Form and Structure Factors: Modeling and Interactions Jan Skov Pedersen, Department of Chemistry and Inano Center University of Aarhus Denmark SAXS Lab

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

ii σ []i = ,...3,2,1)(),(, NiqIqIq

qi calibrated

qI i )( calibrated, i.e. on absolute scale - noisy, (smeared), truncated

σ []qI i )( Statistical standard errors: Calculated from counting 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

dσ(q)/dΩ : 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 dσ(q) 2 2 2 dΩ = I(q) = n Δρ V P(q)S(q) = c M Δρm P(q)S(q)

n is the number density of particles, Δρ 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 • Δρ 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 Af monomers: 29. Polycondensates of ABf 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: 34. Arbitrarily branched self-avoiding polymers: (Block copolymer micelle) 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) = Pcross-section(q) Plarge(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 ∞ qr)sin( R qr)sin( qA = ρπr)(4)( 2 drr = 4π 2 drr ∫ qr ∫ qr 0 0 r 4π R = ∫ )sin( rdrqr = q 0 ()∫∫' []−= 'dxfgfgdxgf (partial integration)… R π ⎛ cos4 qRR ⎡sin qr ⎤ ⎞ ⎜−= + ⎟ q ⎜ q ⎢ q ⎥ ⎟ ⎝ ⎣ ⎦ 0 ⎠ π ⎛ qRR sincos4 qr ⎞ ⎜−= + ⎟ ⎜ 2 ⎟ q ⎝ q q ⎠ 4π = ()sin − cos qRqRqR q 3 4 − qRqRqR )]cos()[sin(3 = π R 3 spherical Bessel function 14 3 qR)( 3 Form factor of sphere P(q) = A(q)2/V2

1/R

q-4

Porod

15 C. Glinka Measured data from solid sphere (SANS)

2 dσ 22 ⎡ − qRqRqR )]cos()[sin(3 ⎤ )( Δ= ρ Vq ⎢ 3 ⎥ dΩ ⎣ qR)( ⎦

105

SANS from Latex spheres 104 Smeared fit Ideal fit 103 Instrumental smearing is 102 I(q) routinely included

101 in SANS data analysis 100

10-1 0.000 0.005 0.010 0.015 0.020 0.025 0.030

-1 q [Å ] 16 Data from Wiggnal et al. Ellipsoid

Prolates (R,R,εR) Oblates (R,R,εR) ε > 1 ε < 1

π 2/ 2 qP []Φ= qR sin)'()( dαα ∫0

R’ = R(sin2α + ε2cos2α)1/2

[ − xxx )cos()sin(3 ] )( x)( =Φ x 3

17 P(q): Ellipsoid of revolution

Glatter 18 Lysosyme Lysozyme 7 mg/mL

100 Ellipsoid of revolution + background

R = 15.48 Å 2 ε = 1.61 (prolate) χ =2.4 (χ=1.55) 10-1 ] -1

I(q) [cm 10-2

10-3

0.0 0.1 0.2 0.3 0.4 q [Å-1] 19 Core-shell particles: Rout Rin

= −+

−shellcore qA )( []Δ= ρ outshell Φ qRV out − ()( Δρshell − Δρ incore Φ qRV in )() where 3 3 Vout= 4π /3 and Vin= 4πRin /3.

Δρcore is the excess scattering length density of the core,

Δρshell is the excess scattering length density of the shell and:

[]− cossin3 xxx x)( =Φ x3

20 101

100

10-1

10-2 P(q)

10-3

10-4

10-5 0.0 0.1 0.2 0.3 0.4 0.5

q [Å-1] Rout =30Å Rcore =15Å Δρshell =1 Δρcore =−1. 21 SDS micelle Hydrocarbon core

Headgroup/counterions

20 Å

www.psc.edu/.../charmm/tutorial/ mackerell/membrane.html mrsec.wisc.edu/edetc/cineplex/ micelle.html 22 SDS micelles: prolate ellispoid with shell of constant thickness 0.1

χ2 =2.3

I(0) = 0.0323 ± 0.0005 1/cm

Rcore = 13.5 ± 2.6 Å ε =1.9 ± 0.10 ] 0.01 dhead = 7.1 ± 4.4 Å -1 ρhead/ρcore = − 1.7 ± 1.5 backgr = 0.00045 ± 0.00010 1/cm I(q) [cm

0.001

0.0 0.1 0.2 0.3 0.4 23 q [Å-1] Molecular constraints:

Z e Z e ρ e tail −=Δ 2OH tail core = VNV tailagg Vtail V 2OH

e e n water molecules in headgroup shell e head + nZZ 2OH Z 2OH ρ head =Δ − head + nVV 2OH V 2OH shell = agg ( head + nVVNV 2OH )

c nmicelles = agg MN surfactant

24 Cylinder 2R L

π 2/ 2 ⎡ 1 ()qRJ sin2 α qL α 2/cossin( ⎤ qP )( = ∫ ⎢ ⎥ sin dαα 0 ⎣ qR sinα qL α 2/cos ⎦

J1(x) is the Bessel function of first order and first kind.

25 P(q) cylinder

L>> R P(q) ⇒ Pcross-section(q) Plarge(q)

q−1

L = 60 Å R = 30 Å 120 Å 300 Å 3000 Å

26 Glucagon Fibrils

R=29Å

R=16 Å

Cristiano Luis Pinto Oliveira, Manja A. Behrens, Jesper Søndergaard Pedersen, Kurt Erlacher, 27 Daniel Otzen and Jan Skov Pedersen J. Mol. Biol. (2009) 387, 147–161 Primus

28 Form factor for particles with arbitrary shape

Spherical monodisperse particles

n n sin(qrij ) ()= sphere RqPqI ),( ∑∑ i==11j qrij

29 Monte Carlo integration in calculation of form factors for complex structures •Generate points in space by Monte Carlo simulations

•Select subsets by x(i)2+y(i)2+z(i)2 ≤ R2 geometric constraints

•Caclulate histograms p(r) functions

•(Include polydispersity)

•Fourier transform Sphere 30 MC points ISCOM vaccine Protein Micelle particle

31 Immune-stimulating complexes: ISCOMs - Self-assembled vaccine nano-particles

Quil-A

Stained TEM

ISCOMs typically:

60–70wt% Quil-A, 10–15wt% Phospholipids 10-15wt% cholesterol

M T Sanders et al. Immunology and Cell Biology (2005) 83, 119–128 32 Investigation of the structure of SAXS data ISCOM particles by SAXS S. Manniche, H.B. Madsen, L. Arleth, H.M. Nielsen, ISCOMs without toxoid C.L.P. Oliveira, J.S. Pedersen, in preparation. 100

Oscillations ⇒ relatively monodisperse Bump at high q ⇒ core-headgroup structure

10-1 I(q)

10-2

10-3 0.01 0.1 33 q [Å-1] Combination of icosahedrons, tennis balls and footballs

0 10 Fit result:

Icosah: ϕm M = 0.057 Tennis: ϕm M = 0.770 Football: ϕm M= 0.173 10-1

From volumes of cores:

I(q) Icosah: M = 0.557 a.u. Tennis: M = 1.000 a.u. -2 10 Football M= 1.616 a.u

Mass distribution: -3 10 Icosah: ϕm = 0.104 0.01 0.1 Tennis: ϕm = 0.786 Football: ϕm = 0.110 q [Å-1] 34 Icosahedrons, tennis ball, football

Hydrophobic core: (2 x) 11 Å Hydrophilic headgroup region: Width 13 Å

242 Å 335 Å 440 Å

35 Pure SDS micelles in buffer:

C > CMC (∼ 5 mM)

Nagg= 66±1 oblate ellipsoids R=20.3±0.3 Å ɛ=0.663±0.005

C ≤ CMC

36 Polymer chains in solution Gigantic ensemble of 3D random flights - all with different configurations

37 Gaussian polymer chain

Look at two points: Contour separation: L Spatial separation: r

r qr)sin( r Contribution to scattering: qI )( = L 2 qr

[− 2 Lbq 6/exp ] For an ensemble of polymers, points with L has =Lb ⎡ 3r 2 ⎤ and r has a Gaussian distribution: rD exp)( ⎢−∝ 2 ⎥ ⎣ 2 r >< ⎦ Add scattering from all pair of points

’Density of points’: (L -L) o Lo L L 38 Gaussian chains: The calculation

1 ∞ o LL o qr)sin( qP )( = − LLrDdLdLdr |)|,( r 2 1 ∫∫∫ 2 12 Lo 0 0 0 qr 1 ∞ Lo qr)sin( = − LrDLLdLdr ),()( r 2 ∫∫ o Lo 00 qr 1 Lo = []−− 2 LbqLLdL 6/exp)( ∫ o Lo 0 +−− xx ]1)[exp(2 22 2 = = g qRx R = Lb/6 x 2 g

How does this function look?

39 Polymer scattering

Form factor of Gaussian chain 10

≈ /1 Rq g 0.1 1 − ν −2 P(q) = qq 0.01

0.001

0.0001 0.1 1 10 100 40 qRg Gaussian chain+ background R = 21.3 Å g χ2=1.8 α-synuclein Lysozyme in Urea 10 mg/mL 10-1

Gaussian chain+ background

Rg = 21.3 Å

-2

] 10 -1 χ2=1.8 I(q) [cm

10-3

0.01 0.1 1

q [Å-1]

Δρ is low! 41 Self avoidence

No excluded volume No excluded volume => expansion − − + xx ]1)[exp(2 22 no analytical solution! g qRxP )( == 2 x 42 Monte Carlo simulation approach Pedersen and Schurtenberger 1996 (1) Choose model

(2) Vary parameters in a broad range Generate configs., sample P(q) (3) Analyze P(q) using physical insight (4) Parameterize P(q) using physical insight

(5) Fit experimental data using numerical expressions for P(q)

P(q,L,b) L = contour length

b = Kuhn (‘step’) length 43 Expansion = 1.5

Form factor polymer chains

≈ Rq g Gaussian)(/1 1

Excluded volume chains: ≈ volexclRq .).(/1 g /1 ν 0.1 − = qq − 70.1 ν = 588.0 P(q) 0.01 Gaussian chains 1 − ν = qq −2 q-1 local 0.001 ν = 2/1 stiffness

0.0001 0.1 1 10 100

qRg (Gaussian) 44 C16E6 micelles with ‘C16-’SDS

Sommer C, Pedersen JS, Egelhaaf SU, Cannavaciuolo L, Kohlbrecher J, Schurtenberger P.

Variation of persistence length with ionic strength

works also for polyelectrolytes like unfolded proteins

45 Models II Polydispersity: Spherical particles as e.g. vesicles

No interaction effects: Size distribution D(R)

Oligomeric mixture: Discrete particles

σ qd )( 2 Application to insulin: Δ= ρ ∑ iii qPfMc )( dΩ Pedersen, Hansen, Bauer (1994). i European Biophysics Journal 23, 379-389. (Erratum). ibid 23, 227-229.

f = mass fraction Used in PRIMUS i ∑ f i = 1 i ‘Oligomers’ 46 Concentration effects

47 Concentration effects in protein solutions

α-crystallin eye lens protein

= q/2π

48 p(r) by Indirect Fourier Transformation (IFT)

- as you have done by GNOM !

I. Pilz, 1982

At high concentration, the neighborhood is different from the average further away!

(1) Simple approach: Exclude low q data.

(2) Glatter: Use Generalized Indirect Fourier Transformation (GIFT)49 Low concentrations Zimm 1948 – originally for light scattering

P’(q) = - Subtract overlapping configuration

P’(q) = P(q) - υ P(q)2 υ ~ concentration = P(q)[1 - υ P(q)] Higher order terms: P’(q) = P(q)[1 - υ P(q){1 - υ P(q)}] = P(q)[1 - υ P(q)+ υ 2P(q)2] … = P(q)[1 - υ P(q )+ υ 2P(q)2 - υ 3P(q)3…..] qP )( = +υ qP )(1

= Random-Phase Approximation (RPA) 50 Zimm approach qP )( )( = KqI +υ qP )(1

−− 11 +υ qP )(1 −1 ⎡ 1 ⎤ )( = KqI = K ⎢ +υ⎥ qP )( ⎣ qP )( ⎦ 1 With qP )( ≈ −− 11 22 22 [ RqKqI g 3/1)( ++= υ] + Rq g 3/1

Plot I(q)−1 versus q2 + c and extrapolate to q=0 and c=0 !

51 Zimm plot Kirste and Wunderlich PS in toluene

c = 0 I(q)-1 P(q)

q = 0

q2 + c 52 My suggestion: ( - which includes also information from what follows)

• Minimum 3 concentrations for same system. • Fit data simultaneously all data sets

qI i )( Pi = 2 2 c 1 −+ i aqca 2 )3/exp(1

with a1, a2, and Pi as fit parameters

53 Osteopontin (M=35 kDa )in water and 10 mM NaCl

2.5, 5 and 10 mg/mL

1.0000

0.1000 ) -1

0.0100 I(q) (cm

0.0010

0.0001 0.01 0.1 1 q (A-1) 54 Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) 55 56 Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) 57 Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) But now we look at the information content related to these effects…

Understand the fundamental processes and principles governing aggregation and crystallization

Why is the eye lens transparent despite a protein concentration of 30-40% ? 58 Structure factor

Spherical monodisperse particles sin qr )( Δ= ρ 22 qPVqI )( ∑ jk qrjk Δ= ρ 22 qSqPV )()(

22 sin qrj Δ= ρ ∑ rpqPV j )()( qrj

S(q) is related to the probability distribution function of inter-particles distances, i.e. the pair correlation function g(r)

59 Correlation function g(r)

g(r) = Average of the normalized density of atoms in a shell [r ; r+ dr] from the center of a particle

60 g(r) and S(q)

qr)sin( += π rgnqS − )1)((41)( 2drr ∫ qr

61 GIFT

Glatter: Generalized Indirect Fourier Transformation (GIFT)

qr)sin( = π rpqI )(4)( dr ∫ qr

With concentration effects

qr)sin( = ση π rpcZRRqSqI )(4),),(,,,()( dr eff salt ∫ qr

Optimized by constrained non-linear least-squares method

- works well for globular models and provides p(r)

62 A SAXS study of the small hormone glucagon: equilibrium aggregation and fibrillation

29 residue hormone, withHome-written a net charge software of +5 at pH~2-3

10.7 mg/ml

6.4 mg/ml Hexamers p(r) [arb. u.] [arb. p(r)

I(q) [arb. u.] I(q) [arb. 5.1 mg/ml

2.4 mg/ml trimers monomers 1.0 mg/ml 0 10203040506070 0,01 0,1 r [Å] q [Å-1]

63 Osmometry (Second virial coeff A2)

Π/c

Ideal gas A2 = 0

64 S(q), virial expansion and Zimm

From statistical mechanics…:

−1 ⎛ Π∂ ⎞ 1 )0( == RTqS ⎜ ⎟ = 2 ⎝ ∂c ⎠ 2 MAccMA 3 +++ ...321

Î In Zimm approach ν = 2cMA2

65 Isoelectric point A2 in lysozyme solutions

O. D. Velev, E. W. Kaler, and A. M. Lenhoff 66 Colloidal interactions • Excluded volume ‘repulsive’ interactions (‘hard-sphere’) • Short range attractive van der Waals interaction (‘stickiness’) • Short range attractive hydrophobic interactions (solvent mediated ‘stickiness’) •Electrostatic repulsive interaction (or attractive for patchy charge distribution!) (effective Debye-Hückel potential) • Attractive depletion interactions (co-solute (polymer) mediated )

Debye-Hückel hard sphere sticky hard sphere screened Coulomb potential

67 Theory for colloidal stability

DLVO theory: (Derjaguin-Landau-Vewey-Overbeek)

Debye-Hückel screened Coulomb potential + attractive interaction

68 Depletion interactions

π π

Asakura & Oosawa, 58 69 Integral equation theory

Relate g(r) [or S(q)] to V(r)

At low concentration g(r) = exp(− V (r) / kBT) ≈1 − V (r) / kBT Boltzmann approximation (weak interactions) Make expansion around uniform state [Ornstein-Zernike eq.] g(r) = 1 – n c(r) –[3 particle] – [4 particle] - … = 1 – n c(r) – n2 c(r) * c(r) – n3 c(r) * c(r) * c(r) − … [* = convolution] c(r) = direct correlation function 1 S(q) = 1− n c(q) – n2 c(q)2 – n3 c(q)3 − …. = − qnc )(1 ⇒ but we still need to relate c(r) to V(r) !!! 70 Closure relations

Systematic density expansion….

Mean-spherical approximation MSA: c(r) = − V(r) / kBT (analytical solution for screened Coulomb potential - but not accurate for low densities)

Percus-Yevick approximation PY: c(r) = g(r) [exp(V (r) / kBT)−1] (analytical solution for hard-sphere potential + sticky HS)

Hypernetted chain approximation HNC:

c(r) = − V(r) / kBT +g(r) −1 – ln (g(r) ) (Only numerical solution - but quite accurate for Coulomb potential)

Rogers and Young closure RY: Combines PY and HNC in a self-consistent way (Only numerical solution - but very accurate for Coulomb potential) 71 α-crystallin S. Finet, A. Tardieu

DLVO potential

HNC and numerical solution

72 α-crystallin + PEG 8000 S. Finet, A. Tardieu Depletion interactions: DLVO potential

HNC and numerical solution

40 mg/ml α-crystallin solution pH 6.8, 150 mM ionic strength.

73 Anisotropy

Small: decoupling approximation (Kotlarchyk and Chen,1984) :

Measured structure factor: ∂σ q)( qS )( ≡ Ω∂ += β []≠− qSqSq )(1)()(1 !!!!! meas Δρ 2 qPn )(

74 Large anisotropy

Polymers, cylinders…

Anisotropy, large: Random-Phase Approximation (RPA):

dσ qP )( )( Δ= ρ Vnq 22 dΩ +ν qP )(1 υ ~ concentration

Anisotropy, large: Polymer Reference Interaction Site Model (PRISM) Integral equation theory – equivalent site approximation

dσ qP )( )( Δ= ρ Vnq 22 c(q) direct correlation function dΩ − qPqnc )()(1 related to FT of V(r)

75 Empirical PRISM expression

SDS micelle in 1 M NaBr

dσ qP )( )( Δ= ρ Vnq 22 dΩ − qPqnc )()(1

c(q) = rod formfactor - empirical from MC simulation

Arleth, Bergström and Pedersen

76 Overview: Available Structure factors

1) Hard-sphere potential: Percus-Yevick approximation 2) Sticky hard-sphere potential: Percus-Yevick approximation 3) Screened Coulomb potential: Mean-Spherical Approximation (MSA). Rescaled MSA (RMSA). Thermodynamically self-consistent approaches (Rogers and Young closure) 4) Hard-sphere potential, polydisperse system: Percus-Yevick approximation 5) Sticky hard-sphere potential, polydisperse system: Percus-Yevick approximation 6) Screened Coulomb potential, polydisperse system: MSA, RMSA, 7) Cylinders, RPA 8) Cylinders, `PRISM': 9) Solutions of flexible polymers, RPA: 10) Solutions of semi-flexible polymers, `PRISM': 11) Solutions of polyelectrolyte chains ’PRISM’:

77 Summary • 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 78 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 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 79