Tunable Cooperative Motion, Rigidity, and Glassy Dynamics in Knotted Ring Polymer Melts

Abstract

We present a molecular dynamics study of the influence of knot complexity and molecular mass on glass formation upon cooling in knotted ring polymer melts. We find that cooperative motion, rigidity, and glassy dynamics can be tuned over a wide range by knots. By leveraging these knotting constraints, we assess the validity of prevalent models of glass formation, including the string model based on cooperative particle motion, the localization model emphasizing fluctuations in local particle mobility, and the shoving model derived from emergent elastic properties in relation to material stiffness. In line with our previous findings on polymeric and other glass-forming liquids, we demonstrate that all these models of glass formation provide a quantitative description of segmental relaxation as a function of knot complexity, molecular mass, and temperature, despite their apparently distinct conceptual foundations. Our study thus provides additional evidence for an underlying unity among various theoretical frameworks and for the presence of quantitative relations between the characteristic properties emphasized by these models. Furthermore, we discuss dynamic and elastic heterogeneities in relation to fragility and stiffness variations of knotted ring polymer melts, with a focus on how these trends relate to other glass-forming liquids where fragility is tuned over a large range.

Figures (15)

• Figure 1: Example of a knotted ring polymer melt with the minimum crossing number of $m_c = 7$ and the molecular mass of $M = 32$. The left part illustrates the initial configuration of a knotted ring polymer. The right part displays a melt composed of $250$ chains after equilibration, where one of the chains is highlighted and all the others are shown as translucent thin cylinders.
• Figure 2: Structural relaxation time $\tau_{\alpha}$ in variation with temperature $T$, knot complexity $m_c$, and molecular mass $M$. Panels (a) and (b) show $\log (\tau_{\alpha} / \tau)$ versus $\varepsilon / k_{\mathrm{B}} T$ for a range of $m_c$ at $M = 32$ and for a range of $M$ at $m_c = 7$, respectively. Panels (c) and (d) show the variations with $m_c$ and $M$ of $\log (\tau_{\alpha} / \tau)$ over a range of fixed $M$ and $m_c$ at $T = 1.0 \varepsilon / k_{\mathrm{B}}$, respectively. The results for linear polymer melts are included as a reference in panel (d).
• Figure 3: Average string length $L$ in variation with temperature $T$, knot complexity $m_c$, and molecular mass $M$. Panels (a) and (b) show $L$ versus $\varepsilon / k_{\mathrm{B}} T$ for a range of $m_c$ at $M = 32$ and for a range of $M$ at $m_c = 7$, respectively. Panels (c) and (d) show the variations with $m_c$ and $M$ of $L$ over a range of fixed $M$ and $m_c$ at $T = 1.0 \varepsilon / k_{\mathrm{B}}$, respectively. The results for linear polymer melts are included as a reference in panel (d).
• Figure 4: String model description of glass formation. Panels (a) and (b) show $\ln (\tau_{\alpha} / \tau_o)$ versus $\Delta G_o z(T) / k_{\mathrm{B}} T$ for a range of $m_c$ at $M = 32$ and for a range of $M$ at $m_c = 7$, respectively. Dashed lines indicate $\ln (\tau_{\alpha} / \tau_o) = \Delta G_o z(T) / k_{\mathrm{B}} T$, where $\tau_o$, $L_A$, and $\Delta G_o$ are explained in the text. Panels (c--f) show the variations with $m_c$ and $M$ of the activation enthalpy $\Delta H_o$ and entropy $\Delta S_o$ over a range of fixed $M$ and $m_c$, respectively. The results for linear polymer melts are included as a reference in panels (d) and (f).
• Figure 5: Debye-Waller parameter $\langle u^2 \rangle$ in variation with temperature $T$, knot complexity $m_c$, and molecular mass $M$. Panels (a) and (b) show $\langle u^2 \rangle$ versus $k_{\mathrm{B}} T / \varepsilon$ for a range of $m_c$ at $M = 32$ and for a range of $M$ at $m_c = 7$, respectively. Panels (c) and (d) show the variations with $m_c$ and $M$ of $\langle u^2 \rangle$ over a range of fixed $M$ and $m_c$ at $T = 1.0 \varepsilon / k_{\mathrm{B}}$, respectively. The results for linear polymer melts are included as a reference in panel (d).