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

Using molecular dynamics simulations, we show that a widely-accepted theoretical prediction for glassy-polymeric strain hardening moduli ($G_R \propto \rho_e$, where $\rho_e$ 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 $S_{\rm c}$ required to activate Kuhn-segment-scale plastic rearrangements, we develop a scaling theory predicting that $G_R = S_{\rm c}(u_0/\ell_0^3) b^3$ 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.