1. Light Scattering – Theoretical Background 1.1. Introduction Light-wave interacts with the charges constituting a given molecule in remodelling the spatial charge distribution: Molecule constitutes the emitter of an electromagnetic wave of the same wavelength as the incident one (“elastic scattering”) Wave-equation of oscillating electic field of the incident light:
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 ~ N i M i 2 P i (q) (scattering vector q, see below!) Particles smaller than /20 (left picture): - scattered intensity independent of scattering angle, I ~ N i M i 2
Particles in solution show Brownian motion (D = kT/(6 R), and =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, Dynamic Lightscattering S(q,t) (DLS))
2. Lichtstreuung – experimenteller Aufbau Detector (photomultiplier, photodiode): scattered intensity only! detector rDrD I sample I0I0 Scattered light wave emitted by one oscillating dipole: Light source I 0 = 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, n D )
Light Scattering Setup of the F-Practical Course, Phys.Chem., Mainz:
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
Scattering from dilute solutions of very small particles (“point scatterers”) (e.g. nanoparticles or polymer chains smaller than /20) in cm 2 g -2 Mol contrast factor: Absolute scattered intensity of ideal solutions, Rayleigh ratio ([cm -1 ]): For calibration of the setup one uses a scattering standard, I std : Toluene ( I abs = 1.4 e-5 cm -1 ) Reason of “Sky Blue”! (scattering from gas molecules of atmosphere)
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:
q q = inverse observational length scale of the light scattering experiment : q-scaleresolutioninformationcomment qR << 1whole coilmass, radius of gyratione.g. Zimm plot qR < 1topologycylinder, sphere, … qR ≈ 1topology quantitativesize of cylinder,... qR > 1chain conformationhelical, stretched,... qR >> 1chain segmentschain segment density
For large (ca. 500 nm) homogeneous spheres : Minimum bei qR = 4.49
Two different types of Polystyrene nanospheres (R = 130 nm und R > 260 nm) are investigated in the practical course!
Dynamic Light Scattering Brownian motion of the solute particles leads to fluctuations of the scattered intensity mean-squared displacement of the scattering particle: change of particle position with time is expressed by van Hove selfcorrelation function, DLS-signal is the corresponding Fourier transform (dynamic structure factor) Stokes-Einstein-Gl.
The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS, one measures the intensity correlation !) (note: in static light scattering, you measure the average scattered intensity (see dashed line left graph!)) Siegert-Relation:
”Cumulant-Method“: for polydisperse samples F s (q, ) is a superposition of various exponentials 1 st cumulant: yields the average apparent diffusion coefficient 2 nd cumulant: 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) !!! Data analysis for polydisperse (monomodal) samples Note the weighting factor “N i Mi 2 P i (q)“ which is the average static scattered intensity per sample faction ! Taylor series expansion of this superposition leads to: q→0 : D app is the z-average diffusion coefficient, since all P i (q) = 1 !
Cumulant analysis – graphic explanation: Monodisperse sample Polydisperse sample linear slope yields diffusion coefficient slope at =0 yields apparent diffusion coefficient, which is an average weighted with N i M i 2 P i (q) larger, slower particles small, fast particles
Z-average diffusion coefficient is determined by interpolation of D app vs. q 2 -> 0 (straight line only for particles < 100 nm !!!)
Explanation for q-dependence of D app for larger particles due to P i (q): Note the minimum in P(q) for the larger particles, where the average diffusion coefficient will reach a maximum !!!
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 D c (q) = D s /S(q): DLS of concentrated samples – influence of the static structure factor S(q): From: Gapinsky et al., J.Chem.Phys. 126, 104905 (2007) S(q) from SAXS, particle radius ca. 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 !!!) D(q) from XPCS (Xray-correlation), particle radius 80 nm, c = 200, 97 und 75 g/L, in water: left: c(salt) = 0.5 mM, right: c(salt) = 50 mM)