Phase separation between phospholipids and grafted polymer chains onto a fluctuating membrane

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 χ_m > 0 , which scales as χ_m κ^-2 , where κ 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.