Langmuir 1998, 14, 7427-7434
7427
Gaussian Curvature Modulus of an Amphiphilic Monolayer R. H. Templer,* B. J. Khoo, and J. M. Seddon Department of Chemistry, Imperial College, London SW7 2AY, U.K. Received June 16, 1998. In Final Form: October 2, 1998 In this paper the Helfrich ansatz for bending an amphiphilic monolayer is re-expressed in terms of the deviations in curvature away from a preferred value. Using this expression and a simple model of the response of amphiphiles to interfacial bending, it is demonstrated that there are strict limits to the value of the ratio of the Gaussian curvature modulus to the mean curvature modulus (κG/κ). We have made an estimate of κG/κ from measurements of the swelling behavior in water of inverse bicontinuous cubic mesophases in a system composed of 1-monoolein, dioleoylphosphatidylcholine, and dioleoylphosphatidylethanolamine. The estimate, -0.75 ( 0.08, is in agreement with the limits set by our model of -1 < κG/κ < 0. This determination is the first to be made on an inverse bicontinuous cubic phase which is sufficiently swollen to be in a regime where first-order curvature elastic energetics should be sufficient to describe the state of the mesophase and hence provide a reliable estimate of κG/κ.
cal state cr will be
Introduction In 1973 Helfrich developed the following expression for the bending energy per unit area gc of a surfactant-loaded fluid film1
1 gc ) (c1 + c2 - 2c0)2 + κGc1c2 2
(1)
Here c1 and c2 are the principal curvatures at the region of interest on the interface, c0 is called the spontaneous curvature of the interface, κ is the mean curvature modulus, and κG is the Gaussian curvature modulus. Equation 1 may be used to describe either a bilayer or a monolayer interface. We will be dealing with the monolayer case, and hence c0, κ, and κG are all values for the monolayer. The Helfrich ansatz simplifies the physical description of bending a fluid interface by subsuming the details of all of the local molecular interactions into the three curvature elastic parameters c0, κ, and κG of an infinitely thin sheet. The first term in (1) is apparently Hookean; that is, departures in the sum of c1 and c2 away from the spontaneous curvature c0 give rise to a quadratic increase in the curvature elastic energy of the interface. Thus the first bending rigidity κ must be positive. The second term in (1) is less easily understood. The product c1c2, the Gaussian curvature of the interface, will have a nonzero value only when the interface has elliptic or hyperbolic curvature and will only be of energetic consequence when the system undergoes a topological change. This is a clever device for analyzing energy differences between fluid interfaces of different geometry but fixed topology. However, it makes it difficult to see the physical meaning and significance of κG in terms of the localized deformations of the amphiphilic molecules embedded at the interface, but we can examine the consequences of varying the sign and magnitude of κG/κ. In the range of values for κG/κ where
-2 e κG/κ e 0
(2)
the spherical monolayer is at a global energetic minimum. The interfacial curvature in the curvature relaxed spheri(1) Helfrich, W. Z. Naturforsch. 1973, 28c, 693-703.
cr )
2κ c 2κ + κG 0
(3)
with a free energy gc,r of
gc,r )
2κκGc02 (2κ + κG)
(4)
For the case where κG/κ < -2 there is no globally stable state. The spherical interface has the lowest free energy, but it is unstable; that is, it will continue to curl up until higher order terms in the curvature elasticity stop it. When κG/κ > 0, there is also no globally stable monolayer state. Here saddle-shaped monolayer deformations (c1 > 0, c2< 0) are of lower free energy than either spherical or cylindrical deformations but again cannot be stabilized without additional higher order curvature elastic terms. Given the variety of quite different curvature elastic behavior that is possible by varying the relative magnitude of κG, the question arises as to whether such variations are in fact physically possible. This is the central topic of our investigation. We can get some idea about the probable sign of κG from its relationship to the lateral stress profile2 (Figure 1)
κG )
∫0lt(z)(z - ξ)2 dz
(5)
where t(z) is the lateral stress as a function of the depth into the flat monolayer, l is the thickness of the monolayer, and ξ is the distance to the pivotal surface from the chain ends. The pivotal surface is defined to be the position on the molecule whose cross-sectional area is unchanged during isothermal bending;3,4 where the interfacial curvature is not great, this surface is identical to the neutral surface.5 Measurements of the pivotal surface in a variety of systems invariably place it close to the polar/apolar (2) Kozlov, M. M.; Markin, V. S. J. Chem. Soc., Faraday Trans. 2 1989, 85, 277-292. (3) Rand, R. P.; Fuller, N. L.; Gruner, S. M.; Parsegian, V. A. Biochemistry 1990, 29, 76-87. (4) Templer, R. H. Langmuir 1995, 11, 334-340. (5) Kozlov, M. M.; Winterhalter, M. J. Phys. II 1991, 1, 1077-1084.
10.1021/la980701y CCC: $15.00 © 1998 American Chemical Society Published on Web 11/20/1998
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Figure 2. Molecular deformation during interfacial bending. (a) At the global energetic minimum c1 ) c2 ) cr and the average molecular splay is isotropic. (b) We now flatten the interface along c1 but maintain ∆c2 ) 0. The energy required to make this cylindrical interface is gc ) 1/2kcr2, but this energy can be reduced somewhat if we allow molecules to change splay along c2. (c) The coupling between molecular splay deformation along c1 and c2 means the system’s free energy is reduced if the molecular splay is increased along c2. Figure 1. Lateral stress profile t(z) of an amphiphile which wishes to bend toward the water (light gray). A negative stress in the chain region is due to the outward pressure produced by chain collisions, the positive stress at the interface arises because of the polar/apolar tension, and headgroup repulsion produces a small negative stress around the headgroups. Superimposed on t(z) in darker gray is the quadratic function with its origin at the pivotal surface. The product of these two is shown in solid black and shows that the second moment of the lateral stress is negative.
interface.3,4,6,7 This has the effect that the negative contribution from the outward pressure in the chain and headgroup region will be greater than the positive contribution from the interfacial tension. As chain lengths shorten, κG will tend to zero but not go positive. To get a positive value of κG would require either that the pivotal surface was situated closer to the chain ends or that there were positive lateral stresses in the chain region. Such molecules would have a propensity for dilation in their cross-sectional area in the middle of the molecule and area contraction at both ends. Typical amphiphiles do not behave in this way, and we would therefore expect κG to be negative. In response to this observation we have reformulated the curvature elasticity of a monolayer under the assumption that there is a global minimum in the free energy. The new expression provides a simple link between curvature elasticity and the associated molecular deformations that occur during bending. This has been done before,2,8 but we have imposed a simple model of the material properties of the monolayer which set more constrained limits on the value of κG/κ. We have then gone on to measure κG/κ from measurements of the swelling of large inverse bicontinuous cubic phases in a mixture of 1-monoolein, dioleoylphosphatidylcholine, and dioleoylphosphatidylethanolamine (henceforth MO, DOPC, DOPE). Values have been reported before from measurements on inverse bicontinuous cubic phases, but this is the first example of measurements made in a system where the interfacial curvatures are sufficiently gentle that the harmonic approximation of (1) is sufficient to describe the bending energy of the monolayer. Recasting the Helfrich Ansatz Let us consider the bending energetics of an amphiphilic monolayer. In keeping with the measurements we will present, we will deal only with monolayer interfaces which (6) Chung, H.; Caffrey, M. Biophys. J. 1994, 66, 377-381. (7) Chen, Z.; Rand, R. P. Biophys. J. 1997, 73, 267-276. (8) Wennerstro¨m, H.; Anderson, D. M. In Statistical thermodynamics and differential geometry of microstructured materials; Davis, H. T., Nitsche, J. C. C., Eds.; Springer-Verlag: New York, 1993.
want to bend toward the water with curvatures which are gentle. The monolayer will have no constraints set upon its desire for curvature. Experimentally we would do this by mixing the amphiphile in an excess of appropriate polar and nonpolar solvents,3,9,10 thereby removing any packing constraints and allowing the system to achieve a global, curvature elastic free energy minimum. We could then effect interfacial bending away from the minimum by the removal of polar solvent. We then express the energy required to remove polar solvent from the system at equilibrium in terms of the deviations in the interfacial curvatures away from the globally relaxed curvature cr. We will call these ∆ci ) (ci - cr), where i ) 1 and 2. The lowest order terms in the bending energy expansion will be quadratic in ∆ci. Hence,
1 1 gc ) k∆c12 + k∆c22 + k h ∆c1∆c2 2 2
(6)
where k is the splay modulus and has the same value for bending along c1 and c2, since the torque tension is isotropic, and k is the coupling modulus, which is not in general equal to k. At present (6) is quite general and will manifest the same range of behavior as (1) if we vary the sign and magnitude of k. However, the nature of (6) enables us to determine constraints on k as a function of the coupling between orthogonal bending modes. We will assume that the monolayer behaves as if it were an isotropic elastic sheet with a global energetic minimum which, in agreement with our previous discussion, will of course be spherical. To probe the physical meaning of k and determine the limits on its value under these constraints, we undertake a simple thought experiment (Figure 2). For the globally relaxed state c1 ) c2 ) cr and gc ) 0. We now unbend the interface in one direction, so that c1 ) 0. If we constrain the interface along the other direction so that ∆c2 ) 0, the free energy will have risen by gc ) 1/2kcr2 (the splay modulus must be positive). Releasing the externally imposed constraint that ∆c2 ) 0 will result in a change in the molecular splay along c2 until it reaches a local energetic minimum, which we can find by
( ) ∂gc ∂c2
c1)0
) k(c2 - cr) - kcr ) 0
(9) Kirk, G. L.; Gruner, S. M. J. Phys. 1985, 46, 761-769. (10) Khoo, B. J. Ph.D. thesis, Imperial College, London, 1996.
(7)
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Langmuir, Vol. 14, No. 26, 1998 7429
whence
( )
c 2 ) cr 1 +
k k
(8)
Substitution of (8) into (6) yields
( ( ))
1 k (gc)c1)0 ) kcr2 1 2 k
2
(9)
From (9) this means (k/k)2 < 1. Now unbending the monolayer along c1 while maintaining c2 constant reduces the cross-sectional area in the chain region. For the typical lateral stress profile of Figure 1 such compression of the chain region would be expected to increase the outward pressure here. One would therefore expect that reducing chain splay along c1 would increase the desire for chain splay along c2 and that c2/cr g 1. In this case (8) shows us that k/k g 0, and hence for this model of the monolayer
0e
k e1 k
(10)
At one extreme, where k ) 0, inspection of (6) shows that there is no coupling between orthogonal bending modes. That is, there is no transfer of lateral stresses between c1 and c2 when we bend the interface. At the other extreme, where k ) k, the orthogonal bending modes are completely coupled; that is, if we bend the monolayer in any direction, the system will respond in order to maintain c1 + c2 at a constant value of 2cr. Both limits therefore appear to be rather unrealistic and one would anticipate that experimental values would lie well within these extremes. One can identify the curvature elastic parameters of (6) with Helfrich’s expression, under the assumption that there is a global energetic minimum which is a spherical monolayer. This gives
k ) κ; k ) κ + κG; cr )
2κ c 2κ + κG 0
(11)
the last relationship being in agreement with (3). From (11) and (10) we find the following limits on κG/κ
-1 e
κG e0 κ
(12)
which are more constrained than those given in (2), because we have assumed that a compression of the hydrocarbon chains in one direction during unbending tends to increase the splay in the orthogonal direction. Naturally the range of allowed values for k/k sets a strict energetic hierarchy for different types of monolayer interfacial curvature. The same hierarchy exists in the Helfrich Hamiltonian under the constraints in (12) but may be seen most conveniently if we re-express (6) in terms of the mean curvature H ) 1/2(c1 + c2) and the Gaussian curvature, K ) c1c2
gc ) (k + k)(H - Hr)2 + (k - k)(H2 - K)
(13)
For purely spherical deformations, where H2 ) K, the curvature elastic energy is gc ) (k + k)(H - Hr)2. The system has a minimum in the free energy when the mean interfacial curvature is equal to its relaxed value.
Figure 3. Molecular splay deformations become increasingly more asymmetric as we move from amphiphiles on (a) spherical interfaces to amphiphiles on (b) cylindrical ones and then on (c) saddle-shaped interfaces. This increasing asymmetry sets a hierarchy in the curvature elastic energy of interfaces with these shapes.
For a cylindrical deformation, K ) 0, and hence there is an additional, positive contribution to the curvature elastic energy above that of a spherical interface of (k - k)H2. In line with the primary assumptions of our model, we have that at any value of H, as long as k/k < 1, the curvature elastic energy of a cylindrical deformation is always greater than that of a spherical deformation. The additional curvature elastic energy is bound up in the asymmetry of the deformation which is imposed on the molecules embedded on a cylindrical interface (Figure 3). We might aptly term this a curvature elastic frustration. The curvature elastic frustration is even greater for saddle-shaped deformations. Here K e 0, and hence there is yet a further positive contribution to the curvature elastic energy of -(k - k)K. This additional curvature elastic energy is tied up in even further asymmetry in the molecular deformation (Figure 3). Only if k/k ) 1 will the bending energy of all of these deformations be degenerate; otherwise, the curvature elastic costs on a hyperbolic interface are even greater than those on a cylindrical interface. A graphical representation of the free energy surface as a function of c1 and c2 has been used to illustrate this hierarchy (Figure 4). Lyotropic mesomorphism of amphiphile/water mixtures may then be understood in terms of a competition between the monolayer’s desire for isotropic interfacial curvature
7430 Langmuir, Vol. 14, No. 26, 1998
Templer et al.
Experimental Methods
Figure 4. Curvature elastic hierarchy illustrated by mapping the trajectories (white lines) for spherical, cylindrical, and saddle-shaped interfacial deformations on a contour plot of the curvature elastic energy in the space of curvatures c1 and c2. The curvature elastic energy has been calculated for k/k ) 0.3 and cr ) -0.2. The trajectory for a spherical deformation, that is, c1 ) c2, passes through the global minimum. The cylindrical deformation c2 ) 0 does not pass through the global minimum but comes in closer proximity to it than the trajectory for the saddle-shaped interface, that is, c1 ) -c2.
and the need for the monolayer to pack Euclidean space,11,12 which possesses the reverse hierarchical ordering. Therefore, if one wishes to determine curvature elastic parameters, such as κG/κ, it is important that the system has little or no packing frustration. This can of course be done by adding the appropriate alkane, but an alternative is to use the inverse bicontinuous cubic phases.13 Here, although the curvature elastic energy is maximally frustrated, the packing frustration is minimal.14 In this paper we have taken the approach of using such mesophases to estimate κG/κ. As we have shown elsewhere,13 if one is to use the first-order curvature elastic energy as an accurate description of the free energy of these phases, the unit cell dimensions must be sufficiently swollen so that the curvature inhomogeneities of the hyperbolic interface can be ignored and any interactions between bilayers become insignificant compared to the curvature elastic energy. To this end we have used the ternary lipid system MO/DOPC/DOPE. Not only are we able to make inverse bicontinuous cubic phases of the requisite dimensions in this system,15 but the absence of a swollen lamellar phase or an L3 phase in the pseudoternary phase diagram indicates that the bending rigidity of the monolayer is sufficiently high that bilayer fluctuations16 will not be a significant component of the free energy. (11) Kirk, G. L.; Gruner, S. M.; Stein, D. L. Biochemistry 1984, 23, 1093-1102. (12) Duesing, P. M.; Templer, R. H.; Seddon, J. M. Langmuir 1997, 13, 351-359. (13) Templer, R. H.; Turner, D. C.; Harper, P.; Seddon, J. M. J. Phys. II 1995, 5, 1053-1065. (14) Templer, R. H.; Seddon, J. M.; Duesing, P.; Winter, R.; Erbes, J. J. Phys. Chem. B 1998, 102, 7262-7271. (15) Templer, R. H.; Madan, K. H.; Warrender, N. A.; Seddon, J. M. In The structure and conformation of amphiphilic membranes; Lipowsky, R., Richter, D., Kremer, K., Eds.; Springer-Verlag: Berlin, 1992; pp 262-265. (16) Helfrich, W. Z. Naturforsch. 1978, 33a, 305-315.
X-ray diffraction was used to measure the lattice parameter as a function of water content in the inverse bicontinuous cubic phases formed in 58% MO/38% DOPC/4% DOPE. MO was obtained from Larodan AB (Malmo¨, Sweden), with a stated purity >99%, and was used as received. DOPE and DOPC were obtained from Avanti Polar Lipids (Alabaster, Alabama). Their purity was given as >99%, and they were used as received. In our calculations of lipid composition, it was always assumed that, even after extended drying, the DOPC remained dihydrate. The lipids were weighed and handled under dry N2 to prevent the uptake of atmospheric moisture by the lipids (DOPC in particular). They were then codissolved in cyclohexane, concentrated in a stream of N2 gas, pipetted into X-ray capillaries, and lyophilized. Known amounts of triply distilled, deionized water were added to the lipid samples and the capillaries sealed with heat shrink tubing. To mix lipid and water thoroughly, the capillaries were centrifuged and thermally cycled after sealing. Our calculations of the number of water molecules per lipid, ω, treats the MO/DOPC/DOPE mixture as a single molecular entity with a molecular weight given by the sum of the individual molecular weights of each component multiplied by their mole fraction. The samples were reweighed after X-ray measurement, by removing their heat shrink seals, to assess the degree of water loss suffered. The water loss always fell in the range 2-4% and was taken into account in the calculations of ω for each sample. We believe that the loss was due to absorption of water in the heat shrink polymer used to seal the capillary. The error in ω was estimated to be (2 water molecules. The lipid component of our samples was tested for degradation after the final measurement by thin-layer chromatography. There was no evidence of lipid decomposition. We found that it took a month of storage at 4 °C for the samples to be uniformly hydrated. All the samples were examined at 25 °C, after the samples had been brought to room temperature and kept at 25 °C for 10 min or more. X-ray measurements were made with an Elliott GX-20 rotating anode X-ray generator (Nonius, Netherlands) equipped with either double-mirror Franks optics to produce a beam of 150 µm × 150 µm cross section or a single-mirror Franks camera of 150 µm × 10 mm cross section. The latter enabled us to check that samples were uniformly hydrated, while the former improved the accuracy of our lattice parameter determinations. The diffraction was recorded using a CCD-based, two-dimensional detector, capable of quantumlimited X-ray detection. The X-ray repeat distances were calibrated against DOPE in excess water. Most of the samples produced spotty diffraction patterns and were therefore rotated during diffraction. The uncertainty in the lattice parameters for the very large cubics (above 200 Å) was (5 Å, that for those 100 Å and below was (1 Å, and that for those between 100 and 200 Å was (3 Å.
Results and Analysis Our results fall into two parts. First we show by a number of means that the cubic phases we observe are inverse bicontinuous structures based on periodic minimal surfaces. Having ascertained, this we go on to show that the cubic phases are sufficiently swollen that we can use our determination of the excess water composition to calculate the ratio κG/κ with confidence. The data we have been able to collect for the cubic lattice parameters as a function of water composition in our system are presented in Figure 5. An example of the quality of the diffraction patterns upon which the cubic symmetry and the lattice parameters were determined is shown in Figure 6. We have identified diffraction from three phases of cubic symmetry, Ia3d, Pn3m, and Im3m, as well as a fluid lamellar phase. All the data points shown in Figure 5 are, as far as we have been able to determine, monophasic. However, before sample equilibration there were large parts of the phase diagrams which exhibited phase coexistence. Where the system was in excess water, we frequently observed
Gaussian Curvature Modulus of an Amphiphilic Monolayer
Langmuir, Vol. 14, No. 26, 1998 7431
a
Figure 5. Measured variation in the lattice parameter as a function of water composition in 58% MO/38% DOPC/4% DOPE. The black squares are data from the Ia3d cubic phase, dark gray squares are data from the Im3m cubic phase, and light gray squares are data from the Pn3m cubic phase. We do not show the data for the lamellar phase between ω ) 11 and 31. The curves plotted over the data have been calculated from (14) and the data in Table 1.
coexisting Pn3m and Im3m cubic phases. The ratio of the Im3m to the Pn3m lattice parameters was invariably in the range 1.29 ( 0.02. For highly swollen inverse bicontinuous cubic phases based on the P and D minimal surfaces, one would anticipate a ratio of 1.28 in the lattice parameter,17 and this is compelling evidence that these are indeed the structures present in this system. This phase coexistence disappeared completely after one month. The evidence therefore pointed to the Ia3d phase in 58% MO/38% DOPC/4% DOPE being re-entrant, with an intervening fluid lamellar phase in the compositional range 11 < ω < 31. Over this range the lamellar phase swells from 58 to 60 Å. In all other respects the phase ordering of the inverse bicontinuous cubic phases appeared quite normal. The curves superimposed on Figure 5 have been calculated using a model of the swelling behavior of these phases16 which we describe in Appendix A. The model depends on the presence of a pivotal surface parallel to the underlying minimal surface. During isothermal changes in water composition it is then possible to determine the lattice parameter a for given values of the surface averaged area per molecule at the pivotal surface 〈An〉, the surface averaged volume per molecule between the pivotal and minimal surface 〈vn〉, and the average molecular volume 〈v〉
{
(〈v〉 - vwω)
a)
〈An〉
(
〈vn〉
〈v〉 - vwω
( )x (
[
〈vn〉
〈v〉 - vwω
(
〈vn〉
〈v〉 - vwω
) (
2
πχ 8σ03 + 9πχ
21/3 4σ03 + 9πχ 3
c
-2σ0 + 25/3σ02/ 4σ03 + 9πχ
3
[
b
)x (
(
+ 〈vn〉
〈v〉 - vwω
〈vn〉
(
1/3
+
2
〈v〉 - vwω
πχ 8σ03 + 9πχ
)
) )] 2
+
〈vn〉
〈v〉 - vwω
) )] } 2
1/3
(14)
where vw is the volume of a water molecule (29.9 Å3 at 25 °C), σ0 is the dimensionless area, and χ is the Euler characteristic of the underlying minimal surface in the unit cell. (17) Templer, R. H.; Seddon, J. M.; Warrender, N. A. Biophys. Chem. 1994, 49, 1-12.
Figure 6. Typical X-ray diffraction pattern profiles from (a) the Ia3d cubic with a lattice parameter of 247 Å, (b) the Pn3m cubic with a lattice parameter of 160 Å, and (c) the Im3m cubic with a lattice parameter of 227 Å. The indexed peaks and parasitic scatter around the beam stop are indicated in the figures. Spikes which appear in the profile but are not indexed are due to high-energy ionization events on the two-dimensional detector from radioactivity in glass elements next to the detector’s phosphor screen. These radioactive decays are quite easy to discriminate against, since they produce bright spots rather than diffraction rings.
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Table 1. Geometrical Data Used in Calculating a(ω) MOa DOPCb DOPE 58% MO/38% DOPC/ 4% DOPE
〈v〉/Å3
〈vn〉/Å3
〈An〉/Å2
H0/Å-1
612 1300 1220 898
465 950 820 664
33 74 65 50
-2.05 × 10-2 -5.73 × 10-3 -1.70 × 10-2 -1.47 × 10-2
a In the case of MO, 〈v〉 and H come from the work of Khoo,10 0 whilst 〈vn〉 and 〈An〉 come from calculations4 made on data from Chung and Caffrey.6 b For DOPC, 〈v〉 and 〈vn〉 come from Chen and Rand,7 H0 has been measured by Chen and Rand7 and by Keller and co-workers,38 and their values are in good agreement. c The value of 〈An〉 we quote is from our own unpublished measurements of the molecular cross-sectional area in the LR phase at 25 °C. It is in good agreement with the work of Gruner and co-workers,39 who measured the area at 10 °C to be 71 ( 4 Å2, but others have quoted values ranging from 67 to 82 Å.7,40,41 The values for DOPE have less variation, and we use the values quoted by Rand and co-workers.3
The curves drawn in Figure 5 are calculated from (14) using values of 〈An〉, 〈vn〉, and 〈v〉 determined for each individual component and adding them together in their molar proportions. The values we have used and their sources are shown in Table 1. The coincidence between the data and the model is exceptionally good and is further evidence that the cubic phases are indeed based on periodic minimal surfaces. The Ia3d cubic is based on the gyroid minimal surface, the Pn3m cubic on the D surface, and the Im3m cubic on the P surface.18 We can now go on to estimate the value of κG/κ from the excess water equilibrium geometry. It has been shown elsewhere (see Appendix B)13 that the curvature elastic energy of a swollen inverse bicontinuous cubic phase will be given by
[
〈gc〉 Anξ - vn ) 2H02 + (κG/κ - 4H0ξ) 2 κ ξ (vn - Anξ/3)
]
(15)
where ξ is the distance between the minimal and pivotal surfaces. ξ can be calculated from19
(
〈vn〉
) () ()
〈v〉 + ωvw
) 2σ0
4 ξ ξ + πχ a 3 a
Discussion Our re-expression of the Helfrich ansatz is identical to that derived by Wennerstro¨m and Anderson.8 However, our conclusion based on a simple model of the response of amphiphiles to bending, that -1 < κG/κ < 0, is not touched upon by them. The limit to the range of values the model predicts that κG/κ can have is in agreement with monolayer calculations made by Ennis,20 and negative values of the ratio have been calculated by Barneveld et al.,21 but in this case for bilayers (see comments below). This pleasing concordance between theory and computation is not however found with previous experimental investigations, where the values range from approximately -2 to +10. Large and positive values (between 10 and 6) were obtained by Boltenhagen and colleagues22 in the system cetylpyridinium chloride/hexanol/brine. Here polarizing microscopy was used to determine the eccentricity of focal conic defect structures in the region of the phase transition from the lamellar to the L3 sponge phase. A model of the bilayer structure of the focal conic defects was then used to determine the curvature elastic energy tied up in them. A number of pertinent points arise from this work. The curvature elastic energy was determined in terms of the amphiphilic bilayer rather than the monolayer as here. Modeling the bilayer requires that one places the model surface at the bilayer midplane. The bilayer value of the Gaussian modulus κbG and the monolayer value κG are then related by17,23,24
KbG ) 2(κG - 4κH0ξ)
(18)
3
(16)
For (15) to hold true, |〈K〉|ξ2 , 1.13 In this case |K|ξ2 ≈ 0.03 at the excess water point and (15) is therefore a satisfactory approximation. Differentiating (15) with respect to ξ and equating to zero gives us a relationship between κG/κ and H0 at the equilibrium value of ξ, ξ0
[
hydration because the interfacial curvatures in the cubic phases are gentle. This means that our calculation of ξ0 is rather accurate with an error of 4% or less. Our error estimate of H0 depends largely on the uncertainties in the measurement of H0 for MO and DOPC, and this amounts to an error on the order of 10%. We can therefore be confident that κG/κ is less than 0 and greater than -1, consistent with our model of the monolayer curvature elastic behavior.
]
3 - 2〈An〉ξ0/〈vn〉 + (〈An〉ξ0/〈vn〉)2 κG ) 2H0ξ0 κ 3 - 3〈An〉ξ0/〈vn〉 + (〈An〉ξ0/〈vn〉)2
(17)
We can calculate ξ0 from our measurements since we know what the value of ω is at the excess water point. We find ω ) 54.0 ( 0.5 and hence ξ0 ) 12.9 ( 0.5 Å. We can calculate H0 in the same way we determined 〈An〉, 〈vn〉, and 〈v〉 from the values of each individual component (see Table 1). In this way we obtain H0 ) -0.015 ( 0.002 Å-1 and hence from (16) κG/κ ) -0.75 ( 0.08. The error in our estimate of κG/κ is predominantly due to uncertainties in H0. There is little variation in ξ with (18) Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. Chem. Rev. 1988, 88, 221-242. (19) Turner, D. C.; Wang, Z. G.; Gruner, S. M.; Mannock, D. A.; McElhaney, R. N. J. Phys. II 1992, 2, 2039-2063.
As has been pointed out previously,8 this means that κbG is not a constant, since ξ varies as the interface is bent,4 and furthermore that it is possible to have a positive value of κbG, if 4κH0ξ is sufficiently negative. However, inserting reasonable values, one might obtain values for κbG/κb of +2 but not +10 (κb ) 2κ). It might of course be possible that the model used for the defects is incorrect, but it is equally interesting to note that the system is in a swollen lamellar phase. This is indicative of a low bilayer bending rigidity where the thermal fluctuations further reduce the bending rigidity at distances greater than the persistence length.25,26 The Gaussian curvature modulus is similarly renormalized, in this case becoming more positive.27 Effectively the crinkled nature of the bilayer makes it easier to form funnels between bilayers. As a consequence, the determinations of κbG/κb may become large and positive for curvature elastic determinations on systems with fluc(20) Ennis, J. J. Chem. Phys. 1992, 97, 663-678. (21) Barneveld, P. A.; Scheutjens, J. M. H. M.; Lyklema, J. Langmuir 1992, 8, 3122-3130. (22) Boltenhagen, P.; Lavrentovich, O.; Kleman, M. J. Phys. II 1991, 1, 1233-1252. (23) Helfrich, W.; Rennschuh, H. J. Phys. 1990, 51, C7189-C7195. (24) Ljunggren, S.; Eriksson, J. C. Langmuir 1992, 8, 1300-1306. (25) Helfrich, W. J. Phys. (Paris) 1985, 46, 1263-1268. (26) Peliti, L.; Leibler, S. Phys. Rev. Lett. 1985, 54, 1690-1693. (27) Kleinert, H. Phys. Lett. A 1986, 114, 263-268.
Gaussian Curvature Modulus of an Amphiphilic Monolayer
tuations at the same length scales as the measurement. This means of course that such measurements are not simple measurements of the local curvature elasticity. The measurements presented in this report avoid these renormalization effects, because of the greater rigidity of the bilayer. We are therefore measuring the bare rigidities. Other workers have calculated κG/κ in water/oil microemulsions from measurements of size and shape fluctuations of microemulsion droplets or L3 sponges.28-33 A pre´cis of much of this work can be found in the review of Kellay et al.34 This work has been done on di-2-ethylhexyl sulfosuccinate,28-30 a series of alkyl poly(ethylene glycol) ethers,31,33 and sodium dodecyl sulfate.32 The values of κG/κ can never be less than -2 in the theory used for modeling, although a few measurements tend to this value.28 In many cases the values do lie between 0 and -1, but positive values (as high as +8) continue to be found. However, the analyses are particularly sensitive to the particular model used for the free energy of mixing of the droplets, and there is clearly disagreement over which model is appropriate. Furthermore, it has been pointed out32 that the calculations are very sensitive to the determination of κc0. Indeed in the case of the calculations made by Kegel and co-workers,32 positive values of κG/κ arose because the sign of κ was negative, and similar problems have been reported by others.31 There is also evidence reported by Kellay et al.30 that the theories may not be correctly taking into account all of the thermal fluctuations and that oil molecules are present at the interface. This latter observation clearly has an effect on the bending rigidity, which depends on the degree of curvature of the interface.35 It would seem that problems with the underlying theory of fluctuating droplets and the multiplicity of complex measurements which have to be made in this technique render the reliability of the results open to question. Many of the authors have themselves noted these caveats. Finally we should report other estimates of κG/κ made from measurements on the structure and bending energy of inverse bicontinuous cubic mesophases. These have been made on didodecyl-β-D-glucopyranosyl-rac-glycerol,19 monoolein,36 and 2:1 (mole/mole) lauric acid/dilauroylphosphatidylcholine mixtures.14,17 The initial reports on the curvature elasticity of each of these materials were flawed by errors in the theory. After correction,13 values of κG/κ between 0 and -1 have been reported for all of these materials. However, in each case the degree of interfacial curvature in the mesophases is much greater than that reported here, and hence the first-order model used in their analysis is not accurate. Furthermore inter-bilayer forces have been ignored; something that is legitimate where the bilayer interfaces are around 100 Å apart but not at the small separation present in didodecyl-β-Dglucopyranosyl-rac-glycerol and monoolein. (28) Farago, B.; Richter, D.; Huang, J. S.; Safran, S. A.; Milner, S. T. Phys. Rev. Lett. 1990, 65, 3348-3351. (29) Meunier, J.; Lee, L. T. Langmuir 1991, 7, 1855-1860. (30) Kellay, H.; Meunier, J.; Binks, B. P. Phys. Rev. Lett. 1993, 70, 1485-1488. (31) Sicoli, F.; Langevin, D.; Lee, L. T. J. Chem. Phys. 1993, 99, 47594765. (32) Kegel, W. K.; Bodnar, I.; Lekkerkerker, H. N. W. J. Phys. Chem. 1995, 99, 3272-3281. (33) Freyssingeas, E.; Nallet, F.; Roux, D. Langmuir 1996, 12, 60286035. (34) Kellay, H.; Binks, B. P.; Hendrikx, Y.; Lee, L. T.; Meunier, J. Adv. Colloid Interface Sci. 1994, 49, 85-112. (35) Evans, D. F.; Wennerstro¨m, H. The colloidal domain. Where Physics, Chemistry, Biology and Technology meet; VCH Publishers Inc: New York, 1994. (36) Chung, H.; Caffrey, M. Nature 1994, 368, 224-226.
Langmuir, Vol. 14, No. 26, 1998 7433
In conclusion, we would state that the measurements presented here are possibly the first reliable estimate of κG/κ. They do not suffer from the effects of low bending rigidities and the ensuing uncertainties that currently exist as to how we should describe these. The curvatures are gentle, and hence a first-order curvature elastic theory is appropriate. Bilayer separations are sufficiently large that bilayer-bilayer interactions are negligible. The pivotal plane in this system has been independently located and is in agreement with the measurements of others, and our subsequent estimate of κG/κ is simply dependent on our location of the excess water composition of the cubic phase, which is a rather straightforward measurement. Acknowledgment. We wish to thank Drs. Andrew Parry, Peter Duesing, and Misha Kozlov for many fruitful discussions related to curvature elastic theories. This work was funded by EPSRC Grant Number GR/K21054. Appendix A To determine the swelling law for inverse bicontinuous cubic phases (that is a(ω)), we begin by assuming that the pivotal plane is parallel to the underlying periodic minimal surface. From differential geometry we know that the area of a parallel patch A(z) projected away from the minimal surface by a distance z is related to the area of the original patch A(0) by37
A(z) ) A(0)(1 + Kz2)
(A1)
The volume swept out in this projection v(z) is given by
1 v(z) ) A(0)z 1 + Kz2 3
(
)
(A2)
We can use (A1) and (A2) to find An, vn, or v in terms of K and the distances from the minimal surface to the pivotal surface ξ and, in the case of v, from the minimal surface to the polar/apolar interface l. However, because the curvature of the underlying minimal surface is inhomogeneous, we must calculate surface-averaged values of An, vn, and v. In fact, for our purposes it suffices to calculate the ratio An/vn,
〈An〉 〈vn〉
)
(
)
2 1 1 + 〈K〉ξ ξ 1 1 + 〈K〉ξ2 3
(A3)
Using the Gauss-Bonnet theorem, one can relate 〈K〉 to the lattice parameter via
〈K〉 )
2πχ σ0a2
(A4)
where χ is the Euler characteristic of the minimal surface in the unit cell and σ0 is the dimensionless area of the minimal surface, also in the unit cell. With (A3) and (A4) we can find an expression for the lattice parameter as a function of the distance to the pivotal (37) Hyde, S. T. J. Phys. Chem. 1989, 93, 1458-1464. (38) Keller, S. L.; Bezrukov, S. M.; Gruner, S. M.; Tate, M. W.; Vodyanoy, I.; Parsegian, V. A. Biophys. J. 1993, 65, 23-27. (39) Gruner, S. M.; Tate, M. W.; Kirk, G. L.; So, P. T. C.; Turner, D. C.; Keane, D. T.; Tilcock, C. P. S.; Cullis, P. R. Biochemistry 1988, 27, 2853-2866. (40) Lis, L. J.; McAlister, M.; Fuller, N.; Rand, R. P.; Parsegian, V. A. Biophys. J. 1982, 37, 657-665. (41) Marsh, D. Biophys. J. 1996, 70, 2248-2255.
7434 Langmuir, Vol. 14, No. 26, 1998
Templer et al.
surface from the underlying minimal surface. It therefore remains only to determine how ξ varies with ω in order to obtain the expression we require. We do this by noting that the volume fraction in the unit cell between the pivotal surface and minimal surface 〈vn〉/(〈v〉 + ωvw) is related to ξ by19
〈vn〉
ξ 4 ξ ) 2σ0 + πχ a 3 a 〈v〉 + ωvw
()
()
3
(A5)
where vw is the volume of a water molecule. Combining (A3), (A4), and (A5), we obtain a cubic equation in a,
()
()
[
()
An 3 3 A n 2 a2 36πχ vn a + 6σ0 v v (1 - φw) (1 - φw) v
2
+
]
32σ03
(1 - φw)3
We map these integrals on to the minimal surface which lies at the midplane of the inverse bicontinuous cubic bilayer. We assume that the neutral and minimal surfaces are parallel to each other, allowing us to use the following expressions from differential geometry
Kn )
K ; Hn ) Knξ; dSn ) dSu(1 + Kξ2); 2 1 + Kξ Sn ) Su(1 + 〈K〉ξ2) (B2)
where K is the Gaussian curvature on the minimal surface, ξ is the distance between the minimal and pivotal surfaces, Su is the surface area of the minimal surface within the unit cell, and 〈K〉 is the surface-averaged Gaussian curvature in the unit cell. Substituting these expressions into (B1), one obtains an equation which only contains terms in 〈K〉
) 0 (A6) 〈gc〉 ) 2κH02 +
(κG - 4κH0ξ)〈K〉 + 2κξ2〈K2/(1 + Kξ2)〉 1 + 〈K〉ξ2
which when solved gives
a)
[
{
〈v〉 -2σ0 + 25/3σ02/ 4σ03 + 〈An〉(1 - φw) 〈vn〉 2 9πχ(1 - φw)2 + 〈v〉
3(1 - φw)
( )
( )x ( 〈vn〉 〈v〉
[
( )
21/3 4σ03 + 9πχ(1 - φw)2 3(1 - φw)
( )x ( 〈vn〉 〈v〉
( ) )] 〈vn〉
πχ 8σ03 + 9πχ(1 - φw)2 〈vn〉 〈v〉
2
〈v〉
where the surface averages are given by the integral of the quantity in question over the surface area of the unit cell, divided by the unit cell surface area. We now expand the surface averages in the numerator and denominator and note that for very gentle curvatures we only need to retain terms up to and including 〈K〉
1/3
+ 〈gc〉 ) 2κH02 + (κG - 4κH0ξ)〈K〉
2
+
( ) )] }
πχ 8σ03 + 9πχ(1 - φw)2
〈vn〉
2
(B3)
1/3
〈v〉
(A7)
(B4)
We now have an expression for 〈gc〉 which is a function of 〈K〉 and ξ, but 〈K〉 and ξ are not independent variables. As we vary the Gaussian curvature of the minimal surface, that is, change the dimensions of the unit cell, we also change the distance to the pivotal surface. Hyde37 has shown that the relevant relationship is
Appendix B The curvature of hyperbolic surfaces is inhomogeneous, so we must determine the surface-averaged value of the curvature elastic energy for the monolayer of the inverse bicontinuous cubic phase 〈gc〉. If the unit cell surface area at the pivotal surface for a monolayer is Sn, then, from (1), we get
〈gc〉 ) 2κH0 + 2κ 2
∫S Hn2 dSn n
Sn
∫S Hn dSn n
- 4κH0
Sn
+
∫S Kn dSn
κG
n
Sn
(B1)
where Hn and Kn are the mean and Gaussian curvature at the pivotal surface.
〈K〉 )
Anξ - vn
(B5)
2
ξ (vn - Anξ/3)
where vn is the average volume per molecule between the minimal and neutral surfaces and An is the average crosssectional area per molecule at the neutral surface. Since both of these quantities are constant during isothermal bending, we can obtain an expression which is a function of ξ alone
〈gc〉 ) 2κH02 + (κG - 4κH0ξ)
LA980701Y
[
Anξ - vn 2
ξ (vn - Anξ/3)
]
(B6)
