Title:  "Finding the Tube Without Touching the Chains"

Abstract:  The tube is the key concept in modern theories of entangled polymer dynamics.  The tube has been ``destructively'' visualized in simulations, using chain-shrinking methods to reduce the primitive path to a sequence of straight segments between entanglement points.  Instead, we use isoconfigurational averaging to find the tube without touching the chains.  In this way, we can measure properties such as tube diameter, entanglement molecular weight, confining potential, and persistence length.  By applying compression and extension to a melt of long entangled rings (as a proxy for cross-linked rubbers or long entangled melts), we observe that the primitive paths deform almost affinely for moderate strains.  We have also studied the tubes of entangled chains in strong flows by topologically equilibrating a melt of linear chains under constant tension.  We find the tube diameter decreases with increasing tension, which we can account for by extending the Lin-Noolandi ansatz to oriented and stretched chains.  Ultimately, the tube must be of topological origin, resulting from uncross ability of polymer chains.  We propose two definitions of the entanglement length (Ne) in terms of topological properties of topologically equilibrated melts of rings:  1) the probability of a ring of length Ne in a melt being unknotted is a constant; 2) the topological entropy is 3/2kB per Ne for long rings.  To test these ideas, we simulated rings with chain-crossing moves to equilibrate the topological states.  We identified and counted different knotted states by computing the Jones polynomial.  Our topological estimates of Ne are consistent with previous values based on heuristic chain-shrinking methods.