Slide film damping in microelectromechanical system devices

Original Article
Slide film damping in
microelectromechanical system
devices
Proc IMechE Part N:
J Nanoengineering and Nanosystems
227(4) 162–170
Ó IMechE 2013
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DOI: 10.1177/1740349913486097
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Weidong Wang, Jianyuan Jia and Jianwen Li
Abstract
Film damping caused by microfluids has important effects on the dynamic characteristics of moving elements of microelectromechanical system devices. There are two kinds of film damping existing in microelectromechanical system
devices, for instance, slide film damping and squeeze film damping. This article presents an overview on the recent
research progress on the slide film damping in microelectromechanical system devices. Based on the first slip velocity
boundary conditions, this article discusses two kinds of damping models in detail, which commonly used to investigate
on the slide film damping. For the convenience of quick reference in future, laterally moving microstructures, adequate
important equations and applicable conditions are included in this article. Finally, we proposed two unified models for
the analysis and design of laterally moving microstructured devices.
Keywords
Slide film damping, review, microelectromechanical system, laterally oscillation, microfluids
Date received: 14 December 2012; accepted: 18 March 2013
Introduction
For these microelectromechanical system (MEMS)
devices, especially containing moving elements, it is
essential to consider the damping effects of microfluidics produced by the relative motion between the moving elements and the substrates, which have important
influence on the dynamic characteristics of MEMS
devices.1–3 It has been confirmed by many experiments
that the gas behaviors have great effects on the dynamic
characteristics that work at an external environment
with some certain gas pressures, such as micromachined
accelerometers, microgyroscopes, micromirrors and
micromotors.2,4–9
The mechanisms of fluid damping that were received
by moving elements of MEMS devices can be mainly
divided into two categories:9–11 squeeze film damping
and slide film damping. The squeeze film damping
occurs when two parallel plates are in relative perpendicular motion. The fluid pressure difference above and
below the moving microdevices will produce damping
forces. The slide film damping occurs when two parallel
plates are in relative tangential motion. Viscous energy
dissipation in the fluid between the two plates becomes
a representative damping mechanism in laterally driven
microdevices.5,9 Over the past two decades, the squeeze
film damping in MEMS devices has been studied extensively,1–4,6–24 while less attention on slide film damping.5,9,25–31 In 2007, Baoa and Yang17 and Pratap
et al.18 have presented a much comprehensive overview
about squeeze film air damping in MEMS devices.
Moreover, in 2011, Vagia and Tzes22 have also made a
literature review on squeezed film damping in modeling
and control design for electrostatic microactuators.
Herein, we will not investigate on the squeeze film
damping, while we will focus on slide film damping and
discuss its modeling and application in this article.
Two kinds of damping models are commonly used
to study the slide film damping, including the Couettetype model and the Stokes-type model, that is, Stokes’
second problem. The Couette-type model shown in
Figure 1 can be used either when the gap width between
the moving plate and the fixed plate (substrate) is very
School of Electrical and Mechanical Engineering, Xidian University, Xi’an,
PR China
Corresponding author:
Weidong Wang, School of Electrical and Mechanical Engineering, Xidian
University, P.O. Box183, No. 2 South Taibai Road, Xi’an, Shaanxi 710071,
PR China.
Email: [email protected]
Wang et al.
163
and Couette flows considering the effect of the slip condition due to an oscillating wall, but the solutions are
incomplete, and therefore, they did not discuss about
them in the application scope of MEMS. Both the
Couette-type and the Stokes-type models investigated
at microscale in detail will lay a good theoretical foundation for the analysis and design of laterally driven
microstructures.
Couette-type model
Problem solution
Figure 1. Microgap shear flow (Couette-type flow).
Figure 2. Stokes’ second problem (Stokes-type flow).
As shown in Figure 1, a continuum flow is assumed, in
which the minimum size of the structure and the thickness of the interstructure fluid film are at least 10 times
greater than the mean free path of the gas or the intermolecular distance of the liquid. Neglecting the pressure gradients, the Navier–Stokes equations simplify to
equation (1)9,25,32
d2 u
=0
dy2
ð1Þ
At microscale, the slip velocity boundary conditions
should be introduced to solve equation (1)
small or when the vibration frequency is very low; otherwise, the Stokes-type model should be used to study
the slide film damping.9,25,27 The simplest conceptual
configuration finds two infinite, parallel plates separated by a distance h. The top plate translates with a
velocity u = u0cosvt, or u = u0sinvt, in its own plane.
As shown in Figure 2, it is considered that a laterally
oscillating infinite plate is immersed in an incompressible fluid (either a gas or a liquid) with constant transport properties. Due to the liquid viscosity, the plate
drives the fluid to oscillate with itself, forming a kind
of flow named Stokes flow, that is, Stokes’ second
problem. The viscosity damping caused by the relative
motion between the fluid and the plate is parallel to the
motion of the plate, so-called slide film damping. For
the lateral motion of the plate in Figures 1 and 2, the
velocity profile is prescribed as u = u0cosvt, where u0
is the velocity oscillation amplitude and v is the oscillation frequency.
The phenomenon of velocity slip of fluid on the surface of the plate can be neglected at macroscale, but it
plays an important role at microscale. The Stokes-type
model has been used by Cho and coworkers25,26 to
study the slide film damping received by laterally driven microstructures; nevertheless, the effect of velocity
slip has not been considered, which will definitely cause
errors in theoretical analysis and structural design. The
effect of velocity slip of fluid on laterally driven microstructures has been taken into account by Veijola and
Turowski,27 but the Stokes’ second problem has not
been studied. Wang et al.30 have given an analytical
solution for Stokes’ second problem and have obtained
the damping coefficient and spring coefficient. Khaled
and Vafai31 have obtained the exact solutions of Stokes
u(y, t)jy = 0 = u(0, t) = u0 cos vt + Ls1
u(y, t)jy = h = u(h, t) = Ls2
du
j
dy y = h
du
j
dy y = 0
ð2Þ
ð3Þ
where Ls1 and Ls2 are the slip lengths on the moving
plate and the substrate (the stationary plate), respectively. Then, the exact solution of velocity in fluid film
is
u(y, t) =
h + Ls2 y
u0 cos vt
h + Ls1 + Ls2
ð4Þ
In the case of gas as fluid, assuming two different
tangential momentum accommodation coefficients
(TMACs), s1 and s2 of gas molecule on the moving
plate and the substrate (stationary plate), respectively,
the fluid velocity profile in the gap and shear stress can
be deduced as follows
h + Ls2 y
u0 cos vt
h + Ls1 + Ls2
ð5Þ
1 + a2 Kn (y=h)
u0 cos vt
=
1 + (a1 + a2 )Kn
du
u0 cos vt
u0 cos vt
=m
= mefc
t(y, t) = m
dy
h + Ls1 + Ls2
h
ð6Þ
u(y, t) =
where Kn is the Knudsen number, Kn = l/h, here l is
the mean free path of gas molecule; Ls1 = a1l, Ls2 =
a2l, a1 = (2 2s1)/s1, a2 = (2 2s2) /s2, mefc =
m/[1 + Kn(a1 + a2)]. Here, mefc is the slip effective viscosity for Couette flow.
Therefore, the slide film damping coefficient is
164
Proc IMechE Part N: J Nanoengineering and Nanosystems 227(4)
continuity equations result in a one-dimensional diffusion equation9,26,32
∂u
∂2 u
=n 2
∂t
∂y
ð9Þ
where n is the kinematic viscosity of the fluid, defined
as n = m/r, here r is the density of the fluid.
At microscale, the slip boundary conditions can also
be introduced
u(y, t)jy = 0 = u(0, t) = u0 cos vt + Ls1
u(y, t)jy!‘ = u(‘, t) = 0
t(0, t) A
A
= mefc
u0 cos vt
h
ð7Þ
where A is shear area.
At macroscale, the slip lengths Ls1 and Ls2 can be
treated as 0, so that the velocity profile without slip
boundary conditions can be simplified from equation
(2)
u(y) =
hy
u0 cos vt
h
ð10Þ
ð11Þ
where Ls1 is the slip length on the moving plates.
Based on the separation of variables and the integral
transform methods, the steady-state velocity profile,
u(y,t), in the fluid field can be obtained by integrating
twice and solving for the constants using the boundary
conditions,9,28
Figure 3. Effects of the Knudsen number on the velocity
profile (Couette flow).
CCout =
du
j
dy y = 0
ð8Þ
u(y, t) = bu0 exp( ky) cos½vt (ky + u)
ð12Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where b = 1= 2g 2p+
2g + 1, u = arctan[g/(1 + g)],
ffiffiffiffiffiffiffiffiffiffiffi
g = kLs1 and k = v=2n.
Then the shear stress obtained by the moving plate is
given by
pffiffiffi
du(y, t)
jy = 0 = 2kbu0 m cos (vt + p=4 u)
t w =m
dy
pffiffiffi
= 2kbu0 m½cos vt cos (p=4 u)
sin vt sin (p=4 u)
ð13Þ
Discussion
Here, it is assumed that s1 = s2 = 1, a1 = a2 = 1 and
then Ls1 = Ls2 = hKn. Figure 3 shows the effect of the
Knudsen number on the steady periodic velocity profile. As can be seen, for a certain Kn, the velocity profile
of u/u0(cosvt) was keeping straight with a constant
slope at any moment t. Also, the slopes of the velocity
profiles will increase with the increase in the Knudsen
number. For Kn = 0, this situation corresponds to the
case at macroscale, that is, the slip length Ls2 = 0. It
can be found from Figure 3 that the velocity near the
substrate would increase with the increase in the
Knudsen number; moreover, the slip length Ls2 has the
same trend according to Kn. Therefore, one can conclude that due to the slip velocity boundary conditions
at microscale, great changes will happen to the velocity
profile, which will have effects on other mechanical
properties, for instance, sheer stress and damping
coefficient.
Stokes-type model
Typical Stokes’ second problem
Problem solution. As shown in Figure 2, neglecting the
pressure gradient, the Navier–Stokes equations and
It is obvious that the damping force has two components. The phase of the first component is same with
the velocity phase, and the phase of the second component is same with the phase of the displacement.
Therefore, we define the damping coefficient and spring
coefficient shown as follows
pffiffiffi
CStok = vnbmA cos(p4 u)
ð14aÞ
pvffiffiffi
p
KStok = v n bmA sin( 4 u)
ð14bÞ
Similar to equation (5), when Ls1 equals to 0 at
macroscale, the velocity profiles of Stokes flow can be
reduced to equation (15) from equation (12), which
agrees with the expression presented by Wang.9
u(y, t) = u0 exp(ky) cos (vt ky)
ð15Þ
Discussion. The velocity profiles of the typical Stokes’
second problem are shown in Figure 4. Here, the
dotted lines are given for the Stokes flow with no slip
condition, and the solid lines are for the flow with slip
condition, that is, g = kLs1 = 0.215. It is noted that
all of the velocities of the Stokes flow will decay to 0
when Y is large enough, that is to say, the moving plate
would have little effects on the liquid when the distance
Wang et al.
165
Figure 5. Cosine excitation for finite gap height.
Figure 4. Velocity profiles of the Stokes flow at different times.
is larger than some certain values. In order to illustrate
the extent to which the effects of viscosity penetrate
into the fluid film, a concept of penetration depth is
imported. Generally, a penetration depth
dp is
defined as the characteristic thickness of a fluid layer
over which the velocity amplitude has dropped to 1%
of its maximum value u0, that is
u = b exp (kdp )u0 = 0:01u0
∂u
∂2 u
=n 2
∂t
∂§
ð17Þ
u(§, t)j§ = h = u(h, t) = u0 cos vt Ls1
u(§, t)j§ = 0 = u(0, t) = Ls2
du
j
d§ § = h
du
j
d§ § = 0
ð18aÞ
ð18bÞ
Equation (17) can be solved in the frequency domain
for a steady-state cosine velocity excitation, and the
solution can be written as follows
Therefore, dp can be written as follows
dp =
For solving conveniently, a variable § was introduced here, § = h2y. Then equations (2), (3) and (9)
can be written in the following forms
1
ln 100b = 4:605 ln 2g2 + 2g + 1 =k
k
u(§, t) = u0 exp (ivt)½C1 sinh (q§) + C2 cosh (q§)
ð16Þ
It is noted that the penetration depth will decrease with
the increase in the oscillation frequency.
Finite height of fluid gap
The above problem focuses on the typical Stokes
flow with infinite fluid film. However, in practical
MEMS devices, the thickness of the fluid film cannot
be infinite, the analytical results above are not suitable
for actual devices. Therefore, the case that has finite
height of fluid gap will be investigated in the following.
Problem solution. There are two kinds of velocity excitations, including cosine and sine, according to the different velocity phases.
(1) For cosine velocity excitation. As shown in Figure 5,
the governing equation is same with that of the typical
Stokes’ second problem expressed by equation (9),
while the velocity boundary conditions are different
from theirs, that is, equations (10) and (11). The boundary conditions given by equations (2) and (3) should be
introduced here.
ð19Þ
pffiffiffiffiffiffiffiffiffiffi
where q is a complex frequency variable, q = iv=n.
The constants C1 and C2 are determined from the
boundary conditions of equations (18a) and (18b).
Then the velocity function u(§, t) can be obtained as
follows
u(§, t) =
sinh (q§) + qLs2 cosh (q§)
u0 exp(ivt)
(1 + q2 Ls1 Ls2 ) sinh (qh) + q(Ls1 + Ls2 ) cosh (qh)
ð20Þ
When Ls1 = Ls2 = l, the velocity function in equation
(20) agrees with the expression presented by Veijola and
Turowski.27 The velocity profile u(§,t) is the real part of
u(§, t).
The shear force acting on the moving plate at § = h,
with a surface area of A, is given as follows
d
u(§, t)
d§
mAq½1 + qLs2 tanh (q§)
u0 exp (ivt)
=
q(Ls1 + Ls2 ) + (1 + q2 Ls1 Ls2 ) tanh (qh)
t (§, t) = mA
ð21Þ
The damping admittance can be obtained from the
velocity
166
Proc IMechE Part N: J Nanoengineering and Nanosystems 227(4)
CStok (§, t) =
mAq½1 + qLs2 tanh (qh)
q(Ls1 + Ls2 ) + (1 + q2 Ls1 Ls2 ) tanh (qh)
ð22Þ
So the damping coefficient Cstok2 is the real part of the
admittance, and the spring coefficient Kstok2 is the imaginary part
In case of the same slip length on both the moving
plate and the substrate, that is, Ls1 = Ls2 in equations
(2) and (3), this above problem with a sinusoidal velocity excitation was solved in the frequency domain
through the Laplace transformation based on the theory of residues,31 and a steady periodic velocity is
obtained as expressed in equation (24)
U(Y, T)
(M1 M3 + M2 M4 )½f1 (Y) sin (T) + f2 (Y) cos (t) (M1 M4 M2 M3 )½f2 (Y) sin (T) f1 (Y) cos (T)
=
U0
M23 + M24
M23 + M24
(M3 M5 + M4 M6 )½f3 (Y) sin (T) + f4 (Y) cos (T)
(M3 M6 M4 M5 )½f3 (Y) cos (T) f4 (Y) sin (T)
+
+
M23 + M24
M23 + M24
Cstok2 = Re½CStok (§, t)
Kstok2 = Im½CStok (§, t)
ð23aÞ
ð23bÞ
If Ls1 = Ls2, equations (20)–(22) can be simplified as
follows
u(§, t) =
sinh (q§) + qLs1 cosh (q§)
u0 exp (ivt)
(1 + q2 L2s1 ) sinh (qh) + 2qLs1 cosh (qh)
t (z, t) =
mAq½1 + qLs1 tanh (qz)
m exp (ivt)
2qLs1 + (1 + q2 L2s1 ) tanh (qh) 0
CStok (§, t) =
ð24Þ
where U = u/(hv), U0 = u0/(hv), Y = y/h, T = vt and
R = h2v/n; the functions f1, f2, f3 and f4 are defined as
follows
qffiffiffi qffiffiffi 8
R
R
>
>
f
(Y)
=
sinh
4
Y
cos
1
>
2
2Y
>
>
>
qffiffiffi
qffiffiffi >
>
>
R
R
>
< f2 (Y) = cosh
2 Y sin
2Y
qffiffiffi qffiffiffi ð25Þ
>
R
R
>
> f3 (Y) = cosh
Y
cos
Y
>
2
2
>
>
qffiffiffi qffiffiffi >
>
>
>
R
R
: f4 (Y) = sinh
2 Y sin
2Y
mAq½1 + qLs2 tanh (qh)
q(Ls1 + Ls2 ) + (1 + q2 Ls1 Ls2 ) tanh (qh)
The constants M1, M2, M3, M4, M5 and M6 are calculated from the following
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
8
R
R
R
R
R
R
>
)
cos
(
)
+
a
sinh
(
)
cos
(
)
a
cosh
(
)
sin
(
)
M
=
cosh
(
>
1
2
2
2
2
2
>
>
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffi2ffi
>
>
R
R
R
R
R
R
>
>
M2 = sinh ( 2 ) sin ( 2 ) + a cosh ( 2 ) sin ( 2 ) + a sinh ( 2 ) cos ( 2 )
>
>
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
>
>
>
< M3 = sinh ( R) cos ( R) + 2a cosh ( R) cos ( R) 2a2 cosh ( R) sin ( R) 2a sinh ( R) sin ( R)
q2ffiffiffi
qffiffi2ffi
q2ffiffiffi
q2ffiffiffi
qffiffi2ffi
qffiffiffi2
qffiffi2ffi
q2ffiffiffi
ð26Þ
>
R
R
2
R
R
R
R
R
R
>
)
sin
(
)
+
2a
)
cos
(
)
+
2a
sinh
(
)
sin
(
)
+
2a
cosh
(
)
cos
(
)
M
=
cosh
(
sinh
(
4
>
2
2
>
>
qffiffi2ffi
qffiffi2ffi
qffiffiffi 2
qffiffiffi 2
qffiffiffi 2 qffiffiffi 2
>
>
R
R
R
R
R
R
>
M5 = sinh ( 2 ) cos ( 2 ) + a cosh ( 2 ) cos ( 2 ) a sinh ( 2 ) sin ( 2 )
>
>
>
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
qffiffiffi
>
>
: M = cosh ( R) sin ( R) + a sinh ( R) sin ( R) + a cosh ( R) cos ( R)
6
2
2
2
2
2
2
qffiffiffi
where a = Lhs1 R2 .
(2) For sinusoidal velocity excitation. As shown in
Figure 6, the governing equation is same with equation Discussion. As shown in Figures 5 and 6, a laterally
(9), while the velocity boundary conditions are different oscillating plate is immersed in an incompressible fluid,
from equations (10) and (11). The boundary conditions assuming the parameters given in Table 1.
Figure 7 shows the fluid velocity profiles at different
of equations (2) and (3) should also be introduced here.
times under the above two kinds of velocity excitation.
It is found that the velocity curves for cosine excitation,
at times T = 0, p/2, p and 3p/2, are in good agreement with those curves for sinusoidal excitation, at
times T = p/2, p, 3p/2 and 0, respectively. That is to
say, there is a lagging phase p/2 of the velocity profile
under sinusoidal excitation with respect to that under
cosine excitation, which is obvious due to the phase difference between cosine and sinusoidal excitation.
Herein in the following, the case of cosine excitation is
utilized to explain the characteristics of the Stokes flow
with finite height of fluid gap.
Figure 6. Sinusoidal excitation for finite gap height.
(1) Effects of oscillation frequency. The oscillation
frequency is changed from 10 Hz to 10 kHz, and
Wang et al.
167
Table 1. Parameters for the laterally oscillating plate.
h (mm)
u0 (m/s)
v (Hz)
l (mm)
Ls1 (mm)
Ls2 (mm)
n (m2/s)
1000
10
1000
68
68
68
531025
Figure 7. Fluid velocity profiles at different times under cosine
and sinusoidal excitation.
Figure 9. Fluid velocity profiles with different distances (T = 0).
frequencies, the fluid velocities near the substrate are
near 0 as the penetration depths become shorter than
the distance between the moving plate and the
substrate.
Figure 8. Fluid velocity profiles at different oscillation
frequencies (T = 0).
other parameters remain unchanged (Figure 8). It is
found that at different frequencies, the fluid velocities
near the substrate (i.e., the fixed wall) are also different. It is found that all the fluid velocities near the
moving plate are less than the velocity oscillation
amplitude u0. At low frequencies, for instance, 10,
100 and 200 Hz, because of the viscous effect of the
fluid on the fixed wall and the slip conditions,
their velocities near the substrate are not 0. It is noted
that at very low frequency, such as 10 Hz, the
velocity profile is of high linearity, which is in good
agreement with the Couette flow. However, at high
(2) Effects of the distance. Maintaining other parameters unchanged in Table 1, we only change the distance h between the moving plate and the substrate
from 10 to 1000 mm. Figure 9 shows the velocity profiles with different heights of the fluid gap at time T =
0. Because of the slip conditions, all the fluid velocities
near the moving plate are not equal to 0. Supposing the
slip lengths Ls1 and Ls2 are equal to 0, then all the velocities near the moving plate would become 0. Except
the case of h = 2000 mm, most of the velocities near
the fixed wall are not equal to 0 because of the viscous
effect of the fluid. However, it can be predicted that if
the distance h is greater than 2000 mm or the oscillation
frequency v is higher than 1 kHz, then the velocities
near the fixed wall (the substrate) will become 0.
For small distance h, including 10, 100 and 200 mm,
the fluid velocity profiles are of high linearity, which
agree well with the Couette flow. We will discuss these
phenomena in the following section. It is noted that for
the smallest distance of h = 10 mm, the fluid velocity
profile is close to a parallel line, that is to say, the velocity is of little variation between h = 0 and 10 mm. We
think that this phenomenon stems from a large slip
length, Ls1 = Ls2 = 68 mm, and a large Knudsen number, Kn = 6.8. In Figure 9, the hollow square symbols
denotes the fluid velocity profile at a small slip length,
Ls1 = Ls2 = 0.68 mm, and a small Knudsen number,
Kn = 0.068.
168
Figure 10. Fluid velocity profiles for the typical Stokes’ second
problem and the Stokes flow with finite height of fluid gap (T =
0, v = 10 kHz).
Figure 11. Fluid velocity profiles for the Couette flow and the
Stokes flow with finite height of fluid gap (T = 0, v = 1 kHz).
Comparison analysis
In this section, we will compare the above models to
find out the relationships among them, which will lay a
good theoretical foundation for future use.
(1) Typical Stokes’ second problem and Stokes flow with
finite height of fluid gap. For the typical Stokes’ second
problem, without a fixed wall (the substrate), the moving plate is immersed in an incompressible fluid. From
the above analysis in section ‘‘Typical Stokes’ second
problem,’’ it is found that for the typical Stokes’ second
problem, if the distance is longer than the penetration
depth dp, the fluid velocity can be considered as 0, that
is to say, the moving plate has little effects on the fluid
once the distance exceeds dp. In practical application,
the substrate (the fixed wall) always exists, so that the
Stokes flow with finite height of fluid gap should be
used to study it.
Proc IMechE Part N: J Nanoengineering and Nanosystems 227(4)
According to equation (16), the penetration depth of
the typical Stokes’ second problem is about 400 mm at
the oscillation frequency of 10 kHz. Figure 10 shows
the fluid velocity profiles of the above two models at a
varying distance, from 100 to 1000 mm. It is obvious
that once the distances exceed 400 mm, the fluid velocity
profiles of the Stokes flow with finite gap height agree
well with those of the typical Stokes’ second problem.
At small distances, for instance, 100 and 200 mm, the
fluid velocity profiles of the Stokes flow with finite gap
distance do not match with the correspondent velocity
profiles of the typical Stokes’ second problem.
Especially for 100 mm, the corresponding velocity profiles take on a discrete state. It can be drawn a conclusion that as long as the distance is equal or greater than
the penetration depth, the solutions of the typical
Stokes’ second problem can be applied to the Stokes
flow with finite height of fluid gap. However, it needs a
precondition for the above conclusion that the slip
lengths on both the moving plate and the substrate are
equal, that is, Ls1 = Ls2.
(2) Couette flow and Stokes flow with finite height of
fluid gap. According to equations (5) and (20), Figure
11 illustrates the fluid velocity profiles of the Couette
flow and the Stokes flow with finite gap height. It is
found that at the oscillation frequency of v = 10 kHz,
only for the case of h = 100 mm, the velocity profile
of Couette flow fits well with that of the Stokes flow;
for other cases, such as h = 200, 400 and 1000 mm,
their velocity profiles take on discrete states between
the above two models. Meanwhile, the longer the distance, the higher the discrete degree. Through a large
amount of calculation results, it is also found that the
higher the oscillation frequency, the higher the discrete degree. It should be noted that the Stokes-type
model is more suitable than the Couette-type model
for describing the fluid velocity profiles at high oscillation frequencies and larger distances. In other
words, the Couette-type model is fit for illustrating
the fluid velocity profile at low frequencies and with
small distances.
In the following, it is assumed that a laterally
oscillating plate is parallel to a substrate, and both of
them are immersed in an incompressible fluid. How to
deal with this problem? Or, how to model it? Based on
the above analysis, two unified models can be used to
describe this problem, as shown in Figure 12. At low frequency and with small distance, the model shown in
Figure 12(a) should be utilized. The fluid between the
moving plate and the substrate can be described by the
Couette-type model, and the fluid above the moving
plate can be modeled through the Stokes-type model. At
high frequency and with large distance, the model shown
in Figure 12(b) should be utilized. All of the fluid can be
modeled through the Stokes-type model.
Wang et al.
169
Figure 12. Unified models for laterally oscillating plate: (a) at low frequency and with small distance and (b) at high frequency and
with large distance.
Summary
First, the article presents an overview and reports the
recent progress of research on slide film air damping in
MEMS. Then the analytical solutions are discussed in
detail for the Couette-type and Stokes-type models.
Especially for Stokes flow, two kinds of excitation of the
moving plate were investigated in detail. Next, the simulation of slide film air damping is analyzed. For quick
reference in future MEMS research, abundant important
equations on slide film damping are included in this article. Finally, some comparison analyses on both the
Couette-type model and the Stokes-type model are carried out and then two unified models are proposed for
the analysis and design of laterally moving microstructured devices. In order to use the above formulae appropriately, we proposed two unified models for the design
and analysis of MEMS devices, as shown in Figure 12.
2.
Acknowledgements
7.
The authors thank the reviewers for their comments
and their kind suggestions for this article.
3.
4.
5.
6.
8.
Declaration of conflicting interests
The authors declare that there is no conflict of interest.
9.
Funding
This work was supported by the Fundamental
Research Funds for the Central Universities under the
grant number K5051304006 and the Young Scientists
Fund of the National Natural Science Foundation of
China under the grant number 51205302.
10.
11.
References
1. Somà A and De Pasquale G. Numerical and experimental comparison of MEMS suspended plates dynamic
12.
behaviour under squeeze film damping effect. Analog
Integr Circ S 2008; 57(3): 213–224.
Steeneken PG, Rijks Th GSM, van Beek JTM, et al.
Dynamics and squeeze film gas damping of a capacitive
RF MEMS switch. J Micromech Microeng 2005; 15(1):
176–184.
Wang W and Jia J. Analysis on microfluids squeezed film
damping of perforated disks with a centre hole. In: Proceedings of ASME 2005 fluids engineering division summer
meeting, Houston, TX, 19–23 June 2005, pp. 489–494.
ASME.
Veijola T, Kuisma H, Lahdenperéi J, et al. Equivalentcircuit model of the squeezed gas film in a silicon
accelerometer. Sensor Actuat A: Phys 1995; 48(3):
239–248.
Tanaka K, Mochida Y, Sugimoto M, et al. A micromachined vibrating gyroscope. Sensor Actuat A: Phys 1995;
50(1–2): 111–115.
Pan F, Kubby J, Peeters E, et al. Squeeze film damping
effect on the dynamic response of a MEMS torsion mirror. J Micromech Microeng 1998; 8(3): 200.
Chang K-M, Lee S-C and Li S-H. Squeeze film damping
effect on a MEMS torsion mirror. J Micromech Microeng 2002; 12(5): 556.
Vagia M and Tzes A. Modeling and analysis of the
squeezed film damping effect on electrostatic microactuators. In: Proceedings of 16th Mediterranean conference on
control and automation, Ajaccio, France, 25–27 June
2008, pp.469–474. IEEE.
Wang W. Study on flow characteristics of microfluidics
for MEMS design. Doctoral Dissertation, Xidian University, China, 2007.
Wu G, Xu D, Xiong B, et al. Analysis of air damping in
micromachined resonators. In: 7th IEEE international
conference on nano/micro engineered and molecular systems (NEMS), Kyoto, Japan, 5–8 March 2012, pp.469–
472. IEEE.
Li W. Analytical modeling of ultra-thin gas squeeze film.
Nanotechnology 1999; 10: 440–446.
Bao M, Yang H, Yin H, et al. Energy transfer model for
squeeze-film air damping in low vacuum. J Micromech
Microeng 2002; 12(3): 341.
170
13. Bao M, Yang H, Sun Y, et al. Squeeze-film air damping
of thick hole-plate. Sensor Actuat A: Phys 2003; 108(1–3):
212–217.
14. Nayfeh AI and Younis MI. A new approach to the modeling and simulation of flexible microstructures under the
effect of squeeze-film damping. J Micromech Microeng
2004; 14(2): 170.
15. Pandey AK, Pratap R and Chau FS. Analytical solution
of the modified Reynolds equation for squeeze film
damping in perforated MEMS structures. Sens Actuat A:
Phys 2007; 135(2): 839–848.
16. Feng C, Zhao Y-P and Liu DQ. Squeeze-film effects in
MEMS devices with perforated plates for small amplitude
vibration. Microsyst Technol 2007; 13(7): 625–633.
17. Baoa M and Yang H. Squeeze film air damping in
MEMS. Sens Actuat A: Phys 2007; 136(1): 3–27.
18. Pratap R, Mohite S and Pandey AK. Squeeze film effects
in MEMS devices. J Indian I Sci 2007; 87(1): 75–94.
19. Sumali H. Squeeze-film damping in the free molecular
regime: model validation and measurement on a MEMS.
J Micromech Microeng 2007; 17: 2231–2240.
20. Pandey AK and Pratap R. A semi-analytical model for
squeeze-film damping including rarefaction in a MEMS
torsion mirror with complex geometry. J Micromech
Microeng 2008; 18(10): 105003.
21. Altuğ Bıcxak MM and Rao MD. Analytical modeling of
squeeze film damping for rectangular elastic plates using
Green’s functions. J Sound Vib 2010; 329(22): 4617–4633.
22. Vagia M and Tzes A. A literature review on modeling
and control design for electrostatic microactuators with
fringing and squeezed film damping effects. In: Proceedings of American control conference (ACC), Baltimore,
MD, 30 June–2 July 2010, pp.3390–3402. IEEE.
Proc IMechE Part N: J Nanoengineering and Nanosystems 227(4)
23. Leung R, Cheung H, Gang H, et al. A Monte Carlo
Simulation approach for the modeling of free-molecule
squeeze-film damping of flexible microresonators. Microfluid Nanofluid 2010; 9: 809–818.
24. Hong G and Ye W. A macromodel for squeeze-film air
damping in the free-molecule regime. Phys Fluids 2010;
22: 012001.
25. Cho Y-H, Kwak BM, Pisano AP, et al. Slide film damping in laterally driven microstructures. Sensor Actuat A:
Phys 1994; 40(1): 31–39.
26. Lee KB and Cho Y-H. Electrostatic control of mechanical quality factors for surface-micromachined lateral resonators. J Micromech Microeng 1996; 6: 426–430.
27. Veijola T and Turowski M. Compact damping models
for laterally moving microstructures with gas-rarefaction
effects. J Microelectromechan S 2001; 10(2): 263–273.
28. Chen C-S and Chou C-F. Analytical and numerical studies on viscous energy dissipation in laterally driven microcomb structures. Int J Heat Mass Tran 2003; 46(4):
695–704.
29. Ongkodjojo A and Tay FEH. Slide film damping and
squeeze film damping models considering gas rarefaction
effects for MEMS devices. Int J Comput Eng Sci 2003;
4(2): 401–404.
30. Wang W, Niu X, Fan K, et al. Stokes’ second problem
with velocity slip boundary condition. Key Eng Mat 2011;
483: 287–292.
31. Khaled A-RA and Vafai K. The effect of the slip condition on Stokes and Couette flows due to an oscillating
wall: exact solutions. Int J Nonlin Mech 2004; 39:
795–809.
32. Wikipedia. Couette flow, http://en.wikipedia.org/wiki/
Couette_flow (2012).