Published: July 05, 2011

r 2011 American Chemical Society 10475 dx.doi.org/10.1021/la201718a | Langmuir 2011, 27, 10475–10483

ARTICLE

pubs.acs.org/Langmuir

A Bicontinuous Mesophase Geometry with Hexagonal SymmetryGerd E. Schr€oder-Turk,*,† Trond Varslot,‡ Liliana de Campo,‡ Sebastian C. Kapfer,† and Walter Mickel§,||

†Theoretische Physik, Friedrich-Alexander-Universit€at Erlangen-N€urnberg, D-91058 Erlangen, Germany‡Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University,0200 ACT, Australia§Universit�e de Lyon, F-69000, Lyon, France

)CNRS, UMR5586, Laboratoire PMCN, Lyon, France

Examples of materials that exhibit hexagonal symmetryabound. For instance, a hexagonal arrangement of cylinders

is a geometry found in almost all self-assembled soft-mattersystems, for example, in diblock copolymers,1 thermotropic2

and lyotropic surfactant,3 and lipid4 and phospholipid5 phases.Also hexagonally arranged micelles have been observed experi-mentally.6,7 However, a bicontinuous mesophase of hexagonalsymmetry has yet to be reported in experiments. This is despitethe fact that a suitable geometric model, Schwarz’s hexagonalsurface, has been known for a long time8 and has been analyzedfrom a geometric point of view.9�13

Cubic bicontinuous phases, on the other hand, have beenfound in a large variety of soft matter systems, ranging from diblockcopolymers14 or ter-block copolymers,15 lipids,16 thermo-tropic17 and lyotropic surfactants,3 to cell membranes.18 Typi-cally, they are of cubic space group symmetry Ia3d, Pn3m or Im3m,corresponding to Schoen’s G(yroid), Schwarz’s D(iamond), andSchwarz’s P(rimitive) surface as the underlying triply periodicminimal surfaces (TPMS). These surfaces separate two inter-penetrating labyrinth-like channel systems, whereby for lipidand surfactant systems two realizations are possible. In direct(or type 1) systems, a water layer is centered on the minimalsurface, separating two hydrophobic channels. In reverse (or type 2)systems, the surfactant or lipid bilayer is draped onto the minimalsurface, separating two aqueous channels. In type 2 systems,often more than one bicontinuous cubic phase is observed, andtransitions between them can be induced by, for example,changing the water content or temperature,19,20 pressure21,22

or by introducing a third component.19,23 Upon increasing thewater content, the usual phase sequence is G f D f P. The Pbicontinuous cubic phase is only rarely observed in binary lipidwater systems (with some exceptions such as monoelaidin24 or

N-monomethylated dioleoylphosphatitidylethanolamine25,26) butusually needs the presence of a third component like a polymer,27

protein,28 or additional lipid.29,30 Experimentally, these bicontin-uous cubic phases can be tentatively identified because they arehighly viscous, optically isotropic, and lie in the phase diagrambetween the lamellar and the (columnar) hexagonal phase,although the lamellar and hexagonal phases need not necessarilybe present. In addition, the continuity of both the water andthe hydrophobic domain can be shown by nuclear magneticresonance analysis.31 The most widely used technique to estab-lish the symmetry and lattice parameter of lipid and surfactantphases is small-angle X-ray scattering (SAXS). This is alsothe main method by which the various structures were initiallyidentified.32�35

The ubiquity of the cubic bicontinuous structures in experi-ments has led to the widespread assumption that all bicontinuousstructures relevant for soft-matter have cubic symmetry. This isfar from certain as noncubic bicontinuous minimal surfacefamilies with very similar geometric properties to the cubic casesexist. These surfaces are families of TPMS with one or morefree parameters.10�12,36 Often specific members of these families(i.e., the realizations for specific values of the free parameter)correspond to the well-known cubic cases, hence making thesesurfaces ideal candidates for transition structures betweenthe cubic cases. The study of such surface families also allowsto address the fundamental question as to why self-assemblyprocesses appear to have a preference for adopting cubic structures

Received: May 9, 2011Revised: June 27, 2011

ABSTRACT: We report that a specific realization of Schwarz’s triplyperiodic hexagonal minimal surface is isotropic with respect to theDoi�Ohta interface tensor and simultaneously has minimal packing andstretching frustration similar to those of the commonly found cubicbicontinuous mesophases. This hexagonal surface, of symmetryP63/mmc with a lattice ratio of c/a = 0.832, is therefore a likely candidategeometry for self-assembled lipid/surfactant or copolymer mesophases.Furthermore, both the peak position ratios in its powder diffractionpattern and the elastic moduli closely resemble those of the cubicbicontinuous phases. We therefore argue that a genuine possibility of experimental misidentification exists.

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and to the question if other noncubic surfaces similarly suitablefor mesophase self-assembly exist.

This article addresses, in particular, a specific TPMS family,called Schwarz’s hexagonal surface or H-surface for short, thathas hexagonal symmetry and one free parameter;8,10,37 illustra-tions are shown in Figures 1 and 2. AWeierstrass parametrizationfor the H-surface is known, with a single free parameter r0∈ [0,1]that determines the location of surface flat points (those withvanishing Gauss curvature) but that has no immediate physicalmeaning;8,10,37 the Weierstrass parametrization is explicitly de-scribed in the Appendix. If the two sides of the surface areindistinguishable, then all members of this family have symmetryP63/mmc. This is the case when the surface is not oriented andthe symmetry group includes operations that exchange the twodomains. In particular, this is the case for the bilayer mesophasediscussed here. (The orientedH-surfaces, for example, realized ina hypothetical monolayer structure with distinguishable channeldomains, have symmetry P6m2.) There are no isolated membersof this family that have higher symmetries. Both the oriented andnonoriented embedding share the same translational unit cell(with the same orientation of the surface, but with a choice oforigin that differs by a vertical offset of c/4 between the twogroups), given by the two horizontal hexagonal lattice vectors oflength a and the vertical lattice vector of length c; see alsoFigure 2. The value of r0 determines the ratio c/a; see Figure 3(top). Note, however, that c/a is not a monotonous functionof r0; that is, two distinct surfaces (that have different r0 and thatare not congruent) can have the same c/a ratio. Due to this

nonmonotony, the c/a ratio cannot be used as the free parameterof the surface family. Themember with r0 = 0.49701 correspondsto the maximum lattice ratio c/a = 0.8840. The relationshipbetween r0 and c/a is shown in Figure 3 (top).

In this paper, we show that a specific member of the H-surfacefamily has similar morphological and physical properties as thecubic cases, while not itself possessing cubic symmetry.

The free energy differences between the observed bicontin-uous amphiphilic mesophase geometries of cubic symmetry(based on P, D, and G surfaces) are small for various simpleenergy models, all of which incorporate approximations ofbending and stretching terms.11,39�46 The frustrations are quan-tified by variations in curvature or domain thickness. For type 2systems, we use the width of the distribution of Gaussian curva-tures on the midplane of the surfactant bilayer as a measure forthe dominant bending contribution. For type 1 phases, wefurther use the distribution of distances to the centered medialsurface skeleton to quantify the chain stretching contribution.11,47,48

We show that, in terms of thesemeasures, bending and stretchingof a hypothetical hexagonal bicontinuous mesophase contributesonly a small penalty of this structure compared to the cubicprimitive structure.

Curiously, a hexagonal multicontinuous structure, consistingof three intertwined domains and based on constant-mean curva-ture companions of the H-surface, has recently been found in amesoporous silicate templated from a surfactant,49 after earliertheoretical predictions.50 While it is known that the presence ofsilica affects the self-assembly process and while the mesophasestructure of the pure surfactant phase without silica has not beenidentified, this result emphasizes the possibility of bicontinuousminimal surface mesophases with hexagonal symmetry.

Figure 2. Illustration of the centered medial surface skeleton used toquantify geometric packing properties, for the member with r0 = 0.679and c/a = 0.832. The color gradient on the medial surface indicates thedistance to the nearest minimal surface point, with red corresponding tothe minimal and blue to the maximal value. Themaximal distance (at thelocation of the nodes on the vertical edges) is 0.34a; the minimaldistance (at the edge midpoints of the MS triangular patch around thehorizontal node) is 0.18a; the distance at the horizontal vertices is 0.21a.

Figure 1. Illustration of the hypothetical bicontinuous type-2 lipidmesophase with hexagonal symmetry. The lipid bilayer is draped ontoSchwarz’s hexagonal minimal surface with “isotropic” lattice ratioc/a = 0.832, corresponding to r0 = 0.679. Note that this image is not asnapshot of a simulation, but a model obtained by randomly placingmock-molecules on Schwarz’s hexagonal minimal surface. Parametersare chosen to be similar to the equivalent DDAB/cyclohexane meso-phase based on the primitive minimal surface, as described in ref 38. Thewater content is 50%, and the number of the double-chain DDABmolecules is approximately 1600 (i.e., 3200 chains); The cyclohexanechains are not shown; the surface-to-volume ratio of the minimal surfaceand the surface density of molecules match those of ref 38 (see ref 11 forlength-scale conversion factor). The resulting bilayer width is alsosimilar to that of ref 38.

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The key hypothesis of this paper is that a bicontinuous lipid/water mesophase based on the hexagonal surface is energeticallysimilarly favorable to the bicontinuous cubic Im3m mesophasesbased on the primitive surface and is easily misidentified as thiscubic mesophase. This hypothesis is supported by the followinganalyses: First, bending and stretching energies are quantifiedby variations of domain thicknesses and curvature. Second, thedegree of isotropy is analyzed in terms of the interface tensor(which also relates to optical isotropy). These analyses establishthat the most isotropic member of the H-surface family and themembers minimizing bending and stretching energies are almostidentical. Third, we show that the expected small-angle X-raypowder scattering pattern of this member of the hypotheticalHmesophase bears close resemblance to that of the bicontinuouscubic primitive phase. Finally, we also show that, at least in asimplified linear-elastic model, the rheological properties of thisH mesophase show the signature of an isotropic system.

’BENDING FRUSTRATION

Interfacial curvature properties relate to the process of lipidand surfactant self-assembly through concepts such as themolecular shape parameter51 and Helfrich functionals.11,45,52

In the following, we will assume an idealized molecular modelwhere all shapes are identical and where the midlayer of the lipidbilayer traces a periodic minimal surface.

In principle, bending free energies depend on both the meanand Gaussian curvatures of the interfacial surface, that is, thesurface defined by the location of the lipid head groups. However,if this interfacial surface is modeled as a parallel surface to thebilayer midplane, the mean and the Gauss curvatures of theinterfacial surface at any point are a function of the mean and Gausscurvature at the corresponding point on themidplane surface.44,45,53

Since the midplane of the proposed H-surface mesophase isthe minimal H-surface itself, and the mean curvature there-fore zero everywhere, the bending free energy of the lipidbilayer is only a function of the Gaussian curvature K(p) of themidplane.

This motivates the index ΔK = [A�1RS[K(p) � K]2 dA]1/2,

where S is any representative patch of the TPMS (such as atranslational unit cell or the asymmetric unit patch), A =

RS dA is

the surface area, and K = A�1RSK(p) dA is the average Gaussian

curvature. This index quantifies the width of the distribution ofGaussian curvatures. Since the dimension of ΔK is [m�2], it isnot scale invariant. We therefore introduce the scale invariantquantity ΔK/Γ2. Here Γ = A/(V/2) and V is the volume of thetranslational unit cell (both domains together). ΔK/Γ2 is ameasure for the width of the curvature distribution for surfaceswhose length scales are adjusted to give the same surface-to-volume ratio. Figure 3 (bottom) showsΔK/Γ2 as a function of thefree parameter r0 of the H-surface.ΔK/Γ

2 adopts a clear minimumat r0 = 0.621 corresponding to a crystallographic lattice ratio c/a =0.859, hence identifying the hexagonal surface family memberthat minimizes bending frustration. (The length scale of a lipidmesophase geometry is determined by the complex interplaybetween bending, stretching, and packing constraints. The normaliza-tion to constant surface-to-volume ratio is a pragmatic choice.Importantly, this result holds also for other length-scale normal-izations to, e.g., constant average curvature or domain size.11)Furthermore, comparing this hexagonal surface to the three cubicbicontinuous minimal surfaces, gyroid, diamond, and primitive,we find that their respective values for ΔK/Γ2 differ by a rela-tively small amount (see Figure 3, bottom).

’STRETCHING FRUSTRATION

Phases where a layer of water separates two intertwinedhydrophobic channels are referred to as direct or type 1 lyotropicmesophases. For these phases in particular, a secondary source offrustration is the proportionality between the channel thick-nesses and the typical molecular lengths.43 A simple way toquantify this stretching contribution is afforded by the so-calledmedial surface transform.11,47,48

The medial surface is a skeleton consisting of surface patchestracing the centers of the labyrinthine domains and provides ageometric counterpart of the chaotic zones introduced by Luzzatiet al.55 As illustrated in Figure 2, for a channel domain boundedby a TPMS, themedial surface is defined as the set of points in thedomain with the property that each point pMS on the medialsurface has two (or more) nearest points p1 and p2 on theTPMS.47,56�58 It defines a map from points on the TPMS to

Figure 3. (Top) Anisotropy II(w) and crystallographic c/a ratio forSchwarz’s hexagonal surface as function of the free parameter r0.(Bottom) Variations of Gaussian curvature ΔK/Γ2 and of the averagedomain size ΓΔD. The parameter Γ = A/(V/2) sets the length-scale ofthe system such that the ratio of surface-to-volume is constant. V is thetotal volume of the translational unit cell (both domains combined), andA is the surface area of the translational unit cell of the H-surface. Alsoshown is a secondary measure for the variations of the domain sizeΓΔDCRT. The small horizontal lines indicate the values of ΓΔD andΔK/Γ2 for the cubic minimal P-, D-, and G-surfaces. Importantly, theisotropicmember of the surface family, that is, themember with II(w) = 0,is very close in c/a-ratio to the members that minimize bending andpacking frustration.

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corresponding points on the medial surface, such thatMS(p1) =pMS and MS(p2) = pMS. This map can be expressed as MS(p) =D(p)N(p) withD(p) = |MS(p)� p| andN(p) the surface normalvector of the TPMS at p. The point MS(p) is thepoint furthest from p for which p is still the closest point onthe TPMS (more precisely, one of the closest points on theTPMS). The value ofD(p) gives a point-wise defined measure ofdomain width or channel thickness that can be used to assesschain packing properties.

The index ΔD = [A�1RS[D(p) � D]2 dA]1/2, with the aver-

age distance function D = A�1RSD(p) dA, gives a measure for

the variations of the domain thickness throughout the structureand hence for the degree of packing frustration. Figure 3(bottom) shows ΓΔD as function of the free parameter r0 ofthe H-surface; that is, it shows ΔD for surfaces with length-scale adjusted to give constant V/A. Similar to the bendingfrustration, ΔK/Γ2, the packing frustration ΓΔD also adopts aminimum. This minimum is at r0 = 0.735, corresponding to a

crystallographic lattice ratio c/a = 0.7978, and is only a slightlydifferent member of the H family than the one that minimizesbending frustration. As above, this result holds also for otherlength-scale normalizations.11 Notably, the packing frustrationΓΔD adopted by this optimal H-surface is lower than that of thecubic primitive minimal surface. It is larger, however, than thevalue for the diamond or gyroid geometries; see also Figure 3,bottom.

A different approach to quantify domain size variations is theuse of the covering radius transform (CRT).59,60 Figure 3,bottom, shows the width ΔDCRT of this alternative pore sizemeasure. While the absolute values are expectedly different fromΔD, the functional forms of ΔDCRT(r0) and ΔD(r0) are similar.

’ANISOTROPY

The H-surface is embedded in a hexagonal space group, withdifferent values of the free parameter r0 corresponding todifferent c/a ratios. We show here that for the value r0 = 0.679(corresponding to c/a = 0.832), the surface is isotropic withrespect to the Doi�Ohta interface tensor. For this value of r0, thesurface is isotropic in the sense that there is an equal amount ofprojected surface area for any set of three orthogonal directions.

Define the interface tensor w with components wij =(1/A)

RS(ninj � δij/3) dA where ni is the ith component of the

unit normal vector, S is the surface corresponding to thetranslational unit cell, δii = 1 for i = 1,2,3 and δij = 0 for i 6¼ j,and A the surface area of S. (This tensor w corresponds to theinterface tensor qij = A/Vwij, defined in refs 61 and 62 except forthe different normalization; qij has unit [m], whereas wij isdimensionless. V is the total volume of the translational unit cell.)The surface S is said to be isotropic with regard to (wrt) the tensorw ifw= 0.The degree of anisotropy is conveniently quantified by thedimensionless second invariant II(q) = tr(w2)/2g 0, as previouslyproposed by L�opez-Barr�on and Macozko:62 II(w) = 0 correspondsto a surface that is isotropic wrt w and II(w) > 0 otherwise.

Figure 3 (top) shows the anisotropy measure II(w) forSchwarz’s hexagonal surfaces as function of r0, evaluated by usingthe Weierstrass parametrization of Schwarz’s hexagonal surface.For r0 ≈ 0.679 (c/a ≈ 0.832), we find II(w) = 0. Hence, thisspecific member of Schwarz’s H-surface family is isotropic withrespect to the interface tensor.

Table 1. Morphological Properties of the Members of Schwarz’s H-Surface That Have c/a ≈ √3/2, Isotropic w*, Minimal

Variations of Gaussian Curvature, and Minimal Domain Size Variationsa

surface r0 c/a a A ÆKæ ΔK ÆDæ ΔD H

H, � c/a ≈√3/2 0.600 0.866 5.311 56.20 �0.4472 0.256 1.266 0.190 0.7477

H, � w* isotropic 0.679 0.832 5.389 56.37 �0.4459 0.263 1.259 0.181 0.7488

H, � ΔK/Γ2 min 0.621 0.859 5.328 56.27 �0.4467 0.255 1.269 0.190 0.7481

H, � Γ(ΔD) min 0.735 0.798 5.460 56.24 �0.4469 0.286 1.237 0.174 0.7480

P 4.690 51.59 �0.4872 0.228 1.305 0.183 0.7163

D 3.838 28.14 �0.4446 0.208 1.298 0.096 0.7498

G 6.183 118.2 �0.4253 0.199 1.297 0.061 0.7667aThe corresponding values of the cubic P, D, and G surfaces are provided as reference. The lattice parameter a is chosen such that Γ = A/(V/2) equalsunity; that is, the surfaces are scaled to have constant surface to volume ratio. The c/a ratio and hence the lattice parameter c are determined by the valuer0 of the free parameter. S is the fraction of the H surface within the translational periodic unit cell; A is the surface area of S; the total volume of this unitcell is V =

√3a2c/2 (as all surfaces discussed here are balanced, the volume within each channel domain is V/2). H = A3/2/((2πχ)1/2V/2) is the

homogeneity index, defined by Hyde and co-workers,10,42,54 with the Euler number χ =RSK/(2π) dA of the translational unit cell. The data refers to the

hexagonal P63/mmc unit cell with χ = �4 for the H surfaces, and the cubic translational unit cells Im3m with χ = �4 for the P, Pn3m withχ = �2 for the D, and Ia3d with χ = �8 for the G surfaces; these are the space groups of the unoriented minimal surfaces.

Table 2. Systematic Extinction of Peaks Imposed by the Ia3d,Pn3m, and Im3m Space Groups Lead to aCharacteristic Seriesof q-Values Indicating the Scattering Peak Locations for theBicontinuous Cubic G, D, and P Mesophase

hkl q � (a/2π) G (Ia3d, Q230) D (Pn3m, Q224) P (Im3m, Q229)

100 1

110√2

√2

√2

111√3

√3

200√4

√4

√4

210√5

211√6

√6

√6

√6√

7

220√8

√8

√8

√8

300√9

√9

221√9

310√10

√10

√10

√10

311√11

√11

222√12

√12

√12

320√13

321√14

√14

√14

√14

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The study of isotropy or anisotropy using the surface tensor wrelates to the question of optical isotropy. Ignoring the effect ofthermal membrane deformations, we calculate the conventionalorientational order parameter ÆP2(cos θ)æ = Æ(3cos2 θ � 1)/2æbased on the average of the second Legendre polynomial. Weassume that the molecular director is perpendicular to theminimal surface and that the lipid molecules are uniformlydistributed over the minimal surface. It follows from the defini-tion of w that the sample is isotropic with respect to theorientational order parameter, that is, ÆP2(cos θ)æ = 0, if wvanishes (a similar calculation is presented in depth in ref 63). Inliquid crystalline matter, optical isotropy is commonly identifiedwith ÆP2(cos θ)æ = 0. Under the stated assumptions, themesophase on the H-surface with r0 = 0.679 (c/a = 0.832) istherefore also optically isotropic.

Summarizing these geometric findings, the members of theSchwarz hexagonal surface family that minimize bending frustra-tion, stretching frustration, and deviations from isotropy are notidentical but very similar. Their c/a ratios vary between 0.798 and0.859. The (geometrically and optically) isotropic member hasa c/a ratio of 0.832 and the width of the curvature and thick-ness distributions (corresponding to bending and stretchingfrustration) for this member are only slightly larger than thosefor the respective optimal members; see Table 1 for numericalvalues. In particular, the stretching frustration for this memberis still slightly lower than that for the cubic primitive geometry.This suggests that this member of Schwarz’s hexagonal surface isa likely candidate geometry for lipid/surfactant self-assembly.

’SMALL-ANGLE X-RAY SCATTERING

This section shows that the powder scattering pattern ofhypothetical H-surface bilayer mesophases in the vicinity of theisotropic member with r0 = 0.679 (c/a = 0.832) resembles closelythat of a bicontinuous Im3m (Q229) phase based on the P-surface.

Most liquid crystalline samples do not form single crystals ormonodomains, but instead a powder, that is, an ensemble ofsmaller crystal domains where every possible orientation isrepresented. The resulting SAXS measurement is representedas a radially symmetric structure factor, that is, a plot of scatter-ing intensity as function of scattering angle or q-value (spatialfrequency magnitude). The relative location of peaks givesinformation on the symmetry (q-ratio), whereas the absolutevalue gives information on the lattice parameter(s).

For a primitive cubic phase, a scattering pattern accordingto Bragg’s law can have peaks at q-values given by q =(2π/a)(h2 + k2 + l2)1/2. Here a is the lattice parameter and h,k,

l are the Miller indices.64 This relates to the lattice spacing d by q =2π/d. Not all of these peaks can be observed for a given cubic crystalstructure, since the space group symmetries cause systematic extinc-tions of some of the peaks. Therefore, each cubic phase has acharacteristic set of q-ratios. Table 2 shows the characteristicq-ratios of the first peaks of the bicontinuous G, D, and P phases.

For a 3D hexagonal unit cell, the Bragg scattering has allowedpeaks at q-values of q= (2π/c)[(c/a)2(4/3)(h2 + k2 + hk) + l2]1/2

where a and c are the two independent lattice parameters. Inorder to correctly index all peaks which are observed in such ascattering experiment, we need to determine both lattice para-meters. For c/a =

√3/2 = 0.866 (corresponding to the H-surface

with r0 = 0.60 and with only 4% relative difference to the valuec/a = 0.832 of the isotropic member of the H-surface), the q-ratiosbecome q(c/2π) = {1,

√2,√3,√4,√5,√7}. As with the cubic

surfaces, the space group symmetry of the H-phase, P63/mmc,imposes additional systematic extinctions of Bragg peaks in thescattering pattern.64 However, the extinct peaks are at q-valuesthat can also be obtained for combinations of hkl which areallowed, and therefore, the space group does not change the setof observable q-ratios in a powder scattering experiment.

From Table 2, we see that all q-values for the H-surface have acorresponding peak which is at 1/

√2 times the q-value allowed

for the P-surface of cubic symmetry. Indeed, with the exceptionof the sixth peak for the P-surface, at

√12, there is a one-to-one

correspondence between the peaks of the P-surface and theH-surface. This correspondence is illustrated in Table 3. Inparticular, it means that a scattering curve from the H-surfacecan be incorrectly indexed as a P-surface where the sixth peak isassumed to be too weak to be observed (or extinguished by theform factor). Similar geometrical ambiguities in the indexing ofpowder patterns have also been reported for minerals.65,66

Beyond systematic extinctions, the scattering curve of three-dimensional structure can be numerically determined by Four-ier transform.67 Our analysis is based on a voxelized representa-tion of a bilayer of constant thickness D draped onto theminimal surface. A two-phase scattering model is used where

Table 3. Diffraction Peak Correspondence between CubicIm3m and Hexagonal P63/mmc at c/a =

√3/2

H (P63/mmc) f P (Im3m)

q(c/2π) �√2 q(a/2π)

√1 f

√2√

2 f√4√

3 f√6√

4 f√8√

5 f√10√12√

7 f√14

Figure 4. Simulated SAXS curves for the P-phase (black) and thehypothetical H-phase (blue) with r0 = 0.667 (c/a = 0.838) and withvolume fraction approximately 50%. Themaximum scattering intensitieshave been normalized to a peak value of 1 and have been plotted against anormalized q-value. Dashed vertical lines indicate the allowed peakpositions for the P-surface that are not prohibited by systematicextinction. All resolvable peak positions for the hypothetical H-phasecan be indexed to allowed peaks for the P-surface.

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constant scattering density values FBL and FAQ are assigned tothe bilayer and its complement (the aqueous labyrinth-likecompartments), respectively. For the case of the hexagonalmesophases based on the H-surface, the bilayer is discretized ona rectangular grid of size

√3a � a � c. This grid represents a

translational (but not primitive) unit cell of the hexagonal spacegroup (lattice vectors are [100], [�110], and [001]). The X-rayscattering density of the points inside the bilayer (i.e., thosewhose distance to the minimal surface is less than D/2) is FBL;for the remaining grid points representing the aqueous do-mains, F = FAQ.

We can without loss of generality assume that the meanscattering density is 0, as deviations from this will only contributeto the scattering curve at q = 0 (forward scattering). For a two-phase system, linearity of the Fourier transform therefore impliesthat the numerical value for the scattering densities only changesthe absolute scaling of the scattering curves and does not alter thepeak locations or relative intensities. The numerical value istherefore irrelevant for our analysis. From the voxelized repre-sentations of the mesophase, SAXS scattering curves are deter-mined by fast Fourier transform (FFT) using an approach whichis based on that presented in Schmidt-Rohr (in ref 68). (Usingthe FFT to compute the Fourier transform, we inherently assumethat the computational grid is a rectangular lattice. This leadsto some complications with respect to our discretization: bydiscretizing a structure with hexagonal symmetries, we invariablyintroduce discretization errors. These errors do not obey by thesymmetries of the original structure, but are rather forced ontothe rectangular grid used for the computations. As a result, theylead to additional peaks corresponding to symmetries of thecomputational grid. To break the symmetries of the discretiza-tion, we introduce random permutations of the discretization.This ensures that an error which is present in one cell does notpersist periodically across the computational domain and, there-fore, does not contribute additional peak values in the resolvedspectra.)

Figure 4 shows the simulated scattering curve for the hypothe-tical lipid mesophase based on the member of Schwarz’s hex-agonal surface family with free parameter r0 = 0.667, and with c/a =0.838 close to

√3/2 ≈ 0.866. The aqueous volume fraction is

approximately 50%, similar to the volume fraction for whichthe bicontinuous P-phase has been observed for various lipidsystems.24,29,38,69 For comparison, a simulated curve of thisP-phase at the same water content is also shown.

As expected from the symmetry rules, the peaks for both theIm3m P-mesophase and those of this specific hexagonal meso-phase are to a very good approximation at the same normalizedq-values. Though the spectra are not identical, all resolvable peakson the curve for the hexagonal phase correspond to allowedpeaks of the P-phase. Based on q-ratios for the resolved peaks, ourproposed bicontinuous H-phase cannot be distinguished from abicontinuous cubic P-mesophase. In fact, as shown in Table 3,the peaks of anH-phase can be indexed as amesophase based on theP-phase. Only the presence of a clear sixth peak for the P-phase,at qa/(2π) =

√12, would discern the P-phase from the H-phase.

Compared to solid crystals, liquid crystals generally show alower number of reflections. For bicontinuous cubic P-meso-phases, often only the first three peaks are strong enough to beobserved, and the typical peak ratio

√2:√4:√6 is used as a basis

for structure assignment, particularly in dispersions or whenadditional mesophases coexist.26,70�72 In very well-orderedbicontinuous cubic P-mesophases, the number of peaks can be

higher (e.g., 8 clear peaks in Figure 2b in ref 38 and 11 peaksidentified in ref 73).

The relative intensities of the peaks are modulated by theunderlying form factor (bilayer thickness, etc.) and by thermalfluctuations. In established scattering patterns of the Im3mP-mesophase, the intensity can be largest for the first peak (as inFigure 8a of ref 74), for the second peak (as in Figure 2b of ref 38),or the third peak (as in Figure 5c of ref 69). In this respect,there is no significant difference between the P-mesophase and ourhypothetical H-phase.

A significant intensity difference is not expected before thefourth peak. The peak strength of this fourth peak for theP-mesophases (if existent) is generally very low. For theH-phase,the fourth peak corresponds to a combined (200) and (002)reflection, and based on simulated experiments we do not expectthis peak to disappear easily; it seems to be present in systematicsimulations of various c/a ratios and bilayer thicknesses, with andwithout explicit modeling of the head groups (data not shown).Within the validity of the scattering model used here, thisindicates that a fourth peak with substantial intensity may bean indication of it being an H-phase rather than a P-phase. Withrespect to systematic extinctions, however, the sixth peak (

√12)

needs to be resolved for a positive identification of the underlyingstructure as a P-phase rather than a H-phase.

The static two-phase model used here to predict scatteringcurves ignores effects of the finer detail in the molecular structurefactor, and nonuniformity of the structure due to for examplethermal fluctuations. As it has been shown that the amplitude ofthe scattering peaks can be affected by a nonuniform bilayerthickness or fluctuating scattering densities,75�77 this warrantsmore detailed molecular modeling. However, these effects do notchange the peak positions in the small-angle scattering curves and

Figure 5. Elastic moduli of a linearly elastic warped sheet draped ontothe H-surface. Shown are the six eigenvalues Λi of the Mehrabadimatrix82 as function of the free surface parameter r0. For general r0 theeigenvalues show the signature of hexagonal symmetry, with two pairs ofdegenerate eigenvalues corresponding to isochoric (shear) modes,marked “I”, and two nondegenerate eigenvalues corresponding todilatational (compression) modes, marked by the letter “D”. For theH-surface with r0 ≈ 0.7 (those that are approximately isotropic wrtinterface tensor), the Mehrabadi matrix shows the signature of isotropicsymmetry with a 5-fold degenerate isochoric eigenvalue and a singledilatational eigenvalue. The volume fraction occupied by the bilayer is50% and the microscopic Poisson ratio of the linear-elastic constituentmaterial is 1/5. The material bulk modulus k0 is an irrelevant prefactor.

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should not affect the main conclusion presented here that thehypothetical H mesophase has very similar scattering patterns tothe established cubic P-mesophases.

’RHEOLOGY AND RESPONSE TO MECHANICALSHEAR AND COMPRESSION

The response of bicontinuous cubic lipid and surfactant phasesto shear and deformation is nonlinear andgoverned by the interplayof several complex relaxation mechanisms.78�81 Neglecting theseeffects, wemodel the lipid bilayer in a first approximation as a linear-elastic solid composite. In this geometric approximation, thebehavior of the hexagonal phase with c/a = 0.832 cannot bedistinguished from an isotropic mesophase in terms of the defor-mation modes, providing further indication of the similaritybetween this hexagonal phase and the cubic mesophases.

The linear-elastic moduli of a lipid system are analyzedassuming the bilayer to be composed of a homogeneous isotropiclinearly elastic material and the aqueous domains of vacuum(the latter reflects the possibility of change in water content byexchange with excess water). Finite-element calculations83 on thesame voxelized representations described above yield the elasticshear- and bulk-moduli of the system. Figure 5 shows the linear-elastic moduli (given as the eigenvalues of the Cowin�Mehrabadi matrix82) as function of the free parameter r0 ofthe hexagonal bilayer with 50% volume fraction. In the vicinity ofthe members near r0≈ 0.7 that optimize bending, stretching, andisotropy, the character of the deformation modes shows thesignature of an isotropic system: A single dilatational mode hasa clearly distinct eigenvalue from a set of five isochoric modeswith eigenvalues that are, to the precision of our computation,degenerate. For all other values of r0 the Cowin�Mehrabadimatrix has the form expected for a system with hexagonalsymmetry (see Table 3 on page 28 of ref 82).

’CONCLUSION

We have demonstrated that a bicontinuous geometry ofhexagonal symmetry P63/mmc with c/a ≈ 0.83 is a likelycandidate for a novel lipid or surfactant mesophase. This findingis supported by the minimal, or almost minimal, geometricbending and stretching frustration of the corresponding memberof Schwarz’s hexagonal minimal surface, with r0 ≈ 0.68, com-pared to all other members of that family. Their absolute valuesare similar to those of the cubic Im3m mesophase based onSchwarz’ primitive surface. Given that the Im3m primitive phaseis observed experimentally in lipids, we postulate, based on thegeometric arguments given here, that the hexagonal phase shouldalso be found.

The described hexagonal phase is similar to a cubic structure,both with respect to scattering experiments, polarized lightmicroscopy and to rheology and deformations (at least insimplified models thereof). The characteristics of the linearlyelastic deformation modes are the same as that of an isotropic(or cubic) system. The first peaks of the powder scattering func-tion show strong similarities to that of a bicontinuous P-mesophase(therefore, higher reflections need to be resolved in order to discernthese two phases by SAXS). This raises the possibility of experi-mental misidentification of this hexagonal phase as a cubic phase.

This is further supported by explicit anisotropy measuresdemonstrating that in the vicinity of this member the surfaceof the Hexagonal surface family is isotropic in the sense that theDoi�Ohta surface tensor vanishes.61

More detailed molecular modeling of the formation, structure,and properties of the hypothetical hexagonal bicontinuousmesophase is possible and desirable; this should include im-provedmodeling of the free energy contributions, for example, interms of constant mean curvature versus parallel surface modelsand higher-order curvature contributions,84,85 and improvedanalyses of rheology and optical properties observed by polarizedlight microscopy. However, the consistency of the variousfundamental geometric properties discussed here suffices toraise a caution to both experimentalists and simulations that abicontinuous hexagonal mesophase is a genuine likely alternativeto the established cubic phases. In particular, simulations of lipidmesophases should exercise care with respect to the boundaryconditions imposed by the choice of simulation box; a cubicsimulation box, which is not commensurate with the hexagonalmesophase, will prevent the occurrence of this phase even if it isthe true equilibrium phase.

Further work is needed to assign the correct place of thehexagonal mesophase in the phase diagram (we expect theH-surface to occur at a very similar position in the phase diagramas the P-surface as it also has intermediate curvature between alamellar and a hexagonal columnar phase). An understanding ofthe swelling behavior and the thermal membrane fluctuations isalso needed. Our conjecture of a bicontinuous hexagonal form asa novel (or as yet unidentified) form in self-assembly contributesto an understanding of lipid polymorphism, and of the subse-quent implications lipid mesophase shape has for biological,biochemical, and pharmaceutical applications; see, for example,ref 86.

’APPENDIX: WEIERSTRASS PARAMETRIZATION

The Weierstrass parametrization provides a parametrizationof minimal surfaces in general and the H-surface in particular thatinvolves integration of analytic functions in the complex plane. Atranslational unit cell of the nonoriented H-surface with sym-metry group P63/mmc (no. 194 in ref 64) is obtained from theasymmetric surface patch (“Fl€achenst€uck”) of the space group byapplication of all symmetries. Following the notation of ref 64,the asymmetric unit volume of the space group P63/mmc is theprism given by 0e Xe 2/3, 0e Ye 2/3, 0e Ze 1/4, Xe 2Y,and Y e min(1 � X,2X) in crystallographic coordinates X,Y,Z.Crystallographic (hexagonal) coordinates convert to Carte-sian coordinates by (x,y,z) = (Xa)a1 + (Ya)a2 + (Zc)a3 with thethree basis vectors a1 = (

√3/2,�1/2,0), a2 = (0,1,0), and a3 =

(0,0,1) and the crystallographic lattice parameters a and c. Thesurface patch contained in the asymmetric volume is a conve-nient choice for the asymmetric surface patch. A correspondingset of points that locks into this symmetry group is obtained byWeierstrass integration. The Weierstrass parametrization maps apoint ω ∈ C from the complex plane to a corresponding point inEuclidean space E3 on the H-surface, by the following equations

xðωÞ ¼ R eiθZ ω

0ð1� r2Þ RðrÞ dr

� �ð1Þ

yðωÞ ¼ R i eiθZ ω

0ð1 þ r2Þ RðrÞ dr

� �ð2Þ

zðωÞ ¼ R eiθZ ω

02r RðrÞ dr

� �ð3Þ

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with R(ω) = 1/[ω(ω6� (r03 + 1/r0

3)ω3 + 1)]1/2 and the Bonnetangle θ = π/2; the symbol R denotes the real part of a complexnumber, and i the imaginary unit (i2 = �1). r0 ∈ [0,1] is the freereal-valued parameter of the H-surface. The integrals are integra-tion along paths in the complex plane C. (An integration pathconsisting of a straight segment from the origin to a point ω~approximately in the center of G and of a straight segment ω~ to ωavoids numerical issues related to poles and branch cuts. For thepole ω = r0, the exact expressions x = 0, y = a/2, and z = 0 hold.)Application of these formulas to all points of the patch G ⊂ Ccomprising all pointsω = r exp(ij) with r∈ [0,1] andj ∈ [0,π/3](a sixth of the unit circle in the complex plane, in the first quadrant,and adjacent to the real axis) yields points (x(ω),y(ω),z(ω)) thatrepresent the asymmetric unit patch, contained in the asymmetricvolume listed above. The intervalω∈ [0,r0]⊂Rmaps to points x=0, y ∈ [0,�a/2], z = 0 on a 2-fold rotation axis with Wyckoff letter12i, and the remaining three boundary curves (including the interval[r0,1]) to mirror planes 12k and 12j. The crystallographic latticeparameters a and c are obtained as the coordinate values of specificpoints of the complex plane, a(r0) = 2y(1) (with y(1) = y(ω) for allω ∈ [r0,1]⊂ R) and c(r0) = 4z(exp(iπ/3)). The translational unitcell and the orientation of the surface, and hence the parameters aand c, are identical for the space group P63/mmc of the nonorientedsurface and the space group P6m2 of the oriented surface, but con-ventionally the choice of origin differs by a vertical translation by c/4(the origin inP63/mmc is the flat point on the surface correspondingtoω = 0, whereas the origin of P6m2 is the horizontal vertex of theskeletal graph). See refs 10, 11, and 37 for further details.

’AUTHOR INFORMATION

Corresponding Author*E-mail: Gerd.Schroeder-Turk@physik.uni-erlangen.de.

’ACKNOWLEDGMENT

We thank Stephen Hyde for inspiration and advice, andAndrew Christy and Barry Ninham for valuable comments onthemanuscript. G.E.S.-T. and S.K. acknowledge financial supportby the Deutsche Forschungsgemeinschaft (DFG) under GrantSCHR-1148/2-1 and by the GermanAcademic Exchange Service(DAAD) through a joint program with the Australian Group-of-Eight universities. T.V. and L.d.C. gratefully acknowledge sup-port from the Australian Research Council through the Discov-ery Projects scheme.

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