  We study the global influence of curvature on the free energy landscape of
two-dimensional binary mixtures confined on closed surfaces. Starting from a
generic effective free energy, constructed on the basis of symmetry
considerations and conservation laws, we identify several model-independent
phenomena, such as a curvature-dependent line tension and local shifts in the
binodal concentrations. To shed light on the origin of the phenomenological
parameters appearing in the effective free energy, we further construct a
lattice-gas model of binary mixtures on non-trivial substrates, based on the
curved-space generalization of the two-dimensional Ising model. This allows us
to decompose the interaction between the local concentration of the mixture and
the substrate curvature into four distinct contributions, as a result of which
the phase diagram splits into critical sub-diagrams. The resulting free energy
landscape can admit, as stable equilibria, strongly inhomogeneous mixed phases,
which we refer to as antimixed states below the critical temperature. We
corroborate our semi-analytical findings with phase-field numerical simulations
on realistic curved lattices. Despite this work being primarily motivated by
recent experimental observations of multi-component lipid vesicles supported by
colloidal scaffolds, our results are applicable to any binary mixture confined
on closed surfaces of arbitrary geometry.
