“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:
 2 x 2 t 
E  x, t   E0  cos 



 c

Molecule constitutes the emitter of an electromagnetic wave of the same wavelength
as the incident one (“elastic scattering”)
m
E
Es
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 ~ NiMi2Pi(q) (scattering vector q, see below!)
Particles smaller than /20:
- scattered intensity independent of scattering angle, I ~ NiMi2
Particles in solution show Brownian motion (D = kT/(6hR), and <Dr(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


 2m  1
4 2 2 E0
Es   2  2 
exp i 2 t  kr D
2
rD c
 t  rD c
Detector (photomultiplier, photodiode): scattered intensity only!
I0

I s  Es Es  Es
sample

rD
I
detector
Light source I0 = laser: focussed, monochromatic, coherent
Sample cell: cylindrical quartz cuvette, embedded in toluene bath (T, nD)
2
Standard Light Scattering Setup of the Schmidt Group, Phys.Chem., Mainz:
laser
sample,
bath
detector on
goniometer 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)
Fluctuation theory:
c
I : b 2 kT
(  )T , N
c
Ideal solutions, van’t Hoff:
4 2
2 n
b  4 nD ,0 ( D )2  K
c
0 N L
2
contrast factor
in cm2g-2Mol
 kT

c M

1
 kT (  2 A2c  ...)
c
M
Real solutions, enthalpic interactions solvent-solute:
Absolute scattered intensity of ideal solutions, Rayleigh ratio ([cm-1]):
I std ,abs
4 2
rD
2 nD 2
and
R

I

I

R  b  c  M  4 nD,0 (
)  c  M  ( I solution  I solvent )
 solution solvent 
I std
c
V
0 N L
2
2

Kc 1

R M
Scattering standard Istd: Toluene
( Iabs = 1.4 e-5 cm-1 )
04
Reason of “Sky Blue”! (scattering from gas molecules of atmosphere)
I
Real solutions, enthalpic interactions solvent-solute expressed by 2nd Virial coeff.:
Kc 1

 2 A2 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:
k0

k
q
4 nD sin( )
2
q

q = inverse observational length scale of the light scattering experiment:
q
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
A
k0

B
C 
rij
k0
k
k
AB  BC  ???
AB  rij  cos 
rij  k0  rij 
2
 cos  
AB  rij  k0 

2

2

rij  k  rij 
 cos 180  
 BC  rij  k 
BC  rij  cos

2


AB  BC  rij  k0  k 
 rij  q 
2
2 leads to phase difference: D  rij  q
2 interfering waves with phase difference D:



I (q)  Es  exp(ikr )  Es  exp i kr  D

2
 Es  exp(ikr )  1  exp  iD   
2
2
2
1
1

 I s  1  exp  iD   1  exp  iD    I s   exp iqr ij

Scattered intensity due to Z pair-wise intraparticular interferences, N particles:
Z

Z

I (q)  Nb  exp iq r i  r j   Nb2


i 1 j 1
2
Z
Z
 exp iqr
i 1 j 1
ij


orientational average and normalization lead to:
Z Z
1
P(q) 
I  q   1 2  exp iqr ij 
2 2
Z i 1 j 1
NZ b

1

Z Z
 sin  qrij  
1
1 q 2 rij 2  ...

2  
2  1 
6
Z i 1 j 1  qrij 
Z i 1 j 1


Z

Z

replacing Cartesian coordinates ri by center-of-mass coordinates si we get:
Z
Z
 r
i 1 j 1
ij
2
Z
Z

  s j  si
i 1 j 1
    s
2
P(q)  1  1 s 2 q 2  ...
3
Z
Z
i 1 j 1
2
j
 2si s j  si
2
  2Z s
2 2
s2, Rg2 = squared radius of gyration
.
regarding the reciprocal scattered intensity, and including solute-solvent
interactions
Zimm equation:
finally yields the well-known Zimm-Equation (series expansion of P(q), valid for small R):
Kc
R
 1
MP(q)
 2 A2c  ...
Kc
R
1
(
1  1 s 2 q 2 )  2 A2c
M
3
The Zimm-Plot, leading to M, s (= Rg) and A2:
Kc
R
1
(
1  1 s 2 q 2 )  2 A2c
M
3
6,0
example: 5 c, 25 q
5,5
c=0
5,0
4,0
3,5
-7
Kc/R / 10 mol/g
4,5
3,0
2,5
2,0
q=0
1,5
1,0
0,0
5,0
10,0
2
15,0
10
-2
(q +kc) / 10 cm
20,0
Zimm analysis of polydisperse samples yields the following averages:
The weight average molar mass
K
Mw 
N M
k 1
K
k
k
Mk
N M
k 1
k
k
The z-average squared radius of gyration:
K
 s 2  z  Rg
2
z

 N k M k sk
k 1
K
2
 Nk M k
2
2
k 1
Reason: for given species k, Ik ~ NkMk2
Fractal Dimensions
M (R) : R
df
if
q  Rg
1
I (q) : M 2 : q
- 2d f
log I  q   2d f  log q
log P  q   log  I  q   cM    d f  log q
:
topology
df
cylinder, rod
1
thin disk
2
homogeneous sphere
3
ideal Gaussian coil
2
Gaussian coil with excluded volume
5/3
branched Gaussian chain
16/7
Particle form factor for “large” particles
1
P( q ) 
I q  1 2
2 2
Z
NZ b
 exp  iqr 
Z
Z
ij
i 1 j 1
 1
 sin  qrij  


Z 2 


i 1 j 1
 qrij 
Z
Z
for homogeneous spherical particles of radius R:
2
9
P(q ) 
sin
qR

qR
cos
qR






6
 qR 
P(q)
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
first minimum at qR = 4.49
Zimm!
0
2
4
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)
Gs (r , )  n(0, t )n(r , t   ) V ,T

isotropic diffusive particle motion
Fs (q, )   Gs (r , ) exp(iqr )d r
Gs (r , )  [2
3
3
 DR( ) ]
2
2
3r ( )2
exp(
)
2  DR( )2 
mean-squared displacement of the scattering particle:
DR  
2
 6Ds
Ds 
kT
kT

f
6h RH
Stokes-Einstein,
Fluctuation - Dissipation
I(t)
<I(t)I(t+)>T
The Dynamic Light Scattering Experiment - photon correlation spectroscopy
1
I t  I t   
exp  2  ,
  Ds q 2
2
t

Siegert relation:
Fs (q, )  exp( Ds q 2 )  Es (q, t ) Es *(q, t   ) 
note: usually the “coherence factor” fc is
smaller than 1, i.e.:
 I ( q, t ) I ( q, t   ) 
I  q, t 
2
 I (q, t ) I (q, t   ) 
I  q, t 
2
1
 1  f c  Fs (q, )
2
fc increases with decreasing pinhole diameter, but photon count rate decreases!
DLS from polydisperse (bimodal) samples

Fs  q,    P  Ds  exp  q 2 Ds dDs
0
log Fs(q,)
Fs(q,)
Fs  q,   A1  exp  q 2 Ds1   A2  exp  q 2 Ds 2 

log 
Data analysis for polydisperse (monomodal) samples
”Cumulant method“, series expansion, only valid for small size polydispersities < 50 %
ln Fs  q,   1 
1
1
 2 2   3 3  ...
2!
3!
first Cumulant 1  Ds q²
second Cumulant  2 
D
 Ds
2
s
2
q
4
yields polydispersity  D 
for samples with average particle size larger than 10 nm:
n  M  P q  D

q 
 n  M  P q
2
Dapp
i
i
Dapp  q   Ds
i
i
note:
2
i
z
i
1  K
i
Rg
RH
yields inverse average hydrodynamic radius
2
z
q2 

Ii  q 
ni  M i  Pi  q 
2
D
2
s
 Ds
Ds
1
2

2
12
Cumulant analysis – graphic explanation:
polydisperse sample
log(Fs(q,)
log(Fs(q,)
monodisperse sample
Dy/Dx=-Dsq
Dy/Dx=-Dsq
2
large, slow
particles
2
small, fast
particles


linear slope yields diffusion coefficient
slope at =0 yields apparent diffusion
coefficient, which is an average weighted
with niMi2Pi(q)
2,0x10
-14
1,5x10
-14
1,0x10
-14
5,0x10
-15
D/m s
2 -1
Dapp vs. q2:
Ds
z
0,0
0
1x10
10
2x10
2
10
-2
q /cm
3x10
10
4x10
10
Explanation for Dapp(q):
2
for larger particle fraction i, P(q) drops first,
leading to an increase of the average Dapp(q)
 ni  M i  Pi  q 
2
q1
10
P(q)
Dapp  q  
 ni  M i  Pi  q   Di
q2
0
10
-1
10
-2
10
-3
10
-4
R = 60 nm
R = 80 nm
R = 100 nm
-5
10
0,00
0,01
0,02
-1
q [nm ]
0,03
0,04
ln(g1())=P1+P2*+P3/2
*^2
50
PI = SQRT(P3/P2^2)
20
Ni
40
Ni
15
P(Ri)
P(Ri)
30
20
10
5
10
0
0
0
5
10
15
20
0
5
Ri/nm
10
15
20
Ri/nm
0,00
lng1
Data: Data2_lng1
Model: cumulant
Chi^2 = 3.7224E-8
P1
0.00882
±0.00003
P2
-10790.57918 ±0.23957
P3
896471.16145 ±926.09523
ln g1(90°,)
Data: Data2_lng1
Model: cumulant
Chi^2 = 3.3258E-10
P1
0.0079 ±5.5823E-6
P2
-8423.55623 ±0.25513
P3
2723184.05649 ±4894.69843
lng1
-0,04
-0,06
ln g1(90°,)
0,0
-0,02
-0,08
-0,10
-0,12
-0,14
-0,16
2
Dapp(90°)=2.04e-11 m /s, entspr. R = 10.5 nm
PI = 0.09, DR/R=10% (Normalvert.)
-0,2
0,00000
-0,18
0,00002
/s
-0,20
0,000000
2
Dapp(90°)=1.59e-11 m /s, entspr. R = 13.5 nm
PI = 0.20, DR/R=30% (Normalvert.)
0,000004
0,000008
0,000012
/s
0,000016
0,000020
27
100
Ni
80
P(Ri)
60
40
20
0
0
5
10
15
20
25
30
35
40
2,80E-008
1,20E-011
2,78E-008
2,76E-008
1,00E-011
2,74E-008
2,72E-008
8,00E-012
RH/m
Dapp/m s
2 -1
Ri/nm
2,70E-008
2,68E-008
2,66E-008
6,00E-012
2,64E-008
2,62E-008
4,00E-012
0,0001
0,0002
0,0003
0,0004
2
0,0005
q /nm
-2
0,0006
0,0007
0,0008
2,60E-008
0,0001
0,0002
0,0003
0,0004
2
0,0005
-2
q /nm
0,0006
0,0007
28
0,0008
100
Ni
80
P(Ri)
60
40
20
0
0
50
100
150
200
250
300
2,00E-007
2,00E-012
Dapp/m s
2 -1
Ri/nm
1,80E-007
1,50E-012
RH/m
1,60E-007
1,00E-012
1,40E-007
1,20E-007
5,00E-013
0,0001
0,0002
0,0003
0,0004
2
0,0005
q /nm
-2
0,0006
0,0007
0,0008
1,00E-007
0,0001
0,0002
0,0003
0,0004
2
0,0005
-2
q /nm
0,0006
0,0007
29
0,0008
20
Ni
P(Ri)
15
10
5
0
0
200
400
600
800
1000
Ri/nm
5,70E-007
4,40E-013
5,65E-007
5,60E-007
5,55E-007
RH/m
Dapp/m s
2 -1
4,20E-013
4,00E-013
5,50E-007
5,45E-007
5,40E-007
3,80E-013
5,35E-007
5,30E-007
3,60E-013
5,25E-007
0,0001
0,0002
0,0003
0,0004
2
0,0005
-2
q /nm
0,0006
0,0007
0,0008
5,20E-007
0,0001
0,0002
0,0003
0,0004
2
0,0005
-2
q /nm
0,0006
0,0007
30
0,0008
Combining static and dynamic light scattering, the r-ratio:
r
Rg
RH
topology
r-ratio
homogeneous sphere
0.775
hollow sphere
1
ellipsoid
0.775 - 4
random polymer coil
1.505
1
 l

 ln   0.5 
3
D

cylinder of length l,
diameter D
for polydisperse samples:
r
Rg
2
Z

 RH
1

Z
Strategy for particle characterization by light scattering - A
Sample topology (sphere, coil, etc…) is known
yes
no
Static light scattering necessary
Dynamic light scattering
sufficient (“particle sizing“)
Time intensity correlation function
decays single-exponentially
yes
Only one scattering angle needed,
determine particle size (RH) from
Stokes-Einstein-Eq.
(in case there are no particle
interactions (polyelectrolytes!)
no
Sample is polydisperse or shows non-diffusive
relaxation processes!
- to determine “true” average particle size,
extrapolation q -> 0
- to analyze polydispersity, various methods
Applicability of commercial particle sizers!
Strategy for particle characterization by light scattering - B
Sample topology is unknown,
static light scattering necessary
Plot of
2
Kc
vs. q is linear
R
yes
no
Particle radius between 10 and 50 nm:
analyze data following Zimm-eq. to get:
MW
Rg Mz
w
Particle radius larger than 50 nm
and/or very polydisperse sample:
use more sophisticated methods to
evaluate particle form factor
A2
Dynamic light scattering to determine RH 
Estimate (!) particle topology from r 
Rg
RH
R
H
1
z

1
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
10-6 Mw
(g/mol)
Rg
(nm)
104 A2
[(mol cm3)/g2]
LD11
0.92
36
1.00
LD16
1.87
48
0.60
LD12
5.20
70
0.28
LD19
14.5
113
0.13
LD18
43
180
0.082
LD17
64
190
0.060
LD13
97
233
0.025
Normalized particle form factors
universal up to values of qRg = 2
Details at higher q (smaller length scales) – Kratky Plot:
C
form factor fits:
P q 

1 C
qR 
3 
2
g
2
  1  C  
1

qR


g
 

6 
 

2
C related to branching probability,
increases with molar mass
Are the starch samples, although not self-similar, fractal objects?
P q
q
d f
 log P  q   d f  log 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
Data Analysis:

  3 j1  qRo 
3
3 j1  qRi  
P q   3
R
 Ri

3  o
qR
qR
R

R
o
i


i 
 o
2
monodisperse vesicles
Ro  R  t 2
j1  x  
Ri  R  t 2
thin-shell approximation
 sin  qR  
P q  

qR


small values of qR, Guinier approximation
sin x cos x

2
x
x
2

P  q   exp q 2 Rg 3
2

3 2 1   Ri Ro 
2
Rg  Ro
3
5
1   Ri Ro 
5
2
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)
 sin  qu  
P  q, a, b    
 dx
qu 
0 
1
2
u  a 2 x2  b2 1  x2 

x  cos
k
k0
prolate vesicles, surface area 4  (60 nm)2
oblate vesicles, surface area 4  (60 nm)2
anisotropy vs. polydispersity:
monodisperse ellipsoidal vesicles
 sin  qR  
P  q, a, b    G  R  
 dR
 qR 
a
2
b
G  R 
1
R
a 2  b2
R 2  b2
polydisperse spherical vesicles

 M R P  q, R  G  R  dR
2
P q 
0

 M R G  R  dR
2
 sin  qR  
P  q, R   

qR


2
0
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
input for a,b – fits
to SLS data
,
result:
polydisperse (DR = 10%) oblate vesicles,
a : b < 1 : 2.5
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
2
P q 
qL
qL

0
sin  ql 
 sin  qL 2  
dql  4 

ql
qL


 k  1 Lw 
p  L  
asymmetric Schulz-Zimm distribution

polydisperse stiff rods
P q 
 L  p  L  P  q dL
0
k 1
2
Lk exp    k  1 L Lw 
  k  1
k  1  M w M n  1

 L  p  L dL
0
Koyama, flexible wormlike chains P  q  
1
lK
2
 1 2
 sin  q ' xg  x  
0  lK  x  exp   3 q '  xf  x   q ' xg  x  dx
lK
Holtzer-plot of SLS-data :
q
R
 q  I  q  vs. q
Kc
plateau value = mass per length of a rod-like scattering particle
fit results:
(i) polydisperse stiff rods:
Lw  389 nm, M w M n  1.2
(ii) polydisperse wormlike chains:
Lw  380 nm, M w M n  2.0, lK  410 nm
Analysis of DLS-results:
3


g1  q,    S n  2n, qL   exp  DT q 2   2n  2n  1  DR 
n 0
amplitudes depend on the length scale of the DLS experiment:
- long diffusion distances (qL < 4): only pure translational diffusion S0
- intermediate length scales (4 < qL < 15): all modes (n = 0, 1, 2) present
according to Kirkwood and Riseman:
DT 
L
ln   ,
3h L  d 
kT
DR 
polydispersity leads to an average amplitude correlation function!
9 DT
L2
DLS relaxation rates :
linear fit over the whole q-range: significant deviation from zero intercept,
additional relaxation processes or “higher modes” at higher q
results:
D z  4.0  0.4 102 nm2 s 1
RH  60  6 nm
Rg RH  2
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-translationcoupling 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
sample
s2
s3
s4
s5
s6
s7
nominal diameter/nm
19
54
91
19, 91
19, 54
54, 91
diameter ratio
-
-
-
4.8
2.8
1.7
Data analysis of polydisperse samples:
1. Cumulant method (CUM), polynomial series expansion:


ln f 0.5 g1    ln  f 0.5     
polydispersity index
PI   2
 2

   B   d   Dapp  q 2
2
2

0

2     
2
 B  d
2
0
particle diameter is a so-called harmonic z-average:
d 
 ni di
i
n d
i
(only for homogeneous spheres) M i
2
di
6
6
i
5
i
2. non-negatively least squares method (NNLS):
2
N
M


 2    g1  j    bi exp  i j 
j 1 
i 1

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):
See 2., decay rates chosen according to:

n1  1 exp  n

 max 
4. Provencher’s CONTIN algorithm:
2
2
 1 

     
d     LB   
2  g1  i    B     e


i 
i 
Numerical procedure to calculate a continuous decay rate distribution B(), also called
Inverse Laplace Transformation, enclosed in most commercial DLS setups.
2
5. double-exponential method (DE):
g1    b1e1  b2e2
Results:
sample
nominal diameter
s2
s3
s4
19 ± 1.5 54 ± 2.7 91 ± 3
s5
s6
s7
19, 91
19, 54
54, 91
diameter ratio
-
-
-
4.8
2.8
1.7
<d> - CUM (1.)
20.3
55.0
87.0
36.9
29.5
69.0
PI – CUM (1.)
0.029
0.009
0.008
0.248
0.191
0.069
d1,d2 – NNLS (2.)
-
-
-
18, 81
16, 50
-
d1,d2 – ES (3.)
-
-
-
-
19, 54
-
d1,d2 – DE (5.)
-
-
-
-
18, 54
-
Bimodal samples s5, s6, s7: I1(q=0) = I2(q=0)
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 L
[nm]
DL
[nm]
diameter d
[nm]
Dd
[nm]
aspect ratio L/d
Sphere18
18
5
-
-
-
Sphere15
15
3
-
-
-
Rod2.6a
47
17
18
3
2.6
Rod2.6b
39
10
15
3
2.6
Rod8.9
146
37
17
3
8.9
Rod12.6
189
24
15
3
12.6
Rod14
283
22
20
3
14.0
Rod17.2
259
60
15
3
17.2
Rod17.4
279
68
16
3
17.4
Rod39
660
20
17
3
39.0
Rod49
729
-
15
3
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
g 2  q,   y     exp  b  q   c  q  2 
vv-mode (only translation is detected):
b  q   2DT q 2
depolarized dynamic light scattering (vh-mode)
(translation and rotation are detected, no coupling in case qL < 5)
translational diffusion coefficient DT determined from the slope,
rotational diffusion coefficient DR from the intercept
of the data measured in vh –geometry.
b  q   2DT q 2  12DR
Results:
qL < 5
q2 / 1014 m-2
qmaxL > 5 (≈ 9) !
q2 / 1014 m-2
diffusion coefficients according to Tirado and de la Torre,
using as input parameters length and diameter from TEM
2
kT   L 
d
d  
DT 
ln    0.312  0.565  0.100   
3h L   d 
L
 L  
3kT
DR 
h L3
2
 L
d
d  
ln    0.662  0.917  0.050   
L
 L  
  d 
system
10-12 DT, exp
[m2s-1]
10-12 DT, calc
[m2s-1]
DR, exp
[s-1]
DR, calc
[s-1]
Rod8.9
6.0
8.4
306
2238
Rod12.6
4.9
7.4
281
1258
Rod14
2.9
5.2
66
396
Rod17.2
4.0
6.0
177
563
Rod17.4
3.5
5.6
175
452
Rod39
1.2
2.9
14
46
Rod49
0.7
2.8
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
problem
solution
bimodal spheres
size resolution
- double exponential fits
- size distribution fits
- CONTIN ; only if R1/R2 > 2
stiff gold nanorods
length and diameter in solution =?; depolarized DLS (vh) => Drot
standard DLS (vv) => Dtrans;
deviation TEM – DLS ?
deviation TEM-DLS due to
PVP stabilization layer