Thermal mismatch strains in sidewall functionalized carbon nanotube/polystyrene nanocomposites

Raman spectra were recorded on a Jobin Yvon S3000 spectrometers equipped with a LN-cooled charged coupled device (CCD). A long-working distance microscope objective 50× of Olympus 45 microscope was used for both to focus the laser beam to a spot with diameter $∼2μm$ on the sample surface and to collect the scattered light. The laser power density was kept below $104W∕cm2$ to prevent a substantial overheating of the sample at the laser spot.

Functionalized carbon nanotubes: Pristine SWNTs produced by HiPCO process (Ref. 14) were functionalized by reacting with organic diazonium compounds (Ref. 12).

Atactic polystyrene with weight-average molecular weight $Mw=152000Da$⁠, polydispersity ratio $Mw∕Mn<1.05$ was used for preparation of the composites. Composites were prepared by solution mixing appropriate quantities of pristine or functionalized SWNTs and polystyrene in toluene at room temperature. The solutions were dried extensively at room temperature and subsequently annealed at 180 °C in a vacuum oven for 24 h.

The RBM mode frequency in a SWNT is given by $ωRBM(cm−1)=241(cm−1)∕dnt(nm)$⁠. From this expression it directly follows that $ΔωRBM∕ωRBM=−Δdnt∕dnt=−ϵci$⁠, where $ϵci$ is the strain along the circumference. On the other hand, under the reasonable approximation given in Ref. 4: $ΔωG+∕ωG+=−γ(1−νnt)ϵz$⁠, where $γ=1.24$ is the Grüneisen parameter and $νnt=0.16$ to the nanotubes Poisson’s ratio. Given $ϵci=−νntϵz$ we finally obtain $ΔωRBM∕ωRBM=−νntΔωG+∕[γ(1−νnt)ωG+]$⁠.

The axial stress on the nanotube due to the shrinkage of the matrix is $σnt=ΔαΔTEnt∕Ẽ(T)$⁠, where $Ẽ=1+(fntEnt)∕(fmEm)$⁠, and $fnt$ and $fm$ are the volume fractions of the nanotubes and the matrix. The 1.5 wt % HDB-SWNT∕PS nanocomposite was loaded with 1.5 wt % HDB-SWNTs that corresponds to a volume fraction $fnt=0.02$ and $fm=0.98$⁠. For the calculation of $ϵzcal=ΔαΔT∕Ẽ(T)$ we used the following values of the parameters: linear CTE $αm(<Tg)=70×10−6K−1$ (Ref. 26) and $αnt=−1.5×10−6$ (Ref. 27) and the temperature dependence of $Em$ taken from Ref. 28. Note that the volume CTE of PS is three times larger than the linear one (Ref. 26).

The ratio of the number of circumferential SWNTs $(Np)$ and the total number of nanotubes $(N)$ in a bundle is given by $Np∕N=12(Nd−1)∕(3Nd2+1)$⁠, provided the number $Nd=D(nm)∕1.7(nm)$ of the nanotubes across the bundle diameter $D$ is odd (Ref. 29). It is also reasonable to assume that only half of the surface of the nanotubes along the bundle circumference is exposed to the organic modifier. Given these premises it follows that bulk abundance of 1 functionalizing moiety to 66 carbon atom $(1∕66)$ translates to $2N∕66Np≈10∕37$ surface coverage of a SWNT bundle, 60 nm thick. For $Nd$ even, the above expressions have different form (Ref. 29). The estimates for $Nd$ and $Nd′=Nd+1$⁠, however, give very close results.