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

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.

"#$

%&' ()

22 11 )(gB)(G !*!

2

Particles behave like ‘slits’, the orientation of which generates interference patterns

Generates a ‘speckle’ pattern

Various points reflect different scattering angles

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

+ , Dtx )- 2

fluctuations occur on the time-scale that particles move about one wavelength of light…

Assuming Brownian motion of the particles…

The time-scale is:

./-x

+ , secms/cmx.

cmxt 1001052105

28

25// 0

0

D for ~ 200 nm particles

. ~ 500 nmChange on the

msec time frame

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

How is the time scale of the fluctuations related to the particle movement?

Requires several steps:

4. Analyze data using cumulants or CONTIN fitting routines

• Math/Theory

• Application/Optics

• Data Analysis

2 texts:

‘Light scattering by Small Particles’ by van de Hulst

‘Dynamic Light Scattering with applications to Chemistry, Biology and Physics’ by Berne and Pecora

+ , !!! dtI)t(IT

)(G T (1) 021

First, the Intensity Correlation Function, G2(!)

I(t) I(t+!)

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

Mathematically, the correlation function is written as an integral over the product of intensities at some time and with some delay time, !

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…

I(t+!’)

Tau (2sec)

1 10 100 1000

G 2( !

)

8.0e+7

1.0e+8

1.2e+8

1.4e+8

1.6e+8

1.8e+8

Tau (2sec)

0 1000 2000 3000 4000 5000

G 2( !

)

8.0e+7

1.0e+8

1.2e+8

1.4e+8

1.6e+8

1.8e+8

The Intensity Correlation Function has the form of an exponential decay

plot linear in !

plot logarithmic in !

The correlation function typically exhibits an exponential decay

Instead, we correlate the motion of the particles relative to each other

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

Second, Electric Field Correlation Function, G1(!)

+ , !!! dtE)t(ET

)(G T (1) 011

Integrate the difference in distance between particles, assuming Brownian Motion

The electric field correlation function describes correlated particle movement, and is given as:

!! 30) exp)(G1

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

Constructive interference

Destructive interference

The decay constant is re-written as a function of the particle size

ichydrodynammicthermodyna

6))r

kTD62

Boltzmann Constant

temperature

viscosity particle radius

with q2 reflecting the distance the particle travels … and the application of Stokes-Einstein equation

2Dq0)3

The decay constant is related by Brownian Motion to the diffusivity by:

789

:;<)

24 =.6 sinnq

Tau

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

G 2(!

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Tau

0.000 0.002 0.004 0.006 0.008 0.010 0.012

G 2(!

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

large particle

small particle

Rate of decay depends on the particle size

large particles diffuse slower than small particles, and the correlation function decays at a slower rate.

and the rate ofother motions

(internal, rotation…)

"#$

%&' ()

22 11 )(gB)(G !*!

Finally, the two correlation function can be equated using the Seigert Relationship

Intensity Correlation Function(recall: this is measured)

Electric Field Correlation Function(recall: this is what the particles are doing)

where B is the baseline and * is an instrumental response, both of which are constants

The Seigert Relationship is expressed as:

Based on the principle of Guassian randomprocesses – which the scattering light usually is

*2E IIntensity EE >))

• 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

"#$

%&' ()

22 11 )(gB)(G !*!

• 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• Need to calculate

which is approximated by

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

+ , !!! dtI)t(IT

)(G T (1) 021

)()(1)()arg(

12 !! (/ @

)i

elN

ii tItI

NG

Autocorrelation function is collected

2.000000000E+000 1.593461120E+0082.400000095E+000 1.590897440E+0085.000000000E+000 1.582029760E+0081.000000000E+001 1.564198880E+0081.500000000E+001 1.546673760E+0082.000000000E+001 1.529991520E+0082.500000000E+001 1.513296000E+0083.000000000E+001 1.497655360E+0083.500000000E+001 1.482144000E+0084.000000000E+001 1.466891040E+0084.500000000E+001 1.452316800E+0085.000000000E+001 1.438225120E+008

…

6.000000000E+00 4 9.100139200E+007

Tau (2sec)

1 10 100 1000

G 2(!

)

8.0e+7

1.0e+8

1.2e+8

1.4e+8

1.6e+8

1.8e+8

! (2sec) G2+!,

The auto-correlator collects and integrates the intensity at the

different delay times, !, all in real time

Each point is a different !.

Tau (2sec)

1 10 100 1000

G 2(!

)

8.0e+7

1.0e+8

1.2e+8

1.4e+8

1.6e+8

1.8e+8

… then, create the raw correlation function

Evaluate the autocorrelation function from the intensity data

! (2sec)

1 10 100 1000

c(t)

0.0

0.2

0.4

0.6

0.8

… then, normalize the raw correlation function through some simple re-arrangements

+ , !*!! 30)0

) 22 eB

B)(GC

*

B

* is usually less than unity, from measuring more than one speckle

B should go to zero

tau (2sec)

1 10 100 1000 10000

c(t )

0.0

0.2

0.4

0.6

0.8

1.0

100 nm

200 nm300 nm

400 nm

500 nm

General principle: the measured decay is the intensity-weighted sum of the decay of the

individual particles

Recall that different size particles exhibit different decay rates.

Expressed in mathematical terms

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.

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.

+ , + ,@ 3) 30

ii ieGg !!1

tau (2sec)

1 10 100 1000 10000

c(t )

0.0

0.2

0.4

0.6

0.8

1.0

100 nm200 nm300 nm400 nm500 nm

wieghted sum of the individual decay

A sample correlation function would look something like this…

Recall sizes0.30 (100 nm) 0.25 (200 nm) 0.20 (300 nm) 0.15 (400 nm) 0.10 (500 nm)

Short times emphasize the intensity weighted-average

long times reflected the larger particles

• Math/Theory

• Application/Optics

• Data Analysis

r = ??? in cm

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

D = ??? in cm2 sec-1

Finally, calculate the size from the decay constant

DkTr626

)

2qD 3)

789

:;<)

24 =.6 sinnq

Diffusivity is determined… need refractive index, wavelength and angle

Calculate the radius, but need the Boltzmannconstant, temperature and viscosity

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

Monomodal Distribution

Non-Monomodal Distribution

Linear Fit

Exponential Sampling

CONTIN regularization

Cumulant Expansion

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

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

Monomodal Distribution

Non-Monomodal Distribution

Linear Fit

Exponential Sampling

CONTIN regularization

Cumulant Expansion

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

Tau

0 1000 2000 3000 4000 5000

C(! )

1e-4

1e-3

1e-2

1e-1

Tau

0 1000 2000 3000 4000 5000

C(! )

0.0

0.2

0.4

0.6

0.8

!*! DqlnB

B)(Gln 22 20)789

:;< 0

Simplest- the ‘basic’ linear fit

Take the logarithm of the normalized correlation function

ln *

Slope = -2Dq2

C(!)

Assumes that all the particles fall about a relatively tight mean

ln C(!)

Long ! there is just noise…

… but, need Long ! for a good B

Cumulant expansion

+ , + ,1 33)A 30

01 deGg !!

+ , )q(S)q(PMG 2)3

+ , solid : )( 6RRNG /3

+ , (vesicle) shell hollow : )( 4RRNG /3

Integral sum of decay curves

Probability Density Function(Coefficients of Expansion)

Intraparticle Form FactorAnd Interparticle Form Factor

that both DEPEND ON q

Larger particles are ‘seen’ more…

Assumes that the particles distribution is centered on a mean, with a Gaussian-like distribution about the mean.

Where to start…

Then, re-arrange the Seigert Relationship in terms of a cumulant expansion

Recall that the correlation function can be expressed as

+ , + , + ,!*!! 12 2 glnlnB

BGlncln 0)"#$

%&' 0

)

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)

+ , + , + ,1)@BAA

) 001 !!!! d

!nik

!nikgln

nn

n

nn

http://mathworld.wolfram.com/cumulant.html

+ , + ,1 33)A 3

01 dPeg i !!

… need to carry through some mathematics

+ , + ,!!! 303030) eeg1

First, define a mean value

is the mean ‘gamma’3

+ , + , + , + , + ,1 3

""#

$

%%&

'0

303(30303)1 33)

A 30A 30

0

2

01 2

1 d...!

eGdeGg !!! !!

...!x

!xxe x (0(0/0

321

32Note: power series expansion

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

Cumulant Expansion (more)

Working through the integrals…

+ , 778

9::;

<(0(0) 30 ...

!k

!keg 3

332

22

1 3201 !!! !

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

+ ,+ , 33031 3)A

dGk 2

02

+ , + , + , + , + ,1 3

""#

$

%%&

'0

303(30303)1 33)

A30A 30

0

2

01 2

1 d...!

GedeGg !!! !!

+ ,+ , 33031 3)A

dGk 3

03

...xx)xln( (0/( 2211Note: power series expansion

+ ,""#

$

%%&

'778

9::;

<(0(()"#

$%&' 0 30 ...

!k

!kelnln

BBGln 3

332

222

321

21

21 !!*! !

blnaln)abln( ()Note: ln of products

a b x

Note that x terms >> x2 terms, so that x2 are negligible

Cumulant Expansion (even more)

+ , ...KlnB

BGln 0(30)"#$

%&' 0 22

22 2 !!*!

Cumulant Expansion (more…)

average decayintercept polydispersity

Polydispersity index 223

)kC

… and indicates the width of the distribution

005.0)C is mono-dispersed

Note:Multiplied

by 2

Sample of Cumulant Expansion390 nm Beads

+ ,!30*)"#

$%&' 0! 2ln

BBGln 2linear

+ , 222

2 K2lnB

BGln !(!30*)"#$

%&' 0!

quadratic

+ , 33

3222

23

KK2lnB

BGln !0!(!30*)"#$

%&' 0!cubic

quartic + , 44

433

3222

212K

3KK2ln

BBGln !(!0!(!30*)"#$

%&' 0!

Gamma

~ Poly

~ Skew

~ Kurtosis

Examine residuals to the fit

uncorrelated

correlated

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

Monomodal Distribution

Non-Monomodal Distribution

Linear Fit

Exponential Sampling

CONTIN regularization

Cumulant Expansion

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

Multiple modes because of polydispersity, internal modes, interactions… all of what make the sample interesting!

Exponential Sampling for Bimodal Distribution

+ , !! ieag i30@)1

To be reliable the sizes must be ~5X different

e.g. bimodal distribution

Assume a finite number of particles, each with their own decay

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

+ , + ,D E + ,dttfetfLsF st1))A 0

0

Laplace Transform of f(t)

In light scattering regime.

+ , + ,D E + , 31 3)3)A 30 dGeGLg0

1!!

size distribution function

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).

+ , + ,D E!11 gLG 0)3

+ , + ,D E + ,dttfetfG ti1A

A0

F0)G)F

Note: Fourier Transform

CONTIN• Developed by Steve Provencher in 1980’s

• Recognize that

is an example of a “Fredholm Integral” where

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

+ , + ,D E + , 31 3)3)A 30 dGeGLg0

1!!

1) dssAsrKrF )(),()(

measured object of desire defines experiment

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

+ , + ,!! rDDqeg 61

200/3

q2

Slope = Dapp

Diffusion (translation)Finite Rotational Diffusion

3

q2

Slope = Dapp

Dr

Rotational diffusion can change the offset of the decay – can also observe withdepolarized light

First question: How do decay times vary with q?

$= Dappq2 where Dapp is a collective diffusion coefficientthat depends on interactionsand concentration

fkTD ) Shape factor: A hydrodynamic term

that depends on shape

Cylinders

+ ,

+ ,+ ,

7777

8

9

::::

;

<"#$

%&' 0(

"#$

%&' 0

)

abab

ab

ab

f2/12

3/2

2/12

11ln

1Prolate

b

a

Worms

+ ,21 231 DDD ()

D1

D1

+ ,L

KTDLlnD

o6H3)

Not spheres… but still dilute, so D = kT/f

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 + …)

with kD = 2B –kf – %2

like a second virial coefficientfor diffusion

partial molar volume of solute (polymer ormicellar colloid)

Virial Coefficient• Driving force =

1

1

(/

0()778

9::;

<IJI 0

...]1)r-(g(r)4-kT[1

density lowat

]1)((41[

2

12

dr

rgdrrkTT

6K

6KK

1 00)

((/778

9::;

<IJI

drrrg

BkTT

22

2

)1)((4B

where

...]1[

density lowfor so

6

KK

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

SummaryOriented 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 fluctuationsWhich are related to connect the measurement and motion

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

"#$

%&' ()

22 11 )(gB)(G !*!