Thermal noise

While we do not present an explicit analysis of the effects of thermal noise, some general observations seem in order. Classical positional uncertainty is described by the equation (Drexler, 1992):

(1) sigma² = kT/k_s

where:

• sigma is the mean positional uncertainty (meters, m)
• k is Boltzmann's constant (Joules/Kelvin, J/K)
• T is the absolute temperature (Kelvins, K)
• k_s is the stiffness (Newtons/meter, N/m)

Positional uncertainty can be reduced by increasing stiffness and lowering temperature. Obviously, we can reduce temperature if that should prove necessary. We can also increase stiffness: the stiffness of a structure scales favorably with size. If we have two positional devices (e.g., two robotic arms or two Stewart platforms) of otherwise similar design, the larger device will be stiffer. If the larger device is twice the size, and if all its components are scaled to be twice as large in all dimensions, then it will have twice the stiffness of the smaller device.

Once we know how much positional uncertainty we can tolerate -- and for specific molecular tools that are used to cause specific reactions to occur at specific sites, we can compute the maximum positional uncertainty that can be tolerated before something goes wrong -- then we can design a positional device with the required stiffness. This can be done either by scaling the device size, or by specifying the operating temperature. If we specify room temperature operation then we find that the device size is largely dictated to us. (Drexler, 1992) analyzed this problem and concluded that a robotic arm of about 100 nm (nanometers) in height and 30 nm in diameter and made of diamondoid materials would have a positional uncertainty at room temperature that was a modest fraction of an atomic diameter. (Merkle, 1997b) reached substantially the same conclusions. Stiff hydrocarbons should suffice to make both these and many other positional devices.

To provide a numerical example: if a positional device has a stiffness k_s of 10 N/m, then at room temperature (kT ~ 4 × 10^-21 J) equation (1) implies a positional uncertainty sigma of 0.02 nm (0.2 Å). The gaussian fall off implies that positional errors of even a few sigma are of very low probability. A properly designed diamondoid positional device should easily be able to achieve a stiffness much higher than 10 N/m. For comparison, the carbon-carbon bond has a stretching stiffness of about 440 N/m.

Put another way, the energy of a system is:

(2) E = 1/2 k_s x²

For our example k_s of 10 N/m, a 0.154 nm (1.54 Å) deviation (about the length of a carbon-carbon bond) increases the energy of the system by 1.18 × 10^-19 J (17 kcal/mol). Such a positional device is very unlikely to make an error as large as a bond length at room temperature.

The molecular tool itself cannot be scaled. A specific molecular tool operated at a particular temperature will have an upper bound on reliability that cannot be improved regardless of how stiff we make the supporting robotic arm. While the molecular tool cannot be scaled, it can be redesigned. The unmodified hydrogen abstraction tool has an estimated lateral stiffness at the carbon atom at its tip of about 6 N/m (Drexler, 1992, figure 8.2). (This tool is long and thin, which is adverse for stiffness). Drexler suggested buttressing the relatively flexible base of the hydrogen abstraction tool, and proposed one redesign which had an estimated stiffness of 65 N/m.

Ultimately, the room temperature reliability of molecular tools suitable for the synthesis of hydrocarbons depends on the achievable stiffness. Existing work suggests that reliable operation at room temperature can be achieved.