2.5.3. Magnitudes and scaling
With a scale-independent volumetric heat capacity,
A cubic nanometer volume of a material with a (typical) volumetric heat capacity of 10 6 J/m 3ĚK has a heat capacity of 1 maJ/K.
Thermal conductance scales like electrical conductance, with
and a cubic nanometer of material with a (fairly typical) thermal conductivity of 10 W/mĚK has a thermal conductance of 108 W/K.
Characteristic times for thermal equilibration follow from these relationships, yielding
For a cubic nanometer block separated from a heat sink by a thermal path with a conductance of 108 W/K, the calculated thermal time constant is ~ 1013 s, which is comparable to the acoustic transit time. (In an insulator, a calculated thermal time constant approaching the acoustic transit time signals a breakdown of the diffusive model for transport of thermal energy and the need for a model accounting for ballistic transport; in the fully ballistic regime, time constants scale in proportion to L, and thermal energy moves at the speed of sound.)
The scaling relationship for frictional power dissipation, Eq. (2.16), implies a scaling law for the temperature elevation of a device in thermal contact with its environment,
This indicates that nanomechanical systems are more nearly isothermal than analogous systems of macroscopic size.
Table 2.1. Summary of classical continuum scaling laws.
Copyright © 1998 by John Wiley & Sons, Inc.