Given constancy of stress and material strength, both the strength of a structure and the force it exerts scale with its cross-sectional area
Nanoscale devices
accordingly exert only small forces: a stress of 10
and varies less rapidly
with scale; a cubic nanometer block of E = 10
Given the above scaling relationships, the magnitude of the deformation under load
is proportional to scale, and hence the shape of deformed structures is scale invariant. The assumption of constant density makes mass scale with volume,
and the mass of a cubic
nanometer block of density
=3.5×10 The above expressions yield the scaling relationship
A cubic-nanometer mass
subject to a net force equaling the above working stress
applied to a square nanometer experiences an acceleration
of ~3×10 Modulus and density determine the
acoustic speed, a scale-independent parameter [along a
slim rod, the speed is (
The acoustic speed in
diamond is ~1.75×10
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 Characteristic times are inversely proportional to characteristic frequencies
The speed of mechanical motions is constrained by strength and density. Its scaling can be derived from the above expressions
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≈1.7×10 The frequencies characteristic of mechanical motions scale with transit times
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 The above expressions yield relationships for the scaling of mechanical power
and mechanical power density
A 10 nN force and a 1 nm |

Copyright © 1998 by John Wiley & Sons, Inc.