Environmental Engineering Reference
In-Depth Information
3.3 DPD with Hard Potentials
As it concerns the use of DPD with hard potentials, we mention a widely used polymer
coarse-grained model, developed by Grest and Kremer (
1986
) and Kremer and Grest
(
1990
). It has been applied to a variety of thermodynamic conditions and physical
systems as glasses, polymer melts and solutions, and polymer brushes (Grest
1999
;
Baschnagel and Varnik
2005
; Dünweg and Kremer
1993
; Binder and Milchev
2012
;
Kroger
2004
). The bonded interaction along the same polymer is modeled by a finite
extensible non-linear elastic (FENE) potential:
⊧
⊨
2
kR
0
ln
1
R
0
2
1
−
−
r
≤
R
0
U
FENE
=
R
0
,
(10)
⊩
∞
r
>
where the maximum allowed bond length is
R
0
=
1
.
5
˃
, the spring constant
2
, and
r
is
k
denotes the distance between neighboring
monomers. Excluded volume interactions at short distances and van-der-Waals
attractions between beads are described by a truncated and shifted Lennard-Jones
(LJ) potential:
=
30
ʵ/˃
=|
r
i
−
r
j
|
U
(
r
)
=
U
LJ
(
r
)
−
U
LJ
(
r
c
),
(11)
with
r
6
12
r
U
LJ
(
r
)
=
4
ʵ
−
,
(12)
where the LJ parameters,
1, define the units of energy and length,
respectively. The temperature is therefore given in units of
ʵ
=
1 and
˃
=
ʵ/
k
B
, with
k
B
the Boltz-
mann constant.
U
LJ
(
is the LJ potential evaluated at the cut-off radius. There
are typically two values taken as a cut-off distance: (i) the minimum of the
LJ
potential
r
c
r
c
)
2
1
/
6
=
˃
1
.
12
˃
and (ii) twice the minimum of the LJ potential:
2
6
r
c
=
, which allows for poor solvent conditions. In the first case,
the interactions between monomers of different chains are purely repulsive. From the
point of view of polymer solutions, this means good solvent conditions because the
polymer bead is close to the solvent, giving rise to an effective repulsion among poly-
mer beads. In the second case, longer ranged attractions are included (full Lennard-
Jones potential), giving rise to liquid-vapor phase separation and droplet formation
for adequate thermodynamic conditions (Müller and MacDowell
2001
; MacDowell
et al.
2000
; Servantie and Müller
2008
).
2
×
˃
2
.
24
˃
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