Langmuir 1996, 12, 2939-2946
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Surfactant/Water/Oil System with Weakly Charged Films: Dependence on Charge Density Vijay Rajagopalan,* Ha˚kan Bagger-Jo¨rgensen, Keiichi Fukuda,† Ulf Olsson, and Bengt Jo¨nsson Physical Chemistry 1, Chemical Center, Lund University, P.O. Box 124, S221 00, Lund, Sweden Received November 30, 1995. In Final Form: March 11, 1996X The effects on the microstructure and phase equilibria of a nonionic surfactant/water/oil system when replacing one, four, and six per hundred of the nonionic surfactant molecules by the ionic surfactant sodium dodecyl sulfate (SDS) have been studied. The nonionic surfactant system consists of pentaethylene glycol dodecyl ether (C12E5), water, and decane, at a constant surfactant to oil weight ratio of 51.9/48.1. The phase equilibria and the corresponding microstructures were investigated as a function of temperature and water content. As a function of temperature, the initial phase sequence for this composition range was L1 phase-lamellar phase-L3 phase. Three effects on phase behavior are observed when introducing electrostatic interactions. (i) Water rich reverse hexagonal (H2) and reverse micellar (L2) phases replace the two-phase region water + L3 phase. (ii) The crystallization of micellar cubic (I1) and normal hexagonal phases moves to lower particle concentrations. (iii) Phase boundaries at higher water contents are strongly shifted to higher temperatures. The latter effect is analyzed in detail at the emulsification failure boundary where spherical oil-swollen micelles are in equilibrium with excess oil. Using the bending energy approach together with Poisson-Boltzmann calculations within the cell model, we have analyzed the charge dependence of the emulsification failure boundary. The SDS effects can be reproduced with a cationic surfactant. We also show that the electrostatic effects are removed upon addition of salt. The microstructure in the various phases was studied by small angle X-ray scattering (SAXS). A neutral surface located at the polar/apolar interface is identified where the area per surfactant molecule is constant irrespective of the curvature.
1. Introduction Ethylene oxide-based nonionic surfactants show a rich phase behavior in water/oil mixtures, with temperature being a sensitive tuning parameter for the surfactant film spontaneous curvature.1,2 Since microemulsions, sponge phases, and dilute lamellar phases can be formed with the minimum number of components and in particular with a single-component surfactant film, these systems are important model systems to study the thermodynamics of uncharged surfactant films. Within the system pentaethylene glycol dodecyl ether (C12E5)/water/decane we have previously investigated a water dilution line at a constant surfactant to oil ratio (weight ratio: 51.9/48.1).3-5 Particular emphasis has been on the so-called emulsification failure, which corresponds to a droplet microemulsion (here oil-swollen micelles) in equilibrium with excess internal solvent (oil). The micelles on the phase boundary are spherical with a radius dictated by the surfactant to oil ratio, and they interact to a very good approximation as hard spheres.3,5 In a recent paper6 we have also investigated the effects on the phase behavior of introducing long range electrostatic interactions, by replacing one per hundred of the nonionic C12E5 surfactant by an ionic surfactant, sodium dodecyl sulfate, SDS. This corresponds to approximately one charge per 4500 Å2. A striking effect of introducing electrostatic interactions was the formation of a reverse * To whom correspondence should be addressed. † Present address: Tokyo Research Labs, Kao Corporation, Bunka 2-1-3, Sumida-ku, Tokyo 131, Japan. X Abstract published in Advance ACS Abstracts, May 15, 1996. (1) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113. (2) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (3) Olsson, U.; Schurtenberger, P. Langmuir 1993, 9, 3389. (4) Leaver, M. S.; Olsson, U.; Wennerstro¨m, H.; Strey, R. J. Phys. II 1994, 4, 515. (5) Leaver, M. S.; Olsson, U. Langmuir 1994, 10, 3449. (6) Fukuda, K.; Olsson, U.; Wu¨rz, U. Langmuir 1994, 10, 3222.
S0743-7463(95)01094-8 CCC: $12.00
hexagonal (H2) and a reverse micellar phase (L2), replacing a two-phase equilibrium involving excess water and the L3 phase. Interactions in nonionic surfactant systems are weak, and adding charged groups to the surfactant film results in a strong increase of the osmotic pressure, which disfavors phase separation with excess water. This effect is similar to the induced swelling of zwitterionic lipid lamellar phases upon adding ionic surfactants7 and the stabilization of aqueous neutral polymer/polyelectrolyte mixtures.8,9 However new structures are introduced here due to curvature energy constraints.6 The effects of the addition of small amounts of an ionic surfactant have been studied also in other nonionic and zwitterionic surfactant systems. In dilute lamellar phases with nonionic surfactant, the addition of ionic surfactant results in a sharp increase of the first-order quasi Bragg peak intensity.10 Here a shift in peak position is also observed for very dilute samples, which is consistent with a damping of bilayer undulations.11 Transitions from nonswelling to water-swelling lamellar systems can be obtained upon the addition of small amounts of ionic surfactant.7,12-14 Studies on water rich microemulsion and micellar systems show strong effects on the interactions while the microstructure remains essentially unchanged when adding small amounts of ionic surfactant to nonionic or zwitterionic surfactant systems.15-20 (7) Jo¨nsson, B.; Persson, P. J. Colloid Interface Sci. 1986, 115, 507. (8) Iliopoulos, I.; Frugier, D.; Audebert, R. Polym. Prepr. 1989, 30, 371. (9) Perrau, M. B.; Iliopoulos, I.; Audebert, R. Polymer 1989, 1989, 2112. (10) Jonstro¨mer, M.; Strey, R. J. Phys. Chem. 1992, 96, 5993. (11) Schoma¨cker, R.; Strey, R. J. Phys. Chem. 1994, 98, 3908. (12) Gulik-Krzywicki, T.; Tardieu, A.; Luzzati, V. Mol. Cryst. Liq. Cryst. 1969, 8, 285. (13) Larsson, K.; Krog, N. Chem. Phys. Lipids 1973, 10, 177. (14) Rydhag, L.; Gabra´n, T. Chem. Phys. Lipids 1982, 30, 309. (15) Gue´ring, P.; Nilsson, P.; Lindman, B. J. Colloid Interface Sci. 1985, 105, 41.
© 1996 American Chemical Society
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d
Figure 1. Partial phase diagrams of the C12E5/SDS/D2O/decane system, for a constant surfactant to oil weight ratio (Ws/Wo) of 51.9/48.1. The mole fraction of SDS in the surfactant mixture, β ) SDS/(C12E5 + SDS), is (a) β ) 0 (data from ref 3), (b) β ) 0.01 (data from ref 6), (c) β ) 0.04, and (d) β ) 0.06. For the various phases the following notations are used: I1 is a cubic micellar phase. L1 is a liquid microemulsion phase similar to a normal micellar phase. H1 is a normal hexagonal liquid crystalline phase. LR is a lamellar liquid crystalline phase. L3 is a liquid phase having a multiply connected bilayer structure. H2 is a reverse hexagonal liquid crystalline phase. L2 is a microemulsion phase similar to L1, but with a reverse micellar structure. L1 + O is a two-phase region, where we have equilibrium between a pure oil phase (O) and a solution of oil-swollen micelles (L1).
In the present paper, we have extended our study of weakly charged films by gradually increasing the surface charge density to one charge per 1125 and 750 Å2, respectively. We assume here that all ionic surfactant is (16) Gradzieski, M.; Hoffmann, H. Adv. Colloid Interface Sci. 1992, 42, 149. (17) Lindman, B.; Jonstro¨mer, M. In Physics of Amphiphilic Lyaers; Meunier, J., Langevin, D., Boccara, N., Eds.; Springer Verlag: Berlin, 1987; p 235. (18) Nilsson, P.; Lindman, B. J. Phys. Chem. 1984, 88, 5391. (19) Siano, D. B.; Myer, P.; Bock, J. J. Colloid Interface Sci. 1987, 117, 534. (20) Siano, D. B.; Myer, P.; Bock, J. J. Colloid Interface Sci. 1987, 117, 544.
adsorbed at the polar/apolar interface. For SDS, however, this seems to be a good approximation. Schoma¨cker and Strey studied the shift in the Bragg peak position of a dilute lamellar phase with C12E5 upon adding sodium alkyl sulfates with different hydrocarbon chain lengths,11 CmSO4- Na+. A limiting effect was observed for m g 12. Going to higher charge densities, we observe strong effects on phase boundary temperatures. This effect is quantitatively analyzed at the emulsification failure boundary where the electrostatic contribution to the free energy is calculated within the Poisson-Boltzmann cell model. In addition, we present small angle X-ray diffraction data
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from the hexagonal, lamellar, and reverse hexagonal phases and show that the area per surfactant molecule at the polar/apolar interface is approximately equal in all three phases. 2. Experimental Section Materials. The nonionic surfactant pentaethylene glycol dodecyl ether (C12E5) was obtained from Nikkol Ltd. Tokyo; decane (99%) and sodium chloride (NaCl) were from Sigma; sodium dodecyl sulfate (SDS) was from BDH (England); dodecyltrimethylammonium chloride (DOTAC) was from Tokyo Kasei; and D2O (99.8% isotopic purity) was from Dr. Glaser, AG Basel (Switzerland). All chemicals were used as received. Phase Diagram Determination. Samples for the phase diagram were prepared from a stock solution of C12E5 and decane with the appropriate concentration of SDS or DOTAC. The samples were prepared by weighing the desired amounts into screw-capped test tubes or ampules, which were immediately sealed after the addition of small magnetic stirrers. The samples were initially mixed using a vortex mixer in a single phase. The sample tubes were then allowed to equilibrate in a thermostated water bath. The phase boundary temperatures were determined by visual inspection of the transmitted light and between crossed polarizers for birefringence. As the microemulsion-oil equilibrium kinetics are very slow, this phase boundary was determined from the phase separation upon decreasing temperature from the homogeneous microemulsion phase and by solubilization upon increasing the temperature from the microemulsion-excess oil equilibrium. Close to this phase boundary the samples were kept for a few days to check for any phase separation. On the other hand, the kinetics involved in the other phase boundaries are rather fast. The phase boundaries of the anisotropic liquid crystalline phases were determined from the appearance of static birefringence. The volume fractions for the samples were calculated using the following densities (g/cm3): 0.967 (C12E5); 1.105 (D2O); 0.73 (decane). The effect of the added ionic surfactant on the surfactant density has been neglected. SAXS. Small angle X-ray scattering measurements were performed on a Kratky compact small angle system equipped with a position sensitive detector (OED 50M from MBraun, Graz) containing 1024 channels. Cu KR radiation of wavelength 1.542 Å was provided by a Seifert ID-3000 X-ray generator, operating at 50 kV and 40 mA. A 10 µm thick nickel filter was used to remove the Kβ radiation. The sample to detector distance was 277 mm. Each sample for SAXS measurement was placed in a quartz capillary using a syringe. Each sample was drawn into the capillary either in the lamellar phase or in the micellar phase, after confirming the complete homogeneity of the sample. This quartz capillary is glued to an invar steel body, which is designed to make possible simultaneous small and wide angle measurements. The sample stage permits control of temperature between 0 and 70 °C, with an accuracy of 0.1 °C, by using a peltier element. The slit-smeared spectra were desmeared using the direct method of beam height correction.21
3. Phase Behavior In Figure 1c and d we present the partial phase diagram of the four-component system C12E5/SDS/heavy water (D2O)/decane. The phase diagram corresponds to the water rich part of a section defined by a constant surfactant to oil ratio of 51.9/48.1 by weight, and β ) 0.04 and 0.06, respectively, where β is the mole fraction of SDS in the surfactant mixture. The phase diagram is presented as temperature versus the surfactant plus oil concentration measured in weight percent. For comparison, the corresponding phase diagrams for the pure nonionic system (β ) 0) and β ) 0.01 are redrawn from refs 3 and 6, respectively, and are shown in Figure 1a and b. As can be seen there are dramatic changes in the phase diagram when a small charge density is introduced. (21) Singh, M. A.; Ghosh, S. S., Jr. J. Appl. Crystallogr. 1993, 26, 787.
Figure 2. Partial phase diagram of the system C12E5/DOTAC/ D2O/decane. The weight ratio Ws/Wo ) 51.9/48.1, where Ws is the total weight fraction of C12E5 and DOTAC and Wo is the weight fraction of decane, is kept constant. The mole fraction of DOTAC in the surfactant mixture is β ) 0.04. The notations for the various phases are the same as in Figure 1.
Interactions in nonionic systems are very weak (essentially excluded volume only), and the addition of a small amount of surface charge plus counterions dramatically increases the osmotic pressure. An important effect of this pressure is that phase separation with excess water becomes unfavorable. This was discussed in detail in ref 6 and explains why the homogeneous reverse hexagonal and L2 phases are replacing the W + L3 two-phase equilibrium. As the L3 phase cannot swell with water at constant temperature due to curvature energy constraints,22,23 other structures, i.e. reverse cylinders and reverse spheres, which have mean curvatures that better match the spontaneous curvature6 are formed. The introduction of long range interactions also affects the global structure in the water continuous phases. Effects of increased “order” can be seen in the structure factor of for example the lamellar phase (see below). Furthermore, crystallization of the normal hexagonal (H1) and micellar cubic (I1) phases moves to lower concentrations the higher the charge density. An additional effect which becomes important for the higher charge densities is that the phase boundaries are shifted to higher temperatures, in particular at higher water content. This effect, which is due to the curvature dependence of the electrostatic energy, will be discussed separately in section 5 below. In order to see if the phase behavior remains the same if we change the ionic surfactant, we replaced SDS with DOTAC, for the β ) 0.04 system. The partial phase diagram of this system is shown in Figure 2. We see that (22) Anderson, D.; Wennerstro¨m, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243. (23) Daicic, J.; Olsson, U.; Wennerstro¨m, H.; Jerke, G.; Schurtenberger, P. J. Phys. II 1995, 5, 199.
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a
Figure 4. Variation of the first-order peak position q1 as a function of the volume fraction of the surfactant plus oil (φs + φo) in the lamellar phase for 4 mol % SDS, 4 mol % DOTAC, and 6 mol % SDS. The solid line is a least square fit of eq 1, yielding a surfactant length ls ) 15.6 Å.
b
peak, q1, by the following relationship
q1 )
Figure 3. Partial phase diagram of the system C12E5/SDS/0.1 M NaCl/decane. The weight ratio Ws/Wo ) 51.9/48.1, where Ws is the total weight fraction of C12E5 and SDS and Wo is the weight fraction of decane, is kept constant. The notations for the various phases are the same as in Figure 1. The mole fraction of SDS in the surfactant mixture is (a) β ) 0.01 and (b) β ) 0.06. These phase diagrams were made using H2O rather than D2O, which gives approximately a 2 °C upward shift of the phase boundary temperatures.
there is no significant change in the phase behavior, with almost the same phase boundaries as in the corresponding SDS system. Thus we see that the observed phase behavior is not unique only to anionic surfactants. In order to see if the effect produced by the addition of ionic surfactants on the phase behavior is reversible, we have determined the phase behavior by replacing water with 0.1 M NaCl for the β ) 0.01 and 0.06 systems. As can be clearly seen from Figure 3, the pure nonionic phase diagram is essentially recovered when the electrostatics are screened by salt. 4. SAXS Measurements 4.1. Micellar Cubic Phase. This phase occurs at lower temperatures. The bulk samples are highly viscous and optically isotropic. The SAXS pattern from these samples exhibits more than one reflection, indicating the existence of a crystalline-like order. However, because of the difficulty in obtaining a large number of reflections, an unambiguous indexing of the observed reflections has not yet been possible. 4.2. Lamellar Phase. The SAXS pattern for all the samples in the lamellar phase shows a well defined secondorder peak with a ratio of 1:2 in peak position, thus confirming a lamellar structure. The periodicity, d, of the lamellar phase, which is given by the bilayer area per unit volume, is related to the position of the first-order
2π πφs ) d ls
(1)
where φs is the volume fraction of the surfactant. Here we have also introduced the surfactant length, ls, which is another measure of the surfactant area, defined as the surfactant volume to area ratio, i.e. ls ≡ vs/as, where vs ) 702 Å3.24 In Figure 4 we have plotted the first-order peak position q1, as a function of the total volume fraction of surfactant and oil (φs + φo). We observe a linear dependence, indicative of a one-dimensional swelling. From a least squares fit of the plot of q1 versus φs we obtain the length of the surfactant to be 15.6 Å and the corresponding area per head group to be 45 Å2, which is quite consistent with the previously measured value for this surfactant.3,4,6,24,25 An interesting observation in all the systems studied concerns the low-q part of the lamellar phase structure factor. The lamellar phase in the pure ternary system as well as the binary nonionic surfactant water system swells by the undulation force. In general such lamellar systems are rather turbid due to their high osmotic compressibility.26 As a result of this, one observes an intense scattering at low q which in many cases even hides the first-order Bragg peak and a second-order peak is often not observed at all.26-29 On the other hand, in our samples where ionic surfactant has been added the lamellar phase has a very low turbidity. It has been seen that on the introduction of a small charge density the interaction switches over from the dominating undulation force to an electrostatic one.6,10,11 This can also be seen from the well resolved second-order peak that one observes in these systems, even at lower concentrations. 4.3. Hexagonal Phase. The bulk samples of this phase are clear, birefringent, and viscous. In this phase the Bragg reflections correspond to
qhk )
4π xh2 + k2 + hk ax3
(2)
where a is the lattice parameter and h and k are the Miller (24) Olsson, U.; Wu¨rz, U.; Strey, R. J. Phys. Chem. 1993, 97, 4535. (25) Strey, R.; Schoma¨cker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253. (26) Nallet, F.; Roux, D.; Milner, S. T. J. Phys. (Paris) 1990, 51, 2333. (27) Roux, D.; Safinya, C. R. J. Phys. 1988, 49, 307. (28) Porte, G.; Marignan, J.; Bassereau, P.; May, R. Europhys. Lett. 1988, 7, 713. (29) Appell, J.; Bassereau, P.; Marignan, J.; Porte, G. Colloid Polym. Sci. 1989, 267, 600.
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Figure 5. Variation of the first-order peak position q10 as a function of the volume fraction of surfactant plus oil (φs + φo), for the normal hexagonal phase ((0) 4 mol % SDS, (1) 4 mol % DOTAC, (9) 6 mol % SDS) and the reverse hexagonal phase ((+) 1 mol % SDS, (O) 4 mol % SDS, (×) 4 mol % DOTAC, (b) 6 mol % SDS). The solid line and the dashed line are the calculated first-order peak positions for the normal hexagonal and the reverse hexagonal phase using eq 6 and 8, respectively, and a constant surfactant length of 15.6 Å.
Figure 6. Variation of the length of the surfactant as a function of volume fraction of the surfactant plus oil (φs + φo) from the three liquid crystalline phases. The solid line indicates the average length of the surfactant, ls (15.6 Å), obtained from the lamellar phase. (i) Lamellar: (4) 4 mol % SDS, (]) 4 mol % DOTAC, (2) 6 mol % SDS. (ii) Normal hexagonal: (0) 4 mol % SDS, (1) 4 mol % DOTAC, (9) 6 mol % SDS. (iii) Reverse hexagonal: (+) 1 mol % SDS, (O) 4 mol % SDS, (×) 4 mol % DOTAC, (b) 6 mol % SDS.
indices. The first four reflections correspond to (h,k) ) (1,0), (1,1), (2,0), and (2,1) and give the characteristic peak position pattern as 1:x3:x4:x7. If we consider the hexagonal phase to be made up of infinite cylinders, having a cross section radius of Rcyl, then the lattice parameters a and Rcyl are related in the following manner
cylinders. With the previous definition of the polar/apolar interface, the cylinder volume fraction becomes
x( )
1 φhc ) φs + φo 2
(4)
The factor 1/2 comes from the fact that the hydrocarbon block of the C12E5 surfactant makes up 50% of the total surfactant volume. By evaluating the area per molecule at the defined polar/apolar interface, we identify φcyl ) φhc. The radius as the interface then becomes
2lsφhc φs
(5)
Substituting eqs 3-5 in eq 2, we obtain
qhk )
xh2 + k2 + hk
x( (
))
φs x3 φo + 2π 2
(7)
(3)
where φcyl is the volume fraction of the cylinders. For curved interfaces, the area per molecule depends on where the interface is defined. Here we measure the area per molecule at the polar/apolar interface defined as the interface separating the ethylene oxide and the hydrocarbon blocks of the surfactant. This corresponds to a total hydrocarbon volume fraction defined as
Rcyl )
φs 2
and the peak positions are given by
x3 φ 2π cyl
Rcyl ) a
φcyl ) φw +
φs ls
(6)
In Figure 5 we have plotted the first-order peak position q10 as a function of the volume fraction of surfactant plus oil (φs + φo), as obtained from the diffraction patterns. The solid line is the calculated first-order peak position, using eq 6, for a fixed surfactant length of 15.6 Å. We see that the experimentally observed data points lie well along the calculated curve over the entire volume fraction range covered. 4.4. Reverse Hexagonal Phase. In the reverse hexagonal phase water is inside and the oil is outside the
qhk )
xh2 + k2 + hk
φs φs ls x3 φw + 2π 2
x (
)
(8)
A plot of the first-order peak position, q10, versus the volume fraction of surfactant plus oil is shown in Figure 5. The dashed line is the calculated value of the peak position, using eq 8, for a surfactant length of 15.6 Å, and as can be seen, there is a good agreement with the experimental data. 4.5. The Area per C12E5 Molecule is Constant. Using the observed peak positions from the SAXS diffraction patterns in the lamellar, hexagonal, and reverse hexagonal phase, we can calculate the length of the surfactant at the polar/apolar interface of each of the phases. In Figure 6 we have plotted the experimentally obtained length of the surfactant from all these phases as a function of the total volume fraction of surfactant plus oil. We see that in all the lamellar, hexagonal, and reverse hexagonal phases the length of the surfactant is approximately constant, implying a constant area per head group of the surfactant, and this observed value of the length of the surfactant is quite consistent with previously measured values for this surfactant.3,4,6,24,25 As can be seen from the plot the observed length of the surfactant seems to be independent of temperature and aggregate geometry. In addition we also see that the observed length is independent of the charge density of the surfactant film. 5. Emulsification Failure with Charged Films The phase diagrams of nonionic surfactants follow a clear trend of a decreasing mean curvature of the surfactant monolayer film with increasing temperature. It has been found that a useful way to model the thermodynamics of these systems is to consider a curvature elastic energy of the surfactant film. To second
2944 Langmuir, Vol. 12, No. 12, 1996
Rajagopalan et al.
order in the principal curvatures, c1 and c2, the curvature energy (area) density can be written as
gc ) 2κ(H - H0)2 + κjK
(9)
where H ≡ (c1 + c2)/2 is the mean curvature. H0 is the spontaneous curvature, and K ≡ c1c2 is the so-called Gaussian curvature. The two moduli, κ and κj, are called the bending modulus and the saddle splay modulus, respectively. Within eq 9 the phase diagram of the nonionic surfactants can be understood by having H0 vary with temperature. Defining curvature toward oil as positive, H0 decreases with increasing temperature. At lower temperatures the L1 microemulsion phase coexists with excess pure oil. This type of phase separation has been termed emulsification failure30 (EF). The L1 phase has a droplet structure over a large range of concentrations and temperatures; however, the droplet size and shape vary with temperature. The droplets are spherical near the lower phase boundary and grow into prolate shapes when increasing the temperature.3,4,31 For a given area-to-enclosed volume ratio (here constrained by the constant surfactant to oil ratio) the spherical shape corresponds to the maximum mean curvature. Increasing the mean curvature further can only occur by increasing the area to enclosed volume ratio, i.e. by phase separation with excess oil, which explains the L1 + O coexistence at lower temperatures. The radius of the spheres at the EF boundary is for the present system REF ) 80 Å, as measured by SANS32 and by static light scattering.3 This radius is also consistent with the value of ls determined from the lamellar and hexagonal phases above, which can be found by computing REF ) 3(φo + φs/2)ls/φs. The phase equilibrium with excess oil implies33
µo ) 0
(10)
where µo is the oil chemical potential, which defines the EF boundary. The total curvature energy, Gc, is found by integrating the energy density (eq 9) over the total interface, Σ:
Gc )
∫∫ΣdA gc
(11)
which in the case of monodisperse spheres becomes
[(
Gc ) nd4π 2κ 1 -
) ]
R R0
2
+ κj
(12)
where nd is the number of spheres and R0-1 ) H0 is the spontaneous curvature. The number of spheres, nd, is related to the radius R by
nd )
nsas 4πR2
(13)
where ns is the number of surfactant molecules and as is the area per surfactant molecule. There is also a free energy contribution arising from the entropy of mixing of the aggregates with the solvent (30) Turkevich, L. A.; Safran, S. A.; Pincus, P. A. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum Press: New York, 1986; Vol. 6, p 1177. (31) Leaver, M.; Furo´, I.; Olsson, U. Langmuir 1995, 11, 1524. (32) Bagger-Jo¨rgensen, H.; Olsson, U.; Mortensen, K. In preparation. (33) Here we have for simplicity set the standard oil chemical potential µ°0 ) 0.
molecules. An earlier study concerning the L1 phase in the same system without added ionic surfactant showed that the osmotic compressibility is well described by the Carnahan-Starling expression for hard sphres.3 The corresponding entropy of mixing is34
[()
Gmix ) ndkT ln
]
cd 4 - 3φHS - 1 + φHS c°w (1 - φ )2 HS
(14)
which at low volume fractions reduces to the ideal entropy of mixing. Here, cd is the concentration of droplets, c°w is the pure water concentration ()55.5 mol/L), and φHS is the hard sphere volume fraction. In the present system, φHS ≈ 1.14φ, φ being the micellar volume fraction, φ ) (φs + φo), where φs and φo are the volume fractions of the surfactant and oil, respectively.3 The contributions to the oil chemical potential arising from curvature energy and entropy of mixing are then
|
∂Gc ∂no
µco )
[(
6vo )-
ns,nw
R3
2κ 1 -
) ]
R + κj R0
(15)
and
µmix ) o
|
∂Gmix ∂no
ns,nw
{ ()
3kBTvo -
3
4πR
2 ln
}
cd φHS(5 - φHS) + c°w 1 - φHS
(16) respectively, with vo being the molecular volume of the oil and kBT the thermal energy. Solving the equation
)0 µo ) µco + µmix o then gives the equation for the sphere radius at the EF boundary:
( ()
)
κj cd REF φHS(5 - φHS) kBT )1+ 2 ln + + R0 2κ 16πκ c°w 1 - φHS
(17)
This expression, although with an ideal entropy of mixing, was derived previously by Safran.35 After having considered the system with neutral surfactant films, we now turn to the systems with charged films. The increase in phase boundary temperatures when increasing the charge density of the surfactant film is due to the electrostatic energy and can be qualitatively understood from, for example, considering the Born energy of an ion. The electrostatic energy due to an isolated charged sphere of radius R is given by
Uel ∝
q2 R
(18)
where q is the charge. In the present system the charge density σ is kept constant, so the particle charge is proportional to its area and for spheres we have in particular q ∝ σR2. If this is incorporated into eq 18, we find
Uel/A ∝ σ2R
(19)
where A denotes the area. Hence the electrostatic energy, due to the added ionic surfactant, wants to decrease the (34) Overbeek, J. T. G. Faraday Discuss. Chem. Soc. 1978, 65, 7. (35) Safran, S. A. In Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution; Chen, S., Huang, J. S., Tartaglia, P., Eds.; Kluwer: Boston, 1991; Vol. 369.
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Langmuir, Vol. 12, No. 12, 1996 2945
radius of the spheres, which is done by phase separation with excess oil. This electrostatic effect is now balanced by the curvature energy to stabilize a finite radius. By decreasing the spontaneous curvature, i.e. by increasing the temperature, an opposite curvature stress which wants to increase the radius is applied, which balances the electrostatic term. To obtain a quantitative estimate of the electrostatic energy, we also need to take into account the counterions and solve the Poisson-Boltzmann (PB) equation. Electrostatics are also screened by the adjacent charged droplets in the system and their counterions. Hence, the effect on the phase boundary temperatures decreases with increasing surfactant plus oil concentration. The concentration dependence can be taken into account by solving the PB equation within the so-called cell model.36 Here the system is divided into equally sized cells, each containing one spherical aggregate in the center, as illustrated in Figure 7. The PB equation is then solved within the cell, with the boundary condition that the electric field is zero on the cell boundary. This approach has been used previously to investigate phase equilibria in ionic surfactant systems.37-40 We assume that the SDS molecules are randomly distributed within the film and treat the charges as a homogeneously smeared out charge density. No sign of an inhomogeneous distribution of SDS, like for example a rectangular phase, was observed in the phase diagrams. Below we will consider the electrostatic contribution to the EF boundary. Solving for the phase boundary
µco + µmix + µel o o ) 0
Figure 7. Calculation of the electrostatic contribution to the chemical potential of oil performed using the cell model. The charged micellar aggregate with radius R is surrounded by water and counterions. The radius of the cell, Rcell, is determined from (R/Rcell)3 ) φs + φo.
proximately 2 °C. Thus we have used T0 ) 35.9 °C in the calculations below. The electrostatic contribution to the oil chemical potential was obtained from numerical solutions to the (nonlinearized) Poisson-Boltzmann equation in the spherical geometry. The cell radius, Rcell, was chosen according to the micellar size and the volume fraction, such that (REF/Rcell)3 ) φhc, corresponding to the total apolar volume fraction. From calculations performed at various volume fractions, charge densities, and temperatures, we then obtained the following approximate relationship:
where µel o is the electrostatic contribution to the oil chemical potential, we obtain
κj REF )1+ + R0 2κ
( ()
)
cd φHS(5 - φHS) R3EF el kBT + µ (20) 2 ln 16πκ c°w 1 - φHS 12κvo o In order to fit eq 20 to the experimental phase boundary, we also need a relation between the spontaneous curvature and temperature. We will assume that H0 ) R0-1 varies linearly with the temperature, i.e.
1 ) δ(T0 - T) R0
(21)
corresponding to a Taylor expansion to first order around T0, which is the balanced temperature where the spontaneous curvature is zero. As will be seen, this results in an approximately linear temperature dependence also for REF-1 which has been observed experimentally.2 With normal water (H2O) T0 has been determined to be 37.9 °C. When replacing H2O with D2O in C12E5 systems, corresponding temperatures are normally reduced by ap(36) Hill, T. L. Statistical Mechanics; Addison and Wesley: Reading, MA, 1960. (37) Jo¨nsson, B.; Wennerstro¨m, H. J. Colloid Interface Sci. 1981, 80, 482. (38) Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1987, 91, 338. (39) Aamodt, M.; Landgren, M.; Jo¨nsson, B. J. Phys. Chem. 1992, 96, 945. (40) Landgren, M.; Aamodt, M.; Jo¨nsson, B. J. Phys. Chem. 1992, 96, 949.
µel o =
8.68 × 103 (φ -1/3 - 1)kBT 2 hc (Aq + 535)
(22)
where Aq denotes the area per charge (i.e. 45/β Å2) at the micellar interface measured in square angstroms. For our value of REF ) 80 Å and for the respective ranges of the parameters, volume fraction (0.01 e φ e 0.4), charge density (Aq g 750 Å2), and temperature (25-60 °C), considered here, this approximate relation is accurate to within 10%. Inserting eqs 21 and 22 into eq 20, we finally obtain a relation for the temperature of the EF boundary, TEF, as a function of Aq and φhc ) 0.775φ given by
TEF )
T0 -
2κ + κj
/
2REFκδ
[
( ( )
kB 2 ln
1+
3φhcvw 4πR3EF
+
16πREFκδ
(8.68 × 103)R2EFkB 2
12vo(Aq + 535) κδ
)
φHS(5 - φHS) 1 - φHS
]
-
(φHc-1/3 - 1) (23)
Equation 23 was fitted to the experimental EF boundaries using (2κ + κj) and the product κδ as two adjustable parameters, assuming κ and κj to be temperature independent. Here, (2κ + κj)/κδ acts as a temperature offset while κδ determines the curvature of the phase boundaries. The best fit found from visual inspection is shown in Figure 8. Note that the EF boundaries for all charge densities were fitted simultaneously. From the best fit we obtain 2κ + κj ) (1 ( 0.3) × 10-20 J (corresponding to (3 ( 1)kBT
2946 Langmuir, Vol. 12, No. 12, 1996
Figure 8. Fit of eq 23 to the experimental emulsification failure boundaries of various additions of SDS. The symbols are experimental points, and the lines are the calculated phase boundaries with the following notations: no SDS, 2 and solid line; 1 mol % SDS, b and dashed line; 4 mol % SDS, 9 and dash-dotted line; 6 mol % SDS, [ and long-dashed line. The calculated phase boundaries are obtained with 2κ + κj ) (1 ( 0.3) × 10-20 J (corresponding to (3 ( 1)kBT at room temperature) and κδ ) (4 ( 0.5) × 10-24 J K-1 Å-1 (corresponding to ((1 ( 0.1) × 10-3)kBT K-1 Å-1).
Figure 9. 1/REF as a function of temperature calculated from the inverse of eq 23 after omitting the electrostatic term (see text). The values used in the calculation were 2κ + κj ) 3 kT, φS + φo ) 0.10, φHS ) 0.114, and δ ) 0.5 × 10-3 K-1 Å-1.
at room temperature) and κδ ) (4 ( 0.5) × 10-24 J K-1 Å-1 (corresponding to ((1 ( 0.1) × 10-3)kBT K-1 Å-1. The experiments performed here can be seen as applying a curvature stress to the surfactant film at emulsification failure, due to the electrostatic energy, and as monitoring the response of the system. One could also have measured how the radius depends on the charge density. Here we have instead measured how much the spontaneous curvature has to be varied (by varying the temperature), at constant radius, in order to balance the electrostaticly induced curvature stress. The value 2κ + κj ) (3 ( 1)kBT is close to what has been determined recently from analyzing the droplet polydispersity at emulsification failure for the same surfactant with heptane.41 In the calculations above we have assumed that the spontaneous curvature, 1/R0, varies linearly with the temperature. With our obtained values of 2κ + κj and κδ (41) Sicoli, F.; Langevin, D.; Lee, L. T. J. Chem. Phys. 1993, 99, 4759.
Rajagopalan et al.
we have calculated how 1/REF depends on TEF in the uncharged system by inverting eq 23 and omitting the electrostatic term (last term in the denominator). The result is shown in Figure 9 for a volume fraction of φ ) 0.1. As can be seen, 1/REF varies approximately linearly with T with a slope δ ≈ 1 × 10-3 K-1 Å-1. The ratio REF/R0 ≈ δ/δ′, however, depends on the particular value of κ. For example, if κ ) 2kBT, we find δ ) 0.5 × 10-3 K-1 Å-1 and REF/R0 ≈ δ/δ′ ≈ 0.5. A linear temperature dependence of 1/REF has been observed experimentally in the C12E5/ water/octane system by Strey, who measured the micellar radii with small angle neutron scattering.2 The slope found by Strey was 1.2 × 10-3 K-1 Å-1, and a similar value ≈10-3 K-1 Å-1, can also be found by analyzing how the surfactant to oil ratio at the EF boundary varies with T in the phase diagrams on Kunieda and Shinoda of the C12E5/water/tetradecane system.42 Hence, our assumption of a linear temperature dependence of 1/R0 is found to be consistent with experiments. It is interesting to note that the slope seems to be approximately independent of the chain length of the oil, while T0 varies strongly. 6. Conclusions We have studied the phase equilibria of a surfactant/ water/oil system containing weakly charged surfactant films of various charge density. The phases were found to be strongly affected when introducing electrostatic interactions and dependent on the charge density. The weakly charged systems may still be discussed in terms of the flexible surface model where the temperature dependent spontaneous curvature of the nonionic surfactant is an important parameter. The model however has to be extended to include electrostatic contributions to the free energy. The main findings of the present study are summarized below. (i) The crystallization of micellar cubic and normal hexagonal phases moves to lower volume fractions with increasing charge density. (ii) Phase boundaries are shifted due to the electrostatic interactions. The result of Safran30,35 for uncharged films plus an electrostatic part, calculated within the PoissonBoltzmann cell model, was found to describe well the variation of the emulsification failure boundary with the charge density. From the fit to the phase boundaries we obtained 2κ + κj ) (3 ( 1)kBT and κδ ) ((1 ( 0.1) × 10-3)kBT K-1 Å-1, where δ represents the temperature dependence of the spontaneous curvature. (iii) The formation of reverse hexagonal and reverse micellar phases at higher temperatures, which are not present in the uncharged system, is confirmed in the present study, going to higher charge densities. (iv) SAXS data obtained from the normal hexagonal, lamellar, and reverse hexagonal phases at different charge densities all gave the same value, 45 Å2, for the area per C12E5 molecule at the polar/apolar interface. Acknowledgment. This work was supported by The Swedish Natural Science Research Council (NFR) and The Swedish Research Council for Engineering Sciences (TFR). The stay of K.F. in Lund was supported by a grant from the Swedish Institute. LA9510947 (42) Kunieda, H.; Shinoda, K. J. Dispersion Sci. Technol. 1982, 3, 233.
