Part I
Physical Principles
Chapter 2
Classical Magnitudes and
Scaling Laws
2.1. Overview
Most physical magnitudes characterizing
nanoscale systems differ enormously from those familiar
in macroscale systems. Some of these magnitudes can,
however, be estimated by applying scaling laws to the
values for macroscale systems. Although later chapters
seldom use this approach, it can provide orientation,
preliminary estimates, and a means for testing whether
answers derived by more sophisticated methods are in fact
reasonable.
The first of the following sections
considers the role of engineering approximations in more
detail (Section 2.2); the rest present scaling relationships based
on classical continuum models and discuss how those
relationships break down as a consequence of atomicscale
structure, meanfreepath effects, and quantum mechanical
effects. Section 2.3 discusses mechanical systems, where many
scaling laws are quite accurate on the nanoscale.
Section 2.4 discusses electromagnetic systems, where many
scaling laws fail dramatically on the nanoscale.
Section 2.5 discusses thermal systems, where scaling laws
have variable accuracy. Finally,
Section 2.6 briefly describes how later chapters go beyond
these simple models.
2.2.
Approximation and classical continuum models
When used with caution, classical
continuum models of nanoscale systems can be of
substantial value in design and analysis. They represent
the simplest level in a hierarchy of approximations of
increasing accuracy, complexity, and difficulty.
Experience teaches the value of
approximation in design. A typical design process starts
with the generation and preliminary evaluation of many
options, then selects a few options for further
elaboration and evaluation, and finally settles on a
detailed specification and analysis of a single preferred
design. The first steps entail little commitment to a
particular approach. The ease of exploring and comparing
many qualitatively distinct approaches is at a premium,
and drastic approximations often suffice to screen out
the worst options. Even the final design and analysis
does not require an exact calculation of physical
behavior: approximations and compensating safety margins
suffice. Accordingly, a design process can use different
approximations at different stages, moving toward greater
analytical accuracy and cost.
Approximation is inescapable because
the most accurate physical models are computationally
intractable. In macromechanical design, engineers employ
approximations based on classical mechanics, neglecting
quantum mechanics, the thermal excitation of mechanical
motions, and the molecular structure of matter. Since
macromechanical engineering blends into nanomechanical
engineering with no clear line of demarcation, the
approximations of macromechanical engineering offer a
point of departure for exploring the nanomechanical
realm. In some circumstances, these approximations (with
a few adaptations) provide an adequate basis for the
design and analysis of nanoscale systems. In a broader
range of circumstances, they provide an adequate basis
for exploring design options and for conducting a
preliminary analysis. In a yet broader range of
circumstances, they provide a crude description to which
one can compare more sophisticated approximations.
2.3. Scaling
of classical mechanical systems
Nanomechanical systems are fundamental
to molecular manufacturing and are useful in many of its
products and processes. The widespread use in chemistry
of molecular mechanics approximations together with the
classical equations of motion (Sections 3.3, 4.2.3a)
indicates the utility of describing nanoscale mechanical
systems in terms of classical mechanics. This section
describes scaling laws and magnitudes with the added
approximation of continuous media.
2.3.1.
Basic assumptions
The following discussion considers
mechanical systems, neglecting fields and currents. Like
later sections, it examines how different physical
magnitudes depend on the size of a system (defined by a
length parameter L) if all shape parameters and
material properties (e.g., strengths, moduli, densities,
coefficients of friction) are held constant.
A description of scaling laws must
begin with choices that determine the scaling of
dynamical variables. A natural choice is that of constant
stress. This implies scaleindependent ›elastic
deformation, and hence scaleindependent shape; since it
results in scaleindependent speeds, it also implies
constancy of the spacetime shapes describing the
trajectories of moving parts. Some exemplar calculations
are provided, based on material properties like those of
diamond (Table 9.1): density
=
3.5×10^{3} kg /m ^{3};
°Young's modulus E
= 10^{12} N/m^{2}; and a low working
stress (~ 0.2 times tensile strength)
= 10^{10} N/m^{2}. This choice of
materials often yields large parameter values (for
speeds, accelerations, etc.) relative to those
characteristic of more familiar engineering materials.
