dynamic light scattering
correlation spectroscopy inverse problem
Fredholm integral equation
V.B. Lopatko, A.G. Novoselov, E.B. Perminov, A.V. Sumarokov
The program realization of numeric solving of correlation spectroscopy inverse problem for the «LSDRS» dynamic light scattering spectrometer is considered in the paper.
The software of these spectrometers operates in data sampling and mathematical processing. In data sampling filtering, averaging and calculating spectra in frequency domain are performed, so experimentally available are the light intensity fluctuation spectra, the light being scattered by submicron particles, undergoing Brownian motion.
From the mathematical point of view, the spectroscopy inverse problem comes to solving of the first kind Fredholm equation with the Lorenzian kernel. This ill-posed problem is solved in this work by Tikhonov regularization method.
The result of solving this problem in the case of the polydispersity of the system is the reconstructed distribution function А(Г) over the hydrodynamic scatterers’ half widths Г of the fluctuation spectrum, or over these scatterers’ sizes (spherical shape of the scatterer is assuming, as diffusional halfwidth Г = Dq2 is simply related to the scatterer’s size for fixed experimental conditions by the Stokes-Einstein’s formula).
The basic physical and mathematical items of this problem, also as the applied iterative methods are briefly considered in the article. The conditions for the generalized solution for the intensity distribution function over the halfwidths are formulated. The schemes of main calculating moduluses of the computer program are presented.
The realized program has been tested using spectra experimentally obtained by the «LSDRS» spectrometer for one- and two-component water suspensions of calibrated polystyrene latexes with diameters 100 nm, 100 nm and 440 nm, respectively.
The screenshots of the program operation are presented.
The results of calculating for one- and two-component latex suspensions in water have the accuracy <5 % and <20 % respectively, and in some tests < 10 % for two-component suspension.
Authors have made the conclusion that better accuracy can be achieved using additional information; it may be the static scattering data, cross-correlation spectroscopy data etc.