Thermodynamic, Thermal and Elastic Properties of Titanium Nitride TiN: Comparison of Various Data and Determination of the Most Reliable Values

The analysis of literary data on thermodynamic, thermal and elastic properties of titanium nitride TiN which included values Debye temperature θD, volume coefficient of thermal expansion αV and bulk modulus B under standard conditions is carried out. It has been shown that the known data have a significant spread of values from 20 to 43 %. The 8 most rational variants of optimizing calculations are proposed, which make it possible to reveal the most reliable values of some TiN parameters. At the same time, the minimum and maximum values of θD and αV were used from literary sources, as well as the least contradictory data on isobaric heat capacity Cp, melting temperature Tm.p and density d of TiN. To improve the calculated results, the values of θD(TiN) determined using the methods of Magnus ‒ Lindeman and Debye were also used. The Mayer’s relation was the basic test expression. The obtained values of the bulk modulus were compared with the literature data. This made it possible to distinguish the least and most reliable values of αV and θD, as well as make a refinement correction for the last value. As a result, it was found that under standard conditions, the value of θD(TiN) close to the optimal should be within 746‒769 K, and for its isochoric heat capacity CV ‒ in the range 36.55‒37.19 J/(mol×K). The range of values, after optimization, does not exceed 3 %, unlike the 20 % available in the literature. A more accurate definition of Debye temperature for TiN needs to radically refine the values of its αV and B.

It follows from Table 1 that the differences between the minimum and maximum values of Debye temperature θ D are about 20 %, for volume coefficient of thermal expansion α V exceed 32 %, and for bulk modulus B are within 30-43 %. Such differences in the properties of TiN established by different authors [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] can be explained by using various methods of its synthesis and determination of physicochemical parameters. From this it follows that the properties of titanium nitride are more correctly represented not by specific values, but by rather wide ranges of values. However, this approach has some drawbacks. After all, modern materials science needs high-precision data. At the same time, the maximum deviations in the value of a certain parameter should not exceed one or several percent. Based on this, the task of this work was formulated -to carry out optimizing calculations that would minimize existing contradictions in the properties of TiN and could reveal the most reliable values of its physicochemical parameters.
In this work, C p (TiN)=37.50 J/(mol×К) was taken. This is the average among data [28] and [29] at 298.15 K, which differ only by 2 %. Differences between density d of titanium nitride given in different works [6,30], are within 1 %. The value of d from [30] was used in this paper. Values of molar mass of TiN and its melting point T m.p. also borrowed from [30].

Results
As a result of the analysis, the calculation operations were minimized as much as possible. Their optimized number was reduced to the eight most rational variants, which are grouped in Table 2.
It should also be noted that if to use [2] as the minimum value of θ D (700 K) from the operation, then come to a neg- ative size of α V . In such a case, the TiN should taper upon heating rather than expand. However, this is contrary to all known publications [3][4][5][6][7][8] on the thermal expansion of titanium nitride. Variant II: minimum value of θ D (780 K [3]) and maximum value of α V (28×10 -6 K -1 [4,8]). In this case, the difference C p -C V is also 1.25 J/(mol×K), but the values of B T and B S take sizes 450 and 466 GPa, respectively. As it is possible to see from Table 1, these values are already approaching those presented in the literature [21].

THERMODYNAMIC, THERMAL AND ELASTIC PROPERTIES OF TITANIUM NITRIDE TiN: COMPARISON OF VARIOUS DATA AND DETERMINATION OF THE MOST RELIABLE VALUES
Similarly, calculations were implemented for variants III and IV. At the same time, overestimated values of B T and B S were recorded ( Table 2), which makes these variants unlikely. Let's note that the known data on Debye temperature for TiN [2][3][4][5] does not allow to significantly optimize the range of its most likely values. It can only be stated that a more reliable value θ D should exceed 700 K, but not exceed 780 K. In this regard, let's consider an additional 4 variants that can be reached using the methods of Magnus -Lindeman [22] and Debye [22][23][24]. Consider them below.
Variant V: the value of C V set according to the Magnus -Lindeman method at 298.15 K, and the minimum coefficient α V (19×10 -6 K -1 from [4]). For this case, the difference between C p and C V is 0.34 J/(mol×K). Herewith B T =266 and B S =268 GPa. The values obtained are well consistent with the literature data 257-275 GPa [4,[9][10][11][12][13][14] (Table 1). Calculated size C V (TiN)= =37.16 J/(mol×K) corresponds to Debye temperature equal to 747 K. This indicator can be considered as one of the minimum in determining the optimal interval of the values of θ D (TiN).
The remaining variants VI-VIII have lower accuracy of results ( Table 2).
Thus, the best agreement between the various literature data was achieved with the calculated variants II and V. They were used for subsequent optimization. From variant V, it is possible to reach the following result. If in the expression (5) let's substitute the smallest value of B S equal to 245 GPa [4] ( Table 1), then let's obtain C V (TiN)=37.19 J/(mol×K), which corresponds to θ D (TiN)=746 K. It is this value of Debye temperature that is most rational to consider the lower bound of optimal values θ D (TiN). Of variants II and VIII ( Table 2), the upper bound of the most probable value θ D (-TiN) was determined. By optimizing calculations, let's came to the value C V (TiN)=36.55 J/(mol×K), which corresponds to θ D (TiN)=769 K. If to use it with the value α V = 28×10 -6 K -1 from [4,8], then obtain B T =342 and B S =351 GPa. The latter value is in good agreement with the maximum size of parameter B set in [21].
Let's note that 3 of the 4 considered variants (with numbers I, III, VII) with a minimum value of α V (19×10 -6 K -1 [4]) lead to significantly overestimated values of B (Tables 1, 2). At the same time, when calculating using the maximum value α V (28×10 -6 K -1 [8]), only 1 of the 4 variants (variant of number IV) gives significant deviations from the known quantities of B. From this it follows that the most reliable value of thermal expansion TiN is closer to the size 28×10 -6 K -1 [8] than to 19×10 -6 K -1 [4]. It is also noted that for 6 of the 8 variants considered ( Table 2), the values of B T and B S are obtained, which are closer to the maximum sizes of [21] than to the minimum of [4,9]. It follows that the TiN should have a higher resistance to external pressure than was thought in the works [4,9].

Discussion
The results obtained can be explained by the integrated approach applied in this work. The use in Mayer's relation (4) of most of the known values of θ D and α V leads to both significantly overestimated and in some cases underestimated values of the bulk modulus B. Many calculated values are significantly outside the known range B=245-352 GPa (Tables 1, 2). Thanks to the analysis of literary data [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the use of the methods of Magnus -Lindeman and Debye, it was possible to significantly optimize the values of θ D (TiN) from 700-870 K to 746-769 K. This led to a decrease in the existing scatter of literary data from 20 % to 3 %. The carried-out clarification of sizes θ D (TiN) also allowed to reveal the most probable intervals of values α V (TiN) and B(TiN) which don't contradict Mayer's relation (4). So, for titanium nitride, the most reliable are the values α V ≈(25-28)×10 -6 K -1 and B≈310-350 GPa obtained in works [3,4,[6][7][8] and [6,[19][20][21] respectively. A disadvantage of this work is the impossibility of determining the true θ D (TiN) to one degree. Achieving such high accuracy is possible only after radical refinement of the values α V and B.
The results obtained, together with the previously known 28], serve as an additional confirmation of the practical importance of the TiN as a weakly expandable material when heated that can be used at high external pressures. The established range of optimal values of θ D (TiN) is important for subsequent thermodynamic and thermophysical studies and allows in the future to more accurately determine its thermal and elastic properties.
The approach proposed in this work to identify the most reliable parameters of titanium nitride can be used for many similar compounds. These include nitrides of the composition XN, where X -Boron, Aluminum, Scandium, Vanadium, Yttrium, Zirconium, Niobium, Lanthanum, Hafnium, Tantalum and others. Also in the future, it is planned to significantly develop the approach used and identify the most accurate values of θ D , C V , α V and B for XN in a wide temperature range of 300-3200 K.

Conclusions
Analysis of literature data regarding thermodynamic, thermal and elastic properties of titanium nitride TiN was carried out, which included values of its θ D , α V and B under standard conditions. It was shown that the previous results are characterized by a significant spread of values from 20 to 43 %. There are 8 basic variants of optimizing calculations, which made it possible to identify the most reliable values of some parameters of TiN. It has been established that under standard conditions, the size of θ D (TiN) close to the optimal should be within 746-769 K, and for C V (TiN) -in the range of 36.55-37.19 J/(mol×K). The spread of values for optimized parameters does not exceed 3 %.