Introduction

Clay block masonry comfortably combines thermal and mechanical competences, making it one of the most sustainable and sought-after building materials, in particular when it comes to the construction of small storey houses. The recent quest to extend the applicability of this material, both in scope and volume, motivates deeper scientific scrutiny into what actually lies at the origin of the aforementioned comfortable combination of material characteristics. It is well accepted that porosities which are induced in a more or less designed way at different scales into the material, are the key governing factor for both its mechanical and thermal properties [1, 2]. In this context, pore-forming agents are used to increase porosity at scales ranging from micrometres to millimetres, in order to enhance the thermal insulation characteristics of the material. For the time being, the actual effect of this measure can only be determined empirically, through direct macroscopic testing. A first step towards a more scientific exploration of this effect is the quantification of the porosities themselves, and this is the focus of the present paper. Therefore, our preferred method of choice is micro-computed tomography, which, in recent years, has not only revealed microstructural features as represented by voxel-built patterns [3], but also basic compositional information within each and every voxel [4,5,6]. Accordingly, after presenting the investigated material in the “Investigated materials and micro-CT scanning” section, the “Methods for macroporosity determination” section is devoted to the determination of the ceramic macroporosity from thresholding-based image analysis, while the “Methods for microporosity determination” section describes how fundamental relations from X-ray physics, in combination with basic knowledge on fired clay chemistry, allow for the determination of the microporosity within each and every voxel. In order to check this new way of 3D quantification of the dual-scale ceramic porosities, independent experimental access to the (average) porosities is provided by mercury intrusion porosimetry and weighing tests, as described in the “Mercury intrusion porosimetry and weighing tests” section. The corresponding results are presented in the “Results and discussion” section, and discussed thereafter.

Materials and methods

Investigated materials and micro-CT scanning

Ceramic specimens with dimensions \(30 \times 15 \times 125\,\hbox {mm}^{3}\) were fired at \(880\,^{\circ }\hbox {C}\). Their extrusion direction was parallel to the edge measuring 125 mm. These specimens exhibited different concentrations of different pore-forming agents, namely expanded polystyrene (EPS) in mass fractions of 10 and 20%, as well as paper sludge and sawdust in mass fractions of 10, 20, and 40%, respectively. For reference purposes, we also investigated ceramic samples made of pure clay without pore-forming agents, fired at 880 and \(1100\,^{\circ }\hbox {C}\). In order to image the ceramic microstructures by means of micro-computed tomography (\(\upmu \hbox {CT}40\), Scanco, Switzerland), ten smaller samples with dimensions \(6 \times 5 \times 15\,\hbox {mm}^{3}\) were cut out from the aforementioned samples, by means of a distilled water-cooled low-speed diamond saw (Isomet, Buehler, USA). The extrusion direction of the aforementioned smaller samples was parallel to the edge measuring 6 mm. The following settings were used in the scanning process: source current \(114 \,\upmu \hbox {A}\), source voltage 70 kVP, integration time 300 ms. The voxel size of the resulting micro-CT images was \(6 \times 6 \times 6 \,\upmu \hbox {m}^{3}\), and the voxel-specific attenuation was characterized through a 16-bit grey value scale ranging from 0 to 32767.

Methods for macroporosity determination

First of all, the high-resolution micro-CT images were smoothed by means of a 3D median filter, in order to reduce ring artefacts, as it is done with various CT applications [7, 8]. Then, these cleaned image stacks were considered to illustrate two material phases at the observation scale of millimetres: (1) microporous fired clay matrix and (2) macropores. The characteristics of these two phases are quantified from normalized frequency plots (histograms) of the grey values assigned to all voxels making up the three-dimensional image stack of each and every scanned sample. Therefore, the histograms are optimally represented as the superposition of two Gaussian distribution functions, with correspondingly optimized standard deviations and expected values. The latter are denoted as \({\hbox {GV}}_{{\mathrm{air}}}\), the most frequently occurring voxel filled by air, and \({\hbox {GV}}_{{\mathrm{FC}}}^{{\mathrm{peak}}}\), the most frequently occurring grey value filled by fired clay matrix, see Fig. 2. The optimization process is performed on the cumulative distribution function, by means of an evolutionary algorithm, as described elsewhere [9]. The intersection of the two Gaussian distribution functions serves as threshold value \({\hbox {GV}}_{{\mathrm{thr}}}\), which discriminates macropores and fired clay matrix: all voxels with \({\hbox {GV}} > {\hbox {GV}}_{{\mathrm{thr}}}\) represent fired clay matrix, and the remaining ones represent macropores. The latter can be suitably illustrated through an image segmentation process, see Fig. 1. The macropore volume fractions of the samples, \(\phi _{{\mathrm{macro}}}\), was calculated by summing up, from 0 to \({\hbox {GV}}_{{\mathrm{thr}}}\), the normalized frequency values of the grey value histogram, see Fig. 2c,

$$\begin{aligned} \phi _{{\mathrm{macro}}}=\sum _{x=0}^{{\mathrm{GV}}_{{\mathrm{thr}}}}\mathrm{PDF}(x)\, \times 100 \end{aligned}$$
(1)

The macropores are then further analysed by ImageJ, a public-domain, Java-based image processing software developed at the National Institutes of Health [10]. Two additional plugins, 3D object counter [11] and BoneJ particle analyser [12], were used for identifying the desired geometric properties of the macropores: the 3D object counter delivered the volume and the surface of the pores. Furthermore, the Feret diameters [13] of the macropores along the x-, y-, and z-directions were determined, resulting in macropore-specific bounding boxes. In more detail, the smallest possible cuboids enclosing a pore were identified and characterized by the coordinates of their upper left corner, as well as by their widths, heights, and depths. Finally, the particle analyser gave access to the principal direction related to the largest eigenvalue \(I_1\) of the inertia tensor of each pore [12]. The orientation of the macropore was computed in terms of the angle \(\delta \) between the extrusion direction (labelled by coordinate x-direction) and the aforementioned principal direction. This allowed for comparing the orientation of the macropores between the different samples.

Figure 1
figure 1

Micro-computed tomography of sample with 20% EPS (a), and segmentation-based illustration of macropores (b); grey value distributions throughout cross sections indicated in (a): cross section 1 (c), and cross section 2 (d). The x-axis indicates the extrusion direction

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Figure 2
figure 2

Statistical evaluation of voxel-specific grey values of micro-CT-scanned ceramic sample with 10% EPS: a normalized histogram or probability density function (PDF) of all image voxels; b cumulative distribution function (CDF) of all image voxels; c Gaussian PDFs related to macropore voxels and to fired clay matrix voxels, respectively

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Methods for microporosity determination

The microporous fired clay matrix material is considered as consisting of two phases at the single micrometre scale of one scanned voxel: (1) dense aluminium silicate matrix and (2) micropores. In order to determine the microporosity inside each and every voxel, the protocol of Czenek et al. [6] was adopted to the current needs. First of all, the imaging activities described in the “Methods for macroporosity determination” section were complemented by scanning a single-crystal aluminium cylinder with a diameter of 6 mm and a height of 3 mm. The most frequently occurring grey value within this aluminium sample is denoted as \({\hbox {GV}}_{{\mathrm{Al}}}\), and it amounted to \({\hbox {GV}}_{{\mathrm{Al}}}=10544\). This value is instrumental in assigning the attenuation-related grey values to the corresponding actual X-ray attenuation coefficients. The respective linear relation is defined by slope and intercept parameters a and b, and the latter depend on the photon energy \(\epsilon \). Mathematically, this reads as [6]

$$\begin{aligned} \mu \left( \epsilon \right) = a\left( \epsilon \right) \times \hbox {GV} + b\left( \epsilon \right) \end{aligned}$$
(2)

Strictly speaking, Eq. (2) is only valid for monochromatic X-rays as encountered with synchrotron computed tomography. However, Eq. (2) turns out as a suitable approximative description of the polychromatic case when \(\epsilon \) is considered as an average photon energy, see the end of the “Results of microporosity determination” section for further discussion of this issue. Under these conditions, parameters a and b can be retrieved from X-ray physics-supported image analysis, in the line of [6]: the grey values \({\hbox {GV}}_{{\mathrm{Al}}}\) and \({\hbox {GV}}_{{\mathrm{air}}}\), as determined in the “Methods for macroporosity determination” section, are related to the attenuation coefficients of air and aluminium, \(\mu _{{\mathrm{air}}}\) and \(\mu _{{\mathrm{Al}}}\), which are known from the NIST database [14,15,16], see Fig. 3, and this provides two linear equations for the coefficients a and b, reading as

$$\begin{aligned} \mu ^{{\mathrm{NIST}}}_{{\mathrm{Air}}} \left( \epsilon \right)&= a\left( \epsilon \right) \times {\hbox {GV}}_{{\mathrm{Air}}} + b\left( \epsilon \right) \end{aligned}$$
(3)
$$\begin{aligned} \mu ^{{\mathrm{NIST}}}_{{\mathrm{Al}}} \left( \epsilon \right)&= a\left( \epsilon \right) \times {\hbox {GV}}_{{\mathrm{Al}}} + b\left( \epsilon \right) \end{aligned}$$
(4)

Still, a and b are functions of \(\epsilon \), as depicted in Fig. 4, and the actually used (average) energy \(\epsilon \) still needs to be retrieved. This is done based on the grey value which most frequently occurs in the fired clay matrix, \({\hbox {GV}}_{{\mathrm{FC}}}^{{\mathrm{peak}}}\), as described in the “Methods for macroporosity determination” section, and on the corresponding peak attenuation coefficient \(\mu _{{\mathrm{FC}}}^{{\mathrm{peak}}}\), which according to Eq. (2), are related to each other through

$$\begin{aligned} \mu _{{\mathrm{FC}}}^{{\mathrm{peak}}}=a(\epsilon ) \times {\hbox {GV}}_{{\mathrm{FC}}}^{{\mathrm{peak}}} + b(\epsilon ) \end{aligned}$$
(5)

There is an independent alternative access to \(\mu _{{\mathrm{FC}}}^{{\mathrm{peak}}}\), via the volume average rule for attenuation coefficients, reading as [4, 17, 18]

$$\begin{aligned} \mu _{{\mathrm{FC}}}^{{\mathrm{peak}}}=\phi ^{{\mathrm{peak}}} \mu _{{\mathrm{air}}} + \left( 1-\phi ^{{\mathrm{peak}}}\right) \mu _{{\mathrm{Si}}}^{{\mathrm{NIST}}} \end{aligned}$$
(6)

with \(\phi ^{{\mathrm{peak}}}\) as the most frequently occurring value for the microporosity, and \(\mu _{{\mathrm{Si}}}^{{\mathrm{NIST}}}\) as the attenuation coefficient of a theoretically completely dense clay matrix. The latter consists almost exclusively of silicates, as revealed from standard X-ray fluorescence spectroscopy, see Table 1 for the chemical composition. This composition, together with the NIST database [16], provides \(\mu _{{\mathrm{Si}}}^{{\mathrm{NIST}}}\), see dash-dotted line in Fig. 3. In more detail, the attenuation coefficient \(\mu _{{\mathrm{Si}}}^{{\mathrm{NIST}}}\) was computed from the X-ray mass attenuation coefficient of each molecular constituent \((\mu / \rho )_i \) and its weight fraction \(w_i\) [19], see Table 1,

$$\begin{aligned} \mu _{{\mathrm{Si}}}^{{\mathrm{NIST}}}=\sum _i w_i (\mu / \rho )_i \times \rho _{{\mathrm{Si}}} \end{aligned}$$
(7)

with \(\rho _{{\mathrm{Si}}}=2.7\,\hbox {g}/\hbox {cm}^{3}\) being the density of the theoretically completely dense clay matrix.

The aforementioned density was determined by means of Archimedes’ principle [20]. A sample fired at \(880\,^{\circ }\hbox {C}\) without pore-forming agent was weighed, which delivered its dry weight, \(w_{{\mathrm{dry}}}=7.505\,\hbox {mN}\). Then, it was placed in a glass beaker containing liquid xylene, which subsequently filled the pores. The sample was weighed every 4 h for 36 h. After 24 h the weight of the fully saturated sample, \(w_{{\mathrm{wet}}}=8.923\,\hbox {mN}\), did not change any more, indicating that the sample was completely hydrated and that all the pores were filled with xylene. The hydrated sample was then submerged into xylene, and the weight \(w_{{\mathrm{sub}}}\) of the submerged sample was measured, \(w_{{\mathrm{sub}}}=5.121\,\hbox {mN}\). The latter weight and the dry weight then give access to the overall sample volume as

$$\begin{aligned} V_{{\mathrm{sample}}}=\frac{\left( w_{{\mathrm{dry}}}-w_{{\mathrm{sub}}} \right) g}{\rho _{{\mathrm{xylene}}}}=0.282\,\hbox {cm}^{3}, \end{aligned}$$
(8)

with \(\rho _{{\mathrm{xylene}}}=0.861\,\hbox {g}/\hbox {cm}^{3}\) as the mass density of xylene at room temperature (\(20\,^{\circ }\hbox {C}\)) and \(g=9.81 \,\hbox {m/s}\) as the gravitational acceleration. Thus, the density \(\rho _{{\mathrm{Si}}}\) of the theoretically completely dense clay matrix can be given as

$$\begin{aligned} \rho _{{\mathrm{Si}}}=\frac{m_{{\mathrm{dry}}}}{V_s}=2.707\,\hbox {g}/\hbox {cm}^{3}, \end{aligned}$$
(9)

which is equal or close to values given elsewhere [21,22,23].

Table 1 Chemical composition of clay: chemical elements in (mass %), and in parts per million (ppm), as a function of firing temperature, obtained from X-ray fluorescence spectroscopy
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Figure 3
figure 3

X-ray attenuation coefficients of air, aluminium, and the solid fired clay matrix consisting almost exclusively of silicates, as functions of the photon energy \(\epsilon \), according to NIST [16]

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Figure 4
figure 4

Slope and intercept parameters a and b, as functions of the photon energy \(\epsilon \), related to the image stack of ceramic sample with 10% EPS

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Equating the previously described expressions for the peak attenuation coefficient \(\mu _{{\mathrm{FC}}}^{{\mathrm{peak}}}\), namely Eqs. (5) and (6), yields

$$\begin{aligned} a\left( \epsilon \right) \times {\hbox {GV}}_{{\mathrm{FC}}}^{{\mathrm{peak}}} + b\left( \epsilon \right) = \mu ^{{\mathrm{NIST}}}_{{\mathrm{Si}}}\left( 1-\phi ^{{\mathrm{peak}}}\right) +\mu _{{\mathrm{Air}}}\phi ^{{\mathrm{peak}}} \end{aligned}$$
(10)

Equation (10) constitutes a non-bijective function between \(\phi ^{{\mathrm{peak}}}\) and \(\epsilon \), with the characteristic that a specific value of \(\phi ^{{\mathrm{peak}}}\) is related to either none, one, or two values of the (average) photon energy \(\epsilon \). As only one (average) photon energy was used for the scanning process, the one value of \(\phi _{{\mathrm{peak}}}\) which is related to only one photon energy \(\epsilon \), is the only and unique physically admissible (average) energy level. This provides unique values for both \(\epsilon \) and \(\phi ^{{\mathrm{peak}}}\).

Once \(\epsilon \) is known, the grey values of the images can be converted into attenuation coefficients, according to Eq. (2) with the functions retrieved from Eqs. (3) and (4), as depicted in Fig. 4. Subsequently, the voxel-specific microporosity follows from the average rule (6) applied to all voxels representing fired clay matrix,

$$\begin{aligned} \phi =\frac{\mu _{{\mathrm{FC}}}-\mu ^{{\mathrm{NIST}}}_{{\mathrm{Si}}}}{\mu _{{\mathrm{Air}}}-\mu ^{{\mathrm{NIST}}}_{{\mathrm{Si}}}} \end{aligned}$$
(11)

The integral of the voxel-specific microporosity over the volume of the fired clay matrix yields the microporosity volume,

$$\begin{aligned} V_{{\mathrm{micropores}}}=\int _{V_{{\mathrm{FC}}}}\phi (x)\hbox {d}V, \end{aligned}$$
(12)

which is used to calculate the sample-specific microporosity \(\phi _{{\mathrm{micro}}}\) as

$$\begin{aligned} \phi _{{\mathrm{micro}}}=\frac{V_{{\mathrm{micropores}}}}{V_{{\mathrm{sample}}}} \end{aligned}$$
(13)

Accordingly, the ceramic sample-related total porosity derived from micro-CT can be given as

$$\begin{aligned} \phi _{{\mathrm{sample}}}^{{\mathrm{CT}}}=\phi _{{\mathrm{macro}}}+\phi _{{\mathrm{micro}}}, \end{aligned}$$
(14)

with \(\phi _{{\mathrm{macro}}}\) and \(\phi _{{\mathrm{micro}}}\) according to Eqs. (1) and (13), respectively.

Mercury intrusion porosimetry and weighing tests

The total porosity of all samples as determined from (14) was checked by two independent methods, namely mercury intrusion porosimetry and weighing tests. Mercury is a non-wetting liquid that does not spontaneously infiltrate pores, so that pore infiltration requires an external pressure. Using the system Pascal 140-240/440, POROTEC GmbH, Germany, the volume of intruded mercury was measured, as well as the applied pressure [24]. The volume of intruded mercury is equal to the pore volume \(V_{{\mathrm{por}}}\) of the sample. If the mass of the sample is known, also its volume can be determined; therefore, the dilatometer is filled with mercury up to a defined volume and then weighed. Thereafter, the sample is put into the dilatometer, and the rest of the defined volume in the dilatometer is filled with mercury under vacuum. The difference in mass between the dilatometer filled by mercury only and the dilatometer filled with both the sample and mercury, is equal to the mass \(\Delta m\) of mercury filling the volume of the sample, which reads mathematically as

$$\begin{aligned} \Delta m= \left( m_{\mathrm{Dilatometer+Hg}}-m_{{\mathrm{Dilatometer}}}\right) - \left( m_{{\mathrm{Dilatometer+Hg+Sample}}}-m_{{\mathrm{Sample}}}-m_{{\mathrm{Dilatometer}}}\right) \end{aligned}$$
(15)

Knowing also the mass density of mercury at the measured room temperature, \(\rho _{{\mathrm{Hg}}}\), the aforementioned mass difference readily gives access to the volume of the sample [25],

$$\begin{aligned} V_{{\mathrm{sample}}}=\frac{\Delta m}{\rho _{{\mathrm{Hg}}}} \end{aligned}$$
(16)

Finally, the porosity of the sample \(\phi _{{\mathrm{sample}}}^{{\text {Hg-intr}}}\) is obtained through

$$\begin{aligned} \phi _{{\mathrm{sample}}}^{{\text {Hg-intr}}}=\frac{V_{{\mathrm{por}}}}{V_{{\mathrm{sample}}}} \end{aligned}$$
(17)

At higher pressures, the compressibility of mercury was considered according to the algorithm described in [21]. Since mercury intrusion porosimetry is only capable of measuring the volume of open pores with a diameter ranging from 2 nm to \(500\,\upmu \hbox {m}\), also weighing tests were performed, in order to determine the total (potentially partially closed) porosity of the samples. Their weight, \(w_{{\mathrm{sample}}}\), was measured by means of a precision balance (PGH403-S, Mettler-Toledo International Inc., Switzerland), and the exact dimensions of the samples were extracted from the micro-CT images. Considering a cuboid shape, the volume \(V_{{\mathrm{sample}}}\) was computed, which in turn provides access to the sample densities \(\rho _{{\mathrm{sample}}}\),

$$\begin{aligned} \rho _{{\mathrm{sample}}} =\frac{w_{{\mathrm{sample}}}/g}{V_{{\mathrm{sample}}}} \end{aligned}$$
(18)

From the latter and the known theoretical density of the silicate matrix \(\rho _{{\mathrm{Si}}}= 2.7\,\hbox {g}/\hbox {cm}^{3}\), the porosity \(\phi _{{\mathrm{sample}}}^{{\mathrm{weighing}}}\) follows as

$$\begin{aligned} \phi _{{\mathrm{sample}}}^{{\mathrm{weighing}}}=1-\frac{\rho _{{\mathrm{sample}}}}{\rho _{{\mathrm{Si}}}} \end{aligned}$$
(19)

The latter and \(\phi _{{\mathrm{sample}}}^{{\text {Hg-intr}}}\) are compared to the porosity resulting from the micro-CT images, \(\phi _{{\mathrm{sample}}}^{{\mathrm{CT}}}\). The relative errors, \({\hbox {err}}^{{\mathrm{CT}}}\) and \({\hbox {err}}^{{\text {Hg-intr}}}\), are calculated on the basis of \(\phi _{{\mathrm{sample}}}^{{\mathrm{weighing}}}\), so that

$$\begin{aligned} {\hbox {err}}^{{\mathrm{CT}}}&= 100 \times \frac{\phi _{{\mathrm{sample}}}^{{\mathrm{CT}}}-\phi _{{\mathrm{sample}}}^{{\mathrm{weighing}}}}{\phi _{{\mathrm{sample}}}^{{\mathrm{weighing}}}} \end{aligned}$$
(20)
$$\begin{aligned} {\hbox {err}}^{{\text {Hg-intr}}}&= 100 \times \frac{\phi _{{\mathrm{sample}}}^{{\text {Hg-intr}}}-\phi _{{\mathrm{sample}}}^{{\mathrm{weighing}}}}{\phi _{{\mathrm{sample}}}^{{\mathrm{weighing}}}} \end{aligned}$$
(21)

Results and discussion

Results of macroporosity determination

Histogram evaluations as described in the “Methods for macroporosity determination” section reveal macropores to be present in samples processed with EPS and sawdust, while samples processed with paper sludge are free of macropores, see Figs. 5, 6 and 7. In this context, sawdust induces the largest macroporosity, and the largest number of macropores according to the 3D object counter evaluation described in the “Methods for macroporosity determination” section, while EPS is remarkably less effective in its pore-forming ability (see Table 2). Moreover, the sawdust-induced pores are smaller and more homogeneously distributed than the EPS-induced pores (see Fig. 8). Generally, an increase in the pore-forming agents EPS and sawdust causes a higher macroporosity and bigger macropores in the samples. It is also interesting to regard the macropore size distribution (see Figs. 9, 10): as a rule, large macropores (with volumes exceeding \(10^{8}\,\upmu \hbox {m}^{3}\)) make up most of the macropore volume, while the numbers of medium-sized macropores (with volumes ranging from \(10^{5}\) and \(10^{8}\,\upmu \hbox {m}^{3}\)) and of small macropores (with volume smaller than \(10^{5}\,\upmu \hbox {m}^{3}\), but still larger than the voxel size, with \(216\,\upmu \hbox {m}^{3}\)) outnumber, by far, the number of the aforementioned large macropores. This effect is more pronounced with the EPS-processed samples, when compared to the sawdust-processed samples. The principal axes related to the largest eigenvalues of the pore-specific inertia tensor (quantified by the particle analyser as described in the “Methods for macroporosity determination” section) are preferentially oriented in the extrusion direction (i.e. in direction x of Fig. 8), see Fig. 11. The results of the bounding box evaluations also confirm that the macropores show a preferred orientation, as most macropores have their biggest dimension along the x-axis, the extrusion direction, see Table 3.

Figure 5
figure 5

Probability density function (PDF) of attenuation-related grey values from micro-CT scans of samples processed with EPS: a 10% mass fraction, b 20% mass fraction; representation of experimental data through two Gaussians related to macropores and fired clay matrix, respectively

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Figure 6
figure 6

Probability density function (PDF) of attenuation-related grey values from micro-CT scans of samples processed with paper sludge: a 10% mass fraction, b 20% mass fraction, c 40% mass fraction; representation of experimental data through one Gaussian related to the fired clay matrix

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Figure 7
figure 7

Probability density function (PDF) of attenuation-related grey values from micro-CT scans of samples processed with sawdust: a 10% mass fraction, b 20% mass fraction, c 40% mass fraction; representation of experimental data through two Gaussians related to macropores and fired clay matrix, respectively

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Figure 8
figure 8

Segmentation-based illustration of macropores induced by pore-forming agents EPS and sawdust; extrusion direction x is illustrated in the upper left corner. a 10% EPS, b 10% sawdust, c 20% EPS, d 20% sawdust, e 40% sawdust

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Table 2 Characterization of the macropore space of samples with EPS and sawdust
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Figure 9
figure 9

Frequency of macropore size categories in samples processed with sawdust at mass fractions of a 10%, b 20%, c 40%, and with EPS at mass fractions of d 10%, e 20%

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Figure 10
figure 10

Volume fraction of macropore size categories in samples processed with sawdust at mass fractions of a 10%, b 20%, c 40%, and with EPS at mass fractions of d 10%, e 20%

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Figure 11
figure 11

Volume fraction of macropore orientation categories in terms of deviation angle \(\delta \) between the extrusion direction and the principal axis related to the largest eigenvalues of the macropore-specific inertia tensors; for samples processed with sawdust at mass fractions of a 10%, b 20%, c 40%, and with EPS at mass fractions of d 10%, e 20%

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Table 3 Statistical distribution of bounding box dimensions: percentage of bounding boxes with largest edges oriented in x-, y-, and z-directions
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Results of microporosity determination

Evaluation of the implicit energy–microporosity relation Eq. (10), together with the intercept and slope parameters resulting from Eqs. (3) and (4), for all samples described in the “Investigated materials and micro-CT scanning” section, yields the graphs depicted in Fig. 12. The edges in the otherwise smooth curves stem from the so-called K-edge of barium, known to occur at 37.4 keV [16]. The maximum values of the sample-specific graphs of Fig. 12 relate to the actual peak microporosities (as only one average photon energy was used per scanning); these values are summarized in Table 4. The black line in Fig. 12 represents the sample without pore-forming agent fired at \(1100\,^{\circ }\hbox {C}\), which has expectedly the lowest microporosity since most likely the amorphous fraction of the fired clay matrix increases. Consistently, the pure clay sample fired at \(880\,^{\circ }\hbox {C}\), indicated by a red line in Fig. 12, has a higher microporosity. Both samples with EPS are slightly above the red line, which implies that the concentration of EPS does not have a significant influence on the peak microporosity. The samples processed with sawdust show the same behaviour: independent of the sawdust concentration, the microporosity stays nearly constant and close to the value of pure clay fired at \(880\,^{\circ }\hbox {C}\), see Table 4. In contrast, the microporosity of the samples processed with paper sludge increases with increasing concentration of the pore-forming agent, resulting in the overall voxel-related highest peak microporosity for 40% paper sludge, see Table 4. Similar results prevail for the overall microporosities \(\phi _{{\mathrm{micro}}}\) according to Eqs. (13) and (14), see Table 5. The overall microporosities per volume of ceramic sample, as a rule, comprise the by far major portion of the total porosity, namely between 60 and 100% of the latter, see Table 5. Finally, the total porosity derived from micro-CT agrees very well with that derived from mercury intrusion and weighing, with mercury intrusion porosimetry results being slightly worse than those stemming from weighing in combination with chemical analysis, see columns 7 and 8 of Table 5. In contrast to mercury intrusion porosimetry and weighing, our new CT evaluation scheme provides, through Eq. (11), detailed information on the microporosity distributions throughout the scanned ceramic samples, see Figs. 13 and 14. We reiterate from the “Materials and methods” section that we have introduced only one photon energy value as the basis for the quantitative intra-voxel evaluation, in spite of the use of a polychromatic CT device. Given the satisfactory results validated through independent experimental methods such as weighing and mercury intrusion, as shown in Table 5, we consider the aforementioned choice of only one energy value as a posteriori justified. At the same time, we emphasize that typical artefacts stemming from polychromatic light, such as streaks, dark bands, cupping, or flare artefacts [26,27,28], have been largely removed through the use of an aluminium filter with a thickness of 0.1 mm and Scanco’s in-built beam hardening correction. This may lead to grey values of a quality “close to that” of synchrotron CT. Actually, the ability of polychromatic systems to deliver results with a significant quantitative meaning has been explicitly shown in the context of bone density measurements, with errors not exceeding 10% [29,30,31]. Interestingly, this is fully consistent with the results of Table 5. Hence, it appears that low-cost X-ray microtomography systems, which undergo continuous improvement [32], do not only allow for precise reconstruction of geometrical features of several voxels size, but also allow for retrieving reliable quantitative information inside the voxels. The latter is valuable for a deeper multiscale analysis, which may, for example, focus on mechanical properties [33, 34].

Figure 12
figure 12

Voxel-related peak microporosities as a function of average photon energy according to Eq. (10): physically admissible solution at \(\epsilon =38\,\hbox {keV}\)

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Figure 13
figure 13

Distribution of microporosity \(\phi _{{\mathrm{micro}}}\) across different planes cut through the scanned sample without a pore-forming agent fired at \(880\,^{\circ }\hbox {C}\)

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Figure 14
figure 14

Distribution of microporosity \(\phi _{{\mathrm{micro}}}\) across different planes cut through the scanned sample processed with 20 (mass %) paper sludge fired at \(880\,^{\circ }\hbox {C}\)

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Table 4 Peak microporosities \(\phi ^{{\mathrm{peak}}}\) in all investigated samples
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Table 5 Comparison of micro-CT-derived porosities, with those obtained from mercury intrusion and weighing
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Conclusion

Apart from the characteristics of the raw clay, the thermal and mechanical properties of bricks and clay blocks are governed by porosity, pore sizes, and pore morphologies. In this context, a new, validated, micro-CT evaluation technique reveals the following features:

  • The pore size distribution depends critically upon the used pore-forming agent. EPS and sawdust induce macropores with sizes ranging from many micrometres up to millimetres, which paper sludge does not. Hence, paper sludge-processed ceramic samples only exhibit micropores of many nanometres to a few micrometres in size, such pores also contributing to the total porosity of EPS—and sawdust-processed ceramic samples.

  • The macropores are larger and less numerous in the EPS as compared to the sawdust-processed samples, and the pores are typically elongated in shape and oriented towards the extrusion direction.

  • The voxel-specific microporosity (measured per \(6 \times 6 \times 6\,\upmu \hbox {m}^{3}\) reference volume) depends strongly on the firing temperature and moderately on the sawdust and paper sludge content; which it is not affected by the EPS content.