SAXS Evaluation of Size Distribution for Nanoparticles

Abstract

Size distribution is an important structural aspect in order to rationalize relationship between structure and property of materials utilizing polydisperse nanoparticles. One may come to mind the use of dynamic light scattering (DLS) for the characterization of the size distribution of particles. However, only solution samples can be analyzed and even for those, the solution should be transparent or translucent because of using visible light. It is needless to say that solid samples are out of range. Furthermore, the size distribution only in the range of several tens of nanometers can be characterized, so DLS is useless for particles in the range of several nanometers. Therefore, the small-angle X-ray scattering (SAXS) technique is much superior when considering the determination of the size distribution in several nanometers length scale for opaque solutions and for solid specimens. Furthermore, the SAXS technique is applicable not only for the spherical particle but also for platelet (lamellar) and rod-like (cylindrical) particles. In this chapter, we focus on the form factor of a variety of nanostructures (spheres, prolates, core-shell spheres, core-shell cylinders and lamellae). Also getting started with a monodisperse distribution of the size of the nanostructure, to unimodal distribution with a narrow standard deviation or wide-spreading distribution and finally to the discrete distribution can be evaluated by the computational parameter fitting to the experimentally obtained SAXS profile. In particular, for systems forming complicated aggregations, this methodology is useful. Not only the size distribution of 'a bunch of grapes' but also the size distribution of all 'grains of grapes in the bunch' can be evaluated according to this methodology. This is very much contrasted to the case of the DLS technique by which only 'a bunch of grapes' is analyzed but 'grains of grapes in the bunch' cannot be. It is because the DLS technique in principle evaluates diffusion constants of particles and all of the grains in the same bunch of grapes diffuse as a whole. Thus, the methodology is important to highlight versatility and diversity in real materials, especially in soft matter, both in the liquid and in the solid states.

Nano/meso/macroporous materials characterization affected by experimental conditions and features of the used methods

“…The f(r) function could be converted into incremental PSD (IPSD) (r i ) = (f(r i+1 ) + f V (r i ))(r i+1  r i )/2 for better view of the PSD at larger r values (similar to the PSD based on the adsorption data). The main advantage of the SAXS method upon the textural characterization [16,[36][37][38][39] is that all open and closed pores could be analyzed in contrast to the adsorption methods giving the characteristics only of pores accessible for probe molecules (Figs. 6 and 7); therefore, practically always S SAXS > S BET (Figs.…”

“…To calculate the particle size distribution (PaSD) functions based on the SAXS data [39], several models of particles (e.g. spherical, cylindrical, lamellar ones and various blends of them [37]) could be used.…”

“…where C is a constant, r is the radius of particles, f(r) is the distribution function (differential PaSD), and P(r) is the form factor for spherical particles [39] (the kernel of the integral equation):…”

“…is the first-order Bessel function, V = πr 2 H is the cylinder volume, and C is a constant [39]. For lamellar particles [39] max min ( ) ( , ) ( )…”

Applying SAXS to study the structuring of Fe3O4 magnetic nanoparticles in colloidal suspensions

“…The presence of MNP structures in liquid suspensions can be estimated using scattering techniques, where the scattering pattern can be modelled using a form factor that gives information about particle size and morphology, and a structure factor, that gives information about how particles are spatially assembled [9].…”