The Statistical Mechanics of Crumpled Membranes

Abstract

An enterprise of considerable current interest in theoretical physics is the study of interfaces and membranes. In condensed matter physics, an “interface” usually means a boundary between two phases, whose fluctuations can be studied by methods adapted from equilibrium critical phenomena. The statistical mechanics is typically controlled by a surface tension, which insures that such surfaces are relatively flat. Recently, however, there has been increasing interest in membrane-like surfaces. “Membranes” are composed of molecules different from the medium in which they are imbedded, and they need not separate two distinct phases. Because their microscopic surface tension is small or vanishes altogether, membranes exhibit wild fluctuations. New ideas and new mathematical tools are required to understand them.

We first sketch the physics of “flat” interfaces, and then discuss crumpled tethered membranes which are natural generalizations of linear polymers. More generally, the large distance behaviors of membranes fall into a variety of universality classes, depending, for example, on whether the local order is liquid or crystalline. We show that membranes with a nonzero shear modulus differ from their liquid counterparts in that they exhibit a flat phase with long-range order in the normals at sufficiently low temperatures. Because entropy favors crumpled surfaces with decorrelated normals, there must be a transition to a crumpled phase at sufficiently high temperatures. We also discuss the energies of discinations and dislocations in flexible membranes with local crystalline order. Unlike crystalline ifims forced to be flat by a surface tension, it is energetically favorable for membranes to screen out elastic stresses by buckling into the third dimension. Dislocations, in particular, are predicted to have a finite energy. We conclude that a finite density of dislocations must exist at all nonzero temperatures in nominally crystalline but unpolymerized membranes. The result macroscopically is a hexatic membrane, with zero shear modulus, but extended bond orientational order. The elastic energy which controls undulations in hexatic membranes is discussed briefly.

Related problems arise in field theory models of elementary particles.¹ In contrast to these models of quantum mechanical strings, however, most of the models discussed here have explicit experimental realizations in condensed matter physics. Much of the vitality of this subject arises because of a delicate interplay between theory and experiment: theoretical predictions can, in principle, be checked by inexpensive but revealing laboratory experiments in a matter of months.