SCEJ 54th Autumn Meeting
Sept. 12, 2023

Viscoelastic analysis of structural change
in slurry during mixing
混練によるスラリーの構造変化の粘弾性解析○ 辰巳 怜小池 修
山口 由岐夫
辻󠄀 佳子

（PIA）
（PIA）
（PIA）
（東大環安セ/東大院工）Material fabrication from suspensions

Dispersing

CoatingFood Cosmetics PaintMedicine• Dispersion/Aggregation

DryingElectrodeCeramics• Coating properties• Structural change by shear flow

Rheological properties: Viscosity, Viscoelasticity

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Rheological properties(A) Rotation(B) Oscillation(A) Steady shear𝜎Viscosity 𝜂 =𝛾ሶ

Rheometer

Shear rate 𝛾ሶ

Stress 𝜎(B) Oscillatory shear
𝛾 𝑡 = 𝛾0 sin 𝜔𝑡𝛾ሶ 𝑡 = 𝛾0 𝜔 cos 𝜔𝑡
𝜎 𝑡 = 𝛾0 𝐺 ′ sin 𝜔𝑡 + 𝐺 ′′ cos 𝜔𝑡

Shear rate 𝛾ሶ
Loss modulus 𝐺 ′′Storage modulus 𝐺 ′Frequency 𝜔4

Shear thinning/thickening
AggregatedViscosity

Agglomerated

Kneaded

Dispersed

Shear rate
山口 由岐夫 「ものづくりの化学工学」，丸善 （2015）.

5Structure estimation from viscoelasticity
Preparation process(1) In-whole (2) In-parts(2)

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

Structure(3) (2) + high shear
Komoda et al., JCIS Open 5 100038 (2022).

′′
𝐺𝐺𝐺𝐺′′′′

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Objective & Method
Suspensions• Particles• Solvent• Additives

Rheometer?

Viscosity

Structure

Modeling

Numerical simulation

Shear rate
𝐺 ′′𝐺′

Interaction

Structure

Frequency

Relationship?
Analysis• Contact number• Pair distribution function, Etc.

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Objective & Method
Viscoelastic analysis of structural change under shear flow
Numerical simulation
Step-1: Steady shear

Viscosity

Shear rate

Step-2: Oscillatory shear

𝐺 ′ , 𝐺 ′′ vs. 𝜔8Motion of particles
H𝑚𝒗ሶ = 𝑭 + 𝑭Fluidcnt+𝑭DLVO

Inter-particle• Drag: 𝑭H = −𝜁(𝒗 − 𝑽ex ) (Stokes’ law)𝛾ሶ 𝑡 = 𝛾ሶ 0 cos 𝜔𝑡• DLVO force: 𝑭DLVO- ++ + ++++ + - - + +- -• Shear flow
𝑽ex = 𝛾ሶ 𝑡 𝑦 𝒆𝑥

• Contact force: 𝑭cnt

ℎ𝑦𝑧𝑥

• Boundary conditions:
Periodic (𝑥, 𝑧), Lees-Edwards (𝑦)Electric double layer repulsionℎVan der Waals attraction

Rheological evaluation
Viscosity (Steady shear)Stress𝜎 = 𝜎f + 𝜎p1𝑦= 𝜂f 𝜙 𝛾ሶ − ෍ 𝐹𝑖𝑗𝑥 𝑟𝑖𝑗𝑉𝜎𝜂 = = 𝜂f 𝜙 + 𝜂p𝛾ሶ𝑖<𝑗Fluid𝑖

Particle𝑭𝑖𝑗𝑗𝒓𝑖𝑗 = 𝒓𝑖 − 𝒓𝑗
𝜂f 𝜙 = 𝜂051+ 𝜙
2

Dynamic modulus (Oscillatory shear)
Storage modulus
2𝜋/𝜔𝜔𝐺′ 𝜔 =
න𝜎 𝑡 sin 𝜔𝑡 d𝑡 = 𝐺p′𝜋𝛾0 0Loss modulus
2𝜋/𝜔𝜔
𝐺 ′′ 𝜔 =න𝜎 𝑡 cos 𝜔𝑡 d𝑡 = 𝜂f 𝜙 𝜔 + 𝐺p′′
𝜋𝛾0 0

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10Simulation conditions
Particles• Diameter: 𝑑 = 100 nm• Concentration: 45 vol%• Zeta potential: −28 mV

DLVO potential- + + - ++ ++- + + -- ++- -ℎSolvent: Water• Ion concentration: 3.8 mM𝑦𝑧𝑥 Side length: 10.5𝑑

Step-1: Steady shear

Step-2: Oscillatory shear• Shear rate :𝛾𝜏ሶ = 10−3 − 10

• Strain: 𝛾0 = 1 × 10−2
• Frequency:𝜔𝜏 = 3 × 10−3 − 103

Structural change

Viscoelastic analysis

Step-1: Structural change under shear flow
Characteristic time𝑑 3𝜋𝜂𝑑2 ← 3𝜋𝜂𝑑𝑈 = 𝐹
𝜏= =Drag = Adhesion force
𝑈𝐹(a) 𝛾𝜏
ሶ = 0.030(b) 𝛾𝜏ሶ = 0.06

Contact number

9(c) 𝛾𝜏ሶ =1Shear rate

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Viscosity

Dimensionless shear rate

3𝜋𝜂𝛾𝑑ሶ 2𝛾𝜏
ሶ =𝐹(Shear force)=(Adhision force)

Shear rate

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Structure

Dimensionless shear rate

3𝜋𝜂𝛾𝑑ሶ 2𝛾𝜏
ሶ =𝐹(Shear force)=(Adhision force)

Contact number09

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Step-2: Viscoelasticity
(a)

𝐺′𝐺′′
Relaxation(b)(c)

𝜂f 𝜙 𝜔

Structural strength at rest (𝜔𝜏 → 0)
Structure maintained by repulsion (a) / attraction (b, c)Frequency

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Relaxation time𝑚𝒓ሷ 𝑖 = −𝜁 𝒓ሶ 𝑖 + 𝑭𝑖Strain𝑖 = 1, … , 𝑁
Fluid Inter-particle

Relaxation

Ignoring inertia term
Linearization𝒓ሶ = 𝜁 −1 𝑯 ⋅ 𝒓 − 𝒓0
𝒓=𝒓1𝒓2⋮𝒓𝑁

𝑯=𝑯11𝑯21⋮𝑯𝑁1

Equilibrium state: 𝒓0𝑯12𝑯22⋱⋯⋯ 𝑯1𝑁⋱⋮⋱⋮⋯ 𝑯𝑁𝑁𝜕𝑭𝑖𝑯𝑖𝑗 = −𝜕𝒓𝑗

𝑭𝑖 = ෍ 𝑭𝑖𝑗𝑗≠𝑖Hessian matrix: 𝑯Diagonalization

1𝑞ሶ 𝑘 = − 𝑞𝑘𝜏𝑘𝑘 = 1, … , 3𝑁

𝑞𝑘 = 𝒖𝑘 ⋅ 𝒓

Relaxation time: 𝜏𝑘Eigenvector: 𝒖𝑘

Distribution of relaxation frequency

16(a)Relaxation frequency

𝜔𝑘 = 𝜏𝑘−1(b)Distribution of 𝐥𝐨𝐠 𝝎𝒌 : 𝐻 𝜔(Probability density function)(c)

𝐻 𝜔 dlog𝜔 = 𝐷 𝜔 d𝜔1
𝐷 𝜔 =෍ 𝛿 𝜔 − 𝜔𝑘
3𝑁 − 3 𝑘

Relaxation by repulsion / attraction

Distribution of relaxation frequency

17(a)Relaxation frequency

𝜔𝑘 = 𝜏𝑘−1(b)Distribution of 𝐥𝐨𝐠 𝝎𝒌 : 𝐻 𝜔(Probability density function)(c)

𝐻 𝜔 dlog𝜔 = 𝐷 𝜔 d𝜔1
𝐷 𝜔 =෍ 𝛿 𝜔 − 𝜔𝑘
3𝑁 − 3 𝑘

Relaxation by repulsion / attraction

Eigenmodes of particles’ motion

18(a)Participation ratio(b)

σ𝑖 𝒖𝑘 ⋅ 𝒓𝑖 2 2
𝑃𝑘 =𝑁 σ𝑖 𝒖𝑘 ⋅ 𝒓𝑖 4
Ratio of particles participating
in an eigenmode of motion(c)Relaxation frequency:
Repulsion < Attraction
Extent of collective motion:
Repulsion > Attraction

19Summary
Viscoelastic
analysis

Viscosity

Shear rate

Viscoelastic
behavior

Structure

Structure
analysis

Relaxation
frequency

Viscoelastic behavior may show inter-particle interactions determining structure
Knowledge how to prepare slurry to control its viscoelasticity