Atomistic Simulations of Nano-Ind

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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 materials 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 films1-8. With

the development of sensitive atomic force microscopy (AFM)3 and

other techniques2,9, we can now measure the force Pon the indenter

and indenter displacement hwith nanometer scale precision10,11. (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 Newtons

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 Pis calculated by summing the forces acting on

the atoms of the indenter in the indentation direction. Indentation

depth his 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-hresponse 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 response14-19. The role of simulations is not

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 designmaterials 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.

Izabela Szlufarska

Department of Materials Science and Engineering, University of Wisconsin, Madison, 1509 University Ave, Madison, WI 53706-1595, USA

E-mail: [email protected]

ISSN:1369 7021 Elsevier Ltd 2006

Atomistic simulations

of nanoindentation

M AY 20 06 | VOLUME 9 | N UMBER 54 2
mailto:[email protected]:[email protected]


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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)20-22. 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 ofnanomechanics simulations in bulk crystalline materials23-25, 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 groups26-30,

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)31,32. The JC phenomenon (i.e. adhesion at h < 0) manifests

itself as a dip in the P-hcurve.

As a matter of fact, JC is only one of many atomistic events that

can leave a signature on a computer-generated P-hcurve. Another

example is shown inFig. 1a, where the nonmonotonic features of the

P-hcurve are correlated with discrete dislocation bursts in the indented

material33 (Fig. 3). Similar pop-in events have been observed in

experimentally determined P-hcurves34-37.

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 etal.39 observed a localized

crystalline-to-amorphous transition in Si at temperatures close to the

melting point, which is consistent with experiments35,40. 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 simulationsof 3C-SiC13. Defect-stimulated growth and coalescence of dislocation

Fig. 1 (a) Force P versus depth h of the indenter obtained in an MD simulation of crystalline SiC. The load drops in the P-h response correspond to discrete plasticevents in the indented material. (b) Plot of P against h during the unloading phase where the indenter is pulled out from the sample in 0.5 increments. Up toh = 1.83 , the deformation is entirely elastic, i.e. unloading from that depth produces a curve (squares) that retraces the loading curve (diamonds). The onset of

plastic deformation happens at h = 2.33 , reflected in the hysteresis of the second unloading curve (circles). (Reprinted with permission from13

. 2004 AmericanInstitute of Physics.)

Atomistic simulations of nanoindentation REVIEW FEATUR

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Fig. 2 MD simulation of indentation of solid Au with a Ni indenter. Atomicpositions during loading and unloading simulations are shown from top left tobottom right. During unloading a connective neck is formed by Au (yellow)atoms. (Reprinted with permission from32. 1995 Nature Publishing Group.)

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REVIEW FEATURE Atomistic simulations of nanoindentation


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 deformationmechanism 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 statistics33,39,42).

Amorphous and quasicrystallinematerialsAmorphous 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 structures43,44. For example, Gilman45 has conjectured that an

analog to a crystal dislocation exists in noncrystalline solids, and hasdescribed 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

deformation46-49.

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.

Fig. 3 Atomic configurations during indentation of crystalline SiC obtained by MD simulations. The cut of the sample shows the atoms (a) before and (b) after thefirst load drop (compare with the P-h curve inFig. 1). The region marked by yellow rectangles reveals that the load drop is correlated with slipping of atomic layers.(c) and (d) Simultaneously, dislocations are nucleated from under the indenter. Atoms whose local topological network deviates from the perfect crystallographicorder in SiC are shown. (Reprinted with permission from 33. 2005 American Physical Society.)

(a) (b)

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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 inFig. 5, the P-hcurve 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 hsmaller

than those at which the material yields plastically, the materials

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 Falk 53.

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 materials54,55.

Nanostructured materialsIt has been demonstrated in experiments56-58 and simulations59-61 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 regime62,63. (For more detailed descriptions

of these and other properties of nanocrystalline materials, see the

article by Van Swygenhoven on page 24 of this issue.)



Fig. 4 MD simulation of nanoindentation of Si3N4. Slices of the material normal to the indenter reveal cracking in the tensile regions at the indenter corners (left).Atomic configuration near the crack in the vicinity of the indenter (right). The structure of the deformed material was determined by calculating bond angledistribution. (a) Bond angle distribution for bulk crystalline (yellow) and amorphous (green) Si3N4. (b) Local bond angle distribution for region I (yellow) andII (green). Comparison of (a) and (b) indicates that region I is largely crystalline and region II resembles amorphous structure. (Reprinted with permission from41. 2000 American Institute of Physics.)

(a)

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Fig. 6 MD simulation of indentation of nc-Au showing interactions between dislocations and GBs. (a)-(c) Atomic configurations during loading, and (d)-(f)corresponding stress distribution. During the simulation, dislocations are emitted from under the indenter and propagate through the grains until they becomeabsorbed by the GBs. A dislocation is represented by two red lines (two parallel planes20) that mark a stacking fault left behind a propagating partial dislocation.The yellow arrow in (d) marks the region at the GB where a leading partial dislocation arrives. (Reprinted with permission from 67. 2004 Elsevier.)

Fig. 5 P-h response of a-SiC indented in an MD simulation. The curve exhibits a series of load drops, similar in nature to those in Fig. 1for crystalline SiC. Here, theload drops are irregularly spaced as a result of the lack of long-range order in amorphous networks, and correspond to breaking of the local atomic arrangements.The maximum indentation load reached in the a-SiC is lower than in the analogous simulation of crystalline SiC. (Reprinted with permission from 51. 2005American Institute of Physics.)

(a) (d)

(b) (e)

(c) (f)


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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)64,65, 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 hsimilar 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 Youngs 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 notonly 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 thickness 62.

These soft GBs essentially determine the materials 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 hincreases, 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%. Increasedfracture 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

Fig. 7 MD simulation of indentation of nc-SiC. There is a crossover from cooperative response of grains at smaller indentation depths to a decoupled response ofindividual grains at larger depths. (a) Localization of the deformation after the crossover, where the color corresponds to the displacement of the center of mass ofeach grain. (b) Mean displacement of all grains where the crossover at hCR can be clearly seen (coupled at h < hCR and decoupled at larger h). (c)-(e) Examples ofdiscrete plastic events inside the grains, such as sliding at the GB (c2) or propagation of a dislocation (black arrow in (e)) along the stacking fault line (yellow line in(d) and (e)). (Reprinted with permission from68. 2005 American Association for the Advancement of Science.)

(a) (b)

(c)

(d) (e)


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obtained in the MD simulation described above is consistent with theseexperimental measurements.

Challenges in modeling nanoindentationDespite 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 m/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 thermostats70,71). 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 lostbecause 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.73,74 and applied to study

nanoindentation75,76. 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.72,77,78 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 elsewhere79,80.

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

Fig. 8 (a) Atomic configuration of nc-SiC with white grains and yellow GBs.At lower indentation depths h, deformation of the material is dominated byrecrystallization (blue atoms). At depths h > hCR, deformation is dominatedby disordering (red atoms). (b) Percentage of disordered atoms in thematerial as a function of h reflects the crossover. (Reprinted with permissionfrom68. 2005 American Association for the Advancement of Science.)

37.6

37.4

37.2

37.0

36.8

36.6-5 0 5 10 15 20 25 30

(b)

Crossover depthhCR = 14.5

Regimes 1 & 2

Regimes 3 & 4

Indenter depth, h [A]

Percentage

ofdisorderedatoms

(b)

(a)


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accurate quantum mechanical calculations. Current, state-of-the-art

empirical potentials can account for bond formation and breaking,change in hybridization, charge transfer, etc.81,82. 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 andextract 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

grains 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.

OutlookFor the design of materials with superior mechanical properties, a

mutual feedback process between experiments and simulations is



Fig. 9 Combined MD and FE simulations of indentation of Cu. (a) P-h curves obtained from MD (red) and FE (blue) calculations are in good agreement.MD configurations at the beginning of the simulation (b) and (c) after several nucleation events. In (c), a shear band (dashed line) is formed. (d) The initialnucleation event modeled by the FE method, where the color corresponds to the Mises stress. Good agreement is found between MD and FE regarding the predictednucleation site, slip plane, and Burgers vector. (Reprinted with permission from72. 2002 Nature Publishing Group.)

(b)(a)

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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 ScienceFoundation grant DMR-0512228. I am also thankful to D. Stone and D. Morganfor helpful comments on the manuscript.

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