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The contrasting difference in corrugation amplitude distributions between the inward and outward phases is quantified by plotting the deformation amplitudes δ ¯ µ$_{i}$ introduced in Eq. (7). Figure 4 shows the normalized deformation amplitudes, ξ$_{i}$ ≡ | δ ¯ µ$_{i}$/δ ¯ µ$_{N}$ | , of individual concentric walls for N -walled nanotubes with different N and R = 6 . 0 nm be fixed. It follows from the figure that the value of ξ$_{1}$ suddenly drops off from ξ$_{1}$ > 1 . 0 to ξ$_{1}$ < 1 . 0 across the phase boundary ( i.e. , from N = 8 to 9 with R = 6 . 0 nm being fixed). To understand this visually, we show in Fig. 5 the sequential variation of the corrugation amplitude distribution in the cross section near the phase boundary, In the inward mode ( N ≤ 8), the restoring force exerted by the surrounding medium is relatively strong so that it tends to prevent radial de- formation of outer walls; this is why inner walls exhibit significant corrugation amplitudes to lower the energy U of the system. By increasing N , such restoring force ef- fects become ineffective to support the radial instability of outer walls, as a result of which the system falls into the outward phase for N ≥ 9. A similar scenario applies if we fix N and modulate R in the vicinity of the phase boundary. 

The present results are based on the approximation that the radially shrinking part of an MWNT under electron-beam irradiation is mapped onto a continuum elastic medium with homogeneous and isotropic elastic- ity of the modulus E$_{M}$ = 100 GPa. The validity of the homogeneity and isotropy hypothesis depends on the fol- 

## D. Corrugation amplitudes 

## IV. DISCUSSIONS 

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lowing two effects of irradiation on the mechanical stiff- ness of MWNTs: irradiation reduces the axial stiffness because it creates vacancies, 46,47 and simultaneously, it enhances the radial stiffness owing to the production of covalent bonding between adjacent walls. 32 A quantita- tive examination of the degree to which this hypothe- sis holds requires elaborated measurements or large-scale atomistic simulations, and therefore, we have no avail- able data for the proof. Instead, it is possible to gener- alize the theoretical method by considering the possible anisotropic elasticity in the medium. Our preliminary calculations showed that moderate anisotropy causes lit- tle modification in the phase diagram. More detailed results will be shown elsewhere. 

We have also performed corrugation analyses by im- posing other values of E$_{M}$ than E$_{M}$ = 100 GPa. It was found that larger (smaller) values of E$_{M}$ result in more (less) number of corrugation modes observed in the phase diagram with fixed ranges of R and N set in Fig. 3(a). With decreasing E$_{M}$ , the phase boundary shifts down- ward until E$_{M}$ = 0 . 1 GPa, below which the boundary disappears leaving the outward phase as the only possi- bility. Therefore, our predictions of diverse corrugation patterns and the two corrugation phases are possible for any choice of E$_{M}$ as far as E$_{M}$ > 0 . 1 GPa. 

From an engineering perspective, the selectivity of the innermost wall geometry by tuning the material param- eters R , N and p may be useful in developing nanotube- based nanofluidic 24–26 or nanoelectrochemical devices $^{27}$, because both utilize the hollow cavity within the in- nermost tube. A very interesting issue from the aca- demic viewpoint is the effect of the core tube defor- mation on the physical and chemical properties of in- tercalated molecules confined in the hollow cavity. It has thus far been known that various types of interca- lated molecules (diatomic gas, water, organic, and tran- 

