  The behavior of energy minimizers at the boundary of the domain is of great
importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase
transitions, since it describes the effect of the container walls on the
configuration of the liquid. This problem, also known as the liquid-drop
problem, was studied by Modica in [21], and in a different form by Alberti,
Bouchitte, and Seppecher in [2] for a first-order perturbation model. This work
shows that using a second-order perturbation Cahn-Hilliard-type model, the
boundary layer is intrinsically connected with the transition layer in the
interior of the domain. Precisely, considering the energies $$
\mathcal{F}_{\varepsilon}(u) := \varepsilon^{3} \int_{\Omega} |D^{2}u|^{2} +
\frac{1}{\varepsilon} \int_{\Omega} W (u) + \lambda_{\varepsilon}
\int_{\partial \Omega} V(Tu), $$ where $u$ is a scalar density function and $W$
and $V$ are double-well potentials, the exact scaling law is identified in the
critical regime, when $\varepsilon \lambda_{\varepsilon}^{{2/3}} \sim 1$.
