J . Phys. Chem. 1990, 94, 5328-5331
5328
A Structural Model for the Cubic Phases Formed by Surfactants Ciorgio Taddei Department of Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra 2601, Australia (Received: October 12, 1989: In Final Form: January 12, 1990)
A structural model (PSS) for liquid crystalline cubic phases of water-lipid systems is presented. This model describes the crystalline lattice in terms of a periodical succession of spheres interconnected through necks. The surfactant layer occupies the available surfaces, dividing the whole space into two distinct unconnected subspaces so that a bicontinuous structure is formed. The surfaces are smoothed, Le., without cusps, and they can show a variety of shapes with different symmetries and topologies. In various cases PSS surfaces approximate the periodical surfaces with a constant or zero mean curvature. An example of application of the PSS model to a real ternary system is reported.
Introduction Lipid-water systems are well-known for their polymorphism. These systems show a wide variety of phases with different microstructures. In many cases, phases have been isolated which present cubic symmetries, even if some features of X-ray diffractograms suggest that a long-range cubic order coexists with a short-range liquidlike disorder.’-) There is strong experimental evidence that cubic phases are bicontinuous;2 Le., they have crystalline lattices where the lipidwater interfaces divide the space into two unconnected subspaces. Some structural models have been proposed to justify the observed properties of the cubic phase^.^,^ However, the most successful model is that in which the actual interfaces of the system are assumed to coincide with infinitely periodic surfaces having a zero or constant mean curvature.s,6 Symmetry and energetic reasons seem to particularly favor surfaces with H = 0, or “minimal surfaces”, for those systems where a multilayer (with bilateral symmetry) is formed at the interfaces.’ Unfortunately, the methods of analytical geometry for discovering and describing these surfaces are particularly difficult and not yet fully generalized. [The problem involves the generalization of the Weierstrass function R ( w ) . See refs 5 and 8.1 In addition, many of the known minimal surfaces9 lack the necessary data for a quantitative description of the cubic phases, specifically knowledge of the normalized surface to volume ratio
s/ w 3 .
Under these circumstances, it is useful to use simpler mathematical models which permit a reliable quantitative description of some of the more common cubic phases. The model presented here satisfies these demands. Essentially, it differs with respect to similar models3J0 because it describes the interfaces in terms of (infinite) periodical, “smoothed” surfaces (PSS),with a variety of cubic cells (primitive, body-centered, face-centered, diamond-like, etc.). In some limit cases, the PSS model furnishes structures whose interfaces have a vanishing mean curvature (averaged over the unit cell). For this and more general reasons the PSS model may be considered-within certain limits-an approximation of the IPMS model (minimal surfaces).
Description of the PSS Model The main characteristic of the cubic phases is their bicontinuity and the possibility to cross from a bicontinuous structure to “closed” aggregates, with a small change in the composition of the system. One of the simplest geometrical models which shows this behavior is comprised of a sublattice of empty spheres which are interconnected via empty necks (belonging to a second sublattice). The topology of this structure can change with the composition of the system, since the shapes and the relative sizes of the necks and spheres are functions of the concentration of the components. For instance, two opposite limit cases with different topology are (i) a lattice of unconnected spheres (i.e., a nonbi‘Present address: Dipartimento di Chimica, Universiti di Firenze. via G. Capponi 9, 50121 Firenze, Italy
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continuous lattice of spherical micelles or vesicles) and (ii) a lattice of interconnected necks, where the spheres are totally hidden by the necks. In this latter case, the PSS model furnishes crystalline networks of rods, identical with those presented in refs 3 , 4 , and 10.
Still, the symmetry of the lattice can change with the composition, via phase transitions. However, in these cases it is necessary to introduce “ab initio” changes in the unit cell symmetry, the PSS model being unable to describe a phase transition. The spheres (all identical) and the necks (all identical) have a “thickness” which of course corresponds to the thickness of the layer of the surfactant molecules. If the layer is a monolayer, the thickness corresponds to the average length of the hydrophobic chain I ; if the layer is an n layer, the thickness corresponds to n times I plus the total thickness of the components inside the multilayer. The number of the necks for each sphere (Le., its coordination number) is assumed equal to the number of the nearest-neighbor spheres. (For a different assumption, see the Appendix.) The radius, R, of the external spheres is given by
R = [a/(]+ c ) ] F
(1)
where a is the unit cell parameter, c = D / R where 2 0 is the shortest distance between the external surfaces of two interconnected spheres, and F is a numerical coefficient which depends on the symmetry of the unit cell. The surfaces of the spheres and necks are tangential to each other along the line of their connection, so that the unit vectors normal to the surfaces do not present discontinuities (cusps) anywhere along the surface. The necks are obtained from the surfaces of revolution of a semicircle of radius r (the necks are portions of “inverse spheres”). These surfaces are the mathematically simplest “saddle” surfaces. This choice is very useful, since it permits the mean curvature, H (averaged over the unit cell), to change drastically through a variation in some structural parameter. In particular cases, it is possible to change the sign of H (see below). A different assumption can involve the catenary rather than the semicircle. This latter choice furnishes some ( 1 ) Luzzati, V.; Mariani, P.; Gulik-Krzywicki, T. Presented at the Workshop on “Physics and Amphiphilic Layers”, Les Houches, Feb 10-19,
1987. ( 2 ) Fontell, K.; Jansson, M. f r o g . Colloid Polym. Sci. 1988, 76, 169. (3) Mariani, P.; Luzzati, V.; Delacroix, H. J . Mol. Biol. 1988, 204, 165. (4) Luzzati, V.; Tardieu, A,; Gulik-Krzywicki, T.; Rivas, E.; Reiss-Huson, F. Nature 1968, 220, 485. ( 5 ) Anderson, S.; Hyde, S . T.; Larsson, K.; Lidin, S. Chem. Reo. 1988, 88, 22 1. (6) Anderson, D. M.; Davis, H . T.; Nitsche, J. C. C.; Scriven, L. E. Proc. R . SOC.London, in press. ( 7 ) Hyde, S . T. J . Phys. Chem. 1989, 93, 1458. (8) Hyde, S. T. Z . Kristallogr. 1989, 187, 165. (9) von Schnering, H . G.; Nesper, R. Angew. Chem. 1987, 26, 1059. (IO) Gulik, A.; Luzzati, V.; DeRosa, M.; Gambacorta, A. J . Mol. Biol.
1985. 182, 131
0 I990 American Chemical Society
Infinite Periodic Smoothed Surfaces advantages (Hconstant), but it is disadvantageous as the calculations of the surfaces and volumes involve more difficult integrals when overlapping between necks occurs (see below). Three nondimensional structural parameters are introduced: f = r / R , c = D/R, and g = d / R , with f b 0, c # 0, c Cf,and IcI < 1 if c < 0. R, r, and D were defined above, and d is the total thickness of the layer. The areas of surfaces and volumes are written in terms off, c, and g (see the Appendix). The point of introducing the parameters f, c, and g is that isomorphous structures are easily obtained by changing a, withf, c, and g constant. The necessary input data for the calculations are the symmetry of the unit cell, a,f,c, and g. From these data the composition of the system (in terms of volume fractions of the components) is calculated. At the same time the following are calculated: the total head-group area, A , and the volume of hydrophobic chains, Vchains, of the surfactant layer; the normalized surface-to-volume ratio, S / VI3,of the interface; the surfactant parameter, ulal; the mean curvature, H , and the Gaussian curvature, K (both averaged over the unit cell and referred to the interfaces); the Euler-Poincari characteristic, x, and the topological genus, g, of the surfaces; and the packing characteristic,
s3/ vx.
S / v2/3,A, Vchains, and ulal are calculated (i) directly from the geometry of the structure (see the Appendix) and (ii) by using the equations given in ref 7. [In the surfactant parameter ulal, u is the volume of the hydrophobic chain (per molecule), a the head-group area (per molecule), and I the effective chain length." where Vchainr and For N surfactant molecules ulal = Vchchains/Al, A are defined in the text.] The two methods i and ii give the same results only if the surfaces are free of cusps. The equations of ref 7 involve the formalism of the parallel surfaces for the calculation of H and K,which is valid only for surfaces free of cusps. Furthermore, all the quantities considered in ref 7 are expressed in terms of H and K. Cusps occur in the PSS model at high values off and c when the size of the necks becomes so large with respect to the size of the spheres that necks which belong to the same sphere overlap each other in couples (double-neck overlapping). The cusps are created along the curves of intersection of the overlapping necks. Even if the PSS model gives reliable results in the range off and c where no neck overlapping occurs, nevertheless the necessary correction terms to surfaces and volumes due to the (double) neck overlapping were fully introduced in the calculations reported here (see below). However, these calculations do not consider overlapping among three or more necks in the same portion of space. Multiple-neck overlapping involves necks belonging to the same sphere and/or different spheres. This kind of overlapping occurs for values off and c higher than those required by the (double) neck overlapping. The correction terms to surfaces and volumes, due to multiple-neck overlapping, involve the calculations of integrals of surfaces (and volumes) of very complex shapes. It is necessary to adopt the formalism of the parallel surfaces' for the calculations of H and K. Therefore, Occurrence of neck overlapping introduces unreliability in the calculated values of H , K, x, g, and packing characteristic. Fortunately, the value of g for surfaces in absence of cusps is known (from the calculations at lowerf and c) and the topology does not change for small neck overlapping. For these reasons, the problem of eliminating the cusps is not a truly compelling problem for the PSS model. (Nevertheless, the author of the present paper is currently performing calculations concerning the elimination of cusps from the PSS surfaces.) An Application of the PSS Model The PSS model is particularly suitable for ternary systems (such as water-surfactant-oil) where mono- or bilayers (normal or reversed) are formed at the interfaces. As an example of application, the DDAB-water-cyclohexane system was considered for the following reasons: (i) this system belongs to a series of
The Journal of Physical Chemistry, Vol. 94, No. 13, 1990 5329 0.18 4
t ii
1 I' +
0.12
I f
t t
I/ il
0 06
t -
0 0 2
06
I0
f
Figure 1. Relationship between the two structural parameters fand c which are consistent with the observed composition of system 2. Two different hydrophobic chain lengths were used in the calculations (lengths in angstroms).
DDAB-water-hydrocarbon systems whose phase diagrams are well-studied;2 (ii) the components of this system are practically insoluble in each other, so that the PSS model is particularly reliable; (iii) structuralI2 and ~ a l o r i m e t r i cdata ~ ~ are available. Low-angle X-ray diffraction (SAS), obtained in a wide range of compositions, shows that two different cubic structures occur. Both structures depend on the composition, which affects the areas of the interfaces and volumes. The authorsi2 propose that both structures have normal bilayers at the interfaces and that the interfaces coincide with particular minimal surfaces (see below). For this study two samples, among the many reported in refs 12 and 13, were chosen for the calculations. The first one (system 1) presents a primitive cell with a = 124.7 A and composition (w/o/s) 58.1/8.5/33.4 vol %. The second sample (system 2) presents a body-centered unit cell with a = 125-128 A and composition 47.9/10.5/41.6 vol %. Here the compositions of both systems were calculated assuming that the surfactant formed a normal bilayer at the interfaces. Reversed bilayers or monolayers were not allowed in any of the two systems according to the PSS model. Both the parameters f and c had been changed in order to fit the observed composition. Figure 1 shows the relationship between f and c, consistent with the observed composition (system 2) when two different average chain lengths, I, were used. It is important to note here that the hydrophobic chains do not behave at the interfaces as tightly packed bodies. X-ray diffractograms and packing calculations show that the chains behave as units without a precise and constant ~ h a p e . ~The . ~ average ~ chain length undergoes some changes where particular structural constraints are involved; Le., the thickness of the surfactant layer is not to be considered constant for the whole structure. Unfortunately, neither the PSS model nor the IPMS model is able to describe the local behavior of the layer thickness. However, it was possible to some extent to simulate these nonuniformities of thickness by using different values of 1 within a reasonable range. (Moreover, in the PSS model the error affecting the observed average value of I is added to the errors affecting the observed lattice parameter and the composition.) Figure 2 shows the dependence of the mean curvature, H , of the unit cell upon the parametery (for system 2, using different values of I ) . The star off* distinguishes the f values which are consistent with the observed composition (see Figure 1). High (12) Barois, P.; Hyde, S. T.; Ninham,
B. W.; Dowling, T. Langmuir, in
press. ( 1 1 ) Mitchell, D. J.; Ninham, B. W. J . Chem. SOC.,Faraday Trans. 2 1981, 77, 609.
(13) Radlinska, E. Z.; Hyde, S. T.; Ninham, B. W. Langmuir 1989, 5. 1427. (14) Gruen, D.W. R. J . Colloid Interface Sci. 1981,84, 281.
Taddei
5330 The Journal of Physical Chemistry, Vol. 94, No. 13, 1990 /
I
‘,
,
Figure 4. Double-neck overlapping in a body-centered unit cell for different values of the parameter f. Only the external surfaces of three spheres and two necks are drawn. Neck surfaces 1, 2, 3, and 4 correspond t o f = 0.3, 0.4, 0.7, and 1.3, respectively (at c constant). In the limit case f = m (not drawn), the PSS surfaces form a network of interconnected cylindrical rods (see the text).
\4\
0
L----
@
0
A-T4
+---
O b ff
\ I
Figure 2. Dependence of the mean curvature of the interface H (averaged over the unit cell) o n p , for system 2 using different chain lengths: 1 = 9, 8.5, 8.25, and 8 A for curves 1, 2, 3, and 4,respectively. (Forf see the text.)
\
! 1y
A
-5
:i
35
34
its S / PI3= 2.290 and packing characteristic = -2.786 are close to the corresponding values of the P surface. The authors12 propose that system 2 corresponds to the I-WP minimal surface described by S ~ h o e n which ,~ has the Im3m symmetry, g = 7, S / V l 3= 2.75 13, and packing characteristic = -1.4876. The values of ,!?/PI3 and packing characteristic of the PSS surface having the same symmetry and topology were 2.907 and -2.047 (when f* = 0.55). However, the correlation between the PSS Im3m and I-Wp surfaces for system 2 appeared clearly less convincing than the correlation made above for system I.
I
-. ^
^
‘I
Figure 3. Dependence of some bilayer quantities of system 1 upon?.
values off*, occurring when the necks are more elongated and their surfaces wider, correspond to nonzero values of H (if 1 is high). For smaller values of I it was possible to obtain structures whose surfaces had zero or negative H . (These values of I were used only to indicate the possibilities of the PSS model.) Vanishing values of H were attainable since the positive contribution to H of the spherical parts of the surfaces (for normal layers) were of the same magnitude as the negative contribution of the necks. Figure 3 shows the behavior of ulal, S/V2f3,and A versusfr for system 1. The dependence of v/al, S/ PI3,and A upon fr was strong whenf* 5 0.4. This behavior was due to the neck overlapping, which occurred for f* > 0.36. Neck overlapping reduced the areas of the surfaces asf* increased (whereas the total volume of the structure was less affected). It was interesting to make a correlation between PSS surfaces and some minimal surfaces for both structures. System 1, which shows a primitive cell with PmSm (or Pn3) symmetry, has been correlated to the P (Schwartz) minimal surface which has an Im3m symmetry, g = 3, $ / V I 3= 2.3451, and S 3 / v x = -3.2242.12 The P surface presents 2-fold symmetry axes which are due to the occurrence of “flat points” (where K = 0) in particular zones of the surface and which are responsible for the Im3m symmetry. The 2-fold symmetry axes disappear, when (infinitesimal) deformations of the P surface around these flat points are introduced. The PSS PmSm surface (here surface means the bilayer interface) could be considered a distorted P surface. It has the same topology ( g = 3) and symmetry, and
Conclusions The PSS model seems to give a reasonable qualitative and quantitative description of some of the more common cubic phase structures. The model is truly simple, even if the calculations of the correction terms (to surfaces and volumes) involve heavy surface and volume integrals when neck overlapping occurs. The ingredients of the PSS model are elementary surfaces: spheres and inverse spheres (as necks), whose interconnection produces an ordered and infinite lattice. In spite of this simplicity, it is possible to attain-changing some parameters-a variety of complex surfaces in which the predominance of the spheres over the necks or the predominance of the necks over the spheres creates different geometries and topologies. In certain cases, the PSS surfaces have the same symmetry and topology as the surfaces that have zero or constant mean curvature and whose description involves difficult high mathematics. Since minimal or If-constant surfaces seem to occur in nature in a growing number of biological, organic, and inorganic syst e m ~ ,the ~ ~PSS ~ ~model ’ ~ can furnish a reasonable approximation to the IPMS and to the If-constant models by means of a nononerous expense of undergraduate mathematics. Finally, the PSS model directly addresses the open question about the mechanism for which a concentrated solution of micelles or vesicles can transform into a bicontinuous ordered phase. This mechanism could involve more changes in topology than in symmetry of the whole system, the spherical micelles (or vesicles) being packed in a presumable ordered arrangement before they collapse in a bicontinuous ordered structure. The PSS model shows that only a few geometrical parameters are necessary to describe this collapse. The PSS structures can be interpreted as one of the final stages of the collapse in close proximity of the stage of IPMS structures. (This last stage has to be considered an ideal limit description of the reality.) Acknowledgment. This work was performed during a sabbatical year spent at the Department of Applied Mathematics of the Australian National University in Canberra. The author thanks Prof. B. W. Ninham, Dr. S. T. Hyde, and Dr. G.Lundberg for useful discussions and advice. h
15) Larsson, K . J . Phyr. Chem. 1989, 93, 7304
J . Phys. Chem. 1990, 94. 5331-5336
Appendix The fundamental quantities calculated by the PSS model are the surfaces and the volumes, from which all the other quantities are calculated. We report eqs 2 and 3 which calculate the surface areas S and volumes V per unit cell:
5331
correction terms. The correction terms to surfaces are S,,, = - 2 s A B + S'AB where 2sAB are the areas of those parts of the surfaces of necks A and B that are hidden by the overlapping and have to be neglected. These portions of surfaces form a sort of concave-convex "tent". S'AB are the areas of the portions of spherical surfaces that underlie the "tents". The correction terms for volumes are v,,, = -VAB 2v',B where VABare the portions of the volumes that are in common to neck A and neck B (because of the overlapping) and V',, the volumes of the spherical segments that are in common to the same necks. The correction terms are as many as the parallel surfaces and underlying volumes considered. Each correction term has to be multiplied by OSZnm, where m is the number of next-neighbor necks to each neck upon the sphere. The calculations of the correction terms involve surface and volume integrals which are written in cylindrical coordinates and computed numerically. When two necks widely overlap, the overlapping involves about the whole height of both necks and not only their bases. This happens for highfand c values which may be necessary in describing some structures (see Figure 4 ) . The corresponding correction terms which involve a great number of surface and volume integrals and which are very complicated because of the biconical C,, symmetry of the (isolated) necks were calculated. The (double) overlapping occurs when the angle at the center of the sphere 0, which subtends the arc of the maximum overlap on the sphere, changes sign. It is easy to show that 0 = 2ao 6, where a,,is the angle (at the center of the sphere) between the principal axis of symmetry of a neck and the point of tangence of this neck to the sphere and p the angle at the center of the sphere between the two principal symmetry axes of the two nearest-neighbor necks. p depends on the symmetry of the unit cell. The nearest-neighbor assumption which underlies eq 1 can be abandoned for a different assumption which considers a different number of necks allowed around each sphere (Le., n has to be redefined for each symmetry). In such a case only eq 1 has to be modified. The new assumption can be intentionally designed to obtain particular topologies and symmetries otherwise out of reach by the PSS model as described here.
+
r Z n C f + g ) ( t [ 3 ~ +g ) 2 + (f+ g)(2h
+ H ) + hH + hZ +
with
h = (1
+ f, sin cos-] (::;i(f+d -
and
H = ( 1 - g ) sin c0s-l
(:I;:
-
R is given by eq l,fand c are defined in the text, Z is the number of spheres in the unit cell, n is the coordination number of the spheres, and g = d / R with d = 0, d2, d3, ... where dz, d S ,... are the thicknesses by means of which different parallel surfaces, necessary for the description of a multilayer (monolayers included), are created. Five different parallel surfaces are necessary for a bilayer (normal or reversed): one for the interface and four for v / a l , etc. In eqs 2 and 3 the areas and volumes defining Vchains, of the half-necks are considered, since each neck belongs to a couple of spheres. When (double) neck overlapping occurs, the areas and volumes given by eqs 2 and 3 have to be supplemented with the necessary
Micellar Head-Group Size and Anion Nucleophilicity in SN2 Reactions Cristiana Bonan, Raimondo Germani,'" Pier Paolo Ponti, Gianfranco Savelli,*~'P Dipartimento di Chimica, Ingegneria Chimica e Materiali, Universita di I'Aquila. 641 00 I'Aquila, Italy
Giorgio Cerichelli, Centro CNR Sui Meccanismi di Reazione, Dipartimento di Chimica, Universita la Sapienza, P. Aldo Moro, 00185 Rome, Italy
Radu Bacaloglu,'b and Clifford A. Bunton* Department of Chemistry, University of California, Santa Barbara, California 931 06 (Received: December I , 1989)
Rates of reaction of OH- with methyl naphthalene-2-sulfonate (MeONs) in aqueous cationic surfactants (C16H33NR30H, R = Me, Et, n-Pr, n-Bu, and cetylquinuclidinium hydroxide) increase with increasing surfactant and hydroxide ion concentrations. The variations of rate constants can be fitted to an equation that describes the distribution of both reactants between water and micelles in terms of Langmuir isotherms. The Langmuir coefficients for OH- decrease in the head-group sequence Me3N > quinuclidinium > Et3N > Pr,N > Bu3N. The second-order rate constants at the micellar surfaces are not very sensitive to the bulk of the head group, although they increase at high added OH-. Rate constants for attack of water on micellar-bound substrate increase with increasing bulk of the head group.
Rate effects of aqueous colloidal self-assemblies, e.g., micelles, microemulsion droplets, and synthetic vesicles, are generally ex-
plained in terms of pseudophase models.24 Reaction occurs in water or at colloidal surfaces that are treated as distinct reaction
0022-3654/90/2094-5331$02.50/0 0 1990 American Chemical Society
