3 Scaling Laws in the Micro Domain

(1)

Laboratoire de Mécanique Appliquée – UMR CNRS 6604 – Université de Franche Comté Institut des Microtechniques de Franche Comté

3 Scaling Laws in the Micro Domain

A/ Scaling of Mechanical Structures B/ Scaling of Actuators

C/ Comparing Electrostatic & Magnetic Actuation

If a system is reduced isomorphically in size, the changes in length, area, and volume ratios alter the relative influence of various physical

effects which determine the overall operation in unexpected ways.

As objects shrink, the ratio of surface area to volume increases, rendering the surface forces more important.

Shrinking the linear dimensions does not shrink forces in the same way.

- Friction associated with surfaces becomes more important than masses…

- Weight and inertia tend to be negligible in the microworld…

Laboratoire de Mécanique Appliquée – UMR CNRS 6604 – Université de Franche Comté

Section 3 / Slide 3

As the scale of structures decreases, so does the importance of phenomena which vary with the largest power of the linear

dimension (l):

Phenomena that are more weakly dependent on size dominate in small dimensions:

Friction (l

²

), Surface Tension (l), Diffusion (l

^1/2

), van der Waals Forces (l

^1/4

)…

Gravity & Inertia(l

³

), Magnetism (l

²

, l

³

, or l

⁴

), Flow (l

⁴

), Thermal Emission(l

²

to l

⁴

)

Scaling of Mechanical Structures: Mechanical Strength

Cantilever beam loaded by its own weight per length P:

EI P x =

∂

4 4 ξ

- E: Young Modulus

- I= bh

³

/12: Moment of Inertia - ξ = ξ (x): Beam Deflection - b and h: Width and Thickness ξ ~ L

⁴

bh/bh

³

= L

⁴

/h

²

~l

²

A ten times smaller beam bends 100 times less due to its on weight

Deflection under own weight: ξ ~ l

²

Small mechanical structures are stiff

ξ (x) x

L b

h

(2)

Dynamic Properties: Resonant Frequency

Eigen frequency of a beam (where the restoring force is due to stiffness):

Mode and Boundary conditions

S EI f _r L

ρ α

= 2

f r = h/L

²

~ l

^-1

In case of a bending beam having

a rectangular cross section: - S = bh - I = bh ³ /12

Resonant frequencies become large for small systems Eigen frequencies scale like l

^-1

Parasitic vibrations vanish in The micro world

Scaling of Mechanical Structures: Coulomb Friction

In the macroscopic world, friction forces are independent of the contact area (e.g. 2 rough bodies without load touch each other only at 3 points).

Material Related Scaling Effect

In microscopic world, this seems to be different in some cases because silicon is extremely smooth

(The micro surface roughness of silicon wafers is of the order of 1nm)

- Adhesive forces can be very large and lead to process problems (SM) - These surface forces scale with (l

²

)and become large in small systems

Scaling of Mechanical Structures: Coupling with Fluids & Gases The transition from laminar to turbulent flow is around Re = 2300.

Consequently:

There is no turbulence in

microsystems in which liquids flow There can be turbulence in

microsystems with gas flow

Reynolds number as a function of size From Hayashi, 1994

Linear Dimension l ∼ l

¹

Surfaces A ∼ l

²

Volume V ∼ l

³

Electric Energy W

el

∼ l

³

Electric Force F

el

∼ l

²

Magnetic Energy W

mag

∼ l

⁵

Magnetic Force F

mag

∼ l

⁴

Deflection Under Own Height ζ ∼ l

²

Stable Length L

cr

∼ l

^4/3

Diffusion Times τ ∼ l

²

Drift Velocity υ

∞

∼ l

²

Transient Time τ ∼ l

²

Electric Resistance R

el

∼ l

^-1

Hydraulic Resistance R

hy

∼ l

^-3

Reynolds Number Re ∼ l

²

Resonant Frequency υ ∼ l

^-1

Scaling of Various Physical Properties

(3)

Scaling of Actuators

Atmospheric Engines Electric Actuators

Rapid heat loss would prevent the gasoline from exploding

External supply of electric power would be most convenient

Electromagnetics

Electrostatics

Macroscopic World

Microscopic World

Magnetic Circuit

Coil

B

Air-Gap

B

µ 0 The magnetic energy density stored in the

air-gap is:

0 2

2 µ w _mag = B

B ~ 1.5T by pushing the coil into saturation:

/ 3

000 ,

950 J m

w _mag ≈

?

1) Magnetic field (B) remains constant whatever the size 2) Magnetic field depends on

the size of the device

Scaling in Magnetics

Assumption 1: The magnetic field (B) remains constant when devices shrink Magnetic energy density:

l B Cte

w _mag = = ∀

0 2

2 µ

- B: Magnetic Field - µ

0

: Permeability

Magnetic field in the gap:

l i B = µ ₀ n

- n: Number of Turns of Wire - l: Total Length of the Coil - i: Current Through the Coil

- Magnetic energy density scales like l

⁰

- Energy in a gap of a magnetic circuit scales with l

³

Magnetic forces which are given by the derivative of the magnetic energy scale with l

²

This conclusion is too quick

Keeping n constant when shrinking the coil and the magnet:

Cross section S of the wires must shrink Keeping i constant becomes impossible !

l i µ n B = ₀

Magnetic field (B) scales like l S = Cte n

Magnetic field in the air-gap decreases with respect to the

reduction of turns of wire

n = Cte S

Keeping i constant becomes impossible !

l l

i scales like l

²

Scaling in Magnetics

(4)

Scaling in Electromagnetic Actuators

Assuming a single wire of any shape carrying a current of i amperes:

i

B

B l

d i

F

wire mag

r r

r ⁼ ∫ ^∧

Electric motors are arrangements for utilizing magnetic force F

mag

to produce torque in rotating machinery.

dl is a vector representing a small part of the wire , directed along the wire in the sense chosen as positive for i.

F _mag

(l

²

) (l) (l)

Magnetic force F mag scales like l ⁴ …

When downsizing electromagnetic actuators, it is more reasonable to keep the current density constant,which means that:

- Magnetic field (B) scales like l

- Magnetic field energy density (w mag ) scales like l ² - Magnetic energy (W mag ) ~ l ⁵

- Magnetic force (F mag ) ~ l ⁴ (but heat flow scales like l

^–1

F

mag ~

l

²

…)

In practice, magnetic actuators smaller than 1 mm

³

are hardly feasible and it is difficult to keep n constant !

Scaling effects in downsizing electromagnetic actuators strongly depend on design considerations

Increased current density

Magnetic forces scale like l

²

, l

³

or l

⁴

Depending on design rules

Scaling in Electrostatics

The electrostatic force ( F

el

)between capacitor plates is given by the derivative of its electric energy ( W ) with the gap distance ( x ):

Q

el x

F W 



 





∂

− ∂

=

W can be found in the easiest manner by the energy density (w):

0 2

2 E

w ε

=

Assuming that the dielectric is air ( ε

r

=1):

∫ ⁼

= wdV wAx W

For (w) constant (meaning E = cte):

A Q A

wA E F _el

0 2 0 2

2 2 ε

ε = −

−

=

−

=

The charge Q must be held constant

Electric Field Strength

Surface of the electrodes

Scaling in Electrostatics

A Q A

wA E F _el

0 2 0 2

2 2 ε

ε = −

−

=

−

=

If we are able to shrink the whole capacitor at constant energy density(w) meaning at: constant electric field (E)

constant charge density (Q/A)

- The energy of the capacitor scales with the volume: W

el

~ l

³

- The force between electrodes scales like the surface: F

_el

~ l

²

Electrostatic forces increase relative to the volume when the systems shrink

Electrostatics scales well in the micro world

(5)

Breakdown of Continuum Theory for Electrostatics The maximum field in air is not independent of the distance of the capacitor

plates if they approach closely ! If the separation of plates is roughly

the mean free path of molecules in air, the maximum field increases…

Below 5 µm in air, the Pashen curve sharply reverses, leading to high electrical breakdown

Depending on surface roughness, the Electric field can be as high as:

- 10 ⁸ V/m with 1.5 µm air gap - 3 10 ⁸ V/m in vacuum

P

x

d

If the pressure P remains constant at 1 atm.(760mm Hg), the x-axis can be reduced to a simple distance axis

2.5µm 13.16µm

Comparing Electrostatic and Magnetic Actuation

Besides scaling, many other factors need to be considered when deciding upon a certain type of actuation principle

- Thin insulated layers exhibit breakdown voltage as high as 2 MV/cm

- Corresponding energy density : 7 10

⁵

J/m

³

(Equivalent to the power density of 1.3 T magnetic field) - The contracting pressure is 1.3 Mpa

- 100 V driving voltage is sufficient to generate the strong fields mentioned

- Simple actuation (pair of electrodes) - Voltage switching far easier and faster than

current switching

- Low Energy loss through Joules heating - Low Weight and power consumption - IC compatible

Attributes of SM electrostatic actuators (l

²

)

- In spite of scaling effects, the absolute forces achievable are very large

- In conventional motors using iron, the magnetic induction is restricted to 1.5 T - Corresponding energy density : 9 10

⁵

J/m

³

(More than twice the achievable electrostatic energy in a 1 µm gap)

- Thin ferromagnetic films have yielded 2 T fields

- Unfortunately, 10 to 15 T superconducting magnets fall outside the micromachining domain

- Friction easier to avoid in magnetic motors - Less sensitive to dust &humidity (larger gap) Attributes of electromagnetic actuators (l

⁴

3 Scaling Laws in the Micro Domain