## Method

The method adopted here uses a sparsely occupied high coordination (10 i2+2 sites in shell i) lattice for the coarse-grained representation of the system. This lattice is obtained by deletion of every second site from a diamond lattice.5 In the first generation of applications of the simulation to saturated hydrocarbon melts, each occupied bead on the lattice represents a -CH2CH2- unit (for simulations of polyethylene2-467) or a -CH2CH(CH3)- unit (for simulations of atactic, isotactic, or syndiotactic polypropylenes8-10). The step length on the lattice, 0.25 nm, is defined by the length of the C-Cbond and the tetrahedral angle. The bulk density of a typical polyethylene melt is achieved with occupancy of about 1/6 of the sites on this lattice. A newer second generation of the method, in which each bead in a simulation of polyethylene represents a CH2CH2CH2CH2 unit, has been developed recently.11 With the second generation of the simulation, bulk density for a polyethylene melt is achieved with occupancy of only 1/12 of the sites on the high coordination lattice. The computational efficiency of the simulation benefits from the use of a sparsely occupied lattice. The results presented here will focus on amorphous polyethylene as the material, using the first-generation method.

The Hamiltonian contains two parts. The short-range intramolecular contribution comes from the mapping of a classic rotational isomeric state model1213 for the real chain onto the discrete space available to the coarse-grained chain on the sparsely occupied high coordination lattice.28 Specific examples that have been used in the simulations of melts of polymeric hydrocarbons are three-state rotational isomeric state models for polyethylene14 and polypropylene.15 The long-range and intermolecular interactions are handled by invoking self- and mutual exclusion, along with a discretization into interaction energies for successive shells (u„ i = 1, 2, . . .) of a continuous potential energy function,3 such as a Lennard-Jones potential energy function, that describes the pair-wise interaction of small molecules1617 representative of the collection of atoms assigned to each bead on the high coordination lattice. For example, adoption of a Lennard-Jones potential energy function with = 205K and s = 0.44 nm implies that simulations of polyethylene at 443K should use u1. . . u5 of 14.2,0.429, -0.698, -0.172, and -0.045 kJ/mol, respectively.3 The first shell is strongly repulsive because it covers distances smaller than a. The second shell is much less repulsive because it covers the distance at which the Lennard-Jones potential energy function changes sign. The major attraction appears in the third shell. If the system in the simulation is to be cohesive, at least three shells must be retained in the evaluation of the intermolecular interactions.

The simulation of polyethylene melts proceeds by random jumps (of length 0.25 nm) by individual beads to unoccupied nearest-neighbor sites on the high coordination lattice, with retention of all connections to bonded beads. These single-bead moves correspond to a variety of local moves in the underlying fully atomistic model that change the coordinates of 2 or 3 carbon atoms.4 The acceptance or rejection of proposed moves is via the customary Metropolis criteria.18

When the method was applied to one-component9 and two-component10 melts of polypropylene chains of specific stereochemical sequence, reptation was included along with the single-bead moves, in order to achieve equilibration of the melts on an acceptable time scale. The polypropylenes (especially syndiotactic polypropylene) equilibrate slowly if the simulation uses single bead moves only.

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