Environmental Engineering Reference
In-Depth Information
and , the unit normal to the plane dened by each pair of bonds. Usually the tor-
sional potential involves an expansion in periodic functions of order Eq. (109).
A simulation package force-eld will specify the precise form this equation.
Molecular mechanics force fields, aimed at accurately predicting structures and
properties, will include many cross-terms (e.g., stretch-bend). Quantum mechani-
cal calculations may give a guide to the “best” molecular force-eld; also compari-
son of simulation results with thermophysical properties and vibration frequencies
is invaluable in force-eld development and tenement. A separate family of force
elds, such as AMBER, CHARMM and OPLS are geared more to larger molecules
(proteins, polymers) in condensed phases; their functional form is simpler, closer
to that of this equation and their parameters are typically determined by quantum
chemical calculations combined with thermophysical and phase coexistence data.
This eld is too broad to be reviewed here; several molecular modeling texts (albeit
targeted at biological applications) should be consulted by the interested reader.
The modeling of long chain molecules will be of particular interest to us, especial-
ly as an illustration of the scope for progressively simplifying and “coarse-grain-
ing” the potential model. Various explicit-atom potentials have been devised for
the n-alkanes. More approximate potentials have also been constructed in which
the CH
2
and CH
3
units are represented by single “united atoms.” These potentials
are typically less accurate and less transferable than the explicit-atom potentials,
but significantly less expensive; comparisons have been made between the two
approaches. For more complicated molecules this approach may need to be modi-
fied. In the liquid crystal eld, for instance, a compromise has been suggested: use
the united-atom approach for hydrocarbon chains, but model phenyl ring hydro-
gen's explicitly. In polymer simulations, there is frequently a need to economize
further and coarse grain the interactions more dramatically: significant progress
has been made in recent years in approaching this problem systematically. Finally,
the most fundamental properties, such as the entanglement length in a polymer
melt, may be investigated using a simple chain of pseudoatoms or beads (modeled
using the WCA potential and each representing several monomers), joined by an
attractive nitely extensible non-linear elastic (FENE) potential:
•
�
2
�
��
1
r
˜
−
kR
2
0
ln ln 1
−
r
<
R
€
‚
˜
€‚
( )
(112)
FENE
0
v
r
=
2
ƒ„
R
™
ƒ
„
0
˜
∞<
rR
›
0
The key feature of this potential is that it cannot be extended beyond , ensuring
(for suitable choices of the parameters k and R
0
that polymer chains cannot move
through one another [132].
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