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Folie 1 - uni

Transcript Folie 1 - uni

“Light Scattering from Polymer Solutions and Nanoparticle Dispersions”

By: PD Dr. Wolfgang Schaertl

Institut für Physikalische Chemie, Universität Mainz, Welderweg 11, 55099 Mainz, Germany

Parts from the new book of the same title, published by Springer in July 2007 Slides are found at: http://www.uni-mainz.de/FB/Chemie/wschaertl/105.php

1. Light Scattering – Theoretical Background

1.1. Introduction

Light-wave interacts with the charges constituting a given molecule in remodelling the spatial charge distribution:

 



0  cos   2

 

 2

 

c t

  Molecule constitutes the emitter of an electromagnetic wave of the same wavelength as the incident one (“elastic scattering”)

m E E s

Note: usually vertical polarization of both incident and scattered light (vv-geometry)

Particles larger than 20 nm: - 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 ~ N i M i 2 P i (q) (scattering vector q, see below!) Particles smaller than  /20: - scattered intensity independent of scattering angle, I ~ N i M i 2

Particles in solution show Brownian motion (D = kT/(6 h R), and < D r(t) 2 >=6Dt) => Interference pattern and resulting scattered intensity fluctuate with time

1.2. Static Light Scattering

Scattered light wave emitted by one oscillating dipole

E s

    2



2   1

r c D

2   4 2

r c D

0 exp 

 2





kr D

  Detector (photomultiplier, photodiode): scattered intensity only!

I s



 

E s

2 sample

I 0 r D



detector Light source I 0 = laser: focussed, monochromatic, coherent Sample cell: cylindrical quartz cuvette, embedded in toluene bath (T, n D )

Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz: sample, bath laser detector on goniometer arm

Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz:

Scattering volume: defined by intersection of incident beam and optical aperture of the detection optics   Important: scattered intensity has to be normalized

Scattering from dilute solutions of very small particles

(e.g.

nanoparticles or polymer chains smaller than  /20)

(“point scatterers”)

Fluctuation theory:

: (  



c c

)

T N

contrast factor Ideal solutions, van’t Hoff: 







kT M

Real solutions, enthalpic interactions solvent-solute:

2  4





0 4

N L







c n D

,0 2 ( 

n D



) 2 

in cm 2 g -2 Mol  1

(

 2

A c

2  ...

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



Kc R

 1

M I



0  4  4





0 4

N L n D

,0 2 ( 

n D



) 2  (

I solution



I solvent

)

r D

and





I solution



I solvent





I I std

Scattering standard I std : Toluene ( I abs = 1.4 e-5 cm -1 ) Reason of “Sky Blue”! (scattering from gas molecules of atmosphere)

Real solutions, enthalpic interactions solvent-solute expressed by 2nd Virial coeff.:

Kc R

 1

 2

A c

2  ...

Scattering from dilute solutions of larger particles

- scattered intensity dependent on scattering angle (interference) The scattering vector

(in [cm -1 ]) , length scale of the light scattering experiment:

0 

q k q

 4



n D

sin(

 

2 )

= inverse observational length scale of the light scattering experiment :

-scale

<< 1

< 1

≈ 1

> 1

>> 1 resolution whole coil topology topology quantitative chain conformation chain segments information mass, radius of gyration cylinder, sphere, … size of cylinder, ...

helical, stretched, ...

chain segment density comment e.g. Zimm plot

Scattering from 2 scattering centers – interference of scattered waves



B C



r ij k k AB



 ???

AB BC AB

 

cos



 

cos







0 

 0

ij r ij k

 





  



  cos

 

 

 







 leads to phase difference: 2 interfering waves with phase difference D : 

E s

 exp(

ik r

) 

E s

 exp    D   2 

E s

 exp(

ik r

 2

 

 2 2

 

 D   

ij I s

   

2 2  1 1 exp  

iqr ij



Scattered intensity due to Z pair-wise intraparticular interferences, N particles:



i Z

  1

j Z

 1 exp   



r j

 

i Z

  1

j Z

 1 exp  

iqr ij

orientational average and normalization lead to:  1 2  1

i Z

  1

j Z

 1 exp  

iqr ij

  1

i Z

  1

j Z

 1   sin

qr ij

   1

i Z

  1

j Z

 1  1  1 6 2

q r ij

2  ...

 replacing Cartesian coordinates r i by center-of-mass coordinates s i we get:

i Z

  1

j Z

 1

r ij

2 

i Z

  1

j Z

 1 

s j



s i

2  

i Z

  1

j Z

 1 

s j

2  2



s i

2   2 2 2

Z s

  1 3 2

s q

2  ...

s 2 , R g 2 = squared radius of gyration . finally yields the well-known Zimm-Equation (series expansion of P(q), valid for small R):

Kc R

 1  2

A c

2  ...

Kc R

 1

( 1  1 3 2

s q

2 )  2

A c

The Zimm-Plot, leading to M, s (= R g ) and A 2 :

6,0 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,0 5,0

c = 0

Kc R

 q = 0

10,0 15,0

(q

+kc) / 10

cm

-2

( 1  1 3 2

s q

2 )  2

A c

2 example: 5 c, 25 q

20,0

Zimm analysis of polydisperse samples yields the following averages:

The weight average molar mass

M w



k K

  1

k K

  1

N M M k k k N M k k

The z-average squared radius of gyration: 

2  

z R g



k K

  1

k K

  1 2

N M s k k k

N M k k

2 Reason: for given species k, I k ~ N k M k 2

Fractal Dimensions

R d f



R g

 1 log :

2 :

d f

log  log    2

d f

 log



d f

log

: topology cylinder, rod thin disk homogeneous sphere ideal Gaussian coil Gaussian coil with excluded volume branched Gaussian chain d f 1 2 3 2 5/3 16/7

Particle form factor for “large” particles

 1 2

NZ b

2  1

i Z

  1

j Z

 1 exp  

iqr ij

  1

i Z

  1

j Z

 1   sin

qr ij

for homogeneous spherical particles of radius R:  9 6  sin 

cos  2   10 0 10 -1 Zimm!

10 -2 10 -3 10 -4 10 -5 0 2 4 first minimum at

= 4.49

6 qR 8 10 12

1.3. 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)





n t n r t





)   isotropic diffusive particle motion

s s

 

 



)  [ 2



3  D



2  ] 3 2 exp (  2



 D



2 2  ) mean-squared displacement of the scattering particle: D

 

2  6

D s



D s



kT f



h

R H

Stokes-Einstein, Fluctuation - Dissipation

The Dynamic Light Scattering Experiment - photon correlation spectroscopy

 1  

D q s

2 

 

exp

  

,  2 Siegert relation: t



 exp( 

D q s



) 

s s



 

 



 1 note : usually the “coherence factor” f c smaller than 1, i.e.: is 

 



  



 

2 f c increases with decreasing pinhole diameter, but photon count rate decreases!

DLS from polydisperse (bimodal) samples

 

 0   exp   2

q D s





dD s



1  exp   2

q D s



 

2  exp   2

q D s



  log 

Data analysis for polydisperse (monomodal) samples

”Cumulant method“, series expansion, only valid for small size polydispersities < 50 % ln

 

 

 

1  1 2!

 

2 2  1 3!

 

3 3  ...

first Cumulant



1 

D q s

² second Cumulant



2  

D s

2 yields inverse average hydrodynamic radius 

D s

2 

4 yields polydispersity



 

D s

2 

D s D s R H

 1 2  

 

1 2 2 for samples with average particle size larger than 10 nm:

D app

  



2 



 

P q



 

D i

note:

 



2 

 

D app



D s z

 1 

K R g

z q

2  

Cumulant analysis – graphic explanation:

monodisperse sample polydisperse sample

D y/ D x=-D s q 2 large, slow particles D y/ D x=-D s q 2  linear slope yields diffusion coefficient small, fast particles  slope at  =0 yields apparent diffusion coefficient, which is an average weighted with n i M i 2 P i (q)

D app vs. q 2 :

D s z

2,0x10 -14 1,5x10 -14 1,0x10 -14 5,0x10 -15 0,0 0 1x10 10 2x10 10 q 2 /cm -2 3x10 10 4x10 10

Explanation for D app (q):

D app

  



2 



 

P q i



 

D i

for larger particle fraction i, P(q) drops first, leading to an increase of the average D app (q)

10

-1

10

-2

10

-3

10

-4

10

-5

0,00

R = 60 nm R = 80 nm R = 100 nm

0,01 0,02 q [nm

-1

] 0,03 0,04

ln(g1( 50  ))=P1+P2*  +P3/2 *  ^2 PI = SQRT(P3/P2^2) Ni 40 20 Ni 15 30 20 10 10 5 0,0 0 0 lng1 5 10 R i /nm 15 Data: Data2_lng1 Model: cumulant Chi^2 = 3.7224E-8 P1 P2 P3 0.00882

±0.00003

-10790.57918 ±0.23957

896471.16145 ±926.09523

D app (90°)=2.04e-11 m 2 /s, entspr. R = 10.5 nm PI = 0.09, D R/R=10% (Normalvert.) -0,2 0,00000  /s 20 0 0 5 10 R i /nm 15 20 0,00002 0,00 -0,02 -0,04 -0,06 -0,08 -0,10 -0,12 -0,14 -0,16 -0,18 -0,20 0,000000 lng1 D app (90°)=1.59e-11 m 2 /s, entspr. R = 13.5 nm PI = 0.20, D R/R=30% (Normalvert.) 0,000004 0,000008 Data: Data2_lng1 Model: cumulant Chi^2 = 3.3258E-10 P1 P2 P3 0.0079 ±5.5823E-6 -8423.55623

±0.25513

2723184.05649 ±4894.69843

 /s 0,000012 0,000016 27 0,000020

100 80 60 Ni 40 20 0 0 5 10 15 20 R i /nm 25 30 35 40 1,20E-011 1,00E-011 8,00E-012 6,00E-012 4,00E-012 0,0001 0,0002 0,0003 0,0004 0,0005 q 2 /nm -2 0,0006 0,0007 2,80E-008 2,78E-008 2,76E-008 2,74E-008 2,72E-008 2,70E-008 2,68E-008 2,66E-008 2,64E-008 2,62E-008 0,0008 2,60E-008 0,0001 0,0002 0,0003 0,0004 0,0005 q 2 /nm -2 0,0006 0,0007 28 0,0008

100 80 60 Ni 40 20 0 0 2,00E-012 1,50E-012 50 100 150 R i /nm 200 250 300 2,00E-007 1,80E-007 1,60E-007 1,40E-007 1,00E-012 1,20E-007 5,00E-013 0,0001 0,0002 0,0003 0,0004 0,0005 q 2 /nm -2 0,0006 0,0007 0,0008 1,00E-007 0,0001 0,0002 0,0003 0,0004 0,0005 q 2 /nm -2 0,0006 0,0007 29 0,0008

10 5 20 15 Ni 0 0 4,40E-013 4,20E-013 4,00E-013 3,80E-013 3,60E-013 200 0,0001 0,0002 0,0003 400 R i /nm 600 0,0004 0,0005 q 2 /nm -2 800 1000 0,0006 0,0007 5,70E-007 5,65E-007 5,60E-007 5,55E-007 5,50E-007 5,45E-007 5,40E-007 5,35E-007 5,30E-007 5,25E-007 0,0008 5,20E-007 0,0001 0,0002 0,0003 0,0004 0,0005 q 2 /nm -2 0,0006 0,0007 30 0,0008

Combining static and dynamic light scattering, the

-ratio:

r



R g R H

topology homogeneous sphere hollow sphere ellipsoid random polymer coil cylinder of length l, diameter D r -ratio 0.775

1 0.775 - 4 1.505

1 3  ln

l D

 0.5

for polydisperse samples:

r



R g

  

Strategy for particle characterization by light scattering - A

Sample topology (sphere, coil, etc…) is known yes Dynamic light scattering sufficient (“particle sizing“) no Static light scattering necessary Time intensity correlation function decays single-exponentially yes no Only one scattering angle needed, determine particle size (R H ) from Stokes-Einstein-Eq.

(in case there are no particle interactions (polyelectrolytes!) Applicability of commercial particle sizers!

Sample is polydisperse or shows non-diffusive relaxation processes!

to determine “true” average particle size, extrapolation q -> 0 - to analyze polydispersity, various methods

Strategy for particle characterization by light scattering - B

Sample topology is unknown, static light scattering necessary

Kc R q

2 yes no Particle radius between 10 and 50 nm: analyze data following Zimm-eq. to get:

M W R g M z w A

2 Particle radius larger than 50 nm and/or very polydisperse sample: use more sophisticated methods to evaluate particle form factor Dynamic light scattering to determine

R H

 

R H

 1

  1 Estimate (!) particle topology from

r



R g R H

2. Static Light Scattering – Selected Examples

1. Galinsky, G.;Burchard, W. Macromolecules 1997, 30, 4445-4453 Samples:

Several starch fractions prepared by controlled acid degradation of potatoe starch ,dissolved in 0.5M NaOH Sample characteristics obtained for very dilute solutions by Zimm analysis: sample LD11 LD16 LD12 LD19 LD18 LD17 LD13 10 -6

M w

(g/mol) 0.92

1.87

5.20

14.5

43 64 97

(nm) 36 48

70 113 180 190 233 10 4

A 2

[(mol cm 3 )/g 2 ] 1.00

0.60

0.28

0.13

0.082

0.060

0.025

Normalized particle form factors universal up to values of

qR g

= 2

Details at higher q (smaller length scales) – Kratky Plot:

form factor fits:  1      2    1    



1 



6    2    2

related to branching probability, increases with molar mass

Are the starch samples, although not self-similar, fractal objects?



d f

 log

d f

log

- minimum slope reached at qRg ≈ 10 (maximum q-range covered by SLS experiment !) - at higher q values (simulations or X-ray scattering) slope approaches -2.0 - characteristic for a linear polymer chain (C = 1). - at very small length scale only linear chain sections visible (non-branched outer chains)

2. Pencer, J.;Hallett, F. R. Langmuir 2003, 19, 7488-7497 Samples:

uni-lamellar vesicles of lipid molecules 1,2-Dioleoyl-

-glycero-3-phosphocholine (DOPC) and 1-stearoyl-2-oleoyl-

-glycero-3-phosphocholine (SOPC) by extrusion

Data Analysis:

monodisperse vesicles

R o R i

   

R o

3 3 

R i

3 2 2

R o

3 1







R i

qR o

 

i qR i

  2 1

 

 sin

x x

2  cos

x x

thin-shell approximation     sin

   2 small values of

, Guinier approximation  exp   2

q R g

2 3 

R g

2  3

R o

2 5 1  1 

 

R R i o R R i o



typical

-range of light scattering experiments: 0.002 nm-1 to 0.03 nm -1 vesicles prepared by extrusion: radii 20 to 100 nm => first minimum of the particle form factor not visible in static light scattering

particle form factor of thin shell ellipsoidal vesicles, two symmetry axes (a,b,b)

 

 0 1   sin

  2

dx u

 2

a x

2 

2  1 

2 

 cos  

k k

0 prolate vesicles, surface area 4  (60 nm) 2 oblate vesicles, surface area 4  (60 nm) 2

anisotropy vs. polydispersity:

monodisperse ellipsoidal vesicles

 



b a

   sin

  2

 1

2 

R R

2 

2 polydisperse spherical vesicles  0     0



  

   

 sin

  2 static light scattering from monodisperse ellipsoidal vesicles can formally be expressed in terms of scattering from polydisperse spherical vesicles !

=> impossible to de-convolute contributions from vesicle shape and size polydispersity using SLS data alone !

combination of SLS and DLS:

DLS: intensity-weighted size distribution => number-weighted size distribution (fit a,b) => SLS: particle form factor

result:

polydisperse ( D R = 10%) oblate vesicles, a : b < 1 : 2.5 input for a,b – fits to SLS data

3. Fuetterer, T.;Nordskog, A.;Hellweg, T.;Findenegg, G. H.;Foerster, S.; Dewhurst, C. D. Physical Review E 2004, 70, 1-11 Samples:

worm-like micelles in aqueous solution, by association of the amphiphilic diblock copolymer poly-butadiene(125)-b-poly(ethylenoxide)(155)

Analysis of SLS-results:

monodisperse stiff rods asymmetric Schulz-Zimm distribution

polydisperse stiff rods

 2

 0  

0  sin

 

ql dql

   

  sin



 1



qL L w



  2

 1

L k



exp 1







 1



M w



L L w M n

 1



 0 

 

Koyama, flexible wormlike chains  1

l K

2 0

l K





l K





exp  1 3

' 2  sin  

   

Holtzer-plot of SLS-data :



R Kc

 

plateau value = mass per length of a rod-like scattering particle

fit results:

(i) polydisperse stiff rods: (ii) polydisperse wormlike chains:

L w

 389

L w

 380

w w M n

 1.2

M n

 2.0,

l K

 410

Analysis of DLS-results:

 



3   0

S n

 

 exp   

D q T

2  2



 1



D R





 amplitudes depend on the length scale of the DLS experiment: - long diffusion distances (qL < 4): only pure translational diffusion S 0 - intermediate length scales (4 < qL < 15): all modes (

n =

0, 1, 2) present according to Kirkwood and Riseman:

D T



h

  ,

D R

 9

D T L

2 polydispersity leads to an average amplitude correlation function!

DLS relaxation rates : linear fit over the whole q-range:

significant deviation from zero intercept, additional relaxation processes or “higher modes” at higher

results:

D z

  2 2

nm s

 1

R H



nm R g R H

 2 R g from Zimm-analysis and calculations!

4. Wang, X. H.;Wu, C. Macromolecules 1999, 32, 4299-4301 samples:

high molar mass PNIPAM chains in (deuterated) water

reversibility of the coil-globule transition:

molten globule ? surface of the sphere has a lower density than its center

Selected Examples – Static Light Scattering: sample problem solution

branched polymeric nanoparticles vesicles (nanocapsules) worm-like micelles PNIPAM chains in water at different T self-similarity (fractals) ?; distinguish size polydispersity and shape anisotropy in P(q) ?

characterization: length, R (R H coil : no rotation-translation coupling if qL < 4) – globule - transition g /R H details at qR > 2 by Kratky plot (P(q) q 2 vs. q), fitting parameters for branched polymers, simulation of P(q) at qR > 10 (SLS: qR < 10) => not fractal !

combine DLS (only size polydispersity !) and SLS to simulate expt. P(q) details at higher q by Holtzer plot (I(q) q vs. q), fit P(q), R g from Zimm-analysis at small q values R g from Zimm-analysis, R H DLS, decrease in R and R g by / R H

3. Dynamic Light Scattering – Selected Examples

1. Vanhoudt, J.;Clauwaert, J. Langmuir 1999, 15, 44-57

sample: spherical latex particles in dilute dispersions sample nominal diameter/nm diameter ratio s2 19 s3 54 s4 91 s5 19, 91 4.8

s6 19, 54 2.8

Data analysis of polydisperse samples:

1. Cumulant method (

CUM

), polynomial series expansion: ln 

0.5

1   ln polydispersity index





 

2 2 2  2  



2   0  0  

 

D app



2  2

 

particle diameter is a so-called harmonic z-average:

 



i n d i i

n d i i

5 (only for homogeneous spheres)

M i

d i

6 s7 54, 91 1.7

2. non-negatively least squares method (

NNLS



2 

j N

  1  

1  



i M

  1

b i

exp  



   2

exponentials considered for the exponential series, yielding a set of coefficients

b i

defining the particle size distribution for decay rates equally distributed on a log scale.

3. Exponential sampling (

): See 2., decay rates chosen according to:   

 1 1 exp  

 

max  4. Provencher’s CONTIN algorithm:

 

 

  1



2   

 

 



 

 2



2 Numerical procedure to calculate a continuous decay rate distribution B(  ), also called Inverse Laplace Transformation, enclosed in most commercial DLS setups. 5. double-exponential method (

1   1  

b e

 2 

Results:

sample nominal diameter diameter ratio

- CUM (1.)

– CUM (1.)

d1,d2

– NNLS (2.)

d1,d2

– ES (3.)

d1,d2

– DE (5.)

s2 s3 s4 19 ± 1.5 54 ± 2.7 91 ± 3 20.3

0.029

55.0

0.009

87.0

0.008

Bimodal samples s5, s6, s7: I 1 (q=0) = I 2 (q=0) s5 19, 91 4.8

36.9

0.248

18, 81 s6 19, 54 2.8

29.5

0.191

16, 50 19, 54 18, 54 s7 54, 91 1.7

69.0

0.069

Note: bimodal samples with d2/d1 < 2 (s7) beyond resolution of DLS !

2. van der Zande, B. M. I.;Dhont, J. K. G.;Bohmer, M. R.;Philipse, A. P.

Langmuir

2000

, 16, 459-464

sample (TEM-results): colloidal gold nanoparticles stabilized with poly(vinylpyrrolidone) (M = 40000 g/Mol) system length [nm]

D L [nm] diameter [nm]

D d [nm] aspect ratio L/d Sphere18 Sphere15 Rod2.6a

Rod2.6b

Rod8.9

Rod12.6

Rod14 Rod17.2

Rod17.4

Rod39 Rod49 18 15 47 39 146 189 283 259 279 660 729 5 3 17 10 37 24 22 60 68 20 18 15 17 15 20 15 16 17 15 3 3 3 3 3 3 3 3 3 2.6

2.6

8.9

12.6

14.0

17.2

17.4

39.0

49.0

DLS 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 polydispersity

 

exp    







2  vv-mode (only translation is detected):  2

D q T

2 depolarized dynamic light scattering (vh-mode) (translation and rotation are detected, no coupling in case qL < 5)  2

D q T

2  12

D R

translational diffusion coefficient D T determined from the slope, rotational diffusion coefficient D R from the intercept of the data measured in vh –geometry.

Results:

< 5 q

2 / 10 14 m -2 q max L

> 5 ( ≈ 9) !

2 / 10 14 m -2

diffusion coefficients according to Tirado and de la Torre, using as input parameters length and diameter from TEM

D T



h

  ln 

d L

 0.100

2  

D R

 3

h

3    ln 

d L

 0.050

2   system Rod8.9

Rod12.6

Rod14 Rod17.2

Rod17.4

Rod39 Rod49 10 -12 D T , exp [m 2 s -1 ] 6.0

4.9

2.9

4.0

3.5

1.2

0.7

10 -12 D T , calc [m 2 s -1 ] 8.4

7.4

5.2 6.0

5.6

2.9

2.8

D R , exp [s -1 ] 306 281 66 177 175 14 D R , calc [s -1 ] 2238 1258 396 563 452 46 30 values determined by DDLS systematically too small, because PVP-layer (thickness 10 – 15 nm) not visible in TEM !

Selected Examples – Dynamic Light Scattering: sample

bimodal spheres stiff gold nanorods

problem solution

size resolution length and diameter in solution =?; deviation TEM – DLS ?

- double exponential fits - size distribution fits - CONTIN ; only if R 1 /R 2 > 2 depolarized DLS (vh) => D rot standard DLS (vv) => D trans ; deviation TEM-DLS due to PVP stabilization layer