Thermal Time Constants and Temperature Differences Decrease

Consider a body of heat capacity C (per unit volume) at temperature T connected to a large mass of temperature T = 0 by a thermal link of cross section A, length L and thermal conductivity kr. The heat energy flow dQ/dt is krAT/L and equals the loss rate of thermal energy from the warm mass, dQ/dt = CVdT/dt. The resulting equation dT/T = ~(krA/LCV)dt leads to a solution T = T(0)exp(-i/rth), where Tth= LCV/kTA. Under isotropic scaling rth varies as L2 C/kT. Thermal time constants thus strongly decrease as the size is reduced. As an example, a thermal time constant of a few ¡us is stated  for a tip heater incorporated into a 200 kHz frequency AFM cantilever designed for the IBM "Millipede" 1024 tip AFM high density ther-momechanical memory device. The heater located just above the tip, mounted at the vertex of two cantilever legs each having dimensions 50 x 10 x 0.5 micrometers.

In steady state with heat flow dQ/dt, we see that the temperature difference T is T = (dQ/dt) (L/krA). Since the mechanical power dQ/dt scales as I2, this result implies that the typical temperature difference T scales, in three dimensions, as L. Temperature differences are reduced as the size scale is reduced.

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