UNESCO/IUPAC Postgraduate Course in Polymer Science Lecture: Introduction to X-ray and neutron scattering Zhigunov Alexander Institute of Macromolecular Chemistry ASCR, Heyrovsky sq., Prague -16 06 http://www.imc.cas.cz/unesco/index.html unesco.course@imc.cas.cz
Contents Examples of polymeric structures Common principles Wide-angle X-ray scattering Small-angle x-ray and neutron scattering Examples of structural studies
Examples of polymer structures Polymer chain (in solution) LENGTH SCALES 10 0-10 3 Å Chains and particles Polymer particles LATEX MICELLE 10-10 3 10 1-10 Networks ξ 10 1-10 3 CRYST. AM. CRYST. Semicrystalline and organized structures 10 1-10 LONG PERIOD: 10 1-10 Å CRYSTAL STRUCTURE: 10 - - 10 0 Å CUBIC, LAMELLAR, HEXAGONAL
Wave characteristics are: amplitude, frequency, phase, speed Waves λ - Wavelength of a sinusoidal wave is the spatial period of the wave (the distance over which the wave's shape repeats). When passing through media: Absorption Reflection Interference Refraction Diffraction Polarization Picture by Spigget
Diffraction theory When a wave passes through an opening in a barrier, the wave spreads out, or diffracts. When two waves occupy the same location, they interfere. When this interference results in a larger wave, we call it constructive interference. When the size of the wave is reduced, it is called destructive interference.
Waves interactions When x-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as Rayleigh scattering (or elastic scattering). These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. Picture by Christophe Dang Ngoc Chan
Scattering Experiment Scattering vector r r r q = ( π / λ )( s s0 ) q = (4π / λ) sin θ S r Scattering intensity I(q) r X,N SOURCE COLL. θ SAMPLE S r 0 DETECTOR PLANE Wave should be coherent and collimated (parallel waves)
SAXS and WAXS WAS: Wide-Angle Scattering. Single crystal. Material with inhomogeneities with size of inter-atomic distances shows diffraction spots at angles -90º SAS: Small-Angle Scattering. Material, containing inhomogeneities from 10 to 1000 Ǻ scatters radiation into agngles 0- º
Small and Wide Angle Scattering
Scattering of materials Homogenous material Material with inhomogeneities Single crystal Polycrystalline material Primary beam only. No scattering. Vacuum is the only homogenous material. Inhomogenities 10-1000 A Scattering angle ~ 0- o Small Angle Scattering Interatomic distances Scattering angle ~-90 o Wide Angle Scattering Angle depends on disnatces
Polycrystalline sample
Examples of WAXS patterns AMORPHOUS SAMPLE SINGLE CRYSTAL POLYCRYSTALLINE POWDER
Crystal structure Unit cells Miller indices a3 Simple cubic a a1 Body-centered cubic Face-centered cubic
Crystal structure System Triclinic Monoclinic Orthorombic Tetragonal Hexagonal --- Hexagonal division --- Rhombohedral division Cubic Axial Translations (Unit-cell constants) a b c a b c a b c a = b c a = b c a = b = c a = b = c Angels between Crystal Axes (degrees) α β γ 90 β 90 α = γ = 90 α = β = γ = 90 α = β = γ = 90 α = β = 90 γ = 10 α = β = γ 90 α = β = γ = 90 1400 100 p-tsa 1000 WAXS of p-toluenesulfonic acid Intensity, a.u. 800 600 400 00 0 5 10 15 0 5 30 35 40 Θ, degree
Bragg s Law d sin Θ = n λ d = interplanar distance q d = π n n = integer The interference is constructive when the phase shift is a multiple of π Geometry of the Bragg reflection analogy: Lattice planes The waves reflected by the two adjacent planes are in phase at scattering angle Θ given by the Bragg equation. For all values of Θ that do not satisfy this equation the diffracted rays are out of phase with each other and no reflection is observed.
Examples of polymer structures Scattering from a single atom b = scattering length (s. amplitude) b X-ray = 0.8 x 10-1 cm x number of electrons b N = tabulated b N (H) = - 0.374 x 10-1 cm b N (D) = + 0.667 x 10-1 cm r r r q = ( π / λ)( s s0) Scattering vector B r Incident beam θ O Scattering from a group of atoms Scattered S r S r 0 Which technics to use? I = b r I q = TotalAmplitude rr ( ) = b exp( iqr ) k k k I = I(qr) {Short distances high q (WAXS) long distances small q (SAXS, SANS)}
WAXS on Polypropylene + 50 wt% Starch DEGREE OF CRYSTALLINITY x c = 0 0 q q I c ( q)dq I( q)dq CRYSTALLITE SIZE L = Kλ β cosθ β breadth of the reflection
WAXS Degree of crystallinity F Size of crystallites E A Distinguishing between ordered and disordered structures WAXS on Polymers Lattice parameters D C B Identification of crystalline phases Crystal structure (single crystals, fibres)
SCATTERING PATTERN Interpretation of SAS data I(q) q? STRUCTURE Scattering intensity: I(q) = P(q)S(q) Form factor of sphere: 4 3 sin( qr ) q R cos( qr ) P ( q, R ) = πr ρ 3 3 3 ( qr ) Structure factor for N beads: 1 S( q) = N N n j= 1 k = 1 exp [ iq ( r j r k )] Distance distribution function: 0 r sin( qr ) p( r ) = I ( q ) q dq π qr Scattering length density: ρ n Zr i= = 1 V m e where Z is the atomic number r e =.81 x 10-13 cm, is the classical radius of the electron V m is molecular volume
Radius of Gyration Radius of gyration is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the object s parts from its center of gravity. In polymer physics, the radius of gyration is proportional to the root mean square distance between the monomers: R g = 1 N ( ri i, j r j ) Rg = Sphere Thin rod Thin disc Cylinder 3 R 5 L R g = 1 R R g = R R g = + L 1
Interpretation of SAS data Experimental SAS curve I exp (q) Structure parameters (e.g., Rg, V, S) A priori information Other techniques Structure model NO I(experiment) = I(model)? YES STRUCTURE (? )
SAXS vs SANS Range of scattering vectors: q = 10-3 - 10-1 Å -1 Length scale: D = 10 1-10 3 Å Scattering density: ρ = b/v Scattering contrast: ρ(r) = ρ(r) - ρ 0 r r rr Scattering intensity: I( q) = ρ(r)exp( iqr)dv V Contrast variation for Multicomponent Particles???!!! ρ 0 ρ I(q) = I 1 (q) ρ 0 = ρ I(q) = I 1 (q) ρ ρ 0 ρ 1 1 ρ ρ 0 ρ 1 ρ ρ 0
Scattering from a polymer chain I I=I(0) Guinier exp(-q R g /3) Debye ~q - D: Size of chain L p : Persistence length Rod-like ~q -1 0 Length of scattering vector q q -1 qd«1 qd 1 qd ql p Magnification increases
SAXS and SANS on polymers SANS: Contrast variation. Studying of multicomponent particles B Solid polymers: Characterization of heterogeneities (pores, domains in block copolymers, fractal structures,...) A SAXS and SANS on polymers C Polymer particles: Shape, size (distribution), mass, surface, internal structure, degree of swelling Semicrystalline polymers: Degree of crystallinity, long period, size of crystallites. E D Polymer chains: Radius of gyration, mass, persistence length, cross-sectional parameters.
SANS Example
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