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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
27
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
28
Hydrodynamic Radius
Theoretical Examples
Rh RhH2O
H2O
H2O
H2O
H2O
+
+
+_
29
What can QELS Measure?
• Diffusion constant, DT
• Size, rh
• Polydispersity• Conformation, rh vs. rg
30
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!
32
What is translational diffusion ?
Translational diffusions: signal change
Rotational diffusions: no signal change
Diffusion of molecules ---- Brownian Motion
33
Timescale of Motion
kB – Boltzmann’s constantT – temperature (Kelvin) h – viscosity of solvent rh – hydrodynamic radius
34
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’
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