Estimation of magnetic particle size distribution for ferrofluids based on nonlinear programming optimization and an improved Bayesian method

Abstract For the fundamental research and industrial application of ferrofluids, the particle size distribution (PSD) of magnetic nanoparticles is crucial to characterize their material properties. This work aims to develop efficient and robust PSD estimation methods for ferrofluids without interacting particles under two cases, known and unknown distribution types. The main task of the first case is to estimate the distribution parameters for a given PSD. For this purpose, Lp norm method is employed to convert the PSD estimation matrix equation into a constrained nonlinear programming problem. Then the nonlinear programming method (NPM) is presented to estimate the PSD parameters based on the magnetization data. The main tasks of the second case are to reconstruct the PSD curve and estimate the distribution type and parameters. To achieve these goals, an improved Bayesian method is proposed by using the estimation obtained from NPM as its prior information. Then a non-parameter hypothesis test method, Shapiro-Wilk test is introduced to determine the distribution type of the PSD. To illustrate the effectiveness of the proposed methods, theoretical analysis and Monte Carlo simulation are conducted based on the material parameters and experimental results of two ferrofluid samples. Through the comparison, it is found that the NPM is less sensitive to the noise like measurement errors and it can provide more accurate estimations than conventional inversion methods. Moreover, the proposed improved Bayesian method is efficient and robust to the noise, which can be used to reconstruct the PSD curves of ferrofluids without using the conventional time-consuming experimental methods like transmission electron microscopy.