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