Analysis of Small-Angle X-Ray Scattering Patterns

Abstract

Small-angle X-ray scattering (SAXS) patterns are used to identify features of semicrystalline polymers composed of alternating lamellar crystals and amorphous regions. Provided that lamellae are wide and flat, the structure is a one-dimensional stack with an average period L established by distributions of crystal thickness H(y) and amorphous thickness h(z) and with crystallinity α = [ybar]/[xbar], where average crystal thickness is [ybar], and average long period is [xbar]. The one-dimensional intensity I ₁(s ₁) gives L_B , while the one-dimensional correlation function γ₁(r) and the one-dimensional interface distribution function g ₁(r) also provide measures of periodicity r ^* and r ₃, respectively. Obvious features of γ₁(r) and g ₁(r) also permit estimation of crystallinity, designated α_γ and α _g respectively. I ₁(s ₁), γ₁(r) and g ₁(r) are calculated for models having symmetric and positively skewed distributions H(y) and h(z) and for stacks of finite height N[xbar]. For all conditions, it is found that L_B ≥r ^* ≥ r ₃, and that α _g ≥ α ≥ α_γ. With symmetric distributions and infinite N, L_B nearly equals the weight-average period [xbar]_w, and r ₃ = [xbar]. These equalities do not hold when thickness distributions are skewed. Small stack height N[xbar] distorts I ₁(s ₁) and γ₁(r), but the interface distribution function g ₁(r) is scarcely affected, returning r ₃ = [xbar] and α _g = α. It is recommended that all three functions be analyzed to obtain the most complete picture of the semicrystalline microstructure.

Keywords:

• Crystallinity
• Long period
• Semicrystalline polymers
• Small-angle X-ray scattering

Acknowledgments