Comprehensive scaling theory for entanglement in melts and solutions of flexible and stiff polymer chains

The entanglement length Ne is a key parameter for all entangled polymer fluids, for which no comprehensive scaling theory yet exists. We have pieces of a theory; the Lin-Noolandi (LN) argument predicts Ne scaling for flexible chains that agrees with data on melts. There are arguments for how Ne should depend on polymer volume fraction, but which are not obviously consistent with LN. Morse scaling describes entanglement for stiff chains, consistent with data. Everaers proposed an ansatz that Ne depends only on “arclength concentration”, as if chains were uncrossable threads. This ansatz is consistent with simulation of bead-spring chains but not with LN, it has no role for packing length, the central parameter in LN scaling. We propose a comprehensive scaling theory which includes LN in one limit, thread ansatz in another, and reduces to Morse scaling for stiff chains. Our new ingredient is the observation that the typical distance of closest approach between two chains is governed by packing length or chain diameter, whichever is larger. If a chain is sufficiently flexible and bulky, the packing length is relevant; but for stiffened bead-spring chains without sidegroups, the packing length is likely smaller than the chain diameter, so thread scaling applies.