Chapter 3, Problem 3.109P

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 $T_s=2400\text{K,}$ and the particular liquid catalyst that is used must be maintained in the range $\text{2400}\text{K}\le T_c\le 3000\text{K}$ to perform its function. Determine the maximum length of a nanowire that may be grown for conditions characterized by $h={10}^5{\text{W/m}}^{\text{2}}\cdot \text{K}$ and $T_{\infty }=8000\text{K}\text{.}$ Assume properties of the nanowire are the same as for bulk silicon carbide.