Environmental Engineering Reference
In-Depth Information
Whereas the FH mean-field theory considers
ˇ
12
as proportional to
T
−
1
but
independent of the solute concentration
ʶ
, comparisons with experiments show that
the phenomenological
ˇ
12
contains both energetic and entropic contributions; i.e.,
ˇ
12
. A correct parametrisation in our electrostatic DPD system must
therefore take into account the dependence of the repulsive parameters for the sol-
vated ions
a
ij
with the salt concentration
ʶ
. The way to understand this is as follows:
when we perform a coarse graining, the volume of a DPD particle does not usually
encompass a full molecule or polymer; thus, for instance, although for dodecane
our DPD particle contains only a butane fragment, we do not construct dodecane
from the union of butane particles, and the interaction between the DPD dodecane
particles and water does not correspond to the
ˇ
parameter of butane with water; the
ˇ
parameter employed to estimate the DPD repulsive parameter
a
ij
should be that
of the full dodecane molecule because its behaviour is that of the global joined units
which affect the electronic distribution throughout. In this case, the “monomeric”
units, which constitute the dodecane “polymeric” molecule, interact through short-
range (covalent bond) forces. When considering a solvated electrolyte, e.g.,
N
a
+
or
Cl
−
ions, their concentration is given precisely by the amount of solvated ionic par-
ticles present, which corresponds effectively to the amount of “monomeric” solvated
ionic units. These are in effect the individual DPD units, which in this case are not
covalently joined but are subject to long-range electrostatic forces. The presence and
quantity of “monomeric” solvated ions affect the global properties of the network
and their corresponding
ˇ
parameter should take into account the whole electrolytic
entity, and thus a correct parametrisation of the DPD system forces a dependence of
the conservative force parameters
a
ij
on the concentration
ʶ
, through
ˇ(
=
ˇ
12
(
T
, ʶ)
, ʶ)
T
.
3.1 Concentration Dependence of the DPD
Interaction Parameters
For an electrolyte solution in water,
e
+
w
, the chemical potential
μ
w/
e
for each
component (
w/
e
) may be obtained by differentiating the free energy per unit volume
of the mixture
e
+
w
with respect to the number of molecules
N
e
of the component.
w/
Thus,
μ
w
k
B
T
=
μ
e
k
B
T
=
2
2
ln
ˆ
+
ˇ(
1
−
ˆ)
,
ln
(
1
−
ˆ)
+
ˇˆ
,
(21)
where
ˆ
and 1
(solvent) and
e
(electrolyte)
components, respectively. The activity coefficient for the electrolyte
ʱ
e
is defined as
−
ˆ
are the volumetric fractions for the
w
μ
e
−
μ
e
RT
ln
(ʱ
e
)
=
,
(22)
where
μ
e
denotes an arbitrarily chosen zero for the component
e
and is called the
standard chemical potential
of
e
.The
ˇ
-parameter for the solvent and the electrolyte
can be obtained from
ʱ
e
:
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