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Quantifying Packing Frustration Energy in Inverse Lyotropic Mesophases P. M. Duesing, R. H. Templer,* and J. M. Seddon Department of Chemistry, Imperial College, London SW7 2AY, U.K. Received June 18, 1996. In Final Form: September 23, 1996X In this paper we investigate a simple model of inverse lyotropic mesophase energetics. The total energy is constructed by first minimizing the curvature elastic energy for the interface and then calculating the energy tied up in the chain extension variations that result for the various interfacial shapes and crystallographic space groups. We have calculated the chain packing energy in the harmonic approximation and find that we can separate this into two distinct terms. The first of these we call the packing factor, which is a constant for each interfacial shape and its associated crystallographic space group. The second term describes the variation in the packing frustration energy with mean curvature and monolayer thickness for the different interfacial shapes, i.e. spherical, cylindrical, and hyperbolic. Using this formalism and optimizing the mean interfacial curvature, we are able to build a global phase diagram in terms of the spontaneous mean curvature and the molecular length. The phase diagram we construct from the model places the phase boundaries between the inverse bicontinuous cubic, inverse hexagonal, and inverse micellar cubic phases in the expected regions of the diagram. This gives some encouragement to the widely held notion that the competition between interfacial curvature and hydrocarbon packing constraints can be used to explain lyotropic mesomorphism. However, the model is overly simplistic and breaks down in the regimes where the average interfacial curvature is at its greatest. Specifically it predicts that a bodycentered cubic arrangement of inverse micelles is of lower energy than an Fd3m packing, but the latter are the only arrangements which have been found to date.
Introduction The physically observed classes of lyotropic mesophases cover all possible arrangements of repetitive surfaces in three-dimensional space: from the one-dimensional stacking of planes extending in two dimensions, via the twodimensional stacking of cylinders extending in one dimension, to the recently discovered three-dimensional stacking of discontinuous inverse micelles. Another class of topologies not so readily visualized is formed by intertwined labyrinths of surfaces continuous in all three dimensions. Each of these topologies can be characterized conveniently by the midpoints of either the hydrophobic or the hydrophilic regions. For an inverse system, i.e. a system which tends to curve toward the aqueous phase, the midpoints of the water regions will be defined by lines or points, while the midpoints of the hydrophobic region form surfaces. These surfaces divide space symmetrically, with equal thermodynamic properties on either side. For this reason they cannot curve more toward one side than the other. Their mean curvature, the sum of the principal curvatures, must therefore be equal to zero at any point. The only possible solutions allowed by this constraint are either the special case of planes or the more general case of so-called minimal surfaces, which consist of saddle points or flat points everywhere. Any particular topology can then be defined equally by the center of the water phase, which we shall refer to as the skeleton, or the center of the hydrophobic phase, which we shall call the frame. An example of this construction which is straightforward to visualize is to be found in the packing of cylindrical monolayers (Figure 1). Since cylinders pack best in a two-dimensional hexagonal arrangement, they will have skeletons consisting of a hexagonal array of lines and frames reminiscent of a honeycomb in cross-section. Micellar stackings can be reduced to skeletal points and polyhedral planes,1 while the bicontinuous structures can X Abstract published in Advance ACS Abstracts, December 15, 1996.
(1) Charvolin, J.; Sadoc, J. F. J. Phys. (Paris) 1988, 49, 521.
Figure 1. Frame and skeleton of the inverse hexagonal phase.
be characterized by infinite periodic minimal surfaces (IPMSs) and their skeletal graphs.2-5 Skeleton and frame each contain the complete information about the symmetry and lattice parameter of a system, and each can thus be derived from the other. In the cases of uncurved frames, the frame is defined as the loci in space equidistant to unconnected but neighboring equivalent skeletal points. As we will see later, a more complex situation arises in a lattice which contains a number of distinct symmetry points. The interface must of course be located somewhere between the skeleton and frame, but its precise shape cannot be predicted by symmetry conditions alone; the energetics of the physical system will set the interfacial geometry. We are not yet at the stage of being able to determine the interfacial geometry for curved mesophases from the underlying energetics, so we must choose the (2) Schoen, A. H. NASA Technical Report No. D-5541, 1970. (3) Scriven, L. E. Nature 1976, 263, 123. (4) Anderson, S.; Hyde, S. T.; von Schnering, H. G. Z. Krystallogr. 1984, 168, 1. (5) Anderson, D. M.; Davis, H. T.; Scriven, L. E.; Nitsche, J. C. C. Adv. Chem. Phys. 1990, 77, 337.
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geometry first and calculate the energetics afterward. In this paper we will assume that the interface would like to optimize its curvature requirements. We will consider only binary systems, which will mean that as far as possible the interface will wish to have constant and homogeneous curvature. Taking this approach to defining the interfacial geometry has been used widely in the field.6-9 Yet curvature alone is insufficient to explain the mesophase behavior encountered in practice. The “packing constraint”sthe requirement that all hydrophobic volume in binary systems is filled by the lipid hydrocarbon chains, which are anchored at the interfaceshas often been cited as the next most important energetic contribution.8-11 The packing constraint is based on the reasonable assumption that chemically identical molecules would prefer to have identically extended hydrocarbon chains, their length being close to some preferred optimum value, the relaxed chain length. The subtlety of these systems is then brought about by the fact that three-dimensional Euclidean space cannot accommodate a periodic stacking of homogeneously curved parallel surfaces.12 In other words, no inversely curved lyotropic liquid crystalline arrangement can exist, in which the interface curvature and the chain extension are both uniform. The idea is easily visualized with reference to the frames, which define the midpoints between any periodic arrangement of similar surfaces in space. As noted above, they will have to display zero mean curvature everywhere but also either singularities in principal curvatures at the polyhedral edges orsin the case of IPMSssvariations in principal curvatures throughout. In either case, a surface parallel to an inhomogeneously curved surface cannot be homogeneously curved. An inverse binary lyotropic system will thus always display frustration, either in curvature or in chain extension or both, unless it should prefer not to display any curvature and thus be happily stacked in lamellar planes. This paper then is motivated by a desire to formalize the oft expressed, intuitive idea that observed mesophase behavior could be explained in terms of the competition between the desire for curvature and the need to pack Euclidean space. To that end we have constructed a formalism for determining the relative packing energy cost for a particular interfacial topology (hyperbolic, cylindrical, or spherical) enclosed by a packing frame of specified symmetry. In principle this should allow one to determine the lowest energy packing symmetry for each interfacial topology under certain physical constraints. In this article we constrain the interface to have constant and homogeneous mean curvature. This constraint has been much used in previous models, and here we examine the effect of its use on the model’s ability both to correctly predict the lowest energy packing for a particular interfacial topology and to predict the presence of phase transitions. Driving Forces for Phase Transitions As we have already stated in rather general terms, we may quantify the frustration energy by assuming the (6) Stro¨m, P.; Anderson, D. M. Langmuir 1992, 8, 691. (7) Kozlov, M. M.; Winterhalter, M. J. Phys. II 1991, 1, 1077. (8) Kirk, G. L.; Gruner, S. M.; Stein, D. E. Biochemistry 1984, 23, 1093. (9) Anderson, D. M.; Gruner, S. M.; Leibler, S. Proc. Natl. Acad. Sci. U.S.A. 1988, 85, 5364. (10) Tate, M. W.; Gruner, S. M. Biochemistry 1989, 28, 4245. (11) Charvolin, J.; Sadoc, J. F. Colloq. Phys. 1990, C7, 83. (12) Sadoc, J. F.; Charvolin, J. J. Phys. II 1986, 47, 683.
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curvature energy is optimized, which defines the interface geometry, and then determining the variation in the chain length. Noting that the mean chain length must equal its optimum value, we may then determine the variance in this value over the unit cell in order to determine the frustration energy. Had we decided to optimize the chain length instead, we would have found that, in all except the IPMS frames, curvature singularities develop. So in some senses our choice of interfacial geometry is an avoidance of mathematical difficulty, but it nevertheless is of use in obtaining a first understanding of the mechanism behind phase transitions in curved binary inverse lyotropic liquid crystals. The physical implications of our choice that the curvature elastic free energy of the interface may be optimized first and the chain stretching frustration subsequently calculated is that the monolayer bending rigidity is much greater than the chain stretching rigidity. Using Helfrich’s formalism,13 the bending rigidities are defined via the equation for the curvature elastic energy per unit area of interface, gc:
gc ) κ(H - H0)2 + κGK
(1)
where H is the mean curvature, H0 is called the spontaneous mean curvature, K denotes the Gaussian curvature, and κ and κG are the respective bending rigidities. An optimization of this equation under the constraint of a two-dimensional discontinuous stacking will yield a strictly cylindrical interface, regardless of the relative values of κ and κG. Equally it will yield spherical interfaces for a symmetry based on three-dimensional discontinuous stacking. A different situation arises for geometries based on IPMSs. Here the optimization of the curvature requirements will yield different shapes, depending on the respective values of κ and κG. Only if κG should be negligible compared to κ, do we find a situation where the curvature could be optimized equally well in all classes of topology. In such a case, calculations of IPMS-based structures of constant mean curvature are available, which allow us to compare the energetics of constant mean curvature interfaces having hyperbolic, cylindrical, and spherical topology.5,14 Again we follow this approach, in common with previous attempts at such modeling, largely because of the mathematical complexities imposed by any other route. It has been noted that, the greater the desire for interfacial curvature, the more likely it will be that the system will switch from lamellar to IPMS to cylindrical to micellar symmetries.15 Yet all the curved classes of topologies can in principle accommodate any value of mean curvature, within some absolute physical boundaries, e.g., the outer diameter of a cylinder formed by a monolayer can never be smaller than twice the average chain length. Nevertheless, the mean curvature at the interface can in principle go close to infinity. In the low-curvature limit of very large lattice parameters, the assumption of a curvature-homogeneous interface would generally lead to prohibitive extension of the chain length in some regions. Each amphiphile will have a defined maximum extension, given by all carbon-carbon bonds in a trans configuration in a saturated system. Stretching beyond this limitsas implied by a very large lattice parameter in the constant curvature modelswould thus lead to the formation of a substantial vacuum and thus be physically impossible. (13) Helfrich, W. Z. Naturforsch., C 1973, 28, 693. (14) Gackstatter, F. Colloq. Phys. 1990, C7, 163. (15) Seddon, J. M.; Templer, R. H. Philos. Trans. R. Soc. London, Ser. A 1993, 344, 377.
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amphiphile. At this point we imagine an interface either spherical, cylindrical, or hyperbolic of constant mean interfacial curvature where there is no packing frustration. The monolayer width in this hypothetical situation is the relaxed value, lr. Placing the appropriate frame around the monolayer, we can use a harmonic spring model to measure the packing frustration energy, gp, due to departures in chain length from the relaxed value:
gp ) k(l - lr)2 Figure 2. Origins of the packing frustration in the inverse hexagonal phase. The lipid molecules must stretch into the hexagonal corners of the frame and compress against the faces. The ideal frame in which all lipids have the same extension is indicated by the dotted line. In modeling the packing frustration we assume a constant radius of interfacial curvature R and measure the variance in the lipid length l via the skeleton to frame distance ζ.
These physical boundaries vary for the different classes, because, for the same mean curvature, IPMS-, cylinder-, and micelle-based structures will require increasingly larger lattice parameters, respectively. It could thus be argued that with increasing or decreasing curvature some structures can no longer exist, as the resulting lattice would need to be physically too small or too large. Yet we find that many physical systems show phase transitions even outside the regimes where such extreme constraints are likely to hold. We thus need to quantify which geometry should prevail if all the different systems are theoretically accessible. A Formalism for Modeling Chain-Stretching Frustration Given two possible geometries which satisfy the curvature requirements equally well, the phase-determining contribution will arise through some packing factor. The space on one side of each interface will be filled by the hydrocarbon chains and will thus prefer to have uniform thickness: a condition that cannot be fulfilled by homogeneously curved inverse structures. We can most easily illustrate this problem by looking at inverse cylindrical packing (Figure 2), but the problem is similar in all topologies. Cylinders will want to stack on some twodimensional lattice in such a way that the space associated with each cylinder is as similar in shape to the interface as possible. The ideal arrangement would thus be a cylindrical cross-section for the outer end of the hydrocarbon chains, i.e. the frame. Yet such a geometry is not available in Euclidean space, as cylinders cannot fill space. In general, the frustration in any particular lattice will be minimized, the more similar the frame is to the preferred shape of the interface. Given a cylindrical interface, this statement is equivalent to saying that the cylinders will prefer to sit in a lattice, which will offer them as many nearest neighbors as possible, where an infinite number of nearest neighbors would be equivalent to the frustrationless ideal of a cylindrical frame. Yet it can easily be shown that the largest possible number of equal neighbors for cylindrical stacking is afforded by a hexagonal symmetry, the hexagon being the highest order regular polygon that can fill the plane, as implied by the condition that an integer multiple of the outer angle of a space-filling polygon must be 360°. It is thus not surprising that the only cylindrical stacking witnessed experimentally is of hexagonal symmetry. In order to quantify the frustration still inherent in such a system, we have to introduce a model for the energy contained in variations in the chain extension of the
(2)
where k is the stretching rigidity and l is the monolayer width at some point on the interface. Evidently this model does not take account of the absolute limits which restrain chain extension and compression. A more advanced model would thus be given by
gp ) k′
f(l) (l - lmin)(l - lmax)
(3)
where lmin and lmax are the minimum and maximum values of the chain length, k′ is the new chain stretching rigidity, and f(l) must be a function which gives us a minimum in the energy at l ) lr. Upon differentiating and equating eq 3 to zero we still find that the simplest solution is
f(l) ) (l - lr)2
(4)
Hence eq 2 is simply the first term in the Taylor expansion of eq 3, where
k)
( )
1 d2 g 2 dl2 p
)
l)lr
k′ (lr - lmax)(lr - lmin)
(5)
Working with the simple harmonic model and implementing the assumption that the interface has a constant radius of curvature, R, we can write the surface averaged packing energy, 〈gp〉, as
∫A(l - lr)2 dA ∫AdA
〈gp〉 ) k
(6)
where A is the interfacial area over which we must integrate and l is simply given by
l)ζ-R
(7)
and ζ is the corresponding distance between skeleton and frame (e.g. Figure 2). We can also define the relaxed value of ζ, ζr, which we would find without the imposition of the frame
ζr ) R + lr
(8)
For a given interfacial radius of curvature R, the size of the appropriate frame is determined by finding that frame which minimizes the energy of extension around lr. We therefore find
d〈gp〉 ∝ dζr
∫A(ζ - ζr) dA )0 ∫AdA
(9)
leading to another expression for ζr:
ζr )
∫Aζ dA ∫AdA
(10)
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That is, the relaxed value of ζr is equal to the surface average of ζ. Hence we obtain the actual packing frustration energy:
{
〈gp〉 ) k
}
∫Aζ2 dA 2 - ζr ∫AdA
(11)
By simple rearrangement we find a somewhat more elegant expression
[
]
∫A(ζ/ζr)2 dA 〈gp〉 ) k - 1 ζr2 ) kζr2 dA ∫A
(12)
where , the packing factor, is equal to the term in brackets and provides us with a dimensionless figure of merit for the packing of the chains into a particular frame symmetry. It should be appreciated that this is not equal to the geometrical packing factor, which measures the fraction of unfilled volume in a packing of curved objects. Rather the packing factor is the normalized variance in the distance from skeleton to frame. The smaller the value of , the more favorable the energetics of packing are likely to be. For example, is smaller for a two-dimensional hexagonal frame than for a square stacking of cylinders. The term ζr2 tells us about the scaling of the energy with mean curvature and molecular length. As we will go on to show, the behavior of ζr2 with mean curvature is markedly different between the different interfacial types. The packing factor then can be thought of as discriminating between different possible packing symmetries for each interfacial type. Computation of Packing Factors We now proceed to the calculation of these packing factors. Note that it is sufficient to use some primitive sections of any unit cell, which can reproduce the whole lattice through any kind of symmetry operation. Thus from eq 12
∑m∫patchζ2 dA)(∑m∫patchdA) - 1 (∑m∫patchζ dA)2
( )
(13)
where m is the relative number of any one patch in a unit cell and the summation adds all the different patches. In general we need to find an expression for ζ as a function of two parameters to be able to scan the surface. For frames consisting of flat faces, it is convenient to use two orthogonal angles, where these are measured with respect to the line normal to the frame which bisects the skeleton. We can thus write
ζ ) px1 + tan R + tan β
(14)
dA ) R2 dβ dR
(15)
and
where p is the distance between the skeleton and the point of closest approach on the face, while R and β are orthogonal angles scanning the face. Cylindrical Packing. For cylindrical symmetries the problem is reduced to two dimensions. We can thus simplify the procedure by keeping one angle equal to zero, in view of the equivalence of all sections along the axis.
Table 1. Packing Factor E for Various Forms of Cylindrical Packing symmetry triangular square hexagonal (octagonal/square)
45.792 × 10-3 11.043 × 10-3 1.865 × 10-3 (0.561 × 10-3 + 11.043 × 10-3)/2
We then need to construct the frame for each space group. Again this task is simpler in cases of cylindrical stacking, where as noted above we simply need to construct a list of regular polyhedra that fill space. The appropriate packing factors are shown in Table 1. Micellar Packing. In order to analyze the micellar phases, we need to tessellate space for each cubic space group, through the planes which are equidistant to two neighboring symmetry points. The crossing point of three such planes will then yield a vertex. Choosing the appropriate three planes is facilitated by noting that all the relevant symmetry points will be close to each other. The collection of vertices will yield faces which can be reconstructed in two dimensions, by computing the distances between different vertices. This procedure is adequate for cubic lattices of just one type of symmetry point. We give some values for the integration limits in Table 2 and illustrate the basic polyhedra of the frames for the sc, bcc, and fcc lattices in Figure 3. The skeleton is described by the central points of the polyhedra. As an illustration of the packing frustration, we also show the superimposed spheres of radius ζr, which would correspond to a frustrationless frame. Yet the only experimentally witnessed inverse micellar stacking is of Fd3m symmetry. Disconcertingly, this space group has two types of symmetry points, corresponding to two distinct micelles, referred to as “big micelles” and “small micelles” in the literature.15-18 The procedure for tessellating space is thus no longer rigidly defined. While the face between two equivalent symmetry points must be located halfway between the two, there is no apparent answer to how far from each the face between two distinct symmetry points should be located. One possible approach would be to ignore the distinct nature of the micelles and still fix the planes at halfdistance to either. The model here is of micelles, which retain the same interfacial curvature, the differences in shape of the frame simply being accounted for in chainlength variation. This approach is equivalent to the assumption of optimizedsand thus equalscurvature everywhere; i.e., the average value of ζ is the same in the two polyhedral frames. The resultant frames illustrated in Figure 4 are somewhat oblate and hence do not perform very well compared to the basic cubic arrangements (Table 3). As a second approach, we could allow different average radii of curvature for the two distinct micelles, as set by the relative distances of closest neighbors within both sets of symmetry points. Thus the distance from the center of a big micelle to the closest adjacent big micelle is (x3/8)a, while the equivalent distance for small micelles is (x2/8)a. The face between a big micelle and a small micelle could thus be fixed at x3/x2 the distance from the former to the latter. The result is an increase in the sphericity of each polyhedron, as illustrated in Figure 5. (16) Charvolin, J.; Sadoc, J. F. J. Phys. II 1988, 49, 521. (17) Luzzati, V.; Vargas, R.; Gullik, A.; Mariani, P.; Seddon, J. M.; Rivas, E. Biochemistry 1990, 31, 279. (18) Seddon, J. M.; Bartle, E. A.; Mingins, L. J. Phys. Condens. Matter 1990, 2, 285.
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Table 2. Faces and Integration Limits of the Frames in Various Cubic Space Groupsa
a
The small black circles near the center of each force denote the point of closest approach to the skeleton.
By choosing to do this, we accept a difference in preferred curvature in the two types of micelles, where Hsmall ≈ (x3/x2)Hlarge. We note that such a proposition is at odds with our general condition that the curvature should be optimized everywhere; however, the reader should note that within each micelle the curvature remains homogeneous. We
will return to the implications of this problem later. Now both of these polyhedra display promising packing factors individually, as is to be expected for a relatively round dodecahedron and hexadecahedron, and the Fd3m packing produces the lowest packing factor (Table 3). Hybrid Cylindrical Packing. In view of the fact that we have had to use micelles with two differing radii of
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Figure 5. Frame-defining polyhedra of minimum frustration and their ideal spherical frames for (a) large micelles and (b) small micelles of different interfacial curvature in the Fd3m symmetry.
Figure 3. Polyhedral frames for (a) the bcc lattice, (b) the sc lattice, and (c) the fcc lattice, showing the spheres corresponding to the frustrationless frame in each case.
Figure 4. Frame-defining polyhedra and ideal frames for (a) large micelles and (b) small micelles of equal interfacial curvature in an Fd3m packing of micelles. Table 3. Packing Factor E for Various Forms of Micellar Stacking symmetry simple cubic body-centered cubic face-centered cubic Fd3m (equivalent symmetry points) Fd3m big micelle (distinct symmetry points) Fd3m small micelle (distinct symmetry points) Fd3m weighted average (distinct symmetry points)
15.544 × 10-3 4.166 × 10-3 4.445 × 10-3 6.484 × 10-3 2.126 × 10-3 4.196 × 10-3 3.506 × 10-3
curvature to produce a favorable packing factor for the Fd3m phase, it would be reasonable to consider the possibility of cylindrical stackings of two or more types of cylinders. As we can obtain a relatively good idea of the packing factor of any particular cubic symmetry, simply by looking at the number of faces in each polyhedron, we can do the same for the various cross-sectional polygons we might encounter in cylindrical stackings. The obvious case to consider would be a symmetry with a cross-section based partly on octagons, which would be an improvement on the pure hexagons. Yet to fill the void we would require an equal number of squares. Looking at the values in Table 2, the improvement in packing factor brought about by the octagons would clearly not compensate for the deficit induced by the squares, unless the “octagonal” cylinders are of far larger diameter than the “square” cylinders, such that the weighted average per interface area would weight the former far more than the latter. Such a hybrid system would then show a comparable packing factor to that of the hexagonal phase, if the radius in the octagonal frame were 7 times larger than that in the square frame,
neglecting that under such conditions the octagons would no longer be regular. IPMS-Based Phases. Finding an appropriate form for the packing factor calculations in the IPMS phases is somewhat complex. Primarily, it is no longer clear what shape a curvature-optimized interface would yield, as we have already discussed. If κG should be negligible compared to κ, it would be mathematically correct to compute an infinitely periodic surface of constant mean curvature, defined by an average distance lr from the corresponding IPMS. One then integrates the normal distance from the former to the latter across some primitive section. Yet triply periodic surfaces of constant mean curvature are difficult to access and only recently has a procedure for generating them, rather than simply proving their existence, been formulated.5,14 The relevant normals are still not easily available, and indeed it is questionable whether such a molecular orientation is appropriate. Nevertheless this approach is possible and has been followed through by others.9 In order to provide greater flexibility with respect to further modifications to theoretical modeling, especially if one should chose to simultaneously relax curvature and packing frustration, we chose to construct a somewhat artificial but consistent model along the same lines as our treatment of the discontinuous phases. A distribution of ζ can be defined as the distance along the normal to the skeleton of the IPMS, to the point where the normal crosses the actual IPMS, which corresponds to the frame in this class of topologies. We follow through this procedure for the most frequently observed IPMS-based structure, of Pn3m symmetry, based on the Schwarz D-surface. The skeleton is a double-diamond structure, with tetrahedral angles found at the fourfold nodes. A quadrupole primitive patch of the surface and the corresponding part of the skeleton are shown in Figure 6. In a coordinate system and scale as given by the usual Weierstrass representation, the position of the node can be determined by noting that it must coincide with the focus of the extreme points of the primitive patch.19 The relevant part of the skeleton is thus given by
Pframe )
( ) ( ) 0 0.816 0 +λ 0 -0.843 0.578
(16)
where Pframe is the position vector defining the skeleton and λ is a scalar parameter. Normals to the skeleton, n, with components, nx, ny, and nz, have to obey the conditions (19) Terrones, H. Colloq. Phys. 1990, C7, 345.
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Figure 6. Unit patch and corresponding section of the frame for the Schwarz D minimal surface.
()( )
nx 0.816 1 n 0 o )0 y |n| n 0.578 z
(17)
nz ) cos R |n|
(18)
Figure 7. Variation of ζr with mean curvature in the various classes of topology. The spherical interface is represented by the solid line, the cylindrical interface by the long dashed line, and the D minimal surface by the short dashed line.
and
The primitive patch is scanned by varying λ to move along the skeleton and altering the angle R to the vertical. A function ζ(R,λ) can then be computed by an algorithm which measures the length of the normal from the skeleton to the point where it crosses the IPMS. The limits to R are given by the crossing points with the tetrahedral edges of the primitive patch. One limit of λ corresponds to the point on the skeleton, where the normal with R ) 0 intersects the vertex of the primitive patch. At the other limit, the normal goes through the central point of the patch shown in Figure 6, which corresponds to the origin in this representation. Given these limits, the usual integration can then be performed numerically, to obtain a packing factor of 2 × 10-3 for a geometry based on the Schwarz D-surface. We note that this value is in quite good agreement with the result computed by Anderson and co-workers9 (see their Figure 2b curve d), who find the packing factor to be approximately 3 × 10-3. Determination of the Packing Frustration Energy and Phase Transitions
Table 4. Variation of ζr with Mean Curvature in the Different Classes of Topology 1 lr IPMS c1 ) ζr ) + lr2 ζr - lr H 1 c2 ) -ζr - lr 1 1 cylindrical c1 ) ζr ) + lr ζr - lr 2H
x
micellar
To determine the packing frustration for the different interfacial topologies, we note that ζr is related to the mean interfacial curvature in a different way in each case. The relationships (Table 4) are given by finding the solutions to the definition of the mean curvature in terms of the principal curvatures c1 and c2
1 (c + c2) ) H 2 1
Figure 8. Packing frustration energy according to the optimized mean curvature model for the HII phase (solid line) and the Pn3m bicontinuous cubic phase (dot-dashed line). The relaxed monolayer thickness is assumed to be independent of the mean interfacial curvature.
(19)
Using these relationships, we have plotted ζr/lr as a function of |H0|lr (Figure 7). This gives us some indication of the relative effect of the packing frustration energy for a system at its global energetic minimum in our model; i.e., we have reduced the interfacial curvature energy to gc ) κ(H - H0)2 by our preliminary assumptions. We thus see that, ignoring packing factors, the tendency will always be to favor IPMS phases. However, when lr is large
c2 ) 0 c1 ) c2 )
1 ζr - lr
ζr )
1 + lr H
compared to 1/|H0|, i.e. with high curvature, the difference in energy between the IPMS phases and the hexagonal and then even micellar phases becomes increasingly small. If we include the packing factor terms, we find a phase transition between saddles and cylinders for |H0|lr ≈ 1.4 (Figure 8) but no phase transition from cylinders to spheres. We must however treat these results with some caution, because lr is in fact a function of the mean curvature. This fact has often been ignored, and the subsequent energetic modeling must, as a result, be partially flawed.8,20,21 To model the functionality, we assume that the headgroup, of cross-sectional area Ah, is incompressible and that the lipid density in the monolayer is constant and then solve (20) Turner, D. C.; Wang, Z.-G.; Gruner, S. M.; Mannock, D. A.; McElhaney, R. N. J. Phys. II 1992, 2, 2039. (21) Chung, H.; Caffrey, M. Nature 1994, 368, 224.
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Figure 9. Hypothetical phase diagram for an optimized mean curvature interface where the change in monolayer thickness as a function of mean curvature has been modeled; Qm II (bcc) denotes the body-centered cubic, inverse micellar phase, and QbII(Pn3m) denotes the Pn3m bicontinuous cubic phase.
the following differential geometric relationship for each interfacial topology:
lf ) lr + |H|lr2 +
1 Kl 3 3 r
(20)
where lf is the monolayer thickness for the flat interface and is given by lf ) v/Ah, v being the molecular volume. This approach is described in detail elsewhere.22 Here we simply note that we have set the area neutral surface at the headgroup to simplify the mathematics but that this has no effect on the general applicability of our results. In the case of the cylindrical interface, where K ) 0, and the spherical interface, where K ) H2, the solutions to eq 20 are straightforward. For the hyperbolic interface however we have a far more complex task at hand. For a constant mean curvature hyperbolic interface K varies over the interface and hence so must the relaxed chain length; this is not the case for cylindrical and spherical interfaces where the relaxed chain length is uniform. To determine the frustration energy for the IPMS properly, we therefore need to know how K varies as a function of H at each point on the surface, but such computations have yet to be published. Instead we use the approximation that the average value of lr at some H for the constant mean curvature IPMS interface is given by the value of lr for the IPMS of constant thickness having the same average value of H. In this case we have that K ) |H|/lr and the solution to eq 20 is trivial. The additional sophistication of including the functionality of lr with H in our calculations now gives rise to phase transitions between each of the interfacial topologies (Figure 9). The phase diagram has qualitative features which are in agreement with a number of notions about the phase behavior of lyotropic systems. As we increase the magnitude of H0, for example by heating the system, we drive the system from the bicontinuous phases, denoted QbII, through HII to the inverse micellar phases, denoted Qm II . If we increase the hydrocarbon chain length, it is generally believed that we increase the magnitude of the spontaneous mean curvature as well. This being the case, we see that short chain lengths favor the bicontinuous (22) Templer, R. H. Langmuir 1995, 11, 334.
Figure 10. Variation in monolayer thickness with mean curvature. The calculations are made for a molecule with lf ) 20 Å, the solid line being for a spherical interface, the long dashed line for a cylindrical interface, and the short dashed line for a hyperbolic interface.
cubic phases, whilst long chains favor the inverse micellar cubic phases. This again fits in well with experimental results.15,23 The qualitative behavior of the phase diagram seems reasonable, but there are several worrisome features. Most glaringly the optimized mean curvature model predicts that the bcc micellar packing is the stable inverse micellar cubic phase rather than the Fd3m symmetry. There are at least three purely binary amphiphile/water systems now known to form inverse micellar cubic phases,24 and they all do so with Fd3m symmetry. This implies that the presumption that the energy of mean curvature deformations of the interface predominate all other energetic terms is incorrect. The packing factor calculations show that the Fd3m phase only outperforms the fcc and bcc symmetries under the assumption that the interface assumes two different radii in the two distinct sets of micelles. Indeed electron density reconstructions of the Fd3m structure show that the inverse micelles are of different size,17 all of which indicates that the system is reaching an accommodation between curvature and packing energetics, in which the packing energy can no longer be considered to be a small perturbation. It is also notable that the degree of mean curvature needed to reveal the micellar cubic phases is unphysical, the radius of the interface being of the order of 0.1 Å. Experimentally we know that the water core diameter of an Fd3m phase is of the order of 10 Å.17 A detailed examination of the boundary between the bicontinuous cubic and inverse hexagonal phases reveals a similar though less dramatic problem. These quantitative errors occur because we have ignored the contribution to the mesophase energetics of the Gaussian curvature of the interface. This has two effects on the molecular geometry. The first is that in the absence of packing frustrations the relaxed molecular length at any value of the mean curvature, except zero, is different for each interfacial topology (Figure 10). The second is that the average molecular shape is different. For example the molecules on a spherical interface occupy a coneshaped volume, whilst on a cylindrical interface the cone shape must be distorted to have an elliptical cross-section. Such changes in shape have some associated entropic cost of “confining” the chains in anisotropic geometries. We (23) Templer, R. H.; Seddon, J. M.; Warrender, N. A.; Syrykh, A.; Huang, Z.; Duesing, P. M.; Winter, R.; Erbes, J. J. Phys. Chem., submitted. (24) Seddon, J. M.; Zeb, N.; Templer, R. H.; McElhaney, R. N.; Mannock, D. A. Langmuir 1996, 12, 5250.
+
+
Inverse Lyotropic Mesophases
have so far ignored the energy of these molecular deformations because we set κG to zero, which made all interfacial topologies energetically degenerate in the absence of packing frustrations. We have not calculated additional energetic contributions due to κGK. However, it is clear that with κG negative spherical interfaces have the lowest curvature elastic energy, and this will tend to expand the region of stability of the inverse micellar cubic phases to more realistic values. The effects on the inverse bicontinuous cubics will be more subtle, since the value of κG will alter the curvature-optimized shape of the interface and hence the variation in molecular geometry. We show elsewhere23 that, in the case of a constant mean curvature, inverse bicontinuous cubic phase, the interface based on the D minimal surface is always of lower curvature elastic energy than the interface based on the P minimal surface. There thus remains the intriguing possibility that optimizing the curvature elastic energy in order to reduce the variance in chain extension for these two cubics might give rise to a phase transition due to changes in the value of κG. Conclusion We have presented a formalism for calculating the relative energetic packing cost for curvature-optimized monolayers. This has the potential to allow one to examine all possible packings of a monolayer of prescribed interfacial shape and determine the energetically most favorable packing. With the assumption that the mean interfacial curvature is optimized, we have shown that one can derive a qualitatively useful, universal phase diagram. However, ignoring the fact that the Gaussian curvature of the interface affects the molecular geometry leads to unphysical phase transition boundaries and is (25) Duesing, P. M.; Templer, R. H.; Seddon, J. M. In preparation.
Langmuir, Vol. 13, No. 2, 1997 359
inconsistent with a model in which we include chain stretching energetics. Furthermore, throughout this model any interactions between monolayers, e.g. van der Waals forces and the hydration repulsion, have been ignored. It could thus be that, at very low hydration or higher curvatures, the structure based on the surface with the largest average distance between “pores” will be at an advantage, while the opposite might be true in excess water and for very low curvature requirements. It is also clear that continuum theories of highly condensed lyotropic mesophases cease to be meaningful if there are significant inhomogeneities in molecular shape. Thus models, such as the one described here, will break down in extreme cases. Finally, it should be pointed out that in this model the chain extension has been decoupled from the molecular splay. It is clear that such a decoupling cannot occur if the monolayer density is to remain homogeneous. Furthermore, since the chain extension varies on any frame, the chemical potential (at least in terms of the energetic components described here) must as well. A moment’s thought should convince the reader that changing the molecular length can only be accomplished by changing the molecular splay and/or the molecular cross-sectional area at the neutral surface. We are thus left to conclude that a more satisfactory theoretical description requires the computation of compromise solutions to curvature and chain-stretching optimization, where these effects are coupled. The mathematics of such problems are particulary complex, but we have made a first attempt at such modeling, this paper being a prelude to that work.25 Acknowledgment. P.M.D. wishes to thank Unilever plc for a research studentship. LA960602P
