Advanced Size Distribution Determination of Superparamagnetic Nanoparticles

Abstract

Superparamagnetic nanoparticles are drawing increasing interests because of their special physical properties and potential applications in magnetic-supporting catalysts, water purification, ferrofluids, and medical diagnostics [1,2]. For most applications, an accurate determination of the particle size distribution is required since the magnetic properties of the particles directly depend on the particle size. Transmission electron microscopy (TEM) and X-ray diffraction (XRD) are widely used to determine the particle size. However, both methods have limitations. For TEM, normally only a limited amount of particles contribute to the result, and the precise determination of extremely small particles is difficult due to the limit of the TEM image contrast and resolution. For XRD, the evaluation is based on the Scherrer equation, the result of which is the average size of the particles, more detailed information of the size distribution is not available. Furthermore, this method assumes that the peak broadening is solely due to the crystallite size, other effects such as microstrain are neglected. Besides, neither TEM nor XRD are suitable for embedded nanoparticles.Above the blocking temperature TB, the magnetic moment versus external magnetic field curve (m-H curve) of superparamagnetic particles is described by m=mSL(x), with x=μ0mp(d)H/kBT, where m is the magnetic moment, mS is the saturation magnetic moment, μ0 is the vacuum permeability, mp is the magnetic moment of one particle of size d, H is the external field, kB is the Boltzmann constant, T is the temperature in Kelvin, L(x) is the Langevin function: L(x)=coth(x)-1/x. Since the m-H curve is dependent on the size d, the particle size could be determined by fitting the measured m-H curve with the Langevin function. Note that the realistic particle size normally includes a distribution. In 1978, Chantrell et al. [3] presented a method to determine the mean value and standard deviation of the particle size distribution. This method fits the measured m-H curve to a weighted sum of Langevin functions using a pre-assumed lognormal size distribution. Since then, this method has been used in many works for decades [4-6]. However, the assumption that the size distribution is always lognormal might not be accurate for particles synthesized by all the techniques.In this work, we present a fitting method without a pre-assumed distribution. The measured m-H curve is fitted to a superposition of Langevin functions with corresponding fractions, and the combination of these fractions is the size distribution. In the fitting procedure, three constraints are considered: 1) The method of least squares is used to obtain a well fitted m-H curve; 2) The smoothness of the distribution is a criterion to judge whether the fit is valid, since a realistic distribution should be relatively smooth; 3) Saturation magnetic moment should be fitted since the small particles cannot be saturation magnetized. To obtain a fitted distribution under all these three constraints with optimized weights, we apply the method of Lagrange multipliers.This fitting method is used to determine the size distribution of five different superparamagnetic γ-Fe2O3/SiO2 core-shell nanoparticle samples. The m-H curves were measured by VSM and then fitted with a supperposition of Langevin functions. The fitting results of selected samples are shown in Fig. 1 and Fig. 2. The fitted mean values and standard deviations of the particle size distributions agree well with the results of TEM and XRD [1]. **