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.