Ceramic Foams Porous Microstructure Characterization By X-ray Microtomography

Departamento de Física, Universidade Estadual de Londrina C.P. 6001, 86051-990 Londrina PR, Brazil LMPT, Departamento de Engenharia Mecânica, Universidade Federal de Santa Catarina C.P. 476, 88040-900 Florianópolis SC, Brazil Grupo de Engenharia de Microestrutura de Materiais (GEMM), Departamento de Engenharia de Materiais, Universidade Federal de São Carlos SP, Brazil Embrapa Instrumentação Agropecuária C.P. 741, 13560-970 São Carlos SP, Brazil


Introduction
Millimetric scale X and gamma ray computerized tomography (CT) has been used in many fields of application since its development for medical diagnoses in 1973.More recently, the need to look inside porous media at the level of pores and aggregates in order to observe the topology and geometry of the pores, as well as other applications, led to the development of micrometric scale tomography.This implied a specific study of radiation sources, detectors, collimators, mechanics, hardware and software to design dedicated systems for different fields such as archaeometry, soil physics, porous materials and other applications 1,2 .
Knowledge of the porous structure of amorphous materials is required to calculate microstructural parameters, such as pore size distribution and total porosity, and important for deriving the physical parameters of fluid flow in the sample, such as permeability.In this sense, the first possibility is to use 2D images to extract statistical descriptors of the pore space morphology, such as porosity and autocorrelation functions.These descriptors may be used directly to search for empirical correlations with the transport properties 3 .Alternatively, the statistical properties obtained from a cross-section may be used to constrain the generation of a 3D pore space [4][5][6] .Direct simulation of 3D flow in the reconstructed pore space may be performed using Navier-Stokes equations 7 or lattice gas/lattice Boltzmann simulations 8 .
A nondestructive way to obtain the statistical descriptors of porous geometries is to take microtomographic meas-Materials Research urements of the samples and analyze the images to extract the information of interest.The importance of the methodology employed in this work, which is based on image analyses of microtomographic measurements, is that it allows one to measure several morphological and topological descriptors of the microstructure that are difficult to obtain or not accessible by conventional techniques.
The purpose of this work was to obtain the microstructural parameters of ceramic foams using X-ray microtomography and image analysis.

Materials and Methods
The equipment used was a first generation X-ray tomograph with micrometric resolution 9 .The X-ray tube comprised a W anode and 3 kW tube operated at 70 kV/10 mA and at 50 kV/23 mA for two different energy conditions.A transmission energy filter of Sn was employed to obtain almost monochromatic lines 1,10 of 58.5 and 28.3 keV.Collimators of 60 µm and 215 µm were used at the NaI (Tl) detector entrance and the tube output, respectively.The beam width at the sample position was about 80 µm.The linear step of the projection measurements varied from 53.3 to 57.8 µm and the angular step was 0.8°.The number of projections was 225 and the scanning time for each projection was 5 s. Figure 1 shows a block diagram of the CT scanner.
The data matrix was processed by the "Microvis" reconstruction software developed at the Instrumentation Center of the Brazilian Enterprise for Agricultural Research Corporation (EMBRAPA).The Microvis software reconstructs the image by applying the filtered backprojection algorithm, in which the projections are initially transferred to the frequency domain and then bandlimited through the application of a filter.These filtered projections are brought back to the space domain and summed up to give the distribution of the linear attenuation coefficient values, in other words, to compose the image of the sample section 9,11 .
The microtomographies obtained were processed using the "Imago" image analysis software to determine total porosity, the autocorrelation function and pore size distribution.Imago was developed at the Laboratory of Porous Media and Thermophysical Properties (LMPT), Department of Mechanical Engineering, Federal University of Santa Catarina in association with Engineering Simulation and Scientific Software (ESSS).
The original gray-level microtomographies were processed with the "Imago" software to produce binary images after segmentation of the pore and solid phases.This procedure is based on the gray-level histogram, where the user selects the threshold that appears to best separate the graylevel classes associated with solid and porous phases.
The porous media represented in a 2-D binary image can be characterized by the pore phase function Z (x) as follows: (1) where x denotes the position with respect to an arbitrary origin.
The porosity, φ, the autocorrelation function, C (u), and the normalized autocorrelation function, R (u), can be defined, respectively, by the following statistical averages (de-  (2) where u is the displacement in the plane of the image.
Porosity, φ, is obviously a positive quantity limited to the [0-1] interval.When the media is homogeneous, the statistical parameters are independent of position x in space.Thus, the porosity is constant and R (u) depends only upon the vector u being independent of position x.
Moreover, when the porous media is isotropic, R z is a function only of u = |u| and does not depend on the direction of u.
The correlation function can be calculated directly in the image domain 12 .Let S be a section of a porous medium, given by a 2D binary representation, with the porous phase represented in black and the solid matrix in white.The binary image, S, is divided into two halves, S 1 and S 2 .Hence, In order to calculate R z (u), S 1 is first translated by a distance u along the x-axis; yielding S 1 (+ u).The spatial average indicated in Eq. 3 is calculated as an intersection of images, giving the correlation function.
C(u) relates to the probability of finding two points (pixels) separated by u and belonging to the same phase.
Liang et al. 5 calculated the autocorrelation function C (u) as a function of the two-dimensional vector u = (x,y) and then calculated its mean value around a circle with radius u = |u|.This procedure produces more reliable C (u) values because it increases the number of realizations needed to calculate this probability.
For an image f (x,y), the Fourier transform of the autocorrelation function is the power spectrum of f (x,y) (Wiener-Khinchin theorem).Thus, with the Fourier transform of the image, the correlation function can be obtained rapidly performing the inverse Fourier transform of the power spectrum.The 2D autocorrelation function is calculated by Imago using the Fourier transform.Fluctuations in the autocorrelation function are drastically reduced when C (u) is calculated by Fourier transform, compared with above described spatial method.
The pore size distribution is obtained by successive openings derived from mathematical morphology 13 , using balls with increasing radius.After an opening operation with a given ball radius r, the resulting image can be viewed as the union of radius-r balls completely enclosed in the porous phase.In this way, after opening, the porous phase loses all the features that can be eroded by a given radius-r opening ball.The cumulative porous distribution is given by: (7)   where f is the total porosity of the original image and φ (r) is the volume fraction of the porous phase after opening with a radius-r ball.
To reduce the processing time, the opening operation is not applied directly to the binary image but to a transformed image called background distance image.In this image, each pixel is labeled with its smallest distance to the neighboring background.This labeling technique uses a sequential algorithm 14 , in which the Euclidean distance is approximated by a discrete integer distance.The most commonly used discrete distance is the chamfer distance, known as d 3-4 , in which each neighbor from a given point, which is taken following the horizontal and vertical coordinate axis, is considered to be 3 measuring units distant from the starting point.The diagonal neighbors are considered to be at 4 measuring units from that point.Thus, this discrete distance gives the numerical approximation of 4/3 = 1.333...to the square root of 2 (1.4142...).The main advantage of using this discrete distance is that it requires lower computer storage space, since only integers are stored in the resident memory.
The total porosity was also determined by other methods.Through various transept analyses of each microtomography, the total porosity was calculated by means of the linear attenuation coefficients at particle density and average medium, as follows: (8)   where µ m is the average medium linear attenuation coefficient of the transept considered in the tomography and µ p is the average particle linear attenuation coefficient of the transept.
For purposes of comparison, results from conventional methodologies such as Arquimedes, Hg and gas porosimetry were also employed, as well as the single gamma ray transmission technique.
Three kinds of SiC-Al 2 O 3 ceramic foams used for solidfluid separation in the metallurgical, automotive and petrochemical industries were studied: 60 ppi, 75 ppi and 90 ppi (ppi = pores per linear inch).The foams were supplied by the Microstructural Materials Engineering Group (GEMM), Materials Engineering Department, Federal University of São Carlos.The microtomographies of these Materials Research samples were performed at 28.3 keV. Figure 2 illustrates the three types of samples.
Table 1 lists the porosity results.Each of the three images was divided into two parts and the porosity was determined for both the parts and the entire image.The results show that the 60 ppi image is homogeneous, while the 75 ppi image is somewhat heterogeneous and the 90 ppi image presents important heterogeneities.These heterogeneities can be associated to the scale of the image analysis: possibly, the field of the analyzed image is small compared with the pore size.Therefore, for a more accurate calculation of porosity and other geometrical parameters, larger images may be required (retaining the same spatial resolution) in order to reach the scale of homogeneity of the microstructure, if it really exists.Despite their heterogeneities, the 75 and 90 ppi images were used in    this study to determine the geometrical parameters of porosity, autocorrelation function and pore size distribution.This was also done to present the technique of image analysis that allows the microstructure of ceramic samples to be characterized in greater detail.Table 1 shows that, for determining total porosity in the three types of samples, image and transept analysis, gamma ray transmission 15 and the conventional method 15 supplied results in reasonable agreement.For the 60 ppi ceramic foam, only two of the seven total porosity measurements are not within the 95% confidence level interval, and one of these two (89.7%) is very close to the upper level (89.5%).For the 75 ppi ceramic foam, only one of the five measurements is not within the 95% confidence level interval, and even so it is close (92.7%) to the upper level (91.9%).For the 90 ppi ceramic foam, only one of the five measurements is not within ods only allow the sample's average pore radius to be determined.The pore frequency distribution provides a much better characterization of the quality and properties of the ceramic foam.75.5% of the pores of the 60 ppi ceramic foam had radii ranging from 0.26 to 0.64 mm, but 42% of the pores had radii of 0.38 to 0.45 mm.The 75 ppi ceramic foam displayed no clearly centered pore radius distribution, since 94% of its pores had radii ranging from 0.26 to 0.70 mm.76% of the pores of the 90 ppi ceramic foam presented radii ranging from 0.22 to 0.50 mm.The two halves of the 75 and 90 ppi samples showed total porosity differing by 6% and 26%, respectively.Table 2 shows the comparison of present results for ceramic foams samples with data from the literature, which are discussed in an extensive review paper 16 of the area.This review paper, among other data analysis, reports re-   There is very good agreement for total porosity results for the three types of ceramic filters.
Average pore size is also shown at Table 2. But, due to the complexity of the porous space geometry, this parameter is strongly model dependent and hardly could be compared.
Figure 15 shows a comparison of the normalized autocorrelation for the 60, 75 and 90 ppi ceramic foams.As can be seen, the correlation-curves resemble each other closely, denoting a similar spatial arrangement.Correlationlength, as defined by Lanteajoul, C. 17  , was estimated as 0.30 mm for the three samples.

Figure 1 .
Figure 1.Block diagram of the CT scanner.

Figures 9 to 11
show selected transepts of the samples' CT images.Figures 12 to 14 depict the binary images of the samples generated by the Imago software and used for the calculations of porosity and autocorrelation.

Figure 6 .
Figure 6.60 ppi ceramic foam pore size distribution.

Figure 12 .
Figure 12.Binary image pattern of the 60 ppi sample.

Figure 13 .
Figure 13.Binary image pattern of the 75 ppi sample.

Figure 9 .
Figure 9. Selected transept of the 60 ppi ceramic sample's CT image.

Figure 10 .
Figure 10.Selected transept of the 75 ppi ceramic sample's CT image.

Figure 11 .
Figure 11.Selected transept of the 90 ppi ceramic sample's CT image.

Table 1 .
Total Porosity (P t ) of the ceramic foam samples.