Langmuir 1990,6,1055-1062
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Curvature Energy of Surfactant Interfaces Confined to the Plaquettes of a Cubic Lattice S. T.Hyde,’ I. S. Barnes, and B. W. Ninham Department of Applied Mathematics, Research School of Physical Sciences, Australian National Uniuersity, Box 4, Canberra, Australia Received May 22, 1989. In Final Form: January 25.1990 We consider the curvatures of a faceted interface, for which the curvatures are concentrated along the edges and vertices of the underlying lattice. Such a lattice model has been frequently used to generate random monolayer or hilayer surfactant films. We demonstrate that the mean curvature of such an interface is concentrated along the edges, while the Gaussian curvature is confined to the vertices. We analyze in detail all possible interfacial configurations for an interface confined to the plaquettes of a cubic lattice and obtain an exact expression for the curvatures. The contribution of these curvatures to the total free energy of a surfactant solution is calculated, using two different expressions for the bending energy, and the utility of plaquette models is discussed. Introduction In this paper, we calculate the effective curvatures of a faceted interface, built up of cube faces. The solution is complicated and is very different than earlier estimates of the curvature of a stepped surface. The model arises most often in theoretical treatments of the stability of microemulsions, where the surfactant interface, for the sake of analytic tractability, is assumed to lie on the faces of polyhedral partitions of I t is also of direct relevance to the theory of foams. It is well-known that a significant contribution to the free energy of the surfactant solution comes from the bending energy of the surfactant f i h 9 This contribution to the total free energy of the solution is dependent upon the difference between the actual and “preferred”curvatures of the interface. Therefore, accurate estimates of the curvatures of various interfacial configurations are essential in order to determine the free energy of the solution. Various approximations have been made to estimate the curvatures. de Gennes et al. have suggested that the radii of curvatures in the cubic lattice are comparable with the cell size, t,2.6while Safran et al. have assumed that the curvatures are proportional to the probability of having a cube edge on the interface, where radii of curvature equal to t are assigned to each edge.5.7,8 We attempt a full analysis of the contribution of cube edges and vertices to the interfacial curvatures within a cubic model, assuming random filling of the cells (the random mixing approximation). In some cases, the results differ significantly from those generally used. Moreover, the results depend upon the nature of interactions between neighboring surfactant interfaces, as anticipated by Safran et al.*.7~s ( 1 ) Talmon. Y.;Prager, S.J. Chem. Phys. 1978.69.2984. (2) de Gennes,P.-G.; Jouffroy, J.; Levinson. P. J. Phya. (Le8 Ulia. Fr.) 1982.43, 1241. (3) Widam, B.J. Chem. P h m 1984.81,1030. (4) Auvray. L.J. Phya., Left. 1986.46, L163. ( 5 ) Safran, S. A.; Roux. D.;Cates, M. E.; Andelman. D. Phys. Rev. Lett. 1981,57.491. (6) de Gennes. P.-G.; Taupin, C. J. Phys. Chem. I982.86,2294. (1)Andelman. D.:Cam. M. E.:Roux. D.:Ssfran. S. A. J . Chem.
0743-7463/90/2406-1055$02.50/0
a
1
b
Figure 1. (a) Two-dimensional new of the interfacial geomstry in the lattice model. Cells are filled independently with medium A or B,and the interface separates the two media. If A (heavily shaded)is hydrophilicand B (lightlyshaded)is h y d m
phobic, the interface consist8 of a monolayer. (h) Formation of a bilayer interface in this model. Both A and B cells are hydrophilic (hydrophobic),so the surfactant forms a bilayer (reversed bilayer) separating the two media.
Cell Model for S u r f a c t a n t Interfaces We confine our analysis to the case of a cubic lattice, so that space is partitioned into cubes, a situation first
analyzed by de Gennes et a1.2 The cubes are filled independently with one of two possible media, denoted A and B. The surfactant film decorates all faces which separate the two media. In this way, random monolayers (normal or reversed) or bilayers can be generated. For example, if A and B are of hydrophilic and hydrophobic character, respectively, the surfactant film forms a monolayer, with the hydrophobic moiety of the surfactant directed toward the B regions (Figure la). If both A and B are 0 1990 American Chemical Society
1056 ~ongmuir,Vol. 6, No. 6, I990
a
b
C
Figure 2. (a) Cell configurationresulting in an ambiguousinterfacial geometry in the twedimensional lattice model. The directional singularity at the central vertex can be resolved in two ways. (h) One resolution of the ambiguity: the resulting configuration is continuous in the hydrophilic medium (heavily shaded). (c) Other resolution, resulting in a hydrophobic continuous configuration.
hydrophilic, the surfactant film consists of a bilayer (Figure lb). This model gives rise to a surfactant film whose curvatures vary from one lattice vertex to another. The (local) curvatures of the film are set by the filling of adjacent cells, and we can exhaust all possible interfacial geometries by considering the neighborhood of a single vertex of the cubic lattice. A vertex of the underlying lattice is shared by eight neighboringcubes. If @A is the volume fraction of medium A, the probability that a single cube contains A is @A and that it contains B @B (=1 - @A). It follows that the probability that a vertex configuration consists of n cubes of A (and 8 - n cubes of B) is 'C,@A"(1 - @A)'+". (Here 'C, refers to the number of combinations of n objects from a total of eight.) The curvatures of some of the vertex configurations cannot be unambiguously determined, due to the artificial nature of lattice models. This problem arises also with two-dimensional lattice models.' For example, consider the vertex configuration consisting of like diagonal cells in a square lattice (Figure 2a). In the absence of any preferred continuity of medium A or B, the interface can be considered equally well to be curved toward either medium (Figure 2b.c). While the values of these
Hyde et al.
principal curvatures are the same, they are of opposite sign. For a two-dimensional lattice, this ambiguity occurs in those configurations which give rise to a self-intersection in the film, viz., like-filled cells sharing only a vertex. In three dimensions, among the various configurations, this problem is more frequent. Here, problems of interpretation occur for all vertex configurations which consist of like-filled cells sharing common edges and/or vertices (without sharing faces). Where more than one interface can be traced from the vertex configuration, the probability of either individual interfacial geometry being present depends upon the physics of the situation; for example, A rather than B continuous regions may be overwhelminglypreferred. We shall assume that both geometries are equally likely if the interface consists of a bilayer. Note, however, that a monolayer surfactant film would normally exhibit a preference for one geometry over the other, due to the differing relative magnitudes of tail-tail and head-head interactions between opposing surfactant monolayers. For example. if the surfactant heads repel each other, while the intertail interaction is attractive, the geometry shown in Figure 2c will be favored over that of Figure 2b, due to the difference in the average separation of head groups of opposing monolayers in either Configuration. We quantify this ambiguity as follows. We set the probability of a common edge to be A-continuous to be a and the probability of a B-continuous architecture to be 0 (=1 - a). In the case of many vertex configurations, this ambiguity in the structure of the interface is exacerbated by the presence of a number of common edges in the interface. We shall assume that the likelihood of a certain interfacial architecture is equal to the product of the probabilities of each common edge adopting the required continuity. (Thus, the probability that a configuration containing three common edges is A-continuous a t all these edges is equal to a3.) Interfacial C u w a t u r e s We consider the interface associated with a vertex to be that contained within a cube (centered at the vertex) whose volume is the same as those making up the lattice. (This cube is the Voronoi cell of the lattice.) The interfacial area per vertex is then equal to nt2/4, where n is the number of cube faces separating regions A and B which share the vertex and is the cell edge length. Gaussian Curvature. The Gaussian curvature of the interface is defined via the integral (Gaussian) curvature
where the numerator is equal to the surface integral of the Gaussian curvature (the integral curvature). This definition distributes the curvature associated with the facets of the cubic interface over the Voronoi region of this vertex, massaging the film as far as possible into a smoothly curved surface. (Note that the area averaging cannot be done over a larger region without altering the filling pattern of the cells.) I t is well-known from differential geometry that the integral curvature of a surface is equal to the area of the Gauss map of the surface. This mapping transforms each point on the surface to a point on a unit sphere, given by the intersection of the normal vector to the surface with the sphere centered a t the point on the surface. So, the Gaussian curvature can be calculated by using eq 1
Langmuir, Vol. 6, No. 6, 1990 1057
Curvature Energy of Surfactant Interfaces
a
b
e
f
i
C
d
9
h
k
I
Figure 3. Catalog of all filling patterns of eight cubes sharing a central vertex giving rise to distinct interfacial geometries. The curvatures of each interface are given in Table I. edges. Thus, a boundary circuit on the interface traces out a geodesic triangle on the unit sphere whose vertices are all right angles, giving a segment of area ?r/2 (Figure 4). Since the interface is made up of three cube faces, the Gaussian curvature of this vertex configuration is given by
t
Figure 4. Schematic illustration uf the technique Qmplo).Qd 10 calculate the local Gaussian curvature uf a faceted intert'are. In this rase. the interface consiFt\ c f three mutiinlly perpendiruIar cuhe plaquettes (as in Figure 3a,. The nnrmal vertnrs t n the interface (top, are parallel lransporttd onto a unit sphere such that they are normal t u the sphere (the Gauss map). and the area spanned by these normals 1s equal 10 the integral cur. vature uf the interlace !hortom,. The local Gaussion curvature is determined by dividing the integral curvature by the area of the interface. from the area traced out on the unit sphere by the normal vectors to the interface, divided by the area of the interface. If the path traced out on the sphere is oppositely oriented to the corresponding path on the surface itself, the Gaussian curvature is negative; if both paths are similary oriented, the Gaussian curvature is positive. Consider, for example, the vertex configuration made up of seven like cubes and one unlike cube (Figure 3a). Since the cube faces represent a mathematical idealization of the physical interfacial geometry, the normal vectors to the interface must vary smoothly over the cube
This technique is readily applied to all the vertex configurations. We need not calculate explicitly the curvature of all the configurations, since, from the symmetry of the model, the Gaussian curvature for a configuration consisting of x cubes filled with medium A and 8 - x cubes of medium B is equal to that for x cubes of B and 8 - x cubes of A. The various filling patterns for eight cubes sharing a common vertex are illustrated in Figure 3a-I, and the resulting curvatures are tabulated in Table I. Mean Curvature. The calculation of the mean curvature involves the notion of parallel surfaces. Parallel surfaces are formed by sweeping out the interface along its normal vectors by a constant distance everywhere on the interface, so all parallel surfaces share the same normal vectors. For a small interface, the variation of surface area of parallel surfaces is related to the mean and Gaussian curvatures of the interface' (see Appendix I). Defining the mean curvature
we get (3) where Z(*d) denote the areas of parallel surfaces at a distance d on either side of the interface and Z(0) the area of the interface." This technique is valid only for smoothly curved interfaces, so the stepped interface occurring in the cubic lattice model must be smoothed. This can be done exactly (11) Hyde. S.T.;Ninham, B.W.;Zemb, T.J. Phys. Chem. 1989.93, 1464.
1058 Langmuir, Vol. 6, No. 6, 1990
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Table I. Curvatures of the Interfaces Created by All Possible Vertex-Sharing Cube Fillings Assuming Random Filling with Two Media, A and B, Resulting in n(A) A Cubes and n(B)B Cubes. weight of number of interfacial Gaussian mean 1 A:B interface* (w(i)) Figure combinations (zi) area (a;) curvature ( ( K ) d curvature ((HM 1:7 1 3a 8 2ir/3t2 2 2:6 12 3b 0 2:6 12 3c 3 2r/3t2 2:6 12 4 3c -21r/3t2 2:6 4 5 3d 2ir/3t2 4 2:6 6 3d -2r/3t2 3:5 7 24 3e -27155' 3:5 24 3f 8 2*/7t2 3:5 24 9 3f -6s/7t2 3:5 10 8 2r/3F2 3g 3:5 8 11 -2n/9t2 3g 3:5 12 8 -107/9p 3g 3:5 8 -2a/9f2 13 3g 3h 4:4 6 14 0 3i 4:4 32 15 -2n/3E2 4:4 24 0" 16 3j 24 4:4 17 3j -*It2 4:4 24 0' 18 3j 4:4 2 3k 19 21r/3t2 e 3k 4:4 2 Od 20 3k 4:4 2 -2r/3t2 21 2 -2ir/3t2 3k 4:4 22 3k 4:4 2 OC 23 2 3k 4:4 -4r/3f2 24 3k 4:4 2 -2~134~ 25 2 3k 4:4 -2~135~ 26 3k 2 0 4:4 27 3k 2 2n/3t2 e 4:4 28 0" 31 4:4 6 29 31 6 4:4 30 -7152 31 0 4:4 6 31 The interface separates A from B cells. The weighting factor, w, defines the weight of the film geometry, different from one where it is possible to trace a number of geometries for a single cube filling configuration. The number of ways of generating the cube filling is given by 2 , and the film area and surface-averaged Gaussian and mean curvatures are listed as a function of the edge length of the cube, 4. * This column specifies the probability of a certain interfacial geometry, within a configuration whose interface may adopt a number of geometries, depending on the likelihood of joined or isolated adjacent edges (a and 1- a = fl per common edge, respectively). The specific values of this weighting factor depend on the relative energies of the various geometries. c The interfaces for these configurations consist of two unconnected surfaces. The curvatures quoted here are the area-weighted averages of the curvatures of each surface. The interface for this configuration consists of three unconnected surfaces. e This interface consists of four isolated surfaces.
by considering the edges and vertices as smoothly curved regions of vanishingly small radii of curvature. The detailed calculations for each interfacial configuration are tedious and apart from an illustrative example (Appendix I) are not presented in detail. However, one unexpected result of the calculations is worth reporting. Our calculations indicate that vertices do not contribute at all to the mean curvature of the interface. The only contribution to the interfacial mean curvature is from the edges. (However, only the vertices contribute to the interfacial Gaussian curvature.) This has been shown by Santalo to be rigorously correct.2 The results for the mean curvature associated with each vertex configuration are given in Table I. We have adopted the convention that the curvature toward medium A is positive. It should be noted that the mean curvature of a vertex configuration containing x cells of A and 8 - x cells of B is equal in magnitude, but opposite in sign,to the mean curvature of the complementary configuration, viz., 8 - x cells of A and x cells of B. We define the global curvatures (H, K> to be the areaand ( K ) ) . weighted average of the local curvatures ((H) The values of the global Gaussian and mean curvatures depend on the relative densities of cubes filled with media (12) Santalo, L. A. In Studies in Global Geometry and Analysis; Chem, S. S., Ed.; Studies in Mathematics Vol. 4; Mathematical Assoc. of America, 1967; p 170. (13) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981,77,601. (14) Hyde, S. T. J. Phys. Chem. 1989,93,1458.
A or B (parametrized by the filling fraction of A,
$A),
the length scale (given by the cube edge length (t)),and the relative weightings (wi) of A-continuous and B-continuous regions, which determine the geometry of the interface for those configurations which do not trace out a unique interface. It is convenient to define the probability of a certain configuration arising in terms of the number of ways of achieving this configuration (zi), the weighting factor due to ambiguities in the physically realizable architecture of the interface (wi),the filling volume fraction ($), and the numbers of A and B cells, respectively
as well as the "complementary" probability due to the inverse configuration (where A and B media are swapped) where wi* is determined by swappinga and @ in the expressions for wi, listed in Table I. The Gaussian curvature is given by the eighth-order polynomial
In this expression 6i, is the Kronecker 6 function. In order to avoid overcounting of those configurations which con-
Langmuir, Vol. 6, No. 6, 1990 1059
Curvature Energy of Surfactant Interfaces
Where neither A- nor B-continuous geometries are favored (a = 8 = 1/2), the expressions for the film curvatures are relatively simple. The global mean curvature is a linear function of the filling fraction
0.2
0.3 0.4 0.5 0.6 0.7 0.8 Volume fraction of AIB component
and the global Gaussian curvature is given by the polynomial 1
K 4 - 5 =-
( '2'
?r
15365'
+ 1084 + 1509) - 4851
[(l - 24)2(59244- 1184~$~7004'-
Discussion
I
C3
0.2
0.3 0.4 0.5 0.6 0.7 0.8 Volume fraction of A B component
Figure 5. Variation of the global mean curvature and Gaussian curvature with filling fraction for a film with no preferred continuity ( a and @ in Table I are equal to 1/2). The data are
plotted for cells of unit edge length.
0.2
0.3 0.4 0.5 0.6 0.7 0.8 Volume fraction of A/B component
These results are revealing. In the absence of preferred interactions, the mean curvature is a linear function of the filling fraction, and the Gaussian curvature is a symmetric function about 50% filling fraction. It is clear from the graphs that the global curvature of the interface never vanishes. Although the mean curvature always reaches zero at some filling fraction (50% in the absence of interactions favoring A- or B-continuous regions), the Gaussian curvature is far from negligible at this fraction. These calculations allow us to evaluate exactly the bending energy of a surfactant film within this model. Since the local curvatures vary over the interface, we sum the contributions of each vertex configuration to the bending energy, rather than determining this energy from the global values of the curvatures. (It is assumed here that the cell edge length is substantially larger than the "width" of the surfactant head groups, so that the molecules are indeed subject to these local curvatures.) Various forms have been proposed for the bending energy of the surfactant film, following the original proposal of H e l f r i ~ h . ~ The following form for the bending energy is due to Helfrich: 31
Fbnd
= KlCai[Pi((Hi)-Ho)'
+ (1- S,,,,(A))f'?(-(Hi) -
i=l 21
i=l
9
0.2
0.3 0.4 0.5 0.6 0.7 0.8 Volume fraction of A/B component
Figure 6. Variation of the global mean and Gaussian curva-
tures with filling fraction, assuming that all ambiguous film configurations lead exclusively to A-continuous regions (referring to Table I, @ is equal to 1, a to 0). tain equal numbers of A and B cells, we have introduced the 1- b4ni(A) factor. The expression immediately following this factor accounts for the fact that we must swap A-continuous regions for B-continuous (and hence change the weighting factors, wi, to (wi*)) when we count the contributions due to five or more A cells (see Table I). The mean curvature is obtained from the sum
The resulting curvatures are graphed in Figures 5 and 6 for a cubic lattice of unit-cell edge for the various weighting factors favoring A- and B-continuous film geometries.
where K~ is the splay bending rigidity, ~2 the elastic modulus of Gaussian curvature, and HOand KOare the preferred mean and Gaussian curvatures, respectively. (Note that because this form contains the square of deviations from the preferred mean curvature, we must sum over all local curvatures, rather than simply taking the global mean curvature.) We assume that there is no preferred geometry for a bilayer interface, so w = 1/2. (This is reasonable, since the configurations shown in parts b and c of Figure 2 would be energetically equivalent if the opposing monolayers were replaced by opposing bilayers.) We are now in a position to calculate the bending energy for a bilayer whose natural curvature vanishes (Ho= KO= 0) throughout the two-dimensional domain spanned by the cell size (5) and the filling fraction (4). We choose two asymptotic forms for the bending constants obtained by Mitchell and Ninhamls and Winterhalter and Helfrich.'G For membranes of low surface charge density, or charged membranes in strong saline (15) Mitchell, D. J.; Ninham, B. W. Langmuir 1989,5, 1121. (16) Winterhalter, M.; Helfrich, W. J. Phys. Chem. 1988, 92, 6865.
1060 Langmuir, Vol. 6, No. 6, 1990
H y d e et al.
1
3.0 0.2
Volume fraction of AIB component
Figure 7. Bending energy per unit area (in units of kT) for a bilayer of low surface charge density, assuming the Helfrich form of the bending energy. The values of the bending moduli are discussed in the main text. c
$!
. , . , . , . , . ,
10,
1
0.3 0.4 0.5 0.6 0.7 0.8 Volume fraction of A/B component
Figure 9. Bending energy per unit area for a bilayer (arbitrary units), assuming the energy is harmonic about the preferred surfactant parameter (see main text, Appendix B).
of the surfactant molecule (a). The surfactant parameter can be expressed in terms of the Gaussian and mean (global) curvatures of the interface:"J4
,
-u= 1 i t 1 a1
5
0.2
0.3 0.4 0.5 0.6 0.7 0.8 Volume fraction of AIB component
Figure 8. Bending energy per unit area for a highly charged bilayer (units of kT), assuming the Helfrich form of the bending energy. Further details are in the main text.
solution, we have
(We have adopted the convention that the positive sign on the mean curvature is taken for monolayers which are curved toward water, while the negative sign is taken for those interfaces curved toward oil.) For bilayer interfaces, the curvatures of each surfactant monolayer are equal to those of parallel surfaces located on either side of the interface.14 In terms of the curvatures at the center of the bilayer, the surfactant parameter for a monolayer making up the bilayer is given bY u
(7) where t is the thickness of the membrane, t is the dielectric constant of water, u is the surface charge density, and K is the inverse Debye length.15J6 For highly charged membranes, the bending constants are given by15 K1
=-(-) t kT2 KK
KK
- 1f HI + K12/3
(10) a l - 1 f 2H1+ K12 where the choice of positive or negative sign depends upon which monolayer we choose. It is clear from eq 10 that these values differ for each monolayer making up the bilayer unless the mean curvature at the centre of the bilayer (H) vanishes. We can estimate the energy associated with this change in molecular dimensions most simply by the following assumed form:
e
-( --) '[ f + Kt In (-391
= - 2t k T K2
+-K12 3
(8)
where e is the charge on the proton, k is Boltzmann's constant, and T i s the absolute temperature. We choose typical values for the weakly charged membrane thickness, charge density, and the inverse Debye length: 25 A, 500 A2/esu, and 20 A-l, respectively. For the strongly charged case, we choose an inverse Debye length of 20 A-l, room temperature, and a surface charge density of 1 esu/50 A2. The variation of the bending energy according to Helfrich's formulation (eq 4) is plotted in Figures 7 and 8. It is of interest here to compare Helfrich's form for the bending energy with one derived from the recognition of surfactant films as consisting of molecules, so that film bending must be accommodated by a concomitant change in the surfactant tail length and/or head-group area. The dimensions of the individual molecules are most simply defined by the surfactant parameter, which appears to govern phase behavior in some microemulsion and liquid crystalline phases.l' The surfactant parameter relates the volume of the hydrophobic moiety of the surfactant molecule ( u ) to the length (normal to the interface) of this moiety (1) and the area of the hydrophilic head group
where ( u / a l ) o is the preferred surfactant parameter and u l a l is the actual surfactant parameter adopted by the surfactant molecules (here K is a phenomenological constant). The calculation of the bending energy is involved; we have relegated all details to Appendix 11. The energy can be written in the form
Fben,j= 2 K 1 2 ( W 2+ [ 1 4 ( C , ( ~ ) 2 + C
+ C,(W4) + l6 ...I
2 ( W 2 u
(12)
(where the c values are undetermined constants) provided that the radii of curvature are much larger than the tail length and the spontaneous curvatures are small. This equation differs from that of Helfrich in that it does not contain a linear term in the change in Gaussian curvature. The bending energy for a bilayer decorating the cubic lattice, calculated from eq 12, is plotted in Figure 9. Comparison of Figures 7-9 shows major differences between the qualitative behavior of the bending energies for bilayers as functions of the cell size and filling factor for these two forms of the bending energy. In particular, for Helfrich's form, the free energy is maximized at that volume fraction which leads to the preferred mean
Langmuir, Vol. 6, No. 6, 1990 1061
Curvature Energy of Surfactant Interfaces curvature of zero, due to the large negative value of the Gaussian curvature. Assuming that the values of the curvature moduli we have chosen are reasonable, we can infer from this-assuming the Helfrich form for the bending energy-that the stability of bilayers within the plaquette model is critically dependent upon the lateral compressibility of the bilayer, which sets the surface to volume ratio of the film and hence the relation between the cell size of the cubes and the filling fraction. To date, few estimates of the surface tension of bilayers are available," and a proper analysis of this problem must await further work. Nevertheless, it seems that the random-filling model generates films whose undulations are too large to be stable under Helfrich's form of the bending energy.
Conclusions These exact calculations of the curvatures of faceted interfaces allow us to determine the contribution of the film curvature to the total free energy of surfactant solutions with some confidence-assuming that the functional form of the bending energy is correct. These calculations are mathematically rigorous; nevertheless, the fine details of the relation between this model and the real world of surfactant films remain uncertain. A nagging problem in all lattice models is the length scale below which the interfacial geometry is physically irrelevant. Clearly, as the resolution decreases (i.e., the length scale increases), so too do the interfacial area (crucial to predictions of the phase behavior) and the curvature undulations within the film (and the consequent bending energy). This issue has been thoroughly analyzed by de Gennes and Taupin,' who introduced the notion of a characteristic length scale within these systems (the persistence length). This length must represent a lower bound on the cell edge of any lattice model. In order to retain the geometry of the cubic lattice-the filling pattern-we have calculated the interfacial curvatures at the resolution of the cell edge (by integrating over a Voronoi region to obtain the local curvatures). Ideally, the geometry of the underlying lattice should not be reflected in the physical characteristics of the model. Unfortunately, this can only be achieved by reducing the spatial resolution of the interface, thereby deriving unrealistic estimates of the interfacial area and curvature variations. Thus a compromisemust be struck, and the resultant loss of physical insight may be critical. We have suggested that the bending energy may have a less general form than the phenomenological form proposed by Helfrich, due to the fact that surfactant interfaces are built from discrete molecules, which have preferred dimensions, viz., head-group area and tail length. However, it appears that stability of a bilayer of vanishing spontaneous curvatures (e.g., a lamellar mesophase) is very sensitive to the form of the bending energy. Indeed, it seems that if we assume the Helfrich form of the bending energy it is impossible to stabilize bilayer films which decorate a randomly filled cubic lattice, due to the significant contribution of the Gaussian curvature bending energy. This problem suggests that the assumption of random filling may be too crude an approximation. In order to induce stable bilayers under Helfrich's form of the bending energy, some correlations between cell fillings must be introduced. (17)Evans, E.; Needham, D. In Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Eds.; Springer Verlag: Berlin 1987; p 38. (18)Willmore, T.J. An Introduction to Differential Geometry; Oxford University Press: Delhi, 1985; p 40.
L
I -c' E
Figure 10. Example of the smoothing procedure used to calculate the local mean curvature of an interface (here the interface is that due to the vertex configuration shown in Figure 3a). All three edges are smoothed by cylinders, and the vertex by a torus. The major radius of the torus is denoted by 6, which is also of necessity the radius of the cylinder not drawn in the figure. The minor radius of the torus is e, which is also the radii of the other two cylinders.
We hope that these results will permit a more accurate analysis of these problems.
Acknowledgment. We are very grateful to Hannah Cormick for lending us her play blocks, without which we could not have performed this analysis. Appendix I. Mean Curvature of a Faceted Interface These calculations rely on the magnitudes of the surface area of parallel surfaces on either side of the interface (Z(fd)) and the surface area of the interface (Z(0)). In order to apply the formulas, the interfaces must firstly be smoothed. Consider a single cube of medium A, surrounded by seven containing medium B (Figure 3a). The interface consists of one-half of a cube. The vertex is smoothed by a sixteenth of a torus of major radius of curvature 6 and minor radius E (Figure 10). We smooth two edges by cylinders of radius t, the third by a cylinder of radius 6. The length of the cylinders is equal to one-half the cell edge less a small length taken up by the torus (i.e., 512 - f ( a , e ) , where f is some function of the toroidal radii). We obtain the exact values of the local mean curvature by calculating the surface areas in the limit as t and 6 tend to zero. Using the formula of Willmore for the area of a toroidal section,l8 we obtain for the area of the interface Z(0) =
($+$)+ [ ( z f + $ ) ( $ f ( c , 6 ) ) ]
+
(area of flat faces) (AI.1) The parallel surfaces at a distance of Ad on either side
Hyde et al.
1062 Langmuir, Vola6, No. 6, 1990
of the interface are formed by inflating/deflating the torus and cylinders-corresponding to fixing 6 and increasing/ decreasing the radius e by d. So, we have
(AI.2)
tant parameter to be close to one; i.e., the natural aggregation state of the surfactant molecules is close to a planar geometry. Consider the two-step process of bending the bilayer, consisting of a change in the tail length followed by a change in the head-group area. If the (small) change in the tail length alters the spontaneous Gaussian curvature of the bilayer, KO, to curvatures to H1 and K1, from eq 7 , then
From eq 3 in the text
Z ( + d ) - Z(-d) = -T, (AI.3) (H)= 4dZ(O) 25 (This corresponds to entry 1 in Table I.) The mean curvatures of all other vertex configurations have been obtained in a similar manner. In fact, the mean curvature of these faceted interfaces can also be calculated from the dihedral angles of the interface, ai (the angles between faces sharing a common edge) and the associated edge lengths, l i : l 2
E(* -
aivi
( H )=
i
(AI.4) 2(interfacial area) This method is quite general, although it is strictly only valid for convex polyhedra. It can be generalized to any faceted interface, as long as all dihedral angles are measured with respect to the same side of the surface.
Appendix 11. Bending Energy for a Bilayer in Terms of the Surfactant Parameter Suppose that a surfactant molecule has a preferred headgroup area, a, and a preferred tail length, 1. If the volume of the hydrophobic moiety of the molecule is u , we define a spontaneous surfactant parameter by (ulal). If the curvatures of the bilayer change, the tail length and/or head-group area must also adjust, in order to close pack the molecules into the new shape. This new shape is characterized by a new surfactant parameter u/a'l' defined by u Aa A1 AaAl ( a + &)(l + Al) = (;)(l ' e - T - 7 provided the changes in head group area and tail length are small, which are required for the harmonic approximation (eq 8, main text) to be valid. Thus, the bending energy within the harmonic approximation (eq 8) is
1
Fbnd= K(;)'(-
a- -1 - a1
up to quadratic order. Assume now that the spontaneous Gaussian curvature is small compared with the inverse square of the tail length of the surfactant molecule (noting that the spontaneous mean curvature vanishes for a bilayer). This assumption is equivalent to demanding the spontaneous surfac-
1 f H,(I
+ Al) + ( K , 1 2 / 3 ) [ 1+ 2 (Al/Z)]
1 f 2H,(1 + Al)
1 + K012/3
+ K,L2[1+ 2(Al/l)]
1 + K0l2
2 r H , 1 - -12(K,- K O )- H;l2 (AII.2) 3 to quadratic order of the principal curvatures. (The choice of + or - signs in the linear term depends on the monolayer.) Similarly, if the subsequent change in the head-group area induces a change in the bilayer curvatures from H&I to Hz,K2, we get
. - -" 1
x
* *
-"(a
u
) = 1fH2(l + Al) + ( K 2 / 3 ) ( l+
a(l + A')
-
+
+ K2(1+ A1)' 1f H l ( l + Al) + ( K 1 / 3 ) ( 1+ Al)2 1 * 2H,(1 + Al) + K,(1+ A1)2
1 f 2H2(1 Al)
4 2 Lk? = i ( H , - H,)1 T ( H 2- H,)Al - -(K2 - K,)12- -(K2 a 3 3
K,)lAl - 2(H:
- HI2)l2- 4(H22- H12)1A1- H,H212-
2HlH21Al Substituting for A1 (eq AII.2) gives
2 --' a - 7 ( H 2- H I ) ' - -(K2 - K,)12+ 3H;12 - 2H;12 a
3
HlH212 (AII.3) (to quadratic order in the principal curvatures). Combining these equations to yield the bending energy for each monolayer making up the bilayer gives
H:])
(AII.4)
Thus, the total bending energy for the bilayer is
Fbnd= 2~1'H; + ... (AII.5) where subsequent terms are proportional to quartic powers of the principal curvatures and higher (all odd power terms are equal and opposite for opposing monolayers in the bilayer and do not contribute to the total bending energy).
