ABSTRACT
As the orientation characteristic of the polymer chains link with number j, we use the tensor τj ⊗ τj, in which τj denotes a unit vector defining the space orientation direction of the j-th link. Clearly, a change in the direction of the vector τj to the opposite produces no effect on the values of the tensor τj ⊗ τj. The proposed tensor is suitable for our purpose because its averaged value can be found for all possible links of polymer chains and the result of such averaging will not be equal to a zero tensor. It gives an estimate of the oriented state of the polymer. Its eigenvectors and eigenvalues have a clear physical meaning. The eigenvectors of the averaged tensor τj ⊗ τj define the space directions along which the chain links are mainly oriented and the directions along which the links are rarely oriented. The eigenvalues of this tensor give a quantitative estimate of the degree of orientation of polymer chains in corresponding directions. We assume that the energy of interaction between the i-th and j-th links of polymer chains is represented as a potential
uij = u0w1(rij) w2(τi ⋅ τj),
where u0 is the depth of the energy well, and w1(rij) is the Lenard-Jones potential energy function. The dependence of the energy of interaction uij on the angle between the orientation directions of the links of polymeric chains can be written as
3 1 3
( ) ( ) .τ τ τ τ τ τ τ τ⋅ = ⋅ − = ⊗ ⋅ ⊗ −
The energy of interaction of the i-th link with other links is defined as
u u w r w
u C
= ⋅
= − ⊗ ⋅ 〈 ⊗ = ≠ = ≠ ∑ ∑0 1
0 1 3
, , ( ) ( )τ τ
τ τ τ τ j V〉⎛⎝⎜ ⎞ ⎠⎟ ,
where the value of the constant CN is determined by the discrete normalization condition
N ρ ( ) ,
, =
where ρ (rij) = −w1(rij) is the weight factor. We use the space averaging 〈 ⊗ 〉τ τj j V which in
the discrete formulation can be obtained as
〈 ⊗ 〉 = ⊗ = ≠ ∑τ τ ρ τ τj j V N ij
j jC r( ) , ,1
and in the continuum formulation as
〈 ⊗ 〉 = ∫τ τ ρj j V V V
C r dV( ) .O
Here, the value of the constant CV is determined by the continuum normalization condition
C r dVV V
ρ ( ) .=∫ 1 We assume that there is a continuous twice differ-
entiated tensor function O(t, x), which provides calculation of the energy of interaction of the i-th link with the remaining material, as it is usually fulfilled in the context of probability and discrete models.
