DOI 10.1140/epje/i2010-10576-4 Regular Article Eur. Phys. J. E 31, 383–392 (2010) T HE EUROPEAN P HYSICAL JOURNAL E Polymer-decorated tethered membranes under good- and poor-solvent conditions M. Werner a and J.-U. Sommer Leibniz-Institut f¨ ur Polymerforschung Dresden, Hohe Strasse 6, 01069 Dresden, Germany and Institute of Theoretical Physics, Technische Universit¨at Dresden, 01069 Dresden, Germany Received 27 November 2009 Published online: 26 April 2010 – c  EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2010 Abstract. We study tethered membranes grafted by polymer chains on one side. Mean-field and scaling arguments predicting a spontaneous curvature are compared to the results of lattice-based Monte Carlo simulations using the Bond Fluctuation Model, which are carried out for various grafting densities and chain lengths. We show that already slightly overlapping chains bend the membrane significantly. This proves the entropic origin for the bending stiffness, which is of order kT . To understand the membrane curvature under conditions of very small bending stiffness we apply a geometrical model which takes into account the state of chains at the overlap threshold. Applying a thermal solvent model for the grafted chains, we demonstrate that the bending direction of the membrane can be triggered by variation of the solvent quality. This indicates that polymer-decorated membranes may serve as switchable nanoscale devices. 1 Introduction Membranes play an important role in living nature but can also be obtained in synthetic processes, for instance by cross-linking organic films [1,29]. Tethered membranes, which can be viewed as two-dimensional polymers, have attracted much interest by generalizing the statistical physics of linear polymer chains to higher-dimensional ob- jects. In particular, the role of excluded volume or self- avoidance has been discussed controversially in the litera- ture [2,3]. Today, it is well accepted that excluded volume induces an entropic bending stiffness on local scales due to the exclusion of large bending angles [4]. This effect leads to an asymptotic flat state caused by the fixed con- nectivity [5] in contrast to the prediction of a crumpled membrane using a Flory-type argument [2]. The result- ing linear relation between the radius of gyration of the membrane and its linear size was confirmed by computer simulations [3,6–10] and argued using ǫ-expansion [11, 35]. Kantor and Kremer have shown that the asymptotic flat state appears even in the case of locally restricted excluded-volume interactions [3]. We note that tethered membranes, sometimes also called crystalline membranes, are very different from fluid membranes such as lipid bilay- ers. Fluid membranes usually can display isotropic bend- ing behavior related with a Gaussian curvature. The latter is necessary to form vesicles and closed cell membranes. By contrast Gaussian curvature is strongly suppressed in tethered membranes, where bending is restricted to cylin- a e-mail: werner-marco@ipfdd.de drical forms. A good example in our everyday life is a sheet of paper which can be rolled into a cylinder but it can- not be deformed into a sphere. It is interesting to study combined systems made of membranes and linear polymer chains in order to tune the shape and elastic properties of the polymer-decorated membrane. In particular, grafting the chains to one side of the membrane only can induce spontaneous bending depending on grafting density and chain lengths, solvent quality and membrane properties. This effect is driven by a gain of conformational entropy of the grafted chains. The interaction of polymer chains with flexible inter- faces has attracted much interest in the past [12–26]. The- oretical models have been developed for the case of flexible membranes in general [17,26] or, more specifically, for lipid bilayers and biomembranes [20,22,23,25]. Several models predict that polymer chains, both grafted and adsorbed, can change the elasticity of the membrane/interface and can induce spontaneous curvature: Cantor regarded the stiffening and spherical bending of interfaces between im- miscible liquids by surfactants [12] using a Flory-type mean-field approach. An analysis of properties of cylin- drically and spherically curved brushes was given in a self-consistent field approach (SCF) by Milner and Wit- ten [13], where the free energy was expanded in terms of curvature. Hiergeist and Lipowsky considered mem- branes which are grafted by polymers on one side [17]. They applied a scaling ansatz [27] for the free-energy contribution of a curved brush which is balanced with the bending energy of the grafted membrane. Grafting polymer chains on both sides of a membrane have been