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Grain Size Measurements by the Triple Point Count Method


Aside from the well-known grain size measurement techniques using either the planimetric methods of Jeffries or Saltykov, or the intercept method of Heyn, Hilliard and Abrams, one can measure the grain size through a count of grain-boundary triple point intersections within a known area through the use of Euler’s law. This technique has rarely been used but it should be possible to do such a count by image analysis. In general, measurements based on point counts (0 dimensional) are less subject to errors than lineal measurements (one-dimensional) which are less subject to error than areal measurements (two-dimensional).

Triple-Point Count Method

A circle of known size is drawn on a photograph or a transparency that is placed over the image. The number of grain-boundary triple points, P, within the test area is counted. If a four-grain edge junction is observed, which is much less common, it is counted as two. The number of grains per unit area, NA, is calculated from:

                 0.5P + 1
NA =   —————–   (1)

where AT is the test area at 1X. The ASTM grain size, G, is calculated in the usual way from NA using:

G =    (3.322 log NA) – 2.95   (2)

Experimental Program

Two images of a ferritic stainless steel, etched electrolytically with aqueous 60% nitric acid, were used for the experiments. The magnifications were 100 and 400X and they were located centered at the same area. The microstructural grain structure images were printed at a variety of sizes on paper and circles of different sizes were superimposed over the images to generate a very wide variation in counts (P varied from 5 to 860). The triple points were coded as counted to avoid missing some or counting some twice.

Figure 1 shows a plot of all 41 measurements of P, ranging from 5 to 860. The linear curve fit reveals a slight downward trend with increasing P.  Significant data scatter is observed for counts <25 with a range in G from 6.25 to 7.75. Figure 2 shows the data for P >25, where there is little data scatter and a flat, constant trend line for P versus G. The range of G was 6.44 to 6.98. The mean grain size for all 41 measurements was 6.82 and for those counts >25, the mean G was 6.72.

figure-1_lg figure-2_lg
Figure 1: Plot of all 41 measurements of P, with a range of 5 to 860, gave a linear fit with a slight downward trend as P increased.
Figure 2: Flat linear trend for triple point counts above 25 per field (range 35 to 860).

Figures 3 and 4 compare the ASTM grain size determined by the triple-point count method and the grain size determined by the planimetric methods of Saltykov, using rectangles, and the E 112 method of Jeffries, using circles. Figure 3 shows all of the data by the three methods and the linear curve fit shows excellent agreement between the three methods. Figure 4 shows the data for grain counts by the Saltykov method ≥10, for the Jeffries method >30 and for the triple-point count method >25. The agreement between the results is superb. Figures 5 and 6 compare the triple-point count results with the E 112 intercept count results using a single circle. Figure 5 shows all of the data plotted and the two linear fit lines are identical. The data scatter is high for counts <30. Figure 6 shows the comparison for counts >20. The results again are identical. For Figures 5 and 6, the triple-point counts above 90 were not plotted as there were no intercept counts above 89. Table 1 shows data comparing the four test methods as a function of counts.

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Figure 3: Plot of all data from the planimetric measurements using the Saltykov rectangle ab the Jeffries circle compared to the triple point count method yields a slight downward slope in G with increasing counts.
Figure 4: Plot similar to Figure 3, but with low count data removed, yielding a flat linear correlation over the data range and excellent agreement between the methods.
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Figure 5: Comparison of the intercept data with the triple-point count data (over only the same range as obtained with the intercept data (6 to 89 P counts). The data agreement is exceptional in that the linear data fit curves are on top of each other over the full range and the trend is flat. The data scatter is greater for counts <30.
Figure 6: Data similar to Figure 5, except that counts below 20 are not included. The results are virtually identical.


The triple-point count method is easy to use, but does require marking of the grain boundary intersections to obtain an accurate count. The ASTM grain size determined by the triple-point count method agreed perfectly with the results from the planimetric methods of Saltykov and Jeffries and with the Heyn/Hilliard/Abrams intercept method.

Table 1. Comparison of Test Data by the Four Methods.Note the extremely close agreement in the mean grain size for counts above the critical value (boldface data lines), that is, for counts above the low range with data scatter (column 3).

George Vander Voort has a background in physical, process and mechanical metallurgy and has been performing metallographic studies for 45 years. He is a long-time member of ASTM Committee E-4 on metallography and has published extensively in metallography and failure analysis. He regularly teaches MEI courses for ASM International and is now doing webinars. He is a consultant for Struers Inc. and will be teaching courses soon for them. He can be reached at 1-847-623-7648, EMAIL: and through his web site:

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The articles and presentations that can be down-loaded from this web site are based upon work done by GFV while employed at Bethlehem Steel (1967-1983), Carpenter Technology (1983-1996), Buehler Ltd. (1996-2009) and Struers (2009-Present) and from the authors consulting work for companies such as, Latrobe Steel, Scot Forge, etc., and from his litigation work. GFV's bylined articles appearing in various issues of the ASM Handbook series have been listed here courtesy of ASM International, Materials Park, Ohio.