  Structural and thermodynamic consistency of coarse-graining models across
multiple length scales is essential for the predictive role of multi-scale
modeling and molecular dynamic simulations that use mesoscale descriptions. Our
approach is a coarse-grained model based on integral equation theory, which can
represent polymer chains at variable levels of chemical details. The model is
analytical and depends on molecular and thermodynamic parameters of the system
under study, as well as on the direct correlation function in the k --> 0
limit, c0. A numerical solution to the PRISM integral equations is used to
determine c0, by adjusting the value of the effective hard sphere diameter, d,
to agree with the predicted equation of state. This single quantity
parameterizes the coarse-grained potential, which is used to perform mesoscale
simulations that are directly compared with atomistic-level simulations of the
same system. We test our coarse-graining formalism by comparing structural
correlations, isothermal compressibility, equation of state, Helmholtz and
Gibbs free energies, and potential energy and entropy using both united atom
and coarse-grained descriptions. We find quantitative agreement between the
analytical formalism for the thermodynamic properties, and the results of
Molecular Dynamics simulations, independent of the chosen level of
representation. In the mesoscale description, the potential energy of the
soft-particle interaction becomes a free energy in the coarse-grained
coordinates which preserves the excess free energy from an ideal gas across all
levels of description. The structural consistency between the united-atom and
mesoscale descriptions means the relative entropy between descriptions has been
minimized without any variational optimization parameters. The approach is
general and applicable to any polymeric system in different thermodynamic
conditions.
