Langmuir 1990,6, 159-162
159
Undulations of Charged Membranes A. Fogden, D. J. Mitchell, and B. W. Ninham* Department of Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra A.C.T. 2600, Australia Receiued April 19, 1989.I n Final Form: June 14, 1989 The electrostatic contribution to the change in free energy of an isolated charged planar membrane when it undergoes a small sinusoidal undulation is calculated by using the nonlinear Gouy-Chapman theory for the diffuse double layer. The result is compared with Helfrich's phenomenological formula for the elastic free energy of curvature of fluid membranes. The latter is found to hold only for wavelengths large compared to the Debye-Huckel screening length. Consequently, Helfrich's formula is not valid in the limit of counterions only (no added salt). In practice, this is the situation most often encountered with ionic systems. Introduction The observation by Helfrich' that fluctuations in amplitude about a mean position can lead to strong longranged steric forces between membranes has stimulated a good deal of interest. On the theoretical side, the assumption that such forces exist has led to remarkable results ranging from the shapes assumed by red blood cells' to an account of ,unbinding transitions3 adduced to explain the stability and swelling of lamellar phases. A number of sophisticated synchrotron X-ray4 and other experiments have been used to infer the relevant bending moduli that would govern the new fluctuation forces. But the foundations of the theories have remained on unsecure ground. This is partly because first-principle derivations are justified so far only in harmonic approximation and break down precisely in the regime of most interest where steric interactions are large and strictly anharmonic. Apart from this annoyance, the foundations of the plausible formation of the Helfrich theory have not been explored. This is probably not a serious issue for uncharged phospholipids, where hydrocarbon chain interactions dominate in setting the bending moduli of membranes, but it is an issue for ionic systems like highly swollen lamellar phases, where we have the most hope of measuring the moduli. We have reported some results that bear on the issue for some ionic systems. To put our results into perspective, consider the phenomenologicalformula of Helfrich for the bending energy per unit area of a membrane in quadratic approximation, which is' = (1/2)kC(c1+ Cp - C O ) ~+ KcClC2
(1) Here c1 and c2 are the principal curvatures, co is the spontaneous curvature (which vanishes for symmetric membranes), k, is the bending rigidity, and K, is the elastic modulus of Gaussian curvature. In a previous paper,5 we calculated these moduli using the nonlinear Gouy-Chapman theory for the free energy of the diffuse double layer of a charged membrane for planar, cylindrical, and spherical geometry. If we consider a symmetric planar membrane, then the change in free energy per unit area (averaged over the area), when the membrane suffers a sinusoidal undula(1) Helfrich, W. Z. Naturforsch. 1978, 33A, 305. (2) Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, D., Eds.; Proceedings in Physics 21; Springer: Berlin, 1987. (3) Lipowsky, R.; Leibler, S. Phys. Rev. Lett. 1986, 56, 2541. (4) Bassereau, P.; Marignan, J.; Porte, G . J. Phys. (Les Ulis., Fr.) 1987,48,673.
tion of amplitude a and wavelength 2s/k, should be (from the above formula) (gbnd)
= (1 /4)kck4a2
(2)
We wish to test this formula by comparing it to the electrostatic contribution to the bending free energy per unit area of the membrane (assumed to carry an electric charge) as calculated according to the Gouy-Chapman theory.
Results In the Appendix, we consider an interface between an aqueous electrolyte and a medium which does not dissolve ions, carrying uniform surface charge and subject to a sinusoidal undulation. A double layer will form at such an interface. We have calculated the contribution to the bending free energy of this interface due to the double layer using the nonlinear Gouy-Chapman theory. It is assumed that the total area and the surface charge density remain unchanged. The result is given by eq A34. The corresponding result for a symmetric membrane is assumed double this:
s(* - s2/4
+ pl[pl + (1+ s2/4)'/2] ) (3) P1
where a is the amplitude and 2 s / k the wavelength of the undulation, t is the dielectric constant of water, and q is the protonic charge. @ = l/k,T, where k, is Boltzmann's constant and T i s the absolute temperature, and K is the inverse Debye length. Also, pl' = 1 k'/K' and s = 4TO@q/€K. Clearly this expression is a more complex function of k than the expression in eq 2. However, for k > K S we have
The undulating membrane may lower its mean electrostatic free energy of bending by redistributing the uniform surface charge. For a general (periodic) charge distribution, the derivation of the bending energy via solution of this generalized boundary value problem follows by simple extension of the analysis in the Appendix. Optimizing with respect to the nth Fourier coefficient ( n 2 1)of the distribution, the minimum of this quantity (again doubling for a symmetric membrane) is
The asymptotics of this expression are the same as those of its uniform charge counterpart eq 3, with the exception that, for k
