See J96
Kaptay / Day 2 / 1
Day 2. Interfacial forces
acting on phases situated at (or close to) the
interface of other phases and driving them in space
George Kaptay
kaptay@hotmail.com
A 4-day short course
1
Kaptay / Day 2 / 2
Modeling algorithm
Interfacial energies
Interfacial forces
Interfacial phenomena
Complex phenomena
2
Kaptay / Day 2 / 3
Deriving equations for interfacial forces
x
G   AA / B   A / B
A, B
B
A
F , x
F , x
dG ( x)

dx
dAA / B
d A / B
   A / B 
 AA / B 
dx
dx
A, B
A, B
3
Kaptay / Day 2 / 4
The curvature induced interfacial force
A B
F , x
dAA / B
  A / B 
dx
For a spherical B:
AA / B  4    x 2
F , x  8    x   A / B
P , x 
F , x
AA / B
2  A/ B

x
The Laplace equation
for spheres
4
Kaptay / Day 2 / 5
The general Laplace equation
1 1
  A / B    
 r1 r2 
P , x
For cylinders:
A
B
x
Generally:
For a cylinder:
F , x
P , x  
 A/ B
r
dAA / B
  A / B 
dx
AA / B  2  x  L F , x  2     A / B  L
P , x 
F , x
AA / B

 A/ B
x
5
Kaptay / Day 2 / 6
Summary
The curvature induced interfacial force
The Laplace equation:
P , x
1 1
  A / B    
 r1 r2 
F
F
In equilibrium:
F
p1  p2
x
F
F
phase 1
F
p1
p2
F
F
phase 2
2 
p1  p2 
r
6
P2:atmosphere + gravity +...
Kaptay / Day 2 / 7
Laplace  Kelvin
The Laplace equation
The Gibbs energy change
2 
p
r
G  V  p  S  T
2 
1 1 
G  Vm  2        Vm 
r
r 
r 
G
r
Kelvin equation (Day 1 / 17):
 / 
Gi , ,bulk  Gi ,  ,bulk  2  Vm,   7
r
Kaptay / Day 2 / 8
The interfacial gradient force (1)
x
A
B
F , x
d A / B
  AA / B 
dx
d A / B d A / B dT
d A / B dxi d A / B dE






dx
dT
dx
dxi
dx
dE dx
i
8
See J64
Kaptay / Day 2 / 9
Bubbles in a concentration gradient
F , x
d A / B
  k  4  r 
dx
k=0.5 comes from fluid dynamics
for bubbles moving in a C-gradient
2

Measured: Mukai and Lin
8
ML
v, mm/s
6
x
4
C
KK
2
0
0
5
10
15
rg , m
20
25
30
9
x
See J101
Kaptay / Day 2 / 10
Droplets moving in a T-gradient
F , d , x
d L / L dTd
 4    R 

dT
dx
Pötschke J., Rogge V., 1989:
Hadamard, Rybczinski:
veq , drop  
6  Rd
m
2
d
dTd
3  m
dT


dx 2  m  d dx
Fdrag
m
 4    Rd  m  v 
m  d
m
m  d
d L / L dT




2  m  d 2  m  3  d
dT
dx
10
Kaptay / Day 2 / 11
veq , drop  
6  Rd
m
m
m  d
d L / L dT




2  m  d 2  m  3  d
dT
dx
Can you produce monotectic alloys in space (g=0)?
radial cooling
NO
Even
in
space
you
can
not.
Sorry..
Droplets do not sediment
11
But they coalesce too quickly
Kaptay / Day 2 / 12
Interf. gradient force
 d l / g 


 dx 
Bubble movement
Marangoni force
 d l / g

 dx



Liquid convection
12
Kaptay / Day 2 / 13
2
3
The interfacial capillary force (1)
x
1
G  A1 / 2 ( x)   1 / 2  A3o   2 / 3  A1 / 3 ( x)   1 / 3   2 / 3 
F , x   2 / 3
dA1 / 3 ( x)
dA1 / 2 ( x)
  1/ 3  
  1/ 2 
dx
dx
For a solid particle at a liquid/gas interface:
F , x   l / g
dAl / g 
 dAs / l

 cos  

dx 
 dx
13
Kaptay / Day 2 / 14
The interfacial capillary force (2)
gas
h
solid
liquid
Pc 

Al / g
dAl / g
 dAs / l
 
 cos  
dh
 dh



Al / g  r 2  
As / l  2  r    h
2
Pc     cos 
r
The YoungLaplace equation
14
Wetting liquids penetrate into empty cylinders (see also Day 1 / 15)
See J23
Kaptay / Day 2 / 15
The interfacial capillary force (3)
x
Al / g  2  r    x    x 2
F , x
F , x   l / g
dAl / g 
 dAs / l

 cos  

dx 
 dx
As / l  2  r    x
x

 2  r     l / g  1  cos   
r

15
See J23
Kaptay / Day 2 / 16
Particle equilibrium at interface
For a spherical particle of radius r:
F , x  2  r     l / g
x

 1  cos  c / l / g  
r

equilibrium
xeq  r  (1  cos  c / l / g )
F , x  2     l / g  xeq  x 
16
Kaptay / Day 2 / 17
Wettability versus particle position at interface
xeq  r  (1  cos )
a.
b.
l g
s
g
g
g
l
c.
s
l
g
s
X
l
s
l
s
g
s
l
17
Kaptay / Day 2 / 18
The interfacial capillary force in
physical metallurgy
time
If solid particles (droplets) are dragged by the grain
boundary, its movement is slowed down by the
particles (droplets) (the “Zenner force”) and its size
18
stabilizes at a certain value of Req
Kaptay / Day 2 / 19
F , x
How to make nano-crystalline alloys?
x

 2  r     g / g  1  cos  g / s / g  
r

If the grains are identical:
The maximum force at x = 2r:
cos  g / s / g
 g/s  g/s

0
 g/g
F , x ,max  2  r     g / g
3
8  Req     g / g  ns  2  r     g / g n    R
s
r3
2

r
The equilibrium grain-size:
Req 
Equilibrium if:
s
19
For better properties (low Req) precipitate many nano-particles
Kaptay / Day 2 / 20
The condition of flat meniscus around a sphere (1)
s   g
* 
l   g
Depends on the dimensionless density:
x
1
1
2
2
3
4
 *   * flat
r3
3
4
r4
 *   * flat
4
1
2
1
2
3
 *   * flat
20
Kaptay / Day 2 / 21
The condition of flat meniscus around a sphere (2)
The equilibrium condition for interfacial capillary force, only:
xeq  r  (1  cos )
The equilibrium condition for gravity + buoyancy forces, only:
 s   g xeq2  3  r  xeq 
* 

l   g
4 r3
The two equals, if:
 *   * flat
1
 flat *   (1  cos  c / l / g ) 2  (2  cos  c / l / g )21
4
Kaptay / Day 2 / 22
The interfacial meniscus force (1)
x
1
1
2
3
4
2
r3
3
4
r4
4
1
2
3
1
2
dA
dA 
 dA
F , x   1 / 2   2 / 3  cos  3 / 2 / 1  2 / 4  cos  4 / 2 / 1  12 
dx
dx 
 dx
F , x
81 (m3  g )  (m4  g )


 (  3 *   3, flat *)  (  4 *   4, flat *)
32  
x   1/ 2
22
Chan et al, 1980 (exact solution: Paunov et al, 1993)
Kaptay / Day 2 / 23
The interfacial meniscus force (2)
x
4
r4
r3
3
2.a) 1
2
1
2
x
2.b) 1
2
r3
3
4
r4
1
2
Flat meniscus  no interfacial force
23
Kaptay / Day 2 / 24
The interfacial meniscus force (3)
1.a)
1
2
1
3
4
1.b) 1
1
2
3
4
2
2
Similarly curved menisci  attracting interfacial force24
Kaptay / Day 2 / 25
The interfacial meniscus force (4)
4
3.a) 1
2
2
3
4
3.b) 1
2
1
3
1
2
25
Oppositely curved menisci  repulsing interfacial force
Kaptay / Day 2 / 26
The liquid bridge induced interfacial force (1)
x
r3
r4

2
3
4
1
F , x
dA2 / 4
dA12 
 dA2 / 3
  1/ 2  
 cos  3 / 2 / 1 
 cos  4 / 2 / 1 

dx
dx 
 dx
F , x ,max  2    r   l / g  cos 
Valid at x  0, V  0, same as interfacial capillary force for cylinders
26
(see Today, slide 14)
Kaptay / Day 2 / 27
The liquid bridge induced interfacial force (2)
0
Fx, N
-10 0
1
2
3
-20
-30
Kaptay
-40
Naidich
-50
x, m
(1/2 = 1 J/m2, 3/2/1 = 4/2/1 = 30o, V2/V3 = V2/V4 = 0.01,
27
r = 10 m, Fmax = -54.4 N). )
Kaptay / Day 2 / 28
The interfacial adhesion force (1)
x
2
1
3
d2
F  k   
(d  x) m
Phase 1
Plane of area A
Plane of  area
Sphere of radius r1
Phase 2
Plane of area A
Sphere of radius r
Sphere of radius r2
Position
Parallel
-
Cylinder of radius r1
Cylinder of radius r2
Crossing under

k
2 A
2  r 
r r
2   1 2
r1  r2
m
3
2
2
r r
2   1 2
sin  28
2
Kaptay / Day 2 / 29
A simplified derivation
x
1
2
3
G  A13   13  A23   23
dG
F 
dx
A13  A23  A
 d 13 d 23 
F  A


dx 
 dx
29
Kaptay / Day 2 / 30
ij = f (interface separation)
Boundary condition 1: If x  : 13(x)  13, 23(x)  23
Boundary condition 2: If x  0: 13(x)  12, 23(x)  12
 13 ( x)   13  ( 12
 d 
  13 )  

d  x
interfacial energy
n
 d 
 23 ( x)   23  ( 12   23 )  

d  x
13(x)
13
12
23(x)
23
0
separation
x
30
n
Kaptay / Day 2 / 31
See J24
Substituting….
 d 13 d 23 
F  A


dx 
 dx
 d 

d

x


n
 13 ( x)   13  ( 12   13 )  
 d 
 23 ( x)   23  ( 12   23 )  

d

x


dn
F  n  A  (2   12   13   23 ) 
(d  x) n1
2
Literature
d
F  2  A   
( d  x) 3
  2   12   13   23
n2
31
n
Kaptay / Day 2 / 32
Summary
2
1
3
1
1
3
Hamaker, 1937:
   12   13   23
  2   13
Neumann, 1973:
   12   13
   13
Kaptay, 1996:
See J24
  2   12   13   23
  2   13
32
Kaptay / Day 2 / 33
Conclusions
Equations have been obtained for the interfacial forces:
The “curvature induced interfacial force” (Laplace)
The “interfacial gradient force” (Marangoni)
The “interfacial capillary force” (Young-Laplace, Carman, Zener)
The “interfacial meniscus force” (Nicolson, Denkov, White)
The “liquid bridge induced interfacial force” (Naidich)
The “interfacial adhesion force” (Derjaguin, Hamaker)
33
Kaptay / Day 2 / 34
Conditions for the trial calculations
Liquid: steel at 1600 oC, l/g = 1.7 J/m2, 1 = 7000 kg/m3
Solid particle: Al2O3, r = 10 m, s/g = 0.9 J/m2, s = 3600
kg/m3, m = 1.5.10-11 kg, Fg = m.g = 1,5.10-10 N.
Contact angle: 120o. From the Young equation: s/l = 1.75
J/m2, Derivatives by T and weight % of oxygen, dissolved in
liquid steel: dc/l/dT = –2.10-4 J/Km2 and dc/l/dCO = –10
J/m2w%. Temperature gradient: dT/dx = 10K/mm
Gradient of the oxygen concentration: dCO/dx = 0.01 w%/mm
For capillary force, the depth of immersion: x = 20m
For meniscus force between two, equal particles: x = 10 m.
From the densities: * = 0.51, flat* = 0.16, i.e. (* - flat*) =
0.35. For the liquid bridge induced interfacial force we
suppose that the particles are in contact (x = 0), the volume of
34
liquid is negligible.
Kaptay / Day 2 / 35
The curvature induced interfacial force
F
F
F
x
F
F
phase 1
F
p1
p2
phase 2
F
F
F , x  8    x   A / B
-4.4.10-4 N >> gravity force
35
Kaptay / Day 2 / 36
The interfacial gradient force
x
A
B
F , x
d A / B
  AA / B 
dx
d A / B d A / B dT
d A / B dxi




dx
dT
dx
dxi
dx
i
T-induced: 2.5.10-9 N > gravity force
O-concentration induced: 1.3.10-7 N >> gravity
36
Kaptay / Day 2 / 37
The interfacial capillary force
F , x  2  r     l / g
x

 1  cos  c / l / g  
r

F = -1.6.10-4 N >> gravity force
37
Kaptay / Day 2 / 38
The interfacial meniscus force
x
1
1
2
3
F , x
4
2
r3
3
4
r4
4
1
2
3
1
2
81 (m3  g )  (m4  g )


 (  3 *   3, flat *)  (  4 *   4, flat *)
32  
x   1/ 2
F = -1.3.10-16 N << gravity force
(but perpendicular to gravity)
38
Kaptay / Day 2 / 39
The liquid bridge induced interfacial force (1)
x
r3
r4

2
3
4
1
F , x ,max  2    r   l / g  cos 
F = 5.3.10-5 N >> gravity force
39
Kaptay / Day 2 / 40
The interfacial adhesion force
x
2
1
3
2
d
F  k   
(d  x) m
  2   12   13   23
F = 1.1.10-4 N (x = 0) >> gravity force
F=
1.10-12
40
N (x = 1 micron) < gravity force
Thank you for your attention
so far
If not too tired, please, come
again
(tomorrow morning…)
41