Chemistry Reference
In-Depth Information
the spatial arrangement of polymer molecules must, then, occur in order for the
interfacial thickness to become less than the unperturbed chain dimensions. Chain
perturbations will also occur at a polymer air interface for the same reason, i.e., the
thickness of a polymer surface region (the region between unperturbed bulk polymer
molecules and air) will also typically be less than the chain dimensions of the polymer
molecules. Such chain perturbations contribute to the excess energy of surfaces or
interfaces, and are reflected in the values of surface and/or interfacial tension. Since
there is no direct relationship between the chain perturbations that occur at the
polymer air surfaces of the two individual polymers and the perturbations that
would occur at the interface in a demixed polymer blend, there can be no direct
fundamental relationship between the properties of polymer surfaces (surface tension)
and polymeric interfaces (interfacial tension). Therefore, “theories” that attempt to
present relationships for polymeric systems must be looked upon only as empirical.
3.2.2 Microscopic Theories of Polymer Interfaces
A number of thermodynamic theories have appeared that take a more fundamental
approach, and, specifically, address the question of interfacial structure and its
relation to interfacial tension.
Helfand and Tagami [
27
] formulated a statistical mechanical theory of the
interface between two immiscible polymers, A and B. The approach is based on a
self-consistent field, which determines the configurational statistics of the polymer
molecules in the interfacial region. At the interface, energetic forces (determined
essentially by the polymer A/polymer B segmental interaction parameter,
w
) tend to
drive the A and B molecules apart. This separation, however, must be achieved in
such a way as to prevent a gap from opening between the polymer phases. The
energetic force on, say, an A molecule must be balanced by an entropic force
describing the tendency of A molecules to penetrate into the B phase, because of the
numerous configurations of the A molecule which do so.
The theory was originally developed for symmetric systems, i.e., for similar
polymers A and B that possess identical degrees of polymerization (
Z
), effective
lengths of the monomer units (
b
), monomer number densities (
r
0
), and isothermal
compressibilties (
k
). The authors recommended the use of the geometric mean
when these properties are not actually the same.
In the Helfand Tagami mean field formulation, the effective mean field
W
A
(r)
on a segment of polymer A, which is the reversible work of adding the segment at
position r, where the densities are
r
A
(r) and
r
B
(r), less the work of adding the
segment to bulk A, is given by:
W
A
ð
kT
¼ w
r
B
ð
r
Þ
r
0
þ z
r
A
ð
r
Þ
Þ
r
0
þ
r
r
B
ð
Þ
r
0
r
1
(32)
1
Z
1
, where
k
B
is the Boltzmann constant. The first term
arises from the relatively unfavorable interaction of the A polymer segments with
with
z ¼ kr
0
k
B
T
ð
Þ
