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

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

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!

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′′

Laser

Camera Computer

Sample

Acoustic emitter

Laser speckle visibility acoustic spectroscopy

𝒙𝟏

𝒙𝟐 𝒙𝟑

Displacement 𝒙𝟐

𝒙𝟏

Wintzenrieth et al. 2013. PRE. (Submitted)

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

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

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)

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)

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

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

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

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

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

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

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!

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

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.)

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

Questions?

20