This study focuses on exploring the mechanical properties and nonlinear stress-strain behaviors of monoclinic Ni_{3}Sn_{4 } single crystals under uniaxial tensile test and also their size, temperature, and strain-rate dependence through constant temperature molecular dynamics (MD) simulation using Berendsen thermostat. The deformation evolution of the Ni_{3}Sn_{4} atomic nanostructure during the tensile test is observed. In addition, the tensile yield strains of various Ni_{3}Sn_{4 }single crystals at different strain rates and temperatures are characterized through unloading process. At last, by way of linear regression analysis, the corresponding normal elastic stiffness constants are approximated and then compared with the literature theoretical data. The radial distribution function analysis shows that Ni_{3}Sn_{4} single crystal in a one-dimensional nanowire configuration would become a highly disordered structure after thermal equilibration, thereby possessing amorphous-like mechanical behaviors and properties. The initial elastic deformation of Ni_{3}Sn_{4} single crystal is governed by the reconfiguration of surface atoms, and its deformation evolution after further uniaxial tensile straining is characterized by Ni=Sn bond straightening, bond breakage, inner atomic distortion, cross-section shrinking, and rupture. The calculated normal elastic constants of Ni_{3}Sn_{4} single crystal are found to be consistent with the literature theoretical data.
1. Introduction
Green electronic products, especially in lead (Pb-) free and halogen-free manufactures, are extremely popular nowadays due to more and more environmental awareness by consumers and governmental regulations. They have become the main developing trend in the microelectronics community. Various eco- and environmentally-friendly Pb-free substitute solder alloys or compositions, such as Sn-rich alloys including Sn-Ag, Sn-Zn, and Sn-Ag-Cu [1–3], have been proposed in response to the environmental challenges and needs. Despite the fact that Pb-free solders have been proved to be technologically feasible and shown some very promising advantages, many significant technical issues remain to be dealt with before their wide and successful implementation and application. For instance, a Ni layer is typically used in solder interconnects to prevent the Cu metal in the under bump metallurgy (UBM) from chemical reaction with the Sn metal in the Sn2.5Ag solder to form some hard and brittle Cu-Sn intermetallic compounds (IMCs), such as Cu_{3}Sn [4] and Cu_{6}Sn_{5} [5]. Instead, a Ni_{3}Sn_{4} [6] IMC layer will be formed at the interface between the Ni layer and the Sn metal (shown in Figure 1 [7]). It was reported by Chan et al. [8] that an appropriate IMC layer with a few micrometer thicknesses can upgrade metallurgical bonding strength probably because of its hard properties. Nevertheless, Yao and Shang [9] indicated that as the thickness of the IMC layer increases up to a certain level, the shear strength of the microsolder joints would be greatly downgraded owing to its brittle characteristic, which might eventually cause brittle fracture failure in the microsolder joints when subjected to impact loading. Kim et al. [10] also demonstrated that there is an inverse relation between IMC thickness and solder joint reliability. The risk of brittle fracture in solder interconnects would be increased due to high local stress resulting from extreme thermal-mechanical loading during fabrication process, assembly and environmental testing together with the low surface energy of IMCs. The issue becomes more pronounced for next-generation microelectronic packaging technology, such as three-dimensional (3D) chip stacking packaging, where the dimension of solder interconnects is reduced down to 10~15 μm, and even less than 5 μm in future [11–13]. And at or below this length scale, the thickness of the IMC layer can be as small as one to several grains or as large as up to one half of the height of the solder interconnects (e.g., full IMC solder joints). For better grasp of the thermal-mechanical reliability of the solder interconnects, clear comprehension of their mechanical properties and also their strain, strain rate, size, and temperature dependence is critically crucial.
An SEM image of a microsolder joint.
The material behaviors and properties may vary significantly for IMCs with a thickness ranging from tens of nanometers to tens of micrometers, which are thus difficult or even unable to be precisely and accurately characterized through classical elastic-plastic constitutive models. For micro-/nanomaterial characterization, it is now possible to apply advanced nanoscale testing techniques, such as nanoindentation experiment [14, 15], and atomic modeling methods [16–36], such as molecular dynamics (MD) simulation. However, even with the most advanced testing techniques, precisely characterizing the mechanical and thermodynamic properties of nanoscale IMCs remains great difficulty and challenge, not to mention their strain, strain rate, size, and temperature dependences, due to the inherent limitations of the testing techniques and instruments and also high measurement uncertainty. In addition, experimental testing is typically carried out on a test sample in a bulk polycrystalline form such that single-crystalline material properties of nanomaterials are not readily available.
In recent years, the advance in computer hardware and software has made atomic modeling of nanostructures possible. Atomic modeling can limit the shortage of experimental approaches and also supplement laboratory assessment. Over the past years, various computational approaches have been proposed, including equivalent-continuum modeling (ECM) [20–24], molecular dynamics (MD) simulation [4, 25–31], and quantum mechanics (QM) simulation [19, 32–34]. For example, Cheng et al. [35] and Lee et al. [36] evaluated the elastic properties of Ni_{3}Sn_{4} monocrystal by performing density functional theory (DFT) calculations within the generalized-gradient approximation (GGA) and local density approximation (LDA) using the software CASTEP [37, 38]. However, because of limited computer speed and memory, these studies using first-principles calculations generally adopt an atomic unit cell model with periodic boundary condition to determine the bulk-like properties of a larger system.
This study aims at conducting a comprehensive analysis of the mechanical properties and nonlinear stress-strain behaviors of Ni_{3}Sn_{4} single crystal at room temperature under uniaxial tensile test, and their size, temperature, and strain-rate effects. The alternative focus is placed on the deformation evolution characteristics of the Ni_{3}Sn_{4} atomic nanostructure during uniaxial tensile test. To the author’s best knowledge, the present comprehensive analysis is the first to address the issue. The single-crystalline Ni_{3}Sn_{4} systems under investigation are in a one-dimensional (1D) nanowire form. The nanowires are constructed by assembling a number of Ni_{3}Sn_{4} unit cells along the three axes of a prescribed coordinate system. To achieve the goal, an atomic-scale theoretical analysis based on MD simulation is performed. In the MD model, the physical interaction between atoms is simulated by the universal force field [39, 40]. By the proposed tensile MD simulation, the constitutive relationships between stress and strain for Ni_{3}Sn_{4} single crystal are characterized as a function of strain rate. Furthermore, the strain-rate-dependent mechanical properties are examined, including the normal elastic constants, yield strain and strength, ultimate tensile strength, and fracture strain. To demonstrate the effectiveness of the proposed MD model, the calculated normal elastic constants are compared against the literature theoretical results obtained from first-principles calculations.
2. Potential Function of Ni<sub>3</sub>Sn<sub>4</sub>
In the present MD model, the universal force field (UFF) is applied to simulate the bonding forces between atoms, where bond stretching is described by a harmonic term, angle bending by a three-term Fourier cosine expansion, and torsion and inversion by cosine-Fourier expansion terms. In addition, the Lennard-Jones (LJ) potential is used to describe the van der Waals interactions. Basically, UFF is an all-atom method containing parameters for every atom. The potential energy is expressed as sum of bonded interactions and nonbonded interactions [39, 40]:
(1)Epot=∑b{Kb02(b-b0)2}+∑θ{Kθ02([cosθ-cosθ0]2[sinθ0]2)}+∑ϕ{Bj2(1-djcos[njϕ])}+∑ωKω02sin2(ω0)(cosω-cosω0)2+∑RD0[(R0R)12-2(R0R)6]+332.0647∑i>jqiqjεR,
the first four terms of (1) are the energies associated with bond (b), angle (θ), torsion (ϕ), and out-of-plane (ω) internal coordinates; the last terms are the LJ 12–6 potential and Coulombic potential, denoting the van der Waals interaction and electrostatic energy, respectively. In (1), Kb0, Kθ0, and Kω0 stand for force constant, b0, θ0, and ω0 are equilibrium bond distance and angle, Bj represents tensional barrier, dj is phase factor, nj is periodicity, D0 is equilibrium well depth, R0 is equilibrium, and ε is relative dielectric constant. In general, the empirical force field parameters in the UFF are derived from quantum modeling.
3. Details of Molecular Dynamics (MD) Simulation
The atomic structure of Ni_{3}Sn_{4} single crystal was comprehensively clarified by Rappé et al. [41], where it is a monoclinic lattice structure, as displayed in Figure 2. The unit cell of Ni_{3}Sn_{4} single crystal is composed of six Ni atoms and eight Sn atoms; the smaller spheres in the lattice represent the Ni atoms and the larger ones the Sn atoms. The unit lattice cell is parallelepiped and consists of dimensions 12.214 Å, 4.060 Å, and 5.219 Å along the three lattice directions (a, b, and c, shown in Figure 2), respectively. There is a nonright angle (β) observed between the lattice vectors a and c. The nonright angle is around 105°, suggesting a nonparallelism between the lattice vector a and the axis x of the predefined rectangular coordinate system. In other words, the axis b is in line with the lattice direction y; and the axis c the lattice direction z. Three different Ni_{3}Sn_{4} nanowire models, as shown in Figure 3, are developed for tensile MD simulation, which are tailored from a single-crystalline Ni_{3}Sn_{4} substrate along the three axes of the predefined rectangular coordinate system, that is, x, y, and z, rather than the crystallographic directions, that is, a, b, and c. During the uniaxial tensile test, the Ni_{3}Sn_{4} nanowires are subjected to tensile strain at a prescribed strain rate along the x, y, and z axes, respectively. Figure 3(a) presents a schematic of the atomic model of the nanowire with the longitudinal dimension along the x-axis. It is evident that the x axis of the predefined coordinate system is not parallel to the lattice vector a. Unlike the nanowire model in Figure 3(a), the longitudinal axis of the other two nanowire atomic models (i.e., Figures 3(b) and 3(c)), which is along the y and z axis, is parallel to the corresponding lattice directions b and c, respectively. It should be noted that the Ni_{3}Sn_{4} nanowires consist of a circular cross section mainly because they would possess the most stable and natural cross-sectional configuration [42–44]. Under the same length of 100 Å, three different nanowire diameters, namely, 14 Å (14ϕ), 18 Å (18ϕ), and 22 Å (22ϕ), are considered in the tensile MD simulation, in which they are labeled with “14ϕ×100,” “18ϕ×100,” and “22ϕ×100,” respectively. Figure 4 shows the atomic models for these Ni_{3}Sn_{4} nanowires.
Crystal structure of the Ni_{3}Sn_{4} unit cell (grey spheres represent Ni atoms and purple spheres denote Sn atoms).
Schematic of three different Ni_{3}Sn_{4} nanowire models tailored from a single-crystalline Ni_{3}Sn_{4} substrate along the three axes of the predefined rectangular coordinate system.
x-axis
y-axis
z-axis
Atomic structures of the three Ni_{3}Sn_{4} nanowire models of three different sizes.
By using the proposed MD simulation, the stress-strain relations of the 1D nanocrystal structures are simulated by displacing the atoms (the red zone shown in Figure 4) on the top side of the nanowires while fixing those of the bottom (the orange zone shown in Figure 4) at a constant strain rate ε˙. To assess the strain-rate effects, the Ni_{3}Sn_{4} nanowires are loaded at five different strain rates, namely, 10^{−4}% ps^{−1}, 5 × 10^{−4}% ps^{−1}, 10^{−3}% ps^{−1}, 5 × 10^{−3}% ps^{−1}, and 10^{−2}% ps^{−1}. Furthermore, to investigate the influences of temperature on the stress-strain relations of the Ni_{3}Sn_{4} nanowires, Berendsen thermostat [45] is applied for constant temperature MD simulation at three different temperatures, that is, 300 K, 500 K, and 700 K. These temperatures are well below the melting point of Ni_{3}Sn_{4} crystal (i.e., 1096 K) [46], thereby ensuring that Ni_{3}Sn_{4} nanocrystal during the uniaxial tensile test remains in solid phase.
In the MD simulation, a total of 5 × 10^{3} simulation time steps (Nstep) with a time step size (Δt) of 1 × 10^{−3} ps are first employed for simulating the initial equilibrated unstrained configuration. After the equilibration, a time step size of 1 × 10^{−3} ps is also carried out for each incremental strain during the uniaxial tensile test. By varying the total number of time steps from 10^{3} to 10^{5}, five different strain rates, defined below, in the range of 10^{−4}% ps^{−1} to 10^{−2}% ps^{−1} can be achieved and simulated:
(2)ε˙=ΔεNstepΔt.
The stress on a nanostructure is calculated by the following equation [47]:
(3)σ(ε)=1NΩi∑i=1N∑j=1j≠iNFij(ε)rij(ε),
where N is the total number of atoms, Ωi denotes the volume of atom i, Fij represents the pairwise interatomic force between atoms i and j, and rij is the interatomic distance between atoms i and j.
4. Results and Discussion
Figures 5 and 6 reveal the equilibrated atomic structures of the three Ni_{3}Sn_{4} nanowire models and their corresponding radial distribution function (RDF) value after thermal equilibration. The RDF analysis is a key way to disclose the structure features of a nanosystem, particularly for liquids and amorphous structures. The RDF can be calculated as
(4)g(r)=VN2〈∑i=1nn(r)4πr2Δr〉,
where V represents the volume of the system and n(r) the number of particles, which can be found in the shell from r to r+Δr. In the study, for the binary intermetallic compounds system, the RDF for atom α and atom β is derived by
(5)g(r)=VNαNβ〈∑i=1Nαniβ(r)4πr2Δr〉.
According to the calculated results, it is found that the Ni_{3}Sn_{4} nanowires transform homogenously from a crystalline to amorphous-like state after thermal equilibration. It is also noted that the RDF would considerably decrease with the increase of the diameter, suggesting that the degree of amorphization is reduced with an increasing cross-sectional area. This observation is generally applicable for various materials, such as metals, semiconductors, and ceramics [48–50].
Equilibrated atomic structures of the three Ni_{3}Sn_{4} nanowire models of three different sizes.
x-axis
y-axis
z-axis
Radial distribution function (RDF) of the three Ni_{3}Sn_{4} nanowire models after thermal equilibration.
x-axis
y-axis
z-axis
The stress-strain responses of the Ni_{3}Sn_{4} nanowires, starting from an initial unstressed state to a complete rupture state, are exhibited in Figures 7(a)–7(c). Figure 8 further presents the strain-rate effects on the stress-strain response of the Ni_{3}Sn_{4} nanowire with the longitudinal dimension along the x-axis. According to Figures 7 and 8, the strength of the single-crystalline Ni_{3}Sn_{4} nanostrctures (i.e., “14ϕ×100,” “18ϕ×100,” and “22ϕ×100”) would all substantially decrease with an increasing temperature. This could be ascribed to the fact that the increase of temperature tends to weaken the interatomic bonds between Ni and Sn atoms, thereby resulting in reduced strength. Apart from that, as indicated in Figure 8, for all the simulation temperatures and nanowire sizes (diameters), the strength would also increase as the strain rate increases. The strain-rate effect can be elucidated by means of the structural disorder in the Ni_{3}Sn_{4} nanowires due to external loading [48, 49]. In general, a higher strain rate could enhance the degree of structural disorder of the nanostructure, thereby increasing the extent of amorphization [48–50]. For an amorphization system, the disordered nature does not allow the formation of well-defined slip planes. Because of the absence of the dislocation planes, no abrupt transformation of the system during deformation can be detected. Consequently, the strength would be significantly increased as the strain rate increases [48–50]. Finally, the results in Figures 7 and 8 also demonstrate that the strength of the Ni_{3}Sn_{4} nanowires would rise with the increase of the nanowire diameter. This size effect has also been observed in many literature studies [18, 49, 51]. The significant size effect may be mainly due to the surface effects of nanocrystals, including surface energy [52], surface stress [53], and surface relaxation [54], and the effects are strongly dependent on the surface to volume ratio. For nanosize particles, wires, and films, the corresponding surface to volume ratios are significant, and so are the surface or boundary effects.
Stress-strain responses of the three Ni_{3}Sn_{4} nanowire models of three different sizes at 300 K, 500 K, and 700 K and the strain rate of 10^{−4}% ps^{−1}.
x-axis
y-axis
z-axis
Stress-strain responses of the Ni_{3}Sn_{4} nanowire model of three different sizes constructed along the x-axis as a function of temperature and strain rate.
Strain rate = 10^{−4}% ps^{−1}
Strain rate = 5 × 10^{−4}% ps^{−1}
Strain rate = 10^{−3}% ps^{−1}
Strain rate = 5 × 10^{−3}% ps^{−1}
Strain rate = 10^{−2}% ps^{−1}
The deformation characteristics of the nanostructure during the stress-strain evolution are demonstrated using the tensile σ11-ε11 stress-strain response of the “22ϕ×100” Ni_{3}Sn_{4} nanowire (see Figure 4) at 300 K and strain rate 10^{−4}% ps^{−1}, which is shown in Figure 9. Evidently, the system exhibits a continuous mechanical deformation before fracture. The nanowire initially starts a linearly elastic deformation from its unstrained configuration (i.e., point (1) in Figure 9(a)), where a linear stress-strain relation is given, or in other words, the stress is linearly proportional to the applied strain. This relation follows closely the well-known Hooke’s law, and the slope is known as the elastic modulus of Ni_{3}Sn_{4} single crystal in the x-axis. Furthermore, after a critical threshold value of the stress, referred as the proportional limit (i.e., “σpl” in Figure 9(a)), the nanocrystal structure experiences extraordinarily large nonlinearly elastic deformation. After the critical threshold stress value, the deformation is followed by a nonlinear plastic deformation, causing a permanent or residual strain after unloading. The threshold stress value that transits from nonlinear elasticity to plasticity is referred as the yield stress or yield strength σy (i.e., point (2) in Figure 9(a)), and the corresponding strain is known as the yield strain εy. To determine the yield strain, unloading process is carried out at the several different applied strains and the associated residual strains are characterized in Figure 9(c). It is evident to see that no residual strain is observed as the applied strain is below 3.0%, indicating that the tensile yield strain in the x-axis is around 3.0%, and from Figure 9(a), the corresponding tensile yield strength is about 2.01 GPa. Subsequently, the stress increases with an increasing strain until a maximum stress level is reached. The maximum stress is well known as the ultimate tensile stress or ultimate tensile strength σu, and the corresponding strain is referred to as the ultimate tensile strain εu. For the Ni_{3}Sn_{4} nanowire under the σ11-ε11 tensile test, the ultimate tensile strain of Ni_{3}Sn_{4} nanowire with the longitudinal dimension parallel to the x-axis is around 14.5%, and the ultimate tensile strength is about 2.67 GPa. Furthermore, beyond the maximum stress (point “3” in Figure 9(a)), the stress starts to gradually decrease despite an increasing strain. This is mainly ascribed to a homogenous deformation as a result of the absence of dislocation planes. The phenomenon is well known as necking. The ultimate tensile strain is the onset point of the necking process. A snapshot of the onset point of the necking phenomenon (i.e., point (3) in Figure 9(a)) is exhibited in Figure 9(b)-(3). Beyond the point, the necking process continues, further decreasing the cross-sectional area of the neck, as can be seen in Figure 9(b)-(4) corresponding to the point (4) in Figure 9(a) (i.e., ε11=23.0%), and in Figure 9(b)-(5) corresponding to the point (5) in Figure 9(a) (i.e., ε11=25.0%). The Ni_{3}Sn_{4} nanowire becomes fully ruptured at a strain of ε11=31.0%. Accordingly, the fracture strain εf is 31.0%.
(a) The tensile σ11-ε11 stress-strain evolution of the “22ϕ×100” Ni_{3}Sn_{4} nanowire model at 300 K and strain rate = 10^{−4}% ps^{−1}, (b) snapshots of fracture evolution of the Ni_{3}Sn_{4} atomic structure at various strains, and (c) the residual strain after unloading versus the applied strain.
To have a better insight into the mechanism behind the stress-strain behavior, the atomic rearrangement of the nanowire structure during the deformation evolution process is characterized. Figure 10 displays a small section of the [001] crystalline plane of the “22ϕ×100” Ni_{3}Sn_{4} nanowire model with the longitudinal dimension parallel to the x-axis shown in Figure 4, and the section is also indicated in Figure 9(b)-(1). The left side of the section is the free surface of the nanowire, and the right side is toward the centerline (i.e., the central longitudinal axis) of the nanowire. The atoms within the dash box are monitored during the deformation evolution process to highlight the atomic rearrangements. In addition, the Sn atoms in the section are, respectively, labeled from 1 to 9 (i.e., Sn-1~Sn-9), and the Ni atoms from 1 to 6 (i.e., Ni-1~Ni-6), as indicated in Figure 11. Noticeably, four specific stages of atomic arrangements during the deformation evolution process are detected, namely, initial equilibration, yielding, ultimate tensile strength, and necking deformation. Figure 11(a) shows the initial equilibrated nanowire prior to stretching. It is found that the surface atoms (i.e., Sn-1, Sn-2, and Ni-1) form a noticeable relaxed surface configuration due to the free surface condition while the inner atoms (i.e., Sn-7, Sn-8, Ni-5, Ni-6, and Sn-9) would maintain their initial state or position. At the beginning of the tensile loading, stretching causes the surface atoms, Sn-1 and Ni-1, to undergo significant separation. By contrast, as can be seen in Figure 11(b), the inner atoms would only have a small elastic deformation. The bond breakage of the Ni-2=Sn-3 bond reveals the commencement of irreversible deformation or strain, which is considered as the onset of yielding. As the Ni-2 and Sn-3 atoms are further pulled apart, accompanied by the straightening of nanowire, the necking process starts (see Figure 9(b)-(3)), in which, as can be seen in Figure 11(c), the bond breakage of the Ni-4=Sn-6 bond takes place. When the loading exceeds the bond strength, the Sn-3=Ni-3 bond is broken and a new surface is created, which can be seen in Figure 11(d). This leads to the contraction of the cross-sectional area of the necked region (Figure 9(b)-(4)). Figure 11(d) also reveals that the Sn-2, Ni-1, and Sn-3 atoms are separated from the Ni-3 atom. Accordingly, the stretching of the nanowire is characterized by bond straightening, bond breakage, inner atomic distortion, and cross-section shrinking until forming a single-atom neck, and rupture.
Section of [001] crystalline plane of 22ϕ×100 Ni_{3}Sn_{4} nanowire constructed along principle axis x.
Rearrangement of atoms in the red dash box of Figure 9(b)-(1) under tensile loading at (a) initial equilibration, (b) yielding, (c) ultimate tensile strength, and (d) necking.
To assess the temperature dependence, the σ11-ε11 stress-strain behavior at 700 K and the same strain rate as used in Figure 9 (i.e., 10^{−4}% ps^{−1}) is calculated and the result is illustrated in Figure 12. It is noted that the system entropy would increase with an increasing temperature, thereby leading to a much more significant vibration of atoms. The increasing vibration would further result in much larger structure instability, thereby causing atomic structure reconstruction. It can also be observed from Figure 12(b)-(2) that the Ni_{3}Sn_{4} nanowire at a higher temperature is more inclined to undergo an early onset of atomic structure reconstruction, as compared to that at 300 K shown in Figure 9(b)-(2). Consequently, the yield strain of the Ni_{3}Sn_{4} nanowire at 700 K is 50% lower than that at 300 K, and according to Figure 12(a), the corresponding yield strength is 35% less than that at 300 K. In addition, the ultimate tensile strength of the Ni_{3}Sn_{4} nanowire at 700 K occurs at a strain (i.e., ε11=11.5%, as shown in Figure 12(b)-(3)) much less than that at 300 K (i.e., ε11=14.5%). Further increasing the strain would create a more excessive necking behavior, as shown in Figures 12(b)-(4) and 12(b)-(5). At last, the nanowire is completely ruptured at ε11=0.268.
(a) The tensile σ11-ε11 stress-strain evolution of the “22ϕ×100” Ni_{3}Sn_{4} nanowire model at 700 K and strain rate = 10^{−4}% ps^{−1}, (b) snapshots of fracture evolution of the Ni_{3}Sn_{4} atomic structure at various strains and (c) the residual strain after unloading versus the applied strain.
It is worthy of noting that there exhibits a unique phenomenon in the longitudinal section views of the atomic model of the “22ϕ×100” Ni_{3}Sn_{4} nanowireat 300 K and 700 K, as shown in Figure 13. The figure reveals that the Sn atoms tend to be dispelled to the surface of the Ni_{3}Sn_{4} nanowire while the Ni atoms would gather inside as the temperature increases from 300 K to 700 K. This may be due to the fact that Ni atoms would have much larger cohesive energy, as compared to Sn atoms. The hypothesis is confirmed by the literature finding [55], in which the cohesive energy for Ni and Sn atoms is about 4.44 eV/atom and 3.14 eV/atom, respectively.
Longitudinal section of the “22ϕ×100” Ni_{3}Sn_{4} nanowire constructed along the x-axis at (a) 300 K and (b) 700 K.
The σ11-ε11 stress-strain response of the Ni_{3}Sn_{4} nanowire model with the longitudinal dimension parallel to the x-axis, as shown in Figure 4, at 300 K and the strain rate of 10^{−2}% ps^{−1} is predicted to examine the effects of strain rate, and the result is shown in Figure 14. By comparing the result shown in Figure 9 (i.e., at the strain rate of 10^{−4}% ps^{−1} and 300 K), the stress-strain behavior of the Ni_{3}Sn_{4} nanowire is found to be markedly influenced by the applied strain rate. The elastic stiffness would increase with an increasing strain rate, thereby leading to much greater yield strength. Subsequently, the corresponding yield strain is also estimated through unloading process at different applied strains, and the result is given in Figure 14(b). From Figure 14(b), it is found that the yield strain is around 4.5%, and moreover, according to Figure 14(a), the corresponding yield strength is about 4.30 GPa. Comparatively, the yield strength and yield strain of the nanowire at the strain rate of 10^{−2}% ps^{−1} are about 114% and 66% higher than those of the low strain rate (10^{−4}% ps^{−1}) shown in Figure 9, respectively. It is also found that the ultimate tensile strength and strain are all raised by 91% and 15% as the strain rate increases from 10^{−4}% ps^{−1} to 10^{−2}% ps^{−1}.
(a) The tensile σ11-ε11 stress-strain evolution of the “22ϕ×100” Ni_{3}Sn_{4} nanowire model at 300 K and strain rate = 10^{−2}% ps^{−1}, (b) the residual strain after unloading versus the applied strain.
The normal elastic constants of Ni_{3}Sn_{4} single crystal, including C11, C22, and C33, at different strain rates and temperatures are predicted according to the slope of the stress-strain responses before the yield point is reached. To achieve the goal, a linear regression analysis is performed to determine the least squares best fit straight line of the discrete stress-strain data points obtained from the tensile MD simulation, and the slope of the straight lines yields the normal elastic constants. Furthermore, a factor λ, known as the correlation coefficient, is applied to quantify the degree of linearity between the stress-strain data points. In other words, the correlation coefficient measures the degree of linear correlation between stress and strain: in general, the greater the correlation coefficient, the better the least square fit. For instance, λ=1 represents the best fit and vice versa.
By the linear regression analysis, the least squares best fit straight lines of the σ11-ε11 stress-strain response of the “22ϕ×100” Ni_{3}Sn_{4} nanowire model with the longitudinal dimension parallel to the x-axis, as shown in Figure 4, at five different strain rates and at temperatures 300 K and 700 K are shown in Figures 15 and 16. The associated correlation coefficients λ are also shown in the figures, which are roughly in the range of 94–99%, suggesting that there is a significant linear correlation between stress and strain. The results also show that an increasing strain rate would stiffen the elastic properties of the nanocrystal and so upgrade the normal elastic constant C11, probably because of the change in deformation mechanism. The material stiffening behavior has been observed in numerous previous studies on a number of metals (see, e.g., [56–58]). Table 1 further demonstrates the predicted normal elastic constants, namely, C11, C22, and C33, as a function of the nanowire diameter, strain rate, and temperature by carrying out the tensile stress-strain tests on the three Ni_{3}Sn_{4} nanowire models, shown in Figure 4, with the longitudinal dimension parallel to the x-, y-, and z-axis, respectively. Likewise, it is clear to see that the normal elastic constants are likely to be totally independent of the strain rate as the strain rate is below 5 × 10^{−4}% ps^{−1}. Accordingly, the strain rate (5 × 10^{−4}% ps^{−1}) can be considered as the threshold value of the strain-rate dependence of the normal elastic constants of the Ni_{3}Sn_{4} nanowire, and the strain rates below which can be all treated as a quasistatic strain rate. Moreover, according to Table 1, the normal elastic constants of the Ni_{3}Sn_{4} nanowires would increase with an increasing nanowire diameter while considerably decreasing with an increasing temperature. The calculated normal elastic constants are further compared with the published theoretical data [35, 36]. There is a good agreement that can be found between present results and the published theoretical data. For example, Cheng et al. [35] performed the density functional theory calculations to predict the normal elastic constants of Ni_{3}Sn_{4} IMC. Their results showed that the normal elastic constants C11, C22, and C33 of the Ni_{3}Sn_{4} IMC are around 192.2, 176.5, and 204.6 GPa, respectively. In addition, these three normal elastic constants of Ni_{3}Sn_{4} IMC were also assessed by Lee et al. [36] through first-principles calculations. They found that the three normal elastic constants are about 155.0, 155.7, and 180.1 GPa, respectively. By comparing the present calculation results in Table 1 with those literature findings, a satisfactory consistency between them is observed. Figure 17 further plots the ultimate tensile strength of the Ni_{3}Sn_{4} nanowire model, shown in Figure 4, of three different sizes as a function of temperature and strain rate. Evidently, the ultimate tensile strength varies substantially with the temperature and applied strain. More specifically, an increasing temperature would reduce it while an increasing strain rate would upgrade it. This could be due to the fact that as the strain rate is below the threshold value, a decreasing strain rate would slightly lessen the normal elastic constants of the nanocrystal.
The normal elastic constants as a function of size, strain rate, and temperature.
ε˙ = 10^{−4}% ps^{−1}
ε˙ = 5 × 10^{−4}% ps^{−1}
ε˙ = 10^{−3}% ps^{−1}
ε˙ = 5 × 10^{−3}% ps^{−1}
ε˙ = 10^{−2}% ps^{−1}
300 K
700 K
300 K
700 K
300 K
700 K
300 K
700 K
300 K
700 K
14ϕ × 100 Å
C11
134.7
105.8
133.8
106.1
154.8
131.5
193.5
170.1
232.8
214.8
C22
132.5
102.6
133.4
103.3
153.6
129.6
191.3
169.2
230.1
213.7
C33
149.2
121.4
150.1
122.1
176.1
152.3
218.5
196.4
252.1
234.8
18ϕ × 100 Å
C11
156.1
119.6
156.6
120.1
182.3
160.1
220.3
191.0
244.7
211.5
C22
152.6
117.2
153.2
118.0
179.5
157.4
215.2
188.6
242.6
209.3
C33
169.5
139.5
169.8
140.2
201.1
175.6
236.6
209.4
272.5
241.1
22ϕ × 100 Å
C11
168.4
125.2
168.9
126.1
193.1
165.5
238.7
205.3
273.7
241.6
C22
153.7
122.3
154.1
122.5
189.6
164.3
234.1
201.6
268.1
239.4
C33
176.1
140.6
177.0
140.8
217.1
181.6
255.6
223.1
295.4
261.7
Determination of the C11 at 300 K and five different strain rates.
10^{−4}% ps^{−1}
5 × 10^{−4}% ps^{−1}
10^{−3}% ps^{−1}
5 × 10^{−3}% ps^{−1}
10^{−2}% ps^{−1}
Determination of the C11 at 700 K and five different strain rates.
10^{−4}% ps^{−1}
5 × 10^{−4}% ps^{−1}
10^{−3}% ps^{−1}
5 × 10^{−3}% ps^{−1}
10^{−2}% ps^{−1}
Effects of size, temperature, and strain rate on the ultimate tensile strength of the Ni_{3}Sn_{4} nanowire models constructed along the x-axis.
5. Conclusions
In this study, the mechanical properties and nonlinear stress-strain behaviors of Ni_{3}Sn_{4} single crystals and their size, temperature, and strain-rate dependences are extensively addressed through canonical ensemble MD simulation using Berendsen thermostat. The characteristics of the deformation evolution of the Ni_{3}Sn_{4} nanostructure during the tensile test are also addressed according to the stress-strain behavior and atomic rearrangement. Additionally, the tensile yielded strains of Ni_{3}Sn_{4} single crystals at different strain rates and temperatures are characterized through unloading process. Finally, the normal elastic stiffness constants of Ni_{3}Sn_{4} single crystal are characterized through the least squares best fit straight line of the discrete stress-strain data points using linear regression analysis, and most importantly, compared with the literature theoretical findings. It is shown that the present calculations are in good agreement with the literature theoretical data. Some important conclusions are drawn as follows.
From the radial distribution function analysis, it is found that Ni_{3}Sn_{4} single crystal in a one-dimensional nanowire configuration would become a highly disordered structure after thermal equilibration, thereby possessing amorphous-like mechanical behaviors and properties.
The stress-strain behavior of Ni_{3}Sn_{4} single crystal, such as the elastic modulus, yield strength, and strain and ultimate tensile strength and strain, is very sensitive to the nanowire diameter, applied strain rate, and temperature. Specifically, they would increase with an increasing strain rate and nanowire diameter while decreasing with temperature.
From the stress-strain evolution, the Ni_{3}Sn_{4} nanowire under tensile loading would experience linear elasticity, nonlinear elasticity, nonlinear plasticity, necking process, and fracture.
From the fracture evolution of the Ni_{3}Sn_{4} nanowires, the deformation evolution of the single-crystalline Ni_{3}Sn_{4} nanowire under uniaxial tensile stress can be majorly characterized by five major stages: bond straightening, bond breakage, inner atomic distortion, cross-section shrinking until forming a single-atom neck, and rupture.
The Ni_{3}Sn_{4} nanowire at a higher temperature is more inclined to undergo an early onset of atomic structure reconstruction; consequently, the yield strain and strength and ultimate tensile strength and strain would all be considerably reduced.
It is surprising to find that the Sn atoms in the Ni_{3}Sn_{4} nanowire model tend to reside in the surface of the Ni_{3}Sn_{4} nanowire while the Ni atoms would preferably stay inside the nanowire as the temperature increases from 300 K to 700 K, probably due to the larger cohesive energy in Ni atoms.
The normal elastic constants are likely to be totally independent of the strain rate as the strain rate is below 5 × 10^{−4}% ps^{−1}. Accordingly, the strain rate (5 × 10^{−4}% ps^{−1}) can be considered as the threshold value of the strain-rate dependence of the normal elastic constants of the Ni_{3}Sn_{4} nanowire and the strain rates below which can be all treated as a quasistatic strain rate.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work is partially supported by National Science Council, Taiwan, ROC, under Grants NSC101-2221-E-007-009-MY3 and NSC100-2221-E-035-036-MY3. The authors also thank the National Center for High-Performance Computing (NCHC) for computational resources support.
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