2.6. Beyond classical continuum models
This chapter has described the scaling laws
implied by classical continuum models for mechanical,
electromagnetic, and thermal systems, together with the
magnitudes they suggest for the physical parameters of
nanometer scale systems. It has also considered limits to
the validity of these models, imposed by statistical
mechanics, quantum mechanics, the molecular structure of
matter, and so forth. Different classical models fail at
different length scales, with the most dramatic failures
appearing in AC electrical circuits.
The following chapters go beyond
classical continuum models. Chapters 3 and 4 examine
models of molecular structure, dynamics, and statistical
mechanics from a nanomechanical systems perspective.
Chapters 5 and 6 examine the combined effects of quantum
and statistical mechanics on nanomechanical systems,
first analyzing positional uncertainty in systems subject
to a restoring force, and then analyzing the rates of
transitions, errors, and damage in systems that can
settle in alternative states. Chapter 7 examines
mechanisms for energy dissipation. These chapters provide
a foundation for analyzing specific nanomechanical
systems. Later chapters examine not only nanomechanical
systems, but a few specific electrical and fluid systems;
where analysis of the latter must go beyond classical
continuum approximations, the needed principles are
discussed in context.
2.7.
Conclusions
The accuracy of classical continuum
models and scaling laws to nanoscale systems depends on
the physical phenomena considered. It is low for
electromagnetic systems with small calculated time
constants, reasonably good for thermal systems and slowly
varying electromagnetic systems, and often excellent for
purely mechanical systems, provided that the component
dimensions substantially exceed atomic dimensions.
Scaling principles indicate that mechanical components
can operate at high frequencies, accelerations, and power
densities. The adverse scaling of wear lifetimes suggests
that bearings are a special concern. Later chapters
support these expectations regarding frequency,
acceleration, and power density; Chapter 10 describes
suitable bearings. Table 2.1 summarizes many of the
relationships discussed in this chapter.
