Langmuir 1996, 12, 4057-4059
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Experimental Study of Undulation Forces in a Nonionic Lamellar Phase H. Bagger-Jo¨rgensen* and U. Olsson Physical Chemistry 1, Chemical Center, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden Received April 30, 1996. In Final Form: June 11, 1996X The osmotic stress technique was used to measure the force between the surfactant bilayers in a nonionic lamellar phase. The bilayer repeat distance, d, was varied in the range 90 < d < 250 Å, corresponding to volume fractions between 0.12 and 0.33. Since the bilayers are uncharged and sufficiently far apart, the only relevant force is the steric undulation force. The obtained relation between force and bilayer separation was compared to Helfrich’s expression for fluctuating membranes, and a good agreement was found in the whole concentration range. Together with the qualitative proof of Helfrich’s expression we also obtained a realistic value of the bending modulus κ ) (2.5 ( 0.5)kBT.
Introduction Interactions between membranes are of great fundamental as well as practical interest. A nice example of a multimembrane system is a lyotropic lamellar phase made of surfactant, possibly cosurfactant, and water.1-3 Microscopically the lamellar phase is a (on average) planar multilayer system where each membrane is made of two surfactant monolayers, oriented with the polar head groups toward the solvent and the hydrocarbon chains toward each other (or vice versa if the solvent is apolar). For uncharged, flexible membranes sufficiently far apart the steric repulsion originally proposed by Helfrich,4 originating from the undulatory motions of the surfactant bilayers and hence normally referred to as the undulation force, is the main stabilizing mechanism. Previous studies have reported on LR phases with an anomalous water swelling, leading to LR phases with inter-bilayer repeat distances of several thousand angstro¨ms.3,5-7 Treating the bilayers as mathematical surfaces within the framework of the flexible surface model and using the curvature energy concept, the local curvature free energy density, gc, of a planar membrane is simply8,9
gc ) 2κH2
(1)
where H is the mean curvature and κ is the bending rigidity modulus. The surface free energy, Gc, of one surface configuration is obtained as the surface (Σ) integral over the local curvature free energy density, i.e.
Gc )
∫Σ gc dA
(2)
It was first recognized by Porte et al.10 that there exists an exact scaling relation for the free energy, resulting from the scale invariance of Gc, giving a cubic scaling between the free energy per unit volume and the bilayer X
volume fraction, i.e., G/V ∼ φ3. The numerical prefactor was found by Helfrich,4 and the final expression reads 2 G φ 1 3π2 (kBT) ) V 128 κ (1 - φ)2 δ
()
()
3
(3)
where kBT is the thermal energy and δ is the bilayer thickness. In this formula is also included a factor (1 φ)-2 that takes the finite membrane thickness into account.1 The experimental studies on undulation forces have up to now been limited to a few experimental techniques, mostly scattering studies and surface force measurements. X-ray and neutron scattering studies on lamellar phases have been performed on both oriented11 and nonoriented12,13 samples. Although difficulties such as finite size effects and nonperfect orientation are inherent in these studies, analysis of the pseudo-Bragg peak shape has shown a good qualitatively agreement with Helfrich’s expression. Interesting are also the dynamic light scattering study performed on oriented samples,14 also supporting eq 3. Additionally, surface force measurements in lamellar phases have been performed using the surface force apparatus. In these experiments a bulk lamellar sample was squeezed between two solid mica surfaces and the force as function of mica surface separation was measured. Several studies have shown a qualitative agreement with Helfrich’s formula,15-17 and the bilayer bending modulus has also been derived.17 Experimental difficulties are however great and the data are difficult to interpret.15 The bending rigidity has also been determined from the quadrupole splitting in NMR experiments.18 The technique we explore in this study is the osmotic stress technique.19 It has previously been successfully used in the study of short range forces in several lipid systems,20-22 colloid dispersions,23 and a balanced micro-
Abstract published in Advance ACS Abstracts, August 1, 1996.
(1) Helfrich, W. J. Phys.: Condens. Matter 1994, 23A, 6, 79. (2) Porte, G. J. Phys.: Condens. Matter 1992, 4, 8649. (3) Strey, R.; Schoma¨cker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253. (4) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (5) Larche, F. C.; Appell, J.; Porte, G.; Bassereau, P.; Marignan, J. Phys. Rev. Lett. 1986, 56, 1700. (6) Lichterfeld, F.; Schmeling, T.; Strey, R. J. Phys. Chem. 1986, 90, 5762. (7) Satoh, N.; Tsujii, K. J. Phys. Chem. 1987, 91, 6629. (8) Wennerstro¨m, H.; Anderson, D. M. In Statistical Mechanics and Differential Geometry of Micro-Structured Materials; Friedman, A., Nitsche, J. C. C., Davis, H. T., Eds.; Springer Verlag: Berlin, 1991. (9) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (10) Porte, G.; Appell, J.; Bassereau, P.; Marignan, L. J. Phys. (Paris) 1989, 50, 1335.
S0743-7463(96)00425-8 CCC: $12.00
(11) Nallet, F.; Roux, D.; Milner, S. T. J. Phys. (Paris) 1990, 51, 2333. (12) Roux, D.; Safinya, C. R. J. Phys. (Paris) 1988, 49, 307. (13) Safinya, C. R.; Roux, D.; Smith, G. S.; Sinha, S. K.; Dimon, P.; Clark, N. A.; Bellocq, A. M. Phys. Rev. Lett. 1986, 57, 2718. (14) Nallet, F.; Roux, D.; Prost, J. Phys. Rev. Lett. 1989, 62, 276. (15) Ke´kicheff, P.; Christenson, H. K. Phys. Rev. Lett. 1989, 63, 2823. (16) Ke´kicheff, P.; Richetti, P.; Christenson, H. K. Langmuir 1991, 7, 1874. (17) Abillon, O.; Perez, E. J. Phys. (Paris) 1990, 51, 2543. (18) Halle, B.; Quist, P. O. J. Phys. II 1994, 10, 1823. (19) Parsegian, V. A.; Rand, R. P.; Rau, D. C. Methods Enzymol. 1986, 127, 400. (20) Parsegian, V. A.; Rand, R. P.; Fuller, N. L. J. Phys. Chem. 1991, 95, 4777. (21) Rand, P. R.; Fuller, N. L.; Gruner, S. M.; Parsegian, V. A. Biochemistry 1990, 29, 76.
© 1996 American Chemical Society
4058 Langmuir, Vol. 12, No. 17, 1996
emulsion24,25 but has to our knowledge not yet been used to study the undulation force. The basic idea behind osmotic stress is very simple. Two solutions are put in contact with a semipermeable (i.e., permeable only for the solvent) membrane as a dividing wall. One compartment contains a polymer solution with known osmotic pressure while the other compartment contains the lamellar phase under study. Normally the chemical potential of water is different in the two compartments and hence there will be a transport of water across the dividing membrane. When the net water transport across the membrane is zero, the chemical potential of water is identical in the two phases. Provided that the osmotic pressure of the polymer is known as a function of concentration, the actual value of the water chemical potential is found by measuring the polymer concentration in the polymer compartment. In this study we have investigated the lamellar phase formed by pentaethylene glycol dodecyl ether (C12E5), hexanol (C6E0), and water.26 The cosurfactant was added to move the LR phase from ≈60 °C (in the binary C12E5water system) to 25 °C. The weight ratio C6E0/C12E5 ) 25/75 was constant and, since the monomer solubility of C12E5 and C6E0 is negligible at our concentrations, the system may be treated as a pseudobinary mixture of bilayer material (C12E5 and C6E0) and solvent, H2O. As the stress-exerting polymer we have used dextran with a molecular weight of approximately 500 kDa (T500, Pharmacia). The osmotic pressure at 25 °C of this polymer is accurately known up to 5 wt %.27 Results and Discussion Dextran with high molecular weight and surfactant phases have previously been found to phase separate almost totally segregative,24 i.e. the solution demixes into one dextran phase free from surfactant and one surfactant phase with a very low dextran content. This was also the case in our system, where a macroscopically phase separated sample of dextran and bilayer material showed a very low polymer concentration in the LR phase (measured with optical rotation) and practically free from surfactant in the dextran phase (deduced from the fact that the polymer phase not showed any clouding as the temperature was enhanced, indicating a very low surfactant concentration). Hence, no semipermeable membrane was needed in the experiments. Samples were prepared by simply weighing the chemicals into rather large glass ampules (≈5 mL), which were immediately sealed. The samples were gently shaken for several days to ensure complete mixing. Two different methods for analyzing the phase equilibria were used. With the first method the samples were allowed to phase separate macroscopically. Within hours a clear bottom phase (containing dextran) and a turbid top phase (containing LR phase and dextran) developed. With time, the volume of the lower phase increased, the top phase shrank at the same time as it became less turbid. The top phase was an emulsion of LR phase and dextran solution with domain sizes of the order of the wavelength of visible light. For some reason this emulsion is very stable and it takes long time in a centrifuge (weeks), and several (22) Lis, L. J.; McAlister, M.; Fuller, N. L.; Rand, R. P.; Parsegian, V. A. Biophys. J. 1982, 37, 657. (23) Bonnet-Gonnet, C.; Belloni, L.; Cabane, B. Langmuir 1994, 10, 4012. (24) Kabalnov, A.; Olsson, U.; Wennerstro¨m, H. Langmuir 1994, 10, 2159. (25) Kabalnov, A.; Olsson, U.; Thuresson, K.; Wennerstro¨m, H. Langmuir 1994, 10, 4509. (26) Jonstro¨mer, M.; Strey, R. J. Phys. Chem. 1992, 96, 5993. (27) Edsman, K.; Sundelo¨f, L.-O. Polymer 1988, 29, 535.
Letters
Figure 1. Measured osmotic pressure, Πexp, versus bilayer volume fraction, φ. Points to the left (b) are from dextran concentration measurements and points to the right (0) stem from SAXS measurements. The solid line is a least-squares fit of the calculated osmotic pressure, Π ) -µw/vw, where µw is given by eq 5, to all experimental points. Using the bilayer thickness δ ) 30 ( 2 Å the bending modulus is κ ) (2.5 ( 0.5)kBT. The insert shows the same data in a double logarithmic plot.
months just by gravitation, to completely separate the two phases. Since such long equilibration time is not desirable with dextran due to possible degradation, the bottom phase was analyzed while the top phase still was a turbid emulsion. The dextran concentration in the bottom phase was measured with optical rotation. Assuming a similar local concentration in the dextran domains of the emulsion as in the bottom phase, the equilibrium concentration of C12E5 + C6E0 in the LR phase, LR φS+A , was easily evaluated, using
(
R φLS+A ) φinitial S+A 1 -
initial φdex
)
-1
(4)
bottom φdex
initial initial and φdex are the initial volume fractions of where φS+A bottom bilayer material and dextran, respectively, and φdex is the measured dextran concentration in the bottom phase. The second method we used was slightly different. After mixing the sample thoroughly, the biphasic sample was transferred to a capillary and the bilayer repeat distance in the LR phase was measured with small angle X-ray scattering (SAXS). Comparing with a calibration curve (repeat distance versus volume fraction C12E5 + C6E0) done without dextran, the actual bilayer concentration in the LR domains was directly obtained. The concentration of dextran was calculated from this value, in analogy with eq 4. This latter method was limited to the higher concentrations (φ ≈ 0.3) since, although experimentally more simple, the measured repeat distances are too inaccurate at lower concentrations (i.e., at longer repeat distances). The (excess) chemical potential of water in the LR phase was calculated from equation (3), yielding
µw )
(kBT)2 1 φ ∂G 3π2 |ns ) νw ∂nw 64 κ δ3 1 - φ
(
3
)
(5)
where vw is the molecular volume of water. In Figure 1 the measured osmotic pressure, Πexp, is plotted versus the volume fraction of C12E5 + C6E0 in the
Letters
LR phase. Filled circles are from dextran concentration measurements, and open squares stem from SAXS measurements of the bilayer repeat distance. It is satisfying that the two methods obviously produce consistent results. The solid line is least-squares fit of the calculated osmotic pressure, Π ) -µw/vw, where µw is given by eq 5, to the experimental points. As is seen, a very good agreement with the Helfrich formula is obtained in the full concentration regime. In order to obtain a numeric value of the bending modulus, we insert the thickness of the bilayer, δ ) 30 ( 2 Å, obtained from SAXS measurements. This gives a bending modulus of κ ) (2.5 ( 0.5)kBT. The relevance of the fitted bending modulus depends on the accuracy of the numerical prefactor in eq 3, which may be uncertain up to a factor of 2.28 However, this would merely change the value of κ and does not affect the scaling behavior we have experimentally verified. (28) Helfrich, W.; Servuss, R.-M. Nuovo Cimento 1984, 3, 137.
Langmuir, Vol. 12, No. 17, 1996 4059
We note that an additional correction term to the free energy, logarithmic in φ, has been proposed.1,2 However, the existence of such a term is under debate.29,31 Including this factor into our analysis will only slightly reduce the value of the bending modulus. Hence, our concentration interval is too narrow to either verify or rule out the existence of such a logarithmic correction term in the free energy expression. Acknowledgment. This work was supported by the Swedish Natural Science Research Council (NFR). LA960425S (29) Wennerstro¨m, H.; Olsson, U. Langmuir 1993, 365. (30) Pieruschka, P.; Marcelja, S.; Teubner, M. J. Phys. II 1994, 4, 763. (31) Daicic, J.; Olsson, U.; Wennerstro¨m, H.; Jerke, G.; Schurtenberger, P. J. Phys. II 1995, 5, 199.
