Chapter 3, Problem 3.138P

During the initial stages of the growth of the nanowire of Problem 3.109, a slight perturbation of the liquid catalyst droplet can cause it to be suspended on the top of the nanowire in an off-center position. The resulting nonuniform deposition of solid at the solid-liquid interface can be manipulated to form engineered shapes such as a nanospring, that is characterized by a spring radius r. spring pitch s, overall chord length $L_c$ (length running along the spring), and end-to-end length L as shown in the sketch. Consider a silicon carbide nanospring of diameter $D=15\text{nm,}r=30\text{nm,}s=25\text{nm,}$ and $L_c=425\text{nm}\text{.}$ From experiments, it is known that the average spring pitch $\bar{s}$ varies with average temperature $\bar{T}$ by the relation $d\bar{s}/d\bar{T}=0.1\text{nm/K}\text{.}$ Using this information, a student suggests that a nanoactuator can be constructed by connecting one end of the nanospring to a small heater and raising the temperature of that end of the nano spring above its initial value. Calculate the actuation distance $\Delta L$ for conditions where $h={10}^6{\text{W/m}}^{\text{2}}\cdot \text{K,}{T}_{\infty }=T_i=25°\text{C,}$ with a basetemperature of $T_b=50°\text{C}\text{.}$ If the base temperature can be controlled to within $1°\text{C,}$ calculate the accuracy to which the actuation distance can be controlled. Hint: Assume the spring radius does not change when the spring is heated. The overall spring length may be approximated by the formula,

   $L=\frac{\bar{s}}{2\pi }\frac{L_c}{\sqrt{r^2+{\left(\bar{s}/2\pi \right)}^2}}$