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Elastic properties of porous porcelain stoneware tiles
Author’s Accepted Manuscript Elastic properties of porous porcelain stoneware tiles Elisa Rambaldi, Willi Pabst, Eva Gregorová, Francesca Prete, Maria Chiara Bignozzi www.elsevier.com/locate/ceri
PII: DOI: Reference:
S0272-8842(17)30317-6 http://dx.doi.org/10.1016/j.ceramint.2017.02.114 CERI14730
To appear in: Ceramics International Received date: 24 January 2017 Revised date: 22 February 2017 Accepted date: 22 February 2017 Cite this article as: Elisa Rambaldi, Willi Pabst, Eva Gregorová, Francesca Prete and Maria Chiara Bignozzi, Elastic properties of porous porcelain stoneware tiles, Ceramics International, http://dx.doi.org/10.1016/j.ceramint.2017.02.114 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Elastic properties of porous porcelain stoneware tiles Elisa Rambaldi1*, Willi Pabst2, Eva Gregorová2, Francesca Prete1, Maria Chiara Bignozzi1,3 1
Centro Ceramico, via Martelli 26, 40138 Bologna, Italy
2
Department of Glass and Ceramics, University of Chemistry and Technology, Prague, (UCT
Prague), Technická 5, 166 28 Prague, Czech Republic 3
Department of Civil, Chemical, Environmental and Materials Engineering, University of
Bologna, via Terracini 28, Bologna, Italy *
Corresponding author: Tel.:+39 051534015. [email protected]
Abstract Porcelain stoneware tiles are industrially processed by using high sintering temperatures and fast firing cycles that result in products characterized by an almost impervious surface layer surrounding a rather porous bulk material. Since mechanical properties are affected by porosity, the knowledge of the material stiffness is an important parameter to define the service behavior of tiles. In the present investigation, porcelain stoneware samples having different closed porosity were investigated in order to understand the influence of the porosity on the elastic constants of the materials. Based on the quantitative XRD phase composition, elastic constants have been calculated via Voigt-Reuss-Hill averaging, and the influence of porosity has been taken into account via power-law and exponential relations. It is shown that the effective elastic constants predicted by exponential and power-law relations are in agreement with experimental values. It may be concluded that for this class of materials, in the porosity range below 14–16 %, both exponential and power-law relations are helpful tools to design tiles with controlled microstructure and tailored mechanical properties. Keywords: Porcelain stoneware; Porosity; Phase composition; Elastic properties.
1. Introduction Porcelain stoneware tiles, traditionally used as flooring, are nowadays applied also in internal walls and ventilated facades. For these applications slightly different properties are required, including specially designed mechanical ones [1,2,3,4]. The mechanical characteristics of porcelain stoneware tiles can be directly correlated to their microstructure [5]. From the 1
viewpoint of quality, porcelain stoneware, a rather dense material in which crystals of quartz and mullite are embedded in a large amount of glassy matrix, can be considered the best product among ceramic tiles. Even if these tiles exhibit a negligible water absorption (lower than 0.5%: group BIa of ISO 13006 [6]), which is due to a near-zero open porosity [7,8], they retain a noticeable closed porosity, till values higher than 10% [9]. The maximum temperature, 1210-1230°C, reached during firing, favors the sintering and significantly reduces the total porosity, even if the industrial fast firing cycles (40-60 min) prevent the escape, during the cooling step, of gases developed in the firing reactions. The total porosity, as sum of two different fractions, open porosity at the surface and closed porosity trapped in the bulk material, is very close to the latter. That implies an almost exclusive dependence of the mechanical characteristics and service behavior on the closed porosity. Published works dealing with this dependence, report several models and relations for predicting the porosity dependence of elastic properties [10,11,12]. Nevertheless, in spite of the considerable research addressed to this topic, until now very little attention was focused on the key role played by porosity and its effects on the elastic modulus of ceramic tiles. In this work PMMA particles were introduced in different amounts into a porcelain stoneware mix, in order to control the total porosity and evaluate the contribution of the closed porosity on the elastic moduli of the materials, while keeping constant the composition. Values predicted via rigorous bounds and model relations were compared to experimental data. It is suggested that this approach will turn out to be a useful tool to design porcelain stoneware tiles having the proper porosity for different destination environments such as floors with or without heating system, floating floors for walking noise reduction, internal walls or ventilated facades.
2. Porous porcelain stoneware materials Three porcelain stoneware mixes (S0, S5 andS10) were prepared on the basis of an industrial recipe and by adding three different percentages of polymethylmetacrylate (PMMA) particles (0, 5 and 10 vol%, respectively). The PMMA particles, used as pore former, are spherical in shape and exhibit a monodisperse size distribution (6 m in diameter). Samples were prepared by uniaxial pressing (40 MPa) in form of disks (about 40 mm diameter and 7 mm thickness) and bars (70x10x6 mm). All samples were fired at 1140°C for 90 min. At about 400°C the PMMA particles were completely removed leaving voids in the ceramic samples. A complete
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description of the sample preparation with experimental details is reported in a previous work [13]. The quantitative mineralogical compositions, of the as-fired samples, were determined by Xray diffraction analysis (PANalytical, PW3830, NL). Powdered specimens, diluted with 10 wt% of corundum NIST 676 as internal standard, were side loaded to minimize preferred orientation. Data were collected in the angular range 10–80°2θ with steps of 0.02° and 5 s/step and the Rietveld refinements were performed using the software GSAS. [14]. Results are reported in Table I. Because the samples differ only for the amount of PMMA particles, considering the standard deviation, the mineralogical composition is the same for all of them. The microstructure of the fired materials was analyzed by a scanning electron microscope, SEM (Zeiss EVO 40, D) equipped with an energy dispersive X-ray attachment, EDS (Inca, Oxford Instruments, UK), observing suitable specimens polished to mirror like finish and etched, when appropriate, by using a 5% HF solution for 3 min. Micrographs of the disk samples are reported in Fig. 1a-c. It is clear from the micrographs that the porosities are significantly different for the three types of samples. On the other hand, because the composition is the same, the EDS quantitative analysis of the glassy phase was made only on the etched surface (Fig. 2) of S0. At least three different areas of the sample were analyzed and results are listed in Table II.
Table I. Phase composition (in wt.%) determined by quantitative XRD phase analysis. S0 S5 S10 Quartz (pdf n° 46-1045), wt% 18.9±0.1 19.0±0.1 19.2±0.1 Mullite (pdf n° 15-776), wt% 6.2±0.4 6.2±0.4 6.1±0.4 Plagioclase (pdf n° 41-1480), wt% 5.2±0.3 5.3±0.3 5.8±0.3 Glassy phase, wt% 69.7±1.3 69.5±1.3 68.9±1.3
3
(a)
(b)
(c) Figure 1: SEM micrographs of the fired disk samples S0 (a), S5 (b) and S10 (c).
4
Figure 2: SEM micrographs of the etched surface of S0.
Table II. Chemical composition of the glassy phase of S0 sample by EDS analysis. Average area 1 area 2 area 3 composition Na2O, wt% 2.4 2.6 2.5 2.5±0.1 Al2O3, wt% 11.7 6.6 16.6 11.6±5.0 SiO2, wt% 84.1 89.1 79.1 84.1±5.0 K2O, wt% 1.8 1.7 1.8 1.8±1.0
3. Calculation of the density and elastic constants of the dense (i.e. pore-free) solid material The first step is the calculation of the glass phase properties. As usual in glass science, and in the absence of a better alternative, this is made via empirical additive mixture rules[15]. However, the calculation has to take into account the degree of precision to which the glass phase composition is known. With respect to the fact that the SiO2 and Al2O3 contents determined for the glass phase by EDS are much higher than the Na2O and K2O contents, it can be expected that the former affect the properties most significantly. Since they have at the same time the largest errors (much larger than for Na2O and K2O), the calculation has been made not only for the mean values (84.1 wt.% SiO2, 11.6 wt.% Al2O3), but also for the two extreme cases (determined by the errors), i.e. high Si / low Al (89.1 wt.% SiO2 / 6.6 wt.% Al2O3) and low Si / high Al (79.1 wt.% SiO2 / 16.6 wt.% Al2O3). In this way, the corresponding uncertainties (errors) can be obtained also for the density and the Young’s modulus. Tables III and IV list the resulting density and Young’s modulus values, respectively, according to additive mixture rules using the empirical coefficients determined 5
in different works by different authors. The grand average listed in these tables takes into account the values from the three columns listed and four additional values (not listed) obtained when the uncertainty (error) of the K2O value is taken into account as well. The minimum and maximum values (or, equivalently, the deviations from the averages in the negative and positive direction) may be considered as “worst case estimates”. However, extremely “exotic” values, such as the extremely high density and Young’s modulus values by Winkelmann and Schott in Tables III and IV, respectively, and the extremely low Young’s modulus values by Kozlovskaya in Table IV, have been excluded from this evaluation and are listed in Tables III and IV only for completeness (in parentheses).
Table III. Density of the glass phase, [g/cm3], defined in Table II according to additive mixture rules using the empirical coefficients determined in different works by different authors; values in parenthesis have not been taken into account for the calculation of the averages, extreme values (Min, Max) and absolute deviations (Delta min, Delta max). [g/cm3] [g/cm3] [g/cm3] [g/cm3] Average composition High Si / low Al Low Si / high Al Grand average Demkina 1 2.308 2.296 2.369 Ref [16] Demkina 2 2.323 2.312 2.335 Ref [16] Demkina 3 2.264 2.249 2.279 Ref [16] Gan 2.318 2.269 2.369 Ref [17] Huggins and 2.331 2.288 2.352 Sun Ref [18] Priven 2000 2.280 2.260 2.309 Ref [19] Priven 1998 2.297 2.263 2.330 Ref [20] Winkelmann and (2.548) (2.459) (2.636) Schott Ref [21] 2.303 2.277 2.335 Average 2.303 Min 2.264 2.249 2.279 2.244 Max 2.331 2.312 2.369 2.369 0.039 0.028 0.056 Delta min 0.059 0.028 0.035 0.034 Delta max 0.066
6
Table IV. Young’s modulus, E [GPa], of the glass phase defined in Table II according to additive mixture rules using the empirical coefficients determined in different works by different authors; values in parenthesis have not been taken into account for the calculation of the averages, extreme values (Min, Max) and absolute deviations (Delta min, Delta max). E [GPa] E [GPa] E [GPa] E [GPa] Average composition High Si / low Al Low Si / high Al Grand average Appen 70.66 69.86 71.53 Ref [22] Demkina 69.58 67.83 71.38 Ref [16] Gan 71.26 70.77 71.82 Ref [17] Makishima and 72.00 69.79 74.31 Mackenzie Ref [23] Priven 2000 70.20 68.32 71.68 Ref [19] Priven 1998 72.68 68.67 76.78 Ref [20] Kozlovskaya (66.77) (65.26) (68.39) Ref [24] Winkelmann and (80.76) (75.36) (86.16) Schott Ref [21] 71.06 69.21 72.92 Average 71.07 Min 69.58 67.83 71.38 67.38 Max 72.68 70.77 76.78 78.15 1.48 1.38 1.54 Delta min 3.69 1.62 1.56 3.86 Delta max 7.08
In both cases the agreement between most values is satisfactory, when the values calculated based on the old paper by Winkelmann and Schott [21] (and for the Young’s modulus also the values based on Kozlovskaya’s book [24]) are discarded. It is clear that the empirical coefficients are based on interpolations for glasses containing the four components as our glass (SiO2, Al2O3, Na2O, K2O), but not necessarily exclusively and not necessarily in the same or similar proportions. Differences between the estimates are therefore not at all surprising, but on the contrary, with respect to this fact the agreement of the results is surprisingly good. As a consequence, we have the following values for the density and Young’s modulus of the glass phase, respectively:
Density: 2.30 ± 0.06 g/cm3 (range 2.24–2.37 g/cm3),
Young’s modulus: 71.1± 7.1 GPa (range 67.4–78.2 GPa).
The density and elastic constants of plagioclases are relatively well known. In particular, the elastic tensor components (stiffnesses) of plagioclase monocrystals have been investigated, 7
both experimentally and by computer simulation, in several papers, from pure albite (An0) to pure anorthite (An100) [25,26,27]. Based on these monocrystal values, recommended average values have been calculated for the effective Young’s modulus (tensile modulus), shear modulus, bulk modulus and Poisson ratio of dense isotropic plagioclase materials with a random orientation of crystallites [28]. Based on the anorthite content (in mol.%) the following fit relations have been obtained:
Density [g/cm3]: ∙2
Young’s modulus [GPa]: -0.7∙2
Unfortunately, in the present case the exact composition of the plagioclase is not known. However, the plagioclase content in the ceramic is relatively low (about 5 wt%, see Table II), so that the composition of the plagioclase can hardly have any sensible effect on the properties. Therefore, assuming the plagioclase composition to be exactly intermediate between albite and anorthite (An50), the density is 2.68 g/cm3 and the Young’s modulus 94.9± 8.7 GPa (range 86.2–100.1 GPa). Of course, for quartz (low-quartz) and mullite the density and the elastic constants are well known [29,30]. Table V gives a complete overview on the densities and Young’s moduli of all the phases present in the porcelain stoneware tile ceramics (final average values only). Table V. Theoretical (or true) densities and Young’s moduli of all the phases present in the porcelain stoneware tile ceramics (final average values only). E [GPa] [g/cm3] Glass phase 2.30 (min 2.24) 71.1 (min 67.4) Quartz 2.65 95.6 Mullite 3.14 224.8 Plagioclase 2.68 94.9
This input information, together with the quantitative XRD phase composition given in Table I, is all that is needed to calculate the theoretical (or true) density and Young’s modulus of the dense, i.e. pore-free, solid material. The essential first step for this calculation is the transformation of the mass fractions (weight percentages) fractions
i
wi from Table I into volume
via the relation (1)
i
wi 0i wi 0i
(1)
8
In a second step, the density values can be calculated via the usual mixture rule (volumeweighted arithmetic mean) (2)
0 i 0i (2) and the Young’s modulus values as the simple (i.e. unweighted) arithmetic mean (so-called Voigt-Reuss-Hill average) [30,31] (3)
E0 EVRH
EV E R 2 (3)
of the volume-weighted arithmetic mean (Voigt bound) (4)
EV i Ei
(4)
and the volume-weighted harmonic mean (Reuss bound) (5)
E R i Ei
1
(5)
The results of these calculations (either with the average values or the extreme values for the glass phase) are listed in Table VI. As expected, the values are extremely close for the three types of samples. Actually, all three types of samples are based on the same type of solid material (only the porosity has been varied by applying different amounts of pore formers), so that the three values are indicative rather of the batch-to-batch variation than of any real difference between sample types.
Table VI. Theoretical (or true) density and Young’s modulus of the dense, i.e. pore-free, solid material. S0 S5 S10 3 2.432 2.433 2.435 Theoretical [g/cm ] – average 3 2.385 2.386 2.387 Theoretical [g/cm ] – minimum 3 Theoretical [g/cm ] – maximum 2.488 2.489 2.491 E [GPa] – average 82.1 82.1 82.2 E [GPa] – minimum 78.6 78.6 78.7 E [GPa] – maximum 88.0 88.1 88.1
2. Calculation of the porosity based on the measured bulk density The theoretical (or true) density of the dense (pore-free) solid material being known, the total porosity can be calculated as soon as the bulk density has been measured. Table VII lists the 9
bulk density values, calculated from the geometry of the sample (mass-to-volume ratio) and the corresponding porosities determined via the relation (6)
1
0
(6)
from the average, minimum and maximum theoretical density values, respectively.
Table VII. Bulk densities, (measured as mass-to-volume ratios) and calculated total porosities of bar-shaped specimens (maximum and minimum values are based on different theoretical densities and take into account the specimen-to-specimen standard deviation in the bulk density measurement; five bar-shaped specimens of each type) and disk-shaped specimens (maximum and minimum values are based on different theoretical densities and take into account the uncertainty of the dimension in the volume measurement; one diskshaped specimen of each type). Total porosity Total porosity Total porosity Bulk [g/cm3] [%] – average [%] – minimum [%] – maximum S0 – bar 2.375 ± 0.011 0.4 4.5 2.3 S5 – bar 2.203 ± 0.007 7.7 11.5 9.4 S10 – bar 2.093 ± 0.010 12.3 16.0 14.0 S0 – disk 2.7 ± 1.2 S5 – disk 7.2 ± 1.2 S10 – disk 8.9 ± 1.3
Since the bulk densities have been determined via the mass-to-volume ratio, the values may be expected to be systematically slightly lower (and thus the ensuing porosities slightly higher) than the corresponding values obtained via Archimedes measurements. However, due to the very precise sample geometry the deviation from the Archimedes-based values should be low.
3. Measurement of the elastic constants Elastic constants have been measured via the impulse excitation technique according to ASTM 1876-99 [32] using a Resonant Frequency and Damping Analyzer (RFDA) System 23 (IMCE, Belgium). In the case of long bars these measurement provide the Young’s modulus from the fundamental frequency of flexural vibrations, while in the case of thin disks this method allows the determination of two elastic constants, e.g. the Young’s modulus and the Poisson ratio, from which the other elastic constants of isotropic materials can be calculated. In particular, for bars with height h (in the direction of the vibration), width (breadth) b, 10
length, L, and mass, m, the Young’s modulus can be calculated from the resonant frequency, f, (of the fundamental flexural vibration mode) via the relation (7)
E 0.946447 m f
2
L3 h bar v, , 3 L b h
(7)
where Ψbar is a correction factor for the specimen shape. In the case of disks with thickness, t, and diameter, D, the simultaneous measurement of two resonant frequencies (for flexural and anti-flexural vibrations) allows the Poisson ratio to be calculated from the frequency ratio, see [33]. As soon as the Poisson is known, the Young’s modulus can be calculated from either the flexural or the anti-flexural resonant frequency via the relation (8)
2 2 D E n 12 m f n 3 t
1 1 2 2 , Cn
(8)
where the index (subscript), n, denotes the mode of vibration (flexural or anti-flexural) and Cn is the corresponding specimen shape correction factor, which is tabulated in ASTM 1876-99 [33]. The final Young’s modulus is then taken to be the simple arithmetic mean of the two values. Table VIII lists the measured values.
Table VIII. Elastic constants (measured via impulse excitation) of bar-shaped specimens (average values of five specimens; errors are maximum deviations of the five specimens) and disk-shaped specimens (one specimen of each type; errors are here due to the uncertainty of the dimension measurement). Total porosity Young’s modulus Shear modulus Bulk modulus Poisson [%] [GPa] [GPa] [GPa] ratio [-] S0 – bar 2.3 ± 2.2 74.2 ± 1.0 S5 – bar 9.4 ± 2.1 64.0 ± 2.0 S10 – bar 14.0 ± 2.0 57.1 ± 1.3 S0 – disk 30.1 ± 1.0 38.3 ± 1.3 0.189 2.7 ± 1.2 71.5 ± 2.4 S5 – disk 27.7 ± 0.9 35.7 ± 1.1 0.191 7.2 ± 1.2 66.1 ± 2.1 S10 – disk 26.7 ± 1.0 34.2 ± 1.3 0.190 8.9 ± 1.3 63.5 ± 2.4 4. Comparison of measured and predicted Young’s moduli When the Young’s modulus of the dense, i.e. pore-free, solid material, E0, is known, rigorous upper bounds and model relations can be used to predict the effective Young’s modulus of the porous material E with a certain porosity (pore volume fraction) Φ. This predictions may then be compared with the measured values. Among the most popular predictions available in literature [33,34,35,36,37,38,39] are:
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the Voigt bound (upper bound for all microstructures): E≤E0(1- Φ)
1 the Hashin-Shtrikman upper bound (for isotropic microstructures): E E0 1
the power-law relation: E=E0(1- Φ)2
2 the exponential relation: E E0 exp 1
Fig. 3 shows the comparison between the predictions and the measured values, when the average values from Table VI are used for E0. It is evident that all measured values are significantly below the Voigt bound (as expected) and close to – but still clearly below – the other predictive curves. In this context it has to be remembered that – due to its high content in the ceramic material – the glass phase is the most significant factor affecting the E0 value of the dense ceramic material. However, as discussed above, both the density and the Young’s modulus of the glass phase itself cannot be exactly predicted, but varies in a relatively wide range. Therefore it may well be justified to use the lowest (instead of the average) E0 values in Table VI for constructing the predictive curves. If this is done, the measured values are indeed very close to the predictions, see Fig. 4. It seems that the exponential prediction is in fact closest to the measured values, as expected for convex pores made by pore formers, but since the porosity of the present materials does not exceed 14–16 %, these results do not allow definite conclusions concerning that point.
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Figure 3 - Comparison between the predicted and the measured values of Young’s moduli when the average values of E0 are used (Table VI).
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Figure 4 - Comparison between the predicted and the measured values of Young’s moduli when the lowest values of E0 are used (Table VI). 5. Conclusions In the present investigation, porcelain stoneware samples having different bulk densities and similar composition were considered in order to understand the influence of the porosity on the elastic constants of the material. A consistent methodology has been elaborated that consists in the consequent use of XRD-based quantitative phase composition information, including the glass phase content, the use of EDS-based glass phase composition information and bulk density measurements. The methodology proposed can be generally used for similar materials and allows the calculation of the effective Young’s modulus for materials of a given phase composition and porosity. Elastic constants have been determined via the impulse excitation technique (Young’s moduli have been obtained from flexural vibrations of bars, while the other elastic constants have been obtained from flexural and anti-flexural vibrations of disk-shaped samples). The results show that the control of the porosity is very important to determining the behavior of a porous ceramic product as a ceramic tile. For porcelain stoneware with a total porosity of about 15% at maximum, the comparison of predicted and experimentally measured Young’s moduli leads to the conclusion that the Young’s modulus values predicted via the power-law and exponential relations are in satisfactory agreement with the corresponding experimental data. In fact, the prediction becomes excellent when based on the lowest E0 value (Young’s modulus of the dense, i.e. pore-free, solid material). This shows the sensitivity of the prediction on the glass phase composition and the necessity to perform all calculations concerning the glass phase with great care. As far as the difference between power-law and exponential predictions is concerned, in the present context, where the porosity does not exceed 14–16 %, both relations represent equally helpful tools to design tiles with controlled microstructure and tailored mechanical behavior. Poisson ratios have measured for the first time for these materials. The data indicate that the Poisson ratios are very similar for porcelain stoneware with different porosity (0.189–0.191), and no indication of a porosity dependence of the Poisson ratio has been found.
Acknowledgement: W.P. and E.G. gratefully acknowledge funding within the project “Preparation and characterization of oxide and silicate ceramics with controlled microstructure and modeling of microstructure-property relations” (GA15-18513S), supported by the Czech Science Foundation (GAČR). 14
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[21] A. Winkelmann, O. Schott, Über die Elastizität und über die Zug- und Druckfestigkeit verschiedener Gläser in ihrer Abhängigkeit von der chemischen Zusammensetzung, Ann. Phys. Chem., 51, 1894, 697-730. [22] A.A. Appen, Khimiya Stekla. Leningrad 1974. [23] A. Makishima, J.D. Mackenzie, Direct calculation of Young’s modulus of glass, J. NonCryst. Solids, 12, 1973, 35-45. [24] E.J. Kozlovskaya, Stekloobraznoe Sostonyamie. 340 pp. Moscow 1960. [25] J.M. Brown, E.H. Abramson, R.J. Angel, Triclinic elastic constants for low albite, Phys. Chem. Minerals, 33, 2006, 256-265. [26] J.M. Brown, R.J. Angel, N. Ross, Elasticity of plagioclase feldspars, earthweb.ess.washington.edu/brown/resources/PlagioclaseElasticity.pdf (online since 26 Nov 2013, accessed 10 June 2015). [27] P. Kaercher, B. Militzer, H.R. Wenk, Ab initio calculations of elastic constants of plagioclase feldspars, Am. Mineralogist, 99, 2014, 2344-2352. [28] W. Pabst, E. Gregorová, E. Rambaldi, M.C. Bignozzi, Effective elastic constants of plagioclase feldspar aggregates – a concise review, Ceram. Silik. 59(4), 2015, 326-330. [29] W. Pabst, E. Gregorová, Elastic properties of silica polymorphs – a review, Ceram. Silik. 57(3), 2013, 167-184. [30] R. Hill, The elastic behaviour of crystalline aggregate, Proc. Phys. Soc. A, 65, 1952, 349354. [31] W. Pabst, G. Tichá, E. Gregorová, Effective elastic properties of alumina-zirconia composite ceramics Part III: Calculation of elastic moduli for polycrystalline alumina and zirconia from monocrystal data, Ceram. Silik., 48(2), 2004, 41-48. [32] ASTM E 1876-99 (American National Standard): Standard test method for dynamic Young’s modulus, shear modulus, and Poisson’s ratio by impulse excitation of vibration. American Society for Testing Materials, West Conshohocken (PA) 1999. [33] B. Paul, Prediction of elastic constants of multiphase materials, Trans. Metall. Soc. AIME, 218, 1960, 36-41. [34] Z. Hashin, S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids, 11, 1963, 127-140. [35] W. Pabst, E. Gregorová, Effective elastic properties of alumina-zirconia composite ceramics – Part II: Micromechanical modeling, Ceram. Silik., 48(1), 2004, 14-23. [36] W. Pabst, E. Gregorová, Note on the so-called Coble-Kingery formula for the effective tensile modulus of porous ceramics, J. Mater. Sci. Lett., 22, 2003, 959-962. [37] L.J. Gibson, M.F. Ashby, The mechanics of three-dimensional cellular materials, Proc. Roy. Soc. Lond. A, 382, 1982, 43-59. [38] W. Pabst, E. Gregorová, Derivation of the simplest exponential and power-law relations for the effective tensile modulus of porous ceramics via functional equations, J. Mater. Sci. Lett., 22, 2003, 1673-1675. [39] W. Pabst, E. Gregorová, Mooney-type relation for the porosity dependence of the effective tensile modulus of ceramics, J. Mater. Sci., 39, 2004, 3213-3215.
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PII: DOI: Reference:
S0272-8842(17)30317-6 http://dx.doi.org/10.1016/j.ceramint.2017.02.114 CERI14730
To appear in: Ceramics International Received date: 24 January 2017 Revised date: 22 February 2017 Accepted date: 22 February 2017 Cite this article as: Elisa Rambaldi, Willi Pabst, Eva Gregorová, Francesca Prete and Maria Chiara Bignozzi, Elastic properties of porous porcelain stoneware tiles, Ceramics International, http://dx.doi.org/10.1016/j.ceramint.2017.02.114 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Elastic properties of porous porcelain stoneware tiles Elisa Rambaldi1*, Willi Pabst2, Eva Gregorová2, Francesca Prete1, Maria Chiara Bignozzi1,3 1
Centro Ceramico, via Martelli 26, 40138 Bologna, Italy
2
Department of Glass and Ceramics, University of Chemistry and Technology, Prague, (UCT
Prague), Technická 5, 166 28 Prague, Czech Republic 3
Department of Civil, Chemical, Environmental and Materials Engineering, University of
Bologna, via Terracini 28, Bologna, Italy *
Corresponding author: Tel.:+39 051534015. [email protected]
Abstract Porcelain stoneware tiles are industrially processed by using high sintering temperatures and fast firing cycles that result in products characterized by an almost impervious surface layer surrounding a rather porous bulk material. Since mechanical properties are affected by porosity, the knowledge of the material stiffness is an important parameter to define the service behavior of tiles. In the present investigation, porcelain stoneware samples having different closed porosity were investigated in order to understand the influence of the porosity on the elastic constants of the materials. Based on the quantitative XRD phase composition, elastic constants have been calculated via Voigt-Reuss-Hill averaging, and the influence of porosity has been taken into account via power-law and exponential relations. It is shown that the effective elastic constants predicted by exponential and power-law relations are in agreement with experimental values. It may be concluded that for this class of materials, in the porosity range below 14–16 %, both exponential and power-law relations are helpful tools to design tiles with controlled microstructure and tailored mechanical properties. Keywords: Porcelain stoneware; Porosity; Phase composition; Elastic properties.
1. Introduction Porcelain stoneware tiles, traditionally used as flooring, are nowadays applied also in internal walls and ventilated facades. For these applications slightly different properties are required, including specially designed mechanical ones [1,2,3,4]. The mechanical characteristics of porcelain stoneware tiles can be directly correlated to their microstructure [5]. From the 1
viewpoint of quality, porcelain stoneware, a rather dense material in which crystals of quartz and mullite are embedded in a large amount of glassy matrix, can be considered the best product among ceramic tiles. Even if these tiles exhibit a negligible water absorption (lower than 0.5%: group BIa of ISO 13006 [6]), which is due to a near-zero open porosity [7,8], they retain a noticeable closed porosity, till values higher than 10% [9]. The maximum temperature, 1210-1230°C, reached during firing, favors the sintering and significantly reduces the total porosity, even if the industrial fast firing cycles (40-60 min) prevent the escape, during the cooling step, of gases developed in the firing reactions. The total porosity, as sum of two different fractions, open porosity at the surface and closed porosity trapped in the bulk material, is very close to the latter. That implies an almost exclusive dependence of the mechanical characteristics and service behavior on the closed porosity. Published works dealing with this dependence, report several models and relations for predicting the porosity dependence of elastic properties [10,11,12]. Nevertheless, in spite of the considerable research addressed to this topic, until now very little attention was focused on the key role played by porosity and its effects on the elastic modulus of ceramic tiles. In this work PMMA particles were introduced in different amounts into a porcelain stoneware mix, in order to control the total porosity and evaluate the contribution of the closed porosity on the elastic moduli of the materials, while keeping constant the composition. Values predicted via rigorous bounds and model relations were compared to experimental data. It is suggested that this approach will turn out to be a useful tool to design porcelain stoneware tiles having the proper porosity for different destination environments such as floors with or without heating system, floating floors for walking noise reduction, internal walls or ventilated facades.
2. Porous porcelain stoneware materials Three porcelain stoneware mixes (S0, S5 andS10) were prepared on the basis of an industrial recipe and by adding three different percentages of polymethylmetacrylate (PMMA) particles (0, 5 and 10 vol%, respectively). The PMMA particles, used as pore former, are spherical in shape and exhibit a monodisperse size distribution (6 m in diameter). Samples were prepared by uniaxial pressing (40 MPa) in form of disks (about 40 mm diameter and 7 mm thickness) and bars (70x10x6 mm). All samples were fired at 1140°C for 90 min. At about 400°C the PMMA particles were completely removed leaving voids in the ceramic samples. A complete
2
description of the sample preparation with experimental details is reported in a previous work [13]. The quantitative mineralogical compositions, of the as-fired samples, were determined by Xray diffraction analysis (PANalytical, PW3830, NL). Powdered specimens, diluted with 10 wt% of corundum NIST 676 as internal standard, were side loaded to minimize preferred orientation. Data were collected in the angular range 10–80°2θ with steps of 0.02° and 5 s/step and the Rietveld refinements were performed using the software GSAS. [14]. Results are reported in Table I. Because the samples differ only for the amount of PMMA particles, considering the standard deviation, the mineralogical composition is the same for all of them. The microstructure of the fired materials was analyzed by a scanning electron microscope, SEM (Zeiss EVO 40, D) equipped with an energy dispersive X-ray attachment, EDS (Inca, Oxford Instruments, UK), observing suitable specimens polished to mirror like finish and etched, when appropriate, by using a 5% HF solution for 3 min. Micrographs of the disk samples are reported in Fig. 1a-c. It is clear from the micrographs that the porosities are significantly different for the three types of samples. On the other hand, because the composition is the same, the EDS quantitative analysis of the glassy phase was made only on the etched surface (Fig. 2) of S0. At least three different areas of the sample were analyzed and results are listed in Table II.
Table I. Phase composition (in wt.%) determined by quantitative XRD phase analysis. S0 S5 S10 Quartz (pdf n° 46-1045), wt% 18.9±0.1 19.0±0.1 19.2±0.1 Mullite (pdf n° 15-776), wt% 6.2±0.4 6.2±0.4 6.1±0.4 Plagioclase (pdf n° 41-1480), wt% 5.2±0.3 5.3±0.3 5.8±0.3 Glassy phase, wt% 69.7±1.3 69.5±1.3 68.9±1.3
3
(a)
(b)
(c) Figure 1: SEM micrographs of the fired disk samples S0 (a), S5 (b) and S10 (c).
4
Figure 2: SEM micrographs of the etched surface of S0.
Table II. Chemical composition of the glassy phase of S0 sample by EDS analysis. Average area 1 area 2 area 3 composition Na2O, wt% 2.4 2.6 2.5 2.5±0.1 Al2O3, wt% 11.7 6.6 16.6 11.6±5.0 SiO2, wt% 84.1 89.1 79.1 84.1±5.0 K2O, wt% 1.8 1.7 1.8 1.8±1.0
3. Calculation of the density and elastic constants of the dense (i.e. pore-free) solid material The first step is the calculation of the glass phase properties. As usual in glass science, and in the absence of a better alternative, this is made via empirical additive mixture rules[15]. However, the calculation has to take into account the degree of precision to which the glass phase composition is known. With respect to the fact that the SiO2 and Al2O3 contents determined for the glass phase by EDS are much higher than the Na2O and K2O contents, it can be expected that the former affect the properties most significantly. Since they have at the same time the largest errors (much larger than for Na2O and K2O), the calculation has been made not only for the mean values (84.1 wt.% SiO2, 11.6 wt.% Al2O3), but also for the two extreme cases (determined by the errors), i.e. high Si / low Al (89.1 wt.% SiO2 / 6.6 wt.% Al2O3) and low Si / high Al (79.1 wt.% SiO2 / 16.6 wt.% Al2O3). In this way, the corresponding uncertainties (errors) can be obtained also for the density and the Young’s modulus. Tables III and IV list the resulting density and Young’s modulus values, respectively, according to additive mixture rules using the empirical coefficients determined 5
in different works by different authors. The grand average listed in these tables takes into account the values from the three columns listed and four additional values (not listed) obtained when the uncertainty (error) of the K2O value is taken into account as well. The minimum and maximum values (or, equivalently, the deviations from the averages in the negative and positive direction) may be considered as “worst case estimates”. However, extremely “exotic” values, such as the extremely high density and Young’s modulus values by Winkelmann and Schott in Tables III and IV, respectively, and the extremely low Young’s modulus values by Kozlovskaya in Table IV, have been excluded from this evaluation and are listed in Tables III and IV only for completeness (in parentheses).
Table III. Density of the glass phase, [g/cm3], defined in Table II according to additive mixture rules using the empirical coefficients determined in different works by different authors; values in parenthesis have not been taken into account for the calculation of the averages, extreme values (Min, Max) and absolute deviations (Delta min, Delta max). [g/cm3] [g/cm3] [g/cm3] [g/cm3] Average composition High Si / low Al Low Si / high Al Grand average Demkina 1 2.308 2.296 2.369 Ref [16] Demkina 2 2.323 2.312 2.335 Ref [16] Demkina 3 2.264 2.249 2.279 Ref [16] Gan 2.318 2.269 2.369 Ref [17] Huggins and 2.331 2.288 2.352 Sun Ref [18] Priven 2000 2.280 2.260 2.309 Ref [19] Priven 1998 2.297 2.263 2.330 Ref [20] Winkelmann and (2.548) (2.459) (2.636) Schott Ref [21] 2.303 2.277 2.335 Average 2.303 Min 2.264 2.249 2.279 2.244 Max 2.331 2.312 2.369 2.369 0.039 0.028 0.056 Delta min 0.059 0.028 0.035 0.034 Delta max 0.066
6
Table IV. Young’s modulus, E [GPa], of the glass phase defined in Table II according to additive mixture rules using the empirical coefficients determined in different works by different authors; values in parenthesis have not been taken into account for the calculation of the averages, extreme values (Min, Max) and absolute deviations (Delta min, Delta max). E [GPa] E [GPa] E [GPa] E [GPa] Average composition High Si / low Al Low Si / high Al Grand average Appen 70.66 69.86 71.53 Ref [22] Demkina 69.58 67.83 71.38 Ref [16] Gan 71.26 70.77 71.82 Ref [17] Makishima and 72.00 69.79 74.31 Mackenzie Ref [23] Priven 2000 70.20 68.32 71.68 Ref [19] Priven 1998 72.68 68.67 76.78 Ref [20] Kozlovskaya (66.77) (65.26) (68.39) Ref [24] Winkelmann and (80.76) (75.36) (86.16) Schott Ref [21] 71.06 69.21 72.92 Average 71.07 Min 69.58 67.83 71.38 67.38 Max 72.68 70.77 76.78 78.15 1.48 1.38 1.54 Delta min 3.69 1.62 1.56 3.86 Delta max 7.08
In both cases the agreement between most values is satisfactory, when the values calculated based on the old paper by Winkelmann and Schott [21] (and for the Young’s modulus also the values based on Kozlovskaya’s book [24]) are discarded. It is clear that the empirical coefficients are based on interpolations for glasses containing the four components as our glass (SiO2, Al2O3, Na2O, K2O), but not necessarily exclusively and not necessarily in the same or similar proportions. Differences between the estimates are therefore not at all surprising, but on the contrary, with respect to this fact the agreement of the results is surprisingly good. As a consequence, we have the following values for the density and Young’s modulus of the glass phase, respectively:
Density: 2.30 ± 0.06 g/cm3 (range 2.24–2.37 g/cm3),
Young’s modulus: 71.1± 7.1 GPa (range 67.4–78.2 GPa).
The density and elastic constants of plagioclases are relatively well known. In particular, the elastic tensor components (stiffnesses) of plagioclase monocrystals have been investigated, 7
both experimentally and by computer simulation, in several papers, from pure albite (An0) to pure anorthite (An100) [25,26,27]. Based on these monocrystal values, recommended average values have been calculated for the effective Young’s modulus (tensile modulus), shear modulus, bulk modulus and Poisson ratio of dense isotropic plagioclase materials with a random orientation of crystallites [28]. Based on the anorthite content (in mol.%) the following fit relations have been obtained:
Density [g/cm3]: ∙2
Young’s modulus [GPa]: -0.7∙2
Unfortunately, in the present case the exact composition of the plagioclase is not known. However, the plagioclase content in the ceramic is relatively low (about 5 wt%, see Table II), so that the composition of the plagioclase can hardly have any sensible effect on the properties. Therefore, assuming the plagioclase composition to be exactly intermediate between albite and anorthite (An50), the density is 2.68 g/cm3 and the Young’s modulus 94.9± 8.7 GPa (range 86.2–100.1 GPa). Of course, for quartz (low-quartz) and mullite the density and the elastic constants are well known [29,30]. Table V gives a complete overview on the densities and Young’s moduli of all the phases present in the porcelain stoneware tile ceramics (final average values only). Table V. Theoretical (or true) densities and Young’s moduli of all the phases present in the porcelain stoneware tile ceramics (final average values only). E [GPa] [g/cm3] Glass phase 2.30 (min 2.24) 71.1 (min 67.4) Quartz 2.65 95.6 Mullite 3.14 224.8 Plagioclase 2.68 94.9
This input information, together with the quantitative XRD phase composition given in Table I, is all that is needed to calculate the theoretical (or true) density and Young’s modulus of the dense, i.e. pore-free, solid material. The essential first step for this calculation is the transformation of the mass fractions (weight percentages) fractions
i
wi from Table I into volume
via the relation (1)
i
wi 0i wi 0i
(1)
8
In a second step, the density values can be calculated via the usual mixture rule (volumeweighted arithmetic mean) (2)
0 i 0i (2) and the Young’s modulus values as the simple (i.e. unweighted) arithmetic mean (so-called Voigt-Reuss-Hill average) [30,31] (3)
E0 EVRH
EV E R 2 (3)
of the volume-weighted arithmetic mean (Voigt bound) (4)
EV i Ei
(4)
and the volume-weighted harmonic mean (Reuss bound) (5)
E R i Ei
1
(5)
The results of these calculations (either with the average values or the extreme values for the glass phase) are listed in Table VI. As expected, the values are extremely close for the three types of samples. Actually, all three types of samples are based on the same type of solid material (only the porosity has been varied by applying different amounts of pore formers), so that the three values are indicative rather of the batch-to-batch variation than of any real difference between sample types.
Table VI. Theoretical (or true) density and Young’s modulus of the dense, i.e. pore-free, solid material. S0 S5 S10 3 2.432 2.433 2.435 Theoretical [g/cm ] – average 3 2.385 2.386 2.387 Theoretical [g/cm ] – minimum 3 Theoretical [g/cm ] – maximum 2.488 2.489 2.491 E [GPa] – average 82.1 82.1 82.2 E [GPa] – minimum 78.6 78.6 78.7 E [GPa] – maximum 88.0 88.1 88.1
2. Calculation of the porosity based on the measured bulk density The theoretical (or true) density of the dense (pore-free) solid material being known, the total porosity can be calculated as soon as the bulk density has been measured. Table VII lists the 9
bulk density values, calculated from the geometry of the sample (mass-to-volume ratio) and the corresponding porosities determined via the relation (6)
1
0
(6)
from the average, minimum and maximum theoretical density values, respectively.
Table VII. Bulk densities, (measured as mass-to-volume ratios) and calculated total porosities of bar-shaped specimens (maximum and minimum values are based on different theoretical densities and take into account the specimen-to-specimen standard deviation in the bulk density measurement; five bar-shaped specimens of each type) and disk-shaped specimens (maximum and minimum values are based on different theoretical densities and take into account the uncertainty of the dimension in the volume measurement; one diskshaped specimen of each type). Total porosity Total porosity Total porosity Bulk [g/cm3] [%] – average [%] – minimum [%] – maximum S0 – bar 2.375 ± 0.011 0.4 4.5 2.3 S5 – bar 2.203 ± 0.007 7.7 11.5 9.4 S10 – bar 2.093 ± 0.010 12.3 16.0 14.0 S0 – disk 2.7 ± 1.2 S5 – disk 7.2 ± 1.2 S10 – disk 8.9 ± 1.3
Since the bulk densities have been determined via the mass-to-volume ratio, the values may be expected to be systematically slightly lower (and thus the ensuing porosities slightly higher) than the corresponding values obtained via Archimedes measurements. However, due to the very precise sample geometry the deviation from the Archimedes-based values should be low.
3. Measurement of the elastic constants Elastic constants have been measured via the impulse excitation technique according to ASTM 1876-99 [32] using a Resonant Frequency and Damping Analyzer (RFDA) System 23 (IMCE, Belgium). In the case of long bars these measurement provide the Young’s modulus from the fundamental frequency of flexural vibrations, while in the case of thin disks this method allows the determination of two elastic constants, e.g. the Young’s modulus and the Poisson ratio, from which the other elastic constants of isotropic materials can be calculated. In particular, for bars with height h (in the direction of the vibration), width (breadth) b, 10
length, L, and mass, m, the Young’s modulus can be calculated from the resonant frequency, f, (of the fundamental flexural vibration mode) via the relation (7)
E 0.946447 m f
2
L3 h bar v, , 3 L b h
(7)
where Ψbar is a correction factor for the specimen shape. In the case of disks with thickness, t, and diameter, D, the simultaneous measurement of two resonant frequencies (for flexural and anti-flexural vibrations) allows the Poisson ratio to be calculated from the frequency ratio, see [33]. As soon as the Poisson is known, the Young’s modulus can be calculated from either the flexural or the anti-flexural resonant frequency via the relation (8)
2 2 D E n 12 m f n 3 t
1 1 2 2 , Cn
(8)
where the index (subscript), n, denotes the mode of vibration (flexural or anti-flexural) and Cn is the corresponding specimen shape correction factor, which is tabulated in ASTM 1876-99 [33]. The final Young’s modulus is then taken to be the simple arithmetic mean of the two values. Table VIII lists the measured values.
Table VIII. Elastic constants (measured via impulse excitation) of bar-shaped specimens (average values of five specimens; errors are maximum deviations of the five specimens) and disk-shaped specimens (one specimen of each type; errors are here due to the uncertainty of the dimension measurement). Total porosity Young’s modulus Shear modulus Bulk modulus Poisson [%] [GPa] [GPa] [GPa] ratio [-] S0 – bar 2.3 ± 2.2 74.2 ± 1.0 S5 – bar 9.4 ± 2.1 64.0 ± 2.0 S10 – bar 14.0 ± 2.0 57.1 ± 1.3 S0 – disk 30.1 ± 1.0 38.3 ± 1.3 0.189 2.7 ± 1.2 71.5 ± 2.4 S5 – disk 27.7 ± 0.9 35.7 ± 1.1 0.191 7.2 ± 1.2 66.1 ± 2.1 S10 – disk 26.7 ± 1.0 34.2 ± 1.3 0.190 8.9 ± 1.3 63.5 ± 2.4 4. Comparison of measured and predicted Young’s moduli When the Young’s modulus of the dense, i.e. pore-free, solid material, E0, is known, rigorous upper bounds and model relations can be used to predict the effective Young’s modulus of the porous material E with a certain porosity (pore volume fraction) Φ. This predictions may then be compared with the measured values. Among the most popular predictions available in literature [33,34,35,36,37,38,39] are:
11
the Voigt bound (upper bound for all microstructures): E≤E0(1- Φ)
1 the Hashin-Shtrikman upper bound (for isotropic microstructures): E E0 1
the power-law relation: E=E0(1- Φ)2
2 the exponential relation: E E0 exp 1
Fig. 3 shows the comparison between the predictions and the measured values, when the average values from Table VI are used for E0. It is evident that all measured values are significantly below the Voigt bound (as expected) and close to – but still clearly below – the other predictive curves. In this context it has to be remembered that – due to its high content in the ceramic material – the glass phase is the most significant factor affecting the E0 value of the dense ceramic material. However, as discussed above, both the density and the Young’s modulus of the glass phase itself cannot be exactly predicted, but varies in a relatively wide range. Therefore it may well be justified to use the lowest (instead of the average) E0 values in Table VI for constructing the predictive curves. If this is done, the measured values are indeed very close to the predictions, see Fig. 4. It seems that the exponential prediction is in fact closest to the measured values, as expected for convex pores made by pore formers, but since the porosity of the present materials does not exceed 14–16 %, these results do not allow definite conclusions concerning that point.
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Figure 3 - Comparison between the predicted and the measured values of Young’s moduli when the average values of E0 are used (Table VI).
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Figure 4 - Comparison between the predicted and the measured values of Young’s moduli when the lowest values of E0 are used (Table VI). 5. Conclusions In the present investigation, porcelain stoneware samples having different bulk densities and similar composition were considered in order to understand the influence of the porosity on the elastic constants of the material. A consistent methodology has been elaborated that consists in the consequent use of XRD-based quantitative phase composition information, including the glass phase content, the use of EDS-based glass phase composition information and bulk density measurements. The methodology proposed can be generally used for similar materials and allows the calculation of the effective Young’s modulus for materials of a given phase composition and porosity. Elastic constants have been determined via the impulse excitation technique (Young’s moduli have been obtained from flexural vibrations of bars, while the other elastic constants have been obtained from flexural and anti-flexural vibrations of disk-shaped samples). The results show that the control of the porosity is very important to determining the behavior of a porous ceramic product as a ceramic tile. For porcelain stoneware with a total porosity of about 15% at maximum, the comparison of predicted and experimentally measured Young’s moduli leads to the conclusion that the Young’s modulus values predicted via the power-law and exponential relations are in satisfactory agreement with the corresponding experimental data. In fact, the prediction becomes excellent when based on the lowest E0 value (Young’s modulus of the dense, i.e. pore-free, solid material). This shows the sensitivity of the prediction on the glass phase composition and the necessity to perform all calculations concerning the glass phase with great care. As far as the difference between power-law and exponential predictions is concerned, in the present context, where the porosity does not exceed 14–16 %, both relations represent equally helpful tools to design tiles with controlled microstructure and tailored mechanical behavior. Poisson ratios have measured for the first time for these materials. The data indicate that the Poisson ratios are very similar for porcelain stoneware with different porosity (0.189–0.191), and no indication of a porosity dependence of the Poisson ratio has been found.
Acknowledgement: W.P. and E.G. gratefully acknowledge funding within the project “Preparation and characterization of oxide and silicate ceramics with controlled microstructure and modeling of microstructure-property relations” (GA15-18513S), supported by the Czech Science Foundation (GAČR). 14
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