化学工学会第81年会
2016/03/15

Solute adsorption effects on structure formationof colloidal particles during drying○辰巳 怜1，小池 修3，辻 佳子1,2，山口由岐夫31東京大学 環境安全研究センター

2東京大学 大学院工学研究科化学システム工学専攻
3（一般社団法人）プロダクト・イノベーション協会

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Colloidal suspensions Coating, Drying

Addition of solutes
・Dispersant・Binder

Functional thin films

Particle configuration
Physical properties of film

Mesoscale Modeling & Direct Numerical Simulation

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Adsorption layer overlapping

Physical adsorption(Reversible adsorption)

Interparticle force

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Particles: Newton-Euler equations of motion
Fluid: Hydrodynamic equations
Free surface: Advection equation
Solute: Advection-diffusion equation

CoupledFree surface
movement

TransportMotion
Flow5/23

Drying → Recession of free surface
・Arrangement of particles
・Distribution of solute6/23

Square-well potential（井戸型ポテンシャル）

𝑢 𝑟

O𝑟

−𝜀∞
𝑢 𝑟 = −𝜀0𝑟<𝑎𝑎 ≤𝑟 <𝑎+𝑤
𝑟 ≥𝑎+𝑤

Adsorption energy: 𝜀𝑎 𝑤

Adsorption layer thickness: 𝑤

Physical adsorption (Reversible): 𝜀 ~ 𝑅𝑇

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𝑓=0𝜕𝑐+ 𝛁 ⋅ 𝑐𝒗 = −𝛁 ⋅ 𝑱
𝜕𝑡gas

Diffusion flux
𝑱 = −𝑐𝐷𝛁𝜇𝑓=1

Chemical potential𝑐𝜇 = 𝑅𝑇ln 0 + 𝑓𝑐liquid𝑖𝑢𝑖 + 1 − 𝑓 Δ𝜇g

Particle-soluteinteraction

Excess chemical
potential in gas

Δ𝜇g /𝑅𝑇 ≫ 1 → Impermeability to gas

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Fluid𝛁⋅𝒗=0Body force to constrain
particle velocity
Surface tension

𝜕𝒗
𝜌+ 𝒗 ⋅ 𝛁𝒗 = 𝛁 ⋅ 𝝈 + 𝛾𝜅𝛁𝑓 − 𝑐𝜕𝑡𝑖𝛁𝑢𝑖 + 𝜌Φ𝒂Particle-solute interaction

Stress tensor 𝝈 = −(𝑝 + 𝑅𝑇𝑐)𝑰 + 𝜂 𝛁𝒗 + 𝛁𝒗 𝑇

Momentum exchange

Particle

𝑀𝑖 𝑽𝑖 = 𝑭C𝑖 + 𝑭S𝑖 + 𝑭H𝑖

𝑹𝑖 = 𝑽𝑖

Φ=0𝛀𝑰𝑖 ⋅ 𝛀𝑖 = 𝑵C𝑖 + 𝑵H𝑖𝑭S𝑖 =𝑐𝛁𝑢𝑖 d𝒓 𝑭H𝑖 =−𝑵H𝑖 =−𝜌Φ𝒂d𝒓𝒓 − 𝑹𝑖 × 𝜌Φ𝒂 d𝒓

Φ=1𝒗, 𝑝𝑽

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𝜕𝑓+ 𝒗sf ⋅ 𝛁𝑓 = 0𝜕𝑡𝒗sf

Evaporation mass flux on free surface
liquid𝜌 𝒗 − 𝒗sf ⋅ 𝒏 = 1 − Φ 𝜔
𝑓=1

Velocity of free surface
𝜔𝒗sf = 𝒗 − 1 − Φ𝒏𝜌Contact angle

𝛁𝑓⋅ 𝒏p = cos 𝛼
𝛁𝑓𝒏 gas𝑓=0𝒏p𝛼𝛁𝑓/ 𝛁𝑓M. Fujita et al., J. Comput. Phys. 281, 421 (2015).

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Particle diameter𝑑 = 100 nm
Contact angle
𝛼 = 45°Initial volume fraction
𝜙0 = 30 vol. % (3 particle layers)

6𝑑Kinematic viscosity𝜈 = 5.0 × 10−6 m2 /s
Surface tension
𝛾 = 2.0 × 10−2 N/m
Evaporation rate𝑣e = 1.0 × 10−2 m/s
Solute concentration
𝑐0 = 5.0 × 10−3 mol/L

𝑧 10𝑑10𝑑𝑦
Periodic boundaries: 𝑥, 𝑦𝑥

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Péclet number𝑣e 𝑑
Peα =𝐷αDiffusion coefficientParticle: 𝐷prt = 10−12 ~10−11 m2 /s
Solute: 𝐷slt = 10−10 ~10−8 m2 /s

Peprt 𝐷slt=≫1Peslt 𝐷prt

Approximation: Peprt → ∞(Non-Brownian particle)

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Adsorption 𝑤/𝑑

𝜀/𝑅𝑇𝐏𝐞𝐬𝐥𝐭

𝐷slt /[m2 /s]

(a)

No

00𝟎. 𝟎𝟐

5.0 × 10−8(b)

Yes

0.25

0.5𝟎. 𝟎𝟐

5.0 × 10−8(c)

Yes

0.25

0.5𝟎. 𝟒

2.5 × 10−9

Effects of(1) Solute adsorption(a) ↔ (b)(2) Solute Péclet number (b) ↔ (c)13/23(Equilibrium solute distribution)(1) Solute adsorption → Interparticle force
𝜀/𝑅𝑇 = 0.5𝑤/𝑑 = 0.25

Analytical solution

浸透圧斥力(Osmotic repulsion)𝑤𝑑

架橋引力(Bridging attraction)ℎGap width

14/23(without particles)(2) High solute Péclet number → Concentration gradient𝑐substrate
substrate 0ℎ 𝑣

3𝑑𝑡=𝑣ee(c) 𝜏 = 0.3ℎ = 3𝑑(b) 𝜏 = 0.015

Analytical solution𝑧ℎ 𝑡 = 𝑣e 𝑡𝑣e2
𝜏=𝑡4𝐷𝑣e𝜁=𝑧2𝐷

15/23(2) High solute Péclet number →Concentration gradient → Diffusiophoresis

Solute concentration

16/23(1) Solute adsorption → Interparticle force(a) 𝜀/𝑅𝑇 = 0(b) 𝜀/𝑅𝑇 = 0.5

Non-adsorptive

Adsorptive

17/23(2) Concentration gradient → Diffusiophoresis(b) Peslt = 0.02(c) Peslt = 0.4

Concentration gradient below free surface:Shallow
Steep18/232.5𝑑(a) < (b)(b) > (c)(Time)

19/23𝑧𝑦𝑥𝑣e 𝑡/𝑑 = 0.3(a) < (b)Structural order(b) > (c)𝑣e 𝑡/𝑑 = 3.6𝑣e 𝑡/𝑑 = 2.4
Concentration

Assembly𝑧 = 2.5𝑑

Stacking fault

20/23(a) < (b) Attraction by particles(b) < (c) Low diffusivity of solute𝑣e 𝑡/𝑑 = 2.4

21/23(a) > (b) Assembly by interparticle force(b) < (c) Diffusiophoresis

Fraction of area
covered with particles(Time)

22/23(a) > (b)(b) < (c)Dispersibility(Time)

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Solute adsorption effects on structure formationof colloidal particles during drying
・Solute adsorption → Interparticle force
Particle ordering
・High solute Péclet number →Concentration gradient → Diffusiophoresis
Increase in particle migration to free surface→ Decrease in packing efficiency