J. Phys. Chem. 1995, 99, 13291-13300
13291
Aggregation in Oil-Continuous Water/Sodium Bis(2-ethylhexyl)sulfosuccinate/Oil Microemulsions G. J. M. Koper," W. F. C. Sager, J. Smeets, and D. Bedeaux Department of Physical and Macromolecular Chemistry, k i d e n University, Gorlaeus Laboratories, P.O. Box 9502, 2300 RA Leiden, The Netherlands Received: June 15, 1 9 9 9
This paper deals with the aggegation behavior of microemulsion droplets, especially water droplets covered with a monomolecular surfactant layer in oil. In the first part a picture obtained by a number of experimental techniques reported in the literature is reviewed. In the second part we propose a model that describes the aggregation of microemulsion droplets into chainlike aggregates from a thermodynamic point of view. The model is shown to be capable of explaining experimental data measured in a temary microemulsion using sodium bis(2-ethylhexyl)sulfosuccinate as surfactant quite well.
1. Introduction Microemulsions are thermodynamically stable mixtures of water, oil, and surfactant(s), which can be of ionic or nonionic nature, that exhibit a rich phase behavior. They consist of water and oil domains separated by a surfactant monolayer. The structure of the oil and water domains depends crucially on the water-to-oil ratio. Microemulsions containing a comparable amount of oil and water usually form a bicontinuous structure in a certain range of temperatures, whereas in the oil and water comers of the Gibbs' triangle, nanometer-sized droplets of water-in-oil or oil-in-water exist. For a recent overview on the structure of microemulsions, see refs 1 and 2. The equilibrium size and shape of the domains are determined by the elastic properties of the surfactant monolayer separating water and oil. The natural curvature of the surfactant layer depends on the geometric properties of the hydrophilic and hydrophobic moieties of the surfactant molecule^.^ These properties, in tum, depend on temperature, pressure, ionic strength of the water, nature of the oil, etc. For a given surfactant a change of the natural curvature from curved toward water (positive) to curved toward oil (negative) or vice versa can be achieved by changing one or more of the abovementioned formulation variables. The most pronounced effect is temperature for the nonionic surfactants and ionic strength of the water for ionic surfactants. When the conditions are such that the natural curvature is close to zero, the so-called balanced state of the surfactant, a continuous structural inversion from water-in-oil to oil-in-water droplets can be achieved by varying the water-to-oil r a t i ~ . ~ The . ~ size of the droplets in single-phase oil-in-water or water-in-oil microemulsions is largely given by geometrical considerations, i.e., by the molecular ratio of the dispersed phase to the surfactant. Size polydispersity is rather small (of the order of 10-20%), and the size depends only weakly on temperature and droplet volume fraction.'.2 This paper focuses on the interaction between microemulsion droplets. For molecular solutions one has the McMillan-Mayer theory, where the interactions are described in terms of a potential of mean force (ref 6, Chapter 19). The solvent is treated as a continuum, and one usually assumes pairwise additivity of the potential of mean force. In the case of microemulsion droplets, the solvent can certainly be treated as a continuum because the droplets consist of many molecules @
Abstract published in Advance ACS Abstracts, August 1, 1995.
0022-3654/95/2099- 13291$09.00/0
(lo3 or more) and are therefore much larger than the solvent molecules. From the literature, to be discussed in the next section, we summarize the following characteristics of the interaction potential. Although the droplets are transient structures, where the dispersed phase and the surfactant are constantly exchanging between droplets and the solvent, the interaction potential appears to exhibit a distinct hard-core part of which the associated radius is proportional to the hydrodynamic radius of the droplets.' In addition, there is an attractive part to the pair potential to which the van der Waals interactions between the surfactant molecules c o n t r i b ~ t e . ~ , ~ Further, for oil-in-water droplets with ionic surfactants, one has electrostatic interactions between the droplets. The final contribution to the pair potential is an energy barrier. Such an energy barrier has already been postulated by FletcherIo.'I and Almgren and co-workers'* in order to explain the (strongly temperature dependent) rates in kinetic experiments involving microemulsion droplet interactions. The height of the barrier controls the lifetime of the bound state of two droplets, usually through an Arrhenius-type relation. During this lifetime the bound droplets appear as one entity which may eventually bind more droplets to form aggregates. This aggregation process is an equilibrium phenomenon, Le., the fraction of dimers, trimers, etc., can be calculated by means of equilibrium thennodynamics.".'3 There are many physical properties, for example, the rotation correlation time, that are sensitive to the existence of aggregates, and we shall discuss some of these properties in the present paper. The size of the aggregates (or clusters) depends on temperature and droplet volume fraction. Beyond a certain volume fraction, the percolation threshold, the system may form percolating clusters that span the system.l4.l5 The existence of such percolating clusters can easily be evidenced in the case of water-in-oil microemulsions by means of electric conductivity measurements; at the percolation threshold the conductivity increases dramatically. In section 2 we shall review the developments as described in the literature regarding the interactions between microemulsion droplets. Then, in section 3, we shall introduce a simple thermodynamic model with which we capture most of the physical phenomena that can be experimentally assessed in droplet-phase microemulsions. It is demonstrated how aggregation can account for the large values of the virial coefficients of osmotic pressure and diffusion coefficient found in the literature. We discuss, in section 4, some physical experiments,
0 1995 American Chemical Society
13292 J. Phys. Chem., Vol. 99, No. 35, 1995 namely dielectric permittivity, low shear viscosity, and electrooptic birefringence measurements, that were designed to obtain quantitative information on the aggregation process. These experiments were performed in the low droplet volume fraction regime where the aggregation phenomena already contribute significantly to the virial coefficients for the dielectric constant, viscosity, and the specific Kerr constant. From the experimental results we obtain the main parameter of the model, the binding free energy between two microemulsion droplets. In the Discussion we will come back to the nature of the interactions between microemulsion droplets.
Koper et al. the aggregation phenomenon also underlies the percolation-like behavior of the conductivity. The idea of the formation of dimers has, at approximately the same time, been suggested by Eicke and co-workers to explain the exchange of water-soluble fluorescent probes between droplets.23 The homodyning spectroscopy measurements of Graciaa and co-workers revealed a similar behavior of the second virial coefficient for a water-in-oil microemulsion with SDS as the ~urfactant.~ With ~ this technique they also determined the diffusion coefficient as a function of volume fraction. For a system consisting of identical droplets, it is given by
2. Historical Overview
D , = Dy(1
In 1976 Vrij and co-workers proposed applying liquid state theories to droplet-phase micro emulsion^.^^-^^ The droplets are sufficiently small compared to the wavelength of light to be treated as Rayleigh scatterers, and hence the scattering intensity is proportional to the osmotic compressibility [ref 18, section 3.31. For the osmotic pressure TI they used a semiempirical expression with the virial expansion
+
B2 nvd - f$ --4
kBT
2
+ ...
+ k,f$+ ...)
with B2 k, = 2- - K,
,
(2.4)
d'
where K I is due to hydrodynamic interactions and is approximately equal to 7 [ref 25, section 13.51, leading to a value of approximately k1 = 1 for hard spheres. For noninteracting spheres of radius a , the diffusion constant is
*d
where ud is the droplet volume, ksT is the thermal energy, and 4 is the droplet volume fraction. The second virial coefficient consists of the hard-sphere virial coefficient 6, which is equal to 4vd, and a term --f/vd that represents van der Waals-type attraction, Le., B2=b--
,I d'
The main conclusion of their work is that the data could be well described by the given expression for the osmotic pressure. However, the coefficient -fhad to be taken much larger than the van der Waals attraction between the droplets could account for.I6.l7 This work stimulated many researchers to investigate the virial coefficients obtained by both static and dynamic light scattering experiments. In a series of papers, Ober, Taupin, and co-workers report on different experiments performed on a four-component system with sodium dodecyl sulfate and alcohol. The paper of 197819 is probably the first to interpret the dramatic increase of the conductivity with droplet volume fraction for a water-in-oil microemulsion in terms of a percolation model. They were, however, soon followed by Peyrelasse, Boned, and co-workers.20 An equally interesting result, in our opinion, is the experimental observation that the droplet size remains constant while the volume fraction of the droplets is varied, which has been established by extensive small-angle neutron scattering ( S A N S ) experiments using the contrast variation technique.21 The size polydispersity for these systems has been found to be rather small (< 12%). This work is followed by a SANS study into the interactions and aggregation occurring in these microemulsions.22 They applied the above-mentioned theory of Vrij et al. and came to the same conclusion: the attractive term is too large to be due to van der Waals forces between the droplets. However, they find that the scattering curves are similar to those of prolate anisotropic particles. Since their previous work is not in agreement with the droplets themselves being anisotropic, they conclude that a fraction of the droplets is aggregated into dimers, and moreover, they indicate that this aggregation phenomenon could account for the large magnitude (and sign) of the second vinal coefficient. Finally, they suggested that
where 7 is the viscosity of the solvent. Graciaa et al. obtained large negative values for kl and, because of the associated negative value for the second virial coefficient of the osmotic pressure, concluded that the droplets interact strongly. Experiments on similar water-in-oil systems by Cazabat and coworkers confirm these result^.^ These authors also confirm the coincidence of percolative behavior of the conductivity and the presence of interactions between the droplets.26 Many experiments on various systems reported in the early 1980s verify the relations between the virial expansions for the osmotic compressibility and the diffusion coefficient; cf., eq 2.3.27-30 Brouwer et al. compare these findings with results from sedimentation techniques, from which KI can be determined,31and demonstrate a discrepancy with the results from static and dynamic light scattering. This discrepancy can also be found by combining sedimentation data from Ober, Taupin, and co-workers21and light scattering data of Cazabat et al.' One of the first papers investigating the influence of temperature on both the diffusion constant and the turbidity is by Eicke and c o - w o r k e r ~ .The ~ ~ experiments were performed on the watedsodium bis(2-ethylhexy1)sulfosuccinate (AOT)/isooctane system, and the main conclusions that can be drawn from their work are that ( 1) the apparent hydrodynamic radius increases with the water-to-surfactant ratio W O , as expected because this ratio determines the droplet size, and, more importantly, with temperature and ( 2 ) qualitatively the same information can be obtained from static light scattering. The analysis of the static light scattering data is complicated due to the contrast match that occurs around wo = 30: the refractive index increment vanishes for that value of WO. In 1983 Roux, Bothorel, and co-workers proposed an interaction potential for water-in-oil microemulsions on the basis of the idea that regions of the surfactant tails of the droplets overlap.8 This idea initiated many investigations into the effect of co~urfactant,~~ droplet size,34-36oil type,37*38 surfactant type,39 and combinations there~P.~O on the interaction potential. Initially, small-angle neutron (SANS) and X-ray ( S A X S ) scattering experiments provided confusing results, in particular regarding the size p ~ l y d i s p e r s i t y . ~Later, ' . ~ ~ this discrepancy
Oil-Continuous Water/AOT/Oil Microemulsions
J. Phys. Chem., Vol. 99, No. 35, 1995 13293
between light scattering, yielding a size polydispersity of typically 10-15%, and the measurements on a much smaller length scale by S A N S and SAXS, giving values of more than 30%, was attributed to shape fluctuations of the droplet^?^.^ Electrooptic birefringence experiments performed by Cazabat and c o - ~ o r k e r s ~and ~ - by ~ ~ Eicke and c o - ~ o r k e r salmost ~~ unambiguously evidence the presence of aggregates of droplets in certain microemulsions. Cazabat argues, in a 1982 review, that the fluidity of the interfacial surfactant layer is an important parameter:46rigid layers lead to hard-sphere-like behavior, for which no electrical percolation is observed, and fluid layers lead to attractive systems where the droplets aggregate. The lifetime of the aggregates is estimated from the electrooptic birefringence studies to be of the order of 40 ps for the system studied by Cazabat et al.47 and corresponds well to the nonexponential behavior at long times of the relaxation functions obtained by dynamic light ~ c a t t e r i n g . ~In~ 1988 the microstructure of microemulsions was studied by means of freeze-fracture microscopy by Strey and c o - w ~ r k e r sand, ~ ~ under proper conditions, indeed revealed cluster of droplets. In fact these pictures confirm the structural investigation of Chen and coworker~.~'.~~ In order to account for solubilisate exchange between droplets in water-in-oil microemulsions, Fletcher et al. followed Eicke and c o - ~ o r k e r sand ~ ~proposed a model in which droplets not only aggregate into dimers but also fuse or coalesce.I0.l1 The fusion process is very slow and limited to an extremely small portion of the dimers, so that the latter stage is not expected to be detectable in physical experiments. Other mechanisms, whereby the solubilisate is transferred through the continuous oil phase, were considered but proved to be inconsistent with experimental data. I Another probe that provides information on the lifetime of clusters is the dispersion of the sound velocity (as a function of frequency). For low frequencies the droplets appear to be hard spheres, whereas at high frequencies they appear to be clustered.53.54 The relation between the fluidity of the surfactant layer and the attraction between the droplets, discussed by Cazabat et al.55 and by Fletcher et al.,lO." has recently been confirmed by Holzwarth and c o - w o r k e r ~using ~ ~ the iodine temperature jump technique. From the above discussion we conclude that aggregation does occur in droplet-phase microemulsions, and in the next section we propose a model that explains the above-mentioned phenomena, in particular the rather large values obtained for the virial coefficients of the osmotic pressure and for the diffusion coefficient. losl
3. Aggregation Model In this section we put forward a simple model capable of describing the temperature dependent clustering phenomena occurring in droplet-phase micro emulsion^.^^ The analysis below is essentially that of Chapter 16 of ref 3, specialized to the situation at hand. The system under consideration has a volume fraction $ of droplets of which the volume fraction of single (isolated) droplets is 41. A cluster of k droplets is denoted by Ck, and the volume fraction of droplets aggregated into k-clusters, by $k. The total number of droplets is conserved; hence (3.1) k= I
All clusters are in equilibrium with one another; thus one has,
for instance,
nc,= c,
(3.2)
which states that n single droplets can cluster into one single n-cluster. When pk is the chemical potential per droplet in a k-cluster, so that the chemical potential of a k-cluster is given by kpk, the equilibrium condition implies
Pn =PI
( n 2 1)
(3.3)
Le., the chemical potential per droplet is the same for all types of clusters. For dilute systems the relation between the chemical potential per droplet of k-clusters and their volume fractions is 0
pk = p k
+ -1ogk kBT
4k k
(3.4)
( k is~ the Boltzmann constant and Tis temperature). Since the chemical potential per droplet in all types of clusters is identical, the volume fraction of droplets in k-clusters satisfies
In order to proceed, assumptions have to be made about the nature of the clusters. The first assumption is that the droplets aggregate linearly; they do not form (two-dimensional) sheets or (three-dimensional) compact structures. For low volume fractions this assumption is supported by observations with freeze-fracture electron m i c r o s ~ o p y .To ~ ~be~ precise, ~~ a linear k-cluster is formed by k droplets that are connected by k - 1 bonds. Notice that this implies that a linear k-cluster may have branches, as long as it has only k - 1 bonds. Let B be the free energy per bond relative to isolated droplets; then the standard chemical potential per droplet is given by
= Cl;
k- 1
B +7
(3.6)
Using this expression in eq 3.5 for the k-cluster volume fraction yields
The assumption of constant binding free energy per bond can be further supported by a statistical mechanical calculation where the binding energy per droplet is fixed. The entropic part of the binding free energy can then be calculated and turns out to be almost independent of cluster size, as was assumed. The conservation of volume fraction (3.1) can now be used to express the volume fraction of single droplets solely as a function of both the volume fraction $ of droplets and the bond free energy B. The sum over cluster size converges, and one obtains
4 = 4 41 Solving the above equation for
$1
2)
2
gives
The Boltzmann factor $0 plays the role of a "critical droplet volume fraction" much in the same way as there is a critical micelle concentration in micellar systems. A necessary condi-
13294 J. Phys. Chem., Vol. 99, No. 35, 1995
Koper et al.
tion for aggregation to take place is that 40 1, which holds when the bond free energy B is negative and at least of the order of kBT. Some authors have presented an identical picture of the aggregation process in terms of multiple chemical equilibria, a so-called linear aggregation This involves the coupled set of reaction equations
-
c, + c, c, c2+ c, = c, c, + c, c,
In the present case of clusters of aggregated droplets, we have = &/kvd, and hence eq 3.16 becomes (3.18)
With this we find, using eq 3.9, the virial expansion for the osmotic pressure of the droplets (3.19)
(3.10)
The assumption here is that the reaction constant for any of these equilibria, (3.11) is the same for all values of n 2 2. Since [C,] may verify that
GZ
K, = vd eXp( -B/kBT)
$,,In, one (3.12)
where vd is the volume of the droplet. This shows that this model is identical to the one discussed above. For concentrated systems eq 3.4 is no longer valid and the volume fraction 4 k has to be multiplied by an activity coefficient Y k such that Y k 1 as 4 0. It is, in this context, more appropriate to express the activity coefficients in terms of the virial coefficients for the pressure,6 Le.,
-
@k
-
Assuming the interaction between dro.plets is short ranged so that only excluded-volume interactions need to be taken into account, the single droplet virial coefficient is simply given by = 4vd'* and the effective second virial coefficient, as measured by static light scattering and osmotic pressure, is given by BY =4
- exp( -BIkBT)
(3.20)
The origin of a large and negative effective second virial coefficient for a system of aggregates lies in the fact that although all droplets contribute equally to the overall volume fraction (or mass), this is not the case for the osmotic pressure where only the number of clusters (the number of translational degrees of freedom) counts rather than the number of droplets. The situation is similar to that of polymer solutions. 3.2. Diffusion Coefficient. The effectiue diffusion coefficient can be derived in a similar fashion. For a system consisting of clustered particles, one measures the z-averaged diffusion coefficient [ref 25, section 8.1 11. This coefficient is defined by
c k@pk c m
(3.13)
0:
k= I
(3.21)
m
where @jk is the second virial coefficient between speciesj and k. Equation 3.4 then becomes
k@k
k= I
(3.14)
and reduces, by using eqs 2.3, 3.7, and 3.9, after some algebra, to
The expressions for the cluster volume fractions are then more involved. Since we will restrict this analysis to low volume fractions, we shall not pursue this issue any further here. 3.1. Osmotic Pressure. The vinal expansion for the pressure of a multicomponent system is given by ref 18, p 248)
The term kl is of the same origin as kl in eq 2.3, namely from the static structure function. Hence
pk = pi
+
kBT[
@k
log k
-- kBT
+2
@jk@j
-+ ...) j=l
Jvd
+ B2e2+ ...
(3.15)
where
@=z@k
(3.16)
k= 1
with @ k being the number density of the kth component and where (3.17) j=l k=l with the mole fraction xj =
= 2BY - K ,
(3.23)
The linear term in the volume fraction 4 is the effective virial coefficient of the diffusion coefficient. The term with the difference between the diffusion coefficient for a 2-cluster and a single droplet might be the origin of the discrepancy that has been found between the virial coefficients from static, Le., Beff , and dynamic light scattering and sedimentation data (measuring Kl) by Brouwer et aL3I
4. Experimental Results In this section we discuss some experiments that have been performed in order to obtain more detailed information about the clustering phenomenon, in particular about the binding free energy B. First we shall introduce the phase diagram of the
J. Phys. Chem., Vol. 99, No. 35, 1995 13295
Oil-Continuous Water/AOT/Oil Microemulsions
iso- oc tane
80
5”
-
60-
20
I
0
I
I
I
10
20
30
F I
water
’
do
‘160
’
0 100
AOT
Figure 1. Schematic phase diagram of the water/AOT/isooctane system at 25 O C . Data are taken from F0nte11.~~ For clarity, only the singlephase regions are indicated. L I and L2 are oil-in-water and water-inoil droplet phases. D is lamellar liquid crystal, I is cubic liquid crystal, and F is reversed hexagonal liquid crystal. The L I region extends to about 1% AOT in water and 0.2 % in isooctane and is too small to be properly represented on this scale. microemulsion and the conditions for having a single droplet phase. Then we discuss three sets of experiments: first dielectric permittivity measurements by van Dijk et al.59 and then viscosity measurements and electric birefringence experiments performed at our laboratory. The main interest is in determining the binding free energy B that has an enthalpic part AH and an entropic part AS such that
B = A H - TAS
(4.1)
4.1. AOT Microemulsions. AOT forms water-in-oil microemulsions with water and oil in a certain range of concentrations and temperatures, e.g., thermodynamicallystable dispersions of nanometer-sized water droplets surrounded by a surfactant monolayer in the oil. The Gibbs’ triangle (see Figure 1) shows at room temperature a miscibility gap (two-phase region) entering the triangle from its water/oil side in which a waterin-oil microemulsion is in equilibrium with an almost pure water phase (see refs 60-62). A single water-in-oil microemulsion (Lz-phase) forms adjacent to the two-phase region along the oiVAOT side, while liquid crystalline phases extend from the AOT/water side into the triangle (see ref 60). The ternary mixture of water/AOT/oil exhibits a strong temperature dependence due to the almost balanced state of hydrophobic and hydrophilic moieties of the AOT molecule. With incresaing temperature (and thus with increasing ionization of the ionic head group) water gradually becomes a better solvent for the AOT molecules, while its solubility in the oil decreases. At higher temperatures, the sign of the slope of the tie lines in the two-phase region changes, and an oil-in-water microemulsion stays in equilibrium with an almost pure oil phase, which is surrounded by a single-phase oil-in-water microemulsion (L1-phase).6’ The ternary system does not show a three-phase region with water and oil excess phases, both in equilibrium with a surfactant-rich middle phase, while the addition of salt (fourth component) leads at intermediate temperatures to the formation of a three-phase triangle whose extensions increase with increasing amount of added salt.64The influence of impurities such as the hydrolysis products of AOT has been extensively studied in refs 61 and 65.
(G)
1
50
t
100 40
I
Figure 2. concentratiodtemperature phase diagram at constant AOT/ oil ratio with increasing water content. The line results from a polynomial fit to the data points. The molecular water to surfactant ratio w ois defined by eq 4.2. L2 indicates the water-in-oil droplet phase, and 24 indicates a two-phase region.
oil
Figure 3. Schematic model of a water/AOT/isooctane microemulsion droplet.
A concentratiodtemperature phase diagram erected on a line in the Gibbs’ triangle at constant AOT/oil ratio and increasing water content (heading toward the water comer of the Gibbs’ triangle) is given in Figure 2. The purification procedure that we used for the AOT is described in detail by Smeets et a1.66 At the lower boundary of the single-phase water-in-oil microemulsion (solubilization limit), an almost pure water phase separates out. This line does not show in our system for temperatures over 5 “C. At the upper boundary two surfactantrich isotropic phases or an L2-phase in equilibrium with a liquid crystalline phase coexist. The L2-phase is what we will further call the droplet phase. The droplets are fairly monodisperse (polydispersity 12%@)spherical water pockets coated with a monomolecular layer of surfactant molecules (see Figure 3). Their size is determined by the molecular water-to-surfactant ratio (4.2) Using the simple picture in Figure 3, one derives for the hydrodynamic radius of the droplets 3v
r, = - wo a,
+6
(4.3)
where v is the molecular volume of the water (3.0 x m3), a, is the area occupied by one surfactant molecule at the interface between the water core and the surfactant monolayer, and 6 is the thickness of the surfactant monolayer, including possibly entrapped solvent molecules (approximately 1 nm). Because
13296 J. Phys. Chem., Vol. 99, No. 35, I995
Koper et al.
TABLE 1: Summary of Obtained Binding Enthalpies and Entropies technique AH (kJ/mol) AS (J/K mol) dielectric p e r m i t t i ~ i t y ~ ~ 74 f 5 260 3z 30 viscosity" 8 9 f 18 310 f 60 Kerr effect 81 f 4 315 f 30 SAXS88.89 36 f 3
C
with &I = al/3EO&& &2 = (a2 - ~ U I ) / ~ E etc. O U Figure ~, 4 gives a plot of the quantity A(4,T) versus volume fraction as a function 0.0 0.2 0.4 0.6 of temperature calculated from the dielectric permittivity data cp taken by van Dijk et al.59 The infinite dilution extrapolated value (4 0) of A yields the single droplet value &I = 0.74f 2.0 0.02. Within the accuracy of the measurements, &I is found to be independent of the temperature. Using a coated sphere one can from this value for &I calculate the effective thickness of the surfactant layer to be 0.36 f 0.04 nm. This 1.5 value is to be compared to the length of the surfactant tails ( f l F nm in stretched form). For nonvanishing particle volume fractions, A departs linearly 1.0 from its infinite dilution extrapolated value. The slope is strongly temperature dependent, and a van't Hoff plot of the slope versus inverse temperature yields a straight line from which the binding enthalpy can be extracted. Using an 0.5 0.0 0.2 0.4 0.6 estimated value for &2, which is equal to 0.4 (for conducting cp spheres),76we can also estimate the entropy of binding. The Figure 4. Relative dielectric permittivities (top) at different volume obtained values are listed in Table 1. fractions and temperatures for a wo = 25 water/AOT/isooctane 4.3. Low Shear Viscosity. Most of the low shear viscosity microemulsion. The data are taken by van Dijk et al.59 Underneath experiments on microemulsions have been performed in order the function A is plotted (cf. eq 4.6)for the same volume fractions and temperatures. The inset shows the van't Hoff plot of the slope of the to gain more insight into the percolation behavior which occurs curves versus inverse temperature. in the Berg and co-workers tried to model the low shear viscosity increment with increasing volume of the ionic character of the surfactant molecules used, the area fraction but could not distinguish droplet aggregation from other asper surfactant molecule is dependent on the size of the droplet; explanations, such as electroviscous effects.79 Smeets et al. have for this work we have used67 performed low shear viscosity measurements on droplet-phase water/AOT/isooctane microemulsions66in such a way that the results could be compared to those of the dielectric permittivity a,/nm2 = 0.596 - 0.468 exp(-0.401&) (4.4) measurements of van Dijk et al. Figure 5 presents a plot of the experimentally obtained low shear viscosity for a w o = 25 As an example, a wo = 25 microemulsion consists of droplets microemulsion versus volume fraction as a function of temperwith a hydrodynamic radius of approximately 5.5 nm. ature. 4.2. Dielectric Permittivity. The early dielectric permittivity Just like the dielectric permittivity, cf. eq 4.5,the low shear measurements on microemulsions were mainly performed to viscosity of a suspension of noninteracting aggregated droplets obtain structural information about the droplet^.^*-^^ More relative to the oil viscosity is given by the SaitB equation.66 recent investigations consider the behavior of the dielectric permittivity around the conductivity percolation t h r e ~ h o l d . ~ ' - ~ ~The situation is more difficult here due to the long-ranged nature of the hydrodynamic interactions between the particles so that Here we are interested in the behavior for small volume contributions due to correlations to the higher order terms also fractions, well below the conductivity percolation volume need to be taken into account: fraction. Figure 4 gives the experimentally obtained relative dielectric permittivity for a w o = 25 water/AOT/isooctane 0
4
.
8
,
I
.
I
.
-
microemulsion taken by van Dijk e t al.59
Neglecting interparticle interactions, the low-frequency dielectric permittivity er of a suspension of aggregates of droplets relative to the dielectric permittivity of the oil is given by the Clausius-Mossotti relation75 (4.5) where a k is the polarizability of a k-cluster, gk the number density of k-clusters, and EO the dielectric permittivity of the oil. We rewrite eq 4.5,using eq 3.7, in terms of excess polarizabilities per unit volume,
where pk is the fraction moment of a k-cluster, pl = 1, p j k are friction moments due to j - and k-cluster contributions to the two-particle correlation function, etc. Equation 4.7 is, using eq 3.7, rewritten as
where p2 = p2 - pl is the excess friction moment of a 2-cluster,
Oil-Continuous Water/AOT/Oil Microemulsions ....._..T = 30 "C 25 "C - - - TT == 21 "C T = 4.5 "C
'
J. Phys. Chem., Vol. 99, No. 35, 1995 13297 aligned parallel to the electric field and induces an anisotropy of the microemulsion fluid. The Kerr constant, which is associated with this particle deformation, is defined as the ratio of the anisotropy in the refractive index and the magnitude of the electric field squared. When we model a microemulsion droplet as having a core radius rw,with a surfactant layer of thickness d,, the single-droplet Kerr constant per unit volume if given by
I
6
(4.9)
0.8
I
,
I
where the constants a, b, and c depend on the refractive indices and are given explicitly in ref 81. The constant a is inversely proportional to the bending elasticity modulus of the surfactant layer. Due to the intrinsic anisotropy of the surfactant molecules, the single-droplet Kerr constant first becomes negative as a function of core radius, as found for the first time by Hilfiker et aL6' For larger droplet sizes it is proportional to the core radius cubed, independent of the thickness of the surfactant laye?2 (see Figure 6). The value that one obtains for the bending elasticity modulus is l k ~ when T a polydispersity of approximately 10% is a ~ s u m e d . ~This ~ . ~value ~ agrees well with the value of l . l k ~ Tobtained by e l l i p s ~ m e t r y . ~ ~ In the birefringence relaxation, one can recognize two relaxation modes. The relaxation time of the fast mode corresponds well to what can be calculatedx6for the relaxation of the spheroidal form deformation of the droplets. The slow mode has an amplitude that varies with volume fraction squared, whereas the amplitude of the fast mode varies linearly with volume fraction. The time constant of the slow mode is comparable to the longest rotation correlation time for a dumbbell.86 This observation has also been made by other authors to evidence aggregation in different microemulsion syStemS.45.47.4X.55,X7 Recently we have measured the electrooptic birefringence of a water/AOT/isooctane microemulsion of w o = 25. In Figure 7, the measured Kerr constant is plotted versus volume fraction as a function of temperature. For such a system the measured Kerr constant is equal to (4.10) where KOis the Kerr constant of the solvent. The specific Kerr constant is now calculated from eq 4.10, using eq 3.7, as K=
- 4)
Km -
4
=K,
+ k2e-B'kBT4+ ... (4.1 1)
where K2 = K2 - K I is the excess Ken constant for dimers, etc. The above equation (4.11) has been used to fit the data presented in Figure 7, and we obtain the single droplet value KI = -(3 f 1) x m2N2,which is in fair agreement with the value of KI = -(5 f 2 ) x m 2 N 2of van der Linden et aL8I The binding enthalpy and entropy-are given in Table 1; the latter is found by estimating the ratio K ~ / Kto I be of order 1. 4.5. Results. The binding enthalpy and entropy for dropletphase microemulsions of wo = 25, determined by the three techniques that we discussed above, are presented in Table 1. We have added one value that we obtained from SAXS experiments performed by Robertus et a1.,8Eanalyzed using a sticky hard-sphere where the droplet interaction is modeled by a hard-core part with an attractive well next to it.
Koper et al.
13298 J. Phys. Chem., Vol. 99, No. 35, 1995
3-
n
"E N > N
2-
I
1-
N I
0 4 W
G
-2
0
1
2
3
4
5
6
'
7
ruJ(nm) Figure 6. Single-particle K e n constant as a function of the radius of the water core for a water/AOT/isooctane microemulsion of wo = 25 at 23 'C. The data are taken from van der Linden et a1.*'
501
* -250
0.00
9
7
,
0.05
I
/
0.10
cp 0.0 1
5. Discussion
n N
3
\
N~-0.5
2 0-1.0 rl
v
4
-1.5
-2.0
-2.5 0.00
dynamic consequences of aggregation into account. From this table we conclude that the values obtained are, within experimental error, the same, regardless of which experimental technique has been used, except for those of the SAXS experiments. The small difference between the values obtained by dielectric permittivity measurements and viscosity and Kerr effect could well be due to the fact that a different purification procedure for the AOT was used for the dielectric permittivity measurements. The reason for the discrepancy between the SAXS experiments and the other three is still unclear. A possible explanation might be that SAXS experiments are also sensitive to shape fluctuation^.^^ Shape fluctuations have been shown to be responsible for the larger size polydispersity values obtained by SAXS and SANS experiments as compared to contrast-match methodsM Robertus et al. also find the much too large value of 22% for the polydispersity.88
0.05
v
I
0.10
/
Figure 7. Kerr constant (top) at different (effective) volume fractions and temperatures for a wo = 25 water/AOT/isooctane microemulsion. Underneath the function K is plotted (cf. eq 4.1 1) for the same volume fractions and temperatures. The inset shows the van't Hoff plot of the slope versus inverse temperature.
The advantage of this model is that the structure factor can be calculated analytically. It does, however, not take the thermo-
We have shown that a simple model describing the aggregation of microemulsion droplets into chainlike aggregates can be used succesfully to describe the static as well as the dynamic properties of droplet-phase microemulsions. It is demonstrated how aggregation can account for the large values of the virial coefficients of osmotic pressure and diffusion coefficient found in the literature. We discussed the results of some physical experiments, namely dielectric permittivity, low shear viscosity, and electrooptic birefringence measurements, w h i c h were designed to obtain quantitative information on the aggregation process. These experiments were performed in the low droplet volume fraction regime where the aggregation phenomena contribute already significantly to the strongly temperature dependent virial coefficients for dielectric constant, viscosity, and the specific Kerr constant. From the experimental results, we obtained the main parameter of the model, the binding free energy between two microemulsion droplets. The nature of the interaction between the microemulsion droplets is not yet resolved. The relation between the fluidity of the surfactant layer and the attraction between the droplets, as discussed by Cazabat et al.55and by Fletcher et al.,iO-l'points in the direction of the curvature free energy. Indeed, droplet
Oil-Continuous Water/AOT/Oil Microemulsions aggregation increases when the balanced state of the surfactant molecules is approached and when the spontaneous curvature vanishes. Another dependence that has not been discussed in this paper is the dependence of the interaction on droplet size. Both dielectric p e r m i t t i ~ i t yand ~ ~ viscosity measurements66 indicate a linear dependence of the binding enthalpy with droplet size for the AOT water-in-oil system. A few explanations for the interaction energy between microemulsion droplets have been put foward: van der Waals forces between the droplet core^'^*'^ and between the surfactant molecule^?.^ curvatureinduced interaction^,^'-^^ etc. So far, none of the proposed explanations discuss the temperature dependence reported here, and we conclude that more work, both experimental and theoretical, is needed to identify the nature of the interactions between microemulsion droplets.
Acknowledgment. We thank H. Reiss for pointing out the correctness of the assumption of a constant binding free energy per bond, independent of cluster size on the basis of a statistical mechanical model calculation. It is a pleasure to thank P. D. I. Fletcher, A. M. Cazabat, M. Borkovec for discussions and M. van Zelst for performing the electrooptic birefringence measurements. Part of this work has been performed under auspices of the EC Network Thermodynamics of Complex Systems (Contract No. CHRX-CT92-0007). The authors are grateful for a visitor’s grant from the Dutch Science Foundation (NWO) for one of us (W.S.). References and Notes (1) Chevalier, Y.; Zemb, T. Rep. frog. Phys. 1990,53, 279. (2) Cazabat, A. M. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; DeGiorgio, V., Corti, M., Eds.; North Holland: Amsterdam, 1985. (3) Israelachvilli, J. lntramolecular and Surface Forces: Academic Press Ltd.: London, 1992. (4)Nillsson, P.-G.; Lindman, B. J . Phys. Chem. 1982, 86, 271. ( 5 ) Borkovec, M.; Eicke, H.-F.; Hammerich, H.; Das Gupta, B. J. Phys. Chem. 1988, 92, 206. (6) Hill, T. L. An lntroduction to Statistical Thermodynamics;Dover Publishers Inc.: New York, 1986. (7) Cazabat, A. M.; Langevin, D.; Pouchelon, A. J . Colloid lnterface Sci. 1980, 73, 1. (8) Lemaire, B.; Bothorel, P.; Roux, D. J. Phys. Chem. 1983, 87, 1023. (9) Roux, D.; Bellocq, A. M.; Bothorel, P. Progr. Colloid Polym. Sci. 1984, 69, 1. (IO) Fletcher, P. D. I.; Howe, A. M.; Robinson, B. H. J . Chem. Soc., Faraday Trans. I 1987, 83, 985. (11) Fletcher, P. D. I.; Holzwarth, J. F. J. Phys. Chem. 1991, 95, 2550. (12) Johannsson, R.; Almgren, M. Lungmuir 1993, 9, 2879. (13) Koper, G. J. M.; Smeets, J. Progr. Colloid Polym. Sci. 1994, 97, 237. (14) Kirkpatrick, S. Rev. Mod. Phys. 1973.45, 574. (15) Stauffer, D. Phys. Rep. 1979, 54, 1. (16) Agterof, W. G. M.; van Zomeren, J. A. J.; Vnj, A. Chem. Phys. Lett. 1976, 43, 363. (17) Calje, A. A.; Agterof, W. G. M.; Vrij, A. In Physics ofAmphiphiles: Micelles, Vesicles and Microemulsions; DeGiorgio, V., Corti, M., Eds.; North Holland: Amsterdam, 1977. (18) McQuame, D. A. Statistical Mechanics; Harper Collins Publishers Inc.: New York, 1976. (19) Lagues, M.; Ober, R.; Taupin, C. J. Phys. Lett. 1978, 39, 487. (20) Lagourette, B.: Peyrelasse, J.; Boned, C.; Clausse, M. Nature 1979, 281, 60. (21) Dvolaitzky, M.; Guyot, M.; Lagues, M.; LePlesant, J. P.; Ober, R.; Sauterey, C.; Taupin, C. J. Chem. Phys. 1978, 69, 3279. (22) Ober. R.: Tauuin. C. J. Phvs. Chem. 1980, 84. 2418. (23) Eicke, H.-F.; Shepherd, J. C . W.; Steineman, A. J. Colloid Interface Sci. 1976, 56, 168. (24) Graciaa, A,: Lachaise, J.; Chabrat, P.; Letamendia, L.; Rouch, J.; Vaucamps, C.; Bourrel, M.; Chambu, C. J. Phys. Lett. 1977, 38, 253. ( 2 5 ) Beme, B. J.; Pecora, R. Dynamic Light Scattering; Krieger Publishing Company: Malabar, FL, 1990. (26) Cazabat, A. M.; Chatenay, D.;Langevin, D.: Pouchelon, A. J. Phys. Lett. 1980, 41, 441. (27) Finsy, R.; Devriese, A.; Lekkerkerker, H. J . Chem. SOC., Faraday Trans. 2 1980, 76, 767.
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