INTRODUCTION

Face-centered cubic (fcc) titanium carbide TiCx with the NaCl structure (δ-phase) has a wide homogeneity range, considerable hardness, high melting point, and significant corrosion resistance in aggressive media. Owing to its unique properties, titanium carbide TiCx has been widely used in various areas of technology, industry, and medicine both individually, and as an additive to different structural alloys already for more than a century [1–6]. Strongly nonstoichiometric fcc titanium carbide TiCx (δ-phase) at the lower boundary of the homogeneity range has recently found extensive use in medicine thanks to the resistance to biological media, the absence of toxicity and carcinogenicity, the fatigue resistance, the possibility to reach the elastic modulus close to that of bone tissue, and low cost [7]. Nonstoichiometric fcc titanium carbide TiCx is capable of long-term coexistence (biocompatibility) with the living organism; therefore, it is used in medicine to produce implants to treat various injuries [7]. The crystal structure, phase transformations, and physicochemical properties of titanium carbide TiCx under various external actions are currently intensely studied [4, 8–13]. However, some issues remain open. For example, data on the lower boundary of the homogeneity range of the fcc δ-phase of titanium carbide TiCx vary widely and are in a carbon content range of x = 0.33 to 0.52 [14–18]. It was not determined how structural vacancies affect specific features of structural phase transformations at the lower boundary of the homogeneity range of the δ‑phase of titanium carbide TiCx.

These issues can be resolved by neutron diffraction at the lower boundary of the homogeneity range of titanium carbide. Owing to good resolution and high intensity of diffraction maxima, X-ray diffraction is often used for phase analysis and determination of unit cell parameters [12, 13, 19]. In this work, along with X‑ray diffraction, neutron diffraction was also used because of a number of advantages in localizing light elements (nonmetals) against the background of heavy elements (metals) in the crystal lattices of powder samples [20].

The purpose of this work was to determine the lower boundary of the homogeneity range and study the effect of structural vacancies on specific features of structural phase transformations at the lower boundary of the homogeneity range of the fcc δ-phase of TiCx by neutron diffraction.

EXPERIMENTAL

Titanium carbide samples at the lower boundary of the homogeneity range of TiCx were prepared by powder metallurgy (sintering) [21] from a mixture of a PTEM-grade Ti powder (99.76 wt %) and finely divided special-purity grade carbon powder. To obtain samples of a certain composition, the powders of the components were taken in the corresponding proportions by weighing them on an analytical balance. The mixture was thoroughly stirred in an agate mortar for 4 h. The average grain size in the Ti powder was 30 μm. From the thoroughly mixed mixture of titanium and carbon particles, cylindrical samples were compacted under the pressure 10 MPa. Then the samples were annealed in an SShVL-0.6.2 high-temperature vacuum furnace in the temperature range 1375–1475 K in a vacuum of no more than 1.33 × 10–3 Pa for 24 h with sharp turn-off of heating and subsequent cooling of the samples with the furnace. The total carbon content in the end product was found by chemical analysis at the Institute of General and Inorganic Chemistry, Academy Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan. The carbon content was determined by burning a weighed sample in an oxygen flow with subsequent weighing of the CO2 formed in the combustion. The accuracy of the chemical analysis was 0.3%. The neutron diffraction patterns of all the samples did not show even a very weak diffraction reflection of a free carbon crystal despite a higher absolute value of the amplitude of coherent scattering of neutrons from carbon nuclei (bC = 6.65 × 10–12 cm [20]).

The samples were heat-treated in sealed evacuated quartz ampules in a SNOL-type furnace with subsequent quenching into water to freeze the phase composition corresponding to a certain temperature. The error of determining temperature was ±3°C. The neutron diffraction patterns were recorded at room temperature with a neutron diffractometer installed on the thermal column of the VVR-SM nuclear reactor at the at the Institute of Nuclear Physics, Academy Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan (λ = 1.085 Å) [22]. A test sample was placed in a rotating cylindrical vanadium cartridge with a wall thickness of 0.1 mm, a diameter of 6 mm, and a height of 60 mm. Vanadium for the cartridge was because of a low amplitude of coherent scattering of neutrons (bV = –0.38 × 10–12 cm [20]); therefore, the neutron diffraction patterns hardly contained diffraction maxima of the material of the cartridge. The neutron diffraction patterns were processed by the Rietveld method using the FullProf-2013 software with the pseudo-Voigt profile function [23]. In processing the neutron diffraction patterns, the following parameters were refined: the unit cell parameter, the position occupied by atoms, the coordinates of atoms, the carbon atom content, and the effective thermal factor and the uncertainty factors Ri of crystal structure determination. The X-ray powder diffraction patterns were recorded with a DRON-M (CuKα radiation, angle range 2θ = 10°–110°, detector rotation speed 1 deg/min, Ni filter, and λav = 1.5418 Å). To determine the lattice parameters, the recording was carried out within a narrow angle range of 2θ > 101° near the diffraction maximum at a detector rotation speed of 0.25 deg/min.

The homogeneity of the composition of the phases was estimated from the splitting of the α1–α2 doublet of CuKα radiation at large angles (2θ > 70°). Figure 1 presents the X-ray powder diffraction of a TiC0.47 sample. This pattern is indexed in the space group Fm\(\bar {3}\)m (δ-phase). The splitting of the α1–α2 doublet at 2θ > 60° is indicative of the homogeneity of the composition of the δ-phase [24].

Fig. 1.
figure 1

X-ray powder diffraction pattern of a TiC0.47 sample after quenching from 1475 K. The numbers at the diffraction maxima are the Miller indices in the space group Fm\(\bar {3}\)m.

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Table 1 presents the chemical and phase compositions of the obtained samples after quenching from 1475 K.

Table 1.   Chemical and phase compositions of the studied samples after abrupt cooling from 1475 K and stepwise annealing
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RESULTS AND DISCUSSION

The X-ray powder and neutron diffraction studies showed that the titanium carbide samples obtained by quenching from 1475 K, except the TiC0.28 sample, are single-phase and have an fcc structure (of the δ-phase). The TiC0.28 sample is two-phase and consists of the δ‑phase and α-Ti. Thus, the quenching produced the δ-phases of titanium carbide TiCx in the single-phase state at x ≥ 0.33 and in the two-phase state (α-Ti + δ‑phase) at x < 0.33. Figure 2 presents the neutron diffraction patterns of the TiC0.47 and TiC0.28 samples. The neutron diffraction patterns of the single-phase samples TiC0.42 and TiC0.33 are similar to that of the TiC0.47 sample (Fig. 2, curve a). The processing of the neutron diffraction patterns of all the single-phase samples TiC0.47–TiC0.33 by the Rietveld method demonstrated that their crystal structure is described by the space group Fm..m, in which the titanium atoms are in positions 4b, and the carbon atoms are randomly distributed among octahedral interstitial sites 4a. Figure 3 shows the experimental, calculated, and difference neutron diffraction patterns of titanium carbide TiC0.42 quenched from 1475 K. Table 2 presents the experimentally observed and calculated integral intensities of the diffraction maxima of this sample in the space group Fm\(\bar {3}\)m. The absolute intensity of the deformation peaks reflected from the planes with the even Miller indices is so low in comparison with the intensity of the peaks reflected from the planes with the odd Miller indices (e.g., I(002)/I(111) = 0.000) that they are hardly seen in the neutron diffraction pattern. These results agree with the published data [18] that, after quenching from 1475 K, titanium carbide TiС0.33 is the δ-phase with a similar distribution of Ti and C in the lattice. Note that the phase composition of the TiC0.28 sample after quenching does not correspond to the temperature 1475 K because, according to the equilibrium phase diagram of the Ti–C system [1–3, 17], at this temperature, TiC0.28 should have the phase composition β-Ti + δ‑phase. This disagreement is explained by the allotropic transformation of titanium β-Ti \(\overset {1160\,{\text{K}}} \longleftrightarrow \) α‑Ti [25] during the cooling the sample from the temperature 1470 K.

Fig. 2.
figure 2

Neutron diffraction patterns of the (a) TiC0.47 and (b) TiC0.28 samples after quenching from 1475 K. The numbers at the diffraction maxima are the Miller indices of the reflecting planes corresponding to the δ-phase and α-Ti. λ/2 is the second-order reflection from the (111) plane, which is 1.5% of the intensity of the main maximum.

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Fig. 3.
figure 3

Experimental (points) and calculated (space group Fm\(\bar {3}\)m, solid line) neutron diffraction patterns of titanium carbide TiC0.42 quenched from 1470 K. The numbers at the diffraction maxima are the Miller indices hkl in the space group Fm\(\bar {3}\)m.

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Table 2. Experimentally observed and calculated integral absolute intensities (counts) of the diffraction maxima in the space group Fm\(\bar {3}\)m in the neutron diffraction pattern of titanium carbide TiC0.42 quenched from 1475 K
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It was of interest to estimate the minimum amount of titanium that is detected in the neutron diffraction pattern. For this purpose, a cylindrical vanadium cartridge 6 × 60 mm in size was filled with 6 g of a single-phase titanium carbide powder (the δ-phase of TiC0.33), to which a pure PTEM-grade α-Ti powder was sequentially added to a content of 0.25 to 3 wt %, with a neutron diffraction pattern being recorded each time. In these experiments, beginning with 2 wt % Ti, the neutron diffraction pattern at a Bragg angle of 2θ = 27°52′ shows a trace of a diffraction maximum with the Miller indices (101) from the hexagonal lattice of α-Ti, which, at 3 wt % α-Ti, becomes a definite diffraction maximum. It can be concluded that the lower limit of detection of α-Ti is less than 3 wt %. Consequently, the lower boundary of the homogeneity range of the fcc δ-phase of TiCx quenched from 1475 K is at the composition x = 0.33 with a limit of detection of pure α-Ti of less than 3 wt %.

The obtained result and its comparison with the published data [1] suggest that, if the production of stoichiometric or near-stoichiometric fcc titanium carbide TiCx by sintering requires a temperature of T ≥ 1775 K, then a temperature of 1475 K is sufficient to synthesize the fcc δ-phase of TiCx at the lower boundary of the homogeneity range.

The question arises of whether or not the high-temperature fcc δ-phase of TiCx is stable at the lower boundary of the homogeneity range in the thermodynamically equilibrium state. To answer this question, the TiC0.33–TiC0.47 samples were subjected to homogenizing stepwise annealing at 1270 + 1170 + 1070 + 970 + 870 + 770 K for a total of 144 h, for 24 h at each temperature. The indexing of the neutron diffraction patterns showed that such an equilibrium annealing leads to the decomposition of the δ-phase of the TiC0.33–TiC0.47 samples into pure α-Ti (space group Р63/mmc) with the unit cell parameters a = 2.950 Å and c = 4.679 Å [25] and an ordered cubic phase δ'‑Ti2C2x' in the space group Fd\(\bar {3}\)m with the unit cell parameter a ≈ 2a0, where a0 is the unit cell parameter of the initial δ-phase, x' > x (Fig. 4; Table 1). Hence, the single disordered cubic δ-phase (space group Fm\(\bar {3}\)m) of TiC0.33–TiC0.47 titanium carbide is a high-temperature phase, and at room temperature after quenching from 1475 K, this phase is in the metastable state. At the same time, one should note that the annealing of the metastable δ-phases of the TiC0.33–TiC0.47 samples at 875 → 725 → 675 → 625 K for 60 h (for 15 h at each temperature) does not lead to their decomposition. Consequently, the metastable δ-phase in the composition range TiC0.33–TiC0.47 is stable at T ≤ 800 K and can be used for practical applications at these temperatures. The phase compositions of the samples after the equilibrium annealing are also presented in Table 1. As the structural neutron diffraction analysis of the neutron diffraction pattern of the TiC0.33 sample showed, after the equilibrium annealing and after the subtraction of the diffraction reflections from α-Ti (Fig. 5), the crystal structure of the ordered fcc phase (space group Fd\(\overline 3 \)m) is described by the structural formula δ′-Ti2C0.76 (chemical composition TiC0.38). Table 3 presents the structural characteristics of the phase δ′-Ti2C0.76. According to the neutron diffraction patterns, after the equilibrium annealing, with decreasing total carbon content in the initial δ‑phase of TiCx, the amount of the formed α-Ti phase increases (Fig. 3). As judged from the results of processing the neutron diffraction patterns of the samples after the equilibrium annealing, the carbon content in the formed ordered δ'-phase decreases with decreasing total carbon content in the initial δ-phase of TiCx (Table 1). According to the equilibrium phase diagram of the Ti–C system [17], this corresponds to the fact that, with decreasing temperature and increasing total carbon content in the sample, the carbon content in the δ'-phase increases, and the amount of α-Ti decreases. This suggests that, with decreasing (varying) temperature in the annealing as 1270 + 1170 + 1070 + 970 + 870 + 770 K for 144 h, for 24 h at each temperature, the equilibrium weight ratio between α-Ti and the δ'-phase is established at different temperatures. As the structural neutron diffraction analysis of the neutron diffraction patterns of the TiC0.33–TiC0.47 samples showed, after the equilibrium annealing and after the subtraction of the diffraction reflections from α-Ti (Fig. 5), the crystal structure of the ordered fcc δ'-phase (space group Fd\(\bar {3}\)m) is described by the structural formula δ'-Ti2C2x' (x' > x) because the separation of α-Ti leads to an increase in the carbon content in the carbide phase. In the neutron diffraction patterns of the δ'-phases of the samples (Fig. 4), of interest is the increase in the intensity and the decrease in the half-width of the superstructure reflection (111) from the δ'-phase with decreasing carbon content in the samples. This suggests that the δ'-phases being in equilibrium with α-Ti have different compositions, different degrees of long-range order, and different sizes of antiphase domains. The degree of long-range order of the δ'-phases was found from the formula [26]

$$\eta = {\text{ }}{\frac{{p - x{\kern 1pt} '}}{{{\text{1}} - \nu }}}.$$
(1)
Fig. 4.
figure 4

Neutron diffraction patterns of (a) TiC0.47, (b) TiC0.42, (c) TiC0.33, and (d) TiC0.28 after stepwise equilibrium annealing at 1375 + 1275 + 1175 + 975 + 875 + 775 K (for 24 h at each temperature).

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Fig. 5.
figure 5

Neutron diffraction pattern of titanium carbide TiC0.38 after the subtraction of the diffraction reflections from pure α-Ti (δ'-Ti2С0.76 phase) after stepwise annealing at 1070 + 970 + 870 + 770 K (for 24 h at each temperature).

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Table 3.   Structural characteristics of the ordered cubic δ′-Ti2C0.76 phase of titanium carbide TiC0.38 in the space group Fd\(\bar {3}\)m
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Here, x' is the carbon content in the atomic ratio in the δ'-ratio; ν is the relative fraction of sites of sublattice 16c: ν = N1/(N1 + N2), where N1 and N2 are the numbers (multiplicities) of sites of sublattices 16c and 16d, respectively; and p = n/N1 is the degree of occupation of octahedral interstitial sites 16c with n atoms in these octahedral interstitial sites. Figure 6 presents the degrees of long-range order of the δ'-phases that are observed after the equilibrium annealing and the expected maximum possible values. As Fig. 6 shows, at high carbon content in the stable δ'-phase, the experimentally observed degree of long-range order is much lower than the expected maximum value. This difference decreases with decreasing carbon content in the δ'-phase and vanishes at x = 0.33. Probably, the more carbon vacancies in octahedral interstitial sites of titanium carbide TiCx, the higher the diffusion mobility of carbon atoms. This leads to the fact that, the lower the carbon content (the more carbon vacancies) in TiCx, the more rapidly the ordered state with the maximum degree of long-range order is established in annealing.

Fig. 6.
figure 6

Dependences of the degree of long-range order on the carbon content in the δ'-phase: (1) the degree of long-range order that is determined from the neutron diffraction pattern and (2) the expected maximum degree of long-range order.

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The sizes of antiphase domains in the δ'-phases were also determined from the half-width at half of the superstructural maximum (111) by the Scherrer formula [27]

$$D = {\text{ }}0.94{{\lambda /}}b{\kern 1pt} {\kern 1pt} {\text{cos}}{\kern 1pt} {\kern 1pt} {{\theta }},$$
(2)

where λ is the neutron wavelength; β is the half-width at half of the superstructural maximum (111) after subtracting the instrumental broadening, rad; and θ is the Bragg angle.

β was determined from the formula [27]

$$\beta = \frac{1}{2}\left( {B - b + \sqrt {B(B - b)} } \right),$$
(3)

where B is the half-width at half of the superstructural maximum; and b is the instrumental broadening, which is found by extrapolation of the half-widths of the structural reflections of the annealed samples to the angular position of the superstructural maximum (111). The instrumental broadening was b = 8.5 × 10–3 rad. The results of determining the sizes of antiphase domains at various carbon contents in the δ'-phase are illustrated in Fig. 7, which shows that, with decreasing carbon content in the δ'-phase, the size of antiphase domains nonlinearly increases. This is likely to be related to the increase in the degree of long-range order with decreasing carbon content in the δ'-phase. The determined temperature dependence of the lower boundary of the homogeneity range of the equilibrium ordered fcc δ'-Ti2C2x phase is similar to that of the composition of the lower boundary of the δ'-phase in the equilibrium phase diagram of the Ti–C system, which was constructed based on the results of calculations by the order parameter functional method [17].

Fig. 7.
figure 7

Dependences of (1) the half-width at half of the superstructural maximum (111) in the neutron diffraction pattern of the δ'-phase and (2) the size of antiphase domains on the carbon content in the δ'-phase.

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Note that, in the studied concentration range of the nonstoichiometric fcc δ-TiСx phase (x = 0.33–0.42), in equilibrium stepwise annealing, this phase undergoes a decomposition into the ordered δ'-Ti2С2x phase (space group Fd\(\bar {3}\)m) and α-Ti (space group Р63/mmc), which is accompanied by phase separation in a cylindrical sample. A thin pure α-Ti film with a metallic luster forms on the lateral surface of the cylindrical sample. On the bases of the cylindrical samples, no formation of pure titanium was observed. The formation of the δ-phase of TiCx with a carbon content of x = 0.33–0.47 and the decomposition after the stepwise annealing can be represented by the following scheme

$$\begin{gathered} {\text{Ti}} + x{\text{C}} \\ \xrightarrow{{1475\,{\text{K,24}}\,{\text{h}}}}\delta {\text{-phase}}\,\,{\text{Ti}}{{{\text{C}}}_{x}} \\ \xrightarrow{{1375\,{\text{K,}}\,\,{\text{every}}\,\,{\text{100}}\,{\text{K,}}\,\,{\text{24}}\,{\text{h}}}}\delta {\kern 1pt} '{\text{-phase}}\,\,{\text{T}}{{{\text{i}}}_{{\text{2}}}}{{{\text{C}}}_{{{\text{2}}x{\kern 1pt} '}}} + \alpha {\text{-Ti}}, \\ \end{gathered} $$

where x' > x. The thickness of the Ti layer on the surface of a cylindrical sample depends on the total amount of the sample, and at an amount of the TiC0.33 sample of 10 g, it is 200 μm. The observed phenomenon of phase separation is well known in metallurgy as zonal segregation [28].

CONCLUSIONS

A number of samples of titanium carbide TiCx were obtained with the compositions at the lower boundary of the homogeneity range (TiC0.33–TiC0.47) by powder metallurgy at a temperature of 1475 K, which is much lower than the temperature of the formation of stoichiometric fcc titanium carbide (1775 K).

A neutron diffraction study was made of phase transformations at the lower boundary of the homogeneity range in cubic titanium carbide TiCx. It was shown that quenching from the temperature 1475 K produces a metastable disordered fcc δ-phase of titanium carbide in the composition range TiC0.33–TiC0.47. It was determined that the crystal structure of the high-temperature metastable δ-phase in the composition interval TiC0.33–TiC0.47 is stable at temperatures T ≤ 800 K, which allows one to use it for practical applications at these temperatures. The lower boundary of the homogeneity range of the single stable ordered cubic δ′-phase is at the composition TiC0.49 ± 0.02 (structural formula δ'-Ti2C0.98). Below this composition, the stable ordered δ'-phase of the Τi2C0.88, Τi2C0.76, and Τi2C0.66 samples was observed in equilibrium with pure α-Ti. Consequently, the lower boundary of the homogeneity range of the stable ordered δ′-phase is at the composition Τi2C0.66.

It was found that the equilibrium stepwise annealing at the lower boundary of the homogeneity range of the disordered fcc δ-phase of titanium carbide TiCx in the composition range x = 0.28–0.47 at 1270 + 1170 + 1070 + 970 + 870 + 770 K (for 24 h at each temperature) leads to the decomposition into the ordered fcc δ′-phase with the structural formula δ'-Τi2C2x′, where x' > x, and pure α-Ti. On the lateral surface of the cylindrical sample, a pure α-Ti film forms.

For the first time, the structural characteristics of the ordered δ'-phase were observed and studied at the compositions δ'-Τi2C0.98, δ'-Τi2C0.88, δ'-Τi2C0.76, and δ'-Τi2C0.66. In this phase, the degree of long-range order and the size of antiphase domains in the equilibrium state increase with increasing deviation of the composition from the stoichiometric composition Τi2C.