78
4 Nanofiller Dispersion in Rubber as Revealed by 3D-TEM
Fig. 4.21 Relationship
between cross-link density
(N.Nd/TV) and CB loading
(WCB) (from Fig. 7 in Ref.
[34])
If the total number of the primary CB aggregates that become visible under TEM is
A, A and the number of linking (N.Nd), which is called as the number of cross-linking
points from now on, are expressed as follows:
A =
y=n
y=1
N y x y
(4.6)
where N y is the number of x y -mer (y is between 1 and n)
N.Nd = q A
(4.7)
Further, the amount of loaded CB is expressed by
W CB = ξ m
y=n
y=1
N y x y
(4.8)
where ξ is a calibration coefficient for the CB amount under the assumption of a
homogeneous distribution of CB. From Eqs. (4.6), (4.7) and (4.8), the cross-link
density (N.Nd/TV) is expressed by
N.Nd/TV = (q/ξ mTV) W CB
(4.9)
where TV is the total volume under the TEM, and TV is assumed not much dependent
on the CB-loading amount. Thus, the cross-link density would be proportional to the
loaded amount of CB, W CB , if the Charlesley’s theory is applicable to this system.