### Folie 1

```Seminar PCF
“Lightscattering”
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


Wave-equation of oscillating electic field
of the incident light:
Molecule constitutes the emitter of an electromagnetic wave of the same wavelength
as the incident one (“elastic scattering”)
m
E
Es
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 ~ NiMi2Pi(q) (scattering vector q, see below!)
Particles smaller than /20 (left picture):
- 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
 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 <S(q)>, Dynamic Lightscattering S(q,t) (DLS))
2. Lichtstreuung – experimenteller Aufbau
Scattered light wave emitted by one oscillating dipole:


 2m  1
4 2 2 E0
Es   2  2 
exp i 2 t  kr D
2

t
r
c
r
c

 D
D
Detector (photomultiplier, photodiode): scattered intensity only!
I0

I s  Es Es  Es
2
sample

rD
I
detector
Light source I0 = laser: focussed, monochromatic, coherent
Coherent: the light has a defined oscillation phase over a certain distance (0.5 – 1 m)
and time so it can show interference. Note that only laser light is coherent in time, so:
No laser => no dynamic light scattering!
Sample cell: cylindrical quartz cuvette, embedded in toluene bath (T, nD)
Light Scattering Setup of the F-Practical Course, Phys.Chem., Mainz:
Scattering volume:
defined by intersection of incident beam and optical aperture of the detection optics,
varies with scattering angle !

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)
4 2
2 n
b  4 nD,0 ( D )2  K
c
0 N L
2
contrast factor:
in cm2g-2Mol
Absolute scattered intensity of ideal solutions, Rayleigh ratio ([cm-1]):
2
r
R  b  c  M  ( I solution  I solvent ) D
V
2
For calibration of the setup one uses a scattering standard,
Istd: Toluene ( Iabs = 1.4 e-5 cm-1 )
R   I solution  I solvent  
I std ,abs
I std
Reason of “Sky Blue”! (scattering from gas molecules of atmosphere)
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
k0

k
q  k  k0
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
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
For large (ca. 500 nm) homogeneous spheres :
P( q ) 
9
 qR 
6
sin  qR   qR cos  qR 
2
0
P(q)
10
10
-1
10
-2
10
-3
10
-4
10
-5
Minimum bei qR = 4.49
0
2
4
6
qR
8
10
12
Two different types of Polystyrene nanospheres (R = 130 nm und R > 260 nm)
are investigated in the practical course!
1.00E+00
P(q) 130 nm
P(q) 260 nm
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
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

Fs (q, )   Gs (r , ) exp(iqr )d r
mean-squared displacement of the scattering particle:
DR  
2
 6 Ds
Ds 
kT
kT

f
6h RH
Stokes-Einstein-Gl.
The Dynamic Light Scattering Experiment - photon correlation spectroscopy
( in DLS, one measures the intensity correlation <I(t) · I(t+)> !)
I(t)
<I(t)I(t+)>T
(note: in static light scattering, you measure the average scattered intensity
<I(q,t)> (see dashed line left graph!))
1
2
Basislinie : I  t 
t

Siegert-Relation:
Fs (q, )  exp( Ds q 2 )  Es (q, t ) Es *(q, t   ) 
2
 I ( q, t ) I ( q, t   ) 
I  q, t 
2
1
Data analysis for polydisperse (monomodal) samples
”Cumulant-Method“:
for polydisperse samples Fs(q,) is a superposition of various exponentials
Note the weighting factor “Ni Mi2 Pi(q)“ which is the average static scattered intensity
per sample faction ! Taylor series expansion of this superposition leads to:
ln Fs  q,   1 
1
1
 2 2   3 3  ...
2!
3!
1st cumulant: 1  Ds q² yields the average apparent diffusion coefficient
2nd

2
cumulant:  2  Ds  Ds
2
q
4
is a measure for sample polydispersity
Important:
For polydisperse samples of particles > 10 nm, the apparent diffusion coefficient
Is q-dependent due to the weighting-factor P(q) !!!
n  M  P q  D

q

 
 n  M  P q
2
Dapp
i
i
i
i
2
i
i
i
 Ds
z
1  K
Rg
2
z
q2 

q→0 : Dapp is the z-average diffusion coefficient, since all Pi(q) = 1 !
Cumulant analysis – graphic explanation:
Monodisperse sample
log(Fs(q,)
log(Fs(q,)
Polydisperse sample
Dy/Dx=-Dsq
Dy/Dx=-Dsq
2
larger,
slower 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
Z-average diffusion coefficient is determined by interpolation of Dapp vs. q2 -> 0
(straight line only for particles < 100 nm !!!)
Ds
z
0,0
0
1x10
10
2x10
2
10
q /cm
-2
3x10
10
4x10
10
Explanation for q-dependence of Dapp for larger particles due to Pi(q):
n  M  P q  D

q

 
 n  M  P q
2
Dapp
i
i
i
i
2
i
i
i
Note the minimum in P(q) for the larger particles,
where the average diffusion coefficient will reach
a maximum !!!
1.00E+00
P(q) 130 nm
P(q) 260 nm
1.00E-01
1.00E-02
1.00E-03
1.00E-04
1.00E-05
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
DLS of concentrated samples – influence of the static structure factor S(q):
Due to interparticle interactions, the particles not any longer move independently
by Brownian motion, only. Therefore, DLS in this case measures no self-diffusion
coefficient but a collective diffusion coefficient defined as Dc(q) = Ds/S(q):
S(q) from SAXS, particle radius ca. 80 nm, c = 200, 97 und 75 g/L, in water:
left: c(salt) = 0.5 mM,
right: c(salt) = 50 mM)
From: Gapinsky et al., J.Chem.Phys. 126, 104905 (2007)
D(q) from XPCS (Xray-correlation), particle radius 80 nm, c = 200, 97 und 75 g/L, in water:
left: c(salt) = 0.5 mM,
right: c(salt) = 50 mM)
From: Gapinsky et al., J.Chem.Phys. 126, 104905 (2007)
Note: 1. The q-regime of SAXS/XPCS is much larger than in light scattering due
to the shorter wave length of Xrays (lab course: 0.013 nm-1 < q < 0.026 nm-1 !!!)
2. The investigated Ludox particles R = 25 nm are much smaller, therefore the
maximum in S(q) is located at larger q (q(S(q)_max) > 0.1 nm-1 !!!)
```