1. Dynamic Light Scattering
ZetaPALS w/ 90Plus particle size analyzer
Also equipped w/ BI-FOQELS & Otsuka DLS-700 (Rm CCR230)

2. Dynamic Light Scattering (DLS) Photon Correlation Spectroscopy (PCS)
Quasi-Elastic Light Scattering (QELS)
Measure Brownian motion by …
Collect scattered light from suspended particles to …
Obtain diffusion rate to …
Calculate particle size

3. Brownian motion
Velocity of the Brownian motion is defined by the Translational Diffusion Coefficient (D)
Brownian motion is indirectly proportional to size
Larger particles diffuse slower than smaller particles
Temperature and viscosity must be known
Temperature stability is necessary
Convection currents induce particle movement that interferes with size determination
Temperature is proportional to diffusion rate
Increasing temperature increases Brownian motion

4. Brownian motion
Random movement of particles due to bombardment of solvent molecules
Solvent frequency
Gas frequency

5. Stokes-Einstein Equation
dH = hydrodynamic diameter (m)
k = Boltzmann constant (J/K=kg·m2/s2·K)
T = temperature (K)
η = solvent viscosity (kg/m·s)
D = diffusion coefficient (m2/s)

6. Hydrodynamic diameter
Particle diameter
Hydrodynamic diameter
The diameter measured by DLS correlates to the effective particle movement within a liquid
Particle diameter + electrical double layer
Affected by surface bound species which slows diffusion
Double layer is extended in a low ionic strength liquid (DI water) so the diameter appears larger

7. Nonspherical particles=
Rapid
Equivalent sphere
Slow
Hydrodynamic diameter is calculated based on the equivalent sphere with the same diffusion coefficient

8. Experimental DLS
Measure the Brownian motion of particles and calculate size
DLS measures the intensity fluctuations of scattered light arising from Brownian motion
How do these fluctuations in scattered light intensity arise?

9. What causes light scattering from (small) particles?
Explained by JW Strutt (Lord Rayleigh)
Electromagnetic wave (light) induces oscillations of electrons in a particle
This interaction causes a deviation in the light path through an angle calculated using vector analysis
Scattering coefficient varies inversely with the fourth power of the wavelength
Incident photon induces an oscillating dipole in the electron cloud. As the dipole changes, energy is radiated or scattered in all directions
Really based upon the Tyndall effect
Differs from Mie theory in particle size

10. Interaction of light with matter Rayleigh approximation
For small particles (d ≤ λ/10), scattering is isotropic
Rayleigh approximation tells us that
I α d6
I α 1/λ4
where I = intensity of scattered light
d = particle diameter
λ = laser wavelength

11. Mie scattering from large particles
Used for particles where d ~ λ0
Complete analytical solution of Maxwell’s equations for scattering of electromagnetic radiation from spherical particles
Assumes homogeneous, isotropic and optically linear material
Stratton, A. Electromagnetic theory, McGraw-Hill, New York (1941)

12. Brownian motion and scattering
Constructive interference
Destructive interference

13. Brownian motion and scattering Multiple particles

14. Instrument layout

15. Intensity fluctuations
Apply the autocorrelation function to determine diffusion coefficient
Large particles – smooth curve
Small particles – noisy curve

16. Determining particle size
Determine autocorrelation function
Fit measured function to G(τ) to calculate Γ
Calculate D, given n*, θ, and Γ
Calculate dH, given T* and η*
*User defined values.

17. How a correlator works
Small time increments
Random motion of small particles in a liquid gives rise to fluctuations in the time intensity of the scattered light
Fluctuating signal is processed by forming the autocorrelation function
Calculates diffusion

18. How a correlator works
Large particles – the signal will be changing slowly and the correlation will persist for a long time
Small, rapidly moving particles – the correlation will disappear quickly

19. The correlation function
For monodisperse particles the correlation function is
Where
A= baseline of the correlation function
B=intercept of the correlation function
Γ =Dq2
D=translational diffusion coefficient
q=(4πn/λ0)sin(θ/2)
n=refractive index of solution
λ0=wavelength of laser
θ=scattering angle

20. The correlation function
For polydisperse particles the correlation function becomes
where g1(τ) is the sum of all exponential decays contained in the correlation function

21. Broad particle size distribution
Correlation function becomes nontrivial
Measurement noise, baseline drifts, and dust make the function difficult to solve accurately
Cumulants analysis
Convert exponential to Taylor series
First two cumulants are used to describe data
Γ = Dq2
μ2 = (D2*-D*2)q4
Polydispersity = μ2/ Γ2

22. Cumulants analysis
The decay in the correlation function is exponential
Simplest way to obtain size is to use cumulants analysis1
A 3rd order fit to a semi-log plot of the correlation function
If the distribution is polydisperse, the semi-log plot will be curved
Fit error of less than is good.
1ISO 13321:1996 Particle size analysis -- Photon correlation spectroscopy

23. Cumulants analysis Third order fit to correlation function
b = z-average diffusion coefficient
2c/b2 = polydispersity index
This method only calculates a mean and width
Intensity mean size
Only good for narrow, monomodal samples
Use NNLS for multimodal samples

24. Cumulants analysis

25. Polydispersity index
0 to 0.05 – only normally encountered w/ latex standards or particles made to be monodisperse
0.05 to 0.08 – nearly monodisperse sample
0.08 to 0.7 – This is a mid-range polydispersity
>0.7 – Very polydisperse. Care should be taken in interpreting results as the sample may not be suitable for the technique (e.g., a sedimenting high size tail may be present)

26. Non-Negatively constrained Least Squares (NNLS) algorithm
Used for Multimodal size distribution (MSD)
Only positive contributions to the intensity-weighted distribution are allowed
Ratio between any two successive diameters is constant
Least squares criterion for judging each criterion is used
Iteration terminates on its own

27. Correlation function Correlograms
Correlograms show the correlation data providing information about the sample
The shape of the curve provides clues related to sample quality
Decay is a function of the particle diffusion coefficient (D)
Stokes-Einstein relates D to dH
z-average diameter is obtained from an exponential fit
Distributions are obtained from multi-exponential fitting algorithms
Noisy data can result from
Low count rate
Sample instability
Vibration or interference from external source

28. Correlation function Correlograms
What does the intercept tell us?

29. Data interpretation Correlograms
Very small particles
Medium range polydispersity
No large particles/aggregrates present (flat baseline)

30. Data interpretation Correlograms
Large particles
Medium range polydispersity
Presence of large particles/agglomerates (noisy baseline)

31. Data interpretation Correlograms
Very large particles
High polydispersity
Presence of large particles/agglomerates (noisy baseline)

32. Upper size limit of DLS
DLS will have an upper limit wrt size and density
When particle motion is not random (sedimentation or creaming), DLS is not the correct technique to use
Upper limit is set by the onset of sedimentation
Upper size limit is therefore sample dependent
No advantage in suspending particles in a more viscous medium to prevent sedimentation because Brownian motion will be slowed down to the same extent making measurement time longer

33. Upper size limit of DLS
Need to consider the number of particles in the detection volume
Amount of scattered light from large particles is sufficient to make successful measurements, but …
Number of particles in scattering volume may be too low
Number fluctuations – severe fluctuations of the number of particles in a time step can lead to problems defining the baseline of the correlation function
Increase particle concentration, but not too high or multiple scattering events might arise

34. Detection volume
Detector
Laser

35. Lower particle size limit of DLS
Lower size limit depends on
Sample concentration
Refractive index of sample compared to diluent
Laser power and wavelength
Detector sensitivity
Optical configuration of instrument
Lower limit is typically ~ 2 nm

36. Sample preparation
Measurements can be made on any sample in which the particles are mobile
Each sample material has an optimal concentration for DLS analysis
Low concentration → not enough scattering
High concentration → multiple scattering events affect particle size

37. Sample preparation
Upper limit governed by onset of particle/particle interactions
Affects diffusion speed
Affects apparent size
Multiple scattering events and particle/particle interactions must be considered
Determining the correct particle concentration may require several measurements at different concentrations

38. Sample preparation
An important factor determining the maximum concentration for accurate measurements is the particle size

39. Sample concentration Small particles
For particle sizes <10 nm, one must determine the minimum concentration to generate enough scattered light
Particles should generate ~ 10 kcps (count rate) in excess of the scattering from the solvent
Maximum concentration determined by the physical properties of the particles
Avoid particle/particle interactions
Should be at least 1000 particles in the scattering volume

40. Sample preparation When possible, perform DLS on as prepared sample
Dilute aqueous or organic suspensions
Alcohol and aggressive solvents require a glass/quartz cell
to 1%(v/v)
Dilution media (1) should be the same (or as close as possible) as the synthesis media, (2) HPLC grade and (3) filtered before use
Chemical equilibrium will be established if diluent is taken from the original sample
Suspension should be sonicated prior to analysis

41. Checking instrument operation
DLS instruments are not calibrated
Measurement based on first principles
Verification of accuracy can be checked using standards
Duke Scientific (based on TEM)
Polysciences

42. Count rate and z-average diameter Repeatability
Perform at least 3 repeat measurements on the same sample
Count rates should fall within a few percent of one another
z-average diameter should also be with 1-2% of one another

43. Count rate Repeatability problems
Count rate DECREASES with successive measurements
Particle sedimentation
Particle creaming
Particle dissolution or breaking up
Resolution
Prepare a better, stabilized dispersion
Get rid of large particles
Coulter

44. Count rate Repeatability problems
Count rate is RANDOM with successive measurements
Dispersion instability
Sample contains large particles
Bubbles
Resolution
Prepare a better, stabilized dispersion
Remove large particles
De-gas sample

45. Z-average diameter Repeatability problems
Size DECREASES with successive measurements
Temperature not stable
Sample unstable
Resolution
Allow plenty of time for temperature equilibration
Prepare a better, stabilized dispersion

46. Repeatability of size distributions
The sized distributions are derived from a NNLS analysis and should be checked for repeatability as well
If distributions are not repeatable, repeat measurements with longer measurement duration

47. References http://www.bic.com/90Plus.html
Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics, Bruce J. Berne and Robert Pecora, DOVER PUBLICATIONS, INC. Mineola. New York
Scattering of Light & Other Electromagnetic Radiation, Milton Kerker, Academic Press (1969)
Additional references that Riman mentioned

49. Evaluating the correlation function
If the intensity distribution is a fairly smooth peak, there is little point in conversion to a volume distribution using Mie theory
However, if the intensity plot shows a substantial tail or more than one peak, then a volume distribution will give a more realistic view of the importance of the tail or second peak
Number distributions are of little use because small error in data acquisition can lead to huge error in the distribution by number and are not displayed

50. Correlogram from a sample of small particles

51. Correlogram from a sample of large particles

52. Extra
Time-dependent fluctuations in the scattered intensity due to Brownian motion
Constructive and destructive interference of light
Decay times of fluctuations related to the diffusion constants --- particle size
Fluctuations determined in the time domain by a correlator
Correlation – average of products of two quantities
Delay times chosen to be much smaller than the time required for a particle to relax back to average scattering