Entangled-to-packed crossover in nonlinear extensional rheology of entangled polymers

Abstract

Significant challenges exist in the nonlinear extensional rheology of entangled polymers. With simulations, we show that the key to understanding this problem is to recognize the existence and importance of a strain-induced crossover from the entangled state to a packed state. This crossover, following the saturation of primitive chain stretch, takes place with massive release of entanglements via convective flow and progressive chain alignment. After the crossover, the disentangled, fully-aligned chain segments pack similarly to the random packing of rods. Meanwhile, the system enters the steady state. In this state, the tube model, built on the concept of entanglement, fails, while the stress can be quantitatively calculated by combining the intra-chain conformational contribution and a frictional contribution from the inter-chain separation along flow.

Figures (7)

• Figure 1: $Z(\varepsilon_{\rm H})$, normalized by equilibrium value $Z_{\rm equ}$, for the conditions of (a) $k_{\text{bc}}=1.5$, $N=300$ and $500$, ${Wi}_{\rm R}=10 - 40$, and (b) $N=500$, ${Wi}_{\rm R}=40$, $k_{\text{bc}}=0 - 3$. Symbols: MD results; lines: model IannirubertoMarrucci-30 predictions (with CCR parameter $\beta=0.8$). (c) PP length growth $\bar{L}_{\rm pp}(\lambda)$ normalized by equilibrium value $\bar{L}_{\rm pp}(\rm equ)$ for the sample of $N=500$ and $k_{\text{bc}}=1.5$ at ${Wi}_{\rm R}=10 - 40$. Data are vertically shifted for clarity. In (a) – (c), the positions of $\varepsilon_{\rm H1}$ or $\lambda_1$ are denoted by vertical dashed lines. (d) The critical stretch ratio $\lambda_1$ versus $\sqrt{N_{\rm e}}$.
• Figure 2: Properties of EE. (a) Illustration of SEE, MEE and ineffective entanglement in a segment of PP stretched to $\varepsilon_{\rm H1}$. (b) and (c) respectively give the distributions of $K$ and $\theta$ at equilibrium and at $\varepsilon_{\rm H1}$. (d) Evolutions of total number of entanglements ($Z$), EE ($Z_{\rm EE}$), and SEE ($Z_{\rm SEE}$) per chain during extension. (e) and (f) respectively give the density of entanglements $\rho(i, \varepsilon_{\rm H})$ along chain’s monomer coordinate $i$ as a function of $\varepsilon_{\rm H}$ for SEE and MEE. (g) Comparison between the distributions of birth time of SEE and MEE during extension. Data in (a) – (g) are measured with the sample of $N=500$ and $k_{\rm bc}=1.5$ at $Wi_{\rm R}=40$. (h) Decay of SEE at $\varepsilon_{\rm H}\geq\varepsilon_{\rm H1}$ as a function of strain under various conditions.
• Figure 3: Structural evolution of the sample of $N=500$ and $k_{\rm bc}=1.5$ at $Wi_{\rm R}=40$. (a) Evolution of $g_{\rm 2D}(r)$ during extension. (b) $I_{\rm 1s}$ as a function of $\varepsilon_{\rm H}$. Two critical strains, $\varepsilon_{\rm H1}$ and $\varepsilon_{\rm H2}$, are denoted by vertical dashed lines. (c) – (e) respectively show the snapshots of chains at $\varepsilon_{\rm H}=1.0$, 2.5, and 4.4.
• Figure 4: Tensile stress. (a) Total stress $\sigma_{\rm tot}$ and conformational stress $\sigma_{\rm conf}$ as a function of $\varepsilon_{\rm H}$ for the sample of $N=500$ and $k_{\rm bc}=1.5$ at $Wi_{\rm R}=10-40$. DEMG prediction with friction reduction is also given (dashed line). (b) $\sigma_{\rm tot}$, $\sigma_{\rm conf}$, and DEMG predictions (with and without friction reduction) at steady states as a function of $Wi_{\rm R}$ for the sample of $N=500$ and $k_{\rm bc}=1.5$. (c) Scatter plot between $\Delta\sigma=\sigma_{\rm tot}-\sigma_{\rm conf}$ and frictional stress $\sigma_{\rm fr}$ (eq. \ref{['eq:3']}) for various steady-state conditions denoted in the right side of panel. The dashed line denotes $\Delta\sigma=\sigma_{\rm fr}$.
• Figure 5: Geometric mechanism of SEE release for a sample with $N = 500$ and $k_{\mathrm{bc}} = 1.5$. (a) Illustration of SEE sliding and release along the primitive path during affine deformation of a Z-fold chain. (b) Remaining SEE fraction for $\varepsilon_{\mathrm{HR}} = 2.2$: MD results compared to analytic predictions ($\mathrm{Wi}_{\mathrm{R}} = 10\text{--}40$).