Light Scattering static light scattering
S U Egelhaaf Condensed Matter Physics Laboratory Heinrich-Heine-University Düsseldorf, Germany [email protected]
2011 Winter School of the FOR 1394 ʻNonlinear Response to Probe Vitrificationʼ – 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 methods probes time scales (on a given length scale) • particle dynamics (which depend on the interactions) Concept
detector Length Scale L
Rayleigh scattering L << λ
λ Rayleigh-Gans-Debye L << 1− n1 n2
Mie scattering L ~ λ
Fraunhofer regime € L >> λ
€ Rayleigh Scattering L << λ object acts as induced oscillating dipole and re-radiates or scatters light Rayleigh-Gans-Debye Scattering
L << λ / |1-n1/n2 |
Born Approximation: incident wave is (almost) undistorted by particle c L = 0 t n1 n1 c0 c0 ∆L = L - L2 = t - t n1 n2
c0 n1 n2 > n1 = t (1 - ) n1 n2 n = L (1 - 1 ) << λ n2
c0 λ L2 = t n2 Mie Scattering L ~ λ incident wave undergoes significant changes inside particle, i.e. Born approximation is no longer valid need to consider wave inside and outside particle (incident and scattered) strength of electric field depends highly on position electromagnetic radiation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 2002 General scattering theory: Kerker 'The Scattering of Light and Other Electromagnetic Radiation', Academic Press 1969 van de Hulst 'Light Scattering by Small Particles', Dover 1957, 1981 T-matrix method (any shape, but mainly axisymmetric): Barber & Hill 'Light Scattering by Particles: Computational Methods', World Scientific 1990 Discrete dipole approximation (particle = array of point dipoles with d << λ): Draine, Flatau (1994) J. Opt. Soc. Am. A 11, 1491 Modal analysis (multiple reflection): Mackowski (1991) Proc. R. Soc. London Ser. A 433, 598 Fraunhofer Regime L >> 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 Scattering L << λ scattering amplitude (or field) Es depends on: • strength of incident radiation • distance of detector (spherical wave originating from V) • materialʼs scattering ability ρ • amount of material (volume V ~ L3)
∴ dEs ~ ρ dV (small volume dV) Scattering by a Particle large particle = ensemble of small volume elements dV acting like Rayleigh scatterers
iδφ -iQ⋅r ∴ Es = ∫ e dEs = ∫ e Δ ρ(r) dV V V ki Fourier transform ! θ r
δφ = 2π (δL/λ) = (2π/λ) δL = ki⋅r - ks⋅r |ks|=|ki|=|k| = - Q⋅r (quasi) elastic scattering with scattering vector
Q = ks - ki 4π |Q| = 2k sin(θ/2) = sin(θ/2) λ |Q| ~ 1/(length scale) Scattering by many Particles many particles = ensemble of particles j=1..N
-iQ⋅Rj ∴ Es = Σ Es,j e j
-iQ⋅r -iQ⋅Rj Es = Σ ∫ Δρ(r) e dV e r j Vj
-iQ⋅Rj Es = Σ bj(Q) e j Rj
Rj+1 Scattered Intensity usually the time-averaged (= ensemble-averaged) scattered intensity is determined: 2 Is(Q) =
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)
2 -iQ⋅(Rj(t)-Rk(t)) Is(Q) ~ <|b(Q,t)| > Σ Σ
2 2 <|b(Q,t)| > 1 -iQ⋅(Rj(t)-Rk(t)) Is(Q) ~ N<|b(0,t)| > 2 Σ Σ
amplitude scattered = P(Q) = S(Q) by N single particles form factor structure factor (random walk) Form Factor P(q) homogeneous sphere
b(q) = ∫ Δρ(r) e−iq⋅r dV = Δρ ∫ e−iq⋅r dV = Δρ ∫ dφ ∫ r2 dr ∫ e−iqr cosθ sinθ dθ Vsphere
3 ~ 3 (sin(qR) − qrcos(qR)) (qR)
€
4.49 7.73 10.90 Form Factor P(q) polydisperse homogeneous sphere
b(q) = ∫ Δρ(r) e−iq⋅r dV = Δρ ∫ e−iq⋅r dV = Δρ ∫ dφ ∫ r2 dr ∫ e−iqr cosθ sinθ dθ Vsphere
3 ~ 3 (sin(qR) − qrcos(qR)) (qR)
€
4.49 7.73 10.90 Form Factor P(q) polydisperse homogeneous sphere
∞ ∫ N (r)M(r)2P(q,r)dr P(q) 0 form factor = ∞ ∫ N (r)M(r)2 dr 0
∞ € ∫ N (r)M(r)2 dr M 0 molar mass = ∞ ∫ N (r)M(r)dr 0
∞ € N (r)M(r)2R2 dr radius of ∫ g R2 = 0 gyration g ∞ ∫ N (r)M(r)2 dr 0
€ Form Factor P(q)
intraparticle interference depends on particle size and shape
2 b(q) P(q) = 2 b(0) r rj r with b(q) = ∫ Δρ(r)e−iq⋅r dV k V1 ∞ 2 sin qr b(q) = 4π p(r) dr ∫ qr 0
pair distance distribution function € p(r) = r2 ∫ Δρ(r ′) Δρ(r − r ′) d3 r ′ V Indirect Fourier Transform (IFT)
€ Form Factor
mass of ʻfractalʼ M ~ Ldf with length L, ʻfractalʼ dimension df
-df scattering intensity I ~ (QL) ) s s log(I
log(Q) Form Factor P(q) polymers
global structure Mw Q-1.66 (self avoiding) random walk ) s I Q-2 log( cylinder Q-1
1/Rg 2/lp 1/ R g,cs log(Q) Structure Factor S(Q)
1 S(Q) = Σ Σ
N sin(QR) S(Q) - 1 = 4π ∫ (g(R)-1) R2 dr V 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 Structure Factor S(q) crystals
q fraction of 2 X(t) = cI = c S(q)dq crystalline phase hkl ∫ q1 average linear πK L(t) = dimension Δq (t)a € 1/ 2 X(t) number density n (t) = of crystals c L3 (t) € T.Palberg (1999) JPCM 11, R323
€ Structure Factor S(Q)
2 Is(Q) ~ N<|b(0,t)| > P(Q) S(Q) φ
0.5 φ 0.5 0.4
0.4 0.3
0.2 0.3 divide by P(Q) 0.1 0.2
0.1
de Kruif et al., Langmuir 4, 668 Structure Factor S(Q)
2 Is(Q) ~ N<|b(0,t)| > P(Q) S(Q)
φ
φ 0.5 0.00 0.02 0.04 0.06
0.4
0.3
0.2
0.1 osmotic compressibility
kBT ∂ρ/∂Π Structure Factor S(q) →0)
−1 N A d Π S(q → 0) = kT with osmotic compressibility (dΠ/dc)-1 M dc virial expansion of the osmotic pressure Π
€ N S(q → 0) = 1− 2B2 +… with second virial coefficient B V 2
€ example: hard spheres N N 2R S(0) =1+ 4π ∫ (g(r) −1) r2 dr =1+ 4π ∫ (−1) r2 dr V V 0 4π N =1− (2R)3 =1− 8φ 3 V €
€ Dumbbell |r12 | = r
r12
I(Q) ~ P(Q) S(Q) r 1 with P(Q) form factor of a sphere r2 -iQ⋅(rj-rk) S(Q) = (1/N) Σj Σk
€ Dumbbell |r12 | = r
r12
I(Q) ~ P(Q) S(Q) r 1 with P(Q) form factor of a sphere r2 -iQ⋅(rj-rk) S(Q) = (1/N) Σj Σk
€ Core-Shell Particle
Dumbbell
|r12 | = r
r12
r1 r2 Core-Shell Particle
= -
2 P(Q) ~ ((sin(QRs) - QRs cos(QRs)) - (sin(QRc) - QRc cos(QRc))) I(Q) ~ P(Q) S(Q) with here S(Q) = 1
1
0.1
0.01
0.001
0.0001 R=100, Ri=80 R=100, Ri= 0 (R = R ) R=117, Ri= 0 g g 0.00001 0.01 0.1 Q Repulsive and Attractive Glasses
attractive glass F, G )
C, D, E (mg/ml p attraction c
A, B repulsive glass volume fraction rcp
Pham et al. (2002) Science 296, 104 Pham et al. (2004) Phys. Rev. E 69, 011503 Static Structure Factor Light, x-rays and Neutrons
length scales: L ~ 2π/Q with Q = (4π/λ) sin(θ/2)
θ 0.1° 1° 10° 100° 180°
Q (nm-1) 0.3x10-3 3x10-3 0.02 0.03 light λ ≈ 400 nm 2π/Q (nm) 20,000 2000 300 200 x-rays, Q (nm-1) 0.01 0.1 1 10 13 neutrons λ ≈ 1 nm 2π/Q (nm) 630 63 6 0.6 0.5 Light, x-rays and Neutrons flux: photons or neutrons per time
scattering power and contrast: ∆ρ(r)= ρ(r) - ρ0 Neutron Scattering Length neutron scattering lengths
2
1.5
1 cm]
-12
[10 0.5 c b
0
-0.5 0 10 20 30 40 50 60 70 80 atomic number Z (Matrix) Contrast Variation (Matrix) Contrast Variation (Matrix) Contrast Variation (Matrix) Contrast Variation Structure of the Southern Bean Mottle Virus
RNA
protein minimum for a sphere! only an approximation for a hollow sphere
~ 70% D2O - RNA matched, i.e. protein (shell) scattering -2 -1 qmin = 4.49/Rs = 4.49/200 Å = 2.24 x 10 Å
~ 40% D2O - protein matched, i.e. RNA (core) scattering -2 -1 qmin = 4.49/Rc = 4.49/100 Å = 4.49 x 10 Å
C Chauvin, B Jacrot, J Witz (1976) Conformation of Polymer Chains in Bulk
from R.Heenan (ISIS) Conformation of Polymer Chains in Bulk
expect I ~ q-2
I-1 ~ q2 D-PS (21kDa) in H-PS difference direct beam H-PS
q2I = const Instrumentation light scattering
can be replaced by fibre
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)
source
velocity collimation sample detector selector 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 Neutron Scattering
10.0 ] -1 [m scatteringintensity intensity 1.0 d /d [m-1 Ω (q)/d ]
σ Ω σ d scattering scattering crystal structure (yeast) hexamer model 0.1 0.02 0.06 0.1 0.5 scattering vector q [Å -1] Pair Distance Distribution Function human PCNA: distance distribution function
0.02 /kg] 2 0.01 p(r)/c [m
hexamer model
crystal structure (yeast) 0.00 0 20 40 60 80 100 r [Å] Summary Static light scattering provides information on individual particles (shape, size etc.) and particle arrangements. [email protected]