# Scaling law

With scaling laws one can find out which effects get stronger and which get weaker when machine size is shrunken down (like in diamondoid molecular elements). Based on this one can decide which effects are best to use. As an specific example electric motors can be taken. Scaling laws tell us that its better to use electrostatic motors (which become stronger when shrunken) in contrast to conventional magnetic motors (which get weaker). Furthermore scaling laws provide quantitative numbers and can provide some kind of intuitive feel of how the world down there behaves.

Because of their importance scaling laws are presented right at the beginning of the book Nanosystems.

## Contents

# 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.

## 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 seperated Volumes.

# Scaling laws for mechanical quantities

## 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

One says: Volume and mass scale with the cube of length V ~ m ~ L^{3} since and

### Acceleration tolerance

The main effect of this is that smaller things can endure higher accelerations since acceleration forces are
proportional to mass (Simple Newton: F = m*a – dividing mass by 8 divides force by 8).
This is the reason why motion speeds cam be kept constant when making things smaller albeit the turning radii get tighter and tighter
a = ω ^{2} r = (2 π f)^{2} r.

This scaling interplays with two effects that do not scale with size of products:

- planetary gravity (only effecting the macroscale) and
- macroscale crashes creating acceleration spikes that are untypically high for the macroscale

### Interplay with acceleration from planetary gravity

On planets mass causes weight (earth: a = g = 9.81m/s^{2})
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

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

[Todo: add further relevant scaling laws & example calculation]

# Related

- By using super lubricating infinitesimal bearings one can cheat a bit on the naive scaling law for friction.
- Intuitive feel
- The degree of applicability of macro 3D printing for nanomachine prototyping