a) b - École Polytechnique de Montréal

Based on Sterling’s approximation:
Equation 2-6. ln N! = N ln N − N
Installation of Equations 2-5 and 2-6 in the Equation 2-4 yields:
Equation 2-7. Δ = −k
( N ln x + N ln x )
where
x
1
N1
= and
N
x
2
S m
1
1
2
2
N 2
= . For mixing of polymer components, calculation of entropy of
N
mixing can be developed to(Flory, 1942, 1944; Huggins, 1942):
Equation 2-8. ⎟ ⎛ φ1
φ2
⎞
ΔS = − ⎜
m kV ln x1
+ ln x2
⎝V1
V2
⎠
where V is volume of each polymer molecule and φ 1 represents the volume fraction of phases.
The enthalpy of mixing was expressed by Flory(Flory, 1953) as:
Equation
ΔH
m =
φ kTN
2
1
χ
12
where χ12 is the Flory-Huggins interaction parameter and characterizes the interaction energy per
solution molecule divided by kT. Consequently, the free energy of solution for low molecular
weight molecules was expressed by Meyer-Flory-Huggins as:
Equation 2-9. Δ = kT ( N ln φ + N ln φ + χN
φ )
G m
1
1
McMaster found that volume changes during the mixing of polymers should be considered in the
represented formulation(McMaster, 2002).
Scott extended the calculation of free energy to polymer blends by the following relationship:
Equation 2-10. ⎟ ⎛ V ⎞⎛
φ1
φ2
⎞
ΔG =
⎜
⎟
⎜
m
kT lnφ1 + lnφ2
+ χφ1φ
2
⎝Vs
⎠⎝
M1
M 2
⎠
2
2
1
2
11