micro/nano gas flows and their impact on mems/nems wenjing ye mae, hkust
DESCRIPTION
Micro/Nano Gas Flows and Their Impact on MEMS/NEMS Wenjing Ye MAE, HKUST. Micro Resonators. Resonant structure fabricated with microfabrication technology Driven mechanism: electrical, piezoelectric Sensing: capacitive, piezoresistive Applications Sensors Filters, oscillators. - PowerPoint PPT PresentationTRANSCRIPT
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Micro/Nano Gas Flows and Their Impact on MEMS/NEMS
Wenjing YeMAE, HKUST
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Micro Resonators
• Resonant structure fabricated with microfabrication technology • Driven mechanism: electrical, piezoelectric• Sensing: capacitive, piezoresistive
• Applications• Sensors • Filters, oscillators
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Examples - Resonators
Bio sensorTemperature sensor
IF filter or oscillatorDoms, et al. JMM 2005
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Resonator – 1-D Macro Model
• Macro model
meff: effective mass
dashpot damping coefficient
stiffness of the spring:k
:C
meff x + cx + kx = Factuator
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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
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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
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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
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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
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Continuum Regime – Governing Equations and BC
210Kn
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Slip Regime – Governing Equations and BC
12 1010 Kn
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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
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Example 1 – Air Damping on a Laterally Oscillating Resonator
• Damping forces: primarily fluidic– viscous drag force is dominant– Squeeze-film damping is insignificant
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Experimental Measurement:Computer Microvision
Q = 27f0=19200 Hz ;
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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
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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
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1D - Steady (Couette) Theoryvs. Experiment
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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:
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1D - Unsteady (Stokes) Theoryvs. Experiment
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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
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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
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FastStokes: Force Distribution
• Top force:• Bottom force:• Side force (inter-finger + pressure): %33
%12
%55
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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Examples – Thermal sensing AFM
Write
Read
20 µm 200 nm
Tip Indentation
Heater
Lower Thermal Resistance
Higher Thermal Resistance IBM Millipede
AFMTSAFM
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Thermal Sensing AFM
TSAFM Write
Read
20 µm 200 nm
Tip Indentation
Heater
Lower Thermal Resistance
Higher Thermal Resistance
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Heat Transfer Modes
Semi-Infinite
g < 500 nm
Transfer Paths Length Scales
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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
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Multiscale Simulation – Thermal Multiscale Simulation – Thermal Sensing AFMSensing AFM
Coupling Scheme: Alternating Schwarz Coupling
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Multiscale Simulation – Multiscale Simulation – Temperature FieldTemperature Field
Continuum solution
Multiscale solution
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Multiscale Simulation – Heat FluxMultiscale Simulation – Heat Flux
Total heat flux from the cantilever: 84.46 W/m1-D decoupled model: 91.56 W/m
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Multiscale Simulation – Velocity Multiscale Simulation – Velocity Field Near the CantileverField Near the Cantilever
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Noncontinuum Phenomena
• Thermally Induced Gas Flow
• Knudsen Force
THTC
F
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Phenomena
• Crookes Radiometer
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Radiometric Force
James Clerk Maxwell (1831–1879)
A Einstein (1879- 1955)
William Crookes(1832-1919)
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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.
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Thermal Transpiration
Before Collision
THTC
After Collision
ThTc
nonzero net tangential momentum
TwTw
zero tangential momentum
ThTc
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Thermal Transpiration - Velocity
OSIP-DSMC
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Thermal Transpiration - Velocity
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Thermal Transpiration - Pressure
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Knudsen’s Pump
Gianchandani: JMEMS 2005; JMM 2012; JMEMS in press. Gianchandani & Ye, Transducers 2009
162 stages; 760 Torr 0.9 Torr
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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
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Knudsen ForceKnudsen Force
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Kn = 0.5 Kn = 5.0
Temperature ContoursTemperature Contours
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Kn=1.0
Flow Field AnalysisFlow Field Analysis
Thermal edge flow
Thermal stressslip flow
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Knudsen Force – Shape Knudsen Force – Shape Effect Effect
F
F
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Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis
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Governing Equations
Hot ColdFlow
HotCold
Flow
Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis
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Boundary conditions
Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis
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Knudsen force acting on objects:
Thermal creep flow effect Thermal stress slip flow effect
Shape Effect - Asymptotic Shape Effect - Asymptotic AnalysisAnalysis
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Numerical methods
Asymptotic Analysis – Asymptotic Analysis – Solution ApproachSolution Approach
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Asymptotic Analysis – ResultsAsymptotic Analysis – Results
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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
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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
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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
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Rarefied Gas Transport - Results & Discussion
Torque
Force
Potential applications: particle manipulation, thermal motor
Asymptotic Analysis – Asymptotic Analysis – Knudsen TorqueKnudsen Torque
Torque
Force