Small-angleSmall-angle X-rayX-ray scatteringscattering (SAXS)(SAXS) withwith synchrotronsynchrotron radiationradiation Martin Müller Institut für Experimentelle und Angewandte Physik der Christian-Albrechts-Universität zu Kiel • Introduction to small-angle scattering • Instrumentation • Examples of research with SAXS Small-angleSmall-angle X-rayX-ray scatteringscattering (SAXS)(SAXS) withwith synchrotronsynchrotron radiationradiation • Introduction to small-angle scattering • Instrumentation • Examples of research with SAXS WhatWhat isis small-anglesmall-angle scattering?scattering? elastic scattering in the vicinity of the primary beam (angles 2θ < 2°) at inhomogeneities (= density fluctuations) typical dimensions in the sample: 0.5 nm (unit cell, X-ray diffraction) to 1 µm (light scattering!) WhatWhat isis small-anglesmall-angle scattering?scattering? pores fibres colloids proteins polymer morphology X-ray scattering (SAXS): electron density neutron scattering (SANS): contrast scattering length OnOn thethe importanceimportance ofof contrastcontrast …… ScatteringScattering contrastcontrast isis relativerelative Babinet‘s principle two different structures may give the same scattering: 2 I(Q) ∝ (ρ1 − ρ2 ) r 4π scattering vector Q = sinθ λ 2θ DiffractionDiffraction andand small-anglesmall-angle scatteringscattering cellulose fibre crystal structure scattering contrast crystals - matrix M. Müller, C. Czihak, M. Burghammer, C. Riekel. J. Appl. Cryst. 33, 817-819 (2000) DiffractionDiffraction andand small-anglesmall-angle scatteringscattering Diffraction: 2 r r I(Q) = ∑ f (Q) eiQ⋅Rl l atomic form factor, lattice interference electron distribution ⇒ Bragg peaks small-angle scattering: • form factor = Fourier transform of particle shape • structure factor = interparticle interference FormForm factorfactor andand structurestructure factorfactor form factor: particle shape structure factor: (mean) particle distance FormForm factorfactor (dilute(dilute systems)systems) 2 3(sin QR − QR cosQR) sphere: I(Q) ∝ Q3R3 FormForm factorfactor andand structurestructure factorfactor (non-dilute)(non-dilute) basic principle as in diffraction: • form factor (as before): single particles, dilute systems • structure factor: interparticle distances of the order of particle size ⇒ interference simple multiplication only for spherical symmetry! „long-range Bragg peak“ (d =2π/q ≈ 120 Å) PolystyrenePolystyrene spheresspheres (71(71 nmnm radius)radius) inin glycerolglycerol concentrated: also structure factor d = 127 nm dilute: only form factor d: mean distance L. B. Lurio et al., Phys. Rev. Lett. 84, 785 (2000) DataData analysisanalysis ofof aa SAXSSAXS experimentexperiment 2D SAXS pattern r = 15-20 Å flax fibres averaging scattering curve cellulose crystallites ( = microfibrils) in amorphous matrix (density contrast) FormForm factor:factor: fitfit withwith modelmodel functionfunction cellulose microfibrils in flax fibres, long cylinder with radius r (= 15 Å) yields: better fit with assumption of size distribution (± 4 Å) = polydispersity Model-freeModel-free Guinier parameterparameter determinationdetermination single particle scattering, 2 phases; independent of topology and invariant geometry Porod • invariant volume fractions scattering contrast • Porod limit for distances larger than typical distances in the sample and sharp interfaces: specific (inner) surface InvariantInvariant andand PorodPorod scattering scattering • 90 % white • 10 % black different scattered intensity, but same invariant Porod specific (inner) surface: larger for A GuinierGuinier analysis analysis Guinier (1938): For very small angles scattering function independent of particle shape, only dependent on size: for Guinier radius rG (details „not seen“ at low angles) WhyWhy anan indirectindirect method?method? • non-destructive, no tedious sample preparation (sectioning, staining…) • averaging of larger areas • simultaneous information on several length scales in combination with e. g. diffraction • soft matter: liquids, solutions, emulsions, biological samples… Fourier transforms! H. F. Jakob et al., Macromolecules 28, 8782 (1995) Small-angleSmall-angle X-rayX-ray scatteringscattering (SAXS)(SAXS) withwith synchrotronsynchrotron radiationradiation • Introduction to small-angle scattering • Instrumentation • Examples of research with SAXS PinholePinhole SASSAS cameracamera long distance for high resolution L2 LD low-divergence beam sample 2Θmin Detector beamstop defining guard Apertures Ad: 0.01 mm Ag: 0.02 mm beam shaping detector protection without cutting SAXS signal PinholePinhole cameracamera ID02ID02 atat ESRFESRF “Resolution”“Resolution” in in small-anglesmall-angle scatteringscattering SAXS resolution: 2θmin → dmax d = 1000 Å standard calibration material: rat tail collagen (periodic structure: 67 nm) M. Müller, M. Burghammer, C. Riekel Nucl. Instrum. Meth. A 467-468, 958-961 (2001) MicrodiffractionMicrodiffraction and and µSAXSµSAXS atat thethe ESRFESRF MicrofocusMicrofocus Beamline Beamline focussed X-ray beam single fibre ID13 (∅ 20 – 40 µm) video microscope z y xy translation CCD detector stage at sample position: • beam size ∅ 0.5 - 2 μm •flux1010 -1011 ph/s @ 13 keV CombinationCombination with with microfluorescence microfluorescence ID13 Röntec fluorescence detector X-rays samples beamstop Bonse-HartBonse-Hart cameracamera apertures / slits replaced by Si single crystals (very low angular aceptance): + extremely high resolution (dmax = 7 µm) - low flexibility in flux and resolution SAXSSAXS inin thethe laboratorylaboratory very parallel radiation (low divergence) with Göbel mirrors (graded multilayers) but: lower flux, fixed energy Small-angleSmall-angle X-rayX-ray scatteringscattering (SAXS)(SAXS) withwith synchrotronsynchrotron radiationradiation • Introduction to small-angle scattering • Instrumentation • Examples of research with SAXS • SAXS with a microbeam: cellulose • porosity development in carbon fibres MicrofibrilMicrofibril orientation orientation in in cellulosecellulose fibres fibres flax fibres orientation of crystalline microfibrils responsible for mechanical strength and stiffness measurement with µSAXS (2 µm beam size) on single fibres Micro-SAXSMicro-SAXS on on flaxflax cellulose cellulose fibres fibres equator increasing width of streak with Q position dependence of microfibril alignment M. Müller, C. Czihak, G. Vogl, P. Fratzl, H. Schober, C. Riekel Macromolecules 31, 3953-3957 (1998) PorosityPorosity development development in in carboncarbon fibres fibres PAN-based carbon fibre 2D µSAXS with skin-core structure patterns: untreated 1 nm-1 after activation 2 µm with NaOH (2 h at 750 °C) D. Lozano Castelló, J. A. Maciá Agulló, D. Cazorla Amorós, A. Linares Solano, M. Müller, M. Burghammer, C. Riekel. Carbon 44, 1121–1129 (2006) PorosityPorosity development development in in carboncarbon fibres fibres 200 hxna391 hxna391 core hx1 hx1 core 150 hxk281 hxk281 core NaOH activated 100 KOH activated 50 not activated volume corrected invariant core disordered core region 0 -4 -2 0 2 more4 easily activated position (μm) by NaOH than by KOH (beam size 0.5 µm) Micro-SAXSMicro-SAXS on on singlesingle starch starch granules granules dry wet appearance of new reflections at 1.6 nm H. Lemke, C. Riekel (2003) MicrodropletMicrodroplet generator generator Time-resolvedTime-resolved kinetics kinetics (seconds): (seconds): HydratisationHydratisation of of starchstarch granules granules intensities of the 1.6nm peak outer shell inner shell scanning of a single starch grain during the hydratisation reaction ID13, ESRF droplet frequency 1 s-1 time / 4 s Time-resolvedTime-resolved kinetics kinetics (milliseconds): (milliseconds): Self-assemblySelf-assembly of of ionicionic surfactants surfactants ID02, ESRF; 20 ms exposure! AnomalousAnomalous X-ray X-ray scattering scattering energy-dependent atomic scattering factors (up to 20 % variation) close to X-ray absorption edge: f (E) =Z + f ′(E) + if ′′(E) atomic number anomalous scattering factor absorption Mo K-edge ⇒ contrast variation IronIron oxideoxide precipitates precipitates in in coppercopper single single crystal crystal containingcontaining 1 1 at%at% FeFe (2) 2 species of iron oxide precipitates: (1) platelets with 200 Å ∅ on {111} (1) (2) platelets with 330 Å ∅ on {100} (1) contrast variation by measuring at Cu and Fe edges: (1) Fe3O4 (2) γ-Fe2O3 O. Paris et al., Acta metall. Mater. 42, 2019 (1994) HierarchicalHierarchical cellulose cellulose structure structure unit cell microfibrils plant cell walls typical size (nm) Position-resolvedPosition-resolved X-rayX-ray diffractiondiffraction andand small-anglesmall-angle scatteringscattering withwith aa microbeammicrobeam Simultaneous information on three length scales: diffraction (WAXS): SAXS: position resolution: unit cell pores, particles optical microscopy 12 3 typical size (nm) C. Riekel, M. Burghammer, M. Müller, J. Appl. Cryst. 33, 421-423 (2000).