Day 4. Modelling (some) interfacial phenomena George Kaptay A 4-day short course Kaptay / Day 4 / 1

Subjects to be covered today: Kaptay / Day 4 / 2 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

Kaptay / Day 4 / 3 Story 1 See J26, J27 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”

Kaptay / Day 4 / 4 Gas pressure Inductive coil Liquid metal Blown particles (SiC or WC, 50 micron) Rotating, water-cooled Cu-disc Nozzle Crucible with Fe 40 Ni 40 Si 6 B 14 Composite ribbon 12 mm x 50 micron 1/1. The production of AMMC ribbons Testing AMMC ribbons for their abrasive ability (against wood )

Kaptay / Day 4 / 5 1/2. Observations 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 Fe 40 Ni 40 Si 6 B 14 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 ? Definition: Abrasive ability is mass loss of wood per 1 m of path (g/m).

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

Kaptay / Day 4 / 7 1/4. Model explanation (a) Abrasive ability = mass loss of wood per 1 m of path (kg/m): Density of wood (kg/m 3 ) Unit length (m) and width of ribbon (m) Surface concentration of particles (1/m 2 ) Initial surface concentration of particles (1/m 2 ) Probability that particles stay in the matrix (do not fall out) Probability is estimated from an energy balance (see next slide)

Kaptay / Day 4 / 8 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: See J26, J27

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 Poggi et al, 1969: Missing ρ l and Θ ??? Maru et al, 1971:Missing Θ ??? Princen Huh and Mason Rapachietta and Neumann 1977 Detailed solutions, nothing missing, but only numerical results, no useful equation at all, although the problem could be solved since In the literature of colloid chemistry: In the metallurgical literature:

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

Kaptay / Day 4 / 11 2/2. Analysis ii. If Θ = 0 o, then incorporation takes place at any r. iii. If ρ s = ρ l, then incorporation is possible only if Θ = 0 o. i. If ρ s > ρ l, then incorporation is possible at some r > r cr iv. If ρ s 2σ v. If σ = 1 J/m 2, Θ = 90 o, (ρ s – ρ l ) = 1,000 kg/m 3, then r cr = 12.4 mm. vi. If σ = 0.07 J/m 2, Θ = 90 o, (ρ s – ρ l ) = 1,000 kg/m 3, then r cr = 3.3 mm. See J23

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

Kaptay / Day 4 / 132/4. Comparison (b) Huh and Mason 1973: including meniscus effect for 114 combinations:

Kaptay / Day 4 / 14 C KI. w% N 3 mm P 3 mm P 4 mm P 5 mm P 7 mm P 8 mm T 3 mm A 3 mm 0FL SINK FLSINK 15.9FL SINK 26FL SINK 44FL SINK 58FL SINK C KI. ll   P P        NPTA w%kg/m 3 mJ/m Water–KI solution + particle (N = Nylon, P = Polymer, T = Teflon, A = Alumina) 2/5. Comparison with experiments (FL = float) See J93

Kaptay / Day 4 / 15 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): See J93 If We cr is known, the critical R cr or v cr can be found. Boundary condition: We cr = 0, if Θ = 0 o (spontaneous incorporation – see above). Majority of literature models do not satisfy this condition. We cr is inversely proportional to the dimensionless density, as the kinetic energy of the solid particle should be taken:

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

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

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

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

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

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

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

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

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

3.10. In-situ LMI production of surface-composites: Fe + Ti + WC = Fe + TiC + W Kaptay / Day 4 / 25 TiC (TiW)C Fe 3 W 3 C 50 nm

Kaptay / Day 4 / 26 Story 4 Penetration of liquids into porous solids Young, Laplace, 1805 (cylinder of radius r): The pressure due to gravity if liquid is at height h : Lifting pressure: ΔP = P - P th Equilibrium height (at P = 0) P th + P g = 0: For a water/tree system: ρ l = 1000 kg/m 3, σ = J/m 2, Θ = 0 o, r = 1 μm: h eq = 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)

Kaptay / Day 4 / 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): Kaptay-Stefanescu 1992 (for porous bodies sintered from equal spheres): see J19 The threshold contact angle (i.e. below which spontaneous penetration starts): in all above equations Threshold pressure is function of morphology of a porous solid, and thus ( see J97):

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

Pure NaCl, KCl, RbCl and CsCl salts (> %) Carbon plates 13x10x3 mm, > % purity Polycrystalline graphite 1.76 g/cm 3, 16 % open porosity, 12 μm grain size (rounded grains), 250 nm roughness Kaptay / Day 4 / Experimental conditions on the threshold pressure Salts premelted in low-pressure Ar gas 0.6 g Carbon g salt into furnace (V pores >> V salt ). High vacuum + > % Ar gas of 1 bar. Heating and melting at a rate of 10 °C/min. Digital photographs + image analysis software

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

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

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

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

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

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

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

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

Kaptay / Day 4 / 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, P th = -P σ : Same as (Day 4 / 27), White 1982, Mortensen-Cornie 1987:

Kaptay / Day 4 / 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:

Kaptay / Day 4 / 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:

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

Kaptay / Day 4 / 42 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) Conclusions. The limitations of a general equation The threshold contact angle has the following values: 90 o for penetrating into a cylindrical pore, 90 o for infiltration along long axes of cylindrical fibers, less than 45 o for infiltration normal to fibers axes, 50.7 o for penetration in-between closely packed, equal spheres.

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

Kaptay / Day 4 / 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). see J86

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: i.Will be the particle engulfed spontaneously (i.e. even at very low front velocity)? ii.If not (i.e. if it is pushed), what is the critical front velocity of forced engulfment? iii.What is the influence of alloying elements?

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

Kaptay / Day 4 / 47 engulfment of particles (a) 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:

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

Kaptay / Day 4 / 49 ngulfment (a) 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: But how much is the critical separation, where the catastrophic pushing  forced engulfment phenomenon occur ??? At dynamic equilibrium (F = F drag ) the particle is pushed from an equilibrium separation (h), being lower for increased front velocity (v):

Kaptay / Day 4 / 50 ngulfment (b) 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:

Kaptay / Day 4 / 51 ngulfment (c) 5.6. Critical velocity of forced engulfment (c) Substituting:

Kaptay / Day 4 / 52 ngulfment (d) 5.7. Critical velocity of forced engulfment (d) see J57

Kaptay / Day 4 / 53 QuantityUnitthis paperexperiment [Stefanescu et al] in microgravity  clv deg90--  cs J/m  cls J/m h o cr nm73-- v cr  m/s ngulfment (e) 5.8. Critical velocity of forced engulfment (e) The Al/ZrO 2 system, R = 250  m

Kaptay / Day 4 / Concentration dependence (a) In the presence of the interface active solute: with the gradient force (see Day 2 / 9): The concentration gradient of the solute in the liquid metal, during steady state solidification and close to the planar solid/liquid interface The Belton equation (see Day 3 / 38): Distribution coefficient Diffusion coefficient Bulk concentration see J71

Kaptay / Day 4 / Concentration dependence (b) The ceramic/liquid metal interfacial energy in the Fe-S / Al 2 O 3 system, as function of the S-content in steel (left) and the iso-lines of the interfacial energy  cl are shown by dashed lines (right) (R = 15  m >> h o, v = 4  m/s) see J71

Kaptay / Day 4 / Concentration dependence (c) The final theoretical result with Mu = Mukai number: see J71 Dependence of the critical velocity of engulfment on the radius of the alumina particles in the Fe-S(0.01 w%) melt at 1823 K. Comparison of our theoretical results (black: standard critical velocity, pink: the critical velocity with concentration gradient) with the experimental results of Shibata et al (triangles) and Kimura et al (big square). Shift due to S-content Kimura et al Shibata et al

Kaptay / Day 4 / 57 Story 6 Stabilization of foams and emulsions by solid particles The primary reason of stabilization i. in the community of colloid chemistry: wettability, ii. in the community of metallurgy (with some exceptions): viscosity – surface tension. The particle stabilized foams and emulsions are stable „forever”. Therefore, viscosity cannot be the primary reason for their stabilization. Increased viscosity can decrease the rate of drainage, but can not make it zero. In contrary, „normal” (not particle stabilized) foams and emulsions have a constant drainage. Therefore, viscosity is the primary reason for their stabilization for a while. see J84, J102

Kaptay / Day 4 / A cross section of a typical metallic foam Small particles in cell walls cell walls

Kaptay / Day 4 / Stabilization of foams and emulsions by solid particles The role of particles at liquid/gas or liquid/liquid interfaces: In order to do the above, they should be as stable at the interface, as possible, compared to their position in any of the bulk phases. i. They should effectively separate two liquid/gas, or liquid/liquid interfaces, i.e. should ensure the stability of the thin liquid layer between large droplets or bubbles, ii. They should stabilize the thickness of the thin liquid film at a certain value, and not let more liquid to flow out of it (i.e. ensure zero drainage rate),

6.3. The condition of particle stability at the interface (a) The probability of the particle stability at the interface is proportional to: The energy to remove the particle from the interface should be as large, as possible (see Day 2 / 15 for equation of F σ ): Kaptay / Day 4 / 60 (Contact angle is defined through the water phase. For foams there is no „oil” phase. For metallurgy: water = metal, oil = slag).

6.4. The condition of film stability due to monolayer of particles (a) The „maximum capillary pressure” is the pressure by which particles in thin liquid films stabilize foams or emulsions: Kaptay / Day 4 / 61

Kaptay / Day 4 / The condition of film stability due to monolayer of particles (b)

Kaptay / Day 4 / The condition of film stability due to monolayer of particles (c) Area fraction of interface covered by particles

6.7. The joint analysis of the two equations (monolayer) (a) Both quantities must be as much positive as possible for the foam / emulsion to be stabilized by particles Kaptay / Day 4 / 64

6.8. The joint analysis of the two equations (monolayer) (b) Kaptay / Day 4 / 65

Kaptay / Day 4 / The joint analysis of the two equations (monolayer) (c)

Kaptay / Day 4 / The joint analysis of the two equations (monolayer) (d)

Kaptay / Day 4 / The emulsion stability diagram (monolayer)

Kaptay / Day 4 / Doublelayer, or 3-D network of particles (a) (for detalis see J84, J102)

Kaptay / Day 4 / Double-layer, or 3-D network of particles (b) (for detalis see J84, J102)

Kaptay / Day 4 / Conclusions Foams and emulsions can be stabilized by small solid particles in an efficient way, if: i. the contact angle is around 70 deg, ii. the particles are as small as possible (max 10 μm for metals and max. 1 μm for water) iii. the particles form a 3-D network in the thin liquid layer separating the large bubbles / droplets, iv. the aspect ratio of particles is as large as possible. Foams and emulsions can be destabilized by small solid particles, if the contact angle is above 90 (130) deg.

Kaptay / Day 4 / 72 Story 7 Droplet formation due gas blowing GA Brooks et al: ISIJ Int, 2003: introduced the Blowing number, as measure of probability of droplet formation:

Kaptay / Day 4 / Mechanism of droplet formation i. Instability of flow (Kelvin-Helmholtz) ii. Fingering: Pressure requested for droplet detachment due fingering in convective flow:

Kaptay / Day 4 / Pressure, requested to make a droplet

7.3. The critical condition of droplet formation Kaptay / Day 4 / 75

Kaptay / Day 4 / The possible size range of droplets (see Day 4 / 57) rcrrcr

7.5. The size distribution of droplets (a) Kaptay / Day 4 / 77

7.6. The size distribution of droplets (b) Kaptay / Day 4 / 78

Story 8…. Etc…. Kaptay / Day 4 / 79 There is a further endless list of interfacial phenomena in materials processing. I hope you got some feeling and understanding on how to rationalize and model them. Remember: Modeling is a good game. But it has only practical sense, if it tells you clues on new interrelations. It will be able to do so only, if it is based on a solid physical model.

Thanks for your attention I am looking forward to discussions and new problems to be solved now, or at any time… Kaptay / Day 4 / 80 Congratulations, you survived!!!!!