01 Scaling Laws

1

Northwestern University

ME495/ME598ME495/ME598

SCALING LAWSSCALING LAWSLecture #2Lecture #2

October 3, 2006October 3, 2006


IntroductionScaling in MechanicsScaling in Adhesive ForcesScaling in Electromagnetic Forces Scaling in Piezoelectric EffectsScaling in Fluid Mechanics Scaling in Heat TransferThe Trimmer Force Scaling VectorScaling Examples

Manipulation in the Micro-worldActuation Scaling in Machine Tools and Factory

OUTLINE:

2


ScaleMicro-world

Size of Things


Why Miniaturization?

Minimizing energy and materials used for the manufacture of devices

Integration with electronics; simplifying systems

Cost/performance advantages

Faster devices

Increased selectivity and sensitivity

Exploitation of new effects, e.g., the breakdown of continuum theory

3


Definition of Scaling Laws

Scaling laws deal with the structural and functional consequences of changes in size or scale among otherwise similar (isometric) structures/organisms.

The three parameters that can be changed when the size of a structure is increased/decreased are:– Dimensions (e.g., thicker walls)– Materials (e.g., from brick to steel)– Design (e.g., from compression to tension elements)

Linear extrapolation of length comes easy to us, but we are quickly at a loss when considering the implications that shrinking of length has on surface area to volume ratios (S/V) and on the relative strength of external forces (actuation mechanisms), e.g., capillary tubes: weight scales as L3 and surface tension as L.

A 1 µm diameter capillary will raise water 30 m.


BASIC PREMISE

Isomorphic scalingIsomorphic scaling (i.e., all dimensions scale uniformly

CONSEQUENCE:

Scaling will lead to various physical effects that influence overall

system/device operation in unexpected ways

4


Why Do We Need to Study Scaling Laws?In dealing with very small devices, our “macro-intuition” on their operations is often misleading. It is necessary to develop an improved intuition about the likely behavior of the system when downsized.

Dominant physical quantities between different scales change:Gravitational, inertial forces become less effectiveVan der Waals forces, electrostatic forces, surface tension forces become more Important

Better understand the physical consequence of downscaling mechanical, electrostatic, electromagnetic, fluidic, and thermaldevices.

Based on the better understanding, explain the unexpected behaviors of micro machines and better understand why, in some cases, it makes sense to miniaturize a device for reasons beyondeconomics, volume and weight.


(I) Surface ~ (length)2 or S ~ L2

(II) Volume ~ (length)3 or V ~ L3

(III) Surface ~ (volume)2/3 or S ~ V2/3

In the last equation we learn that as the volume of a body is increased, its surface does not increase in the same proportion, but only in proportion to the two-thirds power of the volume (IV) S = k V0.67 or S/V = k V-0.33

The latter expression repeats another well-known fact: smaller bodies have, relative to their volumes, larger surface areas than larger bodies of the same shape

S/V Relations

5


Animal Size

1000

100

10

1

0.1

0.01 0.1 1 10 100 1000 10000Body mass (kg)

Meta

bolic

rate

(wat

ts)

Small mammals must keep on eating to stay warm ((heat loss ~ l2 and heat generation (through eating) is ~ l3))---insects avoid this problem by being cold bloodedA lower limit for lives in dry environment even for cold blooded animals, limiting them to 25~30 μm, because they can NOT retain their vital fluids long enough to survive.

D’Arcy Thompson: “On Growth and Form” and Knut Schmidt-Nielsen: “Why is Animal Size so Important?”


Animal Size

Niches in nature and gravity limit large size

S/V effect, surface tension, viscosity andBrownain motion limit small size tension

Biosensors in the 10 to 100µm range face evaporation of liquids

Number of life species

Size

1-2 mm

5-6 cm

2-3 m

30-40 cm

103-104 µm

104-105 µm

105-106 µm

106-107 µm

> 107 µm

102-103 µm

< 102 µm1

10

100

1,000

10,000

100,000

1,000,000

6


IntroductionScaling in MechanicsScaling in Adhesive ForcesScaling in Electromagnetic Forces Scaling in Piezoelectric effectsScaling in Fluid Mechanics Scaling in Heat TransferThe Trimmer Force Scaling VectorScaling Examples


OUTLINE:


Scaling of Gravity & Pressure

GravityFgr=m*g=L3*g ~ L3

Pressure P= Fgr/S=L*g ~ L

At the microscopic level, adhesion forces dominate, because the details of the forces at the molecular level are much largerthan the gravitational ones

Fgr

g is the gravitational acceleration

7


Scaling of Energy

Kinetic energy: Ek = mv2/2; Thus,

Ek ~ L3 - for constant v

Ek ~ L5 - for v ~ L

Gravitational potential energy: Ep = mgh; Thus,Ep ~ L3 - for constant h

Ep ~ L4 - for h ~ L


Scaling in Springs

Spring Force

EnergyPotential

PeriodOscillation

FrequencyOscillating

2/3~ LTsping

22 ~)( LLKEpot Δ=

~)( LLKFspring Δ−=

2/3~21 L

mk= −

πω

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Scaling in Rotating Bodies

Inertial Momentum

Rotational Kinetic Energy

2 5I ml l∝ ∝

2 512

K I lω= ∝

This implies that, at constant ω, the rotational energy of a small systems decreases rapidly. A small motor will reach top speed in a fraction of a second; large motors may require seconds to reach full speed.

MICRO STRUCTURE BULLETIN NO.2 MAY 1997


Scaling in Cantilever Beams

Maximum Stress

Oscillating Frequency

2max 2 2

12 62FLb F lb l

σ −= = ∝

2 11 4

EI lAL

ω πρ

−= ∝

If the same maximal stress is desired in the beam, then the applied load F must be

reduced as the square of the linear dimension, by assuming simple beam

theory holds at all structural scales and that the material properties are also

constant

Wautelet, Michel. “Scaling laws in the macro-, micro- and nano-world”, European Journal of Physics 22 (2001): 602-611.

A beam a thousand times smaller bends a million times less under its own weight.Resonant frequencies are large in small systems.

9


Scaling in Strength to Weight Ratio

Strength α l2 while Weight α l3

Small things tend to be stronger

21

3

Strength l lWeight l

−∝ =

50 Timeshttp://www.control.hut.fi/Kurssit/AS-74.3136/materials/scaling_s.pdf#search=%22Quan%20Zhou%2C%20scaling_s.pdf%22

VS

Young and small trees appear slender and old and tall ones

appear squat or stunted


Scaling in Strength to Weight Ratio(continued)

Compressed materials may withstand a maximal stress of σmax

If they are submitted to their own weight, mg,σmax ~ mg/A (A – cross section, e.g., ~ d2)

it follows thatσmax ~ L3/L2 ~ L

For a given material σmax is constant, hence when the dimensions increase, the diameter, d, of the supporting material is such that d2 ~ L3, or

d ~ L3/2

CONSIDER: Animal bones!

10


Scaling in Swimming

Skin friction Rα l2

The larger the creature grows, the greater its swimming speed

2E RV∝

3

2

E lV lR l

∝ = =

R – skin frictionV - swimming speedE – Energy ~ mass of creature’s muscles l3

Madou, Marc J. Fundamentals of Microfabrication: The Science of Miniaturization. Boca Raton: CRC Press, 2002.


Flying (birds fly from 10.8 to 20.7 m/sec):– Wing length ~ l ~ M 1/3 and wing area ~ l2 ~ M 2/3

– The characteristic speed for flying varies as l 1/2 or M 0.17

– Drag/lift forces are given by FL = 1/2 CL ρAv2. This expression has an order of 2 + 2v

– Of key importance is the lift-to weight ratio (divide by l3) which is of the order 2v-1

– Since the lift-to-weight ratio should be invariant with scale to achieve flight a zero order scaling law is needed thus v must be 1/2 to achieve sufficient lift (same result as above)

Flying

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Adhesive forces dominate in the micro-worldAdhesive forces are due to the forces between atoms and moleculesThree forces:– Van der Waals force– Surface tension – Electrostatic force

Fadh ~ L2 (In lots of cases)

Adhesive Forces

Feddema, Xavier and Brown, 1999, Micro-Assembly Planning with van der Waals Force


Scaling in Friction

In the macro-world, friction is independent of the contact area.

Ffr = μFgr = μmg ~ L3

In the micro-world, due to surface roughness and large adhesive forces, strictionstriction (i.e., the combination of adhesion and friction) forces, Fstr, has to be taken into account:

Fstr ~ L2

12


Comparison between Gravity & Adhesive Forces

Gravity Fgr ~ L3 and adhesive Fadh ~ L2

Fadh / Fgr ~ L-1

The adhesion force dominates the gravitational force at low L.

The critical value at which both forces are equal depends on x and on the nature of the medium between the two solids. However, below say L = 1 mm, Fgr is much less than Fvdw.

Gravitation may then be neglected at such small dimensions, both in the micro- and the nano-worlds.




OUTLINE:

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Van der Waals Forces

Attraction force between a sphere and surface

H depends on the nature of the medium between the slabs– Order of 10-19 J in air – Order of 10-20 J in water

Relation valid for x between around 2 and 10 nm.

12vdwHRF

x= H- Hamaker constant

x

R

•F. Arai, D.Ando, T. Fukuda, Y. Nonoda, T. Oota, Micro manipulation based on micro physics, 1995.



Two atoms

Two spheres

6iErλ

= −λ - London-van der Waals constant.

H = π2n2λ

Hamaker constant (where n is the number of atoms per cm3).


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Van der Waals energy between a sphere and a rectangular block

Total Energy

Van der Waals force Feddema, Xavier and Brown, 1999, Micro-Assembly Planning with van der Waals Force



http://www.control.hut.fi/Kurssit/AS-74.3136/materials/scaling_s.pdf#search=%22Quan%20Zhou%2C%20scaling_s.pdf%22

Fvdw~L2

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Van der Waals force is very sensitive to surface roughness

Effects of Surface Roughness



Scaling in Surface Tension

The molecules on the surface tend to be pulled away from the surface, and therefore work must be done to bring the molecules from the body of the liquid to the surface.The work required per unit area to bring molecules to the surface (i.e., to create a new surface) is called surface tension.

http://www.me.jhu.edu/~thwang/notes/Scaling-I.pdf#search=%22Jeff%20Wang%20Johns%20Hopkins%20University%2C%20scaling%22

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When humidity is high, or with hydrophilic surfaces, a liquid film between a spherical object and a planar surface contributes a large capillary force

Assume R2 << d,

Scaling in surface tension is as LL11

γ-the surface tension force

d-the object diameter

•F. Arai, D.Ando, T. Fukuda, Y. Nonoda, T. Oota, Micro manipulation based on micro physics, 1995.



Capillary tubes (L3 vs. L1)– weight scales as L3

– surface tension as L

Size of a droplet (L3 vs. L1)

http://www.me.jhu.edu/~thwang/notes/Scaling-I.pdf#search=%22Jeff%20Wang%20Johns%20Hopkins%20University%2C%20scaling%22

A 1 µm diameter capillary will raise water 30 m.

The mass of a liquid in a capillary tube, and hence the weight, scales as L3 and decreases more rapidly than the surface tension, which scales as L as the system becomes smaller. That is why it is more difficult to empty liquids from a capillary compared to spilling coffee from a cup.

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Surface Tension Gripper

Ref: http://www.aolab.mce.uec.ac.jp/AOLAB/Eng/IWMF/IWMF.html

Small object, a metal ball of 0.8mm in diameter, 2.4 mg in weight is successfully picked up

Picking up several small objects from the working area without damage to the surface by using surface tension


Self Assembly Based on Capillary Forces

Xiaorong Xiong, Yael Hanein, Controlled Multibatch Self-Assembly of Microdevices, JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 12, NO. 2, APRIL 2003

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Electrostatic Forces

Definition:– The electrical force of

repulsion or attraction induced by an electric field E

Coulomb’s law

'

2

14

qqFrπε

=

r

q q’

ε-permittivity of the material separating two particles

Madou, Marc J. Fundamentals of Microfabrication: The Science of Miniaturization. Boca Raton: CRC Press, 2002


Scaling in Electrostatic Forces

Electric potential in parallel plates:

ε0, εr α l0

22 01

2 2rWLVE CV

dε ε

= − =

0 0 1 1 1 23

1

( )( )( )( )( )l l l l lE ll

∝ =

ε0 and εr are the permittivity and relative permittivity of the dielectric medium

L


V (breakdown voltage) is assumed proportional to L

19


The forces in the d, W, and L directions

Lateral forces tend to move the plates towards each other

Fd, FW, and FL α L2

– Assume V~L and d>10um

A 10x size reduction of the parallel plates will lead to a 100x decrease in the electrostatic forces

Scaling in Electrostatic Forces

Fd

Fw

FL

Hsu, Tai-Ran MEMS & MICROSYSTEMS: Design and Manufacture. Boston: McGraw Hill, 2002.


Micro-intuitionThe Paschen effects suggest yet another advantage in the non-linear region - the field scales like l -1/2 and the force like l1Understanding scaling allows one to choose an actuator principle with more confidenceThere are many factors though beyond scaling that play a role in the decision which actuator mechanism to use e.g. absolute energy density

Paschen Curve in Air

V

Breakdown Voltage V

Pressure x distance P X d ( mm Hg-mm)

At 1 atmosphere = 760 mm Hg

0

0

200

400

600

800

1000

1200

1400

10 20 30 402

2.6 µm 13. 16 µm

New Physics and Chemistry

Breakdown of Continuum Theory


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Because of non-linear effects electrostatic devices can be operated in air without breaking down (operation on the left side of the Paschen curve). New physics and chemistries to be explored.

Breakdown of Continuum Theory



Left: Vertically driven polysilicon bridgeResonant microstructures/devicesused for accelerometers

Right: Laterally driven electrostatic actuator large displacement devices

Electrostatic Actuation & Sensing


Actuation and Sensing can utilize both the vertical and laterally driven motionsAdvantage of lateral movement over vertical movement– The force changes linearly with

the distance (L)– Dissipative damping is avoided– Larger displacement

Electrical fields can exert great forces but generally across very short distance only

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Electrostatic Applications

Linear Electrostatic Motor (comb driver)

Rotational Electrostatic Motor


Actuators – Optical switches– Micro motors (wobble)– Micro valves– Grippers– X-Y stages for AFM

Sensors– Gyroscopes– Accelerometers

Micro-gripper Ref: Enikov2003


IntroductionScaling in MechanicsScaling in Adhesive ForcesScaling in Electromagnetic ForcesScaling in Piezoelectric effectsScaling in Fluid Mechanics Scaling in Heat TransferThe Trimmer Force Scaling VectorScaling Examples


OUTLINE:

22


Ampere’s law:

Force on a wire in B-field

Electromagnetic Forces

0 0c AB dl J dA Iμ μ⋅ = ⋅ =∫ ∫

u0 -the magnetic permeability of free space

J – current density



Scaling in Electromagnetic Forces

Assume J constant

So a 10x reduction of size leads to a 10,000xreduction of electromagnetic forceScaling results of the interaction between a coil and a permanent magnet is somewhat better

422 ))(( lllF =∝

3F l∝Hsu, Tai-Ran MEMS & MICROSYSTEMS: Design and Manufacture. Boston: McGraw Hill, 2002.

23


Scaling in Electromagnetic Forces

Scaling results of the interaction between a coil and a permanent magnet

Assume constant heat flow per unit area of winding

The assumption of a constant temperature difference between windings and the environment yields l2, yet the power dissipation per unit volume scales as l-1. Superconductors could eliminate this problem.

3F l∝

3F l∝




Manipulation in Micro-worldActuation Scaling in Machine Tools and Factory

OUTLINE:

24


Piezoelectricity

Piezoelectricity is the ability of crystals to generate a voltage in response to applied mechanical stressThe piezoelectric effect is reversible in that piezoelectric crystals, when subjected to an externally applied voltage, can change shape by a small amountPiezoelectric material: crystals

A 1 cm cube of quartz with 500 lbf (2 kN) of correctly applied force upon it, can produce a voltage of 12,500 V.

http://www.stanford.edu/class/me220/data/lectures/lect06/equivale.gif&imgrefurl=http://www.stanford.edu/class/me220/data/lectures/lect06/lect_6.html&h=496&w=784&sz=7&hl=en&start=19&tbnid=01BmrCASO5qtdM:&tbnh=90&tbnw=143&prev=/images%3Fq%3DPiezoelectricity%26svnum%3D10%26hl%3Den%26lr%3D%26sa%3DN


Actuator Types

Bimorph (extension) Bimorph (bending)

Longitudinal WaferTransverse Wafer

Stack Actuator

25


Scaling of Piezoelectric Effects

Hook’s Law

Constitutive equation for dielectrics

Coupled Equations

D is the electric displacement, is permittivity and

E is electric field strengthε

S is strains is complianceT is stress.

The matrix d contains the piezoelectric coefficients for the material

Piezoelectric effect scales down with the bulk of the material, miniaturization opportunities are limited hybrid-type micro actuator are more reasonable


Piezoelectric MaterialsApplications:– Mechanical to Electrical

Force, Pressure, and acceleration sensorsSmart Sensors for Side Impact DiagnosticsHigh Voltage - Low Current Generators: Spark Igniters for Gas grills, small engines, etc.Yaw Rate SensorsPlatform Stabilization Sensors

– Electrical to Mechanical: Ultrasonic motors, Small Vibration ShakersMicroactuators (High Precision Macro actuators)Sonar array arrays for collision avoidancePumps for Inkjet Printers

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Scaling in Electricity

Electric resistance:

Resistive power loss:

Power dissipated by unit area

Reduction in size leads to more dissipated power. One way to diminish this effect is to reduce the applied voltage

1LR lA

ρ −= ∝

ρ - electric resistivity of the conductorL – Length A – cross section area

2vP lR

= ∝ V - the applied voltage

12unit

P lP lArea l

−= ∝ =




Capacitance:

C = ε0A/d C ~ L

Charge:

Q = CVel Q ~ L

Energy stored in the capacitor:

Ecap = Q2/2C Ecap = ~ L(Stored energy decreases with the size of the capacitor)

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Magnetic field in a solenoid:

B = μIeln/L B ~ L

(It is assumeb that n remains constant and that a constant current density is maintained)

Magnetic energy stored in the solenoid:

Emag = B2Vvol/2μ Emag ~ L5

Fmag ~ L4

(Technological problems must also be considered: Number of turns not independent of L since very this wires cannot be yet manufactured; Solenoids smaller than 1 mm3 seem impractical; Maximum current density is limited by energy dissipation (i.e., maximal allowable temperature; Etc.)

n – number of turnsL – wire lengthIel - current

Vvol – Volume of the magnetic field

Vvol ~ L3



Previous laws apply for miniaturizing devices – it is more important to look at the scaling of electric power supplies

– For a system that carries its own power supply, the available power is related to the volume, i.e., Eav α (l)3

– Therefore, P/Eav α (l)-2

This means that a 10x reduction in the size of the power supply system would lead to a 100x greater power loss due to the increase in resistivity


28


General Scaling Laws

Small things are fastSmall things are strongSmall things are not affected much by gravity compared to frictionSmall things do not provide much torque or powerSmall things are dominated by Van der Waals forces, surface tension

Time l0

van der Waals Force lDistance lVelocity lSurface Tension lMuscle Force l2

Friction l2

Mass l3

Gravity l3

Magnetic force l3

Torque l3

Power l3




OUTLINE:

29


Scaling in Fluid MechanicsTwo characteristic parameters– Density &Viscosity Dynamic Viscosity (kg/m s)– Ratio of shear stress/Shear rate– Causes shear when the fluid is moving– Air 1.85 10-5 and Water 10-3

Newtonian flow– Shear stress and velocity is linearly related– Only a function of the nature of the fluid– Water, air

Non-Newtonian flow– Viscosity is also a function of velocity

gradient; decreases as gradient increases – Milk, blood

s

s

FAuh

τη = =Γ

A

u

h

F


Scaling in Fluid Mechanics

Navier-Stokes lawHagen-Poiseuille law– Experimental determination of fluid

viscosity for laminar flow of a fluid through a capillary

4

8r pQ

xπ

ηΔ

=2

8 xuprη−

Δ =Volumetric flowu=Q/A=avg. velocity

30



Scaling in rate of volumetric flow– Assume constant pressure drop per unit

length

– 10x reduction in size leads to 10,000x reduction in rate of volumetric flow

Scaling in pressure– Assume constant u, x

– Large pressures are needed to drive micro-flows

44~

8r pQ l

xπ

ηΔ

=

22

8 ~ [ ]xUp lr

η −Δ = −



Unfavorable Volume & Pressure scaling cause design of conventional volumetric pumping at the micro-scale impractical

Surface-driving forces more useful:– surface-traction – surface-tension flow (capillary effects)– piezoelectric– electro-osmotic– electro-hydrodynamic

31


Scaling in Fluid MechanicsReynold’s number

– Re < 2100 => laminar flow regimeSlow fluid flow, no inertial effects

– Re >4 000 => turbulent flow regime– whale swimming at 10 m/second ~

300,000,000– mosquito larva, moving at 1mm/sec ~ 0.3

Assume U~l =>Re∼ [l]2Always laminar flow!Dominant laminar flow in micro systems makes it difficult to mix liquids in micro-channels



Micro-fluidic mixing:

32



Body falls vertically in a fluid

Viscous forces rapidly damp any motion for objects with small dimensionsSmall "swimming" objects are very rapidly brought to a halt due to viscosity.A very small body remains immobile in air, as experienced by us by looking at dust in sunrays

2r

22

lim

2

4 ~ [ ]18

~ [ ]

grV l

l

ρη

τ

= −




OUTLINE:

33


Scaling in Thermodynamics

Energy required to heat a system to temperature T is proportional to mass

Eth ~ L3

For conduction and radiation power dissipation is proportional to area

Pd ~ L2

The time needed to homogenize the temperature in a system of given shape is

th ~ L2


Scaling in Heat Conduction

Fourier law in heat conduction

Fourier number– commonly used to determine the time increments in a

transient heat conduction analysis:

Easier to remove heat from a smaller object10x reduction in time with a 10x size reduction

– Application to mammals– Efficient micro heat sinks dissipate up to 1 kw/cm2, 40 time

more than a conventional heat sink

Cp-Specific heat

λ−Thermal diffusivity

Q – Rate of heat conduction

~ [ ]TQ kA lx

Δ= −

Δ

Q Q

A

x

K-heat conductivity


34


Scaling in Heat Convection

Newton’s cooling law

– h - Heat transfer coefficient – Q - Heat flow

– Total heat flow primarily depends on the cross-sectional area, A

2~ [ ]Q hA T l= Δ




Nature seems to favor small, e.g., insects are very well adapted:– See species abundance curve

(many niches)– Insects walk on water (surface

tension supports their mass m)– Insects jump very far (E~mh and

muscle for that work is ~m so h is a constant)

– Faster cooling and heating (cold blooded)

– Small thermal stresses (Small Biotnumber, i.e., little thermal stress)

Derivation of the heating/cooling time constant:

A = surface areaV = volumeCp =specific heat capacityλ = heat transfer coefficient

at the surfaceL = characteristic

dimensionκ = thermal conductance of the

solid

τ =ρCpVλA

Biot =λLκ

τ =ρCpVλA

35


By heating a micromachined and thermally isolated Pt line the thermal budget can be reduced drastically because:– Small thermal mass makes the device

consume much less power– Switch on and off much faster– Biot number is small so it does not

crack that easily

B = λLκ



Thermal Actuators

A) ThermopneumaticactuatorB) Bimetallic actuatorC) Shape Memory alloy actuator

A

B

C

36


A molecule diffuses over 10 µm, 1 million times faster than over 1 cm:

A microbattery on a micromachineis usually not a good idea as power scales with volume, a solar cell incorporated in a micromachinemight be a good idea though, i.e., beam the power in rather than generate it on the micromachineThere are advantages in working with arrays of micro-electrodes, for example, an increased sensitivity for micro arrays of amperometricsensors electrodes (see next viewgraph)

x = 2D τ

m 2 to50 from e.g.rn = R :constant remains area Total 22

μππ

n2 π r > 2πR

πR 2 = nπ r 2

n = R 2

r 2 (625 ) or 625.2 > 5

Electrochemistry


A B

C

δ = πD0t( )12

I l = nFAC∞0 D0

πt⎛ ⎝ ⎜ ⎞

⎠ ⎟

12

I l = nFAC ∞0 D 0

πt⎛ ⎝ ⎜ ⎞

⎠ ⎟

12 + AnFD 0

C∞0

r

Nonlinear diffusion and the advantages of using micro-electrodes:

An electrode with a size comparable to the thickness of the diffusion layer

The Cottrell equation is the current-vs.-time on an electrode after a potential step:

For micro-electrodes it needs correction :

I l = nFAD 0C∞

0

δ

Electrochemistry

37




OUTLINE:


The Trimmer Force Scaling Vector

Trimmer has created a matrix to represent force scaling with related parameters of acceleration, time, and power density (P/V) that is required for scaling systems in motion

Force scaling vector: F=[F=[llFF]=[]=[ll11 ll22 ll33 ll44]]TT

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Scaling in Time

Even in the worst case, F~l4, the time required to perform a task remains constant, when the system is scaled down. Under more favorable force scaling, for F~l2, a system 10 times smaller can perform an operation ten times faster. Smaller things tend to be fast.


39


Scaling in Power Density

When F~l2, P/V0~ l-1. When the system is scaled down 10 time, P/V0 increase by a factor of 10.


Wautelet, Michel. “Scaling laws in the macro-, micro- and nano-world”, European Journal of Physics 22 (2001): 602-611.



Example: Scaling effects of reducing an object’s weight by a factor of 10

W=Mg, where M α l3, which means we use Order=3

there is no reduction in acceleration (l0)there is an (l0.5)= (10)0.5= 3.16 reduction in time there is an (l0.5)= 3.16 reduction in power density (P/V)

Finally, since V is reduced by a factor of 10, the power consumption reduces by P = 3.16/10 ~ 0.3 times reduction

∝


40


Questions about Scaling Basics

Question:– If two objects fall from a certain

height, in different scales, how come the time to reach ground is different?

Physical laws will not change at different scales used.

We are not looking at this problem!!!

S


The size of the system is represented by the linear dimension– Choose linear dimension– All dimensions are equally

scaled down– Dimension of the system

decreased by a factor of 10, then l = 0.1

S

l*S

l-linear dimension

Questions about Scaling Basics

Correct interpretation of the problem

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Example

A 1.0 cm cube of a material weighing 1.0N acted upon by gravity.

1μm cube of the same material – Assume the same acceleration, g– F=ma ~ l3Time to travel 1.0 cm in the Macro-world T= (2 (0.01m)/9.8m/s^2)^0.5=0.045s = 45ms

Time to travel 1μm in the Micro-world T= (2 (0.000001m)/9.8m/s^2)^0.5 = 0.45ms

T=0.045s* (10-6/10-2)0.5=0.45ms

Same system of units

S

l*S

l-linear dimension122xt l

a= ∝




OUTLINE:

42


Scaling Effects



Scaling Effects

Adhesive force could be attributed to electrostatic, Van der Waals or surface tension. Proportional to surface area.In the micro-world, adhesive forces dominate gravityIt is typically easy to pick up parts, but very difficult to release parts because of these interactive forcesPart adheres to one finger when the gripper opens Pneumatic probes require reversible pressure to “blow”the part away from the probe, thus releasing itRelease of objects is difficult.


43


Motion Planning in the Micro-world

As parts approach 1-10 μm or less in outside dimensions, interactive forces such as Van der Waals and electrostatic forces become major factors that greatly change the assembly sequence and path plans

Assembly plans in the micro-domain are not reversible, motions required to pick up a part are not the reverse of motions required to release a part.

Investigate how motion planning changes based on the interactive forces in the micro-domain



Van der Waals Forces in MicroOperations

Gravity 3.678093~10-13


44


Van der Waals Forces in MicroOperations



Electrostatic Forces in MicroOperations


45


Electrostatic Forces in MicroOperations



Pick and Place Techniques


46


Micro Operation Example

2 μm diameter sphere of copper– H=3.43774E-20 J

1 μm square block of copperResting surface of aluminum – H=10.5648E-20 J– D=0.1 nm

– Fg =3.678093E-13N– Fsa=0.110μN – Fta=0.116μN (tool centered)– Fta=0.059μN (tool moved to one edge contact)– Fta=0.025μN (tool titled 45 degree)



Handling Skills with Adhesives

Sliding and inclining tools Using dual manipulators

The disadvantage of these strategies is that friction between tool and part may generate micro dust from yhe object.

47


Vacuum Effects

Principle– Force is generated by the pressure difference on

both sides of the micro-components – A simple equation to estimate the suction force

– A better estimate (Qian, 2000)

2

4PdF π

=P: Pressure differenced: aperture dimension of suction hole

2

4Pd

ddDhF ⎟

⎠⎞

⎜⎝⎛ −

+=π


Vacuum Gripper

A vacuum gripper consists of a glass pipette and a vacuum control unit.

Glass pipetteMade from soda or borosilicat glass tubes. Characterized by inner and outer diameter.An optimum tip size can be found in the range between 25 and 50 μm, which is about 25 - 50% of the object size

Vacuum control systemProvides an adequate air flow rate in both directions for the whole range of operationsVacuum must be sufficient to pick up objects.

48


Vacuum Manipulator Operation

Basic SkillsPickHoldPlace

Releasing StrategiesStrip offPushBlow away

Ref: Zeach(1997)




OUTLINE:

49


Comparison of Maximum Energy Density ofVarious Actuation Mechanisms



Advantages of PTZ Motors Over Various Actuation Mechanisms

Low voltages: no air gap is needed; mechanical forces are generated by applying a voltage directly across the piezo-electric film. With a O.3-~m thin film, only a few volts are required, as opposed to hundreds of volts needed in air-gap electrostatic motors.Geardown: motors can be fabricated without the need of a gearbox. Electrostatic wobble motors are also able to produce an inherent gear reduction but do not have the high dielectric advantage.No levitation: with electrostatic motors, levitation and flatness are very important to obtain good sliding motion of the rotor around the bearing. The piezoelectric motor depends on friction so that no levitation is required, and it can be freely sized.Axial coupling: electrostatic motors require axial symmetry around the bearing. Since height is difficult to obtain with most nontraditional micromachining techniques, a limited area is available for energy transduction. With the piezoelectric traveling-wave motors, linear or rotary motors can be built

50


Magnetic vs. Electrostatic Actuation

Electrostatic actuation is preferred over magnetic one for surface micro machined actuators

Thin insulating layers such as SiO2 or Si3N4 exhibit breakdown strengths as high as 2 MVlcm. The power density in this field is 7 x 105 Jlm3; this value isequal to the power density of a 1.3- T magnetic field. The contracting pressure induced by this field is 1.3 MPa. A voltage of about 100 V is sufficient to generate the strong fields mentioned.

The electrostatic force is a surface force exhibiting a favorable scaling law. The actuation is simple, as it involves only a pair of electrodes separated by an insulator.

The electrostatic actuator is driven by voltage, and voltage switching is far easier and faster than current switching (as in electromagnetic actuators). Energy loss through Joule heating is also lower.

Weight and power consumption are low. The following comparison between electrostatic and magnetic micromotors demonstrates that many factors besides scaling need to be considered when deciding upon a certain type of actuation principle. Whereas, in some cases, the magnetic power might scale disadvantageously into the microscale, the absolute forces achievable are larger


Comparison of Electrostatic vs. Magnetic Actuators

Electrostatics is useful in dry environments and over limited distances; eletromagnetics is still difficult to collapse into integrated structures.

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Actuator Scaling IssuesElectromagnetic actuators scale linearly

– Inefficiencies seem to become more tolerableE.g., Active X, passive Y “flattened” voice coils with longer end turn regions

– Enable an XY platten to be driven from X and Y actuators each of which is anchored to ground

Hertz contact drives scaling of ballscrews and friction drivesLeadscrews (sliding contact) scale linearlyWire capstan drives scale non-linearly (in favor of use in smaller machines!)

– E.g., Roland Modella machinesShort stroke “solid state” actuators’ range of motion generally still too small

– Piezoelectric, Electrostatic, Magnetostrictive – “Inchworm” “traveling wave”…designs that use these actuators can provide

long range of motionMuch exciting development work in this area (Kobe, Delft, Philips…)Enables X,Y,Θ motion planar stages

The greatest benefit of small machines is architectures that allow actuators to mostly be anchored to ground thereby minimizing theamount of moving cables

Alexander Slocum, design of Design of Small Precision Machines


Scaling Issues: Sensors

Unfortunately sensors do not really get smaller than they are now – There probably is a lot of opportunity for advancement!

Capacitance sensors on a chip…Scale read heads on a chip…

Accuracy of measurement is another issue– Are micro-sensors more or less accurate than their macro-

counterparts?Think of a small cantilever beam used as a sensing element!! (Beam deformation is ~L2 thus the deflection is much less in a small system! Provided the resolution is the same the it is clear that the micro-sensor is less accurate!)There is not yet a good general response to this issue!

Alexander Slocum, design of Design of Small Precision Machines

52



Manipulation in the Micro-worldActuation Scaling in The Machine Tools and Factory

OUTLINE:


Scaling of Structures

Size of machine/range of motion ratio is generally less of an issue the smaller the machine becomes.Structurally, smaller machines can be made higher performance:– Beam stiffness is proportional to:

– Thickness to the 3rd power– Length to the -3rd power– Width

– Beam mass is proportional to:– Thickness, length, width

– Beam natural frequency is proportional to:– Thickness– Length to the -2nd power

– Because the structure is smaller it is often easier to make it monolithic and hence of a more rigid nature

More material is also often tolerated/allowed to be addedSnap fit structures? ☺

Cabling and wiring can become more problematic because sensor wires do not shrink.

Alexander Slocum, Design of Small Precision Machines

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Scaling of Machine Tools

Trend is towards more small precision manufactured parts for consumer products and medical devices.– Large volumes can be economically made using conventional

machines with mass-fixturing or parallel feeds (e.g., screw machines)Small volumes might be better made on small machines– Can a small precision machine be made less costly than a large

precision machine?– A small machine takes up less space and thus has less overhead cost– The cost to produce a small run of small precision parts on a small

precision machine should thus be lessSmall precision machines might have work volumes on the order of:– Cubic foot, football, grapefruit…– Include benchtop hobby machines to benchtop production machines

There are currently many manufacturers of desktop machine tools– Generally considered to be of modest accuracy (0.1 to 10 microns)– Can we do better?– Have these machines taken full advantage of scaling trends and new

ideas?

Alexander Slocum, Design of Small Precision Machines


Scaling of Machine ToolsScaling of Machine Tools

http://www-ieem.ust.hk/dfaculty/ajay/courses/ieem215/lecs/cnc_f1.jpg

54


-- Enhancement of Precision by Downsizing Enhancement of Precision by Downsizing --

NOTE: Machine MT is N - times bigger than machine mMT

MACHINE: MT mMT

Thermal Expansion N 1

Geometric Errors N 1

Rigidity N 1

Clearance N 1

Forces N 2 1

Static Deformation N 1

Inertial Forces N 3 1

Etc.

ASSUMPTION: Geometric SimilarityASSUMPTION: Geometric Similarity

Scaling of Machine ToolsScaling of Machine Tools


Scaling Laws

Natural Frequency:

Equivalent Rigidity:

4n nEI

mLω α=

E: Elastic Modulus

I: The area moment of inertia

m: Mass per unit length

L: Length of the beam

an: an appropriate coefficient for boundary conditions

3eqEIKL

=

A B

55


Scaling Laws

Comparing Two Beams

4

4

2

BA

BA

BA

I LII

mm

LL

ρ

ρ

ρ

∝

=

=

=

_ _n A n Bω ρω=

__

eq Beq A

KK

ρ=

Natural Frequencies

Equivalent Rigidity


Scaling Laws - Example

Lee et al. (2005)FEM analysis of the dynamics of a conventional machine tool (2100×900×960)Rigid body assumption of the structure

4th

3rd

2nd

1st

Modes

77.8264.4527.1124.85

FE analysisNatural Frequency (Hz)

71.159.628.023.7

Experiment

Comparing the FEM analysis with Experiments

56


Scaling Laws - Example

Lee et al. (2005)Comparing the natural frequencies of a conventional machine with those of a meso-scale machine tool (150×70×140)

Natural frequencies of the mMT derived by the scaling laws: 385Hz

FEM results of natural frequencies of mMT: 408Hz

Measured natural frequency by impact test: 81Hz

Weak joints lower the Natural Frequencies.


Joint Properties

Modeling joints as springs and dampers in the lumped parameter model

Parameter identifications using measured FRFs

Lee et al. (2005)

- Individual joint parameters are identified by separate sets of tests

- Larger damping and smaller rigidity due to bolted and sliding joints were found.

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Modeling & Identifying Joint Properties

Lee et al.(2005)

Lumped Model

Identifying the parameters

m1=4.20kg, k1=1.3×107 N/m, c1=939.0N•s/m


Scaling the Factory

Okazaki, AIST

Milling, Drilling, Turning, Grinding, Polishing, EDM, ECM, Laser machiningLaser treatment, ED grinding, EC grinding, EC polishing, Laser-assisted milling, etc.

Milling, Drilling, Turning, Grinding, Polishing, EDM, ECM, Laser machiningLaser treatment, ED grinding, EC grinding, EC polishing, Laser-assisted milling, etc.

Processes

High-speed spindle(150,000 rpm)

Middle-speed spindle(25,000 rpm)

Laser units

Exchangeable units

Low-speed spindle(with rotary encoder)

Piezo vibration unit

58


Scaling the FactoryDesk top factory in Japan (DTF)Revised Desktop UHS milling machine -spindle rotation at up to 300,000 rpm and higher stage speed/acceleration with linear motors: 400 mm/s-2.1 G (Okazaki at AIST)

2000


Scaling the Factory

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Scaling the Factory

Benefits: – Savings in energy and materials– Easy to move– Easier to keep clean– Higher accuracy– Temperature control is easier– Humans are outside the clean area– Less expensive

Problems: – Technological hurdles– Mass production– How to ensure quality


Conclusions

Due to the scaling down, there are so called “scaling effects” which must be

taken into account in the micro-world. The physical laws remain the same, but their significance at different scales changes

60


ReferencesHsu, Tai-Ran, MEMS & MICROSYSTEMS: Design and Manufacture. Boston: McGraw Hill, 2002.Madou, Marc J., Fundamentals of Microfabrication: The Science of Miniaturization. Boca Raton: CRC

Press, 2002.Wautelet, Michel., “Scaling laws in the macro-, micro- and nano-world”, European Journal of Physics, 22

(2001): 602-611. Feddema, Xavier and Brown, “Micro-Assembly Planning with van der Waals Force”, 1999Yamagata, Y., T., Higuchi, Micropositioning device for precision automatic assembly using

impact force of piezoelectric elements, Proceedings - IEEE International Conference on Robotics and Automation, v 1, pp. 666-671, 1995

Arai, F. , D.Ando, T. Fukuda, Y. Nonoda, T. Oota, Micro manipulation based on micro physics, 1995.

S.W. Lee, R. Mayor. J. Ni, “Dynamic Analysis of a Meso-Scale Machine Tool,” ASME J. Manuf. Sci. Eng., to be published

Alexander Slocum, Design of Small Precision Machines.http://www.aolab.mce.uec.ac.jp/AOLAB/Eng/IWMF/IWMF.html

Xiaorong Xiong, Yael Hanein, Controlled Multibatch Self-Assembly of Microdevices, JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 12, NO. 2, APRIL 2003

http://www.ee.ucla.edu/~wu/ee250b/Magnetic%20Actuators%20(2).pdf#search=%22magnetic%20actuator%22

Zech, W. Brunner, M. Weber, 1997, A Vacuum tool for handling microobjects with a NanoRobot Robotics and Automation, 1997. Proceedings., 1997 IEEE International Conference on

Documents

01 Scaling Laws