EMULSIFICATION IN TURBULENT FLOW

A. GOAL AND PROBLEMS 1. REFERENCES H = Hinze, J. O. Turbulence, McGraw-Hill, New York, 1975. TL = Tennekes, H; Lumley J. L.; A first course in turbulence, The MIT

Press, Cambridge, 1972 B = Batchelor, G. K The theory of homogeneous turbulence, Cambridge,

1953. L = Levich, V. G. Physicochemical Hydrodynamics, Prentice Hall,

Englewood Cliffs, New Jersey, 1962. T = Coulaloglou, C. A.; Tavlarides, L.L Description of interaction

processes in agitated liquid-liquid dispersions. Chemical Engineering Science 1977, 32, 1289.

P = Princen, H. M. The equilibrium shape of interfaces, drops, and bubbles. In Surface and Colloid Science, Matijevic, E. Ed.; Wiley, New York, 1969, Vol. 2, p. 1.

2. NARROW SLIT HOMOGENIZER

( )Lu

VQp

S

3∆==ε

• Isotropic turbulence (chaotic and non-directional)

3. OVERALL GOAL (1) Drop size distribution vs. stirring (ε), interfacial tension (σ),

viscosity ratio (ηd/ηc) and drop life-time (τD) (surfactant adsorption Γ and distribution).

(2) Phase inversion 4. IMMEDIATE GOAL

• Understand the emulsification mechanism. • Describe quantitavely variations of drop size ∆R vs. variation of

life time ∆τD due to surfactants. ∆N

R RR+∆R

τD

τD+∆τD

N 5. APPROACH Grasp the main factors and the overall scheme

(a) Account for polydispersity by considering discrete set of sizes

(b) Neglect dependence of rate constants on size (c) Assume that drops split always on two equal size drops (d) Assume that only equal size drops coalesce

B. DROPS AND FILMS 1. MAIN EVENTS DURING FLOCCULATION OR COALESCENCE

F = Driving Force (Buoyancy F , Brownian, Turbulent, Surface Interactions)

Rd=43

3π ∆ρg

Main parameters

• Drop Shape • Life Time, τ

• Rate of Thinning, V V d hd t

= − ;

VVhdcrh

inith

1∝=τ ∫

• Critical Thickness of Rupture, hcr • Interfacial Mobility (Surfactant)

2. TRANSITION SPHERE - FILM

22

Ta

Re 2 hFV

V

σπ=

Inversion (Film Formation) Thickness, hi:

1Ta

Re =VV

σπ

=2

Fhi

Inversion at the same dissipation of energy •

3. FILM RADIUS ( ) FRPP lf =π− 2

cldf R

PPP σ+== 2

FR

Rc

=σπ 22 σπ

=2

2 cRFR

3

34

cRgF ρ∆π=

4. LIFE TIME OF DROPS DRIVEN BY BUOYANCY

VVhd

h

h

1~final

in

∫=τ

( )52

32

Re2Ta1~

322~

32

ccc

c RRFhVR

RFhV

ηπσπ=

ηπ=

• Increasing drop radius increases the driving force but thereby

increases also the film radius, thus decreasing the drainage rate and increasing life time.

Qb + ll

0

50

100

150

0.00 0.02 0.04 0.06 0.08 0.10

Dwodqhldms S`xknq qdfhld

τ < /-034 . Qb

Life

time,

τ, s

ec

Small drops (no film)

1 / Rc , mm-1

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Life

time,

τ, s

ec

5. ROLE OF THE ADSORPTION

Dis

join

ing

pres

sure

Thickness

Pc'

Pc''

∆P P= c- ( )Π t

2

d

PRσ∆ = − Π 36

hA Beh

−κΠ = − +π

Ads

orpt

ion,

Γ

Concentration, C

Initialconcentration~ 100 CMC CMC

Effectiveconcentration

• The effective concentration is much lower than the real bulk

concentration.

Surface coverage θ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Life

time,

sec

onds

0

20

40

60

80

100

120

140

1.5x10-4 wt.% BLG1.5x10-4 wt.% β−casein1x10-3 wt.% β−casein3x10-2 wt.% β−casein

Colliding Drop Technique

θCR = 0.75

• Variation of bulk concentration, C, and surface age, tW.

Surface potential, ψs, mV

0 20 40 60 80 1

Life

time,

τ, s

econ

ds

0

20

40

60

80

100

120

140

5 µM10 µM25 µM

1 µMBubbles: Drops:

10 µM50 µM100 µM146 µM200 µM

DDBS (12 mM NaCl)• Both lead to change of Γ. 00 • For DDBS there is a rather sharp transition between unstable and

stable films, depending on the surface potential. • The bubbles are stable with surface potential higher than

=70 – 80 mV. thSψ

6. DRAINAGE RATE AND LIFE TIME OF THIN FILMS • Surfactant Balance

Convection + Diffusion = 0 • Stress Balance

Shear = Elastic + Viscous

hh

VV S+= 1Re

23

Re 32

RPhV

η∆=

TkDh SS Γ

η= 6

ScrcrC

hhxx

xxxhP

R /;1ln23 2

2

2

=

+−η=τ

- with θ = 0.1, hS = 300 nm – strong surface mobility - with hcr = 40 nm, τ = 21 s

• Role of the surface viscosity

2~elasticitySurfaceviscositySurface

RhSS

ηη

• The true surface viscosity, ηS, can play a role only for small films and mobile surfaces when hS >> h, but then ηS ≈ 0.

• For dense monolayers hS ~ 0.5 ÷ 1 nm - the surface mobility is not important.

Interfacial Viscosity

2

2

rv

rzv r

Sr

∂∂η+

∂Γ∂

Γ∂σ∂=

∂∂η

( ) SSSSG

rS hR

RDE

rvr

ηη

ηη

η∂∂η∂σ∂ 2

222 ~//~

//~

ViscousElastic

Surface viscosity is coupled with surface mobility (through hS) and can play a role ONLY for mobile surfaces.

•

•

•

But then the TRUE surface viscosity is small.

Even for small films (R = 10 µm) and mobile surfaces (hS = 10 nm) the effect of surface viscosity is comparable to surface elasticity only if ηS > 0.1 sp.

Apparent Dilatational Viscosity

;~~2

2

2

appdG

SS

hR

RR

EhD

hh

ηηη

diffGS

Gappd tED

RE ==η2

The apparent surface viscosity is due to dissipation through diffusion and is NOT related to intermolecular interaction

•

•

•

appdη depends on the film radius, i.e. it is NOT a property of the

adsorbed layer and CANNOT be directly measured in a separate experiment.

It is extremely high even at low concentration:

( ) sp1.0~10

101~ 5

23

−

−ηappd

7. BASIC EQUATIONS FOR FILM DRAINAGE Lubrication

h

z

r

2

2

dzv

rp r∂η=

∂∂

0=∂∂

zp

0=∂∂+

∂∂

zv

rv zr

Boundary Condition at z = ±h/2

dtdhVvUv zr 2

12

=−==

2

2

rv

rzv r

Sr

∂∂η+

∂σ∂=

∂∂η

zcD

rD

rU

S ∂∂−=

∂Γ∂−

∂∂Γ 2

2

0

( ) 22

10 RzCrCc +=

( )∫ −π=FR

drrppF0

02

Emulsions

∂∂+

∂∂η=

∂∂ρ 2

2

2

2

zv

rv

rvv rrrr

z

8. RUPTURE OF THINNING FILMS

Wave pressure:

( ) ζ ′′σ+ζΠ′+=2

hPP plw

9. EMULSION FILMS SURFACTANT IN THE DROP PHASE (SYSTEM II)

Uniform Distribution of the Surfactant on the Film Surface

(a) (b)

0=∆Γ � 0=Γ

∆Γ=σ∆ GE

� ReII VVV pure >>=

• The surfactant has no effect on the rate of thinning.

Chemical Demulsification

DemulsifierEmulsifier Emulsifier

• The demulsifier fills in the spaces between the adsorbed

emulsifier molecules damps the interfacial tension gradient ∆σ and increases many times the rate of thinning and coalescence.

Comparison of the Life Times of Systems I and II

System I

Water

Water

Oil

Oil

System II

∫∞

−=τcrh

Vhd

;s3. O/Wτ0W/O =τ1000100Re

÷=≈VpureV

~O/W

W/O

W/O

O/Wτ

τ

V

V s35=

The ONLY DIFFERENCE between the two systems is the SURFACTANT LOCATION.

•

• The surfactant concentration does not affect the life time of the emulsion W/O (system II).

10. GENERALIZED BANCROFT RULE

System I

Water

Water

Oil

Oil

System II

c1~ ; 1 ~S ReG

h PV VV h

− Π τ = + E

( )ττ

W/O

O/W

O/W

W/O/≈

××

−−

−8 10 52 3E

PPG

c

cHYDRODYNAMICS INTERACTION

124 341 244 344

ΠΠ

If Π in both phases is zero, τW/O

C. DROP SIZE DISTRIBUTION

1. ISOTROPIC TURBULENCE (KOLMOGOROV) ε - rate of energy dissipation per unit mass

lu

uluu

322 ~

/~1 ρρρ

τερ

20

3/23/22 ~;~ uplu ρε

2. PRESENT THEORIES (STEADY DISTRIBUTION)

coalescebreak tN

tN

∆∆=

∆∆

break

coalescence

EE

eNNNσ−

τ=

τ=∆

br

br

br

brbr

ududEdE /~;~;~ br

232 τρσσ

( ) ( )443442143421

efficiency

/

frequencycoll.

22

coalesce

contacteife ττ−×=∆∆ eNdu

tN

3

contact /~;/~ dNud Φτ

2/1eife3/4

3/2 ln ετ+Φ−=

εσ dd

3. APPROXIMATION OF ∆N/N BY δ - FUNCTION ∆N

RK RR t( )c

( )t t+∆

( )t

N

( )( )

( ) dRetN

tRdN cRR 2, −α−πα= ( )tα=α

• Kolmogorov’s drops become important at longer times

( ) ( )∫=KR

K tRdNtN0

,

Since

( ) 1== ∫∫ dRRfNdN

one can set

απ=

−δ= adRa

RRaN

dN c1

• This reduces most equations to the monodisperse case, including

those for coalescence rate:

( )2coal ~~ cRR RNdNdNV ∫ ′

• Justifies assumptions (b) and (d). • Fails at Rc t RK (small drops). • Works better when coalescence is present.

D. TURBULENT PRESSURE AND DROP SHAPE

• Eddies are NOT molecules

[TL] Correlation coefficients

=−

eddiesfor1-0.5moleculesfor10 6

(H., p. 310) For large Reynolds numbers and small distance, r, the pressure gradient does NOT depend on time

( ) ( 3/4222 4 rApp )BA ερ=−

( ) ( ) 3/22 2~~ rApp ερ∆∆

( ) ( ) pprp ∆+= 0

• Laplace equation [P] Outside pressure p = p(r), inside pressure pin = const.

( ) ( ) →×σ=− curvatureinprp “Equilibrium” shape of stable drops.

2l

2Rturb

• Under dynamic conditions – inside pressure is replaced by the viscous normal stress pnn.

• Possible role of time and temporal pressure fluctuations (a) Through the time dependence of the velocity correlation

function Qpp(r, t) (H, p. 308 Eqs. (3.298), (3.296)). (b) Through the time spectra and distribution functions (TL, p. 274

and Chapter 6).

E. DROP BREAKAGE

1. CRITICAL (KOLMOGOROV) SIZE, RK If ( ) Rrp /σ>∆

lcr

2x

• Find local capillary pressure Pc(x). • Determine lcr from drop volume and the maximum Pc:

( ) 0=

∂∂

xxPc

• Determine the drop radius RK, at which the maximum in Pc

disappears – this will give a generalized form of Kolmogorov size with numerical coefficient:

6.04.06.0~ −− ρεσKR

2. BREAKAGE TIME, τB

• Time needed for deformation from sphere to breakage at lcr

2

2

rv

rp

tv

∂∂η+

∂∂−=

∂∂ρ

• Scalling

( ) Rplvlrt B /~/~~~ σττ=τ

• Inertia regime

rp

tv

∂∂

∂∂ρ ~

lRl /~/ σ

ττρ

2321212

in ~~ RRl

σρ

σρτ

• Viscous regime

2

2~

rv

rp

∂∂η

∂∂

2/~/l

llR τησ

Rσητ ~visc

• Limiting radius between the two regimes

( )

( )( )( ) τη

ρτητρ

∂∂η∂∂ρ=

2

2

2

22 ~//~

// l

lll

rvtvRe

From Re = 1 and τvisc (or τin)

→σρ

η=

2

lim

2

Rl

σρη2

lim ~R

• Rayleigh instability – only inertia regime (Shih-I Pai, Fluid dynamics of jets, § 1.9)

R0

l R-2 0

- Determine R0 and l from the conservation of volume - Critical wave with length λcr ≈ (l-2R0)cr = 2πR0

( )( ) ( ) ( )0

2

0

030

2Ray

;11 RkxxxiJ

xiJxiR

=−′

ρσ=

τ

2/30

21

~ R

σρτ

- Formally coincides with the previous result but with different

quantities. - The total breakage time is obtained by adding to τRay the time

for expansion to length l.

• Goal – Solve Stokes equation inside the drop with Laplace equation at least for small deformations (from sphere and/or cylinder) to obtain dynamic shape and breakage time with numerical coefficients.

F. GENERAL KINETIC SCHEME

L

dnj dt¢ j dt¢dq

• “Convective diffusion“ in time ”space”

( ) iii dndqSLdtjjS =+′−′

ni an Ni= total number and concentration of i – drops j = flux = NiV Nif = const – concentration of i-drops in the feed V = linear velocity dqi = production of i per unit volume in the element θ = L/V = residence time in the element

θ== ii N

LVN

SLSj

f

i i idN N N dqdt dtθ

−= + i

• Source terms

coalbreakiii dqdqdq +=

G. KINETIC SCHEME WITH ONLY BREAKAGE

1. MAIN ASSUMPTIONS

• Drops break in half, forming two smaller drops

3/1

11

11

22

== −− iiiiRRvv

• Drops with size smaller than Kolmogorov’s (Ri ≤ RK) do not

break. • The rate constant are assumed eventually independent of Ri

(ki = k). • Since ( )2/3or iiBi RBRa=τ

Bi

, all Ni(t) drops break simultaneously after time τ .

( ) 0=τ+ Bii tN

( ) ( ) 0=τ∂

∂+=τ+ BiiiBii t

NtNtN

iBiiB

i

i NkNt

N −=τ

−=∂

∂ 1 B

i

Bik τ

= 1

• depends weakly on vBiτ i and is assumed constant, τ

B.

31

3/11

3/1

243

−

π==τ ii

Bi

vaRa

8.02 31

1 ==τ

τ −+i

i

2. THREE-SIZE SYSTEM ∆N

vv1v2vk

N

v1 = initial v3 = vk

• Assume that . Dubious approximation even for a monodisperse emulsion, since N and thereby R and τ

BBB kkk == 21B changes

with t, so that τB and kB must depend at least on Rc(t).

11 Nk

dtdN B−=

212 2 NkNk

dtdN BB −=

22BkdN k N

dt=

( ) ( kBB NNkNNkd )tdN −=+= 21 kNNNN ++= 21

• The process stops at R = RK (N=Nk).

• Nk can be determined from the volume fraction Φ:

∑∑−==Φ

ki

k

ii NvvN1

111

12/

dtdN

dtdN ik

i

ikk ∑−

=

−− −=1

1

11 22

• At short times (kt = X Ü 1) Nk Ü N and

( )0/ln NNYXY ==

• Exact solution:

( )[ ]XkXX eXYeXYeY −−− +−=== 114;2; 21

( ) Xk eXYYYY −+−=++= 23421exact

• If Nk can be neglected

1lnorln ==dX

Ydkdt

Yd B

• If the exact function (i.e. experimental data) is approximated by

lnY, the apparent rate “constant” k is much smaller than k

( )tBappB (see the second figure below):

( ) BB kXkdXYd

0 1 2 3 4 5 0

0.66667

1.3333

2

2.6667

3.3333

4

Y

EXP(X)

Y3

Yexact

Y2 Y1

Yexact = 4-(3+2*X)*EXP(-X) Y1 = EXP(-X) Y2 = 2*XEXP(-X) Y3 = 4*(1-(1+X)*EXP(-X)) EXP(X)

0 2 0

0.25

0.5

0.75

1

1.25

Y

0.1

0.3

0.5

0.7

0.9

dY/d

X

X

Yexac = ln(N/N0) =

X = kt

4 6 0

0.25

0.5

0.75

1

1.25

Y

0.1

0.3

0.5

0.7

0.9

dY/d

X

LN(4-(3+2*X)*EXP(-X))

= kt

3. CONTINUOUS PROCESS WITHOUT COALESCENCE For the i-size in moment t:

( ) ( ) ( )KitNkNktNNdt

dNi

Bii

Bi

iifi ,...,12 11 =−+θ−

θ= −−

• Sum up over all i. • Summation stops at Kolmogorov size with Ni = Nk. • Assume kiB = kB = 1/τB. • For the n-th pass:

( ) ( )tNtN nK

i

ni =∑

=1

( ) ( )[ ]tNtNNdt

dN nk

nB

nf

n−

θ−

τ+

θ= 11

(a) Slow Breakage (τB > θ)

( )011 >−=θ

−τ

slslB kk

• Neglect nkN

( )

τθ−

τθ−= − tkBB

nfn sle

NtN 1

/1

( )

τ+=→ B

nf

n tNtN 10

• Steady state kslt → ∞ (follows also from dNn/dt=0):

( ) Bnfn NtNτθ−

=∞→/1

(b) Fast Breakage (τB < θ)

011 >=θ

−τ

fB k

( )

−

τθ

−τθ= 1

1/tk

BB

nfn fe

NtN

( )

τ+=→ B

nf

n tNtN 10

t

Nfn

Nk

N≤( )∞

N

slow

fast

• In the fast regime there can be also a plateau if the Kolmogorov size is reached.

H. COALESCENCE RATE

tZ

tZNkNkN

Ndt

dN iiiBi

Biifi

∂∂−

∂∂+−+

θ−

θ= +− 22 11

tZi∂

∂ = number of coalescence events between drop of volume vi in unit time.

1. COALESCENCE OF SPHERICAL DROPS (BIMOLECULAR REACTION)

212 AACk

i →

tZ

∂∂

= number of successful collisions

ENdtZ 23/73/1επ=

∂∂

E = efficiency factor, analog of activation energy F = driving force for coalescence Fext = external (turbulent) force Fint = interaction (repulsive) force between the drops

F = Fext - Fint

• Since both Fext and Fint depend on R, some collisions (for which F < 0) will NOT be efficient.

F

ξ = reaction coordinate (distance )-1

Fext≤

Fext ≤ - efficientFext

¢

Fext¢ - non-efficient

Fint

2. CONSECUTIVE IRREVERSIBLE REACTIONS FOR DROPS FORMING

A FILM

212 AZADc kk →→

τDτc Z = intermediate complex F coalescence

flocculation

ξ

Fext≤

Fext≤

FextFint

¢

SINGULAR PERTURBATION

L

dnj dt¢ j dt¢dq

dtdqN

dtdN i

ii +∆

θτ=

θτ

• Macroscale

• Microscale

• Back to macroscale – coupling of macro and micro events: - The rocket slows down the train - The speed of the train affects the taking aim

• The barrier depends also on the size, since the driving pressure is ∆P = 2σ/R - Π.

• Successful coalescence corresponds to

02 max >Π−σ=∆

RP

Kinetic Scheme

CI

Bii

iii dqdqdqdtdqN

dtdN +=+

θ∆= ;

00

t0 and θ = macroscopic time and time scale τ = microscopic time scale scaled times: t = t0/θ and t = t0/τ

dtdqN

dtdN i

ii +∆

θτ=

θτ

If τ/θ → 0, then →= 0dtdqi blow up of the micro-events.

• Micro-kinetic scheme in microtime t (scaling time is the collision

time – scale, τc).

211

1 21 Nkdt

dN c−=τ

0

211

1tZNk

dtdZ c

∂∂+=

τ

0

21tZ

dtdN

∂∂−=

τ

Z = number of complexes (2 drops with film which can rupture). They must rupture at time t0 + τD, where τD is the drainage time:

( ) ( ) 00

0 =τ∂∂+=τ+ DDtZtZtZ

ZkZtZ c

D 20

1 −=τ

−=∂∂

Estimate of τ c Scale N1 by N10, the number of drops 1 at t = 0 (N1 = N10N1′)

( ) →′−=′τ

2

12101

110 2 NNkdt

dNN c 101/1 Nkc

c =τ

21

10

1 2 NNdt

dN −= (a)

ZNNdt

dZD

c

ττ−= 21

10

1 (b)

with N1(0) = N10 from (a) and (b):

tNN

2110

1 +=

ZNK

dNdZ

211 2

1 −−= D

c NKτ

τ=2

10

Introduce x = K/N1

The solution of the linear equation is

xexxxx

Z −

++

×+++−= const...

!22ln21

2

Keep only 1/x. With Z = 0 at x = x0 = K/N10 (i.e. at t = 0) one obtains

For x → 0 it becomes

tt

KN

xxZ

21211 10

0 +=+−=

Going back to macro-time t0 by substituting t = t0/τc, one obtains for any i (Note that Ni0 is Ni(t0))

( )

00

020

0

1

200

202

0

2141

2122

tkNtNkZ

dtdN

tkN

NkNkdtdN

cii

icii

D

i

cii

icii

ci

i

+=

τ−=

+−=−=

+

(c)

• For simplicity we have implicitly assumed (by taking x > τc, corresponds to the outer solution and to scaling t = t0/τD. The determination of the integration constants for this solution requires matching of the inner and outer solution (A.H. Nayfeh, Perturbation methods, John Wiley, 1973, p.114).

• The coalescence contribution to the balance of Ni is:

×−

+=

i

i

Zidtdq

complexesofdeathofrate

21twoof

ecoalescencofratecoal

( ) ( )

dttdNx

dttdN

dtdq iii 21

coal−= +

Here t is macro-time and the two derivatives are given by (c).

3. TURBULENT FORCE F ext AND FILM RADIUS R F (a) Collision probability

• The coalescence process in the emulsion is an ensemble of parallel events, occurring between different pairs of drops. Hence, the overall speed will be determined by the fastest ones.

• Since τD ~ 1/R , the most favorable collisions between the deformed drops are head on (corresponding to smaller RF):

2Rturb

• If Zsph are the collisions between spherical drops, the number of favorable collisions, Z, will be roughly

2

=R

RZZ turbsph

(b) Force Balance on Film

2RF

Rturb

∆p r( )

∆p

( ) drrrpF π∆= ∫ 2ext

(area outside film)

2ext

2FRR

F πσ≈

EMULSIFICATION IN TURBULENT FLOWA. GOAL AND PROBLEMSGrasp the main factors and the overall scheme

B. DROPS AND FILMS8. Rupture of Thinning FilmsUniform Distribution of the Surfactant on the Film SurfaceChemical DemulsificationC. Drop Size Distribution1. Isotropic Turbulence (Kolmogorov)2. Present Theories (Steady Distribution)

Justifies assumptions (b) and (d).D. TURBULENT PRESSURE AND DROP SHAPEEddies are NOT moleculesPossible role of time and temporal pressure fluctuations

E. DROP BREAKAGETime needed for deformation from sphere to breakage at lcrLimiting radius between the two regimes

F. GENERAL KINETIC SCHEMEV = linear velocityG. KINETIC SCHEME WITH ONLY BREAKAGE

Drops break in half, forming two smaller dropsH. COALESCENCE RATE

Singular Perturbation

For simplicity we have implicitly assumed \(by tThe realistic case, (D >> (c, corresponds to the outer solution and to scaling t = t0/(D. The determination of the integration constants for this solution requires matching of the inner and outer solution (A.H. Nayfeh, Perturbation methods, John WileThe coalescence contribution to the balance of Ni is: