Scaling law

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With scaling laws (over size-scales) one can find out:

  • which physical effects get stronger when machine size is shrunken down.
  • which physical effects get weaker when machine size is shrunken down.

Based on whether a particular physical effect gets stronger or weaker one can decide on whether to use it or not.
Or rather: With scaling laws one can take a first educated guess for which effect are best to use on which size scales.

In the context of atomically precise manufacturing the analysis of scaling laws (and further exploratory engineering) leads to the insight that:
Compared to normal macroscale machinery macroscale style machinery at the nanoscale (featuring gemstone-like molecular elements like diamondoid crystolecular machine elements) will perform way better rather than way worse or even not at all. (When all the most relevant effects are taken into account).

This insight was/is for some people (including prominent nanotechnology experts) surprising since

  • natures nanomachinery (molecular biology) and
  • current day limitations in experimental research

both sometimes deceivingly seem to suggest the opposite.
(See: Nature does it differently and Effects of current day experimental research limitations)

Because of scaling laws (over size-scales) being so fundamentally important
for a basic first understanding of the physics down at the nanoscale
scaling laws are presented right at the beginning of the book Nanosystems
(Page 23 Chapter 2 "Classical Magnitudes and Scaling Laws").

Scaling laws can provide ...

  • ... quantitative numbers (sometimes only crude first approximations but still very valuable) and
  • ... some kind of intuitive feel of how the world down there behaves.

As a specific example take electric motors and generators.
Here scaling laws tell us that when going down to the nanoscale it's better to use ...

  • electrostatic motors (which become stronger when shrunken in size) instead of
  • magnetic motors (which get weaker when shrunken down in size).

(For more details see: Electromechanical converter) .

Note: This page is about scaling laws over size-scales.
For scaling laws over other quanities while keeping the size scale fixed see:
See: non size-scale scaling law – and: Scaling law (disambiguation)

Scaling laws involving surface and volume

The most commonly known scaling law.
When halving the size surface area divides by four and volume by eight thus the surface to volume ratio doubles. That is it shrinks linearly with rising size.

Main article: Higher total bearing surface area of smaller machinery

instant heat transfer

The inverse volume to surface ratio shrinks linearly with size. With it the characteristic time of heat transfer shrinks too. This makes it effectively impossible to thermally isolate single nanoscale parts. Thus thermal isolation is best used only between macroscopically separated Volumes.

Scaling laws for mechanical quantities

From Nanosystems:

(2.1) [math] total mechanical strength \propto mechanical force \propto area \propto L^{2} [/math]
=> Even huge macroscale pressues equate to small nanoscale forces (ore vice-versa). – E.g. 1010N/m2 = 10-8N/nm2= 10nN/nm2

(2.2) [math] shear stiffness \propto stretching stiffness \propto area / length \propto L^{1} [/math]
(2.3) [math] bending stiffness \propto radius^4 / length^3 \propto L^{1} [/math]
(2.4) [math] deformation strain \propto force / stiffness \propto L^{1} [/math]

Stiffnesses and strain do not scale as severe as the forces but still significant.
E.g. A cubic nanometer block of E = 1012N/m2 has a stretching stiffness of 1000nN/nm

How mass (and volume) depends on size and the effects of that

  • Halving the size (L) of an object does divide it's volume and mass by eight. (L/2)3 = L/8
  • Doubling the size (L) of an object gives it eight times more volume and mass. (2L)3 = 8L

This allows for big accelerations ...

Formally:
(2.5) [math] mass \propto volume \propto L^{3} [/math] – a pretty well known law
One says: Volume and mass scale with the cube of length V ~ m ~ L3
Example: A cubic nanometer block of 3500kg/m3 (= 3.5kg/liter ~ approximate density of diamond)
has a mass of 3.5*10-24kg = 3.5zg (zg means zeptogram) which is extremely small.
This small: 0.000,000,000,000,000,000,000,003,5 kg

Acceleration tolerance

The main effect of masses at the nanoscale becoming so extremely small is that smaller things can endure way higher accelerations.
This is because acceleration forces are linearly proportional to mass.
That is: Dividing the mass by 8 divides force by 8.
That is basically just Newtons second law: F = m*a

Formally:
(2.6) [math] acceleration \propto force / mass \propto L^{-1} [/math]
With the 10nN/nm2 acting on a cubic nanometer block giving 10nN and the aforementioned mass that gives an
astounding acceleration of 3*1015m/s2

This scaling law interplays with two seperate effects:

  • planetary gravity (only effecting the macroscale – gravitative mass scales just like inertial mass) and
  • macroscale high speed crashes creating acceleration spikes that are untypically high for the macroscale
    (desctructive nanoscale level accelerations at the macroscale)

Interplay with acceleration from planetary gravity

On planets mass causes weight (earth: a = g = 9.81m/s2) With rising sizes weight forces are rising just as the mass does. Structures need to be designed more and more bulky. This can be observed both in architecture and big living things like trees and animals like e.g. elephants. Looking into the other direction there's the common example of ants capable of carrying a multiple of their own body weight.

Since other effects play in too animal legs (from ant to elephant) scale a bit different than expected.

Just as with ants, machinery on the microscale can be rather filigree. Very small manipulators can hold very big chunks for their size, manipulators in the microscale can be way smaller than the building blocks they handle. That is unless one wants to go near the theoretical limits of speed. Getting vibrations very low and efficiency very high can too be a motivation for not building too filigree. (Other parts of a system may have more stringent limits for efficiency though, limiting motivation for raising efficiency via stiffness increase in the microscale to the limits).

Even smaller at the low nanoscale to sub-nanoscale the lower physical size limit comes in the way. A manipulator can't be smaller than the smallest possible building block - an atom - in fact it must be quite a bit bigger.

Weight-forces are constant and unidirected in character. Yes this is obvious. But it's still worth to mentioned here because the character of forces can have a strong effect on the resulting character on the structural design of systems.

In space where in first approximation gravity is not present (that is we don't consider things so incredibly big that tidal forces become relevant) big things can be just as filigree as small things.

Interplay with acceleration from crashes

From everyday experience we know that things that fall down from some height and crash against a hard floor or things that move fast and crash hard against an obstacle usually take severe damage in form of permanent irreversible bending and fractures.

The acceleration levels occurring during such crash induced acceleration spike events is untypically high for the macroscopic size scale. In smaller size scales untypical acceleration spices usually do not occur because:

  • macroscale crash acceleration levels are perfectly normal at smaller scales and occur there all the time in normal operation
  • if something really causes an untipically high acceleration spike it must be so fast and carry so much energy (density) that the result won't be just bending and fractures but melting vaporisation or even ionization (conversion into plasma)

In technical terms: For smaller systems the acceleration spectrum swallows the acceleration spikes stemming from macroscopic crashes.

'In space one has crashes with space debris or (micro)meteorites. The usual impactors have so much energy density that they melt/vaporize/ionize the target and not much energy is left over for the acoustic shock-wave that propagates away from the location of impact. This is somewhat similar to the situation on smaller scales. (TODO: analyze this in more detail)

Combined

Note if inertial acceleration is the limiting factor instead of gravitational acceleration smaller structures can't be made more filigree. Since when speeds are kept constant and turning radii shrink accelerations rise.

Speedup (in terms of frequency)

How natural accelerations grow with shrinking size. To keep waste heat from friction at practical levels it is sensible too slow down at the nanoscale that is as one goes from right to left in the diagram one moves down the lines deviating from the natural scaling law.

The much higher acceleration tolerance of smaller things is the reason why motion speeds can be kept constant when making rotating things smaller despite the turning radii getting tighter and tighter and the zentrifugal accelerations correspondingly rising.
In simple math: v = ω r and a = ω2 r gives a = v2 / r –– (ω = 2π f)
In words: Halving the size of a spinning wheel doubles the centrifugal acceleration.

When keeping the speed constant halving the size doubles the frequency (frequency scales up linearly). This could be called the "insect-wing-effect" the reason a fat bumblebee sounds lower than a tiny mosquito.

In an advanced APM system A million-fold reduction in size theoretically lifts throughput a million-fold Practically it makes sense too slow down a bit and use only a thin** layer for mechanosynthesis instead of the whole volume. This prevent excessive waste heat that is the cooling facilities need not to be greater than the production unit.

Organisation in layers makes nanofactory design more scale invariant (2D fractal stack) and thus easier. For rapid assembly of preproduced parts (a lot more rapid than practical necessary) there is the acceleration limit for the macroscopic product to be considered.

[todo: add infographic - typical accelerations over scale for different speeds]

Forces from compressive and tensile stresses

It can be helpful or at least satisfying to get something of an intuitive understanding for the consistence or "feel" of DME components.

As the size of a rod of any material shrinks linearly (in all three dimensions) the area of the cross section shrinks quadratically. Consequently when keeping tension/compression stress constant the forces fall quadratically and one arrives at very low forces. [Sacling law: longitudinal force ~ length^2] This can be seen nicely in the low seeming inter-atomic spring constants. E.g. the equilibrium position spring constant of an bond in diamond (sp3 carbon-carbon-bond) is about 440nN/nm or 0.44daN/cm (1daN~1kg).

In order to get a feel for these forces one can transform atomic spring constants unchanged to the macrocosm. This can be done by letting the number of parallel and serial bonds grow equally so that the changement of stiffness through serial and parallel connection of bonds compensate. Here for convenience 10,000,000,000 bonds are assumed to be chained serially. We must apply this scaling to the number of parallel bonds too but here it divides up in each dimension of the cross-section sqrt(10^10) = 100,000. With the diamond bond (C-C sp3) length of 1.532Amstrong and area per bond of 6.701Amstrong^2 = (2.59Amstrong)^2 one gets a diamond string (with square cross-section) of 1.532m length and 25.9um thickness side to side (half a hair) that retains the atomic spring constant of 440N/m or 0.44daN/cm (1daN~1kg) If you bind up a half liter bottle of water with that (somewhat dangerous knife like) string it will bend around 1cm.

Putting one end of the sting in a vacuum filled square piston that seals tightly shows how little effect everyday pressures have at the micro and nanocosmos. Taking 1bar = 10^5N/m^2 ambient pressure the string experiences a force of only 67.1µN and elongates 0.152µm an invisible amount.

Though as seen bonds are rather compliable DMEs are still hard diamond since hardness is closely related to tensile and compressive stress which is scale invariant. The small force representation of high pressures might be a bit counterintuitive and hard to grasp.

Low stiffness is an important design restriction for nanomanipulators. (see related topic: A New Family of Six Degree Of Freedom Positional Devices)

By making the compliance at the nanolevel experiencable the model with the weight on the one bond equivalent diamond string should make one (maybe obvious) practical thing clear. That it is very effective to focus forces.

In mechanosynthesis conical tips can easily focus forces down to a more compliable size level. Not much of a size difference is needed. Nanoscale manipulators in the machine phase can hold back on their supporting structures they're mounted to. It is easy to create DMEs with high internal strains such as strained shell cylindrical structures, press fittings, structures under high tensile stress and more. Great amounts of elastic energy can be stored (permanently or temporarily).

An example of safely usable pressures from Nanosystems section 2.3.2.:
Assuming ~1% strain the required stress is ~1% of diamonds young modulus. 10nN/nm^2 = 10GPa = 1000daN/mm^2 (1daN~1kg) this is 20% of the tensile strength of macro-scale diamond with natural flaws. Flawless mechanosynthetically assembled diamond will be capable of handling more stress.

Forces from shearing stresses

[Todo: add info about shearing stress]

Surfaces

When viewing the thickness of a surface as the distance from the point of maximally attractive VdW force to the point of equally repulsive VdW force (experienced by some probing tip) the thickness of the surface relative to the thickness of the diamondoid part is enormous. This makes DMEs somewhat soft in compressibility but not all that much as can be guessed by the compressibility of single crystalline graphite which is a stack of graphene sheets.

[Todo: discuss stiffness changing effects of mechanical chaining]

Scaling laws for electrical properties

Taken from Nanosystems (with some liberties to make a direct electrostatic vs magbetostatic comparison)

(2.10) [math] speed \propto acceleration \times time \propto L^{0} = constant [/math]
We'll need the scaling law for speed for calculating the scaling laws for the powers by multiplying the speeds with the various forces.

(2.19) [math] voltage \propto electric–field \times length \propto L^{1} [/math]
(–.––) [math] current \propto magnetic–field \times length \propto L^{2} [/math]

(–.––) [math] electric field \propto voltage / distance \propto L^{0} [/math]
(2.31) [math] magnetic field \propto current / distance \propto L^{1} [/math]

(2.01) [math] total mechanical strength \propto mechanical force \propto area \propto L^{2} [/math]
(2.20) [math] electrostatic force \propto area \times (electrostatic field)^{2} \propto L^{2} [/math]
(2.31) [math] magnetostatic force \propto area \times (magnetic field)^{2} \propto L^{4} [/math]
Example for electrostatic force:

  • 1V/nm => 0.0044nN/nm2 = 4.4*10-12N/nm2 (1000x times lower than a typical covalent bond)

Example for magnetostatic force:

  • two conductors 1nm long each 1nm apart 10nA => 2*10-23N (1011 times lower than the electostatic example above)
  • one conductor 1nm long 10nA in a 1T field => 10-17N (105 times lower than the electostatic example above)

Unaltered citation from Nanosystems:
"Magnetic forces between nanoscale current elements are usually negligible.
Magnetic fields generated by magnetic materials, in contrast, are independent of scale:
forces, energies, and so forth follow the scaling laws described for constant-field electrostatic systems".

(2.11) [math] mechanical power \propto force \times speed \propto L^{2} [/math]
(2.26) [math] electrostatic power \propto electrostatic force \times speed \propto L^{2} [/math]
(–.––) [math] magnetostatic power \propto magnetostatic force \times speed \propto L^{4} [/math]

(2.11) [math] mechanical power density \propto mechanical power / volume \propto L^{-1} [/math]
(2.26) [math] electrostatic power density \propto electrostatic power / volume \propto L^{-1} [/math]
(–.––) [math] magnetostatic power density \propto magnetostatic power / volume \propto L^{+1} [/math]
Halving a systems size:

  • doubles electrostatic power density
  • halves magnetostatic power density

(–.––) [math] mechanical energy \propto volume \times pressure \propto L^{3} [/math]
(–.––) [math] electrostatic energy \propto volume \times electric field^2 \propto L^{3} [/math]
(2.26) [math] magnetostatic energy \propto volume \times magnetic field^2 \propto L^{5} [/math]

(–. ––) [math] capacitance \propto electrostatic energy / (voltage)^2 \propto L^{1} [/math]
(2.34) [math] inductance \propto magnetostatic energy / (current)^2 \propto L^{1} [/math]

(The corresponding integrated quantitative laws: C = 2E/U2; L = 2E/I2)

(wiki-TODO: Add further relevant scaling laws & example calculation)

Related: Non mechanical technology path

Related

In some cases transitions

  • do not follow polynomial laws but exponential ones or other and
  • are different for different systems
  • not only causable by changes in size scale

like e.g. the onset of quantum mechanical behavior.
See: Quantum mechanics

External Links

Wikipedia:

  • Insect_flight Hovering – scaling of flapping frequency with size is not mathematically covered as of yet (state 2021)

Sketches:

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