Propulsion with a Rotating Elastic Nanorod

The dynamics of a rotating elastic filament is investigated using Stokesian simulations. The filament, straight and tilted with respect to its rotation axis for small driving torques, undergoes at a critical torque a strongly discontinuous shape bifurcation to a helical state. It induces a substantial forward propulsion whatever the sense of rotation: a nanomechanical force-rectification device is established.

Figure 1

(a) Sketch in 3D and projected views of the rotating filament before (A, $\tilde{N}=600$) and after (B, $\tilde{N}=800$) the bifurcation ($M=30$ beads, ${\ell }_{\mathrm{p}}/L={10}^3$). (b) Angular velocity $\tilde{\omega }$ across the shape transition as a function of time ($\tilde{N}=1.33\times {10}^3$, ${\ell }_{\mathrm{p}}/L=2\times {10}^4$). (c) Angular velocity as function of external torque, $\tilde{N}$, for various ${\ell }_{\mathrm{p}}/L\mathrm{\text{:}}3333$ (○), 6667 (□), $1.33\times {10}^4$ (◇), and $2\times {10}^4$ (○) [inset: 333 (◇), 1000 (□) and 2500 (○)]. (d) Hysteresis cycle for ${\ell }_{\mathrm{p}}/L=1667$ for increasing (○) and decreasing torque (□) at constant rate $\delta \tilde{N}/(\delta t/\Delta )=1.25\times {10}^{-6}$. (e) Persistence length vs critical torque for increasing (○) and decreasing torque (□) both obeying a linear law according to Eq. (3) (All data for stalled case, $V_z=0$.)

Figure 2

(a) Propulsion velocity parallel to the rotation axis vs $\tilde{N}L/{\ell }_{\mathrm{p}}$ in the $z$-moving case for ${\ell }_{\mathrm{p}}/L=2500$ ($\times$), 5000 (□), and $2\times {10}^4$ (○). The inset shows the variations of the ratio of velocities after and before the transtion vs the ratio of angular velocities (◇, ${\ell }_{\mathrm{p}}/L=1667$). (b) ${\tilde{V}}_z$ vs the external force ${\tilde{F}}_{\mathrm{ext}}$ applied on monomer 2 $({\ell }_{\mathrm{p}}/L=1667)$ before the bifurcation for $\tilde{N}=0.78{\ell }_p/L$ (□) and after $\tilde{N}=0.84{\ell }_p/L$ (○). (c) Efficiency, $\eta$, vs ${\tilde{F}}_{\mathrm{ext}}$ [same parameters as (b)]. The solid line is a parabolic fit.

Figure 3

(a) Sketch of the self-propelling nanomachine whose base is made of 3 arms of length $L_a$. (b) Propulsion velocity ${\tilde{V}}_p$ (○) and ratio of the base, $\Omega$, and filament, $\omega$, angular velocites (■) against $L/L_a$ for $L=30a$, ${\ell }_{\mathrm{p}}/L=33$, ${\ell }_{\mathrm{p}}^a/{\ell }_{\mathrm{p}}=10$, and $N=2000$. The dashed line is the asymptotic velocity reached by an isolated filament.