Graphical representation of the state of orientation of polymer systems

The orientation distribution function for noncrystalline structural units in polymer systems cannot be determined completely from any experimental source; only the second and/or fourth moments of the distribution function, i.e., the second and/or fourth orders of the generalized orientation factors F_lm^j, can be evaluated. It is there-fore necessary to estimate the distribution function from F_2m^j and F_4m^j. In this paper, a graphical representation of the state of orientation is first discussed in terms of plots of F₄₀^j against F₂₀^j for several types of distribution functions for uniaxial orientation. These are three types of extreme concentration of the distribution at particular polar angles θ₀ given by θ₀ = 0, 0<θ₀<π/2, and θ₀ = π/2; five types of rather realistic distributions having single maxima at θ_j = 0, θ₀, π/2 and double maxima at θ_j = 0, π/2, and a single minimum at θ_j = θ₀; and four types of more realistic distributions including Kratky's floating rod model in an affine matrix. Second, estimation of the distribution function for uniaxial orientation from F₄₀^j and F₂₀^j is discussed quantitatively in terms of the mean-square error by three approximation methods: (a) expansion of the distribution function in finite series of spherical harmonics through the fourth order, (b) approximation of the distribution function as a composite of two components, random orientation and a particular orientation distribution given by N_a (cos²θ_j)^a, N_a being a constant, and (c) approximation of the distribution function by N_a (cos²θ_j)^a alone. It is concluded that when the orientation distribution is sharp, estimation by the second method of approximation gives a smaller error than the first.