c) Typical fluorescence snapshots of a pillar's summit at rest (left panel) and in steady flow (right panel). The green dashed line circles delimit the perimeter of the pillar. (d) Semilogarithmic plot of the displacement $\delta$ of a single pillar versus time during a typical experiment after a sudden start of the rheometer's tool at $t=0$ s. Inset: sketches of sectional views of a pillar at rest (left panel) and subject to a steady shear liquid flow (right panel).

Figure 2

Log-log plot of the normalized displacement, $\delta /{\kappa }_0$, of a single pillar's tip in steady state versus the local shear stress $\sigma$ for four different pillars number densities, $n=0.04{\mathrm{mm}}^{-2}$ (referred to as an isolated pillar; diamonds), $n=1{\mathrm{mm}}^{-2}$ (circles), $n=4{\mathrm{mm}}^{-2}$ (squares), and $n=10{\mathrm{mm}}^{-2}$ (stars). For these experiments, the angular velocity $\omega$ was varied and different pillar's radial positions $\rho$ and viscosities $\eta$ were used ($\rho =5.6,7.0,7.5$ mm and $\eta =0.458,0.320,0.468$ Pa s for $n=1,4,10{\mathrm{mm}}^{-2}$, respectively). Solid lines are linear fits.

Figure 3

(a) Dimensionless pillar tip deformation $\delta /{\delta }_0$ versus normalized pillars density $n/n_0$ with $n_0=5.3{\mathrm{mm}}^{-2}$ (squares). Each point is an average over nine experiments and error bars are taken as the standard deviation over these nine measurements. The green dashed (respectively, blue dash-dotted) line is the prediction of Eq. (6) with the five nearest neighbors [respectively, Eq. (7) in the continuum limit]. The red solid curve is a fit with Eq. (9) based on the phenomenological model of [23] where the value of $\alpha$ has been fitted. Points surrounded by dashed line circles correspond to those in Fig. 2 with the same color code. Inset: Log-linear plot of the main graph. (b) Sketch of the hair assembly system with the geometrical characteristics used in the dilute model. The upper rigid plate positioned at a distance $H$ from the bottom of the pool is sheared along the $x$ direction with a velocity $\mathbf{V}$.