Fig. 4: Relationship between the maximum damping limit (\({b}_{\max }^{2}\)) and spring constants (\({\omega }_{o}^{2}\)) for a series of DWNTs.

a The parameters (Supplementary Table 12-3) from the regression of the series of scans in Fig. 2a are seen to disperse approximately linearly (\({R}^{2}=0.80\)) for the CNT G (29,1)@(35,6). b The linearity is more pronounced (\({R}^{2}=0.96\)) for CNT X (28,5)@(28,11). Stage pressure spans \({10}^{-8}\) to \({10}^{-3}\) bar with no apparent trend (Supplementary Table 12-1), except a set (red filled diamonds) after 12 h of baking out at 100 °C and \(3.2\times {10}^{-5}\) bar (Supplementary Table 12-4). c The linearity is also pronounced (\({R}^{2}=0.99\)) for CNT F (19,3)@(22,11), again showing no apparent trend with stage pressure but with points from the same location clustering along the line represented as different data sets (Supplementary Table 12-2). The different datasets and symbols in Fig. 4a–c represent values from a series of repeated scans at different locations on the same CNT (Supplementary Table 12-1–12-4). d Values for \(\varDelta b\) (\({b}_{\max }-{b}_{\min }\)) for all three DWNTs approximate a normal distribution (dash line) with a mean 41.8 (cm⁻¹) and standard deviation of 11.5 (cm⁻¹)². e Thermal annealing or baking out for 12 h at 100 °C and \(3.2\times {10}^{-5}\) bar had the only demonstrable effect on the dispersion of these two mechanical parameters with a significant reduction in \({b}_{\max }\) and \({\omega }_{{{{{{\rm{o}}}}}}}\) after processing. The data before (Pre-bake, \({R}^{2}=0.99\)) and after (Post-bake, \({R}^{2}=0.97\)) still fall identically on the line for CNT X in b, suggesting only a reduction in the number of graphitic tethers. The different symbols in a–d represent Raman scans at different spots in the same CNT and vacuum conditions. f FWHM at \({T}_{\max }\) and \({\omega }_{{{{{{\rm{RBM}}}}}},\min }\) of CNT X change upon thermally annealing. Scans 1-5 are pre-bake (as-grown) CNT X. After 12-hr annealing of CNT X (scans 6-11), \({\omega }_{{{{{{\rm{RBM}}}}}},\min }\) remains nearly invariant and FWHM becomes narrow. This result is consistent with a decrease in the inhomogeneity of the system and in the number of carbonaceous tethers (Supplementary Table 12-1 to 12-4 and Supplementary Fig. 12-2). Source data are provided as a Source Data File.