Microscopy, SAXD, and NMR Studies of Phase Behavior of the

Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. Chem. Rev. 1988, 88, 221. ... Ericsson, B.; Eriksson, P. O.; Löfroth, J.; Engström, S. ACS Symp. ...
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Langmuir 2000, 16, 10044-10054

Microscopy, SAXD, and NMR Studies of Phase Behavior of the Monoolein-Diolein-Water System Johanna Borne´,* Tommy Nylander, and Ali Khan Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. 124, SE-221 00 Lund, Sweden Received April 28, 2000. In Final Form: October 2, 2000 The phase behavior of the ternary monoolein (MO)-diolein (DO)-water (2H2O) system is presented. The experimental phase behavior and microstructure are studied by a combination of polarizing microscopy, small-angle X-ray diffraction, and NMR methods. Monoolein forms extensive reversed bicontinuous cubic liquid crystalline phases (C) that are in equilibrium with a lamellar liquid crystalline phase (LR) on the water-poor side and with excess water on the other side. The presence of small amounts of DO in the MO-water system is sufficient to destabilize the C and LR liquid crystalline phases. Formation of a reversed hexagonal (HII) phase from the cubic phase occurs at a lower transition temperature than that reported for the MO-water system. Within the cubic region, the diamond cubic phase, CD, is less stable than the gyroid type, CG. The solubility of DO increases within this phase when the MO content increases, and the phase reaches its maximum stability at 4 wt % DO. The large HII-phase formed in the ternary system is in equilibrium with water, and it solubilizes about 30 wt % DO within its stability range. A stable dispersion is formed at even higher DO concentrations. An ideal swelling of the HII-phase with increasing polar volume fraction is observed, whereas the length of the hydrocarbon chains along the hexagonal faces is constant. We measure a slight change of the average area per molecule in the HII-phase with DO concentration. The formation and stability of the liquid crystalline phases can be qualitatively understood from the self-aggregation model, using the geometrical packing parameter of the lipids.

Introduction Lipids are known for their rich polymorphism. Therefore, knowledge of the phase behavior of different glycerides, especially monoglycerides, is of great interest in the food industry,1,2 where they are used as emulsifiers, for example, in margarine, low-calorie spreads, dairy products, and dressings. Since the work of Lutton,3 the binary phase equilibrium of the polar lipid monoolein (MO), a metabolite in fat digestion, and water has been extensively studied.4-10 In the presence of ca. 20-40 wt % water, MO forms an extensive region of cubic liquid crystalline phases (C) with water. At a higher water content, the cubic phase is in equilibrium with excess aqueous solution. A lamellar liquid crystalline phase (LR) exists at a low water content (ca. 5-20 wt %). On heating, both the C and LR phases are destabilized and a reversed hexagonal phase (HII) is formed. The cubic structures are suggested to have a significant role in biological systems. The three-dimensional cubic liquid crystalline (lc) phases, which have been studied extensively, can be formed either by closely packed micelles or from the bicontinuous-type structure consisting of open water channels separated by infinite curved lipid bi* Corresponding author. Fax: +46 46 2224413. Phone: +46 46 2228204. E-mail: [email protected] se. (1) Krog, N. In Food Emulsions; Larsson, K., Friberg, S., Eds.; Marcel Dekker: New York, 1997; p 141. (2) Krog, N.; Riisom, T. H.; Larsson, K. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York, 1983; Vol. 2, p 321. (3) Lutton, E. S. J. Am. Chem. Soc. 1965, 42, 1068. (4) Lindblom, G.; Larsson, K.; Johansson, L.; Fontell, K.; Forse´n, S. J. Am. Chem. Soc. 1979, 101, 5465. (5) Larsson, K. Nature 1983, 304, 664. (6) Hyde, S. T.; Andersson, S.; Ericsson, B.; Larsson, K. Z. Kristallogr. 1984, 168, 213. (7) Chung, H.; Caffrey, M. Biophys. J. 1994, 66, 377. (8) Landh, T. J. Phys. Chem. 1994, 98, 8453. (9) Briggs, J.; Chung, H.; Caffrey, M. J. Phys. II 1996, 6, 723. (10) Qui, H.; Caffrey, M. Biomaterials 2000, 21, 223.

layers.4,11-13 The cubic phases formed in the MO-water system are bicontinuous and consist of curved bilayers organized to form two unconnected systems of water channels. Two microstructures have been identified in the binary system. The curved bilayer in the diamond (CD) type can be described by a diamond IPMS (infinite periodic minimal surface), and a gyroid IPMS describes the gyroid (CG) type of cubic phase.12,14,15 These structures correspond to a primitive lattice (space group Pn3m) and a body-centered lattice (space group Ia3d), respectively. The monoolein cubic phases have been suggested for use in the pharmaceutical industry as carriers of biologically active compounds, drug delivery devices,16,17 and biosensors.18 Over the years, the possibility of obtaining cubic phases with a large variation of entrapped proteins has also been reported.19-23 In industry, monoglycerides are usually produced on a large scale by chemically catalyzed glycerolysis of triglycerides.24 The use of enzymes to catalyze reactions has (11) Lindblom, G.; Rilfors, L. Biochim. Biophys. Acta 1989, 988, 221. (12) Larsson, K. J. Phys. Chem. 1989, 93, 7304. (13) Luzzati, V. Curr. Opin. Struct. Biol. 1997, 7, 661. (14) Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. Chem. Rev. 1988, 88, 221. (15) Hyde, S.; Andersson, S.; Larsson, K.; Blum, Z.; Landh, T.; Lindin, S.; Ninham, B. W. Physics, Chemistry and biology; Elsevier: Amsterdam, 1997. (16) Ericsson, B.; Eriksson, P. O.; Lo¨froth, J.; Engstro¨m, S. ACS Symp. Ser. 1991, 469, 251. (17) Engstro¨m, S.; Ljusberg-Wahren, H.; Gustafsson, A. Pharm. Technol. 1995, 7, 14. (18) Razumas, V.; Kanapieniene´, J.; Nylander, T.; Engstro¨m, S.; Larsson, K. Anal. Chim. Acta 1994, 289, 213. (19) Gulik-Krzywicki, T.; Shechter, E.; Luzzati, V. Nature 1969, 223, 1116. (20) Larsson, K.; Linblom, G. J. Dispersion Sci. Technol. 1982, 3, 61. (21) Ericsson, B. L. K.; Fontell, K. Biochim. Biophys. Acta 1983, 729, 23. (22) Razumas, V.; Larsson, K.; Miezis, Y.; Nylander, T. J. Phys. Chem. 1996, 100, 11766. (23) Nylander, T.; Mattisson, C.; Razumas, V.; Miezis, Y.; Håkansson, B. Colloids Surf., A 1996, 114, 311.

10.1021/la000619e CCC: $19.00 © 2000 American Chemical Society Published on Web 12/01/2000

Monoolein-Diolein-Water System

opened many new synthetic possibilities and applications. Other lipids may be present as biproducts from these processes. The aggregation of monoolein in the presence of these lipids has not been studied in any detail. We have undertaken a systematic study of the aggregation of monoolein in the presence of several lipids by following the phase behavior over a wide concentration range. Here, we present the phase behavior of the binary monooleinwater (composition vs temperature) and isothermal ternary monoolein-diolein-water systems at 25 °C. The experimental phase diagrams and microstructures of phases are studied by a combination of microscopy, SAXD (small-angle X-ray diffraction), and NMR techniques. Experimental Section Materials. Monoolein, Rylo Mg 90-glycerol monooleate (TSED 173, Journal No. 1876-88) and diolein, glycerol dioleate (TSED 175), were kindly provided by Danisco Ingredients (Braband, Denmark). These are commercially available products, and it is important to note that they are not single-component products but also contain other polar lipids. The monoolein samples consist of 95.7 wt % monoglycerides, 3.8 wt % diglycerides, 0.4 wt % free fatty acids, and 0.1 wt % free glycerol. The fatty acid composition was 90.0 wt % oleic acid, 5.0 wt % linoleic acid, 2.7 wt % stearic acid, 1.0 wt % palmitic acid, 0.3 wt % linolenic acid, and 1.0 wt % other fatty acids. Diolein (DO, glycerol dioleate) consists of 98.3 wt % diglycerides, 1.3 wt % triglycerides, and 0.4 wt % monoglycerides. The fatty acid composition was >92 wt % oleic acid, 1. The packing parameters for the LR-, C-, and HII-phases are shown to be 1.0, 1.3, and 1.7, respectively, for the MO-H2O system.12 The cpp value for the different cubic phases increases in the order CG < CD, and therefore the CG-phase and not the CD-phase borders the LR-phase. These cpp values are fairly high for the maximum stability (35) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (36) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601.

Borne´ et al.

of the respective phases. Thus, a slight increase of the cpp value may lead to a phase transition. An increase in temperature can both dehydrate the polar headgroups and increase the volume of the alkyl chains. Both effects increase the cpp value. Moreover, a significant amount of DO (3.4 wt %) is present in our MO batch. The presence of the more wedge-shaped DO molecules is expected to push the cubic phases close to the phase transition to the ΗII-phase. This can be achieved by only a slight increase in temperature. Consequently, the CG-phase, which has a lower cpp value, was observed to have a higher thermal stability than the CD-phase. The ternary phase diagram presented confirms that the large hydrophobic part of DO increases cpp. This favors the phase transition from LR (zero curvature) to C (slightly more negative curvature) to HII (large negative curvature) with increasing content of DO (or temperature). Phase Structure. 2H NMR of Liquid Crystalline Phases. In the course of phase diagram determination, we have recorded 2H NMR quadrupolar splitting for a large number of samples in both the HII and LR lc-phases. The magnitude of the quadrupolar splitting is shown to follow eq 1.37,38

∆)|

∑Pi(νQi)Si|

(1)

where Si is the order parameter describing the orientation of the fraction of 2H present at site i (Pi). νQ is the quadrupolar coupling constant (220 kHz).39 A simplification of the equation can be done by applying the two-site model.40 According to this model, the heavy water molecules can either be “bound” on the surface of the aggregate or exist as “free” water molecules without restrictions in the bulk. In addition, there is a fast exchange of water molecules between the two sites. These assumptions mean that the order parameter for the free water molecules disappears, no splitting of the free water signal occurs, and eq 1 is simplified according to

∆ ) |P(νQ)S|

(2)

where P is the fraction of bound water molecules. In a lamellar phase, the director is perpendicular to the lamellae, and in a hexagonal phase it is perpendicular to the axes of the cylinders. In the HII-phase, the rapid diffusion of the deuterons in the aqueous cylinders causes a further averaging of the expression in eq 2. Because of this effect, the absolute value of the order parameter for the HII-phase is reduced by half upon transition from a lamellar to a hexagonal phase and the measured splitting for the lamellar phase is 2 times larger than that for the hexagonal phase at an identical composition. Further information from the measured 2H NMR splitting values is achieved by changing Pi to the molar ratio of components as

∆obs ) (xlipid/xwater)nvQS

(3)

where xlipid and xwater are the mole fractions of lipid and water, respectively, and n is the (average) number of water molecules bound per lipid molecule.41 (37) Wennerstro¨m, H.; Linblom, G.; Lindman, B. Chem. Scr. 1974, 6, 97. (38) Khan, A.; Fontell, K.; Lindblom, G.; Lindman, B. J. Phys. Chem. 1982, 86, 4266. (39) Glasel, J. A. In Water, a Comprehensive Treatise; Franks, F., Ed.; Plenum Press: New York, 1972; Vol. 1, p 215. (40) Persson, N. O.; Lindman, B. J. Phys. Chem. 1975, 79, 1410. (41) Wennerstro¨m, H.; Persson, N. O.; Lindman, B. ACS Symposium Series 9; American Chemical Society: Washington, DC, 1975; p 253.

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Figure 7. Microscopic image for a stable emulsion of MO-DO-2H2O (30/51/19 wt %); see Figure 6. The largest drop is 300 µm. The magnification is 10×.

Figure 8. The quadrupolar splitting (∆) values of the liquid crystalline phases for the binary MO-water and MO-DOwater systems, presented as a function of (xMO + xDO)/xwater at 25 °C; x ) mole fraction. 9 represents HII, and 2 represents LR.

By plotting ∆ versus x(MO+DO)/xwater (Figure 8), we obtain information on the hydration properties of the lipid molecules as well as on the swelling characteristics of the lc-phases. In Figure 8, ∆ values for the HII-phase are found to increase linearly with decreased water content up to the molar fraction x(MO+DO)/xwater ≈ 0.36, above which the increase of the splitting values is relatively small with decreasing water concentration. For the LR-phase, which exists within a very limited area at the water-poor part, the ∆ values are almost concentration-independent. However, the splitting of the LR-phase is twice that of the HII-phase. This finding confirms the presence and location of the HII and LR liquid crystalline phases in the system. A straight line passing through the origin below x(MO+DO)/ xwater ≈ 0.36 for the HII-phase indicates that the phase shows an ideal swelling behavior. Thus, an increase in the water concentration will not change the hydration of

the lipid molecules but only the amount of free water. The nonlinear behavior of the ∆ values with concentration is a result of the breakdown of the two-site model as the lipid molecules are not fully hydrated within the area. This observation has also been reported for several surfactant systems.42 SAXD Determinations in the Liquid Crystalline Phases. The SAXD data were used to identify or confirm the structures of the different liquid crystalline phases. The data are discussed in relation to the expected dimensions calculated with laws for the ideal swelling of the various liquid crystalline phases, which are given in the Appendix. As discussed in detail by Seddon for hexagonal phases in particular, application of swelling laws requires the definition of the dividing interface between the nonpolar and aqueous regions.43 The author follows three different approaches: (1) the Luzzati approach, where water and lipid components are taken to occupy distinct regions; (2) a portion of the water is considered to be strongly bound to the headgroup, which gives two dividing interfaces; (3) the water and headgroups together compose a polar region, and the dividing interface is between the polar and nonpolar parts of the lipid. Because the 2H NMR quadrupolar splitting data show that the lipid is not fully hydrated at a low water content during the swelling of the LR- and HII-phases, we have applied approach 3. SAXD Analysis of the LR Structure. As the lamellar region in the ternary MO-DO-water system exists in a relatively narrow range of water content, the SAXD data for only three different water contents were recorded. These data are presented in Table 1. The bilayer thickness, calculated by using eq c, is 35.4-33.4 Å. This means that the thickness of the water layer would be 4.2-2.2 Å, which is quite an unrealistic value. It certainly confirms the findings from the 2H NMR quadrupolar splitting data that there is not enough water available to fully hydrate the (42) Edlund, H.; Byde´n, M.; Lindstro¨m, B.; Khan, A. J. Colloid Interface Sci. 1997, 196, 231. (43) Seddon, J. M. Biochim. Biophys. Acta 1990, 1031, 1.

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Table 1. Composition vs Structural Parameters Calculated from SAXD Data for the Lamellar Phase of the Ternary MO-DO-2H2O System at 25 °C composition (weight fraction)

composition (volume fraction)

MO

DO

D2O

lipid

D2O

hc

polar

aLR (Å)

dbilayer (Å)

dhc (Å)

SLR (Å2)

0.879 0.901 0.896

0.000 0.000 0.031

0.121 0.099 0.073

0.895 0.914 0.937

0.105 0.086 0.063

0.700 0.715 0.736

0.300 0.285 0.264

39.5 38.1 35.6

35.3 34.8 33.4

13.8 13.6 13.1

35.6 36.1 38.3

Table 2. Composition vs Structural Parameters Calculated from SAXD Data for the Reverse Hexagonal Phase of the Ternary MO-DO-2H2O System at 25 °C composition(weight fraction)

composition(volume fraction)

MO

DO

D2O

lipid

D2O

hc

polar

aHII (Å)

l1 (Å)

l2 (Å)

SHII (Å2)

0.685 0.700 0.723 0.746 0.700 0.763 0.719 0.638 0.789 0.614 0.597 0.763 0.688

0.040 0.050 0.085 0.067 0.122 0.081 0.132 0.206 0.078 0.266 0.302 0.160 0.244

0.275 0.25 0.192 0.187 0.178 0.156 0.149 0.156 0.133 0.120 0.101 0.077 0.068

0.756 0.779 0.831 0.836 0.844 0.864 0.870 0.864 0.884 0.897 0.914 0.934 0.942

0.244 0.221 0.169 0.164 0.156 0.136 0.130 0.136 0.116 0.103 0.086 0.066 0.058

0.595 0.614 0.659 0.660 0.672 0.684 0.694 0.696 0.699 0.727 0.744 0.746 0.760

0.405 0.386 0.341 0.340 0.328 0.316 0.306 0.304 0.301 0.273 0.256 0.254 0.240

61.4 59.4 52.2 52.0 51.3 48.7 48.5 48.5 47.0 46.3 43.6 42.0 40.9

14.9 14.9 14.1 14.1 14.2 13.7 13.9 13.9 13.6 14.0 13.6 13.1 13.1

10.2 10.3 10.1 10.1 10.2 10.0 10.1 10.2 10.0 10.5 10.2 9.9 9.9

33.7 33.2 33.8 33.3 33.9 33.4 33.8 35.4 32.9 34.8 35.8 33.3 34.5

lipid molecule in the LR-phase. Furthermore, the SAXD pattern showed one sharp reflection at 25 °C, although the quadrupolar splitting on the 2H NMR spectra confirmed the LR-phase. This is probably due to the small content of water in the LR-phase, which reduces the “contrast” between the polar and apolar parts of the lipid bilayer. When the temperature is decreased to 17 °C, four reflections with a Bragg spacing ratio of 1:2:3..., typical for a lamellar type of phase, are observed (Figure 2a). At this temperature, the sample also contains a cubic phase as a minor component and therefore additional reflections are recorded. The 2H NMR for the sample confirms the coexistence of isotropic (cubic) and anisotropic (lamellar) phases by producing a splitting with a central singlet on its spectrum. The interlayer spacing was 47.7 Å, and this value is about 8 Å larger than that recorded at 25 °C for about the same 2H2O concentration. According to a recently published study,10 the LR-phase transforms into lamellar crystals (LC) at 18 °C and below, with a corresponding increase in interlayer spacing, aLC, to 49.3 ( 0.3 Å. This value is shown to remain constant between -13 and 18 °C. In addition, this phase was reported to be able to coexist with a cubic phase. We can therefore conclude that cooling our sample to 17 °C is likely to induce a LR-to-LC transition. The LC-phase is likely to give a better contrast for the SAXD measurements, and hence we observe more reflections even if the water content is low. SAXD Analysis of the HII Structure. This phase consists of water cylinders surrounded by a surfactant monolayer. The hydrocarbon chains of the surfactants are stretched to a length l1 into the hexagonal corners of the unit cell and are somewhat compressed to a length l2 along the hexagonal faces. From the lattice spacing, aHII, obtained from SAXD experiments, the distances and other structural parameters such as the cross section at radius Rcyl and the mean polar headgroup area (SHII) can be obtained. Like the LR-phase, the HII-phase exists at low water content, which also in this case is reflected in the 2H NMR quadrupolar splitting data. The obtained SAXD data for the HII-phase of the ternary MO-DO-water system are shown for several compositions in Table 2, together with the structural parameters calculated from eqs e-g. It is obvious that the length of the hydrocarbon chains along the hexagonal faces, l2, does not change significantly with

Figure 9. The lattice parameter aHII (Å) of the hexagonal phase (HII) plotted versus the apolar volume fraction φhc.

the lipid composition or the water content. This is more evident if we plot aHII versus Φhc and fit eq g to the experimental data (Figure 9). As shown in the figure, the fit is very good and gives an l2 value of 10.1 ( 0.2 Å. This indicates that the HII-phase swells ideally with water, while keeping l2 constant and independent of the lipid composition. The l2 value is comparable to the thickness of the acyl chain region of a MO monolayer at the air/ water interface, which was determined to be 9.3 Å by specular X-ray reflectivity measurements.44 The corresponding value for l1 is 13.9 ( 0.5 Å. The average value of the maximum chain length in the HII-phase is about the same as the average value of the acyl chain length in the LR-phase, which is about 13.5 ( 0.4 Å (Table 1). The average calculated area per molecule of HII is 34.0 ( 0.9 Å2. Here, we can observe a slight dependence on the DO concentration. For instance, the average for those samples with a DO concentration above 20 wt % is 35.1 ( 0.5 Å2, whereas below 20 wt % DO the average area is (44) Jensen, T. R.; Kjær, K.; Howes, P. B.; Svendsen, A.; Balashev, K.; Reitzel, N.; Bjørnholm, T. Model Systems for Biological Membranes Investigated by Grazing-incidence X-ray Diffraction and Specular Reflectivity; Kokotos, G., Constantinou-Kokotou, V., Eds.; Crete University Press: Crete, 1999; p 127.

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Concluding Remarks The C-to-HII phase transition takes place at a much lower temperature in the binary MO-2H2O system (the batch used in this study) than in the corresponding MOaqueous system. The findings in the binary MO-aqueous system parallel those observed on adding DO to the binary system at room temperature. The CG-phase is more stable toward an increase in the temperature than the CD-phase, and the CG-phase can solubilize a larger content of DO than the CD-phase. The hydration of the lipid in the LR- and HII-phases in the MO-based aqueous system changes with the water content at a low water content (lipid-water molar ratio > 0.36). However, if this effect is taken into account the swelling of the HII lc-phase is ideal. A stable emulsion was identified in the diolein-rich area. Figure 10. Volume fraction of lipid φlip versus the lattice parameter acub of the cubic phases, CG and CD. The solid line represents a best fit of the model calculated for the CG cubic phase (eq h). (9) MO-2H2O, (0) MO-DO-2H2O, (2) MO-H2O from ref 32, (b) MO-2H2O, (O) MO-DO-2H2O, ([) from reference for MO-H2O samples (ref 32).

33.5 ( 0.3 Å2. As expected, the headgroup in the LR-phase is larger than that in the HII-phase that has a larger negative curvature. Swelling Behavior of the C Structures. The SAXD data were used to analyze the swelling behavior of the bicontinuous phase of lipids in water. The purpose is to establish that the analyzed samples are one-phase samples and do not contain excess water. Furthermore, it helps us to verify that the indexing of the cubic samples is correct. The solid line in Figure 10 represents a best fit of eq h to the experimental data for the MO-2H2O binary samples in our study together with those reported by Barauskas et al.32 From the fit, the area Ω(t) for the gyroid (CG) phase was determined to 36.4 Å2 at t ) 10.4 Å. These values are close to the values found by Chung and Caffrey7 and Engblom and Hyde.45 Barauskas et al. used the same batch of monoolein as ours, but the sample was prepared with H2O instead of 2H2O. As shown in Figure 10, their data for samples indexed as the CG-phase agree well with ours. In fact, a fit of eq h to our data only gives an area, Ω(t), of 35.9 Å2 for t ) 10.7 Å. It can therefore be concluded that the isotope effect is negligible for the swelling of the cubic phase. The corresponding data for the ternary MO-DO2 H2O samples (unfilled circles) are also inserted but are not used for the fitting. Because the cubic region in the ternary system is relatively small, few one-phase samples were obtained. No conclusions could therefore be drawn, although these data are close to the data for the binary MO-2H2O system. Very few data points were also recorded for CD-phase samples because the one-phase region of this type of cubic phase is limited. To compare the area per molecule in the cubic phase with that in the HII-phase, we need to have the value at about the same distance from the methyl end of the acyl chains. In the HII-phase, the area per molecule reported is taken about 10 Å from the methyl end (Table 2). We can therefore take the area per molecule at the neutral surface of the CG-phase, located at a distance t ≈ 10 Å from the midplane of the bilayer. This gives an area per molecule of about 36.4 Å2, which as expected is slightly higher than the values reported for the HII-phase. (45) Engblom, J.; Hyde, S. T. J. Phys. II 1995, 5, 171.

Acknowledgment. Andrew Fogden is greatly acknowledged for valuable discussions and suggestions concerning the swelling of liquid crystalline phases. This work was financed by the EU Biotech Shared Cost project, Contract No. BIO4-97-2365. Appendix Volume of Polar and Apolar Parts of the Lipids. The densities of MO (FMO ) 0.942 g/cm3 with molecular weight MMO ) 356.55 g/mol), oleic acid (FOA ) 0.8935 with MOA ) 282.47 g/mol), heptadecene (FHD ) 0.7852 g/cm3 with MHD ) 238.46 g/mol), and glycerol (FGL ) 1.2613 g/cm3 with molecular weight MGL ) 92.11 g/mol) can be obtained from the Handbook of Chemistry and Physics,46 and the density of DO (FDO ) 0.9194 g/cm3 with MDO ) 621.02 g/mol) was obtained from the manufacturer. Under the condition that the molar volumes are additive, we obtain densities (molar mass) of the hydrophilic and hydrophobic parts as FhgMO ) 1.1164 g/cm3 (MhgMO ) 92.11 g/mol) and FhcMO ) 0.8935 g/cm3 (MhcMO ) 264.44 g/mol), respectively, for MO, and the corresponding values for DO are FhgDO ) 1.1034 g/cm3 (MhgDO ) 92.11 g/mol) and FhcDO ) 0.8935 g/cm3 (MhcDO ) 528.88 g/mol). The values for the densities obtained for MO and DO are similar and slightly smaller than the value observed for free glycerol. The volume fraction of the apolar part of the total lipid in the ternary system is given by

Φhc )

( ) ( ) ) ( )

MhcMO MhcDO t + t FhcMO MO FhcDO DO MhcMO MhgMO MhcDO MhgDO ww + tMO + + tDO + FhcMO FhgMO FhcDO FhgDO Fw (a)

(

Here, tMO ) wMO/MMO and tDO ) wDO/MDO, and wMO and wDO are the weight fractions of MO and DO, respectively. The expression for the corresponding molecular volume is given by

( ) ( )

MhcMO MhcDO tMO + t FhcMO FhcDO DO 1 × 1024 vhc ) tMO + tDO NA

(b)

where NA is Avogadro’s number. (46) Handbook of Chemistry and Physics; CRC Press: Cleveland, OH, 1974.

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Swelling Laws. The repeat distance of the lamellar phase, aLR, can be divided into the thickness of the apolar part (dhc) (hydrocarbon chain) and the polar part (dhg) of the lipid and the thickness of the water layer (dw). The bilayer thickness, dbilayer, equals 2(dhc + dhg). For the onedimensional ideal swelling, the repeat distance is inversely proportional to the volume fraction of the lipid, Φlip:47

aLR )

dbilayer 2νlip ) Φlip SLRΦlip

(c)

where νlip is the average of molecular volume of the lipid in Å3 and SLR is the average interfacial area per molecule. Equation c can also be expressed in terms of the length and volume, dhc and νhc, respectively, of the hydrocarbon chain:

2dhc 2νhc ) aLR ) Φhc SLRΦhc

( ( ( (

l2 )

)) ))

x3 1 (1 - Φhc) 2π x3 x3 1 (1 - Φhc) 2 2π

1/2

aHII

(e)

1/2

aHII

(f)

(47) Luzzati, V. X-ray diffraction studies of lipid-water systems; Chapman, D., Ed.; Academic Press: New York, 1968; Vol. 2, p 71. (48) Hyde, S. T. Colloids Surf., A 1995, 103, 227.

(

)

1 2π (1 - Φhc) Φhc x3

1/22νhc

aHII

(g)

and vhc is the molecular volume of the hydrocarbon chain in Å3. The swelling of the cubic phases has been modeled by, for instance, Engblom and Hyde.45 A neutral surface is defined at a distance t from the middle of the curved bilayer where the area per molecule (Ω(t)) is constant, independent of the swelling. The volume fraction of lipids Φlip can then be related to the lattice parameter acub (unit cell dimension obtained by SAXD):

Φlip )

c1 c2 + acub a 3

(h)

cub

where

(d)

The structural parameters l1 (maximum chain length), l2 (minimum chain length), Rcyl, and the headgroup area of the lipid, SHII, are obtained from the following equations, which arise from simple geometric considerations:43,47,48

l1 )

SHII )

c1 )

νlip(-16πχH2)1/3 Ω(t)

and

c2 )

+4πχνlipt2 Ω(t)

νlip is the molecular volume of lipid, calculated in analogy with eq b, and χ is a topology index of the surface known as the Euler-Poincare´ characteristic per cubic unit cell. The latter parameter equals -8 and -2 for the global shape of the gyroid and diamond surfaces, respectively. H is the homogeneity index that connects the lattice parameter to topology and equals 0.7665 and 0.7498 for the CG and CD types of cubic phases, respectively. LA000619E