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The crystalline microstructure of mullites obtained by heating monophasic gels has been investigated. Gels with alumina to silica molar ratio of 3:2 (as in secondary mullite) and 2:1 (as in primary mullite) were prepared by gelling mixtures of aluminium nitrate and tetraethylorthosilicate. Phase transformations were induced by heating the gel precursors, with different final treatment temperatures between 1173 and 1873 K. The mullites formed as a result of thermal treatment were studied by means of X‐ray diffraction, scanning electron microscopy and transmission electron microscopy. The crystalline structure (unit‐cell parameters) and microstructure were determined from X‐ray diffraction patterns. The formation of mullites of homogeneous chemical composition and with unit‐cell parameters depending almost linearly on the treatment temperature was found. Their compositions, expressed as alumina to silica molar ratio, were determined from the unit‐cell parameters and were in the range of those characterizing primary and secondary mullites. Mullites processed at lower temperatures were accompanied by small amounts of vitreous phase. The crystalline microstructure of the obtained mullites was interpreted by means of a mathematical model of polycrystalline material, involving prevalent crystallite shape, volume‐weighted crystallite size distribution and second‐order crystalline lattice strain distribution as model parameters. The model parameters were determined for each sample by modelling its X‐ray diffraction pattern and fitting it to a measured pattern. Bimodality of the size distribution was observed and explained as a consequence of two crystallite nucleation and growth processes, which started from small alumina‐rich and alumina‐poor domains, spontaneously formed in a precursor gel at early stages of heating. Images produced by scanning and transmission electron microscopy agreed well with the characteristics obtained from the analysis of the X‐ray diffraction patterns.

Since the ®rst publication of the direct-methods modulus sum function [Rius (1993). Acta Cryst. A49, 406±409], the application of this function to a variety of situations has been shown in a series of seven subsequent papers. In this way, much experience about this function and its practical use has been gained. It is thought by the authors that it is now the right moment to publish a more complete study of this function which also considers most of this practical knowledge. The ®rst part of the study relates, thanks to a new interpretation, this function to other existing phase-re®nement functions, while the second shows, with the help of test calculations on a selection of crystal structures, the behaviour of the function for two different control parameters. In this study, the principal interest is focused on the function itself and not on the optimization procedure which is based on a conventional sequential tangent formula re®nement. The results obtained are quite satisfactory and seem to indicate that, when combined with more sophisticated optimization algorithms, the application ®eld of this function could be extended to larger structures than those used for the test calculations.