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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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
photons
neutrons θ
detector
Scattering vector q = (4π/λ)sin(θ/2)
spatial resolution ~ 1/q
static
SANS (PSI)
Scattering: Basics
Light Scattering
SAXS
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
photons
neutrons θ
detector
Scattering vector q = (4π/λ)sin(θ/2)
spatial resolution ~ 1/q
static
Scattering: Basics
Light Scattering
SAXS
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Single particle shape: the particle form factor P(q)
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)
Single particle shape: the particle form factor P(q)
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
spatial resolution ~ 1/q
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)
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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)
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
photons
neutrons θ
detector
Scattering vector q = (4π/λ)sin(θ/2)
spatial resolution ~ 1/q
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
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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
Interlude: Particle dynamics in real and reciprocal 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)
Interlude: Particle dynamics in real and reciprocal space
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:
Interlude: Particle dynamics in real and reciprocal 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) =
Lund University / Physical Chemistry / The Colloidal Domain - Scattering /
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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