Model of carbon nanofiber internal structure formation and instability of catalytic growth interface

It is well known that the internal structure determines the properties of carbon nanotubes and carbon nanofibers. However, a fundamental understanding of the processes that drive structure formation is missing, hindering the development of controlled synthesis strategies. Here we use theoretical calculations to explore the time evolution of the shape of the interface between the catalyst nanoparticle and its associated graphitic nanofiber at the initial stages of growth. This phenomenological description of the behavior of the catalyst nanoparticle-graphite interface constructed with model parameters provides new understanding of the mechanisms that control the internal structure of carbon nanofibers. We show that if the magnitude of the interface curvature exceeds a critical value ${\kappa }_{\mathrm{crit}}$, the interface loses stability and a cavity forms in the center of the nanofiber.

Figure 1

(Color online) (a) Transmission electron microscopy image of a typical CNF with an empty central cavity. (b) Scanning transmission electron microscopy image of two carbon nanofibers in $z$-contrast mode. The white areas are where the nanofibers are filled with nickel. The nickel catalyst material has been sucked into the internal cavity during growth. (c) High-resolution transmission electron microscopy image of a nanofiber with central cavity. The inset shows where several graphitic planes terminate (see arrows) at the cavity. (d) Transmission electron microscopy image showing the profile of the initial stages of CNF growth. In this fiber the initially flat, continuous planes at the base of the fiber gradually curve upward. Eventually as the slope of the curves increase, the graphitic layers become discontinuous in the center of the nanofiber.

Figure 2

(a) Schematic depicting one curve of the CN-G interface. Here the normal growth velocity, ${\vec{v}}_n$, is shown for point $a$ on the curve. The angle $\theta$ is defined as the angle between the $r$ axis and ${\vec{v}}_n$. (b) Detailed diagram of the translation during time $dt$ of a section of the interface, showing the connection between the ordinary derivatives, partial derivatives, and the angle $\theta$.

Figure 3

(a) Change of shape of the interface at the initial stages of growth without central cavity formation (i.e., continuous solution). The parameter values for the numerical calculation are $v_{n,\max }=5$, $\Theta =0.3\pi$, $\tau =0.4$, and $r_{\max }=2$. The curve number $n$ is related to dimensionless time by the formula $t\left(n\right)=n\bullet 0.1$. (b) Time evolution of the interface tilt angle at the initial stages of growth. The behavior of $\theta (r_{\max },t)$ varies according to Eq. (12), while the angle at the center remains constant, $\theta (0,t)\equiv \pi ∕2$. (c) Time evolution of the first curvature of the interface at the initial stages of growth. At $t⩽2\tau$ (or $n⩽8$), $\mid {\kappa }_1\left(r_{\max }\right)\mid$ grows fast and saturates at $\sim -0.3$. (d) Time evolution of the second curvature at the initial stages of growth. At $t⪡\tau$ (or $n⪡4$), $\mid {\kappa }_2\left(r_{\max }\right)\mid$ increases rapidly, then maximizes and decreases, reaching a steady state of $\sim -0.1$.

Figure 4

Time dependence of curvature at the center of the interface. Dots are results from numerical calculation using Eqs. (3, 4, 5, 6, 7, 8, 9, 10, 11, 12). The solid curve is an approximation of the asymptotic formula, Eq. (13), where ${\kappa }_{\infty }\approx -0.529$, $A\approx 3.58$, ${\tau }_{\kappa }\approx 0.413$.

Figure 5

(a) Change of the interface shape at the initial stages of growth for a nanofiber with central cavity formation (i.e., discontinuous solution). The parameters of the model are $v_{n,\max }=5$, $\Theta =0.1\pi$, $\tau =0.4$, and $r_{\max }=2$. The curve number is related to dimensionless time by the formula $t\left(n\right)=n0.06$. For $t>0.87$ (or $n⩾15$, interface is indicated by the dotted line) an instability begins to develop in the central region of the interface. The growth rate in this region drops suddenly and can even become negative (see dashed line $n=15$). As a result, the deposition of carbon at the center of nanofiber is replaced by dissolution of the graphitic layers. With both positive (at $r_{\max }$) and negative (at $r_0$) interfacial velocities, the surface of catalyst detaches from the surface of graphite and the interface disappears. (b) Time evolution of the interface tilt angle at the initial stages of growth. The behavior of $\theta (r_{\max },t)$ is defined by Eq. (12), while $\theta (0,t)\equiv \pi ∕2$. The instability at $n⩾15$ leads to sharp changes of $\theta (r,t)$ at the central region (see dashed line $n=15$). (c) Time evolution of the first curvature of the interface at the initial stages of growth. At $t⩽\tau$ (or $n⩽7$), $\mid {\kappa }_1\left(r_{\max }\right)\mid$ grows fast and reaches saturation. At the same time, the growth of the first curvature continues in the center of CNF so that ${\kappa }_1\left(r_0\right)$ becomes bigger than ${\kappa }_1\left(r_{\max }\right)$. At $t\approx 0.87$ the dotted line ${\kappa }_1\left(r_0\right)<-1$ and the interface loses its stability. The curvature begins changing very fast and can even change its sign (see dashed line $n=15$). As a result, the interface disappears between catalyst and graphite in the central region of the fiber. (d) Time evolution of the second curvature at the initial growth stages. For $t⪡\tau$, $\mid {\kappa }_2\left(r_{\max }\right)\mid$ grows fast, reaches a maximum and decays reaching steady state value $\mid {\kappa }_2(r_{\max },t\to \infty )\mid ⪡\mid {\kappa }_1(r_{\max },t\to \infty )\mid$. At the center of the nanofiber ${\kappa }_1\left(r_0\right)={\kappa }_2\left(r_0\right)$ and their behavior coincides. In the region of instability ${\kappa }_2\left(r\right)$ depends more strongly on the radius than ${\kappa }_1\left(r\right)$ (see dashed line $n=15$).

Figure 6

Plot showing the dependence of the asymptotic value of the interface curvature at the center of the nanofiber${\kappa }_{\infty }$ on the equilibrium tilt angle $\Theta$ at the outer boundary. The calculation was performed using $v_{n,\max }=5$ and $r_{\max }=2$. The dotted points are obtained using formulas (3, 4, 5, 6, 7, 8, 9, 10, 11, 12). The solid line 1 is calculated using asymptotic formula (15), valid for $r_{\max }\to \infty$. The dashed line 2 is obtained by neglecting the second curvature using formula (16). The dotted line 3 is obtained using Eq. (14), in which the second variable ${\kappa }_2$ is replaced by a constant value at $\Theta =0.3\pi$.