digital image analysis. However, the conversion of deduced two-dimensional into true three-dimensional particle (or pore) size distributions, which represents an ...
2D-3D CONVERSION OF OBJECT SIZE DISTRIBUTIONS IN QUANTITATIVE METALLOGRAPHY Jürgen Gegner SKF GmbH, Department of Material Physics Ernst-Sachs-Strasse 5, D-97424 Schweinfurt, Germany University of Siegen, Institute of Material Engineering Paul-Bonatz-Strasse 9-11, D-57068 Siegen, Germany ABSTRACT In common metallographic laboratory practice, object distributions in discontinuous multi-phase microstructures are characterized by incident light optical or scanning electron microscopy micrographs of polished planar microsections. Precipitation or dispersion particle hardened metal alloys and steel inclusions (e.g. MnS) constitute prominent examples but also sintering, welding or creep pores in metallic or ceramic materials are amongst related structures. The area and from it the volume fraction of these objects can be taken directly from such micrograph series, simplest by applying digital image analysis. However, the conversion of deduced two-dimensional into true three-dimensional particle (or pore) size distributions, which represents an important task of quantitative metallography e.g. for the computation of model distributions of limited extent as representative volume elements or establishing reasonable steel cleanliness specifications in the framework of material quality assurance, requires a mathematical calculation. In the present work, therefore, a simplified correction technique that offers an often-appropriate method for the necessary stereological analysis is derived and exemplarily applied to a known model and a real dispersion microstructure. The usability of this 2D-3D objects size conversion procedure and its restrictions are also discussed. INCLUSIONS IN STEELS – AN INTRODUCING EXAMPLE In practical metallography, object distributions in discontinuous multi-phase (-constituent) microstructures are usually characterized by planar microsections from incident light or scanning electron microscopy (SEM) micrographs. Besides precipitation and dispersion particles e.g. in hardened aluminum alloys or in ODS (oxide-dispersion strengthening) (super) alloys produced by powder metallurgy, which can also be made from dilute solid solutions by controlled internal oxidation (with Ag, Cu, etc. as matrix materials), and pores (resulting from sintering, welding, creep deformation), nonmetallic inclusions in steels caused by manufacturing represent a concrete example that is of particular importance to materials engineering. The basic chemical compositions of such matrix-embedded particles are MnS, Ti(C,N), Al2O3, SiO2, CaO, TiO or CaS. The content of macro-inclusions is measured by the blue fracture test in the framework of a material specification check. Micro-inclusions in steels are usually rated according to chart diagrams for polished unetched samples of the undeformed prematerial at a magnification of 100× as given in the international standard ISO 4967 (1); see also ASTM E 45 and EN ISO 683-17. As an example of a highly clean material grade, Fig. 1 reveals a light-optical micrograph of through-hardenable standard 3-138
rolling bearing steel SAE 52100. Randomly distributed globular oxide-type microinclusions are visible in this planar microsection. The isolated larger particle near the center of the image is rated as DS 0.5 (diameter between 13 and 19 µm). The other still smaller inclusions belong to group D-fine (diameters from 3 to 8 µm), for which the number per reference area is specified. Generally, the sizes range from about 0.1 µm to maximum 100 µm. The standards rate nonmetallic inclusions as globular up to a maximum aspect ratio of 3:1 (ISO 4967) to 5:1 (ASTM E 45) in the cut micrograph.
Fig. 1. Globular oxide-type micro-inclusions in SAE 52100 steel. Figure 2 shows a typical particle size distribution. The probability density function (PDF) of the diameters of the steel inclusions complies with the logarithmic normal distribution, which is most simply expressed in terms of parameters m and s: ⎧ 1 ⋅exp ⎡⎢ − (ln d −m)2 ⎤⎥ , d >0 ⎪⎪ d s 2π ⎢ 2 s 2 ⎥⎦ ⎣ w (d ; m, s ) = ⎨ ⎪ 0 , d ≤0 ⎪⎩
(1)
The length unit µm is used here. Since the k-th moment of a random variable (d) with lognormal distribution is given as exp(km+k2s2/2), the mathematical expectation μ and standard deviation σ are calculated as follows (2):
⎛ 2m + s 2 μ = exp⎜⎜ 2 ⎝
⎞ ⎟ , σ = exp(2m + 2s 2 ) − exp(2m + s 2 ) ⎟ ⎠
(2)
The parameters m and s2 represent the mean value and variance of the logarithmically distributed quantity lnd, respectively. The related data is given in Fig. 2. As a special, positively skewed form of a general non-symmetrical Weibull frequency distribution, the lognormal distribution is typical of the size arrangement of polydisperse particle systems with individual diameters that possess a minimum and a maximum value: a long-known example is fragmented, pulverized or (micro-) milled particles (3). Although rather different in their effect depending upon shape, chemical composition and mechanical properties, especially near-surface nonmetallic inclusions, even from 3-139
dimensions of a few µm on, may considerably reduce the fatigue strength of the steel because of the stress concentration arising there and, hence, can diminish the service life of dynamically highly loaded components (e.g. rolling bearings, toothed gears). Whereas the area fraction, which is most easily deduced directly from a sufficient number of micrographs by means of digital image analysis (4), immediately provides the volume fraction (5, 6), it is necessary to determine the real distribution of particle sizes for a check of the degree of cleanliness in the framework of quality assurance. Therefore, the employment of appropriately automated evaluation tools would allow for significantly facilitate and improve today's industrial practice of testing, specifically in the field of steel making and application.
Fig. 2. Typical size distribution of nonmetallic inclusions in steels. MOTIVATION OF THE WORK
Arising from the discussion above, the translation of a suitable mathematical method for an analysis of cut micrographs into an easy-to-use computer program for serial examinations represents an important object of computer-aided quantitative metallography. Nowadays, the desire to integrate such a technique into current standard software (4), for this reason, is submitted particularly by industrial materials laboratories to the involved electronic data processing (EDP) development companies. However, even advanced commercial image analysis systems (hitherto) are not able to offer this application-relevant option. Therefore, the present paper covers a mathematically rather simple evaluation method and confirms its practical translatability into a computer program. The regarding expansion recommended for the available standard systems is supported this way. On the other hand, the given guidance enables a reader, who must realize this image analysis tool immediately in his metallographic laboratory, to develop an appropriate program code by himself, which task merely requires a commercially available compiler (e.g. FORTRAN or C). For a qualified evaluation of the possible options and the limitations of the technique employed for a stereological analysis of planar microsections, a critical check of the results is of particular interest. Thus, a realistic computer test on a modeled object system is performed. In addition, a real particle dispersion microstructure is analyzed using this mathematical tool. A detailed survey of important textbooks on such statistical tasks in quantitative metallography is already available in the literature (7). 3-140
PROBLEM DEFINITION
When a planar microsection is prepared, particles embedded in the matrix are only rarely cut through their centers but mostly at any other position. Accordingly, the probability of a certain particle to be detected at all is obviously reduced with its volume. The two-dimensional particle size distribution in the plane of the microsection face can be deduced directly from microscopic images. It is expressed in m–2 and designated n A si if this image analysis is evaluated in a suitably discretized form by means of an interval subdivision having the measure classes si. The problem to be solved now is that the real size distribution of the particles in the volume (unit m–3), n s j , which is the search quantity, differs from n A si for the mentioned reasons and V
that the probability densities must not simply be identified with each other. Rather, the conversion requires a stereological evaluation. The spatial measure class is called sj in the applied discretized representation. STEREOLOGICAL SCHEIL-SALTYKOV CORRECTION TECHNIQUE
The shape of the embedded particles can frequently be assumed as (roughly) spherical in real precipitation or particle dispersion microstructures. This approximation holds, for instance, for nonmetallic globular-type inclusions in steels. Considering the orientation of the particles (e.g. rods) towards the plane of the microsection face then becomes unnecessary. In this case, the Scheil-Saltykov technique can be applied for the required stereological object analysis since it permits the calculation of the real (spatial) size distribution of spheres (e.g. particles, pores) from metallographic cut micrographs (8). Therefore, the method is derived below followed by a critical test and discussion with reference to two application examples. Mathematical Analysis of Planar Microsections
The system under examination is supposed to consist of spherical objects (particles, pores) of various diameters in a statistical arrangement in the unit volume. Thus, any planar microsections (micrographs) result in circles. Assuming that a sufficient number of such microsectioned objects are detected and evaluated, the measured maximum of their diameters, s A max , is equal to the theoretical upper limit, i.e. the value s V max of the largest sphere measure: s A max = s V max
(3)
The circles are now assigned to Ns size classes, which are counted from 0 to Ns–1, between 0 and s A max at a constant interval width Δ. Thus, due to Eq. (3), the corresponding categorization of the original objects in the volume is identical with that. Accordingly, spheres of size class i can result in circles of categories 1 to i in the planar microsection. Therefore, a mathematical analysis of the corresponding probability is necessary. For this purpose, n si , j is at first supposed to designate the numA
ber of sectioned circles of size group i per unit area, which result from spheres of the j class with a diameter (j+1)·Δ. Then, the following simple relationship is valid: 3-141
n A si =
N s −1
∑n j =i
A
(4)
si , j
Figure 3 illustrates the interconnections in a schematic representation: sectioned circles of category i having diameters between i·Δ and (i+1)·Δ are thus be produced whenever the distances between the centers of the spheres belonging to the j size class and the section plane under consideration range from hi,j to hi+1,j. The evaluation presupposes a statistical distribution of the objects in the volume. Since they can be located on either side of the microsection plane, the following relationship holds: n
A
si , j
= 2n
V
sj
(hi , j − hi +1, j )
(5)
The lengths hi,j and hi+1,j are calculated by elemental geometry according to Fig. 3: hi , j =
Δ ( j + 1) 2 − i 2 2
hi +1, j =
(6)
Δ ( j + 1) 2 − (i + 1) 2 2
(7)
In order to simplify the manner of expression, a dimensionless quantity of the distance measure is defined as follows:
Aij =
2 (hi , j − hi +1, j ) Δ
(8)
Substituting Eqs. (6), (7) and (8) into Eq. (5) provides: n
A
si , j
=Δ⋅n
V
sj
(9)
Aij
Thus, Eq. (4) can be rearranged as follows to yield a determination formula for deriving the sought spatial ( n s j ) from the known planar size distribution ( n A si ): V
n A si = Δ ⋅
N s −1
∑n j =i
V
sj
(10)
Aij
Fig. 3. Planar microsections through spheres of size group j for cut circles of class i.
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This set of equations, including the unknown quantities n
V
sj
, is of upper triangular
form and can be solved recursively by inserting, starting with the largest index i:
n A si
⎛ = Δ ⋅ ⎜ n V si Aii + ⎜ ⎝
N s −1
∑
n
j =i +1
V
sj
⎞ Aij ⎟ ⎟ ⎠
(11)
Finally, the sought size distribution of the spherical objects within the volume is given as follows:
n V si
1 = Aii
⎛ n si ⎜ A − ⎜ Δ ⎝
N s −1
∑
n
j =i +1
V
sj
⎞ Aij ⎟ ⎟ ⎠
(12)
The derived stereological analysis technique is now applied to two examples. First, a direct verification is performed by means of the evaluation of a planar microsection for a given particle arrangement. Reverse Calculation of a Lognormal Particle Size Distribution
For the evaluation and the estimation of the effect exerted by the size class correction according to the Scheil-Saltykov technique, a computer simulation is performed for a known model system containing 20000 spherical objects that are statistically arranged in a cuboid of dimensions 500×500×50 µm3. The chosen PDF of the diameters, which is shown in Fig. 4, follows a logarithmic normal distribution according to Eq. (1). The parameters, expressed in µm, are indicated in the diagram; see Eq. (2).
Fig. 4. Two-parameter lognormal distribution of the real sphere sizes in the model object arrangement generated artificially for the computer test.
The presented method test is based on the stereological analysis of a sectional micrograph of 500×500 µm2, which is calculated parallel to the large side faces of the cuboid in the middle of the short edge and comprises 810 circles. The result of this modeling and the mathematical evaluation of the simulated metallographic microsection are summarized in Fig. 5, where the probability densities are illustrated in dis3-143
cretized form, respectively. It can be seen that in the planar microsection the size class distribution of the particle diameters widens and shifts towards lower values. Obviously, this marked change is properly corrected by the performed stereological analysis: the normalized probability density for the spatial arrangement, derived from the planar microsection, reproduces the actual size distribution of the given system of spheres in good agreement. The quality of the correction effect apparently depends upon the number of the considered objects and the examined total area. Therefore, in materials engineering practice, a representative series of micrographs should be evaluated to further improve the reliability of the obtained results.
Fig. 5. Simulated planar microsection (2D) and comparison of the calculated with the actual object size distribution (3D).
It is known from the literature that the correction may assign negative occupation to low-occupied size groups (8). The performed stereological analysis avoids this nonphysical effect by setting the relevant classes to zero. STEREOLOGICAL ANALYSIS OF A REAL PARTICLE DISTRIBUTION
The computer test presented above confirms the applicability of the Scheil-Saltykov technique by the successful evaluation of a planar microsection comprising about 800 objects, which has been derived from the known original three-dimensional distribution of randomly located 20000 spheres. In the following practical example, a particle dispersion microstructure produced by internal oxidation of a Pd-2.0 at.% Fe alloy in air at 1573 K is examined using this stereological analysis tool. The basic SEM micrograph of the polished metallographic cross-section is shown in Fig. 6. This planar microsection contains 742 particles, the chemical composition of which is Fe2O3 (internal oxidation at high oxygen offer). The microsectioned objects of Fig. 6 are of a circular shape in an acceptable approximation so that, as precondition for the application of the Scheil-Saltykov correction technique, spherical particles can be assumed. Thus, by considering circles of equal area at the center of the real sectioned objects, the two-dimensional size distribution is deduced from the SEM micrograph by digital image analysis (4). The obtained probability density function is given in Fig. 7 in discretized representation.
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Fig. 6. SEM micrograph (secondary electron image) of the cross-section of an internally oxidized dilute Pd-Fe alloy.
Figure 7 also shows the real particle size distribution in the volume that is calculated from the 2D graph by means of the introduced stereological correction method. The interval width Δ used here matches the value of the preceding computer test. The derived PDF of the object diameters is an approximately exponential distribution.
Fig. 7. Probability density function of the particle diameters in the planar microsection (2D, cf. Fig. 6) and derived size distribution in the volume (3D). DISCUSSION
In practice, the applied stereological analysis of the particle size from planar microsections (micrographs) according to the Scheil-Saltykov correction technique leads to reasonable results for many discontinuous multi-phase or related microstructures. The performed computer test emphasizes this success and, hence, the applicability to the integration into standard software systems of digital image analysis by means of the evaluation of a modeled known particle arrangement in the volume. However, the correction quality generally decreases with increasing deviation from the lognormal distribution of particle sizes, which is taken as the basis of the presented check. The
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monodisperse system of spheres of same diameter represents an extreme example of this kind that is, on the other hand, rather unrealistic. If the particles noticeably deviate from the presupposed spherical shape resulting in sectioned structures, which are markedly different from circles, it is necessary to apply more advanced stereological analysis tools. The orientation of the objects, however, that then must be taken into account, poses difficult problems. SUMMARY
The mathematical derivation of real particle size distributions for particle dispersion or porous microstructures from planar microsections represents a practically important task of quantitative metallography e.g. in the field of quality control in material technology laboratories. The reason is that in most cases polished micrographs are prepared for the characterization, whereas transmission images (i.e. projections, application of transmission electron microscopy mostly unnecessary due to sufficiently large object dimensions) are only rarely used. The required input data is easily obtained by means of (automated) digital image analysis. In practice, approximately spherical particles (circles as sectional objects) often exist with size distributions between an upper and a lower limiting value. In these cases, the Scheil-Saltykov correction technique presented in this work can be applied successfully for stereological analyses. This conclusion is confirmed by the performed computer test: the logarithmic normal distribution simulated for the object sizes of the three-dimensional model covers a large field of real systems (e.g. nonmetallic inclusions in steels). Also, a microstructure produced by internal oxidation is evaluated using the developed analysis tool. The correction technique, which the present paper proves to be suitable for many practical instances, can be employed by the user, according to the instructions given above, within a computer program for an automated evaluation of polished micrographs. In addition, it is recommended that commercially available analysis systems expand their performance spectrum in the corresponding module by this practically important extra operation. CONCLUSION
The present paper describes a method for the simplified calculation of the real size distribution of objects in particle dispersion or porous microstructures from cut micrographs for the use in quantitative metallography. The stereological analysis of the two-dimensional objects of the planar microsections is performed by applying the Scheil-Saltykov technique for spheres. The reliability of the mathematical tool is critically assessed by a computer test, where both the size distribution and the spatial arrangement of the particles are known. Also, a real discontinuous dual-phase microstructure is evaluated. This method has proved to be successfully applicable to many practical cases and, therefore, is suited for individual use and integration into commercial software systems of digital image analysis. Limits, within which the stereological correction technique is applicable, are discussed.
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REFERENCES
1. International Organization for Standardization (ISO): Steel – Determination of content of nonmetallic inclusions – Micrographic method using standard diagrams, ISO 4967: 1998(E), 2nd ed., October 1998. 2. Aitchison J, Brown J A C: The lognormal distribution. Cambridge, Cambridge University Press, Cambridge, 1957. 3. Kolmogorov A N: 'On the logarithmic normal distribution law of the dimensions of particles during fragmentation'. Dokl. Akad. Nauk SSSR 1941 31 (2) 99-101. 4. Gegner J, Öchsner A: 'Digital image analysis in quantitative metallography'. Pract. Metallogr. 2001 38 (9) 499-513. 5. Underwood E E: Quantitative stereology. Reading, Massachusetts, Addison-Wesley, 1970. 6. Exner H E, Hougardy H P: Quantitative image analysis of microstructures. Oberursel, DGM Informationsgesellschaft mbH, 1988. 7. Gegner J, Öchsner A, Henninger C: '2D and 3D modelling of discontinuous dualphase structures and equivalent microstructures from microscope-made image series'. Pract. Metallogr. 2003 40 (11) 564-81. 8. Saltykov S A: Stereometrische Metallographie. Leipzig, VEB Deutscher Verlag für Grundstoffindustrie, 1974.
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