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L12. Mechanics of Nanostructures: Mechanical
Resonance
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Outline1. Theory
2. Mechanical Resonance Experiments
3. Ruoff group work
SiO2 nanowires
Quartz Fibers
Crystalline Boron Nanowires
4. Summary
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Part One:Mechanical Resonance Method
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Simple Beam TheoryThe mechanical resonance testing of nanostructures has to date been based on simple beam theory.
Natural Frequency
ββββ : eigenvalue Eb : elastic modulusI : moment of inertia L : beam length m : mass per unit length
y
z
b
h
L x
z Assumptions on geometry:• Long and thin ( L >>b,h)• Loading is in Z direction (no axial load)Assumption on deformation:• Plane sections remain plane and perpendicular to the mid-plane after deformation
Morse, P. Vibration and sound, 2nd edition, New York and London, McGraw-Hill,(1948)
4b
2n
n mLIE
�2�
f =
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Resonance Test PrincipleAccording to simple beam theory, the natural frequency of a cantilevered circular cross-section beam is given by:
ρπβ
162 2
2bn
n
ELD
f =
Natural Frequency
ββββ: constant D: diameterL: length ρρρρ: densityEb: elastic modulus
Bending Modulus:
22
4
4
264n
nb f
DL
Eβ
ρπ= 10.996
7.855
694.4
875.1
3
2
1
0
====
ββββ
The bending modulus can be calculated through measurement of dimension and resonance frequency.
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Electrical Excitation
VdcVac
tV
tVVVVVV
tVVV
acacdcacdc
acdc
ωαωαα
ωα
2cos2
cos)(221
)(
)cos(F(t)2
22
2
++∆+++∆=
++∆=
The mechanical resonance of a cantilevered structure can be excited by applying a periodic load whose frequency equals the natural frequency of the structure.
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Mechanical Excitation
Piezoelectric Actuator
The mechanical resonance of a cantilevered structure can also be mechanically excited through the vibration of its substrate.
The working frequency range of the mechanical excitation method is here limited by the frequency response of the piezoelectric actuator.
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Part Two:Mechanical Resonance
Experiments
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Mechanical Resonance: Carbon Nanotube
Poncharal,P., et al, Science, 283, 1513-1516 (1999)
(top) Elastic properties of nanotubes
(left) Electromechanical vibration of a MWCNT (A) thermal vibration (B) Fundamental resonance (C) First overtone resonance
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Mechanical Resonance: DLC Pillar
Fujita,J. et al, J.Vac.Sci.Technol. B 19(6), 2834-2836 (2001)
SEM image of the vibration Schematic of mechanical vibration experimental setup
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Mechanical Resonance: ZnO Nanobelt
Bai et al, App. Phys. Lett. 82(26) 4806-4808 (2003)
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Mechanical Resonance: ZnO nanobelt (con’t)
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Mechanical Resonance: Work Function Measurement
X. D. Bai, E. G. Wang, P.X. Gao and Z. L. Wang, Nano Letters, 3 (2003) 1147-1150.
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• For example: boron nanowire is driven electromechanically• Signal includes DC and AC components• Experimental stage is mounted inside SEM for visualization
Parametric Resonance
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Electromechanical Driving: Mathieu Equation
• The force due to the electrical field is similar to that on a cantilever between capacitor plates:
• The voltage-induced force on the nanowire increases as the tube bends closer to the electrode:
• Equation for each mode takes the form of the damped Mathieu equation
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Stability of the Mathieu Equation
From: Bender and Orszag, “Advanced Mathematical Methods forScientists and Engineers.”McGraw-Hill, p.562
a = 0.2983εεεε = 0.05µµµµ = 0.01
a = 0.2985εεεε = 0.05µµµµ = 0.01
Unstable
Stable
( ) 0cos22
2
=+++ ytadtdY
dtYd εµ
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Parametric Resonance• Mathieu equation has
unstable solutions for:
• Resonance is observed at driving frequencies that give the unstable values of a:
( ) 0cos22
2
=+++ ytadtdY
dtYd εµ
Min-Feng Yu, Gregory J. Wagner, Rodney S. Ruoff, Mark J. Dyer, Realization of parametric resonances in a nanowire mechanical system with nanomanipulation inside scanning electron microscope, Phys. Rev. B 66, 073406 (2002).
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Instability Regions for Vibrating Nanowire• Small shifts in frequency or driving
voltage can cause switch from unstable to stable behavior
• This instability can be used to sense changes in the environment of a vibrating nanowire
From: Bender and Orszag, “Advanced Mathematical Methods forScientists and Engineers.”McGraw-Hill, p.562
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.2494 0.2496 0.2498 0.25 0.2502 0.2504 0.2506 0.2508
a
εεεεV=1.5
V=2.0V=2.5
Theory
( ) 0cos22
2
=+++ ytadtdY
dtYd εµ
Min-Feng Yu, Gregory J. Wagner, Rodney S. Ruoff, Mark J. Dyer, Realization of parametric resonances in a nanowire mechanical system with nanomanipulation inside scanning electron microscope, Phys. Rev. B 66, 073406 (2002).
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Part Three:Ruoff Group Work
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Experiment Tool: Nanomanipulator• Four-degree of freedom (x,y,z linear motion and rotation)• Two separate stages (X-Y stage, Z-θθθθ stage) • Sub-nanometer motion resolution
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Experimental Setup
(b) Mechanical Excitation
(a) Electrical Excitation
VdcVac
Piezo Bimorph
Counter Electrode
Conductive AFM Cantilever
X-Y StageZ Stage
Piezo Bimorph
AFM Cantilever
X-Y Stage Vac
Nanostructure
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Mechanical Resonance of SiO2 Nanowire
D. A. Dikin, X. Chen, W. Ding, G. Wagner, R. S. Ruoff, Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation, Journal of Applied Physics 93, 226 (2003).
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SiO2 Nanowire: Source
Ultrasonically dispersed SiO2nanowire
Synthesized by Z.W. Pan (J.Am.Chem.Soc.’02)
TEM image (inserts: High resolution image and diffraction pattern)
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SiO2 Nanowire: Mechanical Resonance
Electrical Excitation Mechanical Excitation
W wireW wire
W wire (counter electrode)
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SiO2 Nanowire: Charge TrappingExperiment 1
E-beam modes, time between
scanlines:TV mode 0.06 ms
3-d mode 17 ms
4-th mode 50 ms
12345..
512
2µµµµm
Hitachi S4500
D. A. Dikin, X. Chen, W. Ding, G. Wagner, R. S. Ruoff, Resonance vibration of amorphous SiO2 nanowiresdriven by mechanical or electrical field excitation, Journal of Applied Physics 93, 226 (2003).
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SiO2 Nanowire: Charge TrappingExperiment 2
Contr Electrode
120 140 160 180 2000
10
20
30
40
a
Am
pl. o
f vib
ratio
n, µ
m
Length of NW under loading, µm
v (x)p xEd
( a x)ao= −8
34
3
4πD. A. Dikin, X. Chen, W. Ding, G. Wagner, R. S. Ruoff, Resonance vibration of amorphous SiO2 nanowires
driven by mechanical or electrical field excitation, Journal of Applied Physics 93, 226 (2003).
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y x A x x B x xn n n n n n n( ) [sin sinh (cos cosh )]= − − −β β β β
Ax xx xn
n n
n n
= −+
sinh sin(cosh cos )
β ββ β2
Bx xx xn
n n
n n
= +−
cosh cossinh sin
β ββ β
fdL
Ei
i= βπ ρ
2
22 12
x
y
Length, um(± 0.2)
Diameter, nm(± 5)
Naturalfrequency, kHz
E, GPa
17 80 172.0 43.4 ± 7.017.3 88 193.8 48.8 ± 7.818.3 98 190.0 47.4 ± 8.4
The density of the SiO2 is 2200 kg/m3.
SiO2 Nanowire: Bending Modulus
β1 = 1.875
Electrical driving2 µµµµm
Calculated
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Mechanical Resonance of Quartz Fiber
Xinqi Chen, Sulin Zhang, Gregory J. Wagner, Weiqiang Ding, and Rodney S. Ruoff, Mechanical resonance of quartz microfibers and boundary condition effects, Journal of Applied Physics, 95 (9), 4823-4828, 2003
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Quartz Fibers
Typical sample geometry: diameter: 30-100 µm, length: 5-10 mm
Quartz fibers were home-made by pulling a fused quartz rod (GE Quartz, Inc) on a wide flame.
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Quartz Fiber: Mechanically Induced Resonance
Optical microscope pictures of the first four modes of resonance of a quartz microfiber. The insets are the theoretical displacement curves.
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Quartz Fiber: Correction for Non-uniform Diameter
Due to the pulling process, the quartz fiber diameter is not quite uniform. The resonance frequency change for a beam of circular cross-section, with linearly varying diameter, was calculated according to the following equation:
α=(D1-D0)/D0 α is generally small.
n=0: f = f0 × (1 - 0.42 α)n=1: f = f0 × (1 + 0.22 α)n=2: f = f0× (1 + 0.42 α)
f0 is the corresponding resonance frequencies of a beam with uniform diameter D0.
D0
D1
fixed end
free end
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Quartz Fiber: Young’s Modulus# L , mm D 0, um D 1,
um f0, Hz f1, Hz f1/f0 Corrected
E 0 Corrected
E 1
1 8.22 32 27 397 2158 5.43 73.3 67.2 2 7.18 32 33 461 2971 6.45 67.1 68.1 3 3.92 37 32 1733 10307 5.95 54.9 58.7
4 7.71 37 37 490 2964 6.05 73.4 68.4 5 6.79 38 35 571 3531 6.18 53.3 57.3 6 7.61 42 39 537 3039 5.66 61.2 54.7 7 5.53 47 52 997 6821 6.84 54.6 56.7 8 6.35 60 69 987 7042 7.13 59.4 63.3 9 5.58 75 75 1618 -- -- 53.4 --
10 9.58 75 75 504 3274 6.50 45.1 48.4 11 6.53 75 77 1274 8243 6.47 63.6 65.5 12 5.60 77 75 1618 10116 6.25 50.3 51.8 13 6.84 77 102 1036 -- -- 62.9 -- 14 10.45 87 87 496 3131 6.31 45.9 46.6 15 5.70 107 107 2451 -- -- 65.6 --
Note: L: length, D0: the clamped side diameter, D1: the free end diameter, f0: the fundamental resonance frequency, f1: the first overtone resonance frequency.
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Mechanical Resonance of Crystalline Boron Nanowires
W. Ding, L. Calabri, X. Chen, K. Kohlhaas, R.S. Ruoff, Mechanics of Crystalline Boron Nanowires, manuscript in preparation
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Boron Nanowire: Source
SEM image of boron nanowires on alumina substrate TEM image of a boron nanowire
Otten et al, J.Am. Chem. Soc.,124 (17), 2002
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Boron Nanowire: Resonance
First two modes of resonance of a cantilevered BNW
Typical frequency response of the 1st
mode resonance
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Boron Nanowire: Length Determination
4
22
4
4
264
LE
fDL
�
��E
b
nn
b
∝
=
Huang, Dikin, Ding, Qiao, Chen, Fridman, Ruoff (2004), Journal of Microscopy, 216, 206.
Schematic representation of a wire being partitioned into N segments, before (a) and after (b) rotation.
SEM images only give a two-dimensional projection of the cantilevered nanowire.
It is critical to have accurate length measurement:
A parallax method was used to reconstruct the correct three-dimensional representation of the nanowire based on two SEM images taken from different angles.
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Boron Nanowire: Length Determination (con’t)
Top view and 45o tilted view of a nanowire 3-D reconstruction result
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Boron Nanowire: Oxide Layer
( )22
44
2
2
2
2
2
2
)2)((2)(1
8
12
12
DTDDETDEE
L
AIEIE
LAEI
Lf
ooB
oOBn
ooBBnnn
ρρρπβ
ρπβ
ρπβ
+−−+−−=
+==
)1
1())11
(1(1
422 ααρρ
α−+−+== obeam
B
oB EEE
ρπβ
162 2
2Beamn
n
ELD
f = 22
4
4
264n
n
Bbeam f
DL
Eβ
ρπ=
Without considering oxide layers:
Considering oxide layers:
TD
EBEODefine: α=(D-2T)/D
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Boron Nanowire: Results
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Boron Nanowire: Curved Wire
In-plane vibration Out-of-plane vibration
A curved circular cross-section cantilevered beam can vibrate in two perpendicular directions: (1) in plane and (2) out of plane.
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FEA AnalysisThe simple beam theory is based on the assumption that the beam deflection is due to bending only and that transverse shear, rotatory inertia, and axial extension effects are negligible; for curved beams these assumptions are not correct. Modal analysis was performed on several
curved cantilever nanowires with ANSYS. The FEA model was based on the 3-D reconstruction of the nanowire configuration.
Length
(µµµµm)
Diameter
(nm)
Frequency
(kHz)
Modulus assume straight(GPa)
ModulusFEA
modeling(GPa)
16.5 ���� 0.1 75 ���� 2 346.4 (out)378.7 (in)
198 ���� 14237 ���� 17
179.3204.9
7.9 ���� 0.1 70 ���� 2 1295 (in) 168 ���� 19 152
18.3 ���� 0.1 116 ���� 2 440.3 (in) 203 ���� 14 191.4
25.5 ���� 0.1 78 ���� 2 102.1 (out)105.9 (in)
91 ���� 898 ���� 9
86.691.9
7.8 ���� 0.1 64 ���� 2 1332.3 (in) 205 ���� 25 194.2
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Summary1. Mechanical Resonance method based on simple beam
theory, and some work has been done with FEA
2. There are two commonly used methods to excite the mechanical resonance of cantilevered nanostructures: electrical excitation and mechanical excitation.
3. It is critical to have accurate geometry measurement.
4. Mechanical resonance method is an nondestructive and effective way to determine the elastic modulus of nanostructures; one aspect deserving careful scrutiny in the future is the low values for the modulus often obtained compared to the bulk material.