Top Banner
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
52

Light Scattering - Uni Konstanz...Light Scattering static light scattering S U Egelhaaf Condensed Matter Physics Laboratory Heinrich-Heine-University Düsseldorf, Germany...

Feb 06, 2021

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
  • 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    incident wave undergoes significant changes inside particle,i.e. Born approximation is no longer valid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

    ∫ 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

    ∫ 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]