  We present a new multiphase-field theory for describing pattern formation in
multi-domain and/or multi-component systems. The construction of the free
energy functional and the dynamic equations is based on criteria that ensure
mathematical and physical consistency. We first analyze previous
multiphase-field theories, and identify their advantageous and disadvantageous
features. On the basis of this analysis, we introduce a new way of constructing
the free energy surface, and derive a generalized multiphase description for
arbitrary number of phases (or domains). The presented approach retains the
variational formalism; reduces (or extends) naturally to lower (or higher)
number of fields on the level of both the free energy functional and the
dynamic equations; enables the use of arbitrary pairwise equilibrium
interfacial properties; penalizes multiple junctions increasingly with the
number of phases; ensures non-negative entropy production, and the convergence
of the dynamic solutions to the equilibrium solutions; and avoids the
appearance of spurious phases on binary interfaces. The new approach is tested
for multi-component phase separation and grain coarsening.
