Charge–Fluctuation–Induced Non–analytic Bending Rigidity

In this Letter, we consider a neutral system of mobile positive and negative charges confined on the surface of curved films. This may be an appropriate model for: i) a highly charged membrane whose counterions are confined to a sheath near its surface; ii) a membrane composed of an equimolar mixture of anionic and cationic surfactants in aqueous solution. We find that the charge fluctuations contribute a non–analytic term to the bending rigidity that varies logarithmically with the radius of curvature. This may lead to spontaneous vesicle formation, which is indeed observed in similar systems. 87.22.Bt, 61.20.Qg, 87.15.Da Typeset using REVTEX 1 Electrostatics of charged objects such as polyelectrolytes and membranes in aqueous solution plays an important role in many biological systems [1]. The fundamental description of these systems has been the mean–field approaches – the Poisson–Boltzmann (PB) or Debye–Hückel (DH) theory (for a review, see [2]). However, for a highly charged surface, the Manning theory of counterion condensation [3] provides an analytically tractable approximation to the PB theory. Indeed, it has been demonstrated rigorously from the solutions to the PB equation [4] that the electrostatic potential far away from the charged surface is independent of the charge density above a certain critical value, implying that the counterions are confined to a thin layer close to the charged surface. However, like the PB theory, it fails to capture the correlation effects of the counterions since it expressly assumes that the “condensed” counterions are uniformly distributed. On physical grounds, we should expect that at low enough temperatures the fluctuations of these condensed counterions about a uniform density would give rise to new phenomena. Indeed, recent simulations [5,6] show that the effective force between two like–charged rods and planar surfaces actually becomes attractive at short distances. These suprising results shed new light on the understanding of the electrostatic adhesion between cells [7] and the puzzling problem of DNA condensation [8]. In this Letter, we examine the effect of fluctuations of these condensed counterions on the bending rigidity of a charged membrane. The elastic properties of a fluid membrane are characterized by three macroscopic parameters – a bending elastic modulus κ, a Gaussian modulus κG, and a spontaneous curvature H0. The deformation free energy per unit area, expressed in terms of the mean curvature H and Gaussian curvature K may be given by the Helfrich free energy [9,10]: f = κ 2 (H − H0) 2 + κG K. (1) Within an additive constant, the free energy of a sphere with radius R is given by fs = (2 κ+κG)/R −2κH0/R and of a cylinder with radius R by fc = κ/2R −κH0/R. Therefore, the parameters κH0 and κ+ κG may be determined from fs and fc. The problem of the electrostatic contribution to the bending constants of layered mem2 branes within the PB mean field approach has been studied [11]. The electrostatic renormalization of the bending rigidity turns out to be positive; hence electrostatics augments the rigidity of charged membranes. Here we go beyond these PB approaches by assuming that the surface charge density n0 is sufficiently high that the condensed counterions are confined to a layer of thickness λ << L, where λ is the Gouy–Chapman length, which scales inversely with n0 and L is the linear size of the charged membrane. By considering the in–plane fluctuations of the condensed counterions and charges on the membranes, we model the system effectively as a 2–D coulomb gas interacting with a r potential. This model has yet another experimental realization – a neutral membrane composed of a dilute mixture of anionic (–) and cationic (+) surfactants. The electrostatic free energy of the system is the sum of the entropy of the charges and the electrostatic interaction energy among them: