Form and Structure Factors: Modeling and Interactions Jan Skov - - PowerPoint PPT Presentation

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Jan Skov Pedersen, Department of Chemistry and iNANO Center University of Aarhus Denmark

Form and Structure Factors: Modeling and Interactions

SAXS lab

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Outline

• Concentration effects and structure factors

Zimm approach Spherical particles Elongated particles (approximations) Polymers

• 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

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Motivation for ‘modelling’

• not to replace shape reconstruction and crystal-structure based

modeling – we use the methods extensively

• describe and correct for concentration effects
• alternative approaches to reduce the number of degrees of

freedom in SAS data structural analysis

• provide polymer-theory based modeling of flexible chains

(might make you aware of the limited information content of your data !!!)

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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

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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!

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I will outline some calculations to show that it is not black magic !

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Input data: Azimuthally averaged data

[ ]

N i q I q I q

i i i

,... 3 , 2 , 1 ) ( ), ( , = σ ) (

i

q I

[ ]

) (

i

q I σ

i

q calibrated calibrated, i.e. on absolute scale

• noisy, (smeared), truncated

Statistical standard errors: Calculated from counting statistics by error propagation

• do not contain information on systematic error !!!!

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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

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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) dΩ = I(q) = n Δρ2V 2P(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

= c M Δρm

2P(q)S(q)

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Form factors of geometrical objects

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Form factors I

• 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:

Homogenous rigid particles

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Form factors II

• 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:
• 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.

’Polymer models’

(Block copolymer micelle)

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Form factors III

• 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

P(q) = Pcross-section(q) Plarge(q)

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From factor of a solid sphere

R r ρ(r) 1

[ ]

( )

( )

3 3 3 2 2 2

) ( )] cos( ) [sin( 3 3 4 cos sin 4 sin cos 4 sin cos 4 ' ' ) sin( 4 ) sin( 4 ) sin( ) ( 4 ) ( qR qR qR qR R qR qR qR q q qr q qR R q q qr q qR R q dx fg fg dx g f rdr qr q dr r qr qr dr r qr qr r q A

R R R

− = − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + − = − = = = = =

∫ ∫ ∫ ∫ ∫

∞

π π π π π π ρ π

(partial integration)… spherical Bessel function

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q-4 1/R

Form factor of sphere

• C. Glinka

Porod P(q) = A(q)2/V2

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Measured data from solid sphere (SANS)

q [Å-1]

0.000 0.005 0.010 0.015 0.020 0.025 0.030

I(q)

10-1 100 101 102 103 104 105 SANS from Latex spheres Smeared fit Ideal fit

Data from Wiggnal et al.

2 3 2 2

) ( )] cos( ) [sin( 3 ) ( ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − Δ = Ω qR qR qR qR V q d d ρ σ

Instrumental smearing is routinely included in SANS data analysis

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Ellipsoid

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

[ ]

α α

π

d qR q P sin ) ' ( ) (

2 2 /

∫

Φ =

R’ = R(sin2α + ε2cos2α)1/2

[ ]

3

) cos( ) sin( 3 ) ( x x x x x − = Φ

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Glatter

P(q): Ellipsoid of revolution

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Lysosyme

Lysozyme 7 mg/mL

q [Å-1]

0.0 0.1 0.2 0.3 0.4

I(q) [cm-1]

10-3 10-2 10-1 100

Ellipsoid of revolution + background R = 15.48 Å ε = 1.61 (prolate)

χ2=2.4 (χ=1.55)

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Core-shell particles:

[ ]

) ( ) ( ) ( ) (

in in core shell

• ut
• ut

shell shell core

qR V qR V q A Φ Δ − Δ − Φ Δ =

−

ρ ρ ρ

where V

• ut= 4π3/3 and Vin= 4πRin

3/3.

Δρcore is the excess scattering length density of the core, Δρshell is the excess scattering length density of the shell and:

[ ]

3

cos sin 3 ) ( x x x x x − = Φ

= − + Rout Rin

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q [Å-1]

0.0 0.1 0.2 0.3 0.4 0.5

P(q)

10-5 10-4 10-3 10-2 10-1 100 101

Rout = 30 Å Rcore = 15 Å Δρshell = 1 Δρcore = −1.

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www.psc.edu/.../charmm/tutorial/ mackerell/membrane.html mrsec.wisc.edu/edetc/cineplex/ micelle.html

SDS micelle

20 Å Hydrocarbon core Headgroup/counterions

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q [Å-1] 0.0 0.1 0.2 0.3 0.4 I(q) [cm-1] 0.001 0.01 0.1

χ2 = 2.3 I(0) = 0.0323 ± 0.0005 1/cm Rcore = 13.5 ± 2.6 Å ε = 1.9 ± 0.10 dhead = 7.1 ± 4.4 Å ρhead/ρcore = − 1.7 ± 1.5 backgr = 0.00045 ± 0.00010 1/cm

SDS micelles:

prolate ellispoid with shell of constant thickness

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Molecular constraints:

O H e O H tail e tail e tail

V Z V Z

2 2

− = Δρ

tail agg core

V N V =

O H e O H O H head O H e head e head

V Z nV V nZ Z

2 2 2 2

− + + = Δρ

n water molecules in headgroup shell

) (

2O H head agg shell

nV V N V + =

surfactant

M N c n

agg micelles =

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Cylinder

L 2R

( )

α α α α α α

π

d qL qL qR qR J q P sin 2 / cos 2 / cos sin( sin sin 2 ) (

2 1 2 /

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ∫

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

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P(q) cylinder

q−1 R = 30 Å L = 60 Å 120 Å 300 Å 3000 Å L>> R P(q) ⇒ Pcross-section(q) Plarge(q)

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Glucagon Fibrils

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

R=29Å R=16 Å

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Primus

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Form factor for particles with arbitrary shape

Spherical monodisperse particles

( )

( )

∑∑

= =

=

n i n j ij ij sphere

qr qr R q P q I

1 1

sin ) , (

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• Generate points in

space by Monte Carlo simulations

• Select subsets by

geometric constraints

• Caclulate histograms

p(r) functions

• (Include polydispersity)
• Fourier transform

MC points Sphere x(i)2+y(i)2+z(i)2 ≤ R2

Monte Carlo integration in calculation of form factors for complex structures

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Protein Micelle ISCOM vaccine particle

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Immune-stimulating complexes: ISCOMs

• Self-assembled vaccine nano-particles

M T Sanders et al. Immunology and Cell Biology (2005) 83, 119–128

Stained TEM

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

Quil-A

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SAXS data

q [Å-1] 0.01 0.1 I(q) 10-3 10-2 10-1 100

ISCOMs without toxoid

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

Investigation of the structure of ISCOM particles by SAXS

• S. Manniche, H.B. Madsen, L. Arleth, H.M. Nielsen,

C.L.P. Oliveira, J.S. Pedersen, in preparation.

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Combination of icosahedrons, tennis balls and footballs

q [Å-1] 0.01 0.1 I(q) 10-3 10-2 10-1 100 Fit result: Icosah: ϕm M = 0.057 Tennis: ϕm M = 0.770 Football: ϕm M= 0.173 From volumes of cores: Icosah: M = 0.557 a.u. Tennis: M = 1.000 a.u. Football M= 1.616 a.u Mass distribution: Icosah: ϕm = 0.104 Tennis: ϕm = 0.786 Football: ϕm = 0.110

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Icosahedrons, tennis ball, football

242 Å 440 Å 335 Å Hydrophobic core: (2 x) 11 Å Hydrophilic headgroup region: Width 13 Å

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Pure SDS micelles in buffer:

C > CMC (∼ 5 mM) C ≤ CMC Nagg= 66±1

• blate ellipsoids

R=20.3±0.3 Å ɛ=0.663±0.005

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Polymer chains in solution

Gigantic ensemble of 3D random flights

• all with different configurations

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Gaussian polymer chain

Look at two points: Contour separation: L Spatial separation: r Contribution to scattering:

r r

L

qr qr q I ) sin( ) (

2

=

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ > < − ∝

2 2

2 3 exp ) ( r r r D

For an ensemble of polymers, points with L has <r2>=Lb and r has a Gaussian distribution: ’Density of points’: (Lo- L)

Lo L L Add scattering from all pair of points

[ ]

6 / exp

2Lb

q −

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Gaussian chains: The calculation

[ ]

2 2 2 2 2 1 2 2 1 2

] 1 ) [exp( 2 6 / exp ) ( 1 ) sin( ) , ( ) ( 1 ) sin( |) | , ( 1 ) ( q R x x x x Lb q L L dL L r qr qr L r D L L dL dr L r qr qr L L r D dL dL dr L q P

g L

• L
• L

L

• =

+ − − = − − = − = − =

∫ ∫ ∫ ∫ ∫ ∫

∞ ∞

How does this function look?

Rg

2 = Lb/6

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Polymer scattering

Form factor of Gaussian chain

qRg

0.1 1 10 100

P(q)

0.0001 0.001 0.01 0.1 1 10 g

R q / 1 ≈

2 1 − −

= q q

ν

41 Lysozyme in Urea 10 mg/mL

q [Å-1]

0.01 0.1 1

I(q) [cm-1]

10-3 10-2 10-1

Gaussian chain+ background Rg = 21.3 Å

χ2=1.8 α-synuclein Δρ is low!

Gaussian chain+ background Rg = 21.3 Å

χ2=1.8

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Self avoidence

No excluded volume No excluded volume => expansion

2 2 2

] 1 ) [exp( 2 ) ( x x x q R x P

g

+ − − = =

no analytical solution!

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Monte Carlo simulation approach

(1) Choose model (5) Fit experimental data using numerical expressions for P(q) (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 Pedersen and Schurtenberger 1996 P(q,L,b) L = contour length b = Kuhn (‘step’) length

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Expansion = 1.5

Form factor polymer chains

qRg (Gaussian)

0.1 1 10 100

P(q)

0.0001 0.001 0.01 0.1 1 10

2 / 1

2 1

= =

− −

ν

ν

q q

588 .

70 . 1 / 1

= =

− −

ν

ν

q q

Excluded volume chains: Gaussian chains

.) . ( / 1 vol excl R q

g

≈

) ( / 1 Gaussian R q

g

≈

q-1 local stiffness

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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

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Models II

Polydispersity: Spherical particles as e.g. vesicles No interaction effects: Size distribution D(R)

∑

Δ = Ω

i i i i

q P f M c d q d ) ( ) (

2

ρ σ

Oligomeric mixture: Discrete particles

fi = mass fraction ∑

=

i i

f 1

Application to insulin:

Pedersen, Hansen, Bauer (1994). European Biophysics Journal 23, 379-389. (Erratum). ibid 23, 227-229.

Used in PRIMUS ‘Oligomers’

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Concentration effects

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Concentration effects in protein solutions

= q/2π

α-crystallin eye lens protein

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p(r) by Indirect Fourier Transformation (IFT)

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)

• I. Pilz, 1982
• as you have done by GNOM !

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Low concentrations

= Random-Phase Approximation (RPA) P’(q) = - Zimm 1948 – originally for light scattering Subtract overlapping configuration υ ~ concentration P’(q) = P(q) - υ P(q)2 = P(q)[1 - υ P(q)]

) ( 1 ) ( q P q P υ + =

… = P(q)[1 - υ P(q )+ υ 2P(q)2 - υ 3P(q)3…..] 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]

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Zimm approach

) ( 1 ) ( ) ( q P q P K q I υ + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = + =

− − −

υ υ ) ( 1 ) ( ) ( 1 ) (

1 1 1

q P K q P q P K q I 3 / 1 1 ) (

2 2 g

R q q P + ≈

[ ]

υ + + =

− −

3 / 1 ) (

2 2 1 1 g

R q K q I

With

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

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Zimm plot

I(q)-1 q2 + c

Kirste and Wunderlich PS in toluene

q = 0 P(q) c = 0

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My suggestion:

) 3 / exp( 1 ) (

2 2 2 1

a q ca P c q I

i i i

− + =

with a1, a2, and Pi as fit parameters ( - which includes also information from what follows)

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

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

q (A-1)

0.01 0.1 1

I(q) (cm-1)

0.0001 0.0010 0.0100 0.1000 1.0000

2.5, 5 and 10 mg/mL

Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

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Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

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Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

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Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

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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

• f 30-40% ?

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Structure factor

) ( ) ( sin ) ( ) (

2 2 2 2

q S q P V qr qr q P V q I

jk jk

ρ ρ Δ = Δ =

∑

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

j j j

qr qr r p q P V sin ) ( ) (

2 2

∑

Δ = ρ

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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

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g(r) and S(q)

q

dr r qr qr r g n q S

2

) sin( ) 1 ) ( ( 4 1 ) (

∫

− + = π

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GIFT

Glatter: Generalized Indirect Fourier Transformation (GIFT)

∫

= dr qr qr r p q I ) sin( ) ( 4 ) ( π

∫

= dr qr qr r p c Z R R q S q I

salt eff

) sin( ) ( 4 ) , ), ( , , , ( ) ( π σ η

With concentration effects Optimized by constrained non-linear least-squares method

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

63

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

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

10 20 30 40 50 60 70

p(r) [arb. u.] r [Å]

0,01 0,1

10.7 mg/ml 6.4 mg/ml 5.1 mg/ml 2.4 mg/ml

I(q) [arb. u.] q [Å

• 1]

1.0 mg/ml

Hexamers trimers monomers Home-written software

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Osmometry (Second virial coeff A2)

c Π/c Ideal gas A2 = 0

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S(q), virial expansion and Zimm

... 3 2 1 1 ) (

3 2 2 1

+ + + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ Π ∂ = =

−

MA c cMA c RT q S

In Zimm approach ν = 2cMA2 From statistical mechanics…:

66

A2 in lysozyme solutions

• O. D. Velev, E. W. Kaler,

and A. M. Lenhoff A2 Isoelectric point

67

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 )

hard sphere sticky hard sphere Debye-Hückel screened Coulomb potential

68

Theory for colloidal stability

Debye-Hückel screened Coulomb potential + attractive interaction

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

69

Depletion interactions

π

Asakura & Oosawa, 58

π

70

Integral equation theory

Relate g(r) [or S(q)] to V(r) At low concentration g(r) = exp(− V (r) / kBT) Boltzmann approximation c(r) = direct correlation function 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] ≈1 − V (r) / kBT (weak interactions) S(q) = 1− n c(q) – n2 c(q)2 – n3 c(q)3 − …. = ⇒ but we still need to relate c(r) to V(r) !!!

) ( 1 1 q nc −

71

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)

72

α-crystallin

• S. Finet, A. Tardieu

DLVO potential HNC and numerical solution

73

α-crystallin + PEG 8000

• S. Finet, A. Tardieu

DLVO potential HNC and numerical solution Depletion interactions: 40 mg/ml α-crystallin solution pH 6.8, 150 mM ionic strength.

74

Anisotropy

Small: decoupling approximation (Kotlarchyk and Chen,1984) : Measured structure factor:

[ ]

) ( 1 ) ( ) ( 1 ) ( ) ( ) (

2

q S q S q q P n q q Smeas ≠ − + = Δ Ω ∂ ∂ ≡ β ρ σ

!!!!!

75

Large anisotropy

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

) ( 1 ) ( ) (

2 2

q P q P V n q d d ν ρ σ + Δ = Ω

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

) ( ) ( 1 ) ( ) (

2 2

q P q nc q P V n q d d − Δ = Ω ρ σ

c(q) direct correlation function related to FT of V(r) Polymers, cylinders…

76

Empirical PRISM expression

Arleth, Bergström and Pedersen ) ( ) ( 1 ) ( ) (

2 2

q P q nc q P V n q d d − Δ = Ω ρ σ

c(q) = rod formfactor

• empirical from MC simulation

SDS micelle in 1 M NaBr

77

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’:

78

Summary

• Concentration effects and structure factors

Zimm approach Spherical particles Elongated particles (approximations) Polymers

• 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

79

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