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# Nonaffine chain and primitive path deformation in crosslinked polymers

Jacob D. Davidson N. C. Goulbourne Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109
July 18, 2018

Chains in a polymer network deform nonaffinely at small length scales due to the ability for extensive microscopic rearrangement. Classically, the conformations of an individual chain can be described solely by an end-to-end length. This picture neglects interchain interactions and therefore does not represent the behavior of a real polymer network. The primitive path concept provides the additional detail to represent interchain entanglements, and techniques have recently been developed to identify the network of primitive paths in a polymer simulation. We use coarse-grained molecular dynamics (MD) to track both chain end-to-end and primitive path deformation in crosslinked polymer networks. The range of simulated materials includes short chain unentangled networks to long, entangled chain networks. Both chain end-to-end and primitive path length are found to be linear functions of the applied deformation, and a simple relationship describes the behavior of a network in response to large stretch uniaxial, pure shear, and equi-biaxial deformations. As expected, end-to-end chain length deformation is nonaffine for short chain networks, and becomes closer to affine for networks of long, entangled chains. Primitive path deformation is found to always be nonaffine, even for long, entangled chains. A visualization of time-dependent chain conformations and the restraining ”tube” in deformed networks shows that end-to-end length is only an accurate measure of microscopic deformation for networks of short, unentangled chains. In contrast, the primitive path is an accurate representation of chain deformation in both short and long chain networks. This initial investigation demonstrates the viability of using primitive path analysis to quantify micro-macro deformations in crosslinked polymers.

###### keywords:
Primitive path; elasticity; molecular dynamics; crosslinked polymer
journal: Journal of the Mechanics and Physics of Solids
journal: Journal of the Mechanics and Physics of Solids

## Introduction

Chains in a polymer network deform nonaffinely at small length scales due to the ability for extensive microscopic rearrangement, and this is a basic element of any statistical mechanics treatment of soft elastic materials. Many physics-based models of elasticity exist in the literature is describe this behavior; these include the phantom network (James et al., 1943), nonaffine tube (Rubinstein et al., 1997; Rubinstein et al., 2002), extended tube (Kaliske et al., 1999), double-tube (Mergell et al., 2001), micro-sphere \citep{MieheGA?ktepeEtAl-micro-macroapproachto-JotMaPoS-2004}, and maximal advance path constraint (Tkachuk et al., 2012) models, the field-theory approach of Goldbart (2009) and Mao et al. (2009), and the nonaffine network model in our previous work (Davidson et al., 2013; Davidson, 2014). The phantom network model (James et al., 1943) was the first (and still widely used) model of nonaffine deformation of chain ends. The physics behind this model is that chain end-to-end vectors deform nonaffinely because the number of chain ends connected at each crosslink is small, thus allowing for re-orientation of chains at the molecular level. Other models have also considered nonaffine deformation of chain ends \citep{KaliskeHeinrich-extendedtube-modelrubber-RCAT-1999,MieheGA?ktepeEtAl-micro-macroapproachto-JotMaPoS-2004,Tkachuk2012,Davidson2013,Davidson2014} in addition to nonaffine changes in the magnitude of monomer fluctuations \citep{RubinsteinPanyukov-NonaffineDeformationand-M-1997,RubinsteinPanyukov-ElasticityofPolymer-M-2002,KaliskeHeinrich-extendedtube-modelrubber-RCAT-1999,MergellEveraers-TubeModelsRubber-Elastic-M-2001,MieheGA?ktepeEtAl-micro-macroapproachto-JotMaPoS-2004,Davidson2013,Davidson2014}, or a general nonaffine deformation field (Goldbart, 2009; Mao et al., 2009). Although nonaffine chain deformation has been suggested by small angle neutron scattering (Straube et al., 1994) and nuclear magnetic resonance spectroscopy (Gronski et al., 1984) and previously studied via molecular dynamics (MD) simulation (Svaneborg et al., 2004; Hoy et al., 2006; Kim et al., 2008; Baig et al., 2010; Leonforte, 2010), there is still no consensus on how this behavior changes for different materials and how it can be incorporated into a micro-macro model of elasticity.

Recent work on uncrosslinked systems has seen the concepts of the primitive path and the restraining ‘tube’ proposed by Doi et al. (1988) evolve from theoretical abstractions into well-defined and measurable quantities able to describe the extent of intermolecular interactions in polymer systems (Likhtman, 2009). The primitive path is defined as the shortest path from one end of a chain to the other which preserves the topological state of the network (i.e. retaining all inter-chain entanglements). The output of a simulation can be processed to obtain the set of primitive paths representing the network. Primitive path analysis (Shanbhag et al., 2007; Baig et al., 2010a) has become an essential tool in polymer melt rheology \citep{Larson2007,KarayiannisKrA?ger-CombinedMolecularAlgorithms-IJoMS-2009,PaddingBriels-Systematiccoarse-grainingof-JoPCM-2011} since intermolecular entanglements are important in determining the flow properties of uncrosslinked polymer melts. Hoy et al. (2006) analyzed nonaffine displacements in polymer glasses with different applied strain rates. Other studies have examined primitive path deformation and chain end-to-end deformation for uncrosslinked polymer melts in response to different applied strain rates (Kim et al., 2008; Baig et al., 2010; Leonforte, 2010). However, there appears to only be one study to date that considers the primitive path in a crosslinked polymer system: Li et al. (2011) varied crosslink density to show how it affects tube diameter and primitive path step length in polyisoprene, and compared with corresponding results for uncrosslinked systems. This was a characterization of undeformed primitive path characteristics in a crosslinked polymer network; deformed primitive path lengths were not investigated.

In this work we use coarse-grained MD simulations to perform a detailed analysis of nonaffine chain and primitive path deformation for crosslinked polymers with different network topologies. The simulations encompass a wide range of network topologies, ranging from short, unentangled chains to long, entangled chains. The networks are formed by simulating a crosslinking reaction at different system densities for each chain length. This results in networks with constant chain length but different entanglement density, and we use these to study the influence of both chain length and entanglement density on nonaffine deformations. The simulated networks are deformed in uniaxial, equi-biaxial, and pure shear loading configurations, and the statistics of both chain end-to-end deformation and primitive path deformation are compared. Although chain end-to-end length deformation approaches an affine value for long chain systems, our simulations suggest that primitive path deformation is always nonaffine under quasistatic deformation, even for very long, entangled chains.

Details on the simulation methods are in Section \ref{sec:simulation-methods}. The primitive path concept is introduced in Section \ref{sec:ppanalysis}, and undeformed network properties are investigated in Section \ref{sec:undefproperties}. Simulation results for average chain primitive path deformation are in Section \ref{sec:averagedeformation}. It is shown that primitive path deformation is always nonaffine, even for long, entangled chains. Both chain end-to-end length and primitive path length can be quantified as linear functions of the applied deformation, and the level of affineness increases when chains are longer and/or more entangled. The simulation results are presented to highlight differences in behavior for networks of short, unentangled chains in comparison with long, entangled chains. In Section \ref{sec:distributions}, we take a more detailed look at chain deformation by looking at the distribution of chain lengths after a deformation is applied. Individual chains are tracked in time to visualize the available conformational space (the restraining “tube”) at various levels of deformation in Section \ref{sec:phasespace}. These results are used to discuss the observed mechanisms and their relation to multiscale modeling in Section \ref{sec:discussion}, and the paper is concluded in Section \ref{sec:conclusions}.