S19_DLS

Determination of Hydrodynamic Size via Dynamic Light Scattering

Jake Holley and Kyle Carrigan

University of Minnesota - Methods of Experimental Physics II

Spring 2019

Introduction

The hydrodynamic size of a particle is defined as a rigid spherical shell that diffuses identically to the particles under study. When small particles are in solution, like polymers in water, they’re knocked about in a random fashion that’s described by the theory of Brownian motion ^[1]. As a brief demonstration of the theory, bigger particles will move slower due to the fact that they have a larger surface area that drags more solvent molecules with it as it diffuses. The reverse is also true for smaller particles given that both particles are observed in identical solvents at similar temperatures. It is this difference in diffusion speeds that is essential to using Dynamic Light Scattering (DLS) as a particle sizing technique.

DLS is employed by shining laser light into a solution of particles, these particles will scatter the light that hits them where it can be picked up by a photodiode. Since the particles are undergoing Brownian motion while also scattering light there are fluctuations in the intensity of light incident on the detector. These fluctuations are therefore a measure of the random Brownian motion of the particles. These fluctuations can be Fourier analyzed and the resulting spectrum can be used to determine the particles' hydrodynamic radius in tandem with the Stokes-Einstein relation:

DLS is commonly used to determine the size of polymers and macromolecules such as carbohydrates, proteins, nucleic acids. However, DLS only works well as a particle sizing technique in the Rayleigh Scattering regime where the particles are small compared to the wavelength. This is what makes DLS a valuable technique for biologists studying important proteins and macromolecules since they lay nicely in the Rayleigh regime. DLS has even seen application in analyzing protein sizes in diseases where protein aggregation is one of the main suspects such as Alzheimer's and Huntington's disease.

It is a sensitive technique that requires the user to be extremely wary of sources of noise. Any source of outside light from computer monitors to fluorescent lights will cover the signal fluctuations that are caused by the particles' Brownian motion. Even running amplifiers off of wall outlets will cause 60 Hz noise that will cover the fluctuations. It is a difficult experiment to remove all sorts of noise from, but invaluable in the fields that it is used in.

Theory

The polystyrene spheres suspended in solution will be knocked about by the solvent molecules resulting in a random stochastic phenomenon called Brownian motion. Brownian motion (in one dimension) can be described by the following partial differential equation and solution:

ρ, x, N, D, and t are number density, position, number of particles, diffusion coefficient, and time respectively. However, we’re not necessarily interested in the actual equations of Brownian motion as we are interested in the diffusion coefficient that is an integral part of them. From just the differential equation it can be understood as a sort of “diffusive speed”. Since it's already been seen that the diffusion coefficient is related to the particle size, if the Brownian motion can be mathematically related to the fluctuations that are seen in DLS then the physics problem is solved. Fortunately, if the Fourier Transform of the signal is taken a Lorentzian curve of the following form will arise:

Where Γ is the width of the curve which is of great importance in this technique. Essentially we take data that looks very much like noise and get a smooth curve out of it. The first picture is somewhat similar to what would be observed in the time domain and the second after the Fourier Transform in the frequency domain:

It's a very lengthy derivation, but it can be shown that the diffusion coefficient is related to the width of the resulting Lorentzian curve in the following manner:

There is some physical justification to the relation though. The faster the particles diffuse means more fluctuations which implies a wider frequency spectrum. The q in the above equation is something called the wave scattering vector which essentially just means that it keeps track of how much light is scattered in which direction. If more light in the first place is scattered, then the width should also increase. In order to see more detail on this derivation I recommend Clark et. al 1970.[2]

One last item to take care of is the notion of a coherence area. Essentially there are certain areas are and are not available to the detector due to interference. The intensity measured is proportional to the number of coherence areas that the scattered light from the spheres is incident upon^[5]. This is because of the fact that since light is wave-like in nature the scattered light constructively and destructively interferes when photons are incident upon the detector which houses the coherence areas. Reducing the number of coherence areas increases the magnitude of fluctuations that are observed, but the signal will be weak. Increasing the number of coherence areas will strengthen the signal but will average out all the fluctuations that need to be observed. There must be a balance struck between number of fluctuations and signal strength. The coherence area can be adjusted by changing the area of scattered light coming out of the holding cell. The coherence area can be increased or decreased by moving the photodiode further and closer respectively. It is recommended that the experiment be carried out with a few thousand coherence areas. This is entirely based on the geometry of our set-up and can be described by the diagram and equations below:

Apparatus

Figure 3 shows the R-30995 Red Helium-Neon laser with an operating wavelength of 632 nm and an output power of 17 mW that provided the scattering light for the experiment^[6]. The laser beam is reflected off of two mirrors and focused onto the sample with a lens of focal length 30 cm which is placed 30 cm in front of the sample. This specific “U” design is put in place to give a lot of space with “parallel to the table” laser beam to work with. The photodiode is aligned with the scattering sample at a distance of approximately 6” in order to ensure that the number of coherence areas stays at around 10,000. A Faraday cage is placed over the sample and photodiode to minimize noise. The beam passes through the Faraday cage from one end out the other to prevent reflected laser light on the inside hitting the detector.

Essentially, after the spheres are in the holding cell, lights are turned off, and laser firing, the following process occurs. The laser is focused into the holding cell and the light starts scattering off of the polystyrene spheres. The scattered light will hit the photodiode where an output current will be generated and then filtered and amplified by a current and voltage amplifier. This signal is sent to a scope for the experimenters to observe the temporal signal as well as to the DAQ card so that LabVIEW can display the signal in the spectral domain. From that data, the hydrodynamic size of the particle can be calculated. Below is an actual picture of our set-up:

Results

Figures 5 and 6: Experimental data and the Lorentzian fit for 0.173 micron diameter spheres in a sample of 60 L of polystyrene spheres in 3 mL distilled water. Figure 4 has the half-width at 157. 7 Hz.

Figures 5 and 6 show the raw data that was obtained from the Fourier transform and the Lorentzian fit for a solution of 3 milliliters of distilled water and 60 micro liters of 0.173 micron diameter polystyrene spheres. The half width is marked on the fit at 157.68 Hz. You may notice that there is cut-off at the beginning of the data when you start getting down to low frequencies. This is simply a relic of the fact that the computer has a finite sampling frequency so frequencies below a certain threshold give non-physical values that resemble delta-function behavior around zero. For completion here is the calculation for the obtained and experimental :

Conclusion

We have successfully used the method of Dynamic Light Scattering to determine the hydrodynamic size of polystyrene spheres in an aqueous solution. This was accomplished by taking a heating laser and scattering the light off the spheres then Fourier analyzing the resulting signal fluctuations caused by the particles’ Brownian motion producing out of phase photons. The obtained value of 0.171 +/- 0.006 μm agrees with the literature value of 0.173 +/- 0.005 μm.

Future Work

This experiment has been used to measure viscosities as well as hydrodynamic size determination. We're assuming it wouldn’t work very accurately, but we're still curious if it can be used to make a crude thermometer given that a viscosity and particle size are known.

References

1. Mackinnon, Edward. (2005). Einstein's 1905 Brownian Motion Paper. CSI Communications. 29. 6-8.

2. N.A. Clark, J.H. Lunacek, G.B. Benedek, A Study of Brownian Motion Using Light Scattering, American Journal of Physics 38, 575 (1970).

3. Shetty, Abhishek. (2010). Connecting Structure and Dynamics to Rheological Performance of Complex Fluids..

4. P. Koczyk, P. Wiewior, C. Radzewicz, Photon Counting Statistics - Undergraduate Experiment, American Journal of Physics 64, 240 (1996).

5. N.C. Ford, Jr , Light Scattering Apparatus, Dynamic Light Scattering Applications of Photon Correlations Spectroscopy, 9-58 (1985)

6. Newport.com (2019). Red HeNe Lasers. Retrieved from https://www.newport.com/p/R-30995

7. White Paper, Dynamic Light Scattering Common Terms Defined, Malvern Instruments Worldwide, pp 1-6 (2011)

8. H.S. Dhadwal, K. Suh, D.A. Ross, A Direct Method of Particle Sizing Based on the Statistical Processing of Scattered Photons From Particles Executing Brownian Motion, Appl. Phys. B 62, 575-581 (1996).

Special Thanks

Big appreciation to Professor William Zimmerman Jr., Neil Schroeder, and Kevin Booth for their guidance and support with this experiment.