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* These authors contributed equally to this work.
‡
Corresponding authors:
jsader@unimelb.edu.au; roukes@caltech.edu
Inertial Imaging with Nanomechanical Systems
M. Selim Hanay
*
, Scott I. Kelber
*
, Cathal D. O’Connell
*
, Paul Mulvaney, John E. Sader
‡
,
Michael L. Roukes
‡
1. Derivation of Eq. (2) in the main text
Precipitous downward shifts in the modal resonance frequencies of a nanomechanical
device occur upon adsorption of individual analytes [1]. These measured frequency shifts can
be used to calculate the mass, position, and molecular shape of individual analytes that adsorb
upon a NEMS resonator – as described in the main text. Importantly, in the limit where the
particle mass is much less than the device mass, the sequential measurement of multiple particles
is unaffected by the mass loading due to previous particles.
Previous analyses [1,2] have considered the analyte particles to be point masses. In this
work, we model the individual particles as finite-sized objects with a spatial mass distribution
that is initially unknown. We consider a general device of arbitrary composition that is loaded
by an adsorbate with mass,
m
, which is much less than the device mass,
M
. We further assume
that the particle size is small compared to the device dimensions (and the wavelengths of the
vibrational modes) or, if not, that the particle is much more compliant than the device itself.
Under such conditions, which are especially relevant for the case of soft biological molecules (of
special interest to us), the vibrational mode shapes of the device are unaffected by the adsorbed
analyte, and thus the strain energy of the device is also unchanged. It then follows that the
maximum kinetic energy of the device, before and after mass loading, is invariant for the same
oscillation amplitude, i.e.,
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DOI: 10.1038/NNANO.2015.32
NATURE NANOTECHNOLOGY
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© 2015 Macmillan Publishers Limited. All rights reserved
SUPPLEMENTARY INFORMATION
Inertial Imaging with Nanomechanical Systems
M.S. Hanay, S.I. Kelber, C.D. O’Co
nnell, P. Mulvaney, J.E. Sader, M.L. Roukes
unloadedloaded
KE=KE
,
(S1)
where
()
()()
2
2
(0)
unloaded
device
1
KE
2
nn
dV
ωρ
Ω
=
∫
rr
Φ
,
(S2)
()
()()
{}
()
analyte
2
2
loaded
device
1
KE
2
nn
dV
ωρρ
Ω
+
Ω
=+
∫
rrr
Φ
,
(S3)
where
device
()
r
ρ
is the mass density of the device and
()
r
ρ
is the mass density of the analyte
absorbed onto the device surface,
(0)
n
ω
and
n
ω
are the angular resonance frequencies of the
unloaded and loaded devices,
()
r
n
Φ
are the natural (vector) vibrational modes of the device in
the absence of analyte adsorption,
Ω
is the spatial integration domain of the device, and
analyte
Ω
is
the spatial integration domain of the analyte.
We consider an adsorbate that is both thin and compliant with the device surface,
i.e.
, as
the device vibrates, the analyte moves with identical velocity to the device surface at any normal
position to the surface. This is expected to hold for analytes of sufficiently small mass that the
device mode shapes remain unaffected. The volume integral in Eq. (S3) over the analyte’s
volume can then be replaced by a surface integral involving its areal mass density
μ
(evaluated
by integrating
()
r
ρ
normal to the surface),
analyte
22
s
nn
drdS
ρ
μ
ΩΩ
=
∫∫
Φ
Φ
,
(S4)
where
s
Ω
is the surface of the device.
Equations (S1) - (S4) then give the following leading-order result
© 2015 Macmillan Publishers Limited. All rights reserved
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Inertial Imaging with Nanomechanical Systems
M.S. Hanay, S.I. Kelber, C.D. O’Co
nnell, P. Mulvaney, J.E. Sader, M.L. Roukes
2
(0)
2
(0)
device
1
2
s
n
nn
n
n
n
dS
dV
μ
ωω
ω
ρ
Ω
Ω
−
Δ≡
=
−
∫
∫
Φ
Φ
,
(S5)
which is formally valid in the limit of small fractional frequency shifts,
Δ
n
.
We require that the modes satisfy the normalization condition
()()
2
device
n
dVM
ρ
Ω
=
∫
rr
Φ
, (S6)
which, for a device of constant density, coincides with the usual orthonormal condition,
i.e.
,
()
2
1
r
n
dV
Ω
=
∫
Φ
Equations (S4) - (S6) then yield the required result,
F
n
=
−
2
Δ
n
M
,
(S7)
where
2
()()
rr
s
nn
Fd
V
ρ
Ω
=
∫
Φ
.
2.
Evaluating higher moments of the analyte’s mass distribution
In the main text, we examined the zeroth moment of the analyte density distribution for a
doubly
-
clamped beam
–
this gives the analyte mass,
m
. Since a beam is a one
-
dimensional
device, we derive results for the analyte’s linear ma
ss density,
()
x
λ
, i.e., the mass density
integrated over the normal and
lateral (
i.e
., transverse)
directions
of
the beam surface. Here, we
extend this analysis to
obtain
the first three higher
-
order moments:
(i)
the
centre
-
of
-
mass
of the
analyte (position),
(ii)
the analyte’s
average size (standard deviation), and
(iii)
its skewness
(asymmetry).
© 2015 Macmillan Publishers Limited. All rights reserved
SUPPLEMENTARY INFORMATION
Inertial Imaging with Nanomechanical Systems
M.S. Hanay, S.I. Kelber, C.D. O’Co
nnell, P. Mulvaney, J.E. Sader, M.L. Roukes
First moment (position):
To evaluate the position of the analyte, coefficients
(1)
n
α
in Eq. (3) (main text) must be
chosen such that
()()
(1)
(1) 2
1
N
nn
n
gxxx
α
=
=
Φ
=
∑
(S8)
over the spatial domain,
l
Ω
, which for a (one
-
dimensional) beam is
[0,1]
x
∈
;
x
is normalized by
the beam length,
L
. Due to the inherent symmetry in
2
()
n
x
Φ
, about
1/2
x
=
,
Eq.
(S
8
)
can be
applied over either subdomain
1/2
x
≤
, or
1/2
x
≥
, but not both. Satisfaction of Eq.
(S
8
)
over
the subdomain
01/2
x
≤≤
, will thus yield a distribution of
(1)
() 1
gxx
=
−
for
1/2
1
x
≤≤
.
Since the analyte’s size is much smaller than the bea
m length, and the beam is symmetric about
1/2
x
=
,
the choice of subdomain is inconsequential and simply defines the origin.
The particle position is therefore given by:
()
()
()()
()()
(1)
11
(1)
(1)
00
1
11
(0)
(0)
(0)
00
1
m
m
N
nn
n
N
nn
n
xxdxgxxdx
x
xdxgxxdx
α
λλ
λλ
α
=
=
Δ
≡
===
Δ
∑
∫∫
∫∫
∑
,
(S9)
This provides the particle position relative to
the nearest clamped end of the beam.
Note that the
particle position is
defined by a rat
ional function
that involves the fractional
-
frequency shifts,
Δ
n
,
of
N
modes. Once these shifts have been experimentally measured, all that is required are the
coefficients for the zeroth and first moments,
(0)
n
α
and
(1)
n
α
(see main text).
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