Abstract

We perform atomistic simulations of dislocation nucleation in two-dimensional (2D) and three-dimensional (3D) defect-free hexagonal crystals during nanoindentation with circular (2D) or spherical (3D) indenters. The incipient embryo structure in the critical eigenmode of the mesoregions is analyzed to study homogeneous dislocation nucleation. The critical eigenmode or dislocation embryo is found to be localized along a line (or plane in 3D) of atoms with a lateral extent, $ξ$, at some depth, $Y*$, below the surface. The lowest energy eigenmode for mesoregions of varying radius, rmeso, centered on the localized region of the critical eigenmode is computed. The energy of the lowest eigenmode, λmeso, decays very rapidly with increasing rmeso and λmeso ≈ 0 for $rmeso≳ξ$. The analysis of a mesoscale region in the material can reveal the presence of incipient instability even for $rmeso≲ξ$ but gives reasonable estimate for the energy and spatial extent of the critical mode only for $rmeso≳ξ$. When the mesoregion is not centered at the localized region, we show that the mesoregion should contain a critical part of the embryo (and not only the center of embryo) to reveal instability. This scenario indicates that homogeneous dislocation nucleation is a quasilocal phenomenon. Also, the critical eigenmode for the mesoscale region reveals instability much sooner than the full system eigenmode. We use mesoscale analysis to verify the scaling laws shown previously by Garg and Maloney in 2D [2016, “Universal Scaling Laws for Homogeneous Dissociation Nucleation During Nano-Indentation,” J. Mech. Phys. Solids, 95, pp. 742–754.] for the size, ξ, and depth from the surface, Y*, of the dislocation embryo with respect to indenter radius, R, in full 3D simulations.

1 Introduction

In perfect crystals the onset of plasticity is indicated by the events of dislocation nucleation. Instabilities govern the critical behavior of materials during various mechanical processes. For the past several decades, dislocation nucleation has been shown to be the instability mechanism that governs the yield strength of crystal films, initially free of defects. Dislocation nucleation from free surfaces has been shown to determine the strength of micro- and nanopillars [13]. Applications such as chemical mechanical polishing (CMP) [46] have nanoscale interconnects in which dislocation nucleation has been shown as a stress relaxation mechanism [7,8]. In microscale crystals, dislocations are typically produced by pre-existing defects in the material. However, in recent years, tools such as nanoindentation and the atomic force microscopy (AFM) have allowed access to mechanical properties in nanoscale samples [920] where one can obtain practically defect-free crystals. In these scenarios, one is typically interested in how temperature, loading rate, or the sample or probe size affect the mechanical response.

Miller and Rodney (MR) showed that homogeneous dislocation nucleation (HDN) is a mesoscale process involving tens of atoms forming an embryo underneath the indenter [21]. The importance of the nonlocal nature of this problem has also been emphasized in the work of Delph et al. for the development of the Wallace criterion for the prediction of cavitation and crack growth problems in face-centered cubic (FCC) solids [22,23]. Garg and Acharya also proposed a dislocation nucleation criterion based on the kinematic structure of the theory of field dislocation mechanics [24]. For dislocation nucleation, MR developed a predictive criterion based on the mesoscale atomic acoustic tensor [25]. According to this criterion, HDN is indicated by loss of positive definiteness of the mesoscale acoustic tensor. The mesoscale criterion is similar to a previously proposed Λ criterion [2530]. Λ is the minimum eigenvalue of the local acoustic tensor, calculated on an atom-by-atom basis [2528]. On the other hand, MR’s criterion is based on the acoustic tensor of a cluster of atoms.

For the non-local analysis described by MR, one has to judiciously choose the meso-region for calculation of the acoustic tensor. MR did not discuss the appropriate procedure to choose the meso-region. The size and location of the meso-region are needed to perform mesoscale analysis to predict dislocation nucleation in nanoindentation simulations. In this work we show that even if the meso-region size is much smaller than the embryo, it can still reveal the instability. However, the meso-region captures the full spatial extent of the embryo only if the meso-region is bigger than the embryo. It has been shown previously by Garg Maloney [31] that the embryo grows nonlinearly with the radius of curvature of the indenter tip. The length of the embryo, $ξ$, and the depth of the embryo from the surface, $Y*$, exhibit nonlinear scaling with indenter radius, $R:ξ∼R1/2$ and $Y*∼R3/4$. In this work, we use these scaling laws [31] to scale the size of the meso-region, $rmeso$, with the indenter tip radius to capture the defect.

An important question we address in this work is related to the location of the center of the meso-region: how far can the meso-region move from the embryo center to detect the instability? If the instability is local, one could naively assume that if the meso-region contains the minimum $Λ$ location (i.e., the embryo center), it can predict the instability. However, our results contradict this. We determine the critical section of the embryo that the meso-region should necessarily contain to detect the dislocation embryo.

We use the results for size and location of the mesoregion to develop a systematic procedure to use mesoscale analysis for identifying the dislocation embryo in nanoindentation simulations. Since mesoscale analysis can identify instability much sooner than full scale analysis, this highlights the utility of mesoscale analysis in terms of computational time and resources for atomistic simulations. We use the mesoscale analysis in full 3D nanoindentation simulations of FCC crystals to verify the scaling laws given by Garg and Maloney [31] for the embryo size and location with indenter radius.

The rest of this paper is organized as follows. In Sec. 2, we describe the details of our simulations and give a brief kinematic description of the mechanism of homogeneous dislocation nucleation. Section 3 describes our rationale and results of the mesoscale analysis. In particular, we discuss the significance of mesoregion size by fixing the analysis at the center of the embryo, and then vary the mesoregion spatially and temporally. Section 4 describes the three-dimensional nanoindentation simulations of FCC crystals to verify the scaling laws for the embryo size and location for homogeneous dislocation nucleation. Section 5 concludes with a brief summary and discussion on implications of our results, together with an outline of our future work.

2 Modeling and Eigenanalysis

We perform athermal quasistatic nano-indentation simulations for 2D hexagonal Lennard Jones (LJ) crystals via energy minimization dynamics. The large-scale atomic/molecular massively parallel simulator (LAMMPS) molecular dynamics framework [32] is used to perform nonlinear energy minimization with the conjugate-gradient algorithm of the Polak–Ribere type, and a custom python code was developed to perform eigenmode analysis using the sparse matrix routines in SciPy. The indenter moves perpendicular to the nearest neighbor axis in hexagonal lattice for all simulations discussed in this paper. The resulting load–displacement and energy–displacement curves for the indentation process are shown in Fig. 1.

Fig. 1
Fig. 1
Close modal
We configure 2D hexagonal crystals as shown in Fig. 2 with periodic boundaries on the sides, rigid base at the bottom, and a circular indenter on top of the crystal. A large enough Lx was chosen to ensure the results are independent of its value. In the rest of the document, all lengths such as Lx, L, R, and C are measured in units of the lattice constant, a. Energy and forces are measured in LJ units. We use a stiff, featureless, harmonic, repulsive, cylindrical indenter for all our simulations as in Refs. [21,33]. The interaction potential used to model the interaction between the indenter and the atoms is of the form
$ϕ(r)=A(R−r)2ifr≤R0ifr>R$
(1)
Here, R is the radius of the indenter and r is the distance of the particle from the indenter center.
Fig. 2
Fig. 2
Close modal
The total potential energy, U(x, D), is a function of atomic positions, x, and indenter depth, D. The first derivative of energy with respect to the particle position gives the force, F, on each particle as
$Fiα=−∂U∂xiα$
(2)
Latin characters are used to index particle number, and Greek characters are used to index Cartesian components. Then, we use the Lanczos algorithm, as implemented in the SciPy toolkit [34], to compute the lowest four eigenvalues of the Hessian matrix for the relaxed configuration at each indenter step. The Hessian matrix, Hiαjβ, is the second derivative of total potential energy as follows:
$Hiαjβ=∂2U∂xiα∂xjβ$
(3)

The lowest four eigenvalues of the Hessian matrix for the relaxed configuration are calculated at each indenter step to identify the reaction coordinate. The system is driven to instability along a single eigenmode, which shows antiparallel motion of small number of atoms on adjacent crystal planes, resulting in the nucleation of a dislocation dipole as shown in Fig. 3(b). We denote the critical depth, Dc, as the depth of the indenter and the critical force, Fc, as the indenter force at which the dislocation dipole nucleates. Then, δD = DDc is the distance of the indenter from the critical depth, and δF = FFc is the remaining force of the indenter required for nucleation. We call the eigenmode corresponding to the eigenvalue vanishing as δD0.5, the critical mode. In Fig. 3(a), the critical mode is along the dotted line. We analyze the critical mode at δD ≈ 10−6, when the lowest mode is the critical mode as shown in Fig. 3(b).

Fig. 3
Fig. 3
Close modal
For a given scalar quantity, we compute Ω as in Ref. [35]. For a given vector field, Ω is the transverse derivative of the vector field along a particular crystal axis in the deformed configuration. The lattice is first triangulated, and on each triangle, Ω is computed by the linear interpolation of the critical mode vectors. Figure 4(a) shows the Ω field corresponding to the mode shown in Fig. 3(b). Ω is nonzero only at the embryo. The centroid of the triangle with maximum Ω is the center of the embryo. s is defined as the distance of each triangle from the triangle corresponding to the maximum Ω along the crystal axis of interest. Figure 4(b) shows Ω(s) profiles for L = 160 and R = 40. To define the size of the embryo, these Ω(s) profiles are fit to Gaussian functions of the form
$Ω(s)Ωmax=e−s2/ξ2$
(4)
We fit Gaussian profiles to the normalized log Ω(s) curves upto half their maximum value using a generalized least-square method and calculate ξ, the embryo size.
Fig. 4
Fig. 4
Close modal

3 Mesoscale Analysis of Incipient Dislocation

In this section, we study the critical eigenmode of the Hessian matrix of a circular mesoregion of radius, rmeso, of a cluster of atoms in the crystal. For an undeformed configuration, the lowest energy eigenvalue of an undeformed mesoregion scales as $1/rmeso2$, where rmeso is the size of the mesoregion as shown in the top panel of Fig. 5. This is expected for an isotropic, linear, elastic 2D sheet of atoms in a crystal with fixed boundaries. The critical eigenmode for an undeformed region of radius rmeso = 8 is a longwavelength mode as shown in the bottom left panel of Fig. 5.

Fig. 5
Fig. 5
Close modal

Just before nucleation, the critical eigenmode for a mesoregion containing the embryo is localized as shown in the bottom right panel of Fig. 5. The mesoregion critical eigenmode, defined as the mesomode, looks similar to the full system eigenmode shown in Fig. 3(b). The lowest eigenvalue corresponding to the critical eigenmode falls off faster than any power law as shown by the curve in the top panel of Fig. 5.

A mesoregion of radius as small as six interatomic distances can capture the defect as shown by the mesomode in Fig. 6. Above a critical radius of the meso-region centered at the center of the incipient dislocation, defined by hot atom, the structure of the meso-mode saturates. In Fig. 6, the mesomode for rmeso = 32 has the same spatial structure as the full system eigenmode. The energy of the mesoregion or the lowest eigenvalue initially decreases as the mesoregion radius increases and then plateaus similar to the spatial structure of the mesoregion. Figure 7 is a log–lin plot for the mesoregion eigenvalue versus rmeso for different R. The height of the plateau is not a function of R; it is governed by the distance from nucleation, δD, for the mesoscale analysis. On the other hand, the structure of the localized mesomode is independent of the proximity to nucleation. Therefore, the rest of this work is focused on the structure of the mesomode versus the eigenvalue associated with the mesoregion.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

3.1 Mesoregion Centered at the Hot Atom.

We compute the eigenmode corresponding to the lowest energy eigenvalue, defined as mesomode, for various rmeso with mesoregions centered at the embryo. Figure 6 shows the mesomode corresponding to rmeso = 6 and rmeso = 32 for L = 160, R = 120, and the critical portion of the eigenmode for full system (rmeso = ∞). The Ω(s) profiles corresponding to these modes is shown in Fig. 8. The width of the Ω curve for rmeso = 8 is smaller than that of the full system. The Ω curves converge when rmeso is greater than a critical value. As expected, they converge for $rmeso≳ξ$; however, mesoregions smaller than ξ definitely have information about the incipient instability.

Fig. 8
Fig. 8
Close modal

We next measure the spatial extents, ξmeso, of the mesoregion eigenmodes for all our systems. ξ is the spatial extent obtained from the full system eigenmode. Figure 9(a) shows ξmesormeso for different indenter radii. As expected from Fig. 9(a), for small rmeso, ξmeso increases before plateauing at ξ. The increase of ξmeso with rmeso for mesoregions smaller than ξ follows a power law. The insets show the collapse of these curves when rescaled by their corresponding plateau value: ξ. The collapsed rescaled curve in Fig. 9(b) is universal for all indenter-crystal geometries. The plateauing of ξmeso observed for rmeso > ξ can be reasonably expected from the fact that those mesoregions completely encompass the incipient dislocation.

Fig. 9
Fig. 9
Close modal

The power law scaling of $ξmeso$ suggests that for $rmeso$smaller than $ξ∞$, the spatial structure of the lowest meso-mode does reveal the structure of the critical mode. However, for meso-regions of radius of one or two particle spacings, the meso-mode invariably ceases to have the structure typical of an incipient dislocation. Although even small meso-regions host a lowest mode that resembles a dislocation embryo, it is only for $rmeso≥ξ∞$, that the meso-region gives a reliable characterization of the embryo size. It is in this sense that we define that nucleation is a quasi-local phenomenon. We define a quasi-local instability as one for which strictly atom-wise quantities (the atomic level virial or quantities derived from it such as the atomic level acoustic tensor) do not determine the stability of the system, but one for which the analysis of a finite region will asymptotically determine the stability of the system as the spatial extent of the analysis region grows. This scenario, where we can pick up the signature of an incipient dislocation with meso-regions smaller than its spatial extent , but not too small, indicates that homogeneous dislocation nucleation is a quasi-local phenomenon.

3.2 Mesoscale Analysis Centered Off From the Highest Ω Triangle.

In this section, we analyze the lowest eigenmode for the mesoregion, called mesomode, when the mesoregion is not centered at the embryo center. We define critical mesomode as the mode that contains the signature of an incipient dislocation. In Fig. 10, the mesomode is shown when the mesoregion is moved spatially along the slip plane. Upto a critical distance, δc, the mesomode is the critical mesomode. At δc, the long wavelength mode has lower energy than the critical mesomode, and the critical mesomode no longer remains the mesomode. The important question we address here is: How δc scales with the radius of mesoregion, rmeso, and the embryo size, ξ?

Fig. 10
Fig. 10
Close modal

We perform mesoscale analysis centered at atoms around the embryo for different embryo sizes. In Fig. 11, the size of the mesoregion is fixed, rmeso = 40. The colors at each atom represent maximum Ω for the mesomode. The mesomode is found to be the critical mesomode when the mesoregion is centered on the red atoms. The area formed by the red atoms depends on rmeso and ξ.

Fig. 11
Fig. 11
Close modal
Again, δc is the critical distance from the center of the embryo upto which the critical mesomode has a lower energy than the long wavelength mode. If we assume that the mesoregion should contain some critical portion of the embryo as shown in Fig. 12, δc can be written as in Eq. (5), where x is the critical portion of the embryo.
$x+δccosθ=(rmeso2−δc2sin2θ)0.5δc=−xcosθ+rmeso(1−x2sin2θ/rmeso2)0.5$
(5)
Assuming rmesox,
$δc=rmeso−xcosθ−(x2sin2θ)/(2rmeso)$
(6)
Along the slip plane, for θ = 0,
$δc=rmeso−x$
(7)
To compute x, we move our mesoregion along the slip plane for various rmeso and ξ as shown in Fig. 13(a). For rmesoξ, x is constant with varying rmeso and increases with an increase in ξ (or R). From Fig. 13(b), x is found to be 1.2ξ. This value of x is substituted back in Eq. (6). Then, in Fig. 11, δc based on Eq. (6) is shown by the black curves for different R (or ξ). The black curves predict the area formed by red atoms fairly well. This implies if rmeso > ξ, then it must encompass roughly 120$%$ of ξ to be the critcal mesomode.
Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

3.3 Mesoscale Analysis Away From Nucleation (Temporally).

Here, we address at what critical indenter depth, Dmeso, the meso-mode becomes localized. Fmeso is the indenter force at Dmeso, the indenter depth at which the meso-mode can detect the incipient dislocation. In other words, at what Dmeso or Fmeso does the energy of the critical meso-mode become lower than the long wavelength meso-mode. δDmeso is defined as the indenter depth difference between $Dc$ and Dmeso. In other words, higher δDmeso implies sooner prediction of incipient dislocation by the meso scale analysis. We study the dependence of δDmeso with rmeso.

Before nucleation as shown in Fig. 3a, the lowest eigenvalue, λ, of a critical meso-mode decreases with distance to dislocation nucleation, δD, as:
$λ=a1*(δD)−0.5$
(8)
In Sec. III-A, it was described that the eigenvalue corresponding to the long-wavelength meso-mode, λe, is a function of the size of the meso-region, rmeso, given by
$λe=a2*(rmeso)−2$
(9)
At critical distance, Dmeso, or critical indenter force, Fmeso, the energy of the critical meso-mode becomes equal to the long-wavelength mode energy. This gives the analytical relation between δDmeso and rmeso:
$δDmeso=Dc−Dmeso,δDmeso=a(rmeso)−4$
(10)
We compute δDmeso and Fmeso versus rmeso curves from simulations as shown in Fig. 14. The power law derived in Eq. (10) is also observed in simulations. δD versus rmeso curves collapse when rmeso is scaled by the intrinsic embryo size, $ξ$, and Fmeso versus rmeso collapse when rmeso is scaled by $ξ1.5$. This can be derived from the fact that hardness, Fmeso scaled by R, is the critical quantity equivalent to δDmeso and $ξ$ increases as a square root of R. From these curves it can be seen that as the system size approaches infinity, δDmeso goes to zero. In other words, the meso-scale analysis predicts nucleation much earlier than the full eigenmode analysis.
Fig. 14
Fig. 14
Close modal

4 Scaling Laws in Full Three-Dimensional Simulations Using Mesoscale Analysis

We perform computational nano-indentation on a FCC lattice using a spherical indenter of radius R as described in Sec. 2. At each indenter step, the Hessian matrix is computed for the FCC lattice, similar to the two-dimensional simulations. Diagonalization of the Hessian matrix for a three-dimensional system can be computationally very expensive. Some of the systems considered for this work contain million atoms approximately. We use mesoscale analysis described in Sec. 3 to compute the critical mesomode. First, we measure the embryo size, ξ, and the embryo location, Y*, for a smaller system size, L, and a smaller indenter radius, R, using full system kinematic analysis. Then, the results for the smaller R are extrapolated to calculate ξ and Y* for the required R, assuming ξR0.5 and Y* ∝ R0.5. These scaling laws for ξ and Y* were shown previously by Garg and Maloney [30] that the embryo grows in a nonlinear way with the radius of curvature of the indenter tip. The length of the embryo, ξ, and depth of the embryo from the surface, Y*, exhibit nontrivial scaling with indenter radius, R: ξR1/2 and Y* ∼ R3/4. In this section, they are shown to hold true for the FCC lattice in full three-dimensional simulations.

It was observed in Secs. 3.1 and 3.2 that mesoregions, rmeso ≈ 6, can capture the embryo. Also, it was shown in Sec. 3.3 that the mesoregion not centered at the embryo can also capture the incipient dislocation. To search for the embryo, the mesoscale analysis was performed only once in each spherical zone of a radius of six atomic units. Since the mesoregions used were much smaller than full system size (rmeso ≈ 6), and the analysis was performed at a fewer locations, locating the embryo was computationally cheaper than full-scale eigenmode analysis. Once the embryo was located, we used a mesoregion of radius rmeso ≈ 1.5ξ centered at the embryo to capture the structure of the critical eigenmode. We utilized the result obtained in Sec. 3.3 that rmesoξ is needed to capture the structure of the embryo. It was verified that increasing rmeso did not affect the results significantly.

The critical mesomode for L = 65 and R = 65 is shown in Fig. 15. At the onset of instability, there are two planes slipping with respect to each other resulting in the nucleation of a dislocation loop. The critical slip plane is (1 1 1) and the slip direction is 〈1 2 1〉. As shown, the embryo structure located on (1 1 1) is hexagonal. We triangulate the (1 1 1) plane containing the embryo using Delaunay triangulation. Each triangle and the vertex atom on the adjacent plane form a tetrahedron. The critical mesomode is interpolated on these tetrahedrons using linear finite element shape functions. Using this interpolation, Ω is defined as the curl of the critical mesomode resolved in the direction perpendicular to the slip plane, as done in Sec. 3.2 for the two-dimensional simulations.

Fig. 15
Fig. 15
Close modal

We call the three diagonal axes of the hexagonal embryo as n1, n2, and n3, as shown in Fig. 16. Ω versus s is plotted in Fig. 16 to obtain the embryo size (as done in Sec. 3.2). Interestingly, the Ω versus s curves along the three diagonal axes collapse, implying that the embryo is regular hexagon. The collapse of these curves was observed for all indenter radii. The embryo size, Ω, can be calculated from these Ω versus s curves by fitting Gaussian profiles as done in Sec. 3.2.

Fig. 16
Fig. 16
Close modal
We observe the following scaling laws for the embryo location, Y*, and the embryo size, ξ (Fig. 17).
$Y*∼∝R0.75orY*R∼∝R−0.25$
(11)
$ξ∼∝R0.5orξR∼∝R−0.5$
(12)
These scaling laws are consistent with two-dimensional simulations [30]. These scaling laws were shown to be independent of interatomic potentials and crystal orientation for two-dimensional hexagonal crystals [30]. In the previous work [30], it was shown that the hardness, $Fc/R$ approaches well-defined asymptotic values for each system in the large R limit, but there was a systematic indentation size effect observed where $Fc/R$ increases at small R. $ξ/R0.5$ also exhibited the same type of indentation size effect as $Fc/R$ where $ξ/R0.5$ increases at small R for all systems. Similar indentation size effect is also observed in the three-dimensional simulations at small R.
Fig. 17
Fig. 17
Close modal

5 Discussion and Summary

In this work, we studied the nucleation of dislocation dipoles in the bulk of perfect 2D and 3D crystals subjected to nanoindentation with a circular (and spherical in 3D), atomistic indenter under athermal, quasistatic conditions. We performed mesoscale analysis of configurations at the stability threshold, showing that small, nonlocal regions of the crystal, centered at the embryo, contain significant information about an incipient nucleation event. However, unlike previous work that utilized the minimum eigenvalue of the mesoregions as the main analysis tool [2123], we focused our attention on the spatial structure of the lowest mesoregion eigenmode. We found that the relation between ξmeso, the embryonic size inferred from the mesoregion, and rmeso, the size of the mesoregion itself, is universal. The lowest mesomode and eigenvalue were found to provide excellent estimates of the structure and energy of the true critical mode but only for mesoregions larger than rmeso > 1.5ξ. We also showed that the mesoregions not centered at the embryo center also reveal the presence of embryo, provided they encompass 1.2ξ. This scenario leads us to think of homogeneous dislocation nucleation as quasilocal: full information about the nature of the embryo can only be obtained by analyzing sufficiently large regions; however, its existence can be inferred by examining regions much much smaller than its intrinsic size.

We have also understood the effects of the proximity to nucleation, δD, on mesoscale analysis. Mesoscale analysis detects the embryo presence much before that the full system kinematic analysis. Mesoscale analysis can be used to make atomistic simulations computationally efficient, especially for 3D simulations where full system kinematic analysis could be challenging or, in some-cases impossible. We showed an initial study for full 3D FCC crystal nano-indentation simulations using the mesoscale analysis. The critical mesomode was used to calculate the embryo size, ξ, and the embryo location, Y*. ξ and Y* vary as the square-root of indenter radius for 3D FCC lattice. These scaling laws are consistent with the two-dimensional hexagonal lattice simulations for various interatomic potentials and crystal orientations. There are two main points to make regarding the overall utility of our results. As a practical consideration, it is dramatically more numerically expedient to employ a meso-scale eigenmode analysis as opposed to a global one. Miller and Rodney first proposed this approach, but understanding its limits is therefore important. Second, we address a fundamental question about the nature of the critical mode and how it evolves as one approaches the limit of stability. We find it likely that, in the future, when attempts to derive a theory for, e.g., activation rates for thermally assisted nucleation in a crystal in a state of inhomogeneous strain, the meso-scale properties we study here and the associated notion of locality/non-locality should be important.

The ultimate use of the analysis presented here would be to inform coarser-grained models, which do not explicitly take into account the atomic degrees of freedom, about the creation of new dislocations out of the void. For example, in field dislocation mechanics [36,37], one introduces a continuous field to represent the dislocation density. It is our hope that a criterion for dislocation nucleation based on a mesoscale analysis like we presented here could serve as a guide for the introduction of atomistic details at the dislocation embryo in concurrent multiscale schemes built on field theories like field dislocation mechanics.

Acknowledgment

We thank Amit Acharya for useful discussions at various stages during this work. This material is based upon the work supported by the National Science Foundation (Award Number CMMI-1100245).

References

1.
Foss
,
C. A.
,
Hornyak
,
G. L.
,
Stockert
,
J. A.
, and
Martin
,
C. R.
,
1994
, “
Template-Synthesized Nanoscopic Gold Particles: Optical Spectra and the Effects of Particle Size and Shape
,”
J. Phys. Chem.
,
98
(
11
), pp.
2963
2971
. 10.1021/j100062a037
2.
Whitney
,
T. M.
,
Jiang
,
J. S.
,
Searson
,
P. C.
, and
Chien
,
C. L.
,
1993
, “
Fabrication and Magnetic Properties of Arrays of Metallic Nanowires
,”
Science
261
(
5126
), pp.
1316
1319
. 10.1126/science.261.5126.1316
3.
Wu
,
Y.
,
Xiang
,
J.
,
Yang
,
C.
,
Lu
,
W.
, and
Lieber
,
C. M.
,
2004
, “
Single-Crystal Metallic Nanowires and Metal/Semiconductor Nanowire Heterostructures
,”
Nature
,
430
, pp.
61
65
. 10.1038/nature02674
4.
Choi
,
J. H.
, and
Korach
,
C. S.
,
2009
, “
Nanoscale Defect Generation in CMP of Low-k/Copper Interconnect Patterns
,”
J. Electrochem. Soc.
,
156
(
12
), pp.
H961
H970
. 10.1149/1.3243852
5.
Srivastava
,
G.
, and
Higgs
,
C. F.
, III
,
2015
, “
An Industrial-Scale, Multi-Wafer CMP Simulation Using the PAML Modeling Approach
”,
ECS J. Solid State Sci. Technol.
,
4
(
11
), pp.
P5088
P5096
. 10.1149/2.0141511jss
6.
Srivastava
,
G.
, and
Higgs
,
C. F.
, III
,
2015
, “
A Full Wafer-Scale PAML Modeling Approach for Predicting CMP
,”
Tribol. Lett.
,
59
(
2
), p.
32
. 10.1007/s11249-015-0553-y
7.
Zhang
,
J.
,
Zhang
,
J. Y.
,
Liu
,
G.
,
Zhao
,
Y.
, and
Sun
,
J.
,
2009
, “
Competition Between Dislocation Nucleation and Void Formation as the Stress Relaxation Mechanism in Passivated Cu Interconnects
,”
Thin Solid Films
,
517
(
9
), pp.
2936
2940
. 10.1016/j.tsf.2008.12.031
8.
Lee
,
J. H.
, and
Gao
,
Y.
,
2011
, “
Mixed-Mode Singularity and Temperature Effects on Dislocation Nucleation in Strained Interconnects
,”
Int. J. Solids Struct.
,
48
(
7–8
), pp.
1180
1190
. 10.1016/j.ijsolstr.2011.01.001
9.
Gerberich
,
W. W.
,
Nelson
,
J. C.
,
Lilleodden
,
E. T.
,
Anderson
,
P.
, and
Wyrobek
,
J. T.
,
1996
, “
Indentation Induced Dislocation Nucleation: The Initial Yield Point
,”
Acta Mater.
,
44
(
9
), pp.
3585
3598
. 10.1016/1359-6454(96)00010-9
10.
Corcoran
,
S. G.
,
Colton
,
R. J.
,
Lilleodden
,
E. T.
, and
Gerberich
,
W. W.
,
1997
, “
Anomalous Plastic Deformation at Surfaces: Nanoindentation of Gold Single Crystals
,”
Phys. Rev. B
,
55
(
24
), p.
R16057
. 10.1103/PhysRevB.55.R16057
11.
Kelchner
,
C. L.
,
Plimpton
,
S. J.
, and
Hamilton
,
J. C.
,
1998
, “
Dislocation Nucleation and Defect Structure During Surface Indentation
,”
Phys. Rev. B
,
58
(
17
), p.
11085
. 10.1103/PhysRevB.58.11085
12.
Tadmor
,
E. B.
,
Miller
,
R.
,
Phillips
,
R.
, and
Ortiz
,
M.
,
1999
, “
Nanoindentation and Incipient Plasticity
,”
J. Mater. Res.
,
14
(
6
), pp.
2233
2250
. 10.1557/JMR.1999.0300
13.
Zimmerman
,
J. A.
,
Kelchner
,
C. L.
,
Klein
,
P. A.
,
Hamilton
,
J. C.
, and
Foiles
,
S. M.
,
2001
, “
Surface Step Effects on Nanoindentation
,”
Phys. Rev. Lett.
,
87
(
16
), p.
R16057
. 10.1103/physrevlett.87.165507
14.
Lilleodden
,
E. T.
,
Zimmerman
,
J. A.
,
Foiles
,
S. M.
, and
Nix
,
W. D.
,
2003
, “
Atomistic Simulations of Elastic Deformation and Dislocation Nucleation During Nanoindentation
,”
J. Mech. Phys. Solids
,
51
(
5
), pp.
901
920
. 10.1016/S0022-5096(02)00119-9
15.
Van Vliet
,
K. J.
,
Li
,
J.
,
Zhu
,
T.
,
Yip
,
S.
, and
Suresh
,
S.
,
2003
, “
Quantifying the Early Stages of Plasticity Through Nanoscale Experiments and Simulations
,”
Phys. Rev. B
,
67
(
10
), p.
104105
. 10.1103/PhysRevB.67.104105
16.
Mason
,
J. K.
,
Lund
,
A. C.
, and
Schuh
,
C. A.
,
2006
, “
Determining the Activation Energy and Volume for the Onset of Plasticity During Nanoindentation
,”
Phys. Rev. B
,
73
(
5
), p.
054102
. 10.1103/PhysRevB.73.054102
17.
Schall
,
P.
,
Cohen
,
I.
,
Weitz
,
D. A.
, and
Spaepen
,
F.
,
2006
, “
Visualizing Dislocation Nucleation by Indenting Colloidal Crystals
,”
Nature
,
440
(
7082
), pp.
319
323
. 10.1038/nature04557
18.
Wagner
,
R. J.
,
Ma
,
L.
,
Tavazza
,
F.
, and
Levine
,
L. E.
,
2008
, “
Dislocation Nucleation During Nanoindentation of Aluminum
,”
J. Appl. Phys.
,
104
(
11
), p.
114311
. 10.1063/1.3021305
19.
Morris
,
J. R.
,
Bei
,
H.
,
Pharr
,
G. M.
, and
George
,
E. P.
,
2011
, “
Size Effects and Stochastic Behavior of Nanoindentation Pop In
,”
Phys. Rev. Lett.
,
106
(
16
), p.
165502
. 10.1103/PhysRevLett.106.165502
20.
Agrawal
,
V.
, and
Dayal
,
K.
,
2015
, “
A Dynamic Phase-Field Model for Structural Transformations and Twinning: Regularized Interfaces With Transparent Prescription of Complex Kinetics and Nucleation. Part I: Formulation and One-Dimensional Characterization
,”
J. Mech. Phys. Solids
,
85
, pp.
270
290
. 10.1016/j.jmps.2015.04.010
21.
Miller
,
R.
, and
Rodney
,
D.
,
2008
, “
On the Nonlocal Nature of Dislocation Nucleation During Nanoindentation
,”
J. Mech. Phys. Solids
,
56
(
4
), pp.
1203
1223
. 10.1016/j.jmps.2007.10.005
22.
Delph
,
T. J.
,
Zimmerman
,
J. A.
,
Rickman
,
J. M.
, and
Kunz
,
J. M.
,
2009
, “
A Local Instability Criterion for Solid-State Defects
,”
J. Mech. Phys. Solids
,
57
(
1
), pp.
67
75
. 10.1016/j.jmps.2008.10.005
23.
Delph
,
T. J.
, and
Zimmerman
,
J. A.
,
2010
, “
Prediction of Instabilities at the Atomic Scale
,”
Modell. Simul. Mater. Sci. Eng.
,
18
(
4
), p.
045008
. 10.1088/0965-0393/18/4/045008
24.
Garg
,
A.
,
Acharya
,
A.
, and
Maloney
,
C.
,
2014
, “
A Study of Conditions for Dislocation Nucleation in Coarser-Than-Atomistic Scale Models
,”
J. Mech. Phys. Solids
,
75
, pp.
76
92
. 10.1016/j.jmps.2014.11.001
25.
Hill
,
R.
,
1962
, “
Acceleration Waves in Solids
,”
J. Mech. Phys. Solids
,
10
(
1
), pp.
1
16
. 10.1016/0022-5096(62)90024-8
26.
Gouldstone
,
A.
,
Van Vliet
,
K. J.
, and
Suresh
,
S.
,
2001
, “
Nanoindentation: Simulation of Defect Nucleation in a Crystal
,”
Nature
,
411
(
6838
), pp.
656
656
. 10.1038/35079687
27.
Li
,
J.
,
Van Vliet
,
K. J.
,
Zhu
,
T.
,
Yip
,
S.
, and
Suresh
,
S.
,
2002
, “
Atomistic Mechanisms Governing Elastic Limit and Incipient Plasticity in Crystals
,”
Nature
,
418
(
6895
), pp.
307
310
. 10.1038/nature00865
28.
Van Vliet
,
K. J.
,
Li
,
J.
,
Zhu
,
T.
,
Yip
,
S.
, and
Suresh
,
S.
,
2003
, “
Quantifying the Early Stages of Plasticity Through Nanoscale Experiments and Simulations
,”
Phys. Rev. B
,
67
(
10
), p.
104105
. 10.1103/PhysRevB.67.104105
29.
Garg
,
A.
,
2014
, “
Homogeneous Dislocation Nucleation
,” Ph.D. dissertation,
Carnegie Mellon University
,
Pittsburgh, PA
,
Paper 401
.
30.
Garg
,
A.
, and
Maloney
,
C. E.
,
2016
, “
Mechanical Instabilities in Perfect Crystals: From Dislocation Nucleation to Buckling Like Modes
,”
ASME J. Appl. Mech.
,
83
(
12
), p.
121006
. 10.1115/1.4034564
31.
Garg
,
A.
, and
Maloney
,
C.
,
2016
, “
Universal Scaling Laws for Homogeneous Dissociation Nucleation During Nano-Indentation
,”
J. Mech. Phys. Solids
,
95
, pp.
742
754
. 10.1016/j.jmps.2016.04.026
32.
Plimpton
,
S.
,
1995
, “
Fast Parallel Algorithms for Short-Range Molecular Dynamics
,”
J. Comput. Phys.
,
117
(
1
), pp.
1
19
. 10.1006/jcph.1995.1039
33.
Miller
,
R.
,
2004
, “
A Stress-Gradient Based Criterion for Dislocation Nucleation in Crystals
,”
J. Mech. Phys. Solids
,
52
(
7
), pp.
1507
1525
. 10.1016/j.jmps.2004.01.007
34.
Jones
,
E.
,
Oliphant
,
T.
, and
Peterson
,
P.
,
2001
, “
SciPy: Open Source Scientific Tools for Python
,” Online, Accessed August 28, 2014.
35.
Hasan
,
A.
, and
Maloney
,
C. E.
,
2012
, “
Mesoscale Harmonic Analysis of Homogenous Dislocation Nucleation
,” .
36.
Acharya
,
A.
,
2001
, “
A Model of Crystal Plasticity Based on the Theory of Continuously Distributed Dislocations
,”
J. Mech. Phys. Solids
,
49
(
4
), pp.
761
784
. 10.1016/S0022-5096(00)00060-0
37.
Acharya
,
A.
,
2010
, “
New Inroads in an Old Subject: Plasticity, From Around the Atomic to the Macroscopic Scale
,”
J. Mech. Phys. Solids
,
58
(
5
), pp.
766
778
. 10.1016/j.jmps.2010.02.001