radii grow with N , implying the occurrence of outward- deformation modes at large N . It also follows from Fig. 2(a) that a large portion of p$_{c}$ ’s data lies on the order of several GPa, though they occasionally exceed ten GPa or more at certain limited conditions. In the latter conditions, the continuum shell approximation may break down due to interwall sp $^{3}$/sp2 hybridization bonds generated by pressure. In fact, ear- lier MD simulations 40 suggested that radial pressure of approximately 40 GPa (or 30 GPa) destroys the tubular structure of an encapsulated SWNT having the radius ∼ 0 . 7 nm ( ∼ 1.2 nm) through the formation of hybrid sp $^{2}$/sp3 self linkages inside the tube. It is thus con- jectured that similar hybrid bonds between neighboring walls in MWNTs for R ≥ 1 . 0 nm occurs above a threshold pressure comparable to (or maybe less than) tens GPa. Precise determination of the threshold requires atom- istic calculations; hence we proceed arguments bearing in mind that the obtained results are physically relevant only when p$_{c}$ is lower than the threshold (more or less tens GPa) that remains to be determined. 4 Figure 2(b) shows the three-dimensional plot p$_{c}$ on the R - N plane. The plot exhibits a ridge line that ex- tends from the top at ( R,N ) = (1 . 0 , 2) to the skirt at ( R,N ) = (4 . 0 , 10) of the p$_{c}$ -surface. The ridge line corre- sponds to a phase boundary that separates the inward- and outward-deformation phases. In fact, an inward- deformation mode occurs in the region to the left of the ridge line, as we will find in the phase diagram in Fig. 3 (a). Figure 3(a) shows a phase diagram of the radial corru- gation modes in MWNTs observed above p$_{c}$ . A stepwise bright line represents the phase boundary between the inward-deformation phase (below the line) and the out- ward phase (above). Figure 3(b) shows a contour plot of p$_{c}$ ( R,N ) that evidences a strong correlation between the ridge line of p$_{c}$ ( R,N ) ( i.e., the solid slanted line in Fig. 3(b)) and the phase boundary shown in Fig. 3(a). A salient feature of Fig. 3(a) is the absence of an ellip- tic deformation phase ( n = 2) within the ranges of R and N we have considered. The absence of the n = 2 mode is in contrast with the cases of MWNTs and SWNTs under hydrostatic pressure; in fact, the n = 2 mode in the latter two cases is the primary mode observed in a large domain of the R - N space. Furthermore, we have obtained a wide variety of corrugation modes for various values of R and N , where the variation of n is systematic with the changes in R and N . In the inward-deformation phase, for instance, larger R and smaller N favor corru- gation modes with large n . Contrariwise, in the outward phase, smaller R and larger N favor corrugation modes with large n . (one upward and one downward) whose positions shift to larger N with an increase in R . One might expect that a larger N leads to a larger p$_{c}$ , because an increase in the concentric walls would enhance the radial rigidity. This conjecture holds for MWNTs with intermediate values of N between the two shallow peaks; for instance, p$_{c}$ for R = 4 . 0 nm increases slowly with N between N = 4 and 10. However, to the right of the upward peak, p$_{c}$ decreases monotonically with N , in contrast to the conjecture above. Such decay of p$_{c}$ at large N arises from the mechanical instability of outside walls whose FIG. 2: (color online) (a) Critical pressure curve p c ( N ) as a function of the total number of concentric walls N con- tained in a MWNT. The innermost tube radius R ranges from 1.0 nm to 6.0 nm as indicated. (b) Three-dimensional plot of p c ( R,N ) on the R - N plane. A ridge line extending from the top to the skirt of the p c -surface, depicted by a bright dashed curve, corresponds to the phase boundary that separates the inward-deformation phase from the outward- deformation phase (see text for definitions of the two phases). C. Phase diagram GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s53> GLYPH<s50>GLYPH<s48> GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s49>GLYPH<s53> GLYPH<s82> GLYPH<s32>GLYPH<s61> GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s110> GLYPH<s109> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s49> GLYPH<s46>GLYPH<s53> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s52> GLYPH<s46>GLYPH<s48> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s54> GLYPH<s46>GLYPH<s48> GLYPH<s32> GLYPH<s32> GLYPH<s67> GLYPH<s114> GLYPH<s105> GLYPH<s116> GLYPH<s105> GLYPH<s99>GLYPH<s97>GLYPH<s108> GLYPH<s32> GLYPH<s112>GLYPH<s114> GLYPH<s101>GLYPH<s115>GLYPH<s115>GLYPH<s117>GLYPH<s114> GLYPH<s101>GLYPH<s32> GLYPH<s32> GLYPH<s112> GLYPH<s67> GLYPH<s32> GLYPH<s32> GLYPH<s91> GLYPH<s71> GLYPH<s80> GLYPH<s97>GLYPH<s93> GLYPH<s78>GLYPH<s117>GLYPH<s109> GLYPH<s98>GLYPH<s101>GLYPH<s114> GLYPH<s32> GLYPH<s111>GLYPH<s102> GLYPH<s32> GLYPH<s119> GLYPH<s97>GLYPH<s108>GLYPH<s108>GLYPH<s115>GLYPH<s32> GLYPH<s78> GLYPH<s40>GLYPH<s97>GLYPH<s41> GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s51> GLYPH<s46>GLYPH<s48> GLYPH<s52> GLYPH<s46>GLYPH<s48> GLYPH<s49> GLYPH<s53> GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s53> GLYPH<s50> GLYPH<s48> GLYPH<s53> GLYPH<s49> GLYPH<s48> GLYPH<s49> GLYPH<s53> GLYPH<s50> GLYPH<s48> GLYPH<s50> GLYPH<s53> GLYPH<s40>GLYPH<s98>GLYPH<s41> GLYPH<s32> GLYPH<s67> GLYPH<s114> GLYPH<s105> GLYPH<s116> GLYPH<s105> GLYPH<s99>GLYPH<s97> GLYPH<s108> GLYPH<s32> GLYPH<s112> GLYPH<s114> GLYPH<s101> GLYPH<s115>GLYPH<s115>GLYPH<s117> GLYPH<s114> GLYPH<s101> GLYPH<s32> GLYPH<s112> GLYPH<s67> GLYPH<s32> GLYPH<s91> GLYPH<s71> GLYPH<s80> GLYPH<s97> GLYPH<s93> GLYPH<s82>GLYPH<s32>GLYPH<s91>GLYPH<s110>GLYPH<s109> GLYPH<s93> GLYPH<s78>