We directly compute, for the first time, a confining "tube" potential acting on a flexible or semiflexible polymer chain entangled with like chains using equilibrium molecular dynamics simulations and compare these results to the conventional tube diameter inferred from the plateau in the time-dependent relaxation modulus G(t) and to an apparent tube diameter inferred from the crossover time τ e from t 1/2 to t 1/4 scaling in the time-dependent monomer diffusivity g 2 (t). We find that the "narrow" tube diameter obtained at the equilibration time τ e , where the monomer first "feels" that it is in a tube, widens with time as the tube potential softens and develops a nonquadratic tail. Our results help explain why the value of the entanglement spacing inferred by measurements at time scale τ e is only half the conventional value obtained by measurements of the plateau modulus in the vicinity of the Rouse time τ R .

The conjecture that branched polymers relax hierarchically in entangled melts is herein verified using microscopic molecular dynamics (MD) simulations of symmetric and asymmetric star polymers made possible by an efficient equilibration method. At time scales less than the short-arm relaxation time, the branch point remains anchored within a length scale of a tube diameter. Once the short arm relaxes, the branch point takes a random hop along the confining tube where the distinct “hop” can be visualized directly by simulation. At time scales much larger than the short-arm relaxation time, the branch point becomes mobile enough to allow the backbone to reptate. Coarse-grained simulations of a one-dimensional Rouse chain diffusing along a three-dimensional random walk with the short arm replaced by a large friction bead at the branch point capture the dynamics of the asymmetric star polymer at time scales larger than the short-arm relaxation time. Our simulations also suggest that branches that are only one or two entanglements long slow the motion of the branch point more than expected by a naive application of the hierarchical model because on the time scale for relaxation of such short arms the entanglement spacing is effectively smaller than at longer times, making the short arm effectively more entangled than expected.

To identify primitive paths, which are centerlines of confining tubes, two leading methods, namely total quadratic energy minimization and length minimization, are explored and compared in molecular dynamics (MD) simulations of linear pearl-necklace polymer chains. Energy minimization leads to a slightly larger averaged length but much narrower contour length distribution around the average length than does length minimization. Applications of both methods to melts of linear polymers in MD simulations confirm a quadratic entropic potential governing the primitive path length distribution. However, length minimization leads to a prefactor of around 1.5, in agreement with the classical result of Doi and Edwards, while energy minimization gives a prefactor of around 3.0.

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