Presentation on theme: "“Light Scattering from Polymer Solutions and Nanoparticle Dispersions”"— Presentation transcript:
1“Light Scattering from Polymer Solutions and Nanoparticle Dispersions” By: PD Dr. Wolfgang SchaertlInstitut für Physikalische Chemie, Universität Mainz,Welderweg 11, Mainz, GermanyParts from the new book of the same title, published by Springer in July 2007Slides are found at:
21. Light Scattering – Theoretical Background 1.1. IntroductionLight-wave interacts with the charges constituting a given molecule in remodellingthe spatial charge distribution:Molecule constitutes the emitter of an electromagnetic wave of the same wavelengthas the incident one (“elastic scattering”)Note: usually vertical polarization of both incident and scattered light (vv-geometry)
3Particles larger than 20 nm: several oscillating dipoles created simultaneously within one given particleinterference leads to a non-isotropic angular dependence of the scattered light intensityparticle form factor, characteristic for size and shape of the scattering particlescattered intensity I ~ NiMi2Pi(q) (scattering vector q, see below!)Particles smaller than l/20:- scattered intensity independent of scattering angle, I ~ NiMi2
4Particles in solution show Brownian motion (D = kT/(6phR), and <Dr(t)2>=6Dt) => Interference pattern and resulting scattered intensity fluctuate with time
6Standard Light Scattering Setup of the Schmidt Group, Phys. Chem Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz:lasersample,bathdetector ongoniometer arm
7Standard Light Scattering Setup of the Schmidt Group, Phys. Chem Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz:
8Standard Light Scattering Setup of the Schmidt Group, Phys. Chem Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz:
9Important: scattered intensity has to be normalized Scattering volume:defined by intersection of incident beam and optical aperture of the detection optics.Important: scattered intensity has to be normalized
10(e.g. nanoparticles or polymer chains smaller than l/20) Scattering from dilute solutions of very small particles (“point scatterers”)(e.g. nanoparticles or polymer chains smaller than l/20)Fluctuation theory:contrast factorin cm2g-2MolIdeal solutions, van’t Hoff:Real solutions, enthalpic interactions solvent-solute:Absolute scattered intensity of ideal solutions, Rayleigh ratio ([cm-1]):andScattering standard Istd: Toluene( Iabs = 1.4 e-5 cm-1 )Reason of “Sky Blue”! (scattering from gas molecules of atmosphere)
11Real solutions, enthalpic interactions solvent-solute expressed by 2nd Virial coeff.: 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:
12q = inverse observational length scale of the light scattering experiment: q-scaleresolutioninformationcommentqR << 1whole coilmass, radius of gyratione.g. Zimm plotqR < 1topologycylinder, sphere, …qR ≈ 1topology quantitativesize of cylinder, ...qR > 1chain conformationhelical, stretched, ...qR >> 1chain segmentschain segment density
13Scattering from 2 scattering centers – interference of scattered waves leads to phase difference:2 interfering waves with phase difference D:
14orientational average and normalization lead to: Scattered intensity due to Z pair-wise intraparticular interferences, N particles:orientational average and normalization lead to:replacing Cartesian coordinates ri by center-of-mass coordinates si we get:s2, Rg2 = squared radius of gyration.regarding the reciprocal scattered intensity, and including solute-solvent interactionsfinally yields the well-known Zimm-Equation (series expansion of P(q), valid for small R):Zimm equation:
15The Zimm-Plot, leading to M, s (= Rg) and A2: example: 5 c, 25 qc = 0q = 0
16Zimm analysis of polydisperse samples yields the following averages: The weight average molar massThe z-average squared radius of gyration:Reason: for given species k, Ik ~ NkMk2
201.3. Dynamic Light Scattering Brownian motion of the solute particles leads to fluctuations of the scattered intensitychange of particle position with time is expressed by van Hove selfcorrelation function,DLS-signal is the corresponding Fourier transform (dynamic structure factor)isotropic diffusive particle motionmean-squared displacement of the scattering particle:Stokes-Einstein,Fluctuation - Dissipation
21The Dynamic Light Scattering Experiment - photon correlation spectroscopy Siegert relation:note: usually the “coherence factor” fc is smaller than 1, i.e.:fc increases with decreasing pinhole diameter, but photon count rate decreases!
23yields polydispersity Data analysis for polydisperse (monomodal) samples”Cumulant method“, series expansion, only valid for small size polydispersities < 50 %first Cumulantyields inverse average hydrodynamic radiussecond Cumulantyields polydispersityfor samples with average particle size larger than 10 nm:note:
24Cumulant analysis – graphic explanation: large, slowparticlessmall, fastmonodisperse samplepolydisperse samplelinear slope yields diffusion coefficientslope at t=0 yields apparent diffusioncoefficient, which is an average weightedwith niMi2Pi(q)
31cylinder of length l, diameter D Combining static and dynamic light scattering, the r-ratio:topologyr-ratiohomogeneous sphere0.775hollow sphere1ellipsoidrandom polymer coil1.505cylinder of length l, diameter Dfor polydisperse samples:
32Strategy for particle characterization by light scattering - A Sample topology (sphere, coil, etc…) is knownyesnoDynamic light scatteringsufficient (“particle sizing“)Static light scattering necessaryTime intensity correlation function decays single-exponentiallyyesnoOnly one scattering angle needed, determine particle size (RH) from Stokes-Einstein-Eq.(in case there are no particle interactions (polyelectrolytes!)Sample is polydisperse or shows non-diffusive relaxation processes!to determine “true” average particle size,extrapolation q -> 0- to analyze polydispersity, various methodsApplicability of commercial particle sizers!
33Strategy for particle characterization by light scattering - B Sample topology is unknown,static light scattering necessaryPlot of vs is linearyesnoParticle radius between 10 and 50 nm:analyze data following Zimm-eq. to get:Particle radius larger than 50 nm and/or very polydisperse sample:use more sophisticated methods to evaluate particle form factorDynamic light scattering to determineEstimate (!) particle topology from
352. Static Light Scattering – Selected Examples 1. Galinsky, G.;Burchard, W. Macromolecules 1997, 30,Samples:Several starch fractions prepared by controlled acid degradation of potatoe starch,dissolved in 0.5M NaOHSample characteristics obtained for very dilute solutions by Zimm analysis:sample10-6 Mw(g/mol)Rg(nm)104 A2[(mol cm3)/g2]LD110.92361.00LD161.87480.60LD125.20700.28LD1914.51130.13LD18431800.082LD17641900.060LD13972330.025
36Normalized particle form factors universal up to values of qRg = 2
37Details at higher q (smaller length scales) – Kratky Plot: form factor fits:C related to branching probability,increases with molar mass
38Are the starch samples, although not self-similar, fractal objects? - minimum slope reached at qRg ≈ 10 (maximum q-range covered by SLS experiment !)at higher q values (simulations or X-ray scattering) slope approaches -2.0characteristic for a linear polymer chain (C = 1).at very small length scale only linear chain sections visible (non-branched outer chains)
392. Pencer, J.;Hallett, F. R. Langmuir 2003, 19, 7488-7497 Samples:uni-lamellar vesicles of lipid molecules 1,2-Dioleoyl-sn-glycero-3-phosphocholine (DOPC)and 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine (SOPC) by extrusionData Analysis:monodisperse vesiclesthin-shell approximationsmall values of qR, Guinier approximation
40typical q-range of light scattering experiments: 0. 002 nm-1 to 0 typical q-range of light scattering experiments: nm-1 to 0.03 nm-1vesicles prepared by extrusion: radii 20 to 100 nm=> first minimum of the particle form factor not visible in static light scattering
41particle form factor of thin shell ellipsoidal vesicles, two symmetry axes (a,b,b) prolate vesicles, surface area 4 p (60 nm)2oblate vesicles, surface area 4 p (60 nm)2
42anisotropy vs. polydispersity: monodisperse ellipsoidal vesiclespolydisperse spherical vesiclesstatic light scattering from monodisperse ellipsoidal vesicles can formally be expressedin terms of scattering from polydisperse spherical vesicles !=> impossible to de-convolute contributions from vesicle shape and size polydispersityusing SLS data alone !combination of SLS and DLS:DLS: intensity-weighted size distribution => number-weighted size distribution (fit a,b) =>SLS: particle form factor
43polydisperse (DR = 10%) oblate vesicles, a : b < 1 : 2.5 input for a,b – fitsto SLS data,result:polydisperse (DR = 10%) oblate vesicles,a : b < 1 : 2.5
443. Fuetterer, T. ;Nordskog, A. ;Hellweg, T. ;Findenegg, G. H 3. Fuetterer, T.;Nordskog, A.;Hellweg, T.;Findenegg, G. H.;Foerster, S.;Dewhurst, C. D. Physical Review E 2004, 70, 1-11Samples:worm-like micelles in aqueous solution, by association of theamphiphilic diblock copolymer poly-butadiene(125)-b-poly(ethylenoxide)(155)Analysis of SLS-results:monodisperse stiff rodsasymmetric Schulz-Zimm distributionpolydisperse stiff rodsKoyama, flexible wormlike chains
45Holtzer-plot of SLS-data : plateau value = mass per length of a rod-like scattering particlefit results:polydisperse stiff rods:(ii) polydisperse wormlike chains:
46Analysis of DLS-results: amplitudes depend on the length scale of the DLS experiment:- long diffusion distances (qL < 4): only pure translational diffusion S0- intermediate length scales (4 < qL < 15): all modes (n = 0, 1, 2) presentaccording to Kirkwood and Riseman:polydispersity leads to an average amplitude correlation function!
47DLS relaxation rates :linear fit over the whole q-range: significant deviation from zero intercept,additional relaxation processes or “higher modes” at higher qresults:Rg from Zimm-analysis and calculations!
484. Wang, X. H.;Wu, C. Macromolecules 1999, 32, 4299-4301 samples:high molar mass PNIPAM chains in (deuterated) water
49reversibility of the coil-globule transition: molten globule ?surface of the sphere has a lowerdensity than its center
50Selected Examples – Static Light Scattering: sampleproblemsolutionbranched polymeric nanoparticlesself-similarity (fractals) ?;details at qR > 2 by Kratky plot (P(q) q2 vs. q), fitting parameters for branched polymers, simulation of P(q) at qR > 10 (SLS: qR < 10) => not fractal !vesicles (nanocapsules)distinguish size polydispersity and shape anisotropy in P(q) ?combine DLS (only size polydispersity !) and SLS to simulate expt. P(q)worm-like micellescharacterization: length, Rg/RH(RH: no rotation-translation-coupling if qL < 4)details at higher q by Holtzer plot (I(q) q vs. q), fit P(q), Rg from Zimm-analysis at small q valuesPNIPAM chains in water at different Tcoil – globule - transitionRg from Zimm-analysis, RH by DLS, decrease in R and Rg / RH
513. Dynamic Light Scattering – Selected Examples 1. Vanhoudt, J.;Clauwaert, J. Langmuir 1999, 15, 44-57sample: spherical latex particles in dilute dispersionssamples2s3s4s5s6s7nominal diameter/nm19549119, 9119, 5454, 91diameter ratio-18.104.22.168Data analysis of polydisperse samples:1. Cumulant method (CUM), polynomial series expansion:polydispersity indexparticle diameter is a so-called harmonic z-average:(only for homogeneous spheres)
522. non-negatively least squares method (NNLS): M exponentials considered for the exponential series, yielding a set of coefficients bidefining the particle size distribution for decay rates equally distributed on a log scale.3. Exponential sampling (ES):See 2., decay rates chosen according to:4. Provencher’s CONTIN algorithm:Numerical procedure to calculate a continuous decay rate distribution B(G), also calledInverse Laplace Transformation, enclosed in most commercial DLS setups.5. double-exponential method (DE):
53Results:samples2s3s4s5s6s7nominal diameter19 ± 1.554 ± 2.791 ± 319, 9119, 5454, 91diameter ratio-22.214.171.124<d> - CUM (1.)20.355.087.036.929.569.0PI – CUM (1.)0.0290.0090.0080.2480.1910.069d1,d2 – NNLS (2.)18, 8116, 50d1,d2 – ES (3.)d1,d2 – DE (5.)18, 54Bimodal samples s5, s6, s7: I1(q=0) = I2(q=0)Note: bimodal samples with d2/d1 < 2 (s7) beyond resolution of DLS !
542. van der Zande, B. M. I.;Dhont, J. K. G.;Bohmer, M. R.;Philipse, A. P. Langmuir 2000, 16,sample (TEM-results):colloidal gold nanoparticles stabilized with poly(vinylpyrrolidone) (M = g/Mol)systemlength L[nm]DLdiameter dDdaspect ratio L/dSphere18185-Sphere15153Rod2.6a47172.6Rod2.6b3910Rod8.9146378.9Rod12.61892412.6Rod14283222014.0Rod17.22596017.2Rod17.4279681617.4Rod3966039.0Rod4972949.0
55DLS setup and data analysis: Kr ion laser (647.1 nm far from the absorption peak of the gold particles (500 nm))Measurements in vv-mode and vh-mode (depolarized dynamic light scattering DDLS)(v = vertical, h = horizontal polarization)intensity autocorrelation functions were fitted to single exponential decays,including a second Cumulant to account for particle size polydispersityvv-mode (only translation is detected):depolarized dynamic light scattering (vh-mode)(translation and rotation are detected, no coupling in case qL < 5)translational diffusion coefficient DT determined from the slope,rotational diffusion coefficient DR from the interceptof the data measured in vh –geometry.
57diffusion coefficients according to Tirado and de la Torre, using as input parameters length and diameter from TEMsystem10-12 DT, exp[m2s-1]10-12 DT, calcDR, exp[s-1]DR, calcRod8.96.08.43062238Rod126.96.36.199811258Rod142.95.266396Rod17.24.0177563Rod188.8.131.5275452Rod391.21446Rod490.72.830values determined by DDLS systematically too small, because PVP-layer(thickness 10 – 15 nm) not visible in TEM !
58Selected Examples – Dynamic Light Scattering: sampleproblemsolutionbimodal spheressize resolution- double exponential fits- size distribution fits- CONTIN ; only if R1/R2 > 2stiff gold nanorodslength and diameter in solution =?;deviation TEM – DLS ?depolarized DLS (vh) => Drot standard DLS (vv) => Dtrans;deviation TEM-DLS due to PVP stabilization layer