Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 2016, S. Roth

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1 > : Small-Angle X-ray Scattering (SAXS) > : Applications & A short excursion into Polymeric materials > : Grazing incidence SAXS (GISAXS) Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

2 T-SAXS vs GISAXS z y x q q z q y q x Beamstop a i z y x a f q z q q y q x Si Lee et al., Macromolecules, 38, 8991 (005) - Easy measurement - Easy analysis - In-plane information (q y,q z ) - Any possible scattering from substrate - Transparency of substrate - High energy - Strong intensity - Easy preparation of samples - Full information (q x,q y,q z ) - Scattering from surface / internal structure - Scattering from reflected AND transmitted beam - Refraction effects (DWBA) - Special setup Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

Aim > To understand the structure property relation of materials on

pieces & materials - Model systems - Nanotechnology http://news.thomasnet.

GE-Gas-Turbine-Technology-Selectedfor-Pearl-GTL-Project-in-Qatar-495497

3 Aim > To understand the structure property relation of materials on multiple length scales - q-resolution - Maximum q-value - Beam size - Real pieces & materials - Model systems - Nanotechnology GE-Gas-Turbine-Technology-Selectedfor-Pearl-GTL-Project-in-Qatar Courtesy: R. Gilles (TUM) 3 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

4 Outline II - Today > SAXS Introduction > Instrumentation Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter. Eds: P. Lindner, Th. Zemb. North Holland Delta Series, Elsevier, Amsterdam (00) ISBN: PETRA III > Bulk materials Transmission U/SAXS: Porous materials Ni-base superalloys Droplet drying 4 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

5 Cross Section > Differential cross section k f I 0 Detector I dw q Q dσ = I I 0 L dω dσ dω = I L dσ I 0 dω = 1 dσ V dω k i V = Sample volume L q = k f -k i k f k i q = π λ sin (θ) > Scattering occurs due to density differences 5 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

WAX, SAX, GISAX Source: Streumethoden zur Untersuchung kondensierter Materie 1996; ISBN 978-3-89336-180-9 Limit Guinier Porod Bragg SAXS WAXS WAXS: Crystal structure d / resolution [Å] SAXS/GISAXS:

6 WAX, SAX, GISAX Source: Streumethoden zur Untersuchung kondensierter Materie 1996; ISBN Limit Guinier Porod Bragg SAXS WAXS WAXS: Crystal structure d / resolution [Å] SAXS/GISAXS: density fluctuations, precipitates Log I(q) 1 10 =1nm =1µm =10µm Log q > R~particles radius > d~interatomic distance > SAXS: q < 5 λ = d sin θ λ = 1.54Å 6 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth d R

7 Scattering Amplitude > Interference in far field k f q q q q k f k i > Phase difference: Δφ i = k f k i r i = q r i > Scattering amplitude: A q = ρ r e iqr dv = ρ r e iqr d 3 r > Intensity: I q = 1 V A q 7 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

8 Form factor and structure factor: Fourier transform p(r) Single particle: Fourier transformation A q = ρ P r e iqr dv = ρ P r e iqr d 3 r Particle distribution function G(r) Electron density distribution ρ r = ρ P r i = ρ P r G(r r )d 3 r = ρ P r G(r) i Scattering amplitudes of the whole arrangement A q = ρ r e iqr dv = [ρ P r G(r) ]e iqr d 3 r i = ρ P r e iqr dv G r e iqr dv Scattered Intensity I q = 1 V A q = P q S(q) r i Form factor Structure factor 8 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

9 Two phase Model: Dilute systems > Only form of particle relevant > Matrix M, volume fraction F Particles P, volume fraction (1-F) Electron density: M,P =n M,P *f M,P f M,P : atomic form factor ( extension of the electron cloud, resonances) n M,P : number density of atoms > Consider M,P as constant resp. ASAX R R R x >> R 9 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

10 Two phase Model > Scattering amplitude: A q = ρ r e iqr d 3 r= ΦV ρ M r e iqr d 3 r + ρ P r e iqr d 3 r (1 Φ)V A q = Δρ ΦV A q = ρ M ρ P e iqr d 3 r ΦV e iqr d 3 r > I q = 1 V A q ~ ρ > Porod Invariant Q (Porod, 198): Q = I q d 3 q = 4πΦ(1 Φ)Δρ Ableiten! Mittelung <..> erklären S.5, S.51 > Only dependent on density contrast D 10 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

11 Herleitung Porod-Invariante > Siehe Handzettel und Übung > Q-Berechnung Übung 11 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

12 Two phase Model single particle approximation > Amplitude: A q = Δρ e iqr d 3 r ΦV > Intensity: I q = 1 V A q > Closer look at I(q) for dilute systems: N P independent scatterers > Incoherent sum of intensities: I q ~NP V P Δρ 1 V P R n M f M = M n P f P = P V P e iqr d 3 r 1 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth R V P ~R 3 Δρ r n P f P n M f M

13 Two phase Model single particle approximation > Amplitude: A q = Δρ e iqr d 3 r ΦV > Intensity: I q = 1 V A q > Closer look at I(q) for dilute systems: N P independent scatterers > Incoherent sum of intensities: I m q ~NP V P Δρ 1 V P P( q) V P e iqr d 3 r sin( qr) qr cos( qr) 3 3 ( qr) - Form factor of a sphere of radius R - Isotropic scattering 13 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

14 Colloid: homogeneous sphere of radius R A simple, but important calculation: F( q) R iqr 3 iqr ( r) e d r 0 e r sin( q) dqddr V particlevolume R R iqr iqr iqrcos( q ) e e 0 e r sin( q) dqddr 0 r sin( ) qr R R R 4 0 r cos( qr) cos( qr) 0 sin( qr) r dr dr q q q q F( q) q dr F( q) F( q) 4 q 0 Rcos( qr) sin( qr) 3 sin( qr) qr cos( qr) 4R 0 q q qr 3 14 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

15 Colloid:homogeneous sphere of radius R 15 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

16 > Q 0 Guinier radius > Homogenous sphere of radius R P q = 3 sin qr qrcos(qr) qr 3 ~1 1 5 q R ~exp ( 1 5 q R ) Ableiten > Radius of gyration: replace homogenous sphere by shell of same moment of intertia: R g > R g = 3 5 R > P q ~exp ( 1 general form of Guinier law [Guinier (1955)] 3 q R g ) > Independent of particle form 16 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

17 Guinier Approximation lim I( q) q0 D V exp( q R g 3 ) Radius of Gyration R g Monodisperse spheres of radius R: R g 3/ 5 R nm Colloids domains Roth et al., Appl. Phys. Lett. 91, (007) 17 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

18 Porod s law: large q Scattered intensity: ~ 4R I(QR) QR 3 sin( qr) qr cos( qr) 0 3 qr ~ ~ ~ Look at maxima of form factor 4 1 q sin( qr) qrcos( qr) 1 qr 40 R R 3 0 qr qr 6 S ~ V P 4 sin( qr) qr cos( qr) 3 qr q qr 3 ~ 4 Surface of sphere qr 3 18 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

19 log I [a.u.] Porod s Law SAXS q^-4 q^-4 USAX I(qR) R>1µm log q [nm-1] R~18nm qr P qr > 4.5 = π S V P q 4 > Depends only on Surface and particle Volume > No shape dependance 19 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

20 The structure factor many particle, close distance > Real systems: not dilute, many particles > Generalisation of Bragg s Law in crystallography: I(q)= c P(q) S(q) Form factor Structure factor R Interference due to assembly of particles > Periodic ordering with periodicity d,x in the electron density : D max,x > I(q) shows a corresponding maximum at q=/(d max, x) S q 1 exp ( σ q² D ) 1 exp σ q² D cos qd max + exp( σ D q ) Lode (1998) Roth et al., J. Appl. Cryst. 36, 684 (003) Smearing Distance of particles 0 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, R. Roth

21 The Structure factor many particle, close distance > Real systems: not dilute, many particles > Generalisation of Bragg s Law in crystallography: I(q)= c P(q) S(q) > Examples: R=5nm, D max =100nm, 5nm, sd/d max =5% Low F P(q), S(q)=1 High F S(q)P(q) q [nm -1 ] q [nm -1 ] 1 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, R. Roth

22 Structure factor and form factor > D max =5nm D max =10nm > s D = 5nm, 1nm, 0.1nm > S q 1 q well separated particles Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

23 Colloidal System > Latex spheres in water I(q)= c P(q) S(q) Low F High F P(q), S(q)=1 S(q)P(q) > Gaussian distribution of particle sizes > Shift in maximum: Decreasing distance Hu et al., Macromolecules, 41, 5073 (008) q [nm -1 ] 3 Methoden Moderner Röntgenphysik II - Vorlesung im Haupt-/Masterstudiengang, Universität Hamburg, SoSe 016, S. Roth

Methoden moderner Röntgenphysik II Streuung und Abbildung

Methoden moderner Röntgenphysik II Streuung und Abbildung Stephan V. Roth DESY 1.5.15 Outline > 1.5. : Small-Angle X-ray Scattering (SAXS) > 19.5. : Applications & A short excursion into Polymeric materials