Seminar PCF “ Lightscattering ”

1. Seminar PCF “Lightscattering”

2. 1. Light Scattering – Theoretical Background 1.1. Introduction Light-wave interacts with the charges constituting a given molecule in remodelling the spatial charge distribution: Wave-equation of oscillating electic field of the incident light: Molecule constitutes the emitter of an electromagnetic wave of the same wavelength as the incident one (“elastic scattering”)

3. Particles larger than 20 nm (right picture): • several oscillating dipoles created simultaneously within one given particle • interference leads to a non-isotropic angular dependence of the scattered light intensity • particle form factor, characteristic for size and shape of the scattering particle • scattered intensity I ~ NiMi2Pi(q) (scattering vector q, see below!) • Particles smaller than l/20 (left picture): • - scattered intensity independent of scattering angle, I ~ NiMi2

4. Particles in solution show Brownian motion (D = kT/(6phR), and <Dr(t)2>=6Dt) • Interference pattern and resulting scattered intensity fluctuate with time • Change in respective particle positions leads to changes in interparticular (!) interference, and therefore temporal fluctuations in the scattered intensity detected at given scattering angle. (s. Static Structurefactor<S(q)>, Dynamic LightscatteringS(q,t) (DLS))

5. sample I0 I rD detector 2. Lichtstreuung – experimenteller Aufbau Scattered light wave emitted by one oscillating dipole: Detector (photomultiplier, photodiode): scattered intensity only! Light source I0 = laser: focussed, monochromatic, coherent Coherent: the light has a defined oscillation phase over a certain distance (0.5 – 1 m) and time so it can show interference. Note that only laser light is coherent in time, so: No laser => no dynamic light scattering! Sample cell: cylindrical quartz cuvette, embedded in toluene bath (T, nD)

6. Light Scattering Setup of the F-Practical Course, Phys.Chem., Mainz:

7. Scattering volume: defined by intersection of incident beam and optical aperture of the detection optics, varies with scattering angle ! . Important: scattered intensity has to be normalized

8. Scattering from dilute solutions of very small particles (“point scatterers”) (e.g. nanoparticles or polymer chains smaller than l/20) contrast factor: in cm2g-2Mol Absolute scattered intensity of ideal solutions, Rayleigh ratio ([cm-1]): For calibration of the setup one uses a scattering standard, Istd: Toluene ( Iabs = 1.4 e-5 cm-1 ) Reason of “Sky Blue”! (scattering from gas molecules of atmosphere)

9. Scattering from dilute solutions of larger particles - scattered intensity dependent on scattering angle (interference) The scattering vector q (in [cm-1]), length scale of the light scattering experiment:

10. q q = inverse observational length scale of the light scattering experiment:

11. For large (ca. 500 nm) homogeneous spheres : Minimum bei qR = 4.49

12. Two different types of Polystyrene nanospheres (R = 130 nm und R > 260 nm) are investigated in the practical course!

13. Dynamic Light Scattering Brownian motion of the solute particles leads to fluctuations of the scattered intensity change of particle position with time is expressed by van Hove selfcorrelation function, DLS-signal is the corresponding Fourier transform (dynamic structure factor) mean-squared displacement of the scattering particle: Stokes-Einstein-Gl.

14. The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS, one measures the intensity correlation <I(t) · I(t+t)> !) (note: in staticlightscattering, youmeasuretheaveragescatteredintensity <I(q,t)> (seedashedlineleftgraph!)) Siegert-Relation:

15. Data analysis for polydisperse (monomodal) samples ”Cumulant-Method“: for polydispersesamples Fs(q,t) is a superposition of various exponentials Note the weighting factor “Ni Mi2 Pi(q)“ which is the average static scattered intensity per sample faction ! Taylor series expansion of this superposition leads to: yields the average apparent diffusion coefficient 1stcumulant: 2ndcumulant: is a measure for sample polydispersity Important: For polydisperse samples of particles > 10 nm, the apparent diffusion coefficient Is q-dependent due to the weighting-factor P(q) !!! q→0 : Dapp is the z-average diffusion coefficient, since allPi(q) = 1 !

16. larger, slowerparticles small, fast particles Cumulant analysis – graphic explanation: Monodisperse sample Polydispersesample linear slopeyieldsdiffusioncoefficient slopeatt=0 yieldsapparentdiffusion coefficient, whichis an averageweighted with NiMi2Pi(q)

17. Z-averagediffusioncoefficientisdeterminedbyinterpolationofDapp vs. q2 -> 0 (straightlineonlyforparticles< 100 nm !!!)

18. Explanation for q-dependenceofDappfor larger particles due to Pi(q): Note theminimum in P(q) forthe larger particles, wheretheaveragediffusioncoefficient will reach a maximum !!!

19. DLS of concentrated samples – influence of the static structure factor S(q): Due to interparticle interactions, the particles not any longer move independently by Brownian motion, only. Therefore, DLS in this case measures no self-diffusion coefficient but a collective diffusion coefficient defined as Dc(q) = Ds/S(q): S(q) from SAXS, particleradiusca. 80 nm, c = 200, 97 und 75 g/L, in water: left: c(salt) = 0.5 mM, right: c(salt) = 50 mM) From:Gapinsky et al., J.Chem.Phys. 126, 104905 (2007)

20. D(q) from XPCS (Xray-correlation), particleradius 80 nm, c = 200, 97 und 75 g/L, in water: left: c(salt) = 0.5 mM, right: c(salt) = 50 mM) From:Gapinsky et al., J.Chem.Phys. 126, 104905 (2007) Note: 1. The q-regime of SAXS/XPCS is much larger than in light scattering due to the shorter wave length of Xrays (lab course: 0.013 nm-1 < q < 0.026 nm-1!!!) 2. The investigated Ludox particles R = 25 nm are much smaller, therefore the maximum in S(q) is located at larger q (q(S(q)_max) > 0.1 nm-1!!!)