SCEJ 53rd Autumn Meeting
Sept. 14, 2022

Structural change in colloidal suspensions
indicated by nonlinear viscoelasticity○ 辰巳 怜（東大環安セ）小池 修（PIA）
山口 由岐夫 （PIA）
辻 佳子（東大環安セ/東大院工）2
𝜔
Oscillation

Oscillatory shear
𝜔>0SAOSLinear

𝛾0 ≪ 1LAOSNonlinear

𝛾0 ≳ 1𝛾0Rheometer

Oscillatory shear flow

Steady shear 𝜔 = 0

SAOS (Small Amplitude Oscillatory Shear; 𝛾0 ≪ 1) → Linear response
Strain: 𝛾0 sin 𝜔𝑡
Shear rate: 𝛾0 𝜔 cos 𝜔𝑡

Stress: 𝜎0 sin 𝜔𝑡 + 𝛿 = 𝛾0 𝐺 ′ sin 𝜔𝑡 + 𝐺 ′′ cos 𝜔𝑡Storage modulus: 𝐺 ′ Loss modulus: 𝐺 ′′StressStrain

3Slurry (LiCoO2 + acetylene black)

Electrode𝐺′𝐺 ′′Discharge capacity

Kuratani et al., J. Electrochem. Soc. 166 A501 (2019).

4Nonlinear region

Hyun et al., Prog. Polym. Sci. 36, 1697 (2011).

LAOS (Large Amplitude Oscillatory Shear; 𝛾0 ≳ 1)
→ Nonlinear response𝜔Oscillatory shear
𝜔>0𝜎(𝑡) = 𝛾0 ෍ 𝐺𝑘′ sin 𝑘𝜔𝑡 + 𝐺𝑘′′ cos 𝑘𝜔𝑡𝑘: odd

𝜔 2𝜋/𝜔𝐺𝑘′=
න𝜎 𝑡𝐺𝑘′′𝜋𝛾0 0sin 𝑘𝜔𝑡d𝑡cos 𝑘𝜔𝑡

Fourier expansion → Higher harmonic contributions
Related to intracycle variation in viscoelasticity

SAOSLinear

𝛾0 ≪ 1LAOS
Nonlinear𝛾0 ≳ 1

𝛾0Steady shear 𝜔 = 05Functional materials

Suspensions

Dispersing

CoatingFood Cosmetics
Medicine• Dispersion/Aggregation

Paint

DryingElectrodeCeramics• Coating properties• Structural change by shear flow

SAOS (𝛾0 ≪ 1)
Linear viscoelasticity

LAOS (𝛾0 ≳ 1)
Nonlinear viscoelasticity

Viscoelasticity can be a measure to control fabrication processes← Relationship among starting materials, structure, and viscoelasticity

6Structure

Materials

Viscoelasticity
SAOS (𝛾0 = 0.01)

0 mV

Solvent: waterParticles• Diameter: 1 μm
• Concentration:
45 vol%• Zeta potential:
0 mV, 20 mV

0 mV
20 mV

- + - - + ++ + -- + + - + -ℎ + + - -20 mV

0Contact number

12

Tatsumi et al., SCEJ 87th Annual meeting (2022).

7Performing numerical simulations of particle dynamics underoscillatory shear flow to calculate nonlinear viscoelasticity

Investigating the relationship between structural change
and the nonlinear viscoelasticity of aggregated suspensions′𝐺L′ − 𝐺M
𝑆=𝐺L′′𝜂L′ − 𝜂M
𝑇=
𝜂L′Structure

Nonlinear viscoelasticity

8Stress

Linear (𝛾0 ≪ 1)

Strain′𝐺M=d𝜎
Minimum-straind𝛾 𝛾=0 modulus

𝐺L′ =𝜎𝛾𝛾 =𝛾0Large-strain
modulus

Ewoldt et al. J. Rheol. 52, 1427 (2008).

Nonlinear (𝛾0 ≳ 1)
Intracycle change in elasticity′𝐺L′ − 𝐺M
𝑆=𝐺L′

𝑆 > 0: strain stiffening
𝑆 < 0: strain softening

9Stress

Shear rate

Linear (𝛾0 ≪ 1)′𝜂M=d𝜎
Minimum-rate
viscosityd𝛾ሶ 𝛾=0ሶ𝜂L′ =𝜎𝛾ሶ𝛾ሶ =𝛾ሶ 0

Large-rate
viscosity

Ewoldt et al. J. Rheol. 52, 1427 (2008).

Nonlinear (𝛾0 ≳ 1)
Intracycle change in viscosity′𝜂L′ − 𝜂M
𝑇=𝜂L′𝑇 > 0: shear thickening
𝑇 < 0: shear thinning10Fourier expansion of stress response𝜎(𝑡) = 𝛾0 ෍ 𝐺𝑘′ sin 𝑘𝜔𝑡 + 𝐺𝑘′′ cos 𝑘𝜔𝑡
𝑘: odd′𝐺L′ − 𝐺M
−4𝐺3′ +⋅⋅⋅
𝑆== ′′𝐺L
𝐺1 − 𝐺3′ +⋅⋅⋅𝐺3′ < 0: strain stiffening′𝜂L′ − 𝜂M
4𝐺3′′ +⋅⋅⋅
𝑇== ′′′
𝜂L
𝐺1 + 𝐺3′′ +⋅⋅⋅

𝐺3′′ > 0: shear thickening

𝑆 & 𝑇 can be calculated from {𝐺𝑘′ } & {𝐺𝑘′′ }11𝑀𝑽ሶ = −𝜁(𝑽 − 𝑽ex ) + 𝑭cnt + 𝑭DLVO
Interparticle

Fluid• Hydrodynamic drag: −𝜁(𝑽 − 𝑽ex )Oscillatory shear flow
𝑽ex = 𝛾ሶ 𝑡 𝑦 𝒆𝑥
• Contact force: 𝑭cnt𝛾ሶ 𝑡 = 𝛾0 𝜔 cos 𝜔𝑡𝑦𝑧𝑥• DLVO force: 𝑭DLVO- + + - ++ ++- + + -- ++- -ℎ• Boundary conditions
𝑥, 𝑧: Periodic, 𝑦: Lees-Edwards

Electric double layer
repulsionℎVan der Waals attraction→ Dispersion / Aggregation

12
𝑭𝑖𝑗 Interparticle force𝑖Stress1𝑦𝜎 𝑡 = − ෍ 𝐹𝑖𝑗𝑥 𝑅𝑖𝑗𝑉𝑖<𝑗𝑗

𝑹𝑖𝑗 = 𝑹𝑖 − 𝑹𝑗= 𝛾0 ෍ 𝐺𝑘′ sin 𝑘𝜔𝑡 + 𝐺𝑘′′ cos 𝑘𝜔𝑡

𝑦𝑘: odd𝑧𝑥

Intracycle change in elasticity/viscosity′𝐺L′ − 𝐺M−4𝐺3′ +⋅⋅⋅
𝑆== ′′𝐺L
𝐺1 − 𝐺3′ +⋅⋅⋅′
𝜂L′ − 𝜂M
4𝐺3′′ +⋅⋅⋅
𝑇== ′′′
𝜂L
𝐺1 + 𝐺3′′ +⋅⋅⋅

13Fluid: Water𝑈𝑥 𝑡 = 𝛾ሶ 𝑡 𝑦Particles• Diameter: 𝑑 = 100 nm• Concentration: 45 vol%• Zeta potential: 0 mV

Shear flow
• Strain: 𝛾0 = 1 × 10−3 − 3• Frequency: 𝜔𝜏 = 1.2
𝜏 = 3𝜋𝜂𝑑 2 /𝐹
𝐹 : Attractive force between
contacting particles𝑦𝑥𝑧Side length: 22.7𝑑Shear rate: 𝛾ሶ 𝑡 = 𝛾0 𝜔 cos 𝜔𝑡
𝛾0 𝜔0−𝛾0 𝜔𝑡14Contact number0𝛾0 = 0.01

12𝛾0 = 0.1

𝛾0 = 1

15𝜎/(𝐹/𝑑 2 )𝛾0 = 0.01𝛾ሶ

𝛾

Stress𝛾0 = 0.1

Strain𝛾0 = 1Stress𝛾0 = 3

Shear rate

16
Nonlinear region

Dynamic modulus (𝑘 = 1)

Nonlinear region

Higher harmonic contributions(𝑘 = 3, 5)′′′𝜎(𝑡) = 𝛾0 ෍ 𝐺𝑘 sin 𝑘𝜔𝑡 + 𝐺𝑘 cos 𝑘𝜔𝑡
𝑘: odd17′𝐺L′ − 𝐺M
𝑆=𝐺L′′
𝜂L′ − 𝜂M
𝑇=𝜂L′

𝑆 > 0: strain stiffening𝑆 < 0: strain softening

𝑇 > 0: shear thickening
𝑇 < 0: shear thinning

StressStrain stiffening
StrainShear thinning

StressShear rate

18

Δ𝑁cnt
𝛾0 = 0.01𝛾ሶ
𝜎𝛾

Contact number
ഥcnt + Δ𝑁cnt 𝑡
𝑁
MeanVariation

𝛾0 = 0.1
Nonlinear region
𝛾0 = 1ഥcnt
𝑁
𝛾0 = 3

19
Nonlinear region

Δ𝑁cnt (𝑡) = ෍ 𝐴′𝑘 sin 𝑘𝜔𝑡 + 𝐴′′𝑘 cos 𝑘𝜔𝑡𝑘:even

20

StrainShear rate

Contact number increases with increasing strain
Strain stiffening
Contact number decreases with increasing shear rate
Shear thinning21log 𝑔 𝒓

No-shear conditionRadial distribution function

Distribution function on 𝑥𝑦-plane• Number density of particles around a particle• Normalized by the bulk density of particles

22

𝛾0 = 3(a) (b)

(c)

(a)

(b)

(c)

(d)

(d)

𝛾ሶ
𝜎𝛾

𝑦𝑧

𝑥

Anisotropic structure23𝑦𝑧𝑥𝛾0 = 0.01

𝛾0 = 0.1

𝛾0 = 1𝛾0 = 3

24• Nonlinear viscoelasticity describes change in elastic modulus andviscosity due to intracycle variation in shear strain/rate→ Nonlinear viscoelasticity can be a measure to control
coating operation in fabrication processes• Nonlinear viscoelasticity of aggregated suspensions indicatesstrain stiffening and shear thinning• Relationship between structural change and nonlinear viscoelasticity:• Contact number increases with increasing strain
Strain stiffening• Contact number decreases with increasing shear rate
Shear thinning• Decrease in contact number

Anisotropic structural change