-
Light Scattering
static light scattering
S U EgelhaafCondensed Matter Physics Laboratory
Heinrich-Heine-UniversityDüsseldorf, Germany
[email protected]
2011 Winter School of the FOR 1394 ʻNonlinear Response to Probe
Vitrification ̓– Colloidal Dispersions and Rheology –
Konstanz, 3 – 8 March 2011
-
Outline• static scattering methods
probes (scattering) mass distribution on different length
scales• particle size and shape• particle arrangement
(interactions)
• dynamic scattering methodsprobes time scales (on a given
length scale)• particle dynamics (which depend on the
interactions)
-
Concept
detector
-
Length Scale L
Rayleigh scattering
Rayleigh-Gans-Debye
Mie scattering
Fraunhofer regime
€
L λ
-
Rayleigh ScatteringL
-
Rayleigh-Gans-Debye ScatteringL
-
Mie Scattering
incident wave undergoes significant changes inside particle,i.e.
Born approximation is no longer valid
need to consider wave inside and outside particle (incident and
scattered)
strength of electric field depends highly on position
electromagnetic radiation interacts non-linearly with
particles
Gentle introduction: Glatter in ʻNeutrons, X-rays and Light:
Scattering Methods Applied to Soft Condensed Matterʼ, Elsevier
2002General scattering theory: Kerker 'The Scattering of Light and
Other Electromagnetic Radiation', Academic Press 1969 van de Hulst
'Light Scattering by Small Particles', Dover 1957, 1981T-matrix
method (any shape, but mainly axisymmetric): Barber & Hill
'Light Scattering by Particles: Computational Methods', World
Scientific 1990Discrete dipole approximation (particle = array of
point dipoles with d
-
Fraunhofer RegimeL >> λ
wave hardly penetrates particle
∴ scattering process approximated by interaction of wave with
cross-section (aperture)
∴ particle sizing, but no information on shape or internal
structure
-
(1) Static Scattering• static light scattering (SLS)
theoretical background examples
• small angle x-ray (SAXS) and neutron scattering (SANS)
theoretical background comparison of SLS, SAXS and SANS
-
Rayleigh ScatteringL
-
Scattering by a Particle
large particle = ensemble of small volume elements dV acting
like Rayleigh scatterers
∴ Es = ∫ dEs = ∫ ρ(r) dV V V
eiδφ e-iQ⋅r
δφ = 2π (δL/λ) = (2π/λ) δL = ki⋅r - ks⋅r = - Q⋅r
with scattering vectorQ = ks - ki |Q| = 2k sin(θ/2) = sin(θ/2)
4π
λ
Fourier transform !
|Q| ~ 1/(length scale)
θki
|ks|=|ki|=|k| (quasi) elasticscattering
r
Δ
-
Scattering by many Particles
e-iQ⋅Rj
many particles = ensemble of particles j=1..N
∴ Es = Σ Es,j j
Rj
Rj+1
r Es = Σ ∫ Δρ(r) e-iQ⋅r dV e-iQ⋅Rj j Vj
Es = Σ bj(Q) e-iQ⋅Rj
j
-
Scattered Intensityusually the time-averaged (=
ensemble-averaged) scattered intensityis determined:Is(Q) = t = = ~
Σ Σ k j
assumption: all particles are identical, i.e. bj(Q,t) = bk(Q,t)
= b(Q,t) (important: particle properties are not linked to their
positions)
Is(Q) ~ Σ Σ
k j
1 N
amplitude scatteredby N single particles
(random walk)
= P(Q) form factor
= S(Q) structure factor
Is(Q) ~ N Σ Σ
k j
-
Form Factor P(q)
homogeneous sphere
€
b(q) = Δρ(r)∫ e− iq ⋅r dV = Δρ e−iq ⋅r dVVsphere
∫ = Δρ dφ∫ r2 d r∫ e−iqr cosθ∫ sinθ dθ
~ 3qR( )3
sin qR( ) − qrcos qR( )( )
4.49 7.73 10.90
-
Form Factor P(q)
homogeneous sphere
4.49 7.73 10.90
€
b(q) = Δρ(r)∫ e− iq ⋅r dV = Δρ e−iq ⋅r dVVsphere
∫ = Δρ dφ∫ r2 d r∫ e−iqr cosθ∫ sinθ dθ
~ 3qR( )3
sin qR( ) − qrcos qR( )( )
polydisperse
-
Form Factor P(q)
homogeneous spherepolydisperse
€
P(q) =N (r)M (r)2P(q,r)
0
∞
∫ dr
N (r)M (r)20
∞
∫ dr
€
M =N (r)M (r)2 dr
0
∞
∫
N (r)M (r)dr0
∞
∫
€
Rg2 =N (r)M (r)2Rg2 dr
0
∞
∫
N (r)M (r)2 dr0
∞
∫
form factor
molar mass
radius ofgyration
-
Form Factor P(q)
rjrkr
pair distance distribution function
Indirect Fourier Transform (IFT)
intraparticle interferencedepends on particle size and shape
€
P(q) =b(q)
2
b(0)2
with b(q) = Δρ(r )V1
∫ e−iq⋅r dV
b(q)2
= 4π p(r) sin qrqr
dr0
∞
∫
€
p(r) = r2 Δρ( ′ r )V∫ Δρ(r − ′ r ) d3 ′ r
-
Form Factor
mass of ʻfractalʼ M ~ Ldfwith length L, ʻfractalʼ dimension
df
scattering intensity Is ~ (QL)-df
log(Q)
log(
I s)
-
global structureMw
Form Factor P(q)
polymers
log(Q) lo
g(I s)
Q-1.66
Q-2
(self avoiding)random walk
1/Rg
Q-1 cylinder
2/lp
. cross section
1/Rg,cs
-
Structure Factor S(Q) 1 N S(Q) = Σ Σ k j
S(Q) - 1 = 4π ∫ (g(R)-1) R2 dr N V
sin(QR) QR
with the radial distribution function g(R) (N/V) g(R)d3R is the
number of particles in d3R at R Boltzmann distribution suggests
g(R) = e-U(R)/kT
R
-
Structure Factor S(q)crystals
T.Palberg (1999) JPCM 11, R323
€
X(t) = cIhkl = c S(q)dqq1
q2
∫
€
L(t) = πKΔq1/ 2(t)a
€
nc (t) =X(t)L3(t)
fraction ofcrystalline phase
average lineardimension
number densityof crystals
-
Structure Factor S(Q) Is(Q) ~ N P(Q) S(Q)
φ 0.5
0.4
0.3
0.2
0.1 divide by P(Q)
φ 0.5
0.4
0.3
0.2
0.1
de Kruif et al., Langmuir 4, 668
-
Structure Factor S(Q)
φ 0.5
0.4
0.3
0.2
0.1osmotic compressibility
kBT ∂ρ/∂Π
φ0.00 0.02 0.04 0.06
Is(Q) ~ N P(Q) S(Q)
-
Structure Factor S(q)with osmotic compressibility (dΠ/dc)-1
virial expansion of the osmotic pressure Π
with second virial coefficient B2
→0)
€
S(q→ 0) = N AM
kT dΠdc
−1
€
S(q→ 0) = 1− 2B2NV
+…
example: hard spheres
€
S(0) =1+ 4π NV
g(r) −1( ) r2 d r∫ =1+ 4π NV −1( ) r2 d r
0
2R
∫
€
=1− 4π3NV(2R)3 =1− 8φ
-
Dumbbell
r1r2
r12|r12 | = r
I(Q) ~ P(Q) S(Q)with P(Q) form factor of a sphere S(Q) = (1/N)
Σj Σk
€
S(Q) = 12 j=1
2
∑ e− iQ⋅ r j −r k( )k=1
2
∑ = 12 1+ e− iQ⋅ r1−r 2( ) + e− iQ⋅ r 2−r1( ) +1 = 1+ cos Q ⋅
r12( )
with cos α = (eiα + e-iα)/2
spherical averaging (rotational Brownian motion)
€
S(Q) =1+ 14π 0
2π
∫ cos Q ⋅ r12( ) sinθ dθ dφ =0
π
∫ 1+ 2π4π cos Q2rcosθ( ) sinθ dθ0
π
∫
S(Q) =1+ 12
cos 2Qru( ) du =1+sin 2Qr( )2Qr−1
1
∫
-
Dumbbell
r1r2
r12|r12 | = r
I(Q) ~ P(Q) S(Q)with P(Q) form factor of a sphere S(Q) = (1/N)
Σj Σk
€
S(Q) = 12 j=1
2
∑ e− iQ⋅ r j −r k( )k=1
2
∑ = 12 1+ e− iQ⋅ r1−r 2( ) + e− iQ⋅ r 2−r1( ) +1 = 1+ cos Q ⋅
r12( )
with cos α = (eiα + e-iα)/2
spherical averaging
€
S(Q) =1+ 14π 0
2π
∫ cos Q ⋅ r12( ) sinθ dθ dφ =0
π
∫ 1+ 2π4π cos Q2rcosθ( ) sinθ dθ0
π
∫
S(Q) =1+sin 2Qr( )2Qr
0.5
1
1.5
2
0 2 4 6
S(Q)
Qr
-
Core-Shell Particle
Dumbbell
r1r2
r12|r12 | = r
-
Core-Shell Particle
0.00001
0.0001
0.001
0.01
0.1
1
0.01 0.1
R=100, Ri=80R=100, Ri= 0R=117, Ri= 0
Q
(Rg = Rg)
= -P(Q) ~ ((sin(QRs) - QRs cos(QRs)) - (sin(QRc) - QRc
cos(QRc)))2I(Q) ~ P(Q) S(Q) with here S(Q) = 1
-
Repulsive and Attractive Glasses
Arepulsiveglass
attractiveglassF, G
C, D, E
rcpvolume fraction
, B
attra
ctio
nc p
(mg/
ml)
Pham et al. (2002) Science 296, 104Pham et al. (2004) Phys. Rev.
E 69, 011503
-
Static Structure Factor
-
Light, x-rays and Neutronslength scales: L ~ 2π/Q with Q =
(4π/λ) sin(θ/2)
θ 0.1° 1° 10° 100° 180°
lightλ ≈ 400 nm
Q (nm-1) 0.3x10-3 3x10-3 0.02 0.03
2π/Q (nm) 20,000 2000 300 200
x-rays,neutronsλ ≈ 1 nm
Q (nm-1) 0.01 0.1 1 10 13
2π/Q (nm) 630 63 6 0.6 0.5
-
Light, x-rays and Neutronsflux: photons or neutrons per time
scattering power and contrast: ∆ρ(r)= ρ(r) - ρ0
-
Neutron Scattering Length
-0.5
0
0.5
1
1.5
2
0 10 20 30 40 50 60 70 80
neutron scattering lengths
b c [1
0-12
cm
]
atomic number Z
-
(Matrix) Contrast Variation
-
(Matrix) Contrast Variation
-
(Matrix) Contrast Variation
-
(Matrix) Contrast Variation
-
Structure of the
Southern Bean Mottle Virus
C Chauvin, B Jacrot, J Witz (1976)
RNA
protein
~ 70% D2O - RNA matched, i.e. protein (shell) scattering qmin =
4.49/Rs = 4.49/200 Å = 2.24 x 10-2 Å-1
~ 40% D2O - protein matched, i.e. RNA (core) scattering qmin =
4.49/Rc = 4.49/100 Å = 4.49 x 10-2 Å-1
minimum for a sphere!only an approximation for a hollow
sphere
-
Conformation of Polymer Chains in Bulk
from R.Heenan (ISIS)
-
Conformation of Polymer Chains in Bulk
H-PS
D-PS (21kDa)in H-PS
directbeam
difference
expect I ~ q-2
I-1 ~ q2
q2I = const
-
can be replaced by fibre
Instrumentation
light scattering
Laser
-
Instrumentation
small-angle x-ray scattering (central
facility)
-
Instrumentation
small-angle x-ray scattering (central
facility)
-
Instrumentation
small-angle neutron scattering (SANS)
-
Instrumentation
small-angle neutron scattering (SANS)
velocityselector
collimation sample
source
detector
-
Data Analysis
model fitting
indirect Fourier transform/deconvolution
-
Solution Structure of Human PCNA
collaboration with U Hübscher
(Zürich)
J Mol Biol 275, 123-132 (1998)
-
Hexamer Model
-
0.1
1.0
10.0
0.02 0.06 0.1 0.5
scat
terin
g in
tens
ityd σ
(q)/dΩ
[m-1
]
scattering vector q [Å -1]
crystal structure(yeast)
hexamer model
scatteringintensity
dσ/dΩ [m-1]
Neutron Scattering
-
0.00
0.01
0.02
0 20 40 60 80 100
human PCNA: distance distribution function
p(r)/
c [m
2 /kg]
r [Å]
crystal structure (yeast)
hexamer model
Pair Distance Distribution Function
-
SummaryStatic light scattering provides information on
individual particles (shape, size etc.)and
particle arrangements.
-
[email protected]