1. Micro/Nano Gas Flows and Their Impact on MEMS/NEMS Wenjing Ye MAE, HKUST

2. Micro Resonators
Resonant structure fabricated with microfabrication technology
Driven mechanism: electrical, piezoelectric
Sensing: capacitive, piezoresistive
Applications
Sensors
Filters, oscillators

3. Examples - Resonators Temperature sensor Bio sensor
IF filter or oscillator
Doms, et al. JMM 2005

4. Resonator – 1-D Macro Model
meff x + cx + kx = Factuator
meff: effective mass
dashpot damping coefficient
stiffness of the spring

5. Resonator – 1-D Macro Model
Quality factor (Q):
meff x + cx + kx = Factuator
meff: effective mass
dashpot damping coefficient
stiffness of the spring
1-D model

6. 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

7. Fundamentals of Gas Transport
Knudsen number:
mean free path of gas molecules
characteristic length of flow field
Bulk region
Bulk region
e.g., air at room temperature, 1 atm

8. Fundamentals of Micro/Nano Gas Flows - Flow Regimes
Knudsen Number:
Continuum flow with no-slip BCs
Continuum flow with slip BCs
Transition regime
Free-molecule regime

9. Continuum Regime – Governing Equations and BC

10. Slip Regime – Governing Equations and BC
Talk about challenges in modeling MEMS devices, the advantage of BEM.

11. Non-continuum Gas Regime
Boltzmann equation
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
f velocity distribution function 11 11

12. Example 1 – Air Damping on a Laterally Oscillating Resonator
Damping forces: primarily fluidic
viscous drag force is dominant
Squeeze-film damping is insignificant

13. Experimental Measurement: Computer Microvision
f0=19200 Hz ;
Q = 27

14. Air Damping on Laterally Oscillating Micro Resonators
Damping forces: primarily fluidic
Continuum regime
Reynolds number << 1
Navier-Stokes Stoke equations
Boundary condition – non-slip and slip

15. Steady Stokes Flow 1D Couette Model: Tang, et al, 1989, 1990
Governing Equations
where
BC:
1D Couette Model:
Tang, et al, 1989, 1990

16. 1D - Steady (Couette) Theory vs. Experiment

17. Unsteady Stokes Flow 1D Stokes Model: Cho, et al, 1993
Governing Equations
where
BC:
1D Stokes Model:
Cho, et al, 1993

18. 1D - Unsteady (Stokes) Theory vs. Experiment

19. FastStokes Results Number of Panels: 23424
CPU (Pentium III) time: 30 minutes
kinematic viscosity:
density:
Drag Force: nN
Q: 29.1

20. Comparison of Different Models and Experiment

21. FastStokes: Force Distribution
Top force:
Bottom force:
Side force (inter-finger + pressure):

22. Example 2 – Squeeze-film Damping on Micro Plate/Beam Resonator in Partial Vacuum
Free-Molecule Regime
Low pressure: vacuum environment
Small scale: nano devices
Monte Carlo Simulation
Courtesy: Prof. O. Brand
Zook, et al, 1994, measurements on a polysilicon. 22 22

23. Monte Carlo Approach
This picture shows the model we used to simulate the resonator in our MC program. The green structure is a rectangular resonator above the substrate. The algorithm used in this program can handle any arbitrary mode shape and frequency. Denote the interaction region be the region between the green part and the substrate. The program keeps generating new gas molecules in a gas reservoir outside the interaction region. These generated molecules enter through open boundaries. When the molecules enter the interaction region, they will collide with the substrate, close boundaries and also the resonator. The program tracks the motion of each molecule within the interaction region and record any collision. After each collision, the velocity of the molecule are updated and the energy transferred is recorded. When the molecules exit the interaction region through open boundaries, they will be discarded.
In this MC simulation program, there are two key assumptions:
The gas reservoir outside the interaction region is at equilibrium state at ambient pressure and temperature.
The amplitude of vibration of resonator is not affected by the molecule collision because the resonator is assumed to be at quasi-steady state.
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

24. 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
Here is a brief summary of our simulation approach. At the beginning of the simulation, we initialize the molecules inside the interaction region. Molecules are generated based on the ambient pressure and temperature. After the initialization, each oscillation cycle is divided into many small time intervals. In each of these intervals, new gas molecules are generated in the outside gas reservoir and enter the interaction region. Once a molecule is generated, the program tracks its motion until it leaves the interaction domain. The program compare and recorded the Kinetic Energy before and after each collision. By summing up all the energy losses of resonator due to collision within one cycle, we can obtain the total air damping on the resonator per cycle. Finally, ensemble average is performed to reduce noise and obtain converged results.

25. Particle Generation Particle initialization , Ideal gas law
Randomly, uniformly distributed over the entire interaction region
Velocities follow Maxwell-Boltzmann distribution
Next, I will talk about the details of some key areas in the program. First, we will look at the particle generation step. There are two steps, the first one is particle initialization. The number of particle generated in this step is calculated using ideal gas law. The position of these generated particle would be assigned randomly , uniformly over the entire interaction region. Their velocities follow the Maxwell-Boltzmann distribution based on the molecule kinetic theory. This is the MB distribution which is a function of gas temperature and molecular mass.

26. Particle Generation At each small time interval:
Tangential velocities  Maxwell-Boltzmann distribution
Normal velocities  Maxwell-Stream distribution
After initialization, particle are generated at each small time interval. These time interval are very short compared to the oscillation cycle. The number of particles generated is expressed which is a function of time interval, surface area of open boundaries and the average speed of gas molecules. These molecules are randomly, uniformly distributed on the surface of the open boundaries. Their tangential velocities follow the MB distribtion which is shown just previously. The normal velocities follow the MS distribtion, or biased MB distribution. Because we constraint that all particle generated will have a positive normal velocity such that they all enter the IR. Therefore, the distribution used is different.

27. 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
Once a molecule is generated, the program determines the time and position of each subsequent collision until it leave the interaction region. If the molecule collide with the substrate of fixed wall, the time and position can be solved analytically easily.
However, when the molecule collide with the moving resonator, the collision point has to be solved numerically because the position and velocity of the resonator keeps changing as the molecule approaches. Because the program can find the position of resonator and particle at any give time, we can express the distance between the resonator and the particle as a function of time. To ensure stability, this program used method of bisection to search for the root, or the collision point. However, there may be more than one roots available. Therefore, we use another algorithm to search and reject irrevalvent root.

28. Collision Model Maxwell gas-wall interaction model Specular reflection
Mirror-like
Diffuse reflection
Particle accommodated to the wall conditions
Accommodation coefficient s
After detecting the collision, the program will update the particle velocities and record the energy transferred using Maxwell gas-wall interaction model. In this model, there are two extreme cases of collision. The first one is specular reflection. It is mirror like reflection. The incident angle is equal to the reflected angle and the collision is elastic. The tangential velocities of the particles is unchanged which the normal velocity is simply reflected. The other extreme case is diffuse reflection. This models a rough wall conditions. A particle may have multiple collision when it hit the wall and become accommodated to the wall conditions. The reflected velocities of the particle after diffuse reflection is independent of the incident velocity. This equation shows the post-collision velocities of the particle. The Ws are the wall’s velocity in normal and tangential direction. The particle accommodated to thewall’s velcocity. On top of that, random variable follow Maxwell Boltzmann distribution is added to the velocities in the tangential direction and a random variable following Maxwell-stream distribution is added to the velocity in the normal direction.
Since MEMS structures are usually rough and the walls in our MC program are modeled as diffuse type.
Specular reflection
Diffuse reflection

29. Computation of Quality Factor
Quality factor is defined by this equation. Einput is the energy input to the resonator every cycle. It is equal to the max. kinetic energy stored in the resonator. Use a plate resonator as an example, the deformation of the plate can be expressed in this equation. Therefore, the Kinectic energy would be equal to this where the rho is density, W is width, H is thickness of resonator, the bracket terms represent the velocity and it integrate over the length.
Efluid is the fluid damping and Eother is the damping from other sources in one cycle. Efluid can be obtained by summing all the recorded energy transfer. In each collision, the energy transferred can be found using this equation based on the work done by the resonator on the particle. F can be written as the rate of change of momentum of gas molecule and delta s is equal to the distance travelled by resonator within the collision time. Finally, the energy transferred is obtained which depends on the mass, velocity change and resonator velocity.
Since this program only models the fluid damping and ignore Eother. The quality factor obtained using this MC program estimate the Q based on the fluid damping only. When Efluid is much larger that Eother, Q from this program will be very close to the real Q.

30. Sumali’s Resonator 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 , 2007.

31. Minikes’s Micro Mirror
Agree well
Viscous flow
Other losses dominate
The validity of our MC simulation program is examined using 2 test cases. We use our simulation program to simulate device that are published in the literature. The first test case is Minikes’s Device. This is a torsion mirror which vibrate about its central axis. The graph on the right shows a plot of the simulated result and the experimental result. Y axis is the quality factor and x-axis is the pressure. The simulated result fit well with the experimental result in the central part, where pressure is approximated between 0.5 to 100 Torr. In the high pressure region, the simulated result deviate ith the experimental results. It is because the flow is viscous in this pressure range but the model we used in simulation assume free-molecule. Here intermolecular collision take place and air damping is no longer linear with Pressure. As the low pressure end, our program underestimate the damping on resonator. As mention, our program only model the air damping. When the pressure is too low, the air damping is too small compared with other sources of damping such as anchor loss and thermoelasitc loss. Therefore, the quality factor obtained in experiment is smaller than that in simulation.
A. Minikes, I. Bucher and G. Avivi, "Damping of a mirco-resonator torsion mirror in rarefied gas ambient," J.Micromech Microeng., Vol. 15, pp , 2005.

32. Examples – Thermal sensing AFM
TSAFM
IBM Millipede

33. Thermal Sensing AFM
TSAFM
Resolution: 3nm vertical and 100 nm lateral

34. Heat Transfer Modes Transfer Paths Length Scales Semi-Infinite
g < 500 nm

35. 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

36. Multiscale Simulation – Thermal Sensing AFM
Issues: determine coupling region, coupling scheme, stability issue
Coupling Scheme: Alternating Schwarz Coupling

37. Multiscale Simulation – Temperature Field
Continuum solution
Multiscale solution

38. Multiscale Simulation – Heat Flux
Total heat flux from the cantilever: W/m
1-D decoupled model: W/m

39. Multiscale Simulation – Velocity Field Near the Cantilever

40. Noncontinuum Phenomena
Thermally Induced Gas Flow
Knudsen Force TH TC F

41. Phenomena Crookes Radiometer
Over the years, there have been many attempts to explain how a Crookes radiometer works: 1.
Crookes incorrectly suggested that the force was due to the pressure of light. This theory was originally supported by James Clerk Maxwell who had predicted this force. This explanation is still often seen in leaflets packaged with the device. The first experiment to disprove this theory was done by Arthur Schuster in 1876, who observed that there was a force on the glass bulb of the Crookes radiometer that was in the opposite direction to the rotation of the vanes. This showed that the force turning the vanes was generated inside the radiometer. If light pressure was the cause of the rotation, then the better the vacuum in the bulb, the less air resistance to movement, and the faster the vanes should spin. In 1901, with a better vacuum pump, Pyotr Lebedev showed that in fact, the radiometer only works when there is low pressure gas in the bulb, and the vanes stay motionless in a hard vacuum. Finally, if light pressure were the motive force, the radiometer would spin in the opposite direction as the photons on the shiny side being reflected would deposit more momentum than on the black side where the photons are absorbed. The actual pressure exerted by light is far too small to move these vanes but can be measured with devices such as the Nichols radiometer. 2.
Another incorrect theory was that the heat on the dark side was causing the material to outgas, which pushed the radiometer around. This was effectively disproved by both Schuster's and Lebedev's experiments. 3.
A partial explanation is that gas molecules hitting the warmer side of the vane will pick up some of the heat, bouncing off the vane with increased speed. Giving the molecule this extra boost effectively means that a minute pressure is exerted on the vane. The imbalance of this effect between the warmer black side and the cooler silver side means the net pressure on the vane is equivalent to a push on the black side, and as a result the vanes spin round with the black side trailing. The problem with this idea is that while the faster moving molecules produce more force, they also do a better job of stopping other molecules from reaching the vane, so the net force on the vane should be exactly the same — the greater temperature causes a decrease in local density which results in the same force on both sides. Years after this explanation was dismissed, Albert Einstein showed that the two pressures do not cancel out exactly at the edges of the vanes because of the temperature difference there. The force predicted by Einstein would be enough to move the vanes, but not fast enough. 4.
The final piece of the puzzle, thermal transpiration, was theorized by Osborne Reynolds, but first published by James Clerk Maxwell in the last paper before his death in Reynolds found that if a porous plate is kept hotter on one side than the other, the interactions between gas molecules and the plates are such that gas will flow through from the cooler to the hotter side. The vanes of a typical Crookes radiometer are not porous, but the space past their edges behaves like the pores in Reynolds's plate. On average, the gas molecules move from the cold side toward the hot side whenever the pressure ratio is less than the square root of the (absolute) temperature ratio. The pressure difference causes the vane to move cold (white) side forward.
Both Einstein's and Reynolds's forces appear to cause a Crookes radiometer to rotate, although it still isn't clear which one is stronger.
There are also many interesting histories in Loeb’s book.

42. Radiometric Force William Crookes (1832-1919) A Einstein (1879- 1955)
James Clerk Maxwell
(1831–1879)
Over the years, there have been many attempts to explain how a Crookes radiometer works: 1.
Crookes incorrectly suggested that the force was due to the pressure of light. This theory was originally supported by James Clerk Maxwell who had predicted this force. This explanation is still often seen in leaflets packaged with the device. The first experiment to disprove this theory was done by Arthur Schuster in 1876, who observed that there was a force on the glass bulb of the Crookes radiometer that was in the opposite direction to the rotation of the vanes. This showed that the force turning the vanes was generated inside the radiometer. If light pressure was the cause of the rotation, then the better the vacuum in the bulb, the less air resistance to movement, and the faster the vanes should spin. In 1901, with a better vacuum pump, Pyotr Lebedev showed that in fact, the radiometer only works when there is low pressure gas in the bulb, and the vanes stay motionless in a hard vacuum. Finally, if light pressure were the motive force, the radiometer would spin in the opposite direction as the photons on the shiny side being reflected would deposit more momentum than on the black side where the photons are absorbed. The actual pressure exerted by light is far too small to move these vanes but can be measured with devices such as the Nichols radiometer. 2.
Another incorrect theory was that the heat on the dark side was causing the material to outgas, which pushed the radiometer around. This was effectively disproved by both Schuster's and Lebedev's experiments. 3.
A partial explanation is that gas molecules hitting the warmer side of the vane will pick up some of the heat, bouncing off the vane with increased speed. Giving the molecule this extra boost effectively means that a minute pressure is exerted on the vane. The imbalance of this effect between the warmer black side and the cooler silver side means the net pressure on the vane is equivalent to a push on the black side, and as a result the vanes spin round with the black side trailing. The problem with this idea is that while the faster moving molecules produce more force, they also do a better job of stopping other molecules from reaching the vane, so the net force on the vane should be exactly the same — the greater temperature causes a decrease in local density which results in the same force on both sides. Years after this explanation was dismissed, Albert Einstein showed that the two pressures do not cancel out exactly at the edges of the vanes because of the temperature difference there. The force predicted by Einstein would be enough to move the vanes, but not fast enough. 4.
The final piece of the puzzle, thermal transpiration, was theorized by Osborne Reynolds, but first published by James Clerk Maxwell in the last paper before his death in Reynolds found that if a porous plate is kept hotter on one side than the other, the interactions between gas molecules and the plates are such that gas will flow through from the cooler to the hotter side. The vanes of a typical Crookes radiometer are not porous, but the space past their edges behaves like the pores in Reynolds's plate. On average, the gas molecules move from the cold side toward the hot side whenever the pressure ratio is less than the square root of the (absolute) temperature ratio. The pressure difference causes the vane to move cold (white) side forward.
Both Einstein's and Reynolds's forces appear to cause a Crookes radiometer to rotate, although it still isn't clear which one is stronger.
There are also many interesting histories in Loeb’s book.

43. Radiometric Force Experimental data;
The latest results on radiometer effect.
Experimental data;
Numerical Studies by DSMC and ES-BGK Model equation.
N Selden, et al., J Fluid Mech., 2009
N Selden, et al., Phys. Rev. E, 2009

44. Thermal TranspirationTC
After Collision
Before Collision Th Th Tc Tw Tc Tw
nonzero net tangential momentum
zero tangential momentum

45. Thermal Transpiration - Velocity
OSIP-DSMC

46. Thermal Transpiration - Velocity

47. Thermal Transpiration - Pressure
1 um: 3.5kpa difference, maximum velocity m/s
100 nm: 6 Kpa difference and 20 nm 29 Kpa difference

48. Knudsen’s Pump 162 stages; 760 Torr 0.9 Torr
1 um: 3.5kpa difference, maximum velocity m/s
100 nm: 6 Kpa difference and 20 nm 29 Kpa difference
162 stages; 760 Torr Torr
Gianchandani: JMEMS 2005; JMM 2012; JMEMS in press.
Gianchandani & Ye, Transducers 2009

49. Knudsen Force Journal of Applied Physics, 2002 Passian, et al.
Physical Review Letters, 2003
Lereu, et al
Applied Physics Letters, 2004
Wall: 500K
Argon
Symmetric
Wall: 300K

50. Knudsen Force

51. Temperature Contours
Kn = 0.5
Kn = 5.0

52. Flow Field Analysis
Thermal
stress
slip flow
Kn=1.0
Thermal edge flow

53. Knudsen Force – Shape Effect

54. Shape Effect - Asymptotic Analysis

55. Shape Effect - Asymptotic Analysis
Governing Equations
Hot
Cold
Flow
Hot
Cold
Flow

56. Shape Effect - Asymptotic Analysis
Boundary conditions
Heated Microbeam
Governing equations
Boundary conditions

57. Shape Effect - Asymptotic Analysis
Knudsen force acting on objects:
Thermal stress slip flow effect
Thermal creep flow effect

58. 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

59. Asymptotic Analysis – Results

60. Asymptotic Analysis – Results
Rarefied Gas Transport - Results & Discussion

61. Asymptotic Analysis – Results
Rarefied Gas Transport - Results & Discussion

62. Asymptotic Analysis – Results
Rarefied Gas Transport - Results & Discussion C A B A B D C D

63. Asymptotic Analysis – Knudsen Torque
Rarefied Gas Transport - Results & Discussion
Torque
Force
Torque
Force
Potential applications: particle manipulation, thermal motor