Environmental Engineering Reference
In-Depth Information
ʓ
M
=
.
(mg
PA A
/mg
TiO
2
). This value agrees well with the experimental data of
96
(mg
PA A
/mg
TiO
2
) reported by Huldén and Sjöblom (
1990
) and Mayoral et al.
(
2011
).
6
4.3 Disjoining Pressure
Colloid stability strongly depends on the
disjoining pressure
. For a confined fluid,
the pressure component perpendicular to the confining walls
P
N
is different from the
unconfined bulk pressure
P
bulk
. This differential pressure relative to the bulk, which
is a function of the separation
L
z
between the parallel walls is called “disjoining
pressure”. For a wall perpendicular to the
z
-direction
ʠ(
L
z
)
=
P
zz
(
L
z
)
−
P
bulk
.
(42)
While
P
bulk
is obtained from the average of the diagonal components of the pres-
sure tensor (cf. Eq.
36
), the pressure normal to the wall is calculated from the
zz
-
component, averaged over the length
L
z
of the simulation box in the direction per-
pendicular to the walls. The disjoining pressure is a measure of the force, per unit
area, needed to bring 2 particles (or a particle and a substrate) together, thus provid-
ing a criterion for stability. It has been calculated (Gama Goicochea et al.
2009
)for
different types of surfactants (those that graft at one end onto a substrate, and those
that can adsorb onto the substrate along their full length thus acting as surface modi-
fiers) and for different substrates. The results show that the greater stability attained
is not a consequence of the greater molecular weight of the dispersant species itself,
as it is so often misinterpreted, but rather of the greater molecule mobility. That is,
the entropic gain due to monomers with more mobility capable to sample the con-
figurational space more than polymers (at the same monomer concentration) is the
leading mechanism responsible for the higher values of disjoining pressure. This is
shown in Fig.
6
for a surface-modifying polymer. In this figure we observe the typical
oscillations in
ʠ
present in confined fluids (Israelachvili
1992
). While the maxima in
ʠ
correspond to more stable thermodynamic configurations, the minima represent
regions of instability. In this case, molecules with a molecular weight
M
400
were considered, corresponding to 7 DPD-particles joined by springs. Having 20
such molecules amounts to having 140 monomeric units, a concentration that can
also be achieved by considering 10 polymeric molecules of
M
w
=
w
=
800 of the same
chemical type.
Polyethylene glycol (PEG) of
M
w
=
800 were used for the results
of Fig.
6
, with a DPD-particle volume of 90 Å
3
which can accommodate 3 water
molecules. The repulsive wall interaction parameter was chosen as
a
w
−
monomer
=
400 and
M
w
=
60,
when the particle interacting with the wall was a monomer of the polymer molecule,
and as
a
w
−
sol
120 for solvent molecules (see Gama Goicochea et al. (
2009
)for
details). For particles of the same species
a
ii
=
=
78
.
0 and for particles of different
species
a
ij
=
79
.
3. This parameters reproduce the isothermal compressibility of
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