Atomistic Investigation of Titanium Carbide Ti₈C₅ under Impact Loading

Abstract

Titanium carbides attract attention from both academic and industry fields because of their intriguing mechanical properties and proven potential as appealing candidates in the variety of fields such as nanomechanics, nanoelectronics, energy storage and oil/water separation devices. A recent study revealed that the presence of Ti₈C₅ not only improves the impact strength of composites as coatings, but also possesses significant strengthening performance as an interlayer material in composites by forming strong bonding between different matrices, which sheds light on the design of impact protection composite materials. To further investigate the impact resistance and strengthening mechanism of Ti₈C₅, a pilot Molecular Dynamics (MD) study utilizing comb3 potential is carried out on a Ti₈C₅ nanosheet by subjecting it to hypervelocity impacts. The deformation behaviour of Ti₈C₅ and the related impact resist mechanisms are assessed in this research. At a low impact velocity ~0.5 km/s, the main resonance frequency of Ti₈C₅ is 11.9 GHz and its low Q factor (111.9) indicates a decent energy damping capability, which would eliminate the received energy in an interfacial reflection process and weaken the shock waves for Ti₈C₅ strengthened composites. As the impact velocity increases above the threshold of 1.8 km/s, Ti₈C₅ demonstrates brittle behaviour, which is signified by its insignificant out-of-plane deformation prior to crack initiation. When tracking atomic Von Mises stress distribution, the elastic wave propagation velocity of Ti₈C₅ is calculated to be 5.34 and 5.90 km/s for X and Y directions, respectively. These figures are inferior compared with graphene and copper, which indicate slower energy delocalization rates and thus less energy dissipation via deformation is expected prior to bond break. However, because of its relatively small mass density comparing with copper, Ti₈C₅ presents superior specific penetration. This study provides a fundamental understanding of the deformation and penetration mechanisms of titanium carbide nanosheets under impact, which is crucial in order to facilitate emerging impact protection applications for titanium carbide-related composites.

1. Introduction

In recent decades, tremendous efforts on the exploration of titanium matrix composites have been conducted because of their outstanding mechanical properties [1,2] and tuneable potential [2,3,4,5,6]. Titanium carbide is a widely adopted strengthening material in composites and has received immense attention from civil automobile, aerospace as well as military industries, owing to its superior bending rigidity [7], Young’s modulus [8], strength to weight ratio, wear and corrosion resistance [9], Oil/water separation capability [10], etc. To this end, an in situ synthesis of titanium carbide is realized via multiple approaches such as supersonic flame spraying process [11], high temperature and high-pressure environment synthesis [12], low temperature salt bath carburizing [13] and state-of-art high energy beam setups, such as laser cladding [14]. Micro-meter titanium carbide can be achieved via the dissolution-precipitation mechanism utilizing a high-energy laser.

Varieties of titanium carbides have been successfully realized so far; however, constrained by the lateral size of sheets and complex oxygen-containing function groups observed in the state-of-art synthesis process, most studies on monolithic titanium carbides have been carried out via in silico approaches such as Molecular Dynamics (MD) and Density Function Theory (DFT)-based simulation [15]. Deploying an empirical potential energy function and embedded atom method, a MD study suggests the bending rigidity of 2D titanium carbides can be as high as ~49.55 eV, which is much more significant than graphene and MoS₂ (2.3 and 9.61 eV, respectively) [7]. A DFT study on functionalized titanium carbides suggests that the oxygen group not only enhances ideal strength, but also induces a strong anisotropy for titanium carbides as the material subjected to tensile loading [16]. Recently, a follow-up MD study utilizing COMB3 potential confirmed the above findings; moreover, the Young’s modulus of the assessed titanium-based carbides ranged between 133 and 517 GPa, which shows its potential in nanomechanics, nanoelectronics and energy storage applications [8].

In addition to having outstanding mechanical properties, pristine titanium carbides are also known for their strengthening material/matrix in varieties of composites. Titanium carbide (Ti₈C₅) thin film on a Ti₆Al₄V alloy surface realized via low temperature carburizing salt bath possesses significantly improved friction and wear properties and its surface hardness reaches ~953.28HV, which is 1.57 times higher than Ti₆Al₄V [13]. Moreover, Ti₈C₅ nanolayers/nanoparticles in situ formed at an interface of RGO/CuTi composite function as ‘rivets’ show sizes of up to ~100 nm. The presence of these Ti₈C₅ ‘rivets’ raises the interfacial strength, increases load transfer efficiency between layers and clearly presents a 53.9% improvement in tensile strength compared with composites without Ti₈C₅ in their interface [12]. The strengthening behaviour is also confirmed by a study on CNTs/Cu-Ti composites with strong Ti₈C₅ interfacial bonding [17]. Moreover, the quasi-static loading experiments’ shock loading result states that Ti₈C₅ formed between diamond and titanium under a sintering temperature improves the impact strength of diamonds as coatings, which clearly suggests their impact protection potential [18].

Considering the mechanical strengthening behaviour of Ti₈C₅ in composites (such as graphene-copper composite) under quasi-static uniaxial tensile loading, its performance under various impact loading condition is rarely systematically discussed in the literature; however, it is critical to facilitate its application in bullet-proof fields as well as promote the interface design of advanced composites for impact resist applications [12,19,20]. Because of the experimental manipulation complex and small lateral dimensions of available Ti₈C₅ sheets [12], a pilot study on impact resistance of Ti₈C₅ under various impact velocities is carried out utilizing the MD method. In this study, resonance pattern, stress distribution capability as well as the specific penetration energy of Ti₈C₅ are accessed. The performance and phenomenon of a Ti₈C₅ nanosheet under impact loading are mainly compared with a thin copper layer and single-layer graphene in this pilot study.

2. Materials and Methods

The mechanical behaviour of Ti₈C₅ under hypervelocity impact was assessed through a series of large-scale MD simulations performed with the aid of the open-source package LAMMPS [21]. A spherical diamond projectile with a radius of 25 Å and a square 50 × 50 nm² Ti₈C₅ nanosheet [22] were considered. The (010) plane of the Ti₈C₅ nanosheet faced the projectile. Initial velocities ranging from 5 to 70 Å/ps (i.e., 0.5–7 km/s) were assigned to the projectiles. The boundaries of the Ti₈C₅ nanosheets were fixed during the impact process (Figure 1a), whereas the projectile was located above the centre of the Ti₈C₅ sample with an initial distance of ~20 Å. Considering the high thermal fluctuations generated during the impact process, the simulation temperature was set to be 10 K. Simulations with higher environment temperatures up to 700 K were also performed for validation, whereby the obtained penetration energy for the same impact velocities was maintained at the same level (Supplementary Materials, Figure S1). However, noting the negative correlation between temperature and material strength [23,24], a further increase to the environmental temperature may lead to lower penetration energy. A time step of 0.1 fs was selected for the simulations.

The third generation Charge-Optimized Many-Body potential (COMB3) was employed to describe the C-Ti atomic interactions in the Ti₈C₅ nanosheet, as well as the Cu–Cu interaction in copper, as it was fully optimized in order to determine the binding energy of carbon- and metal-based systems [25,26,27,28,29]. The COMB3 potential is defined as follows,

$$E_{COMB3}=V^{ES}+V^{SHORT}+V^{vdW}+V^{CORR}$$

(1)

where $V^{ES}$ is an electrostatic term, which is the sum of atomic ionization energies and energy related to the Coulomb interaction. $V^{SHORT}$, $V^{vdW}$ and $V^{CORR}$ represent the short range, van der Waals and correction energy term, respectively. The Tersoff potential was utilized to simulate the atomic interactions in diamonds, which has been shown to describe the binding energy for carbonaceous system well [30]. The interactions between the diamond projectile and nanosheets were described by Morse potential [31]. Prior to the impact process, a conjugate gradient algorithm was applied to conduct energy minimization, and then the Nose–Hoover thermostat [32] was employed to equilibrate the whole system at 10 K for 2000 fs under the canonical ensemble. The equations of motion were integrated with time using the velocity Verlet algorithm [33] and no periodic boundary conditions were adopted. To reproduce the energy conversion from kinetic energy to potential energy, no thermostat was applied during the impact process. During the simulation, the atomic stress was calculated based on the virial stress ${\mathsf{\Pi }}^{\alpha \beta }$ [34],

$${\mathsf{\Pi }}^{\alpha \beta }=\frac{1}{\mathsf{\Omega }}{{\displaystyle \sum }}_i{\varpi }_i{\pi }_i^{\alpha \beta }, {\pi }_i^{\alpha \beta }=\frac{1}{{\varpi }_i}\left(-m_i{v}_i^{\alpha }{v}_i^{\beta }+\frac{1}{2}{{\displaystyle \sum }}_{j\ne i}{F}_{ij}^{\alpha }{r}_{ij}^{\beta }\right)$$

(2)

Here, ${\pi }_i^{\alpha \beta }$ is the atomic stress associated with atom i. ${\varpi }_i$ is the effective volume of the ith atom and $\mathsf{\Omega }$ is the volume of the whole system. $m_i$ and $v_i$ are the mass and velocity of the ith atom, respectively. $F_{ij}$ and $r_{ij}$ are the force and distance between atoms i and j, respectively, and the indices $\alpha$ and $\beta$ denote the Cartesian components. The volume of the Ti₈C₅ nanosheet is estimated by assuming the Ti₈C₅ as a continuum media with a thickness of h (13.6 Å). Considering that adopting different atomic volumes will alter the magnitude of the stress, it will not however change the trends of the results as focused on in this paper. Considering the complex stress states during impact, we tracked the Von Mises stress ${\mathsf{\sigma }}_{VM}$ in the nanosheet based on the atomic virial stress (Equation (2)), as calculated by

$${\mathsf{\sigma }}_{VM}=\sqrt{({\left({\mathsf{\sigma }}_x-{\mathsf{\sigma }}_y\right)}^2+{\left({\mathsf{\sigma }}_y-{\mathsf{\sigma }}_z\right)}^2+{\left({\mathsf{\sigma }}_y-{\mathsf{\sigma }}_z\right)}^2+6({\mathsf{\sigma }}_{xy}{}^2+{\mathsf{\sigma }}_{xz}{}^2+{\mathsf{\sigma }}_{yz}{}^2))/2}$$

(3)

Initially, we focused on the impact performance of the Ti₈C₅ nanosheet under the impact velocity of 3 km/s. In the ideal vacuum condition, the total energy change in the projectile equals the change in the Ti₈C₅ nanosheet. It is notable that during the impact process, although the projectile exhibited an ignorable deformation, a notable amount of potential energy change in the projectile ($\Delta E_{ball,pe}$) was observed during the impact with Ti₈C₅ (Figure 1b). After perforation, the total energy loss in the projectile ($\Delta E_{ball,tot}$ which is the sum of $\Delta E_{ball,pe}$ and $\Delta E_{ball,ke}$) remains constant (Figure 1c); thus, $\Delta E_{ball,tot}$ is taken as the penetration energy ($E_p$) [35,36]. It was noticed that the NVE ensemble was adopted in the impact process and total energy of the system was constant during the impact process, and the energy loss in the projectile was equal to the energy gain in the nanosheet $\Delta E_{NS,tot}$ (where $\Delta E_{NS,tot}$ equals the sum of nanosheet potential energy gain $\Delta P{e}_{NS}$ and kinetic energy gain of nanosheet $\Delta K{e}_{NS}$). As such, $E_p=\Delta E_{ball,tot}=\Delta E_{ball,pe}+\Delta E_{ball,ke}=\Delta E_{NS,tot}=\Delta P{e}_{NS}+\Delta K{e}_{NS}$.

3. Results and Discussion

3.1. Deformation Characteristics

To acquire the deformation process, Figure 2a–c (top panel) illustrates the atomic configurations of Ti₈C₅ at different deformation stages, namely the crack initiation, crack expansion and fully developed cracks, and the results are compared with the copper nanosheet that has a similar number of atom layers under the same experiment setup (Figure 2d–f). As expected, the crack initiates at the geometry centre, where accumulation of stress is observed. The nanosheet experiences insignificant out-of-plane deformation ~11.49 Å prior to the propagation of cracks and stress (Figure 2a) compared with copper (Figure 2d) and carbonaceous nanosheets [36,37,38], which signifies brittle behaviour. During the crack propagation phase, the stress that accumulates around the impact region starts to re-distribute and the stress distribution in the deformed region of Ti₈C₅ exhibits a circular pattern (Figure 2b,c), whereas copper (lattice orientation is [100], [010], [001] for x, y and z direction respectively) demonstrates an X-shaped pattern in the highlighted region (Figure 2d–f). In addition to the stress accumulation, dislocation is also often observed in this highlighted region. This phenomenon is considered as a result of the lattice structure and lattice orientation. Unlike the armchair and zigzag kicking fracture mechanisms observed in carbonaceous nanosheets [36,37,38], the cracks in Ti₈C₅ and copper propagate along all directions and the same crack shape is roughly maintained during the whole process. Full perforation of Ti₈C₅ takes place at ~4.0 ps, which is significantly earlier than copper and the fully developed cracks for Ti₈C₅ are ~16.8% bigger than copper in terms of radius (Figure 2c).

Higher impact velocity induces more severe local deformation. As such, extensive elastic deformation of the nanosheet is not expected. When tracking the number of discrete debris in nanosheets after the crack propagation stage (Figure 3), it presents a positive correlation with impact velocity. For an ultra-high impact velocity ~6 km/s, the contact region melts immediately as the projectile reaches the nanosheet and creates more discrete debris for all the cases presented in Figure 3. Among all tested samples, Ti₈C₅ generally presents a higher number of discrete debris in all examined impact velocity amplitudes. Moreover, for nanosheets subject to the high impact velocity of 6 km/s, most debris of Ti₈C₅ tends to be comprised of smaller atom groups (compared with copper), whereas the debris of graphene is more likely to be comprised of discrete atoms. Additionally, we conducted additional simulations by varying the time step from 0.1 fs to 0.5 fs, from which the same results were obtained.

3.2. Resonance and Damping

Energy dissipation associated with resonance and damping are rarely discussed in the past shock loading or hyper velocity impact studies; however, these characteristics are essential for strengthening the matrix in laminar composites, as materials with different resonance frequencies and outstanding quality factors ($Q$) could eliminate the energy in the interfacial reflection process and weaken shock waves [20,39]. Utilizing the same experiment set up for all cases, oscillations in a relatively short period of 10 ps generated via various velocity impacts are recorded for amplitude and frequency comparisons. Two close observation points labelled in Figure 1a are selected for all the cases, ensuring that the captured oscillation is not influenced by crack propagation. Generally, increasing the impact velocity, oscillation amplitude and frequency does not cause significant variations in all cases. In addition, compared with graphene (Figure 4c,d) and copper (Figure 4e,f) nanosheets, Ti₈C₅ (Figure 4a,b) has significantly lower vibrational frequencies and higher amplitudes.

For a more quantitative vibrational evaluation and to calculate the energy dissipation capability of the Ti₈C₅ nanosheet due to the damping process, nanosheets are subjected to a low impact velocity of 0.5 km/s, which do not penetrate the nanosheets or damage the lattice structure significantly. In all simulation scenarios, projectiles bounce back after contact. In such assessments, the external energy of nanosheets right after the departure of the projectile is recorded (Figure 5) for the calculation of the $Q$ factor based on estimation schemes derived by previous researchers [40,41]. Here, external energy is the difference of Ti₈C₅ potential energy before and after the separation of the two objects. $Q$ is defined as the ratio between the total system energy and the average energy loss in one radian at resonant frequency, i.e., $Q=2\pi E_v/\Delta E_v$, where $E_v$ is the total energy of the vibrating system and $\Delta E_v$ is donated as the dissipated energy in one cycle of vibration. $Q$ is assumed to be constant during vibration; therefore, the correlation between maximum energy ($E_{v,n}$) after the n vibration cycles and initial energy ($E_{v,0}$) in the first vibration cycle can be denoted as $E_{v,n}=E_{v,0}{\left(1–2\pi /Q\right)}^n$. According to the external energy change of Ti₈C₅ presented in Figure 5a, a clear decay pattern is determined. It was noticed that the NVE ensemble adopted in this study and total energy in the vibration phase is constant, and the loss in potential energy contributes to both kinetic energy and the temperature rise of the system. Because of the remarkable decay of the oscillation amplitude, the $Q$ factor of Ti₈C₅ possesses a small value of~111.9, and the corresponding main resonance frequency for Ti₈C₅ is calculated via discrete Fourier transform, being ~11.9 GHz (Figure 5b). For the graphene and copper nanosheet impact simulation, their main resonance frequencies were ~34.01 GHz and ~43.03 GHz, respectively, whereas the $Q$ factors were 413.37 and 137.17, respectively. Considering the external energy variation and $Q$ Factor in the 600 ps time domain, Ti₈C₅ clearly demonstrates a better damping capability over single layer graphene. Although the $Q$ factor calculation results can be affected by the presence of a fixed boundary, all the simulations were conducted with the same fixed boundary setup, which guarantees a fair comparison.

3.3. Stress Distribution and Propagation

The above discussion illustrates the morphology of Ti₈C₅ under impact. It is of great interest to compare how the stress distribution and propagation pattern in Ti₈C₅ would be altered with the variation velocity amplitudes from a more statistical point of view. In this regard, the Von Mises atomic stress distribution and elastic stress wave propagation under different impact velocities are studied as they largely determine the impact resist performance. For a lower impact velocity of 3 km/s, stress throughout the Ti₈C₅ nanosheet prior to bond break presents a multimodal distribution pattern (Figure 6a). The main body of the data centred around 19 GPa is mainly due to impact whereas the peak on the left side is created by the fixed boundaries. As the impact increases, the peak on the left maintains the same level, whereas the mode of the main body decreases with its tail shifts towards the right (Figure 6b–d). Higher impact velocities lead to more severe stress concentrations around the impact region; moreover, stress distribution prior to bond breakage is dominated by the elastic stress propagation velocity, as the impact velocity rises much higher than the stress propagation rate, and ‘less connection’ between the main body and the tail is observed. Cumulative density function (CDF) presented in Figure 6e summaries the finding: the slope of each curve is inversely correlated with the impact velocity.

A nanosheet with a more significant elastic wave propagation velocity transfers momentum at a superior rate; thus, a more even stress distribution/energy delocalization is expected during ballistic impact. Theoretically, the elastic stress wave velocity $v_s$ in a solid material can be calculated based on its Young’s modulus γ and density $\rho$ according to $v_s=\sqrt{Y/\rho }$ [42,43]. In this study, $v_s$ is estimated via tracking the location of the highest Von Mises atomic stress, as studies suggests decent agreement between the two approaches [38]. The elastic wave propagation velocity for Ti₈C₅ is 5.34 and 5.90 km/s for X and Y directions, respectively, which presents a significant gap considering copper is 7.91 (along [100] direction) and 7.52 km/s (along [010] direction). As a result, a more evenly distributed pattern throughout the nanosheet is captured at the same time instance (Figure 7a) compared with Ti₈C₅ (Figure 6a). Graphene possesses the highest elastic wave propagation velocity, which is 20.11 and 19.84 km/s along the armchair and zigzag directions, respectively; as such, its stress distribution shows a unimodal shape (Figure 7b). Slopes of the three curves in the cumulative probability diagram indicate graphene and copper are likely to bear more loading prior to bond break, and it also suggests that Ti₈C₅ owns the least significant breaking stress among all the three materials.

3.4. Impact Resistance Evaluation

To quantitatively evaluate the impact resistance of Ti₈C₅, its penetration (E_p) and specific penetration energy, $E_p^{*}$, under various velocity amplitudes are calculated and compared with graphene and copper. According to Figure 8a, the penetration energy of both Ti₈C₅ and copper show a consistent rate of growth in the whole velocity domain, whereas graphene experiences a notable slump in E_p at a relatively low impact velocity, which is due to the reduction in overall defamation [36]. After increasing the impact above 3 km/s, the same overall ranking is maintained. For all impact velocities, copper presents the most significant E_p and the gap between each category enlarges with impact velocity. Projectiles are not rigid bodies in this study; as a result, cracks in projectiles are found during the process if they were assigned ultra-high velocities (i.e., copper is impacted at 7 km/s).

Theoretically, for nanosheets with thickness h, which is far less than projectile diameter D, i.e., D/h >> 1 [44], it is regarded as a thin film and its penetration energy can be estimated by $E_p=\Delta P{e}_{NS}+\Delta K{e}_{NS}=\left(\rho A_{s}h\right)v^2/2+E_d$. The term $\left(\rho A_{s}h\right)v^2/2$ refers to the minimum inelastic energy or kinetic energy transferred to the target nanosheet (where $A_s$ represents the strike face area and $v$ stands for the impact velocity) and the second term $E_d$ represents other energy dissipation mechanisms such as elastic deformation or breakage of bonds. Considering the different morphology, mass as well as density differences of various nanosheets, specific or gravimetric penetration energy is adopted to evaluate their impact resistance capability, which is calculated as $E_p^{*}=E_p/\left(\rho A_{s}h\right)$ and it can be further expressed as $E_P^{*}=v^2/2+E_d^{*}$. Apparently, $E_d^{*}$ is a figure of merit to evaluate the impact energy delocalization ability, which is dominated by elastic wave propagation velocity. Better energy delocalization ability prevents the development of local severe stress and helps the material to deform more locally prior to the arrival of bond break stress.

$E_P^{*}$ in each group follows the material-independent energy dissipation baseline (i.e., $v^2/2$), and the overall ranking differs from the penetration energy (Figure 8a,b). At low impact velocities, ~2 km/s graphene shows superiority over the other two materials in terms of specific penetration energy, because of its most significant local deformation prior to bond break and, hence, a larger $E_d^{*}$ term is expected, which is confirmed by the energy breakdown presented in Figure 9. For low impact velocities <3 km/s, graphene shows a more significant $\Delta P{e}_{NS} /\Delta E_{NS,tot}$ ratio compared with the other two nanosheets. As impact velocity increases above 3 km/s, the elastic wave propagation velocity or energy delocalization ability becomes less dominant in determining the impact resistance, as all samples experience less overall deformation prior to bond break with a less significant $\Delta P{e}_{NS} /\Delta E_{NS,tot}$ ratio observed. As a result, an increase in impact velocity closes the specific penetration energy gap between graphene and Ti₈C₅.

4. Conclusions

In summary, the fracture behaviour of monolayer Ti₈C₅ subject to various hypervelocity impacts is explored. For Ti₈C₅ under relatively a low impact velocity ~0.5 km/s, its main resonance frequency is 11.9 GHz, which is different from other matrices in composites such as copper and graphene. A resonance study on Ti₈C₅ also shows a low Q factor of 111.9 indicates decent energy damping capability. The combination of resonance frequency and low Q factor would help to eliminate energy in the interfacial reflection process and weaken shock waves for Ti₈C_5-strengthened composites. When increasing the impact velocity above the threshold of 1.8 km/s, a crack initiates at the geometry centre and the nanosheet experiences significant out-of-plane deformation before the propagation of the crack, demonstrating brittle behaviour. Elastic wave propagation velocity is an important figure in impact resistance assessments, as it positively correlates with the energy delocalization rate, which helps materials to deform more prior to bond break and thus absorb more energy. Tracking atomic Von Mises stress distribution, the elastic wave propagation velocity of Ti₈C₅ is calculated to be 5.34 and 5.90 km/s for X and Y directions, respectively, but these figures are not significant compared with graphene and copper, suggesting inferior energy delocalization rates and thus less energy dissipation via deformation prior to bond break. However, because of its relatively small mass density compared with copper, Ti₈C₅ presents superior specific penetration for all tested velocity amplitudes. This study provides a fundamental understanding of the deformation and penetration mechanisms of titanium carbide nanosheets under impact, which should shed lights on the design of titanium carbide- related composites for bullet-proof applications or shielding structures for aerospace systems’ protection applications for titanium carbide-related composites. We note that although these results are obtained from a Ti₈C₅ nanosheet with a small size, similar failure mechanisms are also expected for monolithic Ti₈C₅ nanosheets with a large scale.