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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.

Physical
quantity
Scaling
exponent
Typical
magnitude
Scaling
accuracy
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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