Day 4.

Modelling (some) interfacial phenomena

George Kaptay

kaptay@hotmail.com

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 Fe40Ni40Si6B14

Composite ribbon 12 mm x 50 micron

Oak sample

Composite ribbon

Load

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 Fe40Ni40Si6B14 on different substrates

On SiC: 135 deg

On WC: 60 deg

3.5cos1

cos1

SiC

WC

SiC

WC

W

W

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 / 71/4. Model explanation (a)

Abrasive ability = mass loss of wood per 1 m of path (kg/m):

pw CwAAbr 1

Density of wood (kg/m3)

Unit length (m) and width of ribbon (m)

Surface concentration of particles (1/m2)

popp PCC

Initial surface concentration of particles (1/m2)

Probability that particles stay in the matrix (do not fall out)

A

Acont

h

x

Probability is estimated from an energy balance (see next slide)

Kaptay / Day 4 / 81/5. 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:

x W

rrxeq )cos1(

W

rr

xrAcont

22 22

4

2

22

2 1

4

5.0 vA

Wr

vm

WA

E

EP

ww

cont

out

inp

ww Am 1

2241

v

WWCwrCwAAbr o

ppw

See J26, J27

Kaptay / Day 4 / 9

Story 2The critical size of the particle, which can be separated from a fluid/fluid

interface by the gravity (buyoancy) force

scr g

R

2

3Poggi et al, 1969: Missing ρl and Θ ???

Maru et al, 1971: )(

27.1ls

cr gR

Missing Θ ???

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.

In the literature of colloid chemistry:

In the metallurgical literature:

Kaptay / Day 4 / 102/1. Derivation

See Day 2 / 15:

r

xrF glx cos12 /,

At the critical state, i.e. when x = 2R:

1cos2 /, glx rF

The interfacial force, pulling the particle into the liquid (if Fσ > 0)

The sum of gravity and buyoancy forces (pulling in, if positive)

lsg grF 3

3

4

The particle will be incorporated in the liquid, if: 0 FFg

)(

)cos1(5.1

lscr g

r

See J23

Kaptay / Day 4 / 112/2. Analysis

)(

)2(5.1

)(

)cos1(5.1

lslscr g

W

gr

ii. If Θ = 0o, then incorporation takes place at any r.

iii. If ρs = ρl, then incorporation is possible only if Θ = 0o.

i. If ρs > ρl, then incorporation is possible at some r > rcr

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.

See J23

Kaptay / Day 4 / 12

scr g

R

2

3Poggi et al, 1969: OK, but missing ρl. Θ

Maru et al, 1971: )(27.1

lscr g

R

OK, but missing Θ

Princen 1969 + Huh and Mason 1973 + 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))(

)cos1(5.1

lscr g

r

The Eötvös number: gr

Eo l

24l

s

Introducing:

Then, the critical Eötvös number: 1

)cos1(6

crEo

Kaptay / Day 4 / 13 2/4. Comparison (b)

)(

)cos1(5.1

lscr g

r

R

zo

h

F F

z

1

3

2

3

2

Huh and Mason 1973: including meniscus effect for 114 combinations: y = 0,7775x

R2 = 0,9912

0

0,5

1

1,5

2

0 0,5 1 1,5 2 2,5Kaptay

Hu

h,

Ma

so

n

Kaptay / Day 4 / 14

CKI.

w%

N3 mm

P3 mm

P4 mm

P5 mm

P7 mm

P8 mm

T3 mm

A3 mm

0 FL FL FL FL FL SINK FL SINK

15.9 FL FL FL FL FL FL FL SINK

26 FL FL FL FL FL FL FL SINK

44 FL FL FL FL FL FL FL SINK

58 FL FL FL FL FL FL FL SINK

CKI. l P N P T A

w% kg/m3 mJ/m2 1140 1230 2200 3970

0 997 72.1 63.5 45.3 100.3 43.5 13.1 7.5 6.6 2.0

15.9 1114 73.2 56.9 59.3 101.3 36 28.0 13.7 7.0 1.7

26 1213 73.8 53.4 65 101.1 32.3 -- 39.2 7.4 1.6

44 1441 74.9 48.3 69.3 99.3 27.5 -- -- 8.4 1.4

58 1664 75.8 45.5 67.3 96.5 25.4 -- -- 9.8 1.4

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 3The 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):

2vRWe l

See J93

If Wecr is known, the critical Rcr or vcr can be found.

Boundary condition: Wecr = 0, if Θ = 0o (spontaneous incorporation – see above). Majority of literature models do not satisfy this condition.

l

s

Wecr is inversely proportional to the dimensionless density, as

the kinetic energy of the solid particle should be taken:

1

crWe

Kaptay / Day 4 / 163/1. Dynamics of particle incorporation into liquids

(1)

Low velocity – no incorporation

See J93

Kaptay / Day 4 / 173/2. 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)

gas

liquid

particle

gasgas

gas

particle

particle

liquidliquid

Stage I.

Stage III.

Stage II.

gasparticle

liquid

Stage IV.a

gas

particle

liquid

Stage IV.b

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

cos4)(4 lg22 rrE sgsl

lg

cos

slsg

lg2

lg

4

rf

AE

pgrfrvrvr lls )1(3

4)cos(4148.0

3

2 422323

Gravity + buyoancy

Kaptay / Day 4 / 21

3/6. A simplified model of incorporation (c)

2vrWe l

l

s

Boundary condition 1: at Eo = 0, Wecr = 0, if Θ = 0o f = 1.

Boundary condition 2: Wecr = 0, if Eo = Eocr p = 2.

pgrfrvrvr lls )1(3

4)cos(4148.0

3

2 422323

222.0*

)1(5.0)cos(6

Eopf

Wecr

gr

Eo l

24

222.0*

)1()cos1(6

EoWecr

Kaptay / Day 4 / 22

3/7. Comparison to experiments

l

s

0

3

6

9

12

15

0 1 2 3*

We c

r

Nylon

Teflon

Nylon

Teflon

See J93

222.0*

)1()cos1(6

EoWecr

grEo l

24

Kaptay / Day 4 / 23

3/8. Bubble co-incorporation with particlesEnergy balance (the condition of incorporation with a

bubble): 2)(

3

4)cos(437.0

3

2 422323 grrvrvr lls

The length of air cavity is longer by about times

0

10

20

30

40

0 0,5 1 1,5 2 2,5 3*

We

incorporation with bubbles

incorporation with no bubbles

no incorporation

57.0*

)()cos(6,

EoWe bcr

A diagram, allowing to design optimum blowing conditions for particles in LMI (Laser Melt Injection) Technology to produce particle reinforced surface composite materials

3

2

1

V

V

3.9. 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

Fe3W3C

50 nm

Kaptay / Day 4 / 26 Story 4Penetration of liquids into porous solids

cos2

rPth

Young, Laplace, 1805 (cylinder of radius r):

The pressure due to gravity if liquid is at height h: ghP lg

Lifting pressure:

ΔP = P - Pth

Equilibrium height (at P = 0) Pth + Pg = 0:

cos2

rgh

leq

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)

h

r

h

prsmQV

cos

4

8)/(

3

43

P

h

PthBPthA

A B

PthC

C

Kaptay / Day 4 / 27

4.1. The threshold pressure of penetration

cos

2

RPth

Young, Laplace, 1805 (cylinder):

1

SPthCarman, 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):

cos1 SPth

Kaptay-Stefanescu 1992 (for porous bodies sintered from equal spheres): see J19

oth 7.50

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):

oth 90

)(1

fSPth

Kaptay / Day 4 / 28

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

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 μm grain size (rounded grains), 250 nm roughness

Kaptay / Day 4 / 29

4.3. Experimental conditions on the threshold pressure

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

4.4. 4.4. Results of penetration experimentsResults of penetration experiments

)cos1(/ vlW

Kaptay / Day 4 / 30

Salt T, oC Θ, deg. σl/v, mJ/m2 W, mJ/m2 Penetration NaCl 810 113o 114 69 No KCl 780 78o 99 120 No RbCl 740 58o 95 145 No CsCl 645 31o 92 171 Yes

Θth = 45o ± 14o

4.5. Penetration into porous graphite

t = 0 min

t = 2 min

t = 4 min

CsCl (31o) RbCl (58o)

Kaptay / Day 4 / 31

4.6. Concentration dependence

30

35

40

45

50

55

60

0 20 40 60 80 100

mol % RbCl

Co

nta

ct

an

gle

, de

g.

CsCl RbCl

no penetrationpenetrationpartial

penetration

criticalcontactangle

Kaptay / Day 4 / 32

Θth = 50o ± 4o

4.7. Closely packed spherical model of penetration (a)

h Rp

Kaptay / Day 4 / 33

cos181.1

pporep R

h

RP

2

2907.01

pppore R

h

R

h

At 0 ≤ h ≤ 1.63 Rp:

At 1.63 Rp ≤ h ≤ 2 Rp:

cos63.281.1cos181.1

pporeppporep R

h

RR

h

RP

22

63.163.12907.02907.01

pppppore R

h

R

h

R

h

R

h

-5

0

5

10

15

20

0 2 4

h/Rp

PR

/

= 180o

= 90o

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 needed for further infiltration.

Critical contact angle = 50.7 degrees

-20

-15

-10

-5

0

5

0 2 4

h/Rp

PR

/

= 50.7o

= 0o

Kaptay / Day 4 / 35

4.9. Closely packed spherical model of penetration (c)

see J19, J90

The largest contact angle for which P is negative at any h: 50.7o

-20

-10

0

10

20

0 45 90 135 180

, deg

Pth

Rp /

Cylinder-model

CPES-model

CPES-model

Cylinder-model

Kaptay / Day 4 / 36

4.10. Closely packed spherical model of penetration (d)

cos20

pth R

P

at Θ > 110o:at Θ < 77o:

cos63.00.4

pth R

P

see J90

Kaptay / Day 4 / 37

4.11. Infiltration of fibers

A paper from the future

see J109

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:

2

4

DN

Consider a liquid at height h (0 > h > 1 m). The total interfacial energy:

sgsl hhDnAcG )1(lg

The capillary force (see Day 2 / 3):

The capillary pressure:

cos1

4

DPthSubstitute + Young-equation, Pth = -Pσ :

lg

cos

slsg dh

dGF

dh

dG

A

FP

1

1

Same as (Day 4 / 27), White 1982, Mortensen-Cornie 1987:

cos1 SPth

DLD

LDS

4

25.0 2

Kaptay / Day 4 / 39

4.3. Infiltration of fibers normal to their axes (a)

Dd

liquid penetration

liquid penetrationliquid penetration

liquid penetration liquid penetration

liquid penetration

xt

The cross section of long, parallel cylinders:

Model structure: fibers of equal diameter D, equal smallest separation δ. Then, the volume fraction of fibers:

D

dd * 2*132 d

Kaptay / Day 4 / 40

4.14. Infiltration of fibers normal to their axes (b)

liquid

gas

xo

unit celllsl

llg

In absence of gravity and pressure difference:

Dxx

oo *

2

cos1* o

eqx

The equilibrium depth of liquid (see Day 2 / 16):

*)1(2

3* dtxThe 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:

1*)1(3cos dth

Kaptay / Day 4 / 41

0

20

40

60

80

100

0 0,2 0,4 0,6 0,8 1

d *

th

, de

gre

e

parallel

normal

1*)1(3cos dth

4.15. 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).

cos1 SPth

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).

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.

Kaptay / Day 4 / 434.17. On pre-penetration (a)

Gas inlet

Gas outlet

Crucible

Metal

Porous refractoryLaCrO3 heaterVideo

recorder

ImageintensifierCamera X-ray generator

Cruciblewith a hole inthe bottom

Load

5 m m

h m a x

P e n e tra tin g m e rc u ry

P o ro u s re fra c to ry

0

10

20

30

40

0 5 10 15 20 25 30

External pressure, Po / kPa

Max

imum

pen

etra

tion

hei

ght,

hm

ax /

mm

1st run exper.

2nd run exper.

Bulk-penetration

Pre-penetration

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 / 444.18. 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).

see J86

liquid metal

refractory wallcylindrical capillaries

liquid metal

rmin

rmax

Ll

r

0

5

10

15

20

25

0 5 10 15 20 25 30

External pressure, Po / kPa

Max

imum

pe

netr

atio

n he

ight

, hm

ax /

mm

Kaptay / Day 4 / 45

Story 5

Pushing or engulfment of particles by a Pushing or engulfment of particles by a solidification frontsolidification 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

5.1. Pushing or engulfment of particlesPushing or engulfment of particles

. A spherical particle in front of the moving solid/liquid interface, having a local curvature Ri at a smallest distance ho from the particle

cR

Ri

2 r

s

l

h O

v

iR

R

Kaptay / Day 4 / 47

5.2. Spontaneous engulfment of particlesengulfment of particles (a) (a)

The interfacial adhesion force between front and particle (see Day 2/28):

mxd

dkF

)(

2

2

2

)(1

2

xd

dRF

Spontaneous engulfment, if the force is attractive, i.e. if Δσ < 0.

slclcs 2

For the ceramic particle (c) / solid metal (s) interface: clvcsv lvsl )05.010.0(

clvlvlvclcs cos11.001.0

)cos22.108.0( clvlvcv

clvlvcgcl cos

Kaptay / Day 4 / 48

Theoretical prediction versus experimental facts (reasonable agreement)

5.3. Spontaneous engulfment of particlesengulfment of particles (b) (b)

System cv, J/m2 lv, J/m2 clv, deg , J/m2 Est. P/E Exp. P/E Al/SiC 1.45..1.9 1.1 30-60* +0.20..+1.14 P P

Al/graph 2.1 1.1 40-60* +0.99..+1.34 P P Al/TiC 2.6 1.1 0 <1.17 P/E E

Al/Al2O3 1.0 1.1 60-90 +0.24+0.91 P P Al/ZrO2 0.8 1.1 90 +0.71 P P Al/SiO2 0.35 1.1 70 -80 0 P P Al/TiB2 3.0 1.1 0 +1.57 P P Zn/ZrO2 0.8 0.82 115-155 +1.15...1.64 P P Si/SiC 1.45..1.9 0.83 40 -0.07...+0.37 E/P ? E Si/SiO2 0.35 0.83 90 -0.12 E E

Si/Al2O3 0.90 0.83 70-90 -0.04...+0.43 P/E ? P Si/ZrO2 0.7 0.83 70 -0.24 E E

Kaptay / Day 4 / 49

5.4. Critical velocity of forced engulfment (a)ngulfment (a)

The interfacial force - pushing the particle away from the front:

2

2

1

16

odrag h

RvF

2

2

)(1

2

xd

dRF

The drag force - pushing the particle towards the front:

eqohR

dv

)1(

3

2

But how much is the critical separation, where the catastrophic pushing forced engulfment phenomenon occur ???

At dynamic equilibrium (F = Fdrag) the particle is pushed from an equilibrium separation (h), being lower for increased front velocity (v):

Kaptay / Day 4 / 50

5.5. Critical velocity of forced engulfment (b)ngulfment (b)

cR

Ri

2 r

s

l

h O

v

RRp sl

i

slL

22

The pressure, acting on the solidification front (due to its curvature):

3

2

)(2

oD hd

dp

The “pushing” pressure (“adhesional”) by the particle:

3

2

dR

hsl

o

When the two pressures equal, the equilibrium separation follows:

Kaptay / Day 4 / 51

5.6. Critical velocity of forced engulfment (c)ngulfment (c)

3/13/43/13/2

)1(3 eqeq

slcls

R

dv

Substituting:

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1

eq

v, m/s

PET

cr

v cr

a.)

25.0cr

Kaptay / Day 4 / 52

5.7. Critical velocity of forced engulfment (d)ngulfment (d)

25.0cr

0

100

200

300

0 0,2 0,4 0,6 0,8 1v, m/s

PET

v cr

hocr

1.

2.

b.)

hoeq, nm

324 dRh

sl

clscro

3/43/13/2157.0

R

dv slclscr

see J57

Kaptay / Day 4 / 53

Quantity Unit this paper experiment [Stefanescu et al]in microgravity

clv deg 90 --

cs J/m2 0.82 --

cls J/m2 0.71 --

hocr nm 73 --

vcr m/s 0.67 0.5 .. 1.0

3/43/13/2157.0

R

dv slclscr

5.8. Critical velocity of forced engulfment (e)ngulfment (e)

The Al/ZrO2 system, R = 250 m

Kaptay / Day 4 / 54

5.9. Concentration dependence (a)

In the presence of the interface active solute: 0 draggrad FFF

dx

dC

dC

dRkF cl

grad 2with the gradient force (see Day 2 / 9):

k

kC

D

v

dx

dCo

x

1

0

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):

Cnmoclcl 1ln

Distribution coefficient

Diffusion coefficient

Bulk concentration

vCnD

CknmRF

o

ograd

)1(

)1(2 2

see J71

Kaptay / Day 4 / 55

5.10. Concentration dependence (b)

h0

R

Ri

solid

particle

liquid

1.584

1.583

1.582

The ceramic/liquid metal interfacial energy in the Fe-S / Al2O3 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 >> ho, v = 4 m/s)

1.55

1.6

1.65

1.7

1.75

1.8

0 0.05 0.1 0.15 0.2Co, w %

cl, J

/m2

see J71

05

101520253035

0 10 20 30

v cr, m

/s

R, m

Kaptay / Day 4 / 56

5.11. Concentration dependence (c)

2

3/1

3/1

6

)1(1

)1(

471.0

ML

ocr

kMu

vv

32

)(

)1(aR

CnkD

CknmMu

slo

o

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 / 58

6.1. A cross section of a typical metallic foam

Small particles in cell walls

cell walls

Kaptay / Day 4 / 59

6.2. 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. 6.3. The condition of particle stability at the interfaceThe condition of particle stability at the interface (a) (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 Fσ):

Kaptay / Day 4 / 60

2cos1

(Contact angle is defined through the water phase. For foams there is no „oil” phase. For metallurgy: water = metal, oil = slag).

22 cos1 RdxFGx

xeq

remove

6.4. 6.4. The condition of The condition of filmfilm stability due to monolayer stability due to monolayer of of particlesparticles (a) (a)

TThe he „„maximum capillary pressure” is the pressure by which particles maximum capillary pressure” is the pressure by which particles

in thin liquid films stabilize foams or emulsionsin thin liquid films stabilize foams or emulsions::

Kaptay / Day 4 / 61

h

Rthin liquid film with P2

large droplet with P1 r

solid particle

liquid/liquid interfacesolid/liquid interface

L

large droplet with P1

H

Kaptay / Day 4 / 62

0

0,5

1

1,5

2

-2 -1 0 1 2Pc*

H*

0o

30o

60o

90o

120o

Pc*max

6.5. 6.5. The condition of The condition of filmfilm stability due to monolayer stability due to monolayer of of particlesparticles (b) (b)

cos2max

RpPc

Kaptay / Day 4 / 63

0

2

4

6

8

10

0 0,2 0,4 0,6 0,8 1

f

p

0.907

cos2max

RpPc

6.6. 6.6. The condition of The condition of filmfilm stability due to monolayer stability due to monolayer of of particlesparticles (c) (c)

Area fraction of interface covered by particles

6.7. 6.7. The joint analysis of the two equations The joint analysis of the two equations (monolayer)(monolayer) (a) (a)

22 cos1 RGremove

cos2max

RpPc

Both quantities must be as much positive as possible

for the foam / emulsion to be stabilized by particles

Kaptay / Day 4 / 64

6.8. 6.8. The joint analysis of the two equations The joint analysis of the two equations (monolayer)(monolayer) (b) (b)

0

0,2

0,4

0,6

0,8

1

0 30 60 90 120 150 180

, deg

p

Kaptay / Day 4 / 65

2cos1

Kaptay / Day 4 / 66

0

0,2

0,4

0,6

0,8

1

0 30 60 90 120 150 180

, deg

w-f

1,

o-f1

water film oil film

6.9. 6.9. The joint analysis of the two equations The joint analysis of the two equations (monolayer)(monolayer) (c) (c)

cos

Kaptay / Day 4 / 67

0

0,05

0,1

0,15

0,2

0 30 60 90 120 150 180

, deg

o/w

, w

/o

70o110o

o/w emulsion + + foams

w/o emulsion

6.10. 6.10. The joint analysis of the two The joint analysis of the two equations (monolayer)equations (monolayer) (d) (d)

Kaptay / Day 4 / 68

0

0,25

0,5

0,75

1

0 30 60 90 120 150 180

, deg

w

o/w emulsion w/o emulsion

no

emulsIon

no

emulsIon

6.11. 6.11. The The emulsion stability diagram emulsion stability diagram (monolaye(monolayer)r)

Kaptay / Day 4 / 69

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 30 60 90 120 150 180

, deg

o/w

w

/o

86o94o

o/w emulsion+ foam

w/o emulsion

6.12. Doublelayer, or 3-D network of particles (a)6.12. Doublelayer, or 3-D network of particles (a) (for detalis (for detalis see J84, J102see J84, J102))

Kaptay / Day 4 / 70

0

0,25

0,5

0,75

1

0 30 60 90 120 150 180

, deg

w

o/w emulsions

w/o emulsions

no

emulsIon

no

emulsIon

6.13. Double-layer, or 3-D network of particles (b)6.13. Double-layer, or 3-D network of particles (b)(for detalis (for detalis see J84, J102see J84, J102))

Kaptay / Day 4 / 71

6.14. Conclusions6.14. Conclusions

Foams and emulsions can be stabilized by small solid particlesFoams and emulsions can be stabilized by small solid particles in an efficin an efficient way, if:ient way, if:

i. the contact angle is around 70 deg,i. the contact angle is around 70 deg,

ii. ii. the particles are as small as possible (max 10 the particles are as small as possible (max 10 μm for metals μm for metals and max. 1 μm for water)and max. 1 μm for water)

iii. iii. the particles form a 3-D network in the thin liquid layerthe particles form a 3-D network in the thin liquid layer separating the large bubbles / droplets,separating the large bubbles / droplets,

iv. the aspect ratio of particles is as large as possible.iv. the aspect ratio of particles is as large as possible.

Foams and emulsions can be Foams and emulsions can be dedestabilized by small solid stabilized by small solid particles, ifparticles, if the the contact angle is contact angle is above 90 (130)above 90 (130) deg deg..

Kaptay / Day 4 / 72

Story 7

Droplet formation due gas blowingDroplet formation due gas blowing

GA Brooks et al: ISIJ Int, 2003: introduced the Blowing number, as measure of probability of droplet formation:

L

GGB

g

uN

2

2

Kaptay / Day 4 / 73

7.1. 7.1. Mechanism of droplet formationMechanism of droplet formation

2r

2r

p

p

2r

a cb

i. i. Instability of flowInstability of flow (Kelvin-Helmholtz) (Kelvin-Helmholtz)

ii. Fingering:ii. Fingering:

3

52 rg

rpD

Pressure requested for droplet detachment due fingering in convective Pressure requested for droplet detachment due fingering in convective

flow:flow:

Kaptay / Day 4 / 74

500

1000

1500

2000

0 5 10 15 20 25 30

r, mm

PD, P

a

rcr

pD,min

pG

rminrmax

p

7.2. Pressure, requested to make a droplet7.2. Pressure, requested to make a droplet

7.3. The critical condition of droplet formation

0drdpD

g

rcr

2.1

gpD 65.3min,

min,2 65.35.0 DGGG pgup

crBGG

B Ng

uN ,

2

3.7

Kaptay / Day 4 / 75

Kaptay / Day 4 / 76

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25NB

r, m

m

NB,cr

max

ave

min

7.4. The possible size range of droplets

(see Day 4 / 57)

rcr

7.5. The size distribution of droplets (a)

max

min2max

2min

maxmin,

max

2max

2

max,

ln2

)(2

ln2

)(2

100%

r

rrrrr

N

N

r

rrrrr

N

N

Cum

crB

B

crB

B

2

,

,maxmin/ 11

B

crB

crB

B

N

N

N

Nr

crrrr

Kaptay / Day 4 / 77

7.6. The size distribution of droplets (b)

Kaptay / Day 4 / 78

0

20

40

60

80

100

0 1 2 3 4r/rcr

Cu

m%

1

2

1.5

NB/NBcr

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.

kaptay@hotmail.com

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!!!!!