Mechanical Response of Mesoporous Amorphous NiTi Alloy to External Deformations

Amorphous metal foams based on Ni and Ti exhibit unique physical and mechanical properties, among which are low specific weight, low stiffness, high corrosion resistance and good biocompatibility [1, 2, 3, 4, 5, 6]. These properties depend primarily on size and geometry of the pores as well as on the stability of an amorphous matrix [7, 8, 9, 10]. Existing methods for production of the porous amorphous metallic alloys, such as the spark plasma sintering and three-stage synthesis method, usually make it possible to obtain the materials with the pores greater than $50$ nm [11, 12]. These methods do not guarantee the complete absence of crystalline nuclei, which can start to grow and destroy the amorphous structure of porous material. The cooling procedure of an equilibrium liquid melt with the ultrafast cooling rates (higher than $10^{6}$ K/s) and ultrafast heat dissipation can be applied to obtain a porous material with the amorphous structure and to prevent the crystal nucleation events [13, 14, 15]. In practice, these conditions are difficult to be realized for the most known metallic alloys [16, 17, 18, 19, 20].

Titanium nickelide alloy (Ni₅₀Ti₅₀ or NiTi) is the most famous functional material among intermetallic compounds especially due to the shape memory effect of this alloy [21, 22, 23, 24, 25, 26, 27]. Laser and spark plasma sintering are the main methods of powder metallurgy for synthesis of crystalline NiTi with micro- and macroporous structures [28, 29, 30]. The relative simplicity and accessibility of these methods make it possible to study the mechanical properties of this alloy. However, it is difficult to obtain the porous amorphous NiTi using the above methods primarily due to the high melting point of titanium nickelide, $T_{m}\simeq 1600$ K [31, 32]. Absence of comprehensive experimental measurements and numerical simulations is the reason why the physical and mechanical properties of porous amorphous NiTi are still poorly studied [33, 34, 35, 36, 37]. Improved computational methods and models are required to reproduce the structure and properties of the porous amorphous NiTi [23, 38].

In the present work, we construct a model of porous amorphous NiTi through rapid cooling of a low-density liquid melt. The effect of uniaxial tension, uniaxial compression and uniform shear on the porous amorphous samples with different porosity is studied. We show that the mechanical characteristics of porous amorphous NiTi differ significantly from the mechanical properties of its crystalline analogue. This is confirmed by comparing our simulation results with the available experimental data.

The ground crystalline state of NiTi at the temperature $T=0$ K is characterized by the B2 type cubic lattice. The Ni and Ti atoms are located inside the simulation box with the same length of edges $L_{x}=L_{y}=L_{z}\simeq 12.4$ nm. The periodic boundary conditions are applied in all the directions. The system consists of $N=109\,744$ atoms: $54\,872$ atoms of Ni and the same number of Ti atoms. Interaction energies between the atoms are determined by the modified embedded-atom method (MEAM) potential developed by Ko et al. for NiTi [39]. As found before in Ref. [40], this potential correctly reproduces the structural and dynamic properties of NiTi.

According to the MEAM formulation [41]], the total energy $E$ of the considered many-particle system is determined as

$$E=\sum_{i=1}\left(F_{i}(\bar{\rho}_{i}(r))+\frac{1}{2}\sum_{j\neq i}^{N}\phi_{ij}(r)\right).$$

(1)

Here, $\phi_{ij}(r)$ is the pair interaction between atoms $i$ and $j$ separated by a distance $r$. The embedding function $F_{i}(\bar{\rho}_{i}(r))$ is defined in the form

$$F_{i}(\bar{\rho}_{i}(r))=AE_{s}\bar{\rho}_{i}(r)\ln\bar{\rho}_{i}(r).$$

(2)

The parameter $A$ is the scaling factor for the embedding function and $E_{S}$ is the sublimation energy. The effective electron density $\bar{\rho}_{i}(r)$ is given by

$$\bar{\rho}_{i}(r)=2\rho_{i}^{(0)}(r)\left[1+\exp\left(-\sum_{l=1}^{3}t^{(l)}\left[\frac{\rho_{i}^{(l)}(r)}{\rho_{i}^{(0)}(r)}\right]^{2}\right)\right]^{-1},$$

(3)

where $\rho_{i}^{(0)}(r)$ is the spherically symmetric partial electron density; $\rho_{i}^{(l)}(r)$ ($l=1,\,2,\,3$) are the angular contributions (see Ref. [41]). According to Ref. [41], the quantities $\rho_{i}^{(0)}(r)$ and $\rho_{i}^{(l)}(r)$ depend on the atomic electron density $\rho_{j}^{a(l)}(r)$ of $j$th atom at a distance $r$ from $i$th atom:

$$\rho_{j}^{a(l)}(r)=\exp\left(-\beta^{(l)}\left[\frac{r}{r_{e}}-1\right]\right).$$

(4)

Here, $t^{(l)}$ and $\beta^{(l)}$ represent the weight factors and the dumping coefficients for the partial electron densities; $r_{e}$ is the equilibrium nearest-neighbor distance in an equilibrium reference structure. In the MEAM, only the first nearest-neighbor interactions are considered. Therefore, the pair potential $\phi(r)$ between two atoms separated by a distance $r$ is defined in the form

$$\phi(r)=\frac{2}{Z}\left[E^{u}(r)-F(\bar{\rho}_{0}(r))\right],$$

(5)

where

$$E^{u}(r)=-E_{S}(1+\alpha+d\alpha^{3})e^{-\alpha},$$

(6)

$$\alpha=\left(\frac{9BV}{E_{S}}\right)^{1/2}\left[\frac{r}{r_{e}}-1\right].$$

(7)

Here, $E^{u}(r)$ is the universal function for a uniform expansion or contraction in the reference structure [42]; $Z$ is the number of nearest-neighbor atoms; $\bar{\rho}_{0}$ is the background electron density for the reference structure; $B$ is the bulk modulus; $V$ is the atomic volume of solid elements; $d$ is an adjustable parameter. The values of the quantities $t^{(l)}$, $\beta^{(l)}$, $r_{e}$, $E_{S}$, $A$, $B$, $V$ and $d$ are given in Ref. [39] for pure Ni and Ti systems as well as for NiTi alloy.

Initially, the crystalline sample has been melted at the constant pressure $p=1$ atm by heating above the melting (liquidus) temperature $T_{m}\simeq 1600$ K. The equilibrated liquid samples at the temperatures $T=3500$, $6500$, $7000$, $7500$, $8000$ and $8600$ K have been prepared. Note that all these temperatures do not exceed the boiling temperature $T_{b}\simeq 8750$ K of NiTi [43].

To generate the homogeneous amorphous sample without pores, a liquid melt at the temperature $T=3500$ K has been isobarically cooled with the rate $10^{13}$ K/s to a state with the temperature $T=300$ K at the pressure $p=1$ atm. The generated amorphous sample has the density $\rho_{0}\simeq 6.17$ g/cm³ that is close to the density of crystalline NiTi, $\simeq 6.45$ g/cm³, at the same temperature [44]. Note that the glass transition temperature $T_{g}$ of titanium nickelide estimated by T. Rouxel and Y. Yokoyama is $\sim 750$ K [45].

Amorphous samples with various porosity have been prepared by isochoric cooling of melt samples equilibrated at the temperatures $T=6500$, $7000$, $7500$, $8000$ and $8600$ K. Each sample has been cooled with the rate $10^{13}$ K/s to the temperature $300$ K. It should be noted that with such ultrafast cooling, a negative internal pressure arises in the melt, and the solidifying samples do not form a dense homogeneous solid phase. As a result, the melt transforms into porous amorphous solid, where the linear size of pores varies within the interval from $2$ nm to $10$ nm [Fig. 1]. After ultrafast cooling, each sample has been aged at the state with the temperature $T=300$ K and the pressure $p=1$ atm over the time $1$ ns. Amorphous samples prepared in this way have the different densities $\rho\simeq 5.71$, $5.24$, $4.87$, $4.5$ and $4.01$ g/cm³. These values of the density $\rho$ are much lower than the density $\rho_{0}$ of the homogeneous amorphous system.

The porosity of the system is estimated by the known ratio [47]

$$\phi=\left(1-\frac{\rho}{\rho_{0}}\right)\times 100\%.$$

(8)

In the case of the homogeneous amorphous system, we have $\rho=\rho_{0}$ and the coefficient of porosity is $\phi=0$. The porosity coefficient $\phi$ takes the values $7.5$, $15$, $21$, $27$ and $35$% for porous amorphous NiTi-samples. Visualization of the porous structure given in Fig. 1 reveals predominance of the pore networks within the samples at high porosity, $\phi>20$% [Figs. 1(a) and 1(b)]. The samples with a low degree of porosity (for example, at the porosity $\phi=7.5$%) contain the isolated pores of a roughly spherical shape [Fig. 1(c)].

It is well-known that the internal elastic stresses in the volume of solid arise as a result of external mechanical deformations. These stresses tend to return a deformed solid to its original shape  [48]. Elastic response is defined by the stress tensor, the nine elements of which are the force components acting on an element of a unit area:

$$\mathbf{\sigma}_{\alpha\beta}(t)\equiv\mathbf{\sigma}_{\alpha\beta}=\begin{bmatrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\end{bmatrix}.$$

(9)

We evaluate the tensor components $\sigma_{\alpha\beta}$ through the Irwin-Kirkwood equation [49, 50]

$$\sigma_{\alpha\beta}=-\frac{1}{V}\left(\sum_{i=1}^{N}m_{i}v_{i\alpha}v_{i\beta}+\sum_{i=1}^{N}\sum_{j>i}^{N}r_{ij\alpha}F_{ij\beta}\right).$$

(10)

Here, the index $\alpha=x,y,z$ characterizes the normal vector of a unit area, and the index $\beta=x,y,z$ corresponds to a force component, $V$ is the volume of a system, $m$ is the mass of the $i$th atom; $v_{i\alpha}$ and $v_{i\beta}$ are the $\alpha$ and $\beta$ components of the peculiar velocity of the $i$th atom; $F_{ij\beta}$ is the $\beta$ component of the force between the particles with the labels $i$ and $j$; $r_{ij\alpha}$ is the $\alpha$ component of the radius-vector between the particles $i$ and $j$. The diagonal components $\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{zz}$ in Eq. (9) are the normal stresses, which characterize the action of an external force to the unit surface of a solid in the orthogonal direction. The upper and lower symmetric triangular parts of the tensor (9) are the shear components that characterize the action of force parallel to the unit surface of solid.

Response of considered system to various mechanical deformations is estimated by analysis of the stress-strain curves. The main elastic moduli such as the tensile modulus (or Young’s modulus) $E_{t}$, the compression modulus $E_{c}$ and the shear modulus $G$ are determined by the slope of the linear (elastic) part of the corresponding stress-strain curve [see schematic Fig. 2(a)]. The slope of this linear part have weak dependence on the strain rate [51, 52]. Note that the high elastic moduli will correspond to materials with more pronounced elasticity.

In the given study, the amorphous NiTi samples are exposed to the following types of deformations:

• $\bullet$

  Shear along the $x$-direction with the constant shear rate $\dot{\gamma}=4\times 10^{10}$ s^-1. For an elastic regime corresponding to small shear deformations [48], the shear modulus $G$ is defined as the ratio of the shear stress $\tau_{xy}\equiv\sigma_{xy}$ to the shear strain $\gamma$ [see Figs. 2(a) and 2(d)]:

  $$G=\frac{\tau_{xy}(t)}{\gamma(t)},$$

  (11)

  where

  $$\gamma(t)=\dot{\gamma}t.$$

  (12)

  The shear stress simulations are realized via the method of non-equilibrium molecular dynamics in Lammps software package [50, 53]. In these simulations, atoms are located inside a triclinic simulation box. The shape of this simulation box changes due to the externally applied stress. For example, the simulation box changes the tilt in the $xy$-plane at shear applied along the $x$-direction. In Lammps, the tilt of the triclinic box is usually limited by half of the length of the box side parallel to $x$-axis. Therefore, the simulation box flips in the $xy$-plane at exceeding this limited length. These flips are cyclically repeated until the simulation completes. During each flip event, atoms are remapped into the flipped simulation box.

• $\bullet$

  Uniaxial tension (along the $x$-axis) with the fixed tensile rate $4\times 10^{10}$ s^-1. The tensile modulus $E_{t}$ is defined as the ratio of the tensile stress $\sigma_{xx}$ to the tensile strain $\epsilon$ [see Figs. 2(a) and 2(b)]:

  $$E_{t}=\frac{\sigma_{xx}(t)}{\epsilon(t)},$$

  (13)

  where

  $$\epsilon(t)=\frac{L_{x}(t)-L_{x}^{(init)}}{L_{x}^{(init)}}.$$

  (14)

  Here, $L_{x}^{(init)}$ is the length of the simulation box before the deformation, and $L_{x}(t)$ is this length at time $t$ after the start of deformation.

• $\bullet$

  Uniaxial compression along the $x$-axis with the fixed rate $2\times 10^{10}$ s^-1. The compression modulus $E_{c}$ is defined by the ratio

  $$E_{c}=\frac{\sigma_{xx}(t)}{\xi(t)},$$

  (15)

  where $\sigma_{xx}$ is the compressive stress [see Figs. 2(a) and 2(c)], and $\xi$ is the compressive deformation:

  $$\xi(t)=\frac{L_{x}^{(init)}-L_{x}(t)}{L_{x}^{(init)}}.$$

  (16)

Note that the strain rates of the order $10^{10}$ s^-1 are achievable experimentally, for example, when a shock wave from an explosion impacts on a solid material [54, 51]. The strain with rate $\sim 10^{10}$ s^-1 makes it possible to identify structural changes in the system at nanosecond time scales accessible to molecular dynamics simulations.

In the present work, the mechanical properties of the mesoporous amorphous NiTi alloy were studied by the method of non-equilibrium molecular dynamics simulations. Samples of the system with different porosity are analysed at uniaxial tension, uniaxial compression, and uniform shear. The stress-strain curves are found, and the values of the main mechanical characteristics as functions of system porosity are calculated. It was shown that the Young’s modulus computed at tension and compression of porous amorphous NiTi exceeds the experimental values for crystalline analogues with the same porosity. This indicates that the resistance of the porous amorphous system to external deformations is higher than for the case of the porous crystalline system. The Young’s modulus tends to decrease with increasing porosity according to Nielsen’s power-law (in the case of tensile stress) and according to power-law of Bruno et al. (in the case of compressive stress). This tendency is in good agreement with the results of the available experimental data obtained for the porous crystalline NiTi and the femoral cortical bone.

We find that the degree of the system porosity is correlated with the pore geometry. The amorphous alloy with the porosity $7.5$%$\leq\phi\leq 15$% contains mainly isolated pores of the spherical shape. At the porosity $\phi>15$%, the pores form a percolating network. It has been established that the amorphous solid NiTi with the isolated spherical pores better resists to compression, while it is less resistant to tensile and shear deformations. The amorphous NiTi alloy with a percolating network of pores is able to withstand large deformations at tension and shear. The results of the present work complement existing studies of the mechanical properties of Ni and Ti-based alloys with the mesoporous structure, and these results can be demanded in the development of design methods of porous amorphous alloys with the necessary characteristics of the porosity.

This study is supported by the Russian Science Foundation (project No. 19-12-00022); the computational part of the study is supported by the Russian Foundation for Basic Research (project no. 18-02-00407-a). AVM acknowledges the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” for support.