Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia and Novosibirsk State University, Novosibirsk 630090, Russia.
J Chem Phys. 2013 Nov 28;139(20):204705. doi: 10.1063/1.4830395.
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.
在经典涨落理论的框架下，推导出了一个关于刚刚从催化剂纳米颗粒（NP）表面分离出来的碳帽可逆形成功的解析公式。这个碳帽被认为是单壁碳纳米管（SWCNT）形成的中心。帽的形成功取决于源碳化学势μC。使用这个公式，确定了 SWCNT 形成的速率。从这个表达式可以得到 SWCNT 直径分布。得到的分布有明显的最大值。结果发现，当源碳化学势在很宽的范围内变化时，模态 SWCNT 直径 d(m) 在 1.0 < d(m) < 1.8nm 的狭窄窗口内，随着μC 的增加而微弱增加。所确定的直径分布与化学气相沉积过程中实验测量的 SWCNT 直径的典型值吻合良好。d(m) 的增加伴随着分布宽度Δd 的增加。选择性 d(m)/Δd 是μC 的函数，μC 越高，选择性越差。尽管无法精确计算 SWCNT 形成速率 I，但可以得到 I 与系统参数（如 NP 半径 R(S)）之间的关系。这个关系是在固-液-固系统中推导出来的。为了确定对于特定直径 d 的纳米管的函数 I(R(S))，在严格的界面 Gibbs 理论框架下，推导出了作为 NP 半径函数的催化剂/非晶碳互溶性的公式。使用推导的公式，得到了一个给出 I(R(S)) 依赖性的表达式。这个表达式预测 I 随着 R(S) 的增加而增加。对于金属/碳界面表面张力为 1000mN/m 的情况进行的估算表明，SWCNT 形成速率随着 NP 半径从 1nm 增加到 10nm，增加了几个数量级。
