化学工学会第48回秋季大会
2016/09/06

Direct numerical simulation of structure formationin drying bimodal colloidal suspensions○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

2/21

Colloidal suspensions Coating, Drying

Functional thin films

Particle configuration
Physical properties of film

Mesoscale Modeling & Direct Numerical Simulation

3/21

Force induced by meniscusℎ𝐹cap /𝛾𝑑0.3

present simulation

𝐹cap0.2

𝛼 = 60°
Contact angle

0.1

001

ℎ/𝑑2

Fujita et al., Appl. Phys. Express (2013).
Fujita et al., J. Comput. Phys (2015).

34/21
Bimodal

20 μm

Jenkins et al., J. Colloid Interface Sci. (2012)

5/21▪ Structure formation in drying bimodal colloidal
suspensions▪ Effects of particle wettability and interactions

Segregation

Mixing▪ Mesoscale model▪ Direct Numerical Simulation (DNS)6/21

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

CoupledEvaporation

Free surface
movementMotion
Flow
SNAP研究会

to get detailed information

7/21
Mass conservation

𝛁⋅𝒗=0
Momentum conservation

Body force to constrainRandom stressparticle velocity

𝜕𝒗
𝜌+ 𝒗 ⋅ 𝛁𝒗 = 𝛁 ⋅ 𝝈 + 𝒔 + 𝛾𝜅𝛁𝑓 + 𝜌Φ𝒂𝜕𝑡Surface tension
Stress tensor

𝝈 = −𝑝𝑰 + 𝜂 𝛁𝒗 + 𝛁𝒗 𝑇Direct Numerical Simulation▪ Discretization▪ Diffused boundary

𝑓 = 0 gas𝛀Φ=0

Φ=1𝑓 = 1 liquid

𝑽𝒗, 𝑝

8/21

Translation
buoyancycontact
DLVO
Hሶ
𝑀𝑖 𝑽𝑖 = 𝑭𝑖+ 𝑭𝑖+ 𝑭𝑖 + 𝑭𝑖𝑹ሶ 𝑖 = 𝑽𝑖𝛀𝑽Rotation
H𝑰𝑖 ⋅ 𝛀ሶ 𝑖 = 𝑵contact+𝑵𝑖𝑖

Hydrodynamic force/torqueContact force/torque-𝑭H𝑖 = − න 𝜌Φ𝑖 𝒂d𝒓- + ++- -𝑵H𝑖 = − න 𝒓 − 𝑹𝑖

Electrical double layer repulsion
Van der Waals attraction(Momentum exchange with fluid)× 𝜌Φ𝑖 𝒂 d𝒓-+ + + DLVO force- - -9/21
Advection eq.

𝒗sf

𝜕𝑓+ 𝒗sf ⋅ 𝛁𝑓 = 0
𝜕𝑡
Evaporation mass flux on free surface

liquidෝ = 1 − Φ 𝜌𝑣e
𝜌 𝒗 − 𝒗sf ⋅ 𝒏

𝑓=1

ෝ gas𝒏𝑓=0

Velocity of free surface

ෝ𝒗sf = 𝒗 − 1 − Φ 𝑣e 𝒏Contact angle𝛁𝑓ෝ p = cos 𝛼⋅𝒏𝛁𝑓DNS: Level set method

ෝp𝒏𝛁𝑓/ 𝛁𝑓𝛼

Fujita et al., J. Comput. Phys. (2015).10/21

Particle diameter Large: 200 nm (𝑑) Small: 100 nm (𝑑/2)
Final substrate coverage with particles L: 0.3 S: 0.3

Fluid: WaterEvaporation rate 2.0 × 10−2 m/s (𝑣e )Salt 1.0 × 10−4 mol/L
Temperature 293 K
1.2𝑑𝑧

𝑦10𝑑𝑥
Periodic boundaries: 𝑥, 𝑦10𝑑

11/21DLVO force
Electrical double layer repulsion𝑛salt 𝑘B 𝑇𝑑
𝐹EDL ~~ 10−12 N𝜅Van der Waals attraction
𝐴𝐹vdW ~ ~ 10−13 N𝑑Capillary force 𝐹cap ~ 𝛾𝑑 ~ 10−8 N𝐹cap ≫ 𝐹EDL ~ 𝐹vdW- ++++++- - - - -12/21

Contact angle

Zeta potential /mV

Large

SmallLarge

Small

(a)

30°

30°

−50−50(b)

90°

30°

−50−50(c)

30°

30°

50−50

(1) Wettability(a) ↔ (b)(2) Interaction: DLVO force(a) ↔ (c)

13/21

Meniscus formation → Capillary force → Aggregation

Segregation

14/21(a) L: 30° S: 30°(b) L: 90° S: 30°

15/21𝑁average
NBL = 1 −𝑁max
𝑁: Coordination number
𝑁max = 6(Close packing in 2D)

0 ≤ NBL ≤ 1
Close-packed

Dispersed

NBL for L-L / S-S coordination → Packing domain size
Segregation

Mixing16/21(b)

LS

(a)

(b)

(a)

[Time]

17/21

Meniscus formation → Capillary force → Aggregation
Meniscus is formed around a large (small) particle
when liquid layer thickness is smaller than ℎL (ℎS ).

(a)ℎL
3+2=𝑑4≈ 0.93(b)ℎL= 0.5𝑑ℎS
3+2=
𝑑8≈ 0.47ℎS≈ 0.47𝑑

18/21

Zeta potential(a) L: −50 mV, S: −50 mV(c) L: 50 mV, S: −50 mV
(a)

L-L

L-S (a): repulsive
S-S

L-S (c): attractive(c)ℎInitial structure19/21(a) L: −50 mV
S: −50 mV(c)

L: 50 mV
S: −50 mV

20/21

(c)

(c)

(a)

LS

(a)

[Time]

21/21structure formation in drying bimodal colloidal
suspensions
Decrease in wettability ofL particles

Segregation
L-S attractive interaction

Mixing