J . Phys. Chem. 1989, 93, 1458-1464
1458
Microstructure of Blcontinuous Surfactant Aggregates S . T. Hyde Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Box 4 , Canberra, A.C.T. 2601, Australia (Received: March 25, 1988; In Final Form: July 13, 1988)
It is known that many bicontinuous periodic minimal surfaces (and related constant average curvature surfaces) exist for any space group symmetry. These surfaces are distinguishable by their topology, characterized by the genus per unit cell of the surface. Structural assignation to bicontinuous ordered liquid crystalline phases (e.g., cubic phases) on the basis of symmetry alone is thus inadequate. We analyze constraints on the microstructure of cubic phases due to the surfactant molecular dimensions (which fix the local geometry of the aggregate) and the sample composition (which determines global characteristics) and derive general equations relating the microstructural topology to these data. The analysis indicates that standard microstructural models of bicontinuous cubic phases-which are reversed or normal bilayers on cubic minimal surfaces of genus three-are consistent with the data in many cases. The microstructuresof cubic phases formed by mixtures of SDS/water and AOT/water appear to be more exotic than those already suggested in the literature.
1. Introduction It is well-known that the amphiphilic nature of surfactant molecules leads to self-aggregation in solution, so that they isolate hydrophobic from hydrophilic regions. Surfactant aggregates typically form a variety of liquid crystalline phases as well as isotropic (microemulsion) phases. The microstructures of lamellar, micellar, and many hexagonal phases are reasonably well-modeled by classical surfaces: viz., planes, spheres, and cylinders, respectively. It is becoming clear that more exotic surface geometries are required to describe bicontinuous phases of surfactant solutions. The earliest descriptions of bicontinuous cubic phases, for example, involve arrays of cylinders1 and interconnected plane-faced polyhedra2 that contain cusps that cannot occur naturally. Much attention has been focused on minimal surfaces as models for bicontinuous microstr~ctures.~" It has also been shown recently that surfaces of constant average curvature furnish useful structural models of micro emulsion^.^ In this paper we derive equations linking the topology and geometry of bicontinuous structures to the geometry of the surfactant solution: the polar and paraffin volumes and the surfactant head-group area and chain length. We focus our attention on bicontinuous cubic phases; nevertheless, the derivations are quite general. They are equally valid for any lyotropic liquid crystalline phase. So far, bicontinuous cubic phases of binary and ternary surfactant solutions have been found that exhibit Im3m,' Pn3m,4s5*9 and Ia3d'*5*9 symmetries. In all these cases it has been suggested that the corresponding microstructures consist of two interpenetrating networks of water (oil) separated by a (reversed) bilayer. The suggested geometry of these netwocks is as follows: (i) two interpenetrating simple cubic nets (Zm3m);(ii) two interpenetrating diamond nets (Pn3m);(iii) two interpenetrating oppositely handed three-connected helices (Laves graphs) ( Z d d ) . For all three microstructural candidates the (reversed) bilayer interface forms a surface of genus three per unit cell,lo the lowest possible ( I ) Luzzati, V.; Tardieu, A,; Gulik-Krzywicki, T.; Rivas, E.; Reiss-Husson, F. Nature (London) 1968, 220, 485. (2) Lindblom, B.; Larsson, K.; Johansson, L.; Fontell, K.; ForsCn, S. J. Am. Chem. SOC.1979, 101, 5465. (3) Scriven, L. E. Nature (London) 1976, 263, 123. (4) Longley, W.; McIntosh, J. Nature (London) 1983, 303, 612. (5) Hyde, S. T.; Andersson, S.; Ericsson, B.; Larsson, K. Z . Krisfallogr. 1984, 168, 213. (6) Charvolin, J.; Sadoc, J. F. J. Phys. (Les Ulis, Fr.) 1987, 48, 1559. (7) Barnes, I . S.; Hyde, S. T.;Ninham, B. W.; Derian, P.-J.;Drifford, M.; Zemb, T. N. J . Phys. Chem. 1988, 92, 2286. (8) Ktkicheff, P.; Cabane, B. J . Phys. (Les Ulis, Fr.) 1987, 48, 1571. (9) Gulik, A.; Luzzati, V.; De Rosa, M.; Gambacorta, A. J . Mol. Biol. 1985, 182, 131. (10) Schoen, A. H. N A S A Techn. Note 1970, D-5541
0022-365418912093-1458$01.50/0
topology for a three-dimensional periodic netw0rk.l' Recent studies indicate the profligacy of bicontinuous structures for all space groups. These are of variable topology, viz., genus three per unit cell and higher.'2*13 The structures are distinguishable by only their symmetry and topology. The symmetry of cubic phases is routinely deciphered from X-ray data. The topology of a bicontinuous structure is less directly amenable. In this paper we outline quantitative techniques for determining the topology of periodic bicontinuous microstructures.
2. Differential Geometric Analysis of Interfacial Topology The role of surfactant molecules as structural barriers between polar and paraffin regions suggests three possible structures for bicontinuous cubic phases of surfactant solutions: (i) curved bilayers of surfactant, separating interpenetrating polar regions (designated by Luzatti "type 11" structures,' Figure 1) (for mixtures containing oil, the bilayers are oil-swollen); (ii) curved, reversed, water-swollen bilayers, separating interpenetrating paraffin regions ("type I" structures, Figure 2); (iii) curved monolayers of surfactant, separating continuous paraffin and water regions (Figure 3 ) . The first two possible microstructures have previously been assumed to be omnipresent in cubic phases. Nevertheless, we have found that ternary microemulsions form surfactant monolayers. Translationally ordered surfaces decorated by monolayers thus also merit consideration as microstructural models for cubic phases. These three possibilities are amenable to quantitative analysis using parallel surface theory. A parallel surface to an interface is formed by translating the interface along its normal vectors by an equal distance everywhere on the interface. The surface metric of the parallel surface is related to the metric of the interface (g(0)) by g(5) = + 2HE + (1)
a21*
where E is the oriented distance along the normal vector and H and K are the mean and Gaussian curvatures of the i n t e r f a ~ e . ' ~ (We adopt the convention that the mean curvature is positive if the interface is curved toward the polar region and negative if curved toward the paraffin region.) Now, the area of a coordinate patch on the parallel surface is related to the corresponding patch on the interface da(0) by15
(11) Meeks, W. H. Bull. A m . Math. SOC.1977.83. (12) Lidin, S.; Hyde, S. T. J . Phys. (Les Ulis, Fr.) 1987, 48, 1585. (13) Hyde, S. T. 2.Kristallogr., in press. (14) Goetz, A. An Introduction to Differential Geometry;Addison-Wesley: Reading, MA, 1970; p 268.
0 1989 American Chemical Society
Bicontinuous Surfactant Aggregates
rhe Journal of Physical Chemisfry,Vol. 93, No. 4, 1989 1459 Chain volume _.I.
---\. --..:.-._. _..' P
~ ~ ~surface I I ~i tI I~
Centre ofbilayer
I
,
,\
hematic view of B cross-section through a curve mer formed I m a water/surfactant mixture. Integrating the area &all parallel surfaces from the interface to the parallel surface traced out by the head-groups gives the chain volume (shaded region). The hatched portion of the interface marks the interfacialarea occupied by each facing pair of surfactant molecules. Figure
Parallel surface (.I
Figure 4. Parallel surface construction far a curved bilayer formed from
a binary mixture of water and surfactant. The hatched surface reprsenui the interface, which is situated between the chains of opping surfactant molecules. The head-groups trace out a pair of parallel surfaces, located at distances fl along the normal vector to the surface (hatched arrows). We define surface-averaged values of the mean and Gaussian curvature by
Finally, we obtain the equation linking the volume per unit cell of a shell of thickness f (bounded on one side by a unit cell of the interface and on the other by a parallel surface at a distance f from the interface) to the surface-averaged mean and Gaussian curvatures of the interface: Figure 2. Cross-section through a curved reversed bilayer formed by a surfactant/water mixture. The shaded region indicates the volume occupied by water, which, together with the head-groups, defines the palar region that lies along the minimal surface (of total thickness 2rJ.
Or, with respect to a surface area located on a parallel surface at distance x from the interface
We can link these equations to the surface topology using the Gauss-Bonnet theorem, which relates the integral (Gaussian) curvature to the Euler-Poincart characteristic, x . of the surface.16
Figure 3. Cross-section through a hypathetical cubic phase of a mixture of surfactant and water consisting of a monolayer of surfactant. The curved interface separates polar from paraffin networks.
Thus, the area of a parallel surface to a translationally ordered interface bounded by a unit cell of the translational lattice is given by
(IS) Goetz, A. Reference 14, p 168.
Strictly, this equation only applies to closed surfaces. We form a closed surface from a single unit cell of the interface by identifying surface elements linked by lattice translation vectors (for further details, see ref 13). (The Euler-Poincart characteristic, x. is then related to the genus per unit cell of the interface, g, by
x=2-2g
(9)
(16) Goetr, A. Reference 14, p 248. (17) Hyde, S. T. Ph.D. Thesis, Monash University. Melbourne, Australia, 1986.
1460
Hyde
The Journal of Physical Chemislry, Vol. 93, No. 4, 1989 Bulk oil laver
5 ) . The nonpolar volume per unit cell is related to this thickness by bilayer interface /
where
Furthermore, the bulk oil (excluding any oil absorbed between the surfactant chains) must lie in a shell located within *(Inp I ) from the bilayer center. Thus Figure 5. Dimensions used to calculate the geometry of an oil-swollen curved bilayer, formed by a ternary mixture of surfactant,water, and oil.
assuming that the interface is orientable.) Equations 4, 5, and 8 give
Equations 7 and IO quantify the packing geometries of surfactant monolayers and bilayers. For example, the surface traced out by surfactant head-groups in a curved bilayer is parallel to that defined by the center of the bilayer (Figures 1 and 4). In the absence of oil, the distance between the surfaces is defined by the surfactant chain length (assuming the surfactant molecules lie normal to the interface), and the area per unit cell is set by the head-group area per surfactant molecule, the composition of the bilayer, and the dimensions of the unit cell. Similarly, the volume of the shell bounded by these parallel surfaces is fixed by the sample composition, the chain volume per surfactant molecule, and the unit cell size. Given that surfactant chains in most liquid crystalline phases are essentially melted hydrocarbons,'8 the average thickness and volume of both monolayers making up the bilayer should be the same. This implies that the mean curvature a t the center of the bilayer is zero, so that the bilayer is equally curved in both directions and lies on a minimal surface. (We ignore the effects of structural fluctuations that lead to undulations of the bilayer about a minimal surface.) If the average chain length is /, the volume of the surfactant chains per unit cell is given by eq 7.
Note that we are dealing with a bilayer, so that A (the total head-group area per unit cell) and the chain volume are twice S and twice V, respectively, in eq 7. The surface-averaged Gaussian curvature is given by
The topology of the curved bilayer-parametrized by the Euler characteristic per unit cell of the hilayer-is thus related to the average chain length of the surfactant molecule and the head-group area and chain volume per unit cell by 3A/ - 3VGhaim
X =
8d'
If the bilayer is swollen by oil, the paraffin thickness of the bilayer (Zt.,) is larger than the average chain length, /(Figure (18) Gruen,
D. W. R. 3. Phys. Chem. 1985.89, 146.
with ( K ) as above. Simultaneous solution of eq 14-16 yields the bilayer topology and thickness. The microstructure of curved (water-swollen) reversed bilayers is similarly constrained. Here, too, the average curvature of the bilayer should vanish over a unit cell since the monolayers are structurally equivalent. If the total width of the polar material (water and head-groups) in the reversed bilayer is 2tp (,Figure 2), the polar volume per unit cell is related to the polar thickness by
The Gaussian curvature is given by
These equations can be simplified to give 3 ~ 4 1 ,X =
v,)
8~1,)
(19)
The volume of the surfactant chains per unit cell (including any absorbed oil) can be described by an equation relating parallel surfaces a t f p and (1, + /) (Figure 2):
Here, simultaneous solution of eq 19 and 20 gives the thickness and topology of the water-swollen reversed bilayer in terms of the chain length, chain volume, and head-group area per unit cell. The third candidate microstructure-a monolayer of surfactant separating interpenetrating water and paraffin networks-is more difficult to analyze. Since there is, in general, no symmetry with respect to sides of the interface, the mean curvature of the surfactant film need not.vanish. An extra constraint is required in order to derive this parameter. A natural requirement of such a structure is that the paraffin volume is equal to the volume of the labyrinth created on one side of the interface. For example, the average width of the labyrinth should equal the average chain length of the surfactant molecules in the absence of oil (Figure 3). It is impossible to derive exact values of the labyrinth volumes and widths for a surface of arbitrary surface-averaged mean and Gaussian curvatures. However, we can estimate these values under the assumption that the volume of the focal surface is vanishingly
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 1461
Bicontinuous Surfactant Aggregates small. (The focal surface of an interface is the surface traced out by the centers of curvature of the interface. Thus, for example, the focal surfaces of a sphere and cylinder are a point and line, respectively.) We have checked this approximation for many minimal and constant-mean-curvature surfaces and found it to be accurate within about 5%. Under the assumption of a focal surface of negligible volume, the entire volume of the surface associated with a patch of the interface can be swept out by a foliation of parallel surfaces from the interface up to the parallel surface of zero area. Since this distance should correspond to the average chain length of the surfactant molecule (from eq 5)
a(l)/a(O) = 1
+ 2 ( H ) l + ( K ) 1 2= 0
From eq 7 , the chain volume per unit cell is related to the mean and Gaussian curvatures by
(Note that the head-groups are located at the interface, rather than at a parallel surface, as for the normal and reversed bilayers.) The surface-averaged Gaussian curvature is related to the topology by eq 23 (see eq 10).
(K)= 2 q / A
(23)
Equations 21-23 give approximate values for the monolayer topology and average curvature:
Fortunately, it turns out that more exact forms are not needed for precise analysis (see later text).
3. Local Surfactant Molecular Geometry The foregoing equations for monolayers and bilayers express the interfacial structure (topology and mean curvature) in terms of global geometrical parameters. The polar and paraffin volumes per unit cell are dependent on only the sample composition and the size of the unit cell, given by the lattice parameter. These equations can be recast to yield the local geometry of the surfactant molecules. It has been frequently demonstrated that the surfactant geometry is most conveniently characterized by the "surfactant parameter", v/al, which relates the chain volume per surfactant molecule, u, to the average chain length, I , and head-group area per molecule, a.19*20 The surfactant parameter for a curved bilayer formed from a mixture of surfactant and water can be shown from eq 14 and 15 to be
If this structure is bicontinuous, the surface-averaged value of the Gaussian curvature must be negative, so that the Euler-Poincar6 characteristic is also negative. In order for the structure to form a three-dimensional connected network, the g e m s per unit cell must be greater than two, so that here, too, the Euler-Poincar6 characteristic must be negative." Consequently, the average surfactant parameter must be greater than 1 if the surfactant is to spontaneously form a bicontinuous curved bilayer. The presence of oil changes the geometry of the surfactant molecule, stretching the chains and resulting in a modification of the natural surfactant parameter for an oil-swollen bilayer. (19) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (20) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 609.
After some algebra, we find that the effective surfactant parameter is related to the interfacial topology by
where
Here, too, the effective surfactant parameter must be greater than 1 in order to form a bicontinuous oil-swollen normal bilayer. These equations can also be modified to express the bilayer geometry in terms of the cross-sectional area of the surfactant molecule at the free-chain end of the molecule (achain)and the head-group ( a ) . From eq 5, we obtain
(where ( K ) is given by eq 26), for an oil-swollen curved bilayer. This ratio is larger than 1 for a bicontinuous structure. This mismatch between the two cross-sectional areas has been described as a "frustration", resolved in Euclidean four-dimensional space6 Equation 29 quantitatively expresses the resolution of this frustration (in curved two-dimensional space). Thus, bicontinuous curved bilayers can be expected to be formed by surfactant molecules with small head-group areas, relative to the cross-sectional area occupied by the surfactant chains (e.g., double and higher chained surfactants). Similar constraints on the surfactant molecular geometry can be derived for bicontinuous, curved reversed bilayers. With the notation introduced earlier
(:Ief
4 q 1 ( 3 t p+ I ) =
+
3A
Consequently, the effective surfactant parameter must be less than 1 for a surfactant solution to spontaneously form reversed bilayers. Similarly, the ratio between head-group and chain cross-sectional areas is given by
(with ( K ) as in eq 18). Thus, for bicontinuous reversed bilayers to form, this ratio must also be less than 1 . The surfactant molecular geometry is not constrained if the molecules self-assemble to form monolayers, since these can curve toward water or oil. The effective surfactant parameter for a monolayer is obtained from eq 6.
4. Bicontinuous Cubic Phases of Surfactant/Water Mixtures Luzzati et al. have argued that cubic phases of surfactant solutions consist of reversed or normal bilayers (types I and 11, respectively),corresponding to two possible phase successions upon water dilution.' Reversed bilayers are claimed to exist for cubic phases of mixtures that neighbor lamellar (surfactant parameter equal to 1) and hexagonal phases (surfactant parameter equal to 0.5) at lower and higher water contents, respectively. It is generally agreed that the head-group area increases monotonically with added water, and the surfactant chain volume and length are not significantly affected by water content, so the effective surfactant parameter should also vary monotonically with composition (at least for binary mixtures). Thus, the claim that reversed bilayers exist for mixtures exhibiting lamellar-cubichexagonal phase behavior upon water dilution is consistent with the result that reversed bilayers can only be formed by surfactants
1462 The Journal of Physical Chemistry, Vol. 93, No. 4, 1989
Hyde
TABLE I: Expected Topologies (Genera Per Unit Cell) of Cubic Phase Mixtures of Various Single-Chained Surfactants with Water, Assuming the Surfactants Form Reversed Bilayers That Lie on Periodic Minimal Surfaces‘
surfactants
A2 I , A 4030 l P o 50 17 4S3* 832 14
ulal T , OC 0.837 100’ 0.47 O.9l2I 2S21 0.52
490
302 32
24
18*
0.87’ 0.61
22’
140.8’ 0.665
360
39’ 42
10’
0.92’
55’
117’
A3
8, cm3 g-l
Vi,,
u,
KC14
0.9930
440
400
C12E06
1.0S3’
780
350
GMO
l.05
590 420
SDS
0.92)?
a,
17
0.50
a,A
@surf
90.7’ 0.62I 11SZ1
V,, A3 A , A2 2tp, A 3.25 X lo5 4.20 X lo4 15 5.26 X lo4 12
7.44 X lo4
28
-5.6 -3.0 -7.4
1.26 X lo6 9.35 X lo4 9.97 X lo4
24
27
-1.8 -3.6
7.41 X lo5 9.35 X IO4 1.01 x 105
15
0.7421 1.10 X IO6
0.63’
x
g
-3.4
26
-4.0
14
-10.0
3.2
* 0.5
* 2,1 * o.7 4,5 * 1,5 3.6
‘ 8 denotes the specific volume of the surfactant, V,,, the volume per surfactant molecule, u the volume per molecule of the hydrophobic chain, a the head-group area, and I the chain length. Upper and lower bounds on these data are estimated from reported data for neighbor phases. T, a, V,, and A denote the temperature, lattice parameter, surfactant volume fraction, polar volume per unit cell, and head-group area per unit cell reported for the cubic phase. The thickness of the polar layer is denoted by 2rp (Figure 2). Sources of experimental data are indicated. Chemical formulas for the surfactants are given in the text. The polar layer width and bilayer Euler characteristic (x) represent upper and lower bounds, calculated from eq 19 and 20. The average value of the genus per unit cell (g) obtained for x from 9 is given, together with the uncertainty obtained from upper and lower bounds for the surfactant parameter.
whose effective surfactant parameter is less than 1 (derived above). We have investigated the cubic phases of two typical singlechained surfactant/water mixtures whose effective surfactant parameters are less than 1 . The surfactants are CH3(CH2)12COOK (KCI4)land the nonionic surfactant CH3(CH2)11(0CH2CH2)60H (C12E06).2’ We have also analyzed the lipid glycerolmonooleate (GMO), which exhibits slightly different phase behavior when mixed with water at room temperature. However, it transforms upon water dilution from inverse micellar (surfactant parameter greater than 1) to lamellar (surfactant parameter equal to 1) to a cubic phase: indicating that it, too, is capable of forming a reversed bilayer in the cubic phase (surfactant parameter less than one). In order to determine the preferred topology of the reversed bilayer (with use of eq 19 and 20), it is necessary to estimate the surfactant chain length, as well as the head-group area per unit cell and the polar and chain volumes per unit cell. The chain volume has been calculated using data collected by Reiss-Hu~son.~~ The head-group area and chain length have been estimated by interpolating between published values for neighboring phases. The polar volume and head-group area per unit cell have been calculated from the published sample composition and lattice parameters. Reasonable upper and lower bounds on the data for all surfactants are presented in Table I, together with the resulting polar layer thickness and the reversed bilayer topologies per unit cell, obtained from eq 19 and 20. We have also calculated the expected structure of the cubic phase of CH3(CH2)110S03Na (SDS) and water, recently reported to be of symmetry Im3m.8 This system undergoes phase transitions from a lamellar phase (surfactant parameter of 1) to a hexagonal phase (surfactant parameter of 0.5)upon water dilution, via a rich collection of ordered phases, including the cubic phase. Assuming monotonic variation of the surfactant dimensions, the surfactant parameter of the cubic phase is less than 1, so that it, too, is capable of forming a reversed bilayer. We have used the reported data on the lamellar and hexagonal phases to estimate upper and lower bounds for the reversed bilayer topology (see Table I). It turns out that the actual structure must have a topology of between genus three and six per unit cell. The phase behavior of the binary mixture of Aerosol OT and water is different from that described above. As water is added to the system, it transforms from a reversed hexagonal phase (surfactant parameter greater than l), through a cubic phase, to a lamellar phase (surfactant parameter of l).23 The cubic phase, consequently, has a surfactant parameter greater than 1, consistent with the formation of a normal rather than reversed bilayer (see previous section). We have used eq 13 to determine upper and lower bounds on the bilayer topology, using the published lattice (21) RanGon, Y . ; Charvolin, J. J . Phys. (Les Ulis, Fr.) 1987, 48, 1067. (22) Ktkicheff, P. These d’Etat, Universite Paris-Sud, 1987. (23) Fontell, K. J. Colloid Interface Sci. 1973, 44,318.
TABLE 11: Upper and Lower Bounds on the Expected Topology of an AOT/Water Mixture That Forms a Cubic Phase, Assuming the Surfactant Forms a Bilayer Lying on a Minimal Surface and the Polar Material Lies in the Interwoven Labyrinths Created by the Curved Bilayer“
AOT 0.8623 640 480 6423 6, 7 1 .25,29 1.07
20 6229 O.7Sz9 1.35 x 105 1.80 x 104 -14.9, -3.1 5.5 & 2.9 “All notation is as described in Table I (here V, denotes the surfactant chain volume per unit cell). The bilayer topologies are calculated from eq 13 in the text.
TABLE 111: Calculated Topologies (Genera Per Unit Cell) for Monolayers of Surfactant That Lie on Surfaces of Constant Average Curvature”
surfactant KCl4 C12E06
GMO SDS
AOT
I, A
ulal
11 8 17 9 6
0.7 0.7
0.7 0.7 1.2
A , A2
4.5 x 7.4 x 1.1 x 1.0 x 1.8 x
104 104 105 105 104
genus 26 112
46 1 I9
170
“The calculations are not exact but are indicative of the genera required to pack the surfactant molecules in a unit cell of dimensions listed in Tables I and 11, from eq 20 in the text (Figure 3). Typical values of the surfactant chain length ( I ) and head-group area are taken from the data listed in Tables I and 11. ulal and A denote the surfactant parameter and head-group area per unit cell, respectively. parameter, composition, and head-group areas of neighboring phases (Table 11). We have suggested earlier that the formation of ordered monolayers cannot be discounted for these phases without further calculations. We have taken typical values of the surfactant molecular dimensions and lattice parameters for all surfactants and calculated the approximate topology of a curved monolayer consistent with these data, using eq 24. The results are displayed in Table 111. For all surfactants it is clear that if these phases do indeed consist of curved monolayers, they must be of extremely high topology. 5. Discussion
In the foregoing analysis the connection between the (reversed) bilayer topology and the geometry of the surfactant molecule
The Journal of Physical Chemistry, Vol. 93, No. 4, 1989 1463
Bicontinuous Surfactant Aggregates TABLE IV: Normalized Surface Area to Volume Ratios ( S / V Y 3 ) and Packing Characteristics (S3/ V 2 x ) for Some Periodic Minimal Surfaces (See Eq 36, Main Text)" surface mace erouD genus S/V2/3 S3/V2x .P34 1m5m36 3" 2.3451" -3.2242 F34 3" 2.4177'' -3.5330 gyr~id~~I~5d'~ 3" 2.4533" -3.6914 T34 P42/nmc37 3'O 2.461434 -3.7554 F-RD" 617 3.005338 -2.7143 Fm3mb" Im3mi7 717 2.751 338 I-WP'O -1.4876 917 3.510535 -2.7039 N e o v i u ~ ~ ~Im3mI7
. - . -
I
OThe space groups and genera per unit cell are also given for each surface. Note that many more periodic minimal surfaces are known, but this table exhausts surfaces whose surface to volume ratios are known a t present. bThese space groups probably refer to the surface with differently colored sides, Le, the monolayer symmetry (subgroup of the full surface symmetry, which is exhibited by a bilayer surface).
(characterized by the effective surfactant parameter) has been established. The surfactant geometry is the result of a balance between the competing repulsive interactions between the headgroups and attractive interactions between the hydrophobic chains in the surfactant molecule^.'^ The preferred local geometry of the surfactant sets the curvature of the interface. In fact, an interface of arbitrary topology can be scaled (via the lattice parameter) to form a structure whose curvature corresponds to that demanded by the surfactant geometry. We have used measured lattice parameters to determine the structure of the interface. So it is natural to ask what determines the lattice parameter. Further, what sets the symmetry of the phase? Once again, geometry clearly plays a part in deciding these issues. For a mixture of number fraction n,, no, and n, molecules of surfactant to oil to water (of molecular volume us, u,, and ow, respectively), the surface to volume ratio of the interface is given by
_S -V
nsas
n,u,
+ novo + n,v,
(33)
where a, is the head-group area of the surfactant. We can estimate the curvature of the bilayer from eq 4, 8, and 13: (34) (Both eq 33 and 34 assume the effective surfactant parameter is close to 1.) For a surface of arbitrary topology, eq 4 and 8 imply cy
=
(3J2
(35)
where LY is the lattice parameter and u the "normalized" surface to volume ratio (Le., the surface area per unit cell for a cell of unit volume). The surface to volume ratio, S / V , is given by u/a. Consequently
Therefore, the actual interface adopted by the surfactant is one whose normalized surface to volume ratio divided by the topology per unit cell satisfies eq 36. Values of this fraction for some periodic minimal surfaces are displayed in Table IV. These data indicate that the normalized surface t o volume ratio depends upon the symmetry of the surface as well as the topology. In fact, this ratio does not vary monotonically with topology. Consequently, more than a single surface may satisfy the geometrical constraints imposed by this equation. However, for reasons outlined below, it is expected that the lowest topology surface to satisfy this equation will be that adopted by the bilayer. These equations demonstrate that both the symmetry and the lattice parameter of the interface are functions of the bulk composition and the surfactant geometry. These results are striking.
It appears that the "structure" of the interface-given by the topology, the symmetry, and the lattice parameter-are all set by geometrical constraints. We have yet to rule out the existence of ordered monolayers in cubic phases. We have found that the formation of ordered monolayers in cubic phases of single-chained surfactants can only occur if the monolayers lie on surfaces of very high topology. Our experience of higher genus periodic minimal surfaces suggests that as the genus of a surface increases, so does the variation in Gaussian curvature over the surface. For example, the Schwarz P-surface is much more uniformly curved than the Neovius surface, which contains large, relatively flat, regions, as well as localized areas of high curvature.24 Since the effective surfactant parameter is dependent upon the Gaussian curvature, packing of surfactant molecules on high-genus surfaces would result in large variations of the surfactant molecular geometry over the surface, resulting in significant deviations from the preferred surfactant parameter. Thus, in general, highly interconnected surfaces (high genus) are less favorable structures than surfaces of lower topology for surfactant/liquid mixtures. Further, for a fixed unit cell size, the radius of the tunnels must decrease as the topology of the interface increases. For the very high topologies required for the formzltion of monolayers for typical cubic phases (Table HI), the tunnels would be too small to accommodate the polar and paraffin regions without unreasonable crowding. On these grounds, it seems that ordered (normal or reversed) bilayers are formed in cubic phases in preference to ordered monolayers. It is important to note that these calculations assume that the surfactant is insoluble in water. More accurate calculations are impossible without detailed measurements of the solubility of the surfactants. However, it is certain that the solubility is low, so the assumption is reasonable. Of course, these calculations are only valid if the bending energy of the surfactant film is larger than the thermal energy of the system. In other words, we assume that the effective surfactant parameter has a well-defined value for the system. Accurate data for the bending energies of various systems are not yet available, but there is ample evidence that this assumption, which underlies the geometric approach, holds. SAXS studies of microemulsions indicate that the high-water region in many ternary mixtures consists of extremely monodisperse spheres, and the upper water limit of the microemulsion region is a line of constant water to surfactant c o m p ~ s i t i o n . These ~ ~ ~ ~facts ~ can only be understood if the surfactant adopts a unique effective surfactant parameter (which may vary with c o m p ~ s i t i o n ) . * ~Furthermore, *~~ measurements of the phase diagrams of many binary systems indicate that the location of phase boundaries is virtually independent of temperature over a wide temperature range.5,22,27,28 Again, this suggests that geometry dominates the phase behavior and microstructure of the system. (The phase behavior of ternary systems (24) For models of this and other periodic minimal surfaces see: Hyde, S. T.; Andersson, S . Z . Kristallogr. 1984, 168, 221. (25) Zemb, T. N.; Hyde, S. T.; Derian, P.-J.; Barnes, I. S.; Ninham, B. W. J. Phys. Chem. 1987, 91, 3814. (26) Hyde, S. T.; Ninham, B. W.; Zemb, T. N. J . Phys. Chem., in press. (27) Fontell, K. Prog. Chem. Fats Other Lipids 1976, 16, 145. (28) Arvidson, G.; Brental, I.; Khan, A,; Lindblom, G.; Fontell, K. Eur. J . Biochem. 1985, 152, 753. (29) Ekwall, P.; Mandell, L.; Fontell, K. J . Colloid Interface Sci. 1970, 33, 215. (30) Gallot, B.; Skoulios, A. E. Kolloid Z . Z . Polym. 1966, 208, 37. (3 1) Christenson, H., private communication. (32) Clunie, J. S.; Goodman, J. N.; Symond, P. C. Trans. Faraday SOC. 1969, 65, 287. (33) Husson, F.;Mustacchi, H.; Luzzati, V. Acta Crystallogr. 1960, 13, 668. (34) Schwarz, H. A. Gesammalte Mathematische Abhandlungen; Springer: Berlin, 1890. (35) Neovius, E. R. Bestimmung Zweier Speciellen Periodischen Minimalfachen; J. C. Frenckel & Son: Helsinki, 1883. (36) Nesper, R.; von Schnering, H.-G. Poster, Tagung AGKr, Koln, 1985. (37) Mackay, A. L.; Klinowski, J. Comp. Math. Appl. 1986, 12B, 803. (38) Anderson, D. M.; Davis, H. T.; Scriven, L. E., in press.
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is more sensitive to temperature due to the temperature sensitivity of oil uptake between the surfactant chains. This means that the effective surfactant parameter is a sensitive function of temperature.) 6. Conclusions This paper indicates the utility of equations and concepts borrowed from differential geometry for the determination of microstructures of bicontinuous cubic phases. The constraints imposed upon the interfacial structure by the composition and the preferred surfactant molecular geometry set the topology and symmetry of the interface. Unfortunately, accurate measurements of the surfactant chain length in cubic phases are not available. These measurements are needed in order to uniquely prescribe the structure. The microstructures of cubic phases of water mixtures with the surfactants KC14, AOT, GMO, SDS, and C12E06 have been analyzed and compared with earlier proposals (which all invoke genus three surfaces). Except for the SDS/water and AOT/water mixtures, previously suggested structures are consistent with these analyses. The SDS mixture forms a reversed bilayer of genus between three and six. Work is under way to derive new interfaces of symmetry Imjm and genus four to six. The microstructure of the AOT/water mixture appears to be of genus five to eight. Again, we are deriving new periodic minimal surfaces of the measured symmetry and higher topologies than genus three. These structures are thus still open. The formulas and analyses presented here suggest that it is unwise to assume that the microstructure is fixed throughout a single phase. It is certainly not fixed for single phases that occur for large ranges of composition. We have already demonstrated
the large variation in microstructure throughout the microemulsion phase of ternary mixtures of DDAB, water, and 0ils.25*26The calculations on ordered phases suggest that the microstructures of these phases are also sensitive to compositional changes. These changes cannot in general be accommodated by a simple change in lattice parameters (conserving the microstructural topology and symmetry), since such a variation would alter the effective shape of the surfactant molecules (characterized by the effective surfactant parameter). In particular, cubic phases that are formed within a large composition (or temperature) range must exhibit a diversity of microstructures. This imparts a new urgency to the determination of new periodic minimal surfaces and surfaces of constant average curvature. We have not explicitly addressed the fundamental issue of what determines the phase of a mixture. While subtle questions of statistical mechanics must be involved (and these are, in some sense, subsumed within the surfactant parameter), it is remarkable how much local and global geometrical demands alone constrain the structure. The approach used here should be combined with a comparison of theoretical SAXS spectra from suspected structures with actual spectra. Work is under way to determine accurate form factors for periodic bicontinuous structures. The rich variety of periodic bicontinuous structures emerging from mathematical studies must surely be reflected in nature.
Acknowledgment. I thank Hugo Christenson, Krister Fontell, KBre Larsson, and Barry Ninham for ideas and advice concerning this subject. Registry No. SDS, 151-21-3; AOT,577-1 1-7; CIZE06,3055-96-7; KCII, 13429-27-1; GMO, 25496-72-4.
Phase Boundaries for Ternary Microemulsions. Predictions of a Geometric Model S. T. Hyde,* B. W. Ninham, and T. Zembt Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 2601, Australia (Received: March 25, 1988; In Final Form: July 13, 1988)
It is shown that analytical geometric arguments can be used to account for the observed existence region of the L2 phase for a variety of ternary systems. For the systems studied, a surfactant parameter that varies slightly with water content is sufficient to determine phase boundaries and the microstructure throughout the L2 phase.
Introduction The assignment and meaning of microstructure for the class of chaotic fluids called microemulsions has presented a problem. In principle small-angle X-ray (SAXS) or neutron scattering (SANS) ought to have resolved the issue long since. But until recently even the existence of observed peaks in spectra, let along their position, remained unexplained. That is no longer an issue. There have been two lines leading to progress. In one the scattering spectra are derived from a phenomenological effective Hamiltonian description of fluctuations. This approach is apparently quite successful in general.’ It accounts for spectra from a wide variety of systems. But it has some difficulties. Among these the arbitrary truncation of a Landau expansion of the assumed Hamiltonian in an order parameter whose physical meaning is unspecified and the necessity to involve three or more unquantified coefficients are the least. More serious is the fact that such a description says nothing that allows any direct conceptualization of microstructure. ‘Permanent address: Department de Physico-chimie, Centre d’Etudes Nucleaires de Saclay, 91 191 Gif sur Yvette Cedex, France.
0022-3654/89/2093-1464$01 S O / O
A second very different approach based on geometry avoids that difficulty, at least for some systems. These are ternary microemulsions formed from water, various alkanes, and the double-chained cationic surfactant didodecyldimethylammonium bromide (DDAB).24 The systems have the special property that the surfactant is insoluble in both oil and water. Hence the surfactant must reside at the oil-water interface with curvature set by component ratios and the balance of interfacial forces. For these mixtures SAXS and SANS spectra as well as conductivity data have been shown to be quantitatively consistent with the predictions of an explicit random geometric model (DOC model).24 In our earlier work we have assumed that the microstructure throughout a single-phase region is characterized by a single (1) Teubner, M.; Strey, R. J. Chem. Phys. 1987, 87, 3195. (2) Zemb, T. N.; Hyde, S. T.; Derian, P.-J.; Barnes, I. S.;Ninham, B. W. J . Phys. Chem. 1987, 91, 3814.
(3) Ninham, B. W.; Barnes, I. S.; Hyde, S. T.; Derian, P.-J.; Zemb, T.N. Europhys. Lett. 1987, 4, 561. (4) Barnes, I. S.;Hyde, S.T.; Ninham, B. W.; Derian, P.-J.; Drifford, M.; Zemb, T. N. J. Phys. Chem. 1988, 92, 2286.
0 1989 American Chemical Society
