Stereological Analysis of the Statistical Distribution of the Size of Graphite Nodules in DI

The estimate of a distribution law of the nodule diameters in a volume of cast iron provides information about the graphite nucleation kinetics, and also about the crystallization kinetics. This information is essential for building more accurate mathematical models of the alloy crystallization. The mapping of a Cumulative Distribution Function (CDF3) of radii for graphite nodules in ductile iron is presented on the base of a Probability Density Function (PDF1) of the chord length distribution for random sections of the sample at the planar cross-section.


Introduction
Ductile Cast Iron (DI) is a widely used alloy of the modern technique due to his high level of functional and technological quality. Depending on the chemical composition and processing method, the mechanical properties of DI can vary within a wide range. Among the factors that influence on the properties of DI are the volume fraction of the graphite nodules, their volumetric quantity and size distribution.
In the paper of Yin at al. [1] the results of the use of X-ray microtomography for estimation a three-dimensional size distribution and quantitative characteristic of the morphology of graphite nodules in the stepwise casting of cast iron from cast iron are presented. Unfortunately, the graphite nodules whose diameter is smaller than 25 μm are ignored because of the resolution of 3D volume and the interference of inclusion and microporosity. Probably, it is for this reason that samples of castings with a wall thickness of less than 15 mm were not analyzed.
The stereological analysis of the 2D image is used most often for the analysis of the characterization of the properties of the set of spherical particles in the nontransparent material like DI [2,3].
The determination of probability density function f 3 (r 3 ) of the radii of graphite nodules (PDF 3 ) occurring in a volume of ductile cast iron using data on the probability density function of the radii of the cross-sections of these nodules f 2 (r 2 ) in a metallographic specimen (PDF 2 ) can be regarded as a special case of Wicksell's corpuscle problem (WCP) [4]. According to Wicksell PDF 2 can be calculated explicitly based on PDF 3 as follow: where R max denotes the maximum radius of the nodules in the sample, and E[r 3 ] is the expected (mean) value of the radius of spherical particles r 3 in this sample.
For the estimation of the PDF 3 based on PDF 2 the inverse solution of the Eq. (1) should be obtained. In practical metallographic tasks, the application of analytical solutions to WCP on the base of PDF 2 gives unsatisfactory results [3], therefore there is a need for the use of implicit numerical solution method of Eq. (1).
Application of the Wicksell equation for the estimation of the size distribution of spherical particles on the base of empirical distribution of the sizes of planar sections has been presented by Sheil [5], Schwartz [6], Saltykov [7,8], Li at al. [9]. A similar solution for the mineralogy task has been presented in [10]. Unfortunately, small numerous errors of the empirically estimated function f 2 (t) result in the "arbitrarily large perturbations of the solution" [11,12].

Research Method
Theoretical base. As it is has been proved in [13] the empirical cumulative distribution function of the spherical particles size (CDF 3 ) can be calculated on the basis of the statistical distribution of the length of the chords (PDF 1 ) obtained by intersecting spherical particles with a system of random secants. CDF 3 can be mapped by equation: where S is the mean external surface (estimated value) of the nodular particles in the sample, and f 1 (t) is the PDF 1 .
The derivations of Eq. (2) with respect t parameter gives: ( This formulation corresponds to known solutions of Cahn and Fullmann [14], Lord and Willis [15], and Spektor [16]: (4) Verification of the method. Correctness of the Eq. (2) application has been verified in [17]. Two sets of virtual spherical particles with inhomogeneous dimensions were generated: normal distribution (2048 particles with mean radius 120 units and standard deviation 15 units) and bimodal one as the superposition of two normal distributions (2048 particles with mean radius / std. deviation 90 / 10 and 1024 particles 150 /150). Comparison of the mapping results of the CDF 3 with the theoretical values for above distributions is presented in Fig. 1.

Sample Preparation and Data Acquisition
Analyzed DI. As the subject of the measurements the samples were using from the industrial casting produced from DI. The chemical composition of the iron was measured by spectrometer LECO type GDS500A and apparatus HORIBA Scientific type EMIA 820 B(W) for carbon and sulfur. Chemical composition of the iron used for casting production is presented in the Table 1. where eutectic saturation ratio S c = C / (4.26 -0.3·Si -0.36·P). Preliminary data processing by using the Leica QWin software. Each analyzed image of a microstructure was transformed into a binary image. The binary images included only these areas from the original image which could be potentially taken for measurements. All pixels belonging to the matrix were discarded. Having the binary images it was necessary to clean them from the areas which were unsuitable for the analysis. In this work all detected cohesive group of pixels with the area less than 15 pixels (3.5 μm 2 ) were rejected from the data set as noise. Also, if the shape of each group was significantly different from the circular one, this group was deleted. Determination of the shape was done by using two factors from the QWin software i.e. AspectRatio and Roundness. For both factor a limit was set from 1 to 1.2. Each group which did not belong to this limit was not taken into consideration. In addition, compact pixel groups were not included in the measurements if their geometric center was located at a distance less than the radius of the largest cross section of particles from the image boundary.

Obtained Results and Discussion
Data processing. The total length of the secants used for measuring the length of the chords was 49,280,000 pix (23,593,046 μm). The calculations using the Eq.
and by the Saltykov's method of invers diameter for N V where: P L -the number of intersections of the surface of graphite nodules per unit length of the secants, F -total surface area of the analyzed metallographic sections, d i -diameters of the 2D sections i. The results of the average surface area estimation for the graphite nodules are presented in the Table 2. The empirical cumulative distribution functions and probability density function prepared on the base of measurement are presented in Fig. 4.  Distribution of the sizes of graphite nodules. Results of mapping of graphite nodules sizes calculated by Eq. 2 are presented in Fig. 5a. As follows from this diagram, the chord method has a low accuracy for the smallest 10% of the spheroids in the analyzed population. For the samples described, this interval includes spherical particles with a radius of less than about 7 µm. Unfortunately, for r 3 < 6 μm the result of CDF 3 evaluation is negative. With the increase of the nodule radius, the accuracy of the CDF 3 mapping by the chord method increases and for the particles with big sizes, the accuracy of mapping is reasonable.
It was assumed that the spherical particles of graphite with a radii smaller than 6 μm are not present in the samples. Dependences of cumulative amount of the nodular graphite particles, whose radius does not exceed a given value, for samples No. I and II are shown in the Fig. 5b

Summary
Empirical form of the cumulative distribution function of the sizes of graphite nodules in the DI can be mapped on the basis of the empirical measurement of probability density function of the statistical distribution of the length of random chords of the nodules by Eq. (2). Results of the experimental data processing are presented for two samples of industrial DI.
The chord method has a low accuracy at the interval nearby to 10% of the smallest spheroids of the analyzed population (like in the case of virtual one [17]). With the increase of the nodule radius, the accuracy of the mapping of the cumulative distribution function of the graphite nodules by the chord method increases and for the particles with big sizes, the accuracy of mapping is reasonable.

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Science and Processing of Cast Iron XI