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1、 Quantification of Microstructure and TextureVolume Fraction from Planar SectionsQuantification of Microstructure and Texture5. Volume Fraction from Planar SectionsVolume FractionThe volume fraction of phases in a microstructure can be very important to assess the effect of processing and the likely
2、 properties. Although it is obvious that microstructures are inherently three dimensional structures, we generally examine them by looking at planar sections. Measurement of volume fraction (VV) of second phase particles may be made from such planar sections using a variety of methods because of the
3、 following fundamental relationships:VV = (1)VV = (2)VV = (3)where is the mean point fraction, the mean line fraction, and is the mean area fraction of a second phase in a planar section. There are a number of possible ways in which one of these 3 parameters might be measured, giving the volume frac
4、tion. One of the earliest microscopists to examine materials, Henry Sorby, who worked in Sheffield, determined volume fractions of different phases in rock samples by cutting up an image and weighing the various parts.The three methods that will be considered in this lecture are point counting, line
5、al analysis and areal analysis. For making measurements “by hand” point counting to obtain a point fraction is generally the most efficient method but either areal or lineal analysis may be carried out using automatic image analysis systems.Point CountingVolume fraction determination can be carried
6、out either in the microscope, or, more commonly, on a micrograph or series of micrographs taken for a particular sample. The method of point counting lends itself particularly well to manual application in either of these cases (although a graticule would be needed to carry out the assessment in the
7、 microscope), and is generally the quickest and most statistically efficient way to collect data.Point counting is based on counting the fraction of points that fall in the phase of interest in a random array of points. This could be, for example, pearlite in steel, b phase grains in a titanium allo
8、y or recrystallised grains in a deformed and annealed structure. In this method points are distributed at the intersection of the horizontal and vertical lines of a grid randomly placed on the image, and it is decided how many fall in each phase. Those in the phase of interest (usually the minor pha
9、se) are counted as 1, those in the other phase as 0. Points lying on the phase boundary are counted as 0.5. Simple calculation of the point fraction gives the volume fraction under Eqn. (1) above. In selecting the grid size, it is important in order to maintain the validity of the statistical approa
10、ch used, that no feature is measured more than once. That is, the grid spacing should be as large as the largest second phase region present. This has the effect that the scale of the structure influences the area of the section needed. For example, a grid containing 1000 points at a separation of 5
11、0 m would require a total area of about 2.5 mm2 to be examined.A worked example of volume fraction determination by the point counting method using an artificial microstructure is given below.Worked Example Part 1Figure 1 shows a simulated material microstructure, with the black circles representing
12、 second phase particles.Figure 1 A simulated 2 phase microstructure.The analysis proceeds as follows, using the procedures shown in Figure 2, and in Table 1.1) Identify an appropriate grid size for the image (with spacing large enough that no feature is sampled more than once), and draw this grid ra
13、ndomly on the image, Figure 2a.2) Taking either the vertical or horizontal lines, go along the line assessing in which phases the intersections of the grid are located, and assign them the values 1 for the second phase, 0 for the primary phase, and 0.5 for the interphase boundary. Total the count fo
14、r the line (Figure 2b).3) Repeat for all of the lines (Figure 2c). The reason for treating each line like this is that we will later use the results from each line as one “measurement” of the volume fraction, which allows us to perform statistical analysis on the result.4) For each line, i, divide t
15、his total by the total number of points per line to get the point fraction, PPi (Table 1, column 3).5) These point fractions are summed, and divided by the total number of lines to get the mean point fraction (Table 1, column 3).6) The difference of the point fraction of each line PPi from the mean
16、point fraction is calculated and squared (Table 1, column 4).7) This data is then used to calculate the standard deviation of the measurements using the equation given in the Statistics lecture (Table 1, column 4).8) From the standard deviation, the standard error can be calculated using:where n is our number of lines.9) From the standard error, the 95% confidence limit can be
