form and structure factors: modeling and interactions
TRANSCRIPT
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Jan Skov Pedersen,
Department of Chemistry and iNANO CenterUniversity of Aarhus
Denmark
Form and Structure Factors:
Modeling and Interactions
SAXS lab
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Outline
• Concentration effects and structure factorsZimm approachSpherical 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
- 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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LiteratureJan 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 PedersenMonte 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 PedersenModelling 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 KleinInteracting 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
[ ] NiqIqIq iii ,...3,2,1)(),(, =σ
)( iqI
[ ])( iqIσ
iq calibrated
calibrated, i.e. on absolute scale - noisy, (smeared), truncated
Statistical standard errors: Calculated from countingstatistics 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 toi.e. statistical agreement between model and data
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Cross sectiondσ(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 particlesdσ(q)dΩ = I(q) = n ∆ρ2V 2P(q)S(q)
n is the number density of particles, ∆ρ is the excess scattering length density,
given by electron density differencesV is the volume of the particles, P(q) is the particle form factor, P(q=0)=1S(q) is the particle structure factor, S(q=∞)=1
• V ∝ M
• n = c/M
• ∆ρ can be calculated from partial specific density, composition
= c M ∆ρm2P(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 Lens10. Cube with terraces: 11. Cylinder: 12. Cylinder with elliptical cross section: 13. Cylinder with hemi-spherical end-caps: 13x Cylinder with ‘half lens’ end caps14. 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 III40. 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)
14
From factor of a solid sphere
R
r
ρ(r)1
0
[ ]( )
( )
33
3
2
0
0
2
0
2
0
)(
)]cos()[sin(334
cossin4
sincos4
sincos4
''
)sin(4
)sin(4
)sin()(4)(
qR
qRqRqRR
qRqRqRq
q
qr
q
qRR
q
q
qr
q
qRR
q
dxfgfgdxgf
rdrqrq
drrqr
qrdrr
qr
qrrqA
R
R
R
−=
−=
+−=
+−=
−=
==
==
∫ ∫
∫
∫∫∞
π
π
π
π
π
πρπ
(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 spheresSmeared fitIdeal fit
Data from Wiggnal et al.
2
322
)(
)]cos()[sin(3)(
−∆=
Ω qR
qRqRqRVq
d
dρ
σ
Instrumental smearing is routinely includedin SANS dataanalysis
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And:
Ellipsoid
18Glatter
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 + backgroundR = 15.48 Åε = 1.61 (prolate) χ2=2.4 (χ=1.55)
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where Vout= 4πRout
3/3 and Vin= 4πRin3/3.
∆ρcore is the excess scattering length density of the core, ∆ρshell is the excess scattering length density of the shell and:
Core-shell particles:
[ ])()()()( inincoreshelloutoutshellcoreshellcore qRVqRVqA Φ∆−∆−Φ∆∆=− ρρρρ
[ ]3
cossin3)(
x
xxxx
−=Φ
= − +
21
Glatter
P(q): Core-shell
22www.psc.edu/.../charmm/tutorial/ mackerell/membrane.htmlmrsec.wisc.edu/edetc/cineplex/ micelle.html
SDS micelle
20 Å
Hydrocarbon core
Headgroup/counterions
23q [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/cmRcore = 13.5 ± 2.6 Åε = 1.9 ± 0.10 dhead = 7.1 ± 4.4 Åρhead/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:
OH
e
OH
tail
e
taile
tailV
Z
V
Z
2
2−=∆ρtailaggcore VNV =
OH
e
OH
OHhead
OH
e
heade
headV
Z
nVV
nZZ
2
2
2
2 −+
+=∆ρ
n water molecules in headgroup shell
)( 2OHheadaggshell nVVNV +=
surfactantMN
cn
agg
micelles =
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Cylinder
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E. Gilbert
P(q) cylinder
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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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Collection of particles with spherical symmetry
))'(exp()exp()( jiijjj srqibrqibqArrrrrr
+⋅−=⋅−= ∑∑
srrrr
+= '
center of sphere
Debye, 1915
self-term interference term phase factor
))''(exp()( lkjiklij srsrqibbqIrrrrr
−−+⋅−=∑
))''(exp()'exp()'exp( kjkkliij ssqirqibrqibrrrrrrr
−⋅−⋅⋅−=∑
jk
jk
kkk
kj
jjjjjjqd
qdqRVqRVqRPV
sin)()()(22 Φ∆Φ∆+∆= ∑∑
≠
ρρρ
jk
jk
kkkjjjqd
qdqRVqRV
sin)()(∑ Φ∆Φ∆= ρρ
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•Generate points in space by Monte Carlo simulations
•Select subsets by geometric constraints
•Caclulate histogramsp(r) functions
•(Include polydispersity)
•Fourier transformMC 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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Pure SDS micelles in buffer:
C > CMC (∼ 5 mM)
C ≤ CMC
Nagg= 66±1 oblate ellipsoidsR=20.3±0.3 Åɛɛɛɛ=0.663±0.005
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Polymer chains in solutionGigantic ensemble of 3D random flights
- all with different configurations
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Gaussian polymer chain
Look at two points: Contour separation: LSpatial separation: r
Contribution to scattering:rr
L qr
qrqI
)sin()(2 =
><−∝ 2
2
23
exp)(r
rrD
For an ensemble of polymers, points withL has <r2>=Lb
and r has a Gaussian distribution:
’Density of points’: (Lo- L) Lo
LL
Add scattering from all pair of points
[ ]6/exp 2Lbq−
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Gaussian chains: The calculation
[ ]
222
0
2
2
00
0
122
0
1
02
]1)[exp(2
6/exp)(1
)sin(),()(
1
)sin(|)|,(
1)(
qRxx
xx
LbqLLdLL
rqr
qrLrDLLdLdr
L
rqr
qrLLrDdLdLdr
LqP
g
L
o
o
L
o
o
LL
o
o
o
oo
=+−−
=
−−=
−=
−=
∫
∫∫
∫∫∫∞
∞
How does this function look?
Rg2 = 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
gRq /1≈
21
−− = qq ν
36
Lysozyme in Urea 10 mg/mL
q [Å-1]
0.01 0.1 1
I(q)
[cm
-1]
10-3
10-2
10-1
Lysozyme in urea
Gaussian chain+ backgroundRg = 21.3 Å
χ2=1.8
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Self avoidence
No excluded volume No excluded volume=> expansion
222 ]1)[exp(2)(
x
xxqRxP g
+−−== no analytical solution!
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Monte Carlo simulation approach
(1) Choose model
(5) Fit experimental data using numerical expressionsfor 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
l
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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
21
=
= −−
ν
ν qq
588.0
70.1/1
=
= −−
ν
νqq
Excluded volume chains:
Gaussian chains
.).(/1 volexclRq g≈
)(/1 GaussianRq g≈
q-1 local stiffness
40
C16E6 micelles with ‘C16-’SDS
Sommer C, Pedersen JS, Egelhaaf SU, Cannavaciuolo L, Kohlbrecher J, Schurtenberger P.
Variation of persistencelength with ionic strength
works also for polyelectrolyteslike unfolded proteins
41
Models IIPolydispersity: Spherical particles as e.g. vesicles
No interaction effects: Size distribution D(R)
∑∆=Ω i
iii qPfMcd
qd)(
)( 2ρσ
Oligomeric mixture: Discrete particles
fi = mass fraction ∑ =i
if 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
43
Concentration effects in protein solutions
= q/2π
α-crystallineye lens protein
44
p(r) by Indirect Fourier Transformation (IFT)
At high concentration, the neighborhood is different from theaverage 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 !
45
Low concentraions
= Random-Phase Approximation (RPA)
P’(q) = -
Zimm 1948 – originally for light scattering
Subtract overlapping configuration
υ ~ concentrationP’(q) = P(q) - υ P(q)2
= P(q)[1 - υ P(q)]
)(1)(qP
qP
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]
46
Zimm approach
)(1)(
)(qP
qPKqI
υ+=
+=
+= −−− υ
υ
)(1
)()(1
)( 111
qPK
qP
qPKqI
3/1
1)(
22gRq
qP+
≈ [ ]υ++= −− 3/1)( 2211gRqKqIWith
Plot I(q)-1 versus q2 + c and extrapolate to q=0 and c=0 !
47
Zimm plot
I(q)-1
q2 + c
Kirste and WunderlichPS in toluene
q = 0
P(q)
c = 0
48
My suggestion:
)3/exp(1)(
22
21 aqca
P
c
qI
i
ii
−+=
With a1, a2, and Pi 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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But now we look at the information content related to
these effects…
Understand the fundamental processes and principlesgoverning aggregation and crystallization
Why is the eye lens transparent despite a protein concentrationof 30-40% ?
50
Structure factor
)()(
sin)()(
22
22
qSqPV
qr
qrqPVqI
jk
jk
ρ
ρ
∆=
∆= ∑
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
jqr
qrrpqPV
sin)()(22 ∑∆= ρ
51
Correlation function g(r)
g(r) =Average of the normalized density of atoms in a shell [r ; r+ dr]from the center ofa particle
52
g(r) and S(q)
q
drrqr
qrrgnqS 2)sin(
)1)((41)( ∫ −+= π
53
GIFTGlatter: Generalized Indirect Fourier Transformation (GIFT)
∫= drqr
qrrpqI
)sin()(4)( π
∫= drqr
qrrpcZRRqSqI salteff
)sin()(4),),(,,,()( πση
With concentration effects
Optimized by constrained non-linear least-squares method
- works well for globular models and provides p(r)
54
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
0 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
trimersmonomers
Home-written software
55
Osmometry (Second virial coeff A2)
c
Π/c
Ideal gas A2 = 0repulsio
n A 2> 0
attraction A2 < 0
56
S(q), virial expansion and Zimm
...3211
)0(3
22
1
+++=
∂
Π∂==
−
MAccMAcRTqS
In Zimm approach ν = 2cMA2
From statistical mechanics…:
57
A2 in lysozyme solutions
O. D. Velev, E. W. Kaler, and A. M. Lenhoff
A2
Isoelectric point
58
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 sphereDebye-Hückelscreened Coulomb potential
59
Depletion interactions
π
Asakura & Oosawa, 58
π
60
Theory for colloidal stability
Debye-Hückelscreened Coulomb potential+ attractive interaction
DLVO theory: (Derjaguin-Landau-Vewey-Overbeek)
61
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) !!!
)(11
qnc−
62
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)
63
α-crystallin S. Finet, A. Tardieu
DLVO potential
HNC and numerical solution
64
α-crystallin + PEG 8000 S. Finet, A. Tardieu
DLVO potential
HNC and numerical solution
Depletion interactions:
40 mg/ml α-crystallinsolution pH 6.8, 150 mM ionic strength.
65
Anisotropy
Small: decoupling approximation (Kotlarchyk and Chen,1984) :
Measured structure factor:
[ ] )(1)()(1)(
)(
)( 2 qSqSqqPn
q
qSmeas ≠−+=∆
Ω∂
∂
≡ βρ
σ
!!!!!
66
Large anisotropy
Anisotropy, large: Random-Phase Approximation (RPA):
)(1)(
)( 22
qP
qPVnq
d
d
νρ
σ
+∆=
Ω υ ~ concentration
Anisotropy, large: Polymer Reference Interaction Site Model (PRISM)Integral equation theory – equivalent site approximation
)()(1)(
)( 22
qPqnc
qPVnq
d
d
−∆=
Ωρ
σ c(q) direct correlation functionrelated to FT of V(r)
Polymers, cylinders…
67
Empirical PRISM expression
Arleth, Bergström and Pedersen
)()(1)(
)( 22
qPqnc
qPVnq
d
d
−∆=
Ωρ
σ
c(q) = rod formfactor- empirical from MC simulation
SDS micelle in 1 M NaBr
68
Overview: Available Structure factors
1) Hard-sphere potential: Percus-Yevick approximation 2) Sticky hard-sphere potential: Percus-Yevick approximation3) 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 approximation5) Sticky hard-sphere potential, polydisperse system: Percus-Yevick approximation6) 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’:
69
Summary
• Concentration effects and structure factorsZimm approachSpherical 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
70
LiteratureJan 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 PedersenMonte 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 PedersenModelling 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 KleinInteracting Colloidal Suspensions
in Neutrons, X-Rays and Light
P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 351