MODELING THE MICROSTRUCTURE OF SINTERED COPPER ¨rkka ¨2 Claudia Lautensack1 , Katja Schladitz1 , Aila Sa 1
Models and Algorithms in Image Processing, Fraunhofer-Insitut f¨ ur Techno- und Wirtschaftsmathematik, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany (
[email protected]) 2 Department of Mathematical Statistics, Chalmers University of Technology, 412 96 Gothenburg, Sweden Abstract. Microfocus computer tomography (µCT) is a promising tool for gaining insight into the microstructure of sintered materials. In this paper, we investigate volume images of sintered copper with spherical grains in the early stages of the sintering process. Methods from spatial statistics are used to analyze the microstructure of the copper samples. Starting from these observations, a model based on a random dense packing of spheres is proposed. Keywords: dense random packing, pair correlation function, porous media, spherical contact distribution, sintering
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Introduction
Sintering is a process for production of porous metals or ceramics in very complex shapes. First, powdered or granular material is compressed in a die. Then, the material is heated to a temperature close to the melting point which makes particles stick together resulting in a porous yet strong material. Since sintered copper combines a high specific surface area with mechanical strength, catalysts as well as filters are typical fields of application. Macroscopic properties (e.g. permeability) of these components highly depend on the microstructure of the material. Thus, mathematical models for the geometry of sintered structures are of interest, since they allow a systematic design of materials and production processes and thus an optimization of properties of materials or components. Modeling the microstructure of the sintered copper is part of a larger project aiming at a better understanding of the sinter process, in particular the relative movements of the particles before forming stable connections, see also [4],[6],[10]. First investigations have shown that microfocus computer tomography (µCT) is a promising tool for gaining insight into the microstructure of sintered materials (see [2],[9],[10]). In this paper, methods from spatial statistics are used to analyze the microstructure of sintered copper with spherical grains in the early stages of the sintering process. The results are used for fitting a model to the observed structures. In the beginning of the sintering, the sinter particles form a random packing. When heated, they start to move about each other until necks form. In order to describe this interaction, a Gibbs point process would be a desirable model for the particle centers. However, we have to deal with too high volume fractions (> 54%), which could be reached neither by classical hard core models such as the random sequential adsorption (RSA) nor by our simulations of Gibbs point processes. Therefore, we consider models based on random dense packings of spheres. [1] used random shifts in a system of non-overlapping 1
Figure 1: Volume renderings of the dataset at the beginning of the process and after 635 minutes. Sinter balls are contained in a cylinder with a half-spherical cap at the bottom. This view is from the bottom. balls to model the geometrical structure of concrete with spherical grains. For the sintered copper, we use a combination of the force biased packing ([3]) with random shifts. Moreover, we introduce methods for modeling of large pores and the distortion of the balls during the sinter process.
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Experimental setup
Spherical copper powder (diameter 100–120µm) was filled into alumina crucibles and fixed in position by an initial sintering step. For each sintering stage, a volume image of the specimen was taken using µCT (at room temperature). Subsequently, the next sintering stage was prepared by heating the specimen with a rate of 5K/min in a hydrogen atmosphere to the sintering temperature, which was equal during all sintering steps. Holding time at sintering temperature was doubled in each sintering step. This results in a sequence of 8 volume images, each 903 × 903 × 721 voxels large with a voxel spacing of 5.4µm containing approximately 10000 sinter particles. Each of the images is binarized and the balls are separated as described in [10]. A slice of the resulting 3d image is shown in Figure 2.
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Analysis
The set of balls is analyzed using methods from spatial statistics, namely summary statistics for point processes and random closed sets. Since the packing of sinter balls is less dense in the top layers of the container and in order to avoid boundary effects caused by balls touching the walls, analysis is restricted to a cube of 240 voxels side length. The number of balls included in the analysis ranges from 1679 (sinter time 5) to 1952 (sinter time 635). First, the pair correlation function g(r) of the ball centers is estimated using the edge corrected kernel estimator described in [5, p. 279]. Heuristically, g(r) measures how often pairs of points at distance r apart occur. See [8, p. 129] for details. The results presented in [10] indicate that irregularities (gaps at some locations, clusters of balls at others) exist within the copper samples. In order to detect them, the spherical contact distribution Hs (r) is used, which is the distribution function of the distance from a random point in void space to the closest point on the surface of the copper grains (see [8, p. 206]). Calculation is done via the Euclidean distance transformation on the background of a binary image of the sinter balls (2403 voxels with a voxel spacing of 5.4µm). 2
Figure 2: Slices of the binarized sinter particles (left), the corresponding force biased packing (middle) and the closing of the shifted packing (right). The third characteristic is the mean coordination number c¯ of the copper balls, i.e. the mean number of grains in contact with a given grain. Since the simulation algorithms produce hardly any grains in exact contact, we adopt the method of [3] and work with a tolerance parameter ε, i.e. we count spheres with surface-to-surface distance smaller than ε. The volume fraction of 54% (compared to 64% in [3]) makes the choice of ε difficult. The minimal value being detectable is the voxel spacing of 5.4µm. Instead of choosing a fixed value of ε, the change of c¯ with increasing ε is studied. All three characteristics show changes over time due to the so-called shrinkage: inter particle distances as well as pore sizes decrease slightly, see Figure 3. Sinter time Number of balls analyzed Volume fraction [%] Mean coordination number
0 5 15 35 75 155 315 635 1679 1786 1781 1804 1828 1847 1909 1952 54.8 54.5 55.0 55.1 55.1 55.5 55.9 55.8 6.82 6.79 6.76 6.79 6.88 6.95 7.12 7.20
Tab. 1: Results for different sinter steps. The values for the mean coordination number are taken from [10].
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Modeling
For the time being, we restrict attention to the situation at time 0, where the structure is closest to a sphere packing. Using the force biased algorithm (see [3]) 10 packings of 1679 balls with volume fraction 54.8% and with the same distribution of radii as in the original data set are generated. However, the force biased algorithm produces structures too regular, which do not cover the irregularities caused by the sinter necks. This is why a modification of the model is considered here. Taking up an idea of [1], after creating the packing as in the previous case, random shifts are applied to the ball centers. For a given number n of steps, a ball is chosen randomly from the packing and a new position is proposed from a uniform distribution in a cube of side length ρ centered at the old position of the ball. The proposal is accepted, if no overlaps occur, otherwise it is rejected. Here, the parameters n = 106 and ρ = 12.96µm are used. This model performs better than packings obtained with the pure force biased algorithm. However, the spherical contact distribution shows a deviation from the sinter data. This might be caused by gaps within the sinter structure as well as by the distortion of the sinter balls. Therefore, further variations of the force biased packing are considered. In order to model gaps, we use the same procedures as in the last model (force biased, then random shift) but include some dummy balls, which are deleted after the shifting operation. After 3
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Figure 3: Pair correlation function (a), spherical contact distribution (b), mean coordination numbers (c) and the tail of the spherical contact density (d) of the ball centers for sinter time 0, 15, 75, 315. some experiments, we used 100 dummy balls of radius 29.7µm at random positions within the packing. In order to model the distortion, the closing operation of mathematical morphology (see [7]) is applied to the image of the shifted force biased packing (2403 voxels with a voxel spacing of 5.4µm). The effect of this operation is illustrated in Figure 4. The structuring element is chosen as a ball of radius 2 voxels, which nearly preserves the volume fraction of the image (54.9%). The second method yields a better result for the spherical contact distribution. For the tail of its density the first method provides a better match. An analysis of the combination of both models and a method for estimating their parameters are subject of further research. The results of the statistical evaluation of the model structures compared to the real data are shown in Figure 5.
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Conclusion
Random close packings obtained by a modified force biased algorithm are a good model for the initial state of the sintering process. The random shifts from [1] already yield stronger variations observed in the data. The newly introduced use of dummy balls combined with morphological transformations might also cover irregularities caused by the growth of 4
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sinter necks. A further step is to examine whether this geometric model can capture the structural differences in the observed time steps of the sintering process, too, and could thus be used for space-time-modeling. Moreover, the model can be used for simulation of macroscopic properties of the sintered material.
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Acknowledgements
We would like to thank the ECKA Granulate GmbH & Co. KG for the supply of copper powder, Michael N¨othe for the experimental setup and the 3d imaging and Oliver Wirjadi for the segmentation of the images. We acknowledge financial support by the PPP Sweden project D/04/04372 of the German Academic Exchange Service. K. Schladitz acknowledges financial support by grant OH 59/5-1 of the German Research Foundation (DFG).
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