Abstract
The total number of
inhomogeneously distributed grains in
solidified metallic samples is essential
data for understanding solidification
mechanisms, but estimating the total
number of grains is challenging when
the number density of grains is spatially
inhomogeneous. This study presents a
method for estimating the total number
of grains from a single central cross
section, by inferring local variations in
grain density through multiple
measurement lines. Based on this
concept, an estimation formula for
spherical samples processed by
electrostatic levitation (ESL) was
developed using Voronoi tessellation
(VT). The formula consists of three
principal components. First, a quartic
relationship between the number of
grains per unit length, NL, and the square of the number of grains per unit area, NA
2, is statistically derived. This
relationship enables the direct estimation of the number of grains per unit volume, NV, in local domains from NL
measurements. Second, the sample is partitioned into spherical segments. The volume of each segment, Vi, is calculated
based on the geometric properties of the sphere. Regional number of grains is then obtained by integrating the number
of grains per unit volume, NV, over each volume of segment. Finally, a correction factor, DC, is introduced to prevent
overcounting when a grain is intersected by more than one measurement line. Validation against VT models
demonstrates that the proposed method reliably estimates the true grain number, with results falling within acceptable
uncertainty ranges. In addition, the proposed formula estimated that an ESL sample contained approximately 989 ± 77
grains under the measurement condition that the interval between measurement lines, ∆d, is less than the average grain
size, NL⊥−1.
