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					            Dynamic Light Scattering
                (aka QLS, PCS)
Oriented particles create interference patterns, each bright spot being a
speckle. The speckle pattern moves as the particle move, creating flickering.

All the motions and measurements are described by correlations functions
     • G2(!)- intensity correlation function describes particle motion
     • G1(!)- electric field correlation function describes measured
     fluctuations

    Which are related to connect the measurement and motion


                   G2 ( ! ) ) B '1 ( * g ( ! ) 2 $
                                %
                                &       1        "
                                                 #

Analysis Techniques:
    • Treatment for monomodal distributions: linear and cumulant fits
    • Treatment for non-monomodal distributions: Contin fits

It is also possible to measure other motions, such as rotation.
Particles behave like ‘slits’, the orientation
 of which generates interference patterns
                               Generates a ‘speckle’ pattern




                          Various points reflect different
                                 scattering angles


                                                               2
Movement of the particles cause fluctuations
              in the pattern

                                The pattern ‘fluctuates’




                          Movement is defined by the rate of
                                     fluctuation
Measure the intensity of one speckle


                 Experimentally, the intensity of one
                 ‘speckle’ is measured
Order of magnitude for time-scale of fluctuations
     fluctuations occur on the time-scale that particles move about one
     wavelength of light…

                    -x / .
     Assuming Brownian motion
     of the particles…               +-x ,2 ) Dt
     The time-scale is:
                                                    Change on the
                                 . ~ 500 nm        msec time frame



             t/
                   +5x10   05
                                cm   ,
                                     2
                                         / 100m sec
                          08     2
                  2.5 x10 cm / s
                                 D for ~ 200 nm particles
How is the time scale of the fluctuations
   related to the particle movement?

Requires several steps:

1. Measure fluctuations an convert
into an Intensity Correlation
Function
2. Describe the correlated movement
of the particles, as related to
particle size into an Electric-Field
Correlation Function.

3. Equate the correlation functions,
with the Seigert Relationship

4. Analyze data using cumulants or
CONTIN fitting routines
• Math/Theory          2 texts:

                       ‘Light scattering by Small Particles’
• Application/Optics   by van de Hulst

                       ‘Dynamic Light Scattering with
                       applications to Chemistry, Biology
• Data Analysis        and Physics’ by Berne and Pecora
First, the Intensity Correlation Function, G2(!)
Describes the rate of change in scattering intensity by comparing the intensity at
time t to the intensity at a later time (t + !), providing a quantitative measurement
of the flickering of the light


                                                  I(t)      I(t+!)      I(t+!’)
Mathematically, the correlation
function is written as an integral
over the product of intensities at
some time and with some delay
time, !


             1 T
G2 ( ! ) )     10 I ( t )I +t ( ! ,d!
             T


Which can be visualized as taking
the intensity at I(t) times the
intensity at I(t+!)- red), followed
by the same product at I(t+t’)-
blue, and so on…
           The Intensity Correlation Function has the
                 form of an exponential decay

                            plot linear in !
        1.8e+8
                                                                 The correlation function typically
        1.6e+8
                                                                  exhibits an exponential decay

        1.4e+8
G2(!)




        1.2e+8

                                                                                  plot logarithmic in !
        1.0e+8
                                                                     1.8e+8

        8.0e+7
                                                                     1.6e+8
                 0   1000     2000    3000     4000   5000

                               Tau (2sec)
                                                                     1.4e+8
                                                             G2(!)
                                                                     1.2e+8


                                                                     1.0e+8


                                                                     8.0e+7
                                                                              1       10         100    1000

                                                                                           Tau (2sec)
Second, Electric Field Correlation Function, G1(!)




 It is Not Possible to Know
 How Each Particle Moves
 from the Flickering



 Instead, we correlate the
 motion of the particles
 relative to each other
Integrate the difference in distance between
     particles, assuming Brownian Motion
The electric field correlation function describes correlated particle movement,
and is given as:

                                                                    Constructive
                                                                    interference
            1 T
G1( ! ) )     10 E( t )E +t ( ! ,d!
            T

G1(t) decays as and
exponential with a decay
constant 345for a system
undergoing Brownian motion



G1( ! ) ) exp 03!                                                    Destructive
                                                                    interference
The decay constant is re-written as a function
             of the particle size
  The decay constant is related by Brownian Motion to the diffusivity by:


                                              2                   46n    <= 9
                                   3 ) 0 Dq                  q)      sin : 7
                                                                   .     ;28
  with q2 reflecting the distance the particle travels … and the application of
  Stokes-Einstein equation


                     Boltzmann Constant

                                                    temperature

                                    kT   thermodynamic
                            D)         )
                                   662r hydrodynamic

                       viscosity              particle radius
              Rate of decay depends on the particle size

        1.0

        0.9

        0.8                                                                           large particles diffuse slower than
        0.7
                                                                                      small particles, and the correlation
        0.6
                                                                                      function decays at a slower rate.
G2(!)




        0.5

        0.4

        0.3

        0.2

        0.1

        0.0
                                                                                     1.0
               0.000   0.002   0.004         0.006   0.008   0.010   0.012
                                                                                     0.9
                                       Tau
                                                                                     0.8                                   large particle


           and the rate of
                                                                                     0.7

                                                                             G2(!)   0.6




            other motions
                                                                                     0.5

                                                                                     0.4     small particle


        (internal, rotation…)
                                                                                     0.3

                                                                                     0.2

                                                                                     0.1

                                                                                     0.0

                                                                                      1e-6       1e-5         1e-4         1e-3      1e-2   1e-1

                                                                                                                     Tau
Finally, the two correlation function can be
   equated using the Seigert Relationship
      Based on the principle of Guassian random
   processes – which the scattering light usually is
                                                                       2
 The Seigert Relationship is expressed as:          Intensity I ) E ) E > E *


                      G2 ( ! ) ) B '1 ( * g ( ! ) 2 $
                                   %
                                   &       1        "
                                                    #

    Intensity Correlation
                                                 Electric Field Correlation
    Function
                                                 Function
    (recall: this is measured)
                                                 (recall: this is what the
                                                 particles are doing)
where B is the baseline and * is an instrumental response, both of which are
constants
• G2(!) intensity correlation function measures
change in the scattering intensity

• G1(!) electric field correlation function describes
correlated particle movements

• The Seigert Relationship equates the functions
connecting the measurable to the motions



             G2 ( ! ) ) B '1 ( * g ( ! ) 2 $
                          %
                          &       1        "
                                           #
• Math/Theory


• Application/Optics

• Data Analysis
So, consider a simple example of the process

    Measure the intensity fluctuations from a dispersion of particles.
              Commercial Equipment
• Need laser, optics, correlator, etc…

• Commercial Sources
   – Brookhaven Instruments
   – Malvern Instruments
   – Wyatt Instruments
      (multiangle measurements, HPLC detectors)
   – ALV (what we have)
• Costs range $50K to $100K
        Instrumental Considerations
• Light Source
  – Monochromatic, polarized and continuous (laser)
  – Static light scattering goes as 1/!4, suggests shorter
    wavelengths give more signal
     • typical Ar+ ion laser at 488 nm
  – Dynamic light scattering S/N goes as !, while detector
    sensitivity goes as 1/!, so wavelength is not too critical.
    HeNe lasers are cheap and compact, but weaker (! = 633 nm)
  – Power needed depends on sample (but there can be heating!)
  – Calculation of G(") depends on two photons, and so on the
    power/area in the cell. Typically focus the beam to about
    200 #m
  – Sample can be as small as 1 mm in diameter and 1 mm high.
    Typical volumes 3-5 ml.
             Instrument Considerations
• Need to avoid noise in the correlation functions
   – Dust!
      • Usually adds an unwanted (slow) component
      • See in analysis – some software help
      • AVOID by proper sample preparation when possible


   – Poisson Noise
      • counting noise, decreases with added counts, important to have
        enough counts; typically 107 over all with 106 at baseline


   – Stray light
      • adds an unwanted heterodyne component (exp (-$) instead of
        exp (-2$). Avoid with proper design
                          Correlators
                                    1
• Need to calculate G2 ( ! ) ) 10 I ( t )I +t ( ! ,d!
                                T
                                    T
                                                  N ( l arg e )
                                              1
which is approximated by            G2 (! ) /
                                              N
                                                     @ I (t ) I (t
                                                      i )1
                                                                  i   i   (! )

so calculate by recording I(t) and sequentially
  multiplying and adding the result. To do in real time
  requires about ns calculations thus specialized
  hardware

• Pike – 1970s (Royal Signals and Radar Establishment,
  Malvern, England)
• Langley and Ford (UMASS) ? Brookhaven
• 1980’s Klaus Schatzel, Kiel University ? ALV
               Autocorrelation function is collected


                                                      The auto-correlator collects and
                                                       integrates the intensity at the
                                                   different delay times, !, all in real time



                                                       ! (2sec)               G2+!,

Each point is a different !.                       2.000000000E+000     1.593461120E+008
                                                   2.400000095E+000     1.590897440E+008
                                                   5.000000000E+000     1.582029760E+008
             1.8e+8
                                                   1.000000000E+001     1.564198880E+008
                                                   1.500000000E+001     1.546673760E+008
             1.6e+8
                                                   2.000000000E+001     1.529991520E+008
             1.4e+8                                2.500000000E+001     1.513296000E+008
                                                   3.000000000E+001     1.497655360E+008
     G2(!)




             1.2e+8                                3.500000000E+001     1.482144000E+008
                                                   4.000000000E+001     1.466891040E+008
             1.0e+8
                                                   4.500000000E+001     1.452316800E+008
                                                   5.000000000E+001     1.438225120E+008
             8.0e+7
                      1   10         100    1000
                                                   …
                               Tau (2sec)

                                                   6.000000000E+00      4 9.100139200E+007
… then, create the raw correlation function

         Evaluate the autocorrelation function from the intensity data

         1.8e+8


         1.6e+8


         1.4e+8
 G2(!)




         1.2e+8


         1.0e+8


         8.0e+7
                   1            10          100          1000

                                     Tau (2sec)
… then, normalize the raw correlation function
    through some simple re-arrangements
          G (! ) 0 B
  C +! , ) 2         ) * e 0 2 3!
              B

                  0.8



                  0.6



                  0.4
           c(t)




                            *
                                * is usually less
                                than unity, from              B should
                  0.2
                                measuring more                go to zero
                                than one speckle
                                                          B
                  0.0

                        1            10             100       1000

                                             ! (2sec)
General principle: the measured decay is the
intensity-weighted sum of the decay of the
             individual particles

              1.0                      200 nm

                        100 nm                  300 nm
              0.8                                   400 nm

                                                         500 nm
              0.6
       c(t)




              0.4



              0.2



              0.0

                    1            10      100              1000    10000

                                      tau (2sec)

      Recall that different size particles exhibit different decay rates.
       Expressed in mathematical terms
g1(!) can be described as the movements from individual particles; where
G(3) is the intensity-weighted coefficient associated with the amount of
each particle.




                       g1 +! , ) @ Gi +3 ,e 03i!
                                 i



For example, consider a mixture of particles:

0.30 intensity-weighted of 100 nm particles,
0.25 intensity-weighted of 200 nm particles,
0.20 intensity-weighted of 300 nm particles,
0.15 intensity-weighted of 400 nm particles,
0.10 intensity-weighted of 500 nm particles.
  A sample correlation function would look
           something like this…
Short times
emphasize the
intensity weighted-                                                 Recall sizes
average                                                             0.30 (100 nm)
               1.0
                                                                    0.25 (200 nm)
                                     wieghted sum of the            0.20 (300 nm)
                                        individual decay            0.15 (400 nm)
               0.8
                                                                    0.10 (500 nm)

               0.6
        c(t)




               0.4
                   100 nm
                   200 nm
               0.2 300 nm

                   400 nm
                   500 nm
               0.0

                     1      10        100          1000     10000

                                 tau (2sec)                long times reflected
                                                           the larger particles
• Math/Theory

• Application/Optics


• Data Analysis
Finally, calculate the size from the decay
                  constant

3 = ??? in sec-1 (experimentally determined)


                                      3        Diffusivity is determined…
                                D) 2             need refractive index,
 D = ??? in cm2 sec-1
                                  q               wavelength and angle

                                                       46n<= 9
                                                 q)   sin : 7
                                                    .     ;28

     r = ??? in cm
                                               Calculate the radius, but
                                     kT
                                 r)            need the Boltzmann
                                    662D       constant, temperature and
                                               viscosity
            What is left?
Need a systematic way to determine 3’s

the distribution of particle sizes defines the approach to fitting
the decay constant


Monomodal Distribution                      Linear Fit

                                            Cumulant Expansion




Non-Monomodal Distribution                  Exponential Sampling

                                            CONTIN regularization
            What is left?
Need a systematic way to determine 3’s

First, consider the monomodal distribution, where the particles have
an average mean with a distribution about the mean (red box, first)



Monomodal Distribution                    Linear Fit

                                          Cumulant Expansion




Non-Monomodal Distribution                Exponential Sampling

                                          CONTIN regularization
                        Simplest- the ‘basic’ linear fit
         Assumes that all the particles fall about a relatively tight mean


                                                               G (! ) 0 B 9
         Take the logarithm of the                         ln< 2
                                                             :
                                                                                       2
                                                                          7 ) ln * 0 2q D!
         normalized correlation function                     ;    B       8


                           C(!)                                                         ln C(!)
                                                          ln *
       0.8


                                                                                       Slope = -2Dq2
       0.6                                                         1e-1



       0.4                                                         1e-2
C(!)




                                                            C(!)
                  … but, need Long ! for
       0.2               a good B                                  1e-3


       0.0
                                                                   1e-4
             0   1000   2000         3000   4000   5000                   0   1000   2000         3000   4000   5000

                               Tau                                                          Tau
                                                     Long ! there is just noise…
                        Cumulant expansion
Assumes that the particles distribution is centered on a mean, with a
Gaussian-like distribution about the mean.

Where to start…

               A                            Integral sum of decay curves
      g1 +! , ) 1 G +3 ,e 03! d3
               0

Larger particles are ‘seen’ more…

                                    Probability Density Function
                                     (Coefficients of Expansion)


                             G +3 , ) M 2 P( q )S ( q )

                                                         Intraparticle Form Factor
     G +3 , / N ( R ) R 6 : solid                      And Interparticle Form Factor
                                                          that both DEPEND ON q
    G +3 , / N ( R ) R 4 : hollow shell (vesicle)
Then, re-arrange the Seigert Relationship in
       terms of a cumulant expansion

 Recall that the correlation function can be expressed as


                                     G +! , 0 B $
                     ln c+! , ) ln ' 2
                                   %            " ) ln * 0 2 ln g1 +! ,
                                   &    B       #

 Cumulant expansion is a rigorous defined tool of re-writing a sum of exponential
 decay functions as a power series expansion… so, that the sum from the
 previous page is replaced by the expansion (GET BACK HERE IN A FEW
 MINUTES)


           A                                        A     +i! ,n ) Ak +i! ,n d!
                  i3 !
  g1 +! , ) 1 e          P+3 ,d3       ln g1 +! , B @ k n          1 n
            0                                      n )0     n!       0  n!


http://mathworld.wolfram.com/cumulant.html
… need to carry through some mathematics
First, define a mean value


                         g1 +! , ) e 0 3 ! e 0+3 0 3 ,!



                                         3       is the mean ‘gamma’


                                                 0x            x 2 x3
     Note: power series expansion            e        / 1 0 x ( 0 ( ...
                                                               2! 3!

Second, substitute the power series for the difference term (second term)

         A                   A               '
g1 +! , ) 1 G +3 ,e 03! d3 ) 1 G +3 ,e 0 3 ! %1 0 +3 0 3 ,! (
                                                              +3 0 3 ,2 ! 0 ...$ d3
                                                                               "
          0                  0               %
                                             &                   2!            "
                                                                               #
                 Cumulant Expansion (more)

         A                           A       '
g1 +! , ) 1 G +3 ,e 03! d3 ) e 0 3 ! 1 G +3 ,%1 0 +3 0 3 ,! (
                                                              +3 0 3 ,2 ! 0 ...$ d3
                                                                               "
          0                          0       %
                                             &                   2!            "
                                                                               #

 Working through the integrals…


                           0 3! <       k 2 2 2 k33 3      9
             g1 +! , ) e        :10 0 (      ! 0    ! ( ...7
                               :         2!      3!        7
                               ;                           8


   Such that k2 is the second moment, k3 is the third moment, …


             A                                       A
      k 2 ) 1 G +3 ,+3 0 3 , d3 2
                                               k3 ) 1 G +3 ,+3 0 3 ,3 d3
             0                                       0
       Cumulant Expansion (even more)


1 ' G2 +! , 0 B $ 1           ' 0 3! < k 2 2 k 3 3       9$
 ln %             ) ln * ( ln %e     :1 ( 2 ! 0 3 ! ( ...7"
2 &     B       " 2
                #                    :    2!    3!       7
                              %
                              &      ;                   8"#

                             a                b                x

 Note: ln of products                 ln( ab ) ) ln a ( ln b


 Note: power series expansion                       1 2
                                   ln( 1 ( x ) / x 0 x ( ...
                                                    2

       Note that x terms >> x2 terms, so that x2 are negligible
               Cumulant Expansion (more…)

                   G +! , 0 B $
              ln ' 2
                                                                     Note:
                 %            " ) ln * 0 2 3 ! ( K 2 2! 2 0 ...     Multiplied
                 &    B       #                                       by 2




                       intercept                   polydispersity
                                   average decay



                                    k2
Polydispersity index         C)
                                   32
… and indicates the width of the distribution


        C ) 0.005       is mono-dispersed
            Sample of Cumulant Expansion
390 nm Beads




                ' G +!, 0 B $                 Gamma
linear       ln % 2         " ) ln * 0 23 !
                &    B      #

                ' G +!, 0 B $                          ~ Poly
quadratic    ln % 2         " ) ln * 0 23 ! ( K22!2
                &    B      #
                                                                 ~ Skew
                                                           3
                ' G2 + ! , 0 B $                   2!2 0 K3 !3
cubic        ln %              " ) ln * 0 23 ! ( K2
               &     B     #                          3
                                                                          ~ Kurtosis

                ' G2 + ! , 0 B $                         K3 3 3 K4 4 4
quartic      ln %              " ) ln * 0 23 ! ( K22!2 0     ! (    !
                &      B       #                          3      12
Examine residuals to the fit



                      uncorrelated




                       correlated
            What is left?
Need a systematic way to determine 3’s

Second, consider the different non-monomodal distribution, where the
particles have a distribution no longer centered about the mean (red
box, next)


 Monomodal Distribution                  Linear Fit

                                         Cumulant Expansion




 Non-Monomodal Distribution              Exponential Sampling

                                         CONTIN regularization



Multiple modes because of polydispersity, internal modes,
interactions… all of what make the sample interesting!
Exponential Sampling for Bimodal Distribution


        g1 +! , ) @ ai e 03i!
                                Assume a finite number
                                of particles, each with
                                their own decay




                                  e.g. bimodal distribution




   To be reliable the sizes must be ~5X different
                         Pitfalls
• Correlation functions need to be measured properly




     a) Good measurements with appropriate delay times

     b) Incomplete, missing the early (fast) decays

     c) Incomplete, missing the long time (slow) decays
    CONTIN Fit for Random Distribution
Laplace Transform of f(t)
                                                    Note: Fourier Transform
                       A                                       A
                                              G +F, ) GDf t E )   0iFt + ,
 F +s , ) LD f +t ,E ) 1 e 0 st f +t ,dt                + ,       1e    f t dt
                        0                                         0A


In light scattering regime.
                                           size distribution function

                            A
 g1 +! , ) LDG +3 ,E ) 1 e 03! G +3 ,d3
                            0

So, to find the distribution function, apply the inverse transformation
which is done by numerical methods, with a combination of minimization of
variance and regularization (smoothing).



 G +3 , ) L01Dg1 +! ,E
                          CONTIN
• Developed by Steve Provencher in 1980’s
                                        A
                                         0 3!
• Recognize that g1+! , ) LDG +3 ,E ) 1 e G +3 ,d3
                                        0

is an example of a “Fredholm Integral” where

                    F (r ) ) 1 K (r , s ) A( s )ds
        measured          object of desire       defines experiment


This is a classic ill-posed problem – which means that in
  the presence of noise many DIFFERENT sets of A(s)
  exist that satisfy the equation
                 CONTIN (cont.)
So how to proceed?
  1. Limit information – i.e., be satisfied with the mean
     value (like in the cumulant analysis)
  2. Use a priori information
     – Non-negative G($) (negative values are not physical)
     – Assume a form for G($) (like exponential sampling)
     – Assume a shape
  3. Parsimony or regularization
     – Take the smoothest or simplest solution
     – Regularization (CONTIN)
       ERROR = (error of fit) +function of smoothness (usually
        minimization of second derivative)
     – Maximum entropy methods (+ p log (p) terms)
             Analysis of Decay Times
First question: How do decay times vary with q?
                                         Finite Rotational Diffusion
    Diffusion (translation)

        Slope = Dapp                      Slope = Dapp


3                                  3
                                                g1 +! , / e    +          ,
                                                              0 Dq 2 06 Dr !
                                   Dr

              q2                                    q2


$= Dappq2 where Dapp is a               Rotational diffusion can
collective diffusion coefficient        change the offset of the
that depends on interactions            decay – can also observe with
and concentration                       depolarized light
Not spheres… but still dilute, so D = kT/f
                   1
  Cylinders   D)     +D1 ( 2 D2 ,                                     D1
                   3
                                               D1

  Worms       D ) ln L + D ,36H L
                            KT
                                 o


                   kT            Shape factor: A hydrodynamic term
              D)
                    f            that depends on shape

  Prolate                                                + , 2 1/ 2
                                                      '1 0 b $
                                                      %
                                                      &     a "
                                                              #
                                         f )
                   b
                                                               + ,
                                                        < ' b 2 $1 / 2 9
                                                        :1( 10         7
                                             + ,
                                             b
                                               a
                                                 2/3
                                                     ln: &
                                                        :
                                                            %
                                                              b
                                                                 a " 7
                                                                   #
                                                                       7
                             a                          :       a      7
                                                        ;              8
          Concentration Dependence
• In more concentrated dispersions (and can only find
  the definition of ‘concentrated’ generally by
  experiment’), measure a proper Dapp, but because of
  interactions Dapp (c)

• Again, D = <thermo>/<fluid> =
              kT(1 + f(B) + …)/fo(1 + kfc + …)

So Dapp= D0 (1 + kDc + …)
                                 like a second virial coefficient
                                 for diffusion
with kD = 2B –kf – %2
                        partial molar volume
                        of solute (polymer or
                        micellar colloid)
                      Virial Coefficient
                  < IJ 9
• Driving force = :    7 ) kT [1 ( 46K 1 drr 2 ( g (r ) 0 1]01
                  : IK 7
                  ;    8T
                          at low density

                          / kT[1 - 46K 1 (g(r) - 1)r 2 dr ( ...]

  so for low density
  < IJ 9
  : IK 7 / kT [1 ( KB2 ( ...]
  :    7
  ;    8T
  where

   B2 ) 046 1 ( g (r ) 0 1)r 2 dr
             Multiple Scattering




 single scattering          multiple scattering


•Three approaches

  • Experimentally thin the sample or reduce contrast
  • Correct for the effects experimentally
  • Exploit it!
  Diffusing Wave Spectroscopy (DWS)
• In an intensely scattering solution, the light is
  scattered so many times the progress of the light is
  essentially a random walk or diffusive process

• Measure in transmission or backscattering mode

• Probes faster times than QLS

• See Pine et al. J. Phys. France 51 (1990) 2101-2127
                                Summary
Oriented particles create interference patterns, each bright spot being a speckle.
The speckle pattern moves as the particle move, creating flickering.

All the motions and measurements are described by correlations functions
     • G2(!)- intensity correlation function describes particle motion
     • G1(!)- electric field correlation function describes measured fluctuations
     Which are related to connect the measurement and motion




                        G2 ( ! ) ) B '1 ( * g ( ! ) 2 $
                                     %
                                     &       1        "
                                                      #
Analysis Techniques:
    • Treatment for monomodal distributions: linear and cumulant fits
    • Treatment for non-monomodal distributions: Contin fits
    • Interactions, polydispersity, require careful modeling to interpret


Other motions, such as rotation, can be measured

				
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