Chemistry Reference
In-Depth Information
flux of macromolecules to the interface as
"
#
;
¼
kAc
i
ðÞ
1
X
i
dc
i
ðÞ
dt
y
i
ð
t
Þ
ð
2
Þ
where
7
=
3
e
1
=
3
k
¼
1
:
4 r
p
;
i
þ
r
d
;
ð
3
Þ
4r
d
and c
i
is the macromolecule concentration for molar mass class i in the solution.
The quantity 1
P
i
y
i
ðÞ
describes the saturation of the interface, where
y
i
¼
(G
i
/G
max,i
). The saturation term can be expressed as
y
i
ð
t
Þ¼
c
i
ð
0
Þ
c
i
ð
t
Þ
½
a
i
=
A
ð
Þ;
ð
4
Þ
where A is the emulsion interfacial area; the quantity a
i
is the projected area of a
macromolecule in class i,
a
i
¼
p
r
p
;
i
N
A
M
i
;
ð
5
Þ
where N
A
is Avogadro's number and M
i
is the molar mass of class i. We can
then model the adsorption in the molar mass range investigated experimentally.
This gives us a set of differential equations that can be solved numerically:
8
<
9
=
8
<
c
n
ðÞ
A
P
i
¼
1
9
=
@
c
n
@
t
@
c
i
@
t
@
c
1
@
t
k
n
r
p
;
n
;
r
d
ð
c
i
ð
0
Þ
c
i
ð
t
Þ
Þ
a
i
c
i
ðÞ
A
P
i
¼
1
¼
k
i
r
p
;
i
;
r
d
ð
c
i
ð
0
Þ
c
i
ð
t
Þ
Þ
a
i
:
ð
6
Þ
c
1
ðÞ
A
P
i
¼
1
:
;
:
;
k
1
r
p
;
1
;
r
d
ð
c
i
ð
0
Þ
c
i
ð
t
Þ
Þ
a
i
The results from the modelling are compared with the experimental results in
Figure 8 in terms of adsorbed ratio. The data show that the ultra-high molar
mass components adsorb preferentially and that the adsorption can be described
theoretically as collisions between particles in turbulent flow. Thus the high
surface loads obtained experimentally may be because of an over-representation
of the ultra-high molar mass components at the interface, which as such may
give rise to higher surface loads than expected. Because kinetics are important,
owing to the short timescales for adsorption during emulsification,
9,31
it is likely
that nonequilibrium structures and jamming will arise at the interface as the
macromolecules will not have sufficient time to optimize their configurations
from a thermodynamic point of view. Furthermore, the surface loads may be
determined by Apollonian packing,
32,33
which means that late arriving mole-
cules find fewer available adsorption sites and so have to 'squeeze in' between
the already adsorbed molecules. This results in a type of close packing which
resembles Apollonian packing, i.e., a space-filling packing circles or spheres
(see Figure 9). To describe this, Douglas et al.
33
have proposed the adap-
tive random sequential adsorption (ARSA) model, where the molecules are
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