  We present a detailed derivation of a new hydrodynamic two-fluid model for
viscoelastic phase separation of polymer solutions. It is directly based upon
the coarse-graining of a well-defined molecular model, such that all degrees of
freedom have a clear and unambiguous molecular interpretation. The
considerations are based upon a free-energy functional, and the dynamics is
split into a conservative and a dissipative part, where the latter satisfies
the Onsager relations and the Second Law of thermodynamics. The model is
therefore fully consistent with both equilibrium and non-equilibrium
thermodynamics. The derivation proceeds in two steps: Firstly, we derive an
extended model comprising two scalar and four vector fields, such that inertial
dynamics of the macromolecules and of the relative motion of the two fluids is
taken into account. In the second step, we eliminate these inertial
contributions and, as a replacement, introduce phenomenological dissipative
terms, which can be modeled easily by taking into account the principles of
non-equilibrium thermodynamics. The final simplified model comprises the
momentum conservation equation, which includes both interfacial and elastic
stresses, a convection-diffusion equation where interfacial and elastic
contributions occur as well, and a suitably convected relaxation equation for
the end-to-end vector field. In contrast to the traditional two-scale
description that is used to derive rheological equations of motion, we here
treat the hydrodynamic and the macromolecular degrees of freedom on the same
basis. Nevertheless, the resulting model is fairly similar, though not fully
identical, to models that have been discussed previously.
