Introduction to Dynamic Light Scattering with Applications. Onofrio Annunziata Department of Chemistry Texas Christian University Fort Worth, TX, USA

Transcription

1 Introduction to Dynamic Light Scattering with Applications Onofrio Annunziata Department of Chemistry Texas Christian University Fort Worth, TX, USA

2 Outline Introduction to dynamic light scattering Particle sizing and particle-particle interactions Particle sizing in Polydisperse systems Mie scattering Comparison with macroscopic-gradient techniques

3 Dynamic light scattering (DLS) Also known as Quasielastic light scattering (QLS) or Photon Correlation Spectroscopy (PCS) A Technique mainly used to determine the diffusion coefficient of macromolecules and colloidal particles in solution. The dominant practical application is in particle sizing ( nm).

4 Light microscopy Electron microscopy (SEM, TEM) Atomic force microscopy (AFM) Techniques for particle size or mass Gel electrophoresis, SDS page Size exclusion chromatography Mass spectrometry (EI, MALDI) Osmotic pressure Light, neutron, X-ray scattering Sedimentation Viscosity Diffusion (DLS, NMR PGSE, Interferometry, Taylor dispersion, etc..)

5 Dynamic light scattering (DLS) Advantages: noninvasive, nondestructive, relatively fast, versatile, sensitive to aggregates Drawbacks: Fairly poor resolution, not sensitive to chemical nature, relatively complex theory, particles are not visualized

6 Instrument scheme Detector

7 Brief History of DLS Lord Rayleigh ( ) Light Scattering Albert Einstein ( ) Brownian Motion Marian Smoluchowski ( ) Stochastic processes Density fluctuations Fluctuations in the density of condensed media result in local inhomogeneities that give rise to light scattered at angles other than the forward direction.

8 Brief History of DLS Pecora (1964) Diffusion processes broadens frequency profile of the scattered electric field Cummins et al. (1964) First experimental report on macromolecular solutions Benedek et al. (1965) First experimental report on pure fluids near critical point K. S. Schmitz, An introduction to dynamic light scattering, Academic Press (1990)

9 DLS instruments commercially available Wyatt Tech. Viscotek Malvern Precision Detectors Brookhaven ALV Photocor Brookhaven IC

10 Important References G. D. J. Phillies, Quasi-elastic Light Scattering, Analytical Chemistry 62, 1049A-1057A (1990). N. C. Santos, M.A.R.B. Castanho Teaching Light Scattering, Biophysical Journal 71, (1996).

11 DLS INSTRUMENT SCHEME THERMOSTATED CELL HOLDER LASER sample IRIS (test tube with filtered solution) D Diffusion Coefficient COMPUTER g( τ ) Correlation function CORRELATOR Scattering at 90 is () t Scattered intensity DETECTOR IRIS OPTICAL FIBER

12 CELL-HOLDER SCHEME Sample Cell holder Incident beam

13 MULTIANGLE SCHEME LASER θ High Angle Low Angle 90 Angle

14 Scattering Vector k 0 wave vector of incident light k S wave vector of scattered light θ q = k k 0 S Scattering Vector 2π θ = = = 2sin ( λ / n) 2 q q k0 k S = = k0 k S λ n 2π ( λ / n) (Elastic Scattering) Wavelength of incident light in vacuum Refractive index of the sample

15 Rayleigh Scattering Elastic scattering of light by particles much smaller than the wavelength of the light. It occurs when light travels in transparent solids and liquids, but is most prominently seen in gases. Rayleigh scattering is more effective at short wavelengths (the blue end of the visible spectrum). i S 1 λ 4 The blue color of the sky is mainly caused by the scattered sunlight The orange of the sky is mainly caused by the transmitted sunlight

16 Rayleigh Scattering of one particle The particle size is assumed small compared to laser wavelength (size < λ / 10) The scattered electric field E S of a single particle is proportional to its number of electrons and consequently to its molecular mass M. Incident field Illuminated Volume E S Scattered field ( particle) ES M i = E M S S 2 2

17 Rayleigh Scattering of many particles The scattered electric field E S of N identical particles must take into account inter-particle interference. E S Scattered field S iq r k k E M e 2 2 iq ( rj rk) 2 is ES M e = M j k N N = number of particles

18 Rayleigh Scattering from particle-solvent mixtures The time-averaged spatial positions of the particles represent the solution structure Ideal Solutions (random structure) < i > N M cm c =NM/ V (mass concentration) * 2 S Real Solutions (structure affected by particle-particle interactions) < i > =< i > Sq ( ) S(q) (Structure Factor) S c 0 * S lim Sq ( ) = 1

19 Second virial coefficient B lim Sq ( ) q 0 RT 1 = = M( Π / c) 1+ 2 B M c+... Π (Osmotic pressure) B is used to characterize particle-particle interactions i S < * S B > 0 particle-particle net repulsion i i S > i * S B < 0 particle-particle net attraction

20 Second virial coefficient and protein crystallization K c / R (10-5 mol/g) Kc R Lysozyme (acetate buffer ph 4.5) NaCl 0% NaCl 2% NaCl 4% NaCl 5% B > 0 soluble c (mg/ml) 1 = + 2 B c +... M B << 0 aggregation R n = nstd B < 0 crystals 2 i i S S, std R std (Rayleigh ratio) K π n dn = 4 NAλ dc

21 Protein crystallization slot Successful Protein Crystallization B (10-4 mol ml g -2 ) Successful crystallization is obtained for B values slightly negative

22 Dynamic light Scattering Particles in solution are moving scatterers particle Illuminated Volume phase difference Incident light (LASER) Time-dependent Scattered intensity 2 iq ( rj rk) is ES e = j k

23 Dynamic light Scattering Particles in solution are moving due to Brownian Motion

24 Brownian Motion r ( τ ) r (0) D = 2 < [ r ( τ ) r (0)] > 6τ

25 Dynamic Light Scattering How the solution structure at time τ correlates with the solution structure at time 0? Dynamic Structure Factor 1 iq [ rj(0) rk( τ )] Fq (, τ) =< ES(0) ES( τ) > < e > N j k Field autocorrelation function Fq (, τ ) g ( q, τ) = = exp q Fq (,0) ( Dτ) (1) 2 D = diffusion coefficient of particles

26 Field autocorrelation function q 2 < [ r( τ ) r(0)] > (1) iq [ r(0) r ( τ)] 6 g (,) qτ =< e >= e 2 Gaussian Random variable D = 2 < [ r( τ ) r(0)] > 6τ r (0) r ( τ ) Fq (, τ ) (, τ) exp Fq (,0) ( τ) g (1) q = = q 2 D

27 Field autocorrelation function g (1) ( τ) < E (0) E ( τ) > S S Strong correlation τ = 0, g (1) =1 g (1) ( τ ) ( τ) (1) 2 g ( q, τ) = exp q D No correlation τ, g (1) = q 2 D τ Short τ Long τ g (1) ( τ ) 1 g (1) ( τ ) 0

28 Field autocorrelation function Fast vs slow particles g (1) ( τ ) fast particles (1) slow particles (2) D 1 = 10 D q 2 D 1 τ ( D1 τ) exp( D2 τ) g = 08. exp q q ( 1) 2 2 Mixture of Slow and Fast particles g (1) ( τ ) q 2 D 1 τ

29 Intensity autocorrelation function 50 The DLS Detector probes i S ( t ) This is a stochastic function i S ( kcounts/s ) The correlator calculates Intensity autocorrelation function t (s) g (2) ( τ) < i (0) i ( τ) > g S ( τ ) = 1 + β g ( τ ) (2) (1) 2 (Siegert equation) β < 1 S Coherence factor g (2) ( τ ) q 2 D τ ( ) 2 τ = + α τ (2) 2 g ( q, ) 1 exp q D

30 Correlation function and Light scattering spectrum Why Light Scattering is quasielastic 1 1 g (1) ( τ ) i S(ν) 2D 2 q 0 0 q 2 D τ ν-ν 0 (1) 2 ( ) 2 i πν τ g e is ( ) d τ = π ν ν 1 i 2 πν τ (1) is ( ν ) = e g ( τ) dτ 2π