Scattering experiments

Transcription

1 Scattering experiments Menu 1. Basics: basics, contrast, q and q-range. Static scattering: Light, x-rays and neutrons 3. Dynamics: DLS 4. Key examples Polymers The Colloidal Domain The Magic Triangle Length- and Timescales Colloids Important quantities: DNA Size, Shape, Mass, Blockcopolymers Structure Interactions fd-virus Dynamics, Diffusion Coefficients Proteins Soft Matter - complex fluids world between fluid and solid Surfactants Equilibriumand Non-equilibrium States Lund University / Physical Chemistry / The Colloidal Domain - Scattering /

2 Characteristic length and time scales colloids: latex, microgels, micelles polymers molecules virus, DNA, vesicles atoms proteins nm 1 nm 1 nm 1 μm 1 ps 1 ns 1μs 1ms 1 s molar mass radius molecular time scales surface 1 m /g 1 m /g 1 m /g microscopic SANS mesoscopic macroscopic systems atomic/molecular colloid physics and chemistry, solid state physics physics and chemistry biology SAXS/WAXS Light scattering USALS CLSM-Videomicroscopy Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 3 Multiscale characterization in Physical Chemistry Zetasizer SAXS/ WAXS 3D light scattering NMR self diffusion Methods nm sec. in-situ, non-invasive time resolved DWS Ares LS 1 Rheometer CLSM-Video-microscopy multi-3d light scattering USALS Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 4

3 Introduction to scattering methods Probe choice: length and time scales, contrast radiation with known wavelength and energy scattered radiation (new) wavelength and energy Source of radiation θ Ensemble of molecules or particles: vibration, rotation, translation and diffusion detector elastic or static conformation, structure, size, interactions Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 5 quasielastic, inelastic or dynamic Local and global dynamics, diffusion, vibrations, rotation, hydrodynamics Scattering: Basics photons neutrons θ Scattering vector q = (4π/λ)sin(θ/) spatial resolution ~ 1/q static detector Light Scattering SAXS SANS (PSI) Lund University / Physical Chemistry / The Colloidal Domain - Scattering /

4 Scattering: Basics photons neutrons θ Scattering vector q = (4π/λ)sin(θ/) spatial resolution ~ 1/q detector Light Scattering static SAXS Lund University / Physical Chemistry / The Colloidal Domain - Scattering / Scattering: Dynamics quasielastic and inelastic scattering experiments: length and time scales, contrast Brownian motion of colloids NSE TOF DLS triple axis time scales vs. energy and frequency Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 8

5 Single particle shape: the particle form factor P(q) 1 Single particle shape: the particle form factor P(q) 1

6 Single particle shape: the particle form factor P(q) 1 Formfactor P(q) for an ideal monodisperse sphere I(q) Formfactor P(q) for a globular Protein (Gamma crystallin) q [nm -1 ] 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, neutron source r k i k = π/λ scattering volume r k s θ detector

7 Interference and scattering vector I Amplitude A i of incoming plane wave at position R: A i ( R )= A e i k i Scattering by a point scatterer fixed in space (1): R = A e iϕ A s product of 3 contributions: Amplitude A of incoming plane wave Scattering length b characteristic spherical wave Rʼ k s R ' A s ( R ') = A b ei R ' Interference and scattering vector II Scattering by two point scatterers fixed in space (1 and ): a Δϕ = π/λ Δs b 1: origin of coordinate system Δϕ 1 = Δϕ given by path length difference Δs Δs = a - b = k i r - k s r A s ( R ') = s A j j =1 A R' ei k s R ' b j j =1 e iδϕ j phase difference interference term

8 Interference and scattering vector III Definition of the scattering vector q: q := k i k s quasielastic scattering k i k s q = 4π λ sinθ a Δϕ = π λ Δs = q r spatial resolution ~ 1/q b A s ( R ') = s A j j =1 A R' ei k s R ' b j j =1 e i q r j Interference and scattering vector IV scattering by N point scatterers at fixed positions A s ( R ') = N s A j j =1 A R' ei k s R ' N b j j =1 e i q r j Fourier transform of b(r) normalized scattering intensity from N mobile point scatterers I s * ( R ') = A s ( R ') A s R ' ( ) = A R' N j;k =1 b j b k e i q r jk r jk = r j - r k, average < > over all possible particle configurations differential scattering cross section, identical particles: dσ dω ( q )= I ( s R ) N R' = b e i I j;k =1 q r jk

9 The particle mass and the form factor an ideal gas of noninteracting particles: I s ( q)= N I p ( q)= N I p () P(q) I p () = V Δρ intensity of single particle at q = Ip() M I() contains information about mass of particle P(q) = I p (q) I p (q ) P(q) contains information about size and structure of particle Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 17 The particle form factor Example: homogeneous sphere ()= 3 Pq sin( qr) ( qr)cos ( qr ) ( qr) 3 function has minima for tan(qr) = qr, or qr = 4.49, 7.73, calculation for sphere with radius R = 6 Å minima at q = 4.49/6 =.75 Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 18

10 The particle form factor Ideal polymers: Debye function dσ dω (q) = N P(q), where P(q) = e q [ R G ] q ( R G ) + q R G 1 P(q) = ( ) f q R G only I(q) R G : radius of gyration with R g = 1 N R CM = 1 N R j N ( R CM ) j =1 N R j j =1 R g = 1 N R j N ( R k ) j;k =1 Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 19 Arbitrary particle shape: The Guinier approximation Guinier approximation: P(q) q R G = q R G direct and model-free determination of R G from small-q scattering comparison of spheres and polymer coils with similar R G Guinier regime Lund University / Physical Chemistry / The Colloidal Domain - Scattering /

11 Interparticle correlation: the structure factor S(q) [ ( )] Sq ( )= FT g r 1 g(r): radial distribution function as a measure of spatial correlation repulsive spheres g(r) S(q) q max = π d char ideal gas interacting particles: the structure factor S(q) I(q) NM P(q)S(q) polystyrene spheres, R = 85 nm in water (added salt hard sphere interactions) I(q) I(q) P(q) S(q)

12 Scattering: Light vs. x-rays vs. neutrons photons neutrons θ Scattering vector q = (4π/λ)sin(θ/) spatial resolution ~ 1/q static detector I(q) Δρ N M P(q) S(q) characteristic properties: probe λ contrast light 5 nm Δn contrast SANS:scattering length SAXS:electron density SLS:polarizability x-rays.1-1 nm Δz neutrons.1-1 nm Δb Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 3 scattering contrast: x-rays vs. neutrons x-rays neutrons H C O Ti Fe Ni U H C O Ti Fe Ni U Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 4

13 Contrast variation - or why neutrons polymer melt Contrast variation - the case of polymer melts Kirste et al. Jülich 1974 P.J. Flory Stanford USA intensity Nobel prize 1974 scattering angle

14 Nucleation and phase separation in microemulsions C 1 E 5 + decane in D O: 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. 16, 6-13 (1997) Neutrons and surfactants - contrast variation again starting point: scattering length 1 H H b in 1-14 m oil-in-water microemulsion h-oil and h-surfactant in D O -> bulk contrast Δρ(r) r

15 Neutrons and surfactants - contrast variation again starting point: scattering length 1 H H b in 1-14 m Δρ(r) r oil-in-water microemulsion h-oil and h-surfactant in D O -> bulk contrast d-oil and h-surfactant in D O -> shell contrast Contrast variation allows to highlight individual parts of complex systems 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

16 Time-resolved SANS experiments time-resolved SANS study (D, ILL) growth of big oil droplets Readjustment of small droplets low polydispersity Key points: contrast variation large q-range large neutron flux S. Egelhaaf, U. Olsson, P. Schurtenberger, J. Morris, and H. Wennerström, Phys. Rev. E (1999) What about dynamics? A short introduction to dynamic light scattering DLS Detector Dynamics Laser Sample Θ D Detector Transmission > 95% Measure fluctuations in light intensity spatial resolution over which we monitor diffusion ~ 1/q

17 A short introduction to dynamic light scattering DLS Detector Laser Sample Transmission > 95% 1.6 μm.1 μm Interlude: Particle dynamics in real and reciprocal space Particle tracking with a microscope Dynamics in reciprocal (Fourier) space

18 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, (199) Interlude: Particle dynamics in real and reciprocal space Dynamics in reciprocal (Fourier) 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.

19 Interlude: Particle dynamics in real and reciprocal space Dynamics in reciprocal (Fourier) space: <I(q,t) I(q,t+τ)> Intensity autocorrelation function <I > <I > ~ T C Delay time Structure of intermediate scattering function, f(q,τ), gives information on scatterer dynamics τ Particle Sizing with DLS Particle Diffusion Stokes-Einstein- Relation f M Correlation Function g(τ) = ( q,τ )= dd PD ( )exp[ Dq τ] Example (Θ=9 ) Numerical Inversion 38

20 Interactions and dynamic light scattering Fq,τ ( )= 1 N j exp[ iq.r j () ] 1 N j exp[ iq.r j ( τ) ] DLS observes stochastic dynamics of sinusoidal density fluctuations of wavelength π/q (spatial Fourier components ) characteristic length D π q >> D π q D π q << D Collective or gradient diffusion collective diffusion coefficient: D C Π ρ = f C Observe dominant structure ( particle and cage of neighbours ) Structural relaxation Local motion of individual particles Self diffusion Lund University / Physical Chemistry / The Colloidal Domain - Scattering / 39 Collective versus self diffusion C 1 E 5 + decane in D O: ΔR() / m φ Hard sphere theory D s /D & D c /D D c /D D s /D φ HS U. Olsson, H. Bagger-Jörgensen, M. Leaver, J. Morris, K. Mortensen, R. Strey, P. Schurtenberger, and H. Wennerström, Prog. Colloid Polym. Sci. 16, 6-13 (1997) 4

21 Summary and conclusions Scattering provides information on: Mass: I()/C M Size, shape and structure: P(q) Interactions: S(q) Diffusion: I(t)I(t+τ) exp(-dq τ) Size and size distribution Light, x-rays and Neutrons Length scales: L ~ π/q with q = (4π/λ) sin(θ/) θ light q (Å -1 ) 3x1-6 3x1-6.3x λ 4 nm π/q (Å) x1 6,, 3,, x-rays, q (Å -1 ) neutrons λ 1 nm π/q (Å)

22 Light, x-rays and Neutrons Summary and conclusions Structure SANS: 1-3 < q < 1 Å -1 SAXS: 1-3 < q < 1 Å -1 ESRF 1 - < q < 1 Å -1 Lab. SANS/SAXS: 1-3 < q < 1 Å -1 USALS/SLS: x1-6 < q <.5x1-3 Å -1 length [Å] Dynamics frequency [Hz] dynamic 1 light XPCS 1 Scattering (DLS) Brillouin/Raman light scattering Neutron inelastic scattering and spin-echo experiment time [s] scattering vector [Å -1 ]