One method that is used to grow nanowires (nanotubes with solid cores) is to initially deposit a small droplet of 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 the 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 Ts = 2400 K, and the particular liquid catalyst that is used must be maintained in the range 2400 K ≤ Tc ≤ 3000 K to perform its function.
Determine the maximum length of a nanowire that may be grown for conditions characterized by h = 10⁵ W/m²⋅K and T[infinity] = 8000 K. Assume properties of the nanowire are the same as for bulk silicon carbide.