SCEJ 88th Annual Meeting
Mar. 17, 2023

Dynamic modulus reflecting aggregation statein aqueous slurries○ 辰巳 怜小池 修
山口 由岐夫
辻 佳子（東大環安セ）（PIA）（PIA）（東大環安セ/東大院工）2

Stress relaxation

Shear deformation𝑙𝑙
𝛾=𝐿𝐿

𝛾

𝜎

Fluid

𝛾!

Viscoelastic body
Elastic body
𝑡𝑡

Fluid
0

Viscoelastic body
Relaxation time

Elastic body∞3𝜔Oscillation

Oscillatory shear
𝜔>0SAOSLinearRheometer𝛾! ≪ 1

Oscillatory shear flow

LAOS
Nonlinear𝛾! ≳ 1𝛾!

Steady shear 𝜔 = 0

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

Stress: 𝜎! sin 𝜔𝑡 + 𝛿 = 𝛾! 𝐺 " sin 𝜔𝑡 + 𝐺 "" cos 𝜔𝑡
Storage modulus: 𝐺 " Loss modulus: 𝐺 ""StressStrain

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Dynamic modulus [Pa]Cathode slurry of LiB (LiCoO2 + Acetylene black + NMP)

𝐺 !!

𝐺!

Komoda et al., JCIS Open 5 100038 (2022).

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Functional materials

Suspensions

Dispersing

Coating

Food Cosmetics
Medicine• Dispersion/Aggregation

Paint

DryingElectrodeCeramics• Coating properties• Structural change by shear flow

SAOS (𝛾! ≪ 1)
Linear viscoelasticity

LAOS (𝛾! ≳ 1)
Nonlinear viscoelasticity

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

6u

Performing numerical simulations of particle dynamics
under oscillatory shear flow to calculate linear viscoelasticityuInvestigating the relationship between structure andlinear viscoelasticity (dynamic modulus) by changing
DLVO interactions between particles- - - + + - + ++ +- + + - - ++- -DLVO potential

InteractionStructure

Dynamic modulus

7𝑀𝑽̇ = 𝑭3 + 𝑭4
Fluid Inter-particle• Contact force: 𝑭"#$• DLVO force: 𝑭%&'(- ++ + ++++ + - - + +- -ℎElectric double layer
repulsionℎVan der Waals attraction→ Dispersion / Aggregation

8𝑀𝑽̇ = 𝑭3 + 𝑭4
Numerical simulations in 2 steps
1. Structure formation： 𝑭, = −𝜁𝑽 + 𝑭*
Brownian motion• Drag: −𝜁𝑽 (Stokes’ law)

• Fluctuations: 𝐹)* 𝑡 ~𝑁 0, 2𝜁𝑘+𝑇Δ𝑡 (Gaussian noise)
2. Rheological evaluation：𝑭, = −𝜁(𝑽 − 𝑽-. )• Oscillatory shear flow
𝑽-. = 𝛾̇ 𝑡 𝑦 𝒆/ 𝛾̇ 𝑡 = 𝛾!𝜔 cos 𝜔𝑡𝑦𝑧𝑥

• Boundary conditions: Periodic (𝑥, 𝑧), Lees-Edwards (𝑦)9Stress15
𝜎 = 𝜎0 + 𝜎1 = 𝜂0 𝜙 𝛾̇ − E 𝐹24/ 𝑅24
𝑉234
FluidParticle𝑖𝑭24
𝑗

𝑹24 = 𝑹2 − 𝑹45𝜂0 𝜙 = 𝜂! 1 + 𝜙 (Einstein’s eq.)
2
Dynamic modulus• Storage modulus• Loss modulus

78/:
𝜔L
𝐺6 𝜔 =𝜎 𝑡 sin 𝜔𝑡 d𝑡 = 𝐺16
𝜋𝛾! !

78/:
𝜔𝐺 66 𝜔 =L
𝜎 𝑡 cos 𝜔𝑡 d𝑡 = 𝜂0 𝜙 𝜔 + 𝐺166
𝜋𝛾! !

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Step-1: Structure formation
Particles• Diameter: 𝑑 = 100 nm• Concentration: 45 vol%• Zeta potential: 0, −20, −28 mV
Fluid: Water• Ion concentration : 3.8 mM
DLVO potential- + + - ++ ++- + + -- ++- -ℎ𝑦𝑧𝑥 Side length: 10.5𝑑

110Contact number

Zeta potential /mV
Average contact number

06.8

12-20
3.7-280.0

12Step-2: Rheological evaluation
Shear flow
• Strain: 𝛾! = 1×10;7• Frequency:𝜔𝜏 = 3×10;< − 10<𝑦𝑧Relaxation time𝑑 3𝜋𝜂𝑑7𝜏= =← 3𝜋𝜂𝑑𝑈 = 𝐹𝑈𝐹
Drag force = Adhesive force𝑥Side length: 10.5𝑑

Shear rate: 𝛾̇ 𝑡 = 𝛾!𝜔 cos 𝜔𝑡𝛾!𝜔
0−𝛾!𝜔

𝑡

130 mV𝜔𝜏 = 1.2

Contact number

12Linear region
0Strain amplitude

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Dominated by fluid contribution𝐺′𝐺′′

𝜂0 𝜙 𝜔Frequency
Structural strength at rest
Structure maintained by attraction (0, −20 mV) / repulsion (−28 mV)

15tan𝛿 = 𝐺 55 /𝐺′

Coating fluidity:
Viscosity at short time

Viscous

Elastic
Shape retention:
Elasticity at long time

Frequency

16• Numerical simulations show dynamic modulus reflecting structure
formed by DLVO potentials• Non-monotonic dependence of dynamic modulus on the degree of
aggregation at low frequency• The non-monotonic dependence is to be noted in rheological control- - - - ++ + - ++ +- ++ - -- ++- - -Interaction

Structure

Dynamic modulus