introduction to SAXS for polymers -a user view-
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1 introduction to SAXS for polymers -a user view- Luigi Balzano DSM Ahead/Material Science Center Geleen, The Netherlands Synchrotron and Neutron Workshop (SyNeW) 2015 Utrecht, June
2 outline basics of X-rays what are X-rays and how they work generation of X-rays interference of waves / Bragg s law the SAXS machinery morphological information from SAXS data interpretation structure and form factor polydispersity, lattice disorder what is in for you? examples summary
3 generation of X-rays lab sources, synchrotron and nebulas
4 history Wilhelm C. Röntgen Nobel Prize in Physics (1901) Hand mit Ringen (Hand with Rings): first "medical" X-ray (of Ms Röntgen)
5 commercial applications material science flight tube (vacuum) contains: sample detector sample/cell handling generator
6 brightest source of X-rays crab nebula synchrotron sources (one of the) brightest source of X-rays that can be focused synchrotron
7 typical experimental setup (WAXD) instrument-independent data data reduction transmission mode non-destructive no sample preparation time resolved (30 frames/s) space resolved (beam spot size limiting, 50nm) statistically sound information combination with other techniques possible
8 interference of waves why do we use X-rays? how are patterns formed? basic interpretation: Bragg s law
9 X-ray patterns: interference of electromagnetic waves X-rays scattering/diffraction patterns are the result of the interference between the incoming X-ray photons and the grating formed by the scatterers in the system (lattice) for positive interference, the wavelength should be of the order of the size/distance between scatterers crystals and molecules often have characteristic sizes ~ Å, just about the X-rays wavelength ( they are suitable for investigation with X-rays)
10 interference of waves single slit interference two slit interference N-slit interference. waves are scattered in phase when the path length difference is an integer number of wavelengths n = d sin is the angle where constructive interference takes place
11 examples FraunhoferDiffractionUsingAFastFourierTransform-author.nb MultipleSlitDiffractionPattern-author.nb
12 Bragg s law example of X-ray wave interference when a sample is irradiated with X-rays: waves are scattered in phase (constructive interference) when the path length difference is an integer number of wavelengths path length difference = n wavelength W.L. Bragg W.H. Bragg dsin n Nobel Prize in Physics (1915)
13 application of Bragg s law length scales in SAXS and WAXD 2 dsin n typical length scales: d 2sin small angles (SAXS) large d wide angles (WAXD) small d SAXS 0.1nm d 60 nm 2sin(0.1/ 2) WAXD 0.1nm d 0. 60nm 2sin(9 / 2) 0.1nm SAXS WAXD
14 macro scale spherulites length scales morphology of semi-crystalline polymers meso scale lamellae micro scale unit cells µm nm 1-10 Å SAXS 0.1nm d 60 nm 2sin(0.1/ 2) WAXD 0.1nm d 0. 60nm 2sin(9 / 2) 0.1nm SAXS WAXD detailed information on the morphology of semicrystalline polymers can be obtained with X-rays
15 morphological information from SAXS modeling with form and structure factors
16 the SAXS signal scattered X-rays X-ray radiation 1 2 information on size, shape, content of the phases scattering is caused by etherogeneties in the electron density I( D scattered intensity is the Fourier transform of the electron density correlation function in real space
17 density profile (x) Fourier transform the SAXS signal scattering amplitude F( (amplitude of the form factor) can be complex, not measurable (X-rays have too high frequency) square scattering intensity (form factor) F( 2 loss of information (phase), intrinsic feature of the data recording process ensemble average (polydispersity) orientation average measured intensity I( I( Stribeck N - X-ray scattering of Soft Matter Schultz JM Diffraction for Material Scientists Glatter O and Kratky O Small Angle X-ray Scattering Balta-Calleja FJ and Vonk CG- X-ray scattering of synthetic polymers and more thanks B. Lotz
18 modelling, how? system with identical particles: = x I( 2 n V 2 P( S( n number of scatterers density difference (contrast) V particle volume P( form factor (often 2 ) S( structure factor P(q=0)=S(q=0)=1 Guinier A and Fournet G Small Angle Scattering of X-rays Pedersen JS Adv Colloid Interface Sci 70 (1997), 171 Förster S et al J Phys Chem B 2005, 109, 1347
19 modelling, how? form factor homogeneous sphere with sharp boundary sin( qx) 2 1 R F( ( x) x dx sin( qx) xdx 0 qx q 0 F( R, sin qr qrcosqr 3 qr 3 P( R, F 2 ( R, R=1 - many form factors available in literature -
20 modelling, how? form factor homogeneous spheres with polydisperse radius P ( x, P( x, h( x) dx h(x) 0 distribution function polydispersity can be introduced by averaging the form factor over the size distribution spheres with Gaussian distribution of radii 2 2 sinqr qr cosqr 1 R R 0 3 PRq (, ) 3 Exp 0.5 dr qr 2 Pxq (, ) hx ( ) for certain h(x), several <P(R,> have analytical expressions, see: Pedersen JS Adv Colloid Interface Sci 70 (1997), 171 Förster S et al J Phys Chem B 2005, 109, 1347.
21 modelling, how? form factor homogeneous spheres with polydisperse radius P ( x, P( x, h( x) dx h(x) 0 distribution function polydispersity can be introduced by averaging the form factor over the size distribution spheres with Gaussian distribution of radii
22 modelling, how? structure factor for disordered systems S( 1 dilute system with randomly distributed particles I( 2 n V 2 P( S( I( n V P( q 2 2 ) the intensity scattered by a dilute system with randomly distributed particles represents the form factor of the scatterers asymptotes (Guinier, Porod) recovered in the low and high q limits
23 modelling, how? structure factor for ordered systems ideal systems lattice factor const 2 Z 0( d mhkl fhkllhkl ( 1 q hkl probability to find a particle in certain (hkl) direction non-ideal systems (, G( 1 (, G( 1 structure factor S( 1 ( [ Z0( 1] G( S ( q ) Z 0( q ) ( G( polydispersity positional disorder Förster S et al J Phys Chem B 2005, 109, 1347
24 modelling, how? example: structure factor of an hexagonal lattice (, G( 1 (, G( 1 non-ideal systems structure factor S( 1 ( [ Z0( 1] G( S ( q ) Z 0( q ) ( G( polydispersity positional disorder Förster S et al J Phys Chem B 2005, 109, 1347
25 what is in it for you? examples: growth of shish-kebabs phase separation of (dissolution type) nucleating agents crazing during cyclic loading spatial heterogeneity in injection molded parts
26 modeling growth of shish-kebabs temperature shear time Hobbs, J.K. et al. Macromolecules 2001 material: ipp isothermal at 165 C shear time s wall stress~ 0.16 MPa wall shear rate~ 750 s -1 Balzano L et al PRL 2008 Balzano L et al Macromolecules 2011
27 modeling growth of shish-kebabs idealization Hobbs, J.K. et al. Macromolecules 2001 all identical particles (monodisperse system) Is F s Zs 2 ( ) ( ) ( ) s { s1, s2, s3} form factor (sharp interfaces) D F( s12, s3) J1( Ds12) sin( Ts3) Ds Ts 12 3 structure factor (perfect orientation) Zs 1 H ( s ) L 3 ( 3) Re 1 HL ( s3 )
28 modeling growth of shish-kebabs idealization Hobbs, J.K. et al. Macromolecules 2001 polydisperse system Is F s Fs Zs 2 2 ( ) ( ) ( ) ( ) 1 J. K. Keum et al. Progress in Colloid and Polymer Science, vol. 130, pp , 2005.
29 modeling growth of shish-kebabs Hobbs, J.K. et al. Macromolecules 2001 =100 detailed kinetic information hint for the formation mechanism
30 Poly(ethylene therephtalate), PET crazing during cyclic loading scattering from fibrils and voids failure initiation under loads reflection from craze-bulk interfaces H.Kausch et al. J. Macromol. Sci. Part B,38, 1999, 803 R.Bubeck, D.Buckley, E.Kramer, H.Brown Journal of Materials Science 26, 1991, 6249
31 Poly(ethylene therephtalate), PET crazing during cyclic loading scattering from fibrils and voids failure initiation under loads reflection from craze-bulk interfaces H.Kausch et al. J. Macromol. Sci. Part B,38, 1999, 803 R.Bubeck, D.Buckley, E.Kramer, H.Brown Journal of Materials Science 26, 1991, 6249
32 phase separation of (dissolution type) nucleating agents ipp+1% DMDBS total scattered intensity (invariant) =2 (1 DMDBS phase separates invariant increases due to the e - density difference DMDBS CH 3 H 3 C H 3 C O O O O OH CH 3 Balzano et al. - Macromolecules 2008, 41, OH
33 phase separation of (dissolution type) nucleating agents ipp+1% DMDBS phase diagram Balzano et al. - Macromolecules 2008, 41,
34 spatial heterogeneity in injection molded parts isotactic polypropylene, ipp cooling rate vs stress as function of the position
35 take away SAXS is a relatively simple technique that can be used to extract morphological information in static and dynamic conditions the SAXS signal comes from e - density differences and can be seen as the Fourier transform of the real-space morphology SAXS data can be modeled in terms of form and structure factors (also in other ways ) mathematical complexity should not scare you! Very many form and structure factors are tabled in literature modeling provides qualitative and quantitative information on size, size distribution and morphology of scatterers results are model dependent model assumptions need to be validated by other techniques
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