018 Finite element analysis of heat transfer in a commercially available conduction cryostage

Towards estimating errors in temperature measurement and control caused by non-collocation of specimen and sensor, we developed a finite-element model of heat transfer in a conduction stage cryomicroscope. The model represented a Linkam Scientific Instruments biological cryostage cooled by a liquid nitrogen pump; finite element analysis was performed using COMSOL Multiphysics® software. The computational domain was 2D axisymmetric and comprised the cryostage chamber interior, which was modeled as a 65-mm diameter cylinder filled with nitrogen gas and having a constant-temperature boundary. The silver heating/cooling element was located in the center of the chamber, and a sample (consisting of a water droplet sandwiched between a pair of circular glass coverslips) was in contact with the top surface of the silver block. Heat transfer processes simulated in the model included heat conduction within domains of silver, glass, water, and nitrogen; joule heating in the heating coil embedded within the silver element; and forced convection cooling from nitrogen vapor that is pumped through an interior channel in the silver block. We measured the nitrogen flow rate corresponding to the temperature profiles used in our experiments, and thus were able to determine the value of the Reynolds number to be >10000; for this regime, the coefficient of forced convection could be estimated using the Dittus–Boelter correlation. Another important heat transfer phenomenon in our model was the contact resistance between the top surface of the silver block and the bottom surface of the glass coverslip. To estimate the contact resistance, we performed experiments in which we used the cryomicroscope to warm frozen water droplets from −10 °C until melting was observed. After correcting for effects of droplet size, the measured difference between the expected and observed melting temperatures provided experimental data on the magnitude of thermal gradients within the cryostage. Thus, finite element simulations of the warming experiment were repeated while iteratively adjusting the assumed value of contact resistance, until there was agreement between the predicted and observed temperature differential. In melting experiments with a warming rate of 80 °C/min, the 95% confidence interval for the temperature error at 0 °C was 1.2 ± 0.4 K when data were extrapolated to droplets of negligible diameter; based on these measurements, the contact resistance was determined to be 2.8 K/W. Temperature differentials were also measured in independent melting experiments at warming rates 0.6, 6, and 150 °C/min; the magnitude of the temperature error was found to increase with increased rate of warming, and was well predicted by the finite element model. In simulations of cooling at a rate of 60 °C/min, the magnitude of the predicted temperature error increased over the initial ∼10 s of the cooling process, reaching a maximum of ∼1 K; simulations also showed that the steady state magnitude of the temperature error decreased with decreasing cooling rate. Finite element analysis also demonstrated that significant thermal gradients develop within the portion of the glass coverslip that is situated above the silver block aperture. This gradient was caused in part by heat transfer from the nitrogen gas above the sample. Source of funding: National Science Foundation Grant CBET-0954587, awarded to JOMK. Conflict of interest: None declared. jens.karlsson@villanova.edu