Physica A 177 ( 1991 ) 207-212 North-Holland Finite-size scaling in layered systems with surface ordering Hao Li, Maya Paczuski, Mehran Kardar and Kerson Huang Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139. USA Motivated by recent experiments on liquid-crystal films, we study finite-size scaling of a layered system in the presence of surface ordering. We find that there is an important length scale ~s, char- acterizing the decay of surface order into the bulk. The usual finite-size scaling is generalized by including n/~s as an additional scaling variable, with n the number of layers. The bulk transition temperature is governed by a new scaling function Fs(n/~s), which decreases with n for n < ~s, but increases with n for n > ~s- Qualitative trends agree well with the experiments. Finite-size scaling and surface phase transitions are just two out of the many aspects of critical phenomena in which Michael Fisher has had a long standing interest, and to which he has contributed many pioneering works and fundamental insights [ 1-4 ]. Finite-size scaling is now an essential tool to any numerical simulation in statistical mechanics [ 5 ] and surface critical phenomena have found diverse applications in surface ordering and wetting transitions [6]. Recent experiments on free-standing liquid crystal films of varying thickness [ 7-9 ] have exhibited critical behavior that is influenced by both surface ordering and finite-size effects. A particularly puzzling feature is a peak in heat capacity due to bulk ordering which moves to lower temper- atures as the thickness is increased. To understand this behavior we examined simple layered spin models with enhanced surface coupling, by mean-field theory in general or by exact solution in two dimensions. We conclude that the shift of the peak to lower temperatures is a pre-asymptotic behavior for films thinner than the length scale as- sociated with the decay of surface order into the bulk, and find a novel scaling func- tion that governs this behavior. Our general approach to this problem, and its resolu- tion, was much influenced by Professor Fisher's general methodology, and his previous results; we hope that he enjoys this small contribution. To study melting and ordering phenomena in two dimensions a number of experi- ments have been carried out on free-standing liquid crystal films; among them precise measurements of heat capacity [ 8,9 ] and diffraction patterns [ 7 ]. While the diffrac- tion experiments reveal suppressed fluctuations in the displacement of surface parti- cles [ 7], the observation of two peaks in heat capacity measurements indicates that the surface of the film develops bond orientational order prior to the bulk [ 8,9 ]. To- gether, these experiments suggest that coupling on and near the surface are stronger 0378-4371/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)