Entanglement Theories: Packing vs. Percolation

There are two emergent theories of polymer entanglements, the Packing Model (Fetters, Lohse, Graessley, Milner, Whitten, $\sim $'98) and the Percolation Model (Wool $\sim $'93). The Packing model suggests that the entanglement molecular weight M$_{e}$ is determined by M$_{e}$ = K p$^{3}$, where the packing length parameter p = V/R$^{2}$ in which V is the volume of the chain (V=M/$\rho $Na), R is the end-to end vector of the chain, and K$\approx $357 $\rho $Na, is an empirical constant. The Percolation model states that an entanglement network develops when the number of chains per unit area $\Sigma $, intersecting any load bearing plane, is equal to 3 times the number of chain segments (1/a cross-section), such that when 3a$\Sigma $ =1 at the percolation threshold, M$_{e}\approx $31 M$_{j}$C$_{\infty }$, in which M$_{j}$ is the step molecular weight and C$_{\infty }$ is the characteristic ratio. There are no fitting parameters in the Percolation model. The Packing model predicts that M$_{e}$ decreases rapidly with chain stiffness, as M$_{e}\sim $1/C$_{\infty }^{3}$, while the Percolation model predicts that M$_{e}$ increases with C$_{\infty }$, as M$_{e}\sim $C$_{\infty }$. The Percolation model was found to be the correct model based on computer simulations (M. Bulacu et al) and a re-analysis of the Packing model experimental data. The Packing model can be derived from the Percolation model, but not visa versa, and reveals a surprising accidental relation between C$_{\infty }$ and M$_{j}$ in the front factor K. This result significantly impacts the interpretation of the dynamics of rheology and fracture of entangled polymers.