Scale-dependent rigidity of polymer-ornamented membranes

Abstract

We study the fluctuation spectrum of fluid membranes carrying grafted polymers. Contrary to usual descriptions, we find that the modifications induced by the polymers cannot be reduced to the renormalization of the membrane bending rigidity. Instead we show that the ornamented membrane exhibits a scale-dependent elastic modulus that we evaluate. In ornamented lamellar stacks, we further show that this leads to a modification of the Caillé parameter characterizing the power-law singularities of the Bragg peaks. PACS numbers: 36.20, 87.16.Dg, 82.35.Gh The characterization of polymer-membrane interactions is a fundamental issue of colloidal science. In cosmetics, pharmaceutics or detergency, many formulations are suspensions of self-assembled surfactant bilayers with polymers added for performance, processing, conditioning or delivery [1]. Likewise, lipid bilayers form the walls of living cells and liposomes, and host a great variety of macromolecules for coating protection, ion exchange and mechanical reinforcement. [2]. In many instances, the polymers are end-tethered to the soft interfaces. Grafting is easily achieved experimentally by using polymer chains that carry hydrophobic groups. Typical examples are provided by the so-called PEG-lipids, hydrophilic chains of polyethylene glycol covalently linked to a double-tail phospholipid molecule. Recent studies have shown that polymers grafted to bilayers can induce gelation [3] or other phase changes [4,5] in liquid lamellar phases. They stabilize monodisperse vesicles [6] and modify the geometry of monolamellar [7,8] and multilamellar [9] cylindrical vesicles. They also lead to drastic changes in the structure and phase behaviour of ternary amphiphilic systems [10].As first explained by Canham and Helfrich, fluid membranes are fluctuating objects [11]. In thermal equilibrium, the surface assumes all the possible shapes allowed by the geometry with the associated Boltzmann probability P ∼ exp{−H 0 /(k B T )}. The Hamiltonian H 0 is a quadratic function of c 1 and c 2 , the two principal curvatures at any given point of the surface,The spontaneous curvature c 0 vanishes for symetric bilayers. The Gaussian rigidityκ plays an important role in the determination of the topology of the system, and the integral dSκc 1 c 2 does not depend on the particular shape of the membrane. The amplitude of the thermal fluctuations is controled by the bending rigidity κ, that ranges from a few k B T to a few tens of k B T . Conversely, the analysis of the height correlations in a membrane system allows for the determination of the constitutive rigidity. Many techniques were specifically developped to extract κ by comparing experiments and theoretical predictions for vesicles, lamellar stacks, bicontinuous phases and other geometries. These methods have so far been of limited application in polymer-membrane systems because of the lack of prediction for the fluctuation spectrum of bilayers in the presence of polymers. In this Letter, we make a first step to reach this gap by investi...

Budding and vesiculation induced by conical membrane inclusions

“…For inclusions on a vesicle, the membrane shape and the minimal bending energy (assuming that the inclusions have maximal mutual distances) can be calculated using Eqs. (5) and (6). For bρ o = aρ i , the membrane around the inclusion has almost catenoid shape [40]; the catenoid is a minimal surface without bendingenergy cost.…”

Nonequilibrium fluctuations of an interface under shear

“…The description of this effect requires to go beyond the linear analysis generally used to derive the dispersion relation for capillary waves. To proceed, we follow a recursive scheme that has proved its worth, e.g., in the context of polymer-membrane interactions [23]. We assume that the deformation can be written h(x, y, t) = εu(x, y, t), with u(x, y, t) ∼ O(1).…”

“…(B4), the coupling between different orders only occurs through the boundary conditions at the interface. This suggests to use a recursive method in order to solve the problem [23].…”

Effect of Bilayer Thickness on Membrane Bending Rigidity

“…17 In contrast to lipid membranes (and perhaps closer to biomembranes), polymer membranes have a dense PEO layer, likely in a brush or partially collapsed brush state. 18 The effect of brushes on membrane elasticity and rigidity has been studied theoretically by sev-eral groups, 19,20 but experimental verifications with lipid-based systems 21 are limited for the reasons mentioned above. Various model scenarios are depicted schematically in Figure 1.…”