Influence of Ionic Charges on the Bilayers of Lamellar Phases

Feb 10, 2007 - The single lamellar phase (5 wt % LA 070 and 240 mM EHG in water) yields a bluish homogeneous solution. With the addition of SDS, the ...
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Langmuir 2007, 23, 2977-2984

2977

Influence of Ionic Charges on the Bilayers of Lamellar Phases Aihua Zou,*,† Heinz Hoffmann,† Norbert Freiberger,‡ and Otto Glatter‡ Bayreuth Center for Colloid and Interface, UniVersity Bayreuth, 95448, Bayreuth, Germany, and Institut fu¨r Chemie, Karl-Franzens UniVersita¨t Graz, 8010 Graz, Austria ReceiVed October 27, 2006. In Final Form: December 20, 2006 The influence of ionic charges on the mesophases in the ternary system of C12-16E6 (LA 070), ethylhexylglycerid (EHG), and water was studied. The charge was introduced by adding the ionic surfactant SDS (sodium dodecyl sulfate). The single lamellar phase (5 wt % LA 070 and 240 mM EHG in water) yields a bluish homogeneous solution. With the addition of SDS, the samples become more and more clear. Rheology measurements indicate that increased charge density increases the storage modulus G′, and the lamellar phases show typical behavior of a viscoelastic fluid with a yield stress at higher SDS concentration. SAXS measurements show that the interlamellar distance D decreases with SDS concentration. The addition of ionic surfactants suppresses the Helfrich undulations, flattens the bilayers, and decreases interbilayer spacing due to electrostatic repulsions of the ionic surfactant head groups. Furthermore, the LR phase transforms into vesicle phases as the SDS concentration is increased. Second, it is shown that with added NaCl electrolyte the phase with charged surfactant behaves again in the same way as the initial uncharged system. The addition of salt screens the electrostatic interaction, which leads to a higher flexibility of the bilayers and a decrease of the storage modulus G′. Theoretical calculations show that the shear moduli of the LR phases are much smaller than the osmotic pressure of the systems. Several models are proposed for the explanation of the shear moduli. The model due to Lekkerkerker for the electric contribution of the bending constant of the bilayer seems to yield good results for the transition to vesicles.

Introduction Many amphiphilic molecules in aqueous solutions self-aggregate to form mesophases. Because of special interest and importance for biological systems, much attention has been focused on the lamellar phases. The most important bilayers are undoubtedly the lipid bilayers found in biological systems. The structure and properties of biological membrane systems as well as those of simple model bilayers have been extensively studied in recent years.1,2 However, the most interesting membrane systems are not the simple binary systems but rather the multicomponent systems for which the specific interactions between the components are known to have a strong impact on the membrane and phase properties.3,4 The lamellar structures are stabilized by a variety of repulsive interactions between the membranes. In particular, individual membranes may undergo strong thermally induced Helfrich undulations.5,6 The phenomenological parameter characterizing the flexibility of surfactant bilayers is the bending modulus kc. If the bending modulus is of the order of the thermal energy kBT, where kB is Boltzmann’s constant and T is the absolute temperature, the bilayers may show strong thermally excited shape fluctuations or undulations. So the total effective area is reduced compared to that of a flat bilayer. Experimentally, it was observed that the undulations of the bilayers consumed a substantial amount up to 30% of the total bilayer area.7 This is the case for very flexible surfactant bilayers, such as those formed by C10E3. It is generally believed that lamellar phases of nonionic surfactants are stabilized by the long-ranged Helfrich8 repulsion resulting from these thermal undulations. But the curvature and * Corresponding author. E-mail: [email protected]. † University Bayreuth. ‡ Karl-Franzens Universita ¨ t Graz. (1) Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Eds.; Springer Press: Berlin, 1987. (2) Handbook of biological physics; Lipowsky, R., Sackmann, E., Eds.; North Holland Press: Amsterdam, 1995; Vol. 1. (3) Strey, R. Ber. Bunsen-Ges Phys. Chem. 1996, 100, 182. (4) Goulian, M. Curr. Opin. Colloid Interf. Sci. 1996, 1, 358. (5) Bagger-Jo¨rgensen, H.; Olsson, U. Langmuir 1996, 12, 413.

bending properties of the uncharged bilayers can be influenced by various additives.9 With the addition of an ionic surfactant, the electrostatic interactions will compete with undulations. Unless the electrostatic interactions are highly screened by the addition of large amounts of electrolyte, one must then consider the interplay between the thermal undulations and the electrostatic interactions.10-18 The electrostatic interactions will combine with entropic interactions in a nontrivial way. It modifies the bending rigidity of each membrane and can thus change the nature of out-of-plane fluctuations that drive the steric repulsions. In general, the addition of an ionic surfactant tends to stabilize the lamellar phase.9 The aim of the present study is to develop a better understanding of the dramatic effects of the charge effect and the shielding for the lamellar structure. With this in mind, we have recently studied the impact of electrostatic interactions on the lamellar phase of a nonionic surfactant, LA 070 (C12-16E6), with a cosurfactant EHG. This is achieved by studying the influence of ionic surfactant SDS and the counteracting effect of inert electrolyte NaCl on the system of LA 070 lamellar phases by phase behavior, rheology, and SAXS measurements. Experimental Section 1. Materials. The investigated surfactant LA 070 was a gift of Clariant Company. Due to the manufacturing process, this surfactant system is a mixture of several ethoxylated carbon alcohols: C12-16E6 (LA 070). EHG, with a content of ethylhexyl(6) Odijk, T. Langmuir 1992, 8, 1690. (7) Schoma¨cker, R.; Strey, R. J. Phys. Chem. 1994, 98, 3908. (8) Helfrich, W. Z. Naturforsch. 1987, 33a, 305. (9) Jonstro¨mer, M.; Strey, R. J. Phys. Chem. 1992, 96, 5993. (10) Bassereau, P.; Marignan, J.; Porte, G. J. Phys. France 1987, 48, 673. (11) Roux, D.; Sanya, C. R. J. Phys. France 1988, 49, 307. (12) Nallet, F.; Roux, D.; Prost, J. Phys. ReV. Lett. 1989, 62, 276. (13) Bassereau, P.; Appell, J.; Marignan, J. J. Phys. II France 1992, 2, 1257. (14) Pincus, P.; Joanny, J.-F.; Andelman, D. Europhys. Lett. 1990, 11, 763. (15) Auguste, F.; Barois, P.; Fredon, L.; Clin, B.; Dufourc, E. J.; Bellocq, A. M. J. Phys. II France 1994, 4, 2197. (16) Li, Z. X.; Lu, J. R.; Thomas, R. K.; Penfold, J. Faraday Discuss. 1996, 104, 127. (17) Podgornik, R.; Parsegian, V. A. Langmuir 1992, 8, 557. (18) Evans, E. A.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7131.

10.1021/la063148q CCC: $37.00 © 2007 American Chemical Society Published on Web 02/10/2007

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Figure 1. Phase behavior of 5 wt % LA 070 with the addition of EHG. The numbers on the photos are the EHG concentrations. glycerid, was a cosmetic additive from Schuelke & Mayr GmbH. Both of the two chemicals were used as received. Sodium dodecyl sulfate (SDS) is from Fluka. Distilled water was used as solvent throughout the study. 2. Methods. 2.1. Phase BehaVior. The phase behavior of LA 070 was established by observing the solutions in test tubes under temperature-controlled conditions for several months. The samples were prepared by shaking and centrifugating the test tubes for the same time to remove the bubbles. The phases were characterized by visual inspection with and without polarizers. 2.2. Rheological Measurements. The rheological measurements were performed by a Haake RS600 with a cone-plate sensor and a Haake RS300 with a double-gap cylinder sensor. The sensor systems can be used properly according to the viscosity of solution: RS600 for highly viscous liquids and RS300 for low viscous liquids ( G′′. From its birefringence pattern it can be concluded that this system is an LRh phase. The rheograms of b, c, and d show the typical behavior of a viscoelastic fluid with a yield stress: both moduli are almost independent of frequency in the measuring range; the storage modulus G′ is by a factor of 10 higher than the loss modulus G′′. The viscosity shows no plateau anymore; it decreases double-logarithmically with increasing frequency in the whole frequency range which appears in typical vesicle systems.23-25 The variation of the charge density on the bilayers can drastically change the rheological properties of these phases. An increase of the charge density reduces the flexibility of the bilayers and then changes the lamellar phase into a vesicle phase, which can be seen from the different birefringence pattern in Figure 2. The vesicle phases are probably formed by shear and mixing during the sample preparation. Without charges the bilayers of the nonionic surfactants are flexible and they form the classic stacked LR phase. Under shear during the sample preparation the bilayers transform to vesicles. However, when the shear is stopped, the vesicles relax back to the LR phase. But (19) Hoffmann, H. Prog. Colloid Polym. Sci. 1990, 83, 16. (20) Platz, G.; Thunig, C.; Hoffmann, H. Ber. Bunsen-Ges. Phys. Chem. 1992, 96, 667. (21) Hoffmann, H.; Thunig, C.; Munkert, U.; Meyer, H. W.; Richter, W. Langmuir 1992, 8, 2629. (22) Valiente, M.; Munkert, U.; Lenz, U.; Hoffmann, H.; Thunig, C. J. Colloid Interface Sci. 1993, 160, 39. (23) Hao, J.; Hoffmann, H.; Horbaschek, K. Langmuir 2001, 17, 4151. (24) Laughlin, R. G.; Munyon, R. L.; Burns, J. L.; Coffindaffer, T. W.; Talmon, Y. J. Phys. Chem. 1992, 96, 374. (25) Gradzielski, M.; Mu¨ller, M.; Bermeier, M.; Hoffmann, H.; Hoinkis, E. J. Phys. Chem. B 1999, 103, 1416.

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Figure 2. Phase behavior of 5 wt % LA 070/240 mM EHG/water/SDS system. The numbers on the photos are the SDS concentrations (mM). Upper photos, in between polarizers; lower, without polarizers.

Figure 3. Dynamic rheological measurements for the LR phase solutions at different SDS concentrations in the 5 wt % LA 070, 240 mM EHG system at 25 °C. SDS concentration: a, 0 mM; b, 2 mM; c, 8 mM; d, 10 mM.

when the bilayers are more and more charged, the bilayers become stiffer and the vesicles, formed during the sample preparation, can no longer relax back to the LR phase. The SAXS measurements further demonstrate the above results. The lattice parameter (the interlamellar distance, D) in Table 1 was obtained directly by the first, and the most intense, diffraction peak position of the SAXS diffraction.26-28 From Table 1, it can be seen that the lattice spacing of the lamellar phase decreases with increasing SDS content. The interlamellar distance D can only be changed by suppressing the bilayer undulation or by the bilayer becoming thinner. Hence, with the addition of SDS the

repulsive Coulombic interactions should make the film thinner and stiffer and therefore enlarge the total bilayer area. The interlamellar distance D is related to the bilayer volume fraction φs and bilayer thickness δ via

φsD ≡ δapp ) δ(1+ ∆A/A)

(1)

where ∆A/A is the fractional area of bilayer stored in thermal (26) Alexandridis, P.; Olsson, U.; Lindman, B. Macromolecules 1995, 28, 7700. (27) Alexandridis, P.; Zhou, D.; Khan, A. Langmuir 1996, 12, 2690. (28) Ivanova, R.; Lindman, B.; Alexandridis, P. Langmuir 2000, 16, 3660.

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Table 1. Interlamellar Distance D and the Apparent Bilayer Thickness δapp for SDS Effect on the Lamellar Phase of the 5 wt % LA 070, 240 mM EHG System

samples

SDS concentration (mM)

D ( 0.1 (nm)

δapp ≈ φsD (nm)

1 2 3 4 5 6

0.5 1 4 6 8 10

31.4 31.3 30.2 29.1 28.9 28.1

3.10 3.10 2.98 2.88 2.85 2.78

undulations.29 Increased undulation amplitudes will be reflected in an increased apparent bilayer thickness δapp. In our system the cosurfactant EHG was assumed to be all in the bilayers when the bilayer volume fraction was φs calculated. As shown in Table 1, the apparent bilayer thickness is largest for the sample with the lowest amount of ionic surfactant concentration, for which undulations are expected to be most pronounced. Recent theoretical effort has focused on determining the electrostatic contribution to the bending modulus in different regimes of membrane surface charge density and aqueous electrolyte strength. It is convenient to characterize these regimes in terms of the characteristic length scales: the interlamellar distance D and Debye screening length L-1 ) (8πn∞Q)-1/2, where n∞ is the bulk electrolyte concentrations, e is the electric unit charge, Q is the Bjerrum length (for water implying Q ) 0.714 nm).7 The calculated Debye screening length L-1 and the interlamellar distance D are shown in Figure 4. This graph demonstrates that the Debye screening length is smaller than the spacing between the amphiphilic bilayers. Accordingly, the planeto-plane repulsion is weak because of the screening of the ionic charges. Thus, it probably can be concluded that the electrostatic effects on the spacing of the lamellar phase are due to the inplane Coulombic repulsion of the ionic head groups rather than the plane-to-plane repulsions. In the densely packed region the modulus depends strongly on the charge density of lamellar phase as demonstrated in Figure 5. With increasing charge density the modulus G′ abruptly increases with the addition of SDS. If the charge density on the vesicles is shielded by excess salt, it will decrease rapidly with the ionic strength, as demonstrated in Figure 10 in section 2. This phenomenon is similar to the previous result in the tetradecyldimethylaminoxide (C14DMAO)/tetradecyl-trimethylammonium

Figure 4. Effect of increasing concentrations of SDS on interlamellar distance D and the Debye screening length L-1 for lamellar phases of 5 wt % LA 070, 240 mM EHG system.

Figure 5. Storage modulus G′ versus frequency of the 5 wt % LA 070, 240 mM EHG system at different SDS concentrations.

Figure 6. Comparison of the magnitude of the complex viscosity |η|* as a function of frequency and the shear viscosity η as a function of the shear rate. The filled and empty symbols are for the complex viscosity and shear viscosity, respectively.

bromide (C14TMABr) system: the moduli G′ were found to increase with increasing charge density.30 The viscosities of many viscoelastic solutions follow the CoxMerz rule, which means that the shear viscosities as a function of the shear rate and the magnitude of the complex viscosities as a function of the angular frequency have the same value in the shear thinning region.31 This, for instance, is the case for viscoelastic solutions of entangled threadlike micelles.32 The LR solutions do not show this simple behavior, as shown in Figure 6 where the shear viscosities and the complex viscosities for two different systems are plotted against the shear rate and the frequency, respectively. Without the SDS, the shear thinning viscosity given by the steady shear measurement does not coincide with that by the dynamic measurement. The relaxation processes of the two measurements are completely distinct from one another. This deviation from the Cox-Merz rule arises from the flow property under an applied stress. With the addition of 10 mM SDS, the two viscosities come nearly to obey the Cox-Merz rule, suggesting that the time constants for the phases for both (29) Helfrich, W. Z. Naturforsch. 1975, 30c, 841. (30) Hoffmann, H.; Thunig, C.; Schmiedel, P.; Munkert, U. Langmuir 1994, 10, 3972. (31) Cox, W. P.; Merz, E. H. J. Polym. Sci. 1958, 28, 619. (32) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933.

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Figure 8. Phase behavior of 5 wt % LA 070, 240 mM EHG (LR) systems (a) LR phase, (b) LR phase with 10 mM SDS, and (c)-(e) LR phase with 10 mM SDS and 10, 15, and 25 mM NaCl.

Figure 7. Steady shear rheograms of the apparent viscosity versus shear rate for the 5 wt % LA 070, 240 mM EHG system at different SDS concentrations.

measurements are equal. These mechanical properties reflect the change in the LR morphologies with charge density. The LR solution without SDS can be expected to consist mostly of stacked bilayers and can be explained on the basis of the low yield stress, and the different relaxation modes between the steady flow and oscillatory flow. As mentioned above, the sufficiently high yield stress and the structural deformation co-incident between the two experiments unambiguously reveal the presence of vesicles at high SDS concentration. The steady shear measurement in Figure 7 indicates that the apparent viscosities increase with the addition of SDS. Figure 7 also demonstrates that the applied shear induces to phase transitions from the bilayer structure (LRh) to the vesicles (LRl) with increasing shear rate.33 Their characteristic slopes of shear thinning manifest the phase transition points. Furthermore, one can observe a plateau range between LRh and LRl,MLV, which corresponds to the coexistence regime of their structures or transformation regimes. The solutions at lower SDS concentration (2 and 4 mM) seem to follow the above transformation. But at higher SDS concentrations, however, the plateau region does not appear because the solutions consist already of the vesicles. The difference between the SDS concentrations demonstrates the structure change with the increase of charge density. 3. Effect of Ionic Strength. The electrostatic nature of the effect, caused by the added ionic surfactant, may be demonstrated by changing the ionic strength of the solution by adding inert electrolyte. We choose NaCl because of the common counterion with the surfactant counterion. After addition of NaCl to the above vesicle phase (5 wt % LA 070, 240 mM EHG, 10 mM SDS), the birefringence pattern for samples changes. Figure 8 shows the pictures in between polarizers. It is clear that, after the addition of NaCl, different birefringence patterns are observed. However, even at the 15 mM NaCl, the pattern has not shifted back to the original pattern (Figure 8a) of the sample without SDS. Figure 9 shows the results of oscillation measurement for the above solutions. In the presence of NaCl at low frequencies the storage modulus is smaller than the loss modulus. At a characteristic frequency from which the structural relaxation time can be derived, both moduli cross each other, and above this frequency the storage modulus reaches a plateau value. In the shown case, only one relaxation is found. The rheograms have (33) Escalante, J. I.; Gradzielski, M.; Hoffmann, H.; Mortensen, K. Langmuir 2000, 16, 8653.

Figure 9. Dynamic rheological measurements for the 5 wt % LA 070, 240 mM EHG, and 10 mM SDS system at different NaCl concentrations at 25 °C. NaCl concentrations: (a) 10 mM; (b) 15 mM.

the typical character of Maxwell fluids. The rheograms are different from the oscillation measurement in Figure 3d, but is similar to the result in Figure 3a. After addition of NaCl to the vesicle phase (5 wt % LA 070, 240 mM EHG, 10 mM SDS), both moduli decrease about 10 times and no yield stress is observed any more. This result shows that the behavior of the viscoelastic LR phase has been shifted back, by the addition of inert electrolyte, to the behavior of an uncharged LR phase. If the charge density on the vesicles is shielded by excess salt, the storage modulus decreases rapidly with the ionic strength, as demonstrated in Figure 10. Figure 10 shows the change in the storage modulus G′ with NaCl concentration in the LA 070/SDS/EHG/water system at 25 °C, where the surfactants concentration and the cosurfactant are fixed at respectively 5 wt %, 10 mM, and 240 mM. The modulus decreases with NaCl concentration. The manner of G′ decay with the salt concentration is coincident with the previous result. The ionic atmosphere around the aggregates is indeed a main contribution to the decreasing G′. The addition of inert electrolyte can be rationalized as a

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Figure 10. Storage modulus G′ versus frequency of the 5 wt % LA 070, 240 mM EHG, 10 mM SDS system at different NaCl concentrations. Figure 12. Phase behavior of 5 wt % LA 070, 140 mM EHG system with the addition of SDS (mM). The numbers on the photos are the SDS concentrations. Upper photos, in between polarizers; Lower, without polarizers. Table 2. SDS Effect on the Upper Phase of the Two-Phase Solution of the 5 wt % LA 070, 140 mM EHG System SDS SDS concentration D ( 0.1 concentration D ( 0.1 (mM) (nm) (mM) (nm) samples samples 1 2 3 4

Figure 11. Complex viscosities versus frequency for 5 wt % LA 070, 240 mM EHG systems.

partial screening of the electrostatic interactions of the ionic head groups within the planes.34 The diffuse electric double layer is compressed by the added salt, and as a consequence, the longrange repulsion force decreases with increasing salt concentration. Therefore, the addition of electrolyte leads to the higher flexibility while an increase of the charge density reduces the flexibility. Changes in the complex viscosities with frequency for different salt concentrations are shown in Figure 11. From Figure 11 it can be seen that the complex viscosity decreases with the addition of salt. Without the addition of NaCl, the vesicle system of 5 wt % LA 070, 240 mM EHG, 10 mM SDS shows the typical shearthinning behavior. While with the addition of NaCl, i.e., in the 10 mM NaCl system, a slight shear thickening is observed which is then followed by shear thinning. This is similar to the system of 5 wt % LA 070, 240 mM EHG without the ionic surfactant SDS. So the addition of NaCl can shift the behavior of the vesicle phase back to the behavior of the bilayer phase. 4. Influence of Charges on the LR/L1 Phase. Figure 12 depicts the charge effect to the two-phase region (LR/L1 phase). It is found that the volume of upper LR phase is swelling and then contracting with the addition of SDS. When the SDS concentration is lower than 1 mM, the upper LR phase swells. However, with further SDS addition, the upper LR phases contract. (34) Hoffmann, H.; Ulbricht, W. Recent Res. Phys. Chem. 1998, 2, 113.

0 0.5 1 2

24.0 35.4 35.5 32.6

5 6 7 8

4 6 8 10

28.1 25.5 22.7 22.0

The effects of SDS observed on the basis of the “macroscopic” information on the general phase behavior are confirmed from the SAXS data. The results for the lattice parameter are summarized in Table 2. From Table 2, it can be seen that SDS first increases and then decreases the lattice spacing with increasing SDS content. Therefore, at low SDS concentration (CSDS < 1 mM) there is mainly the swelling effect and the interlamellar distance D is given by the fluctuation of the bilayer. According to DLVO theory, in the LR phase of the two-phase region, the bilayers have a well-defined spacing which is determined by Van der Waals’s attractive forces and undulation repulsive force. The repulsion force increases on addition of charge, but a small amount of SDS increases the repulsion between the bilayers without any change in its structure. Hence, the LR phase has to swell and thus the interlamellar distance D becomes larger than that without SDS. With a further increase of SDS concentration, the electrostatic repulsion between the bilayers increases but also the ionic strength of the solution increases. Therefore, the bilayers stiffen and the interlamellar distance decreases. Hence, in the two-phase situation we observe the same contraction as in the single lamellar phase, but also the additional swelling of the LR phase. This kind of contraction seems to indicate that there is some attraction between similarly charged bilayers. Such effects have been discussed by Ise and co-worker35,36 on latex dispersions. Our situation is more complicated, however, because the free SDS concentration in the solvent is also increasing, which leads to more shielding. More data are needed to understand the situation in a quantitative way. (35) Ise, N.; Okubo, T. J. Phys. Chem. 1966, 70, 1930. (36) Ise, N. AdV. Polym. Sci. 1971, 7, 536.

Ionic Charges and Bilayers of Lamellar Phases

Langmuir, Vol. 23, No. 6, 2007 2983 Table 3. Electrical Contribution Kel of the Bending Modulus for the 5 wt % LA 070, 240 mM EHG System as a Function of SDS Concentration SDS concentration (mM)

Kel (kBT)

KG,el (kBT)

2Kel + KG,el

0.5 1 4 6 8 10

0.68 1.04 0.73 0.66 0.61 0.57

-2.88 -2.50 -2.23

-1.56 -1.28 -1.09

Boltzmann equation for the current boundary. Lekkerkerker obtains for the electric contribution Kel of kc37

Kel ) Figure 13. Shear modulus G′ versus SDS concentrations of the 5 wt % LA 070, 240 mM EHG system and different SDS concentrations.

5. Theoretical Approach for Understanding the Shear Modulus. 5.1. Modulus and Bending Elasticity. If the lamellar phase is made of ionic surfactants or a mixture of nonionic and ionic surfactants, the repulsive Coulombic interactions contribute to the stability of the structure. It is generally known that when electric charges are placed on the bilayers, the bilayers become stiffer and the undulating modes are suppressed. As a consequence of the stiffening the surface area increases and the interlamellar spacing becomes smaller. The stiffening of the bilayers can be quantitatively understood on the basis of the equation for the elastic energy which has been given by Helfrich8

E ) 1/2kc

∫(C1 + C2 - 2C0)2 dA + kG ∫C1C2 dA

(2)

in which kc is the bending constant, kG is the Gaussian bending constant, C1 and C2 are the principle curvature, and C0 is the spontaneous curvature. Both bending constants increase with the charge density. It has been shown by Lekkerkerker that the Gaussian constant increases faster than kc and the L3 phase and the LRh phase become unstable with respect to the vesicle phase.37 With undulations no longer possible as a result of the stiffening, the system tries other ways of lowering its free energy. This is possible by forming many vesicles from a single bilayer. Obviously, the bending energy of the vesicles is overcompensated by the gain in the entropy. The bending moduli kc and kG in eq 2 consist of at least two contributions: an electric one due to the charge of the bilayers and an intrinsic one. The intrinsic one is due to the interactions of the hydrophobic tails of the surfactant molecules. Both the electric and the intrinsic contribution are reflected by the charge and surfactant concentration dependence of the shear modulus (Figure 13). In Figure 13, the shear modulus increase with the addition of SDS because of the increase of both charges and the total surfactant concentration. This phenomenon is similar to the previous result in the tetradecyl-dimethylaminoxide (C14DMAO)/ SDS system: the moduli G′ were found to increase with increasing charge density and the total surfactant concentration.30 According to Lekkerkerker, the electric contribution of the bending moduli can be calculated by evaluating the mechanical moments resulting from the transverse pressure Profile Π(z). This pressure is provoked by the electric double layer of the lamellae and can be determined by resolving the Poisson(37) Lekkerkerker, H. N. W. Physica A 1989, 159, 319.

kBT (q - 1)(q + 2) 2πQL (q + 1)q

(3)

with

q ) xP2 + 1

(4)

p ) 2πQ|σ|/Le

(5)

and

kB is Boltzmann’s constant, T the absolute temperature, Q the Bjerrum length, L the reciprocal Debye length, and σ the surface charge density. According to eq 3, the dependence of the electric contribution of kc is calculated as a function of the surface charge density σ. This relationship is represented in Table 3. The surface charge density is proportional to the plotted charging degree of the lamellar, i.e., in our case the fraction of the ionic surfactant. The chosen parameters for Table 3 refer to a 5 wt % LA 070 solution, using an area per EO of 8.05 Å2. The bending modulus first increases very steeply at low charge density and then decreases at higher charge densities. This decrease is understandable taking into account that the concentration of counterions grows with increasing charge density and, then therefore, the reciprocal Debye length L in the denominator of eq 3 enlarges. This leads to a reduction of the bending modulus Kel. So the dependence of the electrical contribution of the bending does not agree with that of the shear modulus on the charge density. According to Lekkerkerker, at high surface charge densities and low salt concentrations p = q > 1, the Gaussian modulus can be calculated by the following expression:37

KG,el ) -πkBT/6Qk

(6)

Assuming complete ionization of the SDS, we present values for KG,el in Table 3. At higher SDS concentrations, the KG,el decreased from 2.88 to 2.23 with the SDS concentration. This decrease is also due to the increase of the concentration of counterions when the SDS concentration increases. As discussed by Helfrich and co-workers,1,38 if (2Kel + KG,el) < 0, it might have important implications for spontaneous vesiculation. So from Table 3, it is obvious that vesicles exist at high SDS concentration. In spite of the discrepancy between the charge dependence of the bending modulus and the shear modulus, it seems possible to understand the transition to the vesicle phase by this model. 5.2. Comparison of the Shear Modulus and the Osmotic Pressure. The experimental results demonstrate that the shear (38) Winterhalter, N.; Helfrich, W. J. Phys. Chem. 1988, 92, 6865.

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Table 4. Osmotic Pressure for the 5 wt % LA 070, 240 mM EHG System as a Function of SDS Concentration SDS concentration (mM)

Π × 103 (Pa)

SDS concentration (mM)

Π × 103 (Pa)

4 6

7.6 8.1

8 10

8.9 9.8

modulus is at least to some degree determined by the electrostatic interaction of the charged bilayers. When the charge on the bilayers is shielded by excess salt, the modulus decreases with the salt concentration. It seems, therefore, sensible to explain the value of the modulus on the basis of an electrostatic model. It can be argued that whenever a stress is applied to the solutions, the vesicles are deformed and as a consequence some of the bilayers are pushed closer together. The restoring force for the compression of the system would be given by the osmotic compression modulus. The compression modulus is simply given by the osmotic pressure of the system as has been shown in several experimental investigations.39 We have calculated the osmotic pressure from the equation

Π ) CmkBT

(7)

where Cm is the concentration of ions at the midplane between the bilayers calculated by resolving the Poisson-Boltzmann equation.39 We compared this pressure with the experimental modulus for different charge densities. The results in Table 4 show that the calculated osmotic pressure is, however, much higher (of the order of several thousand Pa) than the measured modulus. This results means that the shear modulus cannot be determined by the osmotic pressure alone. 5.3. Theoretical Model for the Shear Modulus by Van der Linden. van der Linden assumes that vesicles (droplets) are deformed in shear flow from a spherical to an elliptical shape.40,41 When the closed shells are deformed, their energy is changed because both their curvature and the interlamellar distance D is changed. Owing to the interaction of the bilayers, expressed by the bulk compressibility modulus B, the inner shells are deformed in a way that the total deformation energy E of the lamellar droplet is minimized. Assuming that the volume of a droplet is not modified by the deformation, the surface A has consequently increased. It is therefore possible to define an effective surface tension σeff ) E/∆A. van der Linden obtained

σeff ) 1/2(KB)1/2

(8)

where K is the bulk rigidity, which is related to the bilayer’s bending constant k by K ) k/D. We can relate this effective surface tension with the shear modulus of G of a vesicle of radius R. Using the identity G ) 2σeff/R yields

k 1/2 B (KB)1/2 D G) ) R R

( )

(9)

(39) Dubois, M.; Zemb, T.; Belloni, L. J. Chem. Phys. 1992, 96, 2278. (40) van der Linden, E.; DrO ¨ ge, J. H. M. Physica A 1993, 193, 439. (41) Hoffmann, H.; Thunig, C.; Schmiedel, P.; Munkert, U. NuoVo Cimento 1994, 16D, 1373.

Both the bulk compressibility modulus and the bending constant depend on the charge density of the bilayers and the shielding of the charges with excess salt. Strictly speaking, the van der Linden theory results in a calculation of the geometrical average of the compression (EB) and bending energy (EK) per unit volume. Equation 9 can be squared to

k B ( D ) G ) 2

R2

For the bending constant as a function of the charge density we can use the expression given by Lekkerkerker (eq 3). For the compression modulus B, its quantity may be simply given by the osmotic pressure between the bilayers. So now we would like to check whether eq 9 for the modulus gives reasonable results combined with our primary data. We assume the following values: interlamellar distance, D ) 22 nm; radius of a vesicle, R ) 0.2 µm; charging degree, l0%, i.e., a surface charge density on the bilayers of e/480 Å2; electrical contribution of the bending modulus calculated according to eq 3, k ) 0.57kBT; and compression modulus calculated according to eq 7, B ) Π ) 9800 Pa. With these data eq 9 yields G ≈ 144 Pa for the shear modulus. This is almost 3 times higher than the experimental value (55 Pa) and implies that a correct approach to a theoretical description of the solutions’ rheological properties has to be found. At present it is therefore not possible to give a complete quantitative interpretation of the experimental results and fit the data precisely on an exact theoretical model.

Conclusions The effect of added SDS and NaCl on the concentrated system of a polyoxyethylene-type nonionic surfactant LA 070 was studied by means of phase behavior, rheology, and small-angle X-ray scattering. The aim of this study was to further elucidate the undulating nature of the bilayers and the charge effect to the stability of lamellar phases. From the two-phase region LR/L1, the one-dimensional swelling behavior measured of the upper LR phase demonstrates the dilute lamellar nature of these phases. The undulation concept of uncharged bilayers was tested by weakly charging up the bilayers. The response to the rather small addition of ionic surfactants proves that these phases are indeed undulation-stabilized. On the other hand, if electrostatic interactions exist, these will compete with undulations. So with more SDS, the upper LR phase contracts. The observation of a transition from single lamellar phase to an isotropic vesicle phase upon increasing addition of ionic surfactant suggests the charge contribution to the bilayers. An increase of the charge density reduces the flexibility of the bilayers and then changes the lamellar phase into vesicle phase. About the above vesicle phase, they are probably formed by shear and mixing during the sample preparation. When the bilayers are more and more charged, the bilayers become stiffer and the vesicles, formed during the sample preparation, can no longer relax back to the LR phase. Acknowledgment. A. Zou gratefully acknowledges the support of this work by the Alexander von Humboldt Foundation. The authors furthermore would like to thank the Clariant company for the gift of the sample Genapol LA 070. LA063148Q