Our understanding of mechanics is pushed to its limit when the functionality of devices is controlled at the nanometer scale. A fundamental understanding of nanomechanics is needed to design materials with optimum properties. Atomistic simulations can bring an important insight into nanostructure-property relations and, when combined with experiments, they become a powerful tool to move nanomechanics from basic science to the application area. Nanoindentation is a well-established technique for studying mechanical response. We review recent advances in modeling (atomistic and beyond) of nanoindentation and discuss how they have contributed to our current state of knowledge.

Article Outline

Crystalline materials

Amorphous and quasicrystalline materials

Nanostructured materials

Challenges in modeling nanoindentation

Outlook

Acknowledgements

References

Nanotechnology is leading to the development of materials and devices whose structure and function are controlled at the atomic level. In many applications, components experience unintentional or deliberate mechanical contact, and a thorough understanding of a material's mechanical behavior is critical for the design of reliable and durable systems.

Nanoindentation is a widely used technique for probing mechanical properties and stability, especially of surfaces and thin films^{[1], [2], [3], [4], [5], [6], [7] and [8]}. With the development of sensitive atomic force microscopy (AFM)³ and other techniques^{[2] and [9]}, we can now measure the force P on the indenter and indenter displacement h with nanometer scale precision^{[10] and [11]}. (For more details on nanoindentation, see the article by Schuh on page 32 of this issue and reviews elsewhere¹².) From the shape of such P-h curves, one can extract information about elastic moduli or hardness.

In molecular dynamics (MD) simulations, one solves Newton's equation of motion for all the atoms in order to retrace their trajectories while an indenter is being pressed into the material. The load on the indenter P is calculated by summing the forces acting on the atoms of the indenter in the indentation direction. Indentation depth h is calculated as the displacement of the tip of the indenter relative to the initial surface of the indented solid. Fig. 1 shows an example of a P-h response of crystalline silicon carbide (3C-SiC) obtained in a classical MD simulation¹³.

Because of the very complex stress profile generated in the vicinity of the indenter, it is challenging to interpret nanoindentation experiments at a fundamental level. Atomistic computer simulations have been very helpful in unraveling the processes underlying the nanoindentation response^{[14], [15], [16], [17], [18] and [19]}. The role of simulations is not necessarily to reproduce the exact experimental behavior, but rather to identify possible atomistic mechanisms in the early stages of plastic deformation and determine their trends as a function of the nanostructure and the environment (e.g. temperature or surface passivation)^{[20], [21] and [22]}. The goal is to develop robust insights into technologically relevant materials and ultimately design a material with optimum mechanical properties.

Crystalline materials

There are some very good review articles on the subject of nanomechanics simulations in bulk crystalline materials^{[23], [24] and [25]}, where readers can find detailed descriptions of such phenomena as jump-to-contact (JC), pile-up under the indenter, nonmonotonic features in P-h curves, and phase transformation of the indented material. Here, we will provide a brief summary of these phenomena and discuss recent results from nanoindentation simulations of more complex materials, e.g. noncrystalline and nanocrystalline solids.

The JC phenomenon has been observed by a number of groups^{[26], [27], [28], [29] and [30]}, and it describes the ‘bulging up’ of surface atoms to meet the indenter tip (made of a material with higher modulus than that of the indented solid) before the tip makes the actual contact with the surface, i.e. at h < 0. Adhesive forces underlying the JC phenomenon are also responsible for the formation of a connective neck of atoms between the tip and the sample during the retraction of the indenter (Fig. 2)^{[31] and [32]}. The JC phenomenon (i.e. adhesion at h < 0) manifests itself as a ‘dip’ in the P-h curve.

As a matter of fact, JC is only one of many atomistic events that can leave a signature on a computer-generated P-h curve. Another example is shown in Fig. 1a, where the nonmonotonic features of the P-h curve are correlated with discrete dislocation bursts in the indented material³³ (Fig. 3). Similar ‘pop-in’ events have been observed in experimentally determined P-h curves^{[34], [35], [36] and [37]}.

Atomistic simulations have also shed light on the solid-state phase transformations that take place in material in the vicinity of the indenter³⁸. For instance, Kallman et al.³⁹ observed a localized crystalline-to-amorphous transition in Si at temperatures close to the melting point, which is consistent with experiments^{[35] and [40]}. These simulations reveal a dependence of the yield strength of Si on structure, rate of deformation, and temperature. Solid-state amorphization has also been observed in nanoindentation simulations of 3C-SiC¹³. Defect-stimulated growth and coalescence of dislocation loops are found to be the atomistic mechanisms underlying the crystalline-to-amorphous transition. In simulations by Walsh et al.⁴¹, amorphization has been identified as a primary deformation mechanism in indentation of Si₃N₄. During the indentation, amorphization was arrested by cracking at the indenter corners and piling up of material along the indenter sides (Fig. 4). The pile-up material itself has an amorphous structure.

Structural changes, such as amorphization, can be identified in simulations by means of the radial distribution function, which can be directly compared with X-ray diffraction experiments. Other commonly used methods to monitor the evolution of simulated indentation damage are bond angle distribution (Fig. 4); local variation in potential energy, pressure, and shear stress; visual inspection of the computer-generated atomic structure; and changes in local topologies (topological changes can be analyzed, for example, by means of shortest-path ring statistics^{[33], [39] and [42]}).

Amorphous and quasicrystalline materials

Amorphous solids constitute a separate class of materials whose mechanical properties are of great fundamental and technological interest. Because amorphous materials lack a topologically ordered network, analysis of deformations and defects presents a significant challenge. Various models have been proposed to describe defects in such structures^{[43] and [44]}. For example, Gilman⁴⁵ has conjectured that an analog to a crystal dislocation exists in noncrystalline solids, and has described defects as dislocation lines with variable Burgers vectors. Because of the presence of these inhomogenities frozen into the entire material, the nanoindentation damage cannot be readily identified by computational techniques used for crystalline solids. The prevailing theory of plasticity in metallic glasses involves localized flow events in shear transformation zones (STZ). An STZ is a small cluster of atoms that can rearrange under applied stress to produce a unit of plastic deformation^{[46], [47], [48] and [49]}.

Even though these theories provide an essential physical insight, a truly atomistic model of plastic flow in amorphous materials is still lacking. A few atomistic simulations have been performed to tackle this problem. For example, Sinnott et al.⁵⁰ undertook the difficult task of simulating nanoindentation of amorphous carbon (a-C:H) material with an sp³ bonded indenter. The simulations reveal that indentation has little effect on the hybridization of the carbon atoms or the randomly distributed stresses within the material. They also show that while penetrating the amorphous solid, the tip deforms only slightly via shear. This is in contrast to indentation simulations of crystalline diamond, where the tip deformation includes significant shear and twist components.

A series of simulations of nanoindentation of amorphous silicon carbide (a-SiC)⁵¹ point to a noticeable localization of damage in the vicinity of the indenter, however the localization is less pronounced than in the case of 3C-SiC. As shown in Fig. 5, the P-h curve for a-SiC exhibits irregular, discrete load drops similar to 3C-SiC (Fig. 1). Here, the load drops correspond to breaking of the local arrangements of atoms, in analogy to the slipping of atomic layers in 3C-SiC shown in Fig. 3. Simulations also show that, even at indentation depths h smaller than those at which the material yields plastically, the material's response is not entirely elastic. Instead, the amorphous structure, which is metastable by nature, supports a small inelastic flow related to relaxation of atoms through short migration distances.

In metals, there is experimental evidence that the most stable bulk metallic glasses may exhibit a local quasicrystal order⁵². Additional insight has been provided in recent MD simulations by Shi and Falk⁵³. The authors show that in a two-dimensional metallic alloy with quasicrystalline medium-range order, the deformation localization can arise as the result of the breakdown of stable quasicrystal-like atomic configurations. Indentation simulations are of great interest for elucidating the science underlying plasticity in amorphous structures. The advent of these simulations is particularly timely because of the growing potential for structural applications of amorphous materials^{[54] and [55]}.

Nanostructured materials

It has been demonstrated in experiments^{[56], [57] and [58]} and simulations^{[59], [60] and [61]} that nanocrystalline materials exhibit unusual mechanical behavior when compared with their polycrystalline counterparts. For example, normally brittle ceramics are shown to have very high hardness, high fracture toughness, and superplastic behavior, as the grain size is refined to the nanometer regime^{[62] and [63]}. (For more detailed descriptions of these and other properties of nanocrystalline materials, see the article by Van Swygenhoven on page 24 of this issue.)

Mechanical properties of nanocrystalline materials are controlled on the atomic level, and therefore atomistic simulations can bring an invaluable physical insight to experiments and ultimately enable the design of materials with optimum properties.

Since nanocrystalline materials have an increased volume fraction of grain boundaries (GBs)^{[64] and [65]}, it is essential to understand the interactions of GBs with dislocations emitted from under the indenter. This question has been addressed by Feichtinger et al.⁶⁶, who performed MD nanoindentation simulations of nanocrystalline Au (nc-Au) with grains 12 nm in diameter. In the simulation, dislocation nucleation within the grain occurs at the onset of plastic deformation at an indenter depth h similar to that in a perfect crystal. The GBs act as an efficient sink for partial and full dislocations and intergranular sliding is observed. A decrease in Young's modulus is also seen as the grain size is refined to 5 nm in diameter. Recently, the same group reported another simulation⁶⁷, where the indenter size is smaller than the grain size, which shows that the GBs can not only act as a sink for dislocations, but can also reflect or emit dislocations (Fig. 6).

In contrast to metals, GBs in nanocrystalline ceramics form a thicker GB phase that is highly disordered and has a fairly uniform thickness⁶². These soft GBs essentially determine the material's mechanical response. Recent MD simulations of nanoindentation of nc-SiC show how the coexistence of brittle grains and soft amorphous GBs results in unusual deformation mechanisms⁶⁸. The simulated material was sintered from crystalline grains of 8 nm in diameter and consists of about 19 million atoms. As the indenter depth h increases, a crossover is observed from a cooperative deformation mechanism involving multiple grains to a decoupled response of individual grains, e.g. grain rotation and sliding, and intragranular dislocation activity (Fig. 7). The crossover is also reflected in a switch from deformation dominated by crystallization to deformation dominated by disordering, as explained in the caption of Fig. 8. In the early stages of plastic deformation, the soft (amorphous) GB phase ‘screens’ the crystalline grains from deformation, thus making nc-SiC more ductile than its coarse-grained counterpart. Fracture toughness (a measure of how much energy it takes to propagate a crack), measured experimentally in nanocomposites with an nc-SiC matrix and diamond inclusions⁵⁸, exceeds that of a polycrystalline matrix by about 50%. Increased fracture toughness does not necessarily lead to a lower value of hardness. Recent experiments of nanoindentation of nc-SiC with grain sizes of 5-20 nm show quite the opposite trend. Liao et al.⁶³ report nc-SiC to be superhard, i.e. to have a hardness of 30-50 GPa, which is larger than that of crystalline SiC. The hardness value of 39 GPa obtained in the MD simulation described above is consistent with these experimental measurements.

Challenges in modeling nanoindentation

Despite the great advantages of simulating nanoindentation, the technique faces some serious challenges. The first is the limited time scales accessible to simulations because of limited computational resources. For example, the slowest time scales available to MD give 1 m/s indentation velocities⁶⁹, while state-of-the-art AFM systems can only operate at up to 0.001 m/s. Indentation measurements are limited to slower rates of the order of 25 μm/s. As a result, there is a significant disparity between simulated strain rates and those attainable by experiments. The rationale for modeling nanoindentation with MD is that for the simulated solids, all the above speeds are far below the speed of sound in materials (for example, the speed of sound in SiC is 11 000 m/s). For this reason, MD simulations are able to dissipate any reflected waves that arise from the motion of the indenter (this can be done, for example, by coupling the equations of motion to Nose-Hoover thermostats^{[70] and [71]}). There is good reason to believe, therefore, that despite the time-scale problem, simulations can provide understanding of atomistic mechanisms and qualitative trends in nanoindentation response. However, this hypothesis needs to be scrupulously tested.

Limitations in computational resources affect not only the time scale, but also the system dimensions available to simulations. Small system dimensions can introduce unrealistic boundary conditions that, in turn, will artificially alter processes such as dislocation dynamics. For example, Li et al.⁷² studied nucleation and propagation of dislocations in indented solid Al by means of MD simulations. Because the bottom surface of the sample remained unconstrained, a nucleated dislocation loop was able to move all the way to the bottom and leave the sample. On the other hand, in most MD simulations of nanoindentation reported in literature, the bottom layer of the sample is ‘frozen’ and dislocation motion through the material is inhibited. The simplest solution to this problem is to have a good understanding of the implications of given boundary conditions and be aware of the limitations in the conclusions drawn. The case for MD is not lost because even small systems are well suited to studying the effects of boundary conditions on plasticity, and deformation mechanisms in small nanostructures are becoming of increasing interest to experiments and applications. Also, because of the fast development of computer technology as well as new algorithmic optimization methods, MD simulations are now possible for systems consisting of billions of atoms, i.e. for system dimensions in the submicron regime. At this length scale, the artifacts of the boundary conditions can be avoided by smart simulation techniques, such as efficient dissipation of any energy reflected from the system boundaries by strategically distributed thermostats.

In order to extend simulations beyond the micrometer regime, models are being developed that combine direct atomistic simulations with continuum methods. For example, a quasicontinuum model has been developed by Tadmor et al.^{[73] and [74]} and applied to study nanoindentation^{[75] and [76]}. In this approach, a continuum finite element (FE) is employed to characterize the mechanical response of the material, i.e. the positions of the majority of atoms are constrained and determined by the displacement of the nearby node. In contrast to the standard FE method, in the quasicontinuum approach the constitutive response of the system is determined from atomistic calculations based on interatomic potentials. Combined FE and MD simulations of nanoindentation have also been performed by other groups. Li et al.^{[72], [77] and [78]} performed direct FE simulations in which large strain constitutive relations are derived from an interatomic potential (Fig. 9). Unlike the quasicontinuum method, this approach remains fully continuum. A review of simulation work based on FE is beyond the scope of this article but can be found elsewhere^{[79] and [80]}.

Another on-going challenge for MD simulations is the availability of reliable interatomic potentials. Parameters of a (classical) potential function are usually fitted to reproduce empirical data as well as accurate quantum mechanical calculations. Current, state-of-the-art empirical potentials can account for bond formation and breaking, change in hybridization, charge transfer, etc.^{[81] and [82]}. In spite of these developments, there is not one single analytic potential that is capable of describing all possible properties that might be of interest in a particular material. Furthermore, fitting an accurate potential is a difficult and time-consuming process.

An approach that bypasses the need for an interatomic potential is based on combined first-principle and FE calculations. For example, Hayes et al.⁸³ have recently simulated the nanoindentation of Al by means of the orbital-free density functional theory (OFDFT) local quasicontinuum (LC) method. In this OFDFT-LC model, the quasicontinuum approach is adopted but the atomic-scale calculations, based previously on empirical potentials, are now replaced with fast and inexpensive first-principles theory. This method is well suited to study phenomena such as initial dislocation formation; however, it is not capable of treating intermediate length scales (e.g. GBs in nanocrystalline materials). It is clear that with improving computer technology and the development of new algorithms, first-principles-based calculations will play an increasingly important role in nanoindentation modeling.

Fully atomistic simulation of large systems involving many millions of atoms creates another nontrivial challenge, i.e. to seek patterns and extract information from such massive, multivariable datasets. For example, a single nanocrystalline ceramic can contain thousands of randomly oriented crystalline grains surrounded by intergranular regions with various levels of topological disorder. The change in a grain's crystallographic structure and chemical ordering during indentation is a complex phenomenon that depends on many variables. Tracking deformations in a stand-alone amorphous material presents a challenge in itself, let alone as a part of a complex nanocrystalline material. Seeking patterns in such structures requires an extensive, hands-on analysis.

In order to analyze the complicated profiles of indentation damage in structurally advanced materials, there is an urgent need to develop more efficient data-mining techniques. Such developments can be fostered by interdisciplinary collaborations between materials and computer scientists.

Outlook

For the design of materials with superior mechanical properties, a mutual feedback process between experiments and simulations is critical. Because of the transferability of simulation tools and the wide variety of application areas, it is also essential to create a platform for collaboration among scientists from multiple disciplines. New structural applications are being extensively explored for amorphous and nanostructured materials, e.g. superhard coatings, sporting goods, high-speed machining and tooling, and biomaterial implants. If the mechanical properties of these complex materials are to be exploited in industrial applications, a thorough understanding of their mechanical response (e.g. through indentation) is of vital importance.

Acknowledgments

The author gratefully acknowledges support from the US National Science Foundation grant DMR-0512228. I am also thankful to D. Stone and D. Morgan for helpful comments on the manuscript.