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

Authors: Nguyen Quang Hoc; Bui Duc Tinh; Nguyen Duc Hien; Gelu Coman
Journal: Advances in Materials Science and Engineering
 (2021)

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## 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.


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## Body

## 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 AlloysTM.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 alloyAB 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 alloyAC with FCC structure and concentration condition cC << cA (cA=NA/N is the concentration of atoms A, NA is the number of atoms A, cC=NC/N is the concentration of atoms C, NC is the number of atoms C, and N=NA+NCis the total number of atoms of the alloy AC), the cohesive energy u0 and the alloy parameters k,γ1,γ2,γ (k is called the harmonic parameter and γ1,γ2,γ 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](1)u0C=12∑i=1niφACri=3φACr1C+4φACr2C,r2C=3r1C,kC=12∑i∂2φAC∂uiβ2eq=d2φACr1Cdr1C2+2r1CdφACr1Cdr1C+43d2φACr2Cdr2C2+83r2CdφACr2Cdr2C,γC=4γ1C+γ2C,γ1C=148∑i∂4φAC∂uiβ4eq=124d4φACr1Cdr1C4+14r1C2d2φACr1Cdr1C2−14r1C3dφACr1Cdr1C++154d4φACr2Cdr2C4+29r2Cd3φACr2Cdr2C3−29r2C2d2φACr2Cdr2C2+29r2C3dφACr2Cdr2C,γ2C=648∑i∂4φAC∂uiα2∂uiβ2eq=12r1Cd3φACr1Cdr1C3−34r1C2d2φACr1Cdr1C2+34r1C3dφACr1Cdr1C++19d4φACr2Cdr2C4+23r2C2d2φACr2Cdr2C2−23r2C3dφACr2Cdr2C.The cohesive energyu0 and the alloy parameters k,γ1,γ2,γ for main metal atom A1 in body center of cubic unit cell in the approximation of three coordination spheres have the form [5–10](2)u0A1=u0A+φACr1A1,kA1=kA+12∑i∂2φAC∂uiβ2eqr=r1A1=kA+d2φACr1A1dr1A12,γA1=4γ1A1+γ2A1,γ1A1=γ1A+148∑i∂4φAC∂uiβ4eqr=r1A1=γ1A+124d4φACr1A1dr1A14,γ2A1=γ2A+648∑i∂4φAC∂uiα2∂uiβ2eqr=r1A1=γ2A+14r1A1d3φACr1A1dr1A13−12r1A12d2φACr1A1dr1A12+12r1A13dφACr1A1dr1A1.The cohesive energyu0 and the alloy parameters k,γ1,γ2,γ for the main metal atom A2 in corners of cubic unit cell in the approximation of three coordination spheres have the form [5–10](3)uOA2=uOA+φACr1A2,kA2=kA+12∑i∂2φAC∂uiβ2eqr=r1A2=kA+16d2φACr1A2dr1A22+236r1A2dφACr1A2dr1A2,γA2=4γ1A2+γ2A2,γ1A2=γ1A+148∑i∂4φAC∂uiβ4eqr=r1A2=γ1A+154d4φACr1A2dr1A24+29r1A2d3φACr1A2dr1A23−29r1A22d2φACr1A2dr1A22+29r1A23dφACr1A2dr1A2,γ2A2=γ2A+648∑i∂4φAC∂uiα2∂uiβ2eqr=r1A2=γ2A+181d4φACr1A2dr1A24+427r1A2d3φACr1A2dr1A23+1427r1A22d2φACr1A2dr1A22−1427r1A23dφACr1A2dr1A2,where ϕAC is the interaction potential between atoms A and C, r1X=r01X+y0XT 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, r01X 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 u0X,andy0XT is the displacement of atom X from equilibrium position at temperature T. u0A,kA,γ1A,γ2A 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](4)u0A=6φAAr1A+3φAAr2A,r2A=2r1A,kA=2d2φAAr1Adr1A2+4r1AdφAAr1Adr1A+d2φAAr2Adr2A2+2r2AdφAAr2Adr2A,γ1A=124d4φAAr1Adr1A4+14r1Ad3φAAr1Adr1A3−18r1A2d2φAAr1Adr1A2+18r1A3dφAAr1Adr1A+124d4φAAr2Adr2A4+14r2A2d2φAAr2Adr2A2−14r2A3dφAAr2Adr2A,γ2A=148d4φAAr1Adr1A4+78r1Ad3φAAr1Adr1A3−3116r1A2d2φAAr1Adr1A2+3116r1A3dφAAr1Adr1A+12r2Ad3φAAr2Adr2A3−98r2A2d2φAAr2Adr2A2+98r2A3dφAAr2Adr2A.The equations of state for FCC interstitial alloy at temperatureT and pressure P and at 0 K and pressure P are written in the form [5, 6, 8–10, 23](5)Pv=−r116∂uo∂r1+θxcthx12k∂k∂r1,v=2r132,Pv=−r116∂u0∂r1+ℏω04k∂k∂r1.From that, we can calculate the nearest neighbor distancer1XP,0X=A,A1,A2,C, the parameters kXP,0,γ1XP,0,γ2XP,0,γXP,0, the displacement y0XP,T of atom X from equilibrium position as in [21], the nearest neighbor distance r1XP,T, and the mean nearest neighbor distance between two atoms A in alloy r1AP,T¯ as follows [5, 6, 8–10]:(6)r1CP,T=r1CP,0+yA1P,T,r1AP,T=r1AP,0+yAP,T,r1A1P,T≈r1BP,T,r1A2P,T=r1A2P,0+yCP,T,r1AP,T¯=r1AP,0¯+yP,T¯,r1AP,0¯=1−cCr1AP,0+cCr1A′P,0,r1A′P,0=3r1CP,0,yP,T¯=1−15cCyAP,T+cCyCP,T+6cCyA1P,T+8cCyA2P,T.The Helmholtz free energy of FCC interstitial alloyAC with the condition cC << cA is determined by [5–10,21](7)ψAC=1−15cCψA+cCψC+6cCψA1+8cCψA2−TSc,ψX≈U0X+ψ0X+3Nθ2kX2γ2XYX2−2γ1X31+YX2+2θ3kX443γ2XYX1+YX2−2γ1X2+2γ1Xγ2X1+YX21+YX,ψ0X=3NθxX+ln1−e−2xX,YX≡xXcothxX,where ψX is the Helmholtz free energy of one atom X, U0X is the cohesive energy, and Sc is the configurational entropy of FCC interstitial alloy AC.The nearest neighbor distances between two atoms in alloyAC after deformation have the form [9](8)r01XF=r01X1+ε,r1XFP,0=r1XP,01+2ε+ε2r1CFP,T=r01CFP,0+yA1FP,0,r1AFP,0=r01AFP,0+yAFP,T,r1A1FP,T≈r1CFP,T,r1A2FP,T=r01A2FP,0+yCFP,T.The mean nearest neighbor distances between two atomsA in interstitial alloy AC at pressure P and temperature T after deformation have the form [9](9)r1AACFP,T¯=r01AACFP,0¯+yACFP,T¯,r01AACFP,0¯=1−cCr01AFP,0+cCr01A′ACFP,01+2ε+ε2,r01A′FP,0=3r01CFP,0,yACFP,T¯=1−15cCyAFP,T+cCyCFP,T+6cCyA1FP,T+8cCyA2FP,T.The mean nearest neighbor distances between two atomsA 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](10)aABCF=cACaACFBTACFBTbccF¯+cBaBFBTBFBTF¯,BTF¯=cACBTACF+cBBTBF,cAC=cA+cC,aABCF≡r1AABCFP,T¯,aACF≡r1AACFP,T¯,aBF≡r1BFP,T,BTACF=1χTACF=2P+2/aACF1/3N∂2ψACF/∂aACF2T3aACF/a0ACF3,BTBF=1χTBF=2P+2/aBF1/3N∂2ψBF/∂aBF2T3aBF/a0BF3,∂2ψACF∂aACF2T≈1−15cC∂2ψAF∂aAF2T+cC∂2ψCF∂aCF2T+6cC∂2ψA1F∂aA1F2T+8cC∂2ψA2F∂aA2F2T,13N∂2ψXF∂aXF2T=16∂2u0XF∂aXF2+ℏωXF4kXF∂2kXF∂aXF2−12kXF∂kXF∂aXF2,X=A,A1,A2,B,C,a0ABCF=cACa0ACFB0TACFB0TF¯+cBa0BFB0TBFB0TF¯,B0TF¯=cACB0TACF+cBB0TBF,a0ABCF≡r01AABCFP,0¯,a0ACF≡r01AACFP,0¯,a0BF≡r01BFP,0.The Helmholtz free energy of alloy ABC before and after deformation with the conditioncC << cB << cA is determined by [5, 6, 9](11)ψABC=ψAC+cBψB−ψA+TScAC−TScABC,ψAC=1−15cCψA+cCψC+6cCψA1+8cCψA2−TScAC,ψABCF=ψACF+cBψBF−ψAF+TScACF−TScABCF,ψACF=1−15cCψAF+cCψCF+6cCψA1F+8cCψA2F−TScACF.The relationship between the stress and the deformation in nonlinear deformation is given by [9](12)σ1ABC=σ0ABCεαABC1+ε,where σ0ABC and αABC are constants for every alloy.The density of deformation energy can be written in the form [9](13)fABCε=1N1−cAψA1vABCF−1vABC+2εr01AFvABCF∂ψAF∂r1AFT+ε22vABCF∂2ψAF∂r1AF2T2r01AF2+∂ψAF∂r1AFT2r01AF++cCNψB1vABCF−1vABC+2εr01BFvABCF∂ψBF∂r1BFT+ε22vABCF∂2ψBF∂r1BF2T2r01BF2+∂ψBF∂r1BFT2r01BF++cA1NψA11vABCF−1vABC+2εr01A1FvABCF∂ψA1F∂r1A1FT+ε22vABCF∂2ψA1F∂r1A1F2T2r01A1F2+∂ψA1F∂r1A1FT2r01A1F++cA2NψA21vABCF−1vABC+2εr01A2FvABCF∂ψA2F∂r1A2FT+ε22vABCF∂2ψA2F∂r1A2F2T2r01A2F2+∂ψA2F∂r1A2FT2r01A2F++cBNψB1vABCF−1vABC+2εr01BFvABCF∂ψBF∂r1BFT+ε22vABCF∂2ψBF∂r1BF2T2r01BF2+∂ψBF∂r1BFT2r01BF−−cBNψA1vABCF−1vABC+2εr01AFvABCF∂ψAF∂r1AFT+ε22vABCF∂2ψAF∂r1AF2T2r01AF2+∂ψAF∂r1AFT2r01AF.When the deformation rate is constant, the density of deformation energy of alloy is determined by [9](14)fABCε=CABC⋅σABC⋅ε,where CABC is the proportional factor. At the maximum value of the density of deformation energy, we have(15)fABCεABCF=fABCmax=CABCσABCmaxεABCF.The maximum value of stressσABCmax and the maximum real stress σ1ABCmax are [9](16)σABCmax=fABCmaxCABCεABCF,σ1ABCmax=σABCmax1+εABCF=fABCmaxCABCεABCF1+εABCF.From the maximum condition of stress∂σ1ABC/∂εεABCF,=0, we derive the deformation εABCF corresponding to the maximum value of real stress as follows [9]:(17)εABCF=αABC1−αABC⇒σ1ABCmax=σ0ABCεABCFαABC1+εABCF.CABC is determined from the experimental condition of stress σ0,2ABC in alloy in the form [9](18)CABC=fABCε0.2σ0.2ABCε0.2.From the obtained value ofεABCF, we can calculate σ0ABC and αABC. The limit of elastic deformation σe=EABCεe of alloy ABC is determined by [9](19)σe=EABCεe=σ0ABCεeα1+εe.HereEABC is Young’s modulus of alloy ABC and has the form [5, 6](20)EABC=cBEB−EA+EAC,EY=1π⋅r1Y⋅A1Y,A1Y=1kY1+2γY2θ2kY41+12xYcthxY1+xYcthxY,xY=ℏωY2θ,Y=A,B,EAC=EA1−15cC+cC∂2ψC/∂ε2+6∂2ψA1/∂ε2+8∂2ψA2/∂ε2∂2ψA/∂ε2,∂2ψX∂ε2=12∂2U0X∂r1X2+34ℏωXkX∂2kX∂r1X2−12kX∂kX∂r1X24r01X2++12∂U0X∂r1X+xXcthxX3θ2kX∂kX∂r1X2r01X,xX=ℏωX2θ,ωX=kXmX,X=A,A1,A2,C.HereEAC,EA,EB 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](21)φr=Dn−mmr0rn−nr0rm,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](22)DAu−Si=DAu−AuDSi−Si,r0Au−Si=12r0Au−Au+r0Si−Si.We findmAu-Si and nAu-Si by fitting the theoretical result with the experimental data for Young’s modulus of interstitial alloy AuSi3% 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].Table 1
Mie–Lennard-Jones potential parameters for interactions Au-Au, Cu-Cu, and Si-Si.
InteractionD/kB (K)r0 (10−10 m)mnAu-Au [27]7411.52.87511.9615.56Cu-Cu [27]6841.32.54873.038.37Si-Si [25]32701.72.295612From(23)E=σ0,2ε0,2⟹σ0,2=Eε0,2=E×0.2%,the 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 σ0.2=89.1×109×0.002=178.2×106 Pa. Figure 1 shows the density of deformation energy fε and the real stress σ1εof AuCuSi at T = 300 K, P=0, cSi = 1%, and cCu = 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 εF=0.08, and from that, we can find the maximum real stress σ1max,the elastic limit σe, and the elastic strain εe for AuCuSi as shown in Table 2.Figure 1
The density of deformation energyfε (a) and the real stress σ1ε (b) of Au99−xCuxSi1 at T = 300 K and P=0 calculated by the SMM.
(a)(b)Table 2
Values ofεF (f (εF) = fmax), σ1max, σe, and εe of Au99−xCuxSi1 at P=0 and T = 300 K calculated by the SMM.
εF (%)σ1max (MPa)εe (%)σe (MPa)Au98Cu1Si18.0231.790.21190.84Au96Cu3Si18.0230.380.21189.40Au94Cu5Si18.0227.060.20186.38When concentrationscCu,cSi⟶0, we obtain the nonlinear deformation of main metal Au. Figure 2 shows the curve of real stress σ1ε for Au at P=0 and T = 300 K where we have the comparison between the SMM and experiments [11, 12]. The maximum real stress σ1max and the elastic limit σe 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 P=0 and T = 300 K are in good agreement with other calculations and experiments.Figure 2
The real stressσ1ε of Au at T = 300 K and P=0 calculated by the SMM and from EXPT [11] and EXPT [12].Table 3
Values ofεF (f (εF) = fmax), σ1max, εe, and σe for Au at P=0 and T = 300 K calculated by the SMM and from MD [16], EXPT [13], and EXPT [14].
σlmax (MPa)σe (MPa)SMMMD [16]SMMEXPT [13]EXPT [14]εF = 9.6%243.52245εe = 0.19%188.7020045 ÷ 300Figure3 shows the density of deformation energy fε and the real stress σ1ε of AuCuSi at T = 300 K, P=0, cCu = 10%, and cSi = 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 εF, σ1max, εe, and σe for alloys AuSi, AuCu, and AuCuSi in Table 4.Figure 3
The density of deformation energyfε (a) and the real stress σ1ε (b) of Au99Si1,Au90Cu10, and Au89Cu10Si1 at T = 300 K and P=0 calculated by the SMM.
(a)(b)Table 4
Values ofεF (f (εF) = fmax), σ1max, εe, and σe for Au99Si1, Au90Cu10, and Au89Cu10Si1 at P=0 and T = 300 K calculated by the SMM.
AlloyεF (%)σ1max (MPa)εe (%)σe (MPa)Au99Si19.6214.070.18165.49Au90Cu108.3242.950.19196.75Au89Cu10Si18.2216.790.19175.98Figure4 shows the density of deformation energy fε and the real stress σ1ε of AuCuSi at T = 300, 600, and 900 K, P=0, cCu = 10%, and cSi = 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 εF, σ1max, εe, and σe for Au88Cu10Si2 at P = 0 and T = 300, 600, and 900 K in Table 5.Figure 4
The density of deformation energyfε (a) and the real stress σ1ε (b) of Au88Cu10Si2 at P = 0 and T = 300, 600, and 900 K calculated by the SMM.
(a)(b)Table 5
Values ofεF (f (εF) = fmax), σ1max, εe, and σe for Au88Cu10Si2 at P=0 and T = 300, 600, and 900 K calculated by the SMM.
T (K)εF (%)σ1max (MPa)εe (%)σe (MPa)3008.2194.520.18157.366006.7208.290.24179.859005.0210.270.32193.19Figure5 shows the real stress σ1ε of AuCuSi at T = 300 K, P=2, 4, and 6 GPa, cCu = 10%, and cSi = 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 εF, σ1max, εe, and σe for Au87Cu10Si3 at P=2, 4, and 6 GPa and T = 300 K in Table 6.Figure 5
The real stressσ1ε of Au87Cu10Si3 at P=2, 4, and 6 GPa and T = 300 K calculated by the SMM.Table 6
Values ofεF (f (εF) = fmax), σ1max, εe, and σe for Au88Cu10Si2 at P=2, 4, and 6 GPa and T = 300 K calculated by the SMM.
P (GPa)εF (%)σ1max (MPa)εe (%)σe (MPa)28.9175.860.14149.5049.7170.640.10137.75610.2152.060.08123.20


## 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 alloyAB with interstitial atom C and FCC structure under pressure. We apply theoretical results to alloy AuCuSi. The maximum real stress σ1max and the elastic limit σe 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 σ0.2 of alloy AuCuSi at different temperatures, pressures, and concentrations of components, we will obtain better calculations.



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