Confined nanorods: Jamming due to helical buckling

We investigate a longitudinally loaded elastic nanorod inside a cylindrical channel and show within the context of classical elasticity theory that the Euler buckling instability leads to a helical postbuckling form of the rod within the channel. The local pitch of the confined helix changes along the channel and so does the longitudinal force transmitted along the rod, diminishing away from the loaded end. This creates a possibility of jamming of the nanorod within the channel.

Figure 1

(Color online) A long-axis view of helical filaments with lengths $L/R=\left(\text{a}\right)20$ and (b) 640, buckled by a force $\left|F_z\left(0\right)\right|=0.25EI/R^2$, which is approximately (a) 10 and (b) 10 000 times the critical force for Euler buckling, $F_{crit}=EI{\pi }^2/L^2$. There is no torque applied to the ends.

Figure 2

External load dependence of the projection of filament length onto the axis of the confining cylindrical pore. The weak force dependence of length in the straight configuration is due to the small artificial compressibility introduced by the penalty potential (26) (i.e., finite value of the spring constant $k$). In the limit of zero compressibility this part of the functional dependence would be a straight horizontal line.

Figure 3

(Color online) Planar configuration of the confined filament (a) is unstable (b) and is transformed into the helical configuration (c) upon perturbation. The helix can contain metastable irregularities as seen in the equilibrium state (c).

Figure 4

(Color online) Compression force profile $\left|F_z\left(l\right)\right|$ in a filament with length to radius ratio 320. The loading force is $\left|F_z\left(0\right)\right|=0.25EI/R^2$, which is approximately 2600 times the critical force for Euler buckling, $F_{\text{crit}}=EI{\pi }^2/L^2$. Numerical data (black) are excellently fitted (red) with the function $f\left(l\right)=A{\left(l_0+l\right)}^c$ (the curves are perfectly overlapping on the scale of the figure): (a) $A=36.3$, $l_0=285.5$, $c=-0.882$; (b) $A=21.4$, $l_0=131.8$, $c=-0.894$; (c) $A=11.95$, $l_0=44.9$, $c=-0.903$.

Figure 5

Profile of $d\varphi /dl$, the inverse of the helical pitch $q_0=2\pi /\left(d\varphi /dl\right)$, for the filament of Fig. 4c. The boundaries where the pitch is not defined are not displayed. As expected, $\dot{\varphi }$ gets smaller (the pitch gets larger) as the longitudinal force is reduced. Its relative decrease is, however, smaller than that of the force, as hinted by the quadratic dependence $F_z\propto {\dot{\varphi }}^2$ in Eq. (15) of the simple model (which assumes constant pitch).

Figure 6

Compression force $\left|F_z\left(L\right)\right|$ transmitted through filaments of length $L$; the external compressing force is $\left|F_z\left(0\right)\right|=0.25EI/R^2$; $k_{\text{fr}}=0.8$. The dotted line serves as a guide to the eye.

Figure 7

(Color online) Compression force profile $\left|F_z\left(l\right)\right|$ (black) in the filament with length to radius ratio 320 for a smaller loading force $\left|F_z\left(0\right)\right|=0.0125EI/R^2$ and $k_{\text{fr}}=0.8$, fitted (red) with the function $f\left(l\right)=A{\left(l_0+l\right)}^c$; $A=15.5$, $l_0=2010$, $c=-0.942$. The reduction of the force is much smaller than for larger loads in Fig. 4, on account of the singular point $l_c$ in Eq. (19) moving further away. The exponent of the power law, however, remains nearly unchanged. The boundary regions get wider as the load is reduced.

Figure 8

(Color online) If the load is too large, the filament makes a U bend and escapes out of the cylinder, as shown on the subsequent figures instead of forming the helix. $L/R=320$, $\left|F_z\left(0\right)\right|=0.5EI/R^2$; only the upper half of the cylinder is shown.