Ropes of single-walled carbon nanotubes (SWNTs) and individual multi-walled carbon nanotubes (MWNTs) spontaneously form rings, helices, and kinks. We analyze these structures in terms of a balance between elastic strain and surface energy. Using only these two material properties we reproduce the observed structures and provide simple scaling laws for their dimensions. The simplicity of the models suggests their applicability to other nanorods: both tubulin and actin form structures inside cells that are strikingly similar to the structures formed by carbon nanotubes.
Martel and coworkers used ultrasound to form SWNTs into rings. When a cavitation bubble collapses around a nanotube, the nanotube can respond in two ways. Nanotubes below a critical length remain straight and puncture the wall of the bubble. Nanotubes above this critical length buckle because the surface tension of the bubble overcomes the Euler buckling strength of the nanotube. The collapse of the bubble then forces the nanotube into a ring. The observed distribution of ring-radii is consistent with the above mechanism. We propose to form rings of nanotubes in fluid-fluid colloidal dispersions in which the colloidal phase wets the nanotubes better than does the continuous phase. Tuning the wettabilities of the two fluids should provide control over the diameter of the rings. Elbaum and coworkers observed buckling and loop formation of tubulin fibers inside of shrinking vesicles via a mechanism similar to that discussed here.
We predict that ropes of nanotubes spontaneously develop a helical twist. Twisting into a helix costs elastic energy but increases the number of carbon atoms along the contact lines between the nanotubes, and thus increases the number of attractive van der Waals interactions. The balance between elastic energy and surface attraction determines the pitch of the helix. For a pair of single-walled nanotubes, each of diameter 1.4 nm, the equilibrium pitch is predicted to be ~250 nm. Similar coiling instabilities may occur in biological systems that balance attractive interactions with internal strain (e.g. proteins). Spontaneous coiling should affect the mechanical and electrical properties of ropes of NTs.
Individual multi-walled nanotubes (MWNTs) sometimes form segmented helices: straight sections of uniform length are joined by kinks to form an extended helix. An energetic analysis shows that a rod with built-in strain can lower its energy by forming a segmented helix, and predicts the spacing and angles of the kinks. Actin fibers in the sperm of the horseshoe crab form strikingly similar segmented helices.
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