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Lecture 11

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Lecture 11

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

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

  4. T T on p 100 0 2 4 6 8 10 10 3 80 10 2 60 S/cm] s m 10 1 e [ 40 s 10 0 20 10 -1 5 5 10 10 15 15 20 20 25 25 30 30 35 35 40 40 45 45 Temperature ( C ) o 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.

  5. T T on p 100 0 2 4 6 8 10 10 3 80 10 2 60 S/cm] s m 10 1 e [ 40 s 10 0 20 10 -1 5 5 10 10 15 15 20 20 25 25 30 30 35 35 40 40 45 45 Temperature ( C ) o 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 permittivity e' 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 Three dimensional plots of frequency and temperature dependence of the dielectric losses e'' for the AOT/water/decane microemulsion

  7. Permittivity of ionic microemulsions far below percolation 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 e s

  8. DCFs of ionic microemulsions far below percolation AOT-water-decane microemulsion (17.5:21.3:61.2 vol%), W=26.3 Phenomenological fit to the four exponents t 1 = 12 ns ( counterions 2d+ ...) 1% t 2 = 1.4 nsconcentration polarization t 3 = 0.3 nsconcentration polarization t4 = 0.05 ns (bound and bulk water) 44% DCFs at different temperatures

  9. Polarization of ionic microemulsions far below percolation Below percolation, microemulsion is the dispersion of non-interacting water-surfactant droplets Fluctuating dipole moments of the droplets contribute in dielectric permittivity emix : permittivity due to nonionic sources <m2> : mean square dipole moment of a droplet N0 : droplet concentration

  10. Mean square fluctuation dipole moment of a droplet taking square and averaging 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 expanding c(r) at Rwp / lD <<1

  11. Calculation of the counterion density distribution c(r) Distribution of counterions in the droplet interior is governed by the Poisson-Boltzmann equation y = e[Y - Y(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 Solution of the Poisson-Boltzmann equation Counterion concentration

  12. Calculation of the fluctuation dipole moment of a droplet xwp = Rwp /lD (c0 ) Dissociation of surfactant molecules is described by the equilibrium relation Na : micelle aggregation number Ns : number of dissociated surfactant molecules Ks(T) : dissociation constant of the surfactant y(xwp) : dimensionless electrical potential at the surface of the droplet The dissociation constant Ks(T) has an Arrhenius temperature behavior DH : apparent activation energy of the dissociation K0 : Arrhenius pre-exponential factor

  13. Experimental fluctuation dipole moments AOT-water-decane(hexane) microemulsions at W=26.3 with composition (vol%) (1.9:2.4:93.7)  (5.9:7.1:87.0) O (11.7:14.2:74.1) Ñ (17.5:21.3:61.2) D Rwp = ( 1.25 W + 2.7) = 35.6 Ångstrom Temperature dependencies of the apparent dipole moment of a droplet ma = (<m2>)1/2

  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)

  15. Dielectric relaxation in percolation : relaxation laws Yf (t/tf ) : fast processes YR(t/tR) : cluster rearrangements Y(t) = Yf ( t/tf ) + Yc ( t/tc )YR(t/tR) Y(t): total DCF Relaxation laws proposed for description of the Dipole Correlation Functions (DCF) of ionic microemulsions near percolation Yc ( t/tc ) cooperative relaxation Y(t) = At -m exp [- (t/t) n] 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 8 6 7 AOT-water-decane microemulsion 3 2 1 Y(t) ~ At -m

  17. Fitting function Y(t) = At -m exp [- (t/t) n] AOT-water-decane microemulsion (17.5:21.3:61.2 vol%) m A n t

  18. Dielectric relaxation in percolation : model of recursive fractal Feldman Yu, et al (1996) Phys Rev E 54: 5420 t : current time t1 : minimal time z = t /t1 YN(z) : macroscopic relaxation function g*(z) : microscopic relaxation function l : 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 Lj nj = n0 pj Lj = l bj zj = aLjE = a(lbj)E = lEakj k = bE Intermediate asymptotic YN(Z) =A exp [ -B(n)Zn + C(n,N)Z ] n=ln(p)/ln(k), Z=t/alEt1 Df = En = 3n Df : fractal dimension

  19. Recursive fractal model: fitting results Temperature dependence of the macroscopic effective relaxation time tc Temperature dependence of the stretching parameter n and the fractal dimension Df

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

  21. Dielectric relaxation in percolation : statistical fractal description ? g(t,s) : Relaxation function related to s-cluster w(s) : Cluster size probability density distribution function 2 1 3 Dynamic parameters: t1 : minimal time a : scaling parameter Morphology parameters: sm : cut-off cluster size g : polydispersity index h : cut-off rate index Y(t) : relaxation function Asymptotic behavior at z >> 1 , z = t /t1 Y(t) = At -m exp [- (t/t) n] 4

  22. Statistical fractal: results of calculations For h =1

  23. E=3   0.6 Sm~1012 Dd  5 ? sm L Renormalization in the static site percolation model Condition of the renormalization bsL

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

  25. Hyperscaling relationship for dynamic percolation y D' D E C B z L/l bdis an expansioncoefficient LH ./ l Q F Condition of the renormalization m/1 x O A A' L/l sm 

  26. Experimental verification of hyperscaling relationship for dynamic percolation Dd  5 ? If sm<   0.6   0.2 E=3

  27. l The relation between Dd and Ds Ds=2.54 l~110-8 m Lh~2 10-3m m=120 10-9s 1=1 10-9s L