  We extend our previously developed coarse-graining strategy for linear
polymers with a tunable number n of effective atoms (blobs) per chain [D'Adamo
et al., J. Chem. Phys. 137, 4901 (2012)] to polymer systems in thermal
crossover between the good-solvent and the Theta regimes. We consider the
thermal crossover in the region in which tricritical effects can be neglected,
i.e. not too close to the Theta point, for a wide range of chain volume
fractions Phi=c/c* (c* is the overlap concentration), up to Phi=30. Scaling
crossover functions for global properties of the solution are obtained by
Monte-Carlo simulations of the Domb-Joyce model. They provide the input data to
develop a minimal coarse-grained model with four blobs per chain. As in the
good-solvent case, the coarse-grained model potentials are derived at zero
density, thus avoiding the inconsistencies related to the use of
state-dependent potentials. We find that the coarse-grained model reproduces
the properties of the underlying system up to some reduced density which
increases when lowering the temperature towards the Theta state. Close to the
lower-temperature crossover boundary, the tetramer model is accurate at least
up to Phi<10, while near the good-solvent regime reasonably accurate results
are obtained up to Phi<2. The density region in which the coarse-grained model
is predictive can be enlarged by developing coarse-grained models with more
blobs per chain. We extend the strategy used in the good-solvent case to the
crossover regime. This requires a proper treatment of the length rescalings as
before, but also a proper temperature redefinition as the number of blobs is
increased. The case n=10 is investigated. Comparison with full-monomer results
shows that the density region in which accurate predictions can be obtained is
significantly wider than that corresponding to the n=4 case.
