Adsorption of alcohol molecules or other small amphiphilic molecules in the cell membrane can induce significant changes in the structure of the membrane. To understand the molecular mechanisms underlying these structural changes, we developed a mesoscopic membrane model. Molecular simulations on this model nicely reproduce the experimental phase diagrams. We find that alcohol can induce an interdigitated structure in which the normal bilayer structure changes into a monolayer in which the alcohol molecules screen the hydrophobic tails from the water phase. We compute the effect of the chain length of the alcohol on the phase behaviour of the membrane. At low concentrations of alcohol, the membrane has domains of the interdigitated phase that are in coexistence with the normal membrane phase. We use our model to clarify some of the experimental questions related to the structure of the interdigitated phase and put forward a simple model that explains the alcohol chain length dependence of the stability of this interdigitated phase.