probing foam acoustics with coherent lightmesoimage.grenoble.cnrs.fr/img/pdf/wintzenrieth13.pdf ·...
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Light and sound in bubble polycrystals
F. Wintzenrieth, S. Cohen-Addad, M. Le Merrer & R. Höhler
GdR MesoImage December 2013
Probing foam acoustics with coherent light
Institut des Nanosciences de Paris Université Pierre et Marie Curie
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Liquid foam structure and elasticity Shaving foam.
𝑆ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠
𝐺 = 2.8𝛾
𝑑φ φ − 0.64
𝐵𝑢𝑙𝑘 𝑚𝑜𝑑𝑢𝑙𝑢𝑠
𝐵 =1
𝜒𝜑
𝜑 𝑔𝑎𝑠 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝜒 𝑔𝑎𝑠 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦 𝛾 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑡𝑒𝑛𝑠𝑖𝑜𝑛
100 µm
d
𝜑 = 0.9, 𝑅 = 90 µ𝑚
𝐵
𝐺≈105 𝑃𝑎
102Pa ≈ 103 ≫ 1
Cohen-Addad et al. 2013. Annual Review of Fluid Mechanics
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Pentamode materials
3
𝑐11 𝑐12 𝑐12𝑐12 𝑐11 𝑐12𝑐12 𝑐12 𝑐11
0
0𝑐44 0 𝑐44 0 𝑐44 ℬ
3𝑐11 0 0
0
00 0 0 ℬ′
𝑐11 ≫ 𝑐44
𝑐12 = 𝑐11 − 2𝑐44
𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑖𝑠𝑎𝑡𝑖𝑜𝑛
Pentamode: 5 zero eigenvalues
Kadic et al. 2012. APL
𝑐11 𝑐44 = 𝐵 𝐺 ~103
Could foam be used as a self-assembled anisotropic pentamode material?
Milton & Cherkaev. 1995. J. of Eng. Mat. and Tech.
Stiffness tensor c
100 µm
3D pentamode materials with anisotropic acoustic properties would have many applications (lenses, cloaking…)
Polymer structure obtained by laser lithography
Low acoustic attenuation is required!
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Viscoelasticity and shear wave propagation
𝑘∗ = 𝜔𝜌
G∗
Ferry et al. 1947. Journal of Polymer Science
4 High frequency dispersion relation?
Liu et al. 1996. PRL Tighe. 2011. PRL
𝐺∗ 𝑓 = 𝐺 1 +𝑖𝑓
𝑓𝑐+ 2𝑖𝜋𝜂∞𝑓
Krishan et al. PRE 2010
Mechanical measurements for aqueous foam
𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝑤𝑎𝑣𝑒𝑛𝑢𝑚𝑏𝑒𝑟 𝑘∗ =2𝜋
𝜆+ 𝑖1
𝑙𝐴
𝐶𝑜𝑚𝑝𝑙𝑒𝑥 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝐺∗ = 𝐺′ + 𝑖𝐺′′
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Laser
Camera Computer
Sample
Acoustic emitter
Laser speckle visibility acoustic spectroscopy
𝒙𝟏
𝒙𝟐 𝒙𝟑
Displacement 𝒙𝟐
𝒙𝟏
Wintzenrieth et al. 2013. PRE. (Submitted)
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Durian & Bandyopadhyay, RSI 2005
CA
MER
A
𝑆𝑝𝑒𝑐𝑘𝑙𝑒 𝑣𝑖𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦
𝑉 =1
𝛽
Ι2 − Ι 2
Ι 2
Speckle visibility
6
Electric field autocorrelation:
𝑔1 𝑡, 𝜏 =𝐸 𝑡 + 𝜏 𝐸∗ (𝑡)
𝐸(𝑡) 2
𝑉 𝑇, 𝑡 = 2 1 −𝜏
𝑇𝑔1 𝑡, 𝜏
𝑑𝜏
𝑇
𝑇
0
Interfering light paths
LASER
𝑇 𝑒𝑥𝑝𝑜𝑠𝑢𝑟𝑒 𝑡𝑖𝑚𝑒
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Principle of wavelength measurement
𝑥1
𝑆𝑡𝑟𝑎𝑖𝑛 𝜀21
𝑥1
𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑥2
𝑥1
𝑆𝑝𝑒𝑐𝑘𝑙𝑒 𝑣𝑖𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦 𝑉
𝝀/𝟐
1
𝑆𝑛𝑎𝑝𝑠ℎ𝑜𝑡 𝑎𝑡 𝑖𝑛𝑠𝑡𝑎𝑛𝑡 𝑡
7
𝜔Τ ≪ 2𝜋,
𝑉 = 1 −4𝜔𝑇𝜀0𝛾𝜅𝑙
∗
3 10sin 𝑘𝑥1 − 𝜔𝑡
𝑔1 𝑡, 𝜏 = 𝑒−𝛾𝜅𝑙∗ 2 𝑇𝑟 ∆𝜀(𝜏)2 /10
𝜀 𝑠𝑡𝑟𝑎𝑖𝑛 𝑡𝑒𝑛𝑠𝑜𝑟 𝑙∗ 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑚𝑒𝑎𝑛 𝑓𝑟𝑒𝑒 𝑝𝑎𝑡ℎ 𝜅 𝑙𝑎𝑠𝑒𝑟 𝑙𝑖𝑔ℎ𝑡 𝑤𝑎𝑣𝑒𝑛𝑢𝑚𝑏𝑒𝑟
𝐹𝑜𝑟 𝑎 𝑠ℎ𝑒𝑎𝑟 𝑠𝑡𝑟𝑎𝑖𝑛 𝜀21 = 𝜀0cos (𝜔𝑡 − 𝑘𝑥1)
Erpelding et al. 2010. Phys. Rev. E Bicout et al. 1991. J. de Physique Wu et al. 1990. J. Opt. Soc. Am. B
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Time 𝑡
20
mm
𝑥1
10 ms
𝒙𝟏 = 𝒗𝒕 𝑓 = 100 𝐻𝑧
8
Spatio-temporal visibility diagram
𝒗 = 𝟑. 𝟕 𝒎. 𝒔−𝟏 𝜆 = 37 𝑚𝑚
𝛽𝑉
0.02
0.26
Consistent with previous mechanical measurements. (Krishan et al. PRE 2010)
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Measurement of the attenuation length 𝑙𝐴
𝜔Τ > 2𝜋,
V ≅ 1 − 𝛾 𝜅 𝑙 𝜀0 𝑒−𝑥1/𝑙𝐴
9
𝑥1
𝑥1
𝑥
𝑥1
𝑆𝑝𝑒𝑐𝑘𝑙𝑒 𝑣𝑖𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦
𝑇𝑖𝑚𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒
1
0
𝑆𝑡𝑟𝑎𝑖𝑛 𝜀12 𝐹𝑜𝑟 𝑎𝑛 𝑎𝑡𝑡𝑒𝑛𝑢𝑎𝑡𝑒𝑑 𝑠𝑡𝑟𝑎𝑖𝑛
𝜀12 = 𝜀0𝑒
−𝑥1/𝑙𝐴 cos (𝜔𝑡 − 𝑘𝑥1)
Wintzenrieth, Cohen-Addad, Le Merrer & Höhler.
2013. PRE. (Submitted)
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0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
= 1.00 = 0.71 = 0.50 = 0.35
Sp
eckle
vis
ibili
ty
Propagation distance (mm)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30 35 40
Sp
eckle
vis
ibili
ty
Shifted propagation distance (mm)
10
Visibility evolution with propagation distance and displacement amplitude
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑢 𝑥1 = 𝜉 𝑢0 𝑒−𝑥1 𝑙𝐴 cos (𝜔𝑡 − 𝑘𝑥1) = 𝑢0𝑒
−(𝑥1−𝑙𝐴 ln 𝜉 ) 𝑙𝐴 cos (𝜔𝑡 − 𝑘𝑥1)
𝑓 = 100 𝐻𝑧, 𝑙𝐴 = 11.8 𝑚𝑚
Shift distance
Master plot
Consistent with previous mechanical measurements. (Krishan et al. PRE 2010)
𝑢0 = 3.2 µm 𝑢0 = 3.2 µm
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Experimental results
11
100
101
102
103
104
1 10 100 1000
Atte
nu
atio
n le
ng
th l
A (m
m)
Frequency f (Hz)
* Liu et al. 1996. PRL
Mechanical measurements LSVAS
Viscoelastic response of this foam is well described by Liu’s model up to 1 kHz.
Our results validate the LSVAS technique. Can we elaborate foams that are anisotropic, stable and less attenuating?
Predictions*
Mechanical measurements LSVAS
100 µm
d
100
101
102
103
104
1 10 100 1000
Wlg
th 4
5 µ
m (
mm
)
Frequency f (Hz)
Wave
len
gth
(
mm
)
Bubble
diameter (µm)
45
62
75
95
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Confined gelatine crystalline foams are stable for many days
12
5 mm
𝜑 = 0.8𝑑 = 600 µ𝑚𝐺𝑔𝑒𝑙 ≅ 10 𝑘𝑃𝑎
X-ray tomography. (Collaboration: Ovarlez, Lenoir (IFSTAR))
FOAMING GELATINE SOLUTION
GAS MIX (N2, C6F14)
MILLIFLUIDIC GENERATOR
2 mm
106 1 mm 𝑡 (𝑠)
𝐺𝑔𝑒𝑙 (kPa)
10
20
103
GELLIFIED FOAM
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Longitudinal modes in a cylindrical wave guide
𝑢
Displacement profile
𝑘
𝜔
Evanescent regime
𝑘
Propagative regime
Fixed boundary conditions
λ/2
Transducer
𝜔
𝑘= 𝑣𝐿
2R
𝑖𝑘
2.4𝑣𝑇𝑅
5.5𝑣𝑇𝑅
𝒙
𝒛
𝑢𝑧 𝑢𝑥
𝑢𝑧 𝑢𝑥
𝑅
𝑅 𝑅
𝑅
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50100 100 200 300Wavevector 1 m
100
300
500
700
800
1000
1200
Frequency Hz
Predicted dispersion relation for a homogeneous elastic medium
𝑣𝐿 ≫ 𝑣𝑇 , 𝑘2 J1[𝑅𝜔 𝑣𝑇 ] +1
2
𝑅𝜔
𝑣𝑇
𝜔2
𝑣𝐿2− 𝑘2 J0[𝑅𝜔 𝑣𝑇 ] = 0
J1[𝑅𝜔 𝑣𝑇 ] = 0, 𝑘 =𝜔
𝑣𝐿,
𝐵𝑢𝑙𝑘 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑤𝑎𝑣𝑒
J0[𝑅𝜔 𝑣𝑇 ] = 0 𝐸𝑣𝑎𝑛𝑒𝑠𝑐𝑒𝑛𝑡 𝑤𝑎𝑣𝑒
𝑣𝑃 = 27 𝑚/𝑠 𝑣𝑇 = 1.3 𝑚/𝑠
J0, J1 𝐵𝑒𝑠𝑠𝑒𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠
𝑅 = 2 𝑚𝑚
𝜔
𝑘= 𝑣𝐿
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Acoustic pulse propagation in space and time
𝑇𝑖𝑚𝑒 𝑡
𝒙𝟏 = 𝒗 𝒕
𝑓 = 300 𝐻𝑧
𝒙𝟏 = 𝒗𝒈 𝒕
𝑥1
𝑃ℎ𝑎𝑠𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣 = 𝜔 𝑘
𝐺𝑟𝑜𝑢𝑝 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣𝑔 = 𝑑𝜔 𝑑𝑘
Excitation
𝑡
𝑢
50
mm
50 ms
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Dispersion relation in bubble polycrystals
Reduced variables Ω =𝑅𝜔
𝑣𝑇 K = 𝑅𝑘
𝑣𝐿𝑣𝑇, K2 𝐽1 Ω +
1
2Ω Ω2 − K2 𝐽0[Ω] = 0
𝑣𝐿 =𝐵𝑔𝑎𝑠
1 − 𝜑 𝜑 𝜌𝑔𝑒𝑙
𝑣𝑇 =1 − 𝜑 𝐺𝑔𝑒𝑙
𝜌𝑔𝑒𝑙
𝐶𝑢𝑡𝑜𝑓𝑓 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 ω𝐶 𝑦𝑖𝑒𝑙𝑑𝑠 𝑣𝑇
1.0 𝑚/𝑠 ≤ 𝑣𝑇 ≤ 1.8 𝑚/𝑠
Cellular solids. Ashby & Gibsons. 1999.
𝐹𝑖𝑡𝑡𝑒𝑑 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑣𝐿
27 𝑚/𝑠 ≤ 𝑣𝐿 ≤ 33 𝑚/𝑠
0
3
6
9
12
0 2 4 6 8 10 12 14 16
Predictions
Measurements
Red
uce
d a
ng
ula
r fr
eq
ue
ncy
Reduced wavenumber
2 𝑚𝑚 ≤ 𝑅 ≤ 5 𝑚𝑚
𝛺
𝐾= 1
Ω𝐶
𝜆𝐿 > 𝜆𝑇 > 𝑑
A Textbook of Sound. Wood. 1944.
Consistent with Wood’s model!
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0.75 0.80 0.85 0.90 0.95
102
103
104
How can pentamode behaviour be optimized? 𝑃𝑒𝑛𝑡𝑎𝑚𝑜𝑑𝑒 𝑟𝑎𝑡𝑖𝑜:
𝐵𝑓𝑜𝑎𝑚
𝐺𝑓𝑜𝑎𝑚=
𝑣𝐿𝑣𝑇
2
=𝐵𝑔𝑎𝑠
1 − 𝜑 2 𝜑 𝐺𝑔𝑒𝑙
𝐺𝑔𝑒𝑙 = 5 𝑘𝑃𝑎
𝐺𝑔𝑒𝑙 = 10 𝑘𝑃𝑎
𝐺𝑔𝑒𝑙 = 20 𝑘𝑃𝑎
𝐺𝑎𝑠 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝜑
𝐵𝑓𝑜𝑎𝑚
𝐺𝑓𝑜𝑎𝑚
𝑆𝑎𝑚𝑝𝑙𝑒𝑠
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7 10-1
8 10-1
9 10-1
100
0 40 80 120 160
Deviation from continuum model at higher frequencies
VIS
IBIL
ITY
𝑇𝑖𝑚𝑒 𝑡
𝑥1
0
500
1000
1500
2000
2500
3000
3500
0 50 100 150 200 250
Pulse results
Continuous results
Fre
qu
ency (
Hz)
Wavenumber (/m)
𝑓 = 450 𝐻𝑧
𝑓 = 3250 𝐻𝑧
Pulse excitation:
𝑣𝐴𝑖𝑟
Abscissa (mm)
Continuous excitation: standing waves (Collaboration:
A. Spadoni. EPFL.)
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Conclusions
• Laser visibility acoustic spectroscopy is a new method for measuring acoustic dispersion relations in soft turbid materials
• Gelatine foams behave as self-assembled pentamode effective materials in the kHz frequency range
• What is the origin of non-linear dispersion at higher frequencies in crystalline gellified foams?
• Can these foams be made anisotropic?
19
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Questions?
20