1、 Quantification of Microstructure and TextureSize from Planar SectionsR Goodall, October 2010 1Quantification of Microstructure and Texture6. Size from Planar SectionsGrain SizeWhen we measure the size of the grains in a metallic sample, we are almost always looking at a 2D section through the mater
2、ial. It is normal practice to report the grain sizes measured in 2D, without accounting for their 3D nature. The next lecture will consider such conversions in situations where they are applied, but for grain size determination it should just be remembered that the true grain dimensions and distribu
3、tion of sizes may be larger than that reported.In some practical situations, grain size is assessed by comparison of an image with a series of standard images of microstructures. This method is rapid, but it is imprecise and not suitable for detailed quantitative analysis (being rather used for qual
4、ity control in an industrial setting) and is not discussed here.Linear Intercept MethodThe linear intercept method is the main way used to determine the grain size of materials. The mean linear intercept is the average distance between grain boundaries along lines placed at random on the image. The
5、mean linear intercept can be determined using:(1)LTotalNwhere LTotal is the total length of the line, and NL is the number of boundaries crossed.A worked example of grain size determination by the linear intercept method using an artificial microstructure is given below.Worked Example 1Figure 1 show
6、s a simulated material microstructure, where the black lines represent grain boundaries.Figure 1 An artificial microstructure, the black lines representing grain boundaries.Quantification of Microstructure and TextureSize from Planar SectionsR Goodall, October 2010 2Figure 2 The linear intercept met
7、hod for grain size determinationQuantification of Microstructure and TextureSize from Planar SectionsR Goodall, October 2010 3The analysis proceeds as follows, using the procedures shown in Figure 2 and Table 1.1) Draw a series of lines on the image. These lines can be spaced equally, but should be
8、randomly placed and should be spaced sufficiently that no grain is crossed by two or more lines, in order to respect the random sampling criterion for us to be able to statistically analyse our results. See Figure 2a.2) For the first line, identify the number of times that line crosses a grain bound
9、ary and count the total. This is shown in Figure 2b3) Repeat for all of the lines (Figure 2c). The measurement is performed on a line by line basis as this allows each line to be treated as a measurement of the grain size. The results could be put together and treated as one sample, but in this case
10、 we would not be able to use the statistical analysis given here, and would have to estimate the error using the equations discussed later in this lecture. Either method is equally valid.4) Measure the real length of the lines used (using the scale bar or magnification of the image). In the case of
11、this example it is 1mm.5) For each line, i, divide this total length by the number of grain boundaries to get the linear intercept length (Table 1, column 3).6) These linear intercept lengths are summed, and divided by the total number of lines to get the mean linear intercept length (Table 1, colum
12、n 3).7) The difference of the linear intercept length of each line Li from the mean linear intercept length is calculated and squared (Table 1, column 4).8) This data is then used to calculate the standard deviation of the measurements using the equation given in the lecture on statistics (Table 1,
13、column 4).9) From the standard deviation, the standard error can be calculated using:nsSwhere n is our number of lines.10) From the standard error, the 95% confidence limit can be calculated using the relevant t-value (e.g. from the table given in the lecture on Statistics and the result of the meas
14、urement expressed according to: LStn195Line Number, iNo. of Grain Boundaries, NL Linear Intercept Length, Li (m)Difference from sample mean2Li1 20 50.0 38.12 26 38.5 28.73 25 40.0 14.64 21 47.6 14.45 25 40.0 14.66 25 40.0 14.67 26 38.5 28.88 19 52.6 77.59 21 47.6 14.410 23 43.5 0.12Li = 438.3 = 27.3
15、3122nLsiLi / 10 = 43.8 m s = 5.2Table 1 The calculation of grain size using the linear intercept methodIn the case of this example, we can therefore express the mean grain size of the microstructure in Figure 1 with 95% confidence limits as being 43.8 3.7 m (given that S(L)=1.65 and t(95, n-1) for n
16、=10 is 2.262). Unlike when we were measuring the volume fraction, our parameter now has units, and it is important that these be given. Looking at Figure 1 the grain size found seems reasonable, but it is important to note that the range of confidence in our measurement does not relate to the range
17、Quantification of Microstructure and TextureSize from Planar SectionsR Goodall, October 2010 4of grain sizes present in the material; in fact, the method of linear intercepts cannot capture information about the range of grain sizes present.ASTM Grain Size NumberThe ASTM1 grain size number, g, was o
18、riginally defined from the number of grains per square inch at a magnification of 100, but is now related to the number of grains per millimetre squared (NA) measured at any appropriate magnification by the equation:(2)ANlog32.954.Grain Aspect RatioIn cold worked metallic materials, it is common for
19、 grains to be deformed, such that the average grain size measured varies with the direction in which is it measured, see for example Figure 3, an extruded sample of aluminium, showing grains that are elongated along the direction of extrusion. When making grain size measurements on such samples ther
20、e are two options open to us. We could replace the straight lines used in the method described above by circles, which will sample all orientations equally (for this to be valid, the circle should be sufficiently large that on the scale of the grains it approximates to a straight line, something tha
21、t may be difficult to achieve). A better option is to measure the degree of elongation of the grains (as this may have an important effect on the uniformity of properties, or tell us useful information about the process that led to the grains forming in this way). We can do this by measuring the gra
22、in aspect ratio; the ratio between the largest and the smallest grain size when it is measured along three orthogonal directions. The closer this value is to 1 the more equaixed are the grains.In order to determine this parameter, polished sections should be prepared along at least 2 orthogonal dire
23、ctions in the sample. Lines are then placed at right angles on images taken from these surfaces, and used to determine L1, L2 and L3.Figure 3 Extruded 99.5% pure aluminium, showing the elongation of the grains in the extruded direction (horizontal in the image). Polarised light microscopy shows the
24、individual grains in different colours.1 ASTM The American Society for Testing and Materials, an organisation that publishes standard methods for a wide variety of materials testing procedures.Quantification of Microstructure and TextureSize from Planar SectionsR Goodall, October 2010 5If required,
25、the measurements for the separate directions can be combined together to provide a single value giving a measure of the grain size by taking the cube root of the measurements in the three orthogonal directions multiplied together (i.e. ). In the case of plane strain 312Ldeformation (the deformation
26、that occurs in some processing operations such as rolling where the material is under some constrain and shows no strain in the third direction), the parameter can be found by taking the square root of the two measurements in the plane multiplied together (i.e. ).213LSurface Area per Unit VolumeThe
27、grain boundary area per unit volume, SV, is an important parameter involved in the kinetics of many transformations as the grain boundaries can provide sites for nucleation and fast routes for diffusion. Evidently, the grain boundary area per unit volume will be linked to the grain size, smaller gra
28、ins giving more boundaries, and this parameter can also be determined from the number of grain boundary intercepts per unit length, NL. When the grains are equiaxed the relationship is:(3)LVNS2When the grains are not equiaxed, for example, after deformation, the weighted average of the values of NL
29、in each of the three principal directions is used. If the maximum principal strain is in the 1 direction, and the minimum principal strain is in the 3 direction, the relationship below applies2:(4)3212LLV NNS(5)32157.049.LThis reverts to Eqn. 3 when NL1 = NL2 = NL3.Grain Size in Duplex StructuresDup
30、lex structures in materials science are those containing two phases or distinct regions, and these occur frequently in metallurgical systems of practical interest, for example in ferrite / pearlite steels. In these materials, both the size of the pearlite colonies and the ferrite grains are of impor
31、tance in determining the material properties. The determination of grain or colony size in such microstructures requires a simultaneous analysis of the volume fraction and the grain size.A worked example of grain size determination using an artificial duplex microstructure is given below.Worked Exam
32、ple 2Figure 4 shows a simulated material microstructure, where the black lines represent grain boundaries and the green regions represent second phase precipitates or colonies.The analysis proceeds as follows, using the procedures shown in Figure 5 and Table 2.1) Following the point counting method,
33、 draw a grid on the image. The lines of this grid should be spaced sufficiently that no feature is measured twice, in order to respect the random sampling criterion for us to be able to statistically analyse our results. The points where the lines cross on this grid are examined and assigned a value
34、 of 1 if they are in the second phase, 0 if they are in the matrix and 0.5 if they are on the borderline, see Figure 5a and Table 2, column 2.2) Considering the lines on the grid, the number of interphase boundaries (boundaries between the two phases), NL (Inter), is counted, as shown in Figure 5b (
35、for clarity, only the horizontal lines of the grid and the relevant points are shown) and Table 2, column 4.2 E E Underwood, Quantitative Stereology, (1970), Addison-Wesley, PhilippinesQuantification of Microstructure and TextureSize from Planar SectionsR Goodall, October 2010 63) The number of grai
36、n boundaries in the matrix phase (the primary phase), NL (GB), is now counted, Figure 5c and Table 2, column 6.4) The point fraction for each line is then determined, and from these values the overall point fraction (equal to the volume fraction) for the image is found, Table 2.5) We now use the fac
37、t that the fraction of the test lines occupied by each phase is given by the point fraction (i.e. the fact that ) to work out the linear intercept length of each PVLphase: a. Each second phase region has two interphase boundaries, and so the number of regions is half the number of boundaries. For li
38、nes of length L, the linear intercept length of the second phase region (the distance between two interphase boundaries) for each line, LL(Inter) i, is given by:(6)()(2InterLPiInterLNThis is calculated in Table 2, column 5. Lines where no second phase regions were measured give zero and are ignored.
39、b. In a similar way, the linear intercept length of the grains of the matrix phase is given by:(7)21)()()( InterLGBLPiBLNHere we count both the grain boundaries and the interphase boundaries, and we use the volume fraction of the matrix phase.c. From these values for each line, the overall mean inte
40、rcept lengths and errors in the measurement can be determined, as before.d. In the case that there were also grains in the second phase regions, it would be necessary to make a further measurement of the number of intercepts with those grain boundaries, and use a formula like Eqn. (7) counting both
41、these grain boundaries and the interphase boundaries. Should the microstructure also be anisotropic, it may be necessary to perform this analysis along multiple directions to characterise the anisotropy.Figure 4 An artificial 2 phase microstructure, the black lines representing grain boundaries and
42、the green regions representing the second phase.Quantification of Microstructure and TextureSize from Planar SectionsR Goodall, October 2010 7Figure 5 The linear intercept method for grain or colony size determination in a two phase microstructure.Quantification of Microstructure and TextureSize fro
43、m Planar SectionsR Goodall, October 2010 8Line Number, iNo. of Points in Minor PhasePoint Fraction, PPiNo. of Interphase Boundaries, NL (Inter)Linear Intercept Length, L(Inter)i (m)No. of Matrix Grain Boundaries,NL (GB)Linear Intercept Length, L(GB)i (m)1 1 0.125 4 40.6 25 34.02 0 0.000 0 - 33 27.83
44、 0.5 0.063 2 81.3 29 30.64 1 0.125 2 81.3 32 27.85 2 0.250 4 40.6 29 29.66 0.5 0.063 4 40.6 32 27.07 0 0.000 2 81.3 28 31.78 0 0.000 0 - 31 29.69 1.5 0.188 2 81.3 33 27.010 0 0.000 3 54.2 29 30.1= 0.081pP= )(InterL62.6 m= )(GBL29.5 mTable 2 The calculation of grain size and second phase region size
45、using the linear intercept methodExperimental PlanningJust as for the measurements of volume fraction discussed previously, equations have been developed to allow the rapid estimation of the error likely in measurements of the grain size, depending on the number of grains that are analysed.The relat
46、ive standard errors (standard errors as a percentage of the measured values) in measurements of grain size are given by3: (8)nNSLL65.0where n is the number of grains counted. These can be used with the values of t(95, n-1) given in the lecture on statistics to determine 95% confidence limits expecte
47、d for the result. Values of the number of grains that must be counted for a given accuracy are given in Table 3.Relative Error 95% CLNo. of grains, n0.010 2% 42250.025 5% 6760.050 10% 1690.100 20% 42Table 3 Number of grains needing to be measured in order to achieve various estimated relative errors
48、 of volume fraction in linear intercept method, determined from Eqn. (8).A similar equation can be used for the standard error in the ASTM grain size number, g:(9)ngS03.13 R L Higginson and C M Sellars, Worked Examples in Quantitative Metallography, (2003) Maney, LondonQuantification of Microstructu
49、re and TextureSize from Planar SectionsR Goodall, October 2010 9Note that in both of these equations the error will decrease as the root of the number of grains measured. That is, the reductions in error on taking further measurements decrease as the total number of measurements gets larger, in line with the statistical principles discussed before.Worked Example 3In worked example 1, we measured the mean grain size of the example microstructure to be 43.8 3.7 m with 95% confidence limits
