*3.4. Theoretical Considerations of Thermal Expansion and Thermal Expansion Behavior of the Studied Samples*

The coefficient of thermal expansion (CTE) is a property of a substance indicating the degree of material to be expanded by heating; substances expand at different altitudes. In the temperature ranges, the thermal expansion of objects is proportional to temperature alteration. Thermal expansion is an essential property for detecting the targeted application of materials when a structural part is heated and kept at a constant length [32].

**Figure 5.** SEM micrographs of the investigated samples after heat-treatment at 1000 ◦C for 2 h.

**Figure 6.** SEM micrographs of G60 heat-treated at (**a**) 700 ◦C 3 h and 1000 ◦C 1 h; (**b**) 700 ◦C 1 h and 1000 ◦C 1 h.

Most solid substances expand by heating and contract after cooling. The change in length with temperature for a solid material can be expressed as:

$$(\mathrm{L\_f} - \mathrm{L\_0})/\mathrm{L\_0} = \infty \,\, (\mathrm{T\_f} - \mathrm{T\_0}) \tag{1}$$

$$
\Delta \mathcal{L} / \mathcal{L}\_0 = \mathcal{\alpha}\_1 \Delta \mathcal{T} \tag{2}
$$

$$\alpha\_1 = 1/L(\text{dL}/\text{dT})\tag{3}$$

where L0 and Lf stand for the original and final lengths sample with the temperature change from T0 to Tf, respectively. The parameter α<sup>1</sup> (CTE) has units of reciprocal temperature (K–1) such as μm/m K or 10–6/K.

Heating or cooling has a significant effect on the dimensions of a material, with a considerable volume change. Volume changes can be calculated from:

$$
\Delta \mathbf{V} / \mathbf{V}\_0 = \alpha\_\mathbf{v} \Delta \mathbf{T} \tag{4}
$$

where ΔV and V0 are the volume change and original volume of the sample, respectively, and α<sup>v</sup> symbolizes the volume coefficient of thermal expansion. In many materials, the value of α<sup>v</sup> is anisotropic; that is, it depends on the crystallographic orientation along which it is measured. For materials in which the thermal expansion is isotropic, α<sup>v</sup> is approximately 3α<sup>1</sup> [33].

Thermal expansion leads to a change in the space between particles of a substance, which affects the volume of the substance while negligibly varying its mass (the negligible amount comes from energy–mass equivalence).

Depending on the above considerations, the results of thermal expansion coefficients of the studied samples can be discussed as follows:

From Figure 7 and Table 3, it is clear that all coefficients of thermal expansion (CTE) values decrease from G10 to G80 samples, i.e., by increasing the cordierite percent at the expense of wollastonite content.

Coefficient thermal expansion (CTE) values of the glass–ceramic depend on the nature and quantity of the developed crystalline phases and the residual glassy matrix [34,35]. A comprehensive range of thermal expansion coefficients is controlled by the crystal types and proportions of these phases, which are considered the main principle of producing glass–ceramics with restricted thermal expansion coefficients.

**Figure 7.** Thermal expansion coefficient of the studied glass–ceramics after heat-treatment at 1000 ◦C for 2 h.

Low-expansion glass–ceramics usually contain crystalline phases as well as a vitreous phase. Ion exchange always takes place in the vitreous phase. Schairer and Bown [36] found a solid solution between wollastonite and diopside in the CaO–MgO–SiO2 system with the highest 22% diopside at eutectic. The produced Ca–Mg–silicate phase is a solid solution between diopside and wollastonite. Diopside constituent in this solid solution is about 49.6%. However, Omer et al. [37] mentioned that a wollastonite–diopside solid solution phase with about 66 % of diopside might be created. This Mg-rich wollastonite solid solution is eventually transformed by increasing time or raising the heat treatment temperature to wollastonite and diopside [38].

By increasing Al3+ ions (from G10 to G80), the formation of diopside, anorthite, and cordierite phases at the expense of wollastonite is noticed. Thus the values of the expansion coefficients (α) of glass–ceramic samples are decreased. Consequently, higher Tg temperatures could be expected.

The thermal expansion property of the crystalline solids is relatively different from that of the parent glasses. The thermal expansion coefficient (α) of the glass–ceramics is a function of the thermal expansion coefficients, and elastic properties of all precipitated crystalline phases, residual glass, and the resulting microstructure. An extensive range of thermal expansion coefficients is covered by the different crystal types [39].

Wollastonite has an <sup>α</sup>-value of 94 × <sup>10</sup>−<sup>7</sup> ◦C−<sup>1</sup> [40], but anorthite has <sup>α</sup>-values of <sup>51</sup>−<sup>64</sup> × <sup>10</sup>−<sup>7</sup> ◦C−<sup>1</sup> [41].

Cordierite is well known for its very low thermal expansion coefficient of about 2.5 × <sup>10</sup><sup>−</sup>6/◦C [42]. However, the CTE of cordierite may depend on the nature and amount of phases that coexist with it in the glass–ceramic sample [43]. The above α-values give a good explanation of the noticeable decrease in thermal expansion of the studied samples (Table 3 and Figure 7).

The thermal expansion coefficient (CTE) of the diopside is registered to be 83.0 × <sup>10</sup>−<sup>7</sup> ◦C−<sup>1</sup> [44,45].

The co-existence of diopside, which has a high thermal expansion coefficient, with anorthite, which has a comparatively lower value of α, leads to a lowering of the thermal expansion coefficient of the final glass–ceramic product (G50–G70).

In samples (G50–G80), it was observed that the presence of both CaO and Al2O3 containing crystalline phases leads to a decrease in the thermal expansion of the glass– ceramic samples. Al2O3 is mainly efficient in this concept. The α-values of samples G50–G80 (with CaO and Al2O3 containing phases) were lower than those of samples G10– G40, free from Al2O3 containing phases and composed of crystalline phases with high CTE values, which are β-wollastonite, para-wollastonite and diopside.

It is worth mentioning that β-wollastonite (CaSiO3) contains triclinic wollastonite and monoclinic parawollastonite. These forms are not easily differentiated except for single-crystal X-ray investigation [46]. The stabilities of β- and para-wollastonite phases are likely to be very similar; because of the presence and intergrowth of both forms [47].
