With the approximation of scale-independent resistivity,

copacitative time constant resistance · capacitance=constant

(2.36)

This implies that the time required for a capacitor to discharge through a resistor in a pure RC circuit is independent of scale; with the scale dependence of the LR time constant, however, a structure with fixed proportions can change from a nearly pure RC circuit (if built on a small scale) to a nearly pure LR circuit (if built on a large scale). The nanometer-scale RC time constant indicated by this expression is (17 )×(9×10^–21 F) ≈1.5 ×10^–19 s, but this result is nonphysical because it neglects the effects of electron inertia and relaxation time.

The LC product defines an oscillation frequency

oscillation frequency L-1

(2.37)

The characteristic inductance and capacitance calculated above would yield an LC circuit with an angular frequency of ~ 3 ×10¹⁷ rad/s. Alternatively, in structures such as waveguides,

oscillation frequency L-1

(2.38)

To propagate in a hypothetical waveguide 1 nm in diameter, an electromagnetic wave would require a frequency of ~ 9 ×10¹⁷ rad/s or more. Even the lower of the two frequencies just mentioned corresponds to quanta with an energy of ~ 30 aJ, that is, to photons in the x-ray range with energies of ~ 200 eV. These frequencies and energies are inconsistent with physical circuits and waveguides (metals are transparent to x-rays, electrons are stripped from molecules at energies well below 200 eV, etc.). Quantum effects and electron inertia make Eq. (2.38) inapplicable in the nanometer range.

Scale also affects the quality of an oscillator:

Q oscillation frequency L

(2.39)

Since Q is a measure of the damping time relative to the oscillation time, small AC circuits will be heavily damped unless nonclassical effects intervene.

Where the following chapters consider electromagnetic systems at all, they describe systems with currents and fields that are slowly varying by the relevant standards. High-frequency quantum electronic devices, though undoubtedly of great importance, are not discussed here.

2.5. Scaling of classical thermal systems

2.5.1. Basic assumptions •

The classical continuum model assumes that volumetric heat capacities and thermal conductivities are independent of scale. Since heat flows in nanomechanical systems are typically a side effect of other physical processes, no independent assumptions are made regarding their scaling laws.

2.5.2. Major corrections

Classical, diffusive models for heat flow in solids can break down in several ways. On sufficiently small scales (which can be macroscopic for crystals at low temperatures) thermal energy is transferred ballistically by ›phonons for which the mean free path would, in the absence of bounding surfaces, exceed the dimensions of the structure in question. In the ballistic transport regime, interfacial properties analogous to optical reflectivity and emissivity become significant. Radiative heat flow is altered when the separation of surfaces becomes small compared to the characteristic wavelength of blackbody radiation, owing to coupling of nonradiative electromagnetic modes in the surfaces. In gases, separation of surfaces by less than a mean free path again modifies conductivity. The following assumes classical thermal diffusion, which can be a good approximation for liquids and for solids of low thermal conductivity, even on scales approaching the nanometer range.