**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.