Scattering experiments - Lunds universitet · 2011. 10. 7. · Scattering: Light vs. x-rays vs....

Scattering experiments

Menu1. Basics: basics, contrast, q and q-range2. Static scattering: Light, x-rays and neutrons 3. Dynamics: DLS4. Key examples

Lund University / Physical Chemistry / The Colloidal Domain - Scattering /

The Colloidal Domain

Polymers

SurfactantsColloids

Length- and Timescales

Equilibrium- and

Non-equilibriumStates

The Magic Triangle

DNA

fd-Virus

Block-copolymers

Proteins

Soft Matter - „complex fluids“world between fluid and solid

2

Important quantities:• Size, Shape, Mass, Structure

• Interactions

• Dynamics, Diffusion Coefficients


Characteristic length and time scales

3

radius

102100 104 106 108 1010 1012

1 nm 10 nm 100 nm 1 μm

10 ps 1 ns 1μs 1ms 1 s

1000 m2/g 100 m2/g 10 m2/gsurface

molecular time scales

molar mass

atoms proteins

molecules virus, DNA, vesicles

colloids: latex, microgels, micelles

polymers

microscopic mesoscopic macroscopic systemsatomic/molecular colloid physics and chemistry, solid state physicsphysics and chemistry biology

SANS

SAXS/WAXS

Light scattering

USALS

CLSM-Video-microscopy


Multiscale characterization in Physical Chemistry

4

Methods 1-105 nm

10-8-105 sec.in-situ, non-invasive

time resolved

DWS

Zetasizer

CLSM-Video-microscopy

multi-3D light scattering

3D light scattering

SAXS/WAXS

USALS

NMR self diffusionAres LS 1 Rheometer


Probe choice: length and time scales, contrast

Source of

radiation θ

detector

radiation with known wavelength and

energy

Ensemble of molecules or particles: vibration, rotation,

translation and diffusion

scattered radiation(new) wavelength

and energy

elastic or staticconformation, structure, size,

interactions

quasielastic, inelastic or dynamic

Local and global dynamics, diffusion, vibrations, rotation, hydrodynamics

5

Introduction to scattering methods


photons

neutrons θ

detector

Scattering vector q = (4π/λ)sin(θ/2)

spatial resolution ~ 1/q

static

SANS (PSI)

Scattering: Basics

Light Scattering

SAXS


photons

neutrons θ

detector



static

Scattering: Basics

Light Scattering

SAXS


quasielastic and inelastic scattering experiments: length and time scales, contrast

time scales vs. energy and frequency

Brownian motion of colloids

DLS

NSE

triple axis

TOF

Scattering: Dynamics

8

0

0

0

0

1

2

0

0

Single particle shape: the particle form factor P(q)

0

0

0

0

1

2

0

0


0

0

0

0

1

2

0

0

Formfactor P(q)for an ideal

monodispersesphere

0.0001

0.001

0.01

0.1

0.05 0.1 1 5

I(q)

q [nm-1]

Formfactor P(q)for a globularProtein

(Gamma crystallin)


Basic (static) scattering theory: assumptions and definitions

Basic assumptions

The scattering process is fully elastic.The incident primary beam can be described as a plane wave.The scattered probe particles/radiation can be described as spherical waves.The individual scattering centers are small compared to the wave length -> point scatterers.The sample-detector distance is sufficiently large -> far field solution.

photon,neutronsource θ

detector

i

r k

s

r k

scattering volume

k = 2π/λ

Interference and scattering vector I

Ai

R ( ) = A0e

i

k i ⋅

R = A0eiϕ

Amplitude Ai of incoming plane wave at position R:

As(

R ') = A0b

ei

k s ⋅

R '

R '

As product of 3 contributions:

Amplitude A0 of incoming plane

wave

Scattering length b

characteristic spherical wave

Scattering by a point scatterer fixed in space (1):

Rʼ

Interference and scattering vector II

Scattering by two point scatterers fixed in space (1 and 2):

b

a

As(

R ') = A j

s

j=1

2

∑ ≅A0

R'ei

k s ⋅

R ' bj

j=1

2

∑ eiΔϕ j

Δϕ = 2π/λ × Δs

1: origin of coordinate system

→ Δϕ1 = 0

→ Δϕ2 → given by path length

difference Δs

→ Δs = a - b = ki•r - ks•r

phase difference

interference term

Interference and scattering vector III

Definition of the scattering vector q:

b

a

As(

R ') = A j

s

j=1

2

∑ ≅A0

R'ei

k s ⋅

R ' bj

j=1

2

∑ ei

q ⋅

r j

Δϕ2 =

2πλΔs =

q ⋅

r

q :=

k i −

k s

quasielastic scattering |ki| ≅ |ks| →

q =4πλsin

θ2


Interference and scattering vector IV

scattering by N point scatterers at fixed positions

As(

R ') = A j

s

j=1

N

∑ ≅A0

R'ei

k s ⋅

R ' bj

j=1

N

∑ ei

q ⋅

r j

Is

R '( ) = As

R '( ) ⋅ As

*

R '( ) =A02

R'2bjbke

i

q ⋅

r jk

j;k=1

N

∑

normalized scattering intensity from N mobile point scatterers

rjk = rj - rk , average < > over all possible particle configurations

differential scattering cross section, identical particles:

dσdΩ

q ( ) =

Is(

R )

I0R'2 = b2 ei

q ⋅

r jk

j;k=1

N

∑

Fourier transform of b(r)


an „ideal gas“ of noninteracting particles:

Is q( ) = N ⋅ I p q( ) = N ⋅ I p (0) ⋅P(q)

P(q) → contains information about size and structure of particle

P(q) =Ip (q)

Ip (q→ 0)

The particle mass and the form factor

17

Ip(0) = V2Δρ2 intensity of single particle at q = 0 → Ip(0) ∝ M2

I(0) → contains information about mass of particle


Example: homogeneous sphere

function has minima for tan(qR) = qR, or qR = 4.49, 7.73, …

calculation for sphere with radius R = 60 Å → minima at q = 4.49/60 = 0.075

P q( ) = 3sin qR( ) − qR( )cos qR( )

qR( )3⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2

The particle form factor

18


Ideal polymers: Debye function

dσdΩ

(q) = N 2P(q),

where P(q) =2

q2 RG2[ ]2 e

−q 2 RG2( ) + q2 RG

2 −1⎡ ⎣ ⎢

⎤ ⎦ ⎥

I(q) RG: radius of gyration with

Rg2 =

1

N

R j −

R CM( )

j=1

N

∑2

R CM =

1

N

R j

j=1

N

∑

Rg2 =

1

2N 2

R j −

R k( )

j;k=1

N

∑2

P(q) = onlyf q2 RG2( )

The particle form factor

19


Guinier approximation:

direct and model-free determination of RG from small-q scattering

P(q) ≈1−1

3q2RG

2 = 1−1

3q2 RG

2

comparison of spheres and polymer coils with similar RG

Guinier regime

Arbitrary particle shape: The Guinier approximation

20

Interparticle correlation: the structure factor S(q)

0

0

1

2

S q( ) = FT g r( )[ ]g(r): radial distribution function

as a measure of spatial correlation

g(r) S(q)

ideal gas

repulsive spheres

qmax =2πdchar

P(q)

I(q)

S(q)

I(q)

polystyrene spheres, R = 85 nm in water (added salt → hard sphere

interactions)

interacting particles: the structure factor S(q)

I(q) ≈ NM2P(q)S(q)


photons

neutrons θ

detector



static

contrast

SANS:scattering lengthSAXS:electron densitySLS:polarizability

Scattering: Light vs. x-rays vs. neutrons

characteristic properties:

probe λ contrast

light 500 nm Δn

x-rays 0.1 - 1 nm Δz

neutrons 0.1 - 1 nm Δb

I(q) ∼ Δρ2 N M2 P(q) S(q)

23


scattering contrast: x-rays vs. neutrons

24

H C O Ti Fe Ni U

x-rays neutrons

H C O Ti Fe Ni U

Contrast variation - or why neutrons

polymer melt

Nobel prize 1974

P.J. FloryStanfordUSA

scattering angle

inte

nsity

Kirste et al.

Jülich 1974

Contrast variation - the case of polymer melts

C12E5 + decane in D2O:

rapid quench from L -> L+O

main questions and problems:

• droplet growth process?

• very few large droplets

• rapidly very turbid

U. Olsson, H. Bagger-Jörgensen, M. Leaver, J. Morris, K. Mortensen, R. Strey, P. Schurtenberger, and H. Wennerström, Prog. Colloid Polym. Sci. 106, 6 - 13 (1997)

Nucleation and phase separation in microemulsions

Neutrons and surfactants - contrast variation again

starting point:

r

Δρ(r)

oil-in-water microemulsion

• h-oil and h-surfactant in D2O

-> bulk contrast

scattering length 1H 2H

b in 10-14 m -0.38 0.66

Neutrons and surfactants - contrast variation again

starting point:

r

Δρ(r)

oil-in-water microemulsion

• h-oil and h-surfactant in D2O

-> bulk contrast

• d-oil and h-surfactant in D2O

-> shell contrast

Contrast variation allows to highlight individual parts of complex

systems

scattering length 1H 2H

b in 10-14 m -0.38 0.66

Nucleation and phase separation from SANS experiments

main idea: overall contrast match in SANS experiment

forward intensity suppressed

small droplets still visible from core/shell contrast

Time-resolved SANS experiments

time-resolved SANS study (D22, ILL)

• growth of big oil droplets

• Readjustment of small droplets

S. Egelhaaf, U. Olsson, P. Schurtenberger, J. Morris, and H. Wennerström, Phys. Rev. E (1999)

low polydispersity

Key points:• contrast variation• large q-range• large neutron flux

What about dynamics?

A short introduction to dynamic light scattering

Laser

Detector Dynamics

2D – Detector

Measure fluctuations in light intensity

Sample

Transmission > 95%

Θ

spatial resolution over which we monitor diffusion ~ 1/q

DLS

A short introduction to dynamic light scattering

1.6 μm0.12 μm

Laser

Detector

Sample

Transmission > 95%

DLS

Interlude: Particle dynamics in real and reciprocal space

Particle tracking with a microscope

Dynamics in reciprocal (Fourier) space


Particle tracking with a microscope

J. B. Perrin, "Mouvement brownien et réalité moléculaire," Ann. de Chimie et de Physique (VIII) 18, 5-114 (1909)


The typical time scale for the duration of a fluctuation is

determined by the time it takes the relative phase differences between

the two paths to change by approximately unity.

Dynamics in reciprocal (Fourier) space:


Dynamics in reciprocal (Fourier) space:

Structure of intermediate scattering function, f(q,τ), gives information on scatterer dynamics

Delay time τ

Intensity autocorrelation

function

<I(q,t) I(q,t+τ)>

~ TC

<I 2>

<I >2

f M q,τ( ) = dD∫ P D( ) exp −Dq2τ[ ]

Particle Diffusion Stokes-Einstein-Relation

Correlation Function

Example (Θ=90°)

Numerical Inversion

Particle Sizing with DLS

38

g(τ) =


Interactions and dynamic light scattering

39

F q,τ( ) = 1

Nexp −iq.rj 0( )[ ]

j

∑ 1

Nexp iq.rj τ( )[ ]

j

∑

DLS observes stochastic dynamics of sinusoidal density fluctuations of wavelength2π/q (spatial Fourier components )

characteristic length D

Dq >>π2

Collective or gradient diffusion

collective diffusion coefficient:

CC f

D ρΠ ∂∂=

Dq ≈π2

Observe dominant structure( particle and cage ofneighbours )

Structural relaxation

Dq <<π2

Local motion ofindividual particles

Self diffusion

40

U. Olsson, H. Bagger-Jörgensen, M. Leaver, J. Morris, K. Mortensen, R. Strey, P. Schurtenberger, and H. Wennerström, Prog. Colloid Polym. Sci. 106, 6 - 13 (1997)

C12E5 + decane in D2O:

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4

ΔR

(0)

/ m-1

φ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.1 0.2 0.3 0.4

Ds/D

0 & D

c/D0

φHS

Dc/D

0

Ds/D

0

Hard sphere theory

Collective versus self diffusion

Summary and conclusions

Scattering provides information on:

• Mass: I(0)/C ∼ M

• Size, shape and structure: P(q)

• Interactions: S(q)

• Diffusion: 〈I(t)I(t+τ)〉∼ exp(-2Dq2τ)

• Size and size distribution

Light, x-rays and NeutronsLength scales: L ~ 2π/q with q = (4π/λ) sin(θ/2)

θ 0.1° 1° 10° 100° 180°

lightq (Å-1) 3x10-6 30x10-6 0.3x10-3 0.002 0.003

λ ≈ 400 nm2π/q (Å) 2x106 200,000 20,000 3,000 2,000

x-rays, q (Å-1) 0.001 0.01 0.1 1 1.3

neutrons

λ ≈ 1 nm 2π/q (Å) 6300 630 60 6 5

Light, x-rays and Neutrons

Structure

Dynamics

SANS: 10-3 < q < 1 Å-1 SAXS: 10-3 < q < 1 Å-1 ESRF 10-2 < q < 1 Å-1 Lab.

SANS/SAXS: 10-3 < q < 1 Å-1

USALS/SLS: 2x10-6 < q < 2.5x10-3 Å-1

Summary and conclusions

10 -4

10 -2

100

102

104

106

108

1010

1012

1014

1016

104

102

100

10 -2

10 -4

10 -6

10 -8

10 -10

10 -12

10 -14

10 -16

10 -7 10 -6 10 -5 10 -4 10 -3 10-2 10 -1 100 101

108 107 106 105 104 103 102 101 100 10 -1

length [Å]

scattering vector [Å-1]

time [s]

freq

uenc

y [H

z]

dynamic

light

Scattering (DLS)

Brillouin/Raman light scattering

Neutron inelastic scattering

and spin-echoexperiment

XPCS

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Scattering experiments - Lunds universitet · 2011. 10. 7. · Scattering: Light vs. x-rays vs....

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