Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Robert S. Hoy

Abstract

Using molecular dynamics simulations, we show that a widely-accepted theoretical prediction for glassy-polymeric strain hardening moduli ($G_R \propto ρ_e$, where $ρ_e$ 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.

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 (, where is the entanglement density) fails badly for semiflexible polymers with . By postulating that the length, energy and strain scales controlling are the Kuhn length and statistical segment length (where is the backbone bond length), the intermonomer binding energy , and the incremental elastic strain required to activate Kuhn-segment-scale plastic rearrangements, we develop a scaling theory predicting that in the athermal limit. This prediction agrees quantitatively (semi-quantitatively) with simulated values for both flexible and semiflexible polymer glasses subjected to athermal uniaxial-stress extension (constant-volume simple shear), over a range of that is wider than that spanned by real systems.

Paper Structure

This paper contains 8 equations, 3 figures.

Figures (3)

  • Figure 1: Stress-strain curves for uniaxial-stress extension of Kremer-Grest glasses with $0.5 \leq \kappa \leq 5.5$. Colors vary from purple to red with increasing $\kappa$. The inset shows the $G_R(\kappa, 0)$ obtained by fitting these curves to Eq. \ref{['eq:linGRfit']} and a fit of the results for $\kappa \leq 2.0$ to the expected linear scaling with $\rho_e$vanMelick03.
  • Figure 2: Data for the scaled strain hardening moduli $G_R(\ell_K)/(u_0/\ell_0^3)$ (red points) and a fit to $G_R = S_{\rm C}(u_0/ell_0)^3(\ell_K/\ell_0)^{3/2}$ with $S_{\rm C} = 0.08364$ (gray line). The upper and lower insets respectively show the fractional errors associated with this fit and a plot of $\ell_K/\ell_0$ vs. $\rho_e$.
  • Figure 3: Constant-volume simple-shear response for the systems analyzed in Figs. \ref{['fig:1']}-\ref{['fig:2']}. Panel (a) shows the stress-strain curves; the inset schematically illustrates the principal stretch directions. Panel (b) shows the $G_R(\kappa, 0)$ obtained by fitting these curves to Eq. \ref{['eq:GRshear']} along with a fit to $G_R = G_0 (\ell_K/\ell_0)^{3/2}$ scaling. The upper-left inset shows the fractional errors associated with this fit, while the lower-right inset shows a fit of the results for $\kappa \leq 2.0$ to the expected $G_R \sim \rho_e$ scaling vanMelick03.