Under well defined conditions, water and phospholipids are spontaneously assembled into a well-ordered structure, where the lipids form long tubes, of known diameter, filled with water. This structure is referred to as HexII. In our study, we dissolved Pyranine (8-hydroxy 1,3,6 trisulfonate) in the aqueous phase of HexII, and by monitoring the dynamics of proton dissociation, gauged the physical-chemical properties of the aqueous matrix, organized as long tubes with diameter varying between 16 Angstrom < 2r(w) < 50 Angstrom. The measurements, based on real time detection of proton dissociation, indicate that the lipid-water interface is characterized by diminished capacity of water to bind protons and intensified electrostatic interactions. Both properties varied with the diameter of the tubes. These observations are explained by considering the molecular properties of water at the hydration layer and by the effect of dielectric discontinuity at the water-lipid junction.