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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,
                                                              Tirado/de la Torre etc.)
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:
Single Angle Scattering – advantages and disadvantages:



 +:       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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posted:10/17/2012
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