Lecture 1 Physics of miniaturization - ESPCI Paris · 2017. 1. 30. · A list of scaling laws...
Transcript of Lecture 1 Physics of miniaturization - ESPCI Paris · 2017. 1. 30. · A list of scaling laws...
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Lecture 1 – Physics of miniaturization
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Scaling laws relate the properties of a system to its dimensions.
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Example: diffusion law
!~ !!
!
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Example: capillary forces
∆! = !!
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5
100µm
Example: capillary forces
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To establish scaling laws, one estimates terms implied in the equations of the system The same approach applies to partial-derivative equations.
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0 = −∇P +µΔu
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A list of scaling laws relevant to microfluidics
Capillary force
Quantity Scaling
Capillary force Flow speed in a microfluidic system Thermal power transferred by conduction Viscous force in a microfluidic system Electrostatic force Diffusion time Gravity force Magnetic force with an external field Magnetic force without external field
l
l
l
l2
l2
l2
l3
l3
l3
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Scaling laws in nature – Kleibler law
Kleibler
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How to measure the metabolic rate ?
Warburg respirometer
CO2 absorbant
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The consequences of Kleibler law
Métabolism : W ~ l 9/4
Blood volume ~ l 3
Heart size ~ l Lung volume ~ l 3
Heart pulsation ~ l 3/4
Longevity ~ l ¾
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The consequences of Kleibler law
T=1O M1/4
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Muscular force varies as l2
Volume musculaire ~ l3
Energie musculaire E ~ l3
Fmax ~ l2
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Maximum speed of the animals
Running develops a work equal to l3
Kinetic energy is in V2 l3
Conclusion: speed independant of size
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At what speed Tyrannosaurus rex runs ?
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Ostrich (144 pounds, 65.3 kg)—34.5 mph (15.4 m/s) Velociraptor (44 pounds, 20 kg)—24.2 mph (10.8 m/s) Dilophosaurus (948 pounds, 430 kg)—23.5 mph (10.5 m/s) Tyrannosaurus (13,230 pounds, 6,000 kg)—17.9 mph (8 m/s) Human (157 pounds, 71 kg)—17.7 mph (7.9 m/s)
Speeds of two-legs animals
From Manning, Sellers (Manchester Univ.)
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Thermal losses ΔT l . must be compensated by food ~ N l3
This leads to :
l ~ (ΔT )1/2 N-1/2
N cannot be increased indefinitely.There is therefore a minimum size for mammifers
Heat transfer in animals
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Mouse : 1/3 of its own weight per day Elephant: 1/100 of its own weight per day
Heat transfer in animals
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Pygmee shrew is the smallest mammifer on earth
The terrible life of the pygmee shrew
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No problem for the whale to keep its internal heat
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No problem for the whale to keep its internal heat
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Scaling of adhesion
2/3 2/3
Labonte D et al Phil. Trans. R. Soc. B 370 (2015).
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Pros : Jump high (H~ l0), walk on water
Con : Being trapped in a soap bubble
Pro/cons being miniaturized
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Scientific erros in “Honey, I shrunk the kids”
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Scientific errors in “Honey, I shrunk the
kids”
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Scientific errors in
“Honey, I shrunk the kids”
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A useful tool for microfluidics: π theorem
a = f(a1,a2,…..an) and k dimensions. f can be rewritten into a simpler function g : Where ∏n are independant products formed on combination sof a, a1,a2,…..an
€
Π = g Π1,Π2,Π3,..........Πn−k( )
From n+1 variables we move to n+1-k variables
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Exemple 1 : Demonstration of Pythagore theorem
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Exemple 2 : Dynamical symilarity theorem
u = U g(x/l,Re)
Reynolds number = Ul/ν
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Démonstration : u(x) = f(x,U,ρ,µ,l)
n=5 k=3
One can define 6-3 = 3 dimensionless numbers
uU
= g xl,Re
! " #
$ % &
With Re=Ul/ν, the Reynolds number
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Miniaturized jet Ordinary jet
Analysis of a microjet
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Re = Uaν
≈10 ; Ca =µUγ
≈ 10−2 ; Bo =ρa2gγ
≈10−3
Conclusion : jet is laminar, droplets are spherical and gravity deformation of the shape is negligible
3 dimensionless numbers:
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Some vocabulary….
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Vocabulary again ….
109 Giga1012 Tera
1015 Peta
1018 Exa
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Vocabulary again….
1021 Zetta
1024 Yotta
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F = QE ~ CVE ~ ε 0E 2l2
+Q
- QVE
Attraction forces of the planes
Miniaturisation of electrostatic systems
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The micromirror
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The electrokinetic micromotor
Small torque, small power
Torque C ~ Fl ~ l3
P =CΩ ~Ωl3
ERotating electric field
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The MIT Microturbine
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The MIT Microturbine
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The MIT Microturbine
-Diameter/height : 12 mm/3mm- Air flow rate: 0.15 g/s- Température at the outlet: 1600 K- Rotation speed: 2.4 106 tr/mn- Power: 16W- Weight : 1g- Consumption of fuel: 7g/h
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l I
Miniaturisation of electromagnetic systems
F = IBl ~ jBl3
B ~ µIl−1 ~ µjlF ~ µj2l4
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Magnetic/électrostatic
Fe ~ ε0E2l2
Fm ~ µ 0 j2Nl4
Magnetic < Electrostatic
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Scaling law for the vibration frequency of a Cantilever beam
! ≈ 1.8752! !"!!!~ !!!
! = !ℎ!12
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Miniaturisation of RF systems
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q = −k∇T
ΔT
q
For a conducting rod:
Q =KSΔTL
Q ~ l ∆T
Miniaturisation of heat exchangers
Fourier’s law
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Source of heat ~ l3
Conduction losses ~ l ΔT
Temperature control is feasible in microfluidics
Increase of température ΔT ~ l2
Miniaturization of chemical reactors
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Number of molecules = Volume microsystème . Concentration
N ~ l3C
For 10-9 M/litre in a chamber of l ~ 10 mm, N ~1 ymole
Order of magnitude
1st limitation: the small amount of material than can be analyzed
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Flow-rates = speed x area
Q ~ l3
Microfluidics is able to produce ml/hour
Order of magnitude
2nd limitation: small throughput
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From IPI Singapore
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Important to remember
- Table of scaling laws
- Examples in nature
- Examples in MEMS technology and microfluidics
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