One method that is used to grow nanowires (nanotubes with...

One method that is used to grow nanowires (nanotubes with solid cores) is to initially deposit a small droplet of a liquid catalyst onto a flat surface. The surface and catalyst are heated and simultaneously exposed to a higher-temperature, low-pressure gas that contains a mixture of chemical species from which the nanowire is to be formed. The catalytic liquid slowly absorbs the species from the gas through its top surface and converts these to a solid material that is deposited onto the underlying liquid-solid interface, resulting in construction of the nanowire. The liquid catalyst remains suspended at the tip of the nanowire. Consider the growth of a 15 -nm-diameter silicon carbide nanowire onto a silicon carbide surface. The surface is maintained at a temperature of $T_{s}=2400 \mathrm{~K}$, and the particular liquid catalyst that is used must be maintained in the range $2400 \mathrm{~K} \leq T_{c} \leq 3000 \mathrm{~K}$ to perform its function. Determine the maximum length of a nanowire that may be grown for conditions characterized by $h=10^{5} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $T_{\infty}=8000 \mathrm{~K}$. Assume properties of the nanowire are the same as for bulk silicon carbide.

One method that is used to grow nanowires (nanotubes with solid cores) is to initially deposit a small droplet of a liquid catalyst onto a flat surface. The surface and catalyst are heated and simultaneously exposed to a higher-temperature, low-pressure gas that contains a mixture of chemical species from which the nanowire is to be formed. The catalytic liquid slowly absorbs the species from the gas through its top surface and converts these to a solid material that is deposited onto the underlying liquid-solid interface, resulting in construction of the nanowire. The liquid catalyst remains suspended at the tip of the nanowire.

Consider the growth of a 15 -nm-diameter silicon carbide nanowire onto a silicon carbide surface. The surface is maintained at a temperature of $T_{s}=2400 \mathrm{~K}$, and the particular liquid catalyst that is used must be maintained in the range $2400 \mathrm{~K} \leq T_{c} \leq 3000 \mathrm{~K}$ to perform its function. Determine the maximum length of a nanowire that may be grown for conditions characterized by $h=10^{5} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $T_{\infty}=8000 \mathrm{~K}$. Assume properties of the nanowire are the same as for bulk silicon carbide.

Key Concepts

Vapor?Liquid?Solid (VLS) Growth Method

This concept refers to a key mechanism used for synthesizing nanowires wherein a liquid catalyst droplet facilitates the absorption of source materials from a vapor phase. The catalyst actively converts these materials at its interface with the underlying substrate, leading to the formation of a solid nanowire. The process relies on the interplay between liquid and solid phases and is fundamental in the controlled growth of nanostructures.

Heat Conduction in Nanowires

In the context of growing nanowires, heat conduction describes the process by which thermal energy is transferred along the length of the nanowire from the hot ambient or droplet tip to the substrate. The efficiency of this conduction determines how far heat can be removed or maintained along the structure, thereby imposing a limit on the maximum achievable nanowire length when balancing with the catalyst's temperature constraints.

Convection Heat Transfer

This concept involves the transfer of heat from the nanowire's outer surface (or the catalyst droplet) to the surrounding gas environment. Characterized by a heat transfer coefficient, it is crucial in setting the thermal boundary conditions of the growth system. In the growth of nanowires, the ambient temperature and convective heat losses play a key role in determining the temperature gradients along the nanowire.

Thermal Boundary Conditions

Understanding thermal boundary conditions is essential in determining the steady?state temperature profile along a nanowire. These conditions include the fixed temperature at the substrate and the specified temperature at the far-field (due to convection), which together allow for the formulation of a heat conduction problem to predict the maximum sustainable length of the nanowire under the given growth parameters.

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