Langmuir 1994,10, 3972-3981
3972
Surfactant Systems with Charged Multilamellar Vesicles and Their Rheological Properties H. Hoffmann,* C. Thunig, P. Schmiedel, and U. Munkert Universitat Bayreuth, Physikalische Chemie I, 0-95440 Bayreuth, Germany Received April 5, 1994. I n Final Form: August 8, 1994@ A general method is presented for the preparation of a viscoelastic surfactant phase that consists of densely packed multilamellar vesicles in water. The vesicle phase forms spontaneously when ionic surfactants are added to a dilute La-or L3-phase, the bilayers of which consist of mixed uncharged singlechain surfactants and cosurfactants. The investigated phases were prepared from alkyldimethylaminoxides (C,DMAO), n-alcohols (c6-c9), and the ionic surfactant tetradecyltrimethylammonium bromide (c14TMABr)or sodiumdodecyl sulfate (SDS). The structure ofthe vesicles and their dimensionswere determined from freeze-fracture electron micrographs (FF-TEM). For a 100mM surfactant solution the multilamellar vesicles had a diameter in the range of 1pm and an interlamellar spacing of around 800 A. For these conditionsthe vesicles are denselypacked and cannot pass each other. The vesicle phase is highly viscoelastic and has a yield stress value. The viscoelastic properties of the phase were determined from oscillating rheological measurements. The storage modulus was about 1 order of magnitude larger than the loss modulus and was independent offrequency. The moduli were determined as a function ofthe concentration and chain length of the surfactant and cosurfactant, the charge density and ionic strength, the amount of solubilization of hydrocarbon, and the temperature. For a constant charge density the yield stress values and shear moduli increase with the surfactant concentration according to a linear relation G = (co - c,) where co is the total surfactant concentration and ce the surfactant concentration for dense packing of the vesicles. For constant surfactant concentration the moduli increase in an S-shaped form with the charge density and reach saturation for a mole fraction of about 7% ofionic surfactant. The storage moduli and yield values decrease with the addition of excess salt. The storage moduli depend strongly on the chain length of the surfactant. Theoretical calculations show that the shear moduli of the phases are much smaller than the osmotic pressure of the systems. Several models are proposed for the explanation of the shear moduli. The values of the moduli can best be understood on the basis of a hard-sphere model in which the multilamellar vesicles are treated as hard-sphere particles. 1. Introduction It has been known for some time that phospholipids can form uni- and multilamellar vesicles.' The size of these vesicles depends on the method of preparation. One way of preparing vesicles is by sonication of dispersions of phospholipids.2 Vesicles can also be formed by surfactants. Several systems are known in which vesicles have been observed. Kaler e t al. have shown that they are formed when cationic and anionic surfactants are mixed t ~ g e t h e r .Zemb ~ et al. have observed vesicles in dilute h-phases of didodecyldimethylammoniumbromide (DDABr).4 Hoffmann et al. finally have shown very recently that vesicles can be prepared from zwitterionic surfactants (alkyldimethylaminoxides (C,DMAO)) and n-alcohols as cosurfactant^.^ These vesicular solutions can have very interesting rheological properties if the vesicles are charged with ionic surfactants like tetradecyltrimethylammonium bromide (CI4TMAl3r)or sodium dodecyl sulfate (SDS) and the total concentration of surfactant is large enough. It was shown that such systems can even have a yield value when the surfactant concentration is as low as about 2%.5 Such systems are of considerable practical interest because they can be used to suspend oil droplets or solid Abstract publishedinAduanceACSAbstracts, October 1,1994. (1)Hauser, H.; Gains, N.; Muller, M. Biochemistry 1983,22,4775. (2)Hammond, K.;Lyle, J. G.; Jones, M. N. Colloids Surf. 1987,23(3), 241-257. (3)Kaler, E.W . ;Herrington, K. L.; Murthy, A.;Zasadzinski, J. A. N. J. Phys. Chem. 1992,96,6698. (4) (a)Fontell, K.; Ceglie, A.; Lindman, B.; Ninham, B. Acta Chem. Scand. 1986,A40, 247. (b) Zemb, T.; Gazeau, D.; Dubois, M.; GulileKvzywicki, T. Euro. Phys. Lett. 1993,21(7), 759. ( 5 ) (a)Munkert,U.; Hoffmann, H.; Thunig, C . ;Meyer, H. W . ;Richter, W.Progr.ColloidPo1ym.Sci. 1993,93,137.(b)Hohann,H.;Munkert, U.;Thunig, C . ; Valiente, M. J. Colloid Interface Sci. 1994,163,217228. @
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particles which have dimensions of a few tenths of a millimeter. In spite of their yield value and viscoelastic properties, the fluids can be handled with ease because the elastic properties are so low that the fluids cannot withhold gravitational forces resulting from larger bulk forcesU6The fluids, thus, flow easily and can be pumped without problems, and interfaces of the fluids adjust quickly to a horizontal position. It is likely that the microstructure of these fluids is similar to the microstructure in suspending liquids which have been used in the past for the preparation of cosmetic or pharmaceutical preparation^.^ In these systems large volume fractions of cetyl and stearyl alcohol are mixed with surfactant solutions. Because under these conditions the surfactant cannot quantitatively solubilize the alcohol, some of the alcohol is dispersed as large emulsion droplets, and the systems are, therefore, milky white. The systems which will be described in this article are perfectly transparent. The suspending property might come, however, from a network of multilamellar vesicles. Such structures have been made visible in these systems by freeze-fracture electron micrographs.* 2. Experimental Section 2.1 Samples. The surfactants C,DMAO ( x = 10-18) were a gift of the Hoechst AG Gendorf. They were used aRer purification by recrystallization twice from acetone. The surfactants were characterized by their melting points and cmc values. Tetradecyltrimethylammoniumbromide (CI4TMABr)was from Aldrich and was recrystallized from ethedethanol. The anionic surfactant SDS (Serva)and Cl& (Nikko Chemicals)were used without purification. The n-alcohols C,OH (x = 6-9) and decane were (6) Hoffmann, H.; Rauscher, A. Colloid Polym. Sci. 1993,271,390. (7)Barry,B.W . ;Sanders, G . M. J . Colloid Interface Sci. 1970,34, 300. ( 8 )Fukushima, S.;Yamaguchi, M. Cosmet. Toiletries 1983,98,89.
0 1994 American Chemical Society
Langmuir, Vol. 10, No. 11, 1994 3973
Surfactant Systems with Multilamellar Vesicles from Fluka (p.a. quality) and were used without further purification. The double-chain surfactant didodecyldimethylammonium bromide (DDABr)was a commercial product. 2.2 Phase Diagrams. The phase diagrams were established by observing the mixtures in calibrated test tubes under temperature-controlled conditions for several weeks. 2.3 Electron Microscopy. For the freeze-fracturetransmission electron microscopy (FF-TEM)a small amount ofthesample was placed on a 0.1mm thick copper disk covered with a second cooper disk. The sample was frozen by plunging this sandwich into liquid propane, which was cooled by liquid nitrogen. Fracturing and replication were carried out in a freeze-fracture apparatus Bioech 2005 (Leybold-Heraeus, Germany) at a temperature of -100 "C. WC was deposited under an angle of 45". The replicas were examined in a CEM 902 electron microscope (Zeiss, Germany). 2.4 Rheological Measurements. The rheological measurements were performed with a Bohlin CS 10 stress-controlled rheometer. Both a cone plate measuring system and a doublegap system were used. The double gap is applicable for very low viscous liquids and suitable for the detection of very low yield stresses. The lowest possible stress value amounts to 3 mPa. The range ofthe cone plate system starts at 60mPa. This means that it is applicablefor most measurements on the vesicle phases. The very high angular resolution ofthe CS 10 and 1.3prad allows the detection ofthe smallest elastic deformationsas well as very slow viscous flow. The viscoelastic properties, i.e. the dynamic modulus and the magnitude of the complex viscosity,are determinedby oscillatory measurements from 0.001to 10 Hz, whereby alternatively the strain amplitude or the stress amplitude can be kept constant. The latter method can particularly be used on liquids with a yield stress in order not to exceed this value. Further details of the measuring procedure are described in paragraph 3.3.
2100 0
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3. Results and Discussion 3.1 The Phase Diagrams. Most of the rheological results on vesicular systems that will be discussed in this investigation have been prepared from mixtures of singlechain surfactants and cosurfactants. It is generally known that the mean spontaneous curvature on micellar interfaces is continuously lowered when cosurfactants are mixed with single-chain surfactant^.^ The reason for this change in curvature is due to the small area which a cosurfactant occupies a t a micellar interface. As a consequence of this continuous change in the curvature, the systems undergo several phase transitions with increasing cosurfactant concentration.1° The systems try to come as close as possible to the spontaneous mean curvature without causing much bending energy by adjusting the two principle curvatures on the micellar aggregates. By doing this, the systems have to switch the micellar structures from spheres to rods to bilayers and to the L3-phasewith increasing cosurfactant concentration. These changes in the phases and their properties have recently been systematically studied in detail for the alkyldimethylaminoxides and several cosurfactants.11J2 A phase diagram for one dilute system is given in Figure 1. It was found that the La-phase which is first reached with increasing cosurfactant concentration consists of multilamellar vesicles. The moduli for systems which are not charged with ionic surfactants are usually not very high. The viscoelastic properties are, therefore, not very strong, and the systems have no yield value. When the bilayers are charged by adding ionic surfactants to the uncharged systems, the phase diagram is somewhat simpler because some of the mesophases are suppressed ~
~~
(9)Hoffmann, H.Progr. Colloid Polym. Sci. 1990,83,16. (10)Platz,G.;Thunig,C.; Hoffmann, H. Ber.Bunsen-Ges.Phys. Chem. 1992,96,667. (11)Hoffmann, H.; Thunig, C.; Munkert, U.; Meyer, H.W.; Richter, W.Langmuir 1992,8,2629. (12)Valiente, M.; Munkert,U.; Lenz, U.; Hoffmann, H.; Thunig, C. J . Colloid Interface Sci. 1993,160, 39.
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Figure 2. Cut of the quarternary phase diagram C14DMAOI C~~TMAB~/CSOWH~O at 25 "C. The total surfactant concentration is 100 mM. (Details in ref 5b.)
by the influence of charge.sb A cut through such a phase diagram for a total surfactant concentration of 100 mM is shown in Figure 2. The most general result of the replacement of some of the uncharged by the charged surfactant is the shift of the phase boundaries of the LIphase and the La-phase to higher cosurfactant concentrations. The presented phase diagrams are typical for all surfactants. Strey et al. have recently shown that the nonionic alkyl polyglycol surfactants behave in the same way when cosurfactants are added.13 Rheological data for such a system when it is charged will also be given in this investigation. Some surfactants show already bilayer-type structures without the presence of cosurfactants. Micelles of bilayertype structures are generally formed from double-chain surfactants.14J5 The two chains do not have to have the ~
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(13)Jonstromer, M.; Strey, R. J. Phys. Chem. 1992,96,5993. (14)Miller, D.D.; Magid, L. J.; Evans, D. F. J . Phys. Chem. 1990, 94 (15),5921.
3974 Langmuir, Vol. 10, No. 11, 1994
Hoffmann et al.
Figure 3. Electron micrograph of the system C14DMAO/C14TMABr/CsOH. The total surfactant concentration is 100 mM with a surfactant ratio of 9:l;the concentration of C60H is 220 mM. The bar represents 1 pm.
Figure 4. Electron micrograph of a 3 wt % DDABr in water solution. The bar represents 1 pm.
same length. For some systems bilayer structures are already formed when one chain of the surfactant is only half as long as the main chain.16 One ionic double-chain system which has been studied in detail by many groups (15) Kunieda, H.; Shinoda, K. J . Phys. Chem. 1978,82, 1710. (16) Neubauer, G.; Hoffmann, H.; K&s, J.; Schwandner, B. Chem. Phys. 1986,110,247.
is didodecyldimethylammonium bromide. It has been shown by several groups that this system also forms vesicular structures in the dilute range.17 3.2 Freeze-Fracture Transmission Electron Microscopy. We prepared FF-TEM micrographs from all of the investigated systems. In Figure 3 a TEM micro(17) Dubois, M.; Zemb, Th. Langmuir 1991,7(7),1352.
Surfactant Systems with Multilamellar Vesicles
Langmuir, Vol. 10, No. 11, 1994 3975
Figure 5. Tilted samples of a solution containing 100 mM C12E6 and SDS in the ratio 9:l between crossed polarizers. The concentration of C6OH was 250 mM.
graph of a vesicular system is shown which was prepared by the freeze-fracture method for a system which consists of 90 mM tetradecyldimethylaminoxide (C14DMAO), 10 mM tetradecyltrimethylammonium bromide (C14TMAEh-1, and 200 mM hexanol (CGOH). The cationic surfactant in the system can also be replaced by the anionic surfactant SDS without causing a change of the rheological properties. The systems are transparent and have a weak birefringence if the solutions are not under strain. The electron micrograph shows several features which are of relevance for the properties of the systems. The vesicles have a rather large polydispersity. Some vesicles seem to consist of a single shell while others consist of at least 10 bilayers. The interlamellar spacing between the bilayers is rather uniform and is in the range of 800 A. The vesicles are very densely packed, and the whole volume of the system is completely filled with the vesicles. This obviouslymeans that there is no free volume available
and the system is in a single phase situation. Note also that the vesicles are of spherical shape even though the outermost shells can have a radius of several thousand angstroms. One multilamellar vesicle can thus be considered as a small volume element of a La-phase. Note also that some vesicles do not consist of concentric shells but do have defects. The typical size of the larger vesicles is about 1pm, and the wedges which result from the dense packing of these vesicles are filled with smaller vesicles. Each vesicle is completely surrounded by other vesicles and is sitting in a cage from which it cannot escape by a simple diffusion process without deformation of its shells. The systems must, therefore, have viscoelastic properties under deformation. A micrograph for a 3%DDABr system is shown in Figure 4. According to published phase diagrams of this system, a 3% solution is in the single La-phase region. In comparison to that for the multilamellar vesicles from
3976 Langmuir, Vol. 10, No. 11, 1994
Hoffmann et al.
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Figure 6. Oscillatory rheogram of a solution of 90 mM c14DMAO, 10 mM C14TMAl3r, and 220 mM CsOH. The moduli are almost independent of the frequency; G is 1 order of magnitude larger than G and does not vanish for low frequencies. This indicates a yield stress. the mixed surfactantkosurfactant system, the micrograph shows a characteristic difference. For the DDABr system the micrograph shows that all the vesicles fracture in the midplane of the bilayer while for the other systems the fracture is perpendicular through the multilamellar vesicles and shows all the bilayers of the vesicles. The fracture in the DDABr system occurs in the outermost shell of the onion-type vesicles. The vesicles look, therefore, like unilamellar vesicles, and their multilamellar character is not obvious. Contrary to the vesicles which consist of phospholipids, the vesicles from surfactants form spontaneously in the aqueous solution when the components are mixed in the right proportion. The systems may, however, take several days to reach equilibrium. Under small deformations the viscoelastic solutions become strongly birefringent. This is demonstrated in Figure 5 where samples are tilted between crossed polarizers. Note that the birefringence of the sample is reminiscent of the birefringence that occurs if a piece of plexiglass is bent and viewed between cross polarizers. We can see all the colors of the visible spectrum. The areas of the different colors run in broad bands from one side of the cuvette to the other side. The width of the cuvette is 2 cm. A particular color indicates that the stress in the area, showing the same color, is the same. The birefringence is, therefore, very different from the birefringence that is shown in samples with liquid crystalline phases. These samples show patterns with a domainlike structure, and the domains are usually small ( ~ 0 .mm) 1 and independent of the deformation. They are caused by the orientation of the liquid crystal domains and their intrinsic birefringence. In the samples shown the birefringence is correlated over the whole sample over a distance of 2 cm. In accordance with the birefringence of plexiglass, we would like to call it stress birefringence. 3.3 Viscoelastic Properties. 3.3.1 Concentration Dependence. The viscoelastic properties of the systems are demonstrated in Figure 6 and Figure 7. In Figure 6 the storage modulus, the loss modulus, and the magnitude of the complex viscosity are plotted against the oscillating frequency in a double log plot. The storage modulus is much larger than the loss modulus over the whole frequency range and is little dependent on frequency. The systems, thus, behave like a soft solid material. The systems actually have a real yield stress value that is shown in Figure 7. To determine the yield value, we measure the deformation after different times when the systems are exposed to increasing shear stresses. If the system responds only elastically, the deformation depends only on the shear stress and not on the time the stress is applied. Figure 7 shows that the systems indeed do have a yield value. We also made measurements in which the shear stress was recorded for increasing shear rate y . In these
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experiments the samples show flow behavior of a Bingham fluid (Figure 8). The apparent yield stress value can be extrapolated from the shear stress at y = 0. It agreed with the yield value which was determined according to the method demonstrated in Figure 7. The shear modulus and the yield value depend on the total surfactant concentration as demonstrated in Figure 9. There, it is also shown that the modulus and the yield value disappear for concentrations below 1%surfactant. For these concentrations the vesicles are no longer densely packed, and they can easily move around each other under shear flow. The yield value varies linearly with the modulus and is about 10 times smaller than the modulus. This means that when the vesicles are deformed about 10 percent, they can then pass each other under shear. In Figure 10 rheological results are shown for the DDABr system. The rheological properties of this binary system are very much the same as for the mixed surfactant systems. The values of the magnitudes of the complex viscosity and the storage and the loss moduli for both systems are in the same range for the same concentrations even though the charge density of the DDABr system is very much higher than in the mixed surfactant system.
Surfactant Systems with Multilamellar Vesicles
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Figure 13. Storage modulus G against the frequency f for different chain lengths x of the aminoxide. One observes an increase for longer chains but a surprisingdecrease for the CIS chain. All systems had the same composition: co = 100 mM surfactant, ratio of CzDMAO/C14TMAJ3r= 9:1,220mM CeOH.
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This surprising result prompted us to study the influence of the charge density on the storage modulus. 3.3.2 Charge Dependence. In the densely packed region the modulus depends strongly on the charge density of the vesicles as demonstrated in Figure 11. With increasing charge density the modulus has a sigmoidal form and saturates already when the charge density of the bilayers corresponds to less than 10%. If the charge density on the vesicles is shielded by excess salt, it decreases rapidly with the ionic strength, as demonstrated in Figure 12 in a plot of the modulus against the square root of the ionic strength. The last two figures seem to indicate that the shear modulus is mainly determined by the electrostatic repulsion between the bilayers. It could be argued that as for other colloidal systems the shear modulus depends linearly on the compression modulus and that the compression modulus would be given by the osmotic pressure of the charged bilayers. If this would be the case and the modulus would be determined only by the charge density of the system, the shear modulus would be independent
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of the chain length of the surfactant. This is, however, not the case. 3.3.3 Chain Length Dependence. The moduli for five different systems in which the chain length was varied is plotted against the frequency in Figure 13. All the systems have the same concentration and composition. The storage modulus increases about a factor 20 between CIOand C16 but then decreases again for CIS. It is, thus, clear that the modulus depends on the thickness of the bilayers, too. One possible reason for this behavior could be that the modulus depends on the bending constant of the bilayers, and the bending constant depends on their thickness. 3.3.4 Dependence on Solubilization. Surfactant solutions can usually solubilize considerable amounts of hydrocarbon.18 The amount of hydrocarbon that can be solubilized by surfactant depends on the aggregated state of the surfactants. Globular micelles have a low solubilization capacity while rod and bilayer aggregates have a higher capacity. It was expected that the hydrocarbon would solubilize in the interior ofthe vesicles between the monolayers. This being the case, one would expect that the bilayers would become thicker with the solubilized amount of hydrocarbon but the interlamellar spacing should not change very much and the modulus could, therefore, be expected not to change very much. This is, however, not the case as is demonstrated in Figure 14 in a plot of the modulus against the frequency for different hydrocarbon concentrations. The data show an increase of the modulus with solubilization. This result probably indicates that the total bilayer area is increasing and the interlamellar spacing is, therefore, decreasing with the solubilization. It is, therefore, likely that the hydrocarbon is solubilizedinto the palisade layer and is oriented parallel to the surfactant chains. It is conceivable that a hydrocarbon molecule in combination with a cosurfactant molecule acts as a (18)Miller, C.A.; Ghosh, 0.; Benton, W.J.Colloids Surf. 1986,19, 197.
Hoffmannet al.
3978 Langmuir, Vol. 10, No. 11, 1994
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surfactant molecule. Together the two molecules have about the same chain length as a surfactant molecule, and they could, thus, mimic a surfactant molecule in the bilayer. Note also that the results of Figure 14 were obtained with a system in which the bilayers of C14DMAO and CGOH were charged by adding HC1 to the system. The charging up of the bilayers in this way is possible because the aminoxide group has a weak basicity. Practically all the hydrogen ions bind to the oxygen of the surfactant molecules, and the solution remains in the neutral pH range. 3.3.5 Mixed Systems. The method of preparingvesicular systems with a yield value by adding cosurfactants to micellar solutions in the L1-phase is a very general one. We have used this method for the preparation of many different systems, and we always obtained systems with similar properties. Some ofthese results will be presented here to demonstrate the general validity of the method. In Figure 15 rheological results are shown for the system ClzEdSDS hexanol. The total surfactant concentration was again 100 mM, and the mixing ratio was 9:l. The surfactant mixture is somewhat more hydrophilic than the C14DMAO/C14TMA13rmixture, and more hexanol(250 mM) is needed to reach the La-phase. The results in Figure 15 look very much the same as in Figure 6. Again we find that G > G ' over the whole frequency range. This system can also be mixed in the whole mixing ratio with a system which consists of C14DMAO/SDSand hexanol. As shown in Figure 16,the G values vary somewhat with the mixing ratio, but the total changes are small, and the systems always look the same. The existence range of the charged vesicles is not very sensitive to the cosurfactant'surfactant ratio. We varied this ratio and measured the rheological properties. The results are shown in Figure 17. The
+
modulus is increasing with the cosurfactant concentration; the increase is, however, relatively modest in comparison to the strong dependence of G on the surfactant concentration. I t is likely that the increase of the cosurfactant concentration increases the total bilayer interface and hence decreases the interlamellar spacing. This effect should increase the modulus. The increase can be compensated to some degree by an increasing flexibility of the bilayers. The chain length of the cosurfactants can also be varied to a large extent without losing the existence region of the vesicles. We varied the cosurfactant from hexanol to decanol and could always observe a vesicles range. The larger the chain length of the cosurfactant, the less cosurfactant is needed for the formation of the vesicles. Some of the obtained rheological results are shown in Figure 18. The results indicate that the modulus is little dependent on the chain length of the cosurfactant. 3.3.6 Temperature Dependence. For a practical application of the systems, the temperature dependence of the rheological properties is of interest. We measured the temperature dependence of one system and observed only a modest change of the modulus with temperature (Figure 19). The system had a yield value between 10 and 60 "C, and the modulus passed over a maximum with T. 3.3.7 Dependence on Shear. The viscosities of many viscoelastic solutions follow the Cox-Merz rule, which means that the shear viscosities as a function ofthe 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.lg This, for instance, is the case (19)Cox, W. P.;M e n , E.H. J.Polym. Sci. 1968,28, 619.
Surfactant Systems with Multilamellar Vesicles
Langmuir, Vol.10,No.11, 1994 3979
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Figure 20. Comparison of the magnitude of the complex viscosity Iv*l(o)as a function of the angular frequency and the shear viscosity ~ ( pas) a function of the shear rate. for viscoelastic solutions of entangledthreadlike micelles.20 The vesicle solutions do not show this simple behavior, as shown in Figure 20 where the shear viscosities and the complex viscosities for two different systems are plotted against the shear rate and the angular frequency, respectively. Both viscosities are the same for low frequencies. For higher shear rates ~ ( p=.) I~*l(o = y), the C14DMAO/C14TMABrsystem shows two breaks in the plot. It is likely that for shear rates which are larger than the critical values the multilamellar vesicles undergo transformations to new structures. Such transitions have been proposed by D. R o w in vesicular systems of a different composition.21
4. Theoretical Model
4.1 Comparison of the Shear Modulus and the Osmotic Pressure. The experimental results demonstrate that the shear 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 multilamellar vesicles are deformed by the stress 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 compression modulus. The compression modulus is simply given by the osmotic pressure of the system as has been shown in several experimental investigations.22 We have calculated the osmotic pressure from the equation (20)(a) Rehage, H.; H o h a n n , H. Mol. Phys. 1991,74,933. (b) Khatory, A,; Lequeux, F.; Kern,F.; Candau, S. J. Langmuir IWS,9, 1456. (c) Shikata, T.;Hirata, H.; Kotaka, T. Langmuir 1987,3,1081. (21)R o n , D.; Nallet, F.; Diat, 0. Europhys. Lett. 1993,24,53. (22)Dubois, M.; Zemb,Th.; Belloni, L. J . Chem. Phys. 1992,96(3), 2278-2286.
(1)
where cm is the number concentration of ions at the midplane between the bilayers calculated by resolving the Poisson-Boltzmann equation for the current conditions. We compared this pressure with the experimental modulus for different charge densities. The results show that the modulus indeed has the same dependence on charge density as the osmotic pressure. The calculated osmotic pressure is, however, much higher than the measured modulus. Even when taken into account that the shear modulus Go should only be a fraction of the compression modulus and the compression and shear modulus should be related by the equation
which includes the Poisson numberp, the shear modulus cannot be explained on the basis of the osmotic pressure alone. That the modulus is not determined by the osmotic pressure is actually already obvious from the result that the modulus depends very much on the chain length of the surfactant and not only on the charge density. 4.2 The Modulus and the BendingElasticity. From a microscopic point ofview it is obvious that the innermost shell in one onion has to be deformed if a vesicle is deformed. The modulus should, therefore, be related to the totalbending energy of the whole system. The bending energy for the vesicles can be calculated from the Helfrich theory from eq 323
(ci, principle curvatures; CO, spontaneous curvature; k,, bending modulus of the bilayer; k,, Gaussian modulus;A, area of the bilayer). According to this expression, each vesicle would have the same total bending energy independent ofita size. When we assume that the spontaneous curvature co is zero, we obtain the result that
+
E = 4n(2kC k,)
(4)
If we assume further that the bending constants are in the range of kT and the modulus Gois determined by the bending energy per unit volume, we obtain a value for Go of a few Pascals that is in the range of the experimental values. The bending constants are generally considered to depend on the thickness of the bilayers d and should theoretically increase with d3.24 Experimentally, we indeed find an increase of Gowith the chain length from 10 to 16. The value for the c18 system is, however, again lower than for the c16 system. This could perhaps mean that the average size of the vesicles is larger for the C18 system than for the C16 system. In agreement with such an assumption is the birefringence of the systems, which is larger for the cl8 system than for the c16 system. With this model the shear modulus would, thus, depend on the total number of vesicles in the system and the bending constants of the bilayers. The total number of vesicles depends also on the bending energy and on the interaction energy. The systems minimize this total energy by adjusting the radius. As already mentioned, at present no theory seems to be available which has treated this problem. In this model the dependence of Go on charge density has to do with the dependence of the bending constant on charge density.25 Increasing charge (23)Helfrich, W.2. Naturforschung 1987,33,305. (24)Harden, J.L.; Marquee, C.; Joanny,J. F.;Andelman, D. Langmuir 1992,8,1170. (25)Lekkerkerker, H. N.W. Physica A 1989,140,319.
3980 Langmuir, Vol. 10, No. 11, 1994 density stiffens out the bilayers and increases through this effect the bending constants. This problem has been dealt with by H. L~!kkerkerker.~~ It thus seems reasonable t o explain the shear modulus on the basis of the bending energy of the system. However, some results render such a model unlikely. We have noticed that with increasing charge density the shear modulus shows saturation when less than 10%of the surfactant molecules are charged. Furthermore, the modulus increases strongly with increasing surfactant concentration. Both effects would be difficult to understand on the basis of the bending energy. It is mainly for these reasons that we would like to rule out this model at present even though it looks very appealing. 4.3 Vesicle Systems in Comparison with High Internal Phase Emulsions. The moduli of high internal phase emulsions (HIPE) are often described by a theory of Princen26from eq 5
20 G0 -R in which u is the interfacial tension of the aqueous surfactant film against a hydrocarbon and R is the average radius of a sphere which still fits into the polyhedra of the foam structures. For the interfacial tension one usualIy uses the value which is measured on a bulk macroscopic interphase. This model also lends itself to a description of the modulus of the investigated systems. The radius R would now again be the radius of the outermost vesicles that build up the dense packed systems, and for u the interfacial tension can be used which can be determined when the vesicular system is contacted with hydrocarbon. These values have been measured and presented in a previous publication.27 They depend strongly on the charge density and increase from to about 0.1 mNIm with increasing charge density. With a value of 0.1 mN/m and R = 0.5 pm, we obtain for Goa value of around 100 Pa. The value which we thus obtain is at least in qualitative agreement with the experimentalresults. Both models which look as the system from a different viewpoint seem to lead to the same result. Ifthis can be theoretically founded, it would mean that the interfacial tension at the macroscopic interface is a direct measure for the bending energy of the vesicles. The bending constants in eq 3 could, thus, be determined from a value of the interfacial tension and the radius of the outermost vesicle which is obtained from FF-TEM micrographs. It is obvious that this model should not be used for a detailed mathematical description of the systems. The HIPE systems consist of continuous films of an aqueous phase suspended in a hydrocarbon phase while the vesicle phase consists of discrete closed bilayers. The morphologies of the two systems are, thus, very different. 4.4 The Hard-Sphere Model. A dispersion of multilamellar vesicles can be compared to a dispersion of hardsphere particles. This is still the case when the vesicles are in a densely packed arrangement, which is actually a La-phase. The situation with the vesicle system is very similar to that in solutions with globular microemulsion droplets. With increasing concentration such systems form a cubic phase when the effective volume fraction of the droplets is in the range of 0.5. For such a situation it was shown that the shear modulus of a cubic phase could quantitatively be accounted for by the theoretical (26)Princen, H. M.; Kiss, A. D. J. Colloid Interface Sci. 1986,112, 427-437. (27)Hoffmann,H. In Organized Solutions, Friberg, S. E., Lindmann, B., Eds.; Marcel Dekker Inc.: New York,Basel, Hong Kong, 1992.
Hoffmann et al. model of hard-sphere particles.28 The compression modulus for such a system is given by
(6) where v is the number density of the particles and S(0) is the structure factor. It is a thermodynamic quantity and can be expressed by the Percus theory. The shear modulus Gois related to the compression modulus through eq 2. The structure factor for globular particles which are in a densely packed state has values in the range of 1x It is likely that the same is true if the particles are multilamellar vesicles which behave like hard-sphere particles. On the basis ofthis model, we would thus obtain values for the shear modulus in the range of =lo Pa. Each multilamellar vesicle in this model would make a contribution to the bulk modulus independent of the number of bilayers of which the vesicles consist. In this context it is of interest to mention that Mellema et al. actually have treated unilamellar vesicles as hard-sphere particles in order to explain rheological data on a dispersion of vesicles.29 In their system the vesicles were prepared from phospholipids. By a multistep preparation procedure the vesicles were obtained with a narrow size distribution. The vesicles were still in the dilute situation, and the system, therefore, had no yield stress value. They could, however, determine the shear modulus, and it turned out that the values were in the range of vkT. The systems had a finite zero-shear viscosity and, therefore, also a finite structural relaxation time. The authors interpreted this time constant as the relaxation time of the particle distribution function. The experimental shear modulus was discussed on the basis of theories which take account of hydrodynamic interaction of many-body particle interaction. In this model we look at the shear modulus as a thermodynamicquantity which in principle can be obtained from the osmotic compressibility which can be determined from light scattering data. In this context it should be stated that the electrostatic interaction of particles is of little influence for dense systems. The scattering results of such systems can be well understood on the basis of hard-sphere interaction only.28 This seems to be in contrast to the experimental results of this investigation, where it was observed that the modulus decreases strongly with the shielding of the vesicles by excess salt. It is likely that the decrease of the modulus with salt is related to two effects: The vesicles no longer behave as hard particles, and the vesicles become larger, and the number density is, therefore, decreasing. Evidence for these conclusions can be seen in the birefringence of the samples. With increasing salt the birefringence becomes stronger, and a typical domain structure develops. With all the experimental evidence taken together, the last model is likely the best model with which the shear modulus of the particles should be interpreted. According to this model, the modulus ofthe cubic phase with globular microemulsion droplets and the La-phase which consists of multilamellar vesicles can be treated in the same way. Both systems have a yield stress value; that is, their structural relaxation times are infinite, and the shear moduli are determined by the number density of the particles and their thermodynamic interaction. While the number density of microemulsion droplets can simply be calculated from the oil surfactant ratio and the total (28)Gradzielski, M.; H o h a n n , H. J.Phys. Chem. 1994,98,26132623.
(29)Smeulders, J.B. A. F.; Blom, C.; Mellema, J.Phys. Reu. A 1990, 42(6),3483-3498.
Surfactant Systems with Multilamellar Vesicles concentration, the number density of the multilamellar vesicles cannot be calculated at present. There seems to be no theory available which calculates the average size of the multilamellar vesicles. This size, therefore, has to be determined from FF-TEMmicrographs. 5. Conclusions It is shown that the dilute La-phase which is formed in mixtures of single-chain uncharged surfactant and cosurfactant solutions and which consists of extended bilayers is transformed into vesicles by the addition of ionic surfactants. Such systems are highly viscoelastic and have a yield value. The yield values and the values of the storage moduli are a result of the dense packing of the multilamellar vesicles. The moduli depend on the
Langmuir, Vol. 10, No. 11, 1994 3981 charge density of the vesicles, the chain length of the surfactants, and the total concentration of surfactant. Systems with a surfactant concentration of about 2% by weight can already be highly viscoelastic and have a yield value which is large enough to suspend oil droplets or solid particles in the system. The rheological properties of the systems can be understood on a model which considers the multilamellar vesicles as hard-sphere particles.
Acknowledgment. This work has been financially supported by the Deutsche Forschungsgemeinschaft (DFG) through the SFB 213. We thank the DFG for the support. We also thank G. Singer for carrying out some of the rheological measurements.
