We re-examine here the theoretical study of the phase separation between phospholipids and grafted long polymer chains onto a fluid membrane. The polymer chains are assumed to be anchored to the membrane by one extremity (anchor). The anchors are big amphiphile lipid molecules. The anchors and phospholipids forming the bilayer phase separate under the variation of a suitable parameter (temperature, pressure, membrane environment, ...). To investigate the demixtion transition, we elaborate a new approach that takes into account the membrane undulations. We show that these undulations have the tendency to induce additional attractive forces between anchors, and consequently, the separation transition is accentuated and occurs at high temperature. Quantitatively, we show that the membrane undulations contribute with an extra positive segregation parameter chi m > 0 , which scales as chi m approximately kappa(-2) , where kappa is the bending rigidity constant. Therefore, the attraction phenomenon between species of the same kind is significant only for those membranes of small bending rigidity constant. Finally, the study is extended to the case where the lengths of the anchored polymer chains are randomly distributed. To achieve calculations, we choose a length distribution of fractal form. The essential conclusion is that the polydispersity increases the size of domains alternatively rich in phospholipids and anchors.

We re-examine here the phase separation between phospholipids and adsorbed polymer chains on a fluid membrane with a change in some suitable parameter (temperature). Our purpose is to quantify the significant effects of the solvent quality and of the polydispersity of adsorbed loops formed by grafted polymer chains on the segregation phenomenon. To this end, we elaborate on a theoretical model that allows us to derive the expression for the mixing free energy. From this, we extract the phase diagram shape in the composition–temperature plane. Our main conclusion is that the polymer chain condensation is very sensitive to the solvent quality and to the polydispersity of loops of adsorbed chains.