of 6
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,
1
and John E. Sader
1,3
,*
1
Kavli Nanoscience Institute and Departments of Physics, Applied Physics, and Bioengineering, California Institute of Technology,
Pasadena, California 91125, USA
2
School of Chemistry and Bio21 Institute, The University of Melbourne, Victoria 3010, Australia
3
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
(Received 3 July 2013; published 7 January 2014)
Rarefied gas flows generated by resonating nanomechanical structures pose a significant challenge to
theoretical analysis and physical interpretation. The inherent noncontinuum nature of such flows obviates
the use of classical theories, such as the Navier-Stokes equations, requiring more sophisticated physical
treatments for their characterization. In this Letter, we present a universal dynamic similarity theorem: The
quality factor of a nanoscale mechanical resonator at gas pressure
P
0
is
α
times that of a scaled-up micro-
scale resonator at a reduced pressure
α
P
0
, where
α
is the ratio of nanoscale and microscale resonator sizes.
This holds rigorously for any nanomechanical structure at all degrees of rarefaction, from continuum
through to transition and free molecular flows. The theorem is demonstrated for a series of nanomechanical
cantilever devices of different size, for which precise universal behavior is observed. This result is of sig-
nificance for research aimed at probing the fundamental nature of rarefied gas flows and gas-structure
interactions at nanometer length scales.
DOI:
10.1103/PhysRevLett.112.015501
PACS numbers: 81.70.-q, 62.25.-g, 81.07.-b
Miniaturization of resonant mechanical structures has
driven progress across a wide range of advanced technolo-
gies, including sensors for mass detection and imaging with
atomic resolution
[1
3]
, monitoring of biological processes
such as DNA hybridization
[4]
, and mass spectrometry at
the molecular scale
[5]
. While the constituent elastic proper-
ties of mechanical resonators remain identical to bulk values
upon miniaturization to nanometer length scales
[6
9]
,
operation in fluid environments gives rise to physical phe-
nomena not normally seen at macroscopic levels
[10
14]
.
In liquid, these include transport due to the electrical nature
of surfaces
[15]
and the possibility of violation of the usual
no-slip condition
[16]
. Breach of no-slip is exacerbated for
operation in gas, because the relevant molecular length scale
is orders of magnitude larger than in liquid
[17
19]
. This
leads to failure of the continuum hypothesis at relatively
large length scales (microns), obviating the use of classical
theories such as the Navier-Stokes equation.
Continuum theories are used ubiquitously in the design
and characterization of macroscale and microscale mech-
anical structures. In the context of mechanical resonators
in gas, this has provided insight into the fundamental physi-
cal processes underpinning their operation. For example,
it is known that the quality factor (scaled inverse rate-of-
energy dissipation) of a mechanical resonator decreases
as the size of the structure is reduced
[10,20]
. This is
due to growth of the viscous penetration depth, where vor-
ticity is prevalent, relative to the structure size. Yet this
knowledge is predicated on the assumption that the gas
mean free path
λ
is much smaller than the size of the struc-
ture. The mean free path of air at 1 atm and room temper-
ature is approximately 70 nm
[21]
. Thus, miniaturization of
nanomechanical resonators to several hundred nanometers
induces strong noncontinuum effects, which can deviate
from the predictions of the Navier-Stokes equation.
The Knudsen number Kn is often used to characterize gas
flow regimes and select appropriate theories for their calcu-
lation.Thisdimensionlessparameterisdefinedastheratioof
the gas mean free path
λ
to a characteristic length scale of the
mechanical resonator
L
0
, i.e., Kn
λ
=L
0
. While analytical
theories can often be formulated in the continuum
(Kn
<
0
.
01
), slip (
0
.
01
<
Kn
<
0
.
1
), and free molecular
regimes (Kn
>
10
)
[22]
, flows in the transition regime
demand consideration of the kinetic nature of the gas
this
is precisely where many nanoscale resonators operate
[22]
.
The Boltzmann equation dictates the behavior of a gas at
arbitrary degrees of rarefaction, i.e., for all Knudsen
numbers
[17
19,22]
F
t
þ
c
i
F
x
i
þ
a
i
F
c
i
¼

F
t

coll
;
(1)
where
F
is the mass distribution function,
t
is time,
x
i
is posi-
tion,
c
i
is the molecular velocity, and
a
i
is the body force per
unit mass acting on the gas. The right-hand side of Eq.
(1)
is
specified by the nature of intermolecular collisions, which
generally results in a multidimensional phase space integral;
for a hard sphere gas

F
t

coll
¼
d
2
ref
4
m
Z
S
2
d
Ω
Z
R
3
d
c

j
g
F
0
F
0

FF

Þ
;
(2)
wherethesubscript*referstofunctionsofincidentmolecular
velocities, the prime denotes a function of postcollision
velocities,
d
ref
is the reference diameter of gas molecules,
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=
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=
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=
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© 2014 American Physical Society
Ω
is the solid angle,
g
is the relative pre- and post-collision
molecularvelocity,and
m
themolecularmass.Thecomplex-
ity ofthe Boltzmannequationposessignificant challenges to
solutionincomparisontoclassicalcontinuumtreatmentsthat
make use of the Navier-Stokes equations:
ρ
t
þ
x
i
ð
ρ
u
i
Þ¼
0
;
ρ

u
i
t
þ
u
j
u
i
x
j

¼
P
x
i
þ
μ
2
u
i
x
2
j
þ

μ
B
þ
μ
3

2
u
j
x
i
x
j
;
(3)
for which numerical and asymptotic techniques are well
advanced;
u
i
is the mean velocity of the gas,
ρ
is the gas
density,
μ
the shear viscosity,
μ
B
the bulk viscosity, and
P
is
pressure. This situation is especially problematic for nano-
scale resonators, which inherently operate in the transition
regime
oscillatory flows generated by these structures
require solution of the Boltzmann equation for rigorous
analysis.
Advances in developing asymptotic and computational
methods to solve the Boltzmann equation, particularly
for low Mach number flows, have been reported recently
[23
25]
. These methods have been demonstrated for
low-dimensional flows, such as Couette and Poiseuille
flow. However, computational methods must be used for
flows in the transition regime, which can be prohibitively
expensive for practical nanoscale mechanical resonators
with complicated geometries that produce genuinely
three-dimensional time-dependent flows. Their use in pre-
dicting oscillatory flows generated by nanoscale resonators
has been limited
[26,27]
. Characterization of gas-surface
interactions and thus specification of appropriate boundary
conditions can also be problematic, limiting the utility and
accuracy of any such simulation
[17,18,22]
. Consequently,
the characterization of gas flows at nanometer length scales
poses a significant and open challenge in the physical
sciences.
In this Letter, we report a general dynamic similarity
theorem for accurate and routine evaluation of the quality
factor of any nanoscale mechanical resonator in gas. This is
derived rigorously using dimensional analysis
[28]
and
obviates the need for sophisticated computational tech-
niques of rarefied gas flow, which is currently required.
It enables the use of microelectromechanical systems, for
which highly precise fabrication methods are well estab-
lished, in the design and characterization of nanoelectrome-
chanical systems. This approach intrinsically accounts for
all non-continuum processes, including the effects of gas
rarefaction and gas-surface interactions. It also allows mod-
eling of nanoscale mechanical resonators of arbitrary
geometry and structure. The rate-of-energy dissipation in
resonant mechanical structures dictates their sensitivities
to external stimuli, and knowledge of this physical property
is essential for the design and operation of ultrasensitive
devices
[29]
. The dynamic similarity theorem presented
here enables the straightforward characterization of this
fundamental quantity.
Consider a linearly elastic mechanical structure, of arbi-
trary specifications, that is immersed in gas and undergoing
resonant oscillations. Because of its linearity, the rate-of-
energy dissipation by the resonator depends on the square
of its oscillation amplitude. Energy dissipation due to the
gas also depends on the resonator size
L
0
, its radial resonant
frequency in gas
ω
R
, the gas shear viscosity
μ
, gas density
ρ
,
and mean free path,
λ
. The quality factor
Q
of the resonator is
related to its maximum energy stored
E
stored
, and energy dis-
sipated per cycle
E
diss
, via
Q
2
π
ð
E
stored
=E
diss
Þj
ω
¼
ω
R
.
Extending the dimensional analysis of Refs.
[30,31]
to
include an extra physical quantity, the gas mean free path,
yields the scaling law
[32]
k
¼
ρ
L
3
0
Ω
ð
Re
;
Kn
Þ
ω
2
R
Q;
(4)
where,
k
istheresonatorstiffness,
Ω
ð
Re
;
Kn
Þ
isadimension-
lessfunctionthatdependsonlyontheReynoldsnumber
[33]
,
Re
ρ
L
2
0
ω
R
=
μ
, and the Knudsen number, Kn
¼
λ
=L
0
;it
alsointrinsicallydependsontwootherdimensionlessgroups:
the mode shape of resonant oscillations and the geometry of
theresonator.Notethatviscosityisindependentofthedegree
of gas rarefaction
[17
19,30]
.
Next, we study the behavior of the Reynolds and
Knudsen numbers as the resonator size and/or gas density
and pressure are varied
all other properties of the resona-
tor such as its material and geometry are held constant; this
is well satisfied in practice
[6
9]
. The product of these
dimensionless variables is
Re Kn
¼
ð
ρλ
Þð
ω
vac
L
0
Þ
μ
ω
R
ω
vac
;
(5)
where
ω
vac
is the resonant frequency of the resonator in
vacuum. The bracketed terms in Eq.
(5)
do not change
as the gas density and/or resonator dimensions are uni-
formly scaled, i.e., increased or decreased
[19]
. Further-
more, the surrounding gas weakly affects the resonant
frequency of a nanomechanical resonator, because the aver-
age density of the structure is orders of magnitude larger
than the gas density
[34]
; equating these values is a good
approximation. Equation
(5)
immediately establishes that
the product Re Kn is invariant, i.e., Re
¼
a=
Kn where
a
is a constant, as resonator size is uniformly scaled and/or
gas pressure is varied
[35]
, provided the gas used is the
same. Therefore, the dimensionless function
Ω
ð
Re
;
Kn
Þ
is only a function of the Knudsen number.
Equating the maximum potential and kinetic energies of
the resonator gives
k
¼
B
ρ
av
L
3
0
ω
2
vac
, where
ρ
av
is its aver-
age density and
B
is a dimensionless constant that depends
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on its geometry and mode shape. Combining this expres-
sion with Eq.
(4)
gives
Q
¼
ρ
0
ρ

ω
vac
ω
R

2
H
ð
Kn
Þ
;
(6)
where the dimensionless function
H
ð
Kn
Þ
is inversely pro-
portional to
Ω
ð
a=
Kn
;
Kn
Þ
ρ
0
=
ρ
av
. The reference gas density
ρ
0
can be chosen arbitrarily and simply defines the normali-
zation of the dimensionless function
H
ð
Kn
Þ
; see discussion
below. Again, approximating the resonant frequency in gas
is by its value in vacuum (see above) leads to the required
dynamic similarity theorem:
Q
¼
ρ
0
ρ
H
ð
Kn
Þ
:
(7)
Note that the Knudsen number Kn depends explicitly on
the gas mean free path
λ
and thus the gas density
ρ
.
Equation
(7)
is derived formally from dimensional
considerations
[28]
and thus holds for any nanomechanical
resonator, regardless of its geometry and mode shape.
Once the dimensionless function
H
ð
Kn
Þ
is determined for a
particular resonator, the quality factor as the resonator size is
uniformly scaled and/or the gas density adjusted, is immedi-
atelyspecifiedbyEq.
(7)
.Thisisindependentofthedegreeof
gasrarefaction.Weemphasizethatthistheoremappliestothe
quality factor due to the gas only; intrinsic dissipation within
the resonator material is not considered.
Cantilever devices are chosen to assess the validity of
this dynamic similarity theorem and are fabricated from
100, 300, and 500 nm thick SiN films deposited onto sil-
icon wafers. Photolithography followed by a dry etch is
used to define etch windows on the backside of the silicon
wafers. KOH etching is used to create SiN membranes,
which are then patterned with electron-beam lithography
to define the cantilevers. A plasma etch is used to create
the cantilevers, which are subsequently coated with gold
for reflectivity. All devices are characterized by monitoring
their Brownian fluctuations using optical interferometry
over a range of gas pressures, from which the total quality
factor
Q
total
is measured. Dry nitrogen gas at room temper-
ature is used in all measurements. The intrinsic quality
factor
Q
int
of each device is also measured (at very low
pressures) and removed using the standard relation
1
=Q
total
¼
1
=Q
þ
1
=Q
int
to yield the required quality fac-
tor due to the surrounding gas
Q
; only this gas quality fac-
tor is reported in the following figures, table, and
discussion. Details of the experimental methodology are
in the Supplemental Material
[32]
.
Three sets of devices are fabricated with sizes uniformly
scaled in the ratio
1
3
5
, in accord with the above thick-
ness specifications, and include six different cantilever plan
view geometries. This allows for thorough assessment of
the dynamical similarity theorem. The upper row of
Fig.
1
shows SEM micrographs of the triangular shaped
device geometry (device
A
) at all three fabricated sizes
this illustrates the dimensional scaling of these devices.
Importantly, the thicknesses of the silicon nitride and
Cr
=
Au coatings are uniformly scaled in synchrony with
the plan view dimensions to ensure satisfaction of the
underlying assumption of the dynamic similarity theorem;
see Ref.
[32]
. SEM micrographs of the smallest versions of
all other devices (devices
B
,
C
,
D
,
E
, and
F
) are given in
the second and third rows of Fig.
1
. The length scale
L
0
,
chosen for the Knudsen number of each device is as fol-
lows. Devices
A
,
D
,
E
,
F
: leg width at clamp; devices
B
,
C
: width at free end.
We first consider device
E
, which corresponds to a can-
tilever resonator with a rectangular plan view and an aspect
ratio (length to width) of 5. Importantly, analytical theories
exist for the quality factor of this device in the continuum
[10]
and free molecular regimes
[27,32,36,37]
; the effect of
varying the gas-surface interaction on the free molecular
solution is presented in the Supplemental Material
[32]
.
This allows for independent confirmation of the robustness
of the presented measurements and initial validation of the
dynamical similarity theorem. The quality factor of a rec-
tangular cantilever in the continuum limit is
[10]
Q
cont
¼
4
ρ
av
h
πρ
b
þ
Γ
r
ð
ω
R
Þ
Γ
i
ð
ω
R
Þ
;
(8)
where
b
and
h
are the cantilever width and thickness,
respectively,
Γ
ð
ω
R
Þ
is the (complex-valued) hydrodynamic
function
[10]
evaluated at the radial resonant frequency
ω
R
and the subscript
r
and
i
refer to its real and imaginary com-
ponents, respectively. The corresponding result in the free
molecular limit is
[27,32]
FIG. 1 (color online). SEM micrographs of cantilever devices
showing plan view geometries. All devices uniformly scaled in all
dimensions on 100, 300, and 500 nm SiN films. Devices are high-
lighted in yellow. Triangular devices (A) at all three size scales
shown in top row. Smallest versions of all other devices fabricated
from 100 nm thick films shown in middle and bottom rows. Scale
bars for top row (device
A
) are
20
μ
m. Scale bars for all other
devices are
2
μ
m.
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Q
FM
¼
K
ð
ε
Þ
ρ
av
h
ω
vac
ð
2
RT
Þ
1
=
2
P
;
(9)
where
K
ð
ε
Þ¼
ffiffiffi
π
p
=
ð
4
þ
π
þ½
4
π

ε
Þ
,
ε
is the specular
reflection coefficient at the cantilever surface with
ε
¼
1
corresponding to pure specular reflection and
ε
¼
0
for
pure diffuse reflection,
R
is the specific gas constant,
T
the absolute temperature, and
P
the gas pressure.
Figure
2(a)
presents the measured quality factors of three
scaled versions of device
E
(see Fig.
1
) as a function of gas
pressure; the measured intrinsic quality factor of each
device is removed, as discussed
[32]
. At a gas pressure
of 1 atm, the quality factors of these resonators decrease
as their size is uniformly reduced, which is in agreement
with continuum theory, Eq.
(8)
. In contrast, the free
molecular model in Eq.
(9)
predicts that there will be no
change in the quality factor under this uniform scaling.
This is also borne out in the measurements in Fig.
2
that
display a convergence of the measured quality factors of
these resonators as gas pressure is reduced. Interestingly,
we find that the free molecular model incorporating diffuse
reflection at the solid surfaces gives best agreement with
measurements; this is in accord with other studies
[37]
.
In all cases, good agreement is observed between the mea-
surements and predictions of the theoretical models, for
both continuum and free molecular flows. These measure-
ments on device
E
are now used to provide an initial assess-
ment of the dynamic similarity theorem.
Equation
(7)
enables the quality factor of a (unknown)
nanoscale resonator at the reference density
ρ
0
(corresponding to a pressure
P
0
) to be determined from that
of an identical scaled up (test) microscale resonator. This
relationship is obtained by demanding that both resonators
operate at an identical Knudsen number, Kn. In this form,
the dynamic similarity theorem, Eq.
(7)
, becomes
Q
scaled




P
0
α
L
0
¼
α
Q
test




α
P
0
L
0
;
(10)
where
α
is the ratio of the nanoscale (scaled) resonator size
to that of the microscale (test) resonator. The listed super-
scripts of the vertical bars in Eq.
(10)
are the operating gas
pressures, while those in the subscripts are the resonator
sizes. The symbol "0" is used to indicate the reference value
of the indicated variable:
P
0
is gas pressure corresponding
to the reference density
ρ
0
chosen in Eq.
(7)
, and
L
0
is the
test resonator size. The material properties, mode shape,
and geometry of the two resonators are identical, as is
the gas used.
Physically, Eq.
(10)
states that
the quality factor of a
nanoscale resonator immersed in a gas at pressure
P
0
is
α
times that of an identical microscale resonator at the
reduced pressure
α
P
0
.
Importantly, this scaling law is
valid for arbitrary degrees of rarefaction and encompasses
the entire free molecular to continuum flow regimes, i.e.,
arbitrary Knudsen number. This eliminates the (problem-
atic) need to determine the precise nature of gas-surface
interactions, required for robust and accurate computational
modeling
[17,18]
. These properties arise because identical
materials are used for both microscale and nanoscale res-
onators, ensuring gas-surface interactions and the elastic
material properties of their materials are also the same.
Such interactions are automatically included in this theo-
rem of dynamic similarity. As discussed, the elastic proper-
ties of nanomaterials have been found to be identical to the
bulk, down to nanometer length scales
[6
9]
.
Table
I
presents an experimental assessment of this theo-
rem, giving (i) the measured quality factor of the 100 nm
device (2nd column), and (ii) its theoretical prediction as
determined from the 500 nm device at a reduced pressure
(3rd column), see Eq.
(10)
. Quality factors of the 500 nm
device at the listed pressures are given in the rightmost col-
umn of Table
I
note that these values are much larger than
the measured quality factors for the 100 nm device.
Strikingly, the dynamic similarity theorem yields accurate
predictions for the 100 nm device, based on the 500 nm
data at the pressures specified by Eq.
(10)
. This is true
for all gas pressures and over the entire range from con-
tinuum, to transition and free molecular flows, as indicated
by the Knudsen numbers in Table
I
.
In its most general form, Eq.
(7)
requires determination
of the function
H
ð
Kn
Þ
. This can be routinely measured by
first rearranging Eq.
(7)
:
FIG. 2 (color online). Measured quality factors (due to gas) for
three different sizes of device
E
which possess identical rectan-
gular geometries (length to width
¼
5
), as a function of gas pres-
sure. These devices have a nominal thickness 100 (blue), 300
(red), and 500 nm (green), with the plan view dimensions and
coatings scaled accordingly. Continuum theory, Eq.
(8)
(solid
lines at high pressure). Free molecular theory with diffuse reflec-
tion, Eq.
(9)
(dashed lines at low pressure). These complementary
theories are plotted up to their point of intersection, which occurs
in the transition regime, Kn
1
.
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H
ð
Kn
Þ¼
ρ
ρ
0
Q




test
;
(11)
where the subscript "test" refers to measurements on a sin-
gle test resonator, as a function of gas pressure (density).
Since the function
H
ð
Kn
Þ
is dimensionless, it holds for all
resonators of the same geometry, regardless of their size
and the gas density (pressure). With this function deter-
mined, Eq.
(7)
can then be used to calculate the quality fac-
tor of a resonator that has been uniformly resized from the
test device.
For example, to determine the quality factor (at 1 atm) of
a resonator of smaller size to the test resonator, we first set
the reference pressure to 1 atm. Since the quality factor of
the smaller resonator is required at 1 atm, the reference den-
sity
ρ
0
and measured density
ρ
are identical. Equation
(7)
then states that the quality factor of the smaller resonator is
simply
Q
small
¼
H
ð
Kn
small
Þ
, evaluated at the Knudsen num-
ber Kn
small
of the smaller resonator at 1 atm. This provides a
global picture of how the quality factor varies with resona-
tor size, because the Knudsen number is inversely propor-
tional to its size.
Measurements of the function
H
ð
Kn
Þ
for devices
A
,
B
,
C
, and
D
in Fig.
1
are given in Fig.
3
; similar results are
obtained for devices
E
and
F
[32]
. Note that for devices of
identical geometry, this dimensionless function
H
ð
Kn
Þ
is
predicted to be invariant as the device size is uniformly
scaled. Furthermore, the measured quality factors of the
devices differ considerably as their size is varied, as
observed for device
E
in Fig.
2
; see Supplemental
Material for other devices
[32]
. Nonetheless, the data in
Fig.
3
strikingly show that the dynamic similarity theorem
precisely collapses these measurements. This is despite the
minimum and maximum device sizes differing by a factor
of 5. The four geometries used, rectangular, triangular, pad-
dle, and rectangular with legs, highlights the universal
applicability of the dynamic similarity theorem with respect
to device geometry and size variations. The data in Fig.
3
have been plotted on identical axis scales, for both the
scaled quality factor and Knudsen number, to highlight
the effect of resonator geometry. Figure
3
, together with
additional data in the Supplemental Material
[32]
(for all
devices), shows that there are significant variations as a
function of device geometry, with factors of approximately
2
5 observable. This contrasts to the high precision in
which data collapse for different sized devices of identical
geometry.
Data for
H
ð
Kn
Þ
on a single test resonator can be used to
determine the quality factor (at 1 atm) of a resonator of arbi-
trary smaller size, as discussed above. Setting the reference
pressure to 1 atm is required for this implementation.
Evaluating
H
ð
Kn
Þ
at the Knudsen number for the smaller
resonator (at 1 atm) then gives its required quality factor.
FIG. 3 (color online). Scaled quality factor
H
ð
Kn
Þ
for canti-
lever devices with three different sizes: nominal thicknesses
100 (blue), 300 (red), and 500 nm (green). Reference gas density
and pressure specified at 1 atm. Data for devices
A
,
B
,
C
, and
D
.
Other device data display similar agreement; see Supplemental
Material
[32]
. Analytical solutions for device
D
shown for con-
tinuum (solid lines) and free molecular flows (dashed lines), as
per Fig.
2
. Upper horizontal axis gives length of device for uni-
form reduction in size at a gas pressure of 1 atm; setting
ρ
0
¼
ρ
in
vertical axis gives the required quality factor at 1 atm for this size
rescaling. Observed enhanced scatter in
H
ð
Kn
Þ
at high Knudsen
number is due to a finite intrinsic quality factor limiting precision
at low gas pressure. Identical scales are used for the
H
ð
Kn
Þ
and
Kn axes.
TABLE I. Quality factor due to surr
ounding gas for device
E
of
rectangular geometry (asp
ect ratio, length to width
¼
5
)asa
function of gas pressure. Resul
ts for measurements on the 100 nm
device (Measured); dynamic sim
ilarity theorem (DST) using
500 nm data. Quality factors for the 500 nm device are at the
indicated pressures, and provided for comparison only; this is not
used for the DST data on the 100 nm device and their Knudsen
numbers are not specified. The intrinsic quality factors for the 100
and 500 nm devices are
Q
int
¼
479
.
4
and 1648, respectively.
100 nm device
Quality factor
Quality factor
(500 nm device)
P (torr) Meas.
DST
Kn
760
49.2
54.5
0.03
159
310
63.3
68.2
0.08
217
100
96.4
90.6
0.25
305
31
181
172
0.82
400
10
417
426
2.6
615
3.1
1218
1205
8.2
1478
1
3929
3595
25
3903
PRL
112,
015501 (2014)
PHYSICAL REVIEW LETTERS
week ending
10 JANUARY 2014
015501-5
This size scaling is indicated on the upper horizontal axis of
the plots in Fig.
3
, where the resized resonator length is
shown. The data in Table
I
for the 100 nm device, at
1 atm (760 torr), are immediately evident in Fig. S8
(device
E
)
[32]
; see the upper horizontal axis at a device
length of
10
μ
m.
Characterization of gas flows in the transition regime,
which are generated ubiquitously by nanoscale mechanical
resonators, remains a challenging problem in the physical
sciences. In this Letter, we have presented a dynamic sim-
ilarity theorem that facilitates such analysis at arbitrary
degrees of gas rarefaction. This theorem circumvents the
need for advanced computational methods of rarefied flows
that can be prohibitively expensive and are seldom used. It
also permits interrogation of noncontinuum gas flows at the
nanoscale, through use of measurements at the more readily
accessible microscale. This simple yet effective approach is
expected to impact studies aimed at probing the underlying
physics of gas flows and gas-structure interactions at the
nanoscale, and provides a fundamental tool for design
and application.
We thank D. Chi for assistance in electron beam lithog-
raphy and A. Naik for helpful discussions. Experimental
work at Caltech was supported by the U.S. NSF through
Grant No. DBI-0821863, the NIH through Grant
No. R01 GM085666-01AIZ, and a NIH Director's
Pioneer Award (M. L. R.). Device fabrication was carried
out in the Kavli Nanoscience Institute at Caltech. The
authors gratefully acknowledge support of the Australian
Research Council Grants Scheme and a Caltech's
Kavli Nanoscience Institute Distinguished Visiting
Professorship (J. E. S.).
*
To whom all correspondence should be addressed.
jsader@unimelb.edu.au
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015501 (2014)
PHYSICAL REVIEW LETTERS
week ending
10 JANUARY 2014
015501-6