Lecture 1 – Physics of miniaturization

Scaling laws relate the properties of a system to its dimensions.

Example: diffusion law

!~ !!

!

Example: capillary forces

∆! = !!

5

100µm

Example: capillary forces

To establish scaling laws, one estimates terms implied in the equations of the system The same approach applies to partial-derivative equations.

0 = −∇P +µΔu

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

Scaling laws in nature – Kleibler law

Kleibler

How to measure the metabolic rate ?

Warburg respirometer

CO2 absorbant

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

The consequences of Kleibler law

T=1O M1/4

Muscular force varies as l2

Volume musculaire ~ l3

Energie musculaire E ~ l3

Fmax ~ l2

Maximum speed of the animals

Running develops a work equal to l3

Kinetic energy is in V2 l3

Conclusion: speed independant of size

At what speed Tyrannosaurus rex runs ?

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.)

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

Mouse : 1/3 of its own weight per day Elephant: 1/100 of its own weight per day

Heat transfer in animals

Pygmee shrew is the smallest mammifer on earth

The terrible life of the pygmee shrew

No problem for the whale to keep its internal heat

No problem for the whale to keep its internal heat

Scaling of adhesion

2/3 2/3

Labonte D et al Phil. Trans. R. Soc. B 370 (2015).

Pros : Jump high (H~ l0), walk on water

Con : Being trapped in a soap bubble

Pro/cons being miniaturized

Scientific erros in “Honey, I shrunk the kids”

Scientific errors in “Honey, I shrunk the

kids”

Scientific errors in

“Honey, I shrunk the kids”

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

Exemple 1 : Demonstration of Pythagore theorem

Exemple 2 : Dynamical symilarity theorem

u = U g(x/l,Re)

Reynolds number = Ul/ν

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

Miniaturized jet Ordinary jet

Analysis of a microjet

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:

Some vocabulary….

Vocabulary again ….

109 Giga1012 Tera

1015 Peta

1018 Exa

Vocabulary again….

1021 Zetta

1024 Yotta

F = QE ~ CVE ~ ε 0E 2l2

+Q

- QVE

Attraction forces of the planes

Miniaturisation of electrostatic systems

The micromirror

The electrokinetic micromotor

Small torque, small power

Torque C ~ Fl ~ l3

P =CΩ ~Ωl3

ERotating electric field

The MIT Microturbine

The MIT Microturbine

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

l I

Miniaturisation of electromagnetic systems

F = IBl ~ jBl3

B ~ µIl−1 ~ µjlF ~ µj2l4

Magnetic/électrostatic

Fe ~ ε0E2l2

Fm ~ µ 0 j2Nl4

Magnetic < Electrostatic

Scaling law for the vibration frequency of a Cantilever beam

! ≈ 1.8752! !"!!!~ !!!

! = !ℎ!12

Miniaturisation of RF systems

q = −k∇T

ΔT

q

For a conducting rod:

Q =KSΔTL

Q ~ l ∆T

Miniaturisation of heat exchangers

Fourier’s law

Source of heat ~ l3

Conduction losses ~ l ΔT

Temperature control is feasible in microfluidics

Increase of température ΔT ~ l2

Miniaturization of chemical reactors

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

Flow-rates = speed x area

Q ~ l3

Microfluidics is able to produce ml/hour

Order of magnitude

2nd limitation: small throughput

From IPI Singapore

Important to remember

-  Table of scaling laws

-  Examples in nature

-  Examples in MEMS technology and microfluidics