  We develop a segment-scale, force-based theory for the breakdown of the
unentangled Rouse model and subsequent emergence of isotropic mesoscopic
localization and entropic elasticity in chain polymer liquids in the absence of
ergodicity-restoring anisotropic reptation motion. The theory is formulated in
terms of a conformational N-dynamic-order-parameter Generalized Langevin
Equation approach. It is implemented using a field-theoretic Gaussian thread
model of polymer structure and closed in a universal manner at the level of the
chain dynamic second moment matrix. The physical idea is that the isotropic
Rouse model fails due to the dynamical emergence of time-persistent
intermolecular contacts determined by the combined influence of local chain
uncrossability, long range polymer connectivity and a self-consistent treatment
of chain motion and the dynamic forces that hinder it. For long chain melts,
the mesoscopic localization length (tube diameter) and emergent elasticity
predictions are in near quantitative agreement with experiment. Moreover, the
onset chain length scales with the semi-dilute crossover concentration with a
realistic numerical prefactor. Distinctive predictions are made for various
off-diagonal correlation functions that quantify the full spatial structure of
the dynamically localized polymer conformation. As the local uncrossability
constraint and/or intrachain bonding spring are softened, the tube diameter is
predicted to swell until it reaches the chain radius-of-gyration at which point
entanglement localization vanishes in a discontinuous manner. A full dynamic
phase diagram for the destruction of mesoscopic localization is constructed,
which is qualitatively consistent with simulations and the classical concept of
an entanglement degree of polymerization.
