Deformation of Amorphous Polyethylene

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AbstractMethodologyMaterial ModelInput DataResultsAcknowledgmentsReferences

Abstract

Molecular dynamics simulations were used to study deformation mechanisms during uniaxial tensile deformation of an amorphous polyethylene polymer. The stress-strain behavior comprised elastic, yield, strain softening and strain hardening regions that were qualitatively in agreement with previous simulations and experimental results. The chain lengths, number of chains, strain rate and temperature dependence of the stress-strain behavior were investigated. The energy contributions from the united atom potential were calculated as a function of strain to help elucidate the inherent deformation mechanisms within the elastic, yield, and strain hardening regions. The results of examining the partitioning of energy show that the elastic and yield regions were mainly dominated by interchain non-bonded interactions whereas strain hardening regions were mainly dominated by intrachain dihedral motion of polyethylene. Additional results show how internal mechanisms associated with bond length, bond angle, dihedral distributions, change of free volume and chain entanglements evolve with increasing deformation. See our papers for more information [1] [2].

Author(s): Mark A. Tschopp, Delwar Hossain, Mark F. Horstemeyer

Corresponding Author: Mark Tschopp

Figure 1. Movie showing deformation of an amorphous polyethylene structure with 20 chains of 1000 monomers length. The strain rate is 1010 s-1 and the temperature is 100 K.
Figure 2. Stress-strain curve for amorphous polyethylene. (click on the image to enlarge).
Figure 3. Stress-strain curve for amorphous polyethylene at different strain rates. (click on the image to enlarge).

Methodology

Interatomic Potential

The interatomic force field for polyethylene is based on a united atom model using the Dreiding potential. There were several reasons why this potential was chosen. First, this study investigates uniaxial deformation of amorphous polyethylene systems with large numbers of monomers (20,000 to 200,000 united atoms) as a function of strain rates that vary over three orders of magnitude (108 to 1010 s-1). Full atom representations would require approximately three times the number of atoms (i.e., carbon plus two hydrogens) along with a more computationally-expensive force field and, quite possibly, a smaller timestep or different integration scheme due to adding hydrogen. Additionally, the 108 s-1 strain rate in these simulations requires millions of timesteps to reach strains of approximately 100%. Therefore, the use of the united atom model and Dreiding potential was influenced in part by the computational expense. Second, much of the prior literature regarding molecular dynamics simulations of polyethylene have used the Dreiding potential. In this sense, this work can build upon results previously obtained, while also providing guidance as to the influence of chain length, number of chains, temperature, and strain rates when moving to full atom representations with optimized force fields. We leave deformation of full-atom polymer representations for future work.

The united atom model used here considers each methyl group (i.e., the CH2 monomer) as a single atom with a force field that determines the interactions between united atoms. The Dreiding potential is used to describe these interactions. The Dreiding potential has four contributing terms; bond stretching, changes in bond angle, changes in dihedral rotation, and van der Waals non-bonded interactions.

Deformation Simulations

A parallel molecular dynamics code, LAMMPS[3], that incorporates domain decomposition is used to deform the polymer simulation cells. A Nosé-Hoover thermostat was used to regulate the system temperature. The mechanical responses from three different initial configurations were averaged to account for entropic effects due to the high number of possible chain configurations. The link below shows an animation of uniaxial loading of polyethylene.

The initial chain structure was created using a method similar to those previously developed using Monte Carlo self-avoiding random walks. The chain generation step used a face-centered cubic (FCC) lattice superimposed on the simulation cell with the nearest neighbor distance of 1.53 Å. Molecules were added to the lattice in a probabilistic stepwise manner, which based the probability of chain growth in certain directions on the bond angle and the density of unoccupied sites. This chain structure was then formatted into a LAMMPS data file.

This initial structure was then inserted into the molecular dynamics code where an equilibration sequence was performed prior to deforming the amorphous polymer. The equilibration sequence relaxes any high energy configurations that are artificially created due to the FCC lattice used to generate the amorphous polymer structure. The relaxation involves four different steps. Initially, the simulation ran for 10,000 timesteps (Δt=1 fs) using NVT dynamics at 500 K followed by relaxation for 50,000 timesteps (Δt=0.5 fs) using NPT dynamics at 500 K. The next relaxation cooled the structure down to the desired temperature for 50,000 timesteps followed by further relaxation of 50,000 timesteps at the desired temperature.

The amorphous PE system was then deformed under a uniaxial tensile strain applied at a constant strain rate with a zero-pressure condition for the two lateral simulation cell faces. This deformation condition was implemented in LAMMPS by decoupling the boundary in the loading direction from the NPT equations of motion. In addition to calculating the stress as a function of strain, the energy contributions associated with the bond length, bond angles, dihedral angle and non-bonding interactions were tracked as a function of strain. Periodic dumps of the atomic configuration were performed to compute additional microstructure metrics such as chain orientation or the percentage of trans dihedral conformations. Three different strain rates (108, 109 and 1010 s-1) were used with an initial configuration consisting of 20 polymeric chains, each having 1000 monomers, to investigate the effect of strain rate. To investigate the influence of chain length and chain number, polyethylene configurations were uniaxially deformed at a stain rate of 109 s-1 for different numbers of chains (100 and 2000) and chain lengths (20, 200, and 2000 monomers).

Material Model

Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)

Input Data

Running the simulation

  • Download the input script
  • Download the attached LAMMPS datafile,
  • Type 'lmp_exe < in.deform_dreiding.txt'

Results

Molecular dynamics (MD) simulations were used to study the main deformation mechanisms responsible during uniaxial tensile loading of an amorphous polyethylene (PE)-like polymer. These simulations were performed for different strain rates and temperatures under uniaxial tensile loading to investigate the influence of these external factors on mechanical properties of PE polymer. Results show that many mechanical properties of PE can be simulated by MD in qualitative some times in quantitative agreement with experiments.

  • The simulated stress-strain behavior at the strain rate of 1010 s-1 shows the same trends as those observed in experimental testing for amorphous polymers, i.e., an elastic regime followed by a yield peak and subsequent softening and strain hardening regimes. The characteristic yield peak observed in polymers decreases with increasing temperature and decreasing strain rate. This difference provided the contrast between conditions that was necessary for exploring what mechanisms were responsible for the characteristic yield peak in polymers. Overall the strain rate and temperature have much stronger effect on the behavior than chain length and number of chains, which show little effect on the stress-strain behavior.
  • In the elastic and yield peak regimes, the energy evolution plots show that interchain non-bonded (Van der Waal’s) interactions played a significant role, as shown by the amount of the total potential energy increase that is due to the non-bonded energy. This increase in energy is later associated with an increase in the free volume, which is required prior to intrachain torsional processes for some conditions. The evolution of free volume also suggests that Poisson’s ratio evolves within the elastic regime.
  • In the strain softening and hardening regimes, the energy evolution plots show that intrachain torsion mechanisms played a significant role. This decrease in dihedral energy is later associated with the transformation of gauche conformations to lower energy trans conformations. Additionally, an entanglement parameter calculated from the intrachain geometry shows that the majority of disentanglement occurs in this regime.
  • The present simulations clearly show that the bond length and bond angle distributions remain nearly constant over all regimes for all conditions. This is also supported by the minimal change in energy associated with bond length and bond angle compared to the non-bonding and dihedral energy changes.
  • Chain alignment in the direction of loading increases with increasing strain over all regimes.

Future investigation will focus in building a hierarchical multiscale approach using the information obtained from the molecular dynamic scale (e.g. evolution of free volume and entanglement density with deformation, strain rate, temperature) to refine the continuum level constitutive model.

Acknowledgments

The authors would like acknowledge financial support through the Center for Advanced Vehicular Systems at Mississippi State University. Additionally, the authors would like to acknowledge the insightful comments from the reviewers during the review process.

References


  1. Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F., "Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene," Polymer, 51 (2010) 6071-6083.
  2. Tschopp, M.A., Ward, D.K., Bouvard, J.L., Horstemeyer, M.F., "Atomic Scale Deformation Mechanisms of Amorphous Polyethylene under Tensile Loading," TMS 2011 Conference Proceedings, accepted.
  3. S. Plimpton, "Fast Parallel Algorithms for Short-Range Molecular Dynamics," J. Comp. Phys., 117, 1-19 (1995).
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