PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Analytical and computational modeling of viscothermal acoustic
damping in perforated microstructures
Vahid NADERYAN1; Richard RASPET; Craig HICKEY; Mohammad MOHAMMADI
National Center for Physical Acoustics, University of Mississippi, USA

ABSTRACT
Predicting the viscothermal acoustic behavior of microstructures is crucial in the design ofmicro-electro-mechanical systems (MEMS). In MEMS structures the dimensionless shear wave-number istypically smaller than one. Therefore, the viscous effects are larger than inertial effects and are the dominantpart of the impedance. In this work, an analytical solution for the viscothermal acoustic impedance for
perforated microstructures used in MEMS devices is developed based on the low reduced-frequency (LRF)
method. This solution is based on a full-plate approach, as opposed to the single-cell approach, which
includes all viscous and thermal losses as well as compressibility and inertial effects. Additionally, a 3D
viscothermal acoustic model using the finite element method (FEM) is developed to solve the problem
numerically for the case of a MEMS microphone. The results of the analytical LRF solution are in good
agreement with the FEM results. The results can be applied in designing MEMS devices to minimize thedamping and to optimize the acoustic performance of the MEMS devices.
Keywords: MEMS, Squeeze-film damping, Viscothermal acoustics

1. INTRODUCTIONMEMS are micro-dimensional devices and elements that are made using modern micro-fabrication
techniques. In many MEMS devices (e.g. microphones), a diaphragm moves relative to a stationary
back-plate to form a parallel-plate capacitor. Since the gap spacing between the plates is very small
compared to the lateral dimensions, a thin film of air will be squeezed when the diaphragm isoscillating. This results in a complex pressure distribution between the plates that acts as damping andspring forces. This phenomenon called squeeze-film damping is typically the dominant source ofinternal thermal-mechanical noise in MEMS (1). According to the fluctuation-dissipation theorem, thenoise spectral level is proportional to visco-thermal damping in the system (2).
Simulation tools are necessary for modeling the behavior of MEMS devices for designers anddevelopers to advance the technology to the next levels. Evaluation of the squeeze-film and otherviscous and thermal losses of MEMS devices is a crucial and challenging task which requires studying
the behavior of the fluid and its reaction to the diaphragm’s motion. Several analytical models forsqueeze-film damping and spring effects have been developed over the past decades. Most of thesemodels are derived starting from the well-known Reynolds equation (3). In order to improve theaccuracy of the prediction of damping in modern MEMS devices, several geometric or flow conditions
are typically added to the basic models. Modeling the effect of the perforations in the back-plates is themost significant one and has been the center of attention in most of the recent works. Other secondary
effects include compressibility, inertia, gas rarefaction, the end effects of the holes, and edge (border)of the plate effects.
Typically, in many MEMS devices (such as microphones, accelerometers, etc.) perforations are
introduced into one of the parallel plates to reduce the squeeze-film damping. These perforations also
facilitate the etch release of the sacrificial layer between the plates. By means of the per forations, theair flow can move through the holes and consequently, the squeeze-film damping is reduced, however,extra damping due to the motion of the air through the holes is added to the system. Therefore,
modeling the squeeze-film effect for perforated MEMS plates is complex, challenging, and important.
Studies on the modeling of perforated MEMS can be subdivided into two categories based on theapproach used: single-cell approach and full-plate approach. The models consider a uniformly
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Vahid.nad@gmail.com

distributed pattern of holes on the plate. Each hole is surrounded by a cell. In the single-cell approach,the squeeze-film damping of a single-cell is calculated from the Reynolds equation and the properboundary conditions for a cell. Then the damping of the entire plate is calculated by multiplying theresult by the total number of the cells in the plate. Many of the studies in the literature have used thesingle-cell approach, such as references (4-10).
In the full-plate approach, the Reynolds equation is modified for the flow in the gap and through theperforations. The modified Reynolds equation is then solved with the boundary conditions of the plateborder to calculate the total damping of the plates. Most of the existing solutions are provided forrectangular plates. In rectangular MEMS two edges are usually clamped represented by a no -flow wallboundary condition and two edges are open requiring an open venting boundary condition. Although
there are very limited numbers of existing solutions for circular MEMS, circular MEMS are also very
common, specifically in microphones. Several previous studies such as references (11-15) havefollowed the full-plate approach to model the perforated MEMS by solving the modified Reynolds
equation for different plate geometries and boundary conditions.
In this work, a full-plate approach will be followed to model the losses in MEMS structures. All the
full-plate solutions in the literature are based on the modified Reynolds equation based on the
lubrication approximation. In the lubrication approximation, the Navier-Stoke equation is integrated
over the separation distance between the plane surfaces to derive the Reynolds equation, therefore, it isvalid for sufficiently small gaps. For perforated micro-plates, the flow domain is not narrow in theregion of the holes. Hence, the Reynolds equation is not valid unless the pitch of the holes is muchlarger than the gap and the diameter of the holes, which is not the case in many devices (10).
Better theory and calculations will lead to better understanding and better agreement with real
devices. In this work, pressure and velocity fields in the gap region and the hole region will be derived
using the low-reduced frequency (LRF) method that is valid for devices with small transverse
dimensions with respect to a wavelength. A new physical full-plate solution including thermal andviscous losses as well as inertial and compressibility effects will be developed using the properboundary conditions. A 3D finite element model is also developed in order to validate the analyticalmodel. The proposed model will be applicable to various geometries and boundary conditions.

2. LOW REDUCED-FREQUENCY SOLUTION FOR MEMS
2.1 Low Reduced-Frequency Method
In this work, a solution using the LRF method will be developed based on the full-plate approach.
Tijdeman (16) introduced dimensionless parameters to develop a model for viscothermal soundpropagation in cylindrical pores when the dimensions are small with respect to a wavelength. The LRF
solution is based on five fundamental equations that describe the motion of fluids in circular cylinders:the Navier-Stokes equations in both axial and radial directions, the equation of continuity, the equationof state, and the entropy equation. In the LRF model, the system of equations is rewritten in such a waythat equations describing the fluid along the propagation direction are decoupled from the equations
governing cross-sectional behavior. The LRF method is a good method for full analysis of MEMS
devices which includes all viscous and thermal losses as well as compressibility and inertial effects.
2.2

Viscothermal Solution for MEMS

2.2.1 LRF Solution for Parallel Plates
In order to utilize the LRF method to solve for the damping and spring forces in the parallel-plate
MEMS the following dimensionless parameters are defined which are reduced-frequency, axial
variable, radial variable, shear wave-number, and thermal wave-number, respectively:(1)
π ൌ ߱݃଴ Τܿ଴ߦ ൌ ‫ݖ‬Τ݃଴

(2)ߟ ൌ ߱‫ݎ‬Τܿ଴

(3)

ߣ ൌ ݃଴ ඥߩ଴ ߱Τߤ(4)ߣ ் ൌ ݃଴ ටߩ଴ ܿ௣ ߱Τߢ

(5)

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where ɘ is the angular frequency, ଴ is the gap spacing, ଴ is the velocity of sound, ɏ଴ is thedensity of air, Ɋ is the viscosity, ୮ is the specific heat at constant pressure, and Ɉ is the thermal
conductivity. The principle approximation of this method is to require that the reduced-frequency besmall (π ‫)ͳ ا‬. Figure 1 shows a schematic top view of a perforated micro-plate. A cross-section viewis displayed in Figure 2 to show the parameters and the coordinate system.

Figure 1 – Schematic top view of a perforated micro-plate.

Figure 2 – Cross-section view of a perforated MEMS and the coordinate system.
The dimensionless equations are linearized and isothermal boundary condition is applied on the top
and bottom plates ( ߦ ൌ േ ͳΤʹ ). With a velocity boundary condition for the diaphragm and an
impedance boundary condition for the fixed back-plate and venting boundary condition at the plateborder, after algebraic manipulations the damping coefficient, ‫ܦ‬, and the spring coefficient, ܵ, arecalculated:ܴ߱݅ߛߨܴ ଶ ‫݌‬଴ʹܿ଴ ‫ܬ‬ଵ ቀξ‫ܿ ܥ‬଴ ቁ‫ ܦ‬൅ ݅ܵ ൌ
൦ͳ െ
൪
(6)
߱݃଴ ݂ሺߣሻ‫ܥ‬ξ‫ܬ ܴ߱ܥ‬଴ ቀξ‫ ܴ߱ ܥ‬ቁܿ଴where ߛ is the ratio of specific heats, ܴ is the plate radius, ‫݌‬଴ is the ambient pressure, ‫ܬ‬଴ and ‫ܬ‬ଵ arezeroth order and first order Bessel functions of the first kind, respectively, and we have defined:(7)‫ ܥ‬ൌ ‫ ܤ‬െ ݅ Τπ݂ሺߣሻܼ‫ ܤ‬ൌ ߛ െ ሺߛ െ ͳሻ݂ሺߣ ் ሻΤ݂ሺߣሻ

(8)

(9)
݂ሺߣሻ ൌ ͳ െ ʹ ሺξ݅ ߣΤʹሻൗξ݅ߣ ሺξ݅ ߣΤʹሻwhere ܼ is the dimensionless specific acoustic impedance of the perforated back -plate which will becalculated in the next sub-section. The expression of Equation 6 is the principal result of this work that
contains all relevant physics.
2.2.2 Impedance of the Perforated Back-plate
The impedance of the perforated back-plate consists of two parts: the impedance of the holes, ୦ ,and the impedance of the squeeze-film in the gap between two plates, ୥ .
ܼ ൌ ܼ௛ ൅ ܼ௚
(10)
In order to find the impedance of the holes, ୦ , we use Tijdeman’s LRF solution for the flow inside acylindrical tube with zero velocity boundary condition at the walls of the tube s. After algebraic

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manipulations the dimensionless specific acoustic impedance of the perforated plate is :ܼ௛ ൌܼ௜௡௧ ߩ଴ ܿ଴ ሺ݇‫ܪ‬ሻ െ ܼ݅௜௡௧ ሺ݇‫ܪ‬ሻ൤൨െͳɐ ܼ௜௡௧ ሺ݇‫ܪ‬ሻ െ ݅ߩ଴ ܿ଴ ሺ݇‫ܪ‬ሻ

(11)

where
݇ ൌ ሺ߱Τܿ଴ ሻξ‫ܤ‬

(12)

(13)
ܼ௜௡௧ ൌ ͳȀ൫݂ሺߣሻξ‫ܤ‬൯where ɐ is the porosity and ‫ ܪ‬is the plate thickness. This solution assumes a plate with an infinite
thickness which does not include the end effects of the tube. However, for plates with finite thickness,the effect of the ends of the tube has to be taken into account. The end length corrections for theimaginary part (݈௘௜ ) and real part (݈௘௥ ) of the impedance are (17):(14)݈௘௜ ൌ ሺͳ͸‫ݎ‬଴ Τ͵ߨሻሺͳ െ ξߪሻ݈௘௥ ൌ ͲǤͻͶ‫ݎ‬଴

(15)

where ଴ is the radius of the holes.
The impedance of the gap region using the LRF method is solved to be:௜ܼ௚ ൌ π௙ሺఒሻ஻ ቂଶ௥బ ௖బ

ξ஻ఠሺ௥భ మ ି௥బ మ ሻߙ െ ͳቃ

(16)

where ଴ is the equivalent radius of a cell and
ߙൌ

ഘೝభ
ഘೝ
ഘೝ
ഘೝ
ቁ௃భ ቀξ஻ బ ቁି௃భ ቀξ஻ భ ቁ௒భ ቀξ஻ బ ቁ
೎బ
೎బ
೎బ
೎బ
ഘೝ
ഘೝ
ഘೝ
ഘೝ
௒బ ቀξ஻ బ ቁ௃భ ቀξ஻ భ ቁି௃బ ቀξ஻ బ ቁ௒భ ቀξ஻ భ ቁ
೎బ
೎బ
೎బ
೎బ

௒భ ቀξ஻

(17)

where ܻ଴ and ܻଵ are zeroth order and first order Bessel functions of the second kind, respectively.

3. Finite Element Method
In this section, the Finite Element Method (FEM) simulations are conducted to compare with theresults of the analytical model. This is achieved by solving the linearized Navier-Stokes equations,
momentum equation, continuity equation, and energy equation in the Thermoviscous Acoustics(TA) interfaces in the Acoustics Module of COMSOL® software. Both the viscous and heat
conduction effects are included in this interface and it solves for the acoustic perturbations in pressure,velocity, and temperature.
The solution for circular MEMS is solved using a 3D COMSOL® model. Due to the symmetricpattern of the holes on the back-plate a 30-degree pie section of the MEMS is modeled and symmetryboundary condition is used at the edges. The isothermal no-slip wall boundary condition is applied on
the back-plate walls. The diaphragm is driven with a constant harmonic vertical velocity. Open venting
boundary condition is applied at the circular perimeter of the plate. The real and imaginary parts of theforce exerted on the surface of the diaphragm divided by its velocity represent the damping and springforce coefficients, respectively. Figure 3 displays cross-sections of the velocity field in the MEMS atͳ݇‫ݖܪ‬.

Figure 3 – Velocity field cross-sections at ͳ .

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4. RESULTS
In this section the damping force coefficient calculated from the LRF method, Equation 6, iscompared with the FEM results and with two former theoretical models in the literature. The Li (15)model is based on the full-plate approach for a circular MEMS, and the Homentcovschi & Miles (10)model is based on the single-cell approach. Figure 4 shows the damping force coefficient as a functionof pitch. Pitch (݈) is defined as the distance between the centers of the neighboring holes , as displayedin Figure 2. The parameters of Table 1 are used for the calculations.
Table 1 - MEMS parameters used in the calculationsܴ ሺߤ݉ሻ݃଴ ሺߤ݉ሻߪ ሺΨሻ‫ ܪ‬ሺߤ݉ሻ

 ሺ ሻ

400

4

50

3

1000

Figure 4 – MEMS damping force coefficient.
The present model is in a better agreement with the FEM results compared to the previous models,
especially at higher pitch values. At lower pitch values (smaller holes) the damping through the holes
is dominant, but at higher pitch values (larger holes) the gap damping is dominant. The total damping
is a combination of these two. This is the reason that there exists an optimum pitch value. This suggeststhat at a fixed porosity and for a set of given MEMS parameters, an optimum pitch value can becalculated that minimizes the total damping. For the example of Figure 4, the optimum pitch value isaround 22 ߤ݉.

5. CONCLUSIONS
Accurate physical models are crucial for the study of the MEMS devices. Such models can guidethe designers and developers to advance the performance of such devices. In this work, the LRF
method is utilized to develop a new physical model for the viscothermal acoustic mechanism inparallel-plate perforated MEMS. This model is based on a full-plate approach and includes thermaland viscous losses as well as inertial and compressibility effects.
Comparison of the model with the 3D viscothermal acoustic FEM results shows a good agreementcompared with the previous models in the literature. The results suggest that the pitch of theperforations can be optimized in order to minimize the damping of th e MEMS. The accuracy of thisphysical model leads to a better understanding of the phenomena and better agreement with real
devices. The proposed model is applicable to various geometries and boundary conditions.

ACKNOWLEDGMENTS
We acknowledge Knowles Corporation for their support of this work.

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