Determine tube diameter by measuring entropy tensile force
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ORIGINAL PAPER
Determine tube diameter by measuring entropy tensile force Xu‑Chen Yu1 · Ji‑Xuan Hou1 Received: 30 July 2020 / Accepted: 22 September 2020 © The Polymer Society, Taipei 2020
Abstract The macroscopic viscoelastic properties of entangled polymer liquids can be described by the tube theory, and tube diameter is one of the most important quantities in the tube theory. Based on the tube theory, any linear polymer chain immersed in an entangled polymeric liquid is stretched by an entropic force applied on its both chain ends so that the tube surrounding the chain is kept a certain length, and the magnitude of the entropic force is inversely proportional to the tube diameter. We propose a new method to determine tube diameter by measuring the entropic force. This new method shows better accuracy for the grid model compared with other methods. Moreover, this method can be not only used in computer simulations but also achieved in experiments. Keywords Entangled polymeric liquid · Tube theory · Entanglement length
Introduction Dynamics of entangled polymers in mesoscopic and microscopic scale is one of the greatest challenges of modern polymer science. Essentially, entanglements are transient topological constraints arising from the restriction that the backbones of fluctuating polymer molecules cannot cut through each other [1]. During the past several decades, the proposal and development of tube theory based on the principle of mean filed theory make the study of entangled polymer system possible [2, 3]. Tube theory provides systematic methods to explain and describe the rheological properties of a polymer system such as platform modulus and storage modulus. During the early age when the tube theory was first proposed, the tube diameter a was just treated as a fitting parameter to fit the experimental curve [4, 5]. With the development of tube theory, the calculation of tube diameter a becomes one of the most important problems in tube theory since constraints dominate the long-time dynamics of high molecular weight polymers, and entanglements significantly slow down the relaxation [6, 7]. In the purpose of determining precise value of tube diameter, a topological approach called primitive path analysis (PPA) * Ji‑Xuan Hou [email protected] 1
is proposed and is widely used nowadays [8, 9]. PPA is able to provide the statistical information of the primitive path based on tube theory [8–23]. An alternative dynamic method by using the monomer mean-square displacement (MSD) was firstly proposed by Likhtman and McLeish [6]. This method has been used to calculate tube diameter (and also entanglement relaxation time, 𝜏e ) from the MSD results obtained from the molecular dynamics (MD) simulations [24, 25]. However, these two methods give comparable but different values of the tube diameter for the same entangled system [26–28]. Moreover, in the entangled short chain-branching system, the tube diameters obtained using these different methods are not only quantitatively different, but also show different dep
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