L ∼ 1 m
L ∼ 10 nm
Scale factor ∼ 10–8
Molecular dynamics results
fit expectations from scaling laws
I recently posted my MIT doctoral dissertation, which (like my book on Nanosystems) discusses scaling laws in an introductory section.
The scaling laws inherent in geometry, classical mechanics, and the properties of materials can tell us a lot about what to expect from nanoscale systems. In particular, they show why an advanced mechanical nanotechnology can provide startling capabilities, even without startling new materials or physical principles. For example, with a scale factor of factor of 10–6, a mechanical system that handles a power density of 1 W/cm3 corresponds one that handles 10–12 as much power, but at a power density a million times larger, 1 MW/cm3.
The table below surveys several basic mechanical scaling laws, noting how the properties of geometrically similar systems scale with a characteristic length, L. The mechanical scaling laws in the table do, in fact, hold with good accuracy (for a wide range of materials, not all) down to a scale of 10 nm or less, and in this size range, at room temperature, non-classical effects on mechanical behavior are negligible.
A caveat: These scaling laws for mechanical systems apply in the presence of internal friction (an L2 force), but not in the presence of viscous drag from a surrounding medium, which scales as L1, and with less accuracy. The scaling laws that involve both speed and force are therefore of interest primarily in systems that are far from the biomimetic end of the spectrum of technologies that I’ve discussed in many other posts (for example, here and here).
In the table below, the right-hand column lists typical order-of-magnitude values for these properties, all based on a length scale of 10 nm, and assuming rather ordinary material properties. These choices differ from those in the documents referenced above (and in my article in Physics Education); this of course changes the numerical values, most of which are chosen to be very round numbers.
| Physical quantity | Scaling conditions | Typical magnitude | |
| Scaling as L3: | |||
| Volume | — | 10–24 | m3 |
| Mass | Fixed density | 10–21 | kg |
| Gravitational force | Fixed density | 10–20 | N |
| Scaling as L2: | |||
| Area | — | 10–16 | m2 |
| Applied force | Fixed applied stress | 10–8 | N |
| Dynamical force | Fixed speed, density | 10–15 | N |
| Mechanical power | Fixed speed, stress | 10–9 | W |
| Scaling as L1: | |||
| Length | — | 10–8 | m |
| Stiffness | Fixed modulus | 103 | N/m |
| Elastic displacement | Fixed modulus, stress | 10–11 | m |
| Motion time | Fixed speed | 10–7 | s |
| Gravitational stress | Fixed density | 10–4 | N/m2 |
| Scale-independent: | |||
| Density | (Material property) | 3×103 | kg/m3 |
| Young’s modulus | (Material property) | 1011 | N/m2 | Speed | (Design parameter) | 10–1 | m/s | Applied stress | (Design parameter) | 108 | N/m2 | Dynamical stress | fixed speed, density | 101 | N/m2 |
| Dynamical strain | fixed speed, density, modulus | 10–10 | — |
| Scaling as L –1/2: | |||
| Thermal fluctuation amplitude (r.m.s.) |
Fixed stiffness, temperature (here, 300 K) |
2×10–12 | m |
| Scaling as L –1: | |||
| Acceleration | Fixed speed | 106 | m/s2 |
| Stiffness | Fixed modulus | 103 | N/m |
| Motion frequency | Fixed speed | 107 | Hz |
| Mechanical power density | Fixed speed, stress | 1015 | W/m3 |
{ 8 comments… read them below or add one }
Is this how DNA is able to find a matching base pair so quickly?
This is related to other scaling laws that result in a very short time scale for nanoscale Brownian motion. One can think of the molecules as (randomly) “exploring” their surroundings, with significant motions in nanoseconds, and large motions in microseconds. In this connection, you might like to see “Molecular Machine Assembly: The Movie”. Another relevant post is “Motors, Brownian Motors, and Brownian Mechanosynthesis”.
[Comment removed. Please note the policy: “criticize the idea, not the person.”]
@miguel
What the hell does that even mean? And why are you writing like you’re Macbeth on an internet forum?
[Comment removed. Please note the (selectively applied) policy: “criticize the idea, not the person.”]
Well… nanomachines DO exist but they look <http://www.youtube.com/watch?v=D3fOXt4MrOM&feature=related
oops screwed submit…
Well… nanomachines DO exist but they look nothing like cogs, wheels or motors.
@ Kevembuangga — Yes, it’s important not to overgeneralize. As your comment suggests, biological nanomachines are radically different from scaled-down versions of macroscopic machines, and the motion-related scaling laws above don’t link them in any useful way. I’ve blogged about biological machinery with videos and discussion here, here, and here.
Nanomachines that do resemble scaled-down versions of macroscopic machines can’t be fabricated using today’s laboratory techniques, but they can be simulated with good accuracy using molecular dynamics methods. The results fit the scaling laws quite well (high frequency motions, enormous power densities, etc.).
There is a naive mythology that says that machines of this sort are somehow inconsistent with biology, when they are in fact merely different. Regarding the significance of that difference, it’s important to keep in mind that cars and aircraft don’t closely resemble horses and birds, and outperform them as measured by several metrics of practical importance.