2.3.2. Magnitudes and scaling

Given constancy of stress and material strength, both the strength of a structure and the force it exerts scale with its cross-sectional area

 total strength force area L2 (2.1)

Nanoscale devices accordingly exert only small forces: a stress of 1010 N/m2 equals 10 – 8 N/nm2, or 10 nN/nm 2. Stiffness in °shear, like stretching stiffness, depends on both area and length

 shear stiffness stretching stiffness L (2.2)

and varies less rapidly with scale; a cubic nanometer block of E = 1012 N/m2 has a stretching stiffness of 1000 N/m. The bending stiffness of a rod scales in the same way

 bending stiffness L (2.3)

Given the above scaling relationships, the magnitude of the deformation under load

 deformation L (2.4)

is proportional to scale, and hence the shape of deformed structures is scale invariant.

The assumption of constant density makes mass scale with volume,

 mass volume L3 (2.5)

and the mass of a cubic nanometer block of density =3.5×103 kg/m3 equals 3.5×10 – 24 kg.

The above expressions yield the scaling relationship

 accleration L-1 (2.6)

A cubic-nanometer mass subject to a net force equaling the above working stress applied to a square nanometer experiences an acceleration of ~3×1015 m/s2. Accelerations in nanomechanisms commonly are large by macroscopic standards, but aside from special cases (such as transient acceleration during impact and steady acceleration in a small flywheel) they rarely approach the value just calculated. (Terrestrial gravitational accelerations and stresses usually have negligible effects on nanomechanisms.)

Modulus and density determine the acoustic speed, a scale-independent parameter [along a slim rod, the speed is (E/ )1/2; in bulk material, somewhat higher]. The vibrational frequencies of a mechanical system are proportional to the acoustic transit time

 frequency L-1 (2.7)

The acoustic speed in diamond is ~1.75×104 m/s. Some vibrational modes are more conveniently described in terms of lumped parameters of stiffness and mass,

 frequency L-1 (2.8)

but the scaling relationship is the same. The stiffness and mass associated with a cubic nanometer block yield a vibrational frequency characteristic of a stiff, nanometer-scale object: [(1000 N/m)/(3.5×10 – 24 kg)] ½ ≈1.7×1013 rad/s.

Characteristic times are inversely proportional to characteristic frequencies

 time frequency-1 L (2.9)

The speed of mechanical motions is constrained by strength and density. Its scaling can be derived from the above expressions

 speed acceleration · time= constant (2.10)

A characteristic speed (only seldom exceeded in practical mechanisms) is that at which a flywheel in the form of a slim hoop is subject to the chosen working stress as a result of its mass and centripetal acceleration. This occurs when v=( / ) 1/2≈1.7×103 m/s (with the assumed and ). Most mechanical motions considered in this volume, however, have speeds between 0.001 and 10 m/s.

The frequencies characteristic of mechanical motions scale with transit times

 frequency L-1 (2.11)

These frequencies scale in the same manner as vibrational frequencies, hence the assumption of constant stress leaves frequency ratios as scale invariants. At the above characteristic speed, crossing a 1 nm distance takes ~6×10–13 s; the large speed makes this shorter than the motion times anticipated in typical nanomechanisms. A modest 1 m/s speed, however, still yields a transit time of only 1 ns, indicating that nanomechanisms can operate at frequencies typical of modern micron-scale electronic devices.

The above expressions yield relationships for the scaling of mechanical power

 power force · speed L2 (2.12)

and mechanical power density

 power density L-1 (2.13)

A 10 nN force and a 1 nm3 volume yield a power of 17µW and a power density of 1.7×1022 W/m3 (at a speed of 1.7×103 m/s) or 10 nW and 1019 W/m3 (at a speed of 1 m/s). The combination of strong materials and small devices promises mechanical systems of extraordinarily high power density, even at low speeds (an example of a mechanical power density is the power transmitted by a gear divided by its volume).