Study of Nonlinear Deformation of FCC-AuCuSi under Pressure by the Statistical Moment Method

Hoc, Nguyen Quang; Tinh, Bui Duc; Hien, Nguyen Duc; Coman, Gelu

doi:https://doi.org/10.1155/2021/6693326

Advances in Materials Science and Engineering

On this page

Abstract Introduction Model Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6693326 | https://doi.org/10.1155/2021/6693326

Study of Nonlinear Deformation of FCC-AuCuSi under Pressure by the Statistical Moment Method

Nguyen Quang Hoc,¹Bui Duc Tinh,¹Nguyen Duc Hien,²and Gelu Coman³

Academic Editor: Hamed Akhavan

Received19 Oct 2020

Revised08 Feb 2021

Accepted17 Feb 2021

Published05 Mar 2021

Abstract

In this work, we build the model and derive the theory of nonlinear deformation for substitutional alloy AB with interstitial atom C and face-centered cubic structure under pressure from the statistical moment method. The calculation results for FCC-AuCuSi are presented. We obtain the values of density of deformation energy, maximum real stress, limit of elastic deformation, and the stress-strain curve and compare the calculated results with experiments and other calculations.

1. Introduction

The study of elastic and thermodynamic properties of perfect ternary and binary interstitial alloys has also been attractive such as the theory of interstitial alloy [1], calculations from first principles, many-body potentials and dynamical dynamics for defects in metals, alloys and solid solutions [2–4] and elastic and thermodynamic properties of perfect ternary and binary interstitial alloys. We have studied the elastic deformation for body-entered cubic (BCC) and face-centered cubic (FCC) ternary and binary interstitial alloys under pressure by the statistical moment method (SMM) in [5–10].

The dependence of elastic and nonlinear deformations of materials on temperature, pressure, and concentration of components has very important role in predicting and understanding their interatomic interactions, strength, mechanical stability, phase transition mechanisms, and dynamical response. Silicides such as AuCuSi have attracted a lot of attention in recent years because of their functional applications and unusual physical properties. Gold silicide or gold silicon is one of the numerous metal alloys sold by American Elements under the trade name AE Alloys^TM.

The experimental data on the real stress and the limit of elastic deformation in the nonlinear deformation of pure metal Au are presented in [11–14].

Thermodynamic and mechanical properties of metals and interstitial alloys are studied by some theoretical methods and simulations. For example, Mehl and Papaconstantopoulos [15] applied a tight-binding (TB) scheme to extend first-principle calculations (ab initio) to regimes containing 10²–10³ atoms in a unit cell and used two-center, nonorthogonal tight-binding parameters and on-site terms. Numerical calculations for many metals are compared with ab initio calculations and experiments. Deformation mechanisms in the mechanical response of nanoporous gold are investigated by molecular dynamics simulations [16]. In addition, in recent years, some researchers have considered factors affecting the structure, the phase transformation, and the crystallization process of alloys AuCu [17], NiCu [18, 19], and AgCu [20].

In the present paper, we will study nonlinear deformation of FCC ternary alloy (substitutional alloy AB with interstitial atoms C) under pressure by the statistical moment method (SMM) [21–24]. In Section 2, we build the model and theoretical calculations, and in Section 3, we carry out numerical calculations for alloy AuCuSi.

2. Model and Theoretical Calculations

In our model, for interstitial alloy AC with FCC structure and concentration condition c_C << c_A ( is the concentration of atoms A, is the number of atoms A, is the concentration of atoms C, is the number of atoms C, and is the total number of atoms of the alloy AC), the cohesive energy and the alloy parameters (k is called the harmonic parameter and are called anharmonic parameters) for the interstitial atom C in face centers of cubic unit cell in the approximation of two coordination spheres have the form [5–10]

The cohesive energy and the alloy parameters for main metal atom A₁ in body center of cubic unit cell in the approximation of three coordination spheres have the form [5–10]

The cohesive energy and the alloy parameters for the main metal atom A₂ in corners of cubic unit cell in the approximation of three coordination spheres have the form [5–10]where is the interaction potential between atoms A and C, is the nearest neighbor distance between the atom X (X = A, A₁, A₂, C) (A in clean metal and A₁, A₂, and C in interstitial alloy AC) and other atoms at temperature T, is the nearest neighbor distance between the atom X and other atoms at T = 0 K and is determined from the minimum condition of the cohesive energy is the displacement of atom X from equilibrium position at temperature T. are the corresponding quantities in the clean metal A with FCC structure in the approximation of two coordination spheres and have the form [21, 22]

The equations of state for FCC interstitial alloy at temperature T and pressure P and at 0 K and pressure P are written in the form [5, 6, 8–10, 23]

From that, we can calculate the nearest neighbor distance , the parameters , the displacement of atom X from equilibrium position as in [21], the nearest neighbor distance , and the mean nearest neighbor distance between two atoms A in alloy as follows [5, 6, 8–10]:

The Helmholtz free energy of FCC interstitial alloy AC with the condition c_C << c_A is determined by [5–10,21]where is the Helmholtz free energy of one atom X, U_0X is the cohesive energy, and is the configurational entropy of FCC interstitial alloy AC.

The nearest neighbor distances between two atoms in alloy AC after deformation have the form [9]

The mean nearest neighbor distances between two atoms A in interstitial alloy AC at pressure P and temperature T after deformation have the form [9]

The mean nearest neighbor distances between two atoms A in substitutional alloy AB with interstitial atoms C at pressure P and temperature T and at pressure P and temperature T = 0 K after deformation have the form [5, 6, 9]

The Helmholtz free energy of alloy ABC before and after deformation with the condition c_C << c_B << c_A is determined by [5, 6, 9]

The relationship between the stress and the deformation in nonlinear deformation is given by [9]where σ_0ABC and α_ABC are constants for every alloy.

The density of deformation energy can be written in the form [9]

When the deformation rate is constant, the density of deformation energy of alloy is determined by [9]where C_ABC is the proportional factor. At the maximum value of the density of deformation energy, we have

The maximum value of stress σ_ABCmax and the maximum real stress σ_1ABCmax are [9]

From the maximum condition of stress we derive the deformation corresponding to the maximum value of real stress as follows [9]:

C_ABC is determined from the experimental condition of stress σ_0,2ABC in alloy in the form [9]

From the obtained value of , we can calculate σ_0ABC and α_ABC. The limit of elastic deformation of alloy ABC is determined by [9]

Here is Young’s modulus of alloy ABC and has the form [5, 6]

Here are Young’s moduli of alloy AC and metals A, B, respectively.

3. Numerical Results for Alloy AuCuSi

To describe the interaction between atoms Au and Si, we apply the Mie–Lennard-Jones pair interaction potential in the form [25]where D is the depth of potential well corresponding to the equilibrium distance r₀ and m and n are determined empirically. Then, the potential parameters for the interaction Au-Si are determined by [26]

We find m_Au-Si and n_Au-Si by fitting the theoretical result with the experimental data for Young’s modulus of interstitial alloy AuSi_3% at room temperature. The Mie–Lennard-Jones potential parameters for the interactions Au-Au, Si-Si, and Au-Si are given in Table 1. The Poisson ratio is 0.42 for Au [28], 0.34 for Cu [29], and 0.21 for Si [30]. Our investigated range of temperature up to 900 K is below the melting temperature of Au, Cu, and Si in the range of pressure from zero to 6 GPa as studies on the melting curve for these materials [31–33].

Fromthe experimental value of Young’s modulus E = 89.1.10⁹ Pa for Au at T = 300 K and P = 0 [34]. From that, we obtain Pa. Figure 1 shows the density of deformation energy and the real stress of AuCuSi at T = 300 K, , c_Si = 1%, and c_Cu = 1, 3, and 5% calculated by the SMM. For AuCuSi at the same temperature, pressure, and concentration of interstitial atoms, when the concentration of substitutional atoms increases, the maximum real stress decreases by 2% and the elastic limit also decreases by 2%. The density of deformation energy has the maximum value when the strain , and from that, we can find the maximum real stress the elastic limit , and the elastic strain for AuCuSi as shown in Table 2.

(a)

(b)

When concentrations we obtain the nonlinear deformation of main metal Au. Figure 2 shows the curve of real stress for Au at and T = 300 K where we have the comparison between the SMM and experiments [11, 12]. The maximum real stress and the elastic limit calculated by the SMM in Table 3 are in good agreement with the molecular dynamics (MD) results [16] and experiments [13, 14]. The error of the maximum real stress between the SMM calculation and the MD result [16] is 0.6%. The error of the elastic limit between the SMM calculation and the experimental data [13] is 5.65%. Thus, the SMM calculations of the maximum real stress and the elastic limit for Au at and T = 300 K are in good agreement with other calculations and experiments.

Figure 3 shows the density of deformation energy and the real stress of AuCuSi at T = 300 K, , c_Cu = 10%, and c_Si = 0, 1, and 2% calculated by the SMM. For AuCuSi at the same temperature, pressure, and concentration of substitutional atoms, when the concentration of interstitial atoms increases, the maximum real stress and the elastic limit decrease strongly. We have the comparison on values of ε_F, σ_1max, ε_e, and σ_e for alloys AuSi, AuCu, and AuCuSi in Table 4.

(a)

(b)

Figure 4 shows the density of deformation energy and the real stress of AuCuSi at T = 300, 600, and 900 K, , c_Cu = 10%, and c_Si = 2% calculated by the SMM. For AuCuSi at the same pressure, concentration of substitutional atoms, and concentration of interstitial atoms, when the temperature increases, the maximum real stress and the elastic limit increase. We have the comparison on values of ε_F, σ_1max, ε_e, and σ_e for Au₈₈Cu₁₀Si₂ at P = 0 and T = 300, 600, and 900 K in Table 5.

(a)

(b)

Figure 5 shows the real stress of AuCuSi at T = 300 K, , 4, and 6 GPa, c_Cu = 10%, and c_Si = 3% calculated by the SMM. For AuCuSi at the same temperature, concentration of substitutional atoms, and concentration of interstitial atoms, when the pressure increases, the maximum real stress increases and the elastic limit decreases. When the pressure increases from 2 to 4 GPa, the maximum real stress increases by 3% and the elastic limit decreases by 7.86%. When the pressure increases from 4 to 6 GPa, the maximum real stress increases by 11% and the elastic limit decreases by 10.56%. We have the comparison on values of ε_F, σ_1max, ε_e, and σ_e for Au₈₇Cu₁₀Si₃ at , 4, and 6 GPa and T = 300 K in Table 6.

4. Conclusion

In our paper, we derive the analytic expressions of characteristic quantities for the nonlinear deformation such as the density of deformation energy, the maximum real stress, and the limit of elastic deformation depending on temperature, pressure, and concentration of components together with the strain-stress of substitutional alloy AB with interstitial atom C and FCC structure under pressure. We apply theoretical results to alloy AuCuSi. The maximum real stress and the elastic limit calculated by the SMM are in good agreement with MD results [16] and experiments [13, 14]. For AuCuSi at the same temperature, concentration of substitutional atoms, and concentration of interstitial atoms, when the pressure increases, the maximum real stress increases and the elastic limit decreases. When the pressure increases from 2 to 4 GPa, the maximum real stress increases by 3% and the elastic limit decreases by 7.86%. When the pressure increases from 4 to 6 GPa, the maximum real stress increases by 11% and the elastic limit decreases by 10.56%. Our calculated results for alloy AuCuSi are compared with ones for alloys AuCu, AuSi, and metal Au. If we use more coordination spheres and we have exact experimental data for the stress of alloy AuCuSi at different temperatures, pressures, and concentrations of components, we will obtain better calculations.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

A. Smirnov, Theory Of Interstitial Alloys, Nauka, Moscow, Russia, 1979, in Russian.
P. A. Korzhavyi, I. A. Abrikosov, B. Johansson, A. V. Ruban, and H. L. Skriver, “First-principles calculations of the vacancy formation energy in transition and noble metals,” Physical Review B, vol. 59, no. 18, p. 11693, 1999.
View at: Publisher Site | Google Scholar
T. T. Lau, C. J. Först, X. Lin, J. G. Gale, S. Yip, and K. J. Van Vliet, Physical Review Letter, vol. 98, Article ID 215501, 2007.
View at: Publisher Site
M. Li, “Defect-induced topological order-to-disorder transition in two-dimensional binary substitutional solid solutions,” Physical Review B, vol. 62, Article ID 13979, 2000.
View at: Publisher Site | Google Scholar
N. Q. Hoc and N. D. Hien, “42^nd vietnam national Conference on theoretical physics (NCTP-42),” IOP Conference Series: Journal of Physics: Conference Series, vol. 1034, Article ID 012005, 2018.
View at: Publisher Site | Google Scholar
N. Q. Hoc, B. D. Tinh, and N. D. Hien, “Elastic moduli and elastic constants of alloy aucusi with fcc structure under pressure,” High Temperature Materials and Processes, vol. 38, pp. 264–272, 2018.
View at: Publisher Site | Google Scholar
B. D. Tinh, N. Q. Hoc, D. Q. Vinh, T. D. Cuong, and N. D. Hien, Advances in Materials Science and Engineering, vol. 2018, Article ID 5251741, p. 8, 2018.
View at: Publisher Site
N. Q. Hoc, T. D. Cuong, and N. D. Hien, “Proceedings of the ACCMS-theme meeting on “Multiscale modelling of materials for sustainable development”,” VNU Journal of Sciences: Mathematics-Physics, vol. 35, no. 1, pp. 1–12, 2019.
View at: Publisher Site | Google Scholar
N. Q. Hoc, B. D. Tinh, and N. D. Hien, Romanian Journal of Physics, vol. 65, no. 5-6, p. 608, 2020.
N. Q. Hoc, N. T. Hoa, T. D. Cuong, and D. Q. Thang, “On the melting of interstitial alloys FeH, FeSi and FeC with a body-centered cubic structure under pressure Proceedings of the ACCMS-theme meeting on “multiscale modelling of materials for sustainable development”,” Vietnam Journal of Science, Technology and Engineering, vol. 61, no. 2, pp. 17–22, 2019.
View at: Publisher Site | Google Scholar
R. D. Nyilas, M. Kobas, and R. Spolenak, “Synchrotron X-ray microdiffraction reveals rotational plastic deformation mechanisms in polycrystalline thin films,” Acta Materialia, vol. 57, no. 13, pp. 3738–3753, 2009.
View at: Publisher Site | Google Scholar
I. Chasiotis, C. Bateson, K. Timpano, A. S. McCarty, N. S. Barker, and J. R. Stanec, “Strain rate effects on the mechanical behavior of nanocrystalline Au films,” Thin Solid Films, vol. 515, no. 6, pp. 3183–3189, 2007.
View at: Publisher Site | Google Scholar
https://structx.com/Material_Properties_003a.html.
https://www.azom.com/properties.aspx?ArticleID=600.
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 Review B, vol. 54, no. 7, p. 4519, 1996.
View at: Publisher Site | Google Scholar
M. N. Esfahani and M. Jabbari, “Molecular dynamics simulations of deformation mechanisms in the mechanical response of nanoporous gold,” Materials, vol. 13, no. 9, p. 2071, 2020.
View at: Publisher Site | Google Scholar
N. T. Dung, N. C. Cuong, and D. Q. Van, Journal of Multiscale Modelling, vol. 11, no. 2, Article ID 2030001, 2020.
View at: Publisher Site
N. T. Dung and N. T. Phuong, “Molecular dynamics study on factors influencing the structure, phase transition and crystallization process of NiCu6912 nanoparticle,” Materials Chemistry and Physics, vol. 250, Article ID 123075, 2020.
View at: Publisher Site | Google Scholar
T. Q. Tuan and N. T. Dung, “Effect of heating rate, impurity concentration of Cu, atomic number, temperatures, time annealing temperature on the structure, crystallization temperature and crystallization process of Ni1−xCux bulk; x = 0.1, 0.3, 0.5, 0.7,” International Journal of Modern Physics B, vol. 32, no. 26, p. 1830009, 2018.
View at: Publisher Site | Google Scholar
C. L. Van, D. Q. Van, and N. T. Dung, “Ab initio calculations on the structural and electronic properties of AgAu alloy,” ACS Omega, vol. 5, no. 48, pp. 31391–31397, 2020.
View at: Publisher Site | Google Scholar
N. Tang and V. V. Hung, “Investigation of the thermodynamic properties of anharmonic crystals by the momentum method, I. General results for face-centered cubic crystals,” Physica Status Solidi (B), vol. 149, p. 511, 1989.
View at: Google Scholar
N. Tang and V. Van Hung, “Investigation of the thermodynamic properties of anharmonic crystals by the momentum method. II. Comparison of calculations with experiments for inert gas crystals,” Physica Status Solidi (B), vol. 161, no. 1, p. 165, 1990.
View at: Publisher Site | Google Scholar
N. Tang and V. Van Hung, “Investigation of the thermodynamic properties of anharmonic crystals by the momentum method. III. Thermodynamic properties of the crystals at various pressures,” Physica Status Solidi (B), vol. 162, no. 2, p. 371, 1990.
View at: Publisher Site | Google Scholar
N. Tang and V. Van Hung, “Investigation of the thermodynamic properties of anharmonic crystals by momentum method. IV. The limiting of absolute stability and the melting temperature of crystals,” Physica Status Solidi (B), vol. 162, no. 2, p. 379, 1990.
View at: Publisher Site | Google Scholar
M. N. Magomedov, “On calculating the Debye temperature and the Gruneisen parameter,” Zhurnal Fizicheskoi Khimii, vol. 61, no. 4, pp. 1003–1009, 1987, in Russian.
View at: Google Scholar
R. J. Good and C. J. Hope, “New combining rule for intermolecular distances in intermolecular potential functions,” The Journal of Chemical Physics, vol. 53, no. 2, pp. 540–543, 1970.
View at: Publisher Site | Google Scholar
M. N. Magomedov, “The calculation of the parameters of the Mie-Lennard-Jones potential,” High Temperature, vol. 44, no. 4, pp. 513–529, 2006.
View at: Publisher Site | Google Scholar
E. Goens, Physikal. Z., vol. 37, p. 321, 1936.
F. Birch, “Finite elastic strain of cubic crystals,” Physical Review, vol. 71, no. 11, p. 809, 1947.
View at: Publisher Site | Google Scholar
B. Bayram, O. Akar, and T. Akin, “Plasma-activated direct bonding of diamond-on-insulator wafers to thermal oxide grown silicon wafers,” Diamond and Related Materials, vol. 19, no. 11, pp. 1431–1435, 2010.
View at: Publisher Site | Google Scholar
Z. Y. Zhong, H. Saka, T. H. Kim, E. A. Holm, Y. K. Han, and X. S. Xie, “Effect of pressure on melting temperature of silicon and germanium,” Materials Science Forum, vol. 475–479, pp. 1893–1896, 2005.
View at: Google Scholar
D. Errandonea, Journal of Applied Physics, vol. 108, Article ID 033517, 2010.
View at: Publisher Site
D. Errandonea, “High-pressure melting curves of the transition metals Cu, Ni, Pd and Pt,” Physical Review B, vol. 87, Article ID 054108, 2013.
View at: Publisher Site | Google Scholar
L. V. Tikhonov and G. I. Kononenko, Mechanical Properties of Metals and Alloys, Kiev–Nauka Dumka, Ukraine, 1986.

Copyright

Copyright © 2021 Nguyen Quang Hoc 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.

PDF Download Citation

Download other formats

Order printed copies