Here I summarize the analysis of thermal conduction through 3 ceramic ball bearings supporting a test wafer in Megastat. I considered a baseplate or housing for the ceramic ball bearings at the temperature of the cold plate, the 3 ceramic ball bearings themselves, and the wafer seated on top. This pathway can be broken down into the following components:
1. Thermal resistance across joint from Al baseplate to ball bearing
2. Thermal resistance of bulk of ball bearing
3. Thermal resistance across joint from ball bearing to wafer
These resistances in series can be summed, and there exist 3 of these pathways in parallel (1 for each bearing) and the net resistance can then be found using the corresponding formula.
Attachment 1 is a plot of the thermal conductance of alumina ceramic (Al2O3), and it was taken from from this paper. The peak value of 99% Al2O3 agrees with what is listed in the ceramic ball bearing spec sheet: 28 W/mK. I've used these tabulated values to obtain component 2 above.
Attachment 2 is a plot of the thermal conductances across various joints, taken from Ekin. I could not find data for the conductance across Al--Al2O3 pressure joints or Al2O3--Si pressure joints, but I have used the value corresponding to a stainless steel--stainless steel pressure joint as a conservative upper bound. (*When I was calculating non-negligible conduction last Friday, I had forgotten to update the joint conductance value from an Al--Al pressure joint (orange box in Attachment 2). This greatly overestimated the joint conduction, and the ceramic joints should be a few orders of magnitude below this.)
The results of this modeling show that the maximum cooling power delivered via conductive contact is below 7e-5 W. Compared to the 6e-3 W of radiative cooling power (~1%), this is negligible. Attachment 3 shows the nominal difference in wafer cooldown - the solid and dashed orange curves are barely distinguishable (T difference of ~0.2K at steady state - RTD uncertainty is +- 0.15K). We should proceed with this design plan for emissivity testing in Megastat. |