Concepts on Interfaces in Confined Geometry

In this section we return to the situation sketched in Fig. Id, namely a polymer mixture (A,B) confined between two different planar surfaces, such that the lower surface prefers species A, and the upper surface prefers species B. We are interested in the temperature region in between the critical temperature of phase separation in the bulk and the wetting transition temperatures of the two surfaces (i.e., yw % in Fig 6 in principle, one must for film thicknesses D— 

Starting point is the description of the interface in terms of the drumhead model [273], i.e. we disregard for the moment the fact that the interface has a non-trivial intrinsic profile with width w0, and treat it like a sharp kink in terms of the local interfacial height h(x,y) only, x,y being coordinates along the lower surface (at z=0). Then the effective free energy functional Heff [h] associated with interfacial fluctuations can be written as [220] 

For a short range potential acting on the interface due to the wall, one considers an exponentially decaying form, i.e. for the lower wall 

Note that at TC(D), where the square bracket in Eq. (112) changes sign, a transition from the state of the film as shown in Fig. Id) to the state shown in Fig. le) is predicted to occur [268-271], with interfaces bound to the walls, rather than a freely fluctuating interface unbound in the center of the film. 

Using the potential V(h)°ch2 in Eq. (110), one recognizes that Eq. (110) is formally identical to a Ginzburg-Landau theory of a second-order transition for T TC(D), with h(x,y) the order parameter field [186,216]. Therefore, it is straightforward to read off the correlation length , associated with this transition at TC(D), namely 

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