Mathematical models for the oxidative functionalization of multiwalled carbon nanotubes

A systematic, quantitative study has been conducted to elucidate the effect of oxidation protocol and time on the size, integrity and surface characteristics of multiwalled carbon nanotubes (MWCNTs). Pristine MWCNTs with initial lengths ranging between (i) 2–5 μm and (ii) 1–2 μm were selected for the study. Two oxidation protocols, sonication at 40 °C and reflux at 80 °C in a mixture of concentrated H2SO4 and HNO3 (3:1 v/v) were examined with each variety of MWCNTs. Time dependence of oxidative shortening could be correlated by two empirical, exponential relationships: (i) √L0 − √Lt = k log t (Eq. (1)) and (ii) Lt/L0 = c × exp (kt) (Eq. (2)), where L0 and Lt represented the initial and final length of the nanotubes oxidized separately for t hours under ultrasonication and reflux conditions respectively. Time and length dependence of surface carboxyl density was modeled using the following equations: (iii) COOH density (mmol/g, ultrasonication) = 3.088 + [0.353 × t(h)] – [1.185 × Lt (μm)] (Eq. (3)) (iv) COOH density (mmol/g, reflux) = 2.412 + [0.604 × t(h)] – [0.662 × Lt (μm)] (Eq. (4)), which suggest that oxidative shortening and carboxyl enrichment of MWCNTs is not only time-dependent but equally influenced by the oxidation protocol.

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► MWCNTs of different lengths were oxidized by two methods: reflux and ultrasonication.
► Oxidation was performed for different time periods.
► Empirical equation for MWCNTs length shortening was derived.
► Empirical equation for number of COOH groups induced on MWCNTs was derived.
► These results were illustrated by mathematical models.