# Folie plot ratio by alicejenny

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```									Light Scattering – what you learned so far:

m

E

Es

                    c   
  2 x         2    
E  x, t   E0  sin     sin         
   
                 t
        

                        
Static versus Dynamic Light Scattering

from time-independent intraparticle interferences you determine:
average scattered intensity => particle form factor P(q), radius of gyration Rg,
molar mass M (, A2)

from time-dependent interparticle interferences you determine:
time-intensity correlation function => hydrodynamic radius RH
The scattering vector q (in [cm-1]) , (inverse) length scale of light scattering:

k0                      
q
k
4 nD sin( )
q            2


q
The famous Zimm-Equation (series expansion of P(q), valid for 10 nm < R < 50 nm:

I q
1
Kc
R
1
M
(1  13 s 2 q 2 )  2 A2c
6,0

5,5

5,0

4,5
Kc/R / 10 mol/g

4,0

3,5
-7

3,0

2,5

2,0

1,5

1,0
0,0            5,0                10,0        15,0   20,0
2          10    -2
(q +kc) / 10 cm
Averages determined by SLS (Zimm-Plot) from polydisperse samples:
K                                                              K

 Nk M k M k                                                  N M
2        2
k       k       sk
Mw    k 1
 s  z  Rg              k 1
2         2
K                                                            K

 Nk M k                                                     N M
z                                2
k       k
k 1                                                        k 1

Particle form factor for “large” (> 200 nm) particles, e.g. spheres:

0
10                                          first minimum at qR = 4.49
Zimm!
-1
10
-2
10
P(q)

-3
10
-4
10
-5
10
0           2    4                  6                           8   10   12
qR
The concept of fractal dimensions – analysis of P(q) for qR > 1

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
Dynamic Light Scattering –
time-intensity-correlation function, Stokes-Einstein-eq. and RH
 I ( q, t ) I ( q, t   ) 
g 2 (q, )  exp(2 Ds q 2 )  1 
I  q, t 
2

g1 (q, )  exp( Ds q 2 )  Es (q, t ) Es *(q, t   )  g 2  q,  1  Fs  q, 

kT     kT
Ds       
f   6 RH

q1          q2

DLS from polydisperse samples:
0
10

Fs  q,    P  Dapp  q   exp  q 2 Dapp  q  dD                 10
-1

0                                                                        R = 60 nm
-2
10         R = 80 nm
P(q)

1          1                                                R = 100 nm
ln Fs  q,   1         2 2   3 3  ...                         10
-3
2!         3!

                          
-4
10
Dapp  q   Ds       1  K Rg           q2          1  q 2
2
z                  z                                       -5
10
0,00   0,01       0,02            0,03        0,04
-1
q [nm ]
Combining static and dynamic light scattering, the r-ratio:

Rg
r
RH

topology               r-ratio
homogeneous sphere            0.775
hollow sphere               1
ellipsoid            0.775 - 4
random polymer coil          1.505
cylinder of length l,
diameter D
Selected Examples – Dynamic Light Scattering:
sample                problem                                 solution

bimodal spheres       size resolution                         - double exponential fits
- size distribution fits
- CONTIN ; only if R1/R2 > 2
branched polymeric    besides “simple“ characterization       Dapp vs. q for different
nanoparticles         (Rg,M(Zimm),RH),branchg.degree          fractions => all identical !,
(q-dep. of Dapp: 1.polydisp. 2.local    comparison with known
relaxation (segments!))                 cases (chain, sphere)
DNA as poly-          polyelectrolyte effect, interparticle   add salt;
electrolyte and rod   interactions;                           analyze D by different
rod length =?                           models (Broersma,
stiff gold nanorods   length and diameter in solution =?; depolarized DLS (vh) => Drot
deviation TEM – DLS ?               standard DLS (vv) => Dtrans;
deviation TEM-DLS due to
PVP stabilization layer
spherical nanogels    light absorption, local heating,        convection quantified from
cont. gold clusters   q-dep. oscillations in g2(q,)          oscillations in g2(q,) vs. q
Selected Examples – Static Light Scattering:
sample                 problem                            solution

segmented              polydispersity in length and       (A) SLS-Zimm-plots for 3
copolymers             composition =>                     different polymer-solvent-pairs
by Zimm-plot only Mw,app           (B) DLS for 2 different polymer-
solvent-pairs (P(D) => P(M)) !
branched               1. self-similarity (fractals) ?;   1. details at qR > 2 by Kratky plot
polymeric              2. P(q) from SLS of semi-dilute    (P(q) q2 vs. q), fitting parameters
nanoparticles          solutions ?                        for branched polymers,
simulation of P(q) at qR > 10
(SLS: qR < 10) => not fractal !
2. renormalization of I(q,c) with
I(q=0,c), and q with Rg,app
vesicles               distinguish size polydispersity    combine DLS (only size
(nanocapsules)         and shape anisotropy in P(q) ?     polydispersity !) and SLS to
simulate expt. P(q)
worm-like micelles     characterization: length, Rg/RH    details at higher q by Holtzer plot
(RH: no rotation-translation-      (I(q) q vs. q), fit P(q), Rg from
coupling if qL < 4)                Zimm-analysis at small q values
PNIPAM chains in       coil – globule - transition        Rg from Zimm-analysis, RH by
water at different T                                      DLS, decrease in R and Rg / RH
4. Non-Standard Light Scattering Techniques

4.1. Single Angle Scattering Using Goniometer Setups

Laser

components: 1. laser = light source (coherent, focussed, monochromatic, polarized)
2. goniometer with motor, step-wise adjustment of scattering angle
3. sample bath (thermostate, index match) and cylindrical sample cell
4. detector (photomultiplier or photodiode)
5. computer with A/D-converter and hardware correlator
Goniometer setup of the Schmidt group:

+:       accurate adjustment of scattering angles
+/- :    correlation time scale: 100 ns - < 10 s
-:       scattering angle range: 30° - 150°
-:       series of angular-dependent measurements: long time
-:       DLS for low scattering contrast/slow processes: long time
-:       transparent, purified and highly dilute samples needed
-:       difficulties in analyzing polydisperse samples

technical solutions:
time scale enhancement: simultaneous measurement at multiple q
q-scale enhancement:       optical lenses + area detectors for  < 1° !
turbid samples:            cross correlation or backscattering
polydisperse samples:      combine fractionation and SLS
4.2. Simultaneous Multiangle Scattering (MALS):

Laser

Interface
MALS - setup of the Schmidt group:
Simultaneous Multiangle Scattering combined with GPC or FFF:

Laser

GPC

Interface

flow cell
GPC MALS - setup of the Schmidt group:
Wintermantel, M.;Gerle, M.;Fischer, K.;Schmidt, M.;Wataoka, I.;Urakawa, H.;
Kajiwara, K.;Tsukahara, Y. Macromolecules 1996, 29, 978-983

samples:
polystyrene polymacromonomers, synthesized by radical polymerization of anionic
MMA end-functionalized polystyrene macromonomers (“bottle-brushes”), MW /MN > 2 !

GPC-MALS:
- GPC connected to an
on-line Knauer combined viscometer/RI-detector and an
ALV1800 MALS (19 angles plus one monitor channel)
- home-made cylindrical flow cell with 38 µL total volume
- scattering intensity detected at 19 scattering angles over 2 s at DT = 4 s during elution

wormlike chain model

2     3       4
L                       
Rg  k  k  k  k 2 1  exp  2 L  
2LL L    L
        
6   4 4 L 8L                 Lk  

contour length L, Kuhn length Lk
reduced scattered intensity plotted versus q2 for three sample fractions :
Rg  100.6 nm, M w  1.01107 g mol ,
c  6.22 103 g L

Rg  47.7 nm, M w  3.77 106 g mol ,
c  4.21102 g L

Rg  14.3 nm, M w  7.61105 g mol ,
c  4.14 102 g L
Rg vs. MW for a “homologous series of bottlebrushes”, wormlike chain model :

results:
M(side chains) = 2900 g/mol: Lk = 89 nm
M(side chains) = 5000 g/mol: Lk = 208 nm
4.3. Turbid Samples – (A) Fiber-Optic Quasielastic Light Scattering (FOQELS)

Wiese, H.;Horn, D. Journal of Chemical Physics 1991, 94, 6429-6443 (BASF !)

suppresses multiple scattering, defines the scattering angle as 180°
samples:
aqueous polymer latex dispersions (particle size from 41 nm to 326 nm),
as prepared without further purification, particle concentrations above 1 wt% !
data analysis:

concentrated dispersions:
amplitude autocorrelation function includes
contributions of the static structure factor
(interparticle interferences!)
Fs  q, 
g1   
Fs  q,  0 

g1   exp   Dc q 2 

collective diffusion coefficient
c

Dc  Dc  q, c 
Dc vs. concentration, different particle sizes :
position of the maximum of the structure factor compared to q of FOQELS

q < qm :
length scale > particle dist.,
interparticle interactions
=> collective diffusion
Dc(c) ↑

q

q > qm :
length scale < particle dist.
=> selfdiffusion
Ds(c) ↓
4.4. Turbid Samples - (B) Crosscorrelation Techniques - Dual Color and
3d Dynamic Light Scattering (identical q and scattering volume,
different set of interferences => no multiple scattering)

I. 3d – DLS (Prof. Schurtenberger, Univ.Fribourg, CH)
z
Laser
D1                 y
d

q1  q 2
D2
L1            L2
d  18
k 0,1
k0
d
k 0,2 x
d        q1
k1
g2,cross  q,   I1  q, t   I 2  q, t                     k2
q2

www.lsinstruments.ch/3DDLS.htm
II. Dual Color Dynamic Light Scattering

Stieber, F.;Richtering, W. Langmuir 1995, 11, 4724-4727

sample:
polystyrene latex spheres 2R = 82 nm (q < qm)

light scattering setup:
dual color or two color crosscorrelation (TCC) setup:
- 2 light sources with different wavelengths from 1 argon ion laser in multiline mode
(488 nm and 514.5 nm)
- single-mode optical fibers for the optical alignment
- scattering angles from 15° to 140° (FOQELS: 180° only !)

data analysis:
CONTIN (constrained inverse Laplace transformation)

note: q < qm => at higher particle conc. only Dc
comparison of auto- and crosscorrelation:

filled symbols:   autocorrelation, artefacts and bad resolution (multiple scattering)
open symbols:     crosscorrelation, 2 defined relaxation processes

multiple scattering increases with decreasing q (increasing length scale)
the origin of Dslow ?

=> slow process Dslow is selfdiffusion (strong slowing-down with increasing conc.)

Ac S  q 
sample polydispersity enhances selfcorrelation:         :             Da a
As   2
4.5. Dynamic Light Scattering using CCD area detectors

Laser

CCD

I.   concave lens + CCD area detector (multiple q at once),
simultaneous light scattering at very small scattering angles
improves: measurement time, small-q-scale
Laser

CCD

II.   CCD area detector at one defined q ± Dq
DLS: replace time averaging by ensemble averaging
improves: measurement time, long--scale
Wong, A. P. Y.;Wiltzius, P. Rev.Sci.Instrum. 1993, 64, 2547-2549

sample:
commercial latex particles (diameter 215 nm) in glycerol at T = 51°C,
rectangular cuvette of thickness 1 mm
setup:

detection scheme:
digitized image, 500 x 450 pixel,
10 concentric rings:
radius = 20, 40, ...., 200 pixel
no. of pixels = 80, 160, .... , 800
-range: 10° - 60°, simultaneous
calculation of the intensity correlation function:

average speckle intensity of each ring:         I  q, t    q0

deviation from the average intensity for each pixel:              d I  q, t   I  q , t   I  q , t    q0

from two pictures taken at t = 0 and t = :   g 2  q,    d I  q, 0  d I  q ,      q0
(min = 100 ms)
Note: <> = ensemble-average!

conventional single-angle light scattering: <> = time-average!
g2  q,   I  q, t  I  q, t      t
results:

q = 29748 cm-1, 20834 cm-1 and 14832 cm-1
goniometer setup

relaxation time G-1 > 20 s !

Ds = 1.67e-10 cm2s-1 ~ 1 e-14 m2s-1 :
Stokes-Einstein-equation, viscosity of 0.137 Pa s => R = 107.5 nm
Kirsch, S.;Frenz, V.;Schartl, W.;Bartsch, E.;Sillescu, H.
Journal of Chemical Physics 1996, 104, 1758-1761
“multi speckle correlation spectroscopy” (MSCS)
sample:
spherical latex particles (R = 350) nm in glycerol at T = 10°C !

setup:

data acquisition:
images of 512x256 pixels digitized,
scattered intensity for 50 random speckles (= 2x2 pixel) stored at frame rate 0.33 s
calculation of the intensity correlation function:
data represented 50 individual traces of I(q,t) at (nearly) identical q

1. normalized intensity correlation function for each speckle calculated
by time-averaging
N pic  k                                   N pic  k            N pic  k

N     pic   k     I  nDt  I   n  k  Dt    I  nDt   I   n  k  Dt 
I  t  I  t  k Dt     T , norm
                        n 0
N pic  k           N pic  k
n 0                 n 0

 I  nDt   I   n  k  Dt 
n 0                n 0

total number of pictures Npic: 50.000 - 150.000, n = 0, 1, 2, ….., k = 0, 1, 2, …..

2. time-averaged correlation functions are ensemble averaged over all speckles

N sp

 I  t  I  t  k Dt        T , norm , a
I  t  I  t  k Dt    T , E , norm
   a 1
:   g 2  q, K Dt        T ,E
1
N sp

total number of speckles Nsp: 20 - 50
comparison MSCS – conventional DLS (PCS):

MSCS results (symbols) for 20 speckles, 150.000 pictures (texpt = 50.000 s)
PCS results (lines): texpt = 70.000 s
inset, symbols: filled = 1 speckle correlation, open = 20 speckle ensemble average

improved statistics by combined time-ensemble-averaging !

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