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 = kt4 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: