Nanostructures, Nanomaterials,
and Lattice-Scaled Stiffness

by Eric Drexler on 2009/02/15

Toward Advanced Nanotechnology: Nanomaterials (4)

A schematic illustration of a lattice-quantized structure.
The peg aligns with the hole
if the hole is large enough,
and the fluctuations
are small enough.

In a nanofabrication technology that uses nanomachines to assemble products, the stiffness of the machines is important because it limits the amplitude of thermal fluctuations, yet tolerance for fluctuations is important too. When both nanomachines and their products are made of similar materials, increasing the fluctuation-tolerance by using larger building blocks has strong advantages, and the most attractive materials are no longer the stiffest. Let’s consider the scaling laws and derive a figure of merit before discussing some under-appreciated materials that win high scores.

For newcomers, here’s the story to date:

  1. Modular Molecular Composite Nanosystems
       Biomolecular engineering for atomically precise nanosystems
  2. Toward Advanced Nanotechnology: Nanomaterials (1)
       Why I’ve never advocated starting with diamond
  3. Toward Advanced Nanotechnology: Nanomaterials (2)
       Stiffness matters (and protein isn’t remotely like meat)
  4. Self-Assembly for Nanotechnology
       The virtues of self-assembly and the benefits of external guidance
  5. From Self-Assembly to Mechanosynthesis
       Mechanosynthesis begins with soft machines
  6. Toward Advanced Nanotechnology: Nanomaterials (3)
       Mechanical engineering meets thermal fluctuations

Scale, stiffness, errors, and lattice constants

In my previous post, I discussed four design parameters that determine whether thermal fluctuations will fall within design constraints, describing the exponential role of elastic energy in suppressing fluctuation-induced errors, the linear contribution of machine stiffness to elastic energy, and the quadratic contribution of error tolerance. I also noted that the error tolerance will tend to scale with a, the lattice constant of the material of the product (I use a here as a generic measure of the unit cell size). But what if the material of the product is the same as the material of the machine? There are two key observations here:

Machine stiffness scales with the machine size, L: Stiffness increases with cross-section and decreases with length, and therefore scales as L2/L = L.

Machine size scales with the lattice constant, a: All else being equal, the number of building blocks required to make a structure will scale with its complexity, and the size of a nanomachine of a given complexity will be proportional to the lattice constant of its structural material.

In designing a nanomachine in which a major constraint is tolerance for thermal fluctuations, a useful figure of merit is the elastic energy of the machine structure when at its maximum acceptable displacement. In designing a machine for assembling products made of a material similar to its own, the above considerations yield a related figure of merit for materials, Klm, the “lattice-scaled modulus”, in which the lattice constant appears as a cubic factor:

Klm = Ea3r2err .

This expression introduces rerr, the ratio of the error-tolerance displacement to a, which must be included because error-inducing displacements are in some instances less than the full unit cell size. As a measure of material stiffness, I will be using Young’s modulus, E, (for reasons outlined in Elastic Modulii, Stiffness, and Thermal Fluctuations). Klm has the dimensions of energy; typical values are on the order of attojoules.

[Update, 19 Feb: To clearly separate material properties from operating conditions, I’ve recast the above discussion, defining Klm rather than the previous dimensionless number Kls.]

Since lattice constants tend to scale with atom diameters, the Klm figure of merit highlights the utility of elements from lower rows on the periodic table. In addition, there can be advantages to increasing the number of atoms in a unit cell; this favors compounds over elements, and the use of compounds brings advantages of a different sort because materials chemistry often becomes more tractable as the character of bonds shifts from non-polar and covalent toward more polar or ionic.

Finally, considering the progress and promise of atomically precise fabrication using biomolecular structures, there is great appeal to using materials that can form in the presence of water.

Next in this series: High-Klm materials that you can make in a water-filled beaker.

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