Environmental Engineering Reference
In-Depth Information
where we have denoted by
r
i
,(
=
,
,...,
)
the positions of the chain joints
(i.e., the two ends of the
i
-th bond are
r
i
−
1
and
r
i
). A more useful quantity, however,
is the
radius of gyration R
g
i
0
1
N
of the chain, given by
2
N
1
2
R
g
=
0
(
r
i
−
r
CM
)
,
(44)
N
+
1
i
=
1
i
=
0
r
i
is the centre of mass of the chain. Loosely speaking,
the chain occupies the space of a sphere of radius
R
g
, i.e., it intuitively gives a sense
of the size of the polymer coil. Note that
mR
g
1
where
r
CM
=
N
+
2
(with
m
the mass of the polymer
molecule) is the moment of inertia of the molecule about its centre of mass, and so
we can write the equation above as
2
N
1
2
1
2
R
g
=
0
(
r
i
−
r
j
)
,
(45)
2
(
N
+
1
)
i
,
j
=
which is useful since it allows us to calculate the radius of gyration of the molecule
by using the mean square distance between monomers without calculating
r
CM
.
Note also that we have used averages in all the equations above; this is because the
possible chain conformations are numerous and constantly change in time, thus we
understand the radius of gyration as a mean over time of all the polymer molecules,
which by ergodicity principles we calculate as an ensemble average.
The radius of gyration can be easily determined experimentally through light scat-
tering or other alternative methods (neutron scattering, etc.), allowing one to check a
theoretical model against reality, and this is what makes it an interesting quantity of
study. It has been extensively studied for neutral polymeric species but, as the pres-
ence of charges completely changes the possible configurations of the molecules in
solution, it is interesting to study the behaviour of
R
g
in a polyelectrolyte.
One interesting problem is the pH-dependent conformational change of some
biopolyelectrolites because it affects directly the mechanism of action in differ-
ent situations. An example of this is the poly(amidoamine) (PAA) which is used
as endosomolytic biopolymer for intracellular delivery of proteins and genes. Bio-
responsive behaviour of these kinds of compounds is related to with the structure
and conformation in the medium, which could be estimated by the radius of gyra-
tion. This is modified by pH and ionic strength effects. Experimental studies of
small-angle neutron scattering (SANS) have been published in order to illustrate the
pH-dependence and conformational change of PAA ISA 23 (Griffiths et al.
2004
).
Linear poly(amidoamine) polymers (PAAs) have amido- and tertiary amino-groups
along the main polymer, which gives rise to an interesting pH-dependent conforma-
tional change and thus offers a perfect prospect for devising polymers that present
membrane activity at low pH. The neutral structure of this biopolymer is shown in
Fig.
7
a.
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