*2.1. Synthesis*

The magnesium titanates were prepared by the sol-gel method using metalorganic precursors: diethyl ethoxymagnesiomalonate, prepared by metallation of diethyl malonate (1), and titanium(IV) tetra-tert-butoxide (Aldrich), dissolved in anhydrous 2-propanol. The hydrolysis step was carried out at nearly room temperature, using a stream of hot air (about 100 ◦C) containing water vapor (superheated steam), during about 3 hours, to ensure total hydrolysis. The precipitated solid was filtered, washed with 2-propanol and left to dry in air for several days, until a constant weight was obtained. After analysis of the Mg and Ti content, this solid served as starting material for thermal treatment.

The powders were initially fired at 600 ◦C for 3 h. After the initial treatment, the sample with Mg:Ti 1.1:1 found as a single geikielite phase, where a = 5.0537 (3) Å; c = 13.897 (5) Å; atomic positions: z(Mg) = 0.3563 (2); z(Ti) = 0.1445 (2); x(O) = 0.315(1); y(O) = 0.0225(1); z(O) = 0.247(1). However, the expected Mg:Ti ratio was confirmed by inductively coupled plasma (ICP). The amount of impurities was below 10 ppm. Additional treatments were made at 1200 ◦C for 5 h in order to obtain well-crystallized powders. After the 1200 ◦C treatment, the sample was found with 95% geikielite and 5% qandilite. It was designated as "GQ" (major phase geikielite and minor phase qandilite) and will be referred to as such throughout the manuscript. The qandilite within GQ served as the internal standard.

#### *2.2. RT XRD Measurements*

The samples for HT-XRD studies were characterized by a Rigaku powder X-Ray Diffractometer. Data were collected in the conventional Bragg–Brentano configuration (theta/2theta) by means of Cu Kα radiation at 40 kV and 30 mA. The <sup>K</sup>β was filtered out by graphite monochromator attached to the

detector. Phase characterization from XRD data was made by using public domain FullProf/WinPlotter software [9]).

#### *2.3. HT XRD Measurements*

X-ray diffraction was performed on a Bruker D8 Advance in Bragg–Brentano geometry using an X-ray source with a Cu anode having a Kα<sup>1</sup> emission wavelength of 1.5406 angstroms. Samples were placed in an Anton Paar XRK-900 high-temperature reaction chamber using a Macor sample stage. The influence of the thermal expansion of the stage was measured by calibrating the stage height as a function of temperature using an alignment slit. The temperature of the sample was controlled using and Anton Paar TCU 750 controller by mounting a K-type thermocouple in the sample holder adjacent to the sample. A linear PSD detector (LYNXEYE XE-T) was used with an opening of 2.94 degrees. The diffraction pattern was recorded from two-theta of 10 degrees until 120 degrees using a coupled theta/two-theta scan type. Data points were acquired in increments of 0.02 degrees with an acquisition time of 0.25 s. Lattice parameters were fitted using TOPAS software with a TCHZ function. The refined parameters included the lattice parameters, sample displacement and zero error. The line position and effect of instrumental broadening and asymmetry were calibrated by SRM 660c LaB6.

The diffractograms were also analyzed by the program Powder-Cell [10]. In this step, the phases were easily identified, and the unit cells were verified for the thermal expansion. Grain shape and size was assessed by HR-SEM.

## *2.4. Thermal Expansion Methodology*

It is essential to know the dimensions of ceramic materials as function of temperatures in order to calculate the dimensions of objects working at elevated temperatures and to evaluate thermal stresses during temperature changes. For practical reasons it is suggested to define the relative dimension change of a material by Equation (1):

$$\|L(T) - L\_0\|'L\_0 = \mathbf{f}(T - T\_0) \tag{1}$$

where *L*(*T*) is the size of one dimension at temperature *T*, *L*0 is the size of the same dimension at ambient temperature, *T* is the working temperature and *T*0 is the ambient temperature (usually the ambient temperature is 25 ◦C).

In the case of linear thermal expansion, Equation (1) becomes

$$\|L(T) - L\_0\|\_{L\_0} = \alpha (T - T\_0) \tag{2}$$

where α is the linear thermal expansion coefficient. For a general case it is possible to define

$$
\Delta L/L\_0 = \alpha\_1 \,\Delta T + \alpha\_2 \,\Delta T^2 + \dots + \alpha\_n \,\Delta T^n \tag{3}
$$

or

$$L(T) = L\_0(1 + \alpha\_1 \,\Delta T + \alpha\_2 \,\Delta T^2 + \dots + \alpha\_n \,\Delta T^n) \tag{4}$$

where

$$
\Delta L = L(T) - L\_0 \text{ and } \Delta T = T - T\_0
$$

Neglecting the contribution of enhanced vacancy formation at higher temperatures to the size of a sample of matter, the thermal expansion of a periodical crystalline matter can be modeled by measuring its lattice parameters as function of temperature. By using HT diffraction, each lattice parameter, *Aj*, can be obtained directly from the measurement as

$$A^{\bar{j}}(T) = A^{\bar{j}}\_0 + \mathbf{k}\_1 \,\Delta T + \mathbf{k}\_2 \,\Delta T^2 + \dots + \mathbf{k}\_n \,\Delta T^n \tag{5}$$

In this case, the non-linear thermal coe fficients will be given as

$$
\alpha\_i{}^j = \mathbf{k}\_i / A^j{}\_0 \tag{6}
$$

From Equation (3) it is then possible to define an overall thermal expansion coe fficient along a crystal axis *Aj* as

$$
\alpha\_A{}^j = \Delta A^j / (A^j{}\_0 \,\Delta T) = \alpha\_1 + \alpha\_2 \,\Delta T + \dots + \alpha\_n \,\Delta T^{n-1} \tag{7}
$$

Similarly, for unit cell volume *V* we define the volumetric thermal expansion as

$$
\Delta\gamma = \Delta V / (V\_0 \,\Delta T) = \gamma\_1 + \gamma\_2 \,\Delta T + \dots + \gamma\_n \,\Delta T^{n-1} \tag{8}
$$

It should be noted that this methodology is valid only where there is no phase transformation or significant crystal structure change in the range of measurements.
