70
2 Actual Potentials of Theoretical Chemistry: What Can Be Obtained
L
N
L+ L
N
L+ L
L
(a)
(b)
Δ
Δ
Fig. 2.61 Schematic representation of elongation stress to a polymer. Each rectangular denotes the
unit cell. a Original polymer with the total length L and b elongated polymer by with the total
length L + The number of the unit cell is indicated by N, which is actually infinite
where N designates the number of the unit cells formally being infinite. Hence
/N and stand for the energy change and elongation of the polymer length,
respectively, the both quantities being in the sense of per unit cell (see Fig. 2.61).
The number N is formally infinity but can be reduced to the fraction with size of the
unit cell as in Eq. (2.51). The quantity simply implies the elongation length
per unit cell and, in this subsection, is taken as 0.01L/N, that is, 1% elongation of
the original length is considered as in the usual estimation of elastic constant. Also
note that /N is the change in the energy per unit cell upon the elongation. The
prescription to obtain the f -values are in what follows:
(1) The polymer with infinite chain length is first energetically optimized, from
which the corresponding set of all the atomic coordinates in the unit cell and
the unit-cell length L/N are obtained (see Fig. 2.61a). The most stable energy
per unit cell represented by W /N is thus decided (W < 0).
(2) 1% elongation of the whole polymer length L is introduced as the strain along
the polymer chain due to tensile strength. This is realized by making change of
the optimized unit-cell length L/N obtained in (i) into (L + with =
0.01L. The resulting destabilized energy /N ( > 0) affords the data of
f -values with the use of Eq. (2.51).
In Table 2.18 are shown the calculated results of f -values with respect to three
kinds of polymers as shown in Fig. 2.62 along with the experimental data (Nishino
et al. 2014). Note that polyselenophene has not yet been synthesized but that it can be
treated in the theoretical calculation. It is seen that the f -values obtained by theoretical
calculation are in reasonable agreement as a whole with the experimental values in the
order of size. Accompanied change in structural parameters such as bond lengths and
angles of these polymers upon the elongation are listed in Tables 2.19, 2.20 and 2.21
2 Actual Potentials of Theoretical Chemistry: What Can Be Obtained
L
N
L+ L
N
L+ L
L
(a)
(b)
Δ
Δ
Fig. 2.61 Schematic representation of elongation stress to a polymer. Each rectangular denotes the
unit cell. a Original polymer with the total length L and b elongated polymer by with the total
length L + The number of the unit cell is indicated by N, which is actually infinite
where N designates the number of the unit cells formally being infinite. Hence
/N and stand for the energy change and elongation of the polymer length,
respectively, the both quantities being in the sense of per unit cell (see Fig. 2.61).
The number N is formally infinity but can be reduced to the fraction with size of the
unit cell as in Eq. (2.51). The quantity simply implies the elongation length
per unit cell and, in this subsection, is taken as 0.01L/N, that is, 1% elongation of
the original length is considered as in the usual estimation of elastic constant. Also
note that /N is the change in the energy per unit cell upon the elongation. The
prescription to obtain the f -values are in what follows:
(1) The polymer with infinite chain length is first energetically optimized, from
which the corresponding set of all the atomic coordinates in the unit cell and
the unit-cell length L/N are obtained (see Fig. 2.61a). The most stable energy
per unit cell represented by W /N is thus decided (W < 0).
(2) 1% elongation of the whole polymer length L is introduced as the strain along
the polymer chain due to tensile strength. This is realized by making change of
the optimized unit-cell length L/N obtained in (i) into (L + with =
0.01L. The resulting destabilized energy /N ( > 0) affords the data of
f -values with the use of Eq. (2.51).
In Table 2.18 are shown the calculated results of f -values with respect to three
kinds of polymers as shown in Fig. 2.62 along with the experimental data (Nishino
et al. 2014). Note that polyselenophene has not yet been synthesized but that it can be
treated in the theoretical calculation. It is seen that the f -values obtained by theoretical
calculation are in reasonable agreement as a whole with the experimental values in the
order of size. Accompanied change in structural parameters such as bond lengths and
angles of these polymers upon the elongation are listed in Tables 2.19, 2.20 and 2.21