Mechanical engineering meets thermal fluctuations
Thermal fluctuations distort nanoscale structures, and this makes them an enemy of nanotechnologies that rely on precise mechanical motion. Indeed, if one were to set aside design and calculation and instead substitute guesses (I’m not naming guilty parties here), one might suppose that this would prevent nanomechanical engineers from designing reliable machines.
In reality, thermal fluctuations merely create an important design constraint, one that relates four quantities (for clarity, I include SI units):
- The system temperature, T (K),which determines the mean energy of thermal fluctuations: kBT (J)
- The design tolerances, Δxi (m), which set the thresholds at which a fluctuation along coordinate a xi becomes a fault (a flipped bit, a failed device, a product defect, etc.)
- The xi-coordinate stiffness, ki (N/m), which (in the typical linear, classical approximation) determines the elastic energy at the tolerance threshold, kiΔxi2/2 (J)
- The error rate or mean time to failure required to meet system-level requirements
The probability of reaching a particular configuration with elastic energy ΔE is proportional to exp(–ΔE/kBT), and for a linear elastic system, the probability density is a nice multi-dimensional Gaussian, where part of the niceness is that the probability density along any single coordinate (normal or not) is also Gaussian, and proportional simply to exp(–kiΔxi2/2kBT). For most error and failure mechanisms, the rates fall with this exponential.
This highlights why several parameters and design considerations matter so much:
- Stiffness matters because, all else being equal, failure rates decrease exponentially as the stiffness ki increases.
- Modulus of elasticity matters because, all else being equal, the stiffness of a machine is proportional to the modulus of elasticity of the materials that compose it.
- Good mechanical design matters because it is part of the all-else that need not be equal: Soft materials in a rigid configuration can outperform stiff materials in a less rigid configuration.
- And finally, design tolerances matter greatly, because the failure rate decreases exponentially with the tolerance squared.
From soft to rigid machines
Regarding good configurations, I find it interesting that, when biological systems deploy soft materials to achieve atomically precise alignment, they commonly wrap the structure closely around the crucial contact point: Think of reactants oriented and aligned in the active site pocket of an enzyme. A configuration like this can achieve a higher rigidity than would be possible for a long, slim, jointed structure — even if the structure were made made of the stiffest known materials.
At some point along the spectrum of fabrication technologies that stretches from pure self-assembly, to softly guided self-aligning assembly, through stiffly guided assembly, and on high-throughput molecular manufacturing, it becomes of interest to consider nanoscale, atomically precise machines of stiff materials, and how to use machines of this sort to guide the fabrication of machine-parts made of those self-same materials.
Considering thermal fluctuations and failure rates, what is the right figure of merit to use in judging materials for this purpose? Not elastic modulus alone, because design tolerances enter too, and the most important mechanical tolerances in atomically precise fabrication are (with a caveat or two) proportional to the lattice constant (unit cell size) of the material in question; further, the effect on the crucial elastic energies scales, not merely linearly (like the effect of elastic modulus), but as the square of the of lattice constant.
Following up on this, I’ll show how other considerations elevate the quadratic dependency to cubic, and I’ll compare several materials according to the resulting figure of merit. And as readers of this series may guess, despite its outstanding stiffness, diamond doesn’t earn the highest score.
Title updated 10 Feb 09