Rudra Pratap, Suhas Mohite AND Ashok Kumar Pandeyeprints.iisc.ernet.in/12328/1/RP-web.pdf · REVIEW...

CranesSci MEMS Lab,Department ofMechanical Engineering,Indian Institute of Science,Bangalore 560012,Karnataka, [email protected]@[email protected]

Keywords:MEMS,Squeeze-film damping,Reynolds equation,Rarefaction effect,Compressibility effect, Inertialeffect, Perforations

Quality factor, Q: is 2π timesthe ratio of the total energysupplied divided by theenergy lost in a single cycle. Itis a measure of sharpness ofthe resonance peak.

REVIEWS

SqueezeFilmEffects inMEMSDevices

Rudra Pratap, Suhas Mohite AND Ashok Kumar Pandey

Abstract | Squeeze film effects naturally occur in dynamic MEMS structures because most of

these structures employ parallel plates or beams that trap a very thin film of air or some other

gas between the structure and the fixed substrate. An accurate estimate of the effect of

squeeze film is important for predicting the dynamic performance of such devices. In design,

availability of very good models for squeeze film effects is indispensable. In this paper, we

discuss the development of squeeze film flow modelling, tracking its routes to the air damped

vibrating system studies in the early twentieth century. We try to capture the early

developments in gas lubrication theory and then discuss the current developments motivated

by the complexities in squeeze film flow analysis brought out by the geometries and flow

conditions prevalent in dynamic MEMS devices.

1. Introduction1.1. Dynamic MEMS devicesEver since the realization of a silicon microresonator1, the number of dynamic MEMSstructures and their applications have been steadilygrowing. The impetus for this growth is providedby the technological advances in microfabricationthat enable us to make very fine suspendedmechanical structures with varied topologiesfor realizing various kinds of in-plane, out-of-plane, and torsional oscillations. Typically, suchstructures have fairly high resonant frequencies,ranging from a few kHz to tens of MHz. Suchhigh frequency devices, coupled with the ease oflow-voltage electrostatic actuation and capacitiveor piezoelectric sensing, find applications ininertial sensing, acoustic transduction, opticalsignal manipulation, and RF (radio frequency)components. Some important milestones inthese application areas are the developmentof the Analog Devices integrated polysiliconADXL accelerometers2, flexural polysilicon microgyroscopes3, the 300-kHz micromechanical filters4,MEMS torsional mirrors5,6, Texas InstrumentsDigital Light Processing Chip (DLP)7, AKU2000MEMS microphone8, ultrasonic motors9, etc. All

these devices employ one or the other type ofvibrating micromechanical structures.

The most significant characteristic of a vibratingsystem is perhaps its resonant frequency response. Aspecial measure that captures the most prominentfeature of this response is called the quality factor,generally referred to as the Q factor of a device. TheQ factor measures the sharpness of the resonantpeak — a higher value indicating a sharper peak.It turns out that the Q factor is directly related tothe overall damping present in the system. Thus,manipulation of the Q factor essentially involvesmanipulation of damping in the structure. There areseveral applications, especially in RF MEMS, where avery high Q factor is desired. MEMS technology hasmade it possible to build mechanical resonators withvery high Q factors, so much so that they can replacesome electronic components in RF applicationson this singular merit. It has been shown that thequality factor of a polysilicon resonator can be ashigh as 106 if it is operated in vacuum10. However,not all devices can be vacuum packed. For example,acoustic transducers must work in a fluid medium.For other devices also, if the vacuum can be avoidedwithout degrading the performance, it may bemore cost effective to have the device operate in air

Journal of the Indian Institute of Science VOL 87:1 Jan–Mar 2007 journal.library.iisc.ernet.in 75

REVIEW Rudra Pratap et al.

Figure 1: SEM pictures of various dynamic MEMS devices.

Capacitive ultrasonic transducer array, cMUT:(X. Jin et al., JMEMS 7(3) 1998);

Tuning fork gyroscope: (Draper lab.www.draper.com/publications/draper25);

Micro resonator for detecting mass of biomolwcules:(N. Elejalde et al., MME 1999);

z axis MEMS accelerometer:(MOTOROLA MMA 1220D);

Micromirror,:(Bin Mi et al., JMEMS 14 (1) 2005);

MEMS Microphone:(J. W. Weigold et al., Analog Devices Inc.2006);

or some inert gas filled packaging. However, thesurrounding air or gas has significant effect on thedynamics of the device. What is special about thisfluid interaction in MEMS devices is that the fluidis usually trapped under or around the vibratingmicromechanical structures in extremely narrowgaps that necessitate the consideration of the fluidas a thin film, that is dynamically ‘squeezed’ by thevibrating structure. The complex interaction resultsin both damping and stiffening of the structuredue to the fluid film. Modelling of this interactionand computation of the squeeze film effects undervarious conditions are the main topics of discussionin this paper.

Figure 1 shows pictures of some dynamic MEMSdevices employing the principle of electrostaticsensing/actuation using the out-of-plane motionof a planar MEMS structure. Most of these deviceshave a thin film of air or some inert gas betweenthe vibrating structure and the substrate. As thestructure vibrates, it pushes and pulls the fluid filmcreating complex pressure patterns that dependon the geometry of the structure, the boundaryconditions, perforations in the structure, frequencyof oscillations, and thickness of the fluid film. Aswe show later, a reasonably accurate determinationof the fluid pressure is key to computing squeezefilm damping and stiffness. Although, there are

76 Journal of the Indian Institute of Science VOL 87:1 Jan–Mar 2007 journal.library.iisc.ernet.in

Damping factor, ζ: is the ratioof damping coefficient of asystem to its critical dampingcoefficient.

Squeeze Film Effects in MEMS Devices REVIEW

Figure 2: Schematic of a flexible structure under squeeze film damping.

S1 S2

S1, S2 - Anchors

Substrate

Microplate

Air-gap

other damping mechanisms such as support loss,thermoelastic damping, etc., the squeeze filmdominates when it is present, often by one ortwo orders of magnitude over the other dampingmechanisms. Before we get into the modelling ofsqueeze film damping, we briefly review the relevantbasic concepts of dynamic response and damping.

1.2. Dynamic characterizationAs shown in Fig. 1, most dynamic MEMS devicesemploy microfabricated beam and plate structures.The squeeze film analysis in these structures isgenerally performed using the 2D isothermalcompressible Reynolds equation coupled withequations governing the plate deflection11,12. Letus consider a microplate (Fig. 2) subjected to atime varying pressure Pe(t) causing transversemotion of the plate. A squeeze film pressurePsq(x,y, t) develops underneath the plate due to itstransverse motion pushing and pulling on the fluidfilm. Assuming small displacements and strains,we obtain the following equation governing thetransverse motion of the plate

ρhp∂2w

∂t2+D5

4 w −T 52 w = Pe(t)+Psq (1)

Figure 3: Computation of Q-factor from the frequency response curve ofa resonator.

10

5

0

-5

-10

-15

-20

-25

Mag

nitu

de (

dB)

4.0 5.0 6.36.0 7.9

Frequency (kHz)

where w(x,y, t) is the transverse deflection of apoint (x,y) on the plate at time t , ρ is the plate

material density, D =Eh3

p

12(1−ν2)is the plate flexural

rigidity, hp is the plate thickness, E is the Young’smodulus, ν is the Poisson’s ratio and T is the axialforce per unit length. The pressure distributionPsq(x, y, t) in the squeeze film is governed bythe Reynolds equation. The pressure boundaryconditions for the air-gap in Fig. 2 are zero fluxat the clamped edges of the plate and ambientpressure at the open edges. Thus, this analysisinvolves a coupled solution of equation (1) and theReynolds equation (discussed later). Many MEMSstructures, however, are generally assumed to berigid, simplifying the coupled structural-fluid fieldproblem to a single degree of freedom spring-mass-damper problem. This assumption also decouplesthe Reynolds equation, which leads to analyticalsolutions when linearized for small-amplitudeoscillations. To characterize the performance ofdynamic MEMS devices, an equivalent spring-mass-damper system under harmonic excitation isconsidered. The equation of motion is governed by

md2w

dt2+ c

dw

dt+ kw = F(t) (2)

where, k = ks + ka is the total stiffness—that of thestructure and the air spring—and c = cs + ca is thedamping due to intrinsic losses in the structure andthe viscous losses in the squeeze film. The structuralspring constant, ks, and damping constant, cs,depend on the material and geometry and are,therefore, constant for a given system. In contrast,the air spring constant, ka, and the dampingconstant, ca, due to the fluid flow depend on thegeometry, oscillation frequency and many factorswhich we discuss later.

If m is the equivalent mass (for a single degreeof freedom model) of the oscillating structure andωn is its fundamental natural frequency, then thedamping factor ζ is given by

ζ =c

cc=

c

2mωn(3)

Equivalently, the quality factor Q is 2π times theratio of the total energy supplied (Einput) divided bythe energy lost (1Elost) in a single cycle and is givenby13

Q = 2πEinput

1Elost=

mωn

c'

1

2ζ. (4)

To calculate the quality factor directly fromexperimental results, we apply the half-widthmethod (3 dB under the peak amplitude)14. The



Figure 4: A schematic diagram of squeeze film flow (A) downward normal motion; (B) upward normalmotion.20

(A) (B)

expression for the experimentally measured qualityfactor, Qexp, is given by

Qexp ≈1

2ξ=

fn

f2 − f1(5)

where f1 and f2 are frequencies at which theamplitude of the displacement is equal to 1/

√2

times the maximum amplitude (equivalently, 3dBdown), and fn is the natural frequency in Hz. Figure3 shows a typical frequency response curve of aresonator and the computation of its Q-factor.

1.3. Dissipation mechanismsDifferent damping mechanisms are present inMEMS devices such as thermoelastic damping15,support losses16, and losses due to air flow10,17.The relative dominance of different dampingmechanisms depends on the operating conditionsand dimensions of the MEMS structures. To capturethe cumulative effect of all type of loss mechanisms,the net quality factor Qnet is given by18

1

Qmax≤

1

Qnet=

1

Qted

+1

Qsupport+

1

Qvisc+ ... ≤

1

Qmin. (6)

For most dynamic MEMS structures operating atambient conditions, squeeze film damping is themost dominant damping mechanism10,17–19. Thisdamping, however, can vary widely depending onthe fluid pressure in the gap or the cavity. At very lowpressures other damping mechanisms can becomemore dominant. Based on such considerations,Newell19 has divided the pressure range into threemain regions: the intrinsic, the molecular, and theviscous region. In the intrinsic region the ambientpressure is very low, so that the air damping isneglected compared to the intrinsic damping. In thisregion, damping is largely dependent on the surfaceto volume ratio, the surface quality and the material.In the molecular regime, damping is caused by

independent collisions of non-interacting molecules.In this case, there is little interaction between themolecules. In viscous region, the pressure is highsuch that the air molecules also interact amongthemselves causing viscous effect.

Since the performance of any dynamic device islimited by the dominant damping mechanism inthe system, we now discuss the theory of squeezefilm damping, which is the dominant dampingmechanism in most dynamic MEMS devices notoperating in vacuum.

2. Squeeze film damping mechanismWhen a planar structure oscillates normal to thesubstrate, the air-film between the structure and thesubstrate is squeezed causing a lateral fluid motionin the gap. Therefore, there is a change in pressurein the gap due to the viscous flow of air as shown inFig. 4. The forces due to the built-up pressure arealways against the movement of the structure. Thework done in the viscous flow of air is transformedinto heat. Thus, the air-film acts as a damper andthe phenomenon is called squeeze film damping21.

The squeeze film damping is basically prevalentin systems in which the air-gap (cavity) thickness issufficiently small compared to the lateral dimensionsof the structure. For smaller air-gap thickness,damping is higher while for larger gap it is negligible(viscous drag due to air flow on the surface becomesdominant for larger gaps). For a given value ofair gap thickness, there are various factors thataffect squeeze-damping, for example, the operatingfrequency of oscillations, surrounding pressure,boundary conditions, etc22,20.

2.1. Squeeze film damping in lubrication theoryA large part of squeeze film flow analysis comes fromthe literature on lubrication theory. Researchersin this field were interested in understanding themechanics of very thin fluid films between rotatingor squeezing shafts and journal as well as thrustbearings. This problem was intensively investigatedin sixties. While load bearing capacity of these


Knudsen number, Kn: is theratio of mean free path l ofgas molecules to thecharacteristic flow length. It isa measure of gas rarefaction.


Figure 5: Frequency response curves obtained by analytical (—-) and FEMsimulations (marked) for a 2D squeeze film damper of 80×80 µm squareplate separated by a 5µm air-gap (ha); the response curves without inertialeffects are also shown (– · –) T. Veijola.50

1

0.3

0.1

0.03 0.1 0.3 1.0 3.0 10.0 30.0

0.03

0.01

0.003

F/µN

f / MHz

90

45

0

-45

-90

F/pha

films under dynamic conditions was of centralinterest in these studies, the analysis naturally ledto damping calculations and the effect on thedynamic response of the systems under investigation.Good mathematical models were developed fordetermining the squeeze film damping by manyresearchers23–27. The following two decades sawvery little work or interest in this problem. However,the arrival of dynamic MEMS devices in the lateeighties rekindled the interest in this subject. MEMSdevices presented a new application with some newtwists and turns in the problem. Naturally, there hasbeen a lot of work in this area ever since. Althoughthe central interest in the new studies is on dampingcharacteristics of the squeeze film flow because of itsdirect bearing on the Q factor of a MEMS device, thedynamic spring characteristics of the film occupycenter stage when the frequency of oscillationbecomes considerably high. Several mathematicalmodels exist today for the computation of bothdamping and spring effects of the film for variousgeometric and flow conditions. The applicabilityand validity of these different models are undercontinuous investigation because of continuouschanges in device geometries and differences inexperimentally observed results and predictionsof theoretical models. The governing fluid flowequation used in most of these studies are eitherthe linearized Reynolds equation in continuumregion26,27 or the modified Reynolds equationconsidering slip flow using Burgdorfer model28.

One of the most significant effects presentin squeeze film flow in MEMS structures is thatof rarefaction. This effect arises because of theextremely small gaps in which the air or gas isforced to flow. Knudsen number, Kn, defined as the

ratio of the mean free path of the gas molecules tothe characteristic flow length, provides a very goodmeasure of rarefaction. A very small value of Kn(� 1) usually results from large flow lengths andhence a situation where no molecular attention isrequired; continuum flow theory does very well. Onthe other hand, a large value of Kn (> 1) necessitatesthe consideration of molecular motion and attentionon molecular interactions. In such rarefied flows, theclassical assumption of zero relative velocity betweenthe fluid molecules and the structural walls breaksdown and consideration of slip flow at the wallbecomes necessary. Slip flow conditions, however,are not unique to MEMS devices. These conditionswere also observed and analysed by the researchersinterested in gas lubrication theory.

Burgdorfer28, who first introduced the conceptof kinetic theory of gases to the field of gas-film lubrication, proposed a slip flow boundarycondition at the walls by expanding the velocitynear the wall in Taylor series. He retained the zerothand first order terms to be used in the existingReynolds equation for very small values of Knudsennumber, Kn � 1. Hisa and Domoto29 proposed ahigher order approximation for Knudsen numberslarger than the range of Burgdorfer’s equation byconsidering both the first and second order slipvelocities together. Mitsuya30 proposed a fractional(one and a half) order slip flow model. Thesemodels are based on the assumption of low Knudsennumber, i.e., Kn � 1. Therefore, there was a needfor a more general model to cover a wide range ofKnudsen number that could account for ultra thingaps. Gans31, was the first to modify the existingReynolds equation by describing the flows (i.e.,Poiseuille flow, Couette flow, etc.) in terms ofthe linearized Boltzmann equation. To establishthe validity of Gans’s approximation for arbitraryKnudsen number, Fukui and Kaneko32, analyzedit numerically and examined the lubricationcharacteristics. They found that Burgdorfer’smodification overestimated the load carryingcapacity with the first order velocity slip boundarycondition in the Reynolds equation, and thatunderestimated it with one-and-a-half and second-order velocity slip boundary condition. Aftervalidating the Gans’s approximation, Fukui andKaneko33 proposed the use of a Poiseuille flow ratedatabase that they created by numerical calculationbased on the linearized Boltzmann equation. Theyused the database effectively to reduce enormouscalculations required to solve Boltzmann equationwith the same accuracy (±1%) for a very smallrange of Knudsen number.

Another complication that arises because of theultra thin fluid films is the effect of surface roughness



Figure 6: Variation of non-dimensional damping force (fd/ε) and springforce (fs/ε) with squeeze number σ; the point of intersection of the twocurves (i.e., fd=fs) corresponds to the cut-off squeeze number (σc ).

0.6

0.5

0.4

0.3

0.2

0.1

00 5 10 15

Squeeze number (σ)20 25 30

Non

-dim

ensi

onal

dam

ping

and

spr

ing

forc

es (

f d/ε

, fs/

ε)

on the flow characteristics. To model this effect,Patir and Cheng34, derived the average Reynoldsequation by averaging the flow. The average flowrate is expressed as the rate of flow passing throughaveraged spacing, multiplied by an appropriateflow factor. The flow factor adjusts the Reynoldsequation to give the correct average pressure andflow condition. These flow factors are obtained bynumerical calculations using a small control areawith a given surface roughness. The average flowmodel can be easily applied to distributed roughnessproblems. Elrod35 tried to generalize the flow factorsanalytically. Tripp36 outlined the mathematicalmethod using perturbation approach with Greensfunction technique. Tonder37 has shown analyticallythat the flow factors found to express roughnesseffects for incompressible lubrication are also validfor compressible lubrication, at least approximately.Mitsuya et al.38 have extended the average flowconcept to incorporate slip flow conditions.Almost concurrently, Bhushan and Tonder39,40 haveconsidered the surface roughness and gas rarefactionby using the flow factors derived for incompressiblelubrication34 and modified local flow rates derivedfrom the first order slip flow model28.

Since the effects of roughness and rarefactionare coupled, some additional corrections are needed.Li et al.41 derived the modified Reynolds equationfor arbitrary roughness orientation. Their modelincludes the coupled effects of surface roughness andgas rarefaction. For ultra thin layer, the modifiedReynolds equation was extended42 to include higherorder slip flow models43 and was simplified bydropping the Couette flow term which considers theeffect of cross flow. Hwang et al.43 also proposed

the use of Modified Molecular Gas film Lubrication(MMGL) equation along with the coupled effectsof surface roughness and gas rarefaction. Theinertial effects which become dominant at higherfrequencies are also discussed in the lubricationproblem44,45.

Blech27 and Griffin26 developed formulae forcalculating squeeze film force for rectangular andcircular plates. They linearized the governingequation (the Reynolds equation) of the squeezefilm flow and then calculated the squeeze film forceformula. Their models are applicable to only smallamplitude oscillation devices. They also showed thatthe viscous damping effect of the air film dominatesat low frequencies while spring effect dominates athigher frequencies. Their models have been justifiedexperimentally by Turner et al.17. However, largeamplitude oscillations are also common in severaldevices such as deformable mirror devices (DMD),pull-in pressure sensors, etc. These devices requirea solution of nonlinear Reynolds equation for thecalculation of squeeze film force46. Some of theissues of nonlinearity in the squeeze film dynamicshave been studied by Antunes and Piteau47, buttheir work is limited to one-dimensional flows.

Veijola et al.48 have developed an expressionfor Poiseuille flow rate based on the data madeavailable by Fukui and Kaneko33. Li49 linearized theMMGL equation to find out squeeze film dampingin MEMS structures but under the same slip modelused by Wang and Cheng43.

It is clear from the developments discussedabove that the Reynolds equation forms the basisof almost all models. Therefore, we begin ourdiscussion on mathematical modelling by discussingdifferent variants of Reynolds equation derivedfrom the Navier-Stokes equations under differentassumptions.

2.2. Different models derived from Navier-Stokesequations

The fluid flow in continuum regime is governedby the continuity equation and the Navier Stokesmomentum equations which are valid for unsteady,compressible and viscous flow. The equations interms of 3-D spatial co-ordinates and 1-D temporalco-ordinate are written in concise vector form ascontinuity equation:

∂ρ

∂t+∇ · (ρu) = 0; (7)

the Navier-Stokes equation:

ρ

{∂u

∂t+ (∇u)u

}= F −∇p+ (µ∗

+µ)∇(∇ ·u)

+ µ∇ · (∇u); (8)



where, F represents the body forces and u = iu+

jv + kw the velocity, ρ the density of fluid, µ∗

the coefficient of dilatational viscosity, and µ thecoefficient of shear viscosity or dynamic viscosity.

Let us consider a flow problem illustrated inFig. 4, where a rectangular plate oscillates normal tothe substrate. For small air-gap separating the twoplates the squeeze film flow is predominantly twodimensional (i.e., in the x-y plane). Generally, wecan make the following assumptions:

• No external forces act on the film, thus F = 0

• The structure oscillates with a smallamplitude and the main flow is driven bypressure gradients in the x and y directions.So all the terms with w are small and thereforeneglected

• No slip flow occurs at the planar boundaries

• No variation of pressure across the fluid film

• The flow is laminar; no vortex flow and noturbulence occur anywhere in the film

• Fully developed flow is considered within thegap

• Flow is assumed to be isothermal, i.e., p ∝ ρ.

The Navier-Stokes equations (8) then reduce to:

ρ

(∂u

∂t+u

∂u

∂x+ v

∂u

∂y

)= −

∂p

∂x+

∂

∂z

(µ

∂u

∂z

)and (9)

ρ

(∂v

∂t+u

∂v

∂x+ v

∂v

∂y

)= −

∂p

∂y+

∂

∂z

(µ

∂v

∂z

).

Furthermore, when the gap is much smaller thanthe surface dimensions, and velocities u and v arerelatively small, the convective inertial terms u ∂v

∂x ,

v ∂v∂y , u ∂u

∂x and v ∂u∂y can be ignored50. Dropping these

terms, we rewrite eq. (8) as

ρ

(∂u

∂t

)= −

∂p

∂x+

∂

∂z

(µ

∂u

∂z

)and (10)

ρ

(∂v

∂t

)= −

∂p

∂y+

∂

∂z

(µ

∂v

∂z

)If we also neglect the unsteady inertial terms, eq. (10)reduce to

∂p

∂x=

∂

∂z

(µ

∂u

∂z

)and (11)

∂p

∂y=

∂

∂z

(µ

∂v

∂z

)

Using these equations, along with the continuityequation, the generalized Reynolds equation isderived as follows. Equations (10) are first solvedfor u and v under no slip (i.e., u|±h/2 = 0 andv|±h/2 = 0) boundary conditions.

u =1

2µ

∂p

∂x

(z2

−h2

4

)and (12)

v =1

2µ

∂p

∂y

(z2

−h2

4

).

The average radial velocities u and v are thenobtained by integrating equation (12) over theair-gap height as

u = −h2

12µ

∂p

∂xand v = −

h2

12µ

∂p

∂y. (13)

We now integrate the continuity equation (7) acrossthe gap from −h/2 to +h/2 as

h∂ρ

∂t+

∂

∂x

[ρ

∫ h/2

−h/2udz

]

+∂

∂y

[ρ

∫ h/2

−h/2vdz

]+ρ

∂h

∂t= 0. (14)

Note that we have made a substitution for thevelocity in z direction, w =

∂h∂t . The integrals above

represent average velocities multiplied by h andcan be replaced with uh and vh. Furthermore, weassume the flow to be isothermal, i.e., p ∝ ρ and getthe nonlinear compressible Reynolds equation:

∂

∂x

(ph3

µ

∂p

∂x

)+

∂

∂y

(ph3

µ

∂p

∂y

)= 12

∂(ph)

∂t. (15)

In the next subsections, we discuss two importantmodifications to this equation, viz., to account forthe rarefaction effects and the inertial effects.

2.2.1. Rarefaction considerationsFor very small values of Knudsen number

Kn, the first order slip boundary conditionsu|±h/2 =∓l ∂u

∂z and v|±h/2 =∓l ∂v∂z are applied and

the slip-corrected velocity distributions can beobtained from equations (9) as

u =1

2µ

∂p

∂x

(z2

−h2

4−Knh2

)and (16)

v =1

2µ

∂p

∂x

(z2

−h2

4−Knh2

).


The relative flow ratecoefficient, Qpr: whenincorporated in the ordinaryReynolds equation, enablesmodelling the effects ofrarefaction and/or inertia.


Figure 7: Perforation geometries: (A) Condenser microphone: close-up of a thin perforated back-plate, theunderlying membrane and the air-gap in between (A. Kovacs et al., JMM, 5, 1995) suitably modelled as a2-D flow case, (B) Tuning fork gyroscope: close up of a high aspect ratio perforated proof mass, theunderlying substrate and the air-gap in between; (Kwok et al. JMEMS, 14(4) 2005) best modelled as a 3-Dflow case.

(A) Squeeze-film flow in x-y plane, the flow is predominantly planar i.e., 2-D .

ξ0 substrate

perforated proof mass

ha

diaphragmξ0

perforated back-plate

X

X

X

X

(B) Squeeze-film flow in x-y plane and through the perforations along z-axis, 3-D flow.

Cross section XX Cross section XX

We obtain the average radial velocity u and v as

u = −h2(1+6Kn)

12µ

∂p

∂x

and (17)

v = −h2(1+6Kn)

12µ

∂p

∂y;

and substituting these in the continuity equationwe obtain the nonlinear compressible Reynoldsequation with slip correction:

∂

∂x

(ph3(1+6Kn)

12µ

∂p

∂x

)+

∂

∂y

(ph3(1+6Kn)

12µ

∂p

∂y

)=

∂(ph)

∂t. (18)

The factor (1 + 6Kn) in the above equationis referred to as the relative flow rate coefficientQpr = (1+6Kn). The relative flow rate coefficientis generally included in the fluid viscosity and thecombined term is called the effective viscosity, µeff =µ

Qpr. Veijola et al.51 present a simple approximation

for the effective viscosity by fitting the respectiveflow rate coefficients to the experimental valuestabulated by Fukui and Kaneko32. This relationship

is given as µeff =µ

1+9.638Kn1.159 . There are also manyother analytical models to consider rarefactionwhich are compared with the experimental resultslater in subsection 4.2.1.

2.2.2. Inertial effectsA modified Reynolds equation50 is derived from

equations (9) which include the unsteady inertialterms. Thus, including inertial effects and slipcorrection, the velocity profiles for small amplitude

harmonic excitation (i.e., h = δejωt ) are obtained as

u =1

jωρ

(cos(qz)

cos(qh/2)−lqsin(qh/2)−1

)∂p

∂x,

v =1

jωρ

(cos(qz)

cos(qh/2)−lqsin(qh/2)−1

)∂p

∂y,

(19)

where q =√

jωρ/µ. Following the same procedureas before, i.e., substituting the average velocitiesin the continuity equation we get compressibleReynolds equation with inertia and slip correction:

∂

∂x

(ph3

µQpr

∂p

∂x

)+

∂

∂y

(ph3

µQpr

∂p

∂y

)=

∂(ph)

∂t.

(20)


Reynolds number, Re: is theratio of inertial force to theviscous force.

Squeeze number, σ: is themeasure of compressibility ofgas. Low squeeze numbersmean gas escapes readily;high values indicate gas istrapped between thestructures due to its viscosity.


Here, the value of Qpr with the inertialeffects and the gas rarefaction effects, basedon the first order slip, turns out to be Qpr =

12−jRe

[(2−ljRe)tan(√

−jRe/2)−√

−jRe]

[√

−jRe(1−l

√−jRetan(

√−jRe/2))]

where Re =

ρωh2a

µ(= qh) is the Reynolds number. Figure 5

shows the frequency response of a square plateobtained by Veijola50using eq. (19). It is seenthat besides viscous forces that dominate at smallfrequencies, gas compressibility and inertial forcesdetermine the amount of net force at higherfrequencies.

3. Squeeze film damping in MEMS devicesFor many MEMS structures having small amplitudeoscillations, the assumption of a rigid structuresimplifies the model to a one degree of freedomspring-mass-damper. This assumption leads toanalytical solutions when the linearization is appliedto the Reynolds equation.

3.1. Small amplitude approximationsFor small amplitude displacement of the plate,the compressible Reynolds equation (15) can belinearized using the perturbation parameters p

(in pressure) and h (in gap), which were firstintroduced by Griffin26. We substitute p = Pa + p

and h = (ha + h) and neglect the higher orderterms to get the following linearized compressibleReynolds equation:

[∂2 p

∂x2+

∂2 p

∂y2

]=

12µ

Pah3a

[ha

∂p

∂t+Pa

∂h

∂t

]. (21)

For convenience, we will drop the hat in p and hfrom here onwards with an implicit understandingthat we are working with perturbation variablesp and h about the reference pressure Pa andequilibrium gap ha, respectively. Further, if air

is assumed to be incompressible (i.e., ∂p∂t = 0),

eq. (21) can be reduced to the form known as theincompressible Reynolds equation:

∂2p

∂x2+

∂2p

∂y2=

12µ

h3

∂h

∂t. (22)

3.2. Analytical solutionEquation (21) can be non-dimensionalized using

new variables 8 =p

Pa, ε = h

ha, τ = ωt , X = x/L ,

Y = y/L, where ω = the circular frequency and L =

the side of a square plate. In terms of these variables,eq. (21) can be written as[

∂28

∂X2+

∂28

∂Y 2

]= σ

[∂8

∂τ+

∂ε

∂τ

](23)

where σ =12µL2ω

Pa h2a

, is called the squeeze number

which measures the compressibility of thefluid. Its significance is discussed in details insubsection 4.2.2.

Analytical solutions for squeeze film dampingare reported for transverse and tilting motionsin case of simple geometries such as rectangularplates, circular plates and annular plates, etc26. Asearly as 1917, analytical expressions for estimatingthe stiffness and the damping offered by the airfilm between two circular plates for condensertransmitters were given by Crandall52. For rigidlyoscillating rectangular plate, Blech27 solved thelinearized Reynolds equation with ambient pressureat the boundaries and presented a closed formsolution for the damping and spring coefficients as:

ca =64σPaχW 2

π6ωha

×

∑m,n,odd

m2χ2+n2

(mn)2{[m2χ2 +n2]2 +σ2/π4}(24)

ka =64σ2PaχW 2

π8ha

×

∑m,n,odd

1

(mn)2{[m2χ2 +n2]2 +σ2/π4}(25)

where χ = L/W is the length to width ratio ofthe rectangular plate, Pa is the ambient pressure,ha is the air gap height and ω is the frequency ofoscillations of the plate. The central parameterin these expressions is the squeeze number σ,

given by σ =12µL2ω

Pa h2a

which is a measure of the

compressibility. The compressibility is proportionalto the square of the lateral dimension to air-gapratio ( L

ha) and the frequency of oscillation (ω),

and inversely proportional to the ambient pressure(Pa). The variation of damping and spring forceswith squeeze number, plotted as a function offrequency of oscillation σ(ω), is shown in Fig. 6.Initially, the damping force increases steadily withfrequency, reaches a maximum value and thereafterdecreases monotonically. At low frequencies theair can escape easily however, with increase infrequency of oscillation the fluid compression beginsto dominate over the fluid flow and the spring forcegradually increases and reaches a maximum value.The point of intersection of the two curves is knownas the cut-off frequency. Similar analysis is reportedfor circular and annular plates and for the torsionalmotion of plates as well26. Torsional motions areencountered in MEMS torsional mirrors.

These formulae are found to be useful forestimating squeeze film effects in simple MEMS



Figure 8: Pressure contours around the perforations (for air gap = 1µm, frequency f = 10kHz) : (A)staggered hole configuration; (B) non-staggered (matrix) hole configuration54.

(A) (B)

structures. However, many MEMS structures havevery intricate etch hole topography as shown inFig. 7 and the complexity in analysis increasesmany folds when one considers associated fluidflow calculations for damping. Moreover, manyother effects such as rarefaction, compressibilityand even inertia may come into play under differentoperating frequencies. Different approaches havebeen reported over a decade. In the next section, webriefly review various models that deal with theseissues.

4. ComplexitiesThe problem of squeeze film damping in MEMStransducers becomes more involved becauseof the complexities arising from inclusion ofperforations, shrinking length scales and relativelyhigh frequencies of operation. On one hand,scaling down the dimensions makes the validity ofcontinuum assumption questionable and requirescareful modelling, on the other hand, conflictingdemands on damping arise while designingfor desired dynamic characteristics such as thenatural frequency and Q-factor along with theelectromechanical sensitivities.

At this stage it will be important to understandthe complex relationship of device geometry withthe squeeze film damping phenomenon and criticalperformance parameters of the device. Let usconsider a simple device with two parallel plates,one free to vibrate transversely and the other fixed(as a substrate). If we consider this device to be acapacitive sensor or actuator, the most commoncase in dynamical MEMS devices, a reasonabledesign goal will be to maximize the compliance(Cm ∝

At3 ) for higher mechanical sensitivity and

have a high base capacitance (Cb ∝Aha

) for better

electrical sensitivity. This clearly requires sufficientlyhigh surface area (A) and small thickness (t) of thevibrating structure, and a very small air-gap (ha)

separating the two structures. However, the air in thesmall gap between the transversely moving planarstructure and the fixed substrate imparts damping,spring and inertial forces to the structures17,50.These forces have a complex dependence on theair-gap height and the operating frequency. Thisaffects the frequency response of the structure andhence the sensitivity, resolution and the bandwidthof the device. The damping due to the squeeze filmis considerably high for large surface to air-gap ratio(∝ A

h3a). At low frequencies, damping dominates

whereas at high frequencies spring and inertialeffects dominate.

The damping can be minimized if it is possibleto operate the device under vacuum conditionwhich, in turn, requires an expensive packaging.However, in most cases, the Q-factor of a vibratingsystem is still mainly determined by the energylosses to the surrounding air as the vacuum in theencapsulated device can hardly be high enough53.The amount of the squeeze film damping can becontrolled by providing perforations to one of theplanar structures, namely, the back plate or theoscillating proof mass. These perforations alsofacilitate the etch release of the sacrificial layerin surface micromachining but reduce the devicecapacitance which is undesirable. Adequate valueof the base capacitance can be obtained by havingsmaller air-gap spacing. While reducing the airgap, the perforations need to be designed suchthat a constant damping is maintained withoutsacrificing the capacitance. A feasible range ofperforation geometry depends on the chosenmicromachining process. To achieve the design



Figure 9: Schematic showing: (A) staggered hole configuration: hexagonal pattern (A-B-C-D-E-F) andcircular pressure cells within it; (B) non-staggered (matrix) hole configuration: square pattern (L-M-N-O).

(A) Staggered hole configuration (B) Non-staggered hole configuration

goals of minimizing damping and maximizing thecapacitance a trade-off analysis has to be carriedout between the perforation geometry and the air-gap. This necessitates creation of simple analyticalmodels and derivation of closed-form formulae forthe squeeze film analysis. These in turn shouldbe validated using FEM tools or experimentalmeasurements. The forces due to the squeeze filmextracted from these behavioural models are inturn employed in a system-level model for theperformance optimization of a MEMS device.Thus, the air-gap ha, is perhaps the single mostcrucial parameter which affects the squeeze filmbehaviour, base capacitance and hence the overalldevice performance. Clearly, efficient models forthe squeeze film analysis of MEMS structures areindispensable in the design of MEMS devices.

4.1. Modelling strategies for perforated structuresThe squeeze film flow in dynamic MEMS structuresis modelled well by the 2D Reynolds equationobtained from the Navier-Stokes equation byneglecting inertia and assuming the lateraldimension to be an order of magnitude largerthan the air-gap height. Figure 8 shows typicalpressure contours plotted over the fluid domain forcommonly encountered staggered and non staggeredconfigurations of perforations. It is observedthat in both configurations of the holes there isorder and symmetry in the pressure contours.

Various approaches based on analytical, numericaland mixed techniques are found in literaturewhich model perforated structure. The analyticalapproaches employ following two strategies:

1. Pressure cell concept: The inherent symmetryin the distribution of holes leads to theformation of repetitive pressure patternsaround each hole in the entire fluid domain.This particular fact is exploited in thisapproach by analysing the problem for a singlehole embedded in a single ‘cell’, and thenobtaining the result for the entire structureby multiplying with the number of cells (seeFig. 9).

2. Modified Reynolds equation: In this approach,the Reynolds equation is modified to includethe flow leakage at the perforations.

For perforated MEMS structures, the effectivelateral dimension is the pitch of the holes (ξo).Therefore, in the case of perforated structures theassumption of large lateral dimension begins todeviate. Moreover, the flow through the perforationscomes into play adding an extra dimension to themodel. The model dimensionality (2D, or 3D) isgoverned by the relative size of the perforationparameters. The significant perforation parametersare the pitch (ξo), the diameter (d), and the lengthof the holes (l, which is the same as the thicknessof the perforated plate). Various models have been



Figure 10: An overview of the existing squeeze film models for perforated structures (ξo is the pitch of the holes, ha is the air-gapheight, l and d are length and diameter of the holes, respectively).


Deep Reactive Ion Etching(DRIE): is a highly anisotropicetch process developed forMEMS and used to createhigh aspect ratio holes andtrenches in silicon.


Figure 11: Simulated dynamical response of the microaccelerometer withand without air film damping effects (Yang et al. Proc. IEEE Solid StateSensors and Actuators Workshop, SC, USA 1996).

100

50

0

1 10 102 103 104 105 106

-50

-100

-150

-200

-250

Frequency (Hz)

Mag

nitu

de (

dB)

Nominal (low pressure)Proof mass with holesProof mass without holes

Curve 1

Curve 3

Curve 2

presented for the squeeze film analysis of perforatedMEMS structures in the literature spanning over adecade. These models differ from each other in theway they treat the flow (e.g., 2D, or 3D), and theeffects they include in the analysis (e.g., rarefaction,compressibility and inertia etc.). Figure 10 givesan overview of different squeeze film models at aglance. We now discuss some analytical models usedfor various perforation geometries.

When the size of the holes is much largercompared to the air-gap height (d > 10ha) andthe perforations are short (l < d) as shown infigure 10(a), the loss through the holes is notsignificant. In this case, the problem is reducedto solving the 2-D Reynolds equation under thesimplified boundary condition that the acousticpressure vanishes at the edge of a hole. Closed-form solutions derived assuming incompressibleflow exist for such geometries55. This is based onthe concept of pressure cell mentioned above. Thedamping coefficient Ra for a square plate of sidelength L having perforation density n is given by

Ra =12µL2

nh3aπ

(α

2−

α2

8−

ln(α)

4−

3

8

), (26)

where α is the perforation fraction. Solutionsincluding rarefaction, compressibility and inertialeffects are also reported for such geometries54,56–58.

For inertial MEMS sensors (e.g., gyroscope,accelerometer, etc.), high aspect ratio perforatedstructures using deep reactive ion etching (DRIE)process are used as shown in figure 10(b). In thiscase, the loss through the holes can be significantas the flow is a combination of the horizontal flowbetween the planar surfaces and the vertical flowthrough the holes. Bao et al.59,60 have modelled thisproblem by modifying the Reynolds equation using

a pressure leakage term corresponding to the lossthrough the holes (in the z direction). The modifiedincompressible Reynolds equation based on thismodel is given as

∂2p

∂x2+

∂2p

∂y2−

3β2r2o

2h3H

1

η(β)p︸︷︷︸

pressure leakage term

=12µ

h3

∂h

∂t. (27)

This model is suitable for the uniformly distributedholes where the loss through the holes ishomogenized over the entire domain. Forarbitrary perforation having complex shapes,and nonuniform distribution and size, Veijolaet al.61 have suggested another model. Thismodel utilizes a Perforation Profile Reynoldssolver (PPR: a multiphysics simulation software),which includes additional terms in the modifiedReynolds equation that model the leakage flowthrough the perforations, and variable viscosityand compressibility profiles. The frequency domainform of the equation is given as

∇ ·

(Dhh3Qpr

12µ∇p

)−Ch

jωh

Pap−Yhp = w; (28)

where Dh(x, y), Ch(x, y) and Yh(x, y) areextentions that are specific for perforated structures:relative diffusivity, relative compressibility, andperforation admittance profiles, respectively. Widerperforation geometries can be treated with thismodel, however, it requires special computationaltools and takes time to build the model.

Alternatively, the flow problem can be treatedin two distinct parts. The damping effect due tothe lateral flow is modelled by Reynolds equationand the flow through the holes is modelled byPoiseuille equation. However, in the region justabove the hole where the lateral squeeze flow joinsthe poiseuille flow, there is a complex interactioncausing losses which are difficult to estimate.Moreover, the trivial boundary condition that theacoustic pressure vanishes at the edge of a hole doesnot hold. Darling et al.62 used arbitrary pressureboundary conditions based on complex acousticalimpedance of an aperture and treated the squeezefilm problem using Green’s function approach.Kwok et al.63 used the boundary condition onthe hole evaluated for incompressible flow in thenumerical simulation performed using PDEase,a custom made finite element solver. Under theassumption of incompressible flow (i.e., low squeezenumber), the flow through a hole is obtainedmerely by geometrical scaling between the celldiameter to the hole diameter. In a recent study,



Figure 12: The damping, spring and inertial nature exhibited by air atdifferent values of the squeeze number σ and the Reynolds number Re; (A)squeeze film is predominantly viscous, (B) squeeze film is viscous andcompressible, (C) squeeze film besides being viscous and compressible,exerts inertia

Veijola64 used existing analytic models in thesqueeze film and capillary regions, and using FEMsimulations derived approximate flow resistances atthe intermediate region and at the exit flow fromthe hole. The model is valid over a wide rangeof perforation fractions (1% to 90%). The studyalso presents analysis of cut-off frequencies forcompressibility and inertia and specifies an upperlimit on the frequency up to which the model isvalid.

When the pitch of the holes, the air-gap and thediameter of the holes are all of the same order asshown in figure 10(c), the assumptions made inderiving the Reynolds equation do not hold and thecomputationally intensive 3D Navier-Stokes analysisbecomes indispensable. In a recent study, a surfaceextension model is proposed65, where the gap sizeis comparable to the lateral dimensions. In thisapproach it is shown that after extending each openborder by (1.3/2)ha the 2D Reynolds equation cangive accurate results and 3D Navier-Stokes equationcan be dispensed with. This approach could beapplied to the perforated structures as well.

4.2. Criteria for considering Rarefaction,Compressibility and Inertia

In this section, we discuss various phenomenaassociated with the squeeze film analysis in termsof some non-dimensional numbers characterizingthem and highlight how they are related to theperforation geometry and the operating frequencyof a device.

4.2.1. RarefactionIn the presence of perforations, we have to worry

about rarefaction effects in two places— the usualhorizontal expanse between the vibrating plateand the fixed plate, and the airflow space in theholes. Therefore, we use two Knudsen numbers

Kng and Knh, for the usual air-gap and the hole,respectively. We define Kng = l/ha, where ha is theair gap thickness and l is the mean free path of thegas molecules. Similarly, Knudsen number for theflow through a hole is defined as Knh = l/ri, whereri is radius of the hole. The parameter l is related tothe packaging pressure Pa as l =

PoloPa

. At ambientconditions, the mean free path lo for air is 0.064µm.For MEMS devices having very small air-gap andlow packaging pressure, the Knudsen number Knincreases and the fluid flow transits from continuumflow to rarified flow. Based on the values of Kn,the flow regimes can be divided into four differenttypes as shown in Table 166. Thus, it is possible tohave different flow regimes in the air-gap and theholes. As discussed in section 2.2.1, this effect can bemodelled by changing the effective viscosity throughthe use of corresponding flow factor. Dependingon the value of Kn, the value of µeff and hencethe squeeze film damping can vary considerably.For example, the value of damping drops by 30%for 1µm air-gap at 1 atm pressure if slip flow isconsidered. For the rarefied flow regime, analyticalmodel is derived using the slip-flow wall boundaryconditions at the planar surfaces in the air-gap andon the cylindrical wall of the hole. Under normalconditions, the continuum model is used to modelgas flow. But under the condition of extremely lowdensity (or extremely small flow length), discreteparticle effects become significant, and then we haveto adopt discrete model, i.e., the method of rarifiedgas dynamics67.

4.2.2. CompressibilityThe squeeze number σ, which is a measure

of the compressibility, is given by σ =12µL2ω

Pa h20

(and for perforated structures σ =12µωr2

oPa h2

awhere

ro is the radius of an equivalent circular pressurecell which is approximately half the pitch of theholes (ro ≈ ξo/2)). Thus, the compressibility isproportional to the square of the lateral dimensionto air-gap ratio ( L

ha) and the frequency of oscillation

(ω), and inversely proportional to the ambientpressure (Pa). If σ � 1 the compressibility can beneglected and the flow is treated as incompressible.For higher values of σ, the compressibility leadsto a significant air-spring effect which can beundesirable as it can adversely affect the dynamicbehaviour of a device68. Contrary to this, the factthat the perforations increase the cut-off frequency,which is the frequency at which the damping andthe spring forces are equal, can be exploited byvarying the number and size of the perforations69.Blech27 and Allen et al.70 too have reported the useof squeeze film damping to tailor the frequency



Table 1: Knudsen number Range and the corresponding Flow regimes

Knudsen number (Kn) Kn < 0.01 0.01 < Kn < 0.1 0.1< Kn <10 Kn > 10

Flow regime Continuum Flow Slip Flow Transitional Flow Molecular Flow

response of a seismic accelerometer and that ofmicromachined sensors, respectively. Figure 6illustrates the simulated small-amplitude dynamicalresponse of a micro-accelerometer fabricated atMIT46. Curve 1 is a typical response without any airdamping. Curve 2, which rolls off very fast, is theresponse with air damping indicating the severityof damping at 1µm air-gap. In order to reducedamping, a perforated proof mass was used and thecorresponding response is indicated by curve 3 infigure 11. Thus, including the compressibility effectsin the analysis enables the designer to design theperforation geometry such that the compressibilityeffects are either totally ruled out or suitably tunedas the case may be.

4.2.3. InertiaThe small dimensions of MEMS devices

constitute a very small volume which containsa minuscule quantity of air. Hence, fluid inertiamay be neglected at low frequencies (Re � 1).However, for larger air-gap height, or at higherfrequencies of oscillations the inertial effects maynot be negligible. This is done by incorporating thefrequency dependent flow rate coefficient Qpr whichmodifies the velocity profile with frequency50. Forthe squeeze film flow in the air-gap, the modified

Reynolds number is defined as Reg =ρωh2

aµ

and for

the flow through the hole it is defined as Reh =ρωr2

iµ

.The Reynolds number in the two cases increases

Figure 13: (A) 2D pressure cell in cylindrical coordinates, (B) 3D pressure cell including loss through thehole, (C) Frequency response (amplitude) of the full 3D squeeze film model including inertia in comparisonwith the results of 2D model with and without inertia, and numerically simulated model in ANSYS-CFX (×)for ha = 4µm, hole size s = 8µm and l = 20µm.71

102

101

100

10-1

10-2

10-3

104 105 106 107 10810-4

Forc

e (μ

N)

Frequency (Hz)

(C) Response curves using analytical and numerical models.

(A) 2D model of a single circular pressure cell (B) 3D model of a single circular pressure cell

ha = 4μm, s=8μm, l=20μm

offsetloss through

the hole

onset of inertia

Full 3D numerical model

Full 3D analytical model

2D analytical model with inertia (Φb = 0)

2D analytical model without inertia (Φb = 0)



Figure 14: The experimental and theoretical results for the quality factorversus pressure for a torsion mirror given by Minikes et al.83

104

103

102

10-2 100 102

Pressure (Torr)

Qua

lity

fact

or Q

LiVeijolaBao

Exp

with the square of the air-gap height and the radiusof the hole, respectively. Depending on the value ofRe, one can ignore or include the inertia of air inthe analysis and use appropriate form of equationpresented in section 2.2. Fig. 12 shows schematicallythe incompressible, compressible, and compressibleplus inertial nature of squeeze film flow underdifferent values of σ and Re.

4.3. Numerical solutions and analytical solutionsSolution methods based on numerical, analyticalor mixed techniques are developed to suit differentperforation geometries. Numerical methods are bestsuited for modelling nonlinear pressure responsedue to high amplitude of oscillation, arbitraryboundary conditions, complex geometries, andnonuniform size and distribution of the holes68,72,73.Computationally efficient FEM based simulationschemes are reported in the recent past73,74. Ahybrid model combining the Navier-Stokes equationand the Reynolds equation is proposed by DaSilva74. A mixed level approach based on FEMand finite network (FN) simulation is presentedby Schrag et al.75–77. A hierarchical two levelsimulation strategy is employed and coupled,reduced dimensional analysis is performed byRaback et al.78. Another method for arbitraryperforation problem utilizes a Perforation ProfileReynolds (PPR) solver, a multiphysics simulationsoftware73.

The goal of modelling and analyzing squeezefilm behaviour in perforated structures is toperform a reasonably accurate analysis with minimalmodelling and computational effort. In this respect,analytical methods are desirable since they giveclosed-form expressions which can be used directly

in system-level simulation for evaluating designtrade-offs. In these methods, the symmetry ofpressure distributions around each hole is exploitedand the entire perforated domain is discretized intopressure cells under simplifying assumptions64,55,63.However, to set clear limits to these models,extensive and independent numerical simulationsare required. For such computations, one oftenresorts to full 3-D Navier-Stokes solvers available incommercial MEMS packages such as ANSYS-CFX.One such comparative study is shown in figure 1371.One can clearly see both quantitative and qualitativechanges in response when inertia is considered andwhen inertia is ignored.

4.4. Comparison with experimental resultsAs mentioned earlier there are basicallyfour important effects—rarefaction effect,compressibility effect, inertial effect, and perforationeffect that need to be studied through experiments.The compressibility and inertial effects are dominantonly at higher operating frequencies. On the otherhand, rarefaction and perforation effects can beseen even at low frequencies.

Rarefaction can be modelled using twoapproaches. First approach uses the concept ofeffective viscosity as presented by Veijola et al.51

and Li et al.49. In this case the effective viscositybased on the Knudsen number Kn is used. Secondapproach is based on the free molecular dynamicmodels derived for a plate vibrating normal to afixed substrate. Free molecular models are presentedby Christian79, Zook et al.80, Kadar et al.81, Baoet al.53 and Hutcherson and Ye82. Among the firstfour free molecular models, Bao’s model is moreaccurate and handy. Since Bao’s model is valid foronly free molecular regime, Hutchersons modelcan be used for transition regime which is based onmolecular dynamic simulations. By comparing allthe important analytical models with experimentalmeasurements on a MEMS torsion mirror, Minikeset al.83 have shown that the effective viscosity modelpresented by Li et al.49 are most accurate overdifferent flow regimes as shown in Fig. 14.

Andrews et al.17 have performed experimentson a silicon microstructure in which fluid dampingis due to squeeze film flow. In this study, theexperiments are performed for a wide range ofpressures ranging from vacuum to atmospheric andover very low to as high as 50 kHz frequencies.At frequencies less than 10 kHz, the formulagiven by Blech27 gives good agreement while athigher frequencies there is some discrepanciesdue the inertia effect which is not included in theBlech’s formula. Veijola50 has proposed an analyticalformula for inertial and rarefaction effect which hasto be validated with experimental results.



Figure 15: (A) Variation of non-dimensionalized damping and spring force with squeeze number for ε =0.001, 0.05, and 0.1; (B) Percentage change in total non-dimensionalized back force with log(ε).

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 5 10 15 20 25 30 -7 -6 -5 -4 -3 -2 -1

Non

-dim

ensi

onal

ized

for

ces

(fd/

ε, f

s/ε)

Squeeze number σ log(ε = h/h0)

(A) (B)

anal. damp. forceanal. sprg. forcedamp. force at ε = 0.001

damp. force at ε = 0.05sprg. force at ε = 0.001

sprg. force at ε = 0.05damp. force at ε = 0.1sprg. force at ε = 0.1

35

30

25

20

15

10

5

0

% C

hang

e in

net

for

ce (

f/ε)

^

At σ = 25

Kim et al.84 performed experimental studiesto estimate viscous damping offered by planarstructures having different size and number ofperforations. They found that in addition to squeezefilm damping, loss through holes and the lossat the boundaries of the suspended structurestoo are significant. Kowk et al.63 also performedexperiments on perforated MEMS gyroscope tocalculate net squeeze film damping and compared itwith the analytical results. The formulae given bythem is valid for large perforations only. There arevarious other analytical formulae available as givenin Fig. 10 which model perforation and rarefactioneffects simultaneously. All of them need to becompared with experimental results for differentperforation topologies.

5. Open issues and current thrust5.1. NonlinearitiesThe analytical models available in the literatures arederived under the assumptions of small variation inair-gap thickness and pressure. When the amplitudeof vibration increases, the non-linear terms in thegoverning equations start dominating. To study theeffect of non-linear terms in the Reynolds equation,the non-dimensional damping and spring forcesare numerically obtained by solving non-linearReynolds equation for different values of non-dimensionalized amplitude ε (ratio of vibrationalamplitude to the air-gap thickness). The results areobtained for different values of squeeze numberas shown in Fig. 1585. It is found that when the

Figure 16: (A) Effect of isotropic and anisotropic distribution of roughness patterns on squeeze filmdamping and spring forces; (B) Effect of different roughness amplitudes on damping and spring forces.

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

00 5 10 15 20 25 30

Non

-dim

ensi

onal

ized

for

ces

(fd/

ε, f

s/ε)

Squeeze number σ

Amp = 0.005Amp = 0.005Amp = 0.05 -- 0.09

roughness effectrarefaction effectrough + rarefaction

Amp = 0.05 -- 0.09Amp = 0.1 -- 0.25Amp = 0.1 -- 0.25

20

15

21

5-7 -6 -5 -4

log (ε)-3 -2 -1

(con

tinuu

m v

ar/c

ontin

uum

) ×

100

(A) (B)



value of ε varies from 0 to 20% of the initial gap,the absolute back force increases by more than twoorders of magnitude (Fig. 15(B)). Since, some of theMEMS devices such as deformable mirror devicesand RF switches, etc., have large amplitude motion,nonlinear effects of squeeze film damping need tobe investigated well.

5.2. Effect of surface roughnessFor most MEMS devices, air-gap thickness is largeenough to neglect the effect of surface roughnesson squeeze film damping. However, when the sizeof the device reduces to nano scale the relativeamplitude of surface roughness increases andeventually affects the characteristics of squeeze filmdamper86. Theoretical studies show that when theroughness amplitude is about 10% of the air-gapthickness, the surface roughness effect becomespronounced (Fig. 16(A)). Such condition canbe seen in devices with submicron scale air-gapthickness. It is also found that the rarefaction effectobtained by reducing pressure or reducing lengthscale is also affected by surface roughness as shownin Fig. 16(B). This is because, under rarified gas flowthe interaction of gas molecules and irregular surfaceof the oscillating plate plays an important role indeciding the nature of interaction, i.e., diffusiveinteraction, specular interaction or a combinationof diffusive and specular interactions. Althoughmost of the studies done in the literature use surfaceroughness of Gaussian distribution, more accuratemeasurement of surface roughness profile in MEMSstructures are needed for reliable modelling. Themost challenging part is to measure the roughnessprofile of two surfaces facing each other in theair-gap thickness without breaking the structures.

6. ConclusionsSqueeze film damping affects the dynamicperformance of most MEMS devices that employvibrating structures with some gas or other fluidtrapped between the vibrating surface and anarrowly separated fixed substrate. The effect ofthe squeeze film on the dynamics of the structureis found by solving fluid flow equations withappropriate conditions to evaluate the back pressureon the structure which in turn gives dampingand spring force components. Several analyticalsolutions exist that provide very good estimatesof damping and spring forces due to the squeezefilm under various flow conditions (i.e., 1-D,2-D and 3-D flow). These solutions are basedon the mathematical models (mostly variants ofReynolds equation) that differ in their treatmentof rarefaction, compressibility, inertia and flowthrough perforations. Most models compare well

with numerical simulations based on direct solutionof fluid flow equations. Some models have beenexperimentally verified as well. However, morecareful and varied experiments are required forverification of existing models as well as for betterunderstanding of squeeze film effects at extremelysmall scales.

AcknowledgmentsThis study is partially supported by a grant from theDepartment of Science and Technology (DST), NewDelhi, India.

Received 23 December 2006; revised 15 January 2007.

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Prof. Rudra Pratap received his B.Tech. fromthe Indian Institute of Technology (IIT),Kharagpur in 1985; Masters in mechanicsfrom the University of Arizona, Tucson, in1987, and Ph.D. in theoretical and appliedmechanics from Cornell University in 1993.He taught at Cornell in the Sibley Schoolof Mechanical and Aerospace Engineering

during 1993–1996, prior to joining the Indian Institute of Sciencein 1996. He was also an Adjunct Assistant Professor at Cornellfrom 1997 to 2000. Presently, he is an Associate Professor inthe Department of Mechanical Engineering, Indian Instituteof Science, Bangalore, India. His research interests are MEMSdesign, computational mechanics, nonlinear dynamics, structuralvibration and vibroacoustics. Dr. Pratap is a member of IEEE andISSS and on the Editorial Board of CMC (Journal of Computers,Materials and Continua).

Suhas Mohite received his B.E. fromWalchand College of Engineering, Sangli, in1988; Masters in Machine Tools from theIndian Institute of Technology (IIT), Madrasin 1990. After a brief stint in industries andprivate institutes, he joined as faculty inthe Department of Mechanical Engineering,Government College of Engineering, Karad,

(M.S.), in 1996. Currently, he is on deputation at the CranesSciMEMS lab. Department of Mechanical Engineering, IndianInstitute of Science, Bangalore, where he is working toward thePh.D. degree. His research interests include design and modellingof acoustical and inertial MEMS.

Ashok Kumar Pandey received his B.E.degree in Mechanical Engineering fromBhilai Institute of Technology, Durg in 2001.He did his Masters in 2003 and presentlydoing his Ph.D. in MEMS from CranesSciMEMS Lab, Department of MechanicalEngineering, Indian Institute of Science,Bangalore, India. His current research focus is

on the investigation of different energy dissipation mechanisms inmicro and nano devices.


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