Describing microstructure quantitatively leads to precise answers of qualitative problems; that is, which material of several options is better, or which material is adequate for use in a given application. The best way to answer these questions is to describe the microstructure in terms of appropriate numbers.
Generally, in microstructural analysis, no two microstructures are identical, but “sufficiently” similar microstructures represent practically identical material properties. The question is: what is sufficiently similar?
This requires the development of appropriate microstructural measure-ments, or descriptors, that meet the following minimum requirements.
They must be sensitive to changes in microstructural features responsible for a given property, they should be capable of generally describing the representative piece of material, and they should be precisely defined and as simple as possible for practical application and interpretation.
As illustrated previously, microstructure-property relationship models and a set of several quantities (a single one is usually insufficient) are necessary to describe a microstructure. Precise description of single features (grains, for example) produces large data sets, which can be useful for advanced theoretical considerations but not for routine quality control or process optimization.
A large set of stereological measures has been developed that satisfies the above requirements. Taking into account only the most basic set of so-called integral parameters that describe geometrical properties of a system of microstructural elements (e.g., second-phase particles) yields ten descriptors (Ref 3):
O VV: volume fraction, the total volume of features analyzed per unit volume of a material
O SV: specific surface area, total surface area of features analyzed per unit volume of a material
O LV: specific length, total length of lineal features analyzed per unit volume of a material
O NV: numerical density, mean number of features analyzed per unit volume of a material
O AA: area fraction, total surface area of intercepted features per unit test area of a specimen
O LA: total length of lineal features analyzed per unit test area of a specimen
O NA: surface density, mean number of interceptions of features per unit test area of a specimen
O LL: lineal fraction, total length of lineal intercepts analyzed per unit length of a test line
O NL: lineal density, number of interceptions of features per unit length of a test line
O PP: point fraction, mean number of point elements or test points in areal features per test point
V, S, A, L, N, and P are volume, surface area, planar area, length, number of features, and number of points, respectively. Subscripts unequivocally denote the reference space. Image analysis systems offer their own set of parameters, often different from the set used in classical stereology.
Most image analysis programs allow for individual quantification of all specific two-dimensional (2-D) figures (cross sections of three-dimen-sional, or 3-D, microstructural features) visible in the image under consideration. Figure 5 schematically illustrates the basic quantities that can be measured, or rather, computed (Ref 4).
Figure 5(a) shows the initial form of a single particle. This is next converted into binary (black and white) form (Fig. 5b), which is useful for subsequent measurements. The most natural measurement that is easily and quickly evaluated using computerized tools is particle surface area, which is measured by simply counting the pixels forming the particle in the binary image. Another frequently used measure in quantitative analysis is the particle perimeter (Fig. 5c). Unfortunately, the digital, discontinuous nature of computer images implies large errors in evalua-tion of the perimeter, which should be taken into account by the user. Fast, accurate measurements are so-called Feret diameters, characterizing the outer dimension of the particle; horizontal and vertical Feret diameters are shown in Fig. 5(d). Somewhat more difficult for quantification is a user-oriented Feret diameter (Fig. 5e). This yields a set of two numbers describing the length and orientation angle. Figure 5(f) indicates maxi-mum particle width, which is obtained by doubling the maximaxi-mum value of the so called distance function. In a distance function, any pixel has a value proportional to its distance from the particle edge. Among advanced measurements is the very important maximum Feret diameter (Fig. 5g). In the case of a convex particle, this characteristic is identical to the maximum intercept length. This measurement also is characterized by two numbers describing the length and orientation angle. For analysis of spatial distribution and inhomogeneity, the coordinates of the center of gravity (Fig. 5h) can be very useful. A measure of coordinates of the first point (Fig. 5i) has a similar value. First point is selected as the most left-situated pixel from the top row of particle pixels. A characteristic of a concave particle is obtained from the convex hull (Fig. 5j). Note that all the measures presented can be applied to the convex hull as well as to the
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initial particle. Similar to the convex hull is the bounding rectangle (Fig.
5k), suitable for shape characterization. Figure 5(l) illustrates the number of holes. Other characteristics also are available, for example, number of Euler points (suitable for convexity/concavity quantification and devia-tion moments). These are the most common measurements, and most image analysis specific software can be used to obtain their values.
Quantification of some of these measurements is very difficult, if not impossible, without the use of a computer.
One of the most important tasks during microstructure quantification is to develop appropriate links between the classical stereological param-eters (which usually have a well-elaborated theoretical background) with parameters offered by image analysis. Specific problems arise when taking into account the digital (discontinuous) nature of computerized images. Examples of application difficulties are measurements along curvilinear test lines (used in the method of vertical sections) and errors in perimeter evaluation.
Any microstructure should be described in a qualitative way prior to quantitative characterization because the latter simply expresses the former in numbers. Otherwise, large sets of meaningless values are
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
Fig. 5 Basic measures of a single particle: (a) initial particle, (b) area, (c) perimeter, (d) and (e) Feret diameters, (f) maximum width, (g) intercept, (h) coordinates of the center of gravity, (i) coordinates of the first point, (j) convex hull, (k) boundingrectangle, and (l) number of holes
generated that are difficult to interpret. Every material structure consists of basic elements, called structural elements, such as fibers in a compos-ite, chemical compounds, and small precipitates of phases. In practice, four characteristics should be described to fully characterize a microstruc-ture (Fig. 6):
O Amount of all the structural constituents and the following character-istics separately for each constituent
O Size (for example, particles or their colonies) O Shape (for example, of individual particles)
O Spatial distribution or arrangement (form) of particles over the material volume
Figure 6 shows a model of a microstructure and its modifications.
Figure 6(b) shows the structure resulting from doubling the amount of
(a) (b)
(c) (d)
(e) (f)
Fig. 6 Schematic of microstructures containingfour basic characteristics of a microstructure: amount, size, shape, and arrangement
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black phase. Note that there is no difference in size (even the size distribution is identical), shape, and arrangement (in both cases the arrangement is statistically uniform) between the images in Fig. 6(a) and 6(b). The structure in Fig. 6(c) is obtained by changing only the size of the objects, without altering their amount, shape, and arrangement. Changing the arrangement of structure in Fig. 6(c) results in a structure shown in Fig. 6(d), which differs from Fig. 6(a) in size and arrangement of the objects. Keeping the amount, size, and arrangement of the black phase in Fig. 6(a) and changing only the shape yields results shown in Fig. 6(e).
Finally, the structure shown in Fig. 6(f) differs from that in Fig. 6(a) in all the characteristics; that is, amount, size, shape, and arrangement.
Rationale for classifying into four basic characteristics presented above is not proven theoretically. However, in all cases known to the authors, characterization of the structure can be performed within the framework of amount, size, shape, and arrangement. The most important advantage of this simplification is that structural constituents usually require only four characteristics versus a host of parameters (such as those listed at the beginning of this Chapter) supplied by classical stereological methods, and even more parameters offered by contemporary image analysis. The following discussion provides some guidelines both on how to choose the optimal parameter subsets to quantify a structure and how to interpret the results obtained.