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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

schaertl@uni-mainz.de

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