Dynamic Light Scattering Michael M Folkerts and Arielle Yablonovich Kleinfeld Physics 173 Spring 2010 UCSD Physics Department Introduction There are several ways to experimentally determine the sizes of proteins or other small particles Standard optical microscopy is one option although diffraction limits the spatial resolution to half the wavelength of the light While there are other microscopy methods that do not have such limitations a simpler approach to this problem is to take advantage of the Brownian motion of very small particles The probability distribution of the random walk of particles is proportional to their diffusion constant which in turn is related to the particle radii If a monochromatic coherent light source shines on a solution of these particles then the intensity of the scattered light will change continuously with time as the particles diffuse in random directions Detecting and analyzing these changes in intensities as a function of time can lead to the determination of the diffusion coefficients and hydrodynamic radii of the particles This general method known as dynamic light scattering DLS will be used here to determine the sizes of particles While many DLS experimental set ups detect the scattered light with a photomultiplier tube PMT the set up here uses a CCD camera instead The fact that the camera contains many sets of pixels allows us to generate multiple data sets in a short amount of time thus giving us good statistics on the measured particle sizes In contrast only one data point can be taken at a time with a PMT The CCD camera also makes it possible to obtain measurements at multiple scattering angles in a single data acquisition The interference of the waves of the scattered light forms a speckle pattern on the pixels of the CCD The changes in intensity of the speckle pattern are measured as I t for increasing values of This information can be used to graph the autocorrelation function which describes changes in a particular measurement as a function of the time separation between the measurements This function can then be fit to a decaying exponential and the information obtained from this fit will allow us to calculate the diffusion constants and radii of the particles In performing these measurements we sought to see how changing one particular variable at a time changes our output data and figuring out why the variable affects the system in that way Although all measurements were performed with beads this set up could also be used to find the hydrodynamic radii of proteins or to analyze the binding and aggregation of proteins or nanoparticles Methods A 633nm laser with a 1mm diameter beam is used as the monochromatic light source for scattering The CCD camera is a Basler spL2048 140km with two rows of 2048 10 m by 10 m pixels We turned on horizontal and vertical binning to increase our photon count Binning increased the effective size of the detector to one row of 1024 20 m by 20 m pixels The scattering angle was calculated for each pixel using the schematic in Figure 1 Figure 1 Illustration of our experimental setup The scattering angle x1d6f3 for each pixel is defined as the equipment angle 90 degrees in this case minus x1d6fc the angle from the aperture to each of the pixels The scattering volume is defined by the overlap of the projection of each pixel through the aperture with the cylinder of illuminated by the laser beam The speckle size is defined as the location of the first dark fringe in far field diffraction for a given aperture with diameter a When the path length difference becomes 1 2 wavelength the speckle size is equal to L a where is the wavelength of the light and L is the distance between the aperture and the detector We set the speckle diameter equal to the size of a pixel to get a strong signal without mixing of adjacent speckles With a pixel size of 20 m L 11 cm and 633 nm we obtain an aperture diameter of 3 5 mm Due to the refraction that occurs across the cuvette boundary we had to adjust our calculation to back trace the scatting angle Sample Preparation Polystyrene beads diluted in water were used for measurements The bead solutions were put in square plastic cuvettes Prior to taking measurements the bead samples were vortexed for approximately ten seconds In the experimental apparatus the cuvette was placed as close as possible to the aperture while still allowing for the entire laser beam to pass through the sample With this constraint the illumination volume was located about 1mm from the cuvette wall Theory Particles in a mixture will diffuse in random directions According to Brownian motion the probability of finding a particle at position r at time t is where D is the diffusion coefficient of the collection of particles According to the StokesEinstein equation the diffusion coefficient is related to the hydrodynamic radii RH in the following way where kB is the Boltzmann constant T is temperature and is the viscosity of the solvent The change in momentum between the incident light and scattered light is q Here n is in the index of fraction of the solution is the wavelength of the incident light and is the scattering angle The common approach in dynamic light scattering DLS is to auto correlate intensity measurements as a function of offset time Where the intensity I measured by our charged couple device CCD detector is the square of the scattered wave vector Following the derivation set forth in 1 one can equate the the autocorrelation function to an exponential This exponential equation can be used to obtain the diffusion coefficient D and subsequently the hydrodynamic radius of the beads Data Analysis A collection of MATLAB scripts where created to analyze our raw data These scrips can be found in the code folder included with this report Basic outline of code Correlate pixel data Fit autocorrelation function for each pixel in detector Save Dq2 value decay frequency for each pixel Calculate q value for each pixel using Snell s law definition of q equipment and pixel angle Plot Dq2 vs q2 along with theoretical value Average over q values and Dq2 and use these value to compute RH Results Data The following figures and tables include raw data from the CCD camera graphs of the decay rate as a function of bead concentration and equipment angle graphs of the autocorrelation function itself and tables showing our measurements of the bead radii The two bead diameters measured were 214 nm and 2 9 m The different