Fitting Procedure to Reconstruct the Size Distribution and the Concentration of Silver Colloidal Nanoparticles from UV-Vis Spectra

Abstract

In this work, a complete fitting procedure of UV-Vis spectra of silver nanoparticles in colloidal solutions is reported. The fitting function, based on the Beer–Lambert law, Mie theory, and log-normal probability distribution of nanoparticles’ sizes, is developed and confirmed by 33 different independent measurements. In order to validate the accuracy of the function’s behavior on different spectra, freely accessible measurements were used, proving that the fitting function works independently of the method of their production—laser or chemical synthesis of nanoparticles. The developed fitting function is, to the best of our knowledge, novel and not based on any conventional spectral analysis approaches like the Mie–Gans procedure. Furthermore, since fitted parameters are all physical, it allows determination of the mode diameter of nanoparticles as well as the standard deviation of the log-normal distribution of sizes. It enables the reconstruction of size distribution of nanoparticles in colloidal solution. Step-by-step derivation of the fitting function is provided with a physical explanation of all parameters. The importance of Lorentzian dependence emerging at the core of Beer–Lambert law is physically discussed and linked to harmonic oscillator behavior of localized surface plasmon resonance of silver nanoparticles in a colloidal solution. Size distribution reconstruction from fitted parameters according to a log-normal distribution function is provided and a concentration calculation is presented.