  While the excess chemical potential is the key quantity in determining phase
diagrams, its direct computation for high-density liquids of long polymer
chains has posed a significant challenge. Computationally, the excess chemical
potential is calculated using the Widom insertion method, which involves
monitoring the change in internal energy as one incrementally introduces
individual molecules in the liquid. However, when dealing with dense polymer
liquids, inserting long chains requires generating trial configurations with a
bias that favors those at low energy on a unit-by-unit basis: a procedure that
becomes more challenging as the number of units increases. Thus, calculating
the excess chemical potential of dense polymer liquids using this method
becomes computationally intractable as the chain length exceeds $N > 30$. Here,
we adopt a coarse-grained model derived from integral equation theory, for
which inserting long polymer chains becomes feasible. The Integral Equation
theory of Coarse-Graining (IECG) represents a polymer as a sphere or a
collection of blobs interacting through a soft potential. We employ the IECG
approach to compute the excess chemical potential using Widom's method for
polymer chains of increasing lengths, extending up to $N=720$ monomers, and at
densities reaching up to $\rho= 0.767$ g/cm$^{3}$. From a fundamental
perspective, we demonstrate that the excess chemical potentials remain nearly
constant across various levels of coarse-graining, offering valuable insights
into the consistency of this type of procedure. Ultimately, we argue that
current Monte Carlo (MC) algorithms, originally designed for atomistic
simulations, such as Configurational Bias Monte Carlo (CBMC) methods, can
significantly benefit from the integration of the IECG approach, thereby
enhancing their performance in the study of phase diagrams of polymer liquids.
