Interfacial Tension in Binary Polymer Blends and the Effects of Copolymers as Emulsifying Agents - Polymer Thermodynamics - page 206

were equally likely, thus neglecting the fact that bond orientations are inherently
anisotropic. As a measure of the effect of Roe's assumptions on the qualitative
nature of his results, Helfand pointed out that, when a gradient expansion of Roe's
equation was made, the Gaussian random-walk equations [ 27 , 28 ] were not recov-
ered. Experimental verification of the lattice theories, however, has not been
possible, because the lattice parameters a, m , and d are unknown a priori.
Kammer [ 209 ] examined the interfacial phenomena of polymer melts from a
thermodynamic point of view. A system of thermodynamic equations has been
derived to describe the temperature, pressure, and composition dependence of
interfacial structure. Starting from the fundamental equations of Guggenheim
[ 210 ], Kammer employed the Gibbs Duhem equation of intensive parameters
( 13 ) to find that the interfacial composition is given by:

ð d g= dT Þ P þ ð d s 1 = dT Þ P

x 2 ¼ (52)

ð d s 1 = dT Þ P þ 0 : 5 d s 2 = ð dT Þ P

where x 2 is the molar fraction of component 2 at the interfacial region, and s 1 and
s 2 are the surface tensions of the two components against air. Assuming that the
interfacial layer is predominantly occupied by component 2 (i.e., x 1 ! 0), he

obtained:

2
A

m

g ¼

(53)

2 is the chemical potential of component 2 and A is the molar area of the
interface. Use of the Flory-Huggins formula of the chemical potential leads to:

where m

h i

1 2

RT
A

0 S

2

S
1

S

g ¼ g þ

ln f

þ ð r 2 = r 1 Þf þ r 2 wf

1 (54)

S
i are the degrees of polymerization, the Flory-
Huggins interaction parameter, and the volume fraction of component i at the
interphase. The interfacial thickness was shown to be:

0 is a constant, and r i , w , and f

where g

RT g s 2

rwf

a I ¼ (55)

1 2

S S

2

þ ln f

with u the mean molar volume of the polymers.
Hong and Noolandi [ 211 ] have developed a theory for an inhomogeneous
system, starting from the functional integral representation of the partition function
as developed by Edwards [ 212 ], Freed [ 213 ], and Helfand [ 199 ]. The theory has
been used to determine the interfacial properties and microdomain structures of a
combination of homopolymers, block copolymers, monomers, and solvents. In that
approach, the general free energy functional was optimized by the saddle-function
method, subject to constraints of no volume change upon mixing and constant