66
2 Actual Potentials of Theoretical Chemistry: What Can Be Obtained
(Dasgupta et al. 1996), ab initio molecular dynamics (MD) method (Hageman et al.
1997), and CO (see Sect. 3.3) methods using Hessian matrix method (Bartha et al.
2000), and so on.
In the following two subsections, theoretical estimations of the quantities related
to elastic constants of several kinds of oligomers based on the HF method, and those
of polymers with infinite chain length based on the direct estimation using the CO
method are to be described. Each method has its own feature but has common aspect
in directly dealing with deformation modes of the structural parameters such as
bond lengths, bond angles, and dihedral angles due to the elongation of the original
polymers along the longitudinal direction.
2.6.3 Young’s Moduli of Oligomeric Species
Here is discussed the response to mechanical stress by compression or tensile strength
applied to the linear oligomeric chains. In this subsection, the oligomeric materials
are to be called polymers for simplicity, although the MO method is effective only
to oligomers with finite length. In this sense, the obtained result of Young’s modulus
in the below ought to be considered with a certain reservation and/or extrapolated to
that of the infinite polymer chain.
Under small mechanical stress σ caused by the above strength along the mainchain direction of a polymer in general, the response appears as strain ε and is
represented as
σ = E l ε
(2.46)
within the range of linear relationship. The coefficient E l denotes Young’s modulus
with respect to the polymer deformation. Since the stress σ can further be expressed
as F/S with F and S being, respectively, the force applied to the polymer and its
cross-sectional area, E l can be expressed as
E l =
F/S
ε
(2.47)
This equation implies that E l has the same dimension as that of pressure such as
dyn/m
2 or N/m
2 , since strain is a dimensionless quantity. If we consider stretch of
a polymer along its main-chain direction, for instance, ε can be expressed as L/L
(see Fig. 2.58). Hence Young’s modulus E l can also be written as
E l =
F/S
L/L
(2.48)
In the theoretical calculation, one can estimate destabilization energy W caused
by stretch L of the concerning polymer from its original length L at the optimized
2 Actual Potentials of Theoretical Chemistry: What Can Be Obtained
(Dasgupta et al. 1996), ab initio molecular dynamics (MD) method (Hageman et al.
1997), and CO (see Sect. 3.3) methods using Hessian matrix method (Bartha et al.
2000), and so on.
In the following two subsections, theoretical estimations of the quantities related
to elastic constants of several kinds of oligomers based on the HF method, and those
of polymers with infinite chain length based on the direct estimation using the CO
method are to be described. Each method has its own feature but has common aspect
in directly dealing with deformation modes of the structural parameters such as
bond lengths, bond angles, and dihedral angles due to the elongation of the original
polymers along the longitudinal direction.
2.6.3 Young’s Moduli of Oligomeric Species
Here is discussed the response to mechanical stress by compression or tensile strength
applied to the linear oligomeric chains. In this subsection, the oligomeric materials
are to be called polymers for simplicity, although the MO method is effective only
to oligomers with finite length. In this sense, the obtained result of Young’s modulus
in the below ought to be considered with a certain reservation and/or extrapolated to
that of the infinite polymer chain.
Under small mechanical stress σ caused by the above strength along the mainchain direction of a polymer in general, the response appears as strain ε and is
represented as
σ = E l ε
(2.46)
within the range of linear relationship. The coefficient E l denotes Young’s modulus
with respect to the polymer deformation. Since the stress σ can further be expressed
as F/S with F and S being, respectively, the force applied to the polymer and its
cross-sectional area, E l can be expressed as
E l =
F/S
ε
(2.47)
This equation implies that E l has the same dimension as that of pressure such as
dyn/m
2 or N/m
2 , since strain is a dimensionless quantity. If we consider stretch of
a polymer along its main-chain direction, for instance, ε can be expressed as L/L
(see Fig. 2.58). Hence Young’s modulus E l can also be written as
E l =
F/S
L/L
(2.48)
In the theoretical calculation, one can estimate destabilization energy W caused
by stretch L of the concerning polymer from its original length L at the optimized