Sutton-Chen Potential (parameters for gold)
use suttonchen as sc
    e 0.013
    n 10
    a 4.08
    m 8
    c 34.408
    cutoff 7.5
enduse
#- Applying Plugins -#
integrator vv
cellmanager linkedcell
potential sc Au Au
```
Fig. 1. Example of an LPMD control file. The components are loaded (use...enduse) and then applied.

#### **3.1 Structural properties with LPMD**

We will denote by structural property, any quantity *AS* which depends on the instant *t* only through the atomic coordinates,

$$A\_S(t) = A\_S(\vec{r}\_1(t), \dots, \vec{r}\_N(t)),\tag{6}$$

with *N* the number of particles.

LPMD allows the calculation of several structural properties (either as instantaneous values or as averages), including the radial distribution function *g*(*r*) (using the gdr plug-in) and common neighbor analysis (through the cna plug-in), both of which can be used to measure a degree of deviation from an ideal crystal structure.

Fig. 2. Schematic representation of the computation of the radial distribution function *g*(*r*).

The radial distribution function *g*(*r*) represents the probability density for finding a neighboring atom at a distance *r*, normalized to the same probability density in a perfectly uniform distribution of atoms. This ensures that *g*(*r*) goes to unity for large enough *r*, independently of the system. It is formally defined as

$$\log(r) = \frac{V}{N^2} \left\langle \sum\_{i} \sum\_{j \neq i} \delta(\vec{r} - \vec{r\_{ij}}) \right\rangle. \tag{7}$$

Fig. 3. Four common neighbors (green atoms) of the pair *a*-*b* (in blue) in a face-centered cubic structure. The pair depicted as *a*-*b* has indices 1-4-2-1 in CNA notation, and is the only kind

<sup>235</sup> Inelastic Collisions and Hypervelocity Impacts

A typical problem in classical mechanics is the bouncing of a bead in free fall over a surface, due to the action of the force of gravity (Alonso & Finn, 1992; Eisberg & Lerner, 1981). After each bouncing, the body reaches different heights, each one of them less or equal than the previous one. The most common explanation for this phenomenon is the viscoelastic dissipation, which results in an energy loss due to the inelastic collision (Aguirregabiria et al.,

Although there have been many works dealing with the dynamics of inelastic collisions (Goldsmith, 2001; Johnson, 1987; Zukas et al., 1982) and many measurements of the energy loss in such collisions (Bridges et al., 1984; Goldsmith, 2001; Hatzes et al., 1988; Lifshitz & Kolsky, 1964; Lun & Savage, 1986; Raman., 1918; Reed, 1985; Supulver et al., 1995; Tabor, 1948; Tillett, 1954; Tsai & Kolsky, 1967; Zener, 1941), there is a considerable scatter in existing data, and the mechanisms of dissipation and the behavior of the restitution coefficient with the impact velocity are still open problems (Falcon et al., 1998). At high impact velocities, i.e., when fully plastic deformations occur, this behavior is well known both experimentally (Goldsmith, 2001; Raman., 1918; Reed, 1985; Tabor, 1948; Tillett, 1954; Zener, 1941) and theoretically (Goldsmith, 2001; Johnson, 1987; Tabor, 1948), but the mechanisms of

Molecular dynamics allows us to keep track of the position and velocity of every particle in the system at any instant of time. Using statistical mechanics, the calculation of energy, temperature and other thermodynamic properties is straightforward. Moreover, if the target (surface) is considered as being a part of the system, the total energy remains constant, and the "energy loss" that the bead experiments is just a transfer of translational kinetic energy to internal potential and thermal energy, which can be identified with mechanisms of energy

To show how this phenomenon occurs, molecular dynamics simulations were performed using the LPMD program(Davis et al., 2010) (see section 3). A bead is dropped from rest at different heights in a constant force field over a surface. The resulting collisions show two main types of deformation: slight deformation, where the bead remains vibrating after the collision, and substantial deformation, where the bead changes its shape. Then, the evolution of the different energies in time is computed in order to show in detail the energy transfer.

of pair appearing in the FCC structure.

at Nanoscopic Level: A Molecular Dynamics Study

2008; Falcon et al., 1998).

**4. Bouncing of a ball over a surface: atomic level study**

energy loss during a collision are hard to track at a macroscopic level.

loss, such as plastic deformation, vibrational energy and others.

where *N* is the total number of atoms in the system and *V* is the total volume. However, in practice, it is computed from an histogram of the neighbor distribution,

$$g(r) = \frac{V}{N} \frac{n(r)}{\frac{4}{3}\pi((r+\Delta r)^3 - r^3)} \approx \frac{V}{N} \frac{n(r)}{4\pi r^2 \Delta r} \tag{8}$$

where *n*(*r*) is the number of atoms in the spherical shell between *r* and *r* + Δ*r* (see figure 2).

The Common Neighbor Analysis (CNA) (Honeycutt & Andersen, 1987) is a technique used in atomistic simulations to determine the local ordering in a given structure. CNA gives more detailed information than the radial distribution function *g*(*r*), as it considers not only the number of neighbors at a given distance but also their location with respect to other common neighboring atoms. In the CNA method (see figure 3), every pair of atoms is labeled according to four indices (*i*, *j*, *k*, *l*): the first index, *i*, is 1 for nearest neighbor pairs, 2 for next-nearest neighbors, and so on. The second index, *j*, corresponds to the number of *common neighbors* shared by the atoms in the pair. The third index, *k*, corresponds to the number of *bonds* that can be "drawn" between the *j* common neighbors (taking the bond length as the nearest neighbor distance). Finally, the fourth index, *l*, corresponds to the length of the longest chain that connects all the *k* bonds. The different structures have the following distribution of pairs: FCC has only **1-4-2-1** pairs, in hcp the pairs are distributed equally between **1-4-2-1** and **1-4-2-2**, and in bcc there are **1-4-4-4** and **1-6-6-6** present in ratios 3/7 and 4/7, respectively.

6 Will-be-set-by-IN-TECH

Fig. 2. Schematic representation of the computation of the radial distribution function *g*(*r*).

 ∑ *i* ∑ *j*�=*i*

independently of the system. It is formally defined as

*<sup>g</sup>*(*r*) = *<sup>V</sup> N*<sup>2</sup>

practice, it is computed from an histogram of the neighbor distribution,

4

in bcc there are **1-4-4-4** and **1-6-6-6** present in ratios 3/7 and 4/7, respectively.

*<sup>g</sup>*(*r*) = *<sup>V</sup> N*

The radial distribution function *g*(*r*) represents the probability density for finding a neighboring atom at a distance *r*, normalized to the same probability density in a perfectly uniform distribution of atoms. This ensures that *g*(*r*) goes to unity for large enough *r*,

where *N* is the total number of atoms in the system and *V* is the total volume. However, in

<sup>3</sup>*π*((*<sup>r</sup>* <sup>+</sup> <sup>Δ</sup>*r*)<sup>3</sup> <sup>−</sup> *<sup>r</sup>*3) <sup>≈</sup> *<sup>V</sup>*

*n*(*r*)

where *n*(*r*) is the number of atoms in the spherical shell between *r* and *r* + Δ*r* (see figure 2). The Common Neighbor Analysis (CNA) (Honeycutt & Andersen, 1987) is a technique used in atomistic simulations to determine the local ordering in a given structure. CNA gives more detailed information than the radial distribution function *g*(*r*), as it considers not only the number of neighbors at a given distance but also their location with respect to other common neighboring atoms. In the CNA method (see figure 3), every pair of atoms is labeled according to four indices (*i*, *j*, *k*, *l*): the first index, *i*, is 1 for nearest neighbor pairs, 2 for next-nearest neighbors, and so on. The second index, *j*, corresponds to the number of *common neighbors* shared by the atoms in the pair. The third index, *k*, corresponds to the number of *bonds* that can be "drawn" between the *j* common neighbors (taking the bond length as the nearest neighbor distance). Finally, the fourth index, *l*, corresponds to the length of the longest chain that connects all the *k* bonds. The different structures have the following distribution of pairs: FCC has only **1-4-2-1** pairs, in hcp the pairs are distributed equally between **1-4-2-1** and **1-4-2-2**, and

*δ*(*r* − *rij*)

*N*

*n*(*r*) 4*πr*2Δ*r*

. (7)

(8)

Fig. 3. Four common neighbors (green atoms) of the pair *a*-*b* (in blue) in a face-centered cubic structure. The pair depicted as *a*-*b* has indices 1-4-2-1 in CNA notation, and is the only kind of pair appearing in the FCC structure.

### **4. Bouncing of a ball over a surface: atomic level study**

A typical problem in classical mechanics is the bouncing of a bead in free fall over a surface, due to the action of the force of gravity (Alonso & Finn, 1992; Eisberg & Lerner, 1981). After each bouncing, the body reaches different heights, each one of them less or equal than the previous one. The most common explanation for this phenomenon is the viscoelastic dissipation, which results in an energy loss due to the inelastic collision (Aguirregabiria et al., 2008; Falcon et al., 1998).

Although there have been many works dealing with the dynamics of inelastic collisions (Goldsmith, 2001; Johnson, 1987; Zukas et al., 1982) and many measurements of the energy loss in such collisions (Bridges et al., 1984; Goldsmith, 2001; Hatzes et al., 1988; Lifshitz & Kolsky, 1964; Lun & Savage, 1986; Raman., 1918; Reed, 1985; Supulver et al., 1995; Tabor, 1948; Tillett, 1954; Tsai & Kolsky, 1967; Zener, 1941), there is a considerable scatter in existing data, and the mechanisms of dissipation and the behavior of the restitution coefficient with the impact velocity are still open problems (Falcon et al., 1998). At high impact velocities, i.e., when fully plastic deformations occur, this behavior is well known both experimentally (Goldsmith, 2001; Raman., 1918; Reed, 1985; Tabor, 1948; Tillett, 1954; Zener, 1941) and theoretically (Goldsmith, 2001; Johnson, 1987; Tabor, 1948), but the mechanisms of energy loss during a collision are hard to track at a macroscopic level.

Molecular dynamics allows us to keep track of the position and velocity of every particle in the system at any instant of time. Using statistical mechanics, the calculation of energy, temperature and other thermodynamic properties is straightforward. Moreover, if the target (surface) is considered as being a part of the system, the total energy remains constant, and the "energy loss" that the bead experiments is just a transfer of translational kinetic energy to internal potential and thermal energy, which can be identified with mechanisms of energy loss, such as plastic deformation, vibrational energy and others.

To show how this phenomenon occurs, molecular dynamics simulations were performed using the LPMD program(Davis et al., 2010) (see section 3). A bead is dropped from rest at different heights in a constant force field over a surface. The resulting collisions show two main types of deformation: slight deformation, where the bead remains vibrating after the collision, and substantial deformation, where the bead changes its shape. Then, the evolution of the different energies in time is computed in order to show in detail the energy transfer.

#### **4.1 Simulation details**

The system consist of a solid ball that interacts repulsively with a solid surface, both made of argon in the solid state (see figure 4). The interaction between atoms separated by a distance

Fig. 4. Argon ball over a solid argon surface, immersed in a constant force field. Image generated by the LPVisual plugin of the LPMD program.

*r* is modeled using a modified form of the Lennard-Jones potential (Barrat & Bocquet, 1999):

$$V(r) = \begin{cases} 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - c \left( \frac{\sigma}{r} \right)^{6} \right] & r < r\_{\mathcal{L}} = 2.5\sigma \\ 0 & r \ge r\_{\mathcal{L}} \end{cases} \tag{9}$$

this case, the ball was dropped from *z* = 11.03 *σ*. While the ball is hitting the surface, it gets compressed, and then leaves the ground, oscillating harmonically (figures 5(c) and 5(d)).

<sup>237</sup> Inelastic Collisions and Hypervelocity Impacts

at Nanoscopic Level: A Molecular Dynamics Study

(a) (b)

(c) (d)

Fig. 5. Ball dropped from *z* = 11.03 *σ*. (a) The ball is falling towards the surface. (b) The ball hits the surface and gets compressed, inducing an oscillatory movement on it. (c) The ball leaves the surface vibrating. A maximum amplitude is reached. (d) A minimum amplitude is reached due to the induced oscillatory movement. The maximum amplitude is slightly greater than the minimum, so the difference between the size of the ball in 5(c) and its size in 5(d) is not clearly appreciated. You can see the simulations at www.lpmd.cl in the examples section, where these oscillations can be clearly appreciated in the videos.

Figure 6 shows another simulation, where the ball was dropped from *z* = 28.15 *σ*. After the

In this section we present the results of several simulated collisions. The ball was dropped from heights *h*<sup>0</sup> between 11.03 *σ* and 79.53 *σ*. We begin by describing the dynamics of the ball and discuss about the height reached after each bounce. Then, we analyze the deformation

collision, the ball acquires a new shape.

**4.2 Results**

where *rc* is a cut-off chosen here to be 2.5*σ* (*σ* is the Lennard-Jones diameter and *ε* is the depth of the potential well). For our system, the parameters *ε*/*kB* = 119.8 K and *σ* = 3.40 Å are the same for all atoms and correspond to the values for argon (Kittel, 2005), whose atomic mass is *M* = 39.948 amu. In the rest of this section, all quantities are expressed in LJ reduced units, using *σ*, *ε* and *M* as length, energy and mass scales, respectively 2.

The interaction between atoms in the ball and atoms in the surface is given by the equation (9) with *c* = 0 (i.e., purely repulsive), while the interactions between any pair of atoms in the ball is given by the usual Lennard-Jones potential (eq. (9) with *c* = 1). The last also holds for every pair of atoms inside the surface. The Newton equations of motion are integrated using the Beeman algorithm, with a time step <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> 4.651 <sup>×</sup> <sup>10</sup>−4*τ*.

The solid ball and the surface slab were equilibrated at zero temperature for 5 <sup>×</sup> 103 time steps to allow them to adopt relaxed configurations.

The ball, composed of less than 100 atoms, was immersed in a constant force field in the direction of the negative *z*−axis, whose magnitude was 0.026 *F*0, and it was dropped from different heights over the surface, composed of about 1500 atoms. This force produces a constant acceleration of the center of mass of the ball of 0.026 *a*0, which means that it travels 0.49 *<sup>σ</sup>* after a time *<sup>τ</sup>* of being dropped. Each simulation took about 3 <sup>×</sup> 104 time steps (∼<sup>14</sup> *<sup>τ</sup>*).

Different types of collisions were observed. The most representatives are shown in figures 5 and 6. Figure 5 shows one of the simulations where the ball is falling over the surface. In

```
2 Time: τ ≡ σ
             √M/ε = 2.15 ps.
```
Velocity: *<sup>v</sup>*<sup>0</sup> <sup>≡</sup> <sup>√</sup>*ε*/*<sup>M</sup>* <sup>=</sup> 157.91 m/s. Acceleration: *a*<sup>0</sup> = *ε*/*Mσ* = 0.73 Å/ps2. Force: *<sup>F</sup>*<sup>0</sup> <sup>≡</sup> *<sup>ε</sup>*/*<sup>σ</sup>* <sup>=</sup> 3.04 <sup>×</sup> <sup>10</sup>−<sup>3</sup> eV/Å.

8 Will-be-set-by-IN-TECH

The system consist of a solid ball that interacts repulsively with a solid surface, both made of argon in the solid state (see figure 4). The interaction between atoms separated by a distance

Fig. 4. Argon ball over a solid argon surface, immersed in a constant force field. Image

*r* is modeled using a modified form of the Lennard-Jones potential (Barrat & Bocquet, 1999):

where *rc* is a cut-off chosen here to be 2.5*σ* (*σ* is the Lennard-Jones diameter and *ε* is the depth of the potential well). For our system, the parameters *ε*/*kB* = 119.8 K and *σ* = 3.40 Å are the same for all atoms and correspond to the values for argon (Kittel, 2005), whose atomic mass is *M* = 39.948 amu. In the rest of this section, all quantities are expressed in LJ reduced units,

The interaction between atoms in the ball and atoms in the surface is given by the equation (9) with *c* = 0 (i.e., purely repulsive), while the interactions between any pair of atoms in the ball is given by the usual Lennard-Jones potential (eq. (9) with *c* = 1). The last also holds for every pair of atoms inside the surface. The Newton equations of motion are integrated using the

The solid ball and the surface slab were equilibrated at zero temperature for 5 <sup>×</sup> 103 time steps

The ball, composed of less than 100 atoms, was immersed in a constant force field in the direction of the negative *z*−axis, whose magnitude was 0.026 *F*0, and it was dropped from different heights over the surface, composed of about 1500 atoms. This force produces a constant acceleration of the center of mass of the ball of 0.026 *a*0, which means that it travels 0.49 *<sup>σ</sup>* after a time *<sup>τ</sup>* of being dropped. Each simulation took about 3 <sup>×</sup> 104 time steps (∼<sup>14</sup> *<sup>τ</sup>*). Different types of collisions were observed. The most representatives are shown in figures 5 and 6. Figure 5 shows one of the simulations where the ball is falling over the surface. In

 *σ r* 6 

*r* < *rc* = 2.5*σ* <sup>0</sup> *<sup>r</sup> rc* , (9)

generated by the LPVisual plugin of the LPMD program.

*V*(*r*) =

Beeman algorithm, with a time step <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> 4.651 <sup>×</sup> <sup>10</sup>−4*τ*.

to allow them to adopt relaxed configurations.

<sup>√</sup>*M*/*<sup>ε</sup>* <sup>=</sup> 2.15 ps. Velocity: *<sup>v</sup>*<sup>0</sup> <sup>≡</sup> <sup>√</sup>*ε*/*<sup>M</sup>* <sup>=</sup> 157.91 m/s. Acceleration: *a*<sup>0</sup> = *ε*/*Mσ* = 0.73 Å/ps2. Force: *<sup>F</sup>*<sup>0</sup> <sup>≡</sup> *<sup>ε</sup>*/*<sup>σ</sup>* <sup>=</sup> 3.04 <sup>×</sup> <sup>10</sup>−<sup>3</sup> eV/Å.

<sup>2</sup> Time: *<sup>τ</sup>* <sup>≡</sup> *<sup>σ</sup>*

 4*ε <sup>σ</sup> r* <sup>12</sup> <sup>−</sup> *<sup>c</sup>*

using *σ*, *ε* and *M* as length, energy and mass scales, respectively 2.

**4.1 Simulation details**

this case, the ball was dropped from *z* = 11.03 *σ*. While the ball is hitting the surface, it gets compressed, and then leaves the ground, oscillating harmonically (figures 5(c) and 5(d)).

Fig. 5. Ball dropped from *z* = 11.03 *σ*. (a) The ball is falling towards the surface. (b) The ball hits the surface and gets compressed, inducing an oscillatory movement on it. (c) The ball leaves the surface vibrating. A maximum amplitude is reached. (d) A minimum amplitude is reached due to the induced oscillatory movement. The maximum amplitude is slightly greater than the minimum, so the difference between the size of the ball in 5(c) and its size in 5(d) is not clearly appreciated. You can see the simulations at www.lpmd.cl in the examples section, where these oscillations can be clearly appreciated in the videos.

Figure 6 shows another simulation, where the ball was dropped from *z* = 28.15 *σ*. After the collision, the ball acquires a new shape.

#### **4.2 Results**

In this section we present the results of several simulated collisions. The ball was dropped from heights *h*<sup>0</sup> between 11.03 *σ* and 79.53 *σ*. We begin by describing the dynamics of the ball and discuss about the height reached after each bounce. Then, we analyze the deformation

Fig. 7. (Color online) Maximum heights reached after each bounce, by the center of mass of

<sup>239</sup> Inelastic Collisions and Hypervelocity Impacts

decrease is appreciated in the first bounce as well as in the second. For *h*<sup>0</sup> = 79.53 *σ* only two bounces are observed because, after the first bounce, the ball gets deformed and loses its solid structure, which does not allow it to keep bouncing. Nevertheless, a common behavior is observed for all cases: after the third bounce, the height reached is almost the same. At this

We analyze in detail the structure of the ball evaluating the pair-distribution function *g*(*r*) for the atoms that constitute the ball for different time steps (Fig. 8) in different simulations.

Before the first impact, i.e., when the ball is falling, it has a FCC structure (for any *h*0), because the first neighbors are clearly appreciated at 1.09 *σ* for all cases (Fig. 8), which is the known value for first neighbors in argon lattice structure (Kittel, 2005). In Fig. 8(a) it can be seen that for times between *t* = 0 and *t* = 5.581*τ* the initial FCC structure is conserved, despite of the effects of the impacts with the surface, which took place at *t* = 0.71 *τ*, *t* = 2.07 *τ*, *t* = 3.38 *τ*, *t* = 4.52 *τ* and *t* = 5.83 *τ*. The peaks also have similar widths, which means that there are small temperature effects. If *h*<sup>0</sup> is increased, the peaks become wider and smaller (Fig. 8(b) to 8(c)) but still distinguishables, which implies a rising of the temperature inside the ball without melting. We will analyze this subject deeper in the next section (4.2.3). In the latter case (Fig. 8(c)), the ball acquires a new solid structure (Fig. 6), that seems to be, just looking at the *g*(*r*) function, still FCC type. For *h*<sup>0</sup> > 41.47 *σ* (Fig. 8(d) to 8(f)) just one peak is distinguishable as long as the distance *r* to an atom increases, which means that an atom has, in average, a neighbor at 1.09 *σ* and no well defined second neighbors or further. This can be

In an inelastic collision, the dissipation of energy of a body is given by the energy transfer from mechanical energy to internal energies, such as thermal energies and vibrational energies, being the last one the responsible of plastic deformations of the body. Since our simulations consider the surface as part of the system, no dissipation is observed, because the total energy

the ball.

point, the ball remains over the surface.

at Nanoscopic Level: A Molecular Dynamics Study

interpreted as the melting of the ball.

**4.2.3 Energy**

**4.2.2 Deformation of the ball**

Fig. 6. Ball dropped from *z* = 28.15 *σ*. (a) The ball is falling towards the surface. (b) The ball hits the surface and gets compressed. (c) The ball lifts the surface acquiring a new shape, with almost none internal vibrations. (d) The ball keeps its new shape after the collision.

of the ball by evaluating the pair-distribution function *g*(*r*). Finally, we classify kinetic and potential energies in different types and then examine how these energies are transferred from one to another to keep the total energy constant.

#### **4.2.1 Heights reached**

Fig. 7 shows the heights reached by the center of mass of the ball after each bounce. The zeroth bounce represents the initial height *h*0. The case in which the ball is dropped from a height of 11.03 *σ* corresponds to a quasi-elastic bounce because the height of bounces are almost the same. When *h*<sup>0</sup> = 18.64 *σ*, the height reached by the ball after the first bounce is smaller than the initial height from where it was dropped, and something similar happens with the other bounces: the maximum heights reached decrease until the ball remains static over the target. For *h*<sup>0</sup> = 41.47 *σ*, the maximum height reached after the first bounce is considerably smaller than *h*0. Big differences of height are observed also for *h*<sup>0</sup> = 58.6 *σ*, where a notorious 10 Will-be-set-by-IN-TECH

(a) (b)

(c) (d)

Fig. 6. Ball dropped from *z* = 28.15 *σ*. (a) The ball is falling towards the surface. (b) The ball hits the surface and gets compressed. (c) The ball lifts the surface acquiring a new shape, with almost none internal vibrations. (d) The ball keeps its new shape after the collision.

of the ball by evaluating the pair-distribution function *g*(*r*). Finally, we classify kinetic and potential energies in different types and then examine how these energies are transferred from

Fig. 7 shows the heights reached by the center of mass of the ball after each bounce. The zeroth bounce represents the initial height *h*0. The case in which the ball is dropped from a height of 11.03 *σ* corresponds to a quasi-elastic bounce because the height of bounces are almost the same. When *h*<sup>0</sup> = 18.64 *σ*, the height reached by the ball after the first bounce is smaller than the initial height from where it was dropped, and something similar happens with the other bounces: the maximum heights reached decrease until the ball remains static over the target. For *h*<sup>0</sup> = 41.47 *σ*, the maximum height reached after the first bounce is considerably smaller than *h*0. Big differences of height are observed also for *h*<sup>0</sup> = 58.6 *σ*, where a notorious

one to another to keep the total energy constant.

**4.2.1 Heights reached**

Fig. 7. (Color online) Maximum heights reached after each bounce, by the center of mass of the ball.

decrease is appreciated in the first bounce as well as in the second. For *h*<sup>0</sup> = 79.53 *σ* only two bounces are observed because, after the first bounce, the ball gets deformed and loses its solid structure, which does not allow it to keep bouncing. Nevertheless, a common behavior is observed for all cases: after the third bounce, the height reached is almost the same. At this point, the ball remains over the surface.

#### **4.2.2 Deformation of the ball**

We analyze in detail the structure of the ball evaluating the pair-distribution function *g*(*r*) for the atoms that constitute the ball for different time steps (Fig. 8) in different simulations.

Before the first impact, i.e., when the ball is falling, it has a FCC structure (for any *h*0), because the first neighbors are clearly appreciated at 1.09 *σ* for all cases (Fig. 8), which is the known value for first neighbors in argon lattice structure (Kittel, 2005). In Fig. 8(a) it can be seen that for times between *t* = 0 and *t* = 5.581*τ* the initial FCC structure is conserved, despite of the effects of the impacts with the surface, which took place at *t* = 0.71 *τ*, *t* = 2.07 *τ*, *t* = 3.38 *τ*, *t* = 4.52 *τ* and *t* = 5.83 *τ*. The peaks also have similar widths, which means that there are small temperature effects. If *h*<sup>0</sup> is increased, the peaks become wider and smaller (Fig. 8(b) to 8(c)) but still distinguishables, which implies a rising of the temperature inside the ball without melting. We will analyze this subject deeper in the next section (4.2.3). In the latter case (Fig. 8(c)), the ball acquires a new solid structure (Fig. 6), that seems to be, just looking at the *g*(*r*) function, still FCC type. For *h*<sup>0</sup> > 41.47 *σ* (Fig. 8(d) to 8(f)) just one peak is distinguishable as long as the distance *r* to an atom increases, which means that an atom has, in average, a neighbor at 1.09 *σ* and no well defined second neighbors or further. This can be interpreted as the melting of the ball.

#### **4.2.3 Energy**

In an inelastic collision, the dissipation of energy of a body is given by the energy transfer from mechanical energy to internal energies, such as thermal energies and vibrational energies, being the last one the responsible of plastic deformations of the body. Since our simulations consider the surface as part of the system, no dissipation is observed, because the total energy

The velocity of the center of mass of the ball **V**CM *<sup>B</sup>* is calculated for each time-step to compute

<sup>241</sup> Inelastic Collisions and Hypervelocity Impacts

where *MB* = 79 *M* is the mass of the ball. The velocity of the center of mass of the surface **V**CM*<sup>S</sup>* is calculated for each time-step to compute the **translational kinetic energy of the surface**,

where *MS* = 1444 *M* is the mass of the surface. For a given time step, the difference between the velocity of the center of mass **V**CM *<sup>B</sup>* and the velocity of the atom *i*, **v***i*, defines **¯v***<sup>i</sup>* <sup>=</sup> **<sup>v</sup>***<sup>i</sup>* <sup>−</sup> **<sup>V</sup>**CM *<sup>B</sup>*, the velocity of the atom *<sup>i</sup>* relative to the center of mass of the ball, which is used to define the kinetic energy of the ball relative to its center of mass, what we call **thermal**

> 1 2

where *mi* = *M* is the mass of the atom *i*. In the same way, we define the **thermal energy of**

where, in this case, **¯v***<sup>i</sup>* <sup>=</sup> **<sup>v</sup>***<sup>i</sup>* <sup>−</sup> **<sup>V</sup>**CM *<sup>S</sup>*. Now, since the interaction between any pair of atoms is given by equation (9), the total potential energy *U* is given by the sum over all pairs of atoms plus the energy given by the force field, *UF* = *MB a z*CM, where *a* = 0.026 *a*<sup>0</sup> and *z*CM is the *z*−axis coordinate of the center of mass of the ball. We divide the sum in four terms as follows:

The first term corresponds to the potential energy of the ball, which keeps together all the pairs that belong to the ball, and we can associate it to a **vibrational energy of the ball**:

The second term corresponds to the potential energy of the surface, which keeps together all the pairs that belong to the surface, and we can associate it to a **vibrational energy of the**

> all surface atoms

The third term corresponds to the potential energy generated by the interaction of an atom of the ball with an atom of the surface, i.e., between atoms of different types. We will refer to this

> all ball-surface atom pairs

*VB* = ∑ all ball atoms

*VS* = ∑

*Uc* = ∑

1 2

all surface atoms

<sup>2</sup> *MB*�**V**CM *<sup>B</sup>*�2, (10)

<sup>2</sup> *MS*�**V**CM *<sup>S</sup>*�2, (11)

*mi*� **¯v***i*�2, (12)

*mi*� **¯v***i*�2, (13)

*V*(*r*). (15)

*V*(*r*). (16)

*V*(*r*). (17)

*U* = *VB* + *VS* + *Uc* + *UF*. (14)

*KB* <sup>=</sup> <sup>1</sup>

*KS* <sup>=</sup> <sup>1</sup>

*TB* = ∑ all ball atoms

*TS* = ∑

the **translational kinetic energy of the ball**, defined by

at Nanoscopic Level: A Molecular Dynamics Study

defined by

**energy of the ball**, by

**the surface** by

**surface**:

term as collisional energy:

Fig. 8. (Color online) Pair-distribution functions calculated for the atoms in the ball, for different times (see inset in *τ*/10). Each graph corresponds to different initial heights *h*0. The pair distribution function for each *t* at *h*0, has been shifted upwards for clarity.

remains constant. The internal energies of the ball change in each bounce, and the mechanism of loss and gain of energy are explained by considering the following classification of energies: 12 Will-be-set-by-IN-TECH

(a) *h*<sup>0</sup> = 11.03 *σ* (b) *h*<sup>0</sup> = 18.64 *σ*

(c) *h*<sup>0</sup> = 28.15 *σ* (d) *h*<sup>0</sup> = 41.47 *σ*

(e) *h*<sup>0</sup> = 58.6 *σ* (f) *h*<sup>0</sup> = 79.53 *σ*

remains constant. The internal energies of the ball change in each bounce, and the mechanism of loss and gain of energy are explained by considering the following classification of energies:

Fig. 8. (Color online) Pair-distribution functions calculated for the atoms in the ball, for different times (see inset in *τ*/10). Each graph corresponds to different initial heights *h*0. The

pair distribution function for each *t* at *h*0, has been shifted upwards for clarity.

The velocity of the center of mass of the ball **V**CM *<sup>B</sup>* is calculated for each time-step to compute the **translational kinetic energy of the ball**, defined by

$$K\_B = \frac{1}{2} M\_B \|\mathbf{V\_{CM}}^B\|^2 \tag{10}$$

where *MB* = 79 *M* is the mass of the ball. The velocity of the center of mass of the surface **V**CM*<sup>S</sup>* is calculated for each time-step to compute the **translational kinetic energy of the surface**, defined by

$$K\_{\rm S} = \frac{1}{2} M\_{\rm S} \| \mathbf{V\_{CM}}^{\rm S} \| ^2 \,, \tag{11}$$

where *MS* = 1444 *M* is the mass of the surface. For a given time step, the difference between the velocity of the center of mass **V**CM *<sup>B</sup>* and the velocity of the atom *i*, **v***i*, defines **¯v***<sup>i</sup>* <sup>=</sup> **<sup>v</sup>***<sup>i</sup>* <sup>−</sup> **<sup>V</sup>**CM *<sup>B</sup>*, the velocity of the atom *<sup>i</sup>* relative to the center of mass of the ball, which is used to define the kinetic energy of the ball relative to its center of mass, what we call **thermal energy of the ball**, by

$$T\_B = \sum\_{\text{all ball}\atop \text{atoms}} \frac{1}{2} m\_i ||\vec{\mathbf{v}}\_i||^2 \tag{12}$$

where *mi* = *M* is the mass of the atom *i*. In the same way, we define the **thermal energy of the surface** by

$$T\_{\mathbb{S}} = \sum\_{\text{all surface} \atop \text{atoms}} \frac{1}{2} m\_i ||\bar{\mathbf{v}}\_i||^2 \tag{13}$$

where, in this case, **¯v***<sup>i</sup>* <sup>=</sup> **<sup>v</sup>***<sup>i</sup>* <sup>−</sup> **<sup>V</sup>**CM *<sup>S</sup>*. Now, since the interaction between any pair of atoms is given by equation (9), the total potential energy *U* is given by the sum over all pairs of atoms plus the energy given by the force field, *UF* = *MB a z*CM, where *a* = 0.026 *a*<sup>0</sup> and *z*CM is the *z*−axis coordinate of the center of mass of the ball. We divide the sum in four terms as follows:

$$
\mathcal{U} = V\_B + V\_S + \mathcal{U}\_\mathcal{C} + \mathcal{U}\_F. \tag{14}
$$

The first term corresponds to the potential energy of the ball, which keeps together all the pairs that belong to the ball, and we can associate it to a **vibrational energy of the ball**:

$$V\_B = \sum\_{\text{all ball} \atop \text{atoms}} V(r). \tag{15}$$

The second term corresponds to the potential energy of the surface, which keeps together all the pairs that belong to the surface, and we can associate it to a **vibrational energy of the surface**:

$$V\_S = \sum\_{\text{all surface}\atop \text{atoms}} V(r). \tag{16}$$

The third term corresponds to the potential energy generated by the interaction of an atom of the ball with an atom of the surface, i.e., between atoms of different types. We will refer to this term as collisional energy:

$$\mathcal{U}\_{\mathbb{C}} = \sum\_{\substack{\text{all} \\ \text{ball-surface} \\ \text{atom pairs}}} V(r). \tag{17}$$

The last term *UF* was already explained and corresponds to the potential that generates the force field.

ball is zero. This happens at the minima and maxima of *KB*, so the maxima show where the

<sup>243</sup> Inelastic Collisions and Hypervelocity Impacts

When the ball has reached the point of maximum velocity, that is, the impact velocity, the velocity begins to decrease, because now the acceleration (and so the net force) is directed upwards. The ball reaches the point of null velocity (the minima) and then keeps accelerating until a maximum velocity that, for the first bounce, is the same as the impact velocity. In this interval of time, the energy *Uc* becomes greater while the energy *KB* and *TS* decreases, which means that most of the energy was stored in the collision energy and a little bit in the vibrational energy of the surface, with almost no energy transferred to the thermal energy of the ball (*TB*). In the second bounce (see labels above Fig. 9), much of the energy is transferred to the collision energy *Uc*, but now a little bit of energy is now transferred to the thermal energy of the ball *TB*, so the ball lose translational energy and the maximum velocity after the collision (the departure velocity) is smaller than the impact velocity, so it is not able to reach the same height than before the collision. In the third bounce, the thermal energy of the ball is transferred back to kinetic translational energy, and the ball can reach a departure velocity greater than the impact velocity. Something similar happens with the fourth bounce, and what all bounces has in common is that the fluctuations of the thermal energy of the ball *TB* are comparable to the fluctuations of its vibrational energy *VB* and the thermal energy of

Finally, we want to mention that, since *Uc* is non-zero only when atoms of the ball are close to atoms of the surface, this energy gives a reasonable definition of the collision time as the

In conclusion, a molecular dynamics study of the behavior of a ball bouncing repeatedly off a surface, considered as part of the system, has been done. We have observed that a study of the different types of energies of the system clearly shows what may be considered as the duration of a collision which, in contrast with typical macroscopic classical mechanics considerations, is not instantaneous, but a non negligible time interval. We have also shown that, despite of the fact that the collision is actually a continuous process that does not allow us to determine the "instant just before the collision" and the "instant just after the collision", which are always mentioned in macroscopic problems of momentum conservation, the impact velocity (maximum velocity reached before the ball stops) and the departure velocity (maximum velocity reached after the ball stops) can be determined precisely. This makes possible to determine the restitution coefficient in each bounce, a well studied property of bouncing systems, as the usual quotient of these velocities. The study of these energies have also helped to understand the processes of energy loss in inelastic collisions, which are actually not a loss, but a transfer to thermal and vibrational energy, within others. We could conclude that the force exerted by the surface acts as a break for the ball, and this force is the responsible for the decrease of the acceleration of the ball to zero (where the net force is null). So the impact velocity (maximum velocity reached before hitting the surface) is reached after the ball begins its collision with the floor, which can be considered as the moment in which the energy *Uc* becomes relevant. It is clear that *KT* + *UF* is constant when the collision is not taking place, but when it happens, in all bounces, despite of the fact that the collision energy *Uc* behaved similar in every bounce, it was the most important among the energies in the collision, since it stores most of the "dissipated" energy by the ball, more than the vibrational energy transferred to the surface. The other energies, in spite of their changes, do not contribute significantly to

force exerted by the surface equals the force *F* = *MB a* = �∇*UF*�.

at Nanoscopic Level: A Molecular Dynamics Study

width of the peak generated by this energy in each bounce.

the surface *TS*.

the energy transfer.

Fig. 9 shows the evolution of all these energies in time, except for *VS*, which is a negative term that remains almost constant and does not add additional information to the phenomenon. The energies are expressed in the Lennard-Jones energy unit, *ε*. The energy *UF* has been shifted 2 *ε* downwards and *VB*, 400 *ε* upwards to keep all energies in the same range, since what matters is the changes in energy rather than their absolute values. Other cases, where the ball was dropped at different *h*0, are quite similar in shape, but some peaks are bigger than others.

Fig. 9. (Color online) Evolution of the different energies in time for the simulation where the ball was dropped from *h*<sup>0</sup> = 11.03 *σ*.

In the Fig. 9 it can be observed that the translational energy of the ball has several local minima, located between two maxima placed symmetrically around each of them. These minima represent the instant in which the kinetic translational energy *KB* vanishes, i.e., when the ball is completely stopped over the surface during a bounce. The maximum at the left hand of each minimum shows the instant in which the velocity of the center of mass of the ball acquires its maximum value before it begins to stop. The repulsive potential of the surface is equivalent to a force exerted upwards, but this force does not reduce velocity of the ball immediately, in fact, the velocity keeps growing with time before reaching the maximum, but its rate of change, that is, its acceleration, is reduced. Considering just the *z* coordinate of the velocity of the center of mass, the first derivative of *KB* is given by

$$\frac{d\mathcal{K}\_B}{dt} = M\_B V\_{\mathcal{CM}}^B \dot{V}\_{\mathcal{CM}}^B = M\_B V\_{\mathcal{CM}}^B A\_{\mathcal{CM}\prime}^B$$

where *A<sup>B</sup>* CM is the acceleration of the center of mass of the ball. This derivative vanishes whether the velocity or the acceleration is zero, this is, when the net force exerted over the 14 Will-be-set-by-IN-TECH

The last term *UF* was already explained and corresponds to the potential that generates the

Fig. 9 shows the evolution of all these energies in time, except for *VS*, which is a negative term that remains almost constant and does not add additional information to the phenomenon. The energies are expressed in the Lennard-Jones energy unit, *ε*. The energy *UF* has been shifted 2 *ε* downwards and *VB*, 400 *ε* upwards to keep all energies in the same range, since what matters is the changes in energy rather than their absolute values. Other cases, where the ball was dropped at different *h*0, are quite similar in shape, but some peaks are bigger than

1st 2nd 3rd 4th 5th

KB KS TB TS VB Uc UF

0 2000 4000 6000 8000 10000 12000 14000

Time (Δt)

Fig. 9. (Color online) Evolution of the different energies in time for the simulation where the

In the Fig. 9 it can be observed that the translational energy of the ball has several local minima, located between two maxima placed symmetrically around each of them. These minima represent the instant in which the kinetic translational energy *KB* vanishes, i.e., when the ball is completely stopped over the surface during a bounce. The maximum at the left hand of each minimum shows the instant in which the velocity of the center of mass of the ball acquires its maximum value before it begins to stop. The repulsive potential of the surface is equivalent to a force exerted upwards, but this force does not reduce velocity of the ball immediately, in fact, the velocity keeps growing with time before reaching the maximum, but its rate of change, that is, its acceleration, is reduced. Considering just the *z* coordinate of the

velocity of the center of mass, the first derivative of *KB* is given by

*dt* <sup>=</sup> *MBV<sup>B</sup>*

CM*V*˙ *<sup>B</sup>*

whether the velocity or the acceleration is zero, this is, when the net force exerted over the

CM = *MBV<sup>B</sup>*

CM is the acceleration of the center of mass of the ball. This derivative vanishes

CM *A<sup>B</sup>* CM,

*dKB*

force field.

others.


where *A<sup>B</sup>*

ball was dropped from *h*<sup>0</sup> = 11.03 *σ*.



0

5

Energy (ε)

10

15

20

25

ball is zero. This happens at the minima and maxima of *KB*, so the maxima show where the force exerted by the surface equals the force *F* = *MB a* = �∇*UF*�.

When the ball has reached the point of maximum velocity, that is, the impact velocity, the velocity begins to decrease, because now the acceleration (and so the net force) is directed upwards. The ball reaches the point of null velocity (the minima) and then keeps accelerating until a maximum velocity that, for the first bounce, is the same as the impact velocity. In this interval of time, the energy *Uc* becomes greater while the energy *KB* and *TS* decreases, which means that most of the energy was stored in the collision energy and a little bit in the vibrational energy of the surface, with almost no energy transferred to the thermal energy of the ball (*TB*). In the second bounce (see labels above Fig. 9), much of the energy is transferred to the collision energy *Uc*, but now a little bit of energy is now transferred to the thermal energy of the ball *TB*, so the ball lose translational energy and the maximum velocity after the collision (the departure velocity) is smaller than the impact velocity, so it is not able to reach the same height than before the collision. In the third bounce, the thermal energy of the ball is transferred back to kinetic translational energy, and the ball can reach a departure velocity greater than the impact velocity. Something similar happens with the fourth bounce, and what all bounces has in common is that the fluctuations of the thermal energy of the ball *TB* are comparable to the fluctuations of its vibrational energy *VB* and the thermal energy of the surface *TS*.

Finally, we want to mention that, since *Uc* is non-zero only when atoms of the ball are close to atoms of the surface, this energy gives a reasonable definition of the collision time as the width of the peak generated by this energy in each bounce.

In conclusion, a molecular dynamics study of the behavior of a ball bouncing repeatedly off a surface, considered as part of the system, has been done. We have observed that a study of the different types of energies of the system clearly shows what may be considered as the duration of a collision which, in contrast with typical macroscopic classical mechanics considerations, is not instantaneous, but a non negligible time interval. We have also shown that, despite of the fact that the collision is actually a continuous process that does not allow us to determine the "instant just before the collision" and the "instant just after the collision", which are always mentioned in macroscopic problems of momentum conservation, the impact velocity (maximum velocity reached before the ball stops) and the departure velocity (maximum velocity reached after the ball stops) can be determined precisely. This makes possible to determine the restitution coefficient in each bounce, a well studied property of bouncing systems, as the usual quotient of these velocities. The study of these energies have also helped to understand the processes of energy loss in inelastic collisions, which are actually not a loss, but a transfer to thermal and vibrational energy, within others. We could conclude that the force exerted by the surface acts as a break for the ball, and this force is the responsible for the decrease of the acceleration of the ball to zero (where the net force is null). So the impact velocity (maximum velocity reached before hitting the surface) is reached after the ball begins its collision with the floor, which can be considered as the moment in which the energy *Uc* becomes relevant. It is clear that *KT* + *UF* is constant when the collision is not taking place, but when it happens, in all bounces, despite of the fact that the collision energy *Uc* behaved similar in every bounce, it was the most important among the energies in the collision, since it stores most of the "dissipated" energy by the ball, more than the vibrational energy transferred to the surface. The other energies, in spite of their changes, do not contribute significantly to the energy transfer.

x

y

at Nanoscopic Level: A Molecular Dynamics Study

simulation box

sections.

2010).

**5.2 Results**

in the amorphous state.

z

Fig. 10. Cu spherical projectile with velocity −*vz*. Both are sitting in the *xy* plane of the

In the following we present a detailed description of the different processes, emphasizing the structural changes suffered by the target. To perform the analysis of the target in a better way, atoms belonging to the projectile were removed and the target was divided into two radial

<sup>245</sup> Inelastic Collisions and Hypervelocity Impacts

In Figure 11 is shown a general view of the passage of the projectile, in this case corresponding to 1.5 km/s. As can be seen, it melts the sample locally when it is going through. After the passage of the projectile, there are two regimes on the behavior of the target: 1) to certain speeds, including 1.5 and 3.0 km/s, the sample returns to its initial fcc structure, but with at higher temperature and with dislocations of planes, and 2) for speeds equal to or greater than 5.0 km/s, the projectile left a hole in the target and even though the atoms regroup in the same way, the target as a whole can not return its initial fcc structure, resulting a large percentage

Here we will analyze in details only the case at lower velocity, 1.5 km/s. From the snapshots showed in Figure 11, we can see that at 1.2 ps the projectile hits the target producing an increase in temperature at the impact zone. Then the projectile continues to move through the target producing, in addition to local temperature increases (in the vicinity of the projectile trajectory), a wake of disturbed material, which is perceived as temperature fluctuations. At 4.8 ps dislocations appeared. When the projectile begins to leave the target, at 8.4 ps, some target atoms are ejected. At this same time, the area where the bullet impacts begins to become disordered. After a longer time, at 15 ps, the area where the projectile leaves the target (back side) is disordered. Finally, at 28.2 ps, we observe that the whole area which had been disturbed by the projectile is re-ordered, resembling its original structure, but with dislocations. In general, it appears that the projectile disturbs the zone which corresponds to its trajectory and its neighborhood, but it does not causes great impact beyond that (Loyola,

In order to quantify the just described picture, we analyze the change of local temperature, and the atomic order, by means of *g*(*r*) and by the CNA. Temperature profiles are shown in Figs. 12, 13, which correspond to a radial zone close and far to bullet trajectory, respectively. In

#### **5. Hypervelocity impact of projectiles**

Hypervelocity impact of projectiles is of great interest in basic and applied research, and it is present in areas such as engineering and physics of materials, including civilian and military applications, among others. For example, since the development in the 1980's of cluster beam technology, the quality of the beams and the number of applications continues to grow (Jacquet & Beyec, 2002; Kirkpatrick, 2003; Popok & Campbell, 2006), as is the case of the materials which are bombarded with cluster beams in order to clean or smooth their surface or to analyze their composition, as well as to consolidate clusters. In several cases the effect of the cluster beams result from the combined effects of a single impact, which occurs separately and independently (Hsieh et al., 1992). Therefore it is important to understand the dynamics of such a single impact. In the field of space applications, hypervelocity impacts are being studied to see the damage they produce on ceramic tiles when nano and micrometeorites hit satellites, spacecraft and space stations. Because the experimental study at such high velocities (ranging from 3 km/s to 15 km/s approximately) is extremely difficult, computer simulation is an ideal tool to deal with them.

In the following we will study, by classical molecular dynamics simulations, the impact of a cluster composed of 47 atoms of copper (Cu) on a solid target of approximately 50 000 atoms of copper. The main goal is to depict the structural response of the target with respect to three different velocities of impact, 1.5 km/s, 3.0 km/s and 5.0 km/s. There will be a detailed description of the different processes, emphasizing the structural changes suffered by the target.

#### **5.1 Computational procedure**

The impact simulations were performed at high speeds with classical molecular dynamics, using the computer program *LPMD* (Davis et al., 2010). To simulate the impact of a projectile on a target, we initially built a cubic box of edge 86.64 Å containing 55,296 copper atoms in a FCC structure, which is used as target. This target was thermalized to 300 K through rescaling of velocities, for 15,000 time steps with 1Δ*t* = 1 fs. Then it was allowed to evolve without temperature control for another 15,000 time steps. The projectile is spherical in shape with a diameter of approximately 8 Å (one tenth the length of the edge of the target). Both projectile and target were placed in a tetragonal simulation cell length *x* = *y* = 198.55 Å and *z* = 249.09 Å, centered at *x* and *y*, and separated by a distance 11 Å in *z*, as is shown in Figure 10.

The atomic interaction is represented by the empirical many-body Sutton-Chen potential,

$$\phi = \varepsilon \left\{ \sum\_{i=1} \sum\_{j=i+1} \left( \frac{a}{r\_{ij}} \right)^n - c \sum\_{i=1} \left[ \sum\_{j=1, j \neq i} \left( \frac{a}{r\_{ij}} \right)^m \right]^{1/2} \right\},\tag{18}$$

where *ε* = 0.0124 eV, *a* = 3.61 Å (Cu lattice parameter), and *c* = 39.432, *n* = 9, *m* = 6 are adimensional parameters.

The simulations we perform used three different velocities for the projectile: 1.5 km/s, 3.0 km/s and 5.0 km/s, while the target is at rest. The projectile velocity is kept constant during all the simulation, irrespective of the friction or the force exerted by the target. Although this is not a real situation, it represent an extreme condition, where the momentum and hardness of the projectile is much higher than the momentum and hardness of the target.

Fig. 10. Cu spherical projectile with velocity −*vz*. Both are sitting in the *xy* plane of the simulation box

In the following we present a detailed description of the different processes, emphasizing the structural changes suffered by the target. To perform the analysis of the target in a better way, atoms belonging to the projectile were removed and the target was divided into two radial sections.

#### **5.2 Results**

16 Will-be-set-by-IN-TECH

Hypervelocity impact of projectiles is of great interest in basic and applied research, and it is present in areas such as engineering and physics of materials, including civilian and military applications, among others. For example, since the development in the 1980's of cluster beam technology, the quality of the beams and the number of applications continues to grow (Jacquet & Beyec, 2002; Kirkpatrick, 2003; Popok & Campbell, 2006), as is the case of the materials which are bombarded with cluster beams in order to clean or smooth their surface or to analyze their composition, as well as to consolidate clusters. In several cases the effect of the cluster beams result from the combined effects of a single impact, which occurs separately and independently (Hsieh et al., 1992). Therefore it is important to understand the dynamics of such a single impact. In the field of space applications, hypervelocity impacts are being studied to see the damage they produce on ceramic tiles when nano and micrometeorites hit satellites, spacecraft and space stations. Because the experimental study at such high velocities (ranging from 3 km/s to 15 km/s approximately) is extremely difficult, computer simulation

In the following we will study, by classical molecular dynamics simulations, the impact of a cluster composed of 47 atoms of copper (Cu) on a solid target of approximately 50 000 atoms of copper. The main goal is to depict the structural response of the target with respect to three different velocities of impact, 1.5 km/s, 3.0 km/s and 5.0 km/s. There will be a detailed description of the different processes, emphasizing the structural changes suffered by the

The impact simulations were performed at high speeds with classical molecular dynamics, using the computer program *LPMD* (Davis et al., 2010). To simulate the impact of a projectile on a target, we initially built a cubic box of edge 86.64 Å containing 55,296 copper atoms in a FCC structure, which is used as target. This target was thermalized to 300 K through rescaling of velocities, for 15,000 time steps with 1Δ*t* = 1 fs. Then it was allowed to evolve without temperature control for another 15,000 time steps. The projectile is spherical in shape with a diameter of approximately 8 Å (one tenth the length of the edge of the target). Both projectile and target were placed in a tetragonal simulation cell length *x* = *y* = 198.55 Å and *z* = 249.09 Å, centered at *x* and *y*, and separated by a distance 11 Å in *z*, as is shown in Figure 10. The atomic interaction is represented by the empirical many-body Sutton-Chen potential,

**5. Hypervelocity impact of projectiles**

is an ideal tool to deal with them.

**5.1 Computational procedure**

*φ* = *ε*

adimensional parameters.

⎧ ⎪⎨

⎪⎩ ∑ *i*=1

∑ *j*=*i*+1 � *a rij* �*<sup>n</sup>*

− *<sup>c</sup>* ∑ *i*=1

where *ε* = 0.0124 eV, *a* = 3.61 Å (Cu lattice parameter), and *c* = 39.432, *n* = 9, *m* = 6 are

The simulations we perform used three different velocities for the projectile: 1.5 km/s, 3.0 km/s and 5.0 km/s, while the target is at rest. The projectile velocity is kept constant during all the simulation, irrespective of the friction or the force exerted by the target. Although this is not a real situation, it represent an extreme condition, where the momentum and hardness of the projectile is much higher than the momentum and hardness of the target.

⎡ <sup>⎣</sup> ∑ *j*=1,*j*�=*i*

� *a rij* �*m*⎤ ⎦

1/2⎫ ⎪⎬

⎪⎭ , (18)

target.

In Figure 11 is shown a general view of the passage of the projectile, in this case corresponding to 1.5 km/s. As can be seen, it melts the sample locally when it is going through. After the passage of the projectile, there are two regimes on the behavior of the target: 1) to certain speeds, including 1.5 and 3.0 km/s, the sample returns to its initial fcc structure, but with at higher temperature and with dislocations of planes, and 2) for speeds equal to or greater than 5.0 km/s, the projectile left a hole in the target and even though the atoms regroup in the same way, the target as a whole can not return its initial fcc structure, resulting a large percentage in the amorphous state.

Here we will analyze in details only the case at lower velocity, 1.5 km/s. From the snapshots showed in Figure 11, we can see that at 1.2 ps the projectile hits the target producing an increase in temperature at the impact zone. Then the projectile continues to move through the target producing, in addition to local temperature increases (in the vicinity of the projectile trajectory), a wake of disturbed material, which is perceived as temperature fluctuations. At 4.8 ps dislocations appeared. When the projectile begins to leave the target, at 8.4 ps, some target atoms are ejected. At this same time, the area where the bullet impacts begins to become disordered. After a longer time, at 15 ps, the area where the projectile leaves the target (back side) is disordered. Finally, at 28.2 ps, we observe that the whole area which had been disturbed by the projectile is re-ordered, resembling its original structure, but with dislocations. In general, it appears that the projectile disturbs the zone which corresponds to its trajectory and its neighborhood, but it does not causes great impact beyond that (Loyola, 2010).

In order to quantify the just described picture, we analyze the change of local temperature, and the atomic order, by means of *g*(*r*) and by the CNA. Temperature profiles are shown in Figs. 12, 13, which correspond to a radial zone close and far to bullet trajectory, respectively. In

Fig. 12. Local temperature along the *z*−direction for different times, in the zone close to the

<sup>247</sup> Inelastic Collisions and Hypervelocity Impacts

Fig. 13. Local temperature along the *z*−direction for different times, in the zone far the

The structural analysis of the sample was made by the pair distribution function, *g*(*r*) for three different region in the *z* direction: region A, where the bullet hit the sample (Fig. 14 a), region B, in the middle (Fig. 14 b), and region C, at the end of the sample (Fig. 14 c), where the bullet

center of the target, as the 1.5 km/s projectile is moving.

at Nanoscopic Level: A Molecular Dynamics Study

center of the target, as the 1.5 km/s projectile is moving.

Fig. 11. Snapshots of the impact of the projectile traveling at 1.5 km/s over a copper target at different times. The colors are assigned according to the atom's temperature, from low temperature (blue) to high temperature (red) (Loyola, 2010).

general, we can observe a temperature front that is moving in the same direction as the bullet. In Fig. 12 can be seen a temperature maxima of 1760 K at 3.6 ps. After that, at 28.2 ps, this part of the sample is thermalized at 450 K. The temperature, of course, propagates in radial direction from the center outward of the sample. Figure 13 displays the temperature profile beyond the central zone. Although there are not prominent peaks, the information about the passage of the projectile is shown in the 1.2 ps panel as a small peak at 65 Å , indicating that the temperature perturbation propagates at higher velocity than the projectile, ruling out the occurrence of a shock wave.

18 Will-be-set-by-IN-TECH

(a) *t* = 1.2 ps (b) *t* = 2.4 ps (c) *t* = 3.6 ps

(d) *t* = 4.8 ps (e) *t* = 6 ps (f) *t* = 8.4 ps

(g) *t* = 15 ps (h) *t* = 28.2 ps

Fig. 11. Snapshots of the impact of the projectile traveling at 1.5 km/s over a copper target at different times. The colors are assigned according to the atom's temperature, from low

general, we can observe a temperature front that is moving in the same direction as the bullet. In Fig. 12 can be seen a temperature maxima of 1760 K at 3.6 ps. After that, at 28.2 ps, this part of the sample is thermalized at 450 K. The temperature, of course, propagates in radial direction from the center outward of the sample. Figure 13 displays the temperature profile beyond the central zone. Although there are not prominent peaks, the information about the passage of the projectile is shown in the 1.2 ps panel as a small peak at 65 Å , indicating that the temperature perturbation propagates at higher velocity than the projectile, ruling out the

temperature (blue) to high temperature (red) (Loyola, 2010).

occurrence of a shock wave.

Fig. 12. Local temperature along the *z*−direction for different times, in the zone close to the center of the target, as the 1.5 km/s projectile is moving.

Fig. 13. Local temperature along the *z*−direction for different times, in the zone far the center of the target, as the 1.5 km/s projectile is moving.

The structural analysis of the sample was made by the pair distribution function, *g*(*r*) for three different region in the *z* direction: region A, where the bullet hit the sample (Fig. 14 a), region B, in the middle (Fig. 14 b), and region C, at the end of the sample (Fig. 14 c), where the bullet

most atoms in fcc order. Finally, at *t* = 28 ps, the region reach 70% of fcc atoms, 3% of hcp atoms and around 25% of non-crystalline atoms. For the others regions the situation is similar, except it occurs at longer times, when the perturbation and the bullet reach that regions. The only significant difference is that, in the case of region C, the percentage of atoms in a non-crystalline structure is greater than the previous regions, which can be also seen

<sup>249</sup> Inelastic Collisions and Hypervelocity Impacts

Percentage

0 4 8 12 16 20 24 28

FCC HCP

Time (ps)

(c) Region C

In summary, molecular dynamics simulation of hypervelocity projectile impact has been done. The atomic level study allows us to describe several interesting features that are not possible to track by other methods. In particular, two regimes has been identified, in dependence of the projectile initial velocity. At high velocity, the passage of the projectile through the target leaves a hole in the sample, as well as produce structural phase transition. Al low temperature, the case that has been study in detail here, the projectile cause local melting and dislocations as it moves through the sample. At the end, the target recover its original fcc crystalline structure, but with a non-negligible percentage of atoms in hcp structure and amorphous phase.

Descriptions of phenomena far from equilibrium are not an easy task in physics: from the experimental point of view it is required to have both high spatial and temporal resolution in the different variables measured; even worse, often we have destructive experiments, such as

Fig. 15. Common neighbors analysis (CNA) at different times, for three region in the *z*

0 4 8 12 16 20 24 28

FCC HCP

Time (ps)

(b) Region B

directly from the snapshot (Fig. 11(h)).

at Nanoscopic Level: A Molecular Dynamics Study

0 4 8 12 16 20 24 28

FCC HCP

Time (ps)

Percentage

(a) Region A

Percentage

direction.

**6. Conclusions**

Fig. 14. Pair distribution function for different region of the sample. The curves have been shifted for clarity

go out. Figures 14 a, and 14 b show the same phenomenology but at different times. Initially, all sections have fcc structure (the peaks have a finite width because initially the system is at room temperature), then melt, as is appreciated at the times 3.3 ps, 6 ps and 8.5 ps for regions A, B and C, respectively. Finally, all these regions recover their crystalline structure, but at higher temperatures than the initial stage. In the case of the radial zone far from the bullet trajectory the situation is different because this part did not melt at any time, preserving the initial fcc structure but at higher temperatures, around 500 K.

Interestingly, the crystalline structure that the sample recovers after the passage of the bullet is a mixture between fcc and hcp structure. In fact, the high pressure resulting by the impact produce structural transformation, which at the end results in the coexistence of fcc and hcp phases. To quantify its relation, we perform a common neighbors analysis, CNA, for the the three region where the projectile pass, at the beginning (region A), the middle (region B) and the end (region C), with respect to the *z* direction. The CNA calculates the percentage of atoms with structure fcc, hcp, bcc and icosahedral, and the rest is considered as non-crystalline (amorphous) structure. Figure 15 displays the percentage of fcc and hcp atoms (the difference between their sum and 100% corresponds to atoms in amorphous structure). We can see that in the region A, for *t* < 2 ps, all the atoms are still in a fcc order, because the bullet is just hitting. After that, the percentage of fcc atoms decrease to 10% and the hcp atoms appear, reaching also almost 10%. The rest are atoms in a non-crystalline structure. At *t* > 8.5 ps, when the projectile has go out the sample, the region A start to recover its crystalline order, 20 Will-be-set-by-IN-TECH

0

g(r)

Fig. 14. Pair distribution function for different region of the sample. The curves have been

go out. Figures 14 a, and 14 b show the same phenomenology but at different times. Initially, all sections have fcc structure (the peaks have a finite width because initially the system is at room temperature), then melt, as is appreciated at the times 3.3 ps, 6 ps and 8.5 ps for regions A, B and C, respectively. Finally, all these regions recover their crystalline structure, but at higher temperatures than the initial stage. In the case of the radial zone far from the bullet trajectory the situation is different because this part did not melt at any time, preserving the

Interestingly, the crystalline structure that the sample recovers after the passage of the bullet is a mixture between fcc and hcp structure. In fact, the high pressure resulting by the impact produce structural transformation, which at the end results in the coexistence of fcc and hcp phases. To quantify its relation, we perform a common neighbors analysis, CNA, for the the three region where the projectile pass, at the beginning (region A), the middle (region B) and the end (region C), with respect to the *z* direction. The CNA calculates the percentage of atoms with structure fcc, hcp, bcc and icosahedral, and the rest is considered as non-crystalline (amorphous) structure. Figure 15 displays the percentage of fcc and hcp atoms (the difference between their sum and 100% corresponds to atoms in amorphous structure). We can see that in the region A, for *t* < 2 ps, all the atoms are still in a fcc order, because the bullet is just hitting. After that, the percentage of fcc atoms decrease to 10% and the hcp atoms appear, reaching also almost 10%. The rest are atoms in a non-crystalline structure. At *t* > 8.5 ps, when the projectile has go out the sample, the region A start to recover its crystalline order,

2 3 4 5 6 7 8

Initial 6 28.3

Initial 3.3 6.3 9.3 15.6

r(Å)

2 3 4 5 6 7 8

r(Å)

(d) Region A, but far from the center

(b) Region B

5

10

g(r)

15

20

0

shifted for clarity

5

10

g(r)

15

20

g(r)

2 3 4 5 6 7 8

Initial 3.3 6 15

Initial 8.5 28.3

r(Å)

2 3 4 5 6 7 8

r(Å)

initial fcc structure but at higher temperatures, around 500 K.

(c) Region C

(a) Region A

most atoms in fcc order. Finally, at *t* = 28 ps, the region reach 70% of fcc atoms, 3% of hcp atoms and around 25% of non-crystalline atoms. For the others regions the situation is similar, except it occurs at longer times, when the perturbation and the bullet reach that regions. The only significant difference is that, in the case of region C, the percentage of atoms in a non-crystalline structure is greater than the previous regions, which can be also seen directly from the snapshot (Fig. 11(h)).

Fig. 15. Common neighbors analysis (CNA) at different times, for three region in the *z* direction.

In summary, molecular dynamics simulation of hypervelocity projectile impact has been done. The atomic level study allows us to describe several interesting features that are not possible to track by other methods. In particular, two regimes has been identified, in dependence of the projectile initial velocity. At high velocity, the passage of the projectile through the target leaves a hole in the sample, as well as produce structural phase transition. Al low temperature, the case that has been study in detail here, the projectile cause local melting and dislocations as it moves through the sample. At the end, the target recover its original fcc crystalline structure, but with a non-negligible percentage of atoms in hcp structure and amorphous phase.

#### **6. Conclusions**

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#### **7. Acknowledgments**

This work was supported by the Anillo Project ACT-24 *Computer simulation lab for nano-bio systems.* G. Gutiérrez thanks ENL 10/06 VRID-Universidad de Chile. F. González-Cataldo thanks CONICYT-Chile Ph.D fellowship. S. Davis acknowledges Fondecyt grant 3110017.

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