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## Lecture 11

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**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 .**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**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.**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.**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.**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**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**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**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**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**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**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**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**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)**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**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**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**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**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**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 ?**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**E=3** 0.6 Sm~1012 Dd 5 ? sm L Renormalization in the static site percolation model Condition of the renormalization bsL**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**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 **Experimental verification of hyperscaling relationship for**dynamic percolation Dd 5 ? If sm< 0.6 0.2 E=3**l**The relation between Dd and Ds Ds=2.54 l~110-8 m Lh~2 10-3m m=120 10-9s 1=1 10-9s L