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Physical scaling laws enable small machines to be highly productive

Molecular manufacturing will use small machines to assemble small parts, starting at the molecular level and working up. Is this a good idea, or does it have disadvantages, such as inefficiency or low productivity? To examine this question, let’s consider what elementary physical scaling laws (Nanosystems, Chapter 2) say about productivity and efficiency.

   (Note: “zetta” = 1021.)

The goal is to show how various physical parameters vary with scale. As a reference point, the first column (above) describes a hypothetical meter-scale assembly mechanism with round-number parameters: It is a meter-scale, 100-kg device with a single arm that executes one cycle per second, placing a 1 kg part, operating with a friction loss of 100 W. The subsequent columns describe arrays of similar mechanisms, holding total volume constant while varying the size and speed of the individual units. (Please note that these arrays are not complete manufacturing systems, which must include mechanisms for materials transport, power supply, cooling, control, and so forth.)

What are the key scaling laws? Consider the second column, showing 8 mechanisms, each of 1/2 meter size. The overall density of parts is the same, hence the mass is unchanged. Holding material stresses constant (and likewise the ratio of motion frequencies to resonant frequencies), the motion speed (in m/s) is unchanged. Since each motion covers only 1/2 the distance, however, the motion frequency doubles to 2 cycles per second, giving the 8 arms a total assembly rate of 16 parts per second. Each part is 1/8 as large, however, so the throughput is only doubled, to 2 kg/s.

In atomically precise bearings, friction forces (due chiefly to phonon effects) scale like viscous drag: the force per unit area is proportional to speed, and the frictional power per unit area is proportional to speed squared. The 1/2 size mechanisms in the second column operate with unchanged speed but doubled area, hence friction losses double to 200 W.

Another friction loss results from conversion of kinetic energy to heat when arms (and parts) brake to a halt. This is well under 100 W in the meter-scale system. Like bearing losses, braking losses double in the 1/2 size mechanism, but for a different reason — doubling the frequency doubles the number of braking cycles per second. Holding size constant, braking losses scale as the cube of the speed: a square factor from the kinetic energy times a linear factor from the frequency of motion.

The third column (in warning-red) shows the extraordinary results of scaling the mechanisms to 10–7 m, holding speed constant: a throughput of 107 kg and 1028 parts per second. It also shows a fatal flaw common in naïve schemes that fill macroscopic volumes with high-speed nanomachines — the large bearing area results in unacceptably high power dissipation: 1 GW.

A standard engineering response to excessive power dissipation is to reduce speed. The fourth column (in OK-green) shows the effect of dropping the speed of motion (and hence the frequency) by a factor of 10–3. Since bearing friction losses scale as the square of the speed (and braking friction as the cube), a 10–3 reduction in speed cuts power dissipation by at least 10–6, to a modest 1 kW. The throughput is now (merely!) 104 kg and 1025 parts per second. For perspective, total world-wide semiconductor production only recently passed 1018 transistors per year.

Physical scaling laws thus give a clear answer to the question of whether using small machines to assemble small parts makes sense from the perspectives of efficiency and productivity: Efficiency can be good, and productivity can be enormous, whether measured by mass or by numbers of parts. To understand what these simple physical scaling results imply for actual production, however, requires an analysis of complete molecular manufacturing systems.

A similar analysis and illustration appears in:

Drexler, K.E. (1995) “Molecular manufacturing: perspectives on the ultimate limits of fabrication” Phil. Trans. R. Soc. London A 353:323-331.