# dynamiclightscattering by cuiliqing

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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

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

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
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
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

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

' 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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