Catalyst-particle configurations: From nanowires to carbon nanotubes

Surface-energy minimization is used to study capillary effects that determine the stable configurations of (liquid or solid) particles atop tubes or wires. The results give allowable ranges for the volume ($V_c$) of the particles as a function of the inner and outer radii of the tubes, $R_I$ and $R_o$, respectively. When $R_I/R_o=0$, the object is a nanowire. When $R_I/R_o=1$, it is a single-wall carbon nanotube. When $0<R_I/R_o<1$, it can be thought of as a multiwall carbon nanotube. Moreover, the transition paths among different configurations are studied. These results suggest possible mechanisms for the reshaping of the “pear-like” catalyst particles in carbon nanotubes that oscillate across local energy-maximum points with respect to the position of the lower interface in the inner wall.

Figure 1

Typical tip-growth process. (a) A catalyst particle is placed on a substrate. (b–c) As the hydrocarbon decomposes on the catalyst particle and the carbon atoms diffuse to the bottom of the catalyst particle, the whole catalyst particle is pushed off the substrate. (d) The catalyst particle moves up and sits atop the CNT.

Figure 2

(a) Sketch of a catalyst particle at the tip of the growing MW tube; (b) vertical cross section of (a); (c) sketch of the equilibrium Young's angle. Here “wall” includes the outer wall, the inner wall, and the top of the tube.

Figure 3

“Pear-like” shapes of the catalyst particle. (a) P1: Upper meniscus contacts outer edge of tube, and lower contacts the inner wall of the tube. (b) P2: Upper meniscus contacts top of tube, and lower meniscus contacts inner wall of the tube. (c) P3: Upper meniscus contacts inner edge of tube, and lower meniscus contacts inner wall of the tube.

Figure 4

Possible stable shapes of the catalyst particle. (a) C1: Upper meniscus contacts outer edge of tube. (b) C2: Upper meniscus contacts the top of tube, and lower meniscus contacts the inner edge of the tube. (c) C3: Upper meniscus contacts the inner edge of tube, and lower meniscus contacting inner edge of the tube. (d) E: The catalyst particle is encapsulated in the tube.

Figure 5

With initial state shown in Fig. 3 called P1, (a) $({\theta }_Y,R_I/R_o)=({\theta }_Y,\rho )$ map of different situations, when $V_c^{1/3}/R_o\ge V_A({\theta }_Y,R_I/R_o)$. Gray: Evolution to C1 shown in Fig. 4. Purple: Evolution to E shown in Fig. 4. Green: Discussed in detail in Fig. 5. The red line is $R_I/R_o=cot(\pi -{\theta }_Y)$. The white point represents the reference case. (b) $(R_I/R_o,V_c^{1/3}/R_o)=(\rho ,\overline{R})$ map of different states with ${\theta }_Y=2\pi /3$. The line between blue and dark blue regions is Eq. (14). The white point represents the reference case. (c) The representative relative energy ($E=E_{\mathrm{total}}-\text{constant}\text{terms}\text{in}{E}_{\mathrm{total}}$) profiles of different regions shown in Fig. 5. The energy profile with $(V_c^{\frac{1}{3}}/R_o,R_I/R_o)=(\overline{R},\rho )=(3.0,0.8)$ and $(V_c^{\frac{1}{3}}/R_o,R_I/R_o)=(\overline{R},\rho )=(2.8,0.8)$ represents the blue region. The energy profile with $(V_c^{\frac{1}{3}}/R_o,R_I/R_o)=(\overline{R},\rho )=(2.0,0.8)$ represents the dark blue region. The right end of each energy profile is the position $\delta /R_o$ such that $\theta ={\theta }_Y$. The values in the figure are adjusted to the same value at $\delta =0$ for convenience of comparison.

Figure 6

With initial state shown in Fig. 3 called P2, $({\theta }_Y,R_I/R_o)$ map of different states, when $V_c^{1/3}/R_o\ge V_B({\theta }_Y,R_I/R_o)$. Gray: the state of catalyst particle evolves to C2 [Fig. 4] or C1 [Fig. 4]. Green: the state of catalyst particle evolves to P3 [Fig. 3]. Purple: the state of catalyst particle evolves to E [Fig. 4]. Yellow: discussed in Fig. 7. The black line is $R_I/R_o=cot(\pi -{\theta }_Y)$. The white point represents the reference case.

Figure 7

With initial state shown in Fig. 3 named P2, (a) $(R_I/R_o,V_c^{1/3}/R_o)=(\rho ,\overline{R})$ map of different states with ${\theta }_Y=2\pi /3$. (b–d) The representative energy profiles corresponding to different regions shown in Fig. 7. (b) The energy profile with $(V_c^{1/3}/R_o,R_I/R_o)=(\overline{R},\rho )=(3.0,0.5)$ represents the yellow region. (c) The energy profile with $(V_c^{1/3}/R_o,R_I/R_o)=(\overline{R},\rho )=(1.7,0.5)$ represents the blue region. (d) The energy profile with $(V_c^{1/3}/R_o,R_I/R_o)=(\overline{R},\rho )=(1.5,0.5)$ represents the red region. The right end of energy profiles in Fig. 7–7 is the position of $\delta /R_o$ such that $d_i/R_o=0$. The left end of energy profile in Fig. 7 is the position of $\delta /R_o$ such that $d_i/R_o=1-R_I/R_o$. States P1 and P3 are shown in Fig. 3 and 3. State C2 is shown in Fig. 4. The energies in (b,c,d) are the relative energy where $E=E_{\mathrm{total}}-\text{constant}\text{terms}\text{in}{E}_{\mathrm{total}}$

Figure 8

With initial state shown in Fig. 3 called P3, (a) $(R_I/R_o,V_c^{1/3}/R_o)=(\overline{R},\rho )$ map of different states with ${\theta }_Y=5\pi /6$. (b–d) The representative energy profiles corresponding to different regions shown in Fig. 8. (b) Energy profile with $(V_c^{\frac{1}{3}}/R_o,R_I/R_o)=(\overline{R},\rho )=(2,0.5)$ represents the blue region. (c) Energy profile with $(V_c^{\frac{1}{3}}/R_o,R_I/R_o)=(\overline{R},\rho )=(1.1,0.5)$ represents the blue region. (d) Energy profile with $(V_c^{\frac{1}{3}}/R_o,R_I/R_o)=(\overline{R},\rho )=(0.82,0.5)$ represents the yellow region. The right end of energy profiles is the position of $d_i/R_o$ such that $\delta /R_o=0$. The left end of the energy profile in Fig. 8 is the position of $\delta /R_o$ when ${\theta }_i={\theta }_Y$. The right end of the energy profiles are the position of $\delta /R_o$ when ${\theta }_i={\theta }_Y-\pi /2$. States C3 and E are shown in Fig. 4 and 4. State P2 is shown in Fig. 3. The energies in (b,c,d) are the relative energy where $E=E_{\mathrm{total}}-\text{constant}\text{terms}\text{in}{E}_{\mathrm{total}}$

Figure 9

$(R_I/R_o,V_c^{1/3}/R_o)=(\rho ,\overline{R})$ map of having stable state for different possible configurations shown in Fig. 4. (a) ${\theta }_Y=2\pi /3$ is taken as the representative case for $\pi /2<{\theta }_Y\le 3\pi /4$. The white point represents the reference case. (b) ${\theta }_Y=5\pi /6$ is taken as the representative case for $3\pi /4<{\theta }_Y<\pi$.

Figure 10

(a) The upper surface of the catalyst particle contacts the outer edge, whereas the lower surface contacts the inner edge or inner wall of the tube. This configuration is not possible at equilibrium when ${\theta }_Y\le \pi /2$. (b) The catalyst particle is totally encapsulated within the tube. Both the upper and the lower interfaces contact the inner wall of the tube with contact angle ${\theta }_Y$. This is a possible equilibrium state when ${\theta }_Y\le \pi /2$.

Figure 11

(a) The upper surface of the catalyst particle dips below the contact point of the upper surface and the outer edge. This configuration is not possible at equilibrium when ${\theta }_Y>\pi /2$.(b) The upper surface of the catalyst particle contacts the outer edge or the top of the tube, whereas the lower surface of the catalyst particle contacts the inner edge of the tube with $\varphi <\pi /2$. This configuration is not possible at equilibrium when ${\theta }_Y>\pi /2$.

Figure 12

(a) Relative total energy {$E/{\gamma }_{LV}{R}_o^2=[E_{\mathrm{total}}-2\pi {\gamma }_{SV}(R_I+R_o)H-\pi {\gamma }_{SL}(R_o^2-R_I^2)]/{\gamma }_{LV}{R}_o^2$} profile versus $\varphi$. (b) Relative total energy {$E/{\gamma }_{LV}{R}_o^2=[E_{\mathrm{total}}-2\pi {\gamma }_{SV}(R_I+R_o)H-\pi {\gamma }_{SL}(R_o^2-R_I^2)]/{\gamma }_{LV}{R}_o^2$} profile versus $\theta$.