Research Article  Open Access
Thermodynamic and Elastic Properties of Interstitial Alloy FeC with BCC Structure at Zero Pressure
Abstract
The 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. The theoretical results are applied to interstitial alloy FeC in the interval of temperature from 100 to 1000 K and in the interval of interstitial atom concentration from 0 to 5%. In special cases, we obtain the thermodynamic quantities of main metal Fe with BCC structure. Our calculated results for some thermodynamic and elastic quantities of main metal Fe and alloy FeC are compared with experiments.
1. Introduction
Thermodynamic and elastic properties of metals and interstitial alloys are specially interested by many theoretical and experimental researchers [1–21]. 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. The 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 orderdisorder 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 orderdisorder. 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 FinnisSinclair 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) [8–10] and apply the obtained theoretical results to alloy FeC.
2. Content
2.1. Thermodynamic Quantities
The model of interstitial alloy AB with BCC structure in this paper is the same model of interstitial alloy AC in our previous paper [9]. That 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. The cohesive energy and the crystal parameters 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].
The 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 equation (2) permits us to determine the nearest neighbor distance at 0 K and zero pressure. After knowing that, we can determine crystal parameters at 0 K and zero pressure. After that, we can calculate the displacements [8–10].where is determined as in [9]. From that, we derive the nearest neighbor distance at temperature T and zero pressure:Then, we calculate the mean nearest neighbor distance in interstitial alloy AB by the expressions as follows [8–10]:where is the mean nearest neighbor distance between atoms A in interstitial alloy AB at zero pressure and temperature T, is the mean nearest neighbor distance between atoms A in interstitial alloy AB at zero pressure, 0 K, is the nearest neighbor distance between atoms A in clean metal A at zero pressure, 0 K, 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.
The free energy of alloy AB with BCC structure and the condition has the following form:where is the free energy of atom X, is the free energy of interstitial alloy AB, is the configuration entropy of interstitial alloy AB.
The isothermal compressibility of interstitial alloy AB has the following form:
The thermal expansion coefficient of interstitial alloy AB has the following form:
The heat capacity at constant volume of interstitial alloy AB is determined by
The heat capacity at constant pressure of interstitial alloy AB is determined by
2.2. Elastic Quantities
The Young modulus of alloy AB with BCC structure at temperature T and zero pressure is determined as the one of alloy AC at in [9].
The bulk modulus of alloy AB has the following form:
The rigidity modulus of alloy AB at temperature T and zero pressure is as follows:
The elastic constants of alloy AB at temperature T and zero pressure are as follows:
The Poisson ratio of alloy AB is as follows:where and , respectively, are the Poisson ratio of materials A and B and are determined from the experimental data.
2.3. Numerical Results for Interstitial Alloy FeC
For pure metal Fe, we use the m–n potential that the m–n potential parameters between atoms FeFe were given in [12]. For alloy FeC, we use the FinnisSinclair potential as follows:where the FinnisSinclair potential parameters between atoms FeC 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 Figures 1–12. When the concentration 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 coefficient 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 coefficient 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 4.46 × 10^{10} Pa, and C_{44} decreases from 4.87 × 10^{10} to 4.12 × 10^{10} Pa.
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 13.47.10^{10} to 7.21.10^{10} Pa, C_{11} decreases from 23.72.10^{10} to 12.71.10^{10} Pa, C_{12} decreases from 8.33.10^{10} to 4.46.10^{10} Pa, and C_{44} decreases from 7.69.10^{10} to 4.12.10^{10} Pa.
The 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. The 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 , T = 300 K according to the SMM, the other calculation [21] and experiments [18–20] are given in [9]. Our obtained deviations are smaller than 15%.
3. Conclusion
From the SMM, using the minimum condition of cohesive energies and the method of three coordination spheres, we find 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 BCC structure with very small concentration of interstitial atoms. The 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. The 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 coefficient 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, and the elastic constants of alloy FeC with c_{C} = 0.2% and c_{C} = 0.4% are in rather good agreement with experiments.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under Grant No. 103.02–2017.316.
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Copyright © 2018 Bui Duc Tinh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.