View source for Same relative deflections across scales

'''Deflections induced by accelerations from mechanical motions are scale invariant''' <br>
'''This is one important reason for why [[macroscale style machinery at the nanoscale]] works better rather than worse at the nanoscale.''' <br>
See main article: [[How macroscale style machinery at the nanoscale outperforms its native scale]]

= Relevance for scale transposed prototyping =

Scale invariance of deflections from machine motions is a very important result for [[applicability of macro 3D printing for nanomachine prototyping]]. <br>
This is because it means that 3D printed macroscale prototypes will typically vastly underperform nanoscale target systems of equal geometry. <br>
<small>(Slight under-performance is desirable from a [[exploratory engineering]] perspective)</small> <br>
From the stiffness aspect, there is no risk to accidentally prototype a macroscale system that then cannot be ported to the nanoscale. <br>
Heck, if anything this may lead to uncircumventable massive over-engineering in macroscale prototypes. <br>
Over-engineering that ... 
* will surely work at the nanoscale, but 
* will also be be very far from optimal.

See: [[How macroscale style machinery at the nanoscale outperforms its native scale]]

= Same absolute speed across scales =

A necessary assumption to get <br>
[[same relative deflections across scales]] (aka scale invariance of relative deflections) <br>
is to have [[same absolute speed across scales]] (aka scale invariance of absolute speeds). <br>
This is why keeping speeds constant across scales can be seen as a natural choice.

<math>v \propto L^0 \Leftrightarrow \epsilon \propto L^0</math>

When more or less deviating from that natural choice then relative <br>
deflections scale the same as the speeds are scaled. That is: <br>
When slowing down across scales then one (unsurprisingly) reaps the benefit of smaller relative deflections.

Some such deviation seems well motivated by dynamic [[friction in gem-gum technology]] being reducable by <br>
[[deliberate slowdown at the lowest assembly level‎]] & [[increasing bearing area to decrease friction]]. <br>
But this happens in a rather nonlinear fashion across scales. Not like a scaling law. <br>
<small>An issue here is the still unclear [[mesoscale friction]] gap.</small>

Still, when trying to approximate [[deliberate slowdown at the lowest assembly level‎]] with a scaling law <br>
then it might be roughly a square-root law or weaker: <br>
<math>v \propto L^{1/2} \Leftrightarrow \epsilon \propto L^{1/2}</math> <br>
Scaling speeds linear with L would be way too much <br>
as it would fully counter [[higher throughput of smaller machinery]]

'''In all further analysis scale invariance of speeds and deflections is assumed.'''

= Analyzing scaling behavior of relative deflections =

Without loss of generality only the one dimensional case of tensile strain is covered.

== Scaling of deformations from machine motions (spoiler: they are scale invariant) ==

Accelerations from machine motions go up too worsening the situation. <br>
One millionth the size => One million times the accelerations.

But don't fret just yet. We have a quite literally massive compensating factor.
With scaling down machinery to smaller sizes the mass of that machinery is going down. <br>
One millionth the size => (One millionth)³ the mass.

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The critical quantity we want to be preserved across scales is relative strain: <br>
<math>\epsilon = \Delta l / l</math> <br>
Strain epsilon is proportional to stress sigma. <br>
<math>\epsilon = \sigma / E</math> <br>
Stress is given by applied force F. <br>
<math>\sigma = F / A </math> <br>
A natural level of force for a scale is zentrifugal force from rotating a mass at that scale. <br>
<math> F = m \omega^2 r = m (2\pi f)^2 r</math> <br>
Putting it all together we get: <br>
<math>\epsilon = \sigma / E = F / (E A) = m \omega^2 r / (E A)</math> <br>

With:
* mass m scales with <math> L^3 </math>
* frequency f scales <math> L^{-1} </math> (assuming absolute speed v is kept scale invariant)
* radius r scales with <math> L^1 </math>
* Tensile modulus being scale invariant <math> L^0 </math>
* Area scaling with <math> L^2 </math>

We get: <br>
<math> F = \underbrace{m}_{+3} \underbrace{\omega^2}_{-2} \underbrace{r}_{+1} \propto L^2</math> <br>
<math>\epsilon = \underbrace{m}_{+3} \underbrace{\omega^2}_{-2} \underbrace{r}_{+1} / (\underbrace{E}_{0} \underbrace{A}_{+2}) \propto L^0 = const</math> <br>
In words:
* Force F scales with <math> L^2 </math>
* Both stress & strain from machine motions scale with <math> L^0 </math>

In other words when assuming scale invariant speed then: <br>
'''Both stress & strain (relative deflections) from machine motions are scale invariant.'''

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'''Alternative path:''' <br>
From: [[Lower stiffness of smaller machinery]] <br>
We know the that the scaling law for spring constants is: <br>
<math>k \propto L^1</math> <br>
And substituting for E in former result we get the same final result via different path: <br>
<math>E = k ~ (l/A)</math> <br>
<math>\epsilon = \sigma / E = \underbrace{m}_{+3} \underbrace{\omega^2}_{-2} \underbrace{r}_{+1} / (\underbrace{k}_{+1} \underbrace{l}_{+1}) \propto L^0 = const</math> <br>

== Scaling of deformations from gravity ==

<math> F_{grav} = \rho V g \propto L^3</math> <br>
<math>\epsilon_{grav} = \sigma_{grav} / E = F_{grav} / (E A) = \rho V g / (E A) \propto L^1 </math> <br>

'''Both stress & strain from gravity go down linearly with smaller scales.'''

{{wikitodo|Self-gravity of mega-structures should scale differently. Maybe look into that too. But that goes into wild SciFi territory.}}

== Scaling of deformations from thermal motions ==

A natural place to start out seems to be setting the deformation energy equal to the thermal emergy: <br>
<math> E_{therm} = E_{el}</math> <br>
Assigning a block no matter the size just gets one degree of freedom for one thermal energy package (equipartitioning theorem), <br>
and ignoring small constant pre-factors we get: <br>
<math> E_{therm} = k_B T</math> <br>
<math> E_{el} = k_{el} x / 2</math> <br>
Definition of relative deflections is: <br>
<math> \epsilon_{therm} = x / l</math> <br>
Putting all so far together: <br>
<math> k_B T = k_{el} l \epsilon_{therm} / 2</math> <br>
And (with l∝L¹) we finally get: <br>
<math> 2 k_B T / (k_{el} l) = \epsilon_{therm} \propto L^{-1}</math> <br>

That's for the average tensile deflections from just kT worth of flexing energy. <br>
Obviously a [[crystolecule]] rod (e.g. [[Kaehler bracket]]s) <br>
have more internal degrees of freedom (phonon modes) than one <br>
but these shouldn't contribute much. <br>
Especially for free standing small crystolecules allowing only for high energy modes. <br>
{{todo|May be bogus. Investigate that.}}

Note that the scale of thermal motion induced deflections at the atomic scale is already at the atomic scale. <br>
And it only gets smaller from there upwards the scales.

There '''at the atomic scale deflections are big enough though to be of concern for [[piezomechanosynthesis]]''' <br>
and unguided [[covalent welding]]. <br>
For details see: [[Lattice scaled stiffness]] & [[Stiffness]].

== Related ==

* '''[[How macroscale style machinery at the nanoscale outperforms its native scale]]'''
* '''[[Applicability of macro 3D printing for nanomachine prototyping]]'''
* '''[[Lower stiffness of smaller machinery]]''', [[Stiffness]], [[The feel of atoms]]
* [[Scaling law]]
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* [[Deliberate slowdown at the lowest assembly level‎]]
* [[Increasing bearing area to decrease friction]]
* [[Optimal sublayernumber for minimal friction]]


[[Category:Pages with math]]
[[Category:Scaling law]]