Encapsulation of ${\mathrm{C}}_{60}$ fullerenes into single-walled carbon nanotubes: Fundamental mechanical principles and conventional applied mathematical modeling

A well-known self-assembled hybrid carbon nanostructure is a nanopeapod which may be regarded as the prototype nanocarrier for drug delivery. While the investigation of the packing of ${\mathrm{C}}_{60}$ molecules inside a carbon nanotube is usually achieved through either experimentation or large scale computation, this paper adopts elementary mechanical principles and classical applied mathematical modeling techniques to formulate explicit analytical criteria and ideal model behavior for such encapsulation. In particular, we employ the Lennard-Jones potential and the continuum approximation to determine three encapsulation mechanisms for a ${\mathrm{C}}_{60}$ fullerene entering a tube: (i) through the tube open end (head-on), (ii) around the edge of the tube open end, and (iii) through a defect opening on the tube wall. These three encapsulation mechanisms are undertaken for each of the three specific carbon nanotubes (10,10), (16,16), and (20,20). We assume that all configurations are in vacuum and the ${\mathrm{C}}_{60}$ fullerene is initially at rest. Double integrals are performed to determine the energy of the system and analytical expressions are obtained in terms of hypergeometric functions. Our results suggest that the ${\mathrm{C}}_{60}$ fullerene is most likely to be encapsulated by head-on through the open tube end and that encapsulation around the tube edge is least likely to occur because of the large van der Waals energy barriers which exist at the tube ends.

Figure 1

${\mathrm{C}}_{60}$ fullerene encapsulated in carbon nanotube by head-on at open end.

Figure 2

(Color online) Energy profile for ${\mathrm{C}}_{60}$ encapsulated by head-on at open end.

Figure 3

${\mathrm{C}}_{60}$ fullerene encapsulated in carbon nanotube around edge at open end.

Figure 4

(Color online) Energy profile for ${\mathrm{C}}_{60}$ encapsulated into a (10,10) tube.

Figure 5

${\mathrm{C}}_{60}$ fullerene encapsulated in carbon nanotube at defect opening on the tube wall.

Figure 6

(Color online) Potential energy profile along the (10,10) tube.