Adhesion at Entangled Polymer Interfaces: A Unified Approach..

A unified theory of fracture of polymer interfaces was developed which was based on the Rigidity Percolation model of fracture [R.P. Wool, J.Polym.Sci. Part A: Polym Phys., 43,168(2005)]. The polymer fractured critically when the normalized entanglement density p, approached the percolation threshold p$_{c}$. The fracture energy was found to be G$_{1c} \quad \sim $ [p-p$_{c}$]. When applied to interfaces of width X, containing an areal density $\Sigma $ of chains, each contributing L chain entanglements, the percolation term p $\sim \quad \Sigma $L/X and the percolation threshold was related to $\Sigma _{c}$, L$_{c}$, or X$_{c}$. For welding of A/A symmetric interfaces, p = $\Sigma $L/X, and p$_{c} \quad \approx $ L$_{c}$/M $\approx $ 0, such that when $\Sigma $/X $\sim $1/M for randomly distributed chain ends, p$\sim $L $\sim $ (t/M)$^{1/2}$, G/G* = (t/$\tau $*)$^{1/2}$, where the weld time $\tau $* $\sim $ M. When the chain ends are segregated to the surface, $\Sigma $ is constant with time and G/G* = [t/$\tau $*]$^{1/4}$. For sub-T$_{g}$ welding, there exists a surface mobile layer (due to the critical Lindemann Atom fraction) of depth X $\sim $ 1/$\Delta $T$^{\nu }$ such that G $\sim $ $\Delta $T$^{-2\nu }$, where the critical exponent v = 0.8. For incompatible A/B interfaces of Helfand width d, normalized width w = d/R$_{ge}$, and entanglement density N$_{ent} \quad \sim $ d/L$_{e}$, p $\sim $ d such that, G$_{1c} \quad \sim $ [d-d$_{c}$], G$_{1c} \quad \sim $ [w-1], and G $\sim $ [N$_{ent}$-N$_{c}$]. For incompatible A/B interfaces reinforced by an areal density $\Sigma $ of compatibilizer chains, L and X are constant, p $\sim $ $\Sigma $, p$_{c} \quad \sim \Sigma _{c}$, such that G$_{1c} \quad \sim $ [$\Sigma -\Sigma _{c}$], which is in excellent agreement with experimental data.