Static and dynamic light scattering have evolved into powerful methods to investigate a variety of soft and biological matter systems with structures on the nanometer to micrometer scale. They can provide detailed quantitative information on the shape, internal structure, size, and polydispersity of the system as well as interparticle interactions. I will present their fundamentals from a physics and instrumental point of view and also comment on experimental data analysis. The opportunities they offer will be discussed as well as their limits. This will be illustrated by a selection of examples, ranging from colloidal suspensions, detergent, and polymer solutions to proteins and include topics like contrast and absolute intensity, determination of molar mass, polydispersity, and interparticle interactions.

1. Light Scattering:
Fundamentals
Andrea Vaccaro, LS Instruments

2. “Scattering” of Light
particle dispersion
water

3. Light Scattering as Sizing Tool
Nanoparticles: 1-100 nm
Polymers,
Macromolecules
Clays, Oxides, Proteins
etc.

4. Part I: Theory
Fundamentals

5. Static Light Scattering
(SLS)

6. Physical Origin of Scattered Light
• An electron in the atomic cloud is subject
to a force due to the electric field
• The cloud deforms and a dipole is induced
• As the field oscillates so does the dipole
moment
• The resulting charge movement radiates
(“scatters”) light
• “Elastic” scattering: momentum is
preserved, no energy loss ⇒
+
–
Light

7. A Collection of Atoms
Field scattered by a dipole of momentum
Spherical
Wave
Scattering
Geometry

8. A Collection of Atoms
By definition of polarizability
For an object smaller than λ (Clausius-Mossotti Relation):

9. A Collection of Atoms
Spherical
Wave
Contrast:
Ability to scatter of
the material
Piecing everything together:
Scattering
Geometry

10. Origin of the Scattering Contrast
• Interference
• For a larger object it is
possible to find a second
lump that scatters out of
phase and with the same
amplitude
• Completely destructive
interference
• For an infinite object it is
always possible to do this
⇒ No contrast

11. The Measured Quantity:
the Scattered Intensity
• Whatever the detection technology, the observable quantity
is not the electric field but the flux of energy, the so-called
light intensity
• It can be shown that in most conditions
• In practice the intensity
fluctuates in time
• In SLS experiment the
average intensity is measured

12. SLS Theoretical
Approaches
Theory Assumption
Rayleigh (Electrostatic
Approximation)
Rayleigh-Gans-Debye (RGD,
Optically “Soft” Particles)
Mie Scattering None
Fraunhofer (Geometrical
Optics)

13. RGD Theory
• Assumption: The field inside the particle is the incident
field
• To satisfy this assumption we must require for the
incident field:
i) no reflection at the particle/solvent interface
ii) no phase change within the particle
i) ii)
• Every small lump in the particle scatters as if it were
“alone”

14. RGD Interference: The Scattering Vector
Interference
Term
Dropping the dummy time
dependency terms:

15. The Meaning of the Scattering Vector
The module of the scattering vector has dimensions of
inverse of length:
q-1 is the length-scale of the interference phenomenon.
Two material lumps farther than q-1 interfere
destructively.
Closer than q-1 interfere additively
Destructive
Interference,
Smaller
Intensities
No internal
Interference
Maximum
Intensity
q-1 can be interpreted
as a rough measure of
the probed length-scale

16. One Particle: The Scattering
Amplitude
Integrating previous equation over the whole particle:
Labeling the particle with the subscript j and factoring out
its position by means the variable substitution
We obtain
Particle j scattering
amplitude
Interference :
Particle Position
Internal
Interference

17. An Ensemble of Particles
Previous result allows to sum
each contribution
= 0 = 0 !
Independent position and orientation
We measure a time average:

18. RGD Scattered Intensity
Indeed the electric field in not an observable But
intensity is:
Average
Contrast
Structure Factor:
Interparticle
Interference
Form Factor:
Intraparticle
Interference
Scattering
Geometry
• The RGD assumption results in the factorization
of different contributions
• Same factorization for polydisperse systems
After many manipulations:

19. The Ergodic Hypothesis
One of the starting hypotheses od statistical
mechanics is the so called “Ergodic Hypothesis”:
For any system at equilibrium infinite time
averages of observable quantities are equivalent to
ensemble averages, i.e.:
An ensamble average, is an average over the ensable
of all the feasible physical configurations.
Once we know how to construct such an ensemble this
hypothesis enables us to “calculate” observed time
averaged quantities

20. Form Factor
Homogeneous Sphere with radius R
Polydisperse Homogeneous
Sphere

21. Form Factor
Non radially symmetric shapes
Scattering length density weighted
pair distance function:

22. Form Factor at Small q: The
Radius of Gyration
Expanding in series the interference factor...
The form factor becomes
Radius of Gyration:
In a plot of the intensity vs. q2 the extrapolation to zero
yields a size parameter that is model independent

23. Radius of Gyration: Polydisperse
Case

24. Form Factor
Polymers

25. The Structure Factor
In the same vein as the form factor it can be shown that accounting only
for pair interactions (“on the pair level”)
Applying the Ergodic Hypothesis:

26. The Rayleigh Ratio
Scattered
Intensity
The Rayleigh Ratio: Scattered intensity per unit incident
intensity, unit solid angle, and unit scattering volume.
Depends only on the thermodynamic state of the solvent
not on the measuring apparatus
Mass
Concentration
Sample
Contrast
Molar MassInstrumental
Constant
Solvent
Scattering
(Background)

27. Absolute Measurements
• Knowledge of the constant A enables absolute intensity
measurements
• Absolute measurements allow for the determination of
the radius of gyration and the second virial coefficient
but also of the molar mass M, or the particle
concentration
• How do we do it?
Excess Rayleigh Ratio:

28. Absolute Measurements: How
• Scientists have built special devices that allow the
measurement of Rayleigh ratios, values for common
reference solvents are available in literature
• If we measure the same reference solvent in the same
thermodynamic conditions we have:
Substituting back:

29. Data Treatment and Absolute
Intensity
M
R2
g/3
Polydisperse Case:

30. Macromolecular Systems
• The treatment so far was focused on particulate systems
• For macromolecular systems we cannot precisely define a
refractive index, how do we obtain the contrast?
Tabulated or
Measured
Assuming a refractive index mixing rule and in dilute conditions:
RGD Hypothesis: m close to 1

31. Dynamic Light Scattering
(DLS)

32. Intensity Fluctuation
1.6 mm
0.12 mm

33. Particle Dynamics in Real
and Reciprocal Space
Particle tracking with a microscope
Dynamics in reciprocal (Fourier)
space

34. Brownian Motion and Intensity
Fluctuations
Brownian motion
• Particle diffusion due
to thermal motion
• Interference effects on
scattered light
• Stokes-Einstein
equation
RANDOM
FLUCTUATION IN
SCATTERED
INTENSITY

35. Intensity Decorrelation
9
Intensity correlation function
∗
small large
CORRELATION DECORRELATION
>

36. Field Correlation Function
Upon normalization:
Identical Independent
Particles
But cannot be measured!
The electric field correlation function is important as it
is directly connected to colloid dynamics models

37. Intensity Correlation Function
Assuming the field be a Gaussian stochastic variable:
As for SLS we can measure the intensity correlation function
Omitting the scattering amplitudes Fj :

38. Coherence Area, Siegert
Relationship
• In practical implementations the field
is not always a Gaussian stochastic
variable
• This happens since we sometimes
image more than one coherence area
(“speckle”)
• The signal to noise ratio is lowered

39. DLS data treatment
Identical particles (Monodisperse):
Brownian diffusion
theory

40. Monodisperse DLS
Measurement Example

41. Polydisperse Particles: Cumulant
Analysis
Intensity weighted correlation function
Cumulant Expansion:
Typical uncertainty in cv is ± 0.02, i.e. it is hard to determine cv ≤ 0.2 (20%)
For a polydisperse sample
Intensity weighted diffusion coefficient

42. Polydisperse Particles: Cumulant
Analysis
t2
qD
Two species, differing in size by 50%

43. Part II: Applications

44. Static Properties: Zimm Plot
Schmidt M., Macromolecules, 1984, 17 (4)

45. Polymer Properties: Interaction
and Conformation
At the Theta temperature the chain follows the Gaussian chain model,
the second virial coefficient is close but not equal to zero
Berry G. C., J. Chem. Phys. 44, 4550 (1966)

46. Polymer Properties: Chain
Conformation
Dependency of chain size on
solvent “goodness”
Dependency of chain
size on molar mass (at
Theta solvent conditions)
Outer P. et al., J. Chem. Phys. 18, 830 (1950)

47. Dynamic Properties:
DLS Zimm Plot
• Hydrodynamic Radius depends through a certain
scattering/colloid dynamic model on the scattering angle
• DLS Zimm Plot enables the determination of the zero angle, no
interactions hydrodynamic radius
• This quantity is less model dependent
Bantle S. et al., Macromolecules, 1982, 15 (6)

48. Chain Conformation DLS/SLS:
Coil to Globule Transition
Sun S. et al., J. Chem. Phys. 73, 5971 (1980)

49. Kinetic Measurements: DLS
Salt Induced
Polystyrene Latex
coagulation
Fibrillogenesis, Aβ
fibrils elongation rate
𝐼
Holthoff H., Langmuir, 1996, 12 (23)
Lomakin A., PNAS 1996 93 (3) 1125-1129

50. Depolarized DLS: Tobacco Virus
Rotational + Translational
decorrelation rate
Translational
decorrelation rate
• The instantaneous
depolarized intensity
depends on position and
orientation
• At short correlation
times decorrelation is
due to translation, at
larger rimes to
translation and rotation
Wada A. Et al., J. Chem. Phys. 55, 1798 (1971)

51. Depolarized DLS: Tobacco Virus
• Short time decorrelation
rate dependence on q
yields D
• Long time decorrelation
rate dependence on q
yields rotational diffusion
(given the value of D)
• Rotational and
translational diffusion
coefficients yield tobacco
mosaic virus dimensions:
Wada A. Et al., J. Chem. Phys. 55, 1798 (1971)

52. Part III: Concentrated
Systems

53. Traditional Solution: Dilution
Limitations: Time consuming, sample
composition is changed, limited
applicability of methods
Multiple Scattering

54. Solutions:
Multiple Scattering

55. 3D Cross-Correlation
• Two simultaneous scattering experiments with
identical scattering volumes and vectors
• Cross-correlation of the two signals suppresses
multiply-scattered light
Single Scattering:
degenerate single
Fourier component
Multiple Scattering:
different Fourier
components
Laser
Sample in index
matching vat
Photon
Detectors
Cross Correlator
Beam
splitter Lens
Lens
Mirror
y
x
z

56. 32
normalized
3D Cross-Correlation

57. Laser
Sample in index
matching vat
Photon
Detectors
Cross Correlator
Beam
splitter Lens Lens
Mirror
Intensity
Modulator
Modulated 3D Cross-Correlation

58. Laser
Sample in index
matching vat
Photon
Detectors
Cross Correlator
Beam
splitter Lens Lens
Mirror
Intensity
Modulator
Modulated 3D Cross-Correlation

59. 4x
Correlation Function Comparison

60. Improved DLS Particle
Sizing for Turbid Samples
20 kHz Modulation, PS 100nm
max

61. SLS for Turbid Samples
Dilute
Correct scattered intensity
to get single scattering
contribution
Turbid - Modulated 3D DLS
10mm Round Cuvette, Monodisperse
430nm latex beads

62. SLS: Highly Turbid Suspensions of
Small Particles
110nm latex spheres
Intercept and pathlength-
corrected
Uncorrected

63. SLS for Turbid Samples: Square Cell
Corrections
I: intensity
: cross-correlation
intercept
L: path length
l: mean-free-path
Square-cell -2  geometry:
• minimizes path lengths
• self-aligns scattering
volume
Turbidity correctionSingle scattering only

64. Thank you for your attention!