SCEJ 49th Autumn Meeting
Sep. 22, 2017, Nagoya University

Analysis of segregation of additives indrying colloidal suspensions○ 辰巳 怜小池 修
辻 佳子
山口由岐夫

Coating, Drying（東大環安セ）（PIA）（東大環安セ/東大院工）
（PIA）

2
山口由岐夫，「ものづくりの化学工学」 (2015).

Constant rate

Falling rate

Drying rate

Formation of particle
concentrated layer below
free surface

Flow driven bycapillary pressure

Drying of particle layer

Present study

Drying time3
Trueman et al., J. Colloid Interface Sci. (2012).

1D diffusion equation for particle concentration dist.
Large particle

Small particle

Segregation of Larger particles to the top surface← Diffusion rate: (Large particle) < (Small particle)

4Zhou et al., PRL (2017).

Cross-diffusion → Segregation of Smaller particles to the top surface
ve h0
Pe==5Diameter ratio 41
D1

Volume fraction

Diameter ratio

SmallLarge

5Tatsumi et al., SCEJ 82th Annual Meeting (2017).

Pe = 0.3

100

5Diameter ratio: 2Particle drying Péclet number

Pe =Pe = 0.35100

Evaporation rate
Diffusion rate𝑣e 𝑑=𝐷6

Investigation of the segregation of additives in the constantdrying rate period

Analysis using a simple model▪ Particles: Equation of motion (Langevin eq.)
← Force induced by particle-solute interaction▪ Solute concentration dist. : 1D advection-diffusion eq.
(Approximated by the analytical solution without particles)▪ Free surface moving at constant rate

7

Detailed model

Consideration of coupled motion

Particle

SNAP研究会

Fluid flow
Free surface
Solute distribution

8
Langevin equationcplcnt𝑀𝑖 𝑽𝑖 = −𝜉𝑽𝑖 + 𝑭R+𝑭+ 𝑭𝑖𝑖𝑖+ 𝑭slt𝑖

Drag force: −𝜉𝑽𝑖Stokes’ law: 𝜉 = 3𝜋𝜂𝑑Stochastic variables
RRandom force: 𝐹𝑖𝛼 𝑡 ~𝑁 0, 2𝜉𝑘B 𝑇 obeying the Gaussian dist.
𝑘B 𝑇→ Brownian Diffusion: 𝐷p =3𝜋𝜂𝑑Contact force: 𝑭cnt
𝑖

Diffusion coefficientin infinite dilution
cpl

Vertical capillary force: 𝑭𝑖

Contact angle 𝛼 = 0Particle-solute interaction force: 𝑭slt
← Solute concentration dist.
𝑖9𝑣e

Evaporation rate𝑣e𝑣e 𝑑
Pe ===Diffusion rate
𝐷/𝑑𝐷characteristic length = particle diameter 𝑑Particle𝑣e 𝑑
Pep =𝐷pSolute𝑣e 𝑑Pes =𝐷s𝐷/𝑑

Solute Concentration𝑭slt
𝑖 : Force induced by “Solute concentration gradient field”→ Diffusiophoresis (拡散泳動)

10Slip boundary model （𝑤/𝑑 ≪ 1）

𝑭slt 𝒗s (𝜃)Particle-solute interaction𝑢 𝑟

𝜀 < 0 : Repulsion
𝜀 > 0 : Attraction

O
Flow driven by osmoticpressure gradientAnderson & Prieve, Sep. Purif. Meth. (1986)

𝑲slt
𝑭 = −3𝜋𝜂𝑑 𝒗s = Γ𝑘B 𝑇
𝑐0𝑟−𝜀w

Interaction parameter3Γ = 𝜋 𝑒 𝛽𝜀 − 1 𝑐0 𝑤 2 𝑑2

Solute concentration gradient 𝑲← Analytical solution of the concentration dist. without particles

110𝑣e 𝑡

Concentration𝑐 = 𝑐 𝑧, 𝑡 = 𝑐0 𝑓(𝜁, 𝜏)𝜁 = (𝑧 − 𝑣e 𝑡)/𝑙c
𝜏 = 𝑣e 𝑡/𝑙c

Length scale of concentration change 𝑙c = 𝐷s /𝑣e𝑧
1D advection-diffusion eq.
𝜕𝑓 𝜕𝑓 𝜕 2 𝑓=+ 2𝜕𝜏 𝜕𝜁 𝜕𝜁▪ 𝑓 𝜁, 0 = 1 Initial condition▪ 𝜕𝜁 𝑓 0, 𝜏 = −𝑓 0, 𝜏Free surface▪ 𝜕𝜁 𝑓 ∞, 𝜏 = 0
Substrate position ≫ 𝑙c

12Magnitude of gradient|𝑲| 𝜕𝑓 1 𝜕𝑓 Pes 𝜕𝑓===𝑐0
𝜕𝑧 𝑙c 𝜕𝜁𝑑 𝜕𝜁

𝑧 = Center of the particle

Coordinate transform in 𝑓(𝜁)𝑓𝑧 − 𝑣e 𝑡
𝜁=𝑑Pes←𝑧 − 𝑣e 𝑡𝑧 − 𝑣e 𝑡
𝜁== Pes𝑙c𝑑

Increase: Magnitude of gradient ∝ Pes

Increase in Pes
Decrease: Extent of gradient ∝ Pe−1s𝑧 − 𝑣e 𝑡 /𝑑

13Film height ℎ 𝑡 = ℎ0 − 𝑣e 𝑡ℎ/ℎ0 (a)1ℎ0(b)
(25𝑑)
(c)
0.4
O

(a)

(b)

(c)

Time

▪ Particle diameter 𝑑6𝑑▪ Initial volume fraction 0.1▪ Particle Péclet number Pep = 20𝑧𝑦𝑥

Periodic boundaries: 𝑥, 𝑦(1) Interaction parameter Γ = −100~100, Pes = 0.2(2) Solute Péclet number Pes = 0.1~10, Γ = −100

14Γ>0Γ<0Γ
Particle-soluteinteraction

0-100
Attractive−100
Repulsive

15𝑧

Γ>0
Γ<00Γ

0100

−100Particleℎ/ℎ0 = 0.4
Solute16

Deviation from the uniform distribution
Average z-coordinate of particles ⟨𝑧⟩

Free surface

⟨𝑧⟩ 1
Δ≡
−ℎ2𝑧ℎ0Substrate
Segregationℎ/ℎ0 = 0.4

17Γ = −100(Diffusivity of solute)Pes = 0.1

0.2

110

18ℎ/ℎ0 = 0.4

0.1

Particle

0.2
Solute110

19Free surface⟨𝑧⟩ 1Δ≡−ℎ2𝑧ℎ

Substrate

0ℎ/ℎ0 = 0.4

Trade-offConcentration gradient Magnitude ∝ Pes Small ↔ Large
Large ↔ Small
Extent ∝ Pe−1s

20

“Solute concentration gradient field” is generated by drying
and exerts force on the particles. (Diffusiophoresis)▪ Attractive particle-solute interaction→ Migration of particles toward the free surface▪ Repulsive particle-solute interaction→ Migration of particles toward the substrate→ Segregation

Segregation is maximized at Pes ~0.3.
← Trade-off relation about solute concentration gradient:
Magnitude ∝ Pes , Extent ∝ Pe−1s