Environmental Engineering Reference
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adsorption affecting the global stability of a colloidal dispersion, including surface-
modifying polymer chains
versus
end-grafted polymer chains, have been studied by
Gama Goicochea et al. (
2009
). Attempts to measure the forces themselves that act in
a confined complex fluid in thermodynamic equilibrium with its surroundings have
been made using atomic force microscopy (cf., e.g., McNamee et al. (
2004
)), while
it has been argued (Derjaguin and Churaev
1986
) that it is more appropriate to use
the concept of disjoining pressure, which is the difference between the force (per
colloidal particle unit area) normal to the conning surfaces and the fluids bulk pres-
sure. This disjoining pressure allows for a direct determination of the free energy of
interaction, hence its importance.
Polyelectrolyte solutions have very different properties from those observed in
solutions of uncharged polymers, and their behaviour is less well known (de Gennes
1976
; Odijk
1979
; Dobrynin et al.
1995
). In particular, it is not evident that the
scaling of some quantities presents a similar behaviour as that of electrically neutral
solutions, or that they present the same or similar scaling exponents. Calculating
Langmuir isotherms for polyacrylate dispersants adsorbed on metallic oxides, and
their scaling properties as a function of the number of monomeric dispersant units
obtained via Dissipative Particle Dynamics (DPD) simulations, it has been shown
(Mayoral and Nahmad-Achar
2014
; Gonzalez-Melchor et al.
2006
) that the critical
exponent for the renormalized isotherms agree perfectly well with the scaling theory
of Gennes et al. (
1976
) even though polyelectrolytes have been considered.
Due to the long-range Coulombic repulsion produced by the presence of small
mobile counterions in the bulk, the properties of these systems cannot in general be
obtained analytically. The most usual systems are even more complex, encompass-
ing various surfactants of different chemical nature and molecular weight (acting as
dispersants, wetting agents, rheology modifiers, etc.), pigments, “inert” extenders,
and so on. In all these cases there are various different lengths and dynamic scales,
every species interacts with all others at a molecular level, in a way which depends
on temperature and concentration. Furthermore, there is a competitive adsorption
amongst all surfactants present. Ideally, one should have a basic understanding of
all interactions, but the main problem is that all colloidal systems are thermody-
namically unstable. Empirical methods have been used as well as few and greatly
approximated analytic models, and a more recent and promising method is that of
molecular dynamics simulations
. Its basic methodology consists of taking advantage
of the fast computing facilities that are nowadays available, to integrate Newton's
equations of motion for a large number
N
of particle (molecules, atoms, or whatever
the problem in turn calls for). Thus, one sets initial positions
r
i
(
t
)
and momenta
p
i
(
t
)
for each particle
i
at time
t
, and uses the force field felt by each one of them
m
d
2
r
F
(
r
)
=−∇
V
(
r
)
=
dt
2
,
(1)
to find its new position and momentum at time
t
+
ʴ
t
iteratively. The approximation
being made is to consider the potential
V
to be constant during the time step
ʴ
t
which, if taken very small, can make the error negligible. Typical choices for the
(
r
)
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