Biomedical Engineering Reference
In-Depth Information
Fig. 1
Generic isotherms for polymeric monolayers. Surface pressure plotted as a function
of surface mass density (or surface concentration), exhibiting the virial regime giving rise
to molecular weight dependence as in the osmotic virial regime in bulk solution, semi-
dilute regime where the molecular weight dependence is obviated and state of monolayer
collapse.
M
n
is the number average molecular weight and
Γ
∗
is the overlap concentration
of the highest molecular weight sample
density regime is very difficult. Hence, more efforts are directed at the semi-
dilute regime where the surface pressure is more readily determined with
accuracy but the molecular weight dependence for a polymeric system no
longer holds. As with the bulk semi-dilute solution, any attempt to formu-
late an equation of state is abandoned. Instead, a surface density scaling is
adopted for the static characterization [28, 29] for a variety of polymeric sys-
tems, following the scaling argument set forth by de Gennes [30]. Thus, the
surfacepressuredependenceonthesurface density is expressed as a power
law,
y
. Here, y is related to the 2D molecular weight exponent
Π
∼
Γ
ν
(y = 2
- 1)). The analog of the overlap concentration c* for bulk solution
is the cross-over surface density
ν/
(2
ν
Γ
∗
. The relevant length scale is therefore not
the 2D root mean square end-to-end distance R
2
(0) but the distance between
2D density blobs
[31].
This approach leads to values of y that depend on the solvent quality on
the interface. For the case where the interface serves as a good solvent, the
mean-field theory predicts
ξ
= 0.75 [31] while the numerical calculation by Le
Guillou and Zinn-Justin yields
ν
= 0.77 [32], leading to predictions of y = 3
and 2.86, respectively. For cases where the interface gives rise to the theta sol-
ν
