Environmental Engineering Reference
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some simulations of complex fluids, it is sufficient to use the simplest models that
faithfully represent the essential physics. In this chapter we shall concentrate on
continuous, differentiable pair-potentials (although discontinuous potentials such
as hard spheresand spheroids have also played a role). The Lennard-Jones poten-
tial is the most commonly used form:
12
6
s
s
�� ��
=e −
( )
I
vr
.
4
-
€‚ €‚
ƒ„ ƒ„
(105)
r
r
-
with two parameters—the diameter, and the well depth. This potential was used,
for instance, in the earliest studies of the properties of liquid argon. For applica-
tions in which attractive interactions are of less concern than the excluded volume
effects, which dictate molecular packing, the potential may be truncatedat the
position of its minimum, and shifted upwards to give what is usually termed the
model. If electrostatic charges are present, we add the appropriate Coulomb po-
tentials.
QQ
( )
v
Coulomb
r
=
12
(106)
4
π
∈
r
0
where Q
1
, Q
2
are the charges and is the permittivity of free space. The correct
handling of long-range forces in a simulation is an essential aspect of polyelec-
trolyte simulations, which will be the subject of the later chapter of Holm [145].
1.3.2.4 BONDING POTENTIALS
For molecular systems, we simply build the molecules out of site-site potentials
of the form of Eq. (69) or similar. Typically, a single-molecule quantum-chemical
calculation may be used to estimate the electron density throughout the molecule,
whichmay then be modeled by a distribution of partial charges via Eq. (70), or
more accurately by a distribution of electrostatic multipoles. For molecules we
must also consider the intramolecular bonding interactions. The simplest molecu-
lar model will include terms of the following kind:
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