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

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 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) [8-10] 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 [u.sub.0] and the crystal parameters k, [[gamma].sub.1], [[gamma].sub.2], and y for atoms B, [A.sub.1] (atom A in body center), and [A.sub.2] (atom A in peaks) in the approximation of three coordination spheres are determined analogously as for atoms C, [A.sub.1], and [A.sub.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:

Pv = -[r.sub.1] (1/6 [partial derivative][u.sub.0]/[partial derivative][r.sub.1] + [theta]xcthx 1/2k [partial derivative]k/[partial derivative][r.sub.1]). (1)

At 0 K and zero pressure, this equation has the following form:

0 = -[r.sub.1] (1/6 [partial derivative][u.sub.0]/[partial derivative][r.sub.1] + h[[omega].sub.0]/4k [partial derivative]k/[partial derivative][k.sub.1] (2)

If we know the form of interaction potential [[phi].sub.i0], equation (2) permits us to determine the nearest neighbor distance [r.sub.1X] (0,0)(X = B,A,[A.sub.1],[A.sub.2]) at 0K and zero pressure. After knowing that, we can determine crystal parameters [k.sub.X](0,0), [[gamma].sub.1X] (0,0), [[gamma].sub.2X] (0,0), [[gamma].sub.X] (0,0) at 0K and zero pressure. After that, we can calculate the displacements [8-10].

[y.sub.0X] (0,T) = [square root of (2[[gamma].sub.X](0,0)[[theta].sup.2]/ 3[k.sup.3.sub.X] (0,0) [A.sub.X](0,T))], (3)

where [A.sub.X](0,T) is determined as in [9]. From that, we derive the nearest neighbor distance [r.sub.1X](0,T) at temperature T and zero pressure:

[mathematical expression not reproducible] (4)

Then, we calculate the mean nearest neighbor distance in interstitial alloy AB by the expressions as follows [8-10]:

[mathematical expression not reproducible], (5)

where [bar.[r.sub.1A](0, T)] [equivalent to] [a.sub.AB] (0,T) is the mean nearest neighbor distance between atoms A in interstitial alloy AB at zero pressure and temperature T, [bar.[r.sub.1A](0,0)] is the mean nearest neighbor distance between atoms A in interstitial alloy AB at zero pressure, 0 K, [r.sub.1A] (0,0) is the nearest neighbor distance between atoms A in clean metal A at zero pressure, 0 K, [r'.sub.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.sub.B] is the concentration of interstitial atoms B.

The free energy of alloy AB with BCC structure and the condition [c.sub.B] [much less than] [c.sub.A] has the following form:

[mathematical expression not reproducible],

(6)

where [[PSI].sub.X] is the free energy of atom X, [[PSI].sub.AB] is the free energy of interstitial alloy AB, [S.sub.c] is the configuration entropy of interstitial alloy AB.

The isothermal compressibility of interstitial alloy AB has the following form:

[mathematical expression not reproducible].

(7)

The thermal expansion coefficient of interstitial alloy AB has the following form:

[mathematical expression not reproducible]. (8)

The heat capacity at constant volume of interstitial alloy AB is determined by

[mathematical expression not reproducible]. (9)

The heat capacity at constant pressure of interstitial alloy AB is determined by

[mathematical expression not reproducible]. (10)

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 P = 0 in [9].

The bulk modulus of alloy AB has the following form:

[K.sub.AB] = [E.sub.AB]/3(1-2[v.sub.A]) (11)

The rigidity modulus of alloy AB at temperature T and zero pressure is as follows:

[G.sub.AB] = [E.sub.AB]/2(1+[v.sub.A]) (12)

The elastic constants of alloy AB at temperature T and zero pressure are as follows:

[mathematical expression not reproducible], (13)

The Poisson ratio of alloy AB is as follows:

[v.sub.AB] = [C.sub.A][v.sub.A] + [C.sub.B][v.sub.B] [approximately equal to] [v.sub.A], (14)

where [v.sub.A] and [v.sub.B], 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 Fe-Fe were given in [12]. For alloy FeC, we use the Finnis-Sinclair potential as follows:

[mathematical expression not reproducible]. (15)

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

Figures 1-12. When the concentration [c.sub.C] [right arrow] 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 coefficient [[alpha].sub.T] and the heat capacity at constant pressure CP decrease. For example, for FeC at T = 1000 K when [c.sub.C] increases from 0 to 5%, [[alpha].sub.T] decreases from 18.66.[10.sup.-6] to 12.95. [10.sup.-6] [K.sup.-1], and CP 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 [[alpha].sub.T] and the heat capacity at constant pressure [C.sub.P] increase. For example, for FeC at [c.sub.C] = 5% when T increases from 100 to 1000 K, [[alpha].sub.T] increases from 3.23.[10.sup.-6] to 12.95.[10.sup.-6][K.sup.-1], and CP 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.sub.11], [C.sub.12], [C.sub.44] decrease. For example, for FeC at T = 1000 K when [c.sub.C] increases from 0 to 5%, E decreases from 12.28 x [10.sup.10] to 10.39 x [10.sup.10]Pa, G decreases from 4.87 x [10.sup.10] to 4.12 x [10.sup.10] Pa, K decreases from 8.53 x [10.sup.10] to 7.21 x [10.sup.10] Pa, C11 decreases from 15.02 x [10.sup.10] to 12.71 x [10.sup.10]Pa, [C.sub.12] decreases from 5.28 x [10.sup.10] to 4.46 x [10.sup.10]Pa, and [C.sub.44] decreases from 4.87 x [10.sup.10] to 4.12 x [10.sup.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.sub.11], [C.sub.12], and [C.sub.44] also decrease. For example, for FeC at [c.sub.C] = 5% when T increases from 100 to 1000 K, E decreases from 19.39.[10.sup.10] to 10.39.[10.sup.10] Pa, G decreases from 7.69.[10.sup.10] to 4.12.1010 Pa, K decreases from 13.47. [10.sup.10] to 7.21. [10.sup.10]Pa, [C.sub.11] decreases from 23.72.[10.sup.10] to 12.71.[10.sup.10] Pa, [C.sub.12] decreases from 8.33.[10.sup.10] to 4.46.1010 Pa, and [C.sub.44] decreases from 7.69.[10.sup.10] to 4.12.[10.sup.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.sub.11], [C.sub.12], and [C.sub.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.sub.11], [C.sub.12], and [C.sub.44], and the isothermal elastic modulus BT of main metal Fe at P = 0, 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.sub.C] = 0.2% and [c.sub.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.

https://doi.org/10.1155/2018/5251741

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.

References

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[2] V. A. Volkov and S. I. Masharov, "The effect of thermal redistribution of various interstitial impurities on equilibrium vacancy concentration in BCC substitutional-interstitial alloys," Russian Physics Journal, vol. 50, no. 4, pp. 400-404, 2007.

[3] S. E. Andryushechkin and M. G. Karpman, "Calculation of the melting temperature of interstitial alloys of transition metals," Metal Science and Heat Treatment, vol. 41, no. 2, pp. 80-82, 1999.

[4] M. Hirabayashi, S. Yamaguchi, H. Asano, and K. Hiraga, Order-Disorder Transformations in Alloys, Springer-Verlag, Berlin, Germany, 1974.

[5] K. E. Mironov, Interstitial alloy, Plenum Press, New York, USA, 1967.

[6] A. G. Morachevskii and I. V. Sladkov, Thermodynamic Calculations in Metallurgy, Metallurgiya, Moscow, Russia, 1993.

[7] N. Eliaz, D. Fuks, and D. Eliezer, "Non-Arrhenius behavior of the diffusion coefficient of hydrogen in amorphous metals," Materials Letters, vol. 39, no. 5, pp. 255-259, 1999.

[8] N. Tang and V. V. Hung, "Investigation of the thermodynamic properties of anharmonic crystals by the momentum method. I. General results for face-centred cubic crystals," Physica Status Solidi (B), vol. 149, no. 2, pp. 511-519, 1988.

[9] N. Q. Hoc and N. D. Hien, "Study on elastic deformation of substitution AB with interstitial alloy C and BCC structure under pressure," IOP Conf. Series: Journal of Physics: Conference Series, vol. 1034, article 012005, 2018.

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[11] W. B. Pearson, A Handbook of Lattice Spacings of Metals and Alloys, Pergamon, New York, 1958.

[12] M. N. Magomedov, Zhurnal Fizicheskoi Khimii, vol. 61, no. 4, p. 1003, 1987 (in Russian).

[13] N. V. Hung, T. T. Hue, and N.B. Duc, "Calculation of morse potential parameters of bcc crystals and application to anharmonic interatomic effective potential, local force constant," VNU Journal of Science:Mathematical-Physics, vol. 31, no. 3, pp. 23-30, 2015.

[14] J. M. Philibert, Atom Movements-Diffusion and Mass Transport in Solids, EDP Sciences, F-91944 les UlisCedex A, France, 1991.

[15] T. L. Timothy, J. F. Clemens, L. Xi, D. G. Julian, Y. Sidney, and J. V. V. Krystyn, "Many-body potential for point defect clusters in Fe-C alloys," Physical Review Letters, vol. 98, no. 21, article 215501, 2007.

[16] L. S. I. Liyanage, S. G. Kim, J. Houze et al., "Structural, elastic, and thermal properties of cementite ([Fe.sub.3]C) calculated using a modified embedded atom method," Physical Review B, vol. 89, article 094102, 2014.

[17] http://www.engineeringtoolbox.com/young-modulus-d_773. htm. Young's modulus of elasticity for metals and alloys.

[18] D. R. Lide, CRC Handbook of Chemistry and Physics, 86th Ed., Taylor & Francis, Boca Raton London, New York, Singapore, 2005.

[19] L. V. Tikhonov, V. A. Kononenko, G. I. Prokonenko et al., Structure and properties of metals and alloys, Naukova-Dumka, Kiev, 1986 (in Russian).

[20] H. Cyunn and C.-S. Yoo, "Equation of state of tantalum to 174 GPa," Physical Review B, vol. 59, no. 13, p. 8526, 1999.

[21] M. J. Mehl and D. A. Papaconstantopoulos, "Applications of a tight-binding total-energy method for transition and noble metals: elastic constants, vacancies and surfaces of monatomic metals," Physical Riview B, vol. 54, no. 7, p. 4519, 1996.

Bui Duc Tinh, (1,2) Nguyen Quang Hoc, (2) Dinh Quang Vinh, (2) Tran Dinh Cuong, (2) and Nguyen Duc Hien (3)

(1) Institute of Research and Development, Duy Tan University, 3 Quang Trung, Da Nang, Vietnam

(2) Hanoi National University of Education, 136 Xuan Thuy, Cau Giay District, Hanoi, Vietnam

(3) Mac Dinh Chi High School, Chu Pah District, Gia Lai Province, Vietnam

Correspondence should be addressed to Bui Duc Tinh; tinhbd@hnue.edu.vn

Received 25 May 2018; Accepted 7 August 2018; Published 4 October 2018

Academic Editor: Yee-wen Yen

Caption: FIGURE 1: [[alpha].sub.T] (T) for FeC at P = 0, [c.sub.C] = 0, 1, 2, and 4%.

Caption: FIGURE 2: [[alpha].sub.T] ([c.sub.C]) for FeC at P = 0, T = 100, 300, 500, and 700 K.

Caption: FIGURE 3: [C.sub.P] (T) for FeC at P = 0, [c.sub.C] = 0, 1, 3, and 5%.

Caption: FIGURE 4: [C.sub.P] ([c.sub.C]) for FeC at P = 0, T = 100, 200, 300, and 500 K.

Caption: FIGURE 5: [[alpha].sub.T] (T) for Fe at P = 0 from the SMM and the experimental data [11].

Caption: FIGURE 6: [C.sub.P] (T) for Fe at P = 0 from the SMM and the experimental data [11].

Caption: FIGURE 7: E ([c.sub.C]) for FeC at different temperatures T.

Caption: FIGURE 8: E (T) for FeC at different concentrations [c.sub.C].

Caption: FIGURE 9: [C.sub.11], [C.sub.12], and [C.sub.44] ([c.sub.C]) for FeC at T = 300 K.

Caption: FIGURE 10: [C.sub.11], [C.sub.12], and [C.sub.44] (T) for FeC at [c.sub.C] = 3%.

Caption: FIGURE 11: E (T) for alloy FeC with [c.sub.C] = 0.2% from the SMM and alloy FeC with [c.sub.C][less than or equal to]0.3% from EXPT [17].

Caption: FIGURE 12: E (T) for alloy FeC with [c.sub.C] = 0.4% from the SMM and alloy FeC with [c.sub.C][greater than or equal to]0.3% from EXPT [17].
TABLE 1: The Finnis-Sinclair potential parameters between atoms
Fe-C [15].

A (eV)      [R.sub.1]          [t.sub.1]
           ([Angstrom])   [([Angstrom]).sup.-2]

2.958787     2.545937           10.024001

A (eV)                [t.sub.2]                [R.sub.2]
                [([Angstrom]).sup.-3]         ([Angstrom])

2.958787               1.638980                 2.468801

A (eV)             [k.sub.1]                    [k.sub.2]
           (eV [([Angstrom]).sup.-2])   (eV [([Angstrom]).sup.-3])

2.958787            8.972488                    -4.086410

A (eV)             [k.sub.3]
           (eV[([Angstrom]).sup.-4])

2.958787           1.483233

TABLE 2: Dependence of thermal expansion coefficient on temperature
for Fe.

T(K)                        100     200     300     500     700    1000

[[alpha].sub.T]
  ([10.sup.-6][K.sup.-1])
This paper                  5.71   10.93   12.77   14.64   16.15   18.66
EXPT [11]                   5.6    10.0    11.7    14.3    16.3    19.2

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

T(K)              100     200     300     400     500

[C.sub.P] (J/mol*K)

  This paper     10.68   20.13   22.92   24.13   24.77
  EXPT [11]      12.07   21.50   25.13   27.43   29.64

TABLE 4: The dependence of Young modulus E ([10.sup.10]Pa) for alloy
FeC with [c.sub.C] = 0.2% from the SMM and alloy FeC with [c.sub.C]
[less than or equal to] 0.3% from EXPT [17].

T(K)   73      144     200     294     422     533

SMM    22.59   22.03   21.58   20.75   19.49   18.28
EXPT   21.65   21.24   20.82   20.34   19.51   18.82

T(K)   589     644     700     811     866

SMM    17.65   16.96   16.26   14.81   14.06
EXPT   18.41   17.58   16.69   14.07   12.41

TABLE 5: The dependence of Young modulus E (1010 Pa) for alloy FeC
with [c.sub.C] = 0.4% from the SMM and alloy FeC with
[c.sub.C] [greater than or equal to] 0.3% from EXPT [17].

T(K)    73      144     200     294     422     533

SMM    22.46   21.90   21.45   20.62   19.38   18.18
EXPT   21.51   21.10   20.68   20.20   19.37   18.62

T(K)    589     644     700     811     866     922

SMM    17.53   16.87   16.17   14.72   13.98   13.21
EXPT   18.27   17.44   16.55   13.93   12.34   10.62
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Title Annotation:Research Article
Author:Tinh, Bui Duc; Hoc, Nguyen Quang; Vinh, Dinh Quang; Cuong, Tran Dinh; Hien, Nguyen Duc
Publication:Advances in Materials Science and Engineering
Date:Jan 1, 2018
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