Form and Structure Factors: Modeling and Interactions Jan Skov Pedersen, Department of Chemistry and iNANO Center University of Aarhus Denmark SAXS lab 1
Outline • Model fitting and least-squares methods • Available form factors ex: sphere, ellipsoid, cylinder, spherical subunits… ex: polymer chain • Monte Carlo integration for form factors of complex structures • Monte Carlo simulations for form factors of polymer models • Concentration effects and structure factors Zimm approach Spherical particles Elongated particles (approximations) Polymers 2
Motivation for ‘modelling’ - not to replace shape reconstruction and crystal-structure based modeling – we use the methods extensively - alternative approaches to reduce the number of degrees of freedom in SAS data structural analysis (might make you aware of the limited information content of your data !!!) - provide polymer-theory based modeling of flexible chains - describe and correct for concentration effects 3
Literature Jan Skov Pedersen, Analysis of Small-Angle Scattering Data from Colloids and Polymer Solutions: Modeling and Least-squares Fitting (1997). Adv. Colloid Interface Sci. , 70 , 171-210. Jan Skov Pedersen Monte Carlo Simulation Techniques Applied in the Analysis of Small-Angle Scattering Data from Colloids and Polymer Systems in Neutrons, X-Rays and Light P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V. p. 381 Jan Skov Pedersen Modelling of Small-Angle Scattering Data from Colloids and Polymer Systems in Neutrons, X-Rays and Light P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V. p. 391 Rudolf Klein Interacting Colloidal Suspensions in Neutrons, X-Rays and Light P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V. p. 351 4
Form factors and structure factors Warning 1: Scattering theory – lots of equations! = mathematics, Fourier transformations Warning 2: Structure factors: Particle interactions = statistical mechanics Not all details given - but hope to give you an impression! 5
I will outline some calculations to show that it is not black magic ! 6
Input data: Azimuthally averaged data [ ] σ = q , I ( q ), I ( q ) i 1 , 2 , 3 ,... N i i i calibrated q i I ( q ) calibrated, i.e. on absolute scale i - noisy, (smeared), truncated [ ] σ I ( q ) Statistical standard errors: Calculated from counting i statistics by error propagation - do not contain information on systematic error !!!! 7
Least-squared methods Measured data: Model: Chi-square: Reduced Chi-squared: = goodness of fit (GoF) Note that for corresponds to i.e. statistical agreement between model and data 8
Cross section d σ ( q ) / d Ω : number of scattered neutrons or photons per unit time, relative to the incident flux of neutron or photons, per unit solid angle at q per unit volume of the sample. For system of monodisperse particles d σ ( q ) = c M Δ ρ m d Ω = I ( q ) = n Δ ρ 2 V 2 P ( q ) S ( q ) 2 P ( q ) S ( q ) n is the number density of particles, Δ ρ is the excess scattering length density, given by electron density differences V is the volume of the particles, P ( q ) is the particle form factor, P ( q=0 )=1 S ( q ) is the particle structure factor, S ( q= ∞ )=1 • V ∝ M • n = c/M • Δ ρ can be calculated from partial specific density, composition 9
Form factors of geometrical objects 10
Form factors I Homogenous rigid particles 1. Homogeneous sphere 2. Spherical shell: 3. Spherical concentric shells: 4. Particles consisting of spherical subunits: 5. Ellipsoid of revolution: 6. Tri-axial ellipsoid: 7. Cube and rectangular parallelepipedons: 8. Truncated octahedra: 9. Faceted Sphere: 9x Lens 10. Cube with terraces: 11. Cylinder: 12. Cylinder with elliptical cross section: 13. Cylinder with hemi-spherical end-caps: 13x Cylinder with ‘half lens’ end caps 14. Toroid: 15. Infinitely thin rod: 16. Infinitely thin circular disk: 17. Fractal aggregates: 11
Form factors II ’Polymer models’ 18. Flexible polymers with Gaussian statistics: 19. Polydisperse flexible polymers with Gaussian statistics: 20. Flexible ring polymers with Gaussian statistics: 21. Flexible self-avoiding polymers: 22. Polydisperse flexible self-avoiding polymers: 23. Semi-flexible polymers without self-avoidance: 24. Semi-flexible polymers with self-avoidance: 24x Polyelectrolyte Semi-flexible polymers with self-avoidance: 25. Star polymer with Gaussian statistics: 26. Polydisperse star polymer with Gaussian statistics: 27. Regular star-burst polymer (dendrimer) with Gaussian statistics: 28. Polycondensates of A f monomers: 29. Polycondensates of AB f monomers: 30. Polycondensates of ABC monomers: 31. Regular comb polymer with Gaussian statistics: 32. Arbitrarily branched polymers with Gaussian statistics: 33. Arbitrarily branched semi-flexible polymers: (Block copolymer micelle) 34. Arbitrarily branched self-avoiding polymers: 35. Sphere with Gaussian chains attached: 36. Ellipsoid with Gaussian chains attached: 37. Cylinder with Gaussian chains attached: 38. Polydisperse thin cylinder with polydisperse Gaussian chains attached to the ends: 39. Sphere with corona of semi-flexible interacting self-avoiding chains of a corona chain. 12
Form factors III P ( q ) = P cross-section ( q ) P large ( q ) 40. Very anisotropic particles with local planar geometry: Cross section: (a) Homogeneous cross section (b) Two infinitely thin planes (c) A layered centro symmetric cross-section (d) Gaussian chains attached to the surface Overall shape: (a) Infinitely thin spherical shell (b) Elliptical shell (c) Cylindrical shell (d) Infinitely thin disk 41. Very anisotropic particles with local cylindrical geometry: Cross section: (a) Homogeneous circular cross-section (b) Concentric circular shells (c) Elliptical Homogeneous cross section. (d) Elliptical concentric shells (e) Gaussian chains attached to the surface Overall shape: (a) Infinitely thin rod (b) Semi-flexible polymer chain with or without excluded volume 13
From factor of a solid sphere ρ (r) 1 0 R ∞ R sin( qr ) sin( qr ) ∫ ∫ = π ρ = π 2 2 A ( q ) 4 ( r ) r dr 4 r dr qr qr r 0 0 π R 4 ∫ = = sin( qr ) rdr q 0 ( ) [ ] ∫ ∫ (partial integration)… = − f ' g dx fg fg ' dx ⎛ ⎞ R ⎤ π ⎡ 4 R cos qR sin qr ⎜ ⎟ = − + ⎢ ⎥ ⎜ ⎟ q q q ⎣ ⎦ ⎝ ⎠ 0 π ⎛ ⎞ 4 R cos qR sin qr ⎜ ⎟ = − + ⎜ ⎟ 2 q q q ⎝ ⎠ π 4 ( ) = − sin qR qR cos qR 3 q − 4 3 [sin( qR ) qR cos( qR )] 14 spherical Bessel function = π 3 R 3 3 ( qR )
Form factor of sphere P ( q ) = A ( q ) 2 / V 2 1/R q -4 Porod 15 C. Glinka
Measured data from solid sphere (SANS) 2 ⎡ ⎤ σ − d 3 [sin( qR ) qR cos( qR )] = Δ ρ 2 2 ( q ) V ⎢ ⎥ Ω 3 d ( qR ) ⎣ ⎦ 10 5 SANS from Latex spheres 10 4 Smeared fit Ideal fit Instrumental 10 3 smearing is I(q) 10 2 routinely included in SANS data 10 1 analysis 10 0 10 -1 0.000 0.005 0.010 0.015 0.020 0.025 0.030 q [Å -1 ] 16 Data from Wiggnal et al.
Ellipsoid Prolates ( R,R, ε R ) Oblates ( R,R, ε R ) ε > 1 ε < 1 2 [ ] π / 2 ∫ = Φ α α P ( q ) ( qR ' ) sin d 0 R’ = R (sin 2 α + ε 2 cos 2 α ) 1/2 [ ] − 3 sin( x ) x cos( x ) Φ = ( x ) 3 x 17
P(q): Ellipsoid of revolution Glatter 18
Lysosyme Lysozyme 7 mg/mL 10 0 Ellipsoid of revolution + background R = 15.48 Å χ 2 =2.4 ( χ =1.55) ε = 1.61 (prolate) 10 -1 I(q) [cm -1 ] 10 -2 10 -3 0.0 0.1 0.2 0.3 0.4 q [Å -1 ] 19
Core-shell particles: R out R in = − + [ ] = Δ ρ Φ − Δ ρ − Δ ρ Φ A ( q ) V ( qR ) ( ) V ( qR ) − core shell shell out out shell core in in where out = 4 π 3 /3 and V in = 4 π R in V 3 /3. Δ ρ core is the excess scattering length density of the core, Δ ρ shell is the excess scattering length density of the shell and: [ ] − 3 sin x x cos x Φ = ( x ) 3 x 20
10 1 10 0 10 -1 P(q) 10 -2 10 -3 10 -4 10 -5 0.0 0.1 0.2 0.3 0.4 0.5 q [Å-1] R out = 30 Å R core = 15 Å Δρ shell = 1 Δρ core = − 1. 21
SDS micelle Hydrocarbon core Headgroup/counterions 20 Å www.psc.edu/.../charmm/tutorial/ mackerell/membrane.html 22 mrsec.wisc.edu/edetc/cineplex/ micelle.html
SDS micelles: prolate ellispoid with shell of constant thickness 0.1 χ 2 = 2.3 0.0323 ± 0.0005 1/cm I(0) = 13.5 ± 2.6 Å R core = ε = 1.9 ± 0.10 7.1 ± 4.4 Å d head = 0.01 I(q) [cm-1] ρ head / ρ core = − 1.7 ± 1.5 backgr = 0.00045 ± 0.00010 1/cm 0.001 0.0 0.1 0.2 0.3 0.4 23 q [Å-1]
Molecular constraints: e e Z Z = Δ ρ = − e tail H 2 O V N V tail V V core agg tail tail H 2 O n water molecules in headgroup shell + e e Z nZ Z Δ ρ = − e head H 2 O H 2 O head + V nV V head H 2 O H 2 O = + V N ( V nV ) shell agg head H 2 O c micelles = n N M agg surfactant 24
Cylinder 2 R L ( ) 2 π / 2 ⎡ α ⎤ α 2 J qR sin = ∫ sin( qL cos / 2 α α P ( q ) 1 sin d ⎢ ⎥ α α qR sin qL cos / 2 ⎣ ⎦ 0 J 1 ( x ) is the Bessel function of first order and first kind. 25
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