Titanium carbide, titanium nitride and titanium carbo-nitride (TiX with X = N and/or C) are of interest due to their high melting point, high hardness, low density, high flexure strength, superior chemical and thermal stability, and excellent wear resistance.[1,2,3,4,5,6,7,8,9,10,11,12] Molar volume is an important starting parameter to describe thermodynamic, kinetic and transport properties and phenomena such as grain growth,[13] interfacial energies,[14,15] thermal conductivity,[16,17] viscosity,[18] etc. In this paper such a model is presented.

The homogeneity ranges of TiXz compounds with \(0<z<1\) are extended only into the Ti-rich region of the phase diagrams.[19,20,21,22,23,24] That is why their structural[12,21] and thermodynamic[25,26,27] descriptions are based on the two-sub-lattice model Ti1(X, Va)1. The same model is used here to model molar volume of TiXz crystals.

The only paper modeling the volume of the Ti-carbo-nitride before is due to Frisk et al.,[27] who applied 8 semi-empirical parameters at 298 K. Our goal is to develop an improved model with a smaller number of parameters, taking into account new experimental data.

The stoichiometry of TiXz crystals is expressed through the atomic ratio of X atoms to Ti atoms[28,29,30,31,32] denoted as z. The amounts of atoms in the crystal: \({n}_{{\text{Ti}}}=1\) mol/mol-TiXz and \({n}_{{\text{X}}}=z\) mol/mol-TiXz with 0.41 ≤ z ≤ 1.[25,26]

Two independent data should be given to characterize the composition of ternary carbo-nitride Ti1(C, N, Va)1: the site fractions of C and N in the sub-lattice (C, N, Va), denoted by \({y}_{{\text{C}}}\) and \({y}_{{\text{N}}}\). Then, parameters z and \({y}_{Va}\) follow as:

$$ z = y_{{\text{C}}} + y_{{\text{N}}} $$
(1a)
$$ y_{Va} = 1 - y_{{\text{C}}} - y_{{\text{N}}} = 1 - z $$
(1b)

One can also use the mole fractions \({x}_{{\text{C}}}\) and \({x}_{{\text{N}}}\) in Ti1(C, N, Va)1 neglecting vacancies, with the mole fraction of Ti written as:

$$ x_{{{\text{Ti}}}} = 1 - x_{{\text{C}}} - x_{{\text{N}}} $$
(1c)

Then, neglecting the role of the vacancies, parameter z follows as:

$$ z = \frac{{x_{{\text{C}}} + x_{{\text{N}}} }}{{1 - x_{{\text{C}}} - x_{{\text{N}}} }} $$
(1d)

The site fraction of component C: \({y}_{{\text{C}}}={n}_{{\text{C}}}/1={n}_{{\text{C}}}\), where \({n}_{C}\) is the amount of component C in 1 mol of Ti(C, N)z. The mole fraction of component C in Ti(C, N)z: \({x}_{{\text{C}}}={n}_{{\text{C}}}/(1+z)\) where (1 + z) is the total amount of atoms in 1 mole of Ti(C, N)z. Substituting \({n}_{{\text{C}}}={y}_{{\text{C}}}\) from the previous equation into the latter equation:

$$ y_{{\text{C}}} = x_{{\text{C}}} \cdot \left( {1 + z} \right) $$
(1e)

Similarly:

$$ y_{{\text{N}}} = x_{{\text{N}}} \cdot \left( {1 + z} \right) $$
(1f)

Experimental data for TiCz are given in References 28,29,30 and 33,34,35,36. Due to some impurities the data in Reference 28 differ significantly from the results of References 29, 30, and 34,35,36. Experimental data for TiNz are given in References 30, 36, and 37 The estimated data in Reference 31 differ significantly from experimental results.[30,36,37] Experimental data for the ternary Ti(C, N)z crystals were measured in References 38 and 39.

The following model equation is applied for the molar volume of TiXz crystals, taking into account the structural model Ti1(X, Va)1:

$$ V_{{m,{\text{TiX}}_{z} }} = V_{{{\text{Ti}}}} + z \cdot \left( {V_{{{\text{TiX}}}} - V_{{{\text{Ti}}}} } \right) + L_{X - Va} \cdot z \cdot \left( {1 - z} \right) $$
(2a)

where \({V}_{{\text{Ti}}}\) is the molar volume of the Ti1 sub-lattice, \({V}_{{\text{TiX}}}\) is the molar volume of the stoichiometric TiX crystal, \({L}_{X-Va}\) is the interaction volume between atoms X and the vacancies in the (X, Va) sub-lattice. In Eq. [2a] the molar volumes of 1 mole of Ti1 sub-lattice and z moles of the (X, Va) sub-lattice are added, the latter estimated as the difference between the molar volumes of TiX and Ti, while the deviations from ideality within the (X, Va) sub-lattice is modelled by the regular solution model due to interaction between the X atoms and the vacancies. Eq. [2a] is re-arranged as:

$$ V_{{m,{\text{TiX}}_{z} }} = V_{{{\text{Ti}}}} + z \cdot \left( {V_{{{\text{TiX}}}} - V_{{{\text{Ti}}}} + L_{X - Va} } \right) - L_{X - Va} \cdot z^{2} $$
(2b)

If \({V}_{m,{{\text{TiX}}}_{z}}\) is plotted as function of z and fitted as a quadratic polynomial (\({V}_{m}=a+b\cdot z+c\cdot {z}^{2}\)), the three semi-empirical parameters (\({V}_{{\text{Ti}}}\), \({V}_{{\text{TiX}}}\) and \({L}_{X-Va}\)) follow from the fitted parameters a, b and c as:

$$ V_{{{\text{Ti}}}} = a $$
(2c)
$$ L_{X - Va} = - c $$
(2d)
$$ V_{{{\text{TiX}}}} = a + b + c $$
(2e)

Equation [2e] follows from the equality: \(b={V}_{{\text{TiX}}}-{V}_{{\text{Ti}}}+{L}_{X-Va}\), after Eqs. [2c] and [2d] are substituted into it as: \(b={V}_{{\text{TiX}}}-a-c\). Using the molar volume data for TiCz[29,34,35,38] and TiNz,[30,36,37,38] the above semi-empirical parameters were optimized for the two cases, treating parameter a identical, i.e. the consistency of parameter \({V}_{{\text{Ti}}}\) was ensured in molar volume models of TiCz and TiNz. The semi-empirical parameters given in Figures 1 and 2 are substituted into Eqs. [2c] through [2e] and the final results at T = 298 K are given in the captions to Figures 1 and 2.

Fig. 1
figure 1

The molar volume of TiCz at T = 298 K plotted against parameter z, using the experimental data[29,34,35,38] (data[28] are excluded as they differ significantly). Parameters found from here (cm3/mol): \({V}_{{\text{Ti}}}\) = 10.88, \({V}_{{\text{TiX}}}\) = 12.19, \({L}_{{\text{X}}-Va}\) = 1.65

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Fig. 2
figure 2

The molar volume of TiNz at T = 298 K plotted against parameter z, using the experimental data[30,36,37,38] (data[31] are excluded as they differ significantly). Parameters found from here (cm3/mol): \({V}_{{\text{Ti}}}\) = 10.88, \({V}_{{\text{TiX}}}\) = 11.47, \({L}_{{\text{X}}-Va}\) = 0.308

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For the temperature dependence of the parameters in Figures 1 and 2 our earlier model for fcc metals[40] was combined with the experimental data[38] as:

$$ V_{{{\text{Ti}}}} = 10.85 + 2.712 \cdot 10^{ - 6} \cdot T^{1.618} $$
(3a)
$$ V_{{{\text{TiN}}}} = 11.43 + 9.979 \cdot 10^{ - 6} \cdot T^{1.468} $$
(3b)
$$ V_{{{\text{TiC}}}} = 12.14 + 2.050 \cdot 10^{ - 5} \cdot T^{1.360} $$
(3c)

Equations [3a] through [3c] are developed by us for pure fcc metals and reproduce their measured molar volumes from T = 0 K to their melting points with an accuracy of 0.2 pct or better. Equations [3a] through [3c] also obey the boundary condition that the thermal expansion coefficient becomes zero at T = 0 K. We found Eqs. [3a] through [3c] to work with smaller number of fitting parameters compared to alternative models.[41,42,43,44,45,46]

Equations [3a] through [3c] are presented graphically in Fig. 3 together with literature data.[30,34,35,36,37,38] Good agreement can be seen. Let us note that the molar volumes of hcp-Ti and bcc-Ti at 298 K are 10.55 and 10.59 cm3/mol respectively.[41] As follows from Fig. 3, our values for Ti-sublattices within the TiXz crystals are somewhat larger. This is due to the small (less than by 3 pct) expanding influence of the non-metallic sub-lattices.

Fig. 3
figure 3

Temperature dependences of the model parameters calculated by Eqs. [3a] through [3c]

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According to the structural model of titanium carbo-nitride Ti1(C, N, Va)1 its model for molar volume is written as extension to Eq. [2a]:

$$ \begin{gathered} V_{{m,{\text{Ti}}\left( {{\text{C}},{\text{N}}} \right)_{z} }} = V_{{{\text{Ti}}}} + z \cdot \frac{{y_{{\text{C}}} }}{{y_{{\text{C}}} + y_{{\text{N}}} }} \cdot \left( {V_{{{\text{TiC}}}} - V_{{{\text{Ti}}}} } \right) + z \cdot \frac{{y_{{\text{N}}} }}{{y_{{\text{C}}} + y_{{\text{N}}} }} \cdot \left( {V_{{{\text{TiN}}}} - V_{{{\text{Ti}}}} } \right) \hfill \\ + L_{{{\text{C}} - Va}} \cdot y_{{\text{C}}} \cdot \left( {1 - y_{{\text{C}}} - y_{{\text{N}}} } \right) + L_{{{\text{N}} - Va}} \cdot y_{{\text{N}}} \cdot \left( {1 - y_{{\text{C}}} - y_{{\text{N}}} } \right) \hfill \\ \end{gathered} $$
(4)

In principle, further parameters could be taken into account in Eq. [4], such as the interaction volume between components C and N,[27] or even ternary interaction volumes. However, these additional parameters are not needed to reproduce the experimental data by Eq. [4] (see Tables I and II). Indeed, the maximum difference in Table I between the experiments and our model values is ± 0.02 cm3/mol, while the same is ± 0.04 cm3/mol in Table II. This agreement confirms the good physical bases of our models.

Table I Compositions and Molar Volumes for Six Samples Measured by Aigner et al. [38]
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Table II Compositions and Molar Volume Measured for Eight Samples by Saringer et al. [39]
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The only previous paper in which the molar volume of Ti(C, N)z was modeled[27] applied 8 semi-empirical parameters at T = 298 K. As follows from Figures 1 and 2 and Eq. [4], the same goal was achieved here using only 5 binary semi-empirical parameters without any further ternary parameter. Moreover, our model also describes the temperature dependence of molar volume of Ti(C, N)z. Our model is simple and robust enough to be used in modelling further thermophysical properties of Ti(C, N)z crystals.