Environmental Engineering Reference
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and, if correct parametrisations are used, it can reproduce in great detail the scaling
properties of different kinds of real systems.
As an example, the scaling exponent observed for the dependence of the interfacial
tension
ʳ
with temperature
T
, for several liquid-liquid systems, is given by:
)
=
ʳ
o
1
μ
T
T
c
ʳ(
T
−
,
(47)
where
ʳ
0
is a system-dependent constant,
T
c
is the critical temperature at which
the interface becomes unstable, and
μ
is a critical exponent which has been found
experimentally some years ago to be close to 11
9 (Guggenheim
1945
). According
to the hyper-scaling relationship of Widom (
1965
); Fisk and Widom (
1969
), we have
μ
=
ʽ(
/
where
ʽ
is the scaling exponent for the radius of gyration given in
Eq. (
5
) and
d
is the dimensionality of the system. More recently, by renormalization
group calculations (Albert
1982
; Le Guillou and Zinn-Justin
1980
; Moldover
1985
),
more accurate results give
μ
=
d
−
1
)
3 satisfy the hyper-
scaling law. These results have been reproduced by DPD simulations for different
systems (Mayoral and Nahmad-Achar
2013
; Mayoral and Gama Goicochea
2014
)
and are presented in Fig.
11
for a dodecane/water mixture.
Another interesting example is the scaling of
ʳ
max
(maximum adsorption) with the
number
N
of chain units. The number of chains of size
N
per unit area,
1
.
26 and
ʽ
=
0
.
63, which for
d
=
ʓ
max
, needed
to satisfactorily cover some given amount of material, say 1 mol, can be obtained
by performing DPD simulations for the adsorption of polymers with different length
T
T
( T )
T ( K )
Fig. 11
Scaling exponent observed for the dependence of the interfacial tension
ʳ
with temperature
T
for dodecane/water
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