Scaling Properties of Interfaces and Membranes Reinhard Lipowsky Institut f¨ ur Festk¨ orperforschung der KFA J¨ ulich Postfach 1913, D–5170 J¨ ulich, FRG ∗ Source: In Random Fluctuations and Pattern Growth ed. by H. Stanley and N. Ostrowsky, Nato ASI Series E, Vol. 157, pages 227-245 (Kluwer Akad. Publ., Dordrecht 1988) 1 Outline and Summary This review deals with three topics: (i) Fluctuations of interfaces and membranes (Sec. 2); (ii) Interactions of interfaces and membranes (Sec. 3); and (iii) Dy- namics of interfaces and growth of wetting layers (Sec. 4). All three topics are intimately related: fluctuations renormalize the interactions, and the interactions act as a driving force for the dynamics. I will focus on interfaces and membranes which are rough but not (yet) crum- bled. The scaling properties of such surfaces are governed by the roughness exponent, ζ (Sec. 2.2). For interfaces, this exponent depends on the nature of the two phases separated by the interface which can be fluids, periodic crystals, quasicrystals, or random systems (Sec. 2.3). For membranes, ζ depends on the internal membrane structure which may be fluid, crystalline, or hexatic (Sec. 2.4). In all cases, the fluctuations give a singular contribution to the surface free energy, see Sec. 2.5. The effective interaction of surfaces represents a unifying concept by which one can understand many different phenomena, see Sec. 3.1 and 3.2. This point of view is not entirely new: there exists a huge literature on interactions between planar surfaces (Sec. 3.3). However, it has been realized only recently that these interactions are often renormalized by fluctuations. Several scaling regimes for the renormalized interaction must be distinguished, see Sec. 3.4. These different regimes can be studied in a systematic way, starting from an effective Hamiltonian for the interacting surfaces (Sec. 3.5). So far, the most useful method has been a functional renormalization group (RG), see Sec. 3.6. As a result, one finds RG fixed points for the critical behavior associated with the unbinding of surfaces (Sec. 3.7 and 3.8). * Present address: MPI f¨ ur Kolloid- und Grenzfl¨ achenforschung, 14424 Potsdam, Germany. Email: lipowsky@mpikg-golm.mpg.de 1