Glassy polymers' strain-hardening moduli scale with their statistical-segment volumes

Abstract

Using molecular dynamics simulations, we show that a widely-accepted theoretical prediction for glassy-polymeric strain hardening moduli ($G_R\propto \ddot{I}{}_e$, where $\ddot{I}{}_e$ is the entanglement density) fails badly for semiflexible polymers with $N_e\lesssim 4C_{\infty }$. By postulating that the length, energy and strain scales controlling $G_R$ are the Kuhn length ${\ell }_K$ and statistical segment length $b=\sqrt{{\ell }_0{\ell }_K}$ (where ${\ell }_0$ is the backbone bond length), the intermonomer binding energy $u_0$, and the incremental elastic strain $S_{\mathrm{c}}$ required to activate Kuhn-segment-scale plastic rearrangements, we develop a scaling theory predicting that $G_R=S_{\mathrm{c}}(u_0/{\ell }_0^3)b^3$ in the athermal limit. This prediction agrees quantitatively (semi-quantitatively) with simulated $G_R$ values for both flexible and semiflexible polymer glasses subjected to athermal uniaxial-stress extension (constant-volume simple shear), over a range of ${\ell }_K/{\ell }_0$ that is wider than that spanned by real systems.