# Porous Alumina Ceramics with Multimodal Pore Size Distributions

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

_{2}O

_{3}ceramics were manufactured using pyrolyzed cellulose fibers (l = 150 µm, d = 8 µm) and two types of isotropic phenolic resin spheres (d = 30 and 300 µm) as sacrificial templates. The sacrificial templates were homogeneously distributed in the Al

_{2}O

_{3}matrix, compacted by uniaxial pressing and extracted by a burnout and sintering process up to 1700 °C in air. The amount of sacrificial templates was varied up to a volume content of 67 Vol% to form pore networks with porosities of 0–60 Vol%. The mechanical and thermal properties were measured by 4-point-bending and laser flash analysis (LFA) resulting in bending strengths of 173 MPa to 14 MPa and heat conductivities of 22.5 Wm

^{−1}K

^{−1}to 4.6 Wm

^{−1}K

^{−1}. Based on µCT-measurements, the representative volume-of-interest (VOI) of the samples digital twin was determined for further analysis. The interconnectivity, tortuosity, permeability, the local and global stress distribution as well as strut and cell size distribution were evaluated on the digital twin’s VOI. Based on the experimental and simulation results, the samples pore network can be tailored by changing the fiber to sphere ratio and the overall sacrificial template volume. The presence pore formers significantly influenced the mechanical and thermal properties, resulting in higher strengths for samples containing fibrous templates and lower heat conductivities for samples containing spherical templates.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fabrication of Alumina with Multimodal Pore Size Distribution

_{2}O

_{3}powder (CT 3000 SG, Almatis GmbH, Ludwigshafen, Germany, d

_{50}= 400 nm) loaded with varying amounts of pyrolyzed cellulose fibers (anisotropic) and phenolic resin spheres (isotropic). The pyrolyzed cellulose fibers were obtained by pyrolyzing cellulose paper (200 g/m

^{2}, Hahnemühle Fineart GmbH, Dassel, Germany) at 800 °C for 1 h under N

_{2}-atmosphere. The individual fibers were separated by chopping at 20,000 rpm for 15 s (A10, IKA-Werke GmbH&Co. KG, Staufen im Breisgau, Germany). The diameters and lengths of the pyrolyzed cellulose fibers were optically determined by analyzing SEM micrographs (ESEM, Quanta 200 FEG, FEI Company, Peabody, MA, USA) using ImageJ v1.50i [26]. After chopping, fibers with a mean diameter of d

_{50}= 8 µm and length of l

_{50}= 150 µm (l/d = 19) were obtained. Phenolic resin spheres (Brace GmbH, Karlstein am Main, Germany) with a mean size of 30 µm and 300 µm determined by laser light scattering (Mastersizer Hydro 2000S, Malvern Instruments GmbH, Herrenberg, Germany, solvent 2-propanol) were used as isotropic templates (l/d = 1). The SEM-micrographs of the isotropic and anisotropic templates, the corresponding template size distributions and TGA curves for the burnout in air are shown in Figure 1. The templates provide hierarchical stages of porosity in the range 1–400 µm with three discrete, monomodal particle sizes.

_{2}O

_{3}powder blends with mono-, bi-, and trimodal mixtures of pyrolyzed cellulose fibers and phenolic resin spheres were homogenized for 30 min by dry mixing (AR 400, ERWEKA GmbH, Heusenstamm, Germany) using polyethylene glycol as the binder (PEG 1500, Merck KGaA, Darmstadt, Germany), as shown in Table 1. The powder mixtures were then uniaxially compacted to plate geometry (50 mm × 50 mm × 5 mm) using a pressure of 53 MPa (PW 10 E, Paul-Otto Weber GmbH, Remshalden, Germany). The subsequent burnout of the sacrificial templates was carried out up to 570 °C for 2 h in air. The heating rates for the burnout process were established using TGA analysis (STA 429, Netzsch Instruments, Selb, Germany) of the binder and templates applying heating rates of 5 K/min, Figure 1E. Subsequently, the samples were sintered at 1700 °C for 2 h.

#### 2.2. Mechanical and Thermal Properties Characterization

#### 2.3. Microstructural Characterization and Digital Twin

^{3}. The sample was rotated 180° with a rotation step size of 0.2° and an Al filter was used. The 2D sinograms were reconstructed using NRecon (Version 1.6, Skyscan, Kontich, Belgium) and visualized in Amira (Version 2020.2 FEI Imaging Systems, Berlin, Germany). These 3D volume data from the µCT-scans serve as the basis for the creation of the digital twin.

^{3}, the specification in px is retained for simplicity, since the image evaluation with regard to the stereological parameters is based on the image pixels. This 3D volume was now divided into sections with ascending size starting from 200$\times $ 200 $\times $ 200 px in 100 px-steps per axis up to the final size. These sections were then shifted in the x-y direction by 50 px each and also in the z-direction by 50 px up to the point that the section still lies within the original volume (see Figure 2C).

_{2}O

_{3}bulk (Young’s modulus 410 GPa, v = 0.22, r = 3.98 g/cm

^{3}), as boundary conditions a displacement in y = −0.1, which corresponds to a compression (compressive load) of max. 0.224%. The opposite nodes on the model surface were fixed in motion (D

_{x}= D

_{y}= D

_{z}= 0) to represent the loading of a compression test in y-axis. The 4-node tetrahedral elements were assigned the computational properties of 3D solid element No. 157 (MSC.Menat VOL B, Elementlibary, MSC.Software, Munich, Germany). The non-linear calculation is performed using the internal direct solver. From the calculation results, the maximum stress occurring at the applied deformation was determined, where the stress distribution occurring is inversely proportional to the strength: high stress values correspond to low strengths.

## 3. Results and Discussion

#### 3.1. Microstructural Characterization—SEM and µCT

_{50}values) for one sample are summarized in Table 2. The µ-CT derived pore size distributions of the sintered Al

_{2}O

_{3}ceramics partially overlap as a result of template-template interactions. Adjacent pore formers generate interconnected pore channels at high template loadings broadening the particle size distributions. The mean pore sizes of the monomodal fibrous samples (d

_{50}= 9 µm) are in a good agreement with the initial fiber template size of 8 µm. The pore sizes of the monomodal samples with spherical pore formers showed slight deviations from the initial template size with 22 µm and 181 µm for the 30 µm and 300 µm spheres, respectively. This can be attributed to the formation of alumina hollow spheres within the spherical pores, which is significantly pronounced for the large 300 µm spherical templates shown in Figure 8C. Ceramic hollow spheres may form as an effect of the thermal debinding of the phenolic resin spheres. During the initial debinding stage, the phenolic resin spheres slightly expand and embed surrounding the ceramic material. At higher temperatures, they collapse and form a thin ceramic shell during complete burnout (T > 600 °C, see Figure 3), which afterwards shrinks and consolidates to an entrapped hollow sphere in the as-formed pore during the sintering stage. The formation of hollow spheres was also observed by other researchers using expandable microspheres [24]. The entrapped hollow spheres significantly decreased the pore size compared to the original template size. This decrease, however, could be accurately determined by the µCT-evaluation shown in the colored circles of Figure 8C, representing the real determined pore sizes. The multimodal alumina ceramics with bimodal mixtures of pyrolyzed cellulose fibers and 30 µm spheres exhibited a mean pore size of 18 µm due to the formation of interconnected pore channels between the organic templates. Mixtures containing fibers and 300 µm spheres showed bimodal pore size distributions due to the pronounced size difference of the sacrificial templates, characterized by two peaks at d

^{I}

_{50}= 18 µm (fibrous pores) and a d

^{II}

_{50}= 62 µm (spherical pores). For the trimodal mixtures containing all three types of pore formers, bimodal pore size distributions were also observed due to the overlap between the fibers and small spheres. The permeability calculation was performed on the digital twins of the samples.

_{2}O

_{3}matrix leads to a permeability of 4.1 D. For the larger 300 µm spheres a significant higher amount of 52 Vol% was required to obtain a permeable alumina matrix with a permeability of 9.0 D, resulting from the low degree of interconnectivity between the 10-times larger templates. In contrast, a significant increase of the permeability could be obtained for the bimodal samples combining the interconnected pore network of the pyrolyzed cellulose fibers with additional spherical pores for an improved flow-rate. Especially the combination of the 300 µm spheres with the fibers resulted in a permeability of 136.0 D, which is two magnitudes higher than the permeability of the samples with monomodal pore size distributions. The permeability of the multimodal porous alumina far exceeds the requirements for Diesel Particle Filters of 10

^{−11}–10

^{−12}m

^{2}(~1–10 D) [5]. The trimodal combination of templates (fibers, 30 and 300 µm spheres) could not further improve the permeability. The permeability of the multimodal porous alumina ceramics manufactured in this work were mainly dependent on the size of the pore formers as long as the porous matrix provided an interconnection between larger pores.

#### 3.2. Mechanical Properties

_{P}is given by the modulus of dense material M

_{0}, a dimensionless constant b ≈ 2–7 and the pore volume fraction f

_{p}[51,52]. The value of b is predominantly determined by the pore shape and orientation respective to the stress axis [53,54], which also could be confirmed by the results of this work. Both flexural strength and stiffness were mainly influenced by the type and size of the templates since the experimental data could not be well described by single overall curve fits (dash-dot black line with 90% confidence interval, Figure 9). We therefore used separate curve fits for the individual pore formers to highlight the tendencies. All samples containing fibrous pores (non- or half-filled symbols) exhibited higher stiffness and strength compared to the samples containing spherical pores (filled symbols), as shown in Figure 9. The small spherical pores caused significant lower stiffnesses (solid green line) in comparison to samples with fibrous pores (solid red line), while large spherical pores reduced the flexural strength the most (dashed green line). A reduced stiffness with a maintained strength is highly attractive to design strain-tolerant porous ceramics with a higher specific strength or higher thermal shock resistance [3,7]. A pore size dependent strength was frequently reported [12,55,56,57], and high strengths could be achieved with small pore sizes, matching with the results of this work. The origin of material failure is however not predominantly associated with the mean defect size, such as the mean pore size, but occurs at the most critical flaw. Especially under flexural loading, agglomerated pores (“clusters”) frequently caused the catastrophic brittle failure [15,58].

_{yy}in loading direction y. When only fibers are used as pore formers, see Figure 10A, compressive stresses are predominant. As soon as the 30 + 300 µm spheres are used as pore-formers, tensile stresses occur at the pores, see Figure 10B. This was observed for both the 30 µm and the 300 µm pores, see Figure 10B–D. In particular, with the monomodally used spheres, an increase in tensile stresses can be observed here with increasing pore size from 30 to 300 µm in Figure 10C,D. Especially with the 300 µm spheres, which are larger by a factor of 10, tensile stresses occur predominantly here compared to the fiber-only structure.

^{3}or 896 × 896 × 896 µm

^{3}.

#### 3.3. Thermal Properties

_{p}) and dense (λ

_{0}) material and the total porosity f

_{P}. Equation (2) was applied to fit the experimental data of the thermal conductivity as a function of the total porosity. For the fitted λ

_{0}= 27.2 Wm

^{−1}K

^{−1}, an acceptable fit of Equation (2) (R

^{2}= 0.919) shows that corresponding to the mechanical properties, the thermal conductivity was not only dependent on the total porosity but was mainly influenced by the type, volume fractions and orientation of the sacrificial templates. Interestingly, the pore networks generated by monomodal 30 µm spheres (symbol: filled circles) induced significant lower thermal conductivities in the alumina matrix compared to the samples containing other templates or mixtures at similar porosities. This corresponds with the non- or low permeability of those samples. Bi- and trimodal samples containing high amounts of large 300 µm spheres showed increased thermal conductivities (symbols: half-filled diamond, left star), Figure 11. The thermal conductivity of all samples was measured parallel to the pressing direction, to which the tubular pores are perpendicular oriented. Based on the fiber alignment, an anisotropic thermal behavior can be expected with higher conductivities measured parallel to the fibrous pores [21,22].

## 4. Conclusions

_{2}O

_{3}powder mixtures containing different ratios of pyrolyzed cellulose fibers (anisotropic, aspect ratio l/d = 19) and phenolic resin spheres (isotropic, aspect ratio l/d = 1) were homogeneously mixed, uniaxially pressed, and sintered up to 1700 °C to extract the templates. The microstructure analysis revealed tubular and spherical pores in the alumina matrix with porosities of 0 to 60 Vol%. The influence of the pore-former volume fraction, shape and size was investigated and indicates general trends regarding the microstructure and mechanical/thermal properties of the multimodal porous Al

_{2}O

_{3}ceramics:

- The application of the three Minkowski functionals (M1, M2, M3) to the µCT images of the heterogeneous structures with different sample formers in terms of shape and size was successfully applied. Thus, the representative volume-of-interest could be set to a size of 400 $\times $ 400 $\times $ 400 px
^{3}or 896 $\times $ 896 $\times $ 896 µm^{3}at the available resolution of 2.24 µm and thus the digital twin could be defined. The digital twin of each structure enabled the visualization and evaluation of the pore network of the structures and the determination of their connectivity. - The pyrolyzed cellulose fibers show a perpendicular alignment to the pressing direction induced by the uniaxial pressing. The combination of fibrous templates and spherical templates did not interfere with the alignment of the fibers in the samples with multimodal distributions of sacrificial templates. The volumetric evaluation using the digital twin for the orientation of the fibers has confirmed and complemented the 2D analysis.
- The permeability is mainly dependent on the pore size of the spherical pore formers as long as the porous matrix provides an interconnected pore network. The 300 µm phenolic resin spheres provided larger pore channels and thus a higher permeability in comparison to the 30 µm phenolic resin spheres. The tubular pores are essential to connect isolated spherical pores. The identified pore networks and their quantification by segment lengths and connectivity at the branch nodes are consistent with the results of the Euler number.
- The type of the sacrificial templates predominantly influenced the mechanical properties. Small tubular pores lead to a higher stiffness and strength compared to spherical pores, based on the smaller defect size and anisotropic microstructure. Low elastic moduli with higher specific strength were obtained for the samples with a monomodal distribution of 30 µm phenolic resin spheres. The FEM simulations performed on the digital twins agree with the experimental results with respect to the distribution of the stresses.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Scheffler, M.; Colombo, P. Cellular Ceramics: Structure, Manufacturing, Properties and Applications; Wiley-VCH: Weinheim, UK, 2010; ISBN 978-3-527-31320-4. [Google Scholar]
- Green, D.J.; Colombo, P. Cellular Ceramics: Intriguing Structures, Novel Properties, and Innovative Applica-tions. MRS Bull.
**2003**, 28, 296–300. [Google Scholar] [CrossRef] - Greil, P. Advanced Engineering Ceramics. Adv. Mater.
**2002**, 14, 709. [Google Scholar] [CrossRef] - Hammel, E.C.; Ighodaro, O.-R.; Okoli, O.I. Processing and properties of advanced porous ceramics: An applica-tion based review. Ceram. Int.
**2014**, 40, 15351–15370. [Google Scholar] [CrossRef] - Ohji, T.; Fukushima, M. Macro-porous ceramics: Processing and properties. Int. Mater. Rev.
**2012**, 57, 115–131. [Google Scholar] [CrossRef] - Studart, A.R.; Gonzenbach, U.T.; Tervoort, E.; Gauckler, L.J. Processing Routes to Macroporous Ceramics: A Review. J. Am. Ceram. Soc.
**2006**, 89, 1771–1789. [Google Scholar] [CrossRef] - Shigegaki, Y.; Brito, M.E.; Hirao, K.; Toriyama, M.; Kanzaki, S. Strain Tolerant Porous Silicon Nitride. J. Am. Ceram. Soc.
**1997**, 80, 495–498. [Google Scholar] [CrossRef] - Shi, L.; Wang, L.; Duan, Y.; Lei, W.; Wang, Z.; Li, J.; Fan, X.; Li, X.; Li, S.; Guo, Z. The improved biological per-formance of a novel low elastic modulus implant. PLoS ONE
**2013**, 8, e55015. [Google Scholar] [CrossRef] - Ridzwan, M.; Shuib, S.; Hassan, A.Y.; Shokri, A.A.; Mohamad Ib, M.N. Problem of Stress Shielding and Im-provement to the Hip Implant Designs: A Review. J. Med. Sci.
**2007**, 7, 460–467. [Google Scholar] [CrossRef][Green Version] - Ha, J.-H.; Oh, E.; Song, I.-H. The effect of sacrificial templates on the pore characteristics of sintered diatomite membranes. J. Ceram. Soc. Jpn.
**2013**, 121, 940–945. [Google Scholar] [CrossRef][Green Version] - Colombo, P.; Bernardo, E.; Biasetto, L. Novel Microcellular Ceramics from a Silicone Resin. J. Am. Ceram. Soc.
**2004**, 87, 152–154. [Google Scholar] [CrossRef] - Eom, J.-H.; Kim, Y.-W. Effect of template size on microstructure and strength of porous silicon carbide ceram-ics. J. Ceram. Soc. Jpn.
**2008**, 116, 1159–1163. [Google Scholar] [CrossRef][Green Version] - Lopes, R.A.; Segadaĕs, A.M. Microstructure, permeability and mechanical behaviour of ceramic foams. Mater. Sci. Eng. A
**1996**, 209, 149–155. [Google Scholar] [CrossRef] - Schlichting, K.W.; Padture, N.P.; Klemens, P.G. Thermal conductivity of dense and porous yttria-stabilized zirconia. J. Mater. Sci.
**2001**, 36, 3003–3010. [Google Scholar] [CrossRef] - Petit, C.; Meille, S.; Maire, E.; Tadier, S.; Adrien, J. Mechanical behaviour of a β-TCP ceramic with a random porosity: Study of the fracture path with X-ray tomography. J. Eur. Ceram. Soc.
**2016**, 36, 3225–3233. [Google Scholar] [CrossRef] - Lyckfeldt, O.; Ferreira, J. Processing of porous ceramics by ‘starch consolidation’. J. Eur. Ceram. Soc.
**1998**, 18, 131–140. [Google Scholar] [CrossRef] - Zhang, G.-J.; Yang, J.-F.; Ohji, T. Fabrication of Porous Ceramics with Unidirectionally Aligned Continuous Pores. J. Am. Ceram. Soc.
**2001**, 84, 1395–1397. [Google Scholar] [CrossRef] - Isobe, T.; Kameshima, Y.; Nakajima, A.; Okada, K.; Hotta, Y. Gas permeability and mechanical properties of porous alumina ceramics with unidirectionally aligned pores. J. Eur. Ceram. Soc.
**2007**, 27, 53–59. [Google Scholar] [CrossRef] - Okada, K.; Shimizu, M.; Isobe, T.; Kameshima, Y.; Sakai, M.; Nakajima, A.; Kurata, T. Characteristics of mi-crobubbles generated by porous mullite ceramics prepared by an extrusion method using organic fibers as the pore former. J. Eur. Ceram. Soc.
**2010**, 30, 1245–1251. [Google Scholar] [CrossRef] - Okada, K.; Uchiyama, S.; Isobe, T.; Kameshima, Y.; Nakajima, A.; Kurata, T. Capillary rise properties of porous mullite ceramics prepared by an extrusion method using organic fibers as the pore former. J. Eur. Ceram. Soc.
**2009**, 29, 2491–2497. [Google Scholar] [CrossRef] - Eom, J.-H.; Kim, Y.-W.; Jeong, D.-H.; Lee, S.-S. Thermal conductivity and compressive strength anisotropy in vermiculite–ceramic composites with layered structure. J. Ceram. Soc. Jpn.
**2011**, 119, 319–321. [Google Scholar] [CrossRef][Green Version] - Eom, J.-H.; Kim, Y.-W.; Lee, S.-S.; Jeong, D.-H. Processing of Vermiculite-Silica Composites with Prefer-Oriented Rod-Like Pores. J. Korean Ceram. Soc.
**2012**, 49, 347–351. [Google Scholar] [CrossRef] - Suvaci, E.; Seabaugh, M.M.; Messing, G.L. Reaction-based Processing of Textured Alumina by Templated Grain Growth. J. Eur. Ceram. Soc.
**1999**, 19, 2465–2474. [Google Scholar] [CrossRef] - Ciurans Oset, M.; Nordin, J.; Akhtar, F. Processing of Macroporous Alumina Ceramics Using Pre-Expanded Polymer Microspheres as Sacrificial Template. Ceramics
**2018**, 1, 329–342. [Google Scholar] [CrossRef][Green Version] - Andersson, L.; Jones, A.C.; Knackstedt, M.A.; Bergström, L. Permeability, pore connectivity and critical pore throat control of expandable polymeric sphere templated macroporous alumina. Acta Mater.
**2011**, 59, 1239–1248. [Google Scholar] [CrossRef] - Rueden, C.T.; Schindelin, J.; Hiner, M.C.; DeZonia, B.E.; Walter, A.E.; Arena, E.T.; Eliceiri, K.W. ImageJ2: ImageJ for the next generation of scientific image data. BMC Bioinform.
**2017**, 18, 529. [Google Scholar] [CrossRef] - Akolkar, A.; Petrasch, J. Tomography-Based Characterization and Optimization of Fluid Flow through Porous Media. Transp. Porous Med.
**2012**, 95, 535–550. [Google Scholar] [CrossRef] - DeHoff, R.T. Use of the disector to estimate the Euler characteristic of three dimensional microstructures. Acta Stereol.
**1987**, 6, 133–140. [Google Scholar] - Hadwiger, H. Vorlesungen Über Inhalt, Oberfläche und Isoperimetrie; Springer: Berlin/Heidelberg, Germany, 1957; ISBN 978-3-642-94702-5. [Google Scholar]
- DeHoff, R.T. Stereological Characterization of Anisotropy in the Statistical Description of the Spatial Distribu-tion of Scalar Properties of Microstructures. Microsc Microanal
**2004**, 10, 724–725. [Google Scholar] [CrossRef] - Ohser, J.; Nagel, W. The estimation of the Euler-Poincare characteristic from observations on parallel sections. J. Microsc.
**1996**, 184, 117–126. [Google Scholar] [CrossRef] - Gregorová, E.; Uhlířová, T.; Pabst, W.; Diblíková, P.; Sedlářová, I. Microstructure characterization of mullite foam by image analysis, mercury porosimetry and X-ray computed microtomography. Ceram. Int.
**2018**, 44, 12315–12328. [Google Scholar] [CrossRef] - Uhlířová, T.; Gregorová, E.; Pabst, W. Application of Stereological Relations for the Characterization of Porous Materials via Microscopic Image Analysis. KEM
**2015**, 647, 180–187. [Google Scholar] [CrossRef] - Odgaard, A. Three-dimensional methods for quantification of cancellous bone architecture. Bone
**1997**, 20, 315–328. [Google Scholar] [CrossRef] - Hildebrand, T.; Rüegsegger, P. A new method for the model-independent assessment of thickness in three-dimensional images. J. Microsc.
**1997**, 185, 67–75. [Google Scholar] [CrossRef] - Hildebrand, T.; Laib, A.; Müller, R.; Dequeker, J.; Rüegsegger, P. Direct three-dimensional morphometric analy-sis of human cancellous bone: Microstructural data from spine, femur, iliac crest, and calcaneus. J. Bone Miner. Res.
**1999**, 14, 1167–1174. [Google Scholar] [CrossRef] - Fouard, C.; Malandain, G.; Prohaska, S.; Westerhoff, M. Blockwise processing applied to brain microvascular network study. IEEE Trans. Med. Imaging
**2006**, 25, 1319–1328. [Google Scholar] [CrossRef][Green Version] - Kohjiya, S.; Katoh, A.; Suda, T.; Shimanuki, J.; Ikeda, Y. Visualisation of carbon black networks in rubbery ma-trix by skeletonisation of 3D-TEM image. Polymer
**2006**, 47, 3298–3301. [Google Scholar] [CrossRef] - Cignoni, P.; Callieri, M.; Corsini, M.; Dellepiane, M.; Ganovelli, F.; Ranzuglia, G. MeshLab: An Open-Source Mesh Processing Tool. In Proceedings of the The Eurographics Association, Eurographics Italian Chapter Conference, Salerno, Italy, 2–4 July 2008; pp. 129–136. [Google Scholar]
- Biggemann, J.; Hoffmann, P.; Hristov, I.; Simon, S.; Müller, P.; Fey, T. Injection Molding of 3-3 Hydroxyapatite Composites. Materials
**2020**, 13, 1907. [Google Scholar] [CrossRef][Green Version] - El Seoud, O.A.; Fidale, L.C.; Ruiz, N.; D’Almeida, M.L.O.; Frollini, E. Cellulose swelling by protic solvents: Which properties of the biopolymer and the solvent matter? Cellulose
**2008**, 15, 371–392. [Google Scholar] [CrossRef] - Mantanis, G.I.; Young, R.A.; Rowell, R.M. Swelling of compressed cellulose fiber webs in organic liquids. Cellulose
**1995**, 2, 1–22. [Google Scholar] - Heunisch, A.; Dellert, A.; Roosen, A. Effect of powder, binder and process parameters on anisotropic shrinkage in tape cast ceramic products. J. Eur. Ceram. Soc.
**2010**, 30, 3397–3406. [Google Scholar] [CrossRef] - Rutkowski, P.; Klimczyk, P.; Jaworska, L.; Stobierski, L.; Dubiel, A. Thermal properties of pressure sintered alumina–graphene composites. J. Anal. Calorim
**2015**, 122, 105–114. [Google Scholar] [CrossRef][Green Version] - Itatani, K.; Tanaka, T.; Davies, I.J. Thermal properties of silicon carbide composites fabricated with chopped Tyranno® SiAlC fibres. J. Eur. Ceram. Soc.
**2006**, 26, 703–710. [Google Scholar] [CrossRef] - Bradt, R.C.; Evans, A.G.; Hasselman, D.P.H.; Lange, F.F. Fracture Mechanics of Ceramics; Springer: Boston, MA, USA, 1986; ISBN 978-1-4615-7025-7. [Google Scholar]
- Giesche, H. Mercury Porosimetry: A General (Practical) Overview. Part. Part. Syst. Charact.
**2006**, 23, 9–19. [Google Scholar] [CrossRef] - Innocentini, M.D.; Salvini, V.R.; Pandolfelli Victor, C. The Permeability of Ceramic Foams. Am. Ceram. Soc. Bull.
**1999**, 78, 78–84. [Google Scholar] - Gauckler, L.J.; Waeber, M.M. Light Metals. In Proceedings of the 114th Annual Meeting of the Metallurgical Society of AIME, New York, NY, USA, 24–28 February 1985; pp. 1261–1283. [Google Scholar]
- Rice, R.W. Proposity of Ceramics; M. Dekker: New York, NY, USA; Basel, Switzerland; Hong Kong, China, 1998; ISBN 0-8247-0151-8. [Google Scholar]
- Duckworth, W. Discussion of Ryshkewitch Paper by Winston Duckworth. J. Am. Ceram. Soc.
**1953**, 36, 68. [Google Scholar] [CrossRef] - Spriggs, R.M. Expression for Effect of Porosity on Elastic Modulus of Polycrystalline Refractory Materials, Particularly Aluminum Oxide. J. Am. Ceram. Soc.
**1961**, 44, 628–629. [Google Scholar] [CrossRef] - Andersson, C.A. Derivation of the Exponential Relation for the Effect of Ellipsoidal Porosity on Elastic Modulus. J. Am. Ceram. Soc.
**1996**, 79, 2181–2184. [Google Scholar] [CrossRef] - Travitzky, N.; Windsheimer, H.; Fey, T.; Greil, P. Preceramic Paper-Derived Ceramics. J. Am. Ceram. Soc.
**2008**, 91, 3477–3492. [Google Scholar] [CrossRef] - Yoshida, K.; Tsukidate, H.; Murakami, A.; Miyata, H. Influence of Pore Size on Fracture Strength of Porous Ceramics. JMMP
**2008**, 2, 1060–1069. [Google Scholar] [CrossRef][Green Version] - Liu, D.-M. Influence of porosity and pore size on the compressive strength of porous hydroxyapatite ceramic. Ceram. Int.
**1997**, 23, 135–139. [Google Scholar] [CrossRef] - Hu, L.; Wang, C.-A. Effect of sintering temperature on compressive strength of porous yttria-stabilized zirconia ceramics. Ceram. Int.
**2010**, 36, 1697–1701. [Google Scholar] [CrossRef] - Tancret, F.; Bouler, J.-M.; Chamousset, J.; Minois, L.-M. Modelling the mechanical properties of microporous and macroporous biphasic calcium phosphate bioceramics. J. Eur. Ceram. Soc.
**2006**, 26, 3647–3656. [Google Scholar] [CrossRef] - Živcová, Z.; Gregorová, E.; Pabst, W.; Smith, D.S.; Michot, A.; Poulier, C. Thermal conductivity of porous alumina ceramics prepared using starch as a pore-forming agent. J. Eur. Ceram. Soc.
**2009**, 29, 347–353. [Google Scholar] [CrossRef]

**Figure 1.**Characteristics of the utilized isotropic and anisotropic sacrificial templates: SEM micrographs (

**A**–

**C**) of the pyrolyzed cellulose fibers (

**A**), the 30 µm (

**B**) and 300 µm phenolic resin spheres (

**C**); corresponding monomodal particle size distributions with examples for bi-/trimodal mixtures (

**D**) and TGA analysis of the template burn-out in air (

**E**).

**Figure 2.**Schematic characterization techniques utilized to determine the physical properties considering the sample orientation and fibrous pore alignment: flexural strength determined by 4-point bending (

**A**), thermal conductivity by laser-flash analysis (

**B**), and the definition of the analyzed Volume-of-Interests based on the increasing and shifting arguments of the starting volume used for the μCT-analysis (

**C**).

**Figure 3.**Microstructure of the monomodal samples: SEM micrographs of the typical fracture surfaces of sintered Al

_{2}O

_{3}ceramics containing monomodal distributions of pyrolyzed cellulose fibers (

**A**,

**B**), 30 µm phenolic resin spheres (

**C**,

**D**) and 300 µm phenolic resin spheres (

**E**,

**F**).

**Figure 4.**Microstructure of a trimodal sample: SEM micrographs of the fracture surface (

**A**) and polished cross-section (

**B**) of an alumina ceramic with a trimodal distribution of sacrificial templates (batch No. 13); (

**C**) shows the false-color image of the polished cross-section highlighting the interconnected pore network in red (pyrolyzed cellulose fibers), blue (30 µm phenolic resin spheres), and green (300 µm phenolic resin spheres). (

**D**) shows the 2D analysis of the fibrous pore orientation represented by the corresponding angular distribution.

**Figure 5.**Evaluation of the Minkowski functionals and mean cell size for determination of the representative Volume-of-Interest (VOI). Porosity (

**A**), Jeffry size (

**B**), Euler number (

**C**) and Normalized Mean cell size (

**D**) in dependence of the examined VOI between 200 $\times $ 200 and 600 $\times $ 600 px.

**Figure 6.**Orientation of the fibrous pores in the 3D volume analyzed by µCT: (

**A**,

**B**) show the fibrous pore orientation (angle θ) towards the z-axis, (

**C**,

**D**) show the fiber orientation (angle φ) within the xy-plane.

**Figure 7.**Pore network of monomodal samples with fibrous pores (

**A**), 30 µm (

**B**) and 300 µm spherical pores (

**C**) and trimodal sample containing all type of pore formers (Batch No. 13) (

**D**). For figure (

**A**–

**D**), the color indicates the pore diameter, minimum blue ≤5 µm, maximum red ≥60 µm. (

**E**) shows the sum curves of the length distribution of the segments and (

**F**) the amount of connectivity per branch node.

**Figure 8.**Pore size distributions of Al

_{2}O

_{3}ceramics with monomodal (

**A**) and multimodal (

**B**) pore size distributions derived from µ-CT analysis. The formation of hollow spheres inside the spherical pores leads to a reduced pore size and broadening of the pore size distributions (

**C**).

**Figure 9.**Influence of the amount and shape of the sacrificial templates on the mechanical properties of Al

_{2}O

_{3}ceramics with multimodal porosity: Young’s modulus (

**A**) and flexural strength (

**B**) in dependence of the total sample porosity fitted by the exponential model of Equation (1).

**Figure 10.**Stress distribution in σ

_{yy}of the monomodal samples with fibrous (

**A**), 30 µm (

**B**) and 300 µm spherical pores (

**C**) and the multimodal sample (

**D**) with assigned displacement of 0.1% on top face; the corresponding compressive stress is shown in the middle. The legend shows the range from +500 MPa to −500 MPa.

**Figure 11.**Influence of the total porosity (

**A**) and pore former shape (

**B**) on the thermal conductivity.

**Table 1.**Compositions and corresponding target and total porosities of the realized alumina ceramics with monomodal, bimodal, and trimodal mixtures of sacrificial templates.

Batch No. | Distribution Type | Fibers /Vol% | 30 µm Spheres /Vol% | 300 µm Spheres /Vol% | Target Porosity /Vol% | Total Porosity /Vol% |
---|---|---|---|---|---|---|

1 | Reference | 0 | 2.3 | |||

2 | Monomodal | 10 | 10 | 10.0 | ||

3 | 15 | 15 | 13.3 | |||

4 | 15 | 15 | 10.8 | |||

5 | 34 | 34 | 31.0 | |||

6 | 37 * | 37 | 37.5 | |||

7 | 52 * | 52 | 49.8 | |||

8 | Bimodal | 8 | 16 | 25 | 22.2 | |

9 | 8 | 16 | 25 | 24.3 | ||

10 | 22 | 35 * | 57 | 59.1 | ||

11 | 17 | 50 * | 67 | 56.4 | ||

12 | Trimodal | 28 | 8 | 8 | 44 | 47.0 |

13 | 5 | 27 | 27 | 58 | 56.7 |

**Table 2.**µ-CT derived mean pore sizes (d

_{50}) and corresponding permeability of the sintered Al

_{2}O

_{3}ceramics with representative mono-, bi- and trimodal pore distributions in comparison to the initial pore former size. The monomodal pore size distributions are characterized by a single peak, while bi- and trimodal samples containing 300 µm spheres exhibit bimodal pore size distributions (peak I and II).

Sample Type | Ratio of Sacrificial Templates/Vol% | Mean Pore Size ** (d_{50})/µm | Porosity * /Vol% | Permeability ** /D | |||
---|---|---|---|---|---|---|---|

Fibers | 35 µm Spheres | 300 µm Spheres | Peak I (d^{I}_{50}) | Peak II (d^{II}_{50}) | (9.87·10^{−13} m^{2}) | ||

Initial pore formers * | 100 | 8 | - | - | - | ||

100 | 31 | - | - | - | |||

100 | 303 | - | - | - | |||

Monomodal ** | 34 | 9 | - | 31.0 | 0 | ||

37 | 22 | - | 37.5 | 4.1 | |||

52 | 181 | - | 49.8 | 9.0 | |||

Bimodal ** | 22 | 35 | 18 | - | 59.1 | 34.9 | |

17 | 50 | 18 | 62 | 56.4 | 136.0 | ||

Trimodal ** | 5 | 27 | 27 | 18 | 53 | 47.0 | 33.4 |

28 | 8 | 8 | 11 | 69 | 56.7 | 4.6 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Biggemann, J.; Stumpf, M.; Fey, T. Porous Alumina Ceramics with Multimodal Pore Size Distributions. *Materials* **2021**, *14*, 3294.
https://doi.org/10.3390/ma14123294

**AMA Style**

Biggemann J, Stumpf M, Fey T. Porous Alumina Ceramics with Multimodal Pore Size Distributions. *Materials*. 2021; 14(12):3294.
https://doi.org/10.3390/ma14123294

**Chicago/Turabian Style**

Biggemann, Jonas, Martin Stumpf, and Tobias Fey. 2021. "Porous Alumina Ceramics with Multimodal Pore Size Distributions" *Materials* 14, no. 12: 3294.
https://doi.org/10.3390/ma14123294