  The influence of short-range interactions between a multi-phase,
multi-component mixture and a solid wall in confined geometries is crucial in
life sciences and engineering. In this work, we extend the Cahn-Hilliard model
with dynamic boundary conditions from a binary to a ternary mixture, employing
the Onsager principle that accounts for cross-coupling between forces and
fluxes in both bulk and surface. Moreover, we have developed a linear,
second-order, and unconditionally energy stable numerical scheme for solving
the governing equations, utilizing the Invariant Energy Quadratization (IEQ)
method. This efficient solver allows us to explore the impacts of wall-mixture
interactions and dynamic boundary conditions on phenomena like spontaneous
phase separation, coarsening processes, and the wettability of droplets on
surfaces. We observe that wall-mixture interactions influence not only surface
phenomena, such as droplet contact angles, but also patterns deep within the
bulk. Additionally, the relaxation rates control the droplet spreading on
surfaces. Furthermore, the cross-coupling relaxation rates in the bulk
significantly affect coarsening patterns. Our work establishes a comprehensive
framework for studying multi-component mixtures in confined geometries.
