SUPPLEMENTAL
MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by
Nanomechanical Resonators
Elizabeth C. Bullard
1
, Jianchang Li
2
, Charles R. Lilley
3
, Paul Mulvaney
2
, Michael L. Roukes
2
and
John E. Sader
1,3*
1
Kavli Nanoscience Institute and Departments of
Physics, Applied Physics, and Bioengineerin
g
, California
Institute of Technology, Pasadena, California 91125
2
School of Chemistry and Bio21 Inst
itute, The University of Melbourne, Victoria 3010, Australia
3
Department of Mathematics and Statistics, The Univ
ersity of Melbourne, Victoria 3010, Australi
a
1.
Derivation of Equation
(4)
in
the
Main T
ext
The dynamic similarity theorem
in Eq. (7)
is
derived
from
Eq. (4). Equation (4)
is
obtained
by
extending the dimensional analysis of Refs. [
1, 2
] to include an extra physical quantity, the gas mean
free path. Here, we provide the details of
the
derivation
of Eq. (
4)
.
The principal assumptions are:
(1)
The
resonator
behaves as a linearly elastic solid;
(2)
Energy dissipation due to vibration occurs in the gas;
(3)
The oscillation amplitude of the
resonator
is small, so that all nonlinearities due to the
resonator
and
gas
are negligible.
The
maximum energy stored in the
resonator
directly follows:
€
E
stored
=
1
2
k
A
2
,
(
S
1
)
where
€
k
is the
resonator stiffness
, and
A
is the oscillation amplitude. The energy dissipated in the
gas
can be quantified by the (dimensionless) quality factor,
€
Q
≡
2
π
E
stored
E
diss
ω
=
ω
R
,
(
S
2
)
where
€
E
diss
is the energy dissipated per oscillation cycle, at the
radial
resonant frequency
€
ω
R
.
Since the flow is linear, the energy dissipated
in the gas
per oscillation cycle depend
s
on the square
of the oscillation amplitude,
€
A
.
From Eq
s
. (
S
1) and (
S
2
), we
then
obtain
€
k
=
1
2
π
∂
2
E
diss
∂
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
Q
,
(
S
3
)
which is independent of
the oscillation amplitude.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
2
In accord with the above
-
listed
assumptions, the energy dissipated per cycle
€
E
diss
must depend on
(i) the square of the
resonator
’s
oscillation
amplitude,
€
A
, (ii) the
gas
density
€
ρ
and shear viscosity
€
μ
,
(iii)
the gas mean free path
€
λ
, (iv)
the linear dimension (size) of the
resonator
, denoted
€
L
0
, (
v) the
relevant frequency o
f oscillation, which from
Eq. (
S
3
) is the resonant frequency in
gas
,
€
ω
R
, (v
i
) the
mode shape of the
resonator
, and (vi
i
)
its
geometry. Note that the last two quantities are dimensionless.
The functional form of
€
E
diss
, in
terms
of
these
parameters, can be rigorously determined using
dimensional analysis [
3
]. It follows that the product
€
ρ
m
μ
n
L
0
p
ω
R
q
λ
r
1
2
π
∂
2
E
diss
∂
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
,
(
S
4
)
must be dimensionless, where
m
,
n
,
p
,
q
,
r
are constants to be
evaluated
. Equating dimensions in Eq.
(
S
4
) leads to
three
independent dimensionless groups:
€
Π
≡
1
ρ
L
0
3
ω
R
2
1
2
π
∂
2
E
diss
∂
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
,
€
Re
≡
ρ
L
0
2
ω
R
μ
,
€
Kn
=
λ
L
0
.
(
S
5
)
The
second
and third
parameter
s are commonly referred to as the Reynolds and Knudsen number
s
,
respectively.
From
Buckingham’s
π
theorem
[
3
], it follows that there must exist a function
,
€
H
1
ρ
L
0
3
ω
R
2
1
2
π
∂
2
E
diss
∂
A
2
ω
=
ω
R
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
,
ρ
L
0
2
ω
R
μ
,
λ
L
0
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=
0
,
(
S
6
)
which relates the energy dissipated per cycle
€
E
diss
to all
other parameters. Solving Eq. (
S
6
) for the first
argument and rearranging gives
€
1
2
π
∂
2
E
diss
∂
A
2
ω
=
ω
R
=
ρ
L
0
3
ω
R
2
Ω
Re,
Kn
(
)
,
(
S
7
)
where the dimensionless function
€
Ω
Re,
Kn
(
)
is to be determined.
Substituting Eq
.
(
S
7) into Eq. (
S
3
)
gives
€
k
=
ρ
L
0
3
Ω
Re,
Kn
(
)
ω
R
2
Q
,
(
S
8
)
which is the required expression
,
Eq. (4)
,
of the main text.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
3
2
.
Device Fabrication
We used 4" silicon wafers coated with 100nm, 300nm, or 500nm of low stress silicon nitride (SiN)
on both sides to fabricate the devices.
P
hotolithography
was used
to defi
ne etch windows on the
backside of each wafer, followed by a dry etch to remove the Si
N.
A
KOH etch remove
d
the silicon
from the selected areas to form SiN membranes on the front side of the wafer.
E
lectron beam
lithography
was used to
defi
ne cantilevers on the edges of the membranes. We deposited either a layer
of Cr or a Cr/Au layer as an
etch mask. We dry etched the
silicon nitride membranes to defi
ne the
cantilevers. We then removed the metal layer(s) with a wet etch. All of the devices were then coated
with a Cr/Au layer for reflectivity; a thermal evaporator was used for the 100nm and 500nm devices,
while an electron beam evaporator was used for t
he 300nm devices. Prior to deposition of the metal
layer on the devices, we performed a test deposition on a SiN coated SOI wafer, patterned with
rectangles with photolithography. After liftoff, we measured the step height of the metal layer with an
atomic
force microscope. The step height was used to calibrate the subsequent deposition on the
devices. The 500nm devices have a 3nm layer of Cr with a 100
-
110nm thick layer of Au on top. The
300nm devices have a 3nm layer of Cr with a 60
-
66nm layer of Au on to
p. The 100nm devices have a
3nm layer of Cr with a 20
-
22nm layer of Au on top. The reflective layer was not deposited prior to the
etch of the SiN membrane because the etch would have also etched the Au, leaving a layer of unknown
thickness. The thickness
of the metal layers must be with 10% of the desired thickness to ensure
geometric similarity between the devices, as well ensuring that the density of the devices remains
constant. Images of the 100nm set of devices are shown in Figure
S
1.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
4
Figure S1:
SEM micrographs (color
ized for clarity
) of 100nm devices, of length
L
=
10
μ
m. Yellow is gold; purple is SiN. All images are taken at 7500x magnification. Devices
labeled in accordance with Figure 1.
A: Triangular cantilever;
B: (a) Square paddle head
c
antilever;
C: Cantilever with legs;
D:
length/width=10
rectangular cantilever;
E:
length/width=5
rectangular cantilever;
F:
length/width=3
rectangular cantilever.
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
5
3
.
Experimental Setup
We used optical interferometry to measure the quality factors of
the devices with both a network
analyzer and a spectrum analyzer. We placed the devices in chamber and changed the pressure from
3mT to 760T.
Figure S
2
:
Schematic of apparatus used for gas pressure measurements of cantilever
devices.
Optical
Interferometry
A picture of the optical setup is shown in Figure
S
2. The simple
interferometer is very similar
to the
desig
n used by Rugar
et al.
[
4
]. The fi
rst component of the detection system is the laser.
We used an
amplitude stabilized HeNe laser to
m
inimize fluctuations
in the beam intensity.
Fluctuations in the
intensity of the laser are indistinguishable from
fl
uctuations caused by
a change in the path length;
consequently, the amplitude stabilization reduces the noise.
Following the laser is an
optical isolator,
which prevents any light re
fl
ected from components
further along the beam path from entering the
laser; the re
fl
ected light could destabilize
the laser. The next component is a half wave plate in a
T-Cube Power Supply
Sample
Vacuum
Chamber
Achromatic Lens
f=150mm, 2w
0
=12.4um
Motorized
XYZ Stage
Quarter Wave
Plate
Manual
XYZ
Stage
Photodetector
Lens
f=100mm
HeNe Intensity
Stabilized Laser
Optical
Isolator
Beam
Expander
x20
1" Polarizing
Beamsplitter
Cube
d=0.7mm
d=14mm
Neutral
Density
Filter
Half Wave
Plate
Scroll Pump
Gas Inlet
SUPPLEMENTAL MATERIAL
Dynamic Similarity of Oscillatory Flows Induced by Nanomechanical R
esonators
E.C. Bullard,
J. Li, C.R. Lilley, P. Mulvaney, M.L. Roukes and
J.E. Sader
SI
-
6
rotation mount; the polarization is
adju
sted for the maximum amount of light to enter the interferometer.
Following the
half
w
ave plate is a neutral density fi
lter, used to attenuate the beam power. We used a
ND=1.5
fi
lter while making measurement for the 500nm
and 300nm devices and a ND=1.0 fi
l
ter while
making measurements for the 100nm devices. We chose these levels of attenuation to limit
heating of
the device under measurement, in order to prevent drift in the resonant frequency. A 20
×
beam
expander follows. When the beam exits the laser, it
has a
€
1
/
e
2
diameter of 0.7mm. The 20
×
beam
expander increases the
€
1
/
e
2
diameter to 14mm; the beam expander is required to minimize the spot size
at the sample. The minimum spot size at the sample,