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Scaling in the Microworld

Dr. Thara SrinivasanLecture 6

Picture credit: Irene Tsai, MITPolymer thin film after electric field and reactive ion etching, 200x

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Lecture Outline

• Reading• From reader: W.S.N Trimmer,

“Microrobots and Micromechanical Systems,” pp. 267-87.

• H. Fujita, “Microactuators and Micromachines,” pp. 1721-32.

• Watch the movie Microcosmos

• Today’s Lecture• Scaling in Nature• Trimmer’s Matrix Notation • Scaling of Forces• Scaling of Mechanical, Electrical,

and Fluidic Systems

Nikon Small World Photo Competition Winners

Marine diatom 160x

Embryo seeds within fruit capsule (25x)

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Scaling• Why is scaling important for MEMS?

• MEMS are often >1000× smaller than macro counterparts.

• We need to develop new intuition of microscale phenomena.

• Otherwise, different scaling of any one property can be a big roadblock!

• Dependence on length scale s• Length [s1] →• Surface area [s2] →• Volume [s3] →

• Constraints on life• Land-based life contends with gravity at large scale,

drying out at small scale.• Water-based life increases range of sizes by evading

gravity and drying out.

8V

4A

2L2

s

3.8 m

20 m

Mite on Sandia microengine

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Swimming and Flying2

21 AvcF DD ρ=

2

21 AvcF LL ρ=

Ian Thorpe

• Swimming• Mass of muscles ~ [s3]• Drag force FD ~ [s2]• Larger creatures have greater

swimming speed

• Flying is more complex• Mass of muscles ~ [s3]• Weight ~ [s3]• Drag force FD ~ [s2]• Lift force FL ~ [s2]• Larger means faster flight but

more power to keep weight aloft

Dic

kins

on g

roup

, UC

B

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5

Bug’s Life• Most abundant creatures are 1-2

mm in size.• Walking on water is possible as

surface tension supports small weights, but swimming is not fun.

• Bugs are cold-blooded to manage faster cooling and heating.

• Bugs are not easily injured. • They can lift 10-50× their weight.• They jump roughly as high as

people do!• Work = weight × height• Force ~ muscle mass

Water strider

Antz

Mexican ant, Cornell Integrated Microscopy Center

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Why Miniaturize?

• Motivation• Batch fabrication, lower cost per device• Less energy, less material consumed, disposable (or better,

recyclable!)• Arrays of sensors possible, minimally invasive• Similar size scale as individual cells• Can take advantage of different scaling laws (e.g., electrostatic

forces)• Breakdown of macroscale laws of physics

• Performance• Integration with circuitry can reduce noise and improve sensitivity• Yield and reliability may be improved, fewer defects per chip

• 106 defects/cm3 → 1 defect for every 106 µm3

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Challenges to Miniaturization

• Harder to interface with the macroscopic world • Fragility• Interconnect issues

• Smaller device requires higher sensitivity to sense smaller input• Chemical sensors, accelerometers, gyros

• May need to take into account • Molecular forces (i.e., Brownian motion) • Quantum mechanical effects (i.e., phonons)

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Matrix Notation• Matrix shows dependence on length scale [s] for

different cases in simple format, Trimmer 1989.

[ ][ ]3/ −== ssmFa F

( ) 2/12/1 )/2(/2 Fxmaxt ==

[ ][ ][ ]( ) 2/131 Fssst −=

tFxP /=

=

4

3

2

1

ssss

F

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Scaling Results

• Scaling of forces• [s1] ~ surface tension,

electrostatic I• [s2] ~ pressure, muscle,

electrostatic II, magnetic I• [s3] ~ gravitational, magnetic II• [s4] ~ magnetic III

=

4

3

2

1

ssss

F

=−

−

1

0

1

2

ssss

a

=

0

5.0

1

5.1

ssss

t

=−

−

2

5.0

1

5.2

ssss

VP

volume forces (s3)

surface forces (s2)

line forces (s1)Microscale ~µm

Mesoscale ~mm

Log Dimension

Log Force

Nanoscale ~nm

Judy

Macroscale ~m

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Power

• Power generated• Force laws with scaling higher than

s2, power generated per volume degrades as scale decreases

=−

−

−

0

5.0

1

5.1

1

1

1

1

4

3

2

1

ssss

ssss

ssss

P

=

5

5.3

2

5.0

ssss

P

=−

−

2

5.0

1

5.2

ssss

VP

MIT

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Electrostatic Forces

xUF∂∂

−=

20

0

2

21

21

wldEU

EdVdwlC

CVU

ε

ε

=

==

=

• Calculate the force exerted between the plates of a parallel plate capacitor

Fig A6 Trimmer

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Electrostatic Forces

[ ]202

1 wldEx

F∂∂

−= ε

[ ] 22 EsF =

[ ] [ ]05.0 sorsEb−=

[ ] [ ]21 sorsF =

• Two regimes in breakdown field Eb vs. spacing d curve• As d approaches mean free

path λ of insulator molecules, fewer molecules are around to be ionized

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Magnetic Forces

[ ] [ ] [ ]220 sssJAdAJI ===∫ ⋅= •

[ ]40

2s

dIIF ba ==l

πµ

Fig A1, A2 Trimmer

...2

0 +=dlIIF baπ

µ

• Constant current density ~ [s1]

−

= −

−

0

5.0

1

s

s

sJmagnetic

−

=

4

3

2

s

s

sFmagnetic

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Electrostatic vs. Magnetic Microactuation

• Electrostatics + Generally better scaling at microscale+ Simple actuation with pair of electrodes separated by

insulator+ Voltage switching easier than current switching+ Energy loss through Joule heating is lower+ High-force short-range motion concatenated, as in stepper

motor

• Magnetics+ Absolute forces, displacements larger+ Can operate in harsh environments – Magnetic materials not standard– 3D magnets harder to microfabricate using planar IC

processes– High currents, power dissipation

−

=

4

3

2

s

s

sFmagnetic

−−

=2

1

s

s

F ticelectrosta

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Electrostatic Actuators

Offenberg et al., Bosch Brosnihan et al.

x

2

22

22 dwlVFand

dwVF yx ∝∝

• Laterally driven resonators• Electrostatic force

proportional to number of comb fingers

• For largest deflection operate at resonant frequency

y

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Surface Tension• Surface tension (γwater ~ 72 mN/m)

• 20 µm hydrophilic channel filled with water, ∆P across meniscus is 12.5 kPa

• Capillary condensation: d ~ 3 nm for 50% humidity

• Surface tension or capillary forces scale with perimeter of wetted area ~ [s1]• Bug (10 mg) needs 1 mm of foot edge to walk on

water• Human (60 kg) would need feet with 8000 m

perimeter

• Implications for MEMS• Release and in-use stiction are major challenges• Can it be harnessed?

+=∆

21

11RR

P γ

[ ][ ][ ]1

21

sFssPAF

=

=∆= −

Fig

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Surface Tension for Self-Assembly• Use surface tension of liquid polymer and molten metal droplets to

self-assemble hinged MEMS into desired positions.

• γwater = 72 mN/m, γpolystyrene = 39 mN/m at 25°C

• γSn-Bi solder = 319 mN/m at 188°C, γAu = 1070 mN/m at 1200°C, γsodium silicate glass = 286 mN/m at 1000°C

Syms group, Imperial College Bright group, Univ. of Colorado

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Friction• Friction, arising from

• Capillary forces~ [s1]• Adhesive surface forces such

as van der Waals, hydrogen bonding, electrostatic ~ [s2]

• While macroscale contact occurs between few rough protrusions, in MEMS, surfaces may be highly smooth (e.g. SCS) so contact occurs over larger area

• Early micromotors had friction coefficients as high as brake materials on cast iron!

0

0.2

0.4

0.6

0.8

1

100 1000 10 4 10 5

Fric

tion

Coe

ffici

ent

Number of Cycles

µk = 0.23

onset of sticking

µk = 0.02

Friction measurements from Sandia microengine

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Friction in MEMS

• Friction, the bane of micromotors• Side-drive motors use electrostatic force between edges of poly

rotor and stator, limited by friction to 500 rpm (Fan et al.)• Improvements enabled 15000 rpm, operation for a week.

(Mehregany et al.)• Wobble motor uses rolling instead of sliding motion.

• Use friction for good• Arrays of cilia-like actuators use friction for in-plane conveyance• Friction holds hinged MEMS in place once assembled

• Avoid friction! Suspend!

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Mechanical Strength• Strength-to-weight ratio = area/weight ~ [s-1]• Stiffness ~ [s1]

• MEMS are relatively more stiff than macromechanical systems

3

3

4lEwtkandzkF zzz ==

EIP

x=

∂∂

4

4ζ ( ) gwhPwhIx ρζζ === ,12/, 3

• Deflection under own weight ~ [s2]

[ ]23

4

swh

whl ==ζ

x

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Resonant Frequency

• Resonant frequency ~ [s-1]• MEMS cantilever 100 × 3 × 0.1 µm3, f0=12 kHz• NEMS cantilever 0.1 × 0.01 × 0.01 µm3, f0=1.2 GHz

mkf

π21

0 =

ρπππE

lt

MlEwt

MlEIf 23

3

30 44213

21

===

3

3lEIk =

12

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Torque

• Torque• Surface micromachined

electrostatic micromotorsgive pN-m to nN-m torque

• Dissipation to friction is high

• Can only drive small loads, i.e. microshutters

rFrT ×=

θ∂∂

−= er

EF

fTTP πω 2==

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Electrical Properties

• Electrical resistance ~ [s-1]

• Current ~ [s2]

• Capacitance ~ [s1]

[ ] [ ] [ ]220 sssJAdAJI ===∫ ⋅= •

[ ]1−== sAlRσ

[ ]sdwlC == 0ε

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• The dimensionless Reynolds number Re represents ratio of inertial to viscous forces (drag)• fluid density ρ• object velocity v• characteristic

object length or diameter D

• fluid viscosity µ

Inertia

• Flow regimes • Re<<1: laminar flow, following smooth

streamlines • MEMS regime• Viscous forces dominate and inertial

forces can be neglected• Re < 2300: laminar flow• Re > 2300: turbulent flow, complex

eddies

[ ]2Re svd ==µρ

v = 500 µm/sd = 50 µm

µair = 1 × 10-6 m2/sµwater = 15 × 10-6 m2/s

Reair = 0.0016Rewater = 0.025

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Reynolds Numbers

turb

ulen

t flo

wla

min

ar fl

ow

• Earth’s tectonic plates 10-23

• Glacier 10-11

• Bacteria in water 10-5

• Marble falling in honey 10-2

• Tropical fish 102

• Ian Thorpe 105

• Car 106

• Airplane 107

• Whale 108

• Implications for MEMS• Squeezed film

damping• Parallel plate flow

damping• Pressure-driven

flow in microchannel

• No turbulence for mixing, only diffusion

[ ]s

ml

PwhQ33

12=

∆=

η

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Mixing at the Microscale

Schulte et al., Micronics

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Diffusion• Diffusion times (particle and thermal) ~

[s2]• Mixing at microscale mediated only by

diffusion• Time to diffuse over 10 µm million times

faster than over 1 cm• Heat is conducted out of small

structures quickly, so good thermal isolation for microstructures possible

rkTDπη6

=D

x6

2

=τ

Length of cube side x

Time to diffuse x

Volume 1 µL

1 mm

500 s

1 nL

100 µm

5 s

1 pL

10 µm

0.05 s

1 fL

1 µm

0.5 ms

1 aL

100 nm

0.05 ms

Whitesides group

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Microreactors• Leaf model

• Photosynthesis is the world’s most massive chemical operation ~ 300 billion tons of sugar per year (Biomimicry, J. Benyus)

• Heat transfer and mass transfer ~ [s2]• Higher surface area to

volume ratios give microreactors higher yields.

Jensen group, MIT

S. Bohn et al., Physical Review E, June 2002

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Analytical Sample Size• Scaling of analytical

chemical systems is limited • For a fixed

concentration of target molecules, total number of target molecules in sample is reduced as the sample is miniaturized

• Detector with greater sensitivity needed with a cut-off at detecting a single molecule

Number of molecules in 1 µM solution

Volume 1 µL

6×1011

1 nL

6×108

1 pL

6×105

1 fL

600

1 aL

6

100

103

106

109

1012

1015

1018

1021

10-18 10-15 10-12 10-9 10-6 10-3 100

Volume (L)

Con

cent

ratio

n (m

olec

ules

/mL)

1 Molecule1000 Molecules1000000 Molecules

Region representing concentrations of

less than one molecule

per sample

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Summary

• Miniaturization to microscale • Use forces that scale as [s1]

• Electrostatic forces at small separations• Surface tension

• Don’t fight forces that scale as [s2]• Friction

• Can use [s3] forces, but remember to consider other forces involved