Study of the Physical and Mechanical Properties of Titanium in Volume by the MEAM Method

Yahnn J. Mighensle Mimboui^1,2,3*, Alain S. Dzabana Honguelet^1,2,3, Timothée Nsongo^1,2,4

¹Faculty of Science and Technology, Marien Ngouabi University, Brazzaville, Congo.
²Research Group on Physical and Chemical Properties of Materials, Brazzaville, Congo.
³Association Alpha Sciences Beta Technologies, Brazzaville, Congo.
⁴Geological and Mining Research Centre, Brazzaville, Congo.

DOI: 10.4236/mnsms.2024.141003                

Abstract

In this work we present the results of our study on the physical and mechanical properties of titanium in volume. The work consisted in determining its physical and mechanical properties under different crystallographic structures (HCP, FCC, BCC and SC) using the Modified Embedded Atom Method (MEAM) and the MEAM potential of titanium. We used the LAMMPS calculation code, based on classical molecular dynamics, to determine the most stable structure of titanium, which is the hexagonal compact structure (HCP) with crystal parameters a = 2.952 Å and c = 4.821 Å and a cohesion energy of -4.87 eV. This structure is seconded by the cubic centred structure (BCC) with a lattice parameter a = 3.274 Å and a cohesive energy of -4.84 eV. It was shown that titanium can crystallise into a third structure which is the face-centred cubic (FCC) structure with a lattice parameter a = 4.143 Å and a cohesive energy of -4.82 eV. The results obtained in this study were compared with the theoretical results and showed considerable agreement.

Keywords

MEAM Potential, LAMMPS Code, Molecular Dynamics, Elastic Constants, Modules, Ovito

1. Introduction

Titanium is a material whose market is perceived quite differently from other industrial materials. Its use implies high technology, purity, rarity, quality and unequalled strength. It represents 0.6% of the earth’s crust and is ranked 4th among the most abundant metals after iron, aluminum and magnesium [1] .

Titanium as a chemical element has been known for more than 200 years, but it is only since 1920 that its real properties were discovered by Van Arckel and De Boer [2] .

It was in 1791 that William Gregor, a British reverend, mineralogist and chemist discovered titanium. While examining the sand of the Helford River in the Menachan Valley in Cornwall, he isolated what he called black sand, now known as ilmenite [3] [4] .

Professor Nsongo Timothée studied the structure and adhesion of thin films of titanium and titanium nitride prepared by sputtering [5] . In this study, he investigated the growth and structure of titanium and titanium nitride layers prepared on molybdenum, copper and tantalum by radio frequency magnetron and direct current triode sputtering. He also characterised the adhesion of titanium and titanium nitride to the different substrates used by the “scratch” method in order to relate the growth conditions and the resulting structural characteristics to the adhesion.

Stéphane GROSSO worked on Ti, TiN and TiOx architectural coatings developed by sputtering on stainless steel wires [6] . The main objective of his thesis was to understand the effects of the architecture of the coatings and of the processing parameters on the physicochemical characteristics of the Ti, TiN and TiO coatings and thus on their properties of use.

Abel Dominique EBOUNGABEKA, under the direction of Professor Timothée NSONGO, recently conducted a study on the reactivity of titanium dental implants in a salivary environment in the presence of food consumed in the Republic of Congo [7] .

The aim of this work is to carry out a study of the physical and mechanical properties of titanium using the MEAM method, under the LAMMPS code, in order to determine its most stable crystal structure.

2. Methodology

We ran a simulation under the LAMMPS code version 2020 with the executable lmp_mpi, under the Windows operating system, using the MEAM potentials found in the database at https://www.ctcms.nist.gov/potentials/system (Table 1).

The MEAM potential of Titanium used in this work was developed by Y.-M. Kim, B.-J. Lee, and M.I. Baskes (2006), and its parameters are aligned in the following table [8] .

This potential was used to calculate the cohesive energies under different crystallographic structures using the MPCV4 application.

The calculations were simulated for periodic crystallographic structures for 2 × 2 × 2 mesh under Lamps (Figure 1).

3. Results and Discussion

In the following paragraphs we present the results obtained under the LAMMPS

(a) (b)

Table 1. Titanium potential and screening parameters.

Figure 1. Rose energy of titanium in HCP structure.

code using the MEAM potentials, these results were also calculated by the MPCV4 application under Windows 10, 64 bits.

The crystalline parameters, elastic constants and structural stability moduli for different titanium structures are presented.

3.1. Crystalline Parameters

The presented crystal parameters and volumes have been calculated for HCP, CFC, BCC, SC structures and are presented in Table 2.

Table 2. Crystal parameters by structure and crystallographic volume.

The most voluminous structure is the HCP structure followed by the BCC structure, SC and finally the FCC structure.

3.2. Cohesive Energy

The MEAM potential of Titanium has allowed us, thanks to the MPCV4 application, to represent the cohesion energies for different structures as shown in Figure 2 and Figure 3.

We also calculated the cohesion energies in different crystallographic structures for titanium; we realised that the HCP structure of the Ti type is the most stable with a cohesion energy of −4.87 eV followed by the BCC structure with a cohesion energy of −4.84 eV (Table 3).

The evolution of the cohesion energy per structure is shown in Figure 4 and Figure 5.

Although the number of atoms per cell increases in some structures, the cohesive energy becomes very large, reflecting the instability of the structure; the HCP and FCC structures are very similar.

3.3. Mechanical Properties

For the temperature of 298 Kelvin, we calculated the elastic constants of titanium in the hexagonal phase and the corresponding elastic moduli.

3.3.1. Elastic Constants

Elastic constants $C_{ij}$ are essential parameters for predicting the physical properties and mechanical stability of materials [9] [10] .

For elements with a cubic structure, the criterion for predicting structural stability is as follows [11] [12] :

$C_{11}-C_{12}>0,C_{11}>0,C_{44}>0,C_{11}+2C_{12}>0$ (1)

For elements that have a tetragonal structure, we have this [13] [14] :

$C_{11}>\left|C_{12}\right|,C_{44}>0,C_{66}>0,C_{33}\left(C_{11}+C_{12}\right)>2C_{13}^2$ (2)

And for elements with a hexagonal structure, we have [15] [16] :

$C_{44}>0,C_{11}-C_{12}>0,C_{33}\left(C_{11}+C_{12}\right)>2C_{13}^2$ (3)

However, the matrix linking the deformations and the elastic constants is represented by the following relation.

Figure 2. Cohesive energies of titanium in HCP and FCC.

Figure 3. Cohesive energies of BCC and SC titanium.

Table 3. Cohesive energy per structure.

Figure 4. Cohesion energy by crystal structure.

Figure 5. Cohesive energy per atom number.

$\left(\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\end{array}\right)=\left(\begin{array}{cccc}{C}_{11}& C_{12}& C_{13}& C_{14}\\ C_{21}& C_{22}& C_{23}& C_{24}\\ C_{31}& C_{32}& C_{33}& C_{34}\\ C_{41}& C_{42}& C_{43}& C_{44}\end{array}\right)\left(\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\end{array}\right)$ (4)

The elastic constants for the hexagonal structure of titanium have been calculated and are presented in Table 4.

The exploitation of these data allows us to compare the elastic constants between them and between the theoretical and experimental data through the Figure 6 and Figure 7.

The order of magnitude is well defined by C33 > C11 > C12 > C13 > C44 and is preserved in the theory as well as for experimental data.

3.3.2. Elastic Modules

In this section we present the elastic moduli according to the Voigt-Reuss-Hill scheme. These moduli reflect the stiffness and flexibility of the material against external excitations and they reflect the direction of the resulting deformation or the resulting stress. They are also referred to as elastic coefficients, which are simply the engineer’s moduli for the properties of nanomaterials. We give here the mathematical relations between Young’s modulus, Poisson’s ratio and shear coefficient with the elastic constants [19] [20] .

• The H-symbol stiffness is an important parameter for materials that can be estimated as the ability to resist localized deformation [21] [22] . Defects and grain size of materials are of great importance with regard to hardness. The hardness H is given by the following semi-empirical formula [23] :

$H=\frac{\left(1-2v\right)}{6\left(1+v\right)}E$ (5)

• The elastic moduli (Bulk modulus B, Shear modulus G and Young’s modulus E) are estimated by the Voigt-Reuss-Hill method. Generally, the larger the B, the greater the resistance of the material to change in volume.

Table 4. Comparison of theoretical and experimental constants.

Figure 6. Elastic constants of titanium in point cloud.

Figure 7. Values of Elastic constants in a histogram.

• The elastic anisotropy $A^U$ plays a vital role in physical/mechanical processes such as fracture behavior and phase transformations [24] . It is given by the following formula [25] :

$A=U5\frac{G_v}{G_R}+\frac{B_v}{B_R}-6$ (6)

where $G_v$ and $G_R$ represent Voigt and Reuss’ Shear modulus, $B_v$ and $\frac{B_v}{B_R}$ are the Bulk modulus of Voigt and Reuss.

Table 5. Mathematical expressions of elastic moduli for a hexagonal structure [26] [27] [28] .

Table 6. Elastic modules.

In the table below, we present all the elastic moduli in different systems of approaches for a crystallographic structure (Table 5).

The results obtained from the expressions of the previous modules are presented in Table 6.

It can be seen from this table that the elastic moduli calculated from the Lammps code using the MEAM potential is in good agreement with the theoretical and experimental results. Hence the MEAM potential used is an appropriate potential for the study of metals as it takes into account steric and angular effects in the system studied.

4. Conclusion

During this study on the physical and mechanical properties of titanium in volume using the Lammps calculation code and the MEAM potential, we studied this metal in the four different structures namely BCC (cubic centred), FCC (face centred cubic), HCP (hexagonal compact) and SC (simple cubic) in order to determine its most stable structure and its physical and mechanical properties. We found that the most stable structure is the compact hexagonal structure with a mesh parameter a = 2.97 Å, c = 4.821 Å and a cohesive energy Ecoh = −4.87 eV followed by the face-centred cubic (BCC) structure and then the face-centred cubic (FCC) structure which is a transient structure. Finally, we calculated the elastic constants and moduli of elasticity and it was found that our results are in accordance with the theoretical results and this proves sufficiently the effectiveness of the MEAM potential used to carry out this study.

Acknowledgements

We would like to thank the Research Group on Physical, Chemical and Mefchanical Properties of Materials (GRPPMM) for allowing us to finalize this work and we would like to thank the Alpha Sciences Beta Technologies association for its support from the very beginning of the writing of this article.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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