1 Introduction

The topic of physical modeling in the framework of the graduate school Physical Modeling for Virtual Manufacturing Systems and Processes spans many spatial and temporal scales. On the finest scale, atomistic modeling techniques can be used, of which the method of molecular dynamics simulation is an important example. Atomistic modeling allows one to obtain insights into the basic processes underlying manufacturing. In this chapter, we present an example of the potential of atomistic simulation to identify basic processes underlying machining of materials.

In materials-science simulation studies, surface machining processes are often subdivided into the classes of indentation, scratching and cutting. Of these, indentation is the most basic process, as it only requires penetration of a tool in normal direction into the workpiece, but not any lateral movement. We therefore focus in this chapter on the indentation of a tool into the workpiece. In order to allow comparison with the majority of other studies, the tool will be assumed to be spherical; or, in other words, the part of the tool that actually penetrates the surface is considered to be curved with a fixed radius of curvature. Also, the tool will be considered to be rigid; this is a model of a hard diamond indenter and simplifies the discussion, since all aspects of tool wear are excluded.

Indentation processes have been simulated for many materials using molecular dynamics, including metals [1,2,3,4,5,6,7,8,9,10,11] and ceramics [12,13,14,15,16,17,18]. Such studies provided information on the mechanisms underlying material plasticity during indentation, see Refs. [19, 20] for a review. However, the plasticity in composite materials differs from that in homogeneous materials and still poses open questions, such as to the pile-up, transmission and absorption of dislocations at interfaces. We therefore investigate here graphene-reinforced metal-matrix materials, focusing on Ni-graphene nanocomposites. Such graphene-metal composites have excellent mechanical properties [21,22,23,24,25,26,27,28] as they combine the superior mechanical properties of graphene [29, 30] with the ductility of the matrix metal.

The plasticity in these nanocomposites is based on dislocation nucleation and migration in the matrix metal and on how these dislocation-based processes are influenced by the graphene flakes [31,32,33]. These issues are well suited to investigation by molecular dynamics simulation [25, 34,35,36,37,38,39,40,41]. Previous studies focused on the improvement of the composite hardness in the metal matrix [34, 36, 39, 42, 43]. In recent work [37, 38, 44], we studied the effects of the graphene interfaces in absorbing dislocations and hindering their propagation; the present chapter reviews these studies.

2 Method

The Ni blocks used for the simulations have typical sizes of 40 nm × 40 nm in lateral directions and a depth of 30 nm, containing around 5 × 106 atoms. They are filled with graphene flakes of various sizes and morphologies; the exact configurations vary with each simulation and are described in the following sections. The indenter is modeled as a rigid, hard and frictionless sphere of radius R = 5 nm which penetrates in normal direction into the sample with a velocity of 20 m/s.

Ni atoms interact with each other by a many-body interaction potential developed by Mishin et al. [45]. The interaction among C atoms is modeled by the so-called adaptive intermolecular reactive empirical bond order (AIREBO) potential of Stuart et al. [46]. The interaction between C and Ni is described by a pairwise Lennard-Jones potential,

$$ V\left( r \right) = {4}\varepsilon \left[ {\left( {\sigma /r} \right)^{{12}} - \left( {\sigma /r} \right)^{6} } \right], $$
(1)

with length parameter σ and energy parameter ε. These parameters are provided by Huang et al. [47] as ε = 23.049 meV and σ = 2.852 Å. Finally, the interaction of the indenter with the sample (Ni or C) atoms is modeled by a repulsive potential as proposed by Kelchner et al. [48],

$$ V\left( r \right) = k\left( {R - r} \right)^{3} , $$
(2)

for all atoms with distance r to the indenter center smaller than the indenter radius R. The indenter stiffness has been set to k = 10 eV Å−3, following Refs. [5, 48].

The simulations are performed with the open-source code LAMMPS [49] using a constant time step of 1 fs. The indentation is performed in the so-called displacement-controlled mode; i.e., the indenter position is advanced each time step according to its velocity; then the resulting forces on the sample atoms are calculated and the atom positions are updated. Further details on the molecular dynamics method used in nanoindentation are found in Refs. [19, 37, 38, 44].

Common-neighbor analysis [50] is used to identify the crystalline structure and the dislocation detection algorithm DXA [50,51,52] to determine the length of dislocations. The software OVITO [53] is used to visualize the simulation results.

3 Ni Single Crystal

In a first step of our molecular dynamics simulations, a Ni single crystal with a (111) surface is indented with an indenter of radius R = 5 nm. Figure 1a shows the corresponding force-depth curve. It is characterized by an elastic part extending up to around 0.7 nm indentation following the Hertzian F ~ d3/2 behavior; the ensuing load drop is caused by the formation of dislocations which relieves the stress exerted on the substrate. With continuing indentation, the force increases; the fluctuations are caused by the stochastic nature of dislocation generation and emission. The defects generated in the Ni material after an indentation to 4.1 nm are displayed in Fig. 1b. The stacking faults visible in the environment of the indentation pit are the traces left over after the movement of dislocations away from the high-stress environment of the indenter.

We contrast these findings with the indentation of a Ni (111) single crystal into which a graphene flake is embedded at a depth of 3 nm. The flake extends on all sides 17 nm from the indentation point, such that its extension can be considered ‘infinite’ in lateral direction in the sense that dislocations created by the indentation process cannot interact with the flake boundaries. The force-depth curve, Fig. 1a, shows that almost at all indentation depths, less force is required to indent the graphene-loaded Ni matrix than the pure Ni crystal; in other words, the Ni-graphene nanocomposite is weaker than the pure Ni crystal. Figure 1b shows that the graphene layer prevents dislocations to be created in the sub-graphene region and even prevents the migration of dislocations into that region. We conclude that the blocking of dislocation transmission reduces the force necessary for indentation.

Fig. 1.
figure 1

Comparison of the indentation into single-crystalline (SC) Ni with Ni containing a graphene flake (g) at a depth of 3 nm. (a) Dependence of the indentation force on depth. (b) Microstructure formed at an indentation depth of d = 4.1 nm, immediately before dislocation nucleation in the lower Ni block. Atoms are colored according to common-neighbor analysis. Purple: fcc; red: stacking faults; cyan: other defects; brown: graphene. Data taken from Ref. [38] under CC BY 4.0.

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The role of the edges of the graphene flakes is investigated in a further series of simulations. As Fig. 2 shows, here 9 different systems are studied. All of them are based on a Ni single-crystalline matrix with a (100) surface. The depth below the surface at which the flake is situated is varied from 3 nm (‘top’) to 5 nm (‘middle’) and 10 nm (‘bottom’). Also the lateral position of the flake is varied; if it ends below the center of the indenter, it is denoted as system ‘1’; if it extends beyond the center by 5.3 nm or 10.7 nm, as system ‘2’ or ‘3’.

Fig. 2.
figure 2

Side view (left) and top view (right) of the setup of the simulation system. x and y are the cartesian directions parallel to the surface, and z normal to the surface. 9 different positions of the graphene flake (brown) are simulated, which differ in their depth below the surface (‘top’, ‘middle’, and ‘bottom’) as well as in their lateral extension in x direction, designated by ‘1’, ‘2’, and ‘3’. Taken with permission from Ref. [37].

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Fig. 3.
figure 3

Side views onto the y–z plane of the indented pure Ni sample, as well as the bot2, mid2, and top2 Ni-graphene systems for indentation depths of d = 1.5, 3, 4, and 5 nm; the view direction is along the − x axis, see Fig. 2. The volume is cut immediately under the indenter such that only half of the plastic zone is visible for clarity. The light blue color highlights the graphene flake as well as the indentation pit. Dislocations are colored according to their Burgers vector. Green: 1/2 <112>; dark blue: 1/2 <110>; pink: 1/6 <110>; yellow: 1/3 <001>; red: other. Taken with permission from Ref. [37].

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Fig. 4.
figure 4

Evolution of the (a) total dislocation length, (b) indentation force, and (c) contact pressure with indentation depth. The first load drops are indicated in (b) and (c) by arrows. Taken with permission from Ref. [37].

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Figure 3 exemplifies the evolution of the dislocation network for the systems ‘2’, in which the flake extends roughly by the size of the indenter radius beyond the indenter and compares it with that for a pure Ni matrix. The reference case, pure Ni, follows that of previous studies of indentation into face-centered cubic (fcc) metals [5, 54,55,56]. The generated dislocation network is characterized by the ejection of loops in the <110> glide directions of the fcc matrix; in addition, a dense network adherent to the indent pit forms.

The insertion of a graphene flake stops the emission of dislocation loops and also constrains the evolution of the network adherent to the indent pit. A novel situation arises for the ‘top’ conformation of the flake, where it is situated only 3 nm below the surface. After the indenter is pushed deeper into the material – d = 4 and 5 nm in Fig. 3 – dislocations are generated also in the lower part of the Ni matrix. Note that the flake does not rupture; the indenter does never touch the Ni material below the flake, but the stresses it generates are sufficient to lead to dislocation formation and even loop emission into the lower Ni matrix. These simulations thus show that the main role of the graphene flake consists in constraining the dislocation networks building up in the Ni matrix. The dislocation network, i.e., the plastic zone generated by the indenter, adapts its geometry to the non-transmitting flake.

Figure 4a demonstrates that not only the geometry, but also the total length of dislocations in the plastic zone is affected by the presence of the flake. In all cases, the total length of the dislocations is reduced with respect to that in the pure Ni matrix; the amount of reduction is determined by the distance of the flake from the surface. For the ‘bottom’ geometry, the influence of the flake is only noticeable for indentation depths beyond 3 nm, while for the’top’ geometry, already at d = 1 nm, a reduction in dislocation length is visible. These data quantify the qualitative information provided in Fig. 3 of the influence of the flake position on the generated dislocation network.

Figures 4b and c show how the normal force and the build-up of contact pressure during indentation are affected by the presence of the graphene flake. The contact pressure is determined as the ratio of the normal force to the contact area of the tip. The normal forces show a clear ordering with the position of the flake, at least for indentation depths up to 3 nm: the farther the flake is buried inside the matrix, the higher the load that the system can carry. These data thus confirm and extend the conclusions made with the discussion of Fig. 1b. Since we use the same indenter in all these studies, this result also carries through to the contact pressure, Fig. 4c, and as the average contact pressure can be identified with the hardness of the sample, we find that the hardness of the nanocomposite material is decreased if the flake is positioned closer to the surface.

4 Ni Bi-crystal

In real composites, graphene flakes will usually not be incorporated within a single-crystalline matrix, but favor sites at grain boundaries. We investigated such a scenario by studying Ni bi-crystals and inserting a graphene flake in the grain boundary. To be specific, we stayed with a Ni(111) crystal; a twist grain boundary is inserted at a depth of 3 nm, such that the Ni crystal in the bottom grain also has a (111) surface, which is, however, twisted with respect to the top grain by an angle θ. We investigated the cases of θ = 30◦ and 60◦ and also compared to the reference case of θ = 0◦ (no grain boundary). The θ = 60◦ tilt boundary was selected as it is the lowest-energy grain boundary in fcc Ni [57]; it constitutes actually a coherent Σ3 twin boundary and has a specific energy of only 0.06 Jm−2. On the other hand, the θ = 30◦ twist boundary has the highest energy among all (111) twist boundaries; its specific energy amounts to 0.49 Jm−2 [57]. In the following, we will also denote the θ = 30◦ grain boundary as the high-energy grain boundary, since its defect energy is high, and the θ = 60◦ grain boundary as the low-energy grain boundary since its defect energy is low and the material hence more closely resembles an ideal Ni crystal.

Fig. 5.
figure 5

Comparison of the force-depth curves for indentation into bi-crystals containing a 30â—¦ or a 60â—¦ twist grain boundary (hm) with that of a single crystal (SC). Data taken from Ref. [38] under CC BY 4.0.

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Let us first focus on the effect of a graphene-free grain boundary on indentation into pure Ni; these pure Ni bi-crystals will be denoted as hm (homointerface) systems. Figure 5. Shows the force-depth curve for the three systems and Fig. 6 visualizes the dislocations generated. Clearly, the high-energy θ = 30◦ grain boundary prevents dislocations from crossing, while dislocation transmission is possible for the low-energy θ = 60◦ grain boundary. These findings are in agreement with earlier simulation work on fcc metals [58,59,60,61]. Interestingly, the system with the low-energy grain boundary, θ = 60◦, requires the highest load for indentation. This observed ‘hardening’ may be attributed to the fact that dislocation transmission through the twin boundary requires external strain [58, 59, 61, 62].

Fig. 6.
figure 6

Dislocation network building up after indentation to a depth of 2.1 nm into a Ni bi-crystal containing (a) a 60◦ and (b) a 30◦ twist grain boundary at 3 nm depth. Atoms are colored according to common-neighbor analysis. Red: stacking faults; yellow (cyan): defective atoms in the upper (lower) Ni grain. Fcc atoms have been removed for clarity. Green lines show Shockley partials. Data taken from Ref. [38] under CC BY 4.0.

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In contrast, the high-energy θ = 30◦ grain boundary requires less force for indentation, as observed in Fig. 5, at least until the indenter touches the grain boundary at d = 3 nm. Here, dislocation absorption at the grain boundary decreases the hardness of the Ni matrix. This behavior is in agreement with previous simulations [63, 64].

The effect of filling the low-energy θ = 60◦ grain boundary with a graphene flake is shown in Fig. 7. Figure 7a demonstrates that the effect of graphene is to considerably decrease the force necessary for indentation; the material is thus weakened. As the graphene flake blocks the transmission of dislocations through the grain boundary, Fig. 7b, this effect is even stronger than what was observed for the Ni single-crystal, Fig. 1. The blocking of dislocation transmission by the graphene sheet is responsible for the reduced force necessary for indentation of the graphene-Ni composite.

Fig. 7.
figure 7

Comparison of the indentation into a Ni bi-crystal with a 60◦ twist boundary without (hm) and with (g) graphene. (a) Evolution of force with indentation depth. (b) Snapshots showing the microstructure at an indentation depth of d = 4.2 nm. Atoms are colored according to common-neighbor analysis. Purple: fcc (lower Ni block); gold: fcc (upper Ni block); red: stacking faults; cyan: other defects; brown: graphene. Data taken from Ref. [38] under CC BY 4.0.

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For the high-energy θ = 30◦ grain boundary, the effect of graphene coating the grain boundaries is negligible, see Fig. 8a. The reason hereto is that dislocations cannot cross this grain boundary even if graphene is absent, cf. Figure 8b; hence the effect of graphene on dislocation transmission is absent and consequently also the effect on the force-depth curve and the material hardness.

Fig. 8.
figure 8

Comparison of the indentation into a Ni bi-crystal with a 30◦ twist boundary without (hm) and with (g) graphene. (a) Evolution of force with indentation depth. (b) Snapshots showing the microstructure immediately before dislocation nucleation in the lower Ni block: at d = 3.4 nm (hm) and at d = 4.0 nm (g). Atoms are colored as in Fig. 7. Data taken from Ref. [38] under CC BY 4.0.

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5 Ni Polycrystal

Finally, molecular dynamics simulation was used to study the effects of coating grain boundaries with graphene in a Ni polycrystal. The polycrystal was created using the method of rapid quenching from the melt [65, 66]; by choosing the quench rate of 1 K/ps, we obtain a Ni polycrystal with an average grain size of 4.1 nm. Figure 9 shows the final structure of the polycrystalline Ni matrix after quenching.

In order to create a Ni-graphene polycrystalline nanocomposite, 64 square graphene flakes with a side length 2.6 nm were added to the melt before quenching. We observed that the morphology of these flakes in the melt changed considerably during the 14 ns equilibration time at 2300 K. During this time, the flakes are rather mobile and change their curvature and their positions in the Ni melt. They tend to roll up and assume a wrinkled or folded structure as shown in Fig. 10 right; the C-C attraction of flake atoms stabilizes the tubular structures similar as in nanotubes. We denote this structure as the wrinkled morphology; it survives the quench process and is also seen in the nanocomposite.

Fig. 9.
figure 9

Polycrystalline structure of Ni after quenching from the melt. Colors represent different grains. Taken with permission from Ref. [44].

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We created another type of nanocomposite denoted as flat morphology, see Fig. 10 left. This was possible by increasing the C-Ni attraction of the flake edge atoms by a factor of around 10; more precisely, we chose ε = 200 meV and σ = 1.514 Å in Eq. (1). This is justified by the fact that the twofold coordinated C atoms at the flake edges are free to form covalent bonds with surrounding Ni atoms; the Lennard-Jones parameters used here were provided by Tavazza et al. [67] in a density-functional-theory (DFT) study of the Ni-C interaction. As Fig. 10 left shows, this modification lets the flakes keep a more planar structure in the melt and the creation of tubular structures is avoided.

Fig. 10.
figure 10

Crystallographic structure of Ni-graphene composites with (left) flat morphology and (right) wrinkled morphology. Atoms are colored according to common-neighbor analysis. Green: fcc nickel; dark blue: stacking faults in nickel; yellow: grain boundaries; light blue: graphene; red: edge atoms of graphene flakes. Taken with permission from Ref. [44].

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Fig. 11.
figure 11

Dependence of composite hardness on the number of carbon atoms in the plastic zone. Lines are to guide the eye. The gray line gives the average hardness of polycrystalline Ni. Taken with permission from Ref. [44].

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We determine the hardness in these nanocomposites using nanoindentation with an R = 4 nm indenter. In order to take the spatial inhomogeneity of the systems into account, 5 replicas of the nanocomposite systems were created containing free surfaces; these free surfaces were indented at a total of 50 points. These 50 indentations thus varied in the local stoichiometry of the C content, in the local grain size and orientation and in the vicinity and structure of the grain boundaries. For reference, we note that our polycrystalline Ni exhibits a hardness of 9.95 ± 0.13 GPa.

Fig. 12.
figure 12

Dislocation interaction with a graphene flake during indentation of the wrinkled composite. Light blue atoms represent folded graphene flakes. Atoms of Ni are removed for clarity. The gray shades represent grain boundaries. Dislocations are colored according to their Burgers vector. Green: 1/2 <112>; dark blue: 1/2 <110>; red: other. The black loop highlights a set of dislocations that are eventually absorbed at the graphene flake. Taken with permission from Ref. [44].

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Figure 11 shows the result of these hardness measurements. The main observation is that nanocomposite systems with flat graphene are systematically stronger than those with wrinkled graphene flakes; the difference is sizable and amounts to 10%. In other words, wrinkled graphene offers less resistance against deformation than flat graphene. This is plausible, since out-of-plane deformations, i.e., curving, folding and crumpling, can be effected with small forces while in-plane deformation of graphene is considerably harder. Note that the hardness of all flat nanocomposites (average of 10.86 GPa) lies above that of pure polycrystalline Ni with a comparable grain size, while the hardness of wrinkled graphene (average of 10.04 GPa) exhibits considerably more spread and shows data points both above and below the value of polycrystalline pure Ni.

Figure 11 provides more information as it correlates the measured hardness with the amount of C found in the plastic zone generated by the indentation. The latter zone is identified as a hemisphere containing all dislocations nucleated inside the grains and shear activation in grain boundaries induced by the indentation. Its radius amounts to roughly 10–15 nm for the indenter radius used in the simulation and the composites studied here. For both morphologies, we find that the hardness decreases with the local graphene content. This effect is caused by the absorption of dislocations at the flakes discussed above.

Such an absorption event is shown in detail in Fig. 12. A set of dislocations moves under the inhomogeneous stress field of the indenter towards a graphene flake and is eventually absorbed there. Due to the small grain sizes, the change of the dislocation network – nucleation, migration and eventual absorption – is quite fast such that the dislocations highlighted in Fig. 12 vanish within a time scale faster than 0.5 ps.

6 Summary

In this chapter, we gave an example of how the method of molecular dynamics simulation allows to obtain detailed insights into the plastic processes underlying nanoindentation. Even for a complex material – in this example a Ni-graphene nanocomposite – the generation of dislocations in the metal matrix and their propagation could be studied in detail and the effects on the material hardness could be quantified.

For this example, a series of increasingly more complex simulation scenarios was established, starting from a single-crystalline matrix over bi-crystal samples to fully polycrystalline arrangements. This series of simulations allowed us to draw the following conclusions.

  1. 1.

    Single-crystalline Ni is harder than a Ni-graphene nanocomposite. The nanocomposite hardness decreases as the graphene flake is situated closer to the surface.

  2. 2.

    The graphene flake is opaque to dislocation transmission and thus constrains the size of the dislocation network produced by the indenter. This is the cause for the decreased hardness of the nanocomposite.

  3. 3.

    Grain boundaries in the metal matrix generally hinder dislocation transmission and therefore reduce the material hardness. An exception is given by low-energy grain boundaries – such as twin boundaries – which are partially transparent to dislocations.

  4. 4.

    If graphene coats a single grain boundary – such as in a bi-crystal matrix – the dislocation transmission is further reduced resulting in a decrease in hardness.

  5. 5.

    In a polycrystalline metal matrix, coating of the grain boundaries leads to an increase of the material hardness. This is caused by dislocation pile-up in front of the flakes [31, 32, 68]. For high C concentrations in the plastic zone, also dislocation absorption (annihilation) by the graphene flakes is observed.

  6. 6.

    The morphology of the graphene flakes has a strong effect on the nanocomposite hardness. While flat flakes generally increase the hardness, wrinkled flakes have less effect or even reduce the composite hardness.

Further studies are required in particular in order to relate the nanocomposite hardness to the graphene concentration. Experiments find that there is an optimum graphene concentration for improving the composite hardness [69, 70], while in simulations the hardness generally decreases with graphene content, cf. Figure 11. The verification and explanation of the optimum graphene concentration thus poses an interesting problem for atomistic simulation.