SCEJ 87th Annual Meeting
Mar. 17, 2022

Mechanism of the viscoelastic behavior of slurry○ 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）2Functional materials

SlurryDispersing

CoatingDrying

Viscoelasticity can be an index of process• Structure of particles (aggregation/dispersion)• Coating properties• Influence of dispersing/coating on post-process

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𝜔

Oscillation

Oscillatory shear
LAOS
Nonlinear

SAOS
Linear(Present study)

Rheometer

Oscillatory shear
Strain: 𝛾0 sin 𝜔𝑡
Shear rate: 𝛾0 𝜔 cos 𝜔𝑡
Stress: 𝜎0 sin 𝜔𝑡 + 𝛿= 𝛾0 𝐺 ′ sin 𝜔𝑡 + 𝐺 ′′ cos 𝜔𝑡
Storage modulus: 𝐺 ′Loss modulus: 𝐺 ′′𝛾0𝜔 = 0 Steady shear

StressStrain

4Gelation

谷本守正, ミルクサイエンス 69, 83 (2020).

5

佐野康, 日本写真学会誌 77, 65 (2014).

大坪泰文, 成形加工 22, 392 (2010).

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Coating of silica slurry

Shear rate

StrainKomoda et al., J. Coat. Technol. Res. 19, 61 (2022).

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

Constructing a simple particle-scale model that
describes the viscoelastic behavior of slurryVisualizing the motion of particles that is related tothe viscoelastic behavior

Investigating the effects of interparticle interactions on
the viscoelastic behavior

8𝑀𝑽ሶ = −𝜁(𝑽 − 𝑽ex ) + 𝑭cnt + 𝑭DLVO
InterparticleFluid• Hydrodynamic drag: −𝜁(𝑽 − 𝑽ex )Oscillatory shear flow
𝑽ex = 𝛾ሶ 𝑡 𝑦 𝒆𝑥
• Contact force: 𝑭cnt

𝛾ሶ 𝑡 = 𝛾0 𝜔 cos 𝜔𝑡𝑦𝑧𝑥• DLVO force: 𝑭DLVO Potential- + + - ++ ++- + + -- ++- -ℎ• Boundary conditions
𝑥, 𝑧: Periodic, 𝑦: Lees-Edwards

Electric double layer
repulsionℎVan der Waals
attraction→ Dispersion / Aggregation

9Input
Shear strain 𝛾 𝑡 = 𝛾0 sin 𝜔𝑡𝜔Oscillatory shear

Shear rate 𝛾ሶ 𝑡 = 𝛾0 𝜔 cos 𝜔𝑡Output
Stress 𝜎 𝑡 = 𝜎0 sin 𝜔𝑡 + 𝛿= 𝛾0 𝐺 ′ sin 𝜔𝑡 + 𝐺 ′′ cos 𝜔𝑡
𝜎0𝜎0′′′
𝐺 = cos 𝛿 𝐺 = sin 𝛿
𝛾0𝛾0LAOS
Nonlinear

SAOS
Linear(Present study)𝛾0

𝜔 = 0 Steady shear

Dynamic modulus• Storage modulus (Elastic)• Loss modulus (Viscous)2𝜋/𝜔𝜔𝐺′ 𝜔 =න𝜎 𝑡 sin 𝜔𝑡 d𝑡
𝜋𝛾0 02𝜋/𝜔𝜔𝐺 ′′ 𝜔 =න𝜎 𝑡 cos 𝜔𝑡 d𝑡
𝜋𝛾0 01051𝑦𝜎 = 𝜎f + 𝜎p = 𝜂 1 + 𝜙 𝛾ሶ − ෍ 𝐹𝑖𝑗𝑥 𝑅𝑖𝑗2𝑉𝑖<𝑗Fluid(Einstein’s viscosity equation)

Interparticle𝑖𝑭𝑖𝑗𝑗• Storage modulus

𝐺′ = 𝐺p′

𝑹𝑖𝑗 = 𝑹𝑖 − 𝑹𝑗𝑦𝑧𝑥• Loss modulus

𝐺 ′′ = 𝐺f′′ + 𝐺p′′ = 𝜂𝜔

51 + 𝜙 + 𝐺p′′2

11Particles• Diameter: 𝑑 = 1 μm
• Concentration: 45 vol%• Zeta potential: 0 mV, 20 mV

𝑈𝑥 𝑡 = 𝛾ሶ 𝑡 𝑦ℎ𝑦𝑧
Fluid: Water• Ionic strength: 0.23 mM
Shear flow
• Strain: 𝛾0 = 1 × 10−3 − 3• Frequency: 𝜔𝜏 = 3.6 × (10−3 − 102 )𝜏 = 3𝜋𝜂𝑑 2 /𝐹𝐹 : Force between contacting particles

𝑥Side length: 22.7𝑑

Shear rate 𝛾ሶ 𝑡 = 𝛾0 𝜔 cos 𝜔𝑡𝛾0 𝜔0−𝛾0 𝜔Time

12Aggregation

Contact number2

𝛾0 = 0.01

11𝛾0 = 0.1

𝛾0 = 1

13Aggregation

Structure change
Linear region𝜔𝜏 = 3.6 × 10−2

Nonlinear region

𝜎0 = 𝜂𝛾0 𝜔

14Dispersion
𝑙av

Coordination number0𝛾0 = 0.01

11𝛾0 = 0.1

𝑙av /𝑑 = Volume fraction −1/3

𝛾0 = 115
Dispersion

Structure change
Linear region

Nonlinear region

𝜎0 = 𝜂𝛾0 𝜔

16Aggregation
Interparticle force = Drag force𝐹 = 3𝜋𝜂𝑑𝑈
Velocity of adhesion
𝐹𝑈=3𝜋𝜂𝑑𝛾0 = 0.01

Larger interparticle force
Lower frequency

Relaxation time𝑑 3𝜋𝜂𝑑 2
𝜏= =𝑈𝐹17

Aggregation

Dispersion

18

Aggregation

Dispersion

5= 𝜂𝜔 1 + 𝜙
2Decomposition of loss modulus: 𝐺 ′′ = 𝐺f′′ + 𝐺p′′

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Aggregation

Dispersion

20• Viscoelasticity can be described by a simple model that only considers
drag force and interparticle force• Both attractive and repulsive interaction can provide elasticity• Strain dependence reflects the structure change of particles• Frequency dependence:• Aggregated particles provide a finite value of 𝐺 ′ in low frequency limit• 𝐺 ′′ is dominated by fluid part 𝐺f′′ at high frequency• Non-monotonic behavior of 𝐺 ′′ would appear for concentrated slurry