Transcript Rarefied Gas Transport

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
Examples - Resonators
Bio sensor
IF filter or oscillator
Temperature sensor
Doms, et al. JMM 2005
Resonator – 1-D Macro Model
• Macro model
meff x + cx + kx = Factuator
meff: effective mass
C : dashpot damping coefficient
k : stiffness of the spring
Resonator – 1-D Macro Model
• Macro model
meff x + cx + kx = Factuator
•
meff: effective mass
C : dashpot damping coefficient
k : stiffness of the spring
• Quality factor (Q):
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:
Kn 
mean free path of gas molecules

L
characteristic length of flow field

Bulk region

Bulk region
e.g., air at room temperature, 1 atm
L  1 m
Kn  0 . 065
Fundamentals of Micro/Nano
Gas Flows - Flow Regimes
Knudsen Number:
Kn 

L
• Continuum flow with no-slip BCs
• Continuum flow with slip BCs
• Transition regime
• Free-molecule regime
10
1
10
Kn  10
2
2
 Kn  10
1
 Kn  10
Kn  10
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Continuum Regime – Governing
Equations and BC
Kn  10
2
9
Slip Regime – Governing
Equations and BC
10
2
 Kn  10
1
10
Non-continuum Gas Regime
Kn  10
1
Boltzmann equation
f
t
 v
f
r
 Q ( f , f* )
f velocity distribution function
Analytical methods - Moment methods, etc
Numerical methods – Discrete velocity method, etc
Kinetic methods
Particle methods
Molecule Dynamics – Free-molecule flows
Direct Simulation Monte Carlo – Flows in the
transition regime
11 11
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
f0=19200 Hz ; Q = 27
Air Damping on Laterally Oscillating
Micro Resonators
Damping forces: primarily fluidic
Kn 
Re 

 0 . 03
L
 UL

Continuum regime
 0 . 02
Navier-Stokes
Reynolds number << 1
Stoke equations
Boundary condition – non-slip and slip
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Steady Stokes Flow
Governing Equations
 u   p  0
2
u is the velocity
of the fluid
p is the pressure
 u  0
BC:
where
 is the viscosity


ug  uw
u  u0
of the fluid
1D Couette Model:
Tang, et al, 1989, 1990
1D - Steady (Couette) Theory
vs. Experiment
Unsteady Stokes Flow
Governing
Equations


u
t
2 
  u  p
where
u is the velocity
of the fluid
p is the pressure
 u  0
 is the viscosity


BC: u g  u w
of the fluid
 is the density of the fluid
u  u 0 cos  t
1D Stokes Model:
Cho, et al, 1993
1D - Unsteady (Stokes) Theory
vs. Experiment
FastStokes Results
• Number of Panels: 23424
• CPU (Pentium III) time: 30
minutes
• kinematic viscosity:   0 .145
3


1
.
225
kg
m
• density:
• Drag Force: 207.58 nN
• Q: 29.1
cm
2
sec
Comparison of Different Models and
Experiment
C ouette M odel
1D Stokes M odel
FastStokes
M easurem en t
D rag Force (n N )
110.7
123.2
207.6
224
Q
54.5
49
29.1
27
FastStokes: Force Distribution
12 %
• Top force:
55 %
• Bottom force:
• Side force (inter-finger + pressure): 33 %
Example 2 – Squeeze-film Damping on
Micro Plate/Beam Resonator in Partial
Vacuum
Free-Molecule Regime
Kn   L  10
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
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
–
n
p
k bT
, Ideal gas law
– Randomly, uniformly
distributed over the
entire interaction region
– Velocities follow
Maxwell-Boltzmann
f M B  vi  
distribution
  m p v i2
exp 
 2k T
2 k b T
b

mp



Particle Generation
• At each small time interval:
–
K T
N b  nAb


f M S  vi 
b
2 m p
t
Tangential velocities  Maxwell-Boltzmann
distribution
Normal velocities  Maxwell-Stream distribution
  m p v i2

v i exp 
 2K T
K bT
b

mp



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 s
Specular reflection
Diffuse reflection
Computation of Quality Factor
Q 
2  E input
E fluid  E other
y ( x , t )  A ( x ) sin( t )
E input 
E fluid 
E tran
1
L
WH
2
E
2
  A (x )  d x
2
0
tran
 m p v p  m pu p
 F · s  
t


·w  t  m p  v p  u p ·w

Sumali’s Resonator
1.E+06
Sumali's measurement
1.E+05
Hong&Ye's Simulation
veijola's model
Quality factor
1.E+04
Bao's model
1.E+03
1.E+02
1.E+01
1.E+00
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
P(Pa)
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
Agree
well
Viscous
flow
Other losses
dominate
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.
Examples – Thermal sensing
AFM
AFM
TSAFM
W rite
H e a te r
T ip
In d e n ta tio n
Read
20 µm
L ow e r
T h e rm a l
R e s is ta n c e
H ig h e r
T h e rm a l
R e s is ta n c e
IBM Millipede
2 00 n m
Thermal Sensing AFM
TSAFM
p
20 µm
2 00 n m
H ig h e r
T h e rm a l
R e s is ta n c e
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Heat Transfer Modes
Transfer Paths
Length Scales
Semi-Infinite
g < 500 nm
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Multiscale Modeling
• Path 1 – Continuum
• Path 2 – Continuum
• Path 3 – Direct
Simulation Monte Carlo
(DSMC)
– Stochastic method
– Particle motions and
collisions are decoupled
over small time intervals
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Multiscale Simulation – Thermal
Sensing AFM
Coupling Scheme: Alternating Schwarz Coupling
36
Multiscale Simulation –
Temperature Field
Continuum solution
Multiscale solution
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Multiscale Simulation – Heat Flux
Total heat flux from the cantilever: 84.46 W/m
1-D decoupled model: 91.56 W/m
38
Multiscale Simulation – Velocity
Field Near the Cantilever
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Noncontinuum Phenomena
• Thermally Induced Gas Flow
TH
TC
• Knudsen Force
F
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Phenomena
• Crookes Radiometer
41
Radiometric Force
William Crookes
(1832-1919)
A Einstein (1879- 1955)
James Clerk Maxwell
(1831–1879)
42
Radiometric Force
N Selden, et al., J Fluid Mech., 2009
N Selden, et al., Phys. Rev. E, 2009
1. Experimental data;
2. Numerical Studies by DSMC
and ES-BGK Model equation.
43
Thermal Transpiration
TH
TC
After Collision
Before Collision
Th
Tc
nonzero net tangential momentum
Th
Tw
Tw
Tc
zero tangential momentum
44
Thermal Transpiration Velocity
OSIP-DSMC
45
Thermal Transpiration Velocity
46
Thermal Transpiration Pressure
47
Knudsen’s Pump
162 stages; 760 Torr
0.9 Torr
Gianchandani: JMEMS 2005; JMM 2012; JMEMS in press.
Gianchandani & Ye, Transducers 2009
48
Knudsen Force
Passian, et al.
Journal of Applied Physics, 2002
Physical Review Letters, 2003
Lereu, et al
Applied Physics Letters, 2004
Wall: 500K
Argon
Symmetric
Wall: 300K
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Knudsen Force
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Temperature Contours
Kn = 0.5
Kn = 5.0
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Flow Field Analysis
Kn=1.0
Thermal
stress
slip flow
Thermal edge flow
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Knudsen Force – Shape
Effect
F
F
53
Shape Effect - Asymptotic
Analysis
54
Shape Effect - Asymptotic
Analysis
 Governing Equations
Flow
Hot
Hot
Flow
Cold
Cold
55
Shape Effect - Asymptotic
Analysis
 Boundary conditions
Heated Microbeam
Governing equations
Boundary conditions
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Shape Effect - Asymptotic
Analysis
 Knudsen force acting on objects:
Thermal creep flow effect
Thermal stress slip flow
effect
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Asymptotic Analysis – Solution
Approach
 Numerical methods
Temperature
Field
• Finite Element
Method / Boundary
Element Method
• Laplace equation for
Steady Heat transfer
Problem
Velocity Field
• Boundary Element
Method
• Bi-harmonic equation
for stream function
Knudsen Force
• Finite Difference
Method
• first order Partial
differential equation
for Momentum
conservation
58
Asymptotic Analysis – Results
59
F ra m e 0 0 1  2 2 A p r 2 0 1 3 
Rarefied Gas Transport - Results &
4
Asymptotic
Analysis – Results
Discussion
3
S pe e d
2
0 .0 1 9
0 .0 1 7
1
0 .0 1 5
Y
0 .0 1 3
0 .0 1 1
0
0 .0 0 9
0 .0 0 7
0 .0 0 5
-1
0 .0 0 3
0 .0 0 1
-2
-1
0
1
X
2
3
T e m p e ra ture
-0 .0 0 5
-0 .0 1 5
-0 .0 2 5
-0 .0 3 5
-0 .0 4 5
-0 .0 5 5
-0 .0 6 5
-0 .0 7 5
-0 .0 8 5
-0 .0 9 5
4
60
F ra m e 0 0 1  2 2 A p r 2 0 1 3 
Rarefied Gas Transport - Results &
16
Asymptotic
Analysis – Results
Discussion
14
12
10
S p e e d : 0 .0 0 5
0 .0 2
0 .0 3 5
0 .0 5
0 .0 6 5
0 .0 8
F ra m e 0 0 1  0 4 Ju n 2 0 1 3 
8
Y
6
F ra m e 0 0 1  2 1 M a y 2 0 1 3 
T e m p e ra tu re : -0 .0 9 5 -0 .0 8 -0 .0 6 5 -0 .0 5 -0 .0 3 5 -0 .0 2 -0 .0 0 5
4
2
0
-2
-1 0
-5
0
5
10
X
61
F ra m e 0 0 1  2 2 A p r 2 0 1 3 
Rarefied Gas Transport - Results &
16
Asymptotic
Analysis – Results
Discussion
14
12
10
S p e e d : 0 .0 0 5
0 .0 2
0 .0 3 5
0 .0 5
0 .0 6 5
0 .0 8
8
6
T e m p e ra tu re : -0 .0 9 5 -0 .0 8 -0 .0 6 5 -0 .0 5 -0 .0 3 5 -0 .0 2 -0 .0 0 5
Y
C
4
A
B
A
B
2
0
D
-2
D
C
-1 0
-5
0
5
10
X
62
Asymptotic
Analysis
–&
Rarefied
Gas Transport
- Results
Discussion
Knudsen Torque
Torque
Torque
Force
Force
Potential applications: particle manipulation, thermal motor
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