Diel_Lecture11_ Emulsions _Biphasic DFS .ppt

1
Lecture 11
Emulsions and Microemulsions.
The dielectric properties of heterogeneous substances.
Polarization of double layer,
Polarization of Maxwell Wagner.
Nonionic microemulsions.
Zwiterionic microemulsions.
Anionic microemulsions.
Dielectrics with conductive paths.
Percolation phenomena .

2
Microemulsion: A macroscopic, single-phase,
thermodynamically stable system of oil
oil and water
stabilized by surfactant molecules.
surfactant molecules.
ionic
microemulsion
Water-in-oil
microemulsion region
W : molar ratio [water] / [surfactant]
Rwp : radius of water core of the droplet
Rwp = ( 1.25 W + 2.7) Å
n-Decane
AOT
Water
AOT-water-decane microemulsion (17.5:21.3:61.2 vol%),
AOT-water-decane microemulsion (17.5:21.3:61.2 vol%),
W = 26.3, R
W = 26.3, Rwp
wp = 35.6 Angstrom
= 35.6 Angstrom
What is microemulsion?
What is microemulsion?

3
• Interfacial polarization (Maxwell-Wagner,
Interfacial polarization (Maxwell-Wagner,
Triphasic Model)
Triphasic Model)
• Ion diffusion polarization(O’Konski, Schwarz,
Ion diffusion polarization(O’Konski, Schwarz,
Schurr models)
Schurr models)
• Mechanism of charge density fluctuation water
Mechanism of charge density fluctuation water
• bound water,
bound water,
• polar heads of surfactants and
polar heads of surfactants and
• cosurfactants.
cosurfactants.
• In the case of ionic microemulsions the
In the case of ionic microemulsions the
cooperative processes of polarization and
cooperative processes of polarization and
dynamics can take place.
dynamics can take place.
The nature of dielectric polarization in ionic microemulsions
The nature of dielectric polarization in ionic microemulsions

Percolation: The transition associated with the
formation of a continuous path spanning an arbitrarily
large ("infinite") range.
The percolation cluster is a self-similar fractal.
5 10 15 20 25 30 35 40 45

s
Temperature (oC )
0 2 4 6 8 10
10-1
100
101
102
103

[

S/cm]
5 10 15 20 25 30 35 40 45
20
40
60
80
100
Tp
Ton
What is the percolation phenomenon?

5
5 10 15 20 25 30 35 40 45

s
Temperature (oC )
0 2 4 6 8 10
10-1
100
101
102
103

[

S/cm]
5 10 15 20 25 30 35 40 45
20
40
60
80
100
Tp
Ton
T<T
p
o
Percolation: The transition associated with the formation of a
Percolation: The transition associated with the formation of a
continuous path spanning an arbitrarily large ("infinite") range.
continuous path spanning an arbitrarily large ("infinite") range.
 
  p
t
p
p
s
p
T
T
T
T
T
T
T
T




~
~


  p
s
p
s T
T
T
T 

~


6
Three dimensional plots of frequency and
temperature dependence of the dielectric
losses '' for the AOT/water/decane
microemulsion
Three dimensional plots of frequency and
temperature dependence of the dielectric
permittivity ' for the AOT/water/decane
microemulsion
Three-dimensional plots of the time and
temperature dependence of the macroscopic
Dipole Correlation Function for the AOT-
water-decane microemulsion
(t)
 
  
  
M M t
M M
( ) ( )
( ) ( )
,
0
0 0

5 10 15 20 25 30 35
4
8
12
16
20
24
4
3
2
1

s
Temperature (
o
C )
10 20 30 40 50
4
8
12
16
Temperature (
o
C )

s
3'
3
AOT-water-decane(hexane) microemulsions at W=26.3 with composition (vol%)
1) 17.5:21.3:61.2 , 2)11.7:14.2:74.1, 3) and 3’hexane)5.9:7.1:87.0 , 4)1.9:2.4:93.7
Low-frequency
permittivity s
Permittivity of ionic microemulsions far below percolation

8
10 20 30 40 50 60
10
-10
10
-8
10
-6
10
-4
10
-2
10
0
3
2
1
2
o
C - (1)
8
o
C - (2)
12
o
C - (3)
Dipole
correlation
function
t (ns)
DCFs of ionic microemulsions far below
percolation
ns counterions 2d
nsconcentration polarization
nsconcentration polarization
 4 = 0.05 ns (bound and bulk water) 44%
DCFs at different temperatures
AOT-water-decane microemulsion (17.5:21.3:61.2 vol%), W=26.3
( ) A exp( t/ )
i i
i 1
N
t  

 
Ai = 1
i=1
N

Phenomenological fit to the four exponents

9
Polarization of ionic microemulsions far below percolation
( )( )( )
k
     

 
+ +
=
< >
2
 m ix mix w
B
2 2 12 N
T
2
0
 mix : permittivity due to nonionic sources
<2
> : mean square dipole moment of a droplet
N0 : droplet concentration
Below percolation, microemulsion is the
dispersion of non-interacting water-surfactant
droplets
Fluctuating dipole moments of the droplets
contribute in dielectric permittivity

10
Mean square fluctuation dipole moment of a droplet
Rd
Rwp
e : ion charge
Ns : number of dissociated surfactant molecules per droplet
Rwp : radius of droplet water pool
c(r) : counterion concentration at distance r from center
As : area of surfactant molecule in interface layer
Ks : equilibrium dissociation constant of surfactant
l : Debye screening length
 










2 2 5
1 2
32 3
15
e R
K
A R
wp
s
s wp
/
taking square and averaging
expanding c(r) at Rwp / lD <<1
  

2 2 4 2
4
e r c r dr N R
o
R
s wp
wp
{ ( ) }

  
 


e i i
i
Ns
( )
r r
1

11
Calculation of the counterion density distribution
c(r)
Distribution of counterions in the droplet interior is governed by the Poisson-Boltzmann
equation

  
 








 



2
0
0
x
e-
= e[ - (0)]/ kBT : dimensionless potential with respect
to the center
x = r /lD : the dimensionless distance,
lD : the characteristic thickness of the counterion layer,
c0 : the counterion concentration at x=0
l
k T
4 e c
D
w B
2
( ) /

 0
1 2
Solution of the Poisson-
Boltzmann equation
a a a
0 1 2
1
1
6
1
45
  
, , ,...
Counterion concentration
c x c a x
j
j
j
( ) 



0
2
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
-6
-5
-4
-3
-2
-1
0

(x)
x
 (x) = - ln ( a x )
j
2j
j=0


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
-6
-5
-4
-3
-2
-1
0

(x)
x
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
1
10
100
c(x)
/
c
d
x

12
Calculation of the fluctuation dipole moment of a droplet
 

 




 
2
4 k T R
a (j 2)
(2j 3)(2j 5)
x
w B wp
3 j
wp
2(j 1)
j 0
xwp = Rwp /lD (c0 )
Dissociation of surfactant molecules is described by the equilibrium relation
 
K T c e
N
N N
s
x s
a s
wp
( )


0

The dissociation constant Ks(T) has an Arrhenius temperature behavior
K T K
H
k T
s o
B
( ) exp( )
 

Na : micelle aggregation number
Ns : number of dissociated surfactant molecules
Ks(T) : dissociation constant of the surfactant
(xwp) : dimensionless electrical potential at the
surface of the droplet
H : apparent activation energy of the dissociation
K0 : Arrhenius pre-exponential factor

AOT-water-decane(hexane) microemulsions at W=26.3 with composition (vol%)
(1.9:2.4:93.7)
(5.9:7.1:87.0) 
(11.7:14.2:74.1) 
(17.5:21.3:61.2) 
2 4 6 8 10 12
300
400
500
Dipole
M
oment
(
Debye
)
Temperature (
o
C )
Experimental fluctuation dipole moments
Temperature dependencies of the
apparent dipole moment of a droplet
a = (<2
>)1/2
Rwp = ( 1.25 W + 2.7) = 35.6 Ångstrom

14
Modeling of the permittivity
Experimental and calculated (solid line) static dielectric permittivity
versus temperature for the AOT-water-decane microemulsions for
various volume fractions of the dispersed phase:
0.39 (curve 4); 0.26 (curve 3); 0.13 (curve 2); 0.043 (curve 1)
5 10 15 20 25 30
2
4
6
8
10
12
14
16
18
20
22
Dielectric
permittivity
Temperature (oC)

15
c ( t/c ) cooperative relaxation
f (t/f ) : fast processes
(t): total DCF (t) = f ( t/f ) + c ( t/c ) R(t/R)
R(t/R) : cluster rearrangements
Dielectric relaxation in percolation : relaxation laws
Dielectric relaxation in percolation : relaxation laws
where = 0.41, = 0.39
e
1 - c t t < t
xp[- c t ] t > t
1 c
2 c


 
(t) ~ where 0.8
 exp[- c t + c t],
1 2
  



(t) = At
(t) = At -
-


 exp [-
exp [-
(t/
(t/


]
]
Relaxation laws proposed for description of
the Dipole Correlation Functions (DCF) of
ionic microemulsions near percolation
Our suggestion for fitting at
the mesoscale region
(t) ~

16
0.01 0.1 1 10 100
-3
-2
-1
0
C
AA
wt %
1.7
2.1
2.5
2.9
3.3
3.7
4.14
4.7
log(DCF
)
time ( ns )
Macroscopic dipole correlation function behavior at
percolation
AOT/Acrylamide-water-toluene AOT-brine-decane
0.01 0.1 1 10 100
-3
-2
-1
0
Temperature 50 oC
A = 80%
A = 82.5%
A = 85%
A = 87.5%
A = 90%
A = 93.5%
log(
DCF
)
time ( ns )
Percolation is caused by
cosurfactant fraction
brine fraction
temperature
0.01 0.1 1 10 100 1000
-3
-2
-1
0
8
7
6
5
4
3
2
1
T oC
1 - 14
2 - 16
3 - 18
4 - 20
5 - 22
6 - 24
7 - 26
8 - 28
log
(
DCF
)
time (ns)
10
-3
-2
100
-3
-2
8
7
1
2
3
6
AOT-water-decane
microemulsion
(t) ~ At -


17
Fitting function
10 15 20 25 30 35 40
0.0
0.2
0.4
0.6
0.8
1.0
1.2

Temperature ( oC )
10 15 20 25 30 35 40
0.2
0.4
0.6
0.8
1.0
1.2

Temperature ( oC )
A
A 







AOT-water-decane microemulsion (17.5:21.3:61.2 vol%)
10 15 20 25 30 35 40
0.5
0.6
0.7
0.8
0.9
1
2
3
A
Temperature ( oC )
10 15 20 25 30 35 40
0
20
40
60
80
100
120
140

,
ns
Temperture (
o
C)
(t) = At - exp [- (t/]

18
Dielectric relaxation in percolation : model of recursive
Dielectric relaxation in percolation : model of recursive
fractal
fractal
nj = n0 pj
Lj = bj
zj = aLj
 = a(bj) = akj
k = b
N * ( / )
(z) = [ ]
g z zj
n
j
N
j


0
t : current time
1 : minimal time
z = t /1
(z) : macroscopic relaxation function
g*(z) : microscopic relaxation function
: minimal spatial scale
j : current self-similarity stage
N : maximal self-similarity stage
nj : number of monomers on the j-th stage
Lj : spatial scale related to j-th stage
zj : temporal scale related to j-th stage
n0 ,a : proportionality factors
b,p,k : scaling parameters
E = 3 : Euclidean dimension
D
Df
f =
=
E
E


Feldman Yu, et al (1996)
Feldman Yu, et al (1996) Phys Rev E 54: 5420
Phys Rev E 54: 5420
Df : fractal dimension
Intermediate asymptotic
N(Z) =A exp [ -B()Z + C()Z ]
ln(p)/ln(k), Z=t/a
1
Lj

0.1
1
10
100
10 15 20 25 30 35 40

c
(ns)
Temperature ( o
C)
10 15 20 25 30 35 40
0.5
1.0
1.5
2.0
2.5
2
1

D f
Temperature
o
C
Temperature dependence of the
stretching parameter
stretching parameter 

and the fractal dimension
and the fractal dimension D
Df
f
macroscopic effective relaxation
macroscopic effective relaxation
time
time 
c
c
Recursive fractal model: fitting results

10 15 20 25 30 35 40
102
103
104
105
106
107
108
109
1010
L (Å)
Temperature ( o
C)
101
102
103
10 15 20 25 30 35 40
Temperature ( o
C)
N
The effective length of
the percolation cluster LN
versus the temperature
number of droplets in the typical
percolation cluster
?
?

21
Dielectric relaxation in percolation : statistical fractal description ?
Dielectric relaxation in percolation : statistical fractal description ?
   




1
ds
s
w
s
t
g
t )
(
,







1
m
m
ds
s
s
s
s
s
s
s
w
]
)
/
(
exp[
]
)
/
(
exp[
)
(



    
)
(
/
exp
, s
t
s
t
g 


  

 s
s 1

   












 














 z
s
B
z
s
A
z m
2
2
2
m )
,
,
(
exp
)
,
,
,
(
Morphology parameters:
sm : cut-off cluster size
 : polydispersity index
: cut-off rate index
w(s) : Cluster size probability density
distribution function
g(t,s) : Relaxation function
related to s-cluster
Asymptotic behavior at
z >> 1 , z = t /1
Dynamic parameters:
 1 : minimal time
scaling
parameter
1
(t) : relaxation function
2
3
4

(t) = At
(t) = At -
-


 exp [- (t/
exp [- (t/


]
]

22
For
For 


Statistical fractal: results of
Statistical fractal: results of
calculations
calculations
16 18 20 22 24 26 28 30 32 34 36 38 40
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
10
12
10
13
S
m
Temperature
0
C
1
1




2
1





  



















1
1
1
1
1
m
S
10 15 20 25 30 35 40
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0

Temperature
o
C
12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42
0
1
2
3
4
5
6

Temperature
o
C

23

E
Ds  E=3
E=3

 
 0.6
0.6
S
Sm
m~10
~1012
12
D
Dd
d 
 5
5 ?
?
   
m
D
E
m s
s
w
b
s
s
w s
s
~
,
~
, 
 
 
.
~
,
~ s
s D
m
m
D
b
s
s
and
b
s
s 

1

 s
D
E
s
b 
Condition of the
Condition of the
renormalization
renormalization
Renormalization in the static site percolation model
Renormalization in the static site percolation model
L
sm
bsL
m
S
~

24
Percolation cluster s
Percolation cluster sm
m
Occupied sites and the
percolation backbone on
the effective square lattice
Visualization of the dynamic percolation
Visualization of the dynamic percolation



m
1
m
s

A/
D/
O/
E/
y
z

m


1
B
C
E D
L
H
/
l
Q
F
L/l A
O
L/l
x
2
1
2

















m
H
l
L
l
L
d
m
D
s
l
L
1


25
Hyperscaling relationship for dynamic
percolation












 d
D
m
H
d s
L
L
b


1
1
2
1
b
bd
d is an
is an expansion
expansion coefficient
coefficient
1

 d
D
E
d
b 
1
s
1
2
γD
E
αD
1
1
α
2
m
d
d



























0
0
d
d
D
1
D
E


A O
F
B
D
C
E
L/l
L/l
L
H
.
/
l
x
y
z
m
/1
D'
A'
Q
Condition of the
Condition of the
renormalization
renormalization
s
sm
m
 


26
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42
0
2
4
6
8
10
12 0.1

3*
Temperature
0
C
Experimental verification of hyperscaling relationship for dynamic
percolation

 
 0.6
0.6 
 

0.2
0.2
E




1

d
D
E=3
E=3
D
Dd
d 
 5
5 ?
?












 d
D
1
1
2
m
H
d s
1
L
L
b


If sm <
1
s
l
L
d
m
D
1
1
1
m




























2
1
2

















m
H
l
L
l
L

27
12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
D
s
Temperature
o
C
D
Ds
s=2.54
=2.54
m
d
d
s
s
D
1
D
D
lg
lg


4
1
m
10
6
l
L







5
10
2
l
L








2
1
m
10
2
.
1 











L
l m=120 10-9
s
1=1 10-9
s
l~110-8
m
Lh~2 10-3
m
The relation between Dd and Ds










 s
m
D
1
1
s



10 15 20 25 30 35 40
0.0
0.5
1.0
1.5
2.0
2.5
L
eff
,
mm
Temperature
o
C

Diel_Lecture11_ Emulsions _Biphasic DFS .ppt

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