6th	  Asian	  Coa,ng	  Workshop	  (ACW2014)
May	  8,	  2014,	  Kobe	  Rei	  Tatsumi,	  Osamu	  Koike,	  and	  Yukio	  Yamaguchi	  Dept.	  Chem.	  Sys.	  Eng.,	  The	  University	  of	  Tokyo	  2/14	  Addi,on	  of	  solute	  ↓	  Eﬀects	  on	  dispersion	  stability	  ・Surfactants	  	  	  surface	  modiﬁca,on	  →	  change	  in	  adhesion	  ・Electrolytes	  	  	  	  	  electrical	  double	  layer	  overlap	  (DLVO	  theory)	  ・Polymers	   Modeling	  for	  numerical	  simula,ons	  	  	  adsorp,on	  layer	  overlap	  	  3/14	  Time	  evolu,ons	  of	  par,cle,	  solute,	  ﬂuid	  Solute	  transport	  Fluid	  ﬂow	  Par,cle	  mo,on	  4/14	  Fluids	  (Solu,on)	  Hydrodynamic	  force	  Flow	   Boundary	  condi,ons	   Par,cles	  Osmo,c	   Advec,on	  pressure	  Par,cle-­‐solute	  interac,on	  Solute	  concentra,on	   B.C.	  on	  diﬀusion	  ﬂux	  (solute	  impermeability	  	  	  	  	  	  	  	  and	  adsorp,on	  to	  par,cle)	  5/14	  Concentra,on	  ∂c+ ∇ ⋅ (cv) = −∇ ⋅ J∂tDiﬀusion	  ﬂux	  Flow	   ∇ ⋅ v = 0B.C.	  on	  diﬀusion	  ﬂux	  Impermeability	  Adsorp,on	  to	  par,cles	  Par,cle-­‐solute	  Constraint	  of	  par,cle	  veloci,es	   interac,on	  ⎛ ∂v⎞ρ ⎜ + v ⋅ ∇v ⎟ = −∇p + η∇ 2v + ρΦa −c∇U w − ∇π Osmo,c	  	  pressure	  ⎝ ∂t⎠Total	  momentum	  conserved	  Hydrodynamic	  force/torque	  Par,cles	  d
M i Vi = Fi H + Fi S
dt
dI i ⋅ Ωi = N iH
dt

Fi H = −∫ ρΦadr NiH = −∫ (r − Ri ) × ρΦadr
Par,cle-­‐solute	  interac,on	  Fi S = ∫ c∇U wdr

6/14	  Par,cle-­‐solute	  interac,on:	  Square-­‐well	  poten,al	  u(r)0−εaa+w rwu(r) = uex (r) + uw (r)#∞uex (r) = $%0

Hard-­‐core	  repulsion	  r<ar≥aAcrac,on	  →	  Adsorp,on	  $−εuw (r) = %&0a0 ≤ r < a+w
r ≥ a+w

Adsorp,on	  energy:	   βε = ε / RT

Thickness	  of	  adsorp,on	  layer:	   w7/14	  Concentra,on	  Diﬀusion	  ﬂux	  J = −D(1− Φ)Ξ∇c∗∂c+ ∇ ⋅ (cv) = −∇ ⋅ J∂tOsmo,c	  pressure	  π = RT (Ξ −1)c

Ξ =1
Φ=0Impermeability	  &	  Adsorp,on	  to	  par,cles	  Ξ = exp(−U w / RT )∗U w = ∑ uw ( r − Rk )

Ξ = e2 βε
Φ =1

Φ =1

Ξ=ekc = (1− Φ)Ξc∗

Virtual	  concentra,on	  βεcc∗

8/14	  2F2 /RTc0a2Posi,ve	  adsorp,on	  βε > 01.5`¡ = 0.25`¡ = 0.5

Osmo,c	  pressure	  (Repulsion)	  10.5

Bridging	  (Acrac,on)	  0-0.5

L-1w / a = 0.5
22.2

2.4

2.6

L /a2.83

9/14	  0Nega,ve	  adsorp,on	  -0.2
βε < 0w / a = 0.5-0.6

F2 /RTc0a2-0.4-0.8-1Deple,on	  (Acrac,on)	  -1.2`¡ = <0.25`¡ = <0.5-1.4

L

2

2.2

2.4

2.6

L /a2.83

10/14	  7par,cle	  diameter	  d = 100 nm

system	  size	  7 d × 4d × 4d(x:	  ﬁxed;	  y, z:	  periodic)	  ・adsorp,on	  layer	  thickness	  w = 0.25denergy	  βε = 0.5・concentra,on	  average	  1 mol /Lgradient	  2 ×109 mol m −4

11/14	  par,cle	  diameter	  d = 100 nmsystem	  size	  5d × 5d × 3d(periodic)	  ・adsorp,on	  layer	  thickness	  w = 0.25denergy	  βε = 0.5・concentra,on	  average	  0.1 mol /L

12/14	  Par,cle	  diameter:	   d = 100 nm

βε = 01.6 m/s

No	  adsorp,on	  System	  size:	   5d × 5d × 3dβε = 0.5, w / d = 0.25Adsorp,on	  13/14	  Simula,on	  model	  of	  colloidal	  dispersions	  with	  solute	  transport	  and	  adsorp,on	  onto	  the	  par,cle	  is	  constructed.	  
・Solute	  transport	  B.	  C.	  on	  diﬀusion	  ﬂux	  	  ←	  	  Virtual	  concentra,on	  (solute	  impermeability	  and	  adsorp,on	  to	  par,cle)	  ・Solute	  adsorp,on	  onto	  the	  par,cle	  	  Par,cle-­‐solute	  interac,on	  	  (Square-­‐well	  poten,al)	  →	  interac,ons	  among	  par,cles	  Acrac,ve	  force:	  	  Bridging	  force,	  	  	  Deple,on	  force	  Repulsive	  force:	  	  Osmo,c	  pressure	  	  14/14	  ・Breakup	  process	  of	  par,cle	  aggregates	  	  	  	  with	  polymer	  addi,ves	  ・Drying	  and	  concentra,on	  process	  of	  	  	  par,cle-­‐polymer	  dispersions	  evapora,on	  agita,on	  14/19	  7par,cle	  diameter:	  d = 100 nm

adsorp,on	  layer	  thickness:	  w = 0.25d

energy:	  βε = 0.5

concentra,on	  average:	  1 mol /L

gradient:	  2 ×10 9 mol m −4system	  size:	   7d × 4d × 4d
(x:	  ﬁxed;	  y, z:	  periodic)	  15/19	  par,cle	  diameter:	  d = 100 nm

adsorp,on	  layer	  thickness:	  w = 0.25d

energy:	  βε = 0.5

concentra,on	  average:	  0.1 mol /Lsystem	  size:	  5d × 5d × 3d(periodic)	  16/19	  Par,cle	  diameter:	   d = 100 nm

βε = 01.6 m/s

No	  adsorp,on	  System	  size:	   5d × 5d × 3dβε = 0.5, w / d = 0.25

Adsorp,on