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 ^{3}·K 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/m·K has a thermal conductance of 10^{–8} 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 10^{–8} W/K, the calculated thermal time constant is ~ 10^{–13} 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.
^{a} Values included only to
illustrate the failure of the scaling law. |
Copyright © 1998 by John Wiley & Sons, Inc.