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Interpretation of phase diagrams: topological and thermodynamical constraints
Colloids and Surfaces A: Physicochemical and Engineering Aspects 205 (2002) 3 – 13 www.elsevier.com/locate/colsurfa
Interpretation of phase diagrams: topological and thermodynamical constraints Fabienne Testard, Thomas Zemb * DRECAM/Ser6ice de Chimie Mole´culaire, CEA— Centre d’Etudes de Saclay, Batiment 125, F91191 Gif sur Y6ette cedex, France Received 12 November 2001; accepted 12 December 2001
Abstract Some general principles of surfactant self-assembly are rationalised in order to give a short introduction to the use of phase diagrams for binary (two components) and ternary (three components) systems. Finally models for solubilisation are considered to be understood within the frame of those general packing principles: volume, surface and curvature conservation. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Surfactant; Phase diagram; Curvature; Solubilisation
1. Introduction Our aim is to give a short introduction to the use of phase diagrams in order to rationalise some general principles of surfactant self-assembly. These are summarised in the form of guidelines before using more complete textbooks such as the recent one written by Evans and Wennerstroem [1]. We consider separately binary (two components: surfactant and solvent), then ternary systems (surfactant(s), water, oil) and finally models for solubilisation, to be understood within the frame of some general principles and not as tabulated data. We define as ‘surfactant’ any molecule which induces reduction of surface tension, i.e. positive * Corresponding author. E-mail addresses: [email protected] (F. Testard), [email protected] (T. Zemb).
adsorption at air –water interface. Lipids as well as detergents, most ‘complexing molecules’ as well as associative polymers belong to the ‘surfactant’ class of molecules according to this general definition. Due to lack of space, all figures, especially phase diagrams are not included, they can be found in original references, some of them are in the form of useful ‘Atlas’ of phase diagrams, a book [2] useful to physical chemists like a good map for geographers.
2. Binary phase diagrams This is the case where only ‘surfactant’ and ‘solvent’ are present in the sample. At atmospheric pressure, two thermodynamic variables define the system. Concentration (usually in volume or weight fraction, sometimes as chemical potential if the ordinate axis is plotted in logarith-
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mic scale) and temperature are the two axes of commonly found phase diagrams [3].
2.1. Spontaneous cur6ature Spontaneous curvature [4] is the basic concept rationalising surfactant aggregation. Transition from monomers to multi-molecular aggregates is observed as soon as the chemical potential is high enough to induce the transition from a molecular solution towards an organised molecular system. This transition is called the critical micellar concentration (cmc). Even if numerically, most of the time, equivalent, the cmc should not be systematically identified to the concentration of monomers in dynamic equilibrium with micelles. The curvature of an interface is defined by using two circles tangent at each point to a general surface covered by a surfactant monolayer, as shown in Fig. 1. On each point, there is a maximal value (r1) and a minimal value (r2) for radii of circles tangent to the surface. The curvature (dimension: length − 1) is the inverse of radius of the tangent circle. The average curvature H is the average of the inverse of the two radii of curvature. The Gaussian curvature K is the geometrical mean of the inverse of the two radii of curvature. Fig. 1, from the textbook [4] ‘The language of shape’ by Hyde et al., illustrates the sign of these quantities for the three cases frequently described in local packing topologies: globular, cylindrical and cubic. Average curvature and Gaussian curvature are quantitatively lower in the case of lamellar phase, or locally lamellar structures [5]. We first consider the simple case of ionic spherical micelles. With R the radius of the micelle, the Gaussian as well as average curvature are, respectively, given by: K= 1/R 2 and H = 1/R. In this case, the head-group dissociation can be predicted within the frame of the ‘dressed micelle’ model from first principles and a measure of the aggregation number only [6]. For a given micelle, the effective charge, i.e. the net electrical charge taking into account adsorbed counter-ions and non-linear effects, can be explicitly calculated from molecular volumes, ion size and water– oil interfacial tension. The numerical relation shown by Hayter [6] giving the effective charge, once
volume and surface are known, holds for all micelles and predicts effective charge in all cases where there is no ‘chemical’, i.e. non-electrostatic interaction between the oil–water interface and the counter-ions. The micelles which have a lower charge than calculated involve highly polarisable counter-ions, such as bromide and iodide, or specific receptors, such as cage molecule implying ion head-group specific interaction. A general theory for the origin of the Hofmeister effect has been recently proposed [7]. Consequences for the surfactant aggregation are described by Underwood and Anacker [8]. The maximum number of surfactant per globular micelle, the so-called Tanford [9] limit for spherical micelle, is given by simple geometrical arguments: N max agg 3× (apolar molecular volume) (area per molecule at core– water interface)× R =
where R is the radius of the apolar core of the micelle. This number lies between 20 and 200 for
Fig. 1. Examples of curved interfaces. These schematically represent microstructures in the case of droplets, cylinders and cubic phases. Average curvature is non-zero for droplets and cylinders and zero for the cubic case. Gaussian curvature is positive for droplets, zero for the cylinders and negative in the case of cubic packing.
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‘good surfactants’, i.e. easily soluble amphiphilic molecules commonly used in formulation or research work. If microstructure is not known a priori, self-organisation of surfactant molecules can be rationalised in terms of chemical potential primarily depending on the concept of spontaneous curvatures H0 and K0 of any surfactant monolayer [10]. The spontaneous curvatures H0 and K0 are the curvatures adopted by the surfactant film in the absence of any bending. The competition between the spontaneous curvatures, which depend on molecular shape, temperature, salinity, for instance, and the curvatures imposed by sterical constraints such as micellar volume and surface is the most general principle of surfactant self-assembly [4]. The link between molecular dimensions is given by the ‘coverage relation’, which links the local packing parameter p of neighbouring surfactant molecules to macroscopic curvatures imposed by topology: p = 1+ Hl+ 1/3Kl2
(1)
The packing parameter p is a number which gives first order the shape of one molecule embedded in a surfactant film. p= V/(|l)
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ing constraints at the microscopic scale. A standard expression of the cost in free energy F (per unit area) due to the difference between these two quantities is: F= 1/2Kc(H−H0)2 + s(K− K0)
(3)
Expression (3) presents strong self-consistency problems as shown by Fogden et al. [11] in the case of non-zero spontaneous curvature. An alternative microscopic model of the membrane could be used to express the free bending energy of the surfactant film: F= 1/2K*(p− p0)2
(4)
where K* is a generalised bending constant [11], and p0 is the effective value of the surfactant parameter in the sample calculated with Eq. (1) using H0 and K0. For non-ionic systems, surfactant parameter p is easily varied by using temperature: high temperature dehydrates the ethoxy polar head-group, thus reducing the solubility in water and increasing the solubility in oil of the surfactant. For ionic surfactants, the situation is more complex since the counter-ion density function near the oil–water interface sets the area per head-group and also the elastic bending constant of the interface.
(2)
where at microscopic scale, the volume V is the volume of the apolar part of the surfactant molecule, | is the area per molecule at the interface and l is the average chain length. On a macrocopic scale, H is the average curvature, averaged over the whole sample, and K is the Gaussian curvature, averaged over the whole sample. Reference between constraints at microscopic scale and macroscopic constraints allows to rationalise the formulation, i.e. the effect of co-surfactant, temperature and salt. Relations (1) and (2) are too crude to be used directly, but are the basis of any model evaluating the difference between two samples. They can be used differentially, considering temperature, concentration or salinity variations as well as the solubilisation of a host molecule. H characterises a composition, while the spontaneous curvature, H0 reflects local pack-
2.2. Phase limits The phase limits are set by intermolecular interactions. For example, the ‘cloud point’, i.e. a micelle–water phase separation at high temperature is obtained for some non-ionic surfactants. There is a relation between the intermicellar potential and the clouding point [12–14]. Thus, a variation of clouding behaviour is observed by the presence of a solute. If the increase of apolar volume by solute can be neglected, intermicellar potential variation can be directly derived.
2.3. Binary phase In binary phase diagram, the phase boundaries provide lines of known osmotic pressure. The determination of phase coexistence is a powerful tool for determining molecular forces.
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2.3.1. Lamellar or hexagonal phases When a surfactant in the form of a lamellar or hexagonal phase is in equilibrium with excess pure water, the total balance of attractive and repulsive interactions between surfactant aggregates is zero, since the osmotic pressure of the water phase in excess is zero. Thus, a determination of maximum observed periodicity versus any thermodynamical parameter such as temperature, ionic strength or composition of solvent is equivalent to a very sensitive molecular force balance [15]. Parsegian and co-workers [16] have shown that if the attractive and repulsive interactions are algebraically combined as an approximation, quantitative information about magnitude of interaction between membranes can be obtained. 2.3.2. Binodal lines When two phases coexist in a region of binodal lines, there is ‘competition for water’. Since the osmotic pressures in the two phases have to be equal, the sum of attractions and repulsions in the two coexisting phases have to be equal. In the case of diluted and concentrated lamellar phases, this property allows direct observation of nonmonotonic forces between flat surfactant interfaces, an effect ubiquitous in surfactant systems [17]. Close to the critical point, exploitation of this property is a ‘molecular force balance’ of unprecedented sensitivity. It is only sufficient to measure periodicity and osmotic pressure in the coexistence region of two phases to obtain a differential measure of the interaction potential between oil– water interface, without the complexity of AFM or surface force balances.
3. Ternary phase diagrams Phase diagrams are profoundly different for flexible and rigid surfactant films [10]. Flexible films are films for which the cost in bending energy of local curvature variations is smaller than thermal energy. The microstructure of flexible films is dominated by entropy. Rigid films can be modelled as molten liquid crystals or a glass of small crystallites of liquid crystals and
therefore form more homogeneous microstructures, while in flexible films, fluctuations dominate. Basically, flexible films correspond to kc kBT and rigid films to kc kBT. The intermediate case where kc : kBT and associated microstructure has been examined only recently [18].
3.1. Pseudo-binary systems The simplest case is the historic case of sodium octanoate–water–octanoic acid which is a quasibinary system studied by Ekwall and co-workers [19–21] where the ‘oil’ and ‘surfactant’ are acidic and sodium forms of the same molecule. In this system, the connected lamellae (sponge), i.e. the first non-globular micellar aggregates as well as highly swollen lamellar phases have been discovered.
3.2. Two surfactants in water Cationic systems, when an anionic surfactant is associated to a cationic surfactant in water belong to the class of ternary self-assembling organised molecular systems. Strong ion pairing and low critical micellar concentrations characterise these types of solutions [22].
3.3. Surfactant–water–oil Ternary phase diagrams of ionic and non-ionic species demonstrate the interplay between curvature imposed by volume and surface constraints and spontaneous curvature, a basic property of a surfactant film. Interfacial curvature varies with temperature in opposite direction for ionic and non-ionic systems. The observation of the location of phase limits allows identification of flexible and rigid interfaces.
3.3.1. Fluid case (bending constant kc B kBT) The fluid interface archetype are the non-ionic– water–oil ternary systems, as studied by Strey [23] for which a temperature cut through the phase diagram is a convenient experimental method to establish surfactant efficiency and temperature of zero spontaneous curvature.
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3.3.2. Rigid case (bending constant kc \kBT) In this case, fluctuations in curvature are not dominant: this corresponds to double chain ionic surfactants, or ionic surfactants with co-surfactants and in the absence of salt in the solvent. In the case of rigid interface, an analytic predictive model allowing explicit calculation of the location in phase diagrams of conductivity variation, associated with transition of microstructures between droplets, cylinders and locally flat connected bilayers is available [5]. This Disordered, Open and Connected model [5] only considers conservation of volume, surface and curvature and predicts possible microstructures versus composition. Rigid ternary systems are usually made with charged surfactants, the archetype being the easily hydrolysed and hard to purify AOT. The equilibrium ternary phase diagram of this surfactant is not precisely known at room temperature, but this widely used model ternary system has allowed to identify the two important instability classes for water in oil systems. (A) The emulsification failure, where the ‘inside’ water part is expelled by the oil phase. On titration of water, a final state of a microemulsion coexisting with excess water phase is observed. The driving force for phase separation is then violation of the curvature constraint: H =_/(3) H0 where H again is the average curvature imposed by the molecular volumes and sample content, _ is the area of interface per unit volume and is the ‘inside’ part volume fraction, and H0 is the spontaneous curvature, controlled by counter-ions in ionic surfactants. (B) The liquid–liquid phase separation can be driven by attractive interactions between micelles. The final state after phase separation is a micellar rich and micellar poor water in oil solution. The driving force is solvent– hydrocarbon chain wetting (penetration) additive with general van der Waals attraction between droplets. This is shown in Fig. 2. The most complicated case is the composition region where these two types of instabilities coexist, the so-called Winsor-III microemulsion [24].
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Fig. 2. Schematic representation of the maximum water solubilisation in water in oil microemulsion limited by curvature or by attraction between droplets. For a sufficiently strong attractive droplet interaction, a liquid – gas phase separation between a micellar rich and a micellar poor phase occurs (right). For small attractive interactions, the microemulsion will demix with an excess of water because of curvature effect; this mechanism is called the emulsification failure (left). Reproduced with permission from Ref. [34] ©2001 of J. Colloid Interface Sci.
For samples called ‘Winsor III’, a surfactant solution coexists with excess water and oil. It is usually a common idea that either droplets or ill-defined ‘random bicontinuous dispersions’ are present in this case. However, experimentally a broad scattering peak is observed at: qmax = 2p/D* =2p/5.8|Cs/((1− ))
(6)
with q= 4p/u sin(q) and q, half the scattering angle, with incident wavelength u. D* is the Bragg distance corresponding to the maximum of the scattered intensity, it corresponds to the typical size of dispersion size in real space. is the polar part volume fraction of the sample, |Cs = _ is the oil –water specific interface at the interfacial film, which can be directly measured by the Porod asymptotic law of scattering at large angles:
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Lim I(q)q 4 =2p|CsDz 2
q
(7)
where the scattered intensity is expressed in absolute units (cm − 1) and where Dz is the inside– outside contrast in scattering length density (cm − 2). Older microemulsion models developed in the literature are due to Talmon and Prager [25] which predicts monotonic decrease without scattering peak as well as De Gennes and Taupin [26] or Cates and co-workers [27] who give a peak at twice the observed value, without any shift versus spontaneous curvature. For ionic systems in the low salt case, the major component towards surfactant film rigidity is brought by distribution of counter-ions, universal expressions have been given for the case of monovalent counter-ions [28].
4. Solubilisation of solutes We see in the former paragraphs that spontaneous curvature and surfactant layer bending constants are at the origin of the general layout of the binary or ternary phase diagram. We will see in this part how those quantities allow us to distinguish between the different types of solubilisation observed experimentally. The definition of solubilisation is first given by Mc Bain and Hutchinson [29]: to increase the solubility of an insoluble or slightly soluble substance in a given medium. This phenomenon involves the presence of surfactant molecules or colloidal particles in a single phase which incorporate the insoluble substance totally. We focus on this part for the solubilisation of a solute by the use of surfactant molecule. In binary as well as ternary systems, the amount of added solute dispersed in an organised molecular system can be analysed as the additive conjunction of homogenous solubilisation (the solute is soluble in bulk solvent) and surfactant film solubilisation (the solute is located in the surfactant film). Homogenous solubilisation of a solute is identical to molecular dispersion in homogenous bulk solvent. In the case of surfactant film solubilisation, adsorption isotherms of solutes in a surfactant film can be derived from scattering
experiment and phase diagram determination. The basic variable describing surfactant film part of the solubilisation in surfactant solutions is therefore the solute excess, expressed as a molar ratio u: u= [Excess solute in the film]/[surfactant in film] (8) When the solution is in equilibrium with excess solid solute u= umax. Before this saturation point, adsorption isotherms of a solute on the surfactant interface have to be considered. The maximum amount umax of solute solubilised in a given medium and its localisation depend on the nature of the surfactant and therefore on the spontaneous curvature and bending energy of the film. To study solubilisation, three different approaches can be used: (a) What is the maximum amount of solute which can be solubilised? The particular case of maximum water solubilisation in a given amount of surfactant in oil demonstrates the relative importance of the different effects, which induces a limit of solubilisation and therefore a phase separation. The reverse case of solute in a given amount of surfactant in water is a difficult case, since this case cannot be treated independent of the localisation of the solute. Phase boundaries corresponding to instabilities limiting the solubilisation are shown in Fig. 2. (b) The second case is the determination of the localisation of a solute. By topological demonstration the localisation of a solute can be determined in reverse micelle of water in oil microemulsions (see Fig. 3). In direct ionic micelles, topological arguments alone are not sufficient to determine the localisation of the solute, since electrostatic interactions between surfactants have to be calculated to determine the localisation. (c) The determination of the amount of solute adsorbed in the interfacial film at the water– oil interface in microemulsions systems is the most complete approach. For case (a), Shah and co-workers [30–32] propose a theory based on a phenomenological and qualitative approach to predict the maximum of
F. Testard, T. Zemb / Colloids and Surfaces A: Physicochem. Eng. Aspects 205 (2002) 3–13
water solubilisation in water in oil microemulsions. They demonstrate that solubilisation depends on two parameters: the spontaneous curvature H0 related to the packing parameter, and the elasticity constant of the surfactant film related to the bending energy required to bend the film. The free energy of the interfacial film is calculated by assuming a saturated incompressible interfacial film. The free energy of interdroplet interactions is approximated as the sum of a hard
Fig. 3. Schematic representation of the four possible effects of the presence of a solute on the observed radius of water-in-oil microemulsions (reverse micelles). Since radius is imposed by surface to volume ratio, localisation of solute in surfactant layers shrinks the average radius of micelles while localisation in core increases the radius. Separate micellisation may induce polydispersity, while localisation in bulk oil has no effect on the reverse micelle.
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sphere repulsive interaction (entropic) term and of an attractive perturbation term. The minimisation of the total free energy with the radius or with the volume fraction of dispersed phase gives the maximum capacity for solubilisation of water. This capacity depends on both the size and the stability of the dispersed droplets. Maximum solubilisation is obtained with the optimal value of the interfacial curvature and of the elasticity where the bending stress and the attractive forces between droplets are minimised. For rigid interfaces, phase separation occurs at R= R0 (spontaneous radius) and the solubilisation capacity depends on the spontaneous curvature. For fluid interfaces, the phase separation will occur with a radius inferior to R0. For small attractive interactions, the microemulsion demixes with an excess of water because of large difference between spontaneous and average curvatures. This mechanism is called emulsification failure [33]. For sufficiently strong attractive droplets interaction the droplets are stabilised leading to a phase separation of a microemulsion with smaller droplets; it is a liquid–gas transition. This is illustrated in Fig. 2 reproduced from Ref. [30,31]. Therefore the maximum water capacity results from a compromise between these two opposite effects: curvature effect or stability of droplets. In case (b), the localisation of a solute could be determined by topological argument in the case of a solute in reverse micelles. Zemb and co-workers [34] studied the localisation of a solute in the case of soft charged interface in AOT water in oil microemulsions. The topology of the microemulsion is dominated by geometrical constraint, as the area | per head-group is independent of the composition. The radius R is imposed by the composition of the microemulsion: R= 3/_ if is the internal polar volume and _ the interfacial area of the monodisperse water in oil droplets. In the general case, we have to consider that film solubilisation of a solute itself induces modification of the interfacial area or of the polar volume, resulting from a modification of the rigidity of the interface as well as spontaneous curvature. Experimental determinations of droplet size R, the total surface _ of interface available per volume and
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composition allow the determination of the site of solubilisation. Fig. 3 reproduced from Ref. [34] gives the principle of solute localisation from the perturbations induced by different localisation in a given water–oil microemulsion. Topological arguments are not enough to determine the localisation of a solute when there are lateral electrostatic interactions between ionic surfactant molecules in a direct micelles. Jo¨ nsson and co-workers [35,36] introduce an explicit model using numerical evaluation of the free energy variation associated with solubilisation in order to predict the behaviour of uncharged molecule of different polarity in ionic surfactant aggregate. This model evaluates the thermodynamic properties of the solution from the knowledge of the molecular properties of both surfactant and solute. This model can predict the localisation of a solute in the micelle (core or Palisade layer) but also the extension of different phases in the threecomponent phase diagram: ionic surfactant– water – solute. Self-consistent unambiguous definition of ‘palisade layer’ as the full thickness of a pure surfactant layer is considered. This definition allows calculation, but is slightly different from a more classical concept of the layer where permittivity is different from oil and water. For micelles, the localisation of a solute depends on both the charge density of the micelle and the polarity of the solute. When a solute is solubilised in ionic micelles, there is a competition between the gain in surface charge density and the reduction in interfacial energy. The localisation is decided by the relative magnitude of these two opposite effects. The model proposed by Jo¨ nsson and co-workers thus predicts and quantifies the average curvature change induced by the presence of solute. Apolar solute such as hydrocarbons are solubilised in the interior of micelles with a low charge density, because geometrical packing produces a large reduction in the interfacial energy in these cases. Such solutes are solubilised in the palisade layer of micelles with a high charge density because of the favourable mixing with amphiphile and the reduction of the surface charge density. Polar solutes are also solubilised in the palisade layer of highly charged micelles, whereas in the
case of micelles with a low charge density, most polar solutes are located in the interior of the micelles. In case (c), solubilisation of solutes in binary and ternary systems can be understood as the sum of homogenous solubilisation in oil or water domain and surfactant film solubilisation. Since different environments are available, the interface can be considered as a pseudo-phase or a surfactant monolayer. In this last case, the water–oil interface is supposed to be covered by a monolayer surfactant film into which the solute can be adsorbed. In this case, we have to refer back to the basic variable, i.e. the interfacial composition, expressed in molar ratio u= [excess solute in the film]/[surfactant film]. When the solution is in equilibrium with excess solid solute, the saturation of the solution is obtained and u= umax. At this point, there is an excess of solubilisation since the amount of dispersed solute can be larger than the maximum value of solubility in the same volume of pure oil or/and pure water. Before this saturation point, there is equilibrium of the solute between the bulk and the surfactant film, the adsorption curve (isotherms) of a solute on the surfactant interface can be determined [41]. The interplay between solubilisation expressed by u and spontaneous curvature of the interface, has been discovered independently and rationalised by Jo¨ nsson and co-workers [35,36] for polar and apolar solutes in ionic surfactant–water system and by Leodidis and co-workers [37–39] for amino acid solubilisation in reverse water in oil microemulsions. Fletcher [40] and then Leodidis and co-workers [37–39] indicate a general method to determine the amount of solute solubilised in reverse microemulsions, its localisation and the radius variation of the droplets induced by this solute. The phase transfer method is first used to determine the affinity of the solute towards the interfacial film. A reverse micellar phase in oil is in contact with a solution of amino acids in water. After equilibrium is obtained, reverse micelles containing some amino acid and water are in equilibrium with the remaining amino acid in water. The rate transfer of one species (water or amino acid) is
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obtained by t=Nmwp/Naq.in., where Nmwp is the number of moles of the species in the reverse microemulsion and Naq.in. is the number of moles in the initial aqueous phase. Assuming that the water phase in excess is identical to the water phase composition within dispersed droplets, the rate transfer of amino acid and of water are determined by titration of the two phases in equilibrium. The rate transfer of amino acid is identical to the rate transfer of water when there is no adsorption of the solute at the interface. If there is an excess of solubilisation at the interface, the rate transfer of amino acid is higher that the one of water. An excess of solubilisation can be evidenced by this method and quantified by the mole-fraction-base partition coefficient: Kx =
X int a X mwp a
where X ia is the molar fraction of amino acid at the interface (int) or in the water droplet of the microemulsion (mwp). For example, it is demonstrated that there is an excess of solubilisation for phenylalanine which is dispersed in the bulk water and in the interfacial film. Excess of solubilisation is measured as a function of the nature of the oil, the salinity and the surfactant concentration. Since all these factors affect the bending of the interface, the partition coefficient can therefore be related to the variation of the curvature of the interface. Leodidis et al. [38,39] demonstrate that a local equilibrium of solute adsorption at the interface can be related to the mean curvature of the surfactant film. The variation of Kx with curvature can be obtained from the equality of chemical potential of amino acid in excess water and in the interface: ln Kx =ln K 0x −
2 1 f( aCx kBT Rw
(9)
where K 0x represents the coefficient transfer from water to interfacial film with a zero curvature, C is the elastic constant, f( i represents the partial molar area of the amino acid (a) and x is a dimensionless quantity directly related to the variation of the elastic constant with the amount of solute located at the interface.
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This thermodynamic model describes the ‘squeezing out’ phenomenon experimentally observed: curving the film squeezes out a part of the adsorbed solute (or increasing the curvature induces a decrease of Kx ). This model gives also an explanation for the linear dependence of Kx with mean curvature even if the slope is not exactly equal to the experimental values. Using the same general approach, we have investigated [41] the case of the solubilisation of a hydrophobic solute (lindane: gamma-hexachlorocyclohexane) in oil in water microemulsions using non-ionic surfactant of the family of polyoxyethyleneglycol. By varying the temperature the hydrophilic–hydrophobic balance (expressed as spontaneous curvature or surfactant packing parameter or equivalently the HLB balance [42]) of the non-ionic surfactant used in this study can be modified. Therefore, the spontaneous curvature turns from oil to water as the temperature is increased. We measure separately the amount of solute adsorbed in the interfacial film and the average curvature of the sample in order to establish a link between these two quantities. The solute effect on spontaneous [41,43] curvature is determined from the phase diagram determination. A cut at equal water to oil ratio in the ‘temperature–water–oil –Ci Ej ’ phase diagram gives a pseudo-binary phase diagram: ‘temperature–Ci Ej mass fraction’. This phase diagram is a direct experimental method to obtain the temperature of zero spontaneous curvature of any given ternary system. The influence of an added solute on the curvature of the surfactant film can be obtained by determining the modification of the pseudo-phase diagram when pure oil is replaced by a ‘oil + solute’ pseudo component. We find that increasing the amount of lindane in the oil induces a decrease of the zero spontaneous curvature as it is shown in Fig. 4. Temperature of zero spontaneous curvature decreases when the surfactant is replaced by a less hydrophilic surfactant or when the oil is replaced by a less hydrophobic oil (more penetrating). Therefore we conclude that solubilisation of lindane turns the curvature towards water at constant temperature. By determining the maximum amount of lindane which can be solubilised in water–oil –Ci Ej
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DHl = − 0.4u
Fig. 4. Pseudobinary phase diagram of water –oil(cyclohexane+ lindane)– C6E5 for different initial weight composition i of lindane in the oil: ( ) 0%, (× ) 4%, ( ) 10% and () 15%. The arrow indicates the variation of the fish tail with the increasing content of lindane in the oil. The temperature T0 of the zero spontaneous curvature is given by the ordinate of the fish tail.
microemulsion, we find explicitly the value of excess of solubilisation. Stable microemulsions exist with a larger amount of lindane relative to the oil than the saturation value in the same oil alone at the same temperature. By titration of both microemulsions and oil in excess, we determine the interfacial composition ratio (lindane over surfactant at interface) versus the weight ratio of lindane in the oil. We obtain here the adsorption of lindane at the interface of C6E5 – water – oil microemulsions with a constant mean curvature (equal to zero). Then, by phase diagram determination and titration, we determine a linear relation [44] between the solute molar ratio u of lindane adsorbed at the interface and the curvature variation DH induced by the solute: DHl = − 0.7u
(11)
Comparison between curvature variation induced by solubilisation observed in several cases with the value expected from sterical and topological constraints illustrated in Fig. 3, taken from Ref. [44]. We found in this case also that saturation limit corresponding to the maximum amount of lindane, which can be solubilised, depends on the curvature. In all cases (a), (b) and (c), the link between curvature, area and volume perturbation induced by the presence of the solute in the film and the adsorption isotherm of the solute in the film allow to identify the instability inducing the presence of a phase boundary. Considering these general principles allows a quantitative approach to surfactant formulation in industrial applications.
5. Conclusions Models for solubilisation can be rationalised by using general principles of self-assembly. All general intrinsic variables associated with a surfactant layer are profoundly influenced by the presence of solutes: bending constants, spontaneous radius of
(10)
To rationalise the experimental results, we use a geometric interpretation of the observed curvature variations. We find by SANS experiments that the solute has no effect on the area ‘|’ per polar head of the surfactant in the microemulsion. Therefore the excess solute increases the effective volume p =V/|l at constant | and l. Considering the coverage relation (1) at the first order, we obtain the linear relation (Fig. 5):
Fig. 5. Variation of the reduced average curvature Hl vs. interfacial molar composition u, with l representing the chain length of the surfactant. DHl = H(u "0)− H(u =0). The dashed line is calculated with the wedge model. ( ) Experimental points obtained by phase diagram determination and titration in the Winsor III domain. ( ) Maximum of solubilisation obtained in micellar system. () Obtained in a Winsor I microemulsion.
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curvature and hence the corresponding phase limits. Therefore the general principles of surfactant self-assemblies could be used to understand and predict soluibilisation phenomena. The general principles exposed here are guidelines which allow to understand general trends only. Until now, most of the use of solubilisation properties for formulation has to be derived from tabulated values obtained from systematic titration [45,46].
References [1] D.F. Evans, H. Wennerstroem, The Colloidal Domain: Where Physics, Chemistry and Biology Meet, VCH, Wienheim, 1994. [2] P. Ekwall, L. Mandel, K. Fontell, Mol. Cryst. Liq. Cryst. 157 (8) (1969) 157. [3] R.G. Laughlin, Aqueous Phase Behaviour of Surfactants, Academic Press, New York, 1994. [4] S. Hyde, et al., The Language of Shape, Elsevier, Amsterdam, 1997. [5] Th. Zemb, Colloids Surf. A 129 –130 (1997) 435. [6] J.B. Hayter, Langmuir 8 (1992) 2873. [7] B.W. Ninham, V. Yaminski, Langmuir 13 (1997) 2097. [8] A.L. Underwood, E.W. Anacker, J. Colloid Interface Sci. 117 (1987) 242. [9] C. Tanford, The Hydrophobic Effect, Wiley, New York, 1973. [10] Y. Chevalier, Th. Zemb, Rep. Prog. Phys. 53 (1990) 279. [11] A. Fogden, S.T. Hyde, G. Lundberg, J. Chem. Soc. Faraday Trans. 87 (1991) 949. [12] T. Nakagawa, in: M.J. Schick (Ed.), Nonionic Surfactants. In: Surfactant Science Series, vol. 23, New York, Marcel Dekker, 1967 (Chapter 2). [13] R. Kjellander, E. Florin, J. Chem. Soc. Faraday Trans. 1 77 (1981) 2053. [14] Y.-X. Huang, G.M. Thurston, D. Blankschtein, G.B. Benedek, J. Chem. Phys. 92 (3) (1990) 1956. [15] V.A. Parsegian, R.P. Rand, D.C. Rau, Methods Enzymol. 127 (1986) 400. [16] D.M. Le Neveu, R.P. Rand, V.A. Parsegian, D.L. Gingell, Biophys. J. 18 (1977) 209. [17] M.G. Noro, W.M. Gelbart, J. Chem. Phys. 111 (8) (1999) 3733. [18] L. Arleth, M.S. Marcelja, Th. Zemb, J. Chem. Phys. 115 (2001) 3923.
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[19] P. Ekwall, L. Mandell, Kolloid Z. Z. Polym. 233 (1968) 938 – 944. [20] P. Ekwall, P. Solyon, Kolloid Z. Z. Polym. 233 (1968) 945. [21] S. Friberg, L. Mandell, P. Ekwall, Kolloid Z. Z. Polym. 233 (1968) 955. [22] A. Khan, E.F. Marquis, Curr. Opin. Colloid Interface Sci. 4 (1999) 402. [23] R. Strey, Colloid Polym. Sci. 272 (1994) 1005. [24] P.A. Winsor, Solvent Properties of Amphiphilic Compounds, Butterworths, London. [25] Y. Talmon, S. Prager, J. Chem. Phys. 76 (1982) 1535. [26] P.-G. De Gennes, C. Taupin, J. Phys. Chem. 86 (1982) 2294. [27] D. Andelman, M.E. Cates, D. Roux, S. Safran, J. Chem. Phys. 87 (1987) 7229. [28] A. Fogden, B.W. Ninham, Adv. Colloid Interface Sci. 83 (1999) 85. [29] M.E.L. Mc Bain, E. Hutchinson, Solubilization and Related Phenomena, Academic Press, New York, 1955. [30] R. Leung, D.O. Shah, J. Colloid Interface Sci. 120 (1987) 320. [31] R. Leung, D.O. Shah, J. Colloid Interface Sci. 120 (1987) 330. [32] M.-J. Hou, D.O. Shah, Langmuir 3 (1987) 1086. [33] S.A. Safran, J. Chem. Phys. 78 (1983) 2073. [34] M.P. Pileni, Th. Zemb, C. Petit, Chem. Phys. Lett. 118 (1985) 414. [35] M. Aamodt, M. Landgren, B. Jo¨ nsson, J. Phys. Chem. 96 (1992) 945. [36] M. Landgren, M. Aamodt, B. Jo¨ nsson, J. Phys. Chem. 96 (1992) 950. [37] E.B. Leodidis, A. Hatton, J. Phys. Chem. 94 (1990) 6400 (see also p. 6411). [38] E.B. Leodidis, A.S. Bommarius, T.A. Hatton, J. Phys. Chem. 95 (1991) 5943. [39] E.B. Leodidis, T.A. Hatton, J. Phys. Chem. 95 (1991) 5957. [40] P.D.I. Fletcher, J. Chem. Soc. Faraday Trans. 1 82 (1986) 2651. [41] F. Testard, Th. Zemb, R. Strey, Prog. Colloid Polym. Sci. 105 (1997) 332. [42] K. Shinoda, S. Friberg, Emulsion and Solubilization, Wiley, New York, 1986. [43] F. Testard, Th. Zemb, Langmuir 14 (1998) 3175. [44] F. Testard, Th. Zemb, J. Colloid Interface Sci. 219 (1999) 11. [45] K. Bonkhoff, M.J. Schwuger, G. Subklew, Use of microemulsions for the extraction of contaminated solids, Industrial Applications of Microemulsions, Surfactant Science Series, vol. 66, Marcel Dekker, New York, 1997, p. 355. [46] M.J. Schwuger, Detergent in the Environment, Surfactant Science Series, vol. 65, Marcel Dekker, New York, 1997.
Interpretation of phase diagrams: topological and thermodynamical constraints Fabienne Testard, Thomas Zemb * DRECAM/Ser6ice de Chimie Mole´culaire, CEA— Centre d’Etudes de Saclay, Batiment 125, F91191 Gif sur Y6ette cedex, France Received 12 November 2001; accepted 12 December 2001
Abstract Some general principles of surfactant self-assembly are rationalised in order to give a short introduction to the use of phase diagrams for binary (two components) and ternary (three components) systems. Finally models for solubilisation are considered to be understood within the frame of those general packing principles: volume, surface and curvature conservation. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Surfactant; Phase diagram; Curvature; Solubilisation
1. Introduction Our aim is to give a short introduction to the use of phase diagrams in order to rationalise some general principles of surfactant self-assembly. These are summarised in the form of guidelines before using more complete textbooks such as the recent one written by Evans and Wennerstroem [1]. We consider separately binary (two components: surfactant and solvent), then ternary systems (surfactant(s), water, oil) and finally models for solubilisation, to be understood within the frame of some general principles and not as tabulated data. We define as ‘surfactant’ any molecule which induces reduction of surface tension, i.e. positive * Corresponding author. E-mail addresses: [email protected] (F. Testard), [email protected] (T. Zemb).
adsorption at air –water interface. Lipids as well as detergents, most ‘complexing molecules’ as well as associative polymers belong to the ‘surfactant’ class of molecules according to this general definition. Due to lack of space, all figures, especially phase diagrams are not included, they can be found in original references, some of them are in the form of useful ‘Atlas’ of phase diagrams, a book [2] useful to physical chemists like a good map for geographers.
2. Binary phase diagrams This is the case where only ‘surfactant’ and ‘solvent’ are present in the sample. At atmospheric pressure, two thermodynamic variables define the system. Concentration (usually in volume or weight fraction, sometimes as chemical potential if the ordinate axis is plotted in logarith-
0927-7757/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0927-7757(01)01141-4
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mic scale) and temperature are the two axes of commonly found phase diagrams [3].
2.1. Spontaneous cur6ature Spontaneous curvature [4] is the basic concept rationalising surfactant aggregation. Transition from monomers to multi-molecular aggregates is observed as soon as the chemical potential is high enough to induce the transition from a molecular solution towards an organised molecular system. This transition is called the critical micellar concentration (cmc). Even if numerically, most of the time, equivalent, the cmc should not be systematically identified to the concentration of monomers in dynamic equilibrium with micelles. The curvature of an interface is defined by using two circles tangent at each point to a general surface covered by a surfactant monolayer, as shown in Fig. 1. On each point, there is a maximal value (r1) and a minimal value (r2) for radii of circles tangent to the surface. The curvature (dimension: length − 1) is the inverse of radius of the tangent circle. The average curvature H is the average of the inverse of the two radii of curvature. The Gaussian curvature K is the geometrical mean of the inverse of the two radii of curvature. Fig. 1, from the textbook [4] ‘The language of shape’ by Hyde et al., illustrates the sign of these quantities for the three cases frequently described in local packing topologies: globular, cylindrical and cubic. Average curvature and Gaussian curvature are quantitatively lower in the case of lamellar phase, or locally lamellar structures [5]. We first consider the simple case of ionic spherical micelles. With R the radius of the micelle, the Gaussian as well as average curvature are, respectively, given by: K= 1/R 2 and H = 1/R. In this case, the head-group dissociation can be predicted within the frame of the ‘dressed micelle’ model from first principles and a measure of the aggregation number only [6]. For a given micelle, the effective charge, i.e. the net electrical charge taking into account adsorbed counter-ions and non-linear effects, can be explicitly calculated from molecular volumes, ion size and water– oil interfacial tension. The numerical relation shown by Hayter [6] giving the effective charge, once
volume and surface are known, holds for all micelles and predicts effective charge in all cases where there is no ‘chemical’, i.e. non-electrostatic interaction between the oil–water interface and the counter-ions. The micelles which have a lower charge than calculated involve highly polarisable counter-ions, such as bromide and iodide, or specific receptors, such as cage molecule implying ion head-group specific interaction. A general theory for the origin of the Hofmeister effect has been recently proposed [7]. Consequences for the surfactant aggregation are described by Underwood and Anacker [8]. The maximum number of surfactant per globular micelle, the so-called Tanford [9] limit for spherical micelle, is given by simple geometrical arguments: N max agg 3× (apolar molecular volume) (area per molecule at core– water interface)× R =
where R is the radius of the apolar core of the micelle. This number lies between 20 and 200 for
Fig. 1. Examples of curved interfaces. These schematically represent microstructures in the case of droplets, cylinders and cubic phases. Average curvature is non-zero for droplets and cylinders and zero for the cubic case. Gaussian curvature is positive for droplets, zero for the cylinders and negative in the case of cubic packing.
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‘good surfactants’, i.e. easily soluble amphiphilic molecules commonly used in formulation or research work. If microstructure is not known a priori, self-organisation of surfactant molecules can be rationalised in terms of chemical potential primarily depending on the concept of spontaneous curvatures H0 and K0 of any surfactant monolayer [10]. The spontaneous curvatures H0 and K0 are the curvatures adopted by the surfactant film in the absence of any bending. The competition between the spontaneous curvatures, which depend on molecular shape, temperature, salinity, for instance, and the curvatures imposed by sterical constraints such as micellar volume and surface is the most general principle of surfactant self-assembly [4]. The link between molecular dimensions is given by the ‘coverage relation’, which links the local packing parameter p of neighbouring surfactant molecules to macroscopic curvatures imposed by topology: p = 1+ Hl+ 1/3Kl2
(1)
The packing parameter p is a number which gives first order the shape of one molecule embedded in a surfactant film. p= V/(|l)
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ing constraints at the microscopic scale. A standard expression of the cost in free energy F (per unit area) due to the difference between these two quantities is: F= 1/2Kc(H−H0)2 + s(K− K0)
(3)
Expression (3) presents strong self-consistency problems as shown by Fogden et al. [11] in the case of non-zero spontaneous curvature. An alternative microscopic model of the membrane could be used to express the free bending energy of the surfactant film: F= 1/2K*(p− p0)2
(4)
where K* is a generalised bending constant [11], and p0 is the effective value of the surfactant parameter in the sample calculated with Eq. (1) using H0 and K0. For non-ionic systems, surfactant parameter p is easily varied by using temperature: high temperature dehydrates the ethoxy polar head-group, thus reducing the solubility in water and increasing the solubility in oil of the surfactant. For ionic surfactants, the situation is more complex since the counter-ion density function near the oil–water interface sets the area per head-group and also the elastic bending constant of the interface.
(2)
where at microscopic scale, the volume V is the volume of the apolar part of the surfactant molecule, | is the area per molecule at the interface and l is the average chain length. On a macrocopic scale, H is the average curvature, averaged over the whole sample, and K is the Gaussian curvature, averaged over the whole sample. Reference between constraints at microscopic scale and macroscopic constraints allows to rationalise the formulation, i.e. the effect of co-surfactant, temperature and salt. Relations (1) and (2) are too crude to be used directly, but are the basis of any model evaluating the difference between two samples. They can be used differentially, considering temperature, concentration or salinity variations as well as the solubilisation of a host molecule. H characterises a composition, while the spontaneous curvature, H0 reflects local pack-
2.2. Phase limits The phase limits are set by intermolecular interactions. For example, the ‘cloud point’, i.e. a micelle–water phase separation at high temperature is obtained for some non-ionic surfactants. There is a relation between the intermicellar potential and the clouding point [12–14]. Thus, a variation of clouding behaviour is observed by the presence of a solute. If the increase of apolar volume by solute can be neglected, intermicellar potential variation can be directly derived.
2.3. Binary phase In binary phase diagram, the phase boundaries provide lines of known osmotic pressure. The determination of phase coexistence is a powerful tool for determining molecular forces.
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2.3.1. Lamellar or hexagonal phases When a surfactant in the form of a lamellar or hexagonal phase is in equilibrium with excess pure water, the total balance of attractive and repulsive interactions between surfactant aggregates is zero, since the osmotic pressure of the water phase in excess is zero. Thus, a determination of maximum observed periodicity versus any thermodynamical parameter such as temperature, ionic strength or composition of solvent is equivalent to a very sensitive molecular force balance [15]. Parsegian and co-workers [16] have shown that if the attractive and repulsive interactions are algebraically combined as an approximation, quantitative information about magnitude of interaction between membranes can be obtained. 2.3.2. Binodal lines When two phases coexist in a region of binodal lines, there is ‘competition for water’. Since the osmotic pressures in the two phases have to be equal, the sum of attractions and repulsions in the two coexisting phases have to be equal. In the case of diluted and concentrated lamellar phases, this property allows direct observation of nonmonotonic forces between flat surfactant interfaces, an effect ubiquitous in surfactant systems [17]. Close to the critical point, exploitation of this property is a ‘molecular force balance’ of unprecedented sensitivity. It is only sufficient to measure periodicity and osmotic pressure in the coexistence region of two phases to obtain a differential measure of the interaction potential between oil– water interface, without the complexity of AFM or surface force balances.
3. Ternary phase diagrams Phase diagrams are profoundly different for flexible and rigid surfactant films [10]. Flexible films are films for which the cost in bending energy of local curvature variations is smaller than thermal energy. The microstructure of flexible films is dominated by entropy. Rigid films can be modelled as molten liquid crystals or a glass of small crystallites of liquid crystals and
therefore form more homogeneous microstructures, while in flexible films, fluctuations dominate. Basically, flexible films correspond to kc kBT and rigid films to kc kBT. The intermediate case where kc : kBT and associated microstructure has been examined only recently [18].
3.1. Pseudo-binary systems The simplest case is the historic case of sodium octanoate–water–octanoic acid which is a quasibinary system studied by Ekwall and co-workers [19–21] where the ‘oil’ and ‘surfactant’ are acidic and sodium forms of the same molecule. In this system, the connected lamellae (sponge), i.e. the first non-globular micellar aggregates as well as highly swollen lamellar phases have been discovered.
3.2. Two surfactants in water Cationic systems, when an anionic surfactant is associated to a cationic surfactant in water belong to the class of ternary self-assembling organised molecular systems. Strong ion pairing and low critical micellar concentrations characterise these types of solutions [22].
3.3. Surfactant–water–oil Ternary phase diagrams of ionic and non-ionic species demonstrate the interplay between curvature imposed by volume and surface constraints and spontaneous curvature, a basic property of a surfactant film. Interfacial curvature varies with temperature in opposite direction for ionic and non-ionic systems. The observation of the location of phase limits allows identification of flexible and rigid interfaces.
3.3.1. Fluid case (bending constant kc B kBT) The fluid interface archetype are the non-ionic– water–oil ternary systems, as studied by Strey [23] for which a temperature cut through the phase diagram is a convenient experimental method to establish surfactant efficiency and temperature of zero spontaneous curvature.
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3.3.2. Rigid case (bending constant kc \kBT) In this case, fluctuations in curvature are not dominant: this corresponds to double chain ionic surfactants, or ionic surfactants with co-surfactants and in the absence of salt in the solvent. In the case of rigid interface, an analytic predictive model allowing explicit calculation of the location in phase diagrams of conductivity variation, associated with transition of microstructures between droplets, cylinders and locally flat connected bilayers is available [5]. This Disordered, Open and Connected model [5] only considers conservation of volume, surface and curvature and predicts possible microstructures versus composition. Rigid ternary systems are usually made with charged surfactants, the archetype being the easily hydrolysed and hard to purify AOT. The equilibrium ternary phase diagram of this surfactant is not precisely known at room temperature, but this widely used model ternary system has allowed to identify the two important instability classes for water in oil systems. (A) The emulsification failure, where the ‘inside’ water part is expelled by the oil phase. On titration of water, a final state of a microemulsion coexisting with excess water phase is observed. The driving force for phase separation is then violation of the curvature constraint: H =_/(3) H0 where H again is the average curvature imposed by the molecular volumes and sample content, _ is the area of interface per unit volume and is the ‘inside’ part volume fraction, and H0 is the spontaneous curvature, controlled by counter-ions in ionic surfactants. (B) The liquid–liquid phase separation can be driven by attractive interactions between micelles. The final state after phase separation is a micellar rich and micellar poor water in oil solution. The driving force is solvent– hydrocarbon chain wetting (penetration) additive with general van der Waals attraction between droplets. This is shown in Fig. 2. The most complicated case is the composition region where these two types of instabilities coexist, the so-called Winsor-III microemulsion [24].
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Fig. 2. Schematic representation of the maximum water solubilisation in water in oil microemulsion limited by curvature or by attraction between droplets. For a sufficiently strong attractive droplet interaction, a liquid – gas phase separation between a micellar rich and a micellar poor phase occurs (right). For small attractive interactions, the microemulsion will demix with an excess of water because of curvature effect; this mechanism is called the emulsification failure (left). Reproduced with permission from Ref. [34] ©2001 of J. Colloid Interface Sci.
For samples called ‘Winsor III’, a surfactant solution coexists with excess water and oil. It is usually a common idea that either droplets or ill-defined ‘random bicontinuous dispersions’ are present in this case. However, experimentally a broad scattering peak is observed at: qmax = 2p/D* =2p/5.8|Cs/((1− ))
(6)
with q= 4p/u sin(q) and q, half the scattering angle, with incident wavelength u. D* is the Bragg distance corresponding to the maximum of the scattered intensity, it corresponds to the typical size of dispersion size in real space. is the polar part volume fraction of the sample, |Cs = _ is the oil –water specific interface at the interfacial film, which can be directly measured by the Porod asymptotic law of scattering at large angles:
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Lim I(q)q 4 =2p|CsDz 2
q
(7)
where the scattered intensity is expressed in absolute units (cm − 1) and where Dz is the inside– outside contrast in scattering length density (cm − 2). Older microemulsion models developed in the literature are due to Talmon and Prager [25] which predicts monotonic decrease without scattering peak as well as De Gennes and Taupin [26] or Cates and co-workers [27] who give a peak at twice the observed value, without any shift versus spontaneous curvature. For ionic systems in the low salt case, the major component towards surfactant film rigidity is brought by distribution of counter-ions, universal expressions have been given for the case of monovalent counter-ions [28].
4. Solubilisation of solutes We see in the former paragraphs that spontaneous curvature and surfactant layer bending constants are at the origin of the general layout of the binary or ternary phase diagram. We will see in this part how those quantities allow us to distinguish between the different types of solubilisation observed experimentally. The definition of solubilisation is first given by Mc Bain and Hutchinson [29]: to increase the solubility of an insoluble or slightly soluble substance in a given medium. This phenomenon involves the presence of surfactant molecules or colloidal particles in a single phase which incorporate the insoluble substance totally. We focus on this part for the solubilisation of a solute by the use of surfactant molecule. In binary as well as ternary systems, the amount of added solute dispersed in an organised molecular system can be analysed as the additive conjunction of homogenous solubilisation (the solute is soluble in bulk solvent) and surfactant film solubilisation (the solute is located in the surfactant film). Homogenous solubilisation of a solute is identical to molecular dispersion in homogenous bulk solvent. In the case of surfactant film solubilisation, adsorption isotherms of solutes in a surfactant film can be derived from scattering
experiment and phase diagram determination. The basic variable describing surfactant film part of the solubilisation in surfactant solutions is therefore the solute excess, expressed as a molar ratio u: u= [Excess solute in the film]/[surfactant in film] (8) When the solution is in equilibrium with excess solid solute u= umax. Before this saturation point, adsorption isotherms of a solute on the surfactant interface have to be considered. The maximum amount umax of solute solubilised in a given medium and its localisation depend on the nature of the surfactant and therefore on the spontaneous curvature and bending energy of the film. To study solubilisation, three different approaches can be used: (a) What is the maximum amount of solute which can be solubilised? The particular case of maximum water solubilisation in a given amount of surfactant in oil demonstrates the relative importance of the different effects, which induces a limit of solubilisation and therefore a phase separation. The reverse case of solute in a given amount of surfactant in water is a difficult case, since this case cannot be treated independent of the localisation of the solute. Phase boundaries corresponding to instabilities limiting the solubilisation are shown in Fig. 2. (b) The second case is the determination of the localisation of a solute. By topological demonstration the localisation of a solute can be determined in reverse micelle of water in oil microemulsions (see Fig. 3). In direct ionic micelles, topological arguments alone are not sufficient to determine the localisation of the solute, since electrostatic interactions between surfactants have to be calculated to determine the localisation. (c) The determination of the amount of solute adsorbed in the interfacial film at the water– oil interface in microemulsions systems is the most complete approach. For case (a), Shah and co-workers [30–32] propose a theory based on a phenomenological and qualitative approach to predict the maximum of
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water solubilisation in water in oil microemulsions. They demonstrate that solubilisation depends on two parameters: the spontaneous curvature H0 related to the packing parameter, and the elasticity constant of the surfactant film related to the bending energy required to bend the film. The free energy of the interfacial film is calculated by assuming a saturated incompressible interfacial film. The free energy of interdroplet interactions is approximated as the sum of a hard
Fig. 3. Schematic representation of the four possible effects of the presence of a solute on the observed radius of water-in-oil microemulsions (reverse micelles). Since radius is imposed by surface to volume ratio, localisation of solute in surfactant layers shrinks the average radius of micelles while localisation in core increases the radius. Separate micellisation may induce polydispersity, while localisation in bulk oil has no effect on the reverse micelle.
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sphere repulsive interaction (entropic) term and of an attractive perturbation term. The minimisation of the total free energy with the radius or with the volume fraction of dispersed phase gives the maximum capacity for solubilisation of water. This capacity depends on both the size and the stability of the dispersed droplets. Maximum solubilisation is obtained with the optimal value of the interfacial curvature and of the elasticity where the bending stress and the attractive forces between droplets are minimised. For rigid interfaces, phase separation occurs at R= R0 (spontaneous radius) and the solubilisation capacity depends on the spontaneous curvature. For fluid interfaces, the phase separation will occur with a radius inferior to R0. For small attractive interactions, the microemulsion demixes with an excess of water because of large difference between spontaneous and average curvatures. This mechanism is called emulsification failure [33]. For sufficiently strong attractive droplets interaction the droplets are stabilised leading to a phase separation of a microemulsion with smaller droplets; it is a liquid–gas transition. This is illustrated in Fig. 2 reproduced from Ref. [30,31]. Therefore the maximum water capacity results from a compromise between these two opposite effects: curvature effect or stability of droplets. In case (b), the localisation of a solute could be determined by topological argument in the case of a solute in reverse micelles. Zemb and co-workers [34] studied the localisation of a solute in the case of soft charged interface in AOT water in oil microemulsions. The topology of the microemulsion is dominated by geometrical constraint, as the area | per head-group is independent of the composition. The radius R is imposed by the composition of the microemulsion: R= 3/_ if is the internal polar volume and _ the interfacial area of the monodisperse water in oil droplets. In the general case, we have to consider that film solubilisation of a solute itself induces modification of the interfacial area or of the polar volume, resulting from a modification of the rigidity of the interface as well as spontaneous curvature. Experimental determinations of droplet size R, the total surface _ of interface available per volume and
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composition allow the determination of the site of solubilisation. Fig. 3 reproduced from Ref. [34] gives the principle of solute localisation from the perturbations induced by different localisation in a given water–oil microemulsion. Topological arguments are not enough to determine the localisation of a solute when there are lateral electrostatic interactions between ionic surfactant molecules in a direct micelles. Jo¨ nsson and co-workers [35,36] introduce an explicit model using numerical evaluation of the free energy variation associated with solubilisation in order to predict the behaviour of uncharged molecule of different polarity in ionic surfactant aggregate. This model evaluates the thermodynamic properties of the solution from the knowledge of the molecular properties of both surfactant and solute. This model can predict the localisation of a solute in the micelle (core or Palisade layer) but also the extension of different phases in the threecomponent phase diagram: ionic surfactant– water – solute. Self-consistent unambiguous definition of ‘palisade layer’ as the full thickness of a pure surfactant layer is considered. This definition allows calculation, but is slightly different from a more classical concept of the layer where permittivity is different from oil and water. For micelles, the localisation of a solute depends on both the charge density of the micelle and the polarity of the solute. When a solute is solubilised in ionic micelles, there is a competition between the gain in surface charge density and the reduction in interfacial energy. The localisation is decided by the relative magnitude of these two opposite effects. The model proposed by Jo¨ nsson and co-workers thus predicts and quantifies the average curvature change induced by the presence of solute. Apolar solute such as hydrocarbons are solubilised in the interior of micelles with a low charge density, because geometrical packing produces a large reduction in the interfacial energy in these cases. Such solutes are solubilised in the palisade layer of micelles with a high charge density because of the favourable mixing with amphiphile and the reduction of the surface charge density. Polar solutes are also solubilised in the palisade layer of highly charged micelles, whereas in the
case of micelles with a low charge density, most polar solutes are located in the interior of the micelles. In case (c), solubilisation of solutes in binary and ternary systems can be understood as the sum of homogenous solubilisation in oil or water domain and surfactant film solubilisation. Since different environments are available, the interface can be considered as a pseudo-phase or a surfactant monolayer. In this last case, the water–oil interface is supposed to be covered by a monolayer surfactant film into which the solute can be adsorbed. In this case, we have to refer back to the basic variable, i.e. the interfacial composition, expressed in molar ratio u= [excess solute in the film]/[surfactant film]. When the solution is in equilibrium with excess solid solute, the saturation of the solution is obtained and u= umax. At this point, there is an excess of solubilisation since the amount of dispersed solute can be larger than the maximum value of solubility in the same volume of pure oil or/and pure water. Before this saturation point, there is equilibrium of the solute between the bulk and the surfactant film, the adsorption curve (isotherms) of a solute on the surfactant interface can be determined [41]. The interplay between solubilisation expressed by u and spontaneous curvature of the interface, has been discovered independently and rationalised by Jo¨ nsson and co-workers [35,36] for polar and apolar solutes in ionic surfactant–water system and by Leodidis and co-workers [37–39] for amino acid solubilisation in reverse water in oil microemulsions. Fletcher [40] and then Leodidis and co-workers [37–39] indicate a general method to determine the amount of solute solubilised in reverse microemulsions, its localisation and the radius variation of the droplets induced by this solute. The phase transfer method is first used to determine the affinity of the solute towards the interfacial film. A reverse micellar phase in oil is in contact with a solution of amino acids in water. After equilibrium is obtained, reverse micelles containing some amino acid and water are in equilibrium with the remaining amino acid in water. The rate transfer of one species (water or amino acid) is
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obtained by t=Nmwp/Naq.in., where Nmwp is the number of moles of the species in the reverse microemulsion and Naq.in. is the number of moles in the initial aqueous phase. Assuming that the water phase in excess is identical to the water phase composition within dispersed droplets, the rate transfer of amino acid and of water are determined by titration of the two phases in equilibrium. The rate transfer of amino acid is identical to the rate transfer of water when there is no adsorption of the solute at the interface. If there is an excess of solubilisation at the interface, the rate transfer of amino acid is higher that the one of water. An excess of solubilisation can be evidenced by this method and quantified by the mole-fraction-base partition coefficient: Kx =
X int a X mwp a
where X ia is the molar fraction of amino acid at the interface (int) or in the water droplet of the microemulsion (mwp). For example, it is demonstrated that there is an excess of solubilisation for phenylalanine which is dispersed in the bulk water and in the interfacial film. Excess of solubilisation is measured as a function of the nature of the oil, the salinity and the surfactant concentration. Since all these factors affect the bending of the interface, the partition coefficient can therefore be related to the variation of the curvature of the interface. Leodidis et al. [38,39] demonstrate that a local equilibrium of solute adsorption at the interface can be related to the mean curvature of the surfactant film. The variation of Kx with curvature can be obtained from the equality of chemical potential of amino acid in excess water and in the interface: ln Kx =ln K 0x −
2 1 f( aCx kBT Rw
(9)
where K 0x represents the coefficient transfer from water to interfacial film with a zero curvature, C is the elastic constant, f( i represents the partial molar area of the amino acid (a) and x is a dimensionless quantity directly related to the variation of the elastic constant with the amount of solute located at the interface.
11
This thermodynamic model describes the ‘squeezing out’ phenomenon experimentally observed: curving the film squeezes out a part of the adsorbed solute (or increasing the curvature induces a decrease of Kx ). This model gives also an explanation for the linear dependence of Kx with mean curvature even if the slope is not exactly equal to the experimental values. Using the same general approach, we have investigated [41] the case of the solubilisation of a hydrophobic solute (lindane: gamma-hexachlorocyclohexane) in oil in water microemulsions using non-ionic surfactant of the family of polyoxyethyleneglycol. By varying the temperature the hydrophilic–hydrophobic balance (expressed as spontaneous curvature or surfactant packing parameter or equivalently the HLB balance [42]) of the non-ionic surfactant used in this study can be modified. Therefore, the spontaneous curvature turns from oil to water as the temperature is increased. We measure separately the amount of solute adsorbed in the interfacial film and the average curvature of the sample in order to establish a link between these two quantities. The solute effect on spontaneous [41,43] curvature is determined from the phase diagram determination. A cut at equal water to oil ratio in the ‘temperature–water–oil –Ci Ej ’ phase diagram gives a pseudo-binary phase diagram: ‘temperature–Ci Ej mass fraction’. This phase diagram is a direct experimental method to obtain the temperature of zero spontaneous curvature of any given ternary system. The influence of an added solute on the curvature of the surfactant film can be obtained by determining the modification of the pseudo-phase diagram when pure oil is replaced by a ‘oil + solute’ pseudo component. We find that increasing the amount of lindane in the oil induces a decrease of the zero spontaneous curvature as it is shown in Fig. 4. Temperature of zero spontaneous curvature decreases when the surfactant is replaced by a less hydrophilic surfactant or when the oil is replaced by a less hydrophobic oil (more penetrating). Therefore we conclude that solubilisation of lindane turns the curvature towards water at constant temperature. By determining the maximum amount of lindane which can be solubilised in water–oil –Ci Ej
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F. Testard, T. Zemb / Colloids and Surfaces A: Physicochem. Eng. Aspects 205 (2002) 3–13
DHl = − 0.4u
Fig. 4. Pseudobinary phase diagram of water –oil(cyclohexane+ lindane)– C6E5 for different initial weight composition i of lindane in the oil: ( ) 0%, (× ) 4%, ( ) 10% and () 15%. The arrow indicates the variation of the fish tail with the increasing content of lindane in the oil. The temperature T0 of the zero spontaneous curvature is given by the ordinate of the fish tail.
microemulsion, we find explicitly the value of excess of solubilisation. Stable microemulsions exist with a larger amount of lindane relative to the oil than the saturation value in the same oil alone at the same temperature. By titration of both microemulsions and oil in excess, we determine the interfacial composition ratio (lindane over surfactant at interface) versus the weight ratio of lindane in the oil. We obtain here the adsorption of lindane at the interface of C6E5 – water – oil microemulsions with a constant mean curvature (equal to zero). Then, by phase diagram determination and titration, we determine a linear relation [44] between the solute molar ratio u of lindane adsorbed at the interface and the curvature variation DH induced by the solute: DHl = − 0.7u
(11)
Comparison between curvature variation induced by solubilisation observed in several cases with the value expected from sterical and topological constraints illustrated in Fig. 3, taken from Ref. [44]. We found in this case also that saturation limit corresponding to the maximum amount of lindane, which can be solubilised, depends on the curvature. In all cases (a), (b) and (c), the link between curvature, area and volume perturbation induced by the presence of the solute in the film and the adsorption isotherm of the solute in the film allow to identify the instability inducing the presence of a phase boundary. Considering these general principles allows a quantitative approach to surfactant formulation in industrial applications.
5. Conclusions Models for solubilisation can be rationalised by using general principles of self-assembly. All general intrinsic variables associated with a surfactant layer are profoundly influenced by the presence of solutes: bending constants, spontaneous radius of
(10)
To rationalise the experimental results, we use a geometric interpretation of the observed curvature variations. We find by SANS experiments that the solute has no effect on the area ‘|’ per polar head of the surfactant in the microemulsion. Therefore the excess solute increases the effective volume p =V/|l at constant | and l. Considering the coverage relation (1) at the first order, we obtain the linear relation (Fig. 5):
Fig. 5. Variation of the reduced average curvature Hl vs. interfacial molar composition u, with l representing the chain length of the surfactant. DHl = H(u "0)− H(u =0). The dashed line is calculated with the wedge model. ( ) Experimental points obtained by phase diagram determination and titration in the Winsor III domain. ( ) Maximum of solubilisation obtained in micellar system. () Obtained in a Winsor I microemulsion.
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curvature and hence the corresponding phase limits. Therefore the general principles of surfactant self-assemblies could be used to understand and predict soluibilisation phenomena. The general principles exposed here are guidelines which allow to understand general trends only. Until now, most of the use of solubilisation properties for formulation has to be derived from tabulated values obtained from systematic titration [45,46].
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