Chapter 5, Problem 5.13P

A tool used for fabricating semiconductor devices consists of a chuck (thick metallic, cylindrical disk) onto which a very thin silicon wafer $\left(\rho =2700kg/m^3,c=875\text{J/kg}\cdot \text{K,}k=177\text{W/m}\cdot \text{K}\right)$ is placed by a robotic arm. Once in position, an electric field in the chuck is energized, creating an electrostatic force that holds the wafer firmly to the chuck. To ensure a reproducible thermal contact resistance between the chuck and the wafer from cycle to cycle, pressurized helium gas is introduced at the center of the chuck and flows (very slowly) radially outward between the asperities of the interface region.

An experiment has been performed under conditions for which the wafer, initially at a uniform temperature $T_{w,i}=100°\text{C,}$ is suddenly placed on the chuck. which is at a uniform and constant temperature $T_c=23°\text{C}\text{.}$ With the wafer in place, the electrostatic force and the helium gas flow are applied. After 15 s, the temperature of the wafer is determined to be $33°\text{C}\text{.}$ What is the thermal contact resistance $R_{t,c}^{"}\left({\text{m}}^{\text{2}}\cdot \text{K/W}\right)$ between the wafer and chuck? Will the value of $R_{t,c}^{"}$ increase, decrease, or remain the same if air, instead of helium, is used the purge gas?