1

Day 2. Interfacial forces

acting on phases situated at (or close to) the interface of other phases and driving them in space

A 4-day short course

George Kaptay

kaptay@hotmail.com

Kaptay / Day 2 / 1 See J96

2

Interfacial energies Interfacial forces

Interfacial phenomena

Complex phenomena

Modeling algorithmKaptay / Day 2 / 2

3

Deriving equations for interfacial forces

BA

BABAAG,

//

dx

xdGF x

)(,

BA

BABA

BA

BABAx dx

dA

dx

dAF

,

//

,

//,

x

A

B

Kaptay / Day 2 / 3

4

The curvature induced interfacial force

dx

dAF BA

BAx/

/, A B

BAx xF /, 8

xA

FP BA

BA

xx

/

/

,,

2

For a spherical B:

The Laplace equation for spheres

Kaptay / Day 2 / 4

2/ 4 xA BA

5

The general Laplace equation

Kaptay / Day 2 / 5

21/,

11

rrP BAx

xBA

For cylinders:

rP BA

x/

,

dx

dAF BA

BAx/

/, Generally:

For a cylinder: LxA BA 2/ LF BAx /, 2

xA

FP BA

BA

xx

/

/

,,

6

Summary

The curvature induced interfacial force

The Laplace equation:

x

FF

F

FF

F

FF

phase 2phase 1 p2

p1

Kaptay / Day 2 / 6

21/,

11

rrP BAx

21 pp

rpp

221

In equilibrium:

P2:atmosphere + gravity +...

7

Laplace Kelvin

The Laplace equation

The Gibbs energy change

rp

2

TSpVG

r rG

rV

rVG mm

2112

r

VGG mbulkibulki/

,,,,, 2

Kelvin equation (Day 1 / 17):

Kaptay / Day 2 / 7

8

The interfacial gradient force (1)

xA B

dx

dAF BA

BAx/

/,

dx

dE

dE

d

dx

dx

dx

d

dx

dT

dT

d

dx

d BA

i

i

i

BABABA ////

Kaptay / Day 2 / 8

9

dx

drkF BA

x/2

, 4

0

2

4

6

8

0 5 10 15 20 25 30rg , m

v, m

m/s

ML

KK

See J64Bubbles in a concentration gradient

Kaptay / Day 2 / 9

k=0.5 comes from fluid dynamics for bubbles moving in a C-gradient

Measured: Mukai and Lin

x

x

C

10

See J101

Droplets moving in a T-gradient

Kaptay / Day 2 / 10

dx

dT

dT

dRF dLL

dxd /2,, 4

dx

dT

dx

dT

dm

md

2

3

Hadamard, Rybczinski:dm

mmddrag vRF

4

dx

dT

dT

dRv LL

dm

dm

dm

m

m

ddropeq

/, 322

6

Pötschke J., Rogge V., 1989:

11

Can you produce monotectic alloys in space (g=0)?

Kaptay / Day 2 / 11

dx

dT

dT

dRv LL

dm

dm

dm

m

m

ddropeq

/, 322

6

radial cooling

Droplets do not sediment But they coalesce too quickly

NO Even

in space you can not.

Sorry..

12

Interf. gradient force Marangoni force

dx

d gl /

dx

d gl /

Bubble movement Liquid convection

Kaptay / Day 2 / 12

13

The interfacial capillary force (1)

For a solid particle at a liquid/gas interface:

x

1

23

3/23/13/13/232/12/1 )()( xAAxAG o

dx

xdA

dx

xdAF x

)()( 2/12/1

3/13/13/2,

dx

dA

dx

dAF glls

glx//

/, cos

Kaptay / Day 2 / 13

14

The interfacial capillary force (2)

hrA ls 2/

dh

dA

dh

dA

AP glls

glc

//

/

cos

2/ rA gl

cos2 r

PcThe Young-

Laplace equation

solid

liquid

h

gas

Kaptay / Day 2 / 14

Wetting liquids penetrate into empty cylinders (see also Day 1 / 15)

15

The interfacial capillary force (3)

x

dx

dA

dx

dAF glls

glx//

/, cos

2/ 2 xxrA gl xrA ls 2/

r

xrF glx cos12 /,

Kaptay / Day 2 / 15 See J23

16

Particle equilibrium at interface

r

xrF glcglx ///, cos12

For a spherical particle of radius r:

)cos1( // glceq rx

xxF eqglx /, 2

equilibrium

Kaptay / Day 2 / 16 See J23

17

s

l g

l g

s

ls

g

g

g

g

l

l

l

s

s

s

Xa.

b.

c.

Wettability versus particle position at interface

)cos1( rxeq

Kaptay / Day 2 / 17

18

The interfacial capillary force in physical metallurgy

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 stabilizes at a certain value of Req

time

Kaptay / Day 2 / 18

19

ggx rF /max,, 2

r

xrF gsgggx ///, cos12

If the grains are identical: 0cos/

////

gg

sgsggsg

The maximum force at x = 2r:

Kaptay / Day 2 / 19

Equilibrium if: ggsggeq rnR // 28 3

3

r

Rn s

The equilibrium grain-size:

seq

rR

2

For better properties (low Req) precipitate many nano-particles

How to make nano-crystalline alloys?

20

The condition of flat meniscus around a sphere (1)

flat** flat**

flat**

gl

gs

*Depends on the dimensionless density:

Kaptay / Day 2 / 20

21

The condition of flat meniscus around a sphere (2)

)cos2()cos1(4

1* //

2// glcglcflat

The equilibrium condition for gravity + buoyancy forces, only:

flat** The two equals, if:

3

2

4

3*

r

xrx eqeq

gl

gs

The equilibrium condition for interfacial capillary force, only:

)cos1( rxeq

Kaptay / Day 2 / 21

22

The interfacial meniscus force (1)

dx

dA

dx

dA

dx

dAF x

121/2/4

4/21/2/3

3/22/1, coscos

*)*(*)*()()(

32

81,44,33

2/1

43, flatflatx x

gmgmF

Kaptay / Day 2 / 22

Chan et al, 1980 (exact solution: Paunov et al, 1993)

23

The interfacial meniscus force (2)

Flat meniscus no interfacial force

Kaptay / Day 2 / 23

24

The interfacial meniscus force (3)

Similarly curved menisci attracting interfacial force

Kaptay / Day 2 / 24

25

The interfacial meniscus force (4)

Oppositely curved menisci repulsing interfacial force

Kaptay / Day 2 / 25

26

The liquid bridge induced interfacial force (1)

x

3 42

1

r3 r4

dx

dA

dx

dA

dx

dAF x

121/2/4

4/21/2/3

3/22/1, coscos

cos2 /max,, glx rF

Kaptay / Day 2 / 26

Valid at x 0, V 0, same as interfacial capillary force for cylinders (see Today, slide 14)

27

The liquid bridge induced interfacial force (2)

-50

-40

-30

-20

-10

0

0 1 2 3

x, m

F x

, N

Kaptay

Naidich

(1/2 = 1 J/m2, 3/2/1 = 4/2/1 = 30o, V2/V3 = V2/V4 = 0.01,r = 10 m, Fmax = -54.4 N). )

Kaptay / Day 2 / 27

28

The interfacial adhesion force (1)

mxd

dkF

)(

2

3

21x

Kaptay / Day 2 / 28

Phase 1 Phase 2 Position k m Plane of area A Plane of area A Parallel A2 3 Plane of area Sphere of radius r - r2 2

Sphere of radius r1 Sphere of radius r2 -

21

212rr

rr

2

Cylinder of radius r1 Cylinder of radius r2 Crossing under

sin

2 21 rr

2

29

A simplified derivation

dx

dGF

3

x1 2

23231313 AAG

dx

d

dx

dAF 2313

AAA 2313

Kaptay / Day 2 / 29

30

n

xd

dx

)()( 13121313

Boundary condition 1: If x : 13(x) 13, 23(x) 23

Boundary condition 2: If x 0: 13(x) 12, 23(x) 12

n

xd

dx

)()( 23122323

x separation

interfacial energy

13

23

12

0

13(x)

23(x)

Kaptay / Day 2 / 30

ij = f (interface separation)

31

Substituting….

dx

d

dx

dAF 2313

n

xd

dx

)()( 13121313 n

xd

dx

)()( 23122323

1231312 )()2(

n

n

xd

dAnF

2n2313122

3

2

)(2

xd

dAF

Literature

Kaptay / Day 2 / 31 See J24

32

Summary

Hamaker, 1937:

Neumann, 1973:

Kaptay, 1996: 2313122

1312

231312

1

3

21

3

1

132

13

132

Kaptay / Day 2 / 32

See J24

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)

Kaptay / Day 2 / 33

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: dc/l/dT = –2.10-4 J/Km2 and dc/l/dCO = –10 J/m2w%. Temperature gradient: dT/dx = 10K/mmGradient of the oxygen concentration: dCO/dx = 0.01 w%/mm For capillary force, the depth of immersion: x = 20mFor 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 liquid is negligible.

Kaptay / Day 2 / 34

35

The curvature induced interfacial force

BAx xF /, 8

-4.4.10-4 N >> gravity force

x

FF

F

FF

F

FF

phase 2phase 1 p2

p1

Kaptay / Day 2 / 35

36

The interfacial gradient force

xA B dx

dAF BA

BAx/

/,

i

i

i

BABABA

dx

dx

dx

d

dx

dT

dT

d

dx

d ///

T-induced: 2.5.10-9 N > gravity force

O-concentration induced: 1.3.10-7 N >> gravity

Kaptay / Day 2 / 36

37

The interfacial capillary force

r

xrF glcglx ///, cos12

F = -1.6.10-4 N >> gravity force

Kaptay / Day 2 / 37

38

The interfacial meniscus force

*)*(*)*()()(

32

81,44,33

2/1

43, flatflatx x

gmgmF

F = -1.3.10-16 N << gravity force

(but perpendicular to gravity)

Kaptay / Day 2 / 38

39

The liquid bridge induced interfacial force (1)

x

3 42

1

r3 r4

cos2 /max,, glx rF

F = 5.3.10-5 N >> gravity force

Kaptay / Day 2 / 39

40

The interfacial adhesion force

mxd

dkF

)(

2

3

21x

2313122

F = 1.1.10-4 N (x = 0) >> gravity force

F = 1.10-12 N (x = 1 micron) < gravity force

Kaptay / Day 2 / 40

41

Thank you for your attention

so far

If not too tired, please, come again

(tomorrow morning…)