The Cilas Nano DS instrument combines static and dynamic light scattering for improving the reliability of results for bigger particles and polydisperse samples. The static light scattering is based on a spatial dependence of the scattered light. In DLS mode, the scattered light is time-dependent. These different approaches of the scattered light are based on two theoretical aspects which we develop in this document.
Brownian Motion Origin
The motion of the molecules of the fluid, due to the fact that the fluid contains heat, causes the molecules to strike the suspended particles at random. The impact makes the particles move. The molecules of the liquid will move faster when the liquid is heated, causing more agitated Brownian movement of the big particles. Similarly, if you make the liquid less viscous, the molecules can move more easily, also resulting in more particle motion. If the particle is too big, the random bumps by molecules won't be noticed at all.
Particles illuminate by a laser source scatter light. This scattered light then undergoes either constructive or destructive interference by the surrounding particles and within this intensity fluctuation. The sum of these constructive and destructive interferences generates a speckle pattern. The rate of intensity fluctuation occurs depends on the particle size.
Figure 1: Constructive and destructive interferences and speckle pattern
The dynamic information of the particles is derived from the autocorrelation of the intensity trace recorded during the experiment.
Figure 2: Time-dependence of the scattered intensity for large and small particles
At short time delays, the correlation is high because the particles do not have a chance to move to a great extent from the initial state they were in. The two signals are thus essentially unchanged when compared after only a short time interval.
As the time delays become longer, the correlation decays exponentially, meaning that after a long time period has elapsed, there is no correlation between the scattered intensity of the initial and final states.
The second order autocorrelation curve is generated from the intensity trace as follows:
where g2(q, t) is the autocorrelation function at a particular wave vector, q, and time delay, t, and I is the intensity.
where the parameter b is a correction factor that depends on the geometry and alignment of the laser beam in the light scattering setup.
where Dt is the translational diffusion coefficient
where no is the refractive index of the sample, l is the incedent laser wavelength, and q is the angle at which the detector is located.
g2(q, t) is the auto-correlation function, an exponential decaying function of the correlator time delay, t.
The correlogram can give a lot of information about the sample. The time at which the correlation starts to significantly decay is an indication of the mean size of the sample. The steeper the line, the more mondisperse the sample is.
Figure 3 : Normalized auto-correlation function, from 22412 ISO Standard
Stockes Einstein equation
The determination of the translation diffusion coefficient thanks to the auto-correlation function allows to determine the particle size.
Algorithms used for particle size analysis in DLS
One of the most common methods is the cumulant method, from which in addition to the sum of the exponentials above, more information can be derived about the variance of the system as follows:
Where t is the average decay rate and µ2 is the second order polydispersity index (or an indication of the variance). A third order polydispersity index may also be derived but this is only necessary if the particles of the sstem are highly polydisperse.
One must note that the cumulant method is valid for small t and sufficiently narrow G(t). The cumulant method is far less affected by experimental noise than other methods.
A method for analyzing the autocorrelation function can be achieved through an inverse Laplace transform known as CONTIN developed by Steven Provencher.
The particle size distribution is obtained thanks to fit a multiple exponential to the correlation function.
The CONTIN analysis is ideal for heterodisperse, polydisperse and multimodal systems which cannot be resolved with the cumulant method. The resolution for separating two different particle populations is approximately a factor of five or higher and the difference in relative intensities between two different populations should be less than 1 : .