In this work, the three-dimensional fluid–solid interaction vibration of particle in the oscillating resonator and its effect on the dynamic characteristics are analyzed and discussed. It demonstrates that the displacement of a particle is composed of two components, one is in phase with the acceleration of resonator and the other is out of phase. The former is responsible for the added mass effect and the latter results in a small damping. A modified measurement principle for detecting the buoyant mass is then presented by considering the in-phase component. The three-dimensional (3D) fluid–solid interaction problem involving the particle, fluid, and resonator is numerically solved, and the effects of density ratio, inverse Stokes number, and the ratio of channel height to particle diameter are studied. Based on the numerical results, a function characterizing the in-phase component is identified through a fitting procedure. According to the modified measurement principle and the analytical expression for the in-phase component, a calibration method is developed for measuring buoyant mass. Using this calibration method, the systematic measurement error induced by the vibration of particles can be effectively reduced.

References

References
1.
Burg
,
T. P.
,
Godin
,
M.
,
Knudsen
,
S. M.
,
Shen
,
W.
,
Carlson
,
G.
,
Foster
,
J. S.
,
Babcock
,
K.
, and
Manalis
,
S. R.
,
2007
, “
Weighing of Biomolecules, Single Cells and Single Nanoparticles in Fluid
,”
Nature
,
446
(
7139
), pp.
1066
1069
.
2.
Barton
,
R. A.
,
Ilic
,
B.
,
Verbridge
,
S. S.
,
Cipriany
,
B. R.
,
Parpia
,
J. M.
, and
Craighead
,
H. G.
,
2010
, “
Fabrication of a Nanomechanical Mass Sensor Containing a Nanofluidic Channel
,”
Nano Lett.
,
10
(
6
), pp.
2058
2063
.
3.
Weng
,
Y.
,
Delgado
,
F. F.
,
Son
,
S.
,
Burg
,
T. P.
,
Wasserman
,
S. C.
, and
Manalis
,
S. R.
,
2011
, “
Mass Sensors With Mechanical Traps for Weighing Single Cells in Different Fluids
,”
Lab Chip
,
11
(
24
), pp.
4174
4180
.
4.
Vidal-Alvarez
,
G.
,
Marigo
,
E.
,
Torres
,
F.
, and
Barniol
,
N.
,
2016
, “
Fabrication and Measurement of a Suspended Nanochannel Microbridge Resonator Monolithically Integrated With CMOS Readout Circuitry
,”
Micromachines
,
7
(
3
), p.
40
.
5.
Bryan
,
A. K.
,
Hecht
,
V. C.
,
Shen
,
W.
,
Payer
,
K.
,
Grover
,
W. H.
, and
Manalis
,
S. R.
,
2014
, “
Measuring Single Cell Mass, Volume, and Density With Dual Suspended Microchannel Resonators
,”
Lab Chip
,
14
(
3
), pp.
569
576
.
6.
Burg
,
T. P.
, and
Manalis
,
S. R.
,
2003
, “
Suspended Microchannel Resonators for Biomolecular Detection
,”
Appl. Phys. Lett.
,
83
(
13
), pp.
2698
2700
.
7.
Kim
,
J.
,
Song
,
J.
,
Kim
,
K.
,
Kim
,
S.
,
Song
,
J.
,
Kim
,
N.
,
Khan
,
M. F.
,
Zhang
,
L.
,
Sader
,
J. E.
,
Park
,
K.
,
Kim
,
D.
,
Thundat
,
T.
, and
Lee
,
J.
,
2016
, “
Hollow Microtube Resonators via Silicon Self-Assembly Toward Subattogram Mass Sensing Applications
,”
Nano Lett.
,
16
(
3
), pp.
1537
1545
.
8.
Agache
,
V.
,
Blanco-Gomez
,
G.
,
Baleras
,
F.
, and
Caillat
,
P.
,
2011
, “
An Embedded Microchannel in a MEMS Plate Resonator for Ultrasensitive Mass Sensing in Liquid
,”
Lab Chip
,
11
(
15
), pp.
2598
2603
.
9.
Lee
,
J.
,
Shen
,
W.
,
Payer
,
K.
,
Burg
,
T. P.
, and
Manalis
,
S. R.
,
2010
, “
Toward Attogram Mass Measurements in Solution With Suspended Nanochannel Resonators
,”
Nano Lett.
,
10
(
7
), pp.
2537
2542
.
10.
Lee
,
I.
,
Park
,
K.
, and
Lee
,
J.
,
2012
, “
Note: Precision Viscosity Measurement Using Suspended Microchannel Resonators
,”
Rev. Sci. Instrum.
,
83
(
11
),
116106
.
11.
Etayash
,
H.
,
Khan
,
M. F.
,
Kaur
,
K.
, and
Thundat
,
T.
,
2016
, “
Microfluidic Cantilever Detects Bacteria and Measures Their Susceptibility to Antibiotics in Small Confined Volumes
,”
Nat. Commun.
,
7
,
12947
.
12.
Cermak
,
N.
,
Olcum
,
S.
,
Delgado
,
F. F.
,
Wasserman
,
S. C.
,
Payer
,
K. R.
,
Murakami
,
M. A.
,
Knudsen
,
S. M.
,
Kimmerling
,
R. J.
,
Stevens
,
M. M.
, and
Kikuchi
,
Y.
,
2016
, “
High-Throughput Measurement of Single-Cell Growth Rates Using Serial Microfluidic Mass Sensor Arrays
,”
Nat. Biotechnol.
,
34
(
10
), pp.
1052
1059
.
13.
Cetin
,
A. E.
,
Stevens
,
M. M.
,
Calistri
,
N. L.
,
Fulciniti
,
M.
,
Olcum
,
S.
,
Kimmerling
,
R. J.
,
Munshi
,
N. C.
, and
Manalis
,
S. R.
,
2017
, “
Determining Therapeutic Susceptibility in Multiple Myeloma by Single-Cell Mass Accumulation
,”
Nat. Commun.
,
8
(
1
), pp.
1
12
.
14.
Grover
,
W. H.
,
Monica Diez-Silva
,
A. K. B.
,
Suresh
,
S.
,
Higgins
,
J. M.
, and
Manalis
,
S. R.
,
2011
, “
Measuring Single-Cell Density
,”
Proc. Natl. Acad. Sci. U.S.A.
,
108
(
27
), pp.
10992
10996
.
15.
Godin
,
M.
,
Delgado
,
F. F.
,
Son
,
S.
,
Grover
,
W. H.
,
Bryan
,
A. K.
,
Tzur
,
A.
,
Jorgensen
,
P.
,
Payer
,
K.
,
Grossman
,
A. D.
, and
Kirschner
,
M. W.
,
2010
, “
Using Buoyant Mass to Measure the Growth of Single Cells
,”
Nat. Methods
,
7
(
5
), pp.
387
390
.
16.
Sarid
,
D.
,
1994
,
Scanning Force Microscopy: With Applications to Electric, Magnetic, and Atomic Forces
,
Oxford University Press
,
Oxford
.
17.
Dohn
,
S.
,
Svendsen
,
W.
,
Boisen
,
A.
, and
Hansen
,
O.
,
2007
, “
Mass and Position Determination of Attached Particles on Cantilever Based Mass Sensors
,”
Rev. Sci. Instrum.
,
78
(
10
),
103303
.
18.
Ansari
,
M.
,
Esmailzadeh
,
E.
, and
Jalili
,
N.
,
2011
, “
Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations
,”
ASME J. Vib. Acoust.
,
133
(
4
), p.
041003
.
19.
Wang
,
C. Y.
, and
Wang
,
C. M.
,
2012
, “
Exact Vibration Solution for Exponentially Tapered Cantilever With Tip Mass
,”
ASME J. Vib. Acoust.
,
134
(
4
), p.
041012
.
20.
Olcum
,
S.
,
Cermak
,
N.
,
Wasserman
,
S. C.
,
Christine
,
K. S.
,
Atsumi
,
H.
,
Payer
,
K. R.
,
Shen
,
W.
,
Lee
,
J.
,
Belcher
,
A. M.
, and
Bhatia
,
S. N.
,
2014
, “
Weighing Nanoparticles in Solution at the Attogram Scale
,”
Proc. Natl. Acad. Sci. U.S.A.
,
111
(
4
), pp.
1310
1315
.
21.
Olcum
,
S.
,
Cermak
,
N.
,
Wasserman
,
S. C.
, and
Manalis
,
S. R.
,
2015
, “
High-Speed Multiple-Mode Mass-Sensing Resolves Dynamic Nanoscale Mass Distributions
,”
Nat. Commun.
,
6
, p.
7070
.
22.
Zhang
,
Y.
, and
Zhao
,
Y.-P.
,
2015
, “
Mass and Force Sensing of an Adsorbate on a Beam Resonator Sensor
,”
Sensors
,
15
(
7
), pp.
14871
14886
.
23.
Belardinelli
,
R.
,
Hauzer
,
L. M. F. R.
,
Siskins
,
M.
,
Ghatkesar
,
M. K.
, and
Alijani
,
F.
,
2018
, “
Modal Analysis for Density and Anisotropic Elasticity Identification of Adsorbates on Microcantilevers
,”
Appl. Phys. Lett.
,
113
(
14
),
143102
.
24.
Zhang
,
Y.
,
2011
, “
Eigenfrequency Computation of Beam/Plate Carrying Concentrated Mass/Spring
,”
ASME J. Vib. Acoust.
,
133
(
2
), p.
021006
.
25.
Coimbra
,
C. F. M.
, and
Rangel
,
R. H.
,
2001
, “
Spherical Particle Motion in Harmonic Stokes Flows
,”
AIAA J.
,
39
(
9
), pp.
1673
1682
.
26.
Coimbra
,
C. F. M.
,
L’Esperance
,
D.
,
Lambert
,
R. A.
,
Trolinger
,
J. D.
, and
Rangel
,
R. H.
,
2004
, “
An Experimental Study on Stationary History Effects in High-Frequency Stokes Flows
,”
J. Fluid Mech.
,
504
, pp.
353
363
.
27.
Schipitsyn
,
V. D.
, and
Kozlov
,
V. G.
,
2018
, “
Oscillatory and Steady Dynamics of a Cylindrical Body Near the Border of Vibrating Cavity Filled With Liquid
,”
Microgravity Sci. Technol.
,
30
(
1–2
), pp.
103
112
.
28.
Tchen
,
C. M.
,
1947
, “Mean Value and Correlation Problems Connected With the Motion of Small Particles Suspended in a Turbulent Fluid,”
Doctoral dissertation
,
Delft University
,
The Hague
.
29.
Lim
,
E. A.
,
Kobayashi
,
M. H.
, and
Coimbra
,
C. F. M.
,
2014
, “
Fractional Dynamics of Tethered Particles in Oscillatory Stokes Flows
,”
J. Fluid Mech.
,
746
, pp.
606
625
.
30.
Pacheco-Martinez
,
H. A.
,
Liao
,
L.
,
Hill
,
R. J. A.
,
Swift
,
M. R.
, and
Bowley
,
R. M.
,
2013
, “
Spontaneous Orbiting of Two Spheres Levitated in a Vibrated Liquid
,”
Phys. Rev. Lett.
,
110
(
15
).
31.
Yan
,
H.
,
Zhang
,
W.-M.
,
Jiang
,
H.-M.
,
Hu
,
K.-M.
,
Hong
,
F.-J.
,
Peng
,
Z.-K.
, and
Meng
,
G.
,
2017
, “
A Measurement Criterion for Accurate Mass Detection Using Vibrating Suspended Microchannel Resonators
,”
J. Sound Vib.
,
403
, pp.
1
20
.
32.
Sader
,
J. E.
,
Burg
,
T. P.
, and
Manalis
,
S. R.
,
2010
, “
Energy Dissipation in Microfluidic Beam Resonators
,”
J. Fluid Mech.
,
650
, pp.
215
250
.
33.
Ekinci
,
K. L.
,
Yang
,
Y. T.
, and
Roukes
,
M. L.
,
2004
, “
Ultimate Limits to Inertial Mass Sensing Based Upon Nanoelectromechanical Systems
,”
J. Appl. Phys.
,
95
(
5
), pp.
2682
2689
.
34.
Sansa
,
M.
,
Sage
,
E.
,
Bullard
,
E. C.
,
Gely
,
M.
,
Alava
,
T.
,
Colinet
,
E.
,
Naik
,
A. K.
,
Villanueva
,
L. G.
,
Duraffourg
,
L.
,
Roukes
,
M. L.
,
Jourdan
,
G.
, and
Hentz
,
S.
,
2016
, “
Frequency Fluctuations in Silicon Nanoresonators
,”
Nat. Nanotechnol.
,
11
(
6
), pp.
552
558
.
35.
Lee
,
J.
,
Bryan
,
A. K.
, and
Manalis
,
S. R.
,
2011
, “
High Precision Particle Mass Sensing Using Microchannel Resonators in the Second Vibration Mode
,”
Rev. Sci. Instrum.
,
82
(
2
), p.
1066
.
36.
Basset
,
A. B.
,
1888
, “
On the Motion of a Sphere in a Viscous Liquid
,”
Philos. Trans. R. Soc. Lond. A
,
179
, pp.
43
63
.
37.
Weinstein
,
J. A.
,
Kassoy
,
D. R.
, and
Bell
,
M. J.
,
2008
, “
Experimental Study of Oscillatory Motion of Particles and Bubbles With Applications to Coriolis Flow Meters
,”
Phys. Fluids
,
20
(
10
), pp.
353
.
38.
Morrison
,
F. A.
, and
Stewart
,
M. B.
,
1976
, “
Small Bubble Motion in an Accelerating Liquid
,”
ASME J. Appl. Mech.
,
43
(
3
), pp.
399
403
.
39.
Abbad
,
M.
, and
Souhar
,
M.
,
2004
, “
Effects of the History Force on an Oscillating Rigid Sphere at Low Reynolds Number
,”
Exp. Fluids
,
36
(
5
), pp.
775
782
.
40.
Saadatmand
,
M.
, and
Kawaji
,
M.
,
2013
, “
Mechanism of Vibration-Induced Repulsion Force on a Particle in a Viscous Fluid Cell
,”
Phys. Rev. E
,
88
(
2
),
023019
.
41.
Favier
,
J.
,
Revell
,
A.
, and
Pinelli
,
A.
,
2014
, “
A Lattice Boltzmann–Immersed Boundary Method to Simulate the Fluid Interaction With Moving and Slender Flexible Objects
,”
J. Comput. Phys.
,
261
(
3
), pp.
145
161
.
42.
Eshghinejadfard
,
A.
,
Abdelsamie
,
A.
,
Janiga
,
G.
, and
Thevenin
,
D.
,
2016
, “
Direct-Forcing Immersed Boundary Lattice Boltzmann Simulation of Particle/Fluid Interactions for Spherical and Non-Spherical Particles
,”
Particuology
,
25
(
2
), pp.
93
103
.
43.
Cate
,
A. T.
,
Nieuwstad
,
C. H.
,
Derksen
,
J. J.
, and
Akker
,
H. E. A. V. D.
,
2002
, “
Particle Imaging Velocimetry Experiments and Lattice-Boltzmann Simulations on a Single Sphere Settling Under Gravity
,”
Phys. Fluids
,
14
(
11
), pp.
4012
4025
.
44.
Gysling
,
D. L.
,
2007
, “
An Aeroelastic Model of Coriolis Mass and Density Meters Operating on Aerated Mixtures
,”
Flow Meas. Instrum.
,
18
(
2
), pp.
69
77
.
45.
Green
,
C. P.
, and
Sader
,
J. E.
,
2005
, “
Small Amplitude Oscillations of a Thin Beam Immersed in a Viscous Fluid Near a Solid Surface
,”
Phys. Fluids
,
17
(
7
), p.
8
.
46.
Lee
,
J.
,
Bryan
,
A. K.
, and
Manalis
,
S. R.
,
2011
, “
High Precision Particle Mass Sensing Using Microchannel Resonators in the Second Vibration Mode
,”
Rev. Sci. Instrum.
,
82
(
2
),
023704
.
47.
Lee
,
J.
,
Chunara
,
R.
,
Shen
,
W.
,
Payer
,
K.
,
Babcock
,
K.
,
Burg
,
T. P.
, and
Manalis
,
S. R.
,
2011
, “
Suspended Microchannel Resonators With Piezoresistive Sensors
,”
Lab Chip
,
11
(
4
), pp.
645
651
.
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