Dielectric study of temperature dependent aerosol ot/water/isooctane microemulsion structure

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Dielectric study of temperature dependent aerosol ot/water/isooctane microemulsion structure

M.A. van Dijk, E. Broekman, J.G.H. Joosten, D. Bedeaux

To cite this version:

M.A. van Dijk, E. Broekman, J.G.H. Joosten, D. Bedeaux. Dielectric study of temperature dependent aerosol ot/water/isooctane microemulsion structure. Journal de Physique, 1986, 47 (5), pp.727-731.

�10.1051/jphys:01986004705072700�. �jpa-00210254�

DIELECTRIC STUDY OF TEMPERATURE DEPENDENT AEROSOL OT/WATER/ISOOCTANE

MICROEMULSION STRUCTURE

M.A. VAN DIJK, ^E. BROEKMAN, J.G.H. JOOSTEN and ^D. BEDEAUX+

Department of Molecular Biophysics, Physics Laboratory, University ^{of Utrecht,} Princetonplein ^{5, 3584 CC Utrecht,}

The Netherlands

+Department ^of Physical ^and Macromolecular Chemistry,

Gorlaeus Laboratories, PoB 9502, ^{2300 RA} Leiden, The Netherlands

(Reçu ^{le 8 novembre 1985,} révisé le 27 janvier 1986, accepté ^{le 19} février ¹⁹⁸⁶⁾

Résumé : On mesure la limite à basse fré- quence de la permittivité ^{d’une microémul-}

sion AOT/eau/isooctane ^{dans la} gamme de

température 10°C-45°C. Dans le cadre d’un modèle d’agrégàtion ^des gouttelettes, ^on

donne des formules explicites pour cette

permittivité. ^Les dépendances expérimentales

en concentration et en température ^{sont bien}

décrites _par ce modèle à partir ^{de la} polarisabilité ^d’une gouttelette, ^d’une énergie d’activation et d’un préfacteur.

Abstract : The low frequency limiting per-

mittivity ^{of an} AOT/water/isooctane ^micro-

emulsion has been measured over the

temperature range 10°C-45°C. Using ^{a cluste-} ring model, explicit ^{formulae are} given ^for

this permittivity. ^We find that a good description of the measured temperature ^and

concentration dependence is obtained on the basis of this model in terms of the polari- zability ^{of a} single microemulsion droplet,

an activation energy and ^a prefactor.

Classification

Physics ^Abstracts

77.20 - 77.90 - 82.70

1. Introduction

A water in oil (W/0) microemulsion is a thermodynamically stable, isotropic, transparent liquid dispersion ^{of small} (N 10nm) spherical ^water droplets, ^surroun-

ded by ^a monomolecular layer of surfactant molecules, ^{in a} continuous oil phase. ^The

molar water/surfactant ^ratio W, essentially

determines the radius of the droplets ^while the molar oil/surfactant ^ratio So ^determines

their concentration in the oil phase [1].

Such systems ^of particles ^with high permittivity dispersed ^{in an} insulating ^low permittivity ^{medium are} amenable to studies with dielectric spectroscopy [2-4]. ^{The main} drawback to these experiments ^is the lack of

a detailed description of the observed mi- croscopic dielectric behaviour in terms of the properties ^{of the} components ^{of the mix-} ture. Rather, ^one has to resort to effective medium theories, ^which furthermore cannot account for the temperature dependence ^of the dielectric properties.

Here we present ^an analysis ^of dielectric permittivity measurements in terms of the polarizability ^which largely

overcomes the problems ^{above. This} descrip-

tion is used for the interpretation ^{of the}

low frequency permittivity ^{of an Aersol} OT/water/isooctane microemulsion measured

over the temperature range 10°C- 45°C and presented ^{in section 3.}

For low volume fractions of parti-

cles one may rigorously derive the so-called Clausius-Mossotti formula for the effective

permittivity. ^{In section 4 we introduce a}

function I which describes the corrections to Clausius-Mossotti (and ⁱⁿ particular ^to the so-called Clausius-Mossotti function) for higher ^{volume fractions} due to correla- tions. Neglecting higher order correlations

an explicit expression is given ^{for I in}

terms of the pair distribution function.

Using the Percus-Yevick correlation function for hard spheres ^{one obtains a} temperature independent ^{value of I} (and ^{thus of} ~) ^which is small compared ^{to the} experimental

values.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004705072700

728

In order to explain ^the experimental

results we assume that the particles ^cluster together ^{with a} certain activation energy per contact. This then leads to an expres- sion for the temperature dependence ^{of I. In}

section 5 we analyse ^the experimental

results which were given in section 3 using

this clustering ^model. We find that a good description ^{of the} temperature ^{and concen-}

tration dependence ^of the microemulsion permittivity is obtained in terms of only three parameters : ^the polarizability ^{of a} single microemulsion droplet, ^{an activation}

energy per contact and ^a prefactor ^{which is}

related to the number of particles ^{in a} cluster. The clustering hypothesis ^is sup-

ported by ^recent experiments with other

techniques [14-21].

2. Materials and methods

2.1 Materials.- The Aerosol OT or AOT (so- dium di-2-ethylhexyl sulfosuccinate) ^was obtained from Fluka AG (purum) ^and inorganic impurities ^{were removed} following [5]. ^Iso- octane (2,2,4 trimethylpentane) ^{was obtained}

from Baker and the water was deionized and

quadruple ^{distilled in an all} quartz distillator. Solutions were freshly prepared by dissolving appropriate weights ^of AOT, water and isooctane. On calculating ^volume

fractions we assumed the bulk density ^for

water and isooctane and a value of 1/14 g/ml

for the density ^{of AOT} [1].

2.2 Methods.- The dielectric permittivity

measurements were carried out in the

frequency range 10 kHz-13 MHz with a

Hewlett-Packard HP4917A impedance analyser with use of a thermostated cell, ^{similar to} the one described in [6]. ^The temperature

was stable to 0.1°C. The permittivity spectra ^of AOT/water/oil microemulsion show

a (temperature dependent) dielectric relaxa- tion in the MHz range [2,4,7]. ^{Within our} experimental frequency range ^we always ^ob- served a frequancy independent portion ^of

the real part ^{~’ of the} permittivity ^{at the}

low frequency side. The corresponding ^value

of ~’ we denote the low frequency limiting permittivity Elf.

3. Experimental ^results

Figure ^{1 shows the} low frequency limiting permittivity f1f ^{as a} function of the disperse ^matter (water ⁺ AOT) ^volume fraction 0 of a microemulsion with Wo = ²⁵

and at various temperatures. ^The f1t ^rises monotonically ^{and non} linearly with 0 for T 30°C. At higher temperatures Elf ^{(0) goes} through ^{a maximum} (not shown). This maximum is associated with a sharp rise of the con-

ductivity and has been attributed to a per- colation transition [8]. For 0 -+ 0 the Elf

aproaches ^the permittivity ^{of isooctane} (1.94) ^as expected. ^For higher values of o the elf ^is strongly temperature dependent

and reaches large ^{values at} high temperatu-

res. Figure 2 shows that similar results are obtained for Wo = ^35.

4. Theory

In this section we will give ^{a theo-}

retical treatment of the dielectric permit- tivity ^{of a} dispersion ^{a small} spherical

water droplets covered with a surfactant layer ^{in a} continuous oil phase. ^{First we} define the "Clausius-Mossotti function" or

alternatively ^{the effective} polarizability a

per unit of ^volume of the mixture.

where E is the (measurable) permittivity ^of

the mixture and ~0 ^the permittivity ^{of the}

continuous oil phase. ^Our definition of po- larizability per unit of ^volume differs a

factor 3£v ^{with the more usual} definition, where £v ^is the absolute vacuum permittivi- ty. ^The polarizability ^{of a} single homoge-

neous sphere ^{with a} permittivity ~ ^{is given}

by

The problem ^of calculating mp ^{for a particle}

with several concentric layers ^{of different} permittivities ^has already been solved by

Maxwell [2,9]. ^{In a} sufficiently dilute dis-

persion ^of spheres ^the Clausius-Mossotti re-

lation is valid :

where Op is the volume fraction occupied by

the particles. Dipole-dipole interactions between the particles increase their polari- zability and contribute to the right ^hand

side of equation (3) ^for larger ^{values of} Op

To account for this we write the effective

polarizability ^{as :}

where I(OPIT) is defined by ^this equation

and is a measure of the contribution ^{to the} effective polarizability from the inerac- tions. T is the temperature. ^{I will in} gene- ral depend ^on the two and more particle ^cor-

relation functions. It can be shown that the contribution due to the increase of the po-

larizability ^{of a} pair ^of particle ^{above the}

value ap ^at large separation ^is given by [10,11].

where s is the ratio of the distance between the centers of the spheres and their diame- ters, g(s) ^is the Op ^{and T} dependent pair distribution function and a(s) ^{is the direc-} tional average of the polarizability per unit of volume of the pair. ^{If one uses the}

41P

^{0 limit of g(s) in equation (5) one}

obtains the second virial coefficient for an

expansion ^{of a in} 0 p . ^{Equation (5) can be}

evaluated if g(s) ^and a(s) ^are known. The latter has been calculated for a system ^of conducting spheres [10,11], while the Percus Yevick hard sphere ^radial distribution function [12] may be used for g(s). ^{We thus}

find that the value of I is independent ^of temperature and ranges from = 0.5 at Op = 0

to N 1.5 at Op = 0.65 . The function a(s) turns out [10,11] ^{to be} strongly peaked ^at

contact distance so that the regions ^{in the} droplets ^{with a} permittivity significantly larger ^than Eo touch. This implies ^{that the}

value of I is dominated by ^the behaviour of

g(s) ^{in this} asymptotic short distance do- main. Larger ^{values of I thus} point ^{to a} peaked distribution function at close con-

tact for instance due to clustering ^of particles.

In order to explain ^the large ^values of I found experimentally, ^see below, ^and their temperature dependence we use a model in which clusters of two and more particles

are formed. This process may be viewed upon

as an aggregation equilibrium. ^The charge

distribution in the two particles ^constitu- ting ^a dimer, placed ^{in an} external electric field, deviates from the charge distribution in the single particle ; ^the charge ^distri-

bution is only significantly modified is a small region around the point of contact.

This then leads to the steep increase of the

polarizability ^"near contact" for a dimer

[10,11]. Similarly ^the charge distribution of a cluster will only ^be significantly ^mo-

dified near points of contact and it appears reasonable to assume that this modification differs very little from the corresponding

modification in a dimer and therefore the effect will be additive.

Referring ^to equations (3) ^and (4) ^{we then} find that the increase of a above the Clausius-Mossotti value a P(OP ^{equals the num-}

ber of contact points ^{times the excess} polarizability ^{of a bound} pair. ^{Based on}

these observations we now propose the follo-

wing expression ^{for I}

where E. ^{is the} activation energy per bond and R is the gas constant. ^The prefactor io

is related to the average number of bonds per particle ^{and one} expects, ^{in the context}

of this clustering ^model, that it is at most weakly dependent ^on Op.

5. Discussion

The only possible approach ^{to fit}

the curves of figure ^{1 and} figure ^{2 to con-}

ventional mixture equations like for instan-

ce Bruggeman’s unsymmetrical ^{effective me-} dium theory [13] would be to assume that the microemulsion droplets ^have anisotropic ^sha-

pes, e.g. spheroids ^{with a} high axial ratio which furthermore is temperature ^{and concen-}

tration dependent [2]. ^{This is} contrary ^to

observations from other experiments ^which

show that, up to very high concentrations, AOT-microemulsion droplets ^{retain a} spheri-

cal shape {15].

Fig. ^{1.- Low} frequency permittivity elf ^as

a function of the disperse ^matter (water ⁺ AOT) ^volume fraction 0 at six different temperatures ^{of a}

microemulsion with Wo = ^25.

The clustering ^model presented ⁱⁿ

section 4 now provides ^{a means for} interpre- tation of our dielectric data. ^To this end

we plot 0153/Øp ^against Op ^{where a is calcula-}

ted using equation (1) ^{with E -} eifand c 0

=

1.94. As is clear from equation (4), ^the intercept ^{with the} øp = 0 axis then yields

the polarizability ^{of a} single droplet ^while

the slopes ^{of the curves} yield ^{the values of}

I (tp,T) . ^Figures 3 and 4 show the results for Wo - ^{25 and} Wo = ³⁵ respectively. ^In

both cases the data are strikingly ^{well des-}

cribed by ^{a set of} straight lines, ^all pass- ing through ^{the same} intercept ^{at the} øp = 0

axis. The linear dependence prevails ^{for all}

temperatures ^{and volume} fractions below the

730

Fig. ^{2.- Low} frequency permittivity 6,..- ^as

a function of the disperse matter

(water ⁺ AOT ) ^volume fractlon 0 at four different temperatures ^{of a}

microemulsion with Wo = ^35.

percolation ^threshold 0. ^[8]. Io(,o p ) ^{is thus}

found to be independent of 0 p in this domain until close to Oc where this function starts to decrease (not ^{shown in} figures ^{3 and} 4).

Figure ^{5 is an Arrhenius} plot ^of I(T) ^for both cases Wo - ^{25 and} Wo = ^35.

It shows that the predicted temperature dependence ^{of I is} excellently ^fulfilled.

For wo = ^{25 we obtain} E. = ^{77 ± 5 kJ/mol}

and for Wo = ^{35 :} Ea - ^{110 ± 7 kJ/mol.}

Figures ³ and 4 show that the polarizability

of the particles ^is independent ^{of the tem-} peratures : ^for Wo= ^{25 we find} ap= ^0.88±0.02

and for Wo= ^{35 :} ap= 0.92±0.03. The relati- vely ^{low value of} ap ^{supports the use of} spherical particles ^{in our model. Values of}

ap ^{larger than one} would necessitate the use

of spheroids ^{as basic} units ; in this case one would also expect ^{that the} shape ^and

therefore ap ^{depends on the} temperature. ^The

notion of a spherical microemulsion droplet

as a basic unit over a large temperature ^and

concentration range is ^also supported by

recent neutron scattering experiments [14, 15] ^and frequency dependent permittivity

measurements [7].

Using equations (1) ^and (4) ^one may also relate the percolation ^threshold 0. ^and

I(T). It follows from equation (1) ^{that a -}

1 if £ - oo. Identifying ^{the volume fraction}

where a = 1 with the percolation ^threshold then gives

Substituting ^the experimental values of ap

and I(T) gives ^values of Or ^{which are}

Fig. ^3.- Polarizability ^{a divided} by ^the

volume fraction 0 of disperse

matter as a function of 0 at six different temperatures ^{of a micro-} emulsion with Wo = 25. ^{Legend of} figure ¹

Fig. ^4.- Polarizability ^{a divided} by ^the

volume fraction 0 of disperse

matter as a function of 0 at four different temperatures ^{of a micro-}

emulsion with Wo = 35. Legend ^of figure ²

slightly ^{above the volume} fractions where the system ^becomes conducting ^and slightly

below the values where flf ^goes through ^a

maximum for all observed temperatures [8].

The microemulsion dielectric proper- ties have been shown to be well described in terms of the polarizability ^{of the} system, equations (1), (4) ^and (6). ^{For a} given ^va-

lue of WO ^{only the} polarizability ^{of a sin-}

gle particle, ^the activation energy ^{and a}

constant are used as model parameters. ^The

Fig. 5.- Arrhenius plot of the correlation

integral I(T) ^for Wo = ^{25 (0) and} Wo ’= 35 ^(0).

underlying assumption ^{of our} description ^is

that the microemulsion droplets may form ^a

(reversible) bond after a collision and that the process is characterized by ^{an activa-}

tion energy Ea . ^{In this way} aggregates ^of particles (droplets) may be formed. This

implies that attractive interactions exist between the droplets. Recent neutron scat-

tering experiments [14,15] provide ^evidence

that short range (z ³ A) ^and strong ^attrac-

tive interactions do indeed exist in AOT microemulsions. It has been proposed ^that

the attraction is due to the mutual interpe-

netration of surfactant tails [16]. ^Evidence for droplet aggregates ^{is also} obtained from various experimental techniques ^{such as SANS} [17], Kerr-effect [18-10], and self diffu- sion measurements [21]. ^At sufficiently high temperatures ^the (reversible), aggregation

process leads to the occurrence of an infi- nite cluster and a percolation ^transition

[8].

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