显微结构的量化处理及组织分析的教程 Lecture Notes 2

1、Quantification 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 properties. Although it is obvious that microstructures are inherently three dim

2、ensional 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 following fundamental relationships:VV = (1)VV = (2)VV = (3)where is the mean po

3、int 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 fraction. One of the earliest microscopists to examine materials, Henry Sorby, who wo

4、rked 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, lineal analysis and areal analysis. For making measurements “by hand” point counting

5、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 out either in the microscope, or, more commonly, on a micrograph or series of mic

6、rographs 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 microscope), and is generally the quickest and most statistically efficient way

7、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 alloy or recrystallised grains in a deformed and annealed structure. In this method p

8、oints 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 phase) are counted as 1, those in the other phase as 0. Points lying on the phase bo

9、undary 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 approach used, that no feature is measured more than once. That is, the grid spacing sh

10、ould 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 50 µm would require a total area of about 2.5 mm2 to be examined.A worked exa

11、mple 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 second phase particles.Figure 1 A simulated 2 phase microstructure.The ana

12、lysis 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 randomly on the image, Figure 2a.2) Taking either the vertical or horizontal

13、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 for the line (Figure 2b).3) Repeat for all of the lines (Figure 2c). The reas

14、on 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 this total by the total number of points per line to get the point fraction,

15、 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 point fraction is calculated and squared (Table 1, column 4).7) This data i

16、s 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 ca

17、n be calculated using the relevant t-value (e.g. from the table given in the Statistics lecture) and the result of the measurement expressed according to:Figure 2 The method of analysis used for point counting.Line Number, iNo. Points in Minor Phase Point Fraction, PPiDifference from sample mean120.

18、2500.000222.50.3130.0062300.0000.0550420.2500.0002510.1250.0120620.2500.000272.50.3130.0062830.3750.0197SPPi = 1.876= 0.0142= SPPi / 8 = 0.2345s = 0.1193Table 1 The calculation of volume fraction by point countingIn this case the actual volume fraction of the second phase in the image is 0.18, so ou

19、r result calculated above is not very accurate. This is reflected in the large value of the standard deviation. Using the equation for standard error, this can be calculated to be S(PP) = 0.0422, which, as t(95, n-1) is 2.365 for n=8, gives us a result that we can express with 95% confidence limits

20、as VV = 0.235 ± 0.100.The true value evidently does lie inside these bounds, but the measurement has not given us a much better idea than we could have obtained from visual estimation of the volume fraction. The reason for this is linked to the small number of lines we looked at, and the small

21、number of test points per line. Later on we will look at how this could have been predicted, and how the number of measurements required to get a certain value of the 95% confidence limit can be calculated.Lineal AnalysisLineal analysis provides an alternative to point counting as a method of determ

22、ining the volume fraction. If the microstructure is anisotropic, then a series of linear traverses can be laid across the image at any angle, and the length of line occupied by the second phase may be identified following a similar procedure to point counting, but where the length of each intersecti

23、on with the second phase is measured. If the microstructure is not isotropic, e.g. as might be found in rolled material, then a series of different images with rows of lines placed at random angles and locations should be used. Calculation of the line fraction gives the volume fraction under Eqn. (2

24、). Just as for the grid with point counting, care must be taken that the lines are not so close that the same region of second phase is not intersected by adjacent lines.Areal AnalysisAreal analysis was one of the earliest methods proposed for volume fraction determination, but is tedious, requiring

25、 detained scrutiny of images. In it the image is divided up into a grid, this time much smaller than the second phase features, and the number of grid squares lying wholly in the second phase are counted as a proportion of the total number. As for point counting, squares that lie on the boundary may

26、 be counted as 0.5. It is now the method most frequently employed by automatic systems, as the pixel format of the image makes this particularly suitable.Experimental PlanningWhen planning to determine the volume fraction of a second phase in a microstructure using one of the above methods, it is im

27、portant to consider the required accuracy, as it will have a big impact on the number of measurements, and therefore the size of the imaged area, that is required. As was seen in the Statistics lecture, the accuracy of experimental measurements will improve as the square root of the number of measur

28、ements made, while the effort to carry out measurements increases linearly with their number. It is therefore necessary that a compromise be found between the accuracy required and the time available to perform the measurement.In order to plan how many measurements need to be made to achieve certain

29、 error tolerances, we need equations that allow us to estimate the likely error. Fortunately, as the errors we are dealing with are random, it is possible to formulate such equations for the different measurement methods from statistical considerations.It should be noted that all of these equations

30、give us an estimated value for the standard error. This quantity was defined in the Statistics lecture as:(4)and is a term giving us information on the variability of the sample mean (the mean of the measurements made relative to the true mean of the population). This is calculated using Eqn. (4) af

31、ter the measurements are made from the standard deviation, s, of the individual measurements made in the sample, and their number, n, and can then be used in the equation:(5)to express the 95% confidence limits on the value of the true population mean, µ.The equations given here provide a metho

32、d of estimating this parameter in advance of performing the measurements.Expected Error in Point CountingThe expected relative standard error (the standard error as a percentage) can be estimated from the relationship of Gladman and Woodhead T Gladman and J H Woodhead, J. Iron Steel Inst. 194 (1960)

33、 189:(6)where P is the total number of points counted (the individual points, not the number of lines), and other terms as usually defined. It is interesting to note that the accuracy of the measurement depends on its result; there is a dependence on the volume fraction. This is logical, as, for a l

34、ower volume fraction second phase, more points will be needed for a statistically significant number of points to be located in that phase. On a practical level, the volume fraction must also be estimated in order to use this equation. This can be done using the phase diagram and the lever rule (see

35、 ). It is also important to note with this equation that VV must always be the minor phase. For example, when examining a phase transition, the phase to which VV refers must be swapped when the second phase passes VV=0.5.In some situations, the absolute standard error (the standard error as a value,

36、 not a percentage) may be important. Rearranging Eqn. (6) gives:(7)Eqn. (6) has been used to generate the data in Table 2 which shows the estimated number of points required for certain 95% confidence limits, for samples of different volume fraction second phase.The values in Table 2 show that the i

37、nfluence of the expected volume fraction is strong, and that in order to achieve a narrow confidence limit, a restrictively large number of points may be required if the point counting method is to be used. Yet, as will be seen, this method is in fact the most statistically efficient of those that w

38、e will examine.Relative Error95% CLNo. of points, PVV = 0.01VV = 0.1VV = 0.50.010± 2%99000090000100000.025± 5%1584001440016000.050± 10%3690036004000.100± 20%9900900100Table 2 Number of measurement points required in order to achieve various estimated relative errors of volume fra

39、ction in the point counting method, determined from Eqn. (6).Worked Example Part 2In our earlier measurement of the volume fraction of second phase in the simulated microstructure in Figure 1, we found that our result had very wide confidence limits, due to the small number of measurements made. Usi

40、ng the equations given above, we can calculate the error that would be estimated for these measurements.In the example above, we used a total of 8 lines with 8 points each, giving us 64 points. Therefore, using Eqn. (7), the predicted absolute standard error for the volume fraction of 0.18 is S(PP)

41、= 0.0480. This compares well with the S(PP) = 0.0422 calculated from the actual results. Using the equations we can calculate how many points we need if we want 95% confidence limits of ± 5%. First, we can assume for large n that t(95, n-1) is approximately equal to 2, and so we want the estima

42、ted standard error, S(PP) = 2.5%. Using Eqn. (6) gives a number of points for VV = 0.18 of P = 7289, considerably more than we would be able to obtain from the micrograph in Figure 1, without violating the one measurement per feature rule.Expected Error in Lineal AnalysisThe expected relative standa

43、rd error can be estimated using the relationship derived by Gladman T Gladman J. Iron Steel Inst. 201 (1963) 906:(8)where n is the number of second phase particles measured. Eqn. (8) has been used to generate the data in Table 3, which shows the estimated number of second phase particles required fo

44、r certain 95% confidence limits, for samples of different volume fraction second phase.Relative Error95% CLNo. of particles, nVV = 0.01VV = 0.1VV = 0.50.010± 2%196021620050000.025± 5%313625928000.050± 10%7846482000.100± 20%19616250Table 3 Number of second phase particles needing

45、to be measured in order to achieve various estimated relative errors of volume fraction in the lineal analysis method, determined from Eqn. (8).It can be seen from Table 3 that the values of n are less than the corresponding values of P for point counting, but n should really be compared to the number of points in the minor phase, Pminor = VVP, and on this basis point counting is statistically m