# Charge–Fluctuation–Induced Non–analytic Bending Rigidity

###### Abstract

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.

###### pacs:

87.22.Bt, 61.20.Qg, 87.15.DaElectrostatics 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 , a Gaussian modulus , and a spontaneous curvature . The deformation free energy per unit area, expressed in terms of the mean curvature and Gaussian curvature may be given by the Helfrich free energy [9, 10]:

(1) |

Within an additive constant, the free energy of a sphere with radius is given by and of a cylinder with radius by Therefore, the parameters and may be determined from and .

The problem of the electrostatic contribution to the bending constants of layered membranes 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 is sufficiently high that the condensed counterions are confined to a layer of thickness , where is the Gouy–Chapman length, which scales inversely with and 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 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:

(2) |

where is the de Broglie wavelength of the charges, is the Bjerrum length for an aqueous solution of dielectric constant (), , is the Boltzmann constant, is the temperature, and is the coarse–grained density of the charges of species . The domain of the integral in Eq. (2) spans the entire charged membrane. In order to calculate the change in the free energy due to fluctuations, we assume that and expand the electrostatic free energy to second order in [12]:

(3) |

where The first term in the bracket is the Coulomb interaction of the charges. The second term comes from the second variation of the ideal gas entropy of the charges. The change in the free energy is obtained by summing all fluctuations weighted by the Boltzmann factor:

(4) |

It should be mentioned that Eq. (4) contains a divergent self–energy term which has to be substracted out. This means that we have to discard the first two terms in the expansion for , as can be seen easily by considering the zero temperature limit. As , the free energy is reduced to the electrostatic energy which is first order in . Since the self–energy is just a constant independent of temperature, it must be linear in . In the following, we employ this “substraction scheme” together with Eq. (4) to calculate the free energy where charges are confined to the surfaces of three geometries: i) a plane, ii) a sphere, and iii) a cylinder.

For the case of charges confined to a plane in Eq. (3) can be diagonalized by Fourier transform and is quadratic in . Performing the Gaussian integrals in Eq. (4) and substracting out the self–energy term, we obtain the free energy per unit area due to fluctuations [12, 13]

(5) |

where and , which scales like the Gouy–Chapman length, is a length scale analogous to the Debye screening length in 3–D. This result can also be obtained by solving the Debye–Hückel equation in 2D [14]. Note that Eq. (5) is ultravioletly divergent because of the infinite energy associated with the collapse of opposite charges. Thus a microscopic cut–off is necessary. In ref.[15], the author shows by partial summation of the Mayer series that the resulting free energy is convergent and indeed equivalent to a microscopic cut–off. From Eq. (5), one can deduce that the screening of the charges is weak and the potential to the lowest order has the distance dependence of a dipolar field, where plays the role of the dipole moment.

For the case of charges confined on a sphere of radius , after following a similar procedure to that described above we obtain

(6) |

It is easy to show that by setting and taking the limit , we recover the planar result. Equivalently we may write Eq. (5) as

(7) |

The difference can be evaluated as an asymptotic expansion in using the Euler–MacLaurin summation formula [16] with . The result is:

(8) |

In deriving the result above, we have regularized the integral in Eq. (7) and the sum in Eq. (6) by an ultraviolet cut–off . However, the leading term in Eq. (8) is cut–off independent and those higher order cut–off dependent terms tend to zero as .

For the case of a cylinder, we obtain the free energy:

(9) |

where and are modified Bessel functions of order . The evaluation of the integrals here is relatively difficult. However, we argue that has the following asymptotic expansion:

(10) |

for . First, we note that the only relevent contributions to the –integral in Eq. (9) are sharply peaked at with width . Hence, the Bessel functions can be approximated by , yielding

(11) |

Equation (10) can now be obtained by using the Euler–MacLaurin summation formula with

The modificaitons to the bending constants can be obtained from Eqs. (8) and (10) to yield

(12) |

We thus find that the contribution to the membrane elastic contants due to charge fluctuations is non–analytic. This kind of non–analyticity in the bending constants exists in the literature in other situations, for example in a system consisting of a membrane and rod–like cosurfactants[17]. In the present case, this non–analyticity can be considered a signiture of 2–D charged systems. The DH theory in 3–D yields an expression for the change in the free energy per unit volume [18] In constrast, Eq. (5) has a similar expansion for the free energy per unit area but contains a logarithmic term[14]: Therefore, it is not unexpected to find logarithmic corrections to the bending constants.

Secondly, we remark that both and are negative, in contrast to the mean–field PB contributions, where the renormalization of the bending moduli are always positive and the Gaussian moduli may be negative in some cases. In a system in which , is large compared to the mean–field contribution and the membrane becomes more flexible. Therefore, charge fluctuations induce bending of a charged membrane. This conclusion can be seen physically in the light of the recent works [12, 13] on attractive interactions between two planar charge–fluctuating membranes. It is found that for large distance separating the two membranes, the attractive force per unit area scales as . Now, a sphere or a cylinder can be approximated as two flat planar surfaces in the limit and their interaction free energies per unit area therefore should scale like . Hence negative contribution to the bending modulus.

The negative contribution of from charge fluctuations can also be understood physically as follows. Since the potential is rotationally invariant, one might expect that the charged surface has rotational symmetry when the the electrostatic free energy is a minimum. Now, recall that the Gaussian curvature, expressed in terms of the principal radii and , is . Therefore, as can be easily seen, the free energy in Eq. (1) is indeed lowest (since ) if there is rotational symmetry about an axis normal to the charged surface or , in accordance with the rotational symmetry of the potential. Furthermore, the fact that has interesting experimental consequences since strongly negative values of favor the formation of many disconnected pieces with no rims, like spherical vesicles. Therefore, when the surface charge density is made sufficiently large, the membrane might spontaneously form vesicles, due to fluctuations of condensed counterions. Experiments [19] on charged surfactant systems supports this conclusion.

The result presented in this Letter is particularly relevant to recent experiments [20] where the authors find the formation of vesicles by mixing anionic and cationic surfactants. Two aspects of their experiment can be qualitatively accounted for by the present model. They find, in equilibrium, large vesicles with and substantial size polydispersity. Indeed, the vesicle free energy per unit area given by

(13) |

where is the bare value of the bending rigidity, has an equlibrium value , which can be large even for a moderate value of of the order of 3–5 . Furthermore, the second derivative of the free energy is exponentially small. Hence the variance or fluctuations in , is large, implying size polydispersity.

In conclusion, by studying fluctuations of charges on curved films, we have deduced non–analytic contributions to the bending energy of a membrane. Our calculation is applicable to condensed counterions on a highly charged membrane, and mixing of surfactants of opposite charges. For the latter case, we find qualitative agreements with experiments.

We would like to thank S. Safran, G. Rossi, and C. Safinya for stimulating and helpful discussions. AL and PP acknowledge support from NSF grants MRL–DMR–9632716, DMR–9624091, and DMR–9708646.

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