ht Scattering from Polymer Solutions and Nanoparticle Dispers D Dr. Wolfgang Schaertl ut für Physikalische Chemie, Universität Mainz, rweg 11, 55099 Mainz, Germany [email protected]from the new book of the same title, published by Springer in July s are found at: http://www.uni-mainz.de/FB/Chemie/wschaertl/105.php
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“Light Scattering from Polymer Solutions and Nanoparticle Dispersions”
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“Light Scattering from Polymer Solutions and Nanoparticle Dispersions”
By: PD Dr. Wolfgang SchaertlInstitut für Physikalische Chemie, Universität Mainz, Welderweg 11, 55099 Mainz, [email protected]
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
02 2, cos x tE x t E
c
1. Light Scattering – Theoretical Background
1.1. 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 (“elastic scattering”)
E
m
sE
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 ~ NiMi
2Pi(q) (scattering vector q, see below!)
Particles smaller than /20: - scattered intensity independent of scattering angle, I ~ NiMi
2
Particles in solution show Brownian motion (D = kT/(6hR), and <Dr(t)2>=6Dt)=> Interference pattern and resulting scattered intensity fluctuate with time
Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz:
lasersample,bath
detector ongoniometer arm
Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz:
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 (“point scatterers”)(e.g. nanoparticles or polymer chains smaller than /20)
2
,( )T N
cI b kT
c
:
kTc M
21 2 ...( )kT A c
c M
222 2
,040
4 ( )DDL
nb n KcN
in cm2g-2Mol
40I
2222 2
,040
4 ( )( )D DD solution solvent
L
n rR b c M n c M I Ic VN
,std abs
solution solventstd
IR I I
I
Fluctuation theory:
contrast factor
Ideal solutions, van’t Hoff:
Real solutions, enthalpic interactions solvent-solute:
Absolute scattered intensity of ideal solutions, Rayleigh ratio ([cm-1]):
and
Scattering standard Istd: Toluene( Iabs = 1.4 e-5 cm-1 )
Reason of “Sky Blue”! (scattering from gas molecules of atmosphere)
1KcR M
21 2 ...Kc A c
R M
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:
q
0k
k
4 sin( )2Dnq
Real solutions, enthalpic interactions solvent-solute expressed by 2nd Virial coeff.:
q
q = inverse observational length scale of the light scattering experiment:
q-scale resolution information comment
qR << 1 whole coil mass, radius of gyration e.g. Zimm plot
qR < 1 topology cylinder, sphere, …
qR ≈ 1 topology quantitative size of cylinder, ...
qR > 1 chain conformation helical, stretched, ...
qR >> 1 chain segments chain segment density
Scattering from 2 scattering centers – interference of scattered waves
2 2
1 1
1 exp 1 exp exp ijs sI i i I iqr D D
0k
k
0k
k
ijr
A B
C
???AB BC
cosijAB r 02 cosij ijr k r
0 2ijAB r k
cosijBC r 2 cos 180ij ijr k r
2ijBC r k
0 2 2ij ijAB BC r k k r q
leads to phase difference: ijr qD
2 interfering waves with phase difference D:
2 2
( ) exp( ) exp exp( ) 1 exps s sI q E ikr E i kr E ikr i D D
orientational average and normalization lead to:
22 21 1
222 2
1 1 1 1
1 1( ) exp
sin1 1 11 ...6
Z Z
iji j
Z Z Z Zij
iji j i jij
P q I q iqrZNZ b
qrq rZ Zqr
.
22 2
1 1 1 1
2Z Z Z Z
j i i jij ji j i j
r s s s s s
2 2 2
1 1
2Z Z
ii j
s Z s
2 21( ) 1 ...3P q s q
21 2 ...( )
Kc A cR MP q
Zimm equation:
2 22
1 11 23( )Kc s q A cR M
2 2
1 1 1 1
( ) exp expZ Z Z Z
i j iji j i j
I q Nb iq r r Nb iqr
Scattered intensity due to Z pair-wise intraparticular interferences, N particles:
replacing Cartesian coordinates ri by center-of-mass coordinates si we get:
regarding the reciprocal scattered intensity, and including solute-solvent interactions finally yields the well-known Zimm-Equation (series expansion of P(q), valid for small R):
s2, Rg2 = squared radius of gyration
The Zimm-Plot, leading to M, s (= Rg) and A2:
0,0 5,0 10,0 15,0 20,01,0
1,5
2,0
2,5
3,0
3,5
4,0
4,5
5,0
5,5
6,0
Kc/R
/ 10
-7m
ol/g
(q2+kc) / 1010cm-2
2 22
1 11 23( )Kc s q A cR M
q = 0
c = 0example: 5 c, 25 q
Zimm analysis of polydisperse samples yields the following averages:
1
1
K
k k kk
w K
k kk
N M MM
N M
The z-average squared radius of gyration:
2 2
22 1
2
1
K
k k kk
z g Kz
k kk
N M ss R
N M
The weight average molar mass
Reason: for given species k, Ik ~ NkMk2
Fractal Dimensions
( ) fdM R R: 1gq R
:
( ) 22 fdI q M q-: :
log 2 logfI q d q
topology df
cylinder, rod 1thin disk 2
homogeneous sphere 3
ideal Gaussian coil 2Gaussian coil with excluded volume 5/3
branched Gaussian chain 16/7
if
log log logfP q I q cM d q
0 2 4 6 8 10 1210-5
10-4
10-3
10-2
10-1
100
P(q
)
qR
Particle form factor for “large” particles
for homogeneous spherical particles of radius R:
26
9( ) sin cosP q qR qR qRqR
first minimum at qR = 4.49
2 22 2
1 1 1 1
sin1 1 1( ) expZ Z Z Z
ijij
i j i j ij
qrP q I q iqr
Z ZNZ b qr
Zimm!
1.3. 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
isotropic diffusive particle motion232 2
2
3 ( )2( , ) ( ) exp3 2 ( )[ ] ( )s
rG r RR
D D
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 function,DLS-signal is the corresponding Fourier transform (dynamic structure factor)
Stokes-Einstein, Fluctuation - Dissipation
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
<I(t)
I(t+
)>T
I(t
)
t
1
2
The Dynamic Light Scattering Experiment - photon correlation spectroscopy
Siegert relation:
2
exp 2 ,
s
I t I t
D q
note: usually the “coherence factor” fc is smaller than 1, i.e.:
2
2
( , ) ( , ) 1 ( , ),
c sI q t I q t f F q
I q t
fc increases with decreasing pinhole diameter, but photon count rate decreases!
log
F s(q,)
F s(q,)
log
DLS from polydisperse (bimodal) samples
2
0
, exps s s sF q P D q D dD
2 21 1 2 2, exp exps s sF q A q D A q D
”Cumulant method“, series expansion, only valid for small size polydispersities < 50 %
2 31 2 3
1 1ln , ...2! 3!sF q
first Cumulant 1 ²sD q yields inverse average hydrodynamic radius 1HR
second Cumulant 22 42 s sD D q yields polydispersity
22
221
s s
Ds
D D
D
for samples with average particle size larger than 10 nm:
2
2i i i i
appi i i
n M P q DD q
n M P q
2 21app s gz zD q D K R q
Data analysis for polydisperse (monomodal) samples
Time intensity correlation function decays single-exponentially
yes no
Only 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 methods
Strategy for particle characterization by light scattering - A
Applicability of commercial particle sizers!
wM
Sample topology is unknown,static light scattering necessary
yes no
Particle radius larger than 50 nm and/or very polydisperse sample:use more sophisticated methods to evaluate particle form factor
Plot of vs. is linearKcR
2q
Dynamic light scattering to determine 11
H H zR R
Estimate (!) particle topology from g
H
RR
r
Strategy for particle characterization by light scattering - B
Particle radius between 10 and 50 nm:analyze data following Zimm-eq. to get:
g zR 2AWM
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:
Normalized particle form factorsuniversal up to values of qRg = 2
Details at higher q (smaller length scales) – Kratky Plot:
C
form factor fits:
2
22
1 311 6
g
g
C qRP q
C qR
C related to branching probability, increases with molar mass
Are the starch samples, although not self-similar, fractal objects?
log logfdfP q q P q d q
- 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-sn-glycero-3-phosphocholine (DOPC) and 1-stearoyl-2-oleoyl-sn-glycero-3-phosphocholine (SOPC) by extrusion
monodisperse vesicles 2 23 31 1
3 3
3 o io i
o io i
j qR j qRP q R R
qR qRR R
2oR R t
2iR R t 1 2
sin cosx xj xx x
Data Analysis:
thin-shell approximation 2sin qR
P qqR
small values of qR, Guinier approximation 22exp 3gP q q R
52 2
3
135 1
i og o
i o
R RR R
R R
typical q-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)
21
0
sin, ,
quP q a b dx
qu
2 2 2 21u a x b x
cosx
0k k
prolate vesicles, surface area 4 (60 nm)2 oblate vesicles, surface area 4 (60 nm)2
anisotropy vs. polydispersity:
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 !
2sin
, ,b
a
qRP q a b G R dR
qR
2 2 2 2
1 RG Ra b R b
monodisperse ellipsoidal vesicles
2
0
2
0
,R
R
M P q R G R dRP q
M G R dR
2
sin,
qRP q R
qR
polydisperse spherical vesicles
combination of SLS and DLS:DLS: intensity-weighted size distribution => number-weighted size distribution (fit a,b) => SLS: particle form factor
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 q
2 2 14.0 0.4 10z
D nm s 60 6HR nm 2g HR R results:
Rg 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
self-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 micelles characterization: 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 values
PNIPAM chains in water at different T
coil – globule - transition Rg from Zimm-analysis, RH by DLS, decrease in R and Rg / RH
3. Dynamic Light Scattering – Selected Examples
1. Vanhoudt, J.;Clauwaert, J. Langmuir 1999, 15, 44-57
sample: spherical latex particles in dilute dispersions
1. Cumulant method (CUM), polynomial series expansion:
2
0.5 0.5 21ln ln
2f g f 2
0appB d D q
22
0
B d
polydispersity index 2
2PI
particle diameter is a so-called harmonic z-average:
6
5
i ii
i ii
n dd
n d
(only for homogeneous spheres) 2 6i iM d
2. non-negatively least squares method (NNLS):
2
21
1 1
expN M
j i i jj i
g b
M exponentials considered for the exponential series, yielding a set of coefficients bi defining the particle size distribution for decay rates equally distributed on a log scale.
3. Exponential sampling (ES):
1 1max
expnn
See 2., decay rates chosen according to:
4. Provencher’s CONTIN algorithm:
22
22 1
1i
i i
Bg e d LB
Numerical procedure to calculate a continuous decay rate distribution B(), also calledInverse Laplace Transformation, enclosed in most commercial DLS setups.
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
22 , expg q y b q c q
vv-mode (only translation is detected):
depolarized dynamic light scattering (vh-mode) (translation and rotation are detected, no coupling in case qL < 5)
22 Tb q D q
22 12T Rb q D q D
translational diffusion coefficient DT determined from the slope, rotational diffusion coefficient DR from the intercept of the data measured in vh –geometry.
Results:
q2 / 1014 m-2
q2 / 1014 m-2
qmaxL > 5 (≈ 9) !
qL < 5
diffusion coefficients according to Tirado and de la Torre,using as input parameters length and diameter from TEM