a) b - École Polytechnique de Montréal

where μ is the chemical potential of component i, ni is the number of moles, Ai is the interfacial
i
area, and γ ij is the interfacial tension between components i and j. Thus, the phase structure of a
polymer blend can be predicted by comparing the Gibbs free energies of the different structures.
In their work on a HDPE/PS/PMMA blend, numerous conditions were assumed to simplify the
study such as equal value of the first term of the equation for all types of morphology. For
further simplification, they assumed that numbers of droplets of minor phases were equal to each
other, which lead to a big error.
The second term is simplified further and rewritten as:
Equation 2-32.
Equation 2-33.
Equation 2-34.
1 1 2
1
3
( ∑ Aγ
) = ( 4π
) 3 ⎡ n 3 x γ + n 3γ
⎤(
3V
)
i
1 1
2
1
3
( ∑ Aγ
) = ( 4π
) 3 ⎡ n 3 ( 1+
x)
γ + n 3γ
⎤(
3V
)
i
( ) ( ) ( ) 3
1 1 2
1
2
2
3 3 3
3
3
∑ Aγ
= 4π
⎡ n x γ + n ( 1+
x)
γ ⎤ 3V
i
ij
ij
ij
B+
C
B
C
C
B
where Vi is the volume fraction of phase I,
V
V
C
⎢⎣
⎢⎣
⎢⎣
B
B
B
AB
BC
AB
C
C
AC
C
⎥⎦
BC
c
⎥⎦
2
3
AC
⎥⎦
c
24
B x = , nB and nC are numbers of particles of B and
C phases. The most stable morphology is predicted by calculation of the values of ( ∑ Aγ )
which correspond to the value of the Gibbs energy of mixing.
Obviously, the Harkins equation and the Guo model are purely thermodynamic equations in
which some other factors influencing the morphology, such as rheological parameters, have not
been considered. Although in measuring interfacial tensions of polymer pairs by various methods
such as the breaking thread and pendent drop, viscosities of the components are taken into
account, the effect of elasticity is still neglected. In the method developed by Guo to calculate the
free energy compared to Harkins equation, concentration of components is also contributed to
determine the kind of morphology as well and it is more modified. Researchers use both the
spreading coefficients and the minimal free energy surface model to predict the morphologies of
polymer blends (Guo, Gvozdic, et al., 1997; Guo, Packirisamy, et al., 1997; Hara & Sauer, 1998;
Hobbs, et al., 1988; Horiuchi, Matchariyakul, Yase, & Kitano, 1997; Nauman & He, 2001;
i
2
3
ij
c