Colloids and Surfaces, 57 (1991) 343-353

Elsevier Science Publishers B.V., Amsterdam 343

The experimentalists’ kit to describe microemulsions

Wiebke Sager and Hans-Friedrich Eicke

Institute for Physical Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Base&

Switzerland

(Received 2 July 1990; accepted 20 July 1990)

Abstract

Some important concepts are discussed which crystallized from the many theoretical ap- proaches used to describe microemulsions together with the relevant theoretical parameters and their relationship with experimentally accessible data.

INTRODUCTION

Due to the considerable amount of experimental material and the many the- oretical concepts obtained from and applied to thermodynamically stable, gen- erally transparent and low viscous water-in-oil or oil-in-water emulsions, (the so-called microemulsions), a state of knowledge has been achieved where rec- ipes can be extracted, useful for a resonable, i.e., not too crude, treatment of microemulsions. Also predictions of properties and approximate features of phase diagrams from considerations of the components of such emulsions seem feasible without too much sophisticated effort.

The concepts addressed are mainly of a thermodynamic character, i.e., not requiring a detailed knowledge of molecular properties of the components. The material presented in this excursus can be found in a variety of journals and has been discussed in various ways but is, by no means, generally applied to real systems as would be desirable. We would like to emphasize on the following pages what we consider essential for an up-to-date description of microemul- sions and related systems; consequently, we refer to the pioneering ideas and concepts of Bancroft, Schulman, de Gennes and Helfrich to name a few of the many contributors to this active field of research.

SCHULMAN CONDITION AND SCHULMAN LINE

We recall that the macroscopically homogeneous, isotropic and thermo- dynamically stable microemulsions are heterogeneous on a molecular scale.

0166-6622/91/$03.50 0 1991- Elsevier Science Publishers B.V.

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Thus the microemulsion can be considered to consist of water and oil domains with a huge common interface where macroscopic quantities of surfactant are accumulated. Viewed from a thermodynamic standpoint one would think of Gibbs’ adsorption isotherm by following the decrease of the interfacial tension due to the adsorption of surfactant molecules at the oil/water interface [ 11. The drop in the interfacial tension y, eventually to very low values, may cause a spontaneous emulsification of one of the components in the second, obeying Bancroft and Tucker’s rule [ 21. If in a typical plot of Gibbs’ adsorption iso- therm, i.e., y against In s (where s denotes the overall concentration of surfac- tant) &/an s becomes constant, the constancy of the interfacial surfactant concentration has to be concluded. Above the critical aggregation concentra- tion (where micelles or microemulsion droplets are formed) y becomes con- stant; its value depends on the particular surfactant and on the physical con- ditions. The occurrence of a one-phase microemulsion implies a surfactant- saturated state (Fig. l), i.e., according to the “Schulman condition” [3] that y= 0; thus the Langmuir surface pressure of the interfacial film is equal to the interfacial tension between water and oil without surfactant. This simple ap- proach does not take into account entropic effects, curvature energies and in- teractions between different portions of the interface. The Schulman condi- tion allows us to relate the ratio of the interfacial area (S) to the volume (V) (which could also be interpreted as the degree of dispersion) with geometric molecular properties of the surfactant molecule, i.e.,

where C,, v,, I,, & and N, are the equilibrium interfacial area per surfactant molecule in its saturated state, the volume, the length, the volume fraction and the number of surfactant molecules, respectively.

An improved approach by Talmon and Prager [ 41 which has been quantified by de Gennes [ 561 includes entropy effects in order to describe the oil/water

Fig. 1. Large boundaries between water and oil domains covered by a saturated monolayer of surfactants.

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interface. It could be shown [ 41 that the entropy may be responsible for certain phase transitions without regarding other interaction effects when the amount of surfactant decreases. The approach starts by dividing the microemulsion into imaginary cubes of size 5 (volume r3 ) .< is defined in such a way that the interface appears flat at length scales smaller than C$ and curved or wrinkled at larger length scales. Each cube is thought to be filled randomly with oil or water, where the overall proportion of cubes filled with oil (water) is called c$,, (4,). The average interfacial area between two differently filled adjacent cubes, where the surfactant resides, is c” and thus is related to the number of oil-water nearest neighbour pairs. This interfacial area contributes r<” to the free energy. The model predicts that a single phase with oil and water mixed down to scales of the order of {is formed when the coupling between adjacent cubes y{” is weaker than k,T, otherwise the system will separate into two phases.

We are now in a position to describe the microemulsion more precisely: we conclude that the ratio of the average area per cube to its volume is given by

which we can equate with Eqn (1) to yield

(2)

(3)

if z = 6 in a lattice with six nearest neighbours. This is a most interesting relationship which was already obtained by Debye

et al. [ 71 and Porod [ 81 when we identify 4: with the “persistence length”. The latter denotes the length over which a certain property decays to l/e of its original value. If we identify c with the diameter of a water or oil domain (e.g., droplets or characteristic length scales in bicontinuous structures), Eqn (3) can be related to the “Schulman Line” [ 61, which is the line of constant per-

S

Fig. 2. Gibbs phase triangle of a three-component system. The parabolic line represents the Schul- man line, the locus of constant persistence length in a microemulsion. The initial slopes of this line (&+ 1) and (q&+ 1) are a direct measure of the droplet diameter.

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sistence length in the Gibbs’ triangle. We realize at once that the initial slopes of the Schulman line (i.e., at @,+l and &, + 1, respectively) are a direct mea- sure of the diameters (r) of the water or oil droplets (see Fig. 2 ).

In order to understand the phase behaviour of a ternary mixture of surfac- tant, water and oil and to describe its microscopic structure it is essential to introduce curvature energy contributions into the free energy [ 91.

BENDINGENERGY

Since we are concerned with curved interfaces when microemulsion droplets are considered, the existence of rather monodisperse nanodroplets (within a certain region of volume fractions of the the dispersed material and of temper- ature) and the Schulman condition, yz 0, calls for another free energy contri- bution, i.e., the bending energy of the surfactant monolayer as introduced by Helfrich [lo]. According to this author the quadratic form of the curvature- elastic energy per unit area of the fluid interface is composed of two terms, i.e.,

Eb + +c, -c,)2+1cc1c, (4)

where K, rC, are the bending constant or rigidity and the Gaussian or saddle splay constant (which is generally omitted as the integral of the Gaussian cur- vature depends only on the genus of the surface over which it is taken), re- spectively, while c1 and c2 are the local principle curvatures (l/ri) of the curved interface and c, the spontaneous curvature with c, = 2/r,, where r, is the radius of spontaneous curvature. The spontaneous curvature is due to the asymmetry of the polar and apolar portions of the surfactant whose dependence on the physical conditions will be described later. The role played by rigidity and ra- dius of spontaneous curvature may be qualitatively illustrated by comparing the rigidity with the thermal energy. Thus, we expect for K-C kBT disordered microemulsion phases and for K> kBT ordered lamellar phases. While the ra- dius of spontaneous curvature predicts bicontinuous, sponge-like structures if r, < ( and droplet-like phases, if r, > 5.

The sign of K/r, is related to the Bancroft and Tucker rule [ 21 as applied to ordinary emulsions. It predicts the curvature of the droplets from the condition that the dispersion medium is the best solvent for the surfactant. In Fig. 3 we present a schematic picture of the different types of microemulsions which could be obtained in a ternary mixture by varying the temperature, brine con- centration or adding cosurfactants with their corresponding radii of sponta- neous curvatures. 2@ denotes a O/W microemulsion in equilibrium with an excess oil phase (which corresponds to Winsor type I), whereas 24, a W/O microemulsion with an excess water phase (Winsor II) and 3$ a bicontinuous surfactant phase in equilibrium with both water and oil phases (Winsor III).

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r, c 0 l/r,=0 r,=-0

towards oil towards water

Fig. 3. Spontaneous curvature (l/r, ) of surfactant monolayer for different types of microemulsions.

A bicontinuous structure does not necessarily imply a surfactant rich middle phase; it could be also realized in the one phase region at higher surfactant concentrations. For ionic surfactants water becomes an increasingly better sol- vent as one raises the temperature. This is due to an increased dissociation of the ionic head groups which leads to a better hydration of the latter. The radius of spontaneous curvature changes from positive to negative sign. A transition in the opposite direction (2@-+3@-+ 2@) is caused by an increase of ionic strength or an increase of cosurfactant concentration. Nonionic surfactants show an inverse phase behaviour. Due to the dehydration of the head groups with rising temperature, water becomes an increasingly poorer solvent at higher temper- atures [ 11,121.

The dependence of the interfacial tension on the radius of spontaneous cur- vature is given by [ 51

y=$ s

where r corresponds to the radius of a microemulsion droplet in a two-phase system (2@ or 2@). The interfacial tension between the water and the oil rich phases reaches a minimum in the 3@ region, where the mutual solubility of water and oil is maximal. In the 2@ and 2@ microemulsions, the radii of the droplets increase as the 3# region is approached by varying the temperature or the salt concentration. Equation (5) allows one to determine the spontaneous curvature constant x/rs experimentally from measurements of r and the inter- facial tension y. K could be measured independently by ellipsometric [ 131 or Kerr effect techniques [ 141.

From the above it should be clear that the characteristic lengths of our geo- metric entities cannot be obtained from the surface tension alone but is con-

348

trolled by bending energy contributions and entropic terms. Recently the tran- sition of micelles to microemulsion droplets at the c.m.c. has been described in terms of bending energy and spontaneous curvature [ 151. It could be shown - considering shape fluctuations of the aggregates - that spherical micelles may be stabilized by small interfacial tension alone, while the stability of strongly swollen microemulsion droplets requires a finite rigidity of the monolayer. Thus microemulsion droplets whose interfacial tension follows Eqn (5 ) are in con- trast to micelles (where the radius is always close to the packing radius) fa- voured by increasing the rigidity or decreasing the spontaneous curvature.

EXPERIMENTAL ILLUSTRATIONS

The illustrative examples which we want to consider concern: (i) phenom- ena in the so-called oil or water “corners” (@,,&,-+ 1) of the phase triangle, i.e., properties of individual droplets, their radii as a function of the water to sur- factant ratio, the reasons for the polydispersity and the effect of the surfactant monolayer rigidity. In view of the symmetry of the geometric structures in the oil and water corners (to be shown later), we restrict our considerations to the former for which we have more material at hand. (ii) By increasing the volume fraction of the minor component, approaching @, N &,,, phase inversion has to occur, i.e., the frequently discussed “bicontinuous state”. It is rather interest- ing that under special circumstances a continuous variation of the oil (or water) volume fraction from 0 to 1 at constant persistence length is possible, which allows one to follow up the variation of a variety of physical parameters over the whole oil/water composition range.

Experiments with microemulsions (Fig. 4) have clearly demonstrated that

300 l

. 8

oa 8.

&I/

.

100 l O *3

Fig. 4. Persistence length in a water/AOT/alkane system against the volume fraction ratio &.,/q&; filled circles are data obtained from quasi-elastic light scattering [ 161, open circles from small- angle neutron scattering [ 17,18].

349

the postulated droplet structure within certain volume fraction ranges of the dispersed matter and temperature region is manifest, particularly for &,,,$,,-+ 1. If the persistence length < in a water/AOT/alkane system is plotted against the volume fraction ratio &/& the experimental points obtained partially from quasi-elastic light scattering [ 161 and small-angle neutron scattering [ 17,181 measurements can be reasonably fitted by the relation c= 6 1&J@, with l,- 12 A, the length of the AOT molecule, yielding the diameter of the water droplets with increasing amount of water in the system.

The degree of polydispersity of nanodroplets in microemulsions is closely related to the bending energy [ 15,191, Electra-optic Kerr effect measurements allow one to measure the mean-square amplitude of ellipsoidal (1 = 2) shape fluctuations < 1 u2, 1’ > of microemulsion droplets. From this information rig- idities of the surfactant monolayer can be experimentally obtained [ 14,20,21].

The discussion about the unexpected large polydispersity ( - 30% ) of mi- croemulsions which were frequently reported was settled very recently by syn- chroton SAXS measurements [ 221. The analysis of the data reveal that size and shape polydispersities act in the same direction and that experimental results could be definitely better fitted by considering both polydispersity con- tributions. Hence, it has to be concluded that the mass (droplet size ) polydis- persity is indeed smaller as has been anticipated already from theoretical con- siderations [ 231.

From rather general assumptions one has to expect to pass through a bicon- tinuous region as a consequence of the structural inversion from oil-in-water to water-in-oil microemulsions. Moreover, percolation should occur, i.e., infi- nite droplet clusters should be formed, both in the oil and water corners of the Gibbs phase triangle. Fortunately, it has become possible to verify this as- sumption following the Schulman line with a pseudo-ternary microemulsion composed of AOT, 0.5% NaCl, and n-decane where the oil-to-water ratio could be varied continuously from an O/W to a W/O microemulsion without en- countering any phase separation. According to Eqn (3 ), &,&J$s must be kept constant. Figure 5 shows a cut through the Gibbs triangle for r/(6 1,) =2.85=&,&J@, ( w h ere the length of an AOT molecule I- 12 A and thus t= 200 A) as a function of temperature [ 241. The dotted line is the one-phase channel where the contributions of the microemulsions can be varied as de- scribed above. The structure of the phase diagram shows why the temperature could not be kept constant.

All of the measurements (see Ref. [ 241) confirm the existence of two per- colation processes and their occurrence at roughly &,o= 0.2 (percolation of oil droplets) and $0 = 0.8 (percolation of water droplets). The dynamic viscosity, for example, is equally influenced by both percolation processes (Fig. 6). The maxima of this plot are different due to different interactions of oil and water droplets with the respective dispersion media. The temperature coefficients of

Fig. 5. Phase diagram of the pseudo ternary AOT/n-decane/aqueous 0.5% NaCl system along the Schulman line where ~,&,,/~,= 2.85 as a function of the oil volume fraction and temperature; from Ref. [25].

Fig. 6. Semilog plot of the dynamic viscosity of the microemulsion and its temperature coefficient against volume fraction of oil; system: AOT/n-decane/0.5% aqueous NaCl solution; &,&J$.= 2.85; from Ref. [25].

the viscosity display maxima by which the percolation thresholds are more precisely located.

An interesting peculiarity of the AOT system permits one to use electric conductivity measurements to trace the percolation process in the oil corner of the phase diagram. AOT is a strong electrolyte and number fluctuations of

351

Fig. 7. Semilog plot of electric conductivity of the microemulsion and its temperature coefficient as a function of volume fraction of oil. The maximum of the temperature coefficient marks the precise location of the critical fraction of water where percolation starts; from Ref. [ 251.

I

0.02 q/H-’

0.04 006

15. .

Fig. 8. Experimental determination of the surface to volume ratio (S/V) from small angle x-ray scattering. System: C,,E,/tetradecane/water. Top: intensity I(q) against scattering vector (q) (@, = 0.15). Bottom: S/V against the amount of oil in system, according to Lichterfeld et al. [ 29 1.

352

the surfactant monolayer attach, on the average, to every second droplet one elementary charge [ 251. The conductivity against the volume fraction of oil along the Schulman line (Fig. 7) is made up by two contributions: the Stokes transport of charged droplets and an intracluster charge transport which is field independent (for details see Refs. [ 25-271). The conductivity drops rap- idly near &, = 0.8. The accurate location is determined again from the temper- ature coefficient maximum. One finds & - - 0.16 + 0.07. A comparison with the- oretical predictions [ 281 where the thickness of the overlap shell of the droplets has been assumed equal to the thickness of the attractive well shows reasona- ble coincidence for the percolation threshold.

An additional source of information stems from SAXS data obtained from a C&E&etradecane/water system with & = 0.15 [ 291. The data show for small scattering vectors a proportionality between I( Q ) and q -’ (Porod-Debye range ) from which the surface to volume ratio may be determined (Fig. 8). Figure 8 demonstrates a remarkable constancy of S/ V over the whole composition range if one changes the water-to-oil ratio.

ACKNOWLEDGMENTS

The authors thank Dr M. Borkovec for the permission to use unpublished seminar material; they are also grateful to the Swiss National Science Foun- dation and the Ciba Stiftung for financial support.

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