# Diverse corrugation pattern in radially shrinking carbon nanotubes 

Hiroyuki Shima ∗ Division of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan and Department of Applied Mathematics 3, LaC` aN, Universitat Polit` ecnica de Catalunya, Barcelona 08034, Spain 

Motohiro Sato Division of Socio-Environmental Engineering, Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido 060-8628 Japan 

Kohtaroh Iiboshi Division of Socio-Environmental Engineering, Graduate School of Engineering, Hokkaido University, Sapporo, Hokkaido 060-8628 Japan 

Susanta Ghosh and Marino Arroyo Department of Applied Mathematics 3, LaC` aN, Universitat Polit` ecnica de Catalunya, Barcelona 08034, Spain (Dated: October 31, 2018) 

Stable cross-sections of multi-walled carbon nanotubes subjected to electron-beam irradiation are investigated in the realm of the continuum mechanics approximation. The self-healing nature of sp 2 graphitic sheets implies that selective irradiation of the outermost walls causes their radial shrinkage with the remaining inner walls undamaged. The shrinking walls exert high pressure on the interior part of nanotubes, yielding a wide variety of radial corrugation patterns ( i.e., circumferentially wrin- kling structures) in the cross section. All corrugation patterns can be classified into two deformation phases for which the corrugation amplitudes of the innermost wall differ significantly. 

Carbon nanotubes exhibit remarkable flexibility when subjected to cross-sectional deformation. 1–3 Such flexi- bility has been evidenced by spectroscopy and diffrac- tion measurements in which hydrostatic pressure on the order of a few GPa caused flattening and polygoniza- tion in the cross section. 4–6 Molecular-dynamics (MD) 7–9 and density-functional-theory-based simulations 10,11 re- vealed more details about the mechanism of radial defor- mation under pressure. Radial deformation can also be caused by a nanoindentation 12 under which the elastic response of single-walled nanotubes (SWNTs) is deter- mined by a universal law that depends on the tube radius r and applied load. 13 As the radial deformation of nan- otubes is strongly correlated with their electronic 12,14–16 and optical 17 properties, its clarification should be im- portant from the viewpoint of the use of nanotubes as components of nanoscale devices. 

The cross section of an isolated SWNT exhibits a circular-to-elliptic transition at a critical hydrostatic pressure p$_{c}$ . The value of p$_{c}$ decreases with r as 18,19 p$_{c}$ ∝ r − $^{3}$, which agrees with the conclusion of the clas- sical continuum theory. 20 This scenario, however, fails in the case of multi-walled nanotubes (MWNTs). It has been shown that MWNTs consisting of several tens of concentric walls undergo a novel cross-sectional deforma- tion, called radial corrugation, 21 in which the outer walls exhibit wavy structures along the circumferential direc- tion. The radial corrugation originates from the multi- 

PACS numbers: 61.46.Fg, 46.70.De, 62.50.-p, 61.80.Az 

## I. INTRODUCTION 

layered nature, i.e., the competing effects between the mechanical instability of outer walls with large radii and the radial rigidity of inner walls with small radii. 

The corrugation behavior of a MWNT is believed to change if it occurs in an embedding elastic medium. Suppose that a MWNT is embedded in a large elastic medium that is in complete contact with its outermost wall. A uniform shrinkage of the medium exerts an ex- ternal force on the outermost wall, as similar to the case of hydrostatic pressure. But a difference arises when the medium possesses moderate stiffness, since the energy required to deform the medium should be responsible for determining the stable corrugation pattern. In fact, when a thin flat plate is embedded in an elastic medium and comressed, it shows a uni-directional corrugation whose profile is strongly dependent on the medium’s stiffness. 22,23 This fact implies an uncovered corrugation mechanism peculiar to embedded MWNTs, it i.e., din- stinct from the hydrostatic pressure cases, which imposes new challenges. 

In this article, we demonstrate that the presence of a surrounding elastic medium triggers diverse variations of corrugation modes that cannot be observed in MWNTs under hydrostatic pressure. These diverse modes are found to be grouped into two corrugation phases that exhibit a significant difference in the corrugation am- plitude of the innermost wall of the MWNT. These findings will help to tune the core-tube geometry of MWNTs, thus providing useful information for devel- oping nanofluidic 24–26 or nanoelectrochemical 27 devices 

