Inertial imaging with nanomechanical systems
M. Selim Hanay
1,2,†
,
Scott I. Kelber
1,†
,
Cathal D. O’Connell
3,†
,
Paul Mulvaney
3
,
John E.
Sader
1,4,*
, and
Michael L. Roukes
1,*
1
Kavli Nanoscience Institute and Departments of Physics & Applied Physics and Biological
Engineering, California Institute of Technology, Pasadena, California 91125, USA
2
Department of Mechanical Engineering and National Nanotechnology Research Center (UNAM),
Bilkent University, Ankara 06800, Turkey
3
Bio21 Institute & School of Chemistry, The University of Melbourne, Victoria 3010, Australia
4
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
Abstract
Mass sensing with nanoelectromechanical systems has advanced significantly during the last
decade. With nanoelectromechanical systems sensors it is now possible to carry out ultrasensitive
detection of gaseous analytes, to achieve atomic-scale mass resolution and to perform mass
spectrometry on single proteins. Here, we demonstrate that the spatial distribution of mass within
an individual analyte can be imaged—in real time and at the molecular scale—when it adsorbs
onto a nanomechanical resonator. Each single-molecule adsorption event induces discrete, time-
correlated perturbations to all modal frequencies of the device. We show that by continuously
monitoring a multiplicity of vibrational modes, the spatial moments of mass distribution can be
deduced for individual analytes, one-by-one, as they adsorb. We validate this method for inertial
imaging, using both experimental measurements of multimode frequency shifts and numerical
simulations, to analyse the inertial mass, position of adsorption and the size and shape of
individual analytes. Unlike conventional imaging, the minimum analyte size detectable through
nanomechanical inertial imaging is not limited by wavelength-dependent diffraction phenomena.
Instead, frequency fluctuation processes determine the ultimate attainable resolution. Advanced
nanoelectromechanical devices appear capable of resolving molecular-scale analytes.
*
jsader@unimelb.edu.au; roukes@caltech.edu.
†
These authors contributed equally to this work.
Author contributions
M.L.R. and J.E.S. supervised the project. J.E.S. provided the principal mathematical idea for mass measurement using mode
superposition that was extended to imaging by M.L.R. The resulting theory was further developed by M.S.H., S.I.K., J.E.S., and
M.L.R. Droplet measurements were conceived by J.E.S., performed by C.D.O., and supervised by P.M. and J.E.S. The paper was
written by M.S.H., S.I.K., C.D.O., J.E.S., and M.L.R. The FE simulations were executed by M.S.H. and S.I.K. All authors analysed
the data and contributed to the writing of the paper.
Additional information
Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at
www.nature.com/reprints
. Correspondence and requests for materials should be addressed to J.E.S. and M.L.R.
Competing financial interests
The authors declare no competing financial interests.
HHS Public Access
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Nat Nanotechnol
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Over the past decade, mass measurements with nanomechanical devices have systematically
improved to the point where they now offer an interesting capability for engendering a new
approach to mass spectrometry. Nanoelectromechanical systems (NEMS) resonators are
extremely responsive to the added mass of adsorbed particles
1
–
6
and this has led to advances
including the mass detection of individual proteins
7
,
8
, nanoparticles
9
and large
biomolecules
10
,
11
, as well as demonstrations of near-atomic-scale mass resolution
12
–
15
.
Given its capability for measuring neutral, single particles in both the low and high mass
(>500 kDa) domains, NEMS-based mass spectrometry (NEMS-MS) is now being developed
as an analytical approach for proteomics, structural biology and nanoparticle detection
7
.
Here, we propose and demonstrate a new methodology that represents a paradigm shift in
real-time, NEMS-based measurements of individual analytes. It can be implemented using
existing NEMS sensors, or even larger mechanical devices, and simply involves the analysis
of data obtained from standard measurements. We show that this approach enables
simultaneous measurement of the mass, position and molecular size and shape of individual
adsorbates (that is, it enables the acquisition of what we term their ‘inertial image’) through
real time measurements of the frequency shifts of the mechanical sensor’s multiple
vibrational modes, induced by discrete single-molecule adsorption events. We validate our
approach through detailed experimental measurements on a series of adsorbates and by
numerical simulation. We also establish that measurement of the size and shape of
molecular-scale analytes can be achieved using state-of-the-art NEMS devices. In fact, we
find that there are no fundamental wavelength (or diffraction-based) limits to the resolution
of this technique. Instead, it is the mechanical resonator’s intrinsic frequency-fluctuation
processes that impose the ultimate resolution limits.
Calculating spatial moments of a mass distribution
In NEMS-MS, adsorption of an individual analyte on the resonant sensor affects each of its
vibrational modes differently. The mode shapes of the resonator themselves give rise to a
distinct position dependence of these adsorbate-induced modal frequency shifts
7
,
16
,
17
. For
each mode, the induced frequency shift is maximal for analyte adsorption at vibrational
antinodes, whereas it vanishes for an analyte that adsorbs at a node. In a previous study we
used two resonator modes to measure simultaneously the mass and position-of-adsorption of
individual biomolecular analytes, in real time, as they adsorbed on a NEMS resonator
7
.
Deduction of the masses and positions of pairs and triplets of small particles after their
manual attachment to a microcantilever has also been demonstrated using multiple
vibrational modes
18
. In these previous efforts it was assumed that the analytes were point-
like particles. Here we go further and show that our methodology, which involves
measurements of additional adsorbate-induced frequency shifts for higher modes, permits
deduction of all spatial moments of the mass distribution for each adsorbate (which are now
assumed spatially finite), in real time. These moments not only include information about
each analyte’s total mass, but also, importantly, its spatial distribution—its position-of-
adsorption and its size and shape. Once acquired, the measured moments of the mass
distribution can be used to reconstruct an inertial image of the analyte.
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Without loss of generality, we present here the ultimate and practical sensitivity limits to
inertial imaging of small adsorbate masses, which inherently do not perturb resonator mode
shapes. Our focus is on small, soft and compliant adsorbates; this includes biological
molecules and molecular complexes, which are of significant current interest
7
. For such
analytes, effects that may arise from adsorbate stiffness
19
–
21
can be neglected. Our analysis
proceeds by defining coefficients
F
n
, which represent the surface loading of a NEMS device
by a small analyte (Supplementary Section 1):
(1)
Here,
μ
is the areal mass density distribution of the adsorbed analyte (evaluated normal to
the device surface),
Φ
n
(
r
) are the natural (vector) vibrational modes of the device in the
absence of analyte adsorption (normalized such that
∫
Ω
ρ
device
(
r
)|
Φ
n
(
r
)|
2
d
V
=
M
, where
ρ
device
is the mass density of the device and
M
is the device mass),
Ω
represents the region in
space occupied by the device and
Ω
s
is its surface. Implicit in equation (1) is that mass
loading arises from an adsorbate that is thin relative to the dimensions of its contact region
and that the adsorbate is both compliant and strongly adherent to the NEMS surface. When
this holds, the adsorbate’s rotary inertia, induced while following the NEMS vibrations,
exerts a negligible effect. Classical linear elasticity
22
allows these coefficients,
F
n,
to be
related directly to the discrete, experimentally measured fractional-frequency shifts induced
by each analyte adsorption event,
. Here,
is the unperturbed
angular frequency of the
n
th mode and
ω
n
is the shifted angular frequency after adsorption
of the analyte. As detailed in Supplementary Section 1, our analysis yields the connection
between these
F
n
and experimental measurements:
(2)
Equation (1) indicates the relevance of these coefficients:
F
n
are weighted spatial averages of
the analyte’s mass distribution function. It follows then, that moments of the analyte’s mass
distribution, m
(
k
)
, can be determined directly by forming linear combinations of these
functions,
(3)
Here,
are linear superpositions with
N
terms (
N
is the total
number of measured modes), involving the squared magnitudes of the unperturbed mode
shapes evaluated at the device surface. These are, in turn, superposed using coefficients
chosen in a specific manner described immediately below. Superscript (
k
) represents the
moment order. Using equations (2) and (3), we can then relate the moments m
(
k
)
directly to
the experimentally measured set of analyte-induced fractional-frequency shifts {
Δ
n
} through
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(4)
For example, the adsorbate mass
m
can be deduced from experiments by first picking a
particular set of coefficients
to create a superposition that ideally yields
g
(0)
(
r
) = 1 over
the integration region
Ω
s
. In this case
∫
Ω
s
μ
(
r
)d
S
=
m
= m
(0)
=
∫
Ω
s
μ
(
r
)
g
(0)
(
r
)d
S
. Similarly,
by picking a different set of coefficients a different superposition
g
(1)
(
r
) =
x
can be created;
this allows the analyte’s position-of-adsorption (its centre-of-mass along the
x
-direction) to
be deduced. This analysis can be generalized trivially to all higher-order moments.
Case study for a doubly-clamped beam
As an example, we consider a one-dimensional doubly-clamped elastic beam. The position
vector
r
in this case simplifies to the beam coordinate
x
, and we solve for the linear mass
density,
λ
(
x
). We can calculate m
(0)
, m
(1)
and in turn,
〈
x
〉
= m
(1)
/m
(0)
, which is the analyte’s
centre-of-mass along the beam coordinate (its position); this is proportional to
,
where
L
is the beam length. Creating a higher-order expansion along
x
, we form
g
(2)
(
x
) =
x
2
to obtain m
(2)
. This, in turn, allows deduction of the analyte’s standard deviation in
x
; that is,
its average size along the
x
coordinate, (
σ
x
= √m
(2)
/m
(0)
− (m
(1)
/m
(0)
)
2
). Analogous relations
yield even higher spatial moments of the analyte’s mass distribution along
x
, which
represent its width-averaged shape along
x
—for example the analyte’s skewness and
kurtosis, which involve
g
(3)
(
x
) =
x
3
and
g
(4)
(
x
) =
x
4
, respectively (Supplementary Section 2).
With a sufficiently complete set of spatial moments of the analyte’s mass distribution
{m
(
k
)
}, which can be deduced for each adsorbing analyte in real time from experimentally
measured multimodal NEMS frequency shifts, a one-dimensional inertial image (that is, the
width-averaged density distribution of the analyte along the beam’s longitudinal axis) can be
determined
23
,
24
. Equation (3) illustrates that the spatial dimensionality of m
(
k
)
arises directly
from the spatial variation of the vibrational modes used in its expansion. For example, the
out-of-plane displacements of a doubly-clamped beam or a singly-clamped cantilever
change with one spatial variable along the longitudinal axis,
x
. Vibrational mode shapes that
vary with two coordinates, such as those for thin plates or membranes, can provide two-
dimensional moments of the adsorbate distribution. These devices will permit extraction of a
two-dimensional inertial image of the mass distribution of the particle. An example of image
reconstruction using inertial imaging theory is discussed in the following. Note that equation
(2) applies generally to mechanical devices of any size, geometry and composition
(Supplementary Section 1).
Figure 1 presents mode shapes and superpositions for the out-of-plane displacements of an
ideal doubly-clamped beam of length
L
. For such a one-dimensional device the entire
(linear) spatial domain
Ω
l
is
x
∈
[0,
L
], because variations over the transverse direction are
implicitly integrated. Figure 1a depicts the first, second, fourth and tenth mode shapes along
the longitudinal coordinate,
x
. In Fig. 1b we demonstrate use of superpositions to
approximately yield
g
(0)
(
x
) = 1 over the full length of the beam; these involve from one up to
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ten modes. We determine the optimal coefficients
for the expansion of
g
(0)
(
x
) through
least-squares analysis. It can be shown rigorously that a unique analytic solution for these
coefficients exists (Supplementary Section 3).
The fidelity of the adsorbate’s inertial image is determined by how well, over the entire
integration region
Ω
s
, the finite superpositions
g
(
k
)
(
r
) used in equation (3) converge to their
targeted spatial functions. For the specific case illustrated in Fig. 1 involving one-
dimensional modes of a doubly-clamped beam, we seek an expansion
g
(0)
(
x
) = 1 that
converges over
Ω
l
. Figure 1b indicates that a choice of
Ω
l
to span the full beam length
L
is
not ideal and gives poor convergence (over- and undershoot); this choice will yield moments
that poorly approximate those of the analyte mass distribution. Figure 1c shows that
convergence to
g
(0)
(
x
) = 1 over
Ω
l
improves very significantly with the choice of a slightly
foreshortened region of integration. Practically, foreshortening
Ω
l
implies a reduced
measurement zone; in other words, such improvement comes with the cost of excluding a
small fraction of experimental data (from analytes that adsorb outside
Ω
l
). The effect of this
small reduction in the measurement zone,
Ω
l
, is minimal; in fact, adaptive fitting (described
below) can render it inconsequential.
As long as
Ω
l
spans the spatial extent of the analyte, the error for the
k
th moment,
ε
(
k
)
, varies
as a power of the measurement zone:
. Here
N
is the number of modes,
and this expression holds for
N
>
k
+ 1. For an accurate estimation of the
k
th moment, at
least
k
+ 1 modes are needed, and the accuracy obtained increases rapidly as more modes are
employed (Supplementary Section 4). Moreover, in the limit where the adsorbate is
infinitesimal compared to the device size, only
k
+ 1 modes are required to determine the
k
th
moment, and the use of additional modes does not improve resolution (Supplementary
Section 5).
Adaptive fitting to minimize error
To be useful in mass spectrometry and inertial imaging of biomolecules, the NEMS mass
sensor must be of sufficient size that individual analytes are small compared to the device
dimensions. In this case it is possible to markedly enhance resolution by fitting the
measurement zone,
Ω
s
, adaptively to the analyte size using a minimal set of modes (see
above). This straightforward computational procedure can be implemented in real time after
each set of adsorption-induced frequency shifts is acquired; no additional measurements are
involved. Figure 2 demonstrates adaptive fitting with a doubly-clamped beam. The
expansion interval
Ω
l
is progressively shrunk around the analyte’s adsorption position in an
iterative process after the analyte’s location is determined from a first pass of analysis. An
exponential decrease in uncertainty is attainable (Supplementary Sections 6 and 7). Figure 2
demonstrates a decrease in residual error by six orders of magnitude as
Ω
l
is ultimately
reduced to a region somewhat larger than the size of the analyte. We find adaptive fitting
converges (that is, the residual error saturates at a minimum level), typically after less than
ten iterations (Supplementary Section 7). In fact, this procedure enables convergent
measurements over a larger fraction of the beam. Hence, the extent of the region excluded to
obtain convergent expansions of
g
(
k
)
(
r
) can be significantly reduced. In practice, the ultimate
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attainable resolution is determined by the frequency stability of the resonator modes, as
described in the following.
Mass and position validation using published data
We first validate our methodology for mass and position determination using (1) our own
previous study of soft biological adsorbates (IgM antibodies) on NEMS resonators
7
and (2)
an independent study of Au nanoparticles attached to a microcantilever at various positions
along its length
17
. Analysis of these published data permits benchmarking the results of
inertial imaging against reported literature values. In the following section we shall present
new measurements that validate analysis of analyte size and shape by inertial imaging.
The results of our new analysis of the data from ref.
7
are shown in Fig. 3a,b. The two-mode
frequency shift data measured in ref.
7
are used here to independently determine particle
mass (Fig. 3a) and position (Fig. 3b) using inertial imaging. These results are compared to
those obtained from previously validated multimode theory
7
, which implicitly assumes that
analytes are point particles. As seen, the deduced masses and positions from both methods
are in excellent agreement.
Figure 3c,d compares the results of ref.
16
with those obtained from inertial imaging and
direct measurements of the gold bead on the cantilever from optical microscopy. The four-
mode frequency measurements reported in ref.
17
are used to determine the mass and
position of the bead from inertial imaging. This comparison shows excellent agreement for
both mass and position. Although the high-aspect-ratio Au nanoparticles used in ref.
17
are
obviously not thin or compliant with the device surface, this does not compromise the
validity of inertial imaging for mass and position. We find these moments are relatively
unaffected by the rotary inertia of the adsorbate.
Size and shape validation using droplet arrays
With the capabilities of inertial imaging established for both mass and position, we turn to
validation of higher moments—variance and skewness—which represent the size and shape
of the adsorbate. As mentioned, inertial imaging measurements of mass and position are
robust for non-compliant adsorbates, but the same is not true for the higher moments. To
ensure the overarching assumptions of our method are upheld, we used non-volatile liquid
micro-droplet arrays (see Methods). Figure 4a presents optical images of the droplet array
systems studied.
We compare results deduced from inertial imaging to optical measurements of the droplet
arrays in Fig. 4b,c. As shown, inertial imaging of symmetric array distributions provides
excellent agreement with optical measurements for both position and standard deviation.
Also, the results for position are consistent with the analysis of previously published data, as
discussed above.
Our variance measurements of Fig. 4 provide the first validation of the capabilities of inertial
imaging to provide the spatial variance (size) of an adsorbate. Additionally, these data also
deliver a first demonstration of the measurement of mass distribution asymmetry: inertial
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imaging data yield a skewness of −0.453, which is in close agreement with the optically
measured value of −0.537. (Slight non-uniformities in droplet size contribute to the small,
~15% difference in skewness values.)
These experimental validations of inertial imaging are consistent with our full numerical
simulations (Supplementary Sections 8 and 9), which permit absolute specification of
adsorbate properties. In Supplementary Section 8, we also show how the spatial mass
distribution of the adsorbate (that is, its ‘inertial image’) can be reconstructed from the
measured moments.
Practical implications
In frequency-shift-based mass sensing, both the mass distribution function of the analyte and
its mechanical coupling to the surface of the NEMS sensor play important roles in
determining the magnitude of the induced fractional-frequency shifts, {
Δ
n
}. Hence, the
nature of the physical attachment of the analyte to the resonator is also probed. Soft
biological analytes, such as proteins, are ideal targets for inertial imaging. Our standard
NEMS-MS protocol of cooling the sensor below ambient promotes strong physisorption of
the analytes
7
. Accordingly, van der Waals and chemical forces will cause the analyte to
conform to the surface topography of the sensor. However, results inferred from the
frequency shifts induced by rigid analytes, such as for metallic nanoparticles, will instead
yield an inertially imaged size reflecting the region of attachment. In general, this can be
smaller than the diameter of a rigid analyte.
Beyond such experimental details, the primary source of uncertainty—and the ultimate limit
to the resolution of inertial imaging—arises from the frequency instability of the resonator.
This fundamental uncertainty in deducing the moments of the analyte’s mass distribution
function can be evaluated using equation (4) and standard methods of error propagation,
together with knowledge of the spectral density of the resonator’s frequency fluctuations
(Supplementary Section 10). In Table 1 we provide examples of the frequency-noise-limited
resolvable size of an analyte that is attainable with current micro- and nanoscale resonator
technology. These projections are based on strongly driven devices, which provide optimal
frequency resolution. As shown, today’s smallest devices are clearly capable of resolving
molecular-scale analytes.
Inertial imaging enables measurements of both the mass and molecular size and shape of
analytes that adsorb on a nanomechanical resonator. Analogous to the previous
nanomechanical measurements of mass and position-of-adsorption of individual proteins
7
,
inertial imaging is possible in real time as individual analytes adsorb on a NEMS sensor, one
by one. This represents a paradigm shift in the realm of resonator-based particle sensing, in
that it permits spatially resolved imaging of single analytes using current measurement
technology. The ultimate resolution of this technique is not limited by the modal
wavelengths, but instead only by the inherent frequency instability of the nanomechanical
resonators used. Hence, NEMS-based inertial imaging can enable single-molecule mass
spectrometry and, simultaneously, the evaluation of the size and shape of individual
molecules with nanometre-scale dimensions.
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Methods
The liquid droplet arrays were deposited on a tipless reference microcantilever (Bruker,
CLFC-NOBO) using atomic force microscopy (AFM) dip-pen lithography
25
. These ‘soft’
adsorbates are both thin and compliant with the device, which mitigates any effect of the
adsorbates’ rotary inertia, as required with our formulation of inertial imaging. The liquid
droplet arrays also mimic the soft material properties of a biological specimen, while
enabling both precise dimensional control and independent characterization of adsorbate
shape (difficult with a biomolecular test specimen). We used ideally rectangular single-
crystal silicon microcantilevers for these studies. This minimized potential non-idealities in
the mode shapes and thus simplified application of inertial imaging theory.
The droplets were sequentially deposited in rows, beginning from the distal end of the
cantilever. The droplets were deposited in configurations to produce both symmetric and
asymmetric adsorbate mass distributions (the latter permitting an assessment of skewness).
We measured the undriven, thermomechanical response of the cantilever before and after
deposition of the droplet arrays to obtain, with high precision, the induced resonant
frequency shifts for the first four flexural modes (which lie within the bandwidth of our
measurement system). Although this thermal-noise-based measurement scheme limited the
practically achievable frequency resolution (Table 1)
26
, it permitted a controlled, artefact-
free first experimental demonstration of the inertial imaging method. Additional
measurement details are described in Supplementary Section 11.
Supplementary Material
Refer to Web version on PubMed Central for supplementary material.
Acknowledgments
The authors acknowledge support from an NIH Director’s Pioneer award (to M.L.R.), a Caltech Kavli Nanoscience
Institute Distinguished Visiting Professorship (to J.E.S.), the Fondation pour la Recherche et
l’EnseignementSuperieur, Paris (FRES; to M.L.R.), and the Australian Research Council grants scheme (P.M. and
J.E.S.).
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kernel density functions. Probabilist Eng Mech. 2002; 17:273–291.
25. Ginger DS, Zhang H, Mirkin CA. The evolution of dip-pen nanolithography. Angew Chem Int Ed.
2004; 43:30–45.
26. Ekinci KL, Yang YT, Roukes ML. Ultimate limits to inertial mass sensing based upon
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Figure 1. Superposition of resonator mode shapes
a
, Mode shapes of a doubly-clamped beam (from bottom to top) for the first, second, fourth
and tenth out-of-plane flexural modes.
b
, A linear combination of mode shapes intended to
yield a superposition
g
(0)
(
x
) = 1 over an interval
Ω
l
spanning the entire beam [0, 1], using the
first mode (black), first two modes (blue), first four modes (red) and first ten modes (green).
Significant over- and undershoot is evident.
c
, Superpositions with a slightly foreshortened
interval provide greatly improved convergence. Here,
Ω
l
spans [
δ
,1 −
δ
], where
δ
=
N
/(1 +
N
2
). For an expansion involving the first ten modes (green), the interval [0.099, 0.901]
covers about 80% of the beam.
L
, cantilever length.
N
, number of modes used.
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Figure 2. Adaptive fitting for enhanced resolution and accuracy
Superposition fit to
g
(0)
(
x
) = 1 (to calculate mass) using the first four modes of a doubly-
clamped beam, showing the effect of measurement zone reduction.
a
, Initial superposition
(black curve) before the initial values for the location and size of the particle are determined.
After determination of the location and size of the particle, the measurement zone can be
reduced. In this case, the position is determined to be near
x
/
L
= 0.35 (dotted vertical line)
and a measurement zone smaller than the original is chosen (pink shaded area). This
measurement zone (pink shaded area) is centred on the particle position and commensurate
with particle size. With this reduction in measurement zone, a new superposition can be
calculated (blue curve: solid is within zone, dashed is outside).
b
, Zoomed-in view of new
superposition. Further reduction of the measurement zone by a factor of 10 leads to another
superposition (green curve: solid is within zone, dashed is outside). This is again superior to
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the previous superposition (blue curve).
c
, Zoomed-in view of final superposition (dotted
vertical line is the position of the particle). Error is reduced by six orders of magnitude from
a
.
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Figure 3. Mass and position analysis using published experimental data
a
,
b
, Mass (
a
) and position (
b
) calculations for the experimental data from ref.
7
using two
modes of a doubly-clamped beam. The values for mass and position are compared with the
previous values from ref.
7
using multimode theory. Position is scaled to the device length.
Error bars in inertial imaging theory reflect the total error due to both the fitting residual and
frequency fluctuations (2
σ
, 95% confidence level).
c
, Analysis of particle mass for different
positions using the four-mode measurement of the same particle along a cantilever device
16
.
The particle expected mass is estimated from the scanning electron microscopy (SEM)
measurements of ref.
16
(solid red line) with 2% assumed uncertainties in that measurement
(dotted red lines).
d
, Position calculation using the same data, compared with the optically
measured position. Red lines in
a
,
b
,
d
represent curves of exact agreement between the two
methods. Insets in
a
and
c
: Electron micrographs of representative devices used in the two
respective studies. Scale bars, 2 μm (
a
); white, 5 μm and black, 500 nm (
c
). In all figures,
error bars represent the 2
σ
, 95% confidence level.
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Figure 4. Size and shape analysis via frequency-shift measurements of droplet arrays
a
, Optical images of liquid droplet arrays deposited on a silicon microcantilever (397 μm
long, 29 μm wide, 2 μm thick) using AFM dip-pen lithography. Inverted black–white images
serve to highlight the droplets. Inertial imaging using the lowest four vibrational modes
resolves these addenda as constituting a composite, thin, spatially distributed analyte that is
strongly adherent to the cantilever surface. Measurement details are provided in the Methods
and Supplementary Section 11. The numbers of two-droplet rows in the symmetric
distributions are specified, and the asymmetric distribution used to assess skewness is
indicated.
b
, Comparison of mean position of the droplet array distribution, measured using
both inertial imaging and optical microscopy.
c
, Comparison of the variance in the droplet
array distribution measured using both inertial imaging and optical microscopy. Position and
standard deviation are scaled to cantilever length. The single row (denoted ‘1’ in
a
, does not
satisfy the requirement of being a thin and compliant adsorbate and hence is not included in
the comparison of variance. Dashed lines are linear regressions of data centred at the origin.
Slopes of lines and
R
2
values are indicated. Vertical and horizontal error bars in
b
,
c
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represent the 2
σ
uncertainty levels due to frequency noise and droplet mass variability,
respectively.
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