Static correlations functions and domain walls in glass-forming liquids: the case of a sandwich geometry

The problem of measuring nontrivial static correlations in deeply supercooled liquids made recently some progress thanks to the introduction of amorphous boundary conditions, in which a set of free particles is subject to the effect of a different set of particles frozen into their (low temperature) equilibrium positions. In this way, one can study the crossover from nonergodic to ergodic phase, as the size of the free region grows and the effect of the confinement fades. Such crossover defines the so-called point-to-set correlation length, which has been measured in a spherical geometry, or cavity. Here, we make further progress in the study of correlations under amorphous boundary conditions by analyzing the equilibrium properties of a glass-forming liquid, confined in a planar ("sandwich") geometry. The mobile particles are subject to amorphous boundary conditions with the particles in the surrounding walls frozen into their low temperature equilibrium configurations. Compared to the cavity, the sandwich geometry has three main advantages: (i) the width of the sandwich is decoupled from its longitudinal size, making the thermodynamic limit possible; (ii) for very large width, the behaviour off a single wall can be studied; (iii) we can use "anti-parallel" boundary conditions to force a domain wall and measure its excess energy. Our results confirm that amorphous boundary conditions are indeed a very useful new tool in the study of static properties of glass-forming liquids, but also raise some warning about the fact that not all correlation functions that can be calculated in this framework give the same qualitative results.