  Diffuse interface models are widely used to describe evolution of multi-phase
systems of different nature. Dispersed "inclusions", described by the phase
field distribution, are usually three dimensional objects. When describing
elastic fracture evolution, elements of the dispersed phase are effectively 2d
objects. An example of the model which governs evolution of effectively 1d
dispersed inclusions is phase field model for electric breakdown in solids.
Phase field model is defined by appropriate free energy functional, which
depends on phase field and its derivatives. In this work we show that
codimension of the dispersed "inclusion" significantly restrict the functional
dependency of system energy on the derivatives of the problem state variables.
It is shown that free energy of any phase field model suitable to describe
codimension 2 diffuse objects necessary depends on higher order derivatives of
the phase field or need an additional smoothness of the solution - it should
have first derivatives integrable with a power greater then two. To support
theoretical discussion, some numerical experiments are presented.
