Photon Correlation Spectroscopy of Polydisperse Samples: II. Optimal choice of histogram parameters with exponential sampling

Abstract

The use of a histogram model of the distribution function of decay rates in the analysis of light-scattering data from polydisperse samples is investigated. Histograms having a wide range of step sizes and number of steps have been fitted to computer-generated data of various polydispersities and noise levels. These histograms have been compared with the original distributions to determine the optimum histogram parameters in each case. The parameter omax, which determines step size, decreases linearly with increasing In (noise) but only at about half the rate predicted by the theory of McWhirter and Pike. Wmax also decreases with increasing polydispersity. The number of steps is determined, as expected, by the step size and the need for the histogram to span fully that part of the distribution function which contributes to the autocorrelation function at a level above noise. IntroductionIn a recent paper [1], hereafter called I, we described a method of analysing on a desk-top computer (Hewlett-Packard 9835A) data from light-scattering experiments on polydisperse solutions of scatterers, using exponentially spaced histogram steps to describe the distribution function of decay rate y. The method is based on a proposal by McWhirter [2] and Pike [3], subsequently developed and applied by Ostrowsky et al.[4] using a -function representation. In our method different histograms, obtained by systematically varying the step positions, are overlaid to smooth the recovered distribution function. In the present paper we discuss as functions of polydispersity and noise level, the optimal choice of the 'centre' of the histogram (), the number of knots (k) or steps (k -1), and the parameter .)max that determines the step size. These three parameters conveniently describe the span of the histogram that is most appropriate to a particular distribution, We also comment on the choice of a suitable sample time for each channel of the autocorrelator used to collect the data.As in I we have made use of computer-generated data of precisely known characteristics, in order to facilitate a proper comparison between the original distribution of decay rates and the histogram distribution recovered by the fitting procedure. The results of such a comparison are intended to provide a set of working rules which may be used with experimental data of unknown distribution. In the data analysis we again restrict ourselves deliberately to linear routines that can be implemented on a desk-top computer.

Machine Learning for Analysis of Time-Resolved Luminescence Data

“…We implement three simple basis sets: where δ­(Γ) is the Dirac delta function (delta-basis , ), θ h (Γ) is the Heaviside step function (histogram-basis , ), and χ i (Γ) represents the triangular function defined on the region [Γ i –1 , Γ i +1 ] and centered on Γ i (triangular-basis). The decay rate axis Γ ∈ [Γ min , Γ max ] is discretized using exponential sampling, , which puts the individual Γ values equidistant on the logarithmic scale: Γ k = Γ 0 exp­( k π/ω), where ω is a parameter that defines the resolution of sampling.…”

“…16,17 was then extended by introducing the histogram method, which does not need additional normalization of the parameters and offers better resolution of the decay rate distribution. 11,12 The works by Santra et al, 18,19 Regularization combined with the decay rate distribution approach was applied to TRPL data by Petrov et al 10 for analysis of fluorescence dynamics of perylene molecules, while Slavov et al 8 used the regularization in combination with LDA to obtain stable recovery of lifetime distribution. Another way of using regularization to obtain decay rate distribution is the maximum entropy method, which uses Shannon-Janes entropy as a smoothing function (Kumar et al 7 and refs therein).…”

“…The concept of determining a decay rate distribution instead of trying multiexponential fits to describe photophysical processes has been previously applied to analyze particle size distribution of scatterers in polydisperse samples, ,, spin and charge transport dynamics in doped semiconductors, nanoparticle photoluminescence in inverse opal photonic crystals, and the influence of external refractive index on fluorescence kinetics in organics, to name a few. McWhirter and Pike wrote extensively about the numerical inversion of the Laplace transform, which was later applied to other invariant transforms for applications in photon correlation spectroscopy and Fraunhofer diffraction. , Ostrowsky et al introduced the exponential sampling method (ESM) for numerical inversion of the Laplace transform to sample the decay rate distribution of polydisperse scatterers.…”

“…McWhirter and Pike wrote extensively about the numerical inversion of the Laplace transform, which was later applied to other invariant transforms for applications in photon correlation spectroscopy and Fraunhofer diffraction. , Ostrowsky et al introduced the exponential sampling method (ESM) for numerical inversion of the Laplace transform to sample the decay rate distribution of polydisperse scatterers. The ESM was then extended by introducing the histogram method, which does not need additional normalization of the parameters and offers better resolution of the decay rate distribution. , The works by Santra et al, , Kim et al, and Maus et al demonstrate that use of maximum likelihood estimation combined with Poisson instead of Gaussian statistics makes it possible to reliably analyze low signal data. Regularization combined with the decay rate distribution approach was applied to TRPL data by Petrov et al for analysis of fluorescence dynamics of perylene molecules, while Slavov et al used the regularization in combination with LDA to obtain stable recovery of lifetime distribution.…”