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The dynamics of polymer melts at the crossover between unentangled and entangled regimes is formalized here through an extension of the Cooperative Dynamics Generalized Langevin Equation (CDGLE) (J. Chem. Phys. 1999, 110, 7574), by including the constraint to the dynamics due to entanglements through an effective intermonomer potential that confines the motion of the chains. As one polymer chain in a melt interpenetrates with a N other chains, with N the degree of chain polymerization, their dynamics is coupled through their potential of mean-force, leading to chains’ cooperative motion and center-of-mass subdiffusive dynamics. When increasing the degree of polymerization, the extended CDGLE approach describes the dynamical behavior of unentangled to weakly entangled systems undergoing cooperative dynamics. By direct comparison of the CDGLE with data of Neutron Spin Echo (NSE) experiments on polyethylene melts, we find that the cooperative dynamics in entangled systems are confined in the region delimited by entanglements. We extend the CDGLE to describe linear dynamical mechanical measurements and use it to calculate shear relaxation for the polyethylene samples investigated by NSE. The effects of cooperative dynamics, local flexibility, and entanglements in the shear relaxation are discussed. It is noteworthy that the theoretical approach describes with accuracy the crossover from unentangled to entangled-global dynamics for polyethylene melts of increasing chain length, covering the regimes of unentangled and weakly entangled (up to 12 entanglements) dynamics in one approach. 

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 into 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 coarsegrained model derived from the 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 ρ = 0.767 g/cm3. 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 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. 

In the Acknowledgment, the number of the current NSF grant support was omitted. The Acknowledgments section should include: This work was supported by the National Science Foundation (NSF) Grant No. CHE-1665466. Additionally, there were two typo mistakes in eqs 6 and 33. Those did not a!ect any of the results reported in the paper because the typo mistakes were not present in the computa- tional codes that generated the results. The "rst term in the equation of the bond potential, eq 6, should read 9/4 n_b k_B T l_ ij^ 2 / ⟨R^2⟩ Finally, the exponential function in the bond length distribution, eq 33, should read exp(−9n_b r^2/(4⟨R^2⟩)