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 http:/mathworld.wolfram.com/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 p