COMSOL 6.4 - Squeeze-Film Gas Damping in an Accelerometer

Squeeze-Film Gas Damping in an Accelerometer

Micromechanical structures that use capacitance to measure another parameter such as acceleration typically have a very narrow gap between their electrodes. The gap usually contains gas, which damps the movements of the mechanical parts. This model of a microsystem accelerometer shows how to couple squeeze-film gas damping, which you model with the nonlinear Reynolds equation, to displacements in the sensor.

Introduction

Squeeze-film gas damping is a critical aspect of many MEMS transducers and actuators. An example of a microsystem component where gas-damping properties are important is an accelerometer common in vehicle motion-control and safety systems.

In accelerometers, inertia produces a motion that the device detects. A typical structure connects a large proof mass, with dimensions typically in millimeters, to surrounding structures with elastic beams. This combination forms a mechanical oscillator with a specific resonance frequency. However, in accurate motion-detection applications these resonances are unwanted, and the device damps the movements to produce smooth time-step and frequency responses. Such a device can usually achieve suitable damping with a low gas pressure (100 Pa–1000 Pa) that, considering the dimensions of the device, leads to rarefied gas effect in the system.

A narrow gap formed by two solid horizontal plates restricts the displacement of the gas perpendicular to the surfaces. When the sensor squeezes the gap, the gas flows out from its edges. The narrow pathway restricts the flow, which causes gas pressure to increase. This increase in gas pressure, in turn, decelerates the plates’ movement.

You can model the pressure distribution in the narrow gap with the modified Reynolds equation

where the total fluid pressure p is the sum of the initial/ambient pressure, pA and the variation pf; h = h0 + Δh(t) is the gap height consisting of the initial gap and the deformation in the normal direction of the boundary; hs is the location of the solid wall; hb is the location of the channel base; and us and ub define the tangential velocity of the solid wall and the channel base, respectively. Furthermore, the mean film velocity u is given by

where η denotes the fluid viscosity at normal conditions and the term Qch is the relative flow rate function that accounts for the rarefied gas effects. Veijola and others (Ref. 2) have used a simple equation for the relative flow coefficient

which is valid for 0 ≤ Kn ≤ 880. The Knudsen number is the ratio between the gas’ mean free path, λ, and the gap height, h:

The coefficient σP is calculated from the tangential momentum accommodation coefficient, αv:

The mean free path at a pressure p comes from

where λ0 is the mean free path at the reference pressure p0.

Another way to tune the damping is to perforate the structure with holes. By adding a term related to the gas flow through the holes, it is also possible to use the Reynolds equation for perforated plates. For more information about this approach see Ref. 3.

Model Definition

This example models the solid moving parts in the accelerometer using the interface in 3D and using the interface with a plane strain approximation in 2D. This model solves the squeeze-film air damping on the lower and upper surfaces using the multiphysics coupling. The model constrains the film pressure, pf, to 0 at the edges of the boundary.

The following two figures show the accelerometer geometry in 3D and in 2D. The model consists of two thin silicon cantilever beams and a silicon proof mass. The cantilever beams are fixed to the surrounding structures at one end. The proof mass reacts to inertial forces and bends the cantilevers. The external acceleration, a, acts in the z direction and causes a body volume force Fz = ρsolida.

In 2D the two cantilevers are lumped as one structure whose thickness equals the sum of the thicknesses of the two cantilevers. Consequently, the model has two domains with different thicknesses at the connecting boundary. You should therefore be prudent when inspecting stress levels near this area.

Figure 1: Model geometry in 3D.

Figure 2: Model geometry in 2D.

The following tables list the structures’ dimensions as well as pertinent material and gas properties used to calculate the effective viscosity:


Parameter	Cantilevers	Proof mass	GAP
Length	520 μm	1780 μm	1780 μm
Height	40 μm	400 μm	3.95 μm
Width	100 μm	2960 μm	2960 μm


Parameter	Value
Structure material	Silicon
Young’s modulus	170 GPa
Poisson’s ratio	0.28
Density	2329 kg/m3
Viscosity of the gas	22·10-6 Ns/m2
λ0	70 nm
p0	101.325 kPa

Results and Discussion

Figure 3 shows the pressure distribution on the surface of the proof mass after 4 ms of simulation. The ambient pressure, pA, in this case is 50 Pa, and the acceleration switches on at the beginning of the simulation. The acceleration’s magnitude is half that due to gravity, g. In this figure, the maximum displacement at the tip of the proof mass is roughly 0.2 μm, or 0.05% of its thickness.

Figure 4 shows the z displacement of the proof mass tip as a function of time for ambient pressures of 50 Pa, 300 Pa, and 1000 Pa. As ambient pressure increases, the film damping at the upper and lower surfaces increases through the increase in the gas’ effective viscosity and density. This increased damping results in a substantial decrease in oscillation with increasing pressure. At 300 Pa, there is no apparent oscillation, and the proof mass seems asymptotically reaching the value of 0.2 μm in z displacement.

Figure 3: A load on the face of the proof mass in the z direction leads to a deformation.

Figure 4: The z displacement of the proof mass tip at ambient pressures of 3 Pa, 30 Pa, and 300 Pa.

References

1. J.B. Starr, “Squeeze-film Damping in Solid-state Accelerometers,” Technical Digest IEEE Solid-State Sensor and Actuator Workshop, p. 47, 1990.

2. T. Veijola, H. Kuisma, J. Lahdenperä, and T. Ryhänen, “Equivalent-circuit Model of the Squeezed Gas Film in a Silicon Accelerometer,” Sensors and Actuators, vol. A 48, pp. 239–248, 1995.

3. M. Bao, H. Yang, Y. Sun, and P.J. French, “Modified Reynolds’ Equation and Analytical Analysis of Squeeze-film Air Damping of Perforated Structures,” J. Micromech. Microeng., vol. 13, pp. 795–800, 2003.

MEMS_Module/Sensors/squeeze_film_accelerometer

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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In the tree, select > > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Fluid Properties and Loads

In the window, under click .

In the window for , type Fluid Properties and Loads in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
a	g_const/2	4.9033 m/s²	Applied acceleration
mu	22e-6[Pa*s]	2.2E-5 Pa·s	Dynamic viscosity, fluid film
pA	300[Pa]	300 Pa	Ambient gas pressure
h0	3.95[um]	3.95E-6 m	Initial film thickness
Lambda0	70[nm]	7E-8 m	Mean free path
pref	1[atm]	1.0133E5 Pa	Reference pressure

Here, g_const is a predefined COMSOL Multiphysics constant representing the standard acceleration of gravity.

Geometry

In the toolbar, click

and choose > .

In the window for , type Geometry in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
Lpm	1780[um]	0.00178 m	Length of proof mass
Hpm	400[um]	4E-4 m	Height of proof mass
Wpm	2960[um]	0.00296 m	Width of proof mass
Lc	520[um]	5.2E-4 m	Length of cantilevers
Hc	40[um]	4E-5 m	Height of cantilevers
Wc	100[um]	1E-4 m	Width of cantilevers

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Proof Mass

In the toolbar, click

In the window for , type Proof Mass in the text field.

Locate the section. In the text field, type Lpm.

In the text field, type Wpm.

In the text field, type Hpm.

Cantilever 1

In the toolbar, click

In the window for , type Cantilever 1 in the text field.

Locate the section. In the text field, type Lc.

In the text field, type Wc.

In the text field, type Hc.

Locate the section. In the text field, type -Lc.

In the text field, type 2*Wc.

In the text field, type (Hpm - Hc)/2.

Cantilever 2

In the toolbar, click

In the window for , type Cantilever 2 in the text field.

Locate the section. In the text field, type Lc.

In the text field, type Wc.

In the text field, type Hc.

Locate the section. In the text field, type -Lc.

In the text field, type Wpm - 3*Wc.

In the text field, type (Hpm - Hc)/2.

Click

Click the

button in the toolbar.

The geometry is now complete and should look like that in Figure 1.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Silicon (mat1)

By default, the first material you add applies on all domains so you need not alter any settings.

Solid Mechanics (solid)

Body Load 1

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select all domains.

In the window for , locate the section.

Specify the fV vector as


0	x
0	y
-a*solid.rho	z

Fixed Constraint 1

In the toolbar, click

and choose .

Select Boundaries 1 and 6 only.

Thin-Film Flow (tff)

In the window, under click .

Select Boundaries 13 and 14 only.

In the window for , locate the section.

In the pref text field, type pref.

Fluid-Film Properties 1

In the window, under > click .

In the window for , locate the section.

Click in the upper-right corner of the section.

In the pA text field, type pA.

Locate the section. In the hw1 text field, type h0.

Locate the section. From the μ list, choose . In the associated text field, type mu.

Locate the section. From the list, choose .

From the list, choose .

In the λ0 text field, type Lambda0.

Definitions

Next, define nonlocal integration couplings and corresponding variables for later use in the Results section to observe the total damping force.

Bottom Surface Integration Operator

In the toolbar, click

and choose .

In the window for , type Bottom Surface Integration Operator in the text field.

In the text field, type bf.

Locate the section. From the list, choose .

Select Boundary 13 only.

Top Surface Integration Operator

In the toolbar, click

and choose .

In the window for , type Top Surface Integration Operator in the text field.

In the text field, type tf.

Locate the section. From the list, choose .

Select Boundary 14 only.

Variables: Total Forces

In the toolbar, click

In the window for , type Variables: Total Forces in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
F_bottom	-bf(nz*pfilm)	N	Total force, bottom face
F_top	-tf(nz*pfilm)	N	Total force, top face

Mesh 1

Free Tetrahedral 1

In the toolbar, click

In the window for , click

Now, create a simplified 2D model for comparison.

Add Component

In the window, right-click the root node and choose > .

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > > > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Geometry 2

In the window for , locate the section.

From the list, choose .

Proof Mass

In the toolbar, click

In the window for , type Proof Mass in the text field.

Locate the section. In the text field, type Lpm.

In the text field, type Hpm.

Cantilevers

In the toolbar, click

In the window for , type Cantilevers in the text field.

Locate the section. In the text field, type Lc.

In the text field, type Hc.

Locate the section. In the text field, type -Lc.

In the text field, type (Hpm-Hc)/2.

Click

Click the

button in the toolbar.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Solid Mechanics 2 (solid2)

Cantilevers Thickness

In the toolbar, click

and choose .

In the window for , type Cantilevers Thickness in the text field.

Select Domain 1 only.

Locate the section. In the d text field, type 2*Wc.

Proof Mass Thickness

In the toolbar, click

and choose .

In the window for , type Proof Mass Thickness in the text field.

Select Domain 2 only.

Locate the section. In the d text field, type Wpm.

Body Load 1

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both domains.

In the window for , locate the section.

Specify the fV vector as


0	x
-a*solid2.rho	y

Fixed Constraint 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Thin-Film Flow 2 (tff2)

In the window, under click .

Select Boundaries 5 and 8 only.

In the window for , locate the section.

In the pref text field, type pref.

The default point setting, Border, which sets the film pressure to zero, applies to this model.

Next, define fluid-film properties on these boundaries.

Fluid-Film Properties 1

In the window, under > click .

In the window for , locate the section.

Click in the upper-right corner of the section.

In the pA text field, type pA.

Locate the section. In the hw1 text field, type h0.

Locate the section. From the μ list, choose . In the associated text field, type mu.

Locate the section. From the list, choose .

From the list, choose .

In the λ0 text field, type Lambda0.

Definitions (comp2)

Bottom Surface Integration Operator

In the toolbar, click

and choose .

In the window for , type Bottom Surface Integration Operator in the text field.

In the text field, type bf2d.

Locate the section. From the list, choose .

Select Boundary 5 only.

Top Surface Integration Operator

In the toolbar, click

and choose .

In the window for , type Top Surface Integration Operator in the text field.

In the text field, type tf2d.

Locate the section. From the list, choose .

Select Boundary 8 only.

Variables: Total Forces

In the toolbar, click

In the window for , type Variables: Total Forces in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
F_bottom2d	-bf2d(nypfilm2solid2.d)	N	Total damping force, bottom face
F_top2d	-tf2d(nypfilm2solid2.d)	N	Total damping force, top face

Mesh 2

Free Quad 1

In the toolbar, click

Multiphysics

Structure–Thin-Film Flow Interaction 2 (stfi2)

In the window, under > click .

Select Boundaries 5 and 8 only.

Mesh 2

Size

In the window, under > click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type 85.1.

In the text field, type 0.288.

In the text field, type 1.25.

In the text field, type 0.25.

In the text field, type 2.

Study 1

Add a parametric sweep over the ambient pressure.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
pA (Ambient gas pressure)	1000 300 50	Pa

Step 1: Time Dependent

In the window, click .

In the window for , locate the section.

From the list, choose .

In the text field, type range(0,4e-2,4).

If you want smoother plots you can use a time step of 2e-5 (20 μs) or 1e-5 (10 μs) instead. The time step 40 μs gives reasonably smooth plots while keeping the MPH file size down.

Solution 1 (sol1)

In the toolbar, click