Molecular dynamics (MD) is a computational method that allows the study of the behavior of a large number of similar particles or systems at the atomic scale. This technique enables people to obtain the atomic or particle paths (different positions of particles depending on the time), in order to simulate the microscopic system behavior. From this knowledge, several dynamic and static properties of the system can be obtained. Systems can be independent and can interact between them, or with the external environment only by means of instantaneous processes conserving the energy and the momentum. ^{[ 1 – 8 ]}

Nanoindentation and nanoscratch are standard techniques used for analyzing the mechanical properties of materials. In the case of thin films, this technique gives detailed information about the process at low dimension scales, when materials are being indented in accordance with experimental results. In simulations of nanoscratch and nanoindentation processes, longitudinal and cross movements of a nanoindenter are carried out in order to obtain mechanical responses, which are then compared with the experiments.

In published literature, there are some reports of nanoindentation ^{[ 9 , 10 ]} and nanoscratch ^{[ 2 , 11 , 12 ]} processes being carried out for FCC structures of Cu, Al, or Au. In our case, we simulated the BCC-type structure of a generic coating.

MD normally uses interatomic potentials that recreate the atomic vibrations in the crystalline lattice. These potentials, as the Morse and embedded atom method, are well suited for metallic materials. In general, these materials are independent of temperature. Furthermore, it is necessary to consider the particles’ relative positions, as well as the electronic cloud density, for the EAM potential.

Previous simulations have shown information about the elastic-plastic behavior occurring in materials when they are indented. Smith et al . ^{[ 13 ]} described defects and atomic pile-up in Fe samples during the nanoindentation process. Alcalá et al . ^{[ 14 ]} focused on the formation of planar defects and dislocations. Zhu et al . ^{[ 15 ]} studied the influence of the nanoindentation shape on scratch tests.

In this work, the system included an FCC diamond-type structure as the substrate, and a BCC structure as the coating was considered. Several nanoindentation and nanoscratch tests were applied to this system in order to determine its mechanical and tribological properties. The main objective of this work is to present clearly the theoretical process carried out for simulating dynamic systems where external forces are acting.

Molecular dynamics simulations were implemented to study the behavior of monolayers of BCC crystal during nanoindentation and scratch tests. Figure  1 shows a schematic representation of the system. Samples contain monolayers of BCC crystalline structures. This structure is oriented in the plane (100) and placed on a substrate of FCC diamond structure in the [100] direction. These simulations consist of three parts: (i) nanoindentation, during which the indenter is pushed in the z direction into the thin film, (ii) nanoscratching, where the indenter moves at the indentation depth along the x direction, and (iii) retraction of the indenter, where the indenter is moved out of the substrate to return to the initial height. The nanoindenter is built as a rigid sphere with a radius of R = 7.5 Å, and interaction with the sample is by means of a repulsive potential. The thin film in these simulations has dimensions of 60 Å in the x – y plane and 60 Å in the z direction, with a lattice parameter a = 1 Å. Fixed atoms are employed at the bottom and lateral faces of the sample, and free boundary conditions in the top of the sample.

Fig. 1.

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	Fig. 1. Illustration of the system construction including spherical nanoindenter and thin film.

The equipartition energy theorem specifies that in a thermodynamic system, the mean kinetic energy is equivalent to the temperature. This relationship can be expressed in the following way:

The units used in the simulations have to be carefully managed. In order to optimize the computational time, simulations were carried out using reduced units. In this study, reduced units of energy, length and mass, using the corresponding scaling factors for energy ( ε ), length ( σ ), and mass ( μ ) were employed as follows:

Fig. 2.

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	Fig. 2. Cross section in the x – y plane of (a) solid surface penetrated by a spherical indenter and presenting the corresponding effects as pile up and dislocations formation and (b) solid surface after the scratch process by a spherical indenter, presenting atomic agglomeration and elastic recovery.

Mathematically, the expression is represented as follows:

To evaluate a given atom’s embedding function, one needs to compute the electron density at the position of atom i . This is obtained by a superposition of atomic densities, which are described by a density function, ρ _j ( r _{i j} ), as shown in Eq. ( 23 ).

After calculating the force on each atom, the next step is to integrate the Newton movement equation using the time integration algorithm of Verlet. ^{[ 29 – 34 ]} The Taylor expansion of atomic coordinates in a time interval is given by

Summarizing, MD simulations mainly include three stages. (i) Initiation stage. The sample or system is built. Atoms or particles are placed in the phases space. ^{[ 35 – 38 ]} (ii) Once the sample is built (see Fig.  1 ), a velocity field is generated that acts on each particle of the system, meaning that one velocity corresponds to each element. Velocities are obtained using a random number generator and normally take values from 0 to 1 pm/s. One important feature of these velocities is that they possess a continuous distribution. Nevertheless, to represent the real situation it is necessary to carry out a Gaussian-type transformation in order to produce the fortuitous distribution function of N velocities that can occur with a relatively high probability for certain cases. ^{[ 39 – 41 ]} However, it is not relevant during the sample construction since the thermostat is responsible for stabilizing velocities. Depending on the temperature, the system may evolve as a function of time, scaling the particle velocities. The velocity scaling is carried out from the energy equipartition principle to produce linear momentum equal to zero, and velocities coherent with the temperature. ^{[ 42 – 49 ]} The objective of the last step is to relax the system and eliminate energy excesses, allowing the way for temperature control. At this stage, the time for carrying out the simulation is defined. ^{[ 50 – 55 ]} This time is calculated from the semi-empirical equation obtained from the equipartition principle. ^{[ 56 – 59 ]}

After the simulation time calculation, forces that drive the movement of each atom are determined. They are obtained from the partial derivatives of the interatomic potential that depends on the system to be studied. In this case the Morse potential fitted to metallic systems will be used. ^{[ 60 – 67 ]} From the force calculation, new positions for each atom can be obtained using the Verlet time integration algorithm. ^{[ 29 – 34 ]} Finally, the system is stabilized by using the Anderson method for temperature control also known as the stochastic collision method. At this point, the type of distribution used is relevant. In this case, a Boltzmann-type distribution was employed. ^{[ 39 – 41 ]}

To avoid the mechanical response of the substrate, the nanoindentation was carried out by employing low loads for penetrations of the indenter lower than 34% of the coating thickness. In Table  1 , parameters used for Morse potential at different dissociation energies are listed. Figure  3(a) shows curves of the applied force as a function of the depth for several indenter penetrations, ^{[ 35 – 38 ]} using parameters numbered as “1” in Table  1 . The penetrations employed were 8, 10, 12, 14, 16, 18, and 20 Å. At the beginning of the nanoindentation in the load region, atoms are relaxed in their initial position, allowing the indenter to move forward without any problem (zone 1). From a certain depth the applied force begins to increase, generating elastic deformations, after which a continuous increase of the deformation is observed, as the external applied force increases. At this point, a process of dislocation begins, generating an elastic-plastic deformation (zone 2). This behavior is observed in samples indented with depths of 8, 10, 12, and 14 Å. For greater indentations, 16, 18, 20 Å, in the last stage a stabilization of the applied force that corresponds to the maximum load was observed (zone 3). This stabilization is attributed to the greater depth. A more homogeneous response of the material is observed, caused by the strong joining of the material layers simulated, which voids the reorientation or retransformation.

Fig. 3.

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	Fig. 3. (a) Load–unload curve of applied force versus indentation depth for different maximum penetration depths in thin films of a BCC material and (b) maximum load as a function of the maximum depth nanoindentation including the evolution of the imprint.

Table 1.

Table 1.

Table 1. Morse potential parameters for different dissociation energies. . Atomic bonding Dissociation energy D /eV Compressibility parameter/Å –1 Equilibrium radius/Å Lattice parameter/Å 1 BCC-BCC 0,1555 0,5721 0,5594 1,00 2 BCC-BCC 0,5000 0,5721 0,5594 1,00 3 BCC-BCC 1,0000 0,5721 0,5594 1,00 4 BCC-BCC 1,5000 0,5721 0,5594 1,00 5 BCC-BCC 2,0000 0,5721 0,5594 1,00 1 FCC-FCC 0,1555 0,3621 0,8439 1,00 2 FCC-FCC 0,5000 0,3621 0,8439 1,00 3 FCC-FCC 1,0000 0,3621 0,8439 1,00 4 FCC-FCC 1,5000 0,3621 0,8439 1,00 5 FCC-FCC 2,0000 0,3621 0,8439 1,00 1 FCC-BCC 0,1555 0,4671 0,6973 1,00 2 FCC-BCC 0,5000 0,4671 0,6973 1,00 3 FCC-BCC 1,0000 0,4671 0,6973 1,00 4 FCC-BCC 1,5000 0,4671 0,6973 1,00 5 FCC-BCC 2,0000 0,4671 0,6973 1,00

Table 1. Morse potential parameters for different dissociation energies. .

Finally, in the unload region, it is observed that as the indenter is removed from the system, the materials elastically recover (zone 4). The slope of the curve in the graph indicates how much the material can recover. ^{[ 16 ]} In this case, the recovery is of around 10% of the nanoindenter penetration. ^{[ 68 ]} One of the reasons for this recovery occurring in the materials is the spherical form of the indenter. This geometry allows a greater deformation in the elastic regimen since there are no stresses concentrated that locally amplify tensions which generate plastic deformations in the materials, as normally occurs in the case of indenters with different geometries. ^{[ 17 – 25 ]}

Figure  3(b) shows the evolution of the nanoindentation imprint for different depths when the load is at maximum. At this plot, an increase in the maximum load as the indenter reaches greater depths can be observed. This increase occurs with greater speed for depths between 8 and 14 Å, but for depths of 15 Å and deeper the curve slope begins to decrease, indicating a reduction in the increment rate of the external force as the depth is increased. The perpendicular pile-up of material is due to dislocations generated by the elastic-plastic deformations.

Curves shown in Fig.  4(a) present the friction depending on the sliding distance in wear tests for different depths, ^{[ 2 , 69 ]} using parameters numbered as “1” in Table 1 . The analyzed depths are 8, 9, 10, 11, and 12 Å, and the traveled cross distance is 20 nm. The atoms placed at the boundaries are not included in the simulation process, since the scratch test is carried out far enough from the fixed boundary. In these curves, the decrease of the initial coefficient of friction (zone 1) is observed. This is due to the pile-up produced by the indenter in the first stage of the nanoindentation process. The residual stress generated by this perpendicular atomic pile-up produced a decrease in the initial friction force with respect to the maximum normal force as the nanoindenter moves forward. After that, an abrupt change in the mechanical behavior of the material is produced and the friction force becomes greater than the normal force, generating an increase in the coefficient of friction (COF). This behavior is caused by the surface agglomeration of the material, and the atomic pile-up in the scratch direction. The pile-up and agglomeration produce an increase in the opposition to the nanoindenter movement, and then a stabilization of the COF occurs, which are due to the fact that part of the pulled material begins to be laterally detached, allowing the indenter to move forward with an almost constant friction force. ^{[ 70 ]}

Fig. 4.

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	Fig. 4. (a) Coefficient of friction (COF) depending on the sliding distance during the scratch test using a spherical nanoindenter in BCC thin films and (b) friction of coefficient depending on the initial indentation depth, including the evolution of the imprint.

Figure  4(b) shows an image of the scratch imprints after a failure of the coating occurs. In this image, the atomic pile-up caused by dislocations generated during the cross displacement of the nanoindenter is observed, and also an increase in the surface agglomeration as the nanoindenter depth increases is evidenced. ^{[ 2 ]} The failure mode at the critical load was very similar for all the indenter penetrations. The continuous increase of the cross-load during the scratch test produced the coatings detachment into the residual imprint and the region around it, as shown in Fig.  2(b) . In monolayer coatings, an adhesive failure was observed. This failure is a consequence of the traction stresses associated to elastic-plastic deformation induced by the applied load with the spherical indenter on the substrate-coating system. Finally, once the nanoindenter tip is removed, an elastic recovery occurs, which can be observed at the beginning of the scratch. When the indenter begins to move in the sliding direction, around 10% of the deformed material in the first stage of the indentation is elastically recovered.

Curves in Fig.  5 present the friction force depending on the sliding distance in wear tests for different dissociation energies. These dissociation energies take values of 0.15, 0.50, 1.00, 1.50, and 2.00 eV. By definition, as the dissociation energy increases, bonds are stronger. Then, according to these curves, an increase in COF as the dissociation energy increases was evidenced. In the first stage, when the indenter begins the scratch process, similar behavior for all the energies was observed. Nevertheless, as the indenter moves forward, greater energy is accumulated and atoms with lower dissociation energies are delocalized, presenting agglomeration phenomena at the material surface, generating also material detachment.

Fig. 5.

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	Fig. 5. Coefficient of friction depending on the sliding distance at different dissociation energies during the scratch test in BCC thin films.

In this study, a molecular dynamics method was used to determine the mechanical and tribological properties of a generic monolayer with BCC structure, orientation in the (100) direction, and placed on a substrate with diamond-type FCC structure. Nanoindentation and nanoscratch tests were carried out. The following conclusions are obtained.