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Day 4 . Modelling (some) i nterfacial phenomena

Kaptay / Day 4 / 1. Day 4 . Modelling (some) i nterfacial phenomena. George Kaptay kaptay@hotmail.com. A 4-day short course. Kaptay / Day 4 / 2. Subjects to be covered today:. 1. Abrasive ability of composites versus adhesion energy.

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Day 4 . Modelling (some) i nterfacial phenomena

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  1. Kaptay / Day 4 / 1 Day4. Modelling (some) interfacial phenomena George Kaptay kaptay@hotmail.com A 4-day short course

  2. Kaptay / Day 4 / 2 Subjects to be covered today: 1. Abrasive ability of composites versus adhesion energy 2. The critical size of the particle separated from a fluid-fluid interface due to gravity 3. Particle incorporation into liquids (LMI) 4. Penetration of liquids into porous solids (preforms made of particles and fibers) + pre-penetration 5. Pushing-engulfment of particles by solidification fronts 6. Stabilization of foams and emulsions by solid particles 7. Droplet formation by a blowing gas jet

  3. Kaptay / Day 4 / 3 See J26, J27 Story 1 A puzzle on abrasive abilities of AMMCs (AMMC = Amorphous Metal Matrix Composite), reinforced with SiC and WC particles Hard carbide particles (SiC and WC) were incorporated into a relatively soft amorphous metallic matrix to increase the abrasive ability of the matrix against wood samples Unexpectedly, harder SiC particles provided lower abrasive ability compared to less hard WC particles For a wood sample, WC and SiC are similarly hard. However, SiC and WC are kept in the matrix by different adhesion energies. Thus, SiC particles felt out of the matrix, while WC particles stayed there „forever”

  4. Kaptay / Day 4 / 4 1/1. The production of AMMC ribbons Gas pressure Crucible with Fe40Ni40Si6B14 Blown particles (SiC or WC, 50 micron) Nozzle Inductive coil Composite ribbon 12 mm x 50 micron Liquid metal Rotating, water-cooled Cu-disc Testing AMMC ribbons for their abrasive ability (against wood)

  5. Kaptay / Day 4 / 5 1/2. Observations Definition: Abrasive ability is mass loss of wood per 1 m of path (g/m). Expectation: SiC is much harder than WC, so the abrasive ability of SiC-reinforced AMMC will be higher than that of the WC-reinforced AMMC Experimental finding: the abrasive ability of SiC-reinforced AMMC is 5 times lower than that of the WC reinforced AMMC Wettability tests of liquid Fe40Ni40Si6B14 on different substrates On SiC: 135 deg On WC: 60 deg Empirical finding: If adhesion energy is higher, the abrasive ability is also higher WHY ?

  6. Kaptay / Day 4 / 6 1/3. Visual explanation Traces of fallen out (due to poor adhesion) SiC particles from the matrix A WC particle kept strong in the matrix, due to strong adhesion

  7. Kaptay / Day 4 / 7 1/4. Model explanation (a) Abrasive ability = mass loss of wood per 1 m of path (kg/m): Unit length (m) and width of ribbon (m) Density of wood (kg/m3) Surface concentration of particles (1/m2) Initial surface concentration of particles (1/m2) Probability that particles stay in the matrix (do not fall out) Probability is estimated from an energy balance (see next slide)

  8. Kaptay / Day 4 / 8 See J26, J27 1/5. Model explanation (b) Probability that particles stay in the matrix is proportional to the adhesion energy, while the probability the particles fall out of matrix is proportional to the kinetic energy of the turnings: See Kaptay / Day 2 / 15-16:

  9. Kaptay / Day 4 / 9 Story 2 The critical size of the particle, which can be separated from a fluid/fluid interface by the gravity (buyoancy) force In the metallurgical literature: Missing ρl and Θ ??? Poggi et al, 1969: Maru et al, 1971: MissingΘ ??? In the literature of colloid chemistry: Princen 1969 + Huh and Mason 1973 + Rapachietta and Neumann 1977 Detailed solutions,nothing missing, but only numerical results, no useful equation at all, although the problem could be solved since 1805.

  10. Kaptay / Day 4 / 10 See J23 2/1. Derivation The interfacial force, pulling the particle into the liquid (if Fσ> 0) See Day 2 / 15: At the critical state, i.e. when x = 2R: The sum of gravity and buyoancy forces (pulling in, if positive) The particle will be incorporated in the liquid, if:

  11. Kaptay / Day 4 / 11 See J23 2/2. Analysis i. If ρs > ρl, then incorporation is possible at some r > rcr ii. If Θ = 0o, then incorporation takes place at any r. iii. If ρs = ρl, then incorporation is possible only if Θ = 0o. iv. If ρs < ρl, then incorporation is possible only if Θ = 0o and if W > 2σ v. If σ= 1 J/m2, Θ = 90o, (ρs– ρl) = 1,000 kg/m3, then rcr = 12.4 mm. vi. If σ=0.07 J/m2, Θ = 90o, (ρs– ρl) = 1,000 kg/m3, then rcr = 3.3 mm.

  12. Kaptay / Day 4 / 12 2/3. Comparison (a) The Eötvös number: Introducing: Then, the critical Eötvös number: In the metallurgical literature: OK, but missing ρl. Θ Poggi et al, 1969: OK, but missingΘ Maru et al, 1971: In the literature of colloid chemistry: Princen 1969 + Huh and Mason 1973 + Rapachietta and Neumann 1977 numerical results can be converted, but the coefficient is different

  13. Kaptay / Day 4 / 13 2/4. Comparison (b) Huh and Mason 1973: including meniscus effect for 114 combinations:

  14. Kaptay / Day 4 / 14 2/5. Comparison with experiments (FL = float) See J93 Water–KI solution + particle (N = Nylon, P = Polymer, T = Teflon, A = Alumina)

  15. Kaptay / Day 4 / 15 See J93 Story 3 The critical condition of dynamic particle incorporation into liquids, when solid particles are blown on the surface of liquids It is searched in terms of the Weber number (kinetic to surface energy): If Wecr is known, the critical Rcr or vcr can be found. Wecr is inversely proportional to the dimensionless density, as the kinetic energy of the solid particle should be taken: Boundary condition: Wecr = 0, if Θ = 0o (spontaneous incorporation – see above). Majority of literature models do not satisfy this condition.

  16. Kaptay / Day 4 / 16 See J93 3/1. Dynamics of particle incorporation into liquids (1) Low velocity – no incorporation

  17. Kaptay / Day 4 / 17 See J93 3/2. Dynamics of particle incorporation into liquids (2) Medium velocity – incorporation (no bubbles)

  18. Kaptay / Day 4 / 18 See J93 3/3. Dynamics of particle incorporation into liquids (3) High velocity – incorporation with bubbles

  19. Kaptay / Day 4 / 19 3/4. A simplified model of incorporation (a)

  20. Kaptay / Day 4 / 20 3/5. A simplified model of incorporation (b) Energy balance (the condition of incorporation): Surface energy Gravity + buyoancy kinetic energy of particle Energy of deceleration due to drag

  21. Kaptay / Day 4 / 21 3/6. A simplified model of incorporation (c) Boundary condition 1: at Eo = 0, Wecr = 0, if Θ = 0o f = 1. Boundary condition 2: Wecr = 0, if Eo = Eocr p = 2.

  22. Kaptay / Day 4 / 22 See J93 3/7. Comparison to experiments

  23. Kaptay / Day 4 / 23 3/8. Bubble co-incorporation with particles Energy balance (the condition of incorporation with a bubble): The length of air cavity is longer by about times A diagram, allowing to design optimum blowing conditions for particles in LMI (Laser Melt Injection) Technology to produce particle reinforced surface composite materials

  24. Kaptay / Day 4 / 24 1 2 3.9. The LMI technology Verezub – Buza – Kálazi- Kaptay to be published

  25. Kaptay / Day 4 / 25 3.10. In-situ LMI production of surface-composites: Fe + Ti + WC = Fe + TiC + W (TiW)C Fe3W3C 50 nm TiC

  26. Kaptay / Day 4 / 26 Story 4 Penetration of liquids into porous solids Lifting pressure: ΔP = P - Pth Young, Laplace, 1805 (cylinder of radius r): The pressure due to gravity if liquid is at height h: Equilibrium height (at P = 0) Pth + Pg = 0: For a water/tree system: ρl = 1000 kg/m3, σ = 0.072 J/m2, Θ = 0o, r = 1 μm:heq = 14.7 m  interplay between transport rate of water to the upper leaves and desire to grow (trees are higher in high water vapour pressure environment, as evaporation low)

  27. Kaptay / Day 4 / 27 4.1. The threshold pressure of penetration Young, Laplace, 1805 (cylinder): Carman, 1941 (for any perfectly wetted soil, specific surface area S of particles, φ – their volume fraction): White 1982, Mortensen-Cornie 1987 (for different morphologies, any contact angle): The threshold contact angle (i.e. below which spontaneous penetration starts):in all above equations Kaptay-Stefanescu 1992(for porous bodies sintered from equal spheres): see J19 Threshold pressure is function of morphology of a porous solid, and thus (see J97):

  28. Kaptay / Day 4 / 28 4.2. Experiments on the threshold pressure of penetration [Baumli, Kaptay – to be published in MSE A]

  29. Kaptay / Day 4 / 29 4.3. Experimental conditions on the threshold pressure Pure NaCl, KCl, RbCl and CsCl salts (> 99.9.. %) Carbon plates 13x10x3 mm, > 99.99 %purity Polycrystalline graphite 1.76 g/cm3, 16 % open porosity, 12 μmgrain size (rounded grains), 250 nm roughness Salts premelted in low-pressure Ar gas 0.6 g Carbon + 0.02 g salt into furnace (Vpores>> Vsalt). High vacuum + > 99.999 % Ar gas of 1 bar. Heating and melting at a rate of 10 °C/min. Digital photographs + image analysis software

  30. Kaptay / Day 4 / 30 4.4. Results of penetration experiments Θth = 45o± 14o

  31. Kaptay / Day 4 / 31 4.5. Penetration into porous graphite CsCl (31o) RbCl (58o) t = 0 min t = 2 min t = 4 min

  32. Kaptay / Day 4 / 32 4.6. Concentration dependence Θth = 50o± 4o

  33. Kaptay / Day 4 / 33 4.7. Closely packed spherical model of penetration (a) At 0 ≤ h ≤ 1.63 Rp: At 1.63 Rp≤ h ≤2 Rp:

  34. Kaptay / Day 4 / 34 4.8. Closely packed spherical model of penetration (b) At Θ = 90o the liquid penetrates spontaneously only till h = Rp. At h > Rp, some outside pressure is neededfor further infiltration.

  35. Kaptay / Day 4 / 35 see J19, J90 4.9. Closely packed spherical model of penetration (c) The largest contact angle for which P is negative at any h: 50.7o Critical contact angle = 50.7 degrees

  36. Kaptay / Day 4 / 36 see J90 4.10. Closely packed spherical model of penetration (d) at Θ<77o: at Θ> 110o:

  37. Kaptay / Day 4 / 37 see J109 4.11. Infiltration of fibers A paper from the future

  38. Kaptay / Day 4 / 38 4.12. Infiltration of fibers along their axes Consider N fibers of diameter D, volume fraction φ. Consider a unit volume of composite V = 1x1x1 m. Then: Consider a liquid at height h (0 > h > 1 m). The total interfacial energy: The capillary force (see Day 2 / 3): The capillary pressure: Substitute + Young-equation, Pth = -Pσ: Same as (Day 4 / 27), White 1982, Mortensen-Cornie 1987:

  39. Kaptay / Day 4 / 39 4.3. Infiltration of fibers normal to their axes (a) The cross section of long, parallel cylinders: Model structure: fibers of equal diameter D, equal smallest separation δ. Then, the volume fraction of fibers:

  40. Kaptay / Day 4 / 40 4.14. Infiltration of fibers normal to their axes (b) In absence of gravity and pressure difference: The equilibrium depth of liquid (see Day 2 / 16): The distance of the top of the next layer(see previous slide): From the comparison of the two x values, the threshold contact angle is found as:

  41. Kaptay / Day 4 / 41 4.15. Infiltration of fibers normal to their axes (c) Detailed expressions for threshold pressure see J109

  42. Kaptay / Day 4 / 42 4.16. Conclusions. The limitations of a general equation The threshold contact angle has the following values: 90o for penetrating into a cylindrical pore, 90o for infiltration along long axes of cylindrical fibers, less than 45o for infiltration normal to fibers axes, 50.7o for penetration in-between closely packed, equal spheres. White 1982, Mortensen-Cornie 1987: works only, if along infiltration there is no curvature change (parallel to fibers, into cylinders). However, it does not work if along infiltration there is some curvature change (infiltration into preforms made of spheres and made of fibers, normal to fibers axes).

  43. Kaptay / Day 4 / 43 see J86 4.17. On pre-penetration (a) Pre-penetration: penetration of a liquid into a porous refractory at pressures, much below than that of the bulk penetration.

  44. Kaptay / Day 4 / 44 see J86 4.18. On pre-penetration (b) Possible explanation: pores have a periodically changing radii. Based on this, a model was built, the parameters of which were connected to measureable properties of the refractory. This model can help to design anti-penetration refractories (details see J86).

  45. Kaptay / Day 4 / 45 Story 5 Pushing or engulfment of particles by a solidification front Practical interest: the location of particles (precipitates) in solidified alloys have a great influence on their properties. Particles can be inside grains, or at grain boundaries. They are inside grains, if they are engulfed by the solidification front. • Questions to be answered: • Will be the particle engulfed spontaneously (i.e. even at very low front velocity)? • If not (i.e. if it is pushed), what is the critical front velocity of forced engulfment? • What is the influence of alloying elements?

  46. Kaptay / Day 4 / 46 5.1. Pushing or engulfment of particles . A spherical particle in front of the moving solid/liquid interface, having a local curvature Riat a smallest distance ho from the particle

  47. Kaptay / Day 4 / 47 5.2. Spontaneous engulfment of particles (a) The interfacial adhesion force between front and particle (see Day 2/28): Spontaneous engulfment, if the force is attractive, i.e. if Δσ < 0. For the ceramic particle (c) / solid metal (s) interface:

  48. Kaptay / Day 4 / 48 5.3. Spontaneous engulfment of particles (b) Theoretical prediction versus experimental facts (reasonable agreement)

  49. Kaptay / Day 4 / 49 5.4. Critical velocity of forced engulfment (a) The interfacial force - pushing the particle away from the front: The drag force - pushing the particle towards the front: At dynamic equilibrium (F = Fdrag) the particle is pushed from an equilibrium separation (h), being lower for increased front velocity (v): But how much is the critical separation, where the catastrophic pushing  forced engulfment phenomenon occur ???

  50. Kaptay / Day 4 / 50 5.5. Critical velocity of forced engulfment (b) The pressure, acting on the solidification front (due to its curvature): The “pushing” pressure (“adhesional”) by the particle: When the two pressures equal, the equilibrium separation follows:

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