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

3 Interfacial polarization (Maxwell-Wagner, Triphasic Model) Interfacial polarization (Maxwell-Wagner, Triphasic Model) Ion diffusion polarization(O’Konski, Schwarz, Schurr models) Ion diffusion polarization(O’Konski, Schwarz, 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 cooperative processes of polarization and dynamics can take place. In the case of ionic microemulsions the cooperative processes of polarization and dynamics can take place. The nature of dielectric polarization in ionic microemulsions

The percolation cluster is a self-similar fractal. 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. T p T on What is the percolation phenomenon?

5 T p T on What is the percolation phenomenon? 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.

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

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 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 Phenomenological fit to the four exponents

9 Polarization of ionic microemulsions far below percolation  mix : permittivity due to nonionic sources : mean square dipole moment of a droplet N 0 : 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 e : ion charge N s : number of dissociated surfactant molecules per droplet R wp : radius of droplet water pool c(r) : counterion concentration at distance r from center A s : area of surfactant molecule in interface layer K s : equilibrium dissociation constant of surfactant l D : Debye screening length taking square and averaging expanding c(r) at R wp / l D <<1

11 Calculation of the counterion density distribution c(r) Distribution of counterions in the droplet interior is governed by the Poisson-Boltzmann equation  = e[  -  (0)]/ k B T  : dimensionless potential with respect to the center x = r /l D : the dimensionless distance, l D : the characteristic thickness of the counterion layer, c 0 : the counterion concentration at x=0 Solution of the Poisson- Boltzmann equation Counterion concentration

12 Calculation of the fluctuation dipole moment of a droplet x wp = R wp /l D (c 0 ) Dissociation of surfactant molecules is described by the equilibrium relation The dissociation constant K s (T) has an Arrhenius temperature behavior N a : micelle aggregation number N s : number of dissociated surfactant molecules K s (T) : dissociation constant of the surfactant  (x wp ) : dimensionless electrical potential at the surface of the droplet  H : apparent activation energy of the dissociation K 0 : 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)  Experimental fluctuation dipole moments Temperature dependencies of the apparent dipole moment of a droplet  a = ( ) 1/2 R wp = ( 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); (curve 1)

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  (t) = At -   exp [- (t/  ]  (t) = At -   exp [- (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

16 Macroscopic dipole correlation function behavior at percolation AOT/Acrylamide-water-toluene AOT-brine-decane Percolation is caused by cosurfactant fraction brine fraction temperature AOT-water-decane microemulsion  (t) ~ At - 

17 Fitting function A   AOT-water-decane microemulsion (17.5:21.3:61.2 vol%)  (t) = At -   exp [- (t/  ]

18 Dielectric relaxation in percolation : model of recursive fractal n j = n 0 p j L j =  b j z j = aL j  = a( b j )  =  ak j k = b  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 n j : number of monomers on the j-th stage L j : spatial scale related to j-th stage z j : temporal scale related to j-th stage n 0,a : proportionality factors b,p,k : scaling parameters E = 3 : Euclidean dimension D f = E Feldman Yu, et al (1996) Phys Rev E 54: 5420 D f : fractal dimension Intermediate asymptotic  N (Z) =A exp [ -B( )Z + C(  )Z ]  ln(p)/ln(k), Z=t/a   1 LjLj

Temperature dependence of the stretching parameter Temperature dependence of the stretching parameter and the fractal dimension D f and the fractal dimension D f Temperature dependence of the macroscopic effective relaxation time  c Recursive fractal model: fitting results

The effective length of the percolation cluster L N versus the temperature Temperature dependence of the number of droplets in the typical percolation cluster Recursive fractal model: fitting results ?

21 Dielectric relaxation in percolation : statistical fractal description ? Morphology parameters: s m : 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  (t) = At -   exp [- (t/  ]  (t) = At -   exp [- (t/  ]

22 For  For  Statistical fractal: results of calculations

23 E=3   0.6 S m ~10 12 D d  5 ? Condition of the renormalization Renormalization in the static site percolation model L smsm bsLbsL

24 Percolation cluster s m Occupied sites and the percolation backbone on the effective square lattice Visualization of the dynamic percolation A/A/ D/D/ O/O/ E/E/ y z m  1m  1 B C ED L H /l Q F L/lA O x

25 Hyperscaling relationship for dynamic percolation b d is an expansion coefficient A O F B D C E L/l L H./ l x y z m/1m/1 D' A' Q Condition of the renormalization sm sm sm sm 

26 Experimental verification of hyperscaling relationship for dynamic percolation   0.6   0.2 E=3 D d  5 ? If s m < 

27 D s =2.54 L l  m =120  s  1 =1  s l~1  m L h ~2  m The relation between D d and D s