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Small‐angle X‐ray scattering analysis of catalysts: comparison and evaluation of models
Author(s) -
Brumberger H.,
Delaglio F.,
Goodisman J.,
Whitfield M.
Publication year - 1986
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s0021889886089331
Subject(s) - small angle x ray scattering , voronoi diagram , exponential function , scattering , interphase , small angle scattering , correlation function (quantum field theory) , function (biology) , phase (matter) , materials science , debye , computational physics , mathematics , statistical physics , chemistry , physics , mathematical analysis , geometry , optics , condensed matter physics , statistics , spectral density , evolutionary biology , biology , genetics , organic chemistry
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 model‐based 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, five‐parameter 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 S 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 S 23 . In the present systems, the assumption does not seem to be borne out.