Directional motion of forced polymer chains with hydrodynamic interaction

We study the propulsion of a one-dimensional (1D) polymer chain under sinusoidal external forces in the overdamped (low Reynolds number) regime. We show that, when hydrodynamical interactions are included, the polymer presents directional motion which depends on the phase differences of the external force applied along the chain. Moreover, the velocity shows a maximum as a function of the frequency. We discuss the relevance of all these results in light of recent nanotechnology experiments.

Figure 1

Scheme of the chain compound by $N$ monomers. The forces act on every sphere and the last one is shifted in phase with respect to the first one by a phase $\phi$. The $i$th sphere presents a phase shift ${\phi }_i=i\frac{\phi }{N-1}$.

Figure 2

Frequency ($\omega$) and phase ($\phi$) dependence of the velocity for a different number of particles $N$. Note that for reading purposes, the panels with $N=5$ and 10 have the scale of frequencies different from the others.

Figure 3

Phase difference ($\phi$) dependence of the velocity. The vertical lines represent the first phase difference at which the external total force $F_{\mathrm{tot}}$ is 0 at any time. Note that for $N=5$ and 10 only a half period in phase is shown.

Figure 4

Upper panel: Trajectories of the center of mass of the polymer for the case $N=3$ and for different values of the total phase $\phi$. Lower panel: Trajectory of each of the three particles for the phase value $\phi =2.0$. All trajectories are calculated with the frequency $\omega =2.0$.

Figure 5

Frequency ($\omega$) dependence of the velocity for different number of particles and for three different phases.

Figure 6

Modulus of the chain velocity maximum as a function of the number of spheres $N$. Insets: Frequency of the maximum in linear plot (top left) and log-log plot (top right), and phase of the maximum (bottom).