Amount of Microstructural Constituents. Three-dimensional ele-ments, such as particles, can be effectively and precisely expressed in terms of volume fraction, which is easily computed from either binary images (black and white images) or images having gray levels or colors unequivocally coupled with structural features under consideration.
Simple counting of the number of pixels corresponding to the given color or gray level and dividing this number by the total number of pixels in the image yields the estimated volume fraction (see also Chapter 2, “Intro-duction to Stereological Principles,” and Chapter 5, “Measurements”). In contrast to the classical stereological approach, application of image-analysis tools does not require any additional grid of points or test lines.
Volume fraction, even if evaluated using computerized tools, is a stochastic (of a probabilistic nature) measurement. Proper sampling of the images and further computation of the confidence level (CL) provides a complete characteristic of the amount of second-phase particles. Note that this is the only characteristic of the microstructure that can be evaluated in a simple and complete manner regardless of any geometrical details of the structural elements. In other words, there is no obstacle to correctly
quantify the amount of microstructural constituents even if phase distri-bution over the test material volume is highly inhomogeneous.
To avoid misinterpretation of the term amount, consider the following discussion. Any material has a 3-D structure (even if available only in the form of extremely thin layers or fibers), which is described by 2-D or one-dimensional (1-D) geometrical models. Therefore, the term amount is used only to characterize 3-D microstructural constituents. Two-dimen-sional features (e.g., grain boundaries) and 1-D, or linear, features (e.g., dislocation lines) are characterized by means of their densities, LVand SV, respectively.
Size of Structural Constituents. Although there is a clear intuitive understanding of what the term size means, measurement of size is not straightforward. The usual way to measure the size of 3-D objects is by measuring their volume. However, other measurements, such as total surface area, mean section area, largest dimension, mean intercept length, and diameter (for a sphere) also can be used. Thus, in the case of size, choosing a proper parameter is not that easy. Fortunately, there are guidelines for selecting the best size characteristics, which are summa-rized below.
The measurement used to quantify size must be adequate for the process model under consideration. For example, in the case of nodular cast iron, the amount of graphite is approximately stable and (assuming a constant amount of the graphite) its fracture toughness is linearly proportional to the mean graphite nodule diameter—a proper size parameter in this case (Ref 2). Application of the mean nodule volume for this analysis results in highly nonlinear relationships, which can easily lead to false conclu-sions. By comparison, the grain growth process in steel at high tempera-tures is controlled by the presence of small precipitates at the grain boundaries. In this case, characterizing grain size using the surface area of grain boundaries per unit volume, SV, is better than characterizing grain size using any linear dimension of the grain.
Size measurements should be sensitive to changes in the microstructure that can affect the properties studied. For example, especially in the case of recrystallized materials, two entirely different materials could have the same mean values of grain size, but grain size distribution could differ significantly. Thus, even if the grain volume theoretically is the best measure of grain size, it should not be used in this case because it is difficult to evaluate the volumetric distribution of grains (this requires serial sectioning and subsequent 3-D reconstruction). It is recommended in such a case to analyze section areas or intercept lengths because distributions of these variables are easily obtained and compared. Further, a safer approach is to analyze the intercept length distribution because the distribution of grain section areas is much more sensitive to errors in grain boundary detection (Fig. 7). The structure in Fig. 7 consists of grains having partially “lost” grain boundary lines, or segments, which are
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denoted by gray circles. In principle (compare the Jeffries method described in Chapter 5, “Measurements”), there are 24 grains visible;
however, due to the six lost segments, there only are 18 grains (24⫺ 6 ⫽ 18). Consequently, the relative error of grain number evaluation is equal to6⁄24, or 25%. By using intercepts, the relative error in their number (lost intercepts are marked as dashed lines) is equal to 5⁄98, or 5.1%. So the relative error in intercept counts is, in this case, approximately five times smaller. The difference can be smaller or greater for other structures.
However, this does not change the general rule that intercept number is less sensitive than number of grains to lost boundary segments. This solution is a compromise because the distribution of intercept lengths cannot be directly related to distribution of areas of the grains analyzed.
Nevertheless, this is sufficient for comparative purposes.
Image analysis software allows obtaining unbiased results for selected measurements of size. For example, parameters related to individual particles should be measured only on particles totally included in the field of view. Measurements on particles touched by the image edge will be incorrect because it takes into consideration only a part instead of the whole particle. Particles cut by the edge of the image can easily be removed. However, while this is an acceptable solution in the case of small particles, it can introduce large errors in the case of very large particles, because the probability of being cut by the image edge is proportional to the size of the particle section (this problem will be analyzed in more detail later). Quantification of the size of particles or their colonies is relatively easy and efficient following the method described previously.
Fig. 7 Illustration of the effect of the lost grain boundary lines (denoted by circles) on the results of grain size quantification. The lost grain bound-ary segments significantly increase the mean section area (by⬃25%), whereas the mean intercept length (intercepts enlarged due to the absence of some grain boundaries plotted usingbroken lines) is increased approximately 5%.
Shape Quantification and Classification of Particles. The main difficulty in shape quantification is the lack of a precise, universal definition of the term. Intuitively, it seems that any object can be described in terms of its shape and size. Shape can be interpreted as the property of the object that is not connected with its size, but shape often is very difficult to separate from size. For example, small buildings generally have a different shape than big buildings. The same observation is valid for cars, plants, and animals. Another difficulty is that a single number usually cannot describe shape. In contrast, amount and size are limited to a number description; volume fraction (describing amount) always falls in the range from 0 to 1 (0–100%), and the size of any microstructural constituent never exceeds the size of the test specimen.
Having no possibility to describe shape with absolute precision, param-eters called shape factors, which are sensitive to the changes in shape, are defined. Shape factors should have the following common properties to correctly quantify a microstructure:
O Dimensionlessness that keeps their values unaltered in the case of particles of the same shape but different size
O Quantitative descriptiveness that shows how far a given shape deviates from a model, or theoretically ideal shape (shape factors can measure, for example, circularity, elongation, compactness, concavity, or con-vexity)
O Sensitivity to particular shape changes that occur in a process under consideration
It is impossible to define a universal shape factor that is applicable in all the types of microstructure analysis. A sphere is a good reference shape to discuss properties of shape factors because it is the simplest geometri-cal model for 3-D objects. In images of test specimens containing spheres, the cross sections or projections of the spheres are in the form of circles.
Therefore, numerous shape factors are used in image analysis to deter-mine how much a given shape differs from a circle. The shapes observed in practice can deviate significantly from a circle, but finding a “refer-ence” circle is intuitively easy, as illustrated using the series of sketches shown in Fig. 8. Elongation, irregularity, and composition are illustrated in Fig. 8(a), (b), and (c), respectively, and are discussed in more detail below.
Elongation, also known as aspect ratio, commonly is used to describe the shapes of particles after plastic deformation and can effectively be measured using the following shape factor (Fig. 9) (Ref 3, 5):
f1⫽a
b (Eq 1)
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where a and b are the length and width of the minimum bounding rectangle, or a is the maximum Feret diameter, while b is the Feret diameter measured perpendicular to it. These two sets of values used to determine elongation can produce slightly different values of the shape factor due to the digital nature of computer images.
Aspect ratio reaches a minimum value of 1 for an ideal circle or square and has higher values for elongated shapes. Unfortunately, elongation is not useful to assess irregularity, where all particles have an f1value very close to 1, despite the fact that the particles have profoundly different shapes. This is illustrated in Fig. 8(b). Circularity—one of the most popular shape factors—offers a good solution to this situation:
f2⫽ L2
4A (Eq 2)
where L is the perimeter and A is the surface area of the analyzed particle (Fig. 9). The f2 shape factor is very sensitive to any irregularity of the shape of circular objects. It has a minimum value of 1 for a circle and higher values for all other shapes. However, it is much less sensitive to elongation.
Fig. 8 Three families of shapes originating from a circle: ellipses of various elongation (top), shapes having various edge irregularity (middle), and a combination of the two (bottom)
Quantitative characterization of the composition of different shapes in the structure is more complex. It can be treated as a mixture of elongation and irregularity. A practical example of such a situation is the change in the form of graphite precipitates during transition from nodular to flake cast iron. This transition cannot effectively be described using alone any of the shape factors presented previously. Very often in such circum-stances, the weighted average of a few shape factors is applied. While it is possible to successfully manipulate the weights to obtain satisfactory results (i.e., obtain good correlation between the shape factor and properties), the result cannot be directly interpreted. So it is better⫺but also more difficult⫺to construct a new shape factor that has the capability to detect necessary changes in shape.
When analyzing the shapes shown schematically in Fig. 8(c), it is noticeable that all these particles have approximately the same surface area, but when moving across the sequence from right to left, systemati-cally greater objects can be drawn inside the particle. This observation leads to a definition of the new shape factor (Fig. 9) (Ref 4):
f3⫽d2
d1 (Eq 3)
where d1 and d2 are the diameters of the maximum inscribed and circumscribed circles, respectively. This shape factor works very well in the case of complex deviations from ideal circularity. It is sensitive to changes both in particle elongation and irregularity. However, keep in mind that it is not possible to uniquely describe any shape by a single number. Therefore, even the best shape factor available can only quantify elongation or irregularity of a particle, which might be quite insufficient
Fig. 9 Basic measurements used to evaluate shape factors
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for correct particle recognition and/or classification. Usually more than one parameter must be applied for the purpose of classification.
Classification of any object usually is successfully performed by applying the rules of fuzzy logic. In binary logic, a particle can be regular or irregular, whereas in fuzzy logic, the same particle can be, for example, regular, irregular, somewhat regular, or nearly regular. Application of fuzzy logic has some advantages, which are briefly outlined in this section.
Use of fuzzy logic allows quantification of particles that do not fit a selected assumed template, whereas binary logic allows only determina-tion of whether a particle fits or does not fit the template. In addidetermina-tion, it is easy to apply weighted averages or even more sophisticated functions of various quantities for classification needs. Moreover, in the case of fuzzy logic, the results always lie in the 0–100% range. So there is no difficulty in interpreting results; this is relatively close to the way a human would classify the results.
Figure 10 shows how application of the fuzzy logic works on a collection of graphite particle shapes observed in various grades of cast iron. Each particle in this figure is accompanied by two numbers: a value of f3shape factor and a circularity rating, computed using fuzzy logic and expressed as a percentage. The four particles at the right side of Fig. 10 are recognized as fully circular, particles in the middle are rated as partially circular, whereas graphite flakes (shown on the left side of Fig.
10) are judged as not circular. Classification with help from fuzzy logic works somewhat similar to neural networks; classification rules are derived based on test sets of data, but interpretation of the rules applied is not necessary for correct classification.
Fig. 10 Application of fuzzy logic to classify graphite particles. Upper numbers denote the value of the shape factor, and lower numbers (percent) indicate how well the particle fits as a circular shape.
Arrangement of Microstructural Constituents and its Quantifica-tion. Arrangement quantification is discussed on the basis of examples that later are summarized to get a more general understanding of the problem it presents. The first question to answer when dealing with arrangement is how do you quantify inhomogeneity? Note that in any two-phase material observed at high magnification, there are distinct re-gions occupied only by one of the phases. At increasingly higher magnifi-cation, a point is reached where in most fields of view, only a single phase is observed; the second phase is outside the field of view. Generally, the single-phase materials can be treated as being homogeneous, and they dif-fer in homogeneity, but on a microscale, any two-phase material is highly inhomogeneous. (This intuitive meaning of homogeneity often is used in everyday life, especially in the kitchen, where many dishes are prepared using constituents that lead to something homogeneous.)
The difficulty in quantifying homogeneity is somewhat similar to the problem of shape characterization. In both cases, there is no clear definition of the quantified characteristics and no clearly defined mea-surements, and, yet, both characteristics are crucial to define material properties. The most commonly applied solution in quantification of homogeneity is dividing the microstructure into smaller parts—call them cells—specified by their size. It is assumed that some global character-istics of microstructural features (e.g., particle volume fraction) should be approximately stable over randomly selected cells. If a large variation in the value of a selected characteristic is observed during the movement from one cell to another, the material is judged as inhomogeneous at the scale defined by the chosen cell size.
Quantitative characterization of other aspects of arrangement requires similar individual, context-oriented analysis. For example, to determine whether particles tend to concentrate on grain boundaries, detect the boundaries and compare the amounts of particles lying on and away from the grain boundaries.
To summarize, it is impossible, even from the theoretical point of view, to fully characterize in a quantitative way all aspects of arrangement of microstructural constituents. Partial solutions applicable to a limited number of cases can be prepared on the basis of a thorough analysis of the process history of the material being analyzed. Quantification of arrange-ment of microstructural constituents is one of the most important characteristics of the material microstructure even though it is difficult to perform.
Advanced techniques for arrangement quantification are beyond the scope of this text, and, therefore, only the most general characteristics of this problem are outlined. Chapter 5, “Measurements,” and Chapter 6,
“Characterization of Particle Dispersion” describe other methods for quantification in simple problems related to distribution, such as orienta-tion of structural constituents.
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