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Atomistic simulations of nanoindentation
Atomistic simulations of nanoindentation 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. Izabela Szlufarska Department of Materials Science and Engineering, University of Wisconsin, Madison, 1509 University Ave, Madison, WI 53706-1595, USA E-mail: [email protected]
Nanotechnology is leading to the development of materials and
equation of motion for all the atoms in order to retrace their
level. In many applications, components experience unintentional
trajectories while an indenter is being pressed into the material. The
or deliberate mechanical contact, and a thorough understanding of
load on the indenter P is calculated by summing the forces acting on
a material’s mechanical behavior is critical for the design of
the atoms of the indenter in the indentation direction. Indentation
reliable and durable systems. Nanoindentation is a widely used technique for probing mechanical
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
properties and stability, especially of surfaces and thin films1-8. With
example of a P-h response of crystalline silicon carbide (3C-SiC)
the development of sensitive atomic force microscopy (AFM)3 and
obtained in a classical MD simulation13.
other techniques2,9, we can now measure the force P on the indenter
42
In molecular dynamics (MD) simulations, one solves Newton’s
devices whose structure and function are controlled at the atomic
Because of the very complex stress profile generated in the vicinity
and indenter displacement h with nanometer scale precision10,11. (For
of the indenter, it is challenging to interpret nanoindentation
more details on nanoindentation, see the article by Schuh on page 32
experiments at a fundamental level. Atomistic computer simulations
of this issue and reviews elsewhere12.) From the shape of such P-h
have been very helpful in unraveling the processes underlying the
curves, one can extract information about elastic moduli or hardness.
nanoindentation response14-19. The role of simulations is not
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ISSN:1369 7021 © Elsevier Ltd 2006
Atomistic simulations of nanoindentation
(a)
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(b)
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 plastic events 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 Å increments. Up to h = 1.83 Å, 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 Å, reflected in the hysteresis of the second unloading curve (circles). (Reprinted with permission from13. © 2004 American Institute of Physics.)
necessarily to reproduce the exact experimental behavior, but rather
material33 (Fig. 3). Similar ‘pop-in’ events have been observed in
to identify possible atomistic mechanisms in the early stages of plastic
experimentally determined P-h curves34-37.
deformation and determine their trends as a function of the
Atomistic simulations have also shed light on the solid-state phase
nanostructure and the environment (e.g. temperature or surface
transformations that take place in material in the vicinity of the
passivation)20-22. The goal is to develop robust insights into
indenter38. For instance, Kallman et al.39 observed a localized
technologically relevant materials and ultimately design a material
crystalline-to-amorphous transition in Si at temperatures close to the
with optimum mechanical properties.
melting point, which is consistent with experiments35,40. These simulations reveal a dependence of the yield strength of Si on
Crystalline materials
structure, rate of deformation, and temperature. Solid-state
There are some very good review articles on the subject of
amorphization has also been observed in nanoindentation simulations
nanomechanics simulations in bulk crystalline materials23-25, where
of 3C-SiC13. Defect-stimulated growth and coalescence of dislocation
readers can find detailed descriptions of such phenomena as jump-tocontact (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-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
Fig. 2 MD simulation of indentation of solid Au with a Ni indenter. Atomic positions during loading and unloading simulations are shown from top left to bottom right. During unloading a connective neck is formed by Au (yellow) atoms. (Reprinted with permission from32. © 1995 Nature Publishing Group.)
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Atomistic simulations of nanoindentation
(a)
(c)
(b)
(d)
Fig. 3 Atomic configurations during indentation of crystalline SiC obtained by MD simulations. The ¾ cut of the sample shows the atoms (a) before and (b) after the first load drop (compare with the P-h curve in Fig. 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 crystallographic order in SiC are shown. (Reprinted with permission from33. © 2005 American Physical Society.)
loops are found to be the atomistic mechanisms underlying the
challenge. Various models have been proposed to describe defects in
crystalline-to-amorphous transition. In simulations by Walsh et al.41,
such structures43,44. For example, Gilman45 has conjectured that an
amorphization has been identified as a primary deformation
analog to a crystal dislocation exists in noncrystalline solids, and has
mechanism in indentation of Si3N4. During the indentation,
described defects as dislocation lines with variable Burgers vectors.
amorphization was arrested by cracking at the indenter corners and
Because of the presence of these inhomogenities frozen into the entire
piling up of material along the indenter sides (Fig. 4). The pile-up
material, the nanoindentation damage cannot be readily identified by
material itself has an amorphous structure.
computational techniques used for crystalline solids. The prevailing
Structural changes, such as amorphization, can be identified in
theory of plasticity in metallic glasses involves localized flow events in
simulations by means of the radial distribution function, which can be
shear transformation zones (STZ). An STZ is a small cluster of atoms
directly compared with X-ray diffraction experiments. Other commonly
that can rearrange under applied stress to produce a unit of plastic
used methods to monitor the evolution of simulated indentation
deformation46-49.
damage are bond angle distribution (Fig. 4); local variation in potential
Even though these theories provide an essential physical insight, a
energy, pressure, and shear stress; visual inspection of the computer-
truly atomistic model of plastic flow in amorphous materials is still
generated atomic structure; and changes in local topologies
lacking. A few atomistic simulations have been performed to tackle this
(topological changes can be analyzed, for example, by means of
problem. For example, Sinnott et al.50 undertook the difficult task of
shortest-path ring statistics33,39,42).
simulating nanoindentation of amorphous carbon (a-C:H) material with an sp3 bonded indenter. The simulations reveal that indentation has
44
Amorphous and quasicrystalline materials
little effect on the hybridization of the carbon atoms or the randomly
Amorphous solids constitute a separate class of materials whose
penetrating the amorphous solid, the tip deforms only slightly via
mechanical properties are of great fundamental and technological
shear. This is in contrast to indentation simulations of crystalline
interest. Because amorphous materials lack a topologically ordered
diamond, where the tip deformation includes significant shear and
network, analysis of deformations and defects presents a significant
twist components.
MAY 2006 | VOLUME 9 | NUMBER 5
distributed stresses within the material. They also show that while
Atomistic simulations of nanoindentation
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quasicrystalline medium-range order, the deformation localization can
A series of simulations of nanoindentation of amorphous silicon carbide (a-SiC)51 point to a noticeable localization of damage in the
arise as the result of the breakdown of stable quasicrystal-like atomic
vicinity of the indenter, however the localization is less pronounced
configurations. Indentation simulations are of great interest for
than in the case of 3C-SiC. As shown in Fig. 5, the P-h curve for a-SiC
elucidating the science underlying plasticity in amorphous structures.
exhibits irregular, discrete load drops similar to 3C-SiC (Fig. 1). Here,
The advent of these simulations is particularly timely because of the
the load drops correspond to breaking of the local arrangements of
growing potential for structural applications of amorphous materials54,55.
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
Nanostructured materials
than those at which the material yields plastically, the material’s
It has been demonstrated in experiments56-58 and simulations59-61 that
response is not entirely elastic. Instead, the amorphous structure, which
nanocrystalline materials exhibit unusual mechanical behavior when
is metastable by nature, supports a small inelastic flow related to
compared with their polycrystalline counterparts. For example,
relaxation of atoms through short migration distances.
normally brittle ceramics are shown to have very high hardness, high
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
fracture toughness, and superplastic behavior, as the grain size is refined to the nanometer regime62,63. (For more detailed descriptions
Falk53.
of these and other properties of nanocrystalline materials, see the article by Van Swygenhoven on page 24 of this issue.)
The authors show that in a two-dimensional metallic alloy with
(a)
(b)
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 angle distribution. (a) Bond angle distribution for bulk crystalline (yellow) and amorphous (green) Si3N4. (b) Local bond angle distribution for region I (yellow) and II (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.)
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Atomistic simulations of nanoindentation
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. 1 for crystalline SiC. Here, the load 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 from51. © 2005 American Institute of Physics.)
(a)
(d)
(b)
(e)
(c)
(f)
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 become absorbed 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 from67. © 2004 Elsevier.)
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Atomistic simulations of nanoindentation
Mechanical properties of nanocrystalline materials are controlled on
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response. Recent MD simulations of nanoindentation of nc-SiC show
the atomic level, and therefore atomistic simulations can bring an
how the coexistence of brittle grains and soft amorphous GBs results in
invaluable physical insight to experiments and ultimately enable the
unusual deformation mechanisms68. The simulated material was
design of materials with optimum properties.
sintered from crystalline grains of 8 nm in diameter and consists of about 19 million atoms. As the indenter depth h increases, a crossover
Since nanocrystalline materials have an increased volume fraction of grain boundaries (GBs)64,65, it is essential to understand the
is observed from a cooperative deformation mechanism involving
interactions of GBs with dislocations emitted from under the
multiple grains to a decoupled response of individual grains, e.g. grain
indenter. This question has been addressed by Feichtinger et al.66,
rotation and sliding, and intragranular dislocation activity (Fig. 7). The
who performed MD nanoindentation simulations of nanocrystalline
crossover is also reflected in a switch from deformation dominated by
Au (nc-Au) with grains 12 nm in diameter. In the simulation,
crystallization to deformation dominated by disordering, as explained
dislocation nucleation within the grain occurs at the onset of plastic
in the caption of Fig. 8. In the early stages of plastic deformation, the
deformation at an indenter depth h similar to that in a perfect crystal.
soft (amorphous) GB phase ‘screens’ the crystalline grains from
The GBs act as an efficient sink for partial and full dislocations and
deformation, thus making nc-SiC more ductile than its coarse-grained
intergranular sliding is observed. A decrease in Young’s modulus is
counterpart. Fracture toughness (a measure of how much energy it
also seen as the grain size is refined to 5 nm in diameter. Recently,
takes to propagate a crack), measured experimentally in
the same group reported another simulation67, where the indenter
nanocomposites with an nc-SiC matrix and diamond inclusions58,
size is smaller than the grain size, which shows that the GBs can not
exceeds that of a polycrystalline matrix by about 50%. Increased
only act as a sink for dislocations, but can also reflect or emit
fracture toughness does not necessarily lead to a lower value of
dislocations (Fig. 6).
hardness. Recent experiments of nanoindentation of nc-SiC with grain
In contrast to metals, GBs in nanocrystalline ceramics form a thicker
sizes of 5-20 nm show quite the opposite trend. Liao et al.63 report nc-
GB phase that is highly disordered and has a fairly uniform thickness62.
SiC to be superhard, i.e. to have a hardness of 30-50 GPa, which is
These soft GBs essentially determine the material’s mechanical
larger than that of crystalline SiC. The hardness value of 39 GPa
(b)
(a)
(d)
(e)
(c)
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 of individual grains at larger depths. (a) Localization of the deformation after the crossover, where the color corresponds to the displacement of the center of mass of each 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 of discrete 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.)
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Atomistic simulations of nanoindentation
provide understanding of atomistic mechanisms and qualitative trends
(a)
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
Percentage of disordered atoms
37.6
loop was able to move all the way to the bottom and leave the sample.
(b) (b)
On the other hand, in most MD simulations of nanoindentation reported in literature, the bottom layer of the sample is ‘frozen’ and
37.4
dislocation motion through the material is inhibited. The simplest Crossover depth hCR = 14.5 Å
37.2
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
37.0
because even small systems are well suited to studying the effects of Regimes 3 & 4
36.8
Regimes 1 & 2 36.6 -5
0
5
boundary conditions on plasticity, and deformation mechanisms in small nanostructures are becoming of increasing interest to
10
15
20
25
30
Indenter depth, h [A]
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 by recrystallization (blue atoms). At depths h > hCR, deformation is dominated by disordering (red atoms). (b) Percentage of disordered atoms in the material as a function of h reflects the crossover. (Reprinted with permission from68. © 2005 American Association for the Advancement of Science.)
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.
obtained in the MD simulation described above is consistent with these
In order to extend simulations beyond the micrometer regime, models are being developed that combine direct atomistic simulations
experimental measurements.
with continuum methods. For example, a quasicontinuum model has
Challenges in modeling nanoindentation Despite the great advantages of simulating nanoindentation, the
nanoindentation75,76. In this approach, a continuum finite element (FE)
technique faces some serious challenges. The first is the limited time
is employed to characterize the mechanical response of the material,
scales accessible to simulations because of limited computational
i.e. the positions of the majority of atoms are constrained and
resources. For example, the slowest time scales available to MD give
determined by the displacement of the nearby node. In contrast to the
~1 m/s indentation velocities69, while state-of-the-art AFM systems
standard FE method, in the quasicontinuum approach the constitutive
can only operate at up to 0.001 m/s. Indentation measurements are
response of the system is determined from atomistic calculations based
limited to slower rates of the order of ~25 µm/s. As a result, there is a
on interatomic potentials. Combined FE and MD simulations of
significant disparity between simulated strain rates and those
nanoindentation have also been performed by other groups. Li
attainable by experiments. The rationale for modeling nanoindentation
et al.72,77,78 performed direct FE simulations in which large strain
with MD is that for the simulated solids, all the above speeds are far
constitutive relations are derived from an interatomic potential (Fig. 9).
below the speed of sound in materials (for example, the speed of sound
Unlike the quasicontinuum method, this approach remains fully
in SiC is 11 000 m/s). For this reason, MD simulations are able to
continuum. A review of simulation work based on FE is beyond the
dissipate any reflected waves that arise from the motion of the
scope of this article but can be found elsewhere79,80.
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
48
been developed by Tadmor et al.73,74 and applied to study
MAY 2006 | VOLUME 9 | NUMBER 5
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
Atomistic simulations of nanoindentation
(a)
(b)
(c)
(d)
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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 initial nucleation event modeled by the FE method, where the color corresponds to the Mises stress. Good agreement is found between MD and FE regarding the predicted nucleation site, slip plane, and Burgers vector. (Reprinted with permission from72. © 2002 Nature Publishing Group.)
accurate quantum mechanical calculations. Current, state-of-the-art
Fully atomistic simulation of large systems involving many millions
empirical potentials can account for bond formation and breaking,
of atoms creates another nontrivial challenge, i.e. to seek patterns and
change in hybridization, charge transfer, etc.81,82. In spite of these
extract information from such massive, multivariable datasets. For
developments, there is not one single analytic potential that is capable
example, a single nanocrystalline ceramic can contain thousands of
of describing all possible properties that might be of interest in a
randomly oriented crystalline grains surrounded by intergranular
particular material. Furthermore, fitting an accurate potential is a
regions with various levels of topological disorder. The change in a
difficult and time-consuming process.
grain’s crystallographic structure and chemical ordering during
An approach that bypasses the need for an interatomic potential is
indentation is a complex phenomenon that depends on many variables.
based on combined first-principle and FE calculations. For example,
Tracking deformations in a stand-alone amorphous material presents a
Hayes et al.83 have recently simulated the nanoindentation of Al by
challenge in itself, let alone as a part of a complex nanocrystalline
means of the orbital-free density functional theory (OFDFT) local
material. Seeking patterns in such structures requires an extensive,
quasicontinuum (LC) method. In this OFDFT-LC model, the
hands-on analysis.
quasicontinuum approach is adopted but the atomic-scale calculations,
In order to analyze the complicated profiles of indentation damage
based previously on empirical potentials, are now replaced with fast
in structurally advanced materials, there is an urgent need to develop
and inexpensive first-principles theory. This method is well suited to
more efficient data-mining techniques. Such developments can be
study phenomena such as initial dislocation formation; however, it is
fostered by interdisciplinary collaborations between materials and
not capable of treating intermediate length scales (e.g. GBs in
computer scientists.
nanocrystalline materials). It is clear that with improving computer technology and the development of new algorithms, first-principles-
Outlook
based calculations will play an increasingly important role in
For the design of materials with superior mechanical properties, a
nanoindentation modeling.
mutual feedback process between experiments and simulations is
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Atomistic simulations of nanoindentation
critical. Because of the transferability of simulation tools and the wide
industrial applications, a thorough understanding of their mechanical
variety of application areas, it is also essential to create a platform for
response (e.g. through indentation) is of vital importance.
collaboration among scientists from multiple disciplines. New structural applications are being extensively explored for amorphous and
Acknowledgments
nanostructured materials, e.g. superhard coatings, sporting goods, high-
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.
speed machining and tooling, and biomaterial implants. If the mechanical properties of these complex materials are to be exploited in
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79. Mackerle, J., Modell. Simul. Mater. Sci. Eng. (2005) 13, 935
38. Cheong, W. C. D., and Zhang, L. C., Nanotechnology (2000) 11, 173
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40. Minowa, K., and Sumino, K., Phys. Rev. Lett. (1992) 69, 320
82. Brenner, D. W., et al., J. Phys.: Condens. Matter (2002) 14, 783
41. Walsh, P., et al., Appl. Phys. Lett. (2000) 77, 4332
83. Hayes, R. L., et al., Multiscale Model. Simul. (2005) 4, 359
MAY 2006 | VOLUME 9 | NUMBER 5
Nanotechnology is leading to the development of materials and
equation of motion for all the atoms in order to retrace their
level. In many applications, components experience unintentional
trajectories while an indenter is being pressed into the material. The
or deliberate mechanical contact, and a thorough understanding of
load on the indenter P is calculated by summing the forces acting on
a material’s mechanical behavior is critical for the design of
the atoms of the indenter in the indentation direction. Indentation
reliable and durable systems. Nanoindentation is a widely used technique for probing mechanical
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
properties and stability, especially of surfaces and thin films1-8. With
example of a P-h response of crystalline silicon carbide (3C-SiC)
the development of sensitive atomic force microscopy (AFM)3 and
obtained in a classical MD simulation13.
other techniques2,9, we can now measure the force P on the indenter
42
In molecular dynamics (MD) simulations, one solves Newton’s
devices whose structure and function are controlled at the atomic
Because of the very complex stress profile generated in the vicinity
and indenter displacement h with nanometer scale precision10,11. (For
of the indenter, it is challenging to interpret nanoindentation
more details on nanoindentation, see the article by Schuh on page 32
experiments at a fundamental level. Atomistic computer simulations
of this issue and reviews elsewhere12.) From the shape of such P-h
have been very helpful in unraveling the processes underlying the
curves, one can extract information about elastic moduli or hardness.
nanoindentation response14-19. The role of simulations is not
MAY 2006 | VOLUME 9 | NUMBER 5
ISSN:1369 7021 © Elsevier Ltd 2006
Atomistic simulations of nanoindentation
(a)
REVIEW FEATURE
(b)
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 plastic events 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 Å increments. Up to h = 1.83 Å, 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 Å, reflected in the hysteresis of the second unloading curve (circles). (Reprinted with permission from13. © 2004 American Institute of Physics.)
necessarily to reproduce the exact experimental behavior, but rather
material33 (Fig. 3). Similar ‘pop-in’ events have been observed in
to identify possible atomistic mechanisms in the early stages of plastic
experimentally determined P-h curves34-37.
deformation and determine their trends as a function of the
Atomistic simulations have also shed light on the solid-state phase
nanostructure and the environment (e.g. temperature or surface
transformations that take place in material in the vicinity of the
passivation)20-22. The goal is to develop robust insights into
indenter38. For instance, Kallman et al.39 observed a localized
technologically relevant materials and ultimately design a material
crystalline-to-amorphous transition in Si at temperatures close to the
with optimum mechanical properties.
melting point, which is consistent with experiments35,40. These simulations reveal a dependence of the yield strength of Si on
Crystalline materials
structure, rate of deformation, and temperature. Solid-state
There are some very good review articles on the subject of
amorphization has also been observed in nanoindentation simulations
nanomechanics simulations in bulk crystalline materials23-25, where
of 3C-SiC13. Defect-stimulated growth and coalescence of dislocation
readers can find detailed descriptions of such phenomena as jump-tocontact (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-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
Fig. 2 MD simulation of indentation of solid Au with a Ni indenter. Atomic positions during loading and unloading simulations are shown from top left to bottom right. During unloading a connective neck is formed by Au (yellow) atoms. (Reprinted with permission from32. © 1995 Nature Publishing Group.)
MAY 2006 | VOLUME 9 | NUMBER 5
43
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Atomistic simulations of nanoindentation
(a)
(c)
(b)
(d)
Fig. 3 Atomic configurations during indentation of crystalline SiC obtained by MD simulations. The ¾ cut of the sample shows the atoms (a) before and (b) after the first load drop (compare with the P-h curve in Fig. 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 crystallographic order in SiC are shown. (Reprinted with permission from33. © 2005 American Physical Society.)
loops are found to be the atomistic mechanisms underlying the
challenge. Various models have been proposed to describe defects in
crystalline-to-amorphous transition. In simulations by Walsh et al.41,
such structures43,44. For example, Gilman45 has conjectured that an
amorphization has been identified as a primary deformation
analog to a crystal dislocation exists in noncrystalline solids, and has
mechanism in indentation of Si3N4. During the indentation,
described defects as dislocation lines with variable Burgers vectors.
amorphization was arrested by cracking at the indenter corners and
Because of the presence of these inhomogenities frozen into the entire
piling up of material along the indenter sides (Fig. 4). The pile-up
material, the nanoindentation damage cannot be readily identified by
material itself has an amorphous structure.
computational techniques used for crystalline solids. The prevailing
Structural changes, such as amorphization, can be identified in
theory of plasticity in metallic glasses involves localized flow events in
simulations by means of the radial distribution function, which can be
shear transformation zones (STZ). An STZ is a small cluster of atoms
directly compared with X-ray diffraction experiments. Other commonly
that can rearrange under applied stress to produce a unit of plastic
used methods to monitor the evolution of simulated indentation
deformation46-49.
damage are bond angle distribution (Fig. 4); local variation in potential
Even though these theories provide an essential physical insight, a
energy, pressure, and shear stress; visual inspection of the computer-
truly atomistic model of plastic flow in amorphous materials is still
generated atomic structure; and changes in local topologies
lacking. A few atomistic simulations have been performed to tackle this
(topological changes can be analyzed, for example, by means of
problem. For example, Sinnott et al.50 undertook the difficult task of
shortest-path ring statistics33,39,42).
simulating nanoindentation of amorphous carbon (a-C:H) material with an sp3 bonded indenter. The simulations reveal that indentation has
44
Amorphous and quasicrystalline materials
little effect on the hybridization of the carbon atoms or the randomly
Amorphous solids constitute a separate class of materials whose
penetrating the amorphous solid, the tip deforms only slightly via
mechanical properties are of great fundamental and technological
shear. This is in contrast to indentation simulations of crystalline
interest. Because amorphous materials lack a topologically ordered
diamond, where the tip deformation includes significant shear and
network, analysis of deformations and defects presents a significant
twist components.
MAY 2006 | VOLUME 9 | NUMBER 5
distributed stresses within the material. They also show that while
Atomistic simulations of nanoindentation
REVIEW FEATURE
quasicrystalline medium-range order, the deformation localization can
A series of simulations of nanoindentation of amorphous silicon carbide (a-SiC)51 point to a noticeable localization of damage in the
arise as the result of the breakdown of stable quasicrystal-like atomic
vicinity of the indenter, however the localization is less pronounced
configurations. Indentation simulations are of great interest for
than in the case of 3C-SiC. As shown in Fig. 5, the P-h curve for a-SiC
elucidating the science underlying plasticity in amorphous structures.
exhibits irregular, discrete load drops similar to 3C-SiC (Fig. 1). Here,
The advent of these simulations is particularly timely because of the
the load drops correspond to breaking of the local arrangements of
growing potential for structural applications of amorphous materials54,55.
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
Nanostructured materials
than those at which the material yields plastically, the material’s
It has been demonstrated in experiments56-58 and simulations59-61 that
response is not entirely elastic. Instead, the amorphous structure, which
nanocrystalline materials exhibit unusual mechanical behavior when
is metastable by nature, supports a small inelastic flow related to
compared with their polycrystalline counterparts. For example,
relaxation of atoms through short migration distances.
normally brittle ceramics are shown to have very high hardness, high
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
fracture toughness, and superplastic behavior, as the grain size is refined to the nanometer regime62,63. (For more detailed descriptions
Falk53.
of these and other properties of nanocrystalline materials, see the article by Van Swygenhoven on page 24 of this issue.)
The authors show that in a two-dimensional metallic alloy with
(a)
(b)
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 angle distribution. (a) Bond angle distribution for bulk crystalline (yellow) and amorphous (green) Si3N4. (b) Local bond angle distribution for region I (yellow) and II (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.)
MAY 2006 | VOLUME 9 | NUMBER 5
45
REVIEW FEATURE
Atomistic simulations of nanoindentation
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. 1 for crystalline SiC. Here, the load 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 from51. © 2005 American Institute of Physics.)
(a)
(d)
(b)
(e)
(c)
(f)
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 become absorbed 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 from67. © 2004 Elsevier.)
46
MAY 2006 | VOLUME 9 | NUMBER 5
Atomistic simulations of nanoindentation
Mechanical properties of nanocrystalline materials are controlled on
REVIEW FEATURE
response. Recent MD simulations of nanoindentation of nc-SiC show
the atomic level, and therefore atomistic simulations can bring an
how the coexistence of brittle grains and soft amorphous GBs results in
invaluable physical insight to experiments and ultimately enable the
unusual deformation mechanisms68. The simulated material was
design of materials with optimum properties.
sintered from crystalline grains of 8 nm in diameter and consists of about 19 million atoms. As the indenter depth h increases, a crossover
Since nanocrystalline materials have an increased volume fraction of grain boundaries (GBs)64,65, it is essential to understand the
is observed from a cooperative deformation mechanism involving
interactions of GBs with dislocations emitted from under the
multiple grains to a decoupled response of individual grains, e.g. grain
indenter. This question has been addressed by Feichtinger et al.66,
rotation and sliding, and intragranular dislocation activity (Fig. 7). The
who performed MD nanoindentation simulations of nanocrystalline
crossover is also reflected in a switch from deformation dominated by
Au (nc-Au) with grains 12 nm in diameter. In the simulation,
crystallization to deformation dominated by disordering, as explained
dislocation nucleation within the grain occurs at the onset of plastic
in the caption of Fig. 8. In the early stages of plastic deformation, the
deformation at an indenter depth h similar to that in a perfect crystal.
soft (amorphous) GB phase ‘screens’ the crystalline grains from
The GBs act as an efficient sink for partial and full dislocations and
deformation, thus making nc-SiC more ductile than its coarse-grained
intergranular sliding is observed. A decrease in Young’s modulus is
counterpart. Fracture toughness (a measure of how much energy it
also seen as the grain size is refined to 5 nm in diameter. Recently,
takes to propagate a crack), measured experimentally in
the same group reported another simulation67, where the indenter
nanocomposites with an nc-SiC matrix and diamond inclusions58,
size is smaller than the grain size, which shows that the GBs can not
exceeds that of a polycrystalline matrix by about 50%. Increased
only act as a sink for dislocations, but can also reflect or emit
fracture toughness does not necessarily lead to a lower value of
dislocations (Fig. 6).
hardness. Recent experiments of nanoindentation of nc-SiC with grain
In contrast to metals, GBs in nanocrystalline ceramics form a thicker
sizes of 5-20 nm show quite the opposite trend. Liao et al.63 report nc-
GB phase that is highly disordered and has a fairly uniform thickness62.
SiC to be superhard, i.e. to have a hardness of 30-50 GPa, which is
These soft GBs essentially determine the material’s mechanical
larger than that of crystalline SiC. The hardness value of 39 GPa
(b)
(a)
(d)
(e)
(c)
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 of individual grains at larger depths. (a) Localization of the deformation after the crossover, where the color corresponds to the displacement of the center of mass of each 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 of discrete 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.)
MAY 2006 | VOLUME 9 | NUMBER 5
47
REVIEW FEATURE
Atomistic simulations of nanoindentation
provide understanding of atomistic mechanisms and qualitative trends
(a)
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
Percentage of disordered atoms
37.6
loop was able to move all the way to the bottom and leave the sample.
(b) (b)
On the other hand, in most MD simulations of nanoindentation reported in literature, the bottom layer of the sample is ‘frozen’ and
37.4
dislocation motion through the material is inhibited. The simplest Crossover depth hCR = 14.5 Å
37.2
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
37.0
because even small systems are well suited to studying the effects of Regimes 3 & 4
36.8
Regimes 1 & 2 36.6 -5
0
5
boundary conditions on plasticity, and deformation mechanisms in small nanostructures are becoming of increasing interest to
10
15
20
25
30
Indenter depth, h [A]
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 by recrystallization (blue atoms). At depths h > hCR, deformation is dominated by disordering (red atoms). (b) Percentage of disordered atoms in the material as a function of h reflects the crossover. (Reprinted with permission from68. © 2005 American Association for the Advancement of Science.)
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.
obtained in the MD simulation described above is consistent with these
In order to extend simulations beyond the micrometer regime, models are being developed that combine direct atomistic simulations
experimental measurements.
with continuum methods. For example, a quasicontinuum model has
Challenges in modeling nanoindentation Despite the great advantages of simulating nanoindentation, the
nanoindentation75,76. In this approach, a continuum finite element (FE)
technique faces some serious challenges. The first is the limited time
is employed to characterize the mechanical response of the material,
scales accessible to simulations because of limited computational
i.e. the positions of the majority of atoms are constrained and
resources. For example, the slowest time scales available to MD give
determined by the displacement of the nearby node. In contrast to the
~1 m/s indentation velocities69, while state-of-the-art AFM systems
standard FE method, in the quasicontinuum approach the constitutive
can only operate at up to 0.001 m/s. Indentation measurements are
response of the system is determined from atomistic calculations based
limited to slower rates of the order of ~25 µm/s. As a result, there is a
on interatomic potentials. Combined FE and MD simulations of
significant disparity between simulated strain rates and those
nanoindentation have also been performed by other groups. Li
attainable by experiments. The rationale for modeling nanoindentation
et al.72,77,78 performed direct FE simulations in which large strain
with MD is that for the simulated solids, all the above speeds are far
constitutive relations are derived from an interatomic potential (Fig. 9).
below the speed of sound in materials (for example, the speed of sound
Unlike the quasicontinuum method, this approach remains fully
in SiC is 11 000 m/s). For this reason, MD simulations are able to
continuum. A review of simulation work based on FE is beyond the
dissipate any reflected waves that arise from the motion of the
scope of this article but can be found elsewhere79,80.
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
48
been developed by Tadmor et al.73,74 and applied to study
MAY 2006 | VOLUME 9 | NUMBER 5
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
Atomistic simulations of nanoindentation
(a)
(b)
(c)
(d)
REVIEW FEATURE
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 initial nucleation event modeled by the FE method, where the color corresponds to the Mises stress. Good agreement is found between MD and FE regarding the predicted nucleation site, slip plane, and Burgers vector. (Reprinted with permission from72. © 2002 Nature Publishing Group.)
accurate quantum mechanical calculations. Current, state-of-the-art
Fully atomistic simulation of large systems involving many millions
empirical potentials can account for bond formation and breaking,
of atoms creates another nontrivial challenge, i.e. to seek patterns and
change in hybridization, charge transfer, etc.81,82. In spite of these
extract information from such massive, multivariable datasets. For
developments, there is not one single analytic potential that is capable
example, a single nanocrystalline ceramic can contain thousands of
of describing all possible properties that might be of interest in a
randomly oriented crystalline grains surrounded by intergranular
particular material. Furthermore, fitting an accurate potential is a
regions with various levels of topological disorder. The change in a
difficult and time-consuming process.
grain’s crystallographic structure and chemical ordering during
An approach that bypasses the need for an interatomic potential is
indentation is a complex phenomenon that depends on many variables.
based on combined first-principle and FE calculations. For example,
Tracking deformations in a stand-alone amorphous material presents a
Hayes et al.83 have recently simulated the nanoindentation of Al by
challenge in itself, let alone as a part of a complex nanocrystalline
means of the orbital-free density functional theory (OFDFT) local
material. Seeking patterns in such structures requires an extensive,
quasicontinuum (LC) method. In this OFDFT-LC model, the
hands-on analysis.
quasicontinuum approach is adopted but the atomic-scale calculations,
In order to analyze the complicated profiles of indentation damage
based previously on empirical potentials, are now replaced with fast
in structurally advanced materials, there is an urgent need to develop
and inexpensive first-principles theory. This method is well suited to
more efficient data-mining techniques. Such developments can be
study phenomena such as initial dislocation formation; however, it is
fostered by interdisciplinary collaborations between materials and
not capable of treating intermediate length scales (e.g. GBs in
computer scientists.
nanocrystalline materials). It is clear that with improving computer technology and the development of new algorithms, first-principles-
Outlook
based calculations will play an increasingly important role in
For the design of materials with superior mechanical properties, a
nanoindentation modeling.
mutual feedback process between experiments and simulations is
MAY 2006 | VOLUME 9 | NUMBER 5
49
REVIEW FEATURE
Atomistic simulations of nanoindentation
critical. Because of the transferability of simulation tools and the wide
industrial applications, a thorough understanding of their mechanical
variety of application areas, it is also essential to create a platform for
response (e.g. through indentation) is of vital importance.
collaboration among scientists from multiple disciplines. New structural applications are being extensively explored for amorphous and
Acknowledgments
nanostructured materials, e.g. superhard coatings, sporting goods, high-
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.
speed machining and tooling, and biomaterial implants. If the mechanical properties of these complex materials are to be exploited in
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