Report

Seminar PCF “Lightscattering” 1. Light Scattering – Theoretical Background 1.1. Introduction Light-wave interacts with the charges constituting a given molecule in remodelling the spatial charge distribution: 2 x 2 t E x, t E0 cos c 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”) m E Es 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 /20 (left picture): - scattered intensity independent of scattering angle, I ~ NiMi2 Particles in solution show Brownian motion (D = kT/(6hR), 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 Lightscattering S(q,t) (DLS)) 2. Lichtstreuung – experimenteller Aufbau Scattered light wave emitted by one oscillating dipole: 2m 1 4 2 2 E0 Es 2 2 exp i 2 t kr D 2 t r c r c D D Detector (photomultiplier, photodiode): scattered intensity only! I0 I s Es Es Es 2 sample rD I detector 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) 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) 4 2 2 n b 4 nD,0 ( D )2 K c 0 N L 2 contrast factor: in cm2g-2Mol Absolute scattered intensity of ideal solutions, Rayleigh ratio ([cm-1]): 2 r R b c M ( I solution I solvent ) D V 2 For calibration of the setup one uses a scattering standard, Istd: Toluene ( Iabs = 1.4 e-5 cm-1 ) R I solution I solvent I std ,abs I std 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: k0 k0 k q k k0 4 nD sin( ) 2 q q = inverse observational length scale of the light scattering experiment: q q-scale resolution information comment qR << 1 whole coil mass, radius of gyration e.g. Zimm plot qR < 1 topology cylinder, sphere, … qR ≈ 1 topology quantitative size of cylinder, ... qR > 1 chain conformation helical, stretched, ... qR >> 1 chain segments chain segment density For large (ca. 500 nm) homogeneous spheres : P( q ) 9 qR 6 sin qR qR cos qR 2 0 P(q) 10 10 -1 10 -2 10 -3 10 -4 10 -5 Minimum bei qR = 4.49 0 2 4 6 qR 8 10 12 Two different types of Polystyrene nanospheres (R = 130 nm und R > 260 nm) are investigated in the practical course! 1.00E+00 P(q) 130 nm P(q) 260 nm 1.00E-01 1.00E-02 1.00E-03 1.00E-04 1.00E-05 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 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) Gs (r , ) n(0, t )n(r , t ) V ,T Fs (q, ) Gs (r , ) exp(iqr )d r mean-squared displacement of the scattering particle: DR 2 6 Ds Ds kT kT f 6h RH Stokes-Einstein-Gl. The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS, one measures the intensity correlation <I(t) · I(t+)> !) I(t) <I(t)I(t+)>T (note: in static light scattering, you measure the average scattered intensity <I(q,t)> (see dashed line left graph!)) 1 2 Basislinie : I t t Siegert-Relation: Fs (q, ) exp( Ds q 2 ) Es (q, t ) Es *(q, t ) 2 I ( q, t ) I ( q, t ) I q, t 2 1 Data analysis for polydisperse (monomodal) samples ”Cumulant-Method“: for polydisperse samples Fs(q,) 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: ln Fs q, 1 1 1 2 2 3 3 ... 2! 3! 1st cumulant: 1 Ds q² yields the average apparent diffusion coefficient 2nd 2 cumulant: 2 Ds Ds 2 q 4 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) !!! n M P q D q n M P q 2 Dapp i i i i 2 i i i Ds z 1 K Rg 2 z q2 q→0 : Dapp is the z-average diffusion coefficient, since all Pi(q) = 1 ! Cumulant analysis – graphic explanation: Monodisperse sample log(Fs(q,) log(Fs(q,) Polydisperse sample Dy/Dx=-Dsq Dy/Dx=-Dsq 2 larger, slower particles 2 small, fast particles linear slope yields diffusion coefficient slope at =0 yields apparent diffusion coefficient, which is an average weighted with NiMi2Pi(q) 2,0x10 -14 1,5x10 -14 1,0x10 -14 5,0x10 -15 D/m s 2 -1 Z-average diffusion coefficient is determined by interpolation of Dapp vs. q2 -> 0 (straight line only for particles < 100 nm !!!) Ds z 0,0 0 1x10 10 2x10 2 10 q /cm -2 3x10 10 4x10 10 Explanation for q-dependence of Dapp for larger particles due to Pi(q): n M P q D q n M P q 2 Dapp i i i i 2 i i i Note the minimum in P(q) for the larger particles, where the average diffusion coefficient will reach a maximum !!! 1.00E+00 P(q) 130 nm P(q) 260 nm 1.00E-01 1.00E-02 1.00E-03 1.00E-04 1.00E-05 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.024 0.026 0.028 0.030 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, 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) 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) 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 !!!)