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Brief Overview
Dynamic light scattering, the way in which light scatters off of particles in suspension, has the potential to yield a great deal of information about those particles, including their size and some notion of concentration. Such information is useful in many scientific fields, and the techniques of dynamic light scattering can provide such information relatively quickly and inexpensively, compared with other methods. The most direct applications for the method are for those who are interested in diffusive properties of a given solution, such as bacteria or proteins in suspension. Dynamic light scattering is also being used to detect early stages of cataract formation in the eye, detecting the presence of ‘clouding particles’ before they would otherwise be evident. Following is a brief discussion of the concepts that make dynamic light scattering work.

In this experiment we worked to improve on the accuracy and precision of measurements of hydrodynamic radii made by using relatively simple dynamic light scattering techniques. Examination of the physical relationships detailed in the THEORY section and of available literature makes it apparent that there are a number of factors it might be possible to control or measure more carefully then has been the case in past undergraduate laboratories. Correlation problems could be addressed by using cross-correlation (rather than auto correlation) techniques, made possible by simultaneous collection of two different data series at the same scattering angle.


Consequences of Brownian Motion
Brownian motion is a phenomenon that is fundamental to this experiment. It describes the way in which very small particles move in fluid suspension, where the fluid consists of molecules much smaller than the suspended particles. The motion of suspended particles is random in nature, and arises from the cumulative effect of bombardment by the suspending medium’s molecules. Molecules in a liquid are, of course, constantly in motion, randomly bouncing off one another. As these molecules move around in the liquid, they are also bouncing off any suspended particles in a random manner, imparting a momentum to the suspended particles, the magnitude and direction of which fluctuate in time. It is the resulting ‘random walk’ behavior of the suspended particles that is called Brownian motion, and that randomness makes this experiment possible.


Dynamic Light Scattering
When a laser beam is shined through a liquid with suspended particles, the beam scatters off of those particles in all directions, resulting in a scattering-angle-dependent intensity pattern. When the particles are experiencing Brownian motion, the intensity pattern also fluctuates randomly. For the purposes of this experiment, the scattering involved is near the lower threshold of Mie scattering, given that the particle sizes are on the order of the wavelength of the incident light . The resulting scattering pattern is ‘speckled’ in appearance, and the speckles can be divided into two primary categories:
1 – speckles composed of light that is scattered from a single particle, and
2 – speckles composed of light that has scattered off of several particles.
These two types of scattered speckles differ in one very important way – if the incident light is polarized perpendicularly to the plane on which the scattering angle is measured (horizontal plane), singly scattered speckles are tall, narrow vertical streaks intersecting the horizontal plane. Multiply scattered speckles are much smaller, roughly round in shape, and are located randomly with respect to the horizontal plane. The differences in speckle populations arises from the fact that polarization is not well preserved in Mie scattering. Speckles arise from scattered laser light that has remained relatively polarized, and also has not been destructively interfered with by other less well preserved scattered light. In our experiement, the scale of these singly scattered speckles at the distance of our detector (˜0.85m) was on the order of a few cm tall, and approximately 0.5cm wide. Subsequent scattering further destroys the polarization of the incident light, resulting in significantly smaller speckles (<0.2cm in our case), which are relatively short lived and rapidly moving (owing to the complex nature of multiple interactions of light with independently randomly moving particles). Measuring the intensity fluctuations at a given scattering angle can yield a great deal of information about the particles scattering the laser beam, including the ‘hydrodynamic radius’ of the suspended particles.


Hydrodynamic radius
The hydrodynamic radius of a particle is the effective radius of an irregularly shaped particle that is used when describing the manner in which particles in suspension diffuse through the suspending medium. For a hard sphere, like those used in this experiment, the hydrodynamic radius equals the radius of the sphere.


Intensity Cross-Correlation
The randomness of the fluctuation of the intensity of scattered light allows us to use random statistical methods to analyze that scattering pattern. The most important of these is called correlation. If the intensity at a given scattering angle is recorded over a small sample time, the fluctuation of the intensity arising from Brownian motion can be expected also to be small. If two such recordings are made from the same scattering angle simultaneously, the two samples can be compared with one another through cross-correlation, and a measure of, essentially, how quickly the scattered light intensity changes with time can be obtained. Correlation is a mathematical method that essentially evaluates how similar two signals are to one another, and, as it applies to our experiment, it can be reasonably assumed that the two signals will be strongly correlated to one another when they are ‘in synch’ temporally, but grow progressively less correlated as one signal is compared with a time-shifted version of the other.

Here would have liked to take advantage of the differential speckling between singly and multiply scattered light. By carefully placing and aligning two detectors along a perpendicular (but very near) to the scattering plane, we can be confident that both detectors can be illuminated by a single, large, singly scattered speckle, and that they cannot both be illuminated by a single, small, multiply scatted speckle. Accordingly, the two signals will be most strongly correlated when both detectors are illuminated solely by singly scattered light. Reduction of multiple scattering is essential to achieving reliable results. Single scattering is a problem that can be effectively modeled, but the physical complexity of multiple scattering seriously degrades the effectiveness of the model, so screening any secondary effects is desirable.

In a single detector setup (as was used in this experiment), autocorrelation takes the place of cross-correlation. While a single detector is easier to implement in the laboratory, it is far more susceptible to multiply scattered light. Autocorrelation is mathematically identical to cross-correlation, except that rather than comparing two signals with one another, one signal is compared with a time-delayed version of itself. Not surprisingly, cross-correlation offers a particular advantage when concentration levels of suspended particles are relatively high, making secondary scattering more likely.


Experimental Improvements
When trying to establish the size of suspended particles, there are a number of factors that can affect precision and accuracy. As described above, one significant loss of accuracy arises from multiple scatterings. In addition to using cross-correlation as opposed to autocorrelation to limit that effect, minimizing uncertainty in scattering angle can further screen out unwanted scatterings. Any detector that examines the intensity at a given location has a finite aperture, and consequently subtends a finite angle. Minimizing that subtended angle helps to reduce the collection of unwanted light that scatters from particles near the ones being examined. Minimizing the subtended angle also improves the precision of a derived particle radius, since the radius is functionally dependent upon the scattering angle. The most straightforward method for reducing that subtended angle is to both minimize the aperture size and maximize the detector distance. Similarly, very careful attention to detail when placing detectors and measuring scattering angle also serves to improve precision. Calculations of size are particularly sensitive to scattering angle (especially for angles less than about 120 degrees), because it appears in our calculations as the square of the sin of the half-angle.

There is also an important functional dependence upon temperature, and improved precision of temperature measurement corresponds to an improved precision in a determination of radius. A technique that has been used by some to help minimize temperature fluctuations and allow for more precise measurements of temperature is the immersion of the sample solution in a bath. Such a bath acts a heat reservoir, helping to keep sample temperatures stable. As long as the bath has an index of refraction that matches the sample, and the nested containers are carefully aligned, the effect on the laser beam and subsequent scatterings will be negligible for our purposes. The most compelling reason to carefully control temperature for dynamic light scattering is that not only does temperature appear in our calculations of particle size, so do two quantities that are highly temperature dependent: viscosity and index of refraction. Of the three, the strongest dependence is upon index of refraction, which appears in our calculations as a square.