Interfacial Structural Changes and Singularities in Non-Planar Geometries

We consider phase coexistence and criticality in a thin-film Ising magnet with opposing surface fields and non-planar (corrugated) walls. We show that the loss of translational invariance has a strong and unexpected nonlinear influence on the interface structure and phase diagram. We identify 4 non-thermodynamic singularities where there is a qualitative change in the interface shape. In addition, we establish that at the finite-size critical point, the singularity in the interface shape is characterized by two distint critical exponents in contrast to the planar case (which is characterised by one). Similar effects should be observed for prewetting at a corrugated substrate. Analogy is made with the behaviour of a non-linear forced oscillator showing chaotic dynamics. PACS numbers: 68.45.Gd, 68.10.-m, 68.35.Rh Typeset using REVTEX 1 There are a number of well studied examples of fluid interfacial phenomena for planar systems in which surface phases with distinct adsorptions co-exist along a line of first-order phase transitions which terminates at a surface critical point. Examples include the prewetting transition associated with first-order wetting [1] and also interfacial localization in thin-film magnets (with opposite surface fields) associated with confinement effects at critical wetting [2]. In both cases, the difference in adsorption between the two phases vanishes continuously as the critical point, signifying the end of two-phase coexistence, is approached. This second order phase transition is characterized by the critical exponents belonging to the two dimensional Ising universality class (for three dimensional bulk systems) since the adsorption difference acts as a scalar order parameter [3,4]. In this letter, we describe a wealth of new interfacial structural changes and singularities which emerge when the analogous phenomena are considered in slightly non-planar geometries and which are intimately associated with non-linear behaviour. In addition to a shift in the finite-size (FS) critical point (compared to the planar confined system), the shape of the non-planar interface undergoes a number of structural changes as we move along and beyond the line of coexistence. This behaviour has no counterpart in the planar geometry, and has not been reported previously. Moreover, as the shifted surface critical point is approached, the function describing the shape of the non-planar interface shows non-analyticities which are characterised by two critical exponents. Whilst one of these appears to be identical with the usual critical exponent describing the singularity in the total (or average) adsorption, the general identification of the second is a more difficult problem, although scaling arguments (consistent with our explicit results) suggest that its value is related to the energy density. Our predictions are based on a detailed numerical analysis of a simple mean-field (MF) model of interfacial behaviour which we believe is qualitatively correct beyond MF approximation (in three dimensions). These are rather dramatic effects emanating from the introduction of a slight non-planar perturbation to the interface can be viewed profitably by making analogy with the classical mechanics of an extremely sensitive non-linear dynamical system exhibiting chaotic behaviour [5]. As we will see, the interface behaviour may be elegantly 2 portrayed as the temperature evolution of a phase plane plot, similar to that employed in dynamical systems, allowing us to distinguish different qualitative types of interface shape separated by non-thermodynamic singularities. To begin, we recall the relevant properties and phase diagram of the planar system prior to a discussion of the non-planar generalization. The transition that we concentrate on occurs in a thin-film magnet with opposing surface fields but the phenomena are generic to other situations such as prewetting at a planar wall. Consider then an Ising-like thin film magnet of width Lz and infinite transverse area in zero bulk field and below the bulk critical temperature T c with surface fields h1 and h2 =−h1 acing on the spins in the z = 0 and z = Lz planes respectively. We further suppose (through a judicial choice of surface enhancement [4]) that in the semi-infinite limit Lz → ∞ each surface undergoes a critical wetting transition at temperature Tw. For such a system, MF [2] and simulation studies [3] show that the finite size phase diagram is dominated by wetting effects which are able to suppress bulk-like coexistence over a large temperature regime. At sufficiently low temperatures T Tc(Lz) only one phase is possible. Thus, in the temperature window T c > T > Tw, the FS effects suppress bulk-like phase coexistence for all Lz. This temperature range is also characterised by a near soft-mode phase since the transverse correlation length ξ‖ is extremely large due to capillary-wave like excitations. These features can be most easily understood using a simple effective interfacial Hamiltonian model [2]: