Theoretical assessment of systematic errors in volume fraction determinations by microscopy methods
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Theoretical assessment of systematic errors in volume fraction determinations by microscopy methods Edgar Dutra Zanotto Vitreous Materials Laboratory, Department of Materials Engineering, Federal University of S˜ao Carlos, 13565-905, S˜ao Carlos-SP, Brazil (Received 8 July 1997; accepted 23 February 1998)
A set of equations were derived to estimate systematic errors in experimental determinations of volume fractions transformed by microscopy methods. For reactions that occur by continuous nucleation and growth, the experimental values of volume fractions transformed may be subjected to significant errors when the largest grain size of the distribution is close to the microscope resolution limit. For transformations occurring from a fixed number of nuclei, the systematic errors are smaller than those observed in the continuous nucleation case, but can still be significant when reflection methods are used. Transmission methods lead to smaller errors than reflection techniques.
In several problems of phase transformations, such as in nanomaterials synthesis, rapid-solidification studies of metallic alloys, glass-ceramic development, and crystallization research in general, the final grain size can be very small. That fact may render the accurate quantification of important microstructural parameters, such as the number of second phase particles per unit volume, crystal size distributions, and volume fractions transformed, quite difficult. To estimate those microstructural parameters from two-dimensional planar cuts (the specimen cross sections), one often uses microscopy methods, associated with stereological procedures. In most situations one neglects the possible stereological errors and only takes into account the statistical errors due to the limited number of measurements. In a previous paper it was demonstrated that the average number of crystals per unit volume can be heavily underestimated if the microscopes used have a limited resolution power; i.e., no particles whose sizes are below the resolution limit (e) can be detected.1 The problem is amplified when the measurements are carried out by reflected light optical microscopy (RLOM) or scanning electron microscopy (SEM), due to the fact that even particles that are larger than e may show circular cuts smaller than e in the examined cross section. With transmitted light optical microscopy (TLOM) or TEM the problem is minimized, but can also be relevant in some situations because a fraction of particles may be smaller than the resolution limit. Thus, the following question arises: What would be the typical errors in the experimental values of volume fractions transformed for typical cases of phase transformations? In this communication a theoretical assessment of those errors is provided, and demonstrates that the volume fractions transformed can be significantly underestimated in some situations. J. Mater. Res., Vol. 13, No. 8, Aug 1998
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