Cellular Ceramics / p3

3.1 Characterization of Structure and Morphology 255

fixed value of the angle h. A usual measurement can make use of up to several hundred projections.

For a fixed projection angle h the object function can be calculated from all beams that pass through a point (x0,y0):

As only a limited number M of projection angles are measured, the integral must be replaced by a sum:

Points that are not passed by any beam are calculated by interpolation of measured adjacent points. These mathematically extensive Radon backprojection calculations can be carried out in several different ways that are mathematically equivalent but differ in the image quality. The Radon backtransformation, as well as the necessary execution of low-pass, ramp, and window filters to achieve artefact-free two-dimensional images of the sample, is performed by high-performance computers. An example of a two-dimensional reconstruction of a polymer-derived ceramic foam is given in Fig. 28a. Stacking successive two-dimensional reconstructed layers facilitates three-dimensional reconstruction of the volume of interest of the measured specimen (Fig. 28b).

Figure 28 a) Two-dimensional slice and b) three-dimensional reconstruction of an open-cell ceramic foam derived from preceramic polymer.

Measured characteristics

The morphometric structure model index (SMI) is calculated from the three-dimen- sional tomographic data. The SMI was introduced for quantification of bone microarchitecture and can be used in foam structure analysis to characterize the shape and anisotropy of cellular materials. The SMI is calculated according to [47]

256 Part 3 Structure

where S is the strut surface in a volume V, and dS/dr the surface-area derivative. The foam surface area is rendered by triangulating the surface of a specific volume of interest (VOI) with the marching cubes method [48] and the volume is defined by setting up polyhedra inside the VOI that match the bounding surface triangles. An SMI of 4 describes spherical pores; any deviation from spherical cell geometry results in lower values. A rod cell structure is described by an SMI of 3, and lenticular pores have an SMI of 0. Negative SMIs result from cells with concave surfaces, which are common if coalescence of pores occurs.

The surface/volume ratio, mean cell size, and strut thickness were calculated by the distance transformation (DT) method [49]. The connectivity density (CD) was computed from CT data by using the Conn–Eulor principle [50]: reconstructed twodimensional CT data images of two neighboring slices are compared by the Boolean EXCLUSIVE–OR operator. The result is superimposed onto the original pictures and analyzed. All new bridges (B: new connections), holes (H), and islands (I) are counted and CD is calculated:

where h is the distance between the two slices (h = 38 mm), and A the image area. In the case of determining CD of the pores, B and I denote cell windows and cells, respectively, and H, which corresponds to isolated strut material completely surrounded by pores, is unlikely to occur. The value is identical if CD is calculated for the strut material, where B denotes new struts, H cells, and I isolated struts.

A brief overview of CT data-evaluation parameters is given in Fig. 29.

1 >1

strut size/cell diameter

Figure 29 Evaluation parameters from tomography data.

Comparison of ceramic foams from different fabrication methods

Different fabrication methods for ceramic foams result in ceramics with unique structures. Due to the fabrication process, reticulated foams show hollow struts that can be clearly seen both by image analysis and two-dimensional tomography. These foams are characterized by pronounced anisotropy (degree of anisotropy 1.18 compared to 1.08 for polymer-derived ceramic foams) and irregularly shaped foam cells (SMI << 0). Conversely, sintered hollow spheres have a low degree of anisotropy and

3.1 Characterization of Structure and Morphology 257

a moderate SMI. The low SMI is due to the fact that not only is the interior of the sphere taken into account (which would result in an SMI of ca. 4), but also the irregularly shaped gaps between the cells. The polymer-derived ceramic foams that were foamed by in situ formation of gas bubbles had the highest SMI corresponding to nearly spherical foam cells with high interconnectivity. A summary is given in Fig. 30.

Figure 30 Different structural parameters of ceramic foams from different preparation methods.

Figure 31 shows the dependence of strut size and cell window distribution on the preparation method. Foams made from sintered hollow spheres and the polymer precursor have narrow strut thickness distributions. The bimodal distribution for the foam from sintered hollow spheres likely results from two adjacent spheres that are attached to each other (0.45 mm) and thus result in a strut thickness of twice the thickness of the sphere shell (0.25 mm). The reticulated ceramic foam, in contrast, shows a wide distribution of strut thickness. The cell diameter of the ceramic foams differ both in mean value and distribution (Fig. 31b). The polymer-derived ceramic foams have a narrow distribution with a peak at 0.9 mm. The foam made from sintered hollow spheres shows a wider bimodal distribution with peaks at 0.7 and 1.4 mm. The distribution of the reticulated foam has a bimodal curve with a maximum of 1.3 mm.

Specifications

The resolution of tomography is physically limited by the wavelength and energy of the radiation. High-energy synchrotron radiation provides a higher resolution than conventional X-ray sources, which is desirable for the evaluation of microcellular foams with cell diameters in the micrometer and submicrometer range. However, challenging technological difficulties limit the resolution, and the physical limits of

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Results

1)The two-dimensional sectional images made by micro computer tomography offer the opportunity for a visual inspection of the structure in a nondestructive manner. Parameters like homogeneity of the sample, especially the thickness of deposition in the interior of the foam and edge effects can easily be observed from these kinds of images. Figure 32 shows slices through the middle of the sample (scan 150 of about 400 slices) for samples 1, 2, 7, and 8.

Figure 32 Two-dimensional slices made by micro computer tomography for a) sample 1, b) sample 2, c) sample 7, and

d)sample 8.

2)The pore size distribution of the ceramic foams fabricated by replication of reticulated polyurethane is presented as a series of cumulative curves in Fig. 33. The trend of increasing pore diameter for the eight samples, as measured by image analysis and the Visiocell method, is not followed by the micro computer tomography analysis. A complete comparison between the different characterization techniques for these samples and possible fundamental reasons for disagreement are discussed in Section 3.1.2.3.

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3.1.2.2.6Other Techniques

Mercury porosimetry is based on the intrusion of mercury into the pore volume under pressure. The pressure required to force mercury into pores gives the pore diameter and the liquid-intruded volume gives pore volume and pore volume distribution. However, pores larger than a few hundred micrometers are hard to detect [52, 53].

Analogous to capillary flow analysis (see Section 3.1.2.2.4), liquid extrusion porosimetry is based on the extrusion of a wetting liquid from the porous sample that is placed on a membrane. The characteristics of the membrane are such that its largest pore is smaller than the smallest pore in the sample. By gradually increasing the pressure of a nonreactive gas above the sample, the liquid is removed from its pores. However, because of the smaller pore size in the membrane, the pressure is not sufficient to remove the liquid from these pores. As a result, no gas flows through the membrane. Instead, the displaced liquid from the pores of the sample merely passes through the liquid-filled pores of the membrane. By measuring the liquid volume extruded from the sample, this technique has the ability to determine the pore volume distribution, the pore diameter, and the liquid permeability [54, 56].

The BET gas adsorption technique (Brunauer–Emmett–Teller) is based on the principle of measuring the amount of adsorbed gas on the pore surface as a function of the gas pressure below the equilibrium vapor pressure. Measured characteristics are the pore diameter and the combined volume of blind pores and through-pores. The measurable pore size ranges from 0.5 nm to 1 mm. This technique is considered to be one of the most accurate in determining the surface area of samples with high specific surface area [54].

NMR micro-imaging is a three-dimensional, nondestructive technique that can be used to characterize foam materials. Measurement with sufficient resolution is still very time consuming for materials with small pores. Nevertheless, with technical improvements, it could become a promising technique [57–59].

In ultrasound imaging, ultrasonic excitation causes a reflected and backscattered signal in the sample that can be detected for depth-resolved cross sections. This method of nondestructive imaging and noninvasive detection of porosity has been applied to bone samples and aluminum castings. Detailed surface and subsurface images can be obtained, with a resolution from 1 to 100 mm, depending on the frequency of the acoustic microscope. More experimental work has to be done to verify the value of this technique in the characterization of cellular ceramics [60, 61].

Although the theoretical concepts of confocal microscopy date from several decades ago, the actual applicability of this imaging technique is a relatively new development due to the recent advances in optical and electronic technology. Especially in the biology-related sciences, laser scanning confocal microscopy has proven its value in the visualization of micro-organisms after staining with fluorescing dyes (osteoblast cells adhering to porous substrates, protein immobilization, etc.) [62, 63]. Also in geology, the technique is widely applied in the determination of the pore geometry in porous sandstones or for understanding the failure mechanisms in geomaterials [64]. Confocal imaging rejects the out-of-focus information by placing a pinhole in front of the detector. This confocal pinhole is what gives the system its confocal

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slices obtained by micro computer tomography. By assuming a relatively narrow distribution of spherical pores, a stereological correction factor to convert the twodimensional data to the three-dimensions of the actual pore space can be applied to the mean pore diameter. A three-dimensional characterization of the pore architecture was performed by micro computer tomography, from which the mean pore diameter was obtained.

The data from Visiocell and image analysis are in reasonable agreement when corrected for the dimensional aspect of the latter technique. This can be done either by selecting the largest two-dimensional diameter (the d90 value) or by applying a stereological factor to the mean pore diameter. For these samples, the differences between the two methods are 4–7 %.

The mean pore diameters measured by micro computer tomography are for almost all samples in discordance with the results from Visiocell or image analysis. Part of the explanation probably involves differences in threshold procedure, as image analysis on the micro computer tomography slices does not completely match the results on the microscopic value. The image analysis results for the micro computer tomography images are consistently lower than the results from microscopy images. In both methods, the threshold procedure and possible deviations from the actual pore boundary structure will have a significance influence on the measurement.

Thresholding differences cannot account fully for the complete mismatch of the results. Other features come into play. The assumptions made on the shape factor of the individual pores may affect the calculations, both for image analysis and micro computer tomography. With regard to the SMI values of these samples, micro computer tomography detects the pore structures more as rodlike pores, as opposed to the sphere-like porosity in image analysis. Another factor that may contribute to this issue is the fact that every single pore is taken into account equally in the SMI calculation and the pore diameter distribution in the mico computer tomography procedure. This contrary to image analysis, in which artefacts are filtered by imposing a minimum pore area. If small, very irregular pores are present, this might influence the SMI value. Unfortunately, the micro computer tomography software does not enable a visual verification of the identified pores.

A general overview of some of the most common techniques for analysing pore architecture and the specifications and capabilities of each technique are summarised in Tab. 4.

3.1.3

Summary

Due to the complex nature of the three-dimensional pore space in most porous ceramics and the wide variety of internal structures, no single analytical technique so far provides a complete and accurate description of the morphology.