A Multichain Slip-Spring Model with Fluctuating Number of Entanglements Based On Single-Chain Slip-Spring Model

We present a multichain slip-spring model with fluctuating number of entanglements for dynamics of dense polymer melts, which is based on the single chain slip-spring model by Likhtman. By introducing the exclude volume interactions, a multichain version of slip-spring model can be presented with little modification. With this extension, inhomogeneous polymer system can be described properly. The renewal rules for the slip-spring satisfy the detailed balance condition and the number of slip-springs is no longer constant but controlled through a chemical potential. This completion can predict the Possion distribution of Z under equilibrium states and the decrease in the number of entanglements when applied a steady shear flow. The results agree with theoretical predictions and experimental phenomenon. The model bridges the single chain slipspring model and multichain slip-spring model. Introduction Understanding of the dynamics of entangled polymer is very important for many industrial applications. However, for the wide range of spatial and temporal scales involved in the characterization of entangled polymer, theoretical modelling of such material is a difficult challenge. A most successful and advanced models was the tube model of Doi and Edwards[1-4], in which a probe Rouse chain is restricted in a tube formed by the topological constraints imposed by surrounding chains. The chain diffusion along the backbone more easily than lateral to the backbone, which is known as reptation. Based on this reptation idea, many important additional physical ideas, such as constraint release(CR)[5], contour-length fluctuations(CLF)[6], convective constraint release(CCR)[7], have been incorporated into tube model. These improvements have enriched the tube theory to account for many additional relaxation mechanisms. Tube model reveals the dominant relaxation mechanism and could describe numerous characteristic phenomena of entangled polymers successfully. However, it still failed in many respects, such as incorrect prediction for steady shear flow and it's difficult to be extended to various chain architectures. Tube model met these troubles mainly because it employ a single-chain motion to express multichain dynamics, thus introducing excessive assumptions and approximations. For the shortness of tube model, many researchers have transferred to slip-link model, which incorporate constraint release and fluctuations in the tube length in a more natural way. There are several excellent implementations of slip link models in the literature, such as the primitive chain network (PCN) model of Masubuchi and co-workers[8, 9], the TIEPOS approach of RamírezHernández and co-workers[10, 11], the DSM of Schieber and co-workers[12-14]. Likhtman[15] developed a dynamic one-chain slip-spring model of entangled polymers suitable for Brownian dynamics simulations. This model can describe the results from NSE, linear rheology and diffusion experiments properly. Many models extend Likhtman's model for its flexibility, extensibility and implementation simplicity. To simulate large scale or long time macroscopic phenomena, Uneyama slightly modified Likhtman's model to be suitable for simulations on a GPU[16]. The total number of slip-springs could fluctuates and controlled by a pseudo-chemical potential, which is originally introduced by Schieber [17]. However, the model still is single-chain model that it's hard to use it to 515 Advances in Engineering Research (AER), volume 105 3rd Annual International Conference on Mechanics and Mechanical Engineering (MME 2016) Copyright © 2017, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). simulate filled entangled system. Biondo et.al investigated the nonlinear rheological properties of Likhtman's model and extend the model to study inhomogeneous system[18]. The extension is flexible and simply implemented by introducing excluded volume interactions in a mean field. However, the total number of the slip-springs in their model is constant, which seems not true during shear flow. In this work, we will use Biondo's method to achieve a multichain model from Likhtman's model. Meanwhile the fluctuation in the number of slip-springs should be accomplished with Uneyama's method. It is important to highlight that Ramírez-Hernández et al. developed a version slip-spring model refer to as "Theoretically informed entangled polymer simulations" (TIEPOS), which can also describe the dynamics of inhomogeneous systems with fluctuating number of entanglements well[19]. Our work stress the extension from single-chain slip-spring model directly, thus the anchors of each chains are preserved in comparison to their work. With the same entanglement picture we can compare our results with single-chain slip-spring model in the most straightforward way and form a multi-scale computational framework for polymer rheology. Method Original Single-Chain Slip-Spring Model The original single-chain slip-spring model[15] describe the dynamics of a Rouse chain consisting of N+1 beads at position rj connected by N springs. The Rouse chains are constrained by Z=N/Ne evenly distributed slip-springs representing the topological constraints by surrounding chains, where Ne means the average number of Kuhn segments between slip-links. One end of each slipspring defined as a fixed anchoring point at position aj and the other end attached to the Rouse chain at positon sj with a slip ring. The slip-springs can continuously slid from one monomer to another, which imitate the reptation-like motion of the probe chain. Thus the ring's position is      trunc( ) ( ) 1 ( ) r trunc j j j x j j x x x x      j trunc trunc s r r , where xj is the curvilinear abscissa of j-th slip-spring and trunc(x) is the closest integer to x less than or equal to x. The total potential energy of the chain is : 1 2 2 1 2 2 0 1 S 3 3 ( ) ( ) 2 2 N Z B B i i j j i j k T k T U b N b           r r a s , (1) where 2 S 3 2 B k T N b is the slip spring stiffness and Ns is a number of monomers in slip-spring, b is the Kuhn step. With this total potential, the motion of the chain and the slip-springs obeys the following stochastic equation: