Statistical analysis of Dynamic Light Scattering data: from the Schätzel formulas to new approaches based on multi−tau Photon Counting Histogram and variance methods

Dynamic Light Scattering (DLS) is an optical technique aimed at the determination of the dimensions of small particles in a suspension in a range of diameters from about 1 nm to 1 μm. It consists in the measurement, at a fixed angle, of the scattered intensity radiation that, because of the Brownian motion of the particles, fluctuates stochastically with time. A DLS experiment provides the correlation function of the received intensity which is dependent from the coherence time of the fluctuations, that is related to the hydrodynamical radius of the particles that constitute the sample. In this thesis work, we analyzed the error bars associated to a correlation function and proposed a new approach for the analysis of Dynamic Light Scattering data based on multi-tau Photon Counting Histogram and variance methods. We obtained the following results and future perspectives. Statistical analysis of Dynamic Light Scattering data The first goal of this work consists in the analytical determination of the error bars associated to a correlation function. This problem has been analyzed by K. Schätzel in the 1990s. He provided two analytical expressions for the covariance matrix and the variance for the specific case of a Lorentzian spectrum where the correlation function is characterized by a single exponential decay. These formulas do not include the effects due to a triangular averaging and are consequently inaccurate for sampling times higher or similar than the coherence time. In this work, we analyzed these formulas and worked out two exact analytical expressions in which the effects due to the triangular averaging are corrected to all orders. By the use of extensive computer simulations and experimental test carried out on dilute dispersions of calibrated latex spheres, we have shown that the new formula for the variance works quite accurately for sampling times both higher and lower than the coherence time and can be applied well beyond the specific case of a single exponential decay auto-correlation function. We believe that these new covariance and variance formulas would turn out to be a fairly useful tool for the wide community of scientists working in the field of DLS. Nowadays, this technique is ubiquitously based on the use of multi-tau correlators, where, if not properly taken into account, the triangular averaging would introduce huge errors in the estimates of the uncertainties to be associated to the measured correlation function. Photon Counting Histogram applied for the analysis of Dy- namic Light Scattering data The second goal of this thesis work consists in the analysis of a second method for performing particle sizing. In this case the coherence time of the intensity fluctuations will not be recovered from a correlation, but from a series of histograms of the photon counts detected over different integration times. It represents, once normalized, the probability distribution to detect a discrete number of photon counts over a given sam- pling time. In my work we applied the Photon Counting Histogram (PCH) technique for the analysis of DLS data and we analyzed, for a monodisperse sample, the combined use of these two techniques and the advantages it gives with respect to a single one. This technique has been developed for the first time by E. Gratton for the analysis of fluorescent data. His theory works when the sampling time is much smaller than the coherence time. By using the PCH method for the analysis of DLS data, because of the raw data has been taken with the multi-tau method, we analyzed different histograms generated with different sampling times which are both higher and lower than the co- herence time. Moreover, we exploited an alternative method to perform particle sizing by recovering the hydrodynamical diameter by fitting not the histograms themselves, but the variances of the photon count distributions recovered stage by stage as a func- tion of the sampling time. We believe that both the PCH method and the variance analysis applied to DLS data may be useful because they provide a different technique to perform particle sizing, which can work independently from the auto- (or cross-) correlation functions. Both these three techniques can work in parallel and, if they provide the same result, it can be a further proof or cross-check for the correctness of the recovered hydrodynamical diameter of the considered sample