Seminar PCF “ Lightscattering ”

Seminar PCF

ldquoLightscatteringrdquo

02 2 cos x tE x t E

c

1 Light Scattering ndash Theoretical Background

11 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 (ldquoelastic scatteringrdquo)

E

m

sE

Wave-equation of oscillating electic fieldof 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 ~ NiMi

2Pi(q) (scattering vector q see below)

Particles smaller than 20 (left picture) - scattered intensity independent of scattering angle I ~ NiMi

2

Particles in solution show Brownian motion (D = kT(6hR) and ltDr(t)2gt=6Dt)THORN Interference pattern and resulting scattered intensity fluctuate with time

THORN 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 ltS(q)gt Dynamic Lightscattering S(qt) (DLS))

2 22

02 2 2

41 exp 2 DsD D

EmE i t krt r c r c

2 Lichtstreuung ndash experimenteller Aufbau

Detector (photomultiplier photodiode) scattered intensity only 2

s s s sI E E E

detector

rDI

sampleI0

Scattered light wave emitted by one oscillating dipole

Light source I0 = laser focussed monochromatic coherent

Coherent the light has a defined oscillation phase over a certain distance (05 ndash 1 m) and time so it can show interference Note that only laser light is coherent in time soNo laser =gt no dynamic light scattering

Sample cell cylindrical quartz cuvette embedded in toluene bath (T nD)

Light Scattering Setup of the F-Practical Course PhysChem Mainz

Scattering volume defined by intersection of incident beam and optical aperture of the detection opticsvaries with scattering angle

Important scattered intensity has to be normalized

Scattering from dilute solutions of very small particles (ldquopoint scatterersrdquo)(eg nanoparticles or polymer chains smaller than 20)

222 2

040

4 ( )DDL

nb n KcN

in cm2g-2Mol

22 ( ) D

solution solventrR b c M I IV

std abssolution solvent

std

IR I I

I

contrast factor

Absolute scattered intensity of ideal solutions Rayleigh ratio ([cm-1])

For calibration of the setup one uses a scattering standard Istd Toluene ( Iabs = 14 e-5 cm-1 )

Reason of ldquoSky Bluerdquo (scattering from gas molecules of atmosphere)

Scattering from dilute solutions of larger particles

- scattered intensity dependent on scattering angle (interference)

0q k k

0k

k

4 sin( )2Dnq

0k 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-scale resolution information comment

qR ltlt 1 whole coil mass radius of gyration eg Zimm plot

qR lt 1 topology cylinder sphere hellip

qR asymp 1 topology quantitative size of cylinder

qR gt 1 chain conformation helical stretched

qR gtgt 1 chain segments chain segment density

0 2 4 6 8 10 1210-5

10-4

10-3

10-2

10-1

100

P(q

)

qR

For large (ca 500 nm) homogeneous spheres

26

9( ) sin cosP q qR qR qRqR

Minimum bei qR = 449

Two different types of Polystyrene nanospheres (R = 130 nm und R gt 260 nm) are investigated in the practical course

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

0030

100E-05

100E-04

100E-03

100E-02

100E-01

100E+00P(q) 130 nm

P(q) 260 nm

Dynamic Light ScatteringBrownian motion of the solute particles leads to fluctuations of the scattered intensity

( ) (0 ) ( ) ( ) ( ) exp( )s V T s sG r n t n r t F q G r iqr dr

mean-squared displacement of the scattering particle

2 6 sR D D 6s

H

kT kTDf Rh

change of particle position with time is expressed by van Hove selfcorrelation functionDLS-signal is the corresponding Fourier transform (dynamic structure factor)

Stokes-Einstein-Gl

2

2

( ) ( )( ) exp( ) ( ) ( ) 1

s s s sI q t I q tF q D q E q t E q t

I q t

ltI(t)

I(t+

)gtT

I(t)

t

1

2

The Dynamic Light Scattering Experiment - photon correlation spectroscopy ( in DLS one measures the intensity correlation ltI(t) I(t+)gt )

(note in static light scattering you measure the average scattered intensity ltI(qt)gt (see dashed line left graph))

Siegert-Relation

2Basislinie I t

rdquoCumulant-Methodldquo for polydisperse samples Fs(q) is a superposition of various exponentials

2 31 2 3

1 1ln 2 3sF q

1st cumulant 1 sup2sD q yields the average apparent diffusion coefficient

2nd cumulant 22 42 s sD D q is a measure for sample polydispersity

ImportantFor polydisperse samples of particles gt 10 nm the apparent diffusion coefficientIs q-dependent due to the weighting-factor P(q)

22 2

2 1i i i iapp s gz z

i i i

n M P q DD q D K R q

n M P q

Data analysis for polydisperse (monomodal) samples

Note the weighting factor ldquoNi Mi2 Pi(q)ldquo which is the average static scattered intensity per sample faction Taylor series expansion of this superposition leads to

qrarr0 Dapp is the z-average diffusion coefficient since all Pi(q) = 1

Cumulant analysis ndash graphic explanationMonodisperse sample Polydisperse sample

linear slope yields diffusion coefficient slope at =0 yields apparent diffusioncoefficient which is an average weightedwith NiMi

2Pi(q)

log(

F s(q)

DyDx=-Dsq2

log(

F s(q)

DyDx=-Dsq2

larger slower particles

small fast particles

Z-average diffusion coefficient is determined by interpolation of Dapp vs q2 -gt 0 (straight line only for particles lt 100 nm )

0 1x1010 2x1010 3x1010 4x1010

00

50x10-15

10x10-14

15x10-14

20x10-14

Dm

2 s-1

q2cm-2

s zD

Explanation for q-dependence of Dapp for larger particles due to Pi(q)

2

2i i i i

appi i i

n M P q DD q

n M P q

Note the minimum in P(q) for the larger particleswhere the average diffusion coefficient will reacha maximum

0010

0011

0012

0013

0014

0015

0016

0017

0018

0019

0020

0021

0022

0023

0024

0025

0026

0027

0028

0029

0030

100E-05

100E-04

100E-03

100E-02

100E-01

100E+00P(q) 130 nm

P(q) 260 nm

Due to interparticle interactions the particles not any longer move independentlyby Brownian motion only Therefore DLS in this case measures no self-diffusioncoefficient but a collective diffusion coefficient defined as Dc(q) = DsS(q)

DLS of concentrated samples ndash influence of the static structure factor S(q)

From Gapinsky et al JChemPhys 126 104905 (2007)

S(q) from SAXS particle radius ca 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)

From Gapinsky et al JChemPhys 126 104905 (2007)

Note 1 The q-regime of SAXSXPCS is much larger than in light scattering dueto the shorter wave length of Xrays (lab course 0013 nm-1 lt q lt 0026 nm-1 )2 The investigated Ludox particles R = 25 nm are much smaller therefore themaximum in S(q) is located at larger q (q(S(q)_max) gt 01 nm-1 )

D(q) from XPCS (Xray-correlation) particle radius 80 nm c = 200 97 und 75 gL in waterleft c(salt) = 05 mM right c(salt) = 50 mM)