Langmuir 1993,9, 2761-2763
2761
Elastic Modulus of Gaussian Curvature of Partially Polymerized Surfactant Membranes M. M. Kozlov* and W . Helfrich Fachbereich Physik, Freie Universitdt Berlin, Arnimallee 14, 1000 Berlin 33, Germany Received April 12, 1993. In Final Form: September 3 , 1 9 9 9
The effectof partial polymerizationof surfactant membraneson the elasticmodulus of Gaussian curvature,
K, is considered. We show that the steric interaction within a polymer-causes a local increase of this
modulus. For a dilute polymer solution,this results in a nonuniformity of K which may influence the shape of the membrane. The increase of the modulus of Gaussian curvature is probably weak for linear polymers, reaching an order of kT in favorable conditions, but it may be strong for branched polymers. There are two obvious reasons to investigate polymerized membranes. The first is the practical importance of this phenomenon for the stabilization of liposomes used in drug de1ivery.l The second consists in the fascinating physical phenomena observed on partially or fully polymerized membranes.24 Polymerization is achieved by the photochemical formation of bonds between adjacent surfactant The polymerized bilayers tend to wrinkle,= and a wrinkling transition has been found in a partially polymerized bilayer when the temperature is Another interesting property of polymerized or polymerizable membranes is their tendency to form toroidal vesicles? The present paper deals with the effects of partial polymerization on one of the elastic moduli of fluid membranes. Our calculations will be for monolayers, but the results are easily applied to bilayers. The results should bevalid alsofor comblikepolymers, the backbones of which are outside the membrane. The bending elastic energy per unit area of monolayer may be written as
Here K is the bending rigidity, K is the modulus of Gaussian curvature, and J, is the spontaneous curvature. The bending is characterized by the sum of principal curvatures, J,and by the product of principal curvatures, the Gaussian curvature K. In the following, we willstudythe changes of the modulus of Gaussian curvature, K , which result from partial polymerization. It will be shown that_the repulsive selfinteractiop of the polymer increases K. The new contribution to K appears in the regions of the monolayer occupied by polymer chains and does not exist elsewhere. This may have interesting consequences since a nonuniform K can influence the shape of a membrane, while a uniform K drops out of shape equations, coming into play only in changes of topology such as fusion and fission. We do not deal here with the effects of polymerization on the spontaneous curvature which can be large. In other words, we consider monolayers of uniform spontaneous curvature.
* Abstract published in Advance ACS Abstracts, October 1,1993.
(1) Ringsdorf, H.; Schlarb, B.; Venzmer, J. Angew. Chem. 1988,100, 117. (2)Sackmann, E.; Eggl, P.; Fahn, C.; Beder, H.; Ringsdorf, H.; Schollmeier, M. Ber. Bunsen-Ges. Phys. Chem. 1985,89,1198. (3) Mutz, M.; Bensimon, D.; Brienne, M. J. Phys. Rev. Lett. 1991,67, 923. (4) Fourcase, B.; Mutz, M.; Bensimon, D. Phys. Rev. Lett. 1992,68, 2551. (5) Dvolaitzky, M.; Guedeau-Boudeville, M. A.; LBger, L. Langmuir 1992, 8, 2595.
Figure 1. Two-dimensionalpolymer in the partially polymerized membrane.
Possible effects of partial polymerization on the spontaneous curvature of membranes will be treated elsewhere.6 Let us consider partially polymerized surfactant monolayers, which contain side by sidepolymer chainsand single monomers (Figure 1). If not stated otherwise,the polymers will be taken to be linear chains, i.e., without branching. The monolayer is a two-dimensional liquid so that the polymers and the surfactant molecules can move freely in lateral direction. The conformations of the polymer chain in lateral direction are thought to be those of a two-dimensional self-avoiding walk (SAW), since the chemical bonds are stable and the chain ia assumed not to intersect itself. The persistence length E of the chain depends on the nature of the chemical bonds between monomers. We will assume the bonds to be fully flexible so that $, coincides with the molecular diameter (see Figure 1). The bending energy of a piece of monolayer containing a polymer chain is influenced by the effect of curvature on the conformations of the chain. Let us imagine practically uniform bending of a piece of monolayer with a single polymer chain made up of X monomers. When the monolayer is flat, the conformations of the chain are those of a self-avoiding walk in two dimensions. The extension of the polymer chain may be expressed by its radius of g y r a t i ~ n : ~ * ~ R, = bX* (2) where b is related to the persistence length, b N 0.36f.* The value of the critical exponent is v = 0.75 for linear polymer^.^^^ It may be noted that v = 5/8 holds for a particular kind of branched polymer in two dimensions? The energy of a self-avoiding walk derives from ita conformational entropy. The bending of the monolayer can change the area available to the SAW as seen from a given center. The change is produced by Gaussian curvature and varies with the squared distance from the center. In particular, spherical bending (K> 0 ) decreases (6) Helfrich, W.; Kozlov, M. M. J. Phys. (Paris), submitted for publication. (7) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (8)des Cloizeaux, J.; Jannink, G. Polymers in Solutions; Claredon Press: Oxford; 1990. (9) Daoud, M.; Joanny, J. F. J. Phys. (Paris) 1981, 42, 1359.
0743-7463/93/2409-2761$04.00/00 1993 American Chemical Society
Letters
2762 Langmuir, Vol.9,No. 11,1993
the area, cylindrical bending (K = 0) does not change it, and saddle bending (K C 0) increases it. The conformational chain entropy should rise with the available area. Accordingly, we may infer that the contribution Fplof the polymer chain to the bending free energy should obey, to lowest order in K, Fpl
= BplK
(4)
The coefficient KP1 is the contribution of the polymer to the modulus of Gaussian curvature. An important feature of this contribution is its nonuniformity: KP1 is different from zero only in the monolayer pieces containing a polymer, and is equal to zero outside them. For the sake of simplicity, we assume here sharp boundaries of such pieces and a uniform KP1 > 0 inside them. Scaling arguments seem to confirm (3) and provide the order of magnitude of BPI. In order to show this, we extend well-known scaling concepts' from a polymer on a plane to the same polymer on a uniformly bent surface. Accordingly, we assume the energy Fplto be invariant when all lengths, not only the radius of gyration of the chain but also the principal radii of curvature, are multiplied by the same factor. Moreover, we expect this energy to be of the order k T for AplK = 1, i.e., for a Gaussian curvature at which the geometry of the polymer becomes seriously nonplanar. In this spirit we may write (5)
where a is a numerical factor on the order of unity. Comparison with (4) leads to ~
~
4 kl T=)
c-
(3)
where the coefficient Bpl must be positive. Note that Fpl does not depend on the sum of principal curvatures, J,or its square since cylindrical bending does not change the conformational entropy. Factoring out the areaAplcovered by the polymer chain, one can rewirte (3) as
BPI= a7r(kT)RG2= a7r(kT)b2X2"
R
(6)
if we put Apl = rRc2. I t is difficult to predict the practical importance of the effect of Gaussian curvature on linear polymers. Much depends on the size of a,whose determination requires numerical methods. The average of taken over the whole monolayer cannot exceed the local value (6). For maximum effect, it should not matter whether we have complete coverage of the monolayer with essentially noninteracting chains or a semidilute solution of polymers which has to be analyzed in terms of blobs. The maximum increase of K in bilayers is twice as large as in monolayers. Despite this limitation, linear polymers could promote the formation and stabilization of bicontinuous microemulsions for two reasons: First, a negative contribution to K lowers the bending energy of connections of the monolayer with itself, each connection (or handle) being associated with the energy -hi. Second, the monolayer bending rigidity of such a system is also very small, being k T or less.1° A possible manifestation of the nonuniformity of the modulusof Gaussian curvature is the formation of toroidal vesicles in the systems of polymerized or polymerizable lipid^.^ In fact, it seems plausible to assume that the membrane made up of a polymerizable lipid contains some (10) Binks, B. P.;Meunier, J.; Abillon, 0.;Langevin, D.Langmuir 1989,5,425.
I
Figure 2. Tentative decomposition into blobs of a 6-fold star in the plane. The number of outermost blobs will increase in the presence of Gaussian curvature K = 1/R2.This is because the six chains have to decrease their spacing and increase their extension. The effect increases with the number of chains.
amount of polymers even without UV treatment. The polymers can promote the formation of fusion necks and other structures of strong saddle curvature by accumulating in regions of negative Gaussian curvature. A polymer in a neck could in favorable cases be decomposed into blobs, thus loweringthe bending energy by more than 1 times a(kT). We note that branched polymers should have a stronger effect than linear ones on the modulus of Gaussian curvature. This becomes clear if one decomposes them, however imperfectly, into blobs.ll The breakup into blobs does not change KP1 by itself. However, the number of blobs can now rise as Gaussian curvature increases because of a crowding of the ensemble of branched chains. The formation of each new blob requires additional free energy of the order of kT. An illustration is given in Figure 2. To calculate the change of the number of blobs, let us consider a star polymer made up of n equal chains (Figure 2). The blobs are taken to be arranged equally along all the chains. Their density Pb as a function of the distance r from the center, measured in arc length, may be expressed by
n pb
= 27rr(l- (1/6)r2K)
(7)
to first order in K. The full number of blobs is equal to
while the number of monomers in one chain may be written as
x = JRpb [%(1 - i r 2 K ) ]""dr where R is the radial extension of the chains. We assume here that the relationship between the number of monomers in one blob and its radius is given by (2). Equation 9 gives R as a function of K. This is inserted into (8) to obtain the change of the number of blobs caused by bending. The resulting change of the polymer free energy is, for v = 3/4, if the energy of a blob is taken to be kT
AF,, = (1/207r)(kT)n2R2K (10) Accordingly, the additional contribution to the modulus of Gaussian curvature in the case of a star polymer is (11) Daoud, M.; Cotton, J. P. J. Phys. (Paris) 1982,43,531.
Langmuir, Vol.9,No.11, 1993 2763
Letters
Kbr
= (1/20?r2)(k~n2
It becomes larger than kT only for n > 14. A further remark concerns a subtle effect of Gaussian curvature which applies to diffusion and self-avoiding random walks alike. If the walk proceeds in steps of a fixed length, Gaussian curvature changes the length of the ring, plane for K > 0 but bent for K < 0, which is the locus of the end points of all steps from a given origin. This is similar to a (fractional) change of the coordination number in a lattice model. For a step of length I, the relative change ALII in arc length of the ring is found to be
The associated change in the free energy of a random walk of X steps is, to lowest order in K ,
AF
= (kT)(b2/6)XK
(12)
Dividing this by R c ~N b2X3I2,i.e., the area occupied by a linear polymer, we realize that the new correction of the modulus of Gaussian curvature decreases with the size of the polymer, in contrast to in (6) and (10). Obviously, the additional correction of K is negligible except, perhaps, for very short polymer chains or very small blobs. Finally, let us emphasize again that the physical reason for the effect considered in the present paper is the selfavoidance of the polymer chain. The entropy of the twodimensional self-avoiding walk depends considerably on the Gaussian curvature of the surface, while the entropy
of a self-intersecting random walk is much less influenced by the curvature. Our theory applies to the case of partial polymerization; it includes polymers with a backbone outside the membrane or adsorbed, over their whole length, to the membrane. Linear nonintersecting polymers in membranes will favor necks over flat and cylindrical membrane configurations. They may transform planar bilayers into a cubic phase or, in the case of monolayers, convert a microemulsion from a droplet phase into a bicontinuous one. The effect of linear polymers on bending elasticity, Le., K , is possibly rather weak, its strength depending on the unknown numerical factor a. However, phase transitions and equilibria often are sensitiveto small changes. The local action of the polymers may enhance their effectiveness through an accumulation at places where they lower the bending energy of the membrane. Other researchers have consideredsurfaces with weakly adsorbed or grafted polymer^'^-'^ and membranes made of diblock copolymers15to calculate the curvature energy and the bendingelasticity of such layers. All these theories deal with polymers in three-dimensional space, while in our model the polymer configurations are restricted to a surface. Acknowledgment. We are grateful to the Deutsche Forschungsgemeinschaft for supporting this work through Grant He 952/15-1. (12) Pincus, P. A.; Sandroff, C. J.; Witten, T. A., Jr. J.Phys. (Paris) 1981,45,725. (13) Hone, D.; Ji, H.; Pincus, P. A. Macromolecules 1987, 20, 2643. (14) Brooks,J. T.;Marques, C. M.; Cates, M. E. Europhys.Lett. 1991, ,1
"-0
14, 1 1 0 .
(15) Wang, Z.-G., and Safran, S.
9. J. Chem. Phys. 1991,94, 679.
