3.3. Phase Chemistry
The phase compositions are shown as mass% in
Table 3 and as phase formulas in
Table 4. The latter are shown as the average structural formulas normalized to three cations and four oxygen atoms, together with their standard deviations. This normalization method was used to calculate the Fe
2+ and Fe
3+ contents, similar to the method used by Quintiliani et al. [
22] and Ferracutti et al. [
23]. An example of the method is given as
Table S2 in the Supplementary Information. The periclase compositions were normalized to one cation. From the formula calculations, the Fe in the spinels appears to be predominantly trivalent.
From the analyzed spinel compositions, it is apparent that two major spinel types are present in all of the CMD samples (apart from the fine-grained exsolved spinel present in sample CMD0). In
Table 3 and
Table 4, the spinel compositions labeled as spinel A represent the compositions of the coarse relict original chrome spinel. The spinels labeled as spinel B are the spinels that reacted with the magnesia and TiO
2 to form new spinels with increased Ti contents. The fine-grained inclusions in the periclase show higher iron contents than those of the coarser spinel B inclusions and are labeled as spinel C. The compositions of spinels B and C (in sample CMD0) were grouped together to reconcile them with the XRD analyses (which show only two distinct spinels) and with the ICP analyses.
In all samples, the Mg occupancy in the relict chrome spinel A is unity (based on three cations) within its standard deviation (see
Table 4). This means that the tetrahedral sites are fully occupied with Mg, and that the octahedral sites must be exclusively occupied by the trivalent cations due to the octahedral site preference of Cr and Al.
The Mg contents in the secondary B spinels increase concurrently with the increase in Ti content. This is apparent as the Mg occupancies that are in excess of 1.0 are equal to their Ti contents (within the errors of analysis). For example, spinel B in sample CMD3 has an Mg content of 1.27 atoms per formula unit and a Ti content of 0.22 atoms per formula unit. This means that the Mg
2+ is in excess of 1.0 resides in the octahedral site and is balanced by an equal amount of octahedral Ti
4+ to have an effective valency of (Mg + Ti)
3+, resulting in a spinel that is charge-balanced. This holds for all newly formed spinel B grains that were analyzed. The difference in Ti content of spinels A and B in sample CMD3 is apparent in their EDS spectra shown in
Figure S1 in the Supplementary Information.
The average Ti occupancy in the secondary spinels (B) gradually increases from 0.011 to 0.365 (based on three cations) as the TiO
2 addition increases from zero to 7 percent. The Ti occupancy of the relict spinels varies randomly between 0.004 and 0.057. This clearly indicates that the relict spinels react with the TiO
2 to a limited extent. This is shown in
Figure 2. Likewise, the periclase compositions do not change significantly as TiO
2 is added to the refractory mix.
3.4. Mass and Volume Percentages of the Phases
The variation in phase contents and unit cell volumes are given in
Table 5. Using XRD, only two spinel types show clearly resolved peaks, and they were quantified as spinels A and B (see
Figure 3). For each of the samples, the average elemental occupancies given in
Table 4 were used in the Rietveld refinement. The preferred orientation of the periclase affected the accuracy of the phase quantities but the values given in
Table 5 are the averages of four different XRD determinations (two different samples mounted and measured twice). This enabled the calculation of accurate densities as well as the volume percentages of the phases in the different samples.
The Rietveld results of CMD0, CMD1, and CMD3 are shown in
Figure 3 together with the peak allocations of spinel A, spinel B, and periclase. The correspondence between the calculated and experimental diffraction data is given in the difference curve (bottom curve). The small difference in the unit cells of spinels A and B in CMD1 is apparent in their almost coincident peaks at 75 degrees 2-theta.
A mass balance calculation was done to compare the compositions calculated from the XRD phase quantification (
Table 5) and the average phase compositions in
Table 3 with that of the ICP analysis given in
Table 1. The comparison is given in
Table S3 in the Supplementary Information. Obviously, this is dependent on the accuracy of the XRD analysis (2% absolute) and whether the phase compositions are representative, based on limited sampling. The TiO
2 and Fe
2O
3 compositions calculated from XRD and EDS analysis result in overdetermination in the samples, whereas the MgO and Cr
2O
3 values are comparable.
The unit cell volumes of spinels A and B are the most similar in CMD1 (
Table 5). The cell volumes of spinel A are similar in samples CMD1 to CMD7, whereas those of spinel B gradually increase with higher substitutions of TiO
2. The differences in unit cell dimensions between spinels A and B in CMD0 to CMD7 are 0.0785 Å, 0.0328 Å, 0.0605 Å, 0.0663 Å, and 0.0949 Å, respectively.
Because the volume percentages of the phases do not differ markedly from the mass percentages, and to illustrate the volume changes in the samples, the volume fractions were also calculated. These relate to the volumes per 100 g sample and were calculated by dividing the mass percentages of each phase by their respective densities. The latter were calculated directly from the average formulas and the refined lattice parameters, and will differ from sample to sample for each phase. This is shown in
Table 6, and the respective mass percentages, volume fractions of the individual phases, and total volume changes are shown in
Figure 4b.
3.5. Bulk Density and Porosity
The theoretical bulk densities
of the samples were calculated using their mass percentages and respective densities:
where
is the mass fraction of the phase and
is its density.
The bulk density and apparent porosity values of five specimens from each composition were measured using the standard water immersion method, and average values were taken. The theoretical bulk densities were compared with the measured bulk densities, and from these the porosities were calculated using the following formula,
where
is the measured bulk density and
is the theoretical density.
The calculated porosities were then compared to the measured apparent porosities. These comparisons are shown in
Table 7. The calculated porosities agree well with the measured ones, with CMD0 and CMD1 slightly underdetermined.
The theoretical bulk density of CMD1 is the minimum compared to the other compositions, but not significantly different from the other samples.
3.6. Thermal Expansion
Theoretical composite linear thermal expansion coefficients were calculated using the formula given by Turner [
24], which was used by Dealmeida [
25] for ceramic materials and for complex heterogeneous materials, as summarized by Karch [
26]:
where
is the coefficient of linear thermal expansion of phase
i, is the bulk modulus of phase
i, is the mass fraction of phase
i, and
is the density of phase
i.The standard formula for the calculation of
was used:
where
is the lattice parameter of phase
i at 298 K.
Because both monticellite and forsterite are minor components, only the linear expansions of their intermediate crystallographic axes were measured and used in the calculation of the bulk expansion coefficients.
The bulk moduli of the spinels were calculated using the formula given by Zhongying et al. [
27] that relates the bulk moduli of the spinel end members to the bulk modulus of the solid solution:
where
Kss is the bulk modulus of the solid solution and
Ki is the bulk modulus of the spinel end members.
The spinel end members of the spinels in samples CMD0 to CMD7 were calculated using the end member generator (EMG) (Ferracutti et al. [
23]) and the oxide analyses as determined by EDS analysis. These are given in
Table 8, together with their calculated bulk moduli. The bulk moduli of the end members of MgAl
2O
4, MgFe
2O
4, and MgCr
2O
4 were taken from Zhongying et al. [
27] and that of Mg
2TiO
4 from Mingda et al. [
28]. For periclase and forsterite, the bulk moduli were taken from Shanker et al. [
29], and for monticellite, from Sharp et al. [
30]. The densities of the phases were determined in this study.
The individual linear thermal expansion coefficients were determined from high-temperature powder XRD measurements of the lattice parameters of the individual phases as determined by Rietveld analysis. The temperatures of the experiments were calibrated using pure MgO periclase as an external standard and calibration every 100 degrees, using the thermal expansion data of Touloukian et al. [
21].
The lattice parameter data of the refractory phases were fitted to a second-order polynomial and the linear thermal expansion coefficients determined by differentiating the polynomial at two temperatures, 1392 K and 1840 K.
The individual and composite linear thermal expansion coefficients are tabulated in
Table 9 for the two temperatures mentioned. Because of the predominance of the spinel and periclase phases, these phases, and especially the spinel B phase, affect the bulk or composite thermal expansion coefficients the most. The expansion coefficients of the minor phases monticellite and forsterite are not very well determined because of the errors in lattice constant refinements. The linear expansion coefficients for all the phases are shown in
Table 9 and
Figure 5, and for the composite samples (
αcomp) at two temperatures, in
Table 9 and
Figure 6. The actual thermal expansion data as well as their second-order polynomial fits are given as
Figure S3 in the Supplementary Information. The reason for calculating
αcomp at 1392 K and 1840 K was to calculate it at higher temperatures than 1273 K (1000 °C), which is a low temperature for refractory usage.
In
Figure 5, the linear expansion coefficients of spinel B are markedly different from the others, because it becomes negative in both CMD0 and CMD7 and levels off in the other samples. The reason for the decreases in thermal expansion coefficients of spinel B is most likely that the inclusions of spinel B in periclase are affected by the one order of magnitude larger thermal expansion coefficient of periclase. This results in compression of the spinel B inclusions.
The linear thermal expansion coefficient of spinel B also has the highest value in sample CMD1 and is closest to the band defined by the other phases. This implies that the stress that develops in the refractory when the temperature increases will be the lowest for sample CMD1, due to the smallest differences in linear expansion coefficients.
The composite linear expansion coefficients for the samples are shown in
Figure 6. The 1392 K calculated values are similar to the values measured from 293 to 1273 K using the BS 1902-5.3:1900 method. Detailed differences are probably because porosity was not taken into account in the calculations.
To determine the stresses resulting from the difference in thermal expansion, the formula derived by Turner [
22] was used:
where
and
are the volume expansion coefficients (
β = 3
α for cubic compounds) of the total sample and of the individual phases, respectively;
is the temperature interval; and
is the bulk modulus of phase
i. The results are shown in
Figure 7. The stress calculation was done for only the two spinels as well as for periclase in the temperature interval 1392–1840 K. This was because of the predominance of these phases in the samples and the fact that the volume expansion coefficients of monticellite and forsterite are poorly defined.