显微结构的量化处理及组织分析的教程LectureNotes.docx

Quantification of Microstructure and TextureTrue Size and Size Distributions of Second PhasesQuantification of Microstructure and Texture7. True Size and Size Distributions of Second PhasesMetallographic images are a 2D representation of a 3D structure, and as such the measurements of size that we make using the image may not be the same as the sizes of the features in reality. Conventionally, this is not a concern for the grain size, where values reported are those measured in 2D, and generally, the mean intercept length is a good way of characterising the dimensions of features that will be sampled by many processes (e.g. by dislocations moving during plastic deformation) in the material. There remain, however, some situations where the precise true size is of interest. The first part of this lecture is concerned with the relationship between the 2D sections of features observed in metallographic images and the true size of such features. The second part considers what information can be extracted from the measurements concerning the distribution of sizes. Note that, while the linear intercept method can give enough information to allow the determination of the true size of features, individual measurements of features (such as by the method of equivalent circles) is necessary in order to obtain size distributions.Determination of True SizesAlthough grain sizes are by convention reported as the sizes measured in the plane of the section, there may be occasions when we need to know the true size of grains or, more commonly, second phase particles in a microstructure. It is clear on considering a spherical particle, Figure 1, that random sections are unlikely to reveal a section with diameter equal to the real diameter of the sphere. Rather, the mean diameter of spherical inclusions as determined by the linear intercept method will be somewhat less than the true value.Figure 1 Possible sections through a spherical particles and the apparent particle diameter measured as a result.The relationship between the true size of a particle and the mean linear intercept length is given by S I Tomkeieff, Nature, 155 (1945) 24:(1)where V and S are the volume and surface area of the particle respectively. How this is applied can be demonstrated by considering the sphere in Figure 1. This has V = 4/3 p r3 and S = 4 p r2. Substituting into Eqn. (1) gives:(2)where r and d are the radius and diameter of the sphere. Thus for spherical inclusions, the average value of the diameter measured on the section will be 2/3 that of the true diameter.Similar calculations can be performed for other shapes of particle. Example results are given in Table 1.ShapeVolume, VSurface Area, SMean Intercept LengthSphereRadius, rDiskRadius, rThickness, tr tCylinderRadius, rHeight, hr hRodRadius, rLength, ll rHemisphereRadius, rProlate Spheroid*Radius 1, aRadius 2, ca cTable 1 The relationship between the size of features of different shapes measured on 2D sections and the true size in 3D (*Spheroids are formed by the rotation of an ellipse, and have 2 defined radii; the equatorial radius, a, and the radius along the axis of rotation, c. See /Spheroid.html).Analysis of Size Distribution in Planar SectionsIf we have measured the sizes of a number of individual features, for example by the method of equivalent circles, we will be able to plot a histogram of their sizes and thereby obtain some information about the distribution of feature size. However, from what we have learned above, we cannot expect this distribution to be identical to the true distribution of feature sizes in 3D. As indicated by the schematic diagram in Figure 2, random planar sections through a microstructure containing spherical particles are more likely to intersect with the larger ones (from now on in this lecture, it will be assumed that we are working with second phases that approximate in shape to spheres, as this is simpler, although the same reasoning and similar analysis apply to particles of other shapes).Figure 2 A schematic diagram of the intersection of various planar sections through a 3D distribution of spheres of different diameter.Scheil Z Scheil, Z. Metallk. 27 (1935) 199 developed a method of converting the observed distribution of circles in a planar section into the volume distribution of spheres. The central theory was that the observed circular section diameters will range in size from 0 to D, the diameter of the largest sphere. Circles of diameter D could only be observed when the section cut through the centre of the largest sphere. The probability that this would occur can be calculated for different distributions of different sized spheres, and the residual probability can be assigned to the appearance of the smaller size groups of circles. The process can then be repeated for the next largest circles and so on until a theoretical distribution of spheres is found that matches the observed distribution of circles.This method is effective, but leads to large uncertainties for the smallest size groups of spheres. Improved versions of the method have been developed by Schwartz H A Schwartz, Metals and Alloys, 5 (1934) 139, Saltykov S A Saltykov, Stereometric Metallurgy, (1952) Ind. Ed. Metallurgizdat, Moscow and Woodhead (reported in R L Higginson and C M Sellars, Worked Examples in Quantitative Metallography (2003) Maney, London). These methods develop a matrix of coefficients a(i, j) for the number of circles in size group i arising from spheres in size group j from probability distributions of randomly sectioning spheres. By matrix inversion coefficients a(j, i) are formed, which can be used to derive the number of spheres in size group j from the numbers of circles in the various size groups i. Values of these coefficients are given in Tables 5.1 and 5.2 in Higginson and Sellars book. There is a slight difference in the coefficients given by Scwartzs and Saltykovs method and that of Woodhead, as in the former case the spheres are assumed to be of discrete sizes, whereas in the latter their sizes are spread over the range. These coefficients are applied to determine the number of spheres per unit volume in the size range j, NV (j).(3)where D is the size interval, NA (i) is the number of circles per unit area in the size range i and k is the number of groups. The effect of applying these methods to experimental data is shown in Figure 3.Figure 3 The effect of the Schwartz-Saltykov (S-S) and Woodhead (W) analyses applied to experimentally measured distributions of particle diameters in a planar section.From the histogram in Figure 3 it is clear that the experimental measurements underestimate the true size of the features (as was seen at the start of this lecture) and also give the incorrect distribution. The Schwartz-Saltykov analysis gives some improvement, but the best relationship with the true data (which was obtained on material containing particles of mean size 30m) is given by using the Woodhead analysis.Analysis of Size Distribution in Extraction ReplicasExtraction replicas are prepared in order to facilitate the imaging of second phase particles or inclusions in a matrix by removing them from that matrix for examination in the microscope. One way in which this can be done is shown in Figure 4.Figure 4 Schematic diagram of the method of producing an extraction replica.Figure 4 shows a specimen containing spherical inclusions being first polished and then etched (a), to a depth sufficiently deep to remove any particles that have been sectioned by the polish. A material such as carbon that will hold the particles but still allow them to be imaged is then deposited onto the surface revealed (b). Finally, a second etch (b) is used to remove enough matrix phase so that the replica can be floated off, taking with it just the particles that were revealed by the first etch (c).This method can be afflicted by certain systematic errors, such as the failure to capture particles that only just impinge on the first etch surface, or the failure to remove particles released during the first etch from the surface before deposition and particles released in the second etch before imaging. It can also be difficult to ensure that the etched surface is flat; an uneven surface will sample a greater volume and may impinge on more particles. The fact that it is also necessary to work with relatively thin samples makes the process more complicated. Set against this is the fact that as the particles are imaged in their entirety, there is no need to correct for their diameter. As larger spheres are more likely to be trapped than smaller spheres, the number of results in each size range needs to be corrected using:(4)where D(j) is the mean diameter for that size range.Analysis of Size Distribution in Thin FoilsWhen the features of interest are less than about 50 nm in diameter, they will be most easily investigated in the TEM. Here the sample is thinned until it is an electron transparent foil and contrast will be generated by the difference in electron diffraction properties of the matrix and the precipitates or particles.Figure 5 The imaging of particles in a thin foil in the TEMAs indicated in the schematic diagram in Figure 5, the images of the particles produced in the TEM will show their true diameter. However, unlike in previous methods the situation may arise that one particle is wholly or partly obscured by another. This is accounted for by using Eqn (5):(5)where NV (j) is the number of particles per unit volume in the size group j, NA (j) is the number per unit area of size group j measured on the image, D(j) is the mean diameter for that size range, t is the foil thickness and MA is a correction factor for the particles lost by overlap. The diameter appears in the equation due to the fact that particles are generally more resistant to etching than the matrix and can protrude outside the foil such that the particle centre is up to the particle radius beyond the surface. This is indicated in Figure 5.The correction factor MA is a complex parameter important in research, and is discussed in detail in J E Hilliard, Trans. Met. Soc. AIME 224 (1962) 906. It can be ignored with less than 5% error provided the particles are spherical and:(6)Thus lower volume fractions (lower probability of particles being close enough to overlap) and foil thicknesses closer to the mean particle diameter (less space for particles to be positioned in overlapping configurations) mean that it is less inaccurate to ignore MA.R Goodall, October 20106Quantification of Microstructure and TextureMeasurement of Dislocation DensityQuantification of Microstructure and Texture8. Measurement of Dislocation DensityWhy are we Interested in Measuring Dislocation Density?During plastic deformation, the operation of dislocation sources increases the dislocation density in a material. These dislocations will interact with each other via the creation of locks, jogs, pile ups, etc, and thus further plastic deformation will become more difficult. This is the process of work hardening, and the increase in flow stress can be described by:(1)where s0 is a threshold stress, k is a constant, and r is the dislocation density (i.e. the length of dislocation line per unit volume, measured in m-2).Equation 1 highlights the importance of the dislocation density in plastic deformation. Once deformation is complete, if the temperature is sufficiently high the dislocations may rearrange themselves to form sub grain boundaries, which will also cause hardening (dependent on r), and these sub grain boundaries will have a stored energy associated with them (dependent on r), and they may act as locations for the nucleation of second phase precipitates, for example the q phase in Al-Cu alloys (with distribution dependent on r). Thus accurate measurement of dislocation density can be important in a range of situations.Dislocations will be imaged in thin foils examined in the TEM due to the strain of the lattice around their core affecting the local diffraction conditions. Usually multi-beam conditions are used to allow the maximum fraction of dislocations to be in contrast. From such images there are two ways we can calculate the dislocation density, end counting and the intercept method.End CountingOn the images taken in the TEM, dislocation lines will be visible as dark lines (if it is a brightfield image). If it is possible to discern the individual dislocations, these lines will have ends where the dislocation exits from the upper or the lower side of the foil (dislocations cannot terminate inside a crystal).Assuming the orientation of the dislocations is random, the dislocation density (the line length per unit volume) is given by:(2)Where PA is the number of dislocation intersections with the plane per unit area. As the number of ends we count (n) will come from intersections on the top and bottom surface of the foil, the area sampled is effectively 2A, and PA is found from:(3)Hence:(4)Of course, A in these equations is the real area covered by the image, and so the magnification will have to be taken into account. A worked example of this calculation is given below.Intercept MethodThis method is useful in thicker foils where dislocation ends are no so easily discerned. It does however require that the foil thickness is known. This can be achieved using standard methods described in the literature on TEM measurements.The image is taken and a circle of known circumference is randomly placed on it. The number of intercepts of dislocations with the circle, n, is then counted, and this procedure is repeated for a series of micrographs. The dislocation density is given by:(5)where L is the length of the circle (remembering to convert for the magnification) and t is the foil thickness. This derives from the same basic equation as end counting (Eqn (2), except now PA = n/A and the area we are examining is the surface of the cylinder equal to Lt).Worked ExampleFigure 1 shows dislocations in a hot rolled stainless steel sample. We will now calculate the dislocation density of this sample following the two methods outlined above.End CountingThe image in Figure 1a shows 203 dislocation ends in an area that is 860 nm 680 nm. Therefore, Eqn (4) gives:Intercept MethodThe image in Figure 1b shows the circle drawn on the TEM image. The local foil thickness has been measured as 120 nm. The circle, which has a circumference of 1900 nm, has 29 intercepts with dislocations. From Eqn. (5):The results of the two methods are similar, but not the same. It is likely that the end counting result is the more accurate, because the area and number of dislocations sampled is greater. In either case more images should be examined to give more confidence in the result, and each image could be treated as a separate measurement in order to calculate the standard deviation and confidence limits.Figure 1 A hot rolled steel sample imaged in the TEM. The magnification is 172000, and the local foil thickness is 120 nm. b) shows the circle used for the intercept method.R Goodall, October 20109Quantification of Microstructure and TextureImage Analysis SoftwareQuantification of Microstructure and Texture9. Image Analysis SoftwareAutomated image analysis software is now commonly available, and there are a large number of packages available, including as freeware. Each system will have an individual interface and capabilities, but this lecture covers some of the general principles of their use, before going on to demonstrate some of the features of one of the freely available software programs.Input ImagesIn order to use any automated image analysis software, a digitized image is required. Some microscopy techniques inherently produce a digitized image (such as Scanning Electron Microscopy), and in many cases modern equipment is fitted with CCD cameras that can record an image in digital form. Although most equipment has appropriate digital recording devices, it is important to be aware that systematic errors may be introduced of the pixel size is close to the size of the features being measured or the resolution required. In addition, care should be taken that the digital image is an accurate recording of the microstructure. For example, if the image is so bright or dark that large areas are white or black, then image information will be lost. It is better to have images that range over greyscale, even if these are visually less striking, as the automatic analysis will be more sensitive. Figure 1 Poor images can result from a pixel size that is too large for the features to be analysed or incorrect brightness control losing information. This example shows a micrograph of an Al-Si alloy, treated to simulate these two errors.It is sometimes necessary to process the digital image to enhance it before accurate analysis can be performed. Examples of functions that can be applied are a removal of background, adjusting the contrast or the removal of recognised erroneous features, such as scratches. In all of these operations there is a significant risk of introducing systematic errors and bias can have an effect. The ideal situation is to prepare a sample and perform the microscopy well enough that alterations to the image are not required.ThresholdingNevertheless, many of the general image