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 atomic-scale structure, mean-free-path 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 ) 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 scale-independent elastic deformation, and hence scale-independent shape; since it results in scale-independent speeds, it also implies constancy of the space-time shapes describing the trajectories of moving parts. Some exemplar calculations are provided, based on material properties like those of diamond (Table 9.1): density ; Young's modulus ; and a low working stress ( times tensile strength) . This choice of materials often yields large parameter values (for speeds, accelerations, etc.) relative to those characteristic of more familiar engineering materials.
2.3.2. Magnitudes and scaling
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 equals , or . Stiffness in shear, like stretching stiffness, depends on both area and length
and varies less rapidly with scale; a cubic nanometer block of has a stretching stiffness of . The bending stiffness of a rod scales in the same way
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 equals .
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 . Accelerations in nanomechanisms commonly are large by macroscopic standards, but aside from special cases (such as transient acceleration during impact and steady acceleration in a small flywheel) they rarely approach the value just calculated. (Terrestrial gravitational accelerations and stresses usually have negligible effects on nanomechanisms.)
Modulus and density determine the acoustic speed, a scale-independent parameter [along a slim rod, the speed is ; in bulk material, somewhat higher]. The vibrational frequencies of a mechanical system are proportional to the acoustic transit time
The acoustic speed in diamond is . Some vibrational modes are more conveniently described in terms of lumped parameters of stiffness and mass,
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: .
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 (with the assumed and ). Most mechanical motions considered in this volume, however, have speeds between 0.001 and .
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 distance takes ; the large speed makes this shorter than the motion times anticipated in typical nanomechanisms. A modest speed, however, still yields a transit time of only , indicating that nanomechanisms can operate at frequencies typical of modern micron-scale electronic devices.
The above expressions yield relationships for the scaling of mechanical power
and mechanical power density
A force and a volume yield a power of and a power density of (at a speed of ) or and (at a speed of ). The combination of strong materials and small devices promises mechanical systems of extraordinarily high power density, even at low speeds (an example of a mechanical power density is the power transmitted by a gear divided by its volume).
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:
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 across a 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 film of low-viscosity fluid , the viscous shear stress at a speed of is ; the shear stress at a speed of , is still large, dissipating energy at a rate of .
The problems of liquid lubrication motivate consideration of dry bearings (as suggested by Feynman, 1961). Assuming a constant coefficient of friction,
and both stresses and speeds are once again scale-independent. The frictional power,
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,
and a centimeter-scale part having a ten-year lifetime would be expected to have a 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 , although such results are meaningless because a single atom of iron is over 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
where the thermal energy measures the characteristic energy in a single degree of freedom, not in the object as a whole. For , the mean thermal speed of a cubic nanometer object at is . 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 ( , the mean energy of a thermally excited harmonic oscillator at a temperature , if ). The ratio varies directly with the frequency of vibration, that is, as . An object of cubic nanometer size with has . The associated quantum mechanical effects (e.g., on positional uncertainty) are smaller than the classical thermal effects, but still significant (see Figure 5.2).
2.4. Scaling of classical electromagnetic systems
2.4.1. Basic assumptions
In considering the scaling of electromagnetic systems, it is convenient to assume that electrostatic field strengths (and hence electrostatic stresses) are independent of scale. With this assumption, the above constant-stress, constant-speed scaling laws for mechanical systems continue to hold for electromechanical systems, so long as magnetic forces are neglected. The onset of strong field-emission currents from conductors limits the electrostatic field strength permissible at the negative electrode of a nanoscale system; values of can readily be tolerated (Section 11.6.2).
2.4.2. Major corrections
Chapter 11 describes several nanometer scale electromechanical systems, requiring consideration of the electrical conductivity of fine wires and of insulating layers thin enough to make tunneling a significant mechanism of electron transport. These phenomena are sometimes (within an expanding range of conditions) understood well enough to permit design calculations.
Corrections to classical continuum models are more important in electromagnetic systems than in mechanical systems: quantum effects, for example, become dominant and at small scales can render classical continuum models useless even as crude approximations. Electromagnetic systems on a nanometer scale commonly have extremely high frequencies, yielding large values of . Molecules undergoing electronic transitions typically absorb and emit light in the visible to ultraviolet range, rather than the infrared range characteristic of thermal excitation at room temperature. The mass of an electron is less than that of the lightest atom, hence for comparable confining energy barriers, electron wave functions are more diffuse and permit longer-range tunneling. At high frequencies, the inertial effects of electron mass become significant, but these are neglected in the usual macroscopic expressions for electrical circuits. Accordingly, many of the following classical continuum scaling relationships fail in nanoscale systems. The assumption of scale-independent electrostatic field strengths itself fails in the opposite direction, when scaling up from the nanoscale to the macroscale: the resulting large voltages introduce additional modes of electrical breakdown. In small structures, the discrete size of the electronic charge unit, , disrupts the smooth scaling of classical electrostatic relationships (Section 11.7.2c).
2.4.3. Magnitudes and scaling: steady-state systems
Given a scale-invariant electrostatic field strength,
At a field strength of , a one nanometer distance yields a potential difference. A scale-invariant field strength implies a force proportional to area,
and a field between two charged surfaces yields an electrostatic force of .
Assuming a constant resistivity,
and a cubic nanometer block with the resistivity of copper would have a resistance of . This yields an expression for the scaling of currents,
which leaves current density constant. In present microelectronics work, current densities in aluminum interconnections are limited to or less by electromigration, which redistributes metal atoms and eventually interrupts circuit continuity (Mead and Conway, 1980). This current density equals (as discussed in Section 11.6.1b, however, present electromigration limits are unlikely to apply to well-designed eutactic conductors).
For field emission into free space, current density depends on surface properties and the electrostatic field intensity, hence
and field emission currents scale with ohmic currents. Where surfaces are close enough together for tunneling to occur from conductor to conductor, rather than from conductor to free space, this scaling relationship breaks down.
With constant field strength, electrostatic energy scales with volume:
A field with a strength of has an energy density of per cubic nanometer .
Scaling of capacitance follows from the above,
and is independent of assumptions regarding field strength. The calculated capacitance per square nanometer of a vacuum capacitor with parallel plates separated by is ; note, however, that electron tunneling causes substantial conduction through an insulating layer this thin.
In electromechanical systems dominated by electrostatic forces,
and