slide 1 An Experiment to Determine the Accuracy of Squeeze-Film Damping Models in the Free-Molecule...

An Experiment to Determine the Accuracy An Experiment to Determine the Accuracy of Squeeze-Film Damping Models in the of Squeeze-Film Damping Models in the

Free-Molecule Regime Free-Molecule Regime

Presented atPurdue University

Birck Nanotechnology CenterApril 6, 2007

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration

under contract DE-AC04-94AL85000.

Hartono (Anton) SumaliHartono (Anton) Sumali

1526 Applied Mechanics Development1526 Applied Mechanics Development

Albuquerque, NM

Gas damping is important in MEMS.

Motivation:• Many micro/nano devices need high Q factor. Examples abound in

• MEMS switches need high speed (high Q). • Resonant cantilever sensors need high responses. • MEMS gyroscopes.• MEMS accelerometers need controlled damping.

• Damping can reduce Q from several hundred thousands to several hundreds. • Squeeze-film damping determines the dynamics of plates moving a few microns

above the substrate.

Objective:• Provide experimental validation to squeezed-film damping models for rigid plates.

• Continuum models are not known to be valid in rarefied (free molecule) regime. • Molecular-dynamics-based models for predicting squeezed-film damping give

different results.• So which model should one use?• Need experimental validation!

• Published experimental data were obtained for squeeze-film damping on flexible structures. Have been used to validate theory derived for rigid structures.

Squeeze-film damping determines the performance of MEMS.

Radio-frequency MEMS switch is designed to close upon electrostatic actuation.

In vacuum, it is very difficult to close the switch without bouncing.

In atmospheric air, the right actuation schemes can land the plate softly.

Click below to animate.



Squeezed fluid damps oscillation.

Oscillating plate

Gas gap

Substrate

x

y

z

a

b

h

Gap

thic

knes

s

h

Time

(t) = e0 cos(t)

Plate oscillates at fequency .

The squeezed fluid between the plate and the substrate creates damping forces on the plate.

For sensors, rigid plate parallel to substrate, moving up and down, is preferred over flexible plates. [H. Seidel et al. 1990 Sensors and Actuators, 21 312-315.]

Click plate to animate.

Squeeze-film damping is important in the design of MEMS/NEMS

Finite element models

• Simple.• Gives clearer ideas of how parameters affect

damping. • Well suited for design and optimization.

http://www.comsol.com/showroom/gallery/1432.php

32 /)(42.0 habc Andrews

Reduced-order models

ExampleExample

• High resolution.• High fidelity.

However,

Are models based on continuum mechanics valid for rarefied gas in the free-molecule regime?

• Models are needed for predicting and designing performance of MEMS. • Modeling tools are available, especially based on continuum mechanics.

Knudsen number determines damping regime

http://www.phas.ucalgary.ca/~annlisen/teaching/Phys223/PHYS223-LECT34.html

mm

RT

P

2 R = universal gas constant, T = temperature, Kmm=molecular mass, kg/mol

•Mean free path

h

z

tP

p

tP

pPh 22

12•Continuum models, eg. Are based on spatial derivatives.

•Do they apply when the gap is not much larger than the mean free path?

Many researchers say no.

•Knudsen number Ks = 1.016 mean free path/(gap size).

We need experimental data.

Molecular models for free-space drag were used to calculate squeeze-film damping.

Christian’s model

Kadar et al’s model

R.G. Christian 1966 Vacuum, 16 175

ApTk

mc

TChristian

24

• The models below were derived for free-space drag, not squeeze-film damping. • Have been used to explain data measured on squeeze-film damping.

Improved on Christian’s model by replacing its Maxwell-Boltzman velocity distribution with the Maxwellian stream distribution, resulting in

Z. Kadar et al 1996 Sensors and Actuators A, 53 299-303

Li et al’s model B. Li et al 1999 Sensors and Actuators A, 77 191-194

ChristianTR

MLi cu

c2

3 01

5.1

cKadar = cChristian

Improved on Kadar et al’s model by taking into account the velocity of the moving structure relative to the fluid, resulting in

m = mass of one molecule, kgkT = Boltzman’s constant, J/KT = Temperature, KA = Plate area, m2

p = Pressure, Pa

z

Damping

m

c

Plate

Stiffness

kNo substrate

[S. Hutcherson and W. Ye 2004 J. Micromech. Microeng. 14 1726-1733.]

These are “molecular-based’ models, not molecular dynamic models.

Models for free-space drag underestimate squeeze-film damping. Measured on squeeze-film device [J.D. Zook et al 1992 Sensors and Actuators A, 35 51-59] Original free-space damping

model [R.G Christian 1966 Vacuum, 16 175]

Figure in Li et al 1999 Sensors and Actuators A, 77 191-194

Improved model [B. Li et al 1999 Sensors and Actuators A, 77 191-194]

Improved model [Z. Kadar et al 1996 Sensors and Actuators A, 53 299-303]

Models gave higher Q (lower damping) than measured.

Squeeze-film damping is much larger than free-space drag.

Predicted with free-space drag: Higher Q (lower damping).

Measured on squeeze-film damped MEMS: Lower Q (higher damping).

Figure in C. Gui et al 1995 J. Micromech. Microeng. 5 183-185.

Also concluded in R. Legtenberg and H.A.C. Tilmans 1994 Sensors and Actuators A 45 57-66.

Bao et al 2002 J. Micromech. Microeng. 12 341-346. Bao et al’s model

• The models below are adaptation of the free-space drag model to squeeze-film damping. • Have been used to explain data measured on squeeze-film damping.

Adapts Christian’s model to squeeze-film damping (instead of free-space drag), resulting in

Christianncecircumfere

Bao chL

c16

/

Oscillating plate

Gas gap

Substratea

b

h

Lcircumference = 2a+2bh = Gap thickness

Hutcherson and Ye’s (HY) model

• Refines Bao’s model by taking into account the change of velocity of molecules colliding with the plate and the substrate.

• True molecular model generated by molecular dynamic simulation. • For the case considered in their paper,

cHY ≈ 2.2 cBao

Molecular models for free-space drag were adapted to calculate squeeze-film damping.

Squeeze-film damping models were closer to measured data.

Published molecular models still could not explain published measured damping.

Adapted to squeeze-film gap [M Bao et al

2002 J. Micromech. Microeng. 12 341-346.]

MD-simulation-based model [S. Hutcherson and W. Ye 2004 J. Micromech. Microeng. 14 1726-1733.]

Free-space damping model [R.G Christian 1966 Vacuum, 16 175]

Data measured on squeeze-film device [J.D. Zook et al 1992 Sensors and Actuators A, 35 51-59] • Hutcherson and Ye’s (HY) model

appears to be closest to Zook’s data. • However, Zook’s data have a different

slope than all models. • Zook’s geometry and test conditions

are not well known or modeled.

We need • A model that is validated by measured data. • Measurement data with better characterized test device and conditions.

• Assumptions in the continuum models:

• Forces on moving plate from gas layer can be obtained from the linearized Reynolds equation

Assumptions:1.Rigid plate

2.Small gap

3.Small displacement

4.Small pressure variation

5.Isothermal process

6.Small molecular mean free path

• Blech’s model (7. On edges, pressure jumps to ambient pressure.

8. Inertia of fluid neglected)

•Andrews et al.’s limit (9. Zero Knudsen number limit of Blech’s)

•Veijola’s model Free from 7 and 8.

h

z

tP

p

tP

pPh 22

12

P = ambient pressure, Pah = gap size, m = viscosity, Pa sp = pressure at (x,y), Pat = time, s

“Molecular” models could not be validated with available data. Try continuum models

Blech’s model was derived for continuum regime, low Knudsen number.

e0 = amplitude, mmplate= plate mass, kgt = time, sz = gap displacement, m = frequency, rad/s

Blech, J.J., 1983, “On Isothermal Squeeze Films”, Journal of Lubrication Technology, 105, p 615-620.

mplate

ks cs

Plate

Structural stiffness

Solid damping

fkzzczm plate

~

Blechs cc

Blechs jk

cBlech kBlech

• Squeeze film causes an extra damping coefficient

• and an extra spring stiffness

z(t) = e0 cos(t)

• Oscillating plate has stiffness and intrinsic (non-squeeze-film) damping.

For low-squeeze numbers, Blech’s model reduces to Andrew’s et al.’s model.

• For low squeeze numbers, <<2

32 /)(42.0 habc Andrews

Ph

a

m

2

12• Depends on the squeeze number

odd,42222222

222

3

3

6//

/768

nm

Blech

banmnm

banm

h

bac

a = plate width, mb = plate length, mh = gap height, mP = ambient pressure, Pa = viscosity, Pa s = frequency, rad/s

• Sample applicable range: = 1.82(10)-5Pa.s; a = 144m; h = 4.5m. in atmosphere P=9.3(10)4Pa: <1 for

< 70kHz

• Blech’s damping coefficient

Andrews, M., Harris, I., Turner, G., 1993, “A comparison of squeeze-film theory with measurements on a microstructure”, Sensors and Actuators A, 36, p 79-87.

h = gap size, mp = pressure at (x,y), Pat = time, s = viscosity, Pa s = density, kg/m3

• Taking into account rarefaction and the inertia of the gas flowing in and out of the gap, Veijola (2004) modified Reynolds equation into

t

h

y

pQ

h

yx

pQ

h

x prpr

1212

33

Knudsen numberKs = 1.016 /h

Veijola, T., 2004, “Compact models for squeezed-film dampers with inertial and rarefied gas effects”, Journal of Micromechanics and Microengineering, 14, p 1109-1118.

= mean free path, m

M

m

N

n mnmn CjGQ ,3,1 ,3,1 pr

1

• If the gap oscillation is (t) = e0 exp(jt), then the damping force

complex amplitude is

Pabn

mnhCmn

64

24

2

2

2

2236

768 b

n

a

m

ab

mnhGmn

a = width, mb = length, me0 = amplitude, mj = √-1n = 1 for isothermal,

(= cp/cv for adiabatic). P = ambient pressure, Pa = viscosity, Pa s = frequency, rad/s = gas mass density, kg/m3

0ejFVeij

,...3,1222244pr

6196

30101

96

61

k

s

ss

s

K

KKhkj

k

KQ

Veijola’s model accounts for fluid inertia.

Flexible test devices did not validate model derived for rigid plates.

C.C. Cheng and W. Fang 2005 Microsystem Technologies 11 104–110

• Blech’s model was derived for rigid plates.

• Published test on flexible cantilevers gave lower damping.

We need a rigid plate oscillating up and down while staying parallel to the substrate.

A rigid plate gave correct validation measurement for SFD models.

M. Andrews et al [1993 Sensors and Actuators A, 36 79-81] used a correct test device for validating Blech’s model.

Oscillating plate

Substrate

• Would be more obvious if damping were in log scale, that

• scatter is large, especially at high squeeze numbers (also Knudsen numbers).

But the data scatter is large.

A rigid-plate test device validated Veijola’s model at low Knudsen numbers.

Ph

a 2

12

Squeeze number

• Non-squeeze-film damping accounts for solid structural and other unknown damping.

• At high pressure, nonSFD does not contribute much.

• At near-vacuum: • Gas damping is

negligible. • Assume 90% of

damping is nonSFD.

100

101

102

103

104

10510

-4

10-3

10-2

10-1

Squeeze number

Dam

ping

rat

io

BlechAndrewsVeijola (V)V+NonSFDMeasured

100

101

102

103

104

10510

-4

10-3

10-2

10-1

Squeeze number

Dam

ping

rat

io

100

101

102

103

104

10510

-4

10-3

10-2

10-1

Squeeze number

Dam

ping

rat

io

100

101

102

103

104

10510

-4

10-3

10-2

10-1

Squeeze number

Dam

ping

rat

io

100

101

102

103

104

10510

-4

10-3

10-2

10-1

Squeeze number

Dam

ping

rat

io

100

101

102

103

104

10510

-4

10-3

10-2

10-1

Squeeze number

Dam

ping

rat

io

640T

60T

6T

0.06T

0.006T

H. Sumali and D.S. Epp 2006 Proc. ASME IMECE.

• Damping being plotted in log scale, it is obvious that the scatter is large, especially at high squeeze numbers (high Knudsen numbers).

But the data scatter is large.

Continuum models could not be validated with available data. Try a better model:

1

12 6ˆ 1

UP p G p

G G

n

Instead of the trivial boundary conditions at the plate edges, GT introduced

DSMC simulations were used to determine correlations for the gas-damping parameters

0.634 1.572

1 0.537

G

G

0.445 11.20

1 5.510

G

G

1 8.834

1 5.118

G

G

0 1G G = gas film (gap) thickness. /G is modified Knudsen number

2

= accommodation coefficient. (For this test device a = 1).

H. Sumali et al 2006 Proc. ASME IDETC to appear.

Gallis and Torczynski’s (GT) model Is a Direct Simulation Monte Carlo (DSMC) method.

A rigid-plate test device validated GT model.

H. Sumali et al 2006 Proc. ASME IDETC to appear.

• Again, the scatter is large, especially at low pressures (high Knudsen numbers).

• We need a better-controlled experiment to validate models at high Knudsen numbers.

• 2x3 array of plates

Present measurement was done on an oscillating plate.

Plate width154.3 m

Air gap between plate and substrate Mean thickness = 4.1 m.

• Structure is electro-plated Au. • Thickness around 5.7 m. • Substrate is alumina.

Folded-cantilever springs

Anchored to substrate

• Assumed width a and length b, where ab = true plate area.

b

a

A = 29717(m)2

a = 154.3 m

The test structure can be modeled as SDOF.

zb

zzczzkzm bb

• Equation of motionDamping

mplate

c

Plate

Stiffness

k

z(t) = e0 cos(t)

• Frequency response function (FRF) from base displacement to plate displacement:

1

2

2

gassgassb kkccjm

m

Z

Z

non-squeeze-

film

Substrate

Measured

damping

squeeze-film

Need to compare computed damping factor c with measured damping ratio .

• To compare prediction with measurement, use the relationship between c and

• Models predict damping factor c in the equation of motion

)(tfkzzczm extplate

)(ˆ2 2 tfzzz extnn

nplate

BlechBlech m

c

2

e0 = amplitude, mhplate = plate thickness, mj = √-1mplate = plate mass, kg

n = natural frequency, rad/s

nplated

VeijVeij mej

F

2Re

0

• For Veijola’s model,

• For Blech’s model,

• Measurement method gives damping ratio in the equation of motion

• Quality factor Q is the inverse of damping ratio

Q/5.0

Scanning laser Doppler vibrometer enables high temporal- and spatial resolutions.

http://www.polytec.com/int/158_6800.asp

Laser Doppler vibrometry principle

Scanning laser Doppler vibrometry (scanning LDV) gives spatial vibration shapes.

Scanning LDV has been integrated with a micro-otpical diagnostic instrument.


LDV can provide clear insight into dynamics

Such work as

can benefit a lot from LDV.Click to animate.

Click to animate.

Measurement uses LDV and vacuum chamber.

Microscope

Vacuum chamber

Laser beamDie under test

PZT actuator (shaker)

• Substrate (base) was shaken with piezoelectric actuator.

• Scanning Laser Doppler Vibrometer (LDV) measures velocities at base and at several points on MEMS under test.

Oscillating plate is shaken through its supports.

1. Substrate is shaken up and down.

2. Plate moves up and down. 3. Springs flex.

4. Air gap is compressed and expanded by plate oscillation.

Air gap between plate and substrate. Mean thickness = 4.1 m.

• LDV measured frequency response function (FRF) from base displacement to plate displacement:

12

2

gassgassb kkccjm

m

Z

Z

Transmissibility =

FRFs are curve-fit to give natural frequency, damping and mode shapes

• Tests were repeated at different air pressures from atmospheric (640 Torr) to near-vacuum (<1 milliTorr).

• Frequency response function (FRF) from base displacement to gap displacement:

22

2

2 nnb

gap

jZ

Z

= Transmissibility - 1

• LDV measured transmissibility at 17 points on the plate and springs.

are curve-fit simultaneously using standard Experimental Modal Analysis (EMA) process.

• Commercial EMA software gave natural frequency, damping, and mode shapes.

Experimental modal analysis gives natural frequency, damping and mode shapes.

Measured deflection shape, first mode. Higher modes are not considered.

16910Hz. Up-and-down.

27240Hz

33050Hz

• Tests were repeated at different air pressures from atmospheric (640 Torr) to near-vacuum (<1 milliTorr).

Click above to animate.



Damping from EMA agrees well with damping from free decay.

1. Hilbert transform gives decay envelope. 2. Exponential fit gives damping times natural frequency.

3. Damping is constant with time.

4. For this case (P=3830milliTorr), both experimental modal analysis (frequency domain fit) and free decay curve-fitting (time domain fit) give damping ratio = 0.0011

Non-squeeze-film damping is estimated from zero-pressure asymptote.

• At low pressures, Non-Squeeze-Film Damping (NSFD) is the dominant damping.

• Linear-fit total measured damping at a few lowest-pressures. The zero-pressure intercept is NSFD.

• To obtain squeeze-film damping from measured total damping, subtract NSFD from total measured damping.

Some models predict measured data very well.

Conclusions:

• On rigid plates with width ~150 m, oscillating around 4.1 m above the substrate, squeezed air film can cause large damping.

• Non-molecular models are not necessarily less accurate than current molecular models.

• For the conditions tested here, in atmospheric air the simplest model mentioned by Andrews et al. is as good as any more sophisticated models.

32 /42.0 hAc Andrews

• In the high squeeze number regime (low pressures or high frequencies), Veijola’s model appears to match experimental data accurately.

Acknowledgment

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration

under contract DE-AC04-94AL85000.

The author thanks the following contributors:• Chris Dyck and Bill Cowan’s team for providing

the test structures. • Jim Redmond and Dan Rader for technical

guidance and programmatic support. • David Epp for help with the modal analysis.

slide 1 An Experiment to Determine the Accuracy of Squeeze-Film Damping Models in the Free-Molecule...

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