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

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.


Introduction
e 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][3][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][6][7][8][9][10]. e 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 . e experimental data on the real stress and the limit of elastic deformation in the nonlinear deformation of pure metal Au are presented in [11][12][13][14]. ermodynamic 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 10 2 -10 3 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][22][23][24]. In Section 2, we build the model and theoretical calculations, and in Section 3, we carry out numerical calculations for alloy AuCuSi.

Model and Theoretical Calculations
In our model, for interstitial alloy AC with FCC structure and concentration condition c C << c A (c A � N A /N is the concentration of atoms A, N A is the number of atoms A, c C � N C /N is the concentration of atoms C, N C is the number of atoms C, and N � N A + N C is the total number of atoms of the alloy AC), the cohesive energy u 0 and the alloy parameters k, c 1 , c 2 , c (k is called the harmonic parameter and c 1 , c 2 , c 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][6][7][8][9][10] (1) e cohesive energy u 0 and the alloy parameters k, c 1 , c 2 , c for main metal atom A 1 in body center of cubic unit cell in the approximation of three coordination spheres have the form [5][6][7][8][9][10] 2 Advances in Materials Science and Engineering e cohesive energy u 0 and the alloy parameters k, c 1 , c 2 , c for the main metal atom A 2 in corners of cubic unit cell in the approximation of three coordination spheres have the form [5][6][7][8][9][10] where ϕ AC is the interaction potential between atoms A and C, r 1X � r 01X + y 0X (T) is the nearest neighbor distance between the atom X (X � A, A 1 , A 2 , C) (A in clean metal and A 1 , A 2 , and C in interstitial alloy AC) and other atoms at temperature T, r 01X 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 u 0X , and y 0X (T) is the displacement of atom X from equilibrium position at temperature T. u 0A , k A , c 1A , c 2A 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] Advances in Materials Science and Engineering e 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][9][10]23] Pv � −r 1 1 6 From that, we can calculate the nearest neighbor distance r 1X (P, 0) (X � A, A 1 , A 2 , C), the parameters k X (P, 0), c 1X (P, 0), c 2X (P, 0), c X (P, 0), the displacement y 0X (P, T) of atom X from equilibrium position as in [21], the nearest neighbor distance r 1X (P, T), and the mean nearest neighbor distance between two atoms A in alloy r 1A (P, T) as follows [5,6,[8][9][10]: e Helmholtz free energy of FCC interstitial alloy AC with the condition c C << c A is determined by [5][6][7][8][9][10]21] where ψ X is the Helmholtz free energy of one atom X, U 0X is the cohesive energy, and S c is the configurational entropy of FCC interstitial alloy AC. e nearest neighbor distances between two atoms in alloy AC after deformation have the form [9] e mean nearest neighbor distances between two atoms A in interstitial alloy AC at pressure P and temperature T after deformation have the form [9] r ACF 1A (P, T) � r ACF 01A (P, 0) + y ACF (P, T), 4 Advances in Materials Science and Engineering e 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] e 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] e 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. e density of deformation energy can be written in the form [9] Advances in Materials Science and Engineering When the deformation rate is constant, the density of deformation energy of alloy is determined by [9] f ABC (ε) � C ABC · σ ABC · ε, where C ABC is the proportional factor. At the maximum value of the density of deformation energy, we have e maximum value of stress σ ABCmax and the maximum real stress σ 1ABCmax are [9] σ ABC max � f ABC max From the maximum condition of stress (zσ 1ABC /zε) ε F, ABC � 0, we derive the deformation ε F ABC 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] C From the obtained value of ε F ABC , we can calculate σ 0ABC and α ABC . e limit of elastic deformation σ e � E ABC ε e of alloy ABC is determined by [9] σ e � E ABC ε e � σ 0ABC ε α e 1 + ε e .
Here E ABC is Young's modulus of alloy ABC and has the form [5,6]

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 r 0 and m and n are determined empirically. en, 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. e Mie-Lennard-Jones potential parameters for the interactions Au-Au, Si-Si, and Au-Si are given in Table 1. e 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][32][33]. From the experimental value of Young's modulus E � 89.1.10 9 Pa for Au at T � 300 K and P � 0 [34]. From that, we obtain σ 0.2 � 89.1 × 10 9 × 0.002 � 178.2 × 10 6 Pa. Figure 1 shows the density of deformation energy f(ε) and the real stress σ 1 (ε)of AuCuSi at T � 300 K, P � 0, 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%. e density of deformation energy has the maximum value when the strain ε F � 0.08, and from that, we can find the maximum real stress σ 1 max ,the elastic limit σ e , and the elastic strain ε e for AuCuSi as shown in Table 2.   When concentrations c Cu , c Si ⟶ 0, we obtain the nonlinear deformation of main metal Au. Figure 2 shows the curve of real stress σ 1 (ε) for Au at P � 0 and T � 300 K where we have the comparison between the SMM and experiments [11,12]. e maximum real stress σ 1 max and the elastic limit σ e calculated by the SMM in Table 3 are in good agreement with the molecular dynamics (MD) results [16] and experiments [13,14]. e error of the maximum real stress between the SMM calculation and the MD result [16] is 0.6%. e error of the elastic limit between the SMM calculation and the experimental data [13] is 5.65%. us, the SMM calculations of the maximum real stress and the elastic limit for Au at P � 0 and T � 300 K are in good agreement with other calculations and experiments. Figure 3 shows the density of deformation energy f(ε) and the real stress σ 1 (ε) of AuCuSi at T � 300 K, P � 0, 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. Figure 4 shows the density of deformation energy f(ε) and the real stress σ 1 (ε) of AuCuSi at T � 300, 600, and 900 K, P � 0, 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 Au 88 Cu 10 Si 2 at P � 0 and T � 300, 600, and 900 K in Table 5. Figure 5 shows the real stress σ 1 (ε) of AuCuSi at T � 300 K, P � 2, 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 Au 87 Cu 10 Si 3 at P � 2, 4, and 6 GPa and T � 300 K in Table 6.  : e real stress σ 1 (ε) of Au at T � 300 K and P � 0 calculated by the SMM and from EXPT [11] and EXPT [12].

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. e maximum real stress σ 1 max and the elastic limit σ e 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 σ 0.2 of alloy AuCuSi at different temperatures, pressures, and concentrations of components, we will obtain better calculations.
Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Table 5: Values of ε F (f (ε F ) � f max ), σ 1max , ε e , and σ e for Au 88 Cu 10 Si 2 at P � 0 and T � 300, 600, and 900 K calculated by the SMM.