Engineering size-scaling of plastic deformation in nanoscale asperities

Abstract

Size-dependent plastic flow behavior is manifested in nanoindentation, microbending, and pillar-compression experiments and plays a key role in the contact mechanics and friction of rough surfaces. Recent experiments using a hard flat plate to compress single-crystal Au nano-pyramids and others using a Berkovich indenter to indent flat thin films show size scaling into the 100-nm range where existing mechanistic models are not expected to apply. To bridge the gap between single-dislocation nucleation at the 1-nm scale and dislocation-ensemble plasticity at the 1-μm scale, we use large-scale molecular dynamics (MD) simulations to predict the magnitude and scaling of hardness H versus contact size ℓ_c in nano-pyramids. Two major results emerge: a regime of near-power-law size scaling H ≈ ℓ_c^−η exists, with η_MD ≈ 0.32 compared with η_expt ≈ 0.75, and unprecedented quantitative and qualitative agreement between MD and experiments is achieved, with H_MD ≈ 4 GPa at ℓ_c = 36 nm and H_expt ≈ 2.5 GPa at ℓ_c = 100 nm. An analytic model, incorporating the energy costs of forming the geometrically necessary dislocation structures that accommodate the deformation, is developed and captures the unique magnitude and size scaling of the hardness at larger MD sizes and up to experimental scales while rationalizing the transition in scaling between MD and experimental scales. The model suggests that dislocation–dislocation interactions dominate at larger scales, whereas the behavior at the smallest MD scales is controlled by nucleation over energy barriers. These results provide a basic framework for understanding and predicting size-dependent plasticity in nanoscale asperities under contact conditions in realistic engineered surfaces.

It is now well established that plastic flow behavior in metallic materials is size dependent, with flow stress (1, 2) or hardness (3) increasing with decreasing volume of material under load. Experimental studies of nanoindentation (4–6) and wire and thin-film bending (7–10) and tension (11–17) clearly elucidate the size effects at micrometer scales. The rational for size-dependent plasticity has been attributed to the need for “geometrically necessary” defect structures, typically dislocations, which accommodate the rapid strain gradients that must exist in many small-scale deformation conditions and the influence of these dislocations in hardening the material. Phenomenological strain-gradient-plasticity theories (18–21) and discrete dislocation models (22) both qualitatively exhibit the observed size effects under varying conditions. However, the conceptual bases of these models typically rely on interactions between preexisting, statistically generated, and geometrically necessary dislocations that are not thought to be applicable at nanometer scales where relatively few dislocations exist. Atomistic studies using molecular dynamics (MD) at the nanometer scale can exhibit size effects (23–26), but they are usually associated with nucleation of a single dislocation at stress concentrators in initially dislocation-free single-crystal materials, with predicted stresses much larger than those observed in experiments. Recent indentation experiments at the nanometer scale do show pressures at the onset of plasticity comparable with those found at the point of dislocation nucleation in MD studies, but neither the experiments nor the simulations are extensive enough to elucidate any size scaling (5).

Plastic flow in the range 5–500 nm remains largely unexplored. Some of the present authors have recently performed compression experiments on single-crystal Au pyramids having (114) facets and a rounded top surface with a (100) orientation and with contact surface lengths extending down to 100 nm, where Miller index notation is used to define crystallographic planes and directions with parentheses and brackets, respectively. The hardness (contact pressure) versus contact edge length ℓ_c over the range ℓ_c ≈100–600 nm shows a scaling H ≈ ℓ_c^−0.75 independent of the starting contact edge length and with hardness reaching 2.5 GPa at the smallest scales (27). Recent indentation experiments on flat single crystal Au films with the same [100] orientation and over a similar range of contact areas show a similar scaling H ≈ ℓ_c^−0.5 and magnitude (5) comparable with the data in ref. 27. There are no mechanisms, models, or simulations that make any connections with the experiments in this size range.

An important application of size-dependent plastic response is in the contact between surfaces of deformable materials. Most engineered materials are rough at multiple scales, culminating at the smallest scale with asperities that can be on the order of nanometers in height and/or lateral dimensions (28). The macroscopic behavior of rough surfaces under normal loads and under sliding conditions is determined by the spectrum of local asperity contacts and their deformation, which is thus directly related to the size-dependent plastic response of the asperities. Although some work on nanoscale asperities has been done (27, 29–32), connections between experiments and models have remained elusive.

Here, we bridge the gap in size-dependent plasticity between nano- and micrometer-scale models and experiments through large-scale MD simulations of the compression of Au nano-pyramids identical to those studied in ref. 27 and through a mechanistically based predictive model. Our simulations over contact lengths from 4 to 40 nm show a size scaling of H ≈ ℓ_c^−0.32 and independent of the initial contact size, as seen experimentally, and attain nearly quantitative agreement with experimental hardness at the largest scale. Furthermore, because of the ideal structure of this system, we can compute the number of partial dislocations necessary to accommodate the majority of the plastic deformation and develop a mechanistically based model to predict the hardness versus length scale in terms of fundamental material properties. The model can show quantitative agreement with the larger scale MD data and with the experimental data, rationalizing the transition in scaling with contact edge length and suggesting that the hardness is dominated by dislocation–dislocation interactions in this regime. At very small contact lengths, the model suggests that the MD results are controlled by energy barriers to dislocation nucleation. In total, our simulations, analytic model, and prior experiments provide a comprehensive picture for the size scaling of the deformation in nanoscale asperities.

Simulation Model and Results

We model a fully 3-dimensional Au pyramid asperity having 4 (114) side facets and truncated near the peak to form a square top surface with [001] surface normal, as shown schematically in Fig. 1 with an Inset depicting the maximum resolved shear stress (MRS) during initial contact. The important pyramid dimension is the initial contact length along the top surface, ℓ_ci, whereas the height h plays little role if large enough. Pyramids tested here had sizes (ℓ_ci = 1.7 nm, h = 17 nm), (ℓ_ci = 1.7 nm, h = 40 nm), (ℓ_ci = 4.6 nm, h = 40 nm), and (ℓ_ci = 8.6 nm, h = 60 nm). The total number of atoms in these samples ranges from 2,877,990 to 50,506,734.

Fig. 1.

Schematic of initial pyramid structures. (Inset) y–z plane cross-section of the atomic structure near the contact surface for the smallest contact area, with contours of the maximum resolved shear stress (MRS) on slip planes just before dislocation nucleation.

The pyramid is compressed by a flat structureless indenter (see Methods) at a rate of 0.1 Å/ps = 10 m/s, 2 orders of magnitude smaller than the speed of sound and allowing ample time for near-quasistatic deformation. We measure the total force f_T during indentation as the sum of the forces acting between the indenter and all of the Au atoms. The contact area A is determined by using the projected area of all atoms interacting with the indenter. The hardness is then H = f_T/A. The precise contact area is important when determining hardness, but when making comparisons to the experiments and to explain scaling effects, a more useful measure is the contact edge length ℓ_c. Because the contact area is nearly square at all times, ℓ_c = $\sqrt{A}$ is accurate.

Fig. 2 shows the hardness (contact pressure) vs. contact edge length ℓ_c, in log–log form, for the 4 different pyramid sizes at T = 1 K as well as the (ℓ_ci = 1.7 nm, h = 17 nm) pyramid at T = 300 K and also at T = 1 K by using an interatomic potential with a higher stacking fault energy. Each simulation shows an elastic region of rapidly increasing pressure with no increase in contact length, a first plastic event (denoted by the Xs in Fig. 2), further hardening and then softening behavior at larger contact lengths during which more extensive “plastic flow” occurs. The data demonstrate that the deformation is independent of (i) temperature from 1 K to 300 K, not surprising for Au (33–34), (ii) the pyramid height h (up to loads at which dislocations approach the bottom boundary), and (iii) the precise details of the interatomic potential used. The simulation results are thus robust.

Fig. 2.

Hardness H versus deformed contact edge length ℓ_c for MD simulations and experiments. Simulations with initial contact length ℓ_ci = 1.7 nm were run for T = 1 K, h = 17 nm a; T = 300 K, h = 17 nm b; T = 1 K, h = 40 nm c; and T = 1 K, h = 17 nm d by using a different potential with a higher stacking fault energy. The experiments were performed at room temperature.

The most important feature of the material response in Fig. 2 is that all of the simulations converge to a common magnitude and an apparent power-law size scaling with increasing contact length. Thus, the hardness of the material depends, after a brief initial transient, only on the current contact area or edge length and is independent of the prior history of deformation; i.e., there is a unique size scaling of the hardness. This is the first main result of this study.

To make comparisons with the experiments, Fig. 2 also shows the hardness versus contact edge length of the experimental data. Experimental contact areas and edge lengths were approximated by using a local stiffness technique and assuming perfect plasticity as outlined by Wang (27) (see Methods). The experimental data on hardness H shows the same trend as the MD results: an initial transient rise, during which there is elastic loading and some flattening of the as-fabricated rounded pyramid top that is presumably accompanied by some initial dislocation nucleation, followed by softening with a power-law scaling. The gap in size scale between simulations and experiments is less than a factor of 3: The simulations extend to 37 nm, and the experiments, although starting as small as 36 nm, show power-law softening beginning at ≈100 nm. Not only is the softening trend similar between the simulations and experiments, but the absolute magnitude of the pressure differs by less than a factor of 2 between the end of the simulations and the beginning of the experiments, with the datasets able to be connected by a single smooth curve. This level of agreement between MD and real experiments is unprecedented and is due to 3 factors: (i) the exceptional uniformity of the real fabricated pyramids with regard to facet structure, single-crystal nature and orientation, and absence of preexisting dislocations, all of which are reproduced in the MD model; (ii) the ability to perform normal compression tests on these samples; and (iii) the ability to perform large-size MD simulations with extensive amounts of deformation. This is the second main result of this study.

Comparing the simulations and experiments in more detail, both show power-law scaling, but with different exponents. The experimental hardness scales with a range of H ≈ ℓ_c^−0.55 to H ≈ ℓ_c^−0.91 with an average of H ≈ ℓ_c^−0.35, which is somewhat softer than the MD scaling of H ≈ ℓ_c^−0.32. Power-law scaling is not expected a priori, although a variety of power-law behaviors have been reported in experiments (14–17). A single (non-power-law) smooth curve representing the entire set of experimental and simulation data suggests the need for competing mechanisms and a transition in the dominant mechanism with varying pyramid size. We thus next discuss the deformation mechanisms observed in the MD, which lead us to a model for the size-scaling of the hardness.

Deformation Mechanisms

The first onset of plasticity is always the formation of a stacking fault half-octahedral pyramid (simply abbreviated as s.f.h.o.) created by the nucleation of a loop of four (a/6)[112]-type partial dislocations along the four (111) planes intersecting the 4 contact edges of the specimen (Fig. 3A). Similar structures were recently reported in much smaller atomistic simulations (29–33). The s.f.h.o. consists of four (111) stacking faults that intersect along [110] directions to form junctions with (a/6)[100] Burgers vectors. Thus, each face of the s.f.h.o. consists of a stacking fault in the shape of an equilateral triangle. With increasing pressure and indentation depth, a second s.f.h.o. is nucleated from the now-larger contact edge with the same structure and Burgers vectors.

Fig. 3.

Views of the atomic structure after various depths of compression after the first s.f.h.o. (A); showing early stages of dislocation nucleation from s.f.h.o. junctions (B); after formation of the third s.f.h.o., showing formation of 1 full dislocation loop (C); and deformation at a late stage of compression, as viewed from underneath the pyramid (D). Colors are assigned based on the Centro-Symmetry parameter (35): red, surface atoms; yellow, stacking fault atoms; green, partial dislocation core atoms.

Under further compression, s.f.h.o. nucleation continues, whereas the [100] junctions can act as sources for nucleation of new dislocations along the same slip planes as the faces of the s.f.h.o., and these dislocations spread symmetrically into the body of the pyramid (Fig. 3B). Once expansion begins, there are many complicated interactions between the dislocation structures. The nucleated dislocations and new nucleating s.f.h.o.s can inhibit each other as the deformation proceeds, resulting in incomplete s.f.h.o.s or shrinking of the dislocation loops. Once a new s.f.h.o. begins to form, the previously nucleated s.f.h.o.s are able to rotate and form semitwins such that the previous s.f.h.o. faces opposite to the twin return to a perfect face centered cubic (FCC) orientation. Dislocations nucleated from the s.f.h.o.s can ultimately reach the facet surfaces and cause upward steps on the facets (Fig. 3C), showing that these nucleated dislocations have Burgers vectors differing from those of the s.f.h.o.s. Fig. 3C shows that prior nucleated dislocations (e.g., Fig. 3B) have been suppressed upon formation of the third s.f.h.o., and new dislocations are formed from the new junctions, including 1 full dislocation. The dislocation activity begins to take on 4-fold symmetry as specific (111) planes begin to dominate the slip. Fig. 3D shows the deformation of the atoms with a high Centro-Symmetry parameter (35), i.e., atoms not in a perfect FCC structure, as seen from the bottom of the pyramid once the 4-fold symmetry begins to develop. Several full dislocations are also evident at this stage of the deformation. The Fig. 1 Inset shows that the spatial range of the stresses in the pyramid is quite limited; because of geometric similitude and elasticity, the spatial range of the fields in the Inset scales with the contact size. Hence, there are no driving forces for dislocations to move far from the top surface, and so dislocations move short distances in a fairly symmetric fashion, setting a limit on the length of dislocations present relative to the contact edge length.

Predicting Size-Dependent Hardness

Although the MD observations and quantitative comparisons with experiment are interesting and unprecedented, it is also important to gain insight into the origin of the hardness and its size scaling. Because nanoscale plasticity occurs via the creation, motion, and interaction of limited numbers of dislocations, appropriate models should start at the dislocation scale and move upward as needed. Continuum plasticity concepts based on averages over volumes of deformation are not immediately useful in the current context, particularly with regard to predicting size scaling.

We note that the size scaling of the first s.f.h.o. is ≈ℓ_c^−0.5 (points marked by X in Fig. 2), consistent with predictions of Peierls–Nabarro- and Rice–Thompson-type nucleation models (36, 37) driven by the high local stresses near the contact edges evident in Fig. 1 Inset. However, subsequent s.f.h.o.s require higher pressures initially, suggesting the importance of dislocation interactions, followed by a unique softening curve. We are primarily interested in the larger-size (>10 nm) scaling behavior, and so we pursue an energetic model to provide a lower bound estimate for the pressure P required to nucleate successive s.f.h.o.s. The first s.f.h.o. formation, likely requiring excess energy to overcome a short-range energy barrier, cannot be predicted by this energy model. Our approach bears some similarities to recent work on deformation in nanospheres (38, 39).

We consider the energy inequality that must prevail in the deformed pyramid upon injection of a new s.f.h.o. into a preexisting deformed structure, a process shown schematically in Fig. 4 under constant force loading. The inequality compares the work fΔu done by the indenter to the changes in the stored elastic energy ΔU_e, the dissipation of energy from dislocation motion ΔU_τ, the dislocation self-energies ΔU_self, the interaction energies with previous dislocations ΔU_dis, the stacking fault energy of the s.f.h.o. ΔU_sf, and the loss of atomic ledge length upon injection of an s.f.h.o., ΔU_L. For a constant force during the s.f.h.o. formation (constant displacement is similar), the energy inequality is given by

Equality in Eq. 1 provides a lower bound for the applied force necessary to create a new s.f.h.o. The contact pressure is then P⁺ = f/ℓ_c² before formation and P⁻ = f/(ℓ_c + ΔL)² after s.f.h.o. formation, where ΔL is determined by the pyramid slope. The dependence of various energies on the contact edge length ℓ_c then determines the size scaling.

Fig. 4.

Schematic showing the changes in geometry during the nucleation of a new s.f.h.o. under constant force loading. Red pyramids show prior s.f.h.o.s; green pyramid indicates the new nucleated s.f.h.o.

The elastic energy change becomes small at larger sizes, and the stacking fault and edge length energies are negligible in the range 10–500 nm. Thus, the pressure required to create the s.f.h.o. can be approximately written as (see Methods)

where b_p is the Burger's vector of the partial dislocation and C_θ is a fitting parameter associated with the dislocation dissipation. The scaling of the pressure thus depends on the scaling of the self- and interaction energies. Continuum estimates for these energies can be made following ref. 40. To estimate the interaction energies between s.f.h.o.s, we consider here only the interaction between each (a/3)[100] dislocation line in the new s.f.h.o. and its n nearest parallel (a/3)[100] dislocations in the previous s.f.h.o.s. Our final result is (see Methods)

where we use an average value for ΔL, which actually alternates between 0.87 and 1.44 nm in the (114) pyramid. The interaction energies scale with μb²/π(1 − ν), which is only an isotropic linear elastic approximation, so we have introduced an overall adjustable parameter C_w to scale the energies. We limit the number of interacting dislocations to a maximum of N to account for annihilation that eventually eliminates previously nucleated dislocations. Inserting Eq. 3 into Eq. 2, the predicted pressure then depends on C_w/(1 − C_θ) and N.

Fig. 5 shows the predictions of the peak pressure P⁺ versus contact edge length for C_w/(1 − θ_w) = 0.87, N = 5, and ℓ_ci = 1.7, 4.6, 8.6, 60, and 100 nm along with the MD results and experiments. The model predictions are sensitive to the values of the parameters, but the values here are physically reasonable: C_w/(1 − C_θ) = 0.87 corresponds to small dissipation and a minor correction to the continuum elastic dislocation energies. Although N = 5 is consistent with the MD, which shows some annihilation of inner s.f.h.o.s., the pressure for the first dislocation creation is low in all cases and is also low for very small initial contacts, because these regimes are controlled by nucleation and dissipative energy is relatively large. Otherwise, the model predicts the trends in the MD and experiments, showing an increasing pressure at early stages, because of the buildup of dislocation interaction energies, followed by a transition to a unique softening curve as larger dislocation structures become easier to create. In detail, the model captures the transition from an apparent power law ≈ℓ_c^−0.3 in the MD size range to ≈ℓ_c^{−(0.5–0.7)} in the experimental range. Predictions for large initial contact sizes comparable to those in the experiments also show good agreement with the experiments, with all results converging onto a single curve. This is the third main result of this study.

Fig. 5.

Hardness vs. contact edge length for nanopyramids, as predicted by the energy-based model of Eq. 2 for ℓ_ci = 1.7, 4.6, 8.6, 60, and 100 nm and as obtained by MD and pyramid-compression experiments (Fig. 1). Also shown is Berkovich indentation data on (100) Au films 1,000 nm in thickness (A) and 2,500 nm in thickness (B).

The origin of the universal scaling of the softening can be extracted from the model with some analysis. A unique softening curve emerges after the interactions have saturated, i.e., after n ≥ N s.f.h.o.s have been nucleated. Denoting the contact size at n = N as ℓ_N = ℓ_ci + NΔL, the dominant behavior is given by

where B and D are constants containing the remaining dislocation energies (see Methods). Because 2N/D is sufficiently large (2N/D ≈ 1.25 for ℓ_ci = 10 nm) the pressure decreases much more slowly than ℓ_ci⁻¹ over a wide range of contact lengths (ℓ_c > 100ℓ_N), giving rise to an “apparent” power-law scaling that varies slowly in the 10- to 500-nm size range. Although the quantitative predictions of the model depend on a few fitting parameters, the insights from the MD and experiments guide reasonable choices.

For sizes >500 nm, the stacking fault energy is no longer negligible and becomes the dominate energy (see Eq. 6 in Methods). This sets lower limits to the hardness of 0.075GPa and 0.52 GPa for the 2 different Au potentials used. Furthermore, beyond ≈500 nm, the mechanisms responsible for hardening may change, including interactions with statistically stored dislocations (4).

Finally, compression of (114) pyramids and Berkovich nanoindentation of a flat surface should be quite similar, because the included angle of the Berkovich indenter (65.3°) is nearly the same as that of this particular pyramid (70.7°). This is borne out in Fig. 5, where Berkovich data for indentation of (001) Au films with thicknesses of 1,000 and 2,500 nm is shown in the range of 100-nm contact edge length (5), with H ≈ ℓ_c^−0.5 and H ≈2.5 GPa at the smallest contact size of ℓ_c ≈60 nm. Here, the contact edge lengths are calculated as the diameter of the projected area of the Berkovich indenter with a flat 3-nm tip offset to account for a 48-nm radius roundness. The substrate has little effect on the scaling of hardness at these depths of indentation and film thickness (41). Although the magnitudes differ by a factor of 2, the scaling of the hardness, attributed to similar dislocation interaction mechanisms, agrees well with the pyramid data and is consistent with our model predictions.

Summary

Large-scale MD has revealed key features of the size-dependent plastic deformation of nanoscale asperities under contact compression. We find a near-power-law size scaling of the hardness that depends solely on the current contact area, in agreement with experiments. With simulation and experimental sizes approaching one another, unprecedented quantitative agreement is found.

An analytic model that incorporates the energy cost of nucleating the s.f.h.o. dislocation structures observed in the MD predicts the magnitude and scaling of the hardness and bridges MD and experimental scales. The model predictions suggest that dislocation interactions are responsible for the scaling of the hardness in the nanoscale regime. This is conceptually consistent with the Nix–Gao scaling due to excess Burgers vector generated under local deformation conditions (4) but extended into a regime where precise dislocation structure and interactions may be more important. The structure of the model may thus find broader applicability to other nanoscale deformation experiments. Overall, our results establish a fundamental picture for the origins of size scaling of nanoscale asperities ranging in size from one to several hundred nanometers, and can be used to understand contact hardness, and eventually friction behavior, of realistic multiple-asperity surfaces.

Methods

For the MD simulations, the pyramid is truncated on the lateral sides (Fig. 1), where dislocations never venture, to optimize the volume of material participating in the plastic deformation. Periodic boundary conditions are applied on the lateral (x and y) sides. A small flattened region around the edge of the pyramid is included to eliminate the sharp angles that would otherwise exist because of the periodicity and that would cause spurious dislocation nucleation during testing. The vertical (z) displacements of a region of atoms 0.5 nm thick at the bottom of the pyramid are fixed at zero. The lateral boundaries are periodic with a controlled zero pressure and the top surface is loaded by a nearly flat indenter as described below. We study several different pyramid sizes to check scaling effects and to examine the influence of the fixed boundary location on the pressure and forces.

The Au–Au interatomic interactions are modeled by the embedded atom method (EAM) (42–44). The large-scale atomic/molecular massively parallel simulator (LAMMPS) code is used to implement Newton's equations of motion with a Verlet integrator and a time step of 1 fs (45). Isothermal-isobaric (NPT) dynamics are used to control the temperature (1 K or 300 K) of the entire simulation and maintain zero normal stress in the periodic x–y directions by using the Nose–Hoover thermostat and barostat (46). As-fabricated pyramids are built by using the appropriate temperature-dependent lattice constant and are then relaxed for 200–2,000 ps at temperature until the energies, pressures, and temperatures converge in the absence of any applied loads. The pyramid is compressed by an indenter of radius R through an applied force acting between the indenter and any Au atoms of the form f(r) = −k(r − R)² for r < R and f = 0 for r > R, where r is the distance from the atom to the center of the indenter. A force constant k = 100 (eV/nm) and R = 10,000 nm model a nearly-rigid, nearly-flat indenter. The indenter is displaced downward at a rate of 0.01 nm/ps. For the largest pyramid, the simulation carried out to an indentation depth of 5.7 nm required 320 h of CPU time on 400 Apple G4 processors in the X High-Performance Computational Facility at Virginia Polytechnic Institute and State University.

To make comparison with the experimental data, the depth data from Wang et al. (27) and Nix et al. (5) are converted to edge length as follows. For the pyramid data, the contact areas are determined by using equation 7 from Wang (27) with plastic displacement offsets of d_i = 13.9, 6.5, 8.2, and 6.4 nm, and the edge contact lengths is then the square root of this area. For the Berkovich indentation, the contact area is determined from the projected area A = 3u² tan²φ at displacement u, where φ is the slope of the indenter. Assuming the indenter has a canonical shape with a rounded tip with r = 48 nm, the effective edge length is defined as the diameter of the projected area. This is not as precise as the edge length defined for the pyramids but gives a reasonable approximation for the length of dislocation injected into the film.

We also provide a more detailed description of the analytic model summarized in Eqs. 1–4. The stored elastic energy of the applied load f = Pℓ_c², with the configuration of defects fixed, is (αℓ_c³)(P²/2E) = αf²/2ℓ_cE, where E is the elastic modulus for the (001) crystal orientation and α a dimensionless geometric parameter. The work dissipated by the partial dislocation with Burgers vector b_p while moving because of the applied resolved shear stress (Fig. 1 Inset) is ΔU_τ = C_θfb_pcosθ, where C_θ accounts for averaging of the shear stress over the dislocation slip plane. In the case of a preexisting dislocation without interaction energies U_τ would exactly balance the external work. Each s.f.h.o. has a total dislocation line length of 4ℓ_c, and we can write the self-energy of each s.f.h.o. plus its interactions with prior s.f.h.o.s and the pyramid surface as ΔU_self + ΔU_dis = 4Wℓ_c, which is evaluated below. Each s.f.h.o. generates a total stacking fault energy ΔU_sf = $\sqrt{3}$γℓ_c², where γ = 4.7–32 mJ/m² is the stacking fault energy per unit area and annihilates a ledge energy ΔU_L = 4ℓ_cE_L, where E_L = 0.32 eV/nm is the ledge energy per unit length. The displacement change Δu is composed of a plastic component because of the partial dislocation Burgers vector and an elastic component and is given by

Inserting the above quantities into Eq. 1, solving for the applied force at equality, and converting to the pressure yields the prediction

The solution to Eq. 6 at small scales can be imaginary, depending on parameter values, because of the use of elasticity up to high pressures where the material is nonlinear (uniaxial compression on a block of Au shows a maximum stress of 3 GPa). However, at larger size scales, the elasticity contributions become negligible. Although a more complete analysis is available (47), here we maintain the essence of the model by neglecting the elastic energies. Furthermore, γ and E_L are negligible over the size scales of interest here, allowing Eq. 6 to reduce to Eq. 2.

Contributions to the interaction energy per unit length W are estimated by continuum considerations (40). The self-energy is w_self ≈ μb2/4π(1 − ν) w_self ≈ μb2/4π(1 − ν), with μ = 45 GPa the shear modulus, b = 0.136 nm, the magnitude of the (a/3)[100] Burgers vector, and ν = 0.46 the Poisson's ratio. The image energy of each dislocation is approximated by the image energy for a flat surface as w_im = −(2 − $\sqrt{2}$)w_self. The interaction energy between the 4 dislocations in an s.f.h.o. is w_sfho = 0.583[μb²/4π(1 − ν)] (40). The interaction energies between the dislocations in one s.f.h.o. and all other s.f.h.o.s and their image dislocations decay with dislocation separation for finite length dislocation segments. Thus, we consider here only the interaction between each (a/3)[100] dislocation line in the new s.f.h.o. and its n nearest parallel (a/3)[100] dislocations in the previous s.f.h.o.s, which is accurately expressed as

where we use an average value for ΔL, which actually alternates between 0.87 nm and 1.44 nm in the (114) pyramid. As noted, we introduce an overall adjustable parameter C_w to scale the energies, W = C_w (w_self + w_im + w_sfho + w_‖). We limit the number interacting dislocations to a maximum of N to account for annihilation mechanisms that eventually eliminate previously nucleated dislocations. For the first s.f.h.o., there are no parallel interactions and so w_‖ = 0, after which W becomes increasingly dominated by w_‖. The quantities B and D in Eq. 4 can be derived as B = C_wμb²/[πb_pcosθ(1 − C_θ)(1 − ν) and D = 1/3 + $\sqrt{2}$/2 − π + (2ν − 1)ln3 + 2N[ln(4ℓ_N/NΔl) − 1], where ℓ_N is the edge length after N dislocation nucleations and Δl is the average change in edge length.

Appendix

List of variables: ℓ_c, current contact edge length; ℓ_ci, initial contact edge length; H, hardness; H_MD, Hardness for MD simulations; H_expt, hardness for experiments; η_MD, MD scaling factor; η_expt, experimental scaling factor; h, height of pyramid; f_r, total atomic force acting on the indenter; A, contact area between indenter and pyramid; a, lattice parameter; u, indenter displacement; ΔU_e, change of stored elastic energy; ΔU_τ, change of dissipated energy from dislocation motion; ΔU_self, change of dislocation self-energy; ΔU_dis, change of interaction energies between dislocations; ΔU_sf, change of stacking fault energy; ΔU_L, change of pyramid ledge energy; P⁺, contact pressure before dislocation nucleation; P⁻, contact pressure after dislocation nucleation; ΔL, change in edge length; Δu, change in indenter displacement; b_p, partial dislocation Burger's vector; θ, angle between the normal to the loaded pyramid top and the slip planes; C_θ, dislocation dissipation fitting parameter; C_w, dislocation elastic core energy scaling parameter; n, the number of interacting nearest parallel dislocations; b, Burger's vector for the s.f.h.o. junctions; μ, shear modulus; ν, Poisson's ratio; N, maximum number of interacting nearest parallel dislocations; B, first constant containing some of the elastic dislocation energies; D, second constant containing some of the elastic dislocation energies; k, MD indenter stiffness; R, MD indenter radius; r, atomic distance to center of MD indenter; d_I, plastic displacement of indenter; φ, slope of Berkovich indenter; E, Young's modulus; α, dimensionless parameter for determining effective geometry; γ, stacking fault energy; W, energy per unit length for all the dislocation interaction and self-energies; E_L, ledge energy per unit length; w_self, dislocation self-energy per unit length; w_im, dislocation image energy per unit length; w_sfho, interaction between 4 dislocations in an s.f.h.o.; w_‖, interaction between dislocations in one s.f.h.o. and all other s.f.h.o.s.