Micro/Nano Gas Flows and Their Impact on MEMS/NEMS

Wenjing YeMAE, HKUST

Micro Resonators

• Resonant structure fabricated with microfabrication technology • Driven mechanism: electrical, piezoelectric• Sensing: capacitive, piezoresistive

• Applications• Sensors • Filters, oscillators

Examples - Resonators

Bio sensorTemperature sensor

IF filter or oscillatorDoms, et al. JMM 2005

Resonator – 1-D Macro Model

• Macro model

meff: effective mass

dashpot damping coefficient

stiffness of the spring:k

:C

meff x + cx + kx = Factuator

Resonator – 1-D Macro Model

• Macro model

•

• Quality factor (Q):

meff: effective mass

dashpot damping coefficient

stiffness of the spring:k

:C

meff x + cx + kx = Factuator

1-D model

Influence of Gas on MEMS/NEMS

• Momentum exchange • Damping force (viscous damping, squeeze-film

damping)• Inertia force (added mass)• Knudsen force

• Energy exchange• Heat flux • Damping

Fundamentals of Gas Transport

• Knudsen number: L

Kn

mean free path of gas molecules characteristic length of flow field

e.g., air at room temperature, 1 atm mL 1

065.0Kn

Bulk region

Bulk region

8

Fundamentals of Micro/Nano Gas Flows - Flow Regimes

• Continuum flow with no-slip BCsContinuum flow with no-slip BCs

• Continuum flow with slip BCs Continuum flow with slip BCs

• Transition regimeTransition regime

• Free-molecule regimeFree-molecule regime

210Kn

LKn

Knudsen Number:

12 1010 Kn

1010 1 Kn

10Kn

9

Continuum Regime – Governing Equations and BC

210Kn

10

Slip Regime – Governing Equations and BC

12 1010 Kn

11

Boltzmann equationBoltzmann equation

Analytical methods - Moment methods, etcNumerical methods – Discrete velocity method, etcKinetic methods

Particle methodsParticle methodsMolecule Dynamics – Free-molecule flowsDirect Simulation Monte Carlo – Flows in the transition regime 11

Non-continuum Gas Regime

),( *ffQf

t

f

rv

110Kn

f velocity distribution function

Example 1 – Air Damping on a Laterally Oscillating Resonator

• Damping forces: primarily fluidic– viscous drag force is dominant– Squeeze-film damping is insignificant

Experimental Measurement:Computer Microvision

Q = 27f0=19200 Hz ;

14

Air Damping on Laterally Oscillating Micro Resonators

Damping forces: primarily fluidicDamping forces: primarily fluidic

Navier-Stokes Navier-Stokes Stoke equationsStoke equations

Boundary condition – non-slip and slipBoundary condition – non-slip and slip

Reynolds number << 1

02.0Re UL

03.0L

Kn Continuum regimeContinuum regime

Steady Stokes Flow

Governing Equations

0u

0pu2

where

fluid theof viscosity theis

pressure theis p

fluid theof velocity theis u

0uu 1D Couette Model:Tang, et al, 1989, 1990

BC: wg uu

1D - Steady (Couette) Theoryvs. Experiment

Unsteady Stokes Flow

Governing Equations

0u

2

put

u

where

fluid theofdensity theis

fluid theof viscosity theis

pressure theis p

fluid theof velocity theis u

1D Stokes Model:

Cho, et al, 1993

tuu cos0

wg uu

BC:

1D - Unsteady (Stokes) Theoryvs. Experiment

FastStokes Results

• Number of Panels: 23424• CPU (Pentium III) time: 30 minutes• kinematic viscosity: • density:

• Drag Force: 207.58 nN • Q: 29.1

3 225.1 mkgsec 145.0 2cm

Comparison of Different Models and Experiment

Drag Force (nN) QCouette Model 110.7 54.5

1D Stokes Model 123.2 49FastStokes 207.6 29.1

Measurement 224 27

FastStokes: Force Distribution

• Top force:• Bottom force:• Side force (inter-finger + pressure): %33

%12

%55

22 22

Example 2 – Squeeze-film Damping on Micro Plate/Beam Resonator in Partial

Vacuum

10 LKn

Free-Molecule RegimeFree-Molecule Regime

Low pressure: vacuum environment Small scale: nano devices

Monte Carlo Simulation

Courtesy: Prof. O. Brand

Monte Carlo Approach

• Based on the momentum and energy transfer between the free molecules and the walls

• Assumptions:– Gas reservoir at equilibrium– Oscillation mode shape is not affect by collisions

MC Simulation Approach

• Initialization: Generate Molecules

• At each time interval– Generating new gas molecules entering the

interaction region

– Tracking each gas molecule inside the interaction region

– Detecting collisions and calculating energy change during each collision

• Summing all the energy losses in each cycle

• Ensemble averaging

Particle Generation

• Particle initialization

– , Ideal gas law

– Randomly, uniformly distributed over the entire interaction region

– Velocities follow Maxwell-Boltzmann distribution

b

pn

k T

2

exp2 2

p p iMB i

b b

m m vf v

k T k T

Particle Generation

• At each small time interval:–

Tangential velocities Maxwell-Boltzmann distribution

Normal velocities Maxwell-Stream distribution

2b

b bp

K TN nA t

m

2

2expp p i

MS i ib b

m m vf v v

K T K T

Collision Detection

• Determine the time and position of each collision

• Collide with substrate or fixed walls– Solved analytically

• Collide with the moving resonator– Solved numerically– Stability– Multiple roots

Collision Model• Maxwell gas-wall interaction model• Specular reflection

– Mirror-like• Diffuse reflection

– Particle accommodated to the wall conditions

Accommodation coefficient

Specular reflection

Diffuse reflection

Computation of Quality Factor

2

fluid other

inputEQ

E E

2L

0

21W ( )

2input xE H A x d

· · ·p ptran p

m mt m

tE

p pp p

v us w v uF w

( , ) ( )sin( )y x t A x t

fluid tranE E

30

Sumali’s ResonatorSumali’s Resonator

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

P(Pa)

Qua

lity

fact

or

Sumali's measurement

Hong&Ye's Simulation

veijola's model

Bao's model

Specular reflection; Frequency: 16.91 kHz

H. Sumali, "Squeeze-film damping in the free molecular regime: model validation and measurement on a MEMS," J.Micromech Microeng., Vol. 17, pp. 2231-2240, 2007.

Minikes’s Micro Mirror

A. Minikes, I. Bucher and G. Avivi, "Damping of a mirco-resonator torsion mirror in rarefied gas ambient," J.Micromech Microeng., Vol. 15, pp. 1762-1769, 2005.

Viscous flow

Other losses dominate

Agree well

Examples – Thermal sensing AFM

Write

Read

20 µm 200 nm

Tip Indentation

Heater

Lower Thermal Resistance

Higher Thermal Resistance IBM Millipede

AFMTSAFM

33

Thermal Sensing AFM

TSAFM Write

Read

20 µm 200 nm

Tip Indentation

Heater

Lower Thermal Resistance

Higher Thermal Resistance

34

Heat Transfer Modes

Semi-Infinite

g < 500 nm

Transfer Paths Length Scales

35

Multiscale Modeling

• Path 1 – ContinuumPath 1 – Continuum• Path 2 – ContinuumPath 2 – Continuum• Path 3 – Direct Path 3 – Direct

Simulation Monte Carlo Simulation Monte Carlo (DSMC)(DSMC)– Stochastic method– Particle motions and

collisions are decoupled over small time intervals

36

Multiscale Simulation – Thermal Multiscale Simulation – Thermal Sensing AFMSensing AFM

Coupling Scheme: Alternating Schwarz Coupling

37

Multiscale Simulation – Multiscale Simulation – Temperature FieldTemperature Field

Continuum solution

Multiscale solution

38

Multiscale Simulation – Heat FluxMultiscale Simulation – Heat Flux

Total heat flux from the cantilever: 84.46 W/m1-D decoupled model: 91.56 W/m

39

Multiscale Simulation – Velocity Multiscale Simulation – Velocity Field Near the CantileverField Near the Cantilever

40

Noncontinuum Phenomena

• Thermally Induced Gas Flow

• Knudsen Force

THTC

F

41

Phenomena

• Crookes Radiometer

42

Radiometric Force

James Clerk Maxwell (1831–1879)

A Einstein (1879- 1955)

William Crookes(1832-1919)

43

Radiometric Force

N Selden, et al., J Fluid Mech., 2009N Selden, et al., Phys. Rev. E, 2009

1. Experimental data;2. Numerical Studies by DSMC

and ES-BGK Model equation.

44

Thermal Transpiration

Before Collision

THTC

After Collision

ThTc

nonzero net tangential momentum

TwTw

zero tangential momentum

ThTc

45

Thermal Transpiration - Velocity

OSIP-DSMC

46

Thermal Transpiration - Velocity

47

Thermal Transpiration - Pressure

48

Knudsen’s Pump

Gianchandani: JMEMS 2005; JMM 2012; JMEMS in press. Gianchandani & Ye, Transducers 2009

162 stages; 760 Torr 0.9 Torr

49

Symmetric

Wall: 500K

Argon

Wall: 300K

Knudsen ForceKnudsen Force

Passian, et al.

Journal of Applied Physics, 2002 Physical Review Letters, 2003Lereu, et al Applied Physics Letters, 2004

50

Knudsen ForceKnudsen Force

51

Kn = 0.5 Kn = 5.0

Temperature ContoursTemperature Contours

52

Kn=1.0

Flow Field AnalysisFlow Field Analysis

Thermal edge flow

Thermal stressslip flow

53

Knudsen Force – Shape Knudsen Force – Shape Effect Effect

F

F

54

Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis

55

Governing Equations

Hot ColdFlow

HotCold

Flow

Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis

56

Boundary conditions

Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis

57

Knudsen force acting on objects:

Thermal creep flow effect Thermal stress slip flow effect

Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis

58

Numerical methods

Asymptotic Analysis – Asymptotic Analysis – Solution ApproachSolution Approach

59

Asymptotic Analysis – ResultsAsymptotic Analysis – Results

60

X

Y

-2 -1 0 1 2 3 4

-1

0

1

2

3

4

Speed

0.019

0.017

0.015

0.013

0.011

0.009

0.007

0.005

0.003

0.001

Temperature

-0.005

-0.015

-0.025

-0.035

-0.045

-0.055

-0.065

-0.075

-0.085

-0.095

Frame 001 22 Apr 2013

Rarefied Gas Transport - Results & Discussion Asymptotic Analysis – ResultsAsymptotic Analysis – Results

Frame 001 04 Jun 2013

Frame 001 21 May 2013

61

X

Y

-10 -5 0 5 10-2

0

2

4

6

8

10

12

14

16

Speed: 0.005 0.02 0.035 0.05 0.065 0.08

Temperature: -0.095 -0.08 -0.065 -0.05 -0.035 -0.02 -0.005

Frame 001 22 Apr 2013

Rarefied Gas Transport - Results & Discussion Asymptotic Analysis – ResultsAsymptotic Analysis – Results

62

X

Y

-10 -5 0 5 10-2

0

2

4

6

8

10

12

14

16

Speed: 0.005 0.02 0.035 0.05 0.065 0.08

Temperature: -0.095 -0.08 -0.065 -0.05 -0.035 -0.02 -0.005

Frame 001 22 Apr 2013

B A

C D

A

B

C

D

Rarefied Gas Transport - Results & Discussion Asymptotic Analysis – ResultsAsymptotic Analysis – Results

63

Rarefied Gas Transport - Results & Discussion

Torque

Force

Potential applications: particle manipulation, thermal motor

Asymptotic Analysis – Asymptotic Analysis – Knudsen TorqueKnudsen Torque

Torque

Force