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 Temperature sensor Bio sensor IF filter or oscillator Doms, et al. JMM 2005

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

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  

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:   mean free path of gas molecules   characteristic length of flow field Bulk region Bulk region e.g., air at room temperature, 1 atm

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

Continuum Regime – Governing Equations and BC  

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

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

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 Continuum regime Reynolds number << 1 Navier-Stokes Stoke equations Boundary condition – non-slip and slip

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

1D - Steady (Couette) Theory vs. Experiment

Unsteady Stokes Flow 1D Stokes Model: Cho, et al, 1993 Governing Equations where BC: 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: density: Drag Force: 207.58 nN Q: 29.1

Comparison of Different Models and Experiment

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

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

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

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.

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.

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.

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.

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  

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.

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. 2231-2240, 2007.

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. 1762-1769, 2005.

Examples – Thermal sensing AFM TSAFM IBM Millipede

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

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

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

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

Multiscale Simulation – Temperature Field Continuum solution Multiscale solution

Multiscale Simulation – Heat Flux Total heat flux from the cantilever: 84.46 W/m 1-D decoupled model: 91.56 W/m

Multiscale Simulation – Velocity Field Near the Cantilever

Noncontinuum Phenomena Thermally Induced Gas Flow Knudsen Force TH TC F

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 1879. 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.

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 1879. 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.

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

Thermal Transpiration TC After Collision Before Collision Th Th Tc Tw Tc Tw nonzero net tangential momentum zero tangential momentum

Thermal Transpiration - Velocity OSIP-DSMC

Thermal Transpiration - Velocity

Thermal Transpiration - Pressure 1 um: 3.5kpa difference, maximum velocity 0.872 m/s 100 nm: 6 Kpa difference and 20 nm 29 Kpa difference

Knudsen’s Pump 162 stages; 760 Torr 0.9 Torr 1 um: 3.5kpa difference, maximum velocity 0.872 m/s 100 nm: 6 Kpa difference and 20 nm 29 Kpa difference 162 stages; 760 Torr 0.9 Torr Gianchandani: JMEMS 2005; JMM 2012; JMEMS in press. Gianchandani & Ye, Transducers 2009

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

Knudsen Force

Temperature Contours Kn = 0.5 Kn = 5.0

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

Knudsen Force – Shape Effect

Shape Effect - Asymptotic Analysis      

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

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

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

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

Asymptotic Analysis – Results          

Asymptotic Analysis – Results Rarefied Gas Transport - Results & Discussion          

Asymptotic Analysis – Results Rarefied Gas Transport - Results & Discussion    

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

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