  We present a theoretical description of a mechanism for self assembly in
binary soft nanoparticle systems of the type which were studied experimentally
by Talapin et al [1]. We focus on, in particular, the conditions for formation
of dodecagonal phases, and explain why these can be stabilized by the soft
shells of the nanoparticles. We describe the different types of phase
transition that are possible in terms of an effective free energy derived from
n-body depletion potentials. A large variety of crystalline and several
quasiperiodic structures are predicted to form, depending on the composition of
the binary system and the size ratio of the particles. We show that this theory
can qualitatively explain many of the experimentally seen structures, including
striped, tetragonal, hexagonal and quasiperiodic phases. We also predict
several new square triangle tilings corresponding to different packing
fractions and clusters than the one dodecagonal phase that has been observed so
far. Our theory can be tested in principle by detailed numerical investigations
of the depletion forces in binary systems.
