NANOMACHINERY:
ATOMICALLY PRECISE GEARS AND BEARINGS

K. Eric Drexler

Visiting Scholar, Stanford University
Box 60775, Palo Alto, CA 94306

To appear in the proceedings
IEEE Micro Robots and Teleoperators Workshop
Hyannis, Cape Cod, November, 1987

Microtechnology is based on the small-scale application of bulk materials processing nanotechnology will be based on the construction of objects to complex, atomic specifications. Mechanical scaling laws make small bearings a special concern. Examination of atomically precise beatings of several kinds shows that physical principles not exploited in present macroscopic machinery make possible very low friction. Related analytic methods indicate the performance that can be expected from atomically precise gears. Together, atomically precise gears and bearings will make possible a wide range of nanomachinery.

Nanotechnology and microtechnology

The ability to build objects to complex, atomic specifications will open a new realm of electronic and mechanical technologies.

This paper follows the convention of using nanotechnology to describe technology based on this ability, while using microtechnology to describe technology based on the micron and sub-micron scale manipulation of conventional materials These technologies, though related, will rely on different fabrication techniques and will raise qualitatively different issues in implementation and application [1,2]. Figure 1 illustrates the nanotechnologist's view of a cubic nanometer.

Microtechnology characteristically uses light or particle beams to define patterns on a surface, then uses techniques such as etching, sputtering, and film deposition to shape materials. On an atomic level, these operations are dominated by the statistical processes of thermodynamics and kinetics. They do not enable atomically precise control of structures–or, more accurately, they enable choice only among the structures favored by these statistical processes.

Nanotechnology [1,2] will be based on the use of molecular machines–assemblers-able to position reactive molecules to tenth-nanometer tolerances, directing synthetic reactions in a site-specific way much as enzymes and ribosomes do in biochemistry. These operations (while subject to the laws of statistical mechanics) can be made to proceed in a deterministic, programmed fashion with low error rates [3]. Assemblers will yield atomically-precise control of structures, limited chiefly by the constraint of chemically reasonable bonding.

Several paths lead toward this capability. First-generation assemblers may be developed through protein engineering: biochemical analogies indicate that protein engineering (when sufficiently advanced) will enable the design and fabrication of complex, self-assembling molecular machines [1] (relevant chemistry is discussed in [4,5]). Likewise, first-generation assemblers may be developed through the synthesis of self-assembling sets of non-protein molecules (relevant chemistry is discussed in [6,7]). Alternatively, advances in micromanipulation may enable the construction of first-generation assemblers through mechanically-directed molecular assembly; reports of atomic rearrangement through field evaporation from scanning tunnelling microscope (STM) tips are suggestive in this regard [8]. Actual development may involve a combination of chemical, biochemical, and micromechanical techniques. However assemblers may first be built, later assemblers will be built using assemblers. The nature of nanotechnology and its capabilities will then be independent of the nature of proteins, conventional chemistry, or STM technology.

Figure 1. A block of diamond with a volume slightly less than one cubic nanometer (block surface planes fall midway between atomic planes). Contains 160 atoms.

Nanotechnology and nanomachinery

If molecular assemblers will enable the construction of most chemically stable structures, it makes sense to ask what sorts of machinery can be built subject to chemical constraints. Sizes of assembler-built parts will range down to a fraction of a nanometer; the shapes of small pans will be heavily constrained by the available sizes and bonding patterns of atoms.

Mechanical properties, even of molecular-scale pans, can be divided into internal properties (including measures of strength, modulus, and density) and surface properties (describing inter-actions with other parts). Assemblers will enable construction of carbon-based parts having a diamond or diamond-like structure, with a bulk density of about 3,500 kg/m³, a strength of about 5×10¹⁰ N/m², and a modulus of about 10¹² N/m²[9]. They will also enable construction of elastically-tailored parts with fine-grained patterns of inhomogeneity and anisotropy. Where surfaces are concerned, assemblers will make possible tailored frictional properties. Two major goals are minimizing friction (in bearings) and minimizing slippage (in gears); these issues are the focus of the bulk of this paper.

Mechanical scaling laws and internal friction

Consider scaling a machine by a length parameter L while holding shapes, materials, and stresses constant. Speeds of sound and motion, set by modulus-to-density and stress-to-density ratios, are scale-independent constants. Frequencies of motion, accordingly, scale as L¹. Areas scale as L²: with stresses held constant, forces scale with areas; with materials and coefficients of friction held constant, friction-forces scale with other forces. Since speeds of linear motion are a constant, transmitted and dissipated power likewise scale as L².

A constant coefficient of friction–the standard approximation for dry, sliding surfaces–is, however, a poor description of most real machines. To reduce friction and wear, machines typically use a liquid lubricant. In his 1959 talk on miniaturization (which sketched a path to the threshold of nanotechnology), Feynman observed that "Lubrication involves some interesting points. The effective viscosity of oil would be higher and higher in proportion as we went down (and if we increase the speed (i.e., frequency of motion] as much as we can). If we don’t increase the speed so much, and change from oil to kerosene or some other fluid, the problem is not so bad. But actually we may not have to lubricate at all! We have a lot of extra force. Let the bearings run dry; they won’t run hot because the heat escapes away from such a small device very, very rapidly" [10]

Given newtonian liquids, shear rates and hence shear stresses would indeed scale adversely, as L^-1, while the temperature rise in a dry bearing scales favorably, as L.

Still, bearings remain a problem. Since driving forces and classical frictional forces are proportional, there is no extra force in smaller devices to compensate for the greater friction of a dry bearing. Further, if wear rate (thickness eroded per unit time) is constant at constant stress and speed, and if tolerance for wear is proportional to L, then machine lifetime suffers with shrinkage: in scaling parts from centimeters to nanometers, lifetimes scale from years to seconds.

In moving from microtechnology to nanotechnology, however, new approaches make sense. From the perspective of a typical nanomachine, a kerosene molecule is an object, not a lubricant. Atomically precise structures can yield more orderly motion than that of molecules in a liquid film, producing low friction and avoiding wear.

Atomically precise bearings

Atomic friction

An ideal bearing would have zero static and sliding friction in one or more degrees of freedom and infinite rigidity in all others. Intuition suggests that bearings built on a nanometer scale will experience considerable static and sliding friction as a result of atomic scale bumpiness. This bumpiness is largely described by a van der Waals interaction potential, here taken as

(from MM2 )

where

E_ij = energy of van derWaals interaction between atoms i and j (ij) (joules)
e_i = MM2 van der Waals energy parameter for atom i, times 6.95 ×10^-21
r* = MM2 van der Waals radius of atom i
r_ij = distance between atoms i and j (meters)
A = 2.90×10⁵
B = 2.25

The repulsive, exponential term varies sharply with interatomic distances, hence it contributes strongly both to the bumpiness of interaction potentials between sliding surfaces and to the derivative of force with respect to inter-surface separation. in short, it affects both friction and rigidity. In practical bearings, the attractive term and electrostatic forces are less significant

The maximum derivative of potential energy with respect to position (along a sliding-motion coordinate) can be identified with the static friction of the system. Dynamic friction forces, stemming from phonon radiation and scattering, are harder to estimate; the following will focus on static friction.

Roller bearings

One might deal with bumpiness by using roller beatings and meshing their bumps gear-fashion with bumps on their races (Figure 2). An attractive implementation would use hollow rollers (which deform to reduce contact-point pressure, increasing total load-bearing capacity), perhaps with a structure like that of a thin slab of diamond bent to close on itself. Surfaces of bearings and races might be fluorinated (1 1 1) planes. Rows of atoms could be set at an angle (as in Figure 2); this would smooth motion, as does the helicity in a helical rack-and-pinion gear. Multiple rollers between two parallel races make a linear bearing; multiple rollers between two concentric races make a rotary bearing.

For rollers of a few nanometers diameter, potential energy vs. displacement will be virtually flat, yielding low static friction.

Roller bearings suffer from disadvantages of bulk (they interpose substantial structures between moving surfaces) and comparatively low stiffness and load-bearing capacity (they concentrate compressive forces); they may have advantages in dynamic friction.

Figure 2: Schematic of roller bearing on a race. Lines represent rows of surface atoms, sheared into a helical configuration in the roller structure. Rows of atoms mesh like gear teeth.

Van der Waals bearings

Another approach to the atomic bumpiness problem is to make bumps sum to a smooth potential. The Peierls-Nabarro stress (for motion of a straight dislocation, allowing one crystal plane to slide over another) is proportional to exp(-W), where W is the width of the dislocation zone [9]. This is an instance of the general principle that, where physical symmetry yields similar distributions of atom-pairs on the increasing and decreasing sides of their interaction-potential curves, force cancellation can be nearly exact.

Journal bearings: This principle can apply to dry journal bearings with small numbers of atoms To illustrate this, calculations were performed on model systems that incorporate van der Waals interactions but omit bonding constraints on system geometry (to include them would prevent free choice of an illustrative series of model parameters). In these models, atoms that would interact at various points along the length of two concentric cylinders are projected onto a plane, forming two concentric rings (see Figure 4; Figure 3 illustrates a more rigorously correct use of projection). A further idealization treats all atomic coordinates as fixed with respect to their respective rings. AU atoms are taken as fluorine atoms, with van der Waals parameters (r = 0.165 nanometers; e = 0.078) from the MM2 model [12]. The calculations presented here assume an inner ring of six atoms (with a radius chosen to represent that of a column from the lattice of hexagonal diamond, with dangling bonds fluorinated). The outer-ring radius is chosen to ensure strongly repulsive interactions, and its number of atoms is taken as a free para meter. Similar results follow from a wide range ring of radii and atom types and numbers; relaxing the rigidity assumption would soften effective interaction potentials, lowering both the calculated friction and rigidity.

Figure 3: Atom moving past two rows of atoms. Its van der Waals potential as a function of position is unchanged if the two rows are projected onto a single line, with atoms at half their physical spacing.

Figure 4: An idealized van der Waals bearing (bonding constraints on geometry neglected, atoms projected to a single plane), shown in two rotational configurations. Figure 5 shows the results of these model calculations for outer rings with even atom-numbers between six and thirty-two. For a centered, unloaded rotor, the force (applied tangentially to the inner ring) required to overcome static friction falls along the lower line, rapidly reaching values in the attonewton range and below in a typical nanomachine, asymmetries will stop this fall at more moderate values. The potential is smooth despite an inter-ring contact force chosen to be unrealistically high, on the order of 10 nanonewtons per inner-ring atom. In this set of configurations, the bearing stiffness–resistance to lateral displacement–varies from about 500 to over 1000 newtons/meter (in the rigid-ring approximation).

Figure 5: Static friction of six-member ring against outer rings of varying atom number. Inter-ring gap is 0.18 nanometers, inner-ring radius is 028 nanometers. Peaks on the lower line occur when outer-ring numbers are divisible by six. The periodic peaks in static friction occur when the number of atoms in the outer ring is divisible by that in the inner ring; ideally, these numbers should lack common factors (or be divisible only by two, thus introducing a symmetry plane and centering the inner structure). A common factor of N means that N segments of the inner and outer rings match in the same way, with additive forces.

The upper curve in Figure 5 shows friction forces for a loaded bearing in which a lateral displacement (of 0.05 nanometers) yields lateral forces between about 20 and 80 nanonewtons. These are unrealistically high loads, providing a stringent test of bearing smoothness. Loading reduces the symmetry of the system, disrupting force cancellation; the fall of frictional forces proceeds more slowly, dipping to about 15 piconewtons at an outer-ring number of 28. Taking the ratio of the friction force to the lateral shaft load for this configuration as a measure of the coefficient of static friction yields a value of 0.0002, about 200 times less than that between Teflon surfaces.

The force-cancellation responsible for these low values of friction depends chiefly on the rotational symmetry of the shaft and bearing; this may be achieved in a variety of ways. For small bearings, the choices (particularly for shaft structure) will be limited by the small number of atoms and permissible arrangements. Shaft structures could be based on roughly cylindrical segments from crystal lattices or on stacks of cage compounds (like dodecahedrane); Figure 6 shows one possibility. The surrounding journal box might be based on bent slabs from a crystal lattice. (Note that this structural description implies nothing about fabrication strategies.) Larger bearings–with radii of over a nanometer–could be based on concentric bent slabs; these could have large atom numbers, good symmetry, good force cancellation, and very low static friction even under substantial lateral loads.

Figure 6: Carbon backbone of a possible shaft structure (hidden lines removed).

Screw bearings: Cylinders with a suitable helical twist and interface topography will act as screws. Neglecting end conditions, the above results apply almost directly; indeed, opening rings into a helix increases the atom number, enabling excellent force cancellation. Devices in this class can convert rotary to linear motion with low internal friction (at least at low speeds).

Linear bearings: Generalizing from the helical case to a straight line gives a linear sliding bearing. Again (neglecting edge conditions) avoidance of meshing patterns of bumps can yield excel-lent force cancellation and low static friction. Observations with an atomic force microscope give evidence of a crude example of this effect, with a measured static friction coefficient of about 0.004 [13]. Atomically precise structures tailored to account for edge conditions should yield coefficients orders of magnitude lower. One can take the centered, cylindrical surfaces in the 6-on-28 bearing mentioned above as a model for parallel surfaces; the ratio of the sliding resistance to the contact force (a measure of the static friction) is about 4×10^-16 in this system.

Single-atom "shafts": At the other extreme of the spectrum lies the single-atom van der Waals bearing, in which the inner "ring" achieves complete cylindrical symmetry by being a single atom. One implementation of this would use a large, monovalent atom (such as chlorine, bromine, or iodine) as the "shaft" structure, using a substantial axial load to seat this atom in a hollow on a facing surface. In the approximation used here, this bearing would have zero static friction; in practice, the interactions of the facing surfaces would introduce some bumpiness into the potential energy function. The strength and stiffness of such a bearing would be fairly low.

Sigma and triple-bond bearings

Sigma and triple-bond bearings carry loads not through van der Waals repulsion, but through the forces associated with covalent bonds. Their compactness–sigma-bond bearings contain no atoms at all–makes them desirable for the implementation of nanomechanisms of minimal size.

A sigma bond between two objects can serve as a bearing with a working strength of several nanonewtons. The bumpiness of the potential energy function depends on the structures on either side of the bonds. It can be low: the barrier for rotation of a methyl group with respect to a phenyl group is less than 10^-22 J [14] (kT at room temperature is over 40 times greater).

Greater separation of structures (and hence lower static friction) may be achieved by linking them, not with a single sigma bond, but with an interposed triple-bonded pair. Figure 7 illustrates a small molecular rotor supported on a pair of such bearings. If these bearings are placed under an axial tension of several nanonewtons, the supported rotor will have a lateral displacement stiffness of about 20 newtons/meter.

Figure 7: Small rotor supported by a pair of triple-bond bearings.

Atomically precise gears and chains

In designing van der Waals bearings, one avoids systematic meshing of atomic bumps; in designing gears, the goal is the opposite. For gears, atomic spacings on facing surfaces must roughly match and must be wide enough that bumps on one side fit deeply into hollows on the other. In conventional gears, teeth are given special shapes (such as cycloids and involutes) to arrange for proper contact throughout an arc of rotation. The soft, smooth nature of atoms, however, allows them to act as adequate teeth despite their simple shapes.

Both spur and rack-and-pinion gears are straightforward; helical or staggered teeth will increase smoothness of motion. Combining a gear with a screw structure of the sort described above will yield a worm gear (low static friction would depend on having a large enough contact area to arrange for adequate bump-cancellation).

Nanometer-scale bevel gears cannot be implemented by scaling down conventional gears: teeth implemented as rows of atoms cannot shrink uniformly toward the apex of the cone. All that is required, however, is that bumps and hollows mesh, and this does not require teeth with conventional shapes which evolved, in part, to suit machining technology. Figure 8 illustrates a pair of complementary surfaces (based on the diamond (1 1 1) surface) which could, with modest elastic distortion, be rolled into a pair of cones of 45° half-angle. An approach of this sort seems suitable for implementing bevel gears. (Here, as elsewhere, the notion of rolling a surface describes a structure, not a fabrication technique. Further, it does not specify the internal structure of the part, which may have lower symmetry, increasing the static friction.)

In general, the strength and stiffness of gears with respect to shear stresses in the interface will depend on the contact forces and on the number of teeth engaging (which will in turn depend on the gear radii and widths). Strength can be on the order of a nanonewton per engaged tooth, and stiffness can be on the order of 10 newtons/meter per engaged tooth (for monovalent-atom teeth, limited chiefly by bond-bending).

Figure 8. Bevel gear surfaces (unrolled from cones). Circles represent sizes where one would find dangling bonds in diamond cleaved on a (111 plane with no surface rearrangement. Black circles represent sites where a dangling bond is avoided by adding a fluorine atom; white circles represent sites where a dangling bond is avoided by substituting a nitrogen for a carbon atom. Thus, black circles represent bumps; white, hollows. The two surfaces have complementary bumps and hollows.

Chain drives

Molecular systems also lend themselves to chain drives, in which knobby, flexible polymer chains wrap around sprockets with hollows spaced to receive the knobs. This produces a large contact zone, typically at low pressure. The stiffness of coupling from sprocket to sprocket is potentially fairly high, and may typically be limited by the stretching stiffness of the chain.

Harmonic drives

Harmonic drives (Figure 9) can readily be adapted to nanomachinery they offer an attractive way to achieve large gear ratios in a stiff, low-static-friction system. The inner, elliptical ring has two fewer teeth than the outer, circular ring, allowing engagement in two opposite zones. The inner ring is deformed into this shape by an elliptical core which rotates inside it. The resulting waves of meshing and unmeshing teeth (resembling a pair of mechanically-driven dislocations) turn the inner ring by two teeth per rotation of the core. If each ring is a nanometer or so deep, and the ring diameter is ten nanometers, a reasonable gear ratio is 50:1, and the gear’s overall stiffness can be several thousand newtons per meter. The sliding interface of the elliptical core can have the frictional properties of a good van der Waals journal bearing.

Figure 9: Harmonic drive with atomic teeth; deformed ring creeps counter-clockwise as elliptical core turns.

Conclusion

The ability to build structures to complex, atomic specifications will make possible a wide variety of bearings and gears having low static friction; good estimates of dynamic friction remain to be made. The availability of these components will facilitate the construction of a wide range of power-driven machinery in the nanometer size range, including small-scale robots and teleoperators. Other investigations of nanomachinery indicate the feasibility of mechanical logic elements suited to the construction of computer CPUs with volumes on the order of 0.001 cubic micron [2,3]. Systems based on a combination of sub-micron computers and manipulators using nanometer-scale tools will have a wide range of applications, including the characterization and manipulation of cells and tissues on a molecular scale [1,2].

Although nanotechnology has obvious affinities to microtechnology, the issues addressed in this paper highlight its differences.

Moving from micron (and submicron) parts to nanometer (and sub-nanometer) parts will entail more than just a quantitative change of a billion in volume; it will entail a concern with the atom-by-atom design and construction of complex devices. Such devices cannot be built by lithographic technology Molecular assemblers will be needed to build advanced nanomachines, and assemblers will them-selves be important instances of nanomachines Assemblers and nanotechnology may be developed through continued progress in miniaturizing machines, but they also may be developed through continued progress in building large molecules. Once developed, they may be of use in construction on a wide range of scales [b">15].

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NANOMACHINERY: ATOMICALLY PRECISE GEARS AND BEARINGS