Swelling and Structure. Analysis of the Topology and Geometry of

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Swelling and Structure. Analysis of the Topology and Geometry of Lamellar and Sponge Lyotropic Mesophases S. T. Hyde Applied Mathematics Department, Research School of Physical Sciences, Australian National University, Canberra 0200, Australia Received May 31, 1996. In Final Form: November 18, 1996X

The geometry of an “ideal” sponge is defined and described with reference to single-sheeted hyperbolic surfaces. The expected swelling features of this sponge, which depend on the detailed swelling mechanism, can be used to deduce estimates of the structural parameters of the sponge. The analysis is used to investigate the mesostructure of some sponge mesophases, in bulk and confined between mica sheets, relying on data collected by others. Sponge mesophases formed in sodium dodecyl sulphate-pentanolNaCl-water (Europhys. Lett. 1989, 9, 447-452) and Aerosol OT-NaCl-water (Structure and dynamics of strongly interacting colloids and supramolecular aggregates in solution; Chen, S.-H., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992, pp 351-363) mixtures are analyzed, as well as the sponge phases confined between mica sheets (in surface force experiments) in AOT-NaNO3-water mixtures (J. Phys. II 1995, 5, 103-112 and Langmuir 1995, 11, 3928-3936). In addition, we analyze a cubic mesophase formed in the glycerol monooleate-water system (Nature 1994, 368, 224-226) and the AOT-NaCl-water lamellar mesophase (Structure and dynamics of strongly interacting colloids and supramolecular aggregates in solution; Chen, S.-H., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992, pp 351363). The analyses suggest that the sponge mesophases considered have a structure related to that of bicontinuous cubic mesophases, viz. a hyperbolically warped molecular bilayer, dividing two continuous, but disjointed, water labyrinths. The diffuse scattering peak, observed in small-angle scattering spectra of these sponges, corresponds to the average channel diameter in bulk systems but is more complex in the confined case, which appears to be superstructured. The swelling analysis is shown to be equally useful for detailed investigation of the effective (average) topology of other mesophases (the cubic and lamellar phases). The power of the analysis lies in its ability to detect the average topology of a membrane, provided the membrane is sufficiently ordered to furnish at least one correlation peak in small-angle scattering spectra.

Introduction Bicontinuous mesophases are built of warped bilayers of surfactant (or any amphiphilic) molecules, immersed in (one or two) intertwined and tortuously interconnected water continua.5 The bilayer geometry in these mesophases is hyperbolic, i.e. everywhere saddle-shaped.6 Analysis of the variation with bilayer concentration of the single, broad, pseudo-Bragg peak seen in small-angle X-ray and neutron scattering spectra (SAS) supports the view that the structure of L3 membranes resembles those of hyperbolic cubic membranes. In this article, the general features of the shape of hyperbolic bilayers are described in a mathematically precise fashion, relying largely on the peak position seen in small-angle scattering spectra and the variation of this peak location with concentration. The analysis is a general structural probe: given any set of data linking length dimensions in a structure to concentration, an effective structure giving rise to that length data can be deduced. For example, sponge mesophases confined between mica sheets for differing sponge concentrations are analyzed. In this case, the typical wavelength of the oscillations is a measure of a length scale in the confined sponge. X Abstract published in Advance ACS Abstracts, January 15, 1997.

(1) Gazeau, D.; Bellocq, A. M.; Roux, D.; Zemb, T. Europhys. Lett. 1989, 9, 447-452. (2) Strey, R.; Jahn, W.; Skouri, M.; Porte, G.; Marignan, J.; Olsson, U. In Structure and dynamics of strongly interacting colloids and supramolecular aggregates in solution; Chen, S.-H., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; pp 351-363. (3) Antelmi, D.; Ke´kicheff, P.; Richetti, P. J. Phys. II 1995, 5, 103112. (4) Chung, H.; Caffrey, M. Nature 1994, 368, 224-226. (5) Luzzati, V.; Spegt, P. A. Nature 1967, 215, 701-704. (6) Hyde, S. T. J. Phys., Colloq. 1990, C7, 209-228.

S0743-7463(96)00534-3 CCC: $14.00

En route, a basic geometrical account of sponge-shaped surfaces will be traversed. I define an “ideal” sponge, the hyperbolic counterpart to better-known (two-dimensionally) isotropic surfaces: the sphere and the plane. This ideal form lies between planes and cylinders in many respects. Geometrical measures of a sponge, dependent on topology, will be developed, to allow explication of swelling laws (e.g. variation of surface area with characteristic sponge “wavelength”). Sponge forms immersed in the three-dimensions of Euclidean space are necessarily nonideal and inhomogeneoussthey exhibit curvature variations over their surface and are thus two-dimensionally anisotropic (though perhaps isotropic in the threedimensions of space). The degree of inhomogeneity exhibited by a surfactant bilayer is the result of competing entropic and bending energy demands; the former favoring and the latter limiting inhomogeneity (for a monodisperse surfactant distribution). As the topology of the underlying surface describing the warped bilayer becomes more complexsthe genus per unit volume increasessthe inhomogeneity is also raised. Further, spatial disordering of the crystalline hyperbolic surfaces enhances inhomogeneity. It is not unreasonable then to suppose that higher-genera hyperbolic structures (such as the Neovius surface) are generally less crystalline than their more homogeneous and better-known counterparts, the gyroid, D, P, or I-WP surfaces. That proposition may be manifested in the L3 membranes confined between mica sheets. That, in a nutshell, is the paper. In the next section, the surface geometry and topology of hyperbolic shapes will be discussed in more quantitative terms. Following that, the link between molecular shape and membrane geometry will be discussed. The paper closes with a description of swelling laws as a function of swelling mode and mesostructure. These laws are applied to some cubic, © 1997 American Chemical Society

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sponge (L3), and lamellar systems, leading to estimates of their bilayer geometry and topology. Geometry and Topology of Ideal Sponges The classical structures of amphiphilic assemblies are planes, cylinders, and spheres (or polyhedra). For example, lamellar, hexagonal, and discrete micellar mesophases conventionally display these forms. However, a closer look at many of these molecular autoassemblies is rewarded by indications of lesser understood porous structures, related to sponges. That structural form is not, however, confined to amphiphilic molecular systems. Simple atomic tessellations of precisely those surfaces underlying hyperbolic cubic membranes describe well the structures of a number of zeolites also.7 Similar forms can be seen in copolymer molecular melts,8-10 as well as mesostructured layered inorganic materials, including volcanic minerals and novel synthetic mesoporous oxides.11-13 (The last examples are in fact templated using surfactant-water mixtures, so the once estranged fields of surfactant self-assembly and solid-state crystal chemistry are now dramatically united.) The most symmetric spongelike surfaces are minimal surfaces displaying a three-dimensional lattice, called “triply-periodic” minimal surfaces. These surfaces may contain self-intersections, in which case the resulting morphology is not bicontinuous. Some examples of selfintersecting periodic minimal surfaces can be found in publications devoted to the geometric theory of minimal surfaces.14,15 In this paper we are only concerned with “embedded (intersection-free) triply periodic minimal surfaces” (ETPMS, also called IPMS14). Some examples are illustrated below (Figure 1). The crystalline order of these examples is not an essential feature of sponges, as illustrated by the disordered hyperbolic form shown in Figure 2. The essential structural features of a sponge are captured by its local shapesits curvaturesand its global shapesits topology. A one-dimensional curve has a single measure of curvature at each of its points. Formally, this curvature has a magnitude equal to the reciprocal of the radius of the osculating circle at that point, and we can write

k ) R-1

Figure 1. Portions of three triply periodic minimal surfaces: the D-surface (two conventional unit cells), the P-surface (four conventional unit cells), and the gyroid (single unit cell).

(1)

A surface has two measures of curvature associated with each point. These principal curvatures (k1 and k2) at a point P are the reciprocal radii of orthogonal circles containing the surface normal vector at P, sampling the most concave and most convex tangential curves passing through P. The principal curvatures define the usual (7) Andersson, S.; Hyde, S. T.; von Schnering, H.-G. Z. Kristallogr. 1984, 168, 1-17. (8) Thomas, E. L.; Anderson, D. M.; Henkee, C. S.; Hoffman, D. Nature 1988, 334, 598-601. (9) Hasegawa, H.; Tanaka, H.; Yamasaki, K.; Hashimoto, T. Macromolecules 1987, 20, 1651-1662. (10) Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Fetters, L. J. Macromolecules 1994, 27, 4063. (11) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710. (12) Huo, Q.; Margolese, D.; Ciesla, U.; Feng, P.; Gier, T.; Sieger, P.; Leon, R.; Petroff, P.; Schu¨th, F.; Stucky, G. Nature 1994, 368, 317-321. (13) Hyde, S. T. Phys. Chem. Miner. 1993, 20, 190-200. (14) Schoen, A. In Geometric Analysis and Computer Graphics; Finn, R., Hoffman, D. A., Concus, P., Eds.; Springer-Verlag: New York, 1991; pp 147-158. (15) Fischer, W.; Koch, E. Z. Kristallogr. 1996, 211, 1-3.

Figure 2. Portion of a random hyperbolic surface (picture courtesy of P. Pieruschka).

measures of surface curvature, the mean curvature (M) and Gaussian curvature (K):

M)

k1 + k2 ; K ) k1‚k2 2

(2)

In many respects, Gaussian curvature is the central parameter of form. Three local geometries are accessible to two-dimensional surfaces: elliptic (positive K) and parabolicsincluding flat (K equal to zero) and hyperbolic (K negative). Surface patches of sponges are on average hyperbolic, i.e. synclastic, or saddle-shaped (Figure 3). Among hyperbolic shapes, minimal surfaces are found. A minimal surface has by definition zero mean curvature everywhere, so that its convexity along any sectional direction is constantly balanced exactly by an equally

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Figure 3. Surface patches of (left to right) elliptic, parabolic, and hyperbolic geometry. Table 1. Topological and Geometric Global Data for the Simpler Cubic Embedded Three-Periodic Minimal Surfaces (ETPMS)a ETPMS

χuc

H

gyroid D-surface P-surface Neovius surface

-8 -2 -4 -16

0.7665 0.7498 0.7163 0.6640

a χ denotes the Euler-Poincare ´ characteristic per conventional uc cubic unit cell and H the homogeneity index.

concave (orthogonal) section. ETPMS afford structural models of bicontinuous cubic phases. For example, the “gyroid” surface describes well the bilayer geometry of a number of mesophases displaying symmetry Ia3h d,16-18 and the D-surface mimics the bilayer folding in some examples of the Pn3 h m mesophase.17,19,20 A further bicontinuous mesophase, believed to be of cubic space group Im3h m, is likely to be described by the P-surface.17,21,22 These three ETPMS can be distinguished by their crystallographic topology, which is a measure of the integral (Gaussian) curvature of the mid-surface of the bilayer within a conventional cubic unit cell,

∫∫ K da.

Figure 4. Schematic view of (top) a defect-free stack of bilayers, (middle) a puncture defect within a bilayer membrane, and (bottom) a single channel connecting adjacent membrane sheets.

the surface within that volume:

unit cell

The integral curvature is “quantized” for periodic surfaces in three-dimensional Euclidean space and simply related to an even integer known as the Euler-Poincare´ characteristic, χuc:

∫∫ K da

χuc )

unit cell

2π

(3)

The values of χuc for conventional (cubic) unit cells of the P-surface, the gyroid, and the D-surface are listed in Table 1. These numbers, characterizing the topology of ETPMS, are dependent on the choice of repeat unit for the surfacesthus, for example, the effective value of χuc for n unit cells is equal to nχuc. This leads to a useful relation between topology and crystallography, applicable to bicontinuous crystalline mesophases. A specific Bragg reflection within a mesophase of cubic symmetry, corresponding to a peak of scattering intensity in SAS spectra, can be indexed according to its Bragg indices relative to the unit cell, hkl. The distance between lattice planes associated with this reflection is equal to

d ) xh2 + k2 + l2 We can thus associate a volume, d3hkl, with each reflection and a characteristic, χhkl, with the integral curvature of (16) Hyde, S. T.; Andersson, S.; Ericsson, B.; Larsson, K. Z. Kristallogr. 1984, 168, 213-219. (17) Maddaford, P.; Toprakciogliu, C. Langmuir 1993, 9, 2868-2878. (18) Engblom, J.; Hyde, S. T. J. Phys. II 1995, 5, 171-190. (19) Longley, W.; McIntosh, T. J. Nature 1983, 303, 612-614. (20) Barois, P.; Eidam, D.; Hyde, S. T. J. Phys., Colloq. 1990, C7, 25-34. (21) Larsson, K. J. Phys. Chem. 1989, 93, 7304-7314. (22) Barois, P.; Hyde, S. T.; Ninham, B. W.; Dowling, T. Langmuir 1990, 6, 1136-1140.

χhkl ) χuc(h2 + k2 + l2)3/2

(4)

This relation is a useful one for interpretation of the geometric meaning of a scattering peak in correlated sponges and will be invoked later in the paper. Its importance lies in its quantitative coupling of surface topologysvia the parameter χsto linear dimensions within the phase, which are defined by the spacing between hkl lattice planes in an isotropic ordered sponge. Since the magnitude of χ for a single channel can be calculated (and will be, cf. eq 8) and χ scales linearly with the number of channels, eq 4 defines the quantitative link between a spacing within an isotropic sponge and the number of channels within an associated volume. Experimental measures of the Euler-Poincare´ characteristic of bilayers in lyotropic systems may differ from the mathematically ideal values for hyperbolic surfaces, listed in Table 1. From the classical structural viewpoint, the Euler-Poincare´ characteristic is related to the “defect” structure within the membranesdefective with respect to an ideal posited structure onlysas follows. Two types of topological defects can arise: “punctures” within the bilayer and “channels” connecting bilayers (Figure 4). From standard topology theory, the ideal, endless sheet characteristic of a lamellar structure (of zero integral curvature) has an Euler-Poincare´ characteristic equal to zero. Each puncture within the membrane decreases the value of the characteristic by one, while channels decrease it by two. (Thus, ETPMS, which are riddled with three-dimensional arrays of channels, have negative Euler-Poincare´ characteristics.) The Euler-Poincare´ characteristic of a hyperbolic membrane (per specified volume), which is an average measure over the entire membrane, thus includes contributions from defects within the membrane and may vary continuously with the defect density. One further global index of form is required in order to proceed. This is a dimensionless number which involves

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the surface area (A) of the surface of specific characteristic χV enclosed within a volume V, called the “homogeneity index”:6

H≡

A3/2 (-2πχV)1/2V

(5)

This number is a useful dimensionless measure of surface to volume ratio, as it is independent of the choice of repeat unit, in contrast to other dimensionless indices (a good discussion of the subtleties of other measures can be found in ref 23 ). It is exactly 3/4 for an idealized sponge which admits reconstruction of the global sponge form from its local shape alone.6,24 An ideal sponge is defined to be one of uniform (negative) Gaussian curvature everywhere on its surface. The ideal structure is fictitious in threedimensional Euclidean space, as the Gaussian curvature of (singularity-free) hyperbolic surfaces necessarily varies over the surface.25 However, the approximation is not a bad one for simpler three-periodic hyperbolic surfaces, particularly the simpler ETPMS, which have homogeneity indices close to 3/4 (cf. Table 1). It follows from the definitions of Gaussian curvature and homogeneity index, and the link between Gaussian curvature and topology (eq 3), that the radius of curvature of an ideal sponge (by definition both isotropic and homogeneous) scales with any other characteristic length of the sponge, ξ:

R)

(

H -2πχξ

)

1/3

(6)

ξ

For example, if V encloses a single unit cell of the (isotropic) structure, the lattice parameter, R, is related to the radius of curvature by

R)

(

)

H -2πχuc

1/3

(7)

R

These indices of global form also apply to partially molten “correlated” surfaces, which lack a translational unit cell but nevertheless display at least one correlation peak in SAS spectra. In those cases, it is convenient to carry the definition over from the periodic case; here the EulerPoincare´ characteristic refers to the integral curvature of the bilayer mid-surface per “unit” volume. Since the length scale to be adopted is arbitrary, this index need not be an integer for disordered surfaces. Consider, for example, an isotropic sponge membrane of uniform channel diameter, 2R. In that case, the characteristic associated with the volume (2R)3 follows from eq 6:

R)

(

H -2πχV

)

1/3

2R or χV ) -

A

4H ≈ -0.95 π

(8)

(assuming the value of the homogeneity index is equal to 3 /4). The molecular bilayer is constituted of individual molecules, whose chains are liquid-like and splayed to accommodate the bonding requirements of the polar headgroup as well as the preferred relative packing of chains. An average molecular shape is conveniently quantified (23) Grosse-Brauckmann, K. The family of constant mean curvature gyroids; Centre for Geometry, Analysis, Numerics and Graphics, University of Massachusetts, Amherst: 1995. (24) Hyde, S. T. In Defects and processes in the solid state. Some examples in earth sciences; FitzGerald, J. D., Boland, J. N., Eds.; Elsevier: Amsterdam, 1993. (25) Hilbert, D.; Cohn-Vossen, S. Geometry and the Imagination; Chelsea Publishers: New York, 1952.

B

Figure 5. (a) Schematic representation of a “slice” of an ideal, homogeneous sponge structure. The dividing surface (radii of curvature R1 and R2) lines the mid-surface of the bilayers, and the head-groups define two interfaces on either side of the dividing surface. The edges AB and CD lie on the centers of curvature. These edges define the channel axes of the sponge. (The distances x and y are defined in Appendix I). (b) Arrangement of bilayers in type II structures: the mid-surface runs between the chain ends of the constituent monolayers, and the water lies closest to the centers of curvature.

by the “surfactant parameter”, defined by the molecular dimensions volume, v, head-group area, a, and molecular length (normal to the interface), l:26

s≡

v al

(9)

This parameter can be rewritten in terms of the mean and Gaussian curvatures of the interface running through the molecular head-groups (M and K, respectively), assuming these interfaces are parallel to the (ideal) midsurface, running between the monolayers (Figure 5):27

s ) 1 + Ml +

Kl2 3

(10)

Bicontinuous mesophases can be most broadly classified according to the bilayer “polarity”: if the (idealized) minimal surface cleaves the hydrophobic domain, so that water fills the cores of both channel labyrinths, the phase is “reversed” (type II); if the minimal surface lies within the water film, the two intertwined channel labyrinths are hydrophobic (type I). For convenience, consider only type II systems. Here the inner volume contains the water solvent, and the outer volume the molecular chains. The surfactant parameter can be generalized to describe the domain shapes on both sides of any (homogeneous) hyperbolic interface, of curvatures M and K. Two shape parameters can be determined for any interface. These “inner” and “outer” shape parameters (sin, sout) describe the form of typical slices of the subvolumes on either side of the interface. The slices are bounded on one side by the interface and on the other by the center of curvature, (26) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (27) Hyde, S. T.; Ninham, B. W.; Zemb, T. J. Phys. Chem. 1989, 93, 1464-1471.

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some estimates of the functional form of isotropic swelling can be drawn. Two swelling modes can be readily analyzed geometrically: the constant shape parameter (measured at the mid-surface) throughout dilution and the swelling under the constraint of fixed areasretaining an inextensible surface(s), located on (or parallel to) the homogeneous mid-surface.18,28,29 Assume for convenience that the sponge remains isotropic (viz. random or cubic) at all times. Consider first the swelling law fulfilled where the idealized bilayer swells without altering the average molecular shape within the membrane, quantified by the inner shape parameter, sin. In that case, the bilayer concentration (its volume fraction, φ) scales according to the power law28 Figure 6. Variation of the inner and outer shape parameters for ideal sponge-shaped bilayers. Shape parameters for spheres, cylinders, and planes are also shown.

distances R1 and R2 from the interface, where these radii of curvature are related to the curvature by

R1 ) M + xM2 - K, R2 ) M - xM2 - K so that

sin ) 1 - MR1 +

KR12 KR22 , sout ) 1 + MR2 + 3 3

(11)

Assuming a hyperbolic bilayer can be modeled by two parallel interfaces running through the head-groups, on either side of a single-sheeted (idealized) minimal surface, the inner and outer shape parameters vary with the concentration of a type II bilayer, as shown below (Figure 6) (this result is derived in the Appendix). A simple limit case can be read from Figure 6: in very dilute, type II, ideal, hyperbolic membranes, the solvent domain has a shape parameter (sin) approaching 2/3, while the molecular chain domain has a shape parameter (sout) decreasing toward 1. This result follows directly from eq 11. It is important to point out the slow variation of the outer shape parameter with concentrationsthe molecular shape remains close to unity up to ca. 40% (v/v) bilayer in water, while the inner shape parameter approaches 1/2 (at high concentrations) almost linearly. These numbers straddle the (inner) shape parameters for rods (M ) 1/(2R), K ) 0, so that sin ) 1/2) and sheets (M ) K ) 0, sin ) 1). The grossest features of the sponge morphology are thus captured by its inner shape parameter: the sponge is “ideal” if its inner parameter is close to 2/3, sheetlike if its inner shape parameter exceeds 2/3, and increasingly threadlike as the inner parameter decreases to 1/2. If the sponge contains a number of disconnected hyperbolic bilayersssuch as those found in mesocrystalline “mesh” phases6sits average inner shape parameter must lie between 2/3 (one sheet) and 1 (many sheets). Swelling of Sponges Consider adding water to a hyperbolic (type II) molecular bilayer. In order to accommodate the swelling volume required by the water, the membrane must flatten, thereby exposing ever-larger hydrophobic regions to the solvent unless the molecular dimensions of the bilayer adjust correspondingly. If it is assumed that these dimensions are tuned to ensure the polar-apolar interfaces (on both faces of the bilayer) remain parallel to the minimal surface,

(

(1 - φ)sin ∼ 1 -

l R

)

(12)

This equation holds exactly for spheres (sin ) 1/3), cylinders (1/2), and planes (1). Its validity for sponges can be questioned, since their shape parameter varies with concentration. In fact, the swelling laws in eqs 11 and 12 also hold exactly for ideal sponges, analyzed in detail in the Appendix. As noted above, if the mid-surface of the dilute bilayer lies on a homogeneous minimal surface, the exponent sin must be close to 2/3. Thus, fits to the scaling law (eq 12) require two parameters: the inner shape parameter and the ratio between the measured length and the radius of curvature. In general, we are unable to measure directly the average radius of curvature of the mid-surface of the bilayer; however, any characteristic length of the sponge is proportional to that radius, assuming an ideal geometry (eq 7). The swelling law reverts to a polynomial form if the bilayer maintains a surface of fixed area (an inextensible surface, sometimes defined as a “neutral surface”) parallel to the (ideal) minimal surface separating the constituent monolayers. A detailed analysis of these swelling forms lies beyond the scope of this article and can be found elsewhere.18 The simplest law holds if the inextensible surface (of area Ω0 per bilayer molecule) lies on the minimal surface, in which case the bilayer swells without altering the cross-sectional area of the molecular chains at their free ends. In that case the swelling follows a hyperbolic form:

φ)

(-16πχξH2)1/3vs -1 ‚ξ Ω0

(13)

This functional form is identical to that commonly believed to characterize ideal lamellar mesophases and corresponds to the form proposed by Porte et al. to characterize sponge swelling.30 In fact, a hyperbolic dependence of the spacing on concentration is characteristic of any geometry, provided swelling occurs without alteration of the chain-end cross-sectional area.28 The balance of attractive and repulsive forces along the molecules may lead to the maintenance of inextensible surface(s) closer to the polar domains.31 (By symmetry, two surfaces result: one within each monolayer.) If both neutral surfaces, of area Ωt per molecule, are located at a separation t from the mid-surface, the swelling law (28) Hyde, S. T. Colloids Surf., A: Physicochem. Eng. Aspects 1995, 103, 227-247. (29) Templer, R. H. Langmuir 1995, 11, 334-340. (30) Porte, G.; Marignan, J.; Bassereau, P.; May, R. J. Phys. (Paris) 1988, 49, 511-519. (31) Seddon, J. Biochim. Biophys. Acta 1990, 1031, 1-60.

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where

adopts the form

(-16πχξH2)1/3vs -1 4πχξvst2 -3 φ) ‚ξ + ‚ξ Ωt Ωt

(14)

The presence of an inverse quadratic term signals the formation of a mid-surface of nonzero mean curvature (such as expected in an “asymmetric” sponge mesophase):

(

)

-2πχξ H

c1 )

1/3

l

This form is (least squares) fitted to the (ξ, φ) data, yielding estimates of the inner shape parameter, sin, and c1. The neutral surface swelling mechanism results in the slightly different form

18

φ ) c2ξ-1 + c3ξ-3 φ ) cξ

-1

+ c′ξ

-2

+ c′′ξ

(15)

-3

Thus, the assumption of a mid-surface of zero mean curvature (a minimal surface) can be checked by ensuring the second term on the right-hand side of eq 15 is small.

Least-squares fits of the (ξ, φ) data to this formula give estimates of the two fitted parameters, c2 and c3, defined from eq 14. The global structural parameters of the sponge, H and χξ, are given by the expressions

Mesostructure of Sponges: Analysis of Some L3 and Cubic Phases Most sponge mesophases identified to date have been found in dilute mixtures, in contrast to the more concentrated bicontinuous crystalline mesophases. In the dilute case, the molecular dimensions undergo little alteration during swelling, and we may (perhaps) assume that all swelling modes are approximately valid.32 The key to structural determination in reasonably dilute systems rests on this assumption that all swelling modes apply approximately. In that case, the various swelling laws (eqs 12-15) can be fitted to length scale vs concentration (ξ, φ) data, yielding estimates of the structural parameters. Fits of those swelling functions lead to estimates of the homogeneity index of the sponge (H), its topology, χξ contained within the volume ξ3, and the molecular length, l. The details of the fitting procedure are as follows. Four parameters are fitted by two distinct least-squares fits to swelling functions: swelling at constant molecular shape (i.e. fixed shape parameter) and swelling whilst maintaining an inextensible surface at a fixed distance from the mid-surface of the bilayer. The procedure yields estimates of the average shape parameters, the bilayer thickness (twice the molecular length), the homogeneity index (measured at the mid-surface), and the EulerPoincare´ characteristic of the bilayer mid-surface contained within a characteristic volume d3, where d denotes a characteristic length of the bilayer spongessome multiple of the channel radius of the mid-surface. Assuming swelling at fixed shape parameters, formulae 6 and 12 imply

(

(1 - φ)sin ∼ 1 -

)

c1 ξ

(16)

(32) In the dilute limit, the exponential swelling form, eq 12, reduces to sinφ ∼

l R

or (from eq 7)

sinφ ∼

(

)

H -2πχξ

-1/3 l

ξ

The neutral surface swelling form (eq 14) can also be simplified in the dilute approximation, as the neutral area, Ωt, is close to the area at the mid-surface (Ω0) and the shape parameter vs/Ω0l ≈ 1: 2 -1/3

φ≈

2(-2πχξH ) Ω tξ

vs

l ≈ 2(-2πχξH2)-1/3 ξ

Combining these two dilute scaling laws gives the relation 2sin ≈ H-1 which is consistent with the ideal values for the inner shape parameter and homogeneity index, viz. 2/3 and 3/4, respectively. (33) Chung, H.; Caffrey, M. Biophys. J. 1994, 66, 337.

(17)

H) and

χξ )

( )( ) c2 Ωtl 2c1 vs

(18)

( )( ) c2c12 Ωtl 4πl3 vs

(19)

where the postfactor on the right-hand side of these equations (a modified shape parameter) can be determined from the two equations

(

)

vs c2 vs 1 + ξ2 ) Ωtl Ω0l c3

(

( ( ) )(

-1

)

)

c1 2 c1 31+ vs ξ ξ ) sin Ω0l c1 2+ ξ

(20)

(21)

Here vs refers to the volume of an individual surfactant or lipid molecules within the membrane. Thus, the homogeneity index can be fitted from estimates of sin, c1, and c2. The molecular length, l, must be estimated in order to derive an estimate of χξ. This can be done directly from, for example, fits to lamellar spacings. However, these estimates implicitly assume the lamellar structure is ideal, which may be incorrect. A less flexible dimension is the chain area, measured at the free end of the bilayer molecule. I choose a standard value for the cross-sectional area of the bilayer molecules measured at the (minimal) mid-surface, i.e. the area at the chain end of the molecule. The chain length estimate then follows from the estimate for vs/Ω0l (eq A6), as the chain volume, vs, is also known. This somewhat convoluted technique results in physically sensible estimates for the chain length: 20 Å for glycerol monooleate, 8-10 Å for AOT, and 9 Å for SDS (using the data for molecular volume and cross-sectional areas listed below). Mesostructure of Some Cubic and Sponge Phases This assumption of equal adherence to all swelling forms holds exactly only in the limit of infinite dilution. Certainly, the approximation is less valid for (moderately concentrated) bicontinuous cubic mesophases. Therefore it is instructive to assume both swelling laws hold for a cubic mesophase and determine the accuracy of the assumption. To that end, data collected by Chung and

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Figure 7. Fits to the lattice parameter vs concentration swelling data collected by small-angle scattering measurements in glycerol monooleate (GMO)-water mixtures that form the Ia3d bicontinuous cubic mesophase:18,33 top, fit assuming swelling at constant molecular shape; bottom, fit assuming neutral surface swelling. Table 2. Fitted Parameters to Swelling Data for (i) a Bicontinuous Cubic (Ia3 h d) Mesophase in Glycerol Monooleate-Water Mixtures,4,18 (ii) a Sponge Mesophase in SDS-Pentanol-Water Mixtures,1 (iii) a Sponge Mesophase in AOT-NaCl-Water Mixtures,2 (iv) a Confined Sponge Mesophase in the AOT-NaNO3-Water Mixtures,3,34 and (v) a Lamellar Mesophase in AOT-NaCl-Water Mixtures2 a system

mesophase

sin

χ

H

l (Å)

i ii iii iv v

V2 L3 L3 confined L3 LR

0.60 0.68 0.64 0.72 0.82

-5.4 -0.8 -0.8 -5.5 -0.6

0.77 0.80 0.78 0.72 0.47

20 10.4 10 9.2 8.1

a The experimental repeat length (via SAS, surface force measurements) vs concentration data are fitted to two swelling laws, cf. the text. The fitted parameters are sin, the inner shape parameter; χ, the topology of the bilayer within a volume equal to the cube of the repeat length; H, the homogeneity index; and l, the monolayer thickness (Å).

Caffrey4 and Engblom18 in glycerol monoolein-water mixtures that form a bicontinuous cubic mesophase of space group symmetry Ia3h d have been analyzed in this fashion. Assuming specific gravities of unity for all bilayer molecules, the surfactant molecular volume is taken to equal 600 Å3 for glycerol monooleate. The cross-sectional area is taken to equal 35 Å2 per amphiphile for the glycerol monooleate bilayers. The fitting procedure results in estimates for these structural parameters which are surprisingly close to those values expected if the bilayer is folded onto the gyroid (minimal) surface, as well as a physically reasonable estimate of the monolayer thickness (Figure 7, Table 2, cf. Table 1). Comparison of the fitted inner shape parameter with that expected for volume fractions of bilayer between 70

Figure 8. Fits to the characteristic spacing vs. concentration swelling data collected by small-angle scattering measurements in SDS-pentanol-water mixtures that form an L3 (sponge) mesophase:1 top, fit assuming swelling at constant molecular shape; bottom, fit assuming neutral surface swelling.

and 80% (Figure 6) shows that the bilayer shape is close to ideal. Given that a number of investigations have supported the gyroid model as a valid one for this system,4,16,17 the approximation is not a bad one. Clearly, it should be even more accurate for (more dilute) sponge mesophases. The corresponding analyses of (admittedly more meagre) swelling data published in SAS studies of two sponge mesophases, sodium dodecyl sulphate (SDS)-pentanolwater mixtures1 and Aerosol OT-sodium chloride-water mixtures,2 are shown (Figures 8 and 9; Table 2). SAS studies of these sponges indicate that the membranes are spatially correlated but disordered mesostructures, so the analysis involving lattice parameters cannot be invoked. However, the variation in momentum transfer of the single SAS peak as a function of the concentration of the sponge can be used as a structural probe in the same way as the lattice parameter variation is used for crystalline mesophases. (An implicit assumption in this approach is that the form factorsthe scattering motifsremains unchanged with concentration.) The real-space location of the scattering peak offers an estimate of a characteristic length of the sponge, ξ, and fits assuming inextensible surface swelling and swelling at constant molecular shape can be adduced. Notice that physically reasonable values of the bilayer thickness have been deduced from the fits in both cases. Surfactant molecular volumes are taken to equal 600 Å3 for AOT and 360 Å3 for SDS. The parameters for cross-sectional area are set to 60 Å2 for the double-chained AOT molecules and 35 Å2 for SDS molecules. Some idea of the shape of these bilayers can be deduced from the fits. The simplest picture of sponge mesophases as “molten” bicontinuous structures seems a valid one in these cases. They can be modeled reasonably well by an almost-ideal sponge structure, viz. a single-sheeted iso-

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Langmuir, Vol. 13, No. 4, 1997 849

Figure 9. Fits to the characteristic length vs concentration swelling data collected by small-angle scattering measurements in AOT-NaCl-water mixtures that form an L3 (sponge) mesophase:2 top, fit assuming swelling at constant molecular shape; bottom, fit assuming neutral surface swelling.

Figure 10. Swelling plots for sponge mesophase confined between mica sheets. Here the characteristic spacing refers to the wavelength of oscillations in the force measurements, recorded as the sheets are brought together. Data estimated from refs 3 and 34.

tropic hyperbolic bilayer whose mid-surface oscillates about a homogeneous minimal surface to give a slightly higher surface area than that of the underlying minimal surface. This structural interpretation follows from the values of the homogeneity indices, which slightly exceed 3 /4. Further, the Euler-Poincare´ indices are close to -0.95 (cf. eq 8), suggesting that the characteristic lengths associated with the single broad (pseudo-Bragg) peak observed in SAS measurements of these mesophases are indeed equal to a typical channel diameter, as has been widely assumed.

that the mid-surface of the bilayer is either far from minimal or contains a significant density of puncture defects, which lower the effective membrane area. If it is assumed that the sponge-shaped bilayer is isotropic, layers (separated by a single channel) are not expelled between the mica plates one-by-one. If that were so, we expect the fitted value of χ to be close to -0.95; its fitted value is -5.5. However, the sponge is likely to be anisotropic, due to the presence of the confining mica walls, which presumably act to flatten the membrane. In that case, the average radius of curvature of the sponge is an overestimate of the pore radius, so that the characteristic associated with the pore diameter will be less negative than -0.95; thus, the magnitude of χ remains inconsistent with the characteristic length mirroring the pore diameter. The meaning of this discrepancy between average spacing and force-profile wavelength has been the subject of some debate. Antelmi et al.3 have suggested that this is due to the induction of a symmetric sponge, viz. a membrane whose mid-surface is minimal (i.e. nonzero mean curvature). This possibility is consistent with the absence of an inverse quadratic term in the polynomial swelling form (eqs 14 and 15). As explained above, this points to a mid-surface of the membrane of (at least on average) zero mean curvature. It is likely then that the slightly reduced value of H compared with that for an ideal sponge can be ascribed to the presence of puncture defects within the bilayers. Petrov et al. have advanced a different explanation, suggesting that the confined sponge spontaneously orders, to resemble a bicontinuous cubic phase.34 In that case, the effective χ is set by the hkl reticular planes which lie parallel to the mica sheets. It is likely that the ordered sponge will orient such that the densest lattice plane lies parallel to the walls. (This plane contains the highest

Mesostructure of a Confined Sponge Mesophase This analysis can also be applied to “swelling data” collected in surface-force measurements of sponge mesophases spontaneously assembled in AOT-NaNO3-water mixtures confined between crossed mica cylinders in the surface forces apparatus. Two independent studies of this system have been published.3,34 The force profile as a function of separation between the curved mica plates is oscillatory, and the wavelength of the oscillations is believed to be related to sponge dimensions across the bilayer(s). Structural analysis of this system is of some interest, as it may afford insight into the effect of confinement on sponge mesostructure. Fitting of these data to the swelling laws leads to the results shown in Figure 10 and Table 2 below. Here the sponge shape deviates somewhat from the ideal case: both H and the inner shape parameter equal 0.72. The slightly reduced homogeneity index compared with that of an ideal sponge may be due to a loss of homogeneity or decreased bilayer area compared with that of an ideal sponge, so (34) Petrov, P.; Miklavcic, S.; Olsson, U.; Wennerstro¨m, H. Langmuir 1995, 11, 3928-3936.

850

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H ) 0.47 is well below that expected for an ideal sponge and the characteristic χ ) -0.6 differs from the value expected were repeat spacings tuned to the average channel diameter. All three fitted parameters are indicative of a multisheeted stack of slightly “cross-linked” membranes (via channels). Since the SAS spectra reveal uniaxial spatial ordering, the channels must remain disordered, while the sheets order into a stack. (Such a mesostructure has been observed directly in block copolymers, see ref 36 .) The distinctively nonideal topology of this lamellar mesophase cannot be detected by the more usual scaling form relating bilayer volume fraction to spacing (eq 11). This plot is fitted reasonably well by a hyperbolic function,2 as it is for all the systems analyzed here. This result is not surprising, given the insensitivity of the outer shape parameter to concentration. Discussion

It is revealing to apply the same analysis to SAS data collected within a lamellar phase of AOT-NaCl-water mixtures by Strey et al.2 In that case very different parameters emerge to those fitted to sponge mesophases (Figure 11, Table 2). The fitted inner shape parameter, 0.82, exceeds 2/3, indicative of a number of disconnected sheets, consistent with the general topology of lamellar phases. However, the values of the (inner and outer) shape parameters for the “ideal” lamellar geometrysextended planar stackssare exactly unity (Figure 6). Thus this lamellar mesophase is far from ideal, containing a significant density of channels (formally wedge disclinations) between sheets. Further, the fitted characteristic

The technique presented here seems to be a reasonable one for inferring membrane topology in partially ordered lyotropic systems, from which little information can be gleaned via conventional analyses of X-ray or neutron scattering spectra. That conclusion is confirmed by the fits applied to a “known” mesostructure in a bicontinuous cubic mesophase, which yield good agreement with expected structural parameters: the homogeneity index, the Euler-Poincare´ characteristic, bilayer thickness, and shape parameters. The conventional view of sponge mesophases as molten bicontinuous structuresstopologically similar to bicontinuous cubic phasessis supported by the analyses of two examples here. In both cases the structural parameters are close to those expected for “ideal” sponges, i.e. bilayers wrapped onto hyperbolic surfaces of approximately constant negative Gaussian curvature, enclosing intertwined and continuous polar domains. The analysis of confined sponge phases is revealing although far from definitive at this stage. It seems that the confined sponge differs in structure from the bulk case, evidenced by the difference between characteristic lengths detected by SAS techniques and those seen in force measurements. The force data are consistent with the formation of a “superstructured” symmetric (i.e. zero mean curvature at the bilayer mid-surface) sponge, of more complex topology per unit cell than the simplest P-surfaces, D-surfaces, and gyroid surfaces, containing a number of interlayer distances, as in the Neovius surface. (This possibility differs from the conclusions offered by both groups who collected the primary data in this system that structuring, induced by the confining walls, can be understood as a loss of isotropy and ideality.) For example, the Neovius [110] planes are largely flat, offering a favorable orientation parallel to the mica walls, and the fitted data are consistent with a hyperbolic bilayer folded onto the Neovius surface. At this stage it is impossible to determine whether the confined sponge crystallizes; however, its topological features are consistent with the formation of a superstructured sponge. The relevance of this analysis is not confined to spongelike structures alone. Estimation of an inner shape parameter is possible for other structures, such as the “lamellar” mesophase analyzed above. This example forcefully reveals the danger of structural interpretation based only on the pseudo-Bragg peak distribution in single SAS spectra. These peaks are sensitive only to the degree of spatial ordering and indifferent to bilayer topological

(35) Andersson, S.; Hyde, S. T.; Larsson, K.; Lidin, S. Chem. Rev. 1988, 88, 221-242.

(36) Hashimoto, T.; Koizumi, S.; Hasegawa, H.; Izumitani, T.; Hyde, S. T. Macromolecules 1992, 25, 1433-1439.

Figure 11. Fits to the lattice parameter vs concentration swelling data collected by small-angle scattering measurements in AOT-NaCl-water mixtures that form an LR (lamellar) mesophase.2

area of the surface). The densest plane in ETPMS is expected to be parallel to the tangential directions of flat points: i.e. the [111] planes for the D-surfaces, gyroid surfaces, and P-surfaces. From eq 4, it follows that the effective χ should equal -0.4, -0.8, and -1.1 for bilayers folded onto these ETPMS, respectively. These data too are inconsistent with the fitted value of χ, -5.5. (Note that Petrov et al.34 have suggested that alignment of the [111] planes with the walls occurs, and the sponge resembles the gyroid surface. In that case, a fitted χ of -0.8 would result. This possibility is thus unlikely.) Ordering of the hyperbolic bilayer following other, higher topology, ETPMS is, however, possible. Indeed, the data are consistent with alignment of the (flattest) [110] planes of the Neovius (also called the C(P)) surface,35 whose Euler-Poincare´ characteristic per conventional unit cell is equal to -16 (so that χ111 ) -5.6, cf. eq 4). Topological Form of a “Lamellar” Mesophase

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Langmuir, Vol. 13, No. 4, 1997 851

features which remain uncorrelated. In order to infer topological features from SAS spectra, the variation of peak positions with concentration must be analyzed. Appendix The shape parameter for an ideal sponge and its dependence on sponge volume fraction follow directly from analysis of a representative slice of the sponge and the associated volume on either side.6,18 The volumes are assumed to be foliated by a stack of surfaces parallel to the original hyperbolic dividing surface, which runs through the mid-surface of the bilayer. If a rectangular patch of the mid-surface is taken, the slice is tetrahedral in shape, where the edges AB and CD lie along the centers of curvature of the hyperbolic patch (Figure 5). The headgroups of the bilayer constituent molecules are assumed to line the two surfaces parallel to the mid-surface. Define the distance between one of the parallel surfaces and the nearer center of curvature as y and the distance to the farther center of curvature as x. Assume the mid-surface is a minimal surface, with radii of curvature R. This means that the parallel surfaces have nonzero mean curvature and negative Gaussian curvature (M and K, respectively). The inner shape parameter, sinswhich measures the form of the slice from the parallel surface to its nearer center of curvaturesis defined by

sin ) 1 + My +

Ky2 3

Kx2 3

vin vin sin ay (A7) φin ) ) ) x vin + vout vin vout x sin + sout‚ + ‚ y ay ax y where vin and vout are the volumes of the prisms to either side of the parallel surface and a is the area of the quadrilateral parallel surface patch (using the original geometric definitions of the inner and outer shape parameters, cf. eq 9). Combining eqs A5-A7,

φin )

(sin - 1/3)3

y y+x

(A9)

3 - 2λ 6(1 - λ)

(A10)

λ≡ It follows that

sin )

(A2)

∂(ln(φin)) ∂(ln(λ))

(A3)

) sin-1 so that λ ∼ φinsin

λ)

λ) x 1 ) y 6sin - 3 9sin - 4 9(2sin - 1)

(A5)

(A6)

The volume fraction of the quadrilateral prism between the parallel surface and the nearer center of curvature,

(A12)

where φ is the amphiphile concentration (volume fraction). If the mesostructure is type II (curved toward the polar domains),

and

sout )

l 2R

The swelling law then assumes the simple form

These equations imply

(A4)

(A11)

The scaled length, λ, is a simple function of the average radius of curvature (R) of the minimal mid-surface of the hyperbolic membrane, of thickness 2l. For type I bilayers

l ∼ φsin R

My ) 3sin - 2 and Ky2 ) 3 - 6sin

(A8)

A dimensionless length variable, λ, is defined by

The lengths are related to the curvatures by the equations

1 + 2My + Ky2 ) 0 and 1 - 2Mx + Kx2 ) 0

sin(sin - 1/2)2

Straightforward calculus then shows

(A1)

and the outer shape parametersfrom the parallel surface to the farther center of curvaturesby

sout ) 1 - Mx +

φin is given by

R-l 2R

In this case, the inner volume fraction is linked to the amphiphile concentration by φ ) 1 - φin so that the swelling law (eq A11) adopts the form

(1 - Rl ) ∼ (1 - φ)

sin

LA9605347

(A13)