Environmental Engineering Reference
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(ʱ
e
)
−
(
−
ˆ)
ln
ln
1
ˇ
=
,
(23)
ˆ
2
x
y
ʱ
e
=
(
)
(
)
and its explicit concentration-dependence comes out by writing
x
y
z
, where
x
and
y
are the stoichiometric coefficients of the cation and the
anion, and
z
e
m
(ʱ
)
e
denotes the
mean
activity coefficient of the electrolyte, and
m
its molarity. Equation (
23
) allows one to obtain the Flory-Huggins concentration-
dependent parameter if the activity coefficient is known. The scaling of
ˇ
with the
quantity of ions present has been studied by Mayoral and Nahmad-Achar (
2012
).
The behaviour of this quantity as a function of the concentration
ʶ
follows a power
law
ˇ
∼
ʶ
˄
with characteristic scaling exponent
˄
depending on the kind of salt.
Comparing Eqs. (
19
) and (
20
), Groot and Warren (
1997
) proposed that the repul-
sive parameters
a
AB
in the DPD simulation can be obtained using the
=
x
+
y
.
ʱ
ˇ
-Flory-
Huggins parameter as
ʱ (
2
a
AB
−
a
AA
−
a
BB
) ˁ
ˇ
AB
=
,
(24)
k
B
T
and using Eqs. (
24
) and (
23
) the repulsive DPD parameter
a
ij
depending on the
concentration may be obtained as
a
ij
=
a
ii
+
3
.
27
ˇ
ij
,
(25)
with, as before,
(ʺ
−
1
N
m
−
k
B
T
1
)
a
ii
=
.
(26)
2
ʱˁ
DPD
3) and a compressibility of
ʺ
−
1
Thus, for 3 water molecules per particle (
N
m
=
≈
16
for water at 300
ⓦ
K and 1 atm, we have
a
ww
=
78
.
3.
3.2 Temperature Dependence of the DPD
Interaction Parameters
When the heat of mixing is given by the Hildebrand-Scatchard regular solution
theory (Hildebrand and Wood
1933
; Scatchard
1931
; Hildebrand and Scott
1950
;
Barton
1975
), the
ˇ
ij
-parameter can be obtained using the solubility parame-
ters
ʴ
i
(
for the pure components in the mixture, which are themselves
temperature-dependent. We have
T
), ʴ
j
(
T
)
RT
ʴ
i
(
)
2
v
ij
ˇ
ij
(
)
=
)
−
ʴ
j
(
,
T
T
T
(27)
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