S18_Dynamic Light Scattering

Introduction

Dynamic Light Scattering (DLS) is a tool to probe the Brownian Motion of particles suspended in a solution. DLS can be used to ascertain information about the diffusion constant of the motion, from which properties of particle size and the viscosity of the medium can be determined. We induced Rayleigh Scattering through an incident laser light onto a solution of latex spheres and deionized water. In previous years, MXP students performing DLS experiments have analyzed the scattering light with the use of photon-correlation spectroscopy (PCS) and associated autocorrelation functions. In this experiment a different method of analysis is presented through the use of photon counting statistics. The collected scattered light intensity was viewed in Poisson and Bose-Einstein Distribution in hopes of deducing the spheres radial length.

Theory

As the latex spheres move about in the suspension of deionized water, collisions with solvent molecules cause a stochastic motion called Brownian motion. The diffusion constant of the motion can be related to the size of the Brownian particles by the Stokes-Einstein relation

where is the Boltzmann constant, is the absolute temperature of the medium, is the viscosity of the medium, and is the particle radius. The method of analysis in this experiment was photon counting statistics, wherein the fluctuating intensity is viewed as a stream of photons with different probabilities for different numbers of counts based upon the sampling time used. The form of the resulting distribution is dependent on the coherence time of the particles. The notion of coherence time can be presented heuristically as follows: although the intensity fluctuations due to the laser light scattering off of the Brownian particles are stochastic, these fluctuations must necessarily occur at a finite rate. Therefore, if at some time the amplitude and phase of the wave is known, then at some nearby later time the amplitude and phase can be predicted with a probability distribution. However, over a long time period, the stochastic fluctuations in the amplitude and phase make the current values of these parameters completely uncorrelated to their original values. The characteristic time that divides these two regimes is the coherence time

is the count number and is the probability of that count.

Recovering the particle size from the observed statistical distributions can be achieved by calculating the degree of coherence , which can be shown to be

is the count variance, is the mean count, is a dimensionless parameter, is the sampling time, is the characteristic linewidth, is the diffusion constant,

is the scattering wave vector, is the index of refraction of the medium, is the wavelength of the incident laser in vacuum, and is the scattering angle. To calculate the particle size then, the degree of coherence is calculated for several sampling times using the variance and mean count and then fit to the equation on the right side using a nonlinear fit.

Experimental Setup

Figure of Experimental Setup: The red vectors represent the lasers path to the cuvette containing the latex sphere solution.

Using a 10mW, Helium-Neon laser of wavelength 632.8nm, the light was directed off a series of mirrors, for easy laser alignment, and into the sample. The sample was housed in a cuvette and its associated holder. Scattered light was collected at a 90° angle from the light incident onto the sample. The focusing lens was used to narrow the beam in an attempt to minimize the scattering within the sample. The apertures were an iris and a slit opening on the photomultiplier tube (PMT) that were tightly bound to shrink the amount of coherence areas. Data collection was then ran through a Data Acquisition Board and processed as photon counts by a developed LabVIEW program, which generated the relative histogram.

Procedure

For a given sample and sampling rate, a LabVIEW event counter program was used to generate 10000 photon count measurements, and each set of measurements was repeated 10 times. The program then calculated the mean, variance, and degree of coherence and their standard deviations from each set of measurements. Sampling times in the range of 0.1-2.1 ms with intervals of 0.4 ms were tested. This range was chosen based upon a trial-and-error process in which the goal was to be able to see both Bose-Einstein and Poisson statistics for different sampling times in the given range. From the calculated degree of coherence values, a MATLAB script was used to perform a nonlinear regression on the data in order to determine the unknown coefficients

and . The particle size was then calculated from the determined value of using the Stokes-Einstein relation.

Data

Data collection commenced using 0.240 μm manufactured spheres over sampling times of 100μs to 2.1ms as demonstrated by the histograms below. The interval of sampling times increased to 400μs as preliminary runs were demonstrating vastly similar histograms at intervals smaller than this. The calculated amount of coherence areas was found to be 2123 as derived by the equation below, where

as R is the distance between pinholes of 36.5 cm in Figure 2, a is the radius of the iris (the first aperture past the sample), and lambda is the wavelength of the laser.

Poissonian distribution of photon counts collected for a sampling time of 2.1 milliseconds.

Bose-Einstein distribution of photon counts collected for a sampling time of 100 microseconds.

Analysis

Degree of coherence values for various sampling times. The solid curve is a nonlinear fit to the data.

From the nonlinear fit, the calculated particle size was 3.27 nm, compared to a manufacturer provided value of 240 nm. Despite qualitatively confirming the expectation of seeing both Bose-Einstein and Poisson distributions at sampling times less than and greater than the coherence time, respectively, the determined particle size from the fit was markedly inaccurate, being off from the expected value by two orders of magnitude. The exact reason for this discrepancy is unknown, but the most probable explanation for the inaccurate particle size determination is not observing a sufficient amount of fluctuations in the intensity of the scattered light. In the context of photon counting statistics, this corresponds to not observing a large enough variance for a given mean count. Due to time constraints, a full analysis of why the variance-mean ratio was too low could not be performed, but there are a few possible solutions to increase it.

One way to increase the variance-mean ratio would be to reduce additional counts due to extraneous noise that inflates the count number without correspondingly increasing the variance. While measures such as the Faraday cage and turning off computer monitors while taking data were taken to mitigate noise, it is possible that there were other sources of noise not taken into account that could have increased the mean count number without increasing the variance, consequently reducing the degree of fluctuations.

Another way the degree of fluctuations could be improved is to reduce the amount of coherence areas. For the experimental apparatus used, the most convenient way to accomplish this would be to increase the distance between the two apertures or further reduce the area of the exposed PMT photocathode.

Conclusion

The statistical nature of latex spheres undergoing Brownian motion was observed qualitatively. At sampling times less than and greater than the coherence time of the scatter light signal, Bose-Einstein and Poisson distributions were viewed for the collection of photon counts. A numerical calculation was made to determine the radius of the latex spheres within the solution based on the statistical models, resulting in a inaccurate determination of there size. This underestimation, of a calculated value 1.36% of the total value, lead to a quantitative failure for the collected data. If allotted more time, an in-depth look at the effects of the number of coherence areas could have increased the amount of fluctuations seen within the signal and thus increased the accuracy of these results.

Selected References

• • H. S. Dhadwal, K. Suh, and D. Ross, “A direct method of particle sizing based on the statistical processing of scattered photons from particles executing brownian motion,”Applied Physics B, vol. 62, pp. 575–581, 1996.

    • N. Ford, “Light scattering apparatus,” in Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy (R. Pecora, ed.), ch. 2, pp. 7–57, New York: Plenum Press, 1985.

    • P. Koczyk, P. Wiewior, and C. Radzewicz, “Photon counting statistics—undergraduate experiment,” American Journal of Physics, vol. 64, no. 3, pp. 240–245, 1996.