Fluctuation theory of single-walled carbon nanotube formation

In the framework of classical fluctuation theory an analytical formula is derived for the reversible work of formation of just detached carbon cap on the surface of catalyst nanoparticle (NP). This cap is considered as single walled carbon nanotube (SWCNT) formation center. The work of cap formation depends on the source carbon chemical potential μC. Using the derived formula for this work an expression for the rate of SWCNT formation is determined. From this expression the SWCNT diameter distributions can be obtained. The obtained distributions have sharp maxima. It is found that the modal SWCNT diameter d(m) increases weakly with μC being in the narrow window of 1.0 < d(m) < 1.8 nm when changing the source carbon chemical potential in a wide range. The determined diameter distributions proved to be in a good agreement with the typical values of the SWCNT diameters as experimentally measured in the chemical vapor deposition process. The increase of d(m) is accompanied by the increase of the distribution width Δd. The selectivity d(m)/Δd is a function of μC, the higher values of μC the worse selectivity is observed. Although the value of the SWCNT formation rate I cannot be calculated precisely the relationship between I and the system parameters, such as the NP radius R(S), can be obtained. This relationship is derived for the solid-liquid-solid system. To determine the function I(R(S)) for nanotubes of a certain diameter d, formulas for catalyst∕amorphous carbon mutual solubilities as functions of NP radius are derived in the framework of the rigorous Gibbs theory of interface. Using the derived formulas an expression giving the dependence I(R(S)) is obtained. The expression predicts an increase of I with the radius R(S). The estimations carried out for the metal/carbon interface surface tension of 1000 mN/m show that the SWCNT formation rate increases by a few orders of magnitude with the radius increase from 1 to 10 nm.