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 / Day 2 / 1 See J96

2 Interfacial energiesInterfacial forces Interfacial phenomena Complex phenomena Modeling algorithm Kaptay / Day 2 / 2

3 Deriving equations for interfacial forces x A B Kaptay / Day 2 / 3

4 The curvature induced interfacial force AB For a spherical B: The Laplace equation for spheres Kaptay / Day 2 / 4

5 The general Laplace equation Kaptay / Day 2 / 5 For cylinders: Generally: For a cylinder:

6 Summary The curvature induced interfacial force The Laplace equation: Kaptay / Day 2 / 6 In equilibrium: P 2 :atmosphere + gravity +...

7 Laplace  Kelvin The Laplace equation The Gibbs energy change Kelvin equation (Day 1 / 17): Kaptay / Day 2 / 7

8 The interfacial gradient force (1) x AB Kaptay / Day 2 / 8

9 See J64 Bubbles 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

10 See J101 Droplets moving in a T-gradient Kaptay / Day 2 / 10 Hadamard, Rybczinski: Pötschke J., Rogge V., 1989:

11 Can you produce monotectic alloys in space (g=0)? Kaptay / Day 2 / 11 Droplets do not sediment But they coalesce too quickly NO Even in space you can not. Sorry..

12 Interf. gradient forceMarangoni force Bubble movementLiquid convection Kaptay / Day 2 / 12

13 The interfacial capillary force (1) For a solid particle at a liquid/gas interface: x Kaptay / Day 2 / 13

14 The interfacial capillary force (2) The Young- Laplace equation Kaptay / Day 2 / 14 Wetting liquids penetrate into empty cylinders (see also Day 1 / 15)

15 The interfacial capillary force (3) Kaptay / Day 2 / 15 See J23

16 Particle equilibrium at interface For a spherical particle of radius r: equilibrium Kaptay / Day 2 / 16 See J23

17 Wettability versus particle position at interface 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 R eq Kaptay / Day 2 / 18

19 If the grains are identical: The maximum force at x = 2r: Kaptay / Day 2 / 19 Equilibrium if: The equilibrium grain-size: For better properties (low R eq ) precipitate many nano-particles How to make nano-crystalline alloys?

20 The condition of flat meniscus around a sphere (1) Depends on the dimensionless density: Kaptay / Day 2 / 20

21 The condition of flat meniscus around a sphere (2) The equilibrium condition for gravity + buoyancy forces, only: The two equals, if: The equilibrium condition for interfacial capillary force, only: Kaptay / Day 2 / 21

22 The interfacial meniscus force (1) 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) 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) (  1/2 = 1 J/m 2,  3/2/1 =  4/2/1 = 30 o, V 2 /V 3 = V 2 /V 4 = 0.01, r = 10  m, F max =  N). ) Kaptay / Day 2 / 27

28 The interfacial adhesion force (1) x Kaptay / Day 2 / 28

29 A simplified derivation 3 x 12 Kaptay / Day 2 / 29

30 Boundary condition 1: If x   :  13 (x)   13,  23 (x)   23 Boundary condition 2: If x  0:  13 (x)   12,  23 (x)   12 Kaptay / Day 2 / 30  ij = f (interface separation)

31 Substituting…. Literature Kaptay / Day 2 / 31 See J24

32 Summary Hamaker, 1937: Neumann, 1973: Kaptay, 1996: 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 o C,  l/g = 1.7 J/m 2,  1 = 7000 kg/m 3 Solid particle: Al 2 O 3, r = 10  m,  s/g = 0.9 J/m 2,  s = 3600 kg/m 3, m = kg, F g = m.g = 1, N. Contact angle: 120 o. From the Young equation:  s/l = 1.75 J/m 2, Derivatives by T and weight % of oxygen, dissolved in liquid steel: d  c/l /dT = – J/Km 2 and d  c/l /dC O = –10 J/m 2 w%. Temperature gradient: dT/dx = 10K/mm Gradient of the oxygen concentration: dC O /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 *) = 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 N >> gravity force Kaptay / Day 2 / 35

36 The interfacial gradient force x AB T-induced: N > gravity force O-concentration induced: N >> gravity Kaptay / Day 2 / 36

37 The interfacial capillary force F = N >> gravity force Kaptay / Day 2 / 37

38 The interfacial meniscus force F = N << gravity force (but perpendicular to gravity) Kaptay / Day 2 / 38

39 The liquid bridge induced interfacial force (1) F = N >> gravity force Kaptay / Day 2 / 39

40 The interfacial adhesion force x F = N (x = 0) >> gravity force F = 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…)