Light Scattering – what you learned so far: m E Es c 2 x 2 E x, t E0 sin sin t Static versus Dynamic Light Scattering from time-independent intraparticle interferences you determine: average scattered intensity => particle form factor P(q), radius of gyration Rg, molar mass M (, A2) from time-dependent interparticle interferences you determine: time-intensity correlation function => hydrodynamic radius RH The scattering vector q (in [cm-1]) , (inverse) length scale of light scattering: k0 q k 4 nD sin( ) q 2 q The famous Zimm-Equation (series expansion of P(q), valid for 10 nm < R < 50 nm: I q 1 Kc R 1 M (1 13 s 2 q 2 ) 2 A2c 6,0 5,5 5,0 4,5 Kc/R / 10 mol/g 4,0 3,5 -7 3,0 2,5 2,0 1,5 1,0 0,0 5,0 10,0 15,0 20,0 2 10 -2 (q +kc) / 10 cm Averages determined by SLS (Zimm-Plot) from polydisperse samples: K K Nk M k M k N M 2 2 k k sk Mw k 1 s z Rg k 1 2 2 K K Nk M k N M z 2 k k k 1 k 1 Particle form factor for “large” (> 200 nm) particles, e.g. spheres: 0 10 first minimum at qR = 4.49 Zimm! -1 10 -2 10 P(q) -3 10 -4 10 -5 10 0 2 4 6 8 10 12 qR The concept of fractal dimensions – analysis of P(q) for qR > 1 log P q log I q cM d f log q topology df cylinder, rod 1 thin disk 2 homogeneous sphere 3 ideal Gaussian coil 2 Gaussian coil with excluded volume 5/3 branched Gaussian chain 16/7 Dynamic Light Scattering – time-intensity-correlation function, Stokes-Einstein-eq. and RH I ( q, t ) I ( q, t ) g 2 (q, ) exp(2 Ds q 2 ) 1 I q, t 2 g1 (q, ) exp( Ds q 2 ) Es (q, t ) Es *(q, t ) g 2 q, 1 Fs q, kT kT Ds f 6 RH q1 q2 DLS from polydisperse samples: 0 10 Fs q, P Dapp q exp q 2 Dapp q dD 10 -1 0 R = 60 nm -2 10 R = 80 nm P(q) 1 1 R = 100 nm ln Fs q, 1 2 2 3 3 ... 10 -3 2! 3! -4 10 Dapp q Ds 1 K Rg q2 1 q 2 2 z z -5 10 0,00 0,01 0,02 0,03 0,04 -1 q [nm ] Combining static and dynamic light scattering, the r-ratio: Rg r RH topology r-ratio homogeneous sphere 0.775 hollow sphere 1 ellipsoid 0.775 - 4 random polymer coil 1.505 cylinder of length l, diameter D Selected Examples – Dynamic Light Scattering: sample problem solution bimodal spheres size resolution - double exponential fits - size distribution fits - CONTIN ; only if R1/R2 > 2 branched polymeric besides “simple“ characterization Dapp vs. q for different nanoparticles (Rg,M(Zimm),RH),branchg.degree fractions => all identical !, (q-dep. of Dapp: 1.polydisp. 2.local comparison with known relaxation (segments!)) cases (chain, sphere) DNA as poly- polyelectrolyte effect, interparticle add salt; electrolyte and rod interactions; analyze D by different rod length =? models (Broersma, Tirado/de la Torre etc.) stiff gold nanorods length and diameter in solution =?; depolarized DLS (vh) => Drot deviation TEM – DLS ? standard DLS (vv) => Dtrans; deviation TEM-DLS due to PVP stabilization layer spherical nanogels light absorption, local heating, convection quantified from cont. gold clusters q-dep. oscillations in g2(q,) oscillations in g2(q,) vs. q Selected Examples – Static Light Scattering: sample problem solution segmented polydispersity in length and (A) SLS-Zimm-plots for 3 copolymers composition => different polymer-solvent-pairs by Zimm-plot only Mw,app (B) DLS for 2 different polymer- solvent-pairs (P(D) => P(M)) ! branched 1. self-similarity (fractals) ?; 1. details at qR > 2 by Kratky plot polymeric 2. P(q) from SLS of semi-dilute (P(q) q2 vs. q), fitting parameters nanoparticles solutions ? for branched polymers, simulation of P(q) at qR > 10 (SLS: qR < 10) => not fractal ! 2. renormalization of I(q,c) with I(q=0,c), and q with Rg,app vesicles distinguish size polydispersity combine DLS (only size (nanocapsules) and shape anisotropy in P(q) ? polydispersity !) and SLS to simulate expt. P(q) worm-like micelles characterization: length, Rg/RH details at higher q by Holtzer plot (RH: no rotation-translation- (I(q) q vs. q), fit P(q), Rg from coupling if qL < 4) Zimm-analysis at small q values PNIPAM chains in coil – globule - transition Rg from Zimm-analysis, RH by water at different T DLS, decrease in R and Rg / RH 4. Non-Standard Light Scattering Techniques 4.1. Single Angle Scattering Using Goniometer Setups Laser components: 1. laser = light source (coherent, focussed, monochromatic, polarized) 2. goniometer with motor, step-wise adjustment of scattering angle 3. sample bath (thermostate, index match) and cylindrical sample cell 4. detector (photomultiplier or photodiode) 5. computer with A/D-converter and hardware correlator Goniometer setup of the Schmidt group: Single Angle Scattering – advantages and disadvantages: +: accurate adjustment of scattering angles +/- : correlation time scale: 100 ns - < 10 s -: scattering angle range: 30° - 150° -: series of angular-dependent measurements: long time -: DLS for low scattering contrast/slow processes: long time -: transparent, purified and highly dilute samples needed -: difficulties in analyzing polydisperse samples technical solutions: time scale enhancement: simultaneous measurement at multiple q q-scale enhancement: optical lenses + area detectors for < 1° ! turbid samples: cross correlation or backscattering polydisperse samples: combine fractionation and SLS 4.2. Simultaneous Multiangle Scattering (MALS): Laser Interface MALS - setup of the Schmidt group: Simultaneous Multiangle Scattering combined with GPC or FFF: Laser GPC Interface flow cell GPC MALS - setup of the Schmidt group: Wintermantel, M.;Gerle, M.;Fischer, K.;Schmidt, M.;Wataoka, I.;Urakawa, H.; Kajiwara, K.;Tsukahara, Y. Macromolecules 1996, 29, 978-983 samples: polystyrene polymacromonomers, synthesized by radical polymerization of anionic MMA end-functionalized polystyrene macromonomers (“bottle-brushes”), MW /MN > 2 ! GPC-MALS: - GPC connected to an on-line Knauer combined viscometer/RI-detector and an ALV1800 MALS (19 angles plus one monitor channel) - home-made cylindrical flow cell with 38 µL total volume - scattering intensity detected at 19 scattering angles over 2 s at DT = 4 s during elution wormlike chain model 2 3 4 L Rg k k k k 2 1 exp 2 L 2LL L L 6 4 4 L 8L Lk contour length L, Kuhn length Lk reduced scattered intensity plotted versus q2 for three sample fractions : Rg 100.6 nm, M w 1.01107 g mol , c 6.22 103 g L Rg 47.7 nm, M w 3.77 106 g mol , c 4.21102 g L Rg 14.3 nm, M w 7.61105 g mol , c 4.14 102 g L Rg vs. MW for a “homologous series of bottlebrushes”, wormlike chain model : results: M(side chains) = 2900 g/mol: Lk = 89 nm M(side chains) = 5000 g/mol: Lk = 208 nm 4.3. Turbid Samples – (A) Fiber-Optic Quasielastic Light Scattering (FOQELS) Wiese, H.;Horn, D. Journal of Chemical Physics 1991, 94, 6429-6443 (BASF !) suppresses multiple scattering, defines the scattering angle as 180° samples: aqueous polymer latex dispersions (particle size from 41 nm to 326 nm), as prepared without further purification, particle concentrations above 1 wt% ! data analysis: concentrated dispersions: amplitude autocorrelation function includes contributions of the static structure factor (interparticle interferences!) Fs q, g1 Fs q, 0 g1 exp Dc q 2 collective diffusion coefficient c Dc Dc q, c Dc vs. concentration, different particle sizes : position of the maximum of the structure factor compared to q of FOQELS q < qm : length scale > particle dist., interparticle interactions => collective diffusion Dc(c) ↑ q q > qm : length scale < particle dist. => selfdiffusion Ds(c) ↓ 4.4. Turbid Samples - (B) Crosscorrelation Techniques - Dual Color and 3d Dynamic Light Scattering (identical q and scattering volume, different set of interferences => no multiple scattering) I. 3d – DLS (Prof. Schurtenberger, Univ.Fribourg, CH) z Laser D1 y d q1 q 2 D2 L1 L2 d 18 k 0,1 k0 d k 0,2 x d q1 k1 g2,cross q, I1 q, t I 2 q, t k2 q2 www.lsinstruments.ch/3DDLS.htm II. Dual Color Dynamic Light Scattering Stieber, F.;Richtering, W. Langmuir 1995, 11, 4724-4727 sample: polystyrene latex spheres 2R = 82 nm (q < qm) light scattering setup: dual color or two color crosscorrelation (TCC) setup: - 2 light sources with different wavelengths from 1 argon ion laser in multiline mode (488 nm and 514.5 nm) - single-mode optical fibers for the optical alignment - scattering angles from 15° to 140° (FOQELS: 180° only !) data analysis: CONTIN (constrained inverse Laplace transformation) note: q < qm => at higher particle conc. only Dc comparison of auto- and crosscorrelation: filled symbols: autocorrelation, artefacts and bad resolution (multiple scattering) open symbols: crosscorrelation, 2 defined relaxation processes multiple scattering increases with decreasing q (increasing length scale) the origin of Dslow ? => slow process Dslow is selfdiffusion (strong slowing-down with increasing conc.) Ac S q sample polydispersity enhances selfcorrelation: : Da a As 2 4.5. Dynamic Light Scattering using CCD area detectors Laser CCD I. concave lens + CCD area detector (multiple q at once), simultaneous light scattering at very small scattering angles improves: measurement time, small-q-scale Laser CCD II. CCD area detector at one defined q ± Dq DLS: replace time averaging by ensemble averaging improves: measurement time, long--scale Wong, A. P. Y.;Wiltzius, P. Rev.Sci.Instrum. 1993, 64, 2547-2549 sample: commercial latex particles (diameter 215 nm) in glycerol at T = 51°C, rectangular cuvette of thickness 1 mm setup: detection scheme: digitized image, 500 x 450 pixel, 10 concentric rings: radius = 20, 40, ...., 200 pixel no. of pixels = 80, 160, .... , 800 -range: 10° - 60°, simultaneous calculation of the intensity correlation function: average speckle intensity of each ring: I q, t q0 deviation from the average intensity for each pixel: d I q, t I q , t I q , t q0 from two pictures taken at t = 0 and t = : g 2 q, d I q, 0 d I q , q0 (min = 100 ms) Note: <> = ensemble-average! conventional single-angle light scattering: <> = time-average! g2 q, I q, t I q, t t results: q = 29748 cm-1, 20834 cm-1 and 14832 cm-1 goniometer setup relaxation time G-1 > 20 s ! Ds = 1.67e-10 cm2s-1 ~ 1 e-14 m2s-1 : Stokes-Einstein-equation, viscosity of 0.137 Pa s => R = 107.5 nm Kirsch, S.;Frenz, V.;Schartl, W.;Bartsch, E.;Sillescu, H. Journal of Chemical Physics 1996, 104, 1758-1761 “multi speckle correlation spectroscopy” (MSCS) sample: spherical latex particles (R = 350) nm in glycerol at T = 10°C ! setup: data acquisition: images of 512x256 pixels digitized, scattered intensity for 50 random speckles (= 2x2 pixel) stored at frame rate 0.33 s calculation of the intensity correlation function: data represented 50 individual traces of I(q,t) at (nearly) identical q 1. normalized intensity correlation function for each speckle calculated by time-averaging N pic k N pic k N pic k N pic k I nDt I n k Dt I nDt I n k Dt I t I t k Dt T , norm n 0 N pic k N pic k n 0 n 0 I nDt I n k Dt n 0 n 0 total number of pictures Npic: 50.000 - 150.000, n = 0, 1, 2, ….., k = 0, 1, 2, ….. 2. time-averaged correlation functions are ensemble averaged over all speckles N sp I t I t k Dt T , norm , a I t I t k Dt T , E , norm a 1 : g 2 q, K Dt T ,E 1 N sp total number of speckles Nsp: 20 - 50 comparison MSCS – conventional DLS (PCS): MSCS results (symbols) for 20 speckles, 150.000 pictures (texpt = 50.000 s) PCS results (lines): texpt = 70.000 s inset, symbols: filled = 1 speckle correlation, open = 20 speckle ensemble average improved statistics by combined time-ensemble-averaging !
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