Transcript FORCED COALESCENCE OF MICRONSIZE DROPS

Role of Bubble Surface Mobility
for the Foam-Wall Friction:
Experiment and Modeling
N. D. Denkov
Laboratory of Chemical Physics & Engineering
Faculty of Chemistry, Sofia University, Sofia, Bulgaria
Why studying foam-wall friction?
Wall-slip
friction
VW
V0
• Wall-slip is usually significant (including rheo-experiments).
• The research problem is closely related to “bubble in capillary”
problem (Bretherton).
• Very convenient for studying the effect of bubble surface mobility.
Background
Constitutive rheological relation for foams
 F ( )   0 V ( )
 F   0  kF γ n
0 - yield stress (elastic origin)
V - rate-dependent stress
(viscous dissipation)
 - shear rate [1/s]
Princen, 1985
Foam-wall friction stress
W  kW VWm
W – foam-wall viscous friction
VW – relative velocity
Theoretical studies
Schwartz & Princen, Hansen & Kraynik: n = 2/3
(for small oscillations)
Kraynik: m = 2/3 (adapting the Bretherton’s solution)
Experimental measurements
Princen: n = 1/2 (for continuous shear of emulsions)
Mason: n between 1/2 and 0.9 (emulsions with small droplets)
Major Aims
• Foam-wall viscous friction - understand and describe
theoretically.
• Bubble surface mobility - reveale its role for viscous
friction: foam-wall, inside foam, and bubble-in-capillary.
Factors considered
• Relative bubble-wall velocity, VW.
• Liquid viscosity, , and surface tension,  (Ca= VW/).
• Bubble size, RB, and volume fraction, .
• Surface mobility of bubbles (surface elasticity & viscosity).
Contents
1. Experimental methods:
 Parallel-plates rheometry (wall-slip and inside-foam shear).
 Microscope observation of dynamic wetting films.
 Oscillating drop method (solution surface properties).
2. Experimental results:
 Viscous friction in the wall-slip region.
 Viscous friction in sheared foam.
3. Theoretical model of foam-wall friction:
 Viscous friction in the wetting film and in the transition region.
 Role of surface mobility.
4. Conclusions.
1. Experimental methods:
Viscous friction between foam and smooth wall
Parallel-plates rheometry
M
Sand-paper
 = 90 %
Glass plate
  VW
Perfect slip at low velocity
The viscous stress on the wall can be measured
The role of surface mobility can be studied
Microscope observation of dynamic wetting films
Glass plate
VW
y
x
Determination of:
• Profile of bubble surface
• Film thickness vs. VW
VW
Viscous friction inside sheared foam
M
Sand-paper
Sand-paper
  
( )  0  V ( )
Mostly elastic stress at low 
Elastic + viscous stress at high 
Characterizing the surface properties of surfactant solutions
Oscillating Drop
Method
Drop oscillations:
Frequency - 0.125 Hz
Area amplitude - 2 %
15 mM SDS+ 150mM NaCl+0.3mM LA

IFT(mN/m)
30
18
17
28
16
26
SA(mm2)
32
15
24
22
0
20
40
Time(s)
60
A
14
80
ES =  / lnA
= 3 mN/m
2. Experimental results: Foam-wall friction
Effect of bubble surface mobility
10
Stress (Pa)
1wt % K Myristate
immobile surface
1
m = 1/2
m = 2/3
1 wt % Na Laurate
mobile surface
W  kWVW
m
0.1
0.1
1
Shear rate (s-1)
10
Role of Surface Dilatational Modulus
Surfactant
Surface modulus
ES (mN/m)
15 mM SDS
3
15 mM SDS + 0.3 mM LaOH
5
30 mM SDS + Betaine (1:1)
1
3 wt % Betaine
<3
3 wt % SLES
<1
1 wt % Na Laurate
1.5
1 wt % K Myristate
120
K Myristate + K Palmytate
210
Commercial K soap
410
Friction power law
m
2/3  5 %
1/2  5 %
Effect of solution viscosity
1 wt % Na Laurate or 2 wt % K Cocoylglycinate + X wt % Glycerol
101
Stress, Pa
60 % Glycerol
100
0 % Glycerol
m = 2/3
10-1
10-2
10-1
100
-1
Shear rate, s
101
Scaling of the data with solution viscosity
 R32

2/ 3
τW   C2 F2 () Ca * 

 σ

 μV 
Ca *   W 
 σ 
Dimensionless Stress
10-1
1 wt % Na Laurate
2 wt % K Cocoylglycinate
3 wt % Betaine
0 to 60 % Glycerol
10-2
m = 2/3
10-3
10-6
10-5
10-4
Dimensionless shear rate, Ca*
10-3
Thickness and shape of dynamic wetting films
Glass plate
VW
Profile of the upper film surface and film thickness
V0 = 8 mm/s
1.8
Thickness, h, m
1.6
1.4
V0 = 8 mm/s
1.2
1.0
0.8
0.6
V0 = 2 mm/s
0.4
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
 = x/RB
V0 = 2 mm/s
Film thickness vs. Capillary number
hPL/(RFRPB)1/2
Betaine
10-2
Capillary number
Soap
VW
Ca* 

n = 0.60
h
 RPB RF 
12
n = 0.65
10-3
10-5
 A1  Ca* 
10-4
Capillary number, Ca
n
10-3
Viscous friction inside sheared foam
M
Sand-paper
Sand-paper
  
Effect of solution viscosity on viscous friction
3 wt % Betaine + X wt % Glycerol
100
Viscous stress (Pa)
70 % Glycerol
10
0 % Glycerol
1
n = 0.40
0.1
0.01
0.1
1
10
Shear rate
100
1000
Scaling of the data with solution viscosity
R 
n


τ

C
F
(

)
Ca

V 
V
σ


 μγR 
Ca  

 σ 
Dimensionless viscous stress
1
0.1
n = 0.40
0.01
0.001
1E-07
1E-06 0.00001 0.0001 0.001
0.01
Dimensionless shear rate, Ca
0.1
Dimensionless Viscous Stress
Effect of bubble surface mobility
100
10-1
1 wt % Na Laurate
1 wt % K Soap
n = 0.25
3 wt % Betaine
0 to 70 % Glycerol
10-2
n = 0.40
10-3
10-7
10-6
10-5
10-4
10-3
10-2
Dimensionless shear rate, Ca
10-1
Role of Surface Dilatational Modulus
Surface modulus
ES (mN/m)
Friction power law
n
15 mM SDS
3
0.41
15 mM SDS + 0.3 mM LaOH
5
0.42
20 mM SDS + Betaine (1:1)
1
0.44
3 wt % Betaine
<3
0.40
3 wt % SLES
<1
0.41
1 wt % Na Laurate
1.5
0.25
K Myristate + K Palmytate
210
0.27
K Cocoylglycinate
90
0.22
Commercial K soap
410
0.25
Surfactant + 30 wt %
Glycerol
Note: ES is measured at oscillation frequency of 0.125 Hz
n is given at air volume fraction  = 90 %
3. Theoretical modeling of foam-wall friction
2D-Bubbles
VW
Bubble
z
Vx(z)
V0
x
Set of Equations to be Solved
 2 Vx
dP
 2
dx
z
1. Lubrication equation for the liquid flow
2. Liquid conservation along the film
h
Q( x)   dz Vx ( z )  const
0
 P  x  dA  P A
3. Normal force balance
d
Film Area
Bubble
z
Vx(z)
V0
x
C
F
4. Friction Force Bubble-Wall
 Vx 
 Vx 
FFR    
 dA    
 dA
Film Area  z  z  0
PB Area  z  z  0
RPB
RPB
PB
Film Region
0
PB
L
Assumed surface profile (linear, curved)
Wall stress (2D and 3D-bubbles):
x
h( x )  f  x 
W  FFR / AB
Numerical Results - Friction Force
100
FFR(RFRPB)1/2//L
10-1
 = 0.9999 for 2D
( = 0.99 for 3D)
Film, n = 0.5
10-2
10-3
PB, n = 0.6
10
-4
10-5
10-6
10-7
10-6
10-5
10-4
10-3
10-2
Capillary number, Ca
Ca 
VW

Relative contribution of the friction in the PB region
into the total friction force
1
.
0
0
.
8

=
0
.
9
5
fPB
0
.
6
0
.
4

=
0
.
9
9
0
.
2

=
0
.
9
9
9
9
9
0
.
0
71
61
51
41
31
21
1
1
0
0
0
0
0
0
0
C
a
p
i
l
l
a
r
y
n
u
m
b
e
r
,
C
a
Friction force
1/ 2
FFR
1 2  RF 
 2.50 Ca 


R
 PB 
Film
 2.12 Ca 2 3
PB
FFR from empirical relation
101
100
10-1
10-2
10-3
10-3
10-2
10-1
100
FFR from numerical calculations
101
Effect of surface mobility
Surface stress balance
L
d
d u
 Vx 

 S 2
 
dx
 z  z  h dx
Bubble
2
u(x)
z
Vx(z)
V0
Related to
surface
elasticity
d
 d  du
 0
dx
Q dC dx
Mobility factor
Surface
viscosity
x

2u     1 
du
3Q

 2M GQ  

2
d
   
     

 

M G  Ca  FG
FG 

L
EG d  dC
Effect of surface mobility on Film thickness
Mobile
4
FG = 3000
0.87 m
 = h/hC
3
2
FG = 1
1
0
0.2
Immobile
Film region
0.3
0.4
0.5
0.6
0.7
Dimensionless distance,  = x/L
0.8
0.55 m
Summarizing: Two cases are predicted and observed
Friction in the transition region
(Bretherton; Schwartz & Princen)
Friction inside the film
(high surface modulus)

W  C1F1   
Ca1/ 2
R32
FFR 
hF
AFR
 μV 
Ca   W 
  
W  C2 F2   

Ca 2 / 3
R32
V 
 V 
  x  dA    W  AFR
 z  z 0
 hF 
Friction Area

hF
AFR
hF  RPB Ca1/ 2
hF  RPB Ca2 / 3
AFR  AF
2
AFR  RPB
Ca1/ 3
Conclusions
Theoretical model and experiments on foam-wall friction:
• Predominant friction in the film (immobile surfaces), W  Ca1/2
• Friction in the transition region, W  Ca2/3
• The effect of air volume fraction is quantified (RF and RPB).
Experiments on viscous friction in foams
• Two cases (mobile and immobile) are distinguished again
• Effect of solution viscosity - scaling with Can
• Viscous friction inside sheared foams is still poorly understood
N. D. Denkov, V. Subramanian, D. Gurovich, A. Lips, “Wall Slip
and Viscous Dissipation in Sheared Foams: Effect of Surface
Mobility”, Colloids & Surfaces A, in press.
Acknowledgements
Sofia University, Sofia, Bulgaria
• Dr. S. Tcholakova – numerical calculations
• Mr. K. Golemanov – experiments with wetting films
Unilever R&D, Edgewater, USA
• Dr. K. P. Ananth - useful discussions on surfactant properties
• Mr. P. Singh - help in rheological measurements
• Mr. C. Ho - measurements of surface dilatational modulus
Prof. H. Stone and Dr. H. Princen - for the useful
discussions on foam and emulsion rheology
Method for Bubble size determination
P. Garrett et al., 1993
Experimental check for the effect of surface
slip in the experiments with two sand-papers
100
Stress(Pa)
3 wt % Betaine - comparison of
3 mm and 2 mm gaps
10
1
0.01
0.1
10
1
Rate (s-1)
100
1000
2 wt % K Soap solution
Stress (Pa)
100
2 mm gap
10
3 mm gap
1
0.1
1
10
-1
Rate (s )
100
Possible origin of different viscous dissipation
indexes, n, in emulsions and foams
  0.1 to 102 s-1
Hence the contact time C 10 s to 10 ms
Estimate of the time for film drainage by Reynolds equation

τ DR

dh
3 RF2 dh RF2 RB 1



3
2
V
2
P
h

h
C
h DR
h
Foams: DR  100 s > C
Emulsions: DR  50 ms  C