Introduction to Light Scattering

A bulk analytical technique

What is light scattering?

In nature…

blue sky and clouds

red sunsets

What is light scattering?

In the lab…

What can light scattering measure?

• Molar mass, M• Size, rg

• Second virial coefficient, A2

• Translational diffusion coefficient, DT - Can be used to calculate rh

For a solute in solution, light scattering can determine:

Light and its properties

Light is an oscillating wave of electric and magnetic fields

• Polarization: directionof electric field oscillation

• Intensity: 2

EI

How does light scatter?When light interacts with matter, it causes charges to polarize.

The oscillating charges radiate light.

How much the charges move, and hence how much light radiates, depends upon the matter’s polarizability.

Index of refraction n

The polarizability of a material is directly related to its index of refraction n.

The index of refraction is a measure of the velocity of light in a material.

e.g., speed of light

For solutes, the polarizability is expressed as the specific refractive index increment, dn/dc.

dcdn

E scattered

2

scattered

dcdn

I

nvacuum

liquid

vv

Adding light

• Interference:

• Incoherent sum

• Coherent sum

21

2

2

2

1total

II

EEI

21

2

2

2

1

2

21total

termscross

II

EE

EEI

How light scattering measures M

22

total 4 EEEI 222

total 2 EEEI coherent: incoherent:

2

scattered

dcdn

McI

Isotropic scattering

For particles much smaller than the wavelength of the incident light ( <10 nm for l = 690 nm), the amount of radiation scatteredinto each angle is the same in the plane perpendicular to the polarization.

Angular dependence of light scattering

detector at 0°scattered light in phase

detector at , q scattered lightout-of-phase

Intramolecular interference leads to a reduction in scattering intensity as thescattering angle increases.

2

scattered Pdcdn

McI

Definitions

How light scattering measures rg

To calculate the angular distribution of scattered light, integrate over phase shifts from extended particle.

Integrating over extended particle involves integrating over massdistribution.

M

mrr ii

g

22

...2

sin3

161 22

20

20

2

gr

nP

l

Conformation: rh vs. rg

3-arm star polymer

4.1h

g

r

r

solid sphere

77.0h

g

r

r

Molar mass and radius

rg < 10 nm isotropic scatterer

rg > 10 nm

Why isotropic if radius of gyration < 10 nm?

Basic light scattering principles

Principle 1The amount of light scattered is directly proportional to the product of the polymer molar mass and concentration.

Principle 2The angular variation of the scattered light is directly related to the size of the molecule.

2

scattered

dcdn

McI

Basic light scattering equation

In the Rayleigh-Gans-Debye limit, the two light scatteringprinciples are embodied in the equation:

This equation also contains a correction due to concentration c. The correction is due to coherent intermolecular scattering, and contains information on the second virial coefficient.

21 2scattered McPAMcPKRI

Definition of terms 1

2

40

20

24

dcdn

Nn

Al

K* .n0 – solvent refractive indexNA – Avogadro’s numberl0 – vacuum wavelength of incident lightdn/dc - spec. refractive index increment

M – molar mass

R(q) – excess (i.e., from the solute alone) Rayleigh ratio. The ratio of the scattered and incident light intensity,corrected for size of scattering volume and distancefrom scattering volume.

21 2 McPAMcPKR

Definition of terms 2

c – solute concentration (g/ml)

P(q) – form factor or “scattering function”. P( ) relates the angular variation in scattering intensity to the mean square radius rg of the particle. The larger rg, the larger the angular variation. Note that P(0°) = 1.

A2 – second virial coefficient, a measure of solute-solvent interaction. Positive for a “good” solvent.

21 2 McPAMcPKR

Running an experiment 1: Calibration

Why? The detectors output voltages proportional to the light scattering intensities. The voltages must be converted to meaningful units.

How? 1. Flow pure, filtered (0.02 m m) toluene through the flow cell.2. ASTRA software measures the voltages from the 90° and laser

monitor photodiodes with the laser on and off (dark voltages). 3. ASTRA then computes the calibration constant.

Running an experiment 2: Normalization

Why?• detector sensitivities vary.• each detector views a different scattering volume.• scattered light is refracted.• only the 90° detector is calibrated.

How?1. Fill flow cell with isotropic scatterer in actual solvent to be used.2. ASTRA software measures voltages for each angle and:

a. Determines refraction angle from solvent index of refraction.b. Determines angle and scattering volume corrections.c. Normalizes each corrected detector voltage signal to the 90°

detector.

Online Data Collection

Record Rayleigh ratio varying angle (3 or 18 anglesfor miniDAWN or DAWN) but measuring concentration.

Online Data Analysis 21 2 R K McP A McP

1. Perform fit of angular data to retrieve M and rg.2. Assess quality of fit using a Debye plot.

Batch Data Collection

Record Rayleigh ratio varying - angle (3 or 18 angles for miniDAWN or DAWN)- concentration (multiple injections of known c).

excess scattering

solvent scattering+ detector offset

Batch Data Analysis 21 2 R K McP A McP

1. Perform global fit of data to light scattering equation to retrieve M, rg, and A2.

2. Assess quality of fit using a Zimm plot.

Zimm Plot of a Protein

Molar Mass (MM) : (7.714±0.01)e+4 g/mol (0.16%)RMS Radius (Rz) : 2.6±2.2 nm (84%)2nd virial coefficient : (1.413±0.06)e-4 mol mL/g2 (3%)Aqueous microbatch Zimm Plot of BSA monomer

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Radius Results: Light Scattering &Viscometry

Rg or RMS radius – mass average (root mean square) distance of each point in a molecule from the molecule’s center of gravity.

*lower limit 10nm

Rh or Hydrodynamic radius – radius of a sphere with the same diffusion coefficient or viscosity as “our” sample.

*lower limit 1nm

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Hydrodynamic Radius

Theoretical Examples

Rh RhH2O

H2O

H2O

H2O

H2O

+

+

+_

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What can QELS Measure?

• Diffusion constant, DT

• Size, rh

• Polydispersity• Conformation, rh vs. rg

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What is a QELS Experiment?

Scattered light intensity is measured through time

31

How QELS Works: Interference of Light

Constructive interference

Particles diffuse due to Brownian motion, resulting in light intensities which fluctuate with time.

Destructive interference

Diffusion!

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What is translational diffusion ?

Translational diffusions: signal change

Rotational diffusions: no signal change

Diffusion of molecules ---- Brownian Motion

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Timescale of Motion

kB – Boltzmann’s constantT – temperature (Kelvin) h – viscosity of solvent rh – hydrodynamic radius

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DT T

High temperaturespeeds it up

DT 1/R

Small particles move faster

DT 1/fs

Asphericity slows it down

What affects translational diffusion?

DT 1/fh

Attached solvent and/or interparticle interactions create drag

DT 1/

Viscous solvent slows it down.…and if concentration too high, ‘viscosity effects’