  If particles interact according to isotropic pair potentials that favor
multiple length scales, in principle a large variety of different complex
structures can be achieved by self-assembly. We present, motivate, and discuss
a conjecture for the occurrence of non-trivial (i.e., non-triangular) orderings
based on newly-introduced enthalpy-like pair potentials, the capability of
which we demonstrate for the specific example of colloid-polymer mixtures. Upon
examining the phase behavior of two-dimensional colloid-polymer mixtures, which
can also be realized in experiments, we observe that non-trivial structures
only occur in the vicinity of selected densities where triangular ordering is
suppressed by the pair potential. Close to these densities, a large number of
different phases self-assemble that correspond to tilings containing
triangular, rhombic, square, hexagonal, and pentagonal tiles, and including
some of the Archimedean tilings. We obtain the ground-state energies by
minimizing the corresponding lattice sums with respect to particle positions in
a unit cell as well as cell geometry and verify the occurrence of selected
phases at finite temperatures by using Brownian Dynamics simulations. All
reported phases should be accessible in experiments and, in addition, our work
provides a manual on how to find the regions of non-trivial phases in parameter
space for complex pair interactions in general.
