148
6 Micromechanical Models of PVA-Based Bionanocomposite Films
f V
D eq
=
1
InD max eq − InD min eq
D eq
(6.43)
The number dN of particles with various sizes in a range of [D eq , D eq + dD eq ] is
then calculated as:
d N =
f V
D eq
d D eq
V
D eq
(6.44)
where V (D eq ) is the volume of anisotropic particles. Consequently, the number-based
probability density function f N
D eq
for anisotropic particles is expressed as:
f N
D eq
=
f V
D eq
V
D eq
N
(6.45)
and N is define by
N =
D maxeq
D mineq
f V
D eq
V
D eq
d D eq
(6.46)
By substituting Eqs. (6.40) and (6.41) into Eq. (6.46), f N
D eq
is given by:
f N
D eq
=
−q
D
−q
max eq − D
−q
min eq
D
q+1
eq
q = 2.5 → Fuller gradation
q = 3.0 → EVF gradation
(6.47)
Moreover, the kth moment D
k
eq of area of f N
D eq
is given by
D
k
eq
=
D max eq
D min eq
D
k
eq f N
D eq
d D eq
(6.48)
By substituting Eq. (6.47) into Eq. (6.48), D eq , D
2
eq and D
3
eq are obtained to
determine the volume fraction of interphase for an anisotropic polydispersed particle
system. In addition, the parameters required to calculate ∅ Interphase in term of D eq-min ,
D eq-max , D eq-mean , H and t interphase can be then measured for each nanofiller.