Fluctuation spectrum of membranes with anchored linear and star polymers

The effect of linear homopolymers, diblock copolymers, and star polymers anchored to a membrane on the membrane’s fluctuation spectrum is investigated for low grafting densities. Due to the nonlocality of the polymer-membrane interaction, the effective bending rigidity ${\kappa }_{\mathrm{eff}}\left(\mathbf{q}\right)$ of the composite membrane is found to depend strongly on the wave vector $\mathbf{q}$ of the membrane undulations. Analytical calculations for ideal linear chains and simulations for ideal and self-avoiding linear chains as well as for star polymers are presented. The analytical calculations are based on the Green’s function approach of Bickel and Marques [Eur. Phys. J. E 9, 349 (2002)]; for the simulations the Monte Carlo method is used. The functional form of ${\kappa }_{\mathrm{eff}}\left(\mathbf{q}\right)$ differs for end-grafted chains and diblock copolymers. In general, the polymer effect is most pronounced for undulations on length scales larger than or comparable to the polymer size, and decreases rapidly for smaller undulation wavelengths. Anchored linear chains always increase $\kappa$; anchored star polymers may increase as well as decrease $\kappa$, depending on whether they are anchored symmetrically or asymmetrically to the membrane.