Small-angle X-ray scattering analysis of catalysts: comparison and evaluation of models

Abstract

Small-angle X-ray scattering (SAXS) can be used to obtain interphase surface areas of a system, such as a supported-metal catalyst, composed of internally homogeneous phases with sharp interphase boundaries. Measurements of SAXS for samples of porous silica, alumina, platinum on silica, and platinum on alumina are reported. A variety of models and forms for the correlation function, the Fourier transform of which gives the X-ray scattering, are considered, and theoretical and measured intensities are compared. A criterion of fit for comparing models with different numbers of parameters is proposed. For the two-phase (unmetallized) systems the 'Debye-random' model must be rejected. Modifications of the Debye (exponential) correlation function are also not particularly good compared to an exponential-plus-Gaussian form, not derivable from a physical model, and forms based on Voronoi cell models. Since intensities can be fit to experimental error with a five-parameter correlation function, it seems incorrect to ascribe significance to the result of fitting a function with six or more parameters. It is shown that values for the single interphase surface area can be obtained independently of a model. However, fitting intensities using a modelbased correlation function gives information about the structure of the system. The two-cell-size Voronoi and the correlated Voronoi cell models are useful in this regard. For the systems containing metal, fiveparameter correlation functions again suffice to fit intensities. However, for three-phase systems a model or physical assumption is necessary to obtain values for the three surface areas from X-ray scattering intensities. The area of the surface between support and void is quite insensitive to the assumptions employed and the metal-support surface area somewhat less so, but values for the metal-void surface area $23 are consistent only to one significant figure from model to model. If the support in the three-phase catalyst is known to be unchanged from support in the absence of metal, a 'support-subtraction' model can be used to obtain reliable values for $23. In the present systems, the assumption does not seem to be borne out.

On the Porod law

“…* However, the meaning of the law is not always clear. It rests on some assumptions which require interpretation, main-*Porod-law analyses of experimental data are, for example, used by Brumberger, Delaglio, Goodisman & Whitfield (1986), Goodisman, Brumberger & Cupelo (1981) and Ciccariello & Benedetti (1986). ly related to passing from a continuous to a discrete description of a sample.…”

Structure analysis of multiphase systems by anomalous small-angle X-ray scattering

“…It is often treated by subtracting scattering curves measured at two X-ray energies. This is strictly justified when no mutual interference occurs between the catalyst and the pores or the support [50]. Then, 0 = relations were proven to hold for threephase Debye-random system [13].…”

“…Equations (49) and (50) are mathematically equivalent to (8) and can be also presented in the form of (10). Since 49) and (50) show that there are p(p-1)/2 independent () Ŝ functions which is exactly the number of the linearly independent "stick probability functions" [39,41].…”

“…For a completely unknown system under study, both the scattering functions () Sˆ and the atomic number density differences (49) or (50). Alternatively, one may use electron microscopy to determine the geometrical structure and to try to calculate the phase scattering functions () Ŝ .…”

“…In the later case [38,41,50], the most frequently used model involves exponential correlation function , but Gaussian and other functions were also used. The studies went in this direction, since it is not possible to determine by SAXS the phase surface areas in system with more than two phases without assuming a structure model.…”