M ar 2 01 4 Phase separation in a wedge .

The exact theory of phase separation in a two-dimensional wedge is derived from the properties of the order parameter and boundary condition changing operators in field theory. For a shallow wedge we determine the passage probability for an interface with endpoints on the boundary. For generic opening angles we exhibit the fundamental origin of the filling transition condition and of the property known as wedge covariance. Interfacial phenomena at boundaries are a subject of both experimental and theoretical relevance which has received continuous and extensive interest in the last decades [1]-[4]. An aspect particularly important for applications is that the structure and geometry of the substrate can alter the adsorption characteristics of a fluid in an important way (see [5] for a review). Adsorption measurements can then be used, for example, to characterize fractally rough surfaces [6], or the connectivity of porous substrates [7]. The basic case of a wedge-shaped substrate [8] acquired special interest since thermodynamic arguments [9] indicated a specific relation with the adsorption properties of a completely flat surface: the wedge wetting (or filling) transition occurs at the temperature for which the contact angle of a fluid drop on a flat substrate equals the tilt angle α of the wedge, a circumstance that allows to regulate the transition temperature adjusting α. The connections between adsorption characteristics for different opening angles are known as properties of wedge covariance [10, 11, 12] and are experimentally testable [13]. The importance to progress from a thermodynamic to a statistical mechanical description is obvious. In two dimensions the essential role of fluctuations was established by the exact lattice results for the Ising model on the half phane [14, 15], which provided essential support for heuristic statistical descriptions of the wetting of a flat boundary [16]. For the wedge geometry the existence of the filling transition was established for the Ising model on a planar lattice forming a right-angle corner [17], but otherwise theoretical investigation in two and three dimensions has been based on effective interfacial Hamiltonian models [10, 11, 12, 18, 19] or density functional methods [20]. In this paper we derive general exact results for phase separation in a two-dimensional wedge. This is achieved exploiting low energy properties of bulk two-dimensional field theory [21, 22] together with a characterization of the operators responsible for the departure of an interface from a point on a boundary. For a shallow wedge we determine the exact passage probability of an interface with endpoints on the boundary. The theory provides a fundamental meaning to the contact angle and, for generic α, yields the filling transition condition. More generally, wedge covariance is shown to originate from the properties of the boundary condition changing operators in momentum space. We begin the derivation considering a two-dimensional system at a first order transition point, close enough to a second order transition to allow a continuous description in terms of a Euclidean field theory on the plane with coordinates (x, y). We label by an index a = 1, . . . , n the different coexisting phases, denote by σ(x, y) the order parameter operator, and by 〈σ〉a its expectation value in phase a. The Euclidean field theory corresponds to the continuation to imaginary time t = iy of a relativistic quantum theory in one space dimension, for which phase coexistence amounts to the presence of degenerate vacua |Ωa〉 associated to the different phases of the system. The elementary excitations are kink states |Kab(θ)〉 interpolating between different vacua |Ωa〉 and |Ωb〉; the rapidity θ parameterizes energy and momentum of these particles as p0 = mab cosh θ and p 1 = mab sinh θ, respectively, the kink massmab being inversely proportional to the correlation length. The relativistic dispersion relation (p0)2 − (p1)2 = mab is preserved