Thermodynamic and Elastic Properties of Interstitial Alloy FeC with BCC Structure at Zero Pressure

*e analytic expressions for the thermodynamic and elastic quantities such as the mean nearest neighbor distance, the free energy, the isothermal compressibility, the thermal expansion coefficient, the heat capacities at constant volume and at constant pressure, the Young modulus, the bulk modulus, the rigidity modulus, and the elastic constants of binary interstitial alloy with bodycentered cubic (BCC) structure, and the small concentration of interstitial atoms (below 5%) are derived by the statistical moment method. *e theoretical results are applied to interstitial alloy FeC in the interval of temperature from 100 to 1000K and in the interval of interstitial atom concentration from 0 to 5%. In special cases, we obtain the thermodynamic quantities of mainmetal Fe with BCC structure. Our calculated results for some thermodynamic and elastic quantities of main metal Fe and alloy FeC are compared with experiments.


Introduction
ermodynamic and elastic properties of metals and interstitial alloys are specially interested by many theoretical and experimental researchers .For example, in [1,2] the equilibrium vacancy concentration in bcc substitution and interstitial alloys is calculated taking into account thermal redistribution of the interstitial component in different types of interstices.e conditions where this effect gives rise to a decrease or increase in vacancy concentration are formulated.Coatings based on interstitial alloys of transition metals have acquired a wide application range.However, interest in synthesizing coatings from new materials with requisite service properties is limited by the scarceness of data on their melting temperature.In [3], Andryushechkin and Karpman considered the calculation of melting temperature for interstitial alloys of transition metals based on the characteristics of intermolecular interaction.Hirabayashi et al. [4] attempt to present a survey of the order-disorder transformations in the interstitial alloys of transition metals with hydrogen, deuterium, and oxygen.Special attention is given to the formation of interstitial superstructures, stepwise processes of disordering and property changes attributed to order-disorder.Four groups of interstitial alloys are considered: (1) TO, ZrO, and HfO; (2) VO; (3) VH and VD; and (4) TaH and TaD.Characteristic features of the phase transformations in each group and each system are presented and discussed in comparison with others.In [14], Philibert presents the Morse potential and the Finnis-Sinclair for alloys FeH and FeC.In [15], a type of empirical potential for alloy FeC is developed in calculating defects with high energy.Structural, elastic, and thermal properties of alloy FeC are studied by using the modified embedded atom method (MEAM) in [16].
In this paper, we build the thermodynamic and elastic theory for binary interstitial alloy with bcc structure by the statistical moment method (SMM) [8][9][10] and apply the obtained theoretical results to alloy FeC.

ermodynamic Quantities.
e model of interstitial alloy AB with BCC structure in this paper is the same model of interstitial alloy AC in our previous paper [9].at means in this model, the main atoms A are in body center and peaks of cubic unit cell and the interstitial atoms B are in face centers of cubic unit cell.e cohesive energy u 0 and the crystal parameters k, c 1 , c 2 , and c for atoms B, A 1 (atom A in body center), and A 2 (atom A in peaks) in the approximation of three coordination spheres are determined analogously as for atoms C, A 1 , and A 2 in [9].Note that in the expressions of these quantities there are the cohesive energy and the crystal parameters of atoms A in clean metal A in the approximation of two coordination sphere [8].
e equation of state for interstitial alloy AB with BCC structure at temperature T and pressure P is written in the following form: At 0 K and zero pressure, this equation has the following form: If we know the form of interaction potential φ i0 , equation (2) permits us to determine the nearest neighbor distance r 1X (0, 0)(X � B, A, A 1 , A 2 ) at 0 K and zero pressure.After knowing that, we can determine crystal parameters k X (0, 0), c 1X (0, 0), c 2X (0, 0), c X (0, 0) at 0 K and zero pressure.After that, we can calculate the displacements [8][9][10].
where A X (0, T) is determined as in [9].From that, we derive the nearest neighbor distance r 1X (0, T) at temperature T and zero pressure: en, we calculate the mean nearest neighbor distance in interstitial alloy AB by the expressions as follows [8][9][10]: where r 1A (0, T) ≡ a AB (0, T) is the mean nearest neighbor distance between atoms A in interstitial alloy AB at zero pressure and temperature T, r 1A (0, 0) is the mean nearest neighbor distance between atoms A in interstitial alloy AB at zero pressure, 0 K, r 1A (0, 0) is the nearest neighbor distance between atoms A in clean metal A at zero pressure, 0 K, r 1A ′ (0, 0) is the nearest neighbor distance between atoms A in the zone containing the interstitial atom B at zero pressure and 0 K, and c B is the concentration of interstitial atoms B. e free energy of alloy AB with BCC structure and the condition c B << c A has the following form: where ψ X is the free energy of atom X, ψ AB is the free energy of interstitial alloy AB, S c is the configuration entropy of interstitial alloy AB. e isothermal compressibility of interstitial alloy AB has the following form: 2 Advances in Materials Science and Engineering e thermal expansion coefficient of interstitial alloy AB has the following form: e heat capacity at constant volume of interstitial alloy AB is determined by e heat capacity at constant pressure of interstitial alloy AB is determined by

Elastic Quantities.
e Young modulus of alloy AB with BCC structure at temperature T and zero pressure is determined as the one of alloy AC at P � 0 in [9].
e bulk modulus of alloy AB has the following form: e rigidity modulus of alloy AB at temperature T and zero pressure is as follows: e elastic constants of alloy AB at temperature T and zero pressure are as follows: e Poisson ratio of alloy AB is as follows: where ] A and ] B , respectively, are the Poisson ratio of materials A and B and are determined from the experimental data.

Numerical Results for Interstitial Alloy FeC.
For pure metal Fe, we use the m-n potential that the m-n potential parameters between atoms Fe-Fe were given in [12].For alloy FeC, we use the Finnis-Sinclair potential as follows: where the Finnis-Sinclair potential parameters between atoms Fe-C are as shown in Table 1.
Our numerical results for the thermal expansion coefficient and the heat capacity at constant pressure, the Young modulus, the bulk modulus, the rigidity modulus and the elastic constants of alloy FeC are summarized in tables from Tables 2-5 and are described by figures from

Advances in Materials Science and Engineering
Table 1: e Finnis-Sinclair potential parameters between atoms Fe-C [15].
) 600 700 800 900 1000  For alloy FeC at the same temperature when the concentration of interstitial atoms increases, the thermal expansion coe cient α T and the heat capacity at constant pressure C P decrease.For example, for FeC at T 1000 K when c C increases from 0 to 5%, α T decreases from 18.66.10−6 to 12.95.10−6 K −1 , and C P decreases from 26.67 to 25.59 J/(mol K).
For alloy FeC at the same concentration of interstitial atoms when temperature increases, the thermal expansion coe cient α T and the heat capacity at constant pressure C P increase.For example, for FeC at c C 5% when T increases from 100 to 1000 K, α T increases from 3.23.10−6 to 12.95.10−6 K −1 , and C P increases from 9.26 to 25.59 J/(mol K).
For alloy FeC at the same temperature when the concentration of interstitial atoms increases, the elastic moduli E, G, K, and the elastic constants C 11 , C 12 , C 44 decrease.For example, for FeC at T 1000 K when c C increases from 0 to 5%, E decreases from 12.28 × 10 10 to 10.39 × 10 10 Pa, G decreases from 4.87 × 10 10 to 4.12 × 10 10 Pa, K decreases from 8.53 × 10 10 to 7.21 × 10 10 Pa, C 11 decreases from 15.02 × 10 10 to 12.71 × 10 10 Pa, C 12 decreases from 5.28 × 10 10 to   For alloy FeC at the same concentration of interstitial atoms when temperature increases, the elastic moduli E, G, and K, and the elastic constants C 11 , C 12 , and C 44 also decrease.For example, for FeC at c C 5% when T increases from 100 to 1000 K, E decreases from 19.39.10 10 to 10.39.10 10 Pa, G decreases from 7.69.10 10 to 4.12.10 10 Pa, K decreases from 13.47.e calculated values from the SMM for the Young modulus E in Tables 4 and 5 and Figures 11 and 12 and therefore other elastic quantities, such as the elastic moduli G and K, the elastic constants C 11 , C 12 , and C 44 of alloy FeC, are in good agreement with experiments.e nearest neighbor distance, the elastic moduli E, G, and K, the elastic constants C 11 , C 12 , and C 44, and the isothermal elastic modulus B T of main metal Fe at P 0, T 300 K according to the SMM, the other calculation [21] and experiments [18][19][20] are given in [9].Our obtained deviations are smaller than 15%.

Conclusion
From the SMM, using the minimum condition of cohesive energies and the method of three coordination spheres, we nd the mean nearest neighbor distance, the free energy, the isothermal compressibility, the thermal expansion coe cient, the heat capacities at constant volume and at constant pressure, the Young modulus, the bulk modulus, the rigidity modulus, and the elastic constants of binary interstitial alloy with BCC structure with very small concentration of interstitial atoms.e obtained expressions of these quantities depend on temperature and concentration of interstitial atoms.At zero concentration of interstitial atoms, thermodynamic and elastic quantities of interstitial alloy become ones of main metal in alloy.e theoretical results are applied to interstitial alloy FeC.Our calculated results for the nearest neighbor distance, the elastic moduli, the elastic constants, and the isothermal elastic modulus at 300 K, the thermal expansion coe cient in the range from 100 to 1000 K and the heat capacity at constant pressure in the range from 100 to about 450 K of main metal Fe, the Young modulus, the bulk modulus, the rigidity modulus, the elastic constants of alloy FeC with c C 0.2% and c C 0.4% are in rather good agreement with experiments.Advances in Materials Science and Engineering

Figures 1 - 12 .
Figures 1-12.When the concentration c C ⟶ 0, we obtain thermodynamic quantities of Fe.Our calculated results shown in Tables 2-4 and Figures 5, 6, 11, and 12 are in rather good agreement with experiments (the obtained deviations are smaller than 15%).For alloy FeC at the same temperature when the concentration of interstitial atoms increases, the thermal expansion coe cient α T and the heat capacity at constant pressure C P decrease.For example, for FeC at T 1000 K when c C increases from 0 to 5%, α T decreases from 18.66.10−6 to 12.95.10−6 K −1 , and C P decreases from 26.67 to 25.59 J/(mol K).For alloy FeC at the same concentration of interstitial atoms when temperature increases, the thermal expansion

Figure 5 :Figure 6 :
Figure5: α T (T) for Fe at P 0 from the SMM and the experimental data[11].

Figure 7 :
Figure 7: E (c C ) for FeC at di erent temperatures T.

Figure 11 :
Figure 11: E (T) for alloy FeC with c C 0.2% from the SMM and alloy FeC with c C ≤ 0.3% from EXPT [17].

Figure 12 :
Figure 12: E (T) for alloy FeC with c C 0.4% from the SMM and alloy FeC with c C ≥ 0.3% from EXPT [17].

Table 2 :
Dependence of thermal expansion coe cient on temperature for Fe.

Table 3 :
Dependence of heat capacity at constant pressure on temperature for Fe.

Table 4 :
[17]pendence of Young modulus E (10 10 Pa) for alloy FeC with c C 0.2% from the SMM and alloy FeC with c C ≤ 0.3% from EXPT[17].

Table 5 :
[17]pendence of Young modulus E (10 10 Pa) for alloy FeC with c C 0.4% from the SMM and alloy FeC with c C ≥ 0.3% from EXPT[17].