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2.5.3. Magnitudes and scaling

With a scale-independent volumetric heat capacity,

  heat copacity volume L3 (2.40)

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

  thermal conductance L (2.41)

and a cubic nanometer of material with a (fairly typical) thermal conductivity of 10 W/mĚK has a thermal conductance of 10–8 W/K.

Characteristic times for thermal equilibration follow from these relationships, yielding

  thermal time constant L2 (2.42)

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,

  temperature elevation L (2.43)

This indicates that nanomechanical systems are more nearly isothermal than analogous systems of macroscopic size.

Table 2.1. Summary of classical continuum scaling laws.

Area 2 10–18 m2 Definitional
Force, strength 2 10–8 m2 Good
Stiffness 1 103 N/m Good
Deformation 1 10–11 m Good
Mass 3 10–24 kg Good
Acceleration –1 1015 m/s2 Good
Vibrational frequency –1 1013 rad/s Good
Stress-limited speed 0 103 m/s Good
Motion time –1 10–12 to 10–9 s Good
Power 2 10–8 W Good
Power density –1 1019 W/m3 Good
Viscous stress –1 10 6 N/m2 Moderate to poor
Frictional force 2 (see Ch. 10) Moderate to inapplicable
Wear life 1 (see Ch. 6, 10) Moderate to inapplicable
Thermal speed –3/2 102 m/s Good
Voltage 1 1 V Good at small scales
Electrostatic force 2 10–12 N Good at small scales
Resistance –1 10 Ω Moderate to poor
Current 2 10–8 A Moderate to poor
Electrostatic energy 3 10–21 J Good at small scales
Capacitance 1 10–20 F Good
Magnetic field 1 10–6 T Good
Magnetic force 4 10–23 N Good
Inductance 1 10–15 h Good
Inductive time constant 2 < 10–16 s Bada
Capacitive time constant 0 Moderate to poorb
Elect. oscill. frequency –1 > 1018 rad/s Bada
Oscillator Q 1 Moderate to poorb
Heat capacity 3 10–21 J/K Good
Thermal conductance 1 10–8 W/K Good to poor
Thermal time constant 2 10–13 s Good to poor

a Values included only to illustrate the failure of the scaling law.
b Values omitted; realistic geometries would require several arbitrary parameters.


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