By combining molecular dynamics simulations and topological analyses with scaling arguments, we obtain analytic expressions that quantitatively predict the entanglement length $N_e$, the plateau modulus $G$, and the tube diameter $a$ in melts that span the entire range of chain stiffnesses for which systems remain isotropic. Our expressions resolve conflicts between previous scaling predictions for the loosely entangled [Lin-Noolandi: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^3$], semiflexible [Edwards/de Gennes: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^2$], and tightly-entangled [Morse: $G\ell_K^3/k_\textrm{B}T \sim (\ell_K/p)^{1+ε}$] regimes, where $\ell_K$ and $p$ are respectively the Kuhn and packing lengths. We also find that maximal entanglement (minimal $N_e$) coincides with the onset of local nematic order.
Plateau moduli of Kremer-Grest models for commodity polymer melts
University of Southern Denmark（南丹麦大学）；ENSL, CNRS, Laboratoire de Physique (UMR 5672) and Centre Blaise Pascal de l’École Normale Supérieure de Lyon（里昂高等师范学院、法国国家科学研究中心物理实验室（UMR 5672）及里昂高等师范学院布莱兹·帕斯卡中心）
Relations between static and dynamic viscoelastic responses in gels can be very elucidating and may provide useful tools to study the behavior of bio-materials such as protein hydrogels. An important example comes from the viscoelasticity of semisolid gel-like materials, which is characterized by two regimes: a low-frequency regime where the storage modulus $G^{\prime}(ω)$ displays a constant value $G_{\text{eq}}$, and a high-frequency power-law stiffening regime, where $G^{\prime}(ω) \sim ω^{n}$. Recently, by considering Monte Carlo simulations to study the formation of peptides networks, we …
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Stress Relaxation in Monodisperse Entangled Polymer Melts: Correlation Between Viscoelastic Response and Single-Chain Relaxation via Molecular Dynamics Simulations
Johannes Gutenberg-Universität Mainz（约翰内斯·古腾堡美因茨大学）
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We have investigated the reduction of the glass transition temperature, $T_g$, for thin supported films of particularly small molecular weight ($M_W$ = 2 kg/mol) polystyrene, and found good agreement with earlier studies on larger molecules. By combining the eigenmodes of the films with mode coupling theory, we arrive at a simple and predictive model which is in quantitative agreement with the data. Its only fitting parameter is the shear modulus governing the relevant fluctuations. The weak dependence of the latter upon the molecular weight of the polymer is in accordance with the general obs…
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We present atomistic molecular dynamics simulations of two Polyethylene systems where all entanglements are trapped: a perfect network, and a melt with grafted chain ends. We examine microscopically at what level topological constraints can be considered as a collective entanglement effect, as in tube model theories, or as certain pairwise uncrossability interactions, as in slip-link models. A pairwise parameter, which varies between these limiting cases, shows that, for the systems studied, the character of the entanglement environment is more pairwise than collective. We employ a novel metho…
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Dynamical properties of a long polymer ring in a melt of unknotted and unconcatenated rings are calculated. We re-examine and generalize the well known model of a ring confined to a lattice of topological obstacles in the light of the recently developed Flory theory of unentangled rings which maps every ring on an annealed branched polymer and establishes that the backbone associated with each ring follows self-avoiding rather than Gaussian random walk statistics. We find the scaling of ring relaxation time and diffusion coefficient with ring length, as well as time dependence of stress relaxa…
The dynamic-mechanical response of flexible polymer networks is studied in the framework of tube model, in the limit of small affine deformations, using the approach based on Rayleighian dissipation function. The dynamic complex modulus G* is calculated from the analysis of a network strand relaxation to the new equilibrium conformation around the distorted primitive path. Chain equilibration is achieved via a sliding motion of polymer segments along the tube, eliminating the inhomogeneity of the polymer density caused by the deformation. The characteristic relaxation time of this motion <tau>…
Viscoelastic properties of polymer melts are particularly challenging to compute due to the intrinsic stress fluctuations in molecular dynamics (MD). We compared equilibrium and non-equilibrium MD approaches for extracting the storage (G') and loss moduli (G") over a wide frequency range from a bead-spring chain model, in both unentangled and entangled regimes. We found that, with properly chosen data processing and noise reduction procedures, different methods render quantitatively equivalent results. In equilibrium MD (EMD), applying the Green-Kubo relation with a multi-tau correlator method…
The scaling of the shear modulus near the gelation/vulcanization transition is explored heuristically and analytically. It is found that in a dense melt the effective chains of the infinite cluster have sizes that scale sub-linearly with their contour length. Consequently, each contributes k_B T to the rigidity, which leads to a shear modulus exponent dν. In contrast, in phantom elastic networks the scaling is linear in the contour length, yielding an exponent identical to that of the random resistor network conductivity, as predicted by de Gennes'. For non-dense systems, the exponent should c…
We present experimental evidence that the effective medium approximation (EMA), developed by D.C. Morse [Phys. Rev. E {\bf 63}, 031502, (2001)], provides the correct scaling law of the macroscopic plateau modulus $G^{0}\proptoρ^{4/3}L^{-1/3}_{p}$ (where $ρ$ is the contour length per unit volume and $L_{p}$ is the persistence length) of semi-flexible polymer solutions, in the highly entangled concentration regime. Competing theories, including a self-consistent binary collision approximation (BCA), have instead predicted $G^{0}\proptoρ^{7/5}L^{-1/5}_{p}$. We have tested both the EMA and BCA sca…
We present an analytic scheme to connect the fragility and viscoelasticity of metallic glasses to the effective ion-ion interaction in the metal. This is achieved by an approximation of the short-range repulsive part of the interaction, combined with nonaffine lattice dynamics to obtain analytical expressions for the shear modulus, viscosity, and fragility in terms of the ion-ion interaction. By fitting the theoretical model to experimental data, we are able to link the steepness of the interionic repulsion to the Thomas-Fermi screened Coulomb repulsion and to the Born-Mayer valence-electron o…
A long elastic cylinder, radius $a$ and shear-modulus $μ$, becomes unstable given sufficient surface tension $γ$. We show this instability can be simply understood by considering the energy, $E(λ)$, of such a cylinder subject to a homogenous longitudinal stretch $λ$. Although $E(λ)$ has a unique minimum, if surface tension is sufficient ($Γ\equivγ/(aμ)>\sqrt{32}$) it looses convexity in a finite region. We use a Maxwell construction to show that, if stretched into this region, the cylinder will phase separate into two segments with different stretches $λ_1$ and $λ_2$. Our model thus explains w…
Brownian dynamics simulations are used to study highly entangled solutions of semiflexible polymers. Bending fluctuations of semiflexible rods are signficantly affected by entanglement only above a concentration $c^{**}$, where $c^{**}\sim 10^{3}L^{-3}$ for chains of similar length $L$ and persistence length. For $c > c^{**}$, the tube radius $R_{e}$ approaches a dependence $R_{e} \propto c^{-3/5}$, and the linear viscoelastic response develops an elastic contribution that is absent for $c < c^{**}$. Experiments on isotropic solutions of $F$-actin span concentrations near $c^{**}$ for which th…
