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where g and a representthe Lennard-Jones parameters. The parameters for the potential energy functions ofEq.(2)-(5) are shown in Table 1.

Modeling of Polymer Particles

We have developed an efficient method to obtain a desired particle size to model production ofpolymer particles using MD simulations.7 The procedure starts by preparing a set ofrandomly coiled chains with a chain length of 100 beads by propagating a classical trajectory with aperfectlyplanar all-trans zig-zag initialconformation and randomly chosen momentum with a temperature of300 K. The trajectory is terminated at 200 ps, and position and momenta ofthe chain are saved. By repeating this process, a desired set of randomly coiled chains is prepared. From the set, six chains are selected and placed along the Cartesian Axis. To create a collision at the Cartesian origin, these chains are propelled with an appropriate amount ofmomentum, the resulting particle consisting of the six chains is annealed to a desired temperature and rotated through a randomly chosen set of angles in three-dimensional space to create a homogeneous particle. This process is continued until the desired size is achieved for the study. Figure 3 illustrates the process for generating initial conditions of different sized PE particles.

We also generated the PE particles with various chain lengths (50 and 200 beads) based on the initial configurations of the generated amorphous PE particles with a chain length of 100 beads. To do this, we simply expand the space of the particle with a chain length of 100 beads by multiplying the initial configuration by two in Cartesian coordinates. Then, a new atom is inserted between those two atoms. Thus, the total number of atoms is doubled and the new particle consists of chains with a length of 200 beads. For the particle with a chain length of 50 beads, the two, three and four-body bonded interaction are simply turned off every 50 beads. Finally, classical trajectories are propagated with randomly chosen momenta until the density reaches 0.7 g/cm3 and annealed to a desired temperature. It is noted that the density for the particle is low ( 0.05 g/cm3) and the temperature is very high ( 1000 K) at the start of those trajectories, since the space of the original particle is expanded. Using this scheme, we can efficiently create particles with various chain lengths without propelling sets of six chains.

After a desired size PE particle is obtained, classical trajectories are propagated for 50 ps at the above bulk melting point, and then annealed by scaling the Cartesian momenta with a constant scaling factor until the temperature reaches 10 K to find a steady state of the amorphous PE particles. To obtain average values of properties of the particles at a fixed temperature and examine dependence of the conformations on temperature, we have used Nose-Hoover chain (NHC) constant temperature molecular dynamics.16 The initial configurations of the steady state of the amorphous PE particle are used at the start of the NHC simulations; the initial values of the Cartesian momenta are given random orientation in phase space with magnitudes chosen so that the total kinetic energy is the equipartition theorem expectation value.17 The temperature of the particle T is calculated from where k b is the Boltzmann constant and N is the total number of atoms. After we propagate NHC trajectories 10 ps to equilibrate the system at a desired temperature, we begin sampling the molecular positions and momenta at a uniform interval (1.0 ps) until the simulation time reaches 100 ps. Figure 3 d) shows polyethylene particles (60,000 atoms) with chain length of 100 beads at a temperature of 10 K.

Modeling of Bulk Polymer Systems

Modeling of a polyethylene bulk system is achieved by placing 32 CH2 chains with 100 C and 202 H atoms in a cubic box under three-dimensional periodic boundary conditions. The initial configuration of the system was optimized using molecular mechanics calculation. With the initial conformation, MD simulations are propagated by applying external force to the box every 2 fs in order to pack the chains into the box with the desired density. The simulation is terminated when the density reaches 0.85 g/cm3 (bulk amorphous PE density). For the simulations of the bulk system, we considered all atoms of PE polymer in order to compare the computed structural conformations (e.g. radial distribution) with X-ray and neutron scattering experiments. The radial distribution of the simulated bulk PE was in good agreement with the experimental data. By comparing the bulk system with the nano-scale particles, we can study the conformational change of the particles due to the size reduction and the shape.

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