Coarse-Grained Modeling for Macromolecular Chemistry - Multiscale Molecular Methods in Applied Chemistry - page 296

1 Introduction

Polymeric systems are characterized by a wide range of length scales that extend
from ˚ ngstr oms for the distance between the bonded atoms to at least micrometers

for the contour length of the chain. The corresponding time scales associated with
motions on such length scales are even broader: bond vibrations occur on the scale
of picoseconds (10 -13 s) and chain relaxation and morphology formation can occur
over seconds, minutes, or hours, depending on molecular weight, temperature, and
density. For this reason, an equally wide range of simulation methods at different
levels of resolution and, consequently, including differing degrees of freedom is
employed to study them [ 1 ].
Quantum mechanical (QM) methods present the most detailed picture of the
system by using different levels of approximations to solve the Schr

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odinger equa-
tion and evaluate electron wave functions. A work by Martonak et al. [ 2 ] shows that
even at room temperature, quantum effects are crucial to understanding the aniso-
tropic thermal expansion of polyethylene crystals. It is clear that if the number of
details that a simulation technique describes is increased, then the accessibility of
long time scales and large length scales is decreased. Therefore, the QM methods
can be used only for very short time and length scales, which are typically of the
order of ˚ ngstroms and picoseconds, respectively. However, QM methods are
extremely valuable in providing important information for preparation of an atom-
istic model about the basic structure of molecules, i.e., bond lengths, bond angles,
torsion and associated force constant, partial charges, and torsional barriers.
In classical molecular dynamics (MD) simulations, the charge distributions are
approximated either by putting fixed partial charges on interaction sites or by
adding an approximate model for polarization effects. Thus, in MD simulations
the time scale of the system is not dominated by the motion of the electrons, but
mainly by the time of rotational motions and intramolecular vibrations, which are
orders of magnitude slower than those of electron motions. Consequently, the time
step of integration is larger, and trajectory lengths are in the order of nanoseconds
and accessible lengths in the order of 10-100 ˚ . In MD methods, the Hamilton
equations of motion [ 3 ] are integrated to move particles to new positions and to
assign new velocities at these new positions, which is, in principle, probing the
whole phase space. Consequently, MD simulation is a powerful technique for
computing the equilibrium and dynamic properties of classical many-body systems
[ 4 ]. Over the last 20 years, with rapid development of computers, polymeric
systems have been the subject of intense study using MD simulations, but MD
simulations using atomistic force fields are still unable to access the time scales
necessary to achieve chain relaxation for polymeric systems of intermediate or high
molecular weights [ 5 ]. To equilibrate the dense polymeric systems with long
chains, advanced Monte Carlo (MC) methods have been developed that probe the
configuration space by trial moves of particles. Within the so-called Metropolis
algorithm, the energy change between two consecutive steps is used to accept or
reject the new configuration to find the system in its energy-minimum state [ 4 , 6 - 9 ].

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