**Inelastic Collisions and Hypervelocity Impacts at Nanoscopic Level: A Molecular Dynamics Study**

G. Gutiérrez, S. Davis, C. Loyola, J. Peralta, F. González, Y. Navarrete and F. González-Wasaff *Group of NanoMaterials*\**, Departamento de Física, Facultad de Ciencias, Universidad de Chile Chile*

#### **1. Introduction**

228 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

Lim, E. W. C. (2007). Voidage Waves in Hydraulic Conveying through Narrow Pipes. *Chem.* 

Lim, E. W. C. (2008). Master Curve for the Discrete-Element Method. *Ind. Eng. Chem. Res.*,

Lim, E. W. C. (2009). Vibrated Granular Bed on a Bumpy Surface. *Phys. Rev. E*, Vol. 79, pp. 041302 Lim, E. W. C. (2010a). Density Segregation in Vibrated Granular Beds with Bumpy Surfaces.

Lim, E. W. C. (2010b). Granular Leidenfrost Effect in Vibrated Beds with Bumpy Surfaces.

Lim, E. W. C.; Yao, J.; Zhao, Y. (2011). Pneumatic Transport of Granular Materials with Electrostatic Effects. *AIChE J.*, Article in Press, DOI 10.1002/aic.12638 Martinez-Pedrero, F.; El-Harrak, A.; Fernandez-Toledano, J. C.; Tirado-Miranda, M.; Baudry,

Meriguet, G.; Jardat, M.; Turq, P. (2004). Structural Properties of Charge-Stabilized

Meriguet, G.; Jardat, M.; Turq, P. (2005). Brownian Dynamics Investigation of Magnetization and Birefringence Relaxations in Ferrofluids. *J. Chem. Phys.*, Vol. 123, pp. 144915 Miyazima, S.; Meakin, P.; Family, F. (1987). Aggregation of Oriented Anisotropic Particles.

Nandy, K.; Chaudhuri, S.; Ganguly, R.; Puri, I. K. (2008). Analytical Model for the

Promislow, J. H. E.; Gast, A. P.; Fermigier, M. (1995). Aggregation Kinetics of Paramagnetic

Richardi, J.; Pileni, M. P.; Weis, J.-J. (2008). Self-organization of Magnetic Nanoparticles: A

Russel, W. B.; Saville, D. A.; Schowalter, W. R. (1989). *Colloidal Dispersions*, Cambridge

Schaller, V.; Kraling, U.; Rusu, C.; Petersson, K.; Wipenmyr, J.; Krozer, A.; Wahnstrom, G.;

Tsouris, C.; Scott, T. C. (1995). Flocculation of Paramagnetic Particles in a Magnetic Field. *J.* 

Xuan, Y.; Ye, M.; Li, Q. (2005). Mesoscale Simulation of Ferrofluid Structure. *Int. J. Heat Mass* 

Yamada, Y.; Enomoto, Y. (2008). Effects of Oscillatory Shear Flow on Chain-like Cluster Dynamics in Ferrofluids without Magnetic Fields. *Physica A*, Vol. 387, pp. 1–11 Zhu, H. P.; Zhou, Z. Y.; Yang, R. Y.; Yu, A. B. (2008). Discrete Particle Simulation of

Particulate Systems: A Review of Major Applications and Findings. *Chem. Eng. Sci.*,

Sanz-Velasco, A.; Enoksson, P.; Johansson, C. (2008). Motion of Nanometer Sized Magnetic Particles in a Magnetic Field Gradient. *J. Appl. Phys.*, Vol. 104, pp. 093918 Sinha, A.; Ganguly, R.; De, A. K.; Puri, I. K. (2007). Single Magnetic Particle Dynamics in a

J.; Schmitt, A.; Bibette, J.; Callejas-Fernandez, J. (2008). Kinetic Study of Coupled Field-Induced Aggregation and Sedimenation Processes Arising in Magnetic

Ferrofluids under a Magnetic Field: A Brownian Dynamics Study. *J. Chem. Phys.*,

Magnetophoretic Capture of Magnetic Microspheres in Microfluidic Devices. *J.* 

*Eng. Sci.*, Vol. 62, pp. 4529–4543

*AIChE J.*, Vol. 56, pp. 2588–2597

*Eur. Phys. J. E*, Vol. 32, pp. 365–375

Fluids. *Phys. Rev. E*, Vol. 78, pp. 011403

*Phys. Rev. A*, Vol. 36, pp. 1421–1427

*Magn. Magn. Mater.*, Vol. 320, pp. 1398–1405

Pamme, N. (2006). Magnetism and Microfluidics. *Lab Chip*, Vol. 6, pp. 24–38

Colloidal Particles. *J. Chem. Phys.*, Vol. 102, pp. 5492–5498

Monte Carlo Study. *Phys. Rev. E*, Vol. 77, pp. 061510 Rosensweig, R. E. (1985). *Ferrohydrodynamics*, Cambridge University Press

Microchannel. *Phys. Fluids*, Vol. 19, pp. 117102

*Colloid Interface Sci.*, Vol. 171, pp. 319–330

*Trans.*, Vol. 48, pp. 2443–2451

Vol. 63, pp. 5728–5770

Vol. 121, pp. 6078–6085

University Press

Vol. 47, pp. 481–485

In this chapter we present an atomic level study of nano-particle impact using molecular dynamics simulation. Two cases have been considered. First, we simulate the bouncing of a ball over a surface due to a constant force (which mimic the gravity force), modeling the inter-atomic interaction by a modified Lennard-Jones potential, where the ball-surface atom interaction is represented by a purely repulsive term. The analysis of the results makes it possible, among other aspects, to determine the restitution coefficient in each bounce as well as to understand the processes of energy loss in inelastic collisions, which are actually not a loss, but a transfer to thermal and vibrational energy. The second simulation describes the impact mechanisms of a solid projectile hitting a target at high velocity. Both the projectile and the target are made of copper, which is modeled by a realistic many-body tight-binding potential. The projectile velocity is kept constant during all the simulation, representing an extreme condition, where the momentum and hardness of the projectile is much higher than the momentum and hardness of the target. In this regime, we identify two different behavior in dependence of the projectile velocity: at low velocities (less than 4 km/s) the target basically recover its structure after the passage of the projectile, but at higher velocities, the projectile left a permanent hole in the target.

Both problems, inelastic collisions and hypervelocity impacts, are non-equilibrium related phenomena which are important from a basic and applied point of view, in several areas of science: physics, materials science, aeronautics, mechanics, among others. From a theoretical point of view, they have been extensively treated in the macroscopic level, by using continuum hydrodynamic simulation, and only recently researchers are using molecular dynamic simulation, intended to an understanding of these phenomena at the scale of inter-atomic interactions. Besides the calculation of equilibrium properties and their associated fluctuations, molecular dynamics allows for a wider range of problems to be tackled: given that we have access to the atomic trajectories we can study the transit to equilibrium, as well as purely non-equilibrium phenomena (where we are interested not in the final state but in the process itself ), for instance, shock-induced plasticity and fracture of materials. In this regard, Non-Equilibrium Molecular Dynamics (NEMD) has emerged recently as a branch dealing with, and promising to shed light on, the mechanism behind these (and other similar) irreversible processes.

<sup>\*</sup>www.gnm.cl

#### **2. Molecular dynamics in non-equilibrium conditions**

The framework to tackle the problems out of equilibrium is Non-equilibrium Statistical Mechanics. Its concerns the extension of the usual formalism of Statistical Mechanics (microcanonical, canonical and other extended ensembles, partition functions) to systems either approaching thermodynamic equilibrium after a perturbation, or definitely far away from it. So far there is no unified theory we can appropriately call non-equilibrium statistical mechanics 1, only a number of results applicable to processes in the linear response regime (thermodynamic fluxes proportional to the thermodynamic forces), such as the celebrated Onsager regression hypothesis (Callen, 1985) that relates the decay of macroscopic variables in a non-equilibrium setting to the regression of fluctuations in equilibrium. Prigogine's minimum entropy production (Prigogine, 1968) principle, also restricted to the linear response regime, is a possible explanation for the emergence of order in dissipative systems. The fluctuation-dissipation theorem and the Green-Kubo formulas (Zwanzig, 2001) determine transport coefficients from equilibrium measurements. There are also a few results valid arbitrarily far away from equilibrium, such as the family of fluctuation theorems (Evans & Searles, 2002) quantifying the likelihood of instantaneous violations of the Second Law of thermodynamics.

Non-equilibrium Molecular Dynamics (NEMD) is then the natural extension of molecular dynamics techniques to study non-equilibrium problems, and attempts to fill the void left by a missing theoretical framework.

Stationary (or *steady-state*) processes like deformation under shear stress, or a sample submitted under a temperature gradient, among others, require the implementation of NEMD under temperature control. In this case the use of thermostat algorithms is necessary to maintain the steady-state regime, extracting the excess heat generated by the process. However this has the drawback of modifying the equations of motion, introducing friction and noise forces which perturb the original dynamics (energy is not conserved), and affecting the performance of the usual numerical integration methods.

A comprehensive review of thermostat methods and their implementation in the context of NEMD is given by Hoover (Hoover & Hoover, 2007). Briefly, the standard implementation of the thermostat is the Nosé-Hoover equation of motion,

$$\frac{d\vec{p}\_i}{dt} = \vec{F}\_i - \zeta \vec{p}\_{i\nu} \tag{1}$$

the case of shockwave propagation (Holian, 1995), collisions, fast fracture, detonations) where it is not necessary to remove the excess heat. Here energy is conserved, being converted from kinetic or elastic into thermal, vibrational and other forms, the heat eventually produced in the process remains inside the system, causing an increase in temperature and eventually being able to induce local melting. This kind of NEMD simulations are justified because we are

<sup>231</sup> Inelastic Collisions and Hypervelocity Impacts

which describes the evolution of the phase space distribution function *P*(**p**, **q**) of a system of particles obeying Newton's equations, and this is valid arbitrarily away from equilibrium. It is important to consider that, away from the linear response regime, there is no unique definition of thermodynamic intensive variables such as temperature, pressure or chemical potential if those variables are not fixed (Casas-Vásquez & Jou, 2003). However, the usual practice is to take the instantaneous kinetic energy of the system (or even of a region of the

*TK*(*t*) = *<sup>m</sup>*

target cannot be assigned a higher temperature by virtue of its translational speed.

*TC*(*t*) = <sup>1</sup>

3*kB*

When using the instantaneous kinetic energy to evaluate a local instantaneous temperature, it might be required to remove the translational part of the velocities for the atoms in the region, if they happen to have non-zero linear momentum. For instance, a projectile approaching a

It is also possible to evaluate an instantaneous "configurational" temperature (Baranyai, 2000),

*kB*

where Φ is the potential energy function, which depends on *t* only through the atomic positions. Away from equilibrium both definitions (kinetic and configurational) do not necessarily coincide, because an object immersed in the non-equilibrium system and used as a thermometer could equilibrate in different time scales to the configurational and kinetic degrees of freedom and therefore measure different temperatures. In fact "operational" definitions of non-temperature exist that measure the kinetic energy of a tracer (probably heavier) particle placed inside the system, and assumed to be in thermal equilibrium with

In the following, we briefly describe the molecular dynamic method and its implementation our in-house code *Las Palmeras Molecular Dynamics*. Next, the inelastic collisions and hypervelocity impacts simulations are presented, as examples of the potential of an

Although there are many general purpose MD codes, they are usually subjected to design limitations arising mostly due to efficiency considerations. A given code is usually optimized to perform extremely well for one kind of system (for instance bulk systems) but because of

atomic-level description. Finally, general conclusions are drawn.

**3. Las Palmeras Molecular Dynamics**

*N* ∑ *i*=1 *v*2


*<sup>∂</sup><sup>t</sup>* <sup>=</sup> −{*P*, H}, (3)

*<sup>i</sup>* . (4)

<sup>∇</sup>2Φ(*t*) , (5)

*∂P*(**p**, **q**)

system) to evaluate an instantaneous "kinetic" temperature,

implicitly solving the Liouville equation,

at Nanoscopic Level: A Molecular Dynamics Study

it.

where *ζ* is a friction coefficient governed by

$$\frac{d\mathbb{Q}}{dt} = \frac{1}{N\tau^2} \sum\_{i=1}^{N} \left( p\_i^2 / mk\_B T\_0 - 1 \right),\tag{2}$$

*T*<sup>0</sup> the imposed temperature and *τ* is a relaxation time, controlling the degree of coupling of the "thermal bath" with the system.

It is possible, however, to perform NEMD in a completely microcanonical way (i.e. without modifying Newton's equations) for systems outside the steady-state regime (for instance in

<sup>1</sup> The maximum caliber formalism (Jaynes, 1980; Stock et al., 2008), based on information-theoretic ideas, together with the maximum entropy production principle derived from it seem to show promising early results as such a unifying basis (Dewar, 2005; 2003; Kleidon et al., 2005).

2 Will-be-set-by-IN-TECH

The framework to tackle the problems out of equilibrium is Non-equilibrium Statistical Mechanics. Its concerns the extension of the usual formalism of Statistical Mechanics (microcanonical, canonical and other extended ensembles, partition functions) to systems either approaching thermodynamic equilibrium after a perturbation, or definitely far away from it. So far there is no unified theory we can appropriately call non-equilibrium statistical mechanics 1, only a number of results applicable to processes in the linear response regime (thermodynamic fluxes proportional to the thermodynamic forces), such as the celebrated Onsager regression hypothesis (Callen, 1985) that relates the decay of macroscopic variables in a non-equilibrium setting to the regression of fluctuations in equilibrium. Prigogine's minimum entropy production (Prigogine, 1968) principle, also restricted to the linear response regime, is a possible explanation for the emergence of order in dissipative systems. The fluctuation-dissipation theorem and the Green-Kubo formulas (Zwanzig, 2001) determine transport coefficients from equilibrium measurements. There are also a few results valid arbitrarily far away from equilibrium, such as the family of fluctuation theorems (Evans & Searles, 2002) quantifying the likelihood of instantaneous violations of the

Non-equilibrium Molecular Dynamics (NEMD) is then the natural extension of molecular dynamics techniques to study non-equilibrium problems, and attempts to fill the void left by

Stationary (or *steady-state*) processes like deformation under shear stress, or a sample submitted under a temperature gradient, among others, require the implementation of NEMD under temperature control. In this case the use of thermostat algorithms is necessary to maintain the steady-state regime, extracting the excess heat generated by the process. However this has the drawback of modifying the equations of motion, introducing friction and noise forces which perturb the original dynamics (energy is not conserved), and affecting

A comprehensive review of thermostat methods and their implementation in the context of NEMD is given by Hoover (Hoover & Hoover, 2007). Briefly, the standard implementation of

*Fi* − *ζpi*, (1)

, (2)

*dpi dt* <sup>=</sup>

> *N* ∑ *i*=1 *p*2

*T*<sup>0</sup> the imposed temperature and *τ* is a relaxation time, controlling the degree of coupling of

It is possible, however, to perform NEMD in a completely microcanonical way (i.e. without modifying Newton's equations) for systems outside the steady-state regime (for instance in

<sup>1</sup> The maximum caliber formalism (Jaynes, 1980; Stock et al., 2008), based on information-theoretic ideas, together with the maximum entropy production principle derived from it seem to show promising

*<sup>i</sup>* /*mkBT*<sup>0</sup> − 1

**2. Molecular dynamics in non-equilibrium conditions**

the performance of the usual numerical integration methods.

*dζ dt* <sup>=</sup> <sup>1</sup> *Nτ*<sup>2</sup>

early results as such a unifying basis (Dewar, 2005; 2003; Kleidon et al., 2005).

the thermostat is the Nosé-Hoover equation of motion,

where *ζ* is a friction coefficient governed by

the "thermal bath" with the system.

Second Law of thermodynamics.

a missing theoretical framework.

the case of shockwave propagation (Holian, 1995), collisions, fast fracture, detonations) where it is not necessary to remove the excess heat. Here energy is conserved, being converted from kinetic or elastic into thermal, vibrational and other forms, the heat eventually produced in the process remains inside the system, causing an increase in temperature and eventually being able to induce local melting. This kind of NEMD simulations are justified because we are implicitly solving the Liouville equation,

$$\frac{\partial P(\mathbf{p}, \mathbf{q})}{\partial t} = -\{P, \mathcal{H}\},\tag{3}$$

which describes the evolution of the phase space distribution function *P*(**p**, **q**) of a system of particles obeying Newton's equations, and this is valid arbitrarily away from equilibrium.

It is important to consider that, away from the linear response regime, there is no unique definition of thermodynamic intensive variables such as temperature, pressure or chemical potential if those variables are not fixed (Casas-Vásquez & Jou, 2003). However, the usual practice is to take the instantaneous kinetic energy of the system (or even of a region of the system) to evaluate an instantaneous "kinetic" temperature,

$$T\_K(t) = \frac{m}{3k\_B} \sum\_{i=1}^{N} v\_i^2. \tag{4}$$

When using the instantaneous kinetic energy to evaluate a local instantaneous temperature, it might be required to remove the translational part of the velocities for the atoms in the region, if they happen to have non-zero linear momentum. For instance, a projectile approaching a target cannot be assigned a higher temperature by virtue of its translational speed.

It is also possible to evaluate an instantaneous "configurational" temperature (Baranyai, 2000),

$$T\_{\mathbb{C}}(t) = \frac{1}{k\_B} \frac{|\nabla \Phi(t)|^2}{\nabla^2 \Phi(t)},\tag{5}$$

where Φ is the potential energy function, which depends on *t* only through the atomic positions. Away from equilibrium both definitions (kinetic and configurational) do not necessarily coincide, because an object immersed in the non-equilibrium system and used as a thermometer could equilibrate in different time scales to the configurational and kinetic degrees of freedom and therefore measure different temperatures. In fact "operational" definitions of non-temperature exist that measure the kinetic energy of a tracer (probably heavier) particle placed inside the system, and assumed to be in thermal equilibrium with it.

In the following, we briefly describe the molecular dynamic method and its implementation our in-house code *Las Palmeras Molecular Dynamics*. Next, the inelastic collisions and hypervelocity impacts simulations are presented, as examples of the potential of an atomic-level description. Finally, general conclusions are drawn.

#### **3. Las Palmeras Molecular Dynamics**

Although there are many general purpose MD codes, they are usually subjected to design limitations arising mostly due to efficiency considerations. A given code is usually optimized to perform extremely well for one kind of system (for instance bulk systems) but because of said optimization it performs poorly on a different kind of system. This, in practice, only allows the study of certain systems and conditions.

(GPL) version 3. Figure 1 displays an example of the control file. For more information, visit

<sup>233</sup> Inelastic Collisions and Hypervelocity Impacts

#This is a comment. Comments are used usually as a title:

######################################### # System file of Au crystal using LPMD # #########################################

input module=lpmd file=300K-Gold.lpmd level=1 output module=lpmd file=au.lpmd each=15 level=1
