1

Lecture 11Lecture 11Emulsions 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.

Microemulsion: A macroscopic, single-phase, thermodynamically stable system of oil oil and water stabilized by surfactant molecules.surfactant molecules.

ionic microemulsion

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

AOTWater

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, RW = 26.3, Rwpwp = 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, Ion diffusion polarization(O’Konski, Schwarz, Schurr models)Schwarz, Schurr models)

• Mechanism of charge density fluctuation Mechanism of charge density fluctuation waterwater

• 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.

• Interfacial polarization (Maxwell-Wagner, Interfacial polarization (Maxwell-Wagner, Triphasic Model)Triphasic Model)

• Ion diffusion polarization(O’Konski, Ion diffusion polarization(O’Konski, Schwarz, Schurr models)Schwarz, Schurr models)

• Mechanism of charge density fluctuation Mechanism of charge density fluctuation waterwater

• 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 microemulsionsThe 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 The percolation cluster is a self-similar fractal. 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

100TpTon

What is the percolation phenomenon?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

100TpTon

T<T p

o

What is the percolation phenomenon?What is the percolation phenomenon?

Percolation: The transition associated with the Percolation: The transition associated with the formation of a continuous path spanning an arbitrarily formation of a continuous path spanning an arbitrarily large ("infinite") range. large ("infinite") range. The percolation cluster is a self-similar fractal.The percolation cluster is a self-similar fractal.

p

t p

ps

p

TT TT

TT TT

~

~

p

t p

ps

p

TT TT

TT TT

~

~

p

sps TT TT ~ p

sps TT TT ~

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 (t)

M M t

M M

( ) ( )

( ) ( ),

0

0 0

5 10 15 20 25 30 35

4

8

12

16

20

24

4

32

1

s

Temperature ( oC )

10 20 30 40 50

4

8

12

16

Temperature ( oC )

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 6010

-10

10-8

10-6

10-4

10-2

100

3

2

1

2oC - (1)

8oC - (2)

12oC - (3)

Dip

ole

corr

elat

ion

func

tion

t (ns)

DCFs of ionic microemulsions far below percolation

ns counterions 2d

nsconcentration polarization

nsconcentration 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 ii 1

N

t

Ai = 1i=1

N

Phenomenological fit to the four exponents

9

Polarization of ionic microemulsions far below percolation

( )( )( )

k

+ + =

< >2 m ix m ix w

B

2 2 1 2 N

T20( )( )( )

k

+ + =

< >2 m ix m ix w

B

2 2 1 2 N

T20

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

lD : Debye screening length

2 2 5

1 2

32 3

15e R

K

A Rwps

s wp

/

taking square and averaging

expanding c(r) at Rwp / lD <<1

2 2 4 24e r c r dr N Ro

R

s wp

wp

{ ( ) }

e i ii

N s

( )r r1

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

xe-

= 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

lk T

4 e cDw B

2( ) / 0

1 2

Solution of the Poisson-Boltzmann equation

a a a0 1 2116

145 , , , . ..

Counterion concentration

c x c a xjj

j

( )

02

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 )j2j

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)

x0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

1

10

100

c(x)

/ c

dx

12

Calculation of the fluctuation dipole moment of a droplet

2 4 k T R a (j 2)

(2j 3)(2j 5) xw B wp3 j

wp2(j 1)

j 0xwp = Rwp /lD (c0 )

Dissociation of surfactant molecules is described by the equilibrium relation

K T c eN

N Nsx s

a s

wp ( )

0

The dissociation constant Ks(T) has an Arrhenius temperature behavior

K T KH

k Ts oB

( ) 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 12300

400

500

Dip

ole

Mo

men

t (

Deb

ye

)

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 25 302

4

6

8

10

12

14

16

18

20

22

Die

lect

ric

perm

itti

vity

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 lawsDielectric 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/]] (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(

DC

F )

time ( ns )

Macroscopic dipole correlation function behavior at percolation 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

( D

CF

)

time ( ns )

Percolation is caused by

cosurfactant fraction

brine fraction

temperature

0.01 0.1 1 10 100 1000

-3

-2

-1

0

8

76

5

4

321

T oC

1 - 14

2 - 16

3 - 18

4 - 20

5 - 22

6 - 24

7 - 26

8 - 28

log

( D

CF

)

time (ns)

10

-3

-2100

-3

-2

87

12

3

6

AOT-water-decanemicroemulsion

(t) ~ At -

(t) ~ At -

17

Fitting functionFitting 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 )

AAAA

AOT-water-decane microemulsion (17.5:21.3:61.2 vol%)

10 15 20 25 30 35 400.5

0.60.70.80.9

1

2

3

A

Temperature ( oC )

10 15 20 25 30 35 40

0

20

40

60

80

100

120

140

, ns

Temperture (oC)

(t) = At - exp [- (t/] (t) = At - exp [- (t/]

18

Dielectric relaxation in percolation : model of recursive Dielectric relaxation in percolation : model of recursive fractalfractal

nj = n0 pj

Lj = bj

zj = aLj = a(bj) = akj

k = b

N * ( / )(z) = [ ]g z z jn

j

Nj

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 DDff = =

EEDDff = =

EE

Feldman Yu, et al (1996) Feldman Yu, et al (1996) Phys Rev E 54: 5420Phys Rev E 54: 5420

Df : fractal dimension

Intermediate asymptotic

N(Z) =A exp [ -B()Z + C()Z ]

ln(p)/ln(k), Z=t/a1

Lj

0.1

1

10

100

10 15 20 25 30 35 40

c (ns)

Temperature ( oC)

10 15 20 25 30 35 40

0.5

1.0

1.5

2.0

2.5

2

1

D f

Temperature oC

Temperature dependence of the Temperature dependence of the stretching parameter stretching parameter

and the fractal dimension and the fractal dimension DDff

Temperature dependence of the Temperature dependence of the stretching parameter stretching parameter

and the fractal dimension and the fractal dimension DDff

Temperature dependence of the Temperature dependence of the macroscopic effective relaxation macroscopic effective relaxation

time time cc

Temperature dependence of the Temperature dependence of the macroscopic effective relaxation macroscopic effective relaxation

time time cc

Recursive fractal model: fitting Recursive fractal model: fitting resultsresults

10 15 20 25 30 35 40102

103

104

105

106

107

108

109

1010

L (Å)

Temperature ( oC)

101

102

103

10 15 20 25 30 35 40

Temperature ( oC)

N

The effective length of the percolation cluster

LN versus the temperature

The effective length of the percolation cluster

LN versus the temperature

Temperature dependence of the number of droplets in the typical

percolation cluster

Temperature dependence of the number of droplets in the typical

percolation cluster

Recursive fractal model: fitting Recursive fractal model: fitting resultsresults

??

21

Dielectric relaxation in percolation : statistical fractal Dielectric relaxation in percolation : statistical fractal description ?description ?

1

dsswstgt )(,

1 m

m

dssss

ssssw

])/(exp[

])/(exp[)(

)(/exp, ststg

s s 1

zsBzsAz m2

22

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 calculationscalculations

16 18 20 22 24 26 28 30 32 34 36 38 4010

0

101

102

103

104

105

106

107

108

109

1010

1011

1012

1013

Sm

Temperature 0C

11

2

1

1

1

1

11

mS

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 oC

12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

0

1

2

3

4

5

6

Temperature oC

23

E

Ds E

Ds E=3E=3 0.60.6SSmm~10~101212

DDdd 5 5 ??

E=3E=3 0.60.6SSmm~10~101212

DDdd 5 5 ??

mDE

m sswbssw ss

~,~,

.~,~ ss Dmm

D bss and bss

1 sDEsb

Condition of the Condition of the renormalizationrenormalization

Renormalization in the static site percolation modelRenormalization in the static site percolation model

L

sm

bsL

mS~

24

Percolation cluster sPercolation cluster smm

Occupied sites and the percolation backbone on the effective square lattice

Visualization of the dynamic percolationVisualization of the dynamic percolation

m1

m s

m1

m s

A/

D/

O/

E/y

z

m

1

BC

E D

LH

/l

QF

L/l AO

L/l

x

2

1

2

mH

l

L

l

L2

1

2

mH

l

L

l

L

d

m

Dsl

L1

d

m

Dsl

L1

25

Hyperscaling relationship for dynamic percolation

dDm

Hd s

L

Lb

112

1

dDm

Hd s

L

Lb

112

1

bbdd is an is an expansionexpansion coefficientcoefficient

1 dDEdb 1 dDE

db

1s12

γDE

αD

11α2

m

d

d

1s12

γDE

αD

11α2

m

d

d

0

0

d

d

D1

DE

0

0

d

d

D1

DE

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 renormalizationrenormalization

ssmm

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 0.13*

Temperature0C

Experimental verification of hyperscaling relationship for dynamic percolation

0.6 0.6 0.20.2

0.6 0.6 0.20.2

E E1

dD1

dD

E=3E=3E=3E=3

DDdd 5 5 ? ? DDdd 5 5 ? ?

dD

112

mH

d s1L

Lb

dD

112

mH

d s1L

Lb

If sm <If sm <

1s

lL

d

m

D

11

1

m

2

1

2

mH

l

L

l

L

27

12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 420.0

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

Ds

Temperature oC

DDss=2.54=2.54DDss=2.54=2.54

md

ds

sD1

DD

lglg

md

ds

sD1

DD

lglg

41

m

106

lL

41

m

106

lL

5102l

L

2

1

m 102.1

Ll m=120 10-9s1=1 10-9s

m=120 10-9s1=1 10-9s

l~110-8 m

Lh~2 10-3m

l~110-8 m

Lh~2 10-3m

The relation between Dd and Ds

s

m

D

11

s

10 15 20 25 30 35 40

0.0

0.5

1.0

1.5

2.0

2.5

L eff,

mm

Temperature oC