Most mechanical systems use bearings to support moving parts. Macromechanical systems frequently use liquid lubricants, but (as noted by Feynman, 1961), this poses problems on a small scale. The above scaling law ordinarily holds speeds and stresses constant, but reducing the thickness of the lubricant layer increases shear rates and hence viscous shear stresses:

viscous stress at constant speed shear rate L-1

(2.14)

In Newtonian fluids, shear stress is proportional to shear rate. Molecular simulations indicate that liquids can remain nearly Newtonian at shear rates in excess of 100 m/s across a 1 nm layer (e.g., in the calculations of Ashurst and Hoover, 1975), but they depart from bulk viscosity (or even from liquid behavior) when film thicknesses are less than 10 molecular diameters (Israelachvili, 1992; Schoen et al., 1989), owing to interface-induced alterations in liquid structure. Feynman suggested the use of low-viscosity lubricants (such as kerosene) for micromechanisms (Feynman, 1961); from the perspective of a typical nanomechanism, however, kerosene is better regarded as a collection of bulky molecular objects than as a liquid. If one nonetheless applies the classical approximation to a 1 nm film of low-viscosity fluid ( = 10 ^{– 3} N·s/m² ), the viscous shear stress at a speed of 1.7×10³ m/s is 1.7×10⁹ N/m²; the shear stress at a speed of 1 m/s, 10⁶ N/m², is still large, dissipating energy at a rate of 1 MW/m².

The problems of liquid lubrication motivate consideration of dry bearings (as suggested by Feynman, 1961). Assuming a constant coefficient of friction,

frictional force force L2

(2.15)

and both stresses and speeds are once again scale-independent. The frictional power,

frictional power force · speed L2

(2.16)

is proportional to the total power, implying scale-independent mechanical efficiencies. In light of engineering experience, however, the use of dry bearings would seem to present problems (as it has in silicon micromachine research). Without lubrication, efficiencies may be low, and static friction often causes jamming and vibration.

A yet more serious problem for unlubricated systems would seem to be wear. Assuming constant interfacial stresses and speeds (as implied by the above scaling relationships), the anticipated surface erosion rate is independent of scale. Assuming that wear life is determined by the time required to produce a certain fractional change in shape,

ware life L

(2.17)

and a centimeter-scale part having a ten-year lifetime would be expected to have a 30 s lifetime if scaled to nanometer dimensions.

Design and analysis have shown, however, that dry bearings with atomically precise surfaces need not suffer these problems. As shown in Chapters 6, 7, and 10, dynamic friction can be low, and both static friction and wear can be made negligible. The scaling laws applicable to such bearings are compatible with the constant-stress, constant-speed expressions derived previously.

2.3.3. Major corrections

The above scaling relationships treat matter as a continuum with bulk values of strength, modulus, and so forth. They readily yield results for the behavior of iron bars scaled to a length of 10^–12 m, although such results are meaningless because a single atom of iron is over 10^–10 m in diameter. They also neglect the influence of surfaces on mechanical properties (Section 9.4), and give (at best) crude estimates regarding small components, in which some dimensions may be only one or a few atomic diameters.

Aside from the molecular structure of matter, major corrections to the results suggested by these scaling laws include uncertainties in position and velocity resulting from statistical and quantum mechanics (examined in detail in Chapter 5). Thermal excitation superimposes random velocities on those intended by the designer. These random velocities depend on scale, such that

thermal speed L-3/2

(2.18)

where the thermal energy measures the characteristic energy in a single degree of freedom, not in the object as a whole. For = 3.5×10³ kg/m³, the mean thermal speed of a cubic nanometer object at 300 K is ~ 55 m/s. Random thermal velocities (commonly occurring in vibrational modes) often exceed the velocities imposed by planned operations, and cannot be ignored in analyzing nanomechanical systems.

Quantum mechanical uncertainties in position and momentum are parallel to statistical mechanical uncertainties in their effects on nanomechanical systems. The importance of quantum mechanical effects in vibrating systems depends on the ratio of the characteristic quantum energy ( , the quantum of vibrational energy in a °harmonic oscillator of angular frequency ) and the characteristic thermal energy (kT, the mean energy of a thermally excited harmonic oscillator at a temperature T, if kT » ). The ratio /kT varies directly with the frequency of vibration, that is, as L^{– 1}. An object of cubic nanometer size with w = 1.7×10¹³ rad/s has /kT₃₀₀ ≈0.4 (T₃₀₀ = 300 K; kT₃₀₀ ≈ 4.14 maJ). The associated quantum mechanical effects (e.g., on positional uncertainty) are smaller than the classical thermal effects, but still significant (see Figure 5.2).