# Size Dependence of Dislocation-Mediated Plasticity in  Ni Single Crystals: Molecular Dynamics Simulations

Authors: Xiaoyan Li; Wei Yang
Journal: Journal of Nanomaterials
 (2009)

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## Abstract

We investigate the compressive yielding of Ni single crystals by performing atomistic simulations with the sample diameters in the range of 5 nm ∼ 40 nm. Remarkable effects of sample sizes on the yield strength are observed in the nanopillars with two different orientations. The deformation mechanisms are characterized by massive dislocation activities within a single slip system and a nanoscale deformation twining in an octal slip system. A dislocation dynamics-based model is proposed to interpret the size and temperature effects in single slip-oriented nanopillars by considering the nucleation of incipient dislocations.


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## Body

## 1. Introduction

It has been widely  known that the sample size bears
significant effects on the mechanical properties of materials. Early researches
illustrated that the plastic yielding of microscale whiskers was closely related
to the formation of Lüders bands [1, 2]. The theoreticians
then claimed that the smaller was stronger [3].
The tension and torsion testing [4] on thin copper
wires indicated a distinct dependence of yield strength on sample sizes. Recently,
compression experiments on microscale pillars machined by focused-ion-beam
demonstrated that the yield strength of single crystals could rise steadily with the
decrease of sample diameter [5–7]. These size
effects were originated from the microstructural and dimensional constraints
[8]. At present, discrete dislocation simulations
[9–11] were used to investigate the effect of sample size
on strength in the submicron-to-micron range. The relevant results highlighted the
interplay between sample size, internal stress, and length scale inherent to
dislocation generation and motion [9–11].
For metallic nanowires, both atomistic simulations
[12] and experimental observations
[13] revealed that the size strengthening was closely associated
with a phase transformation induced by surface stress [14]. By scanning tunneling microscope
[15, 16], it was found
that a single atom chain has or may have extremely strong strength attributed
to quantized plasticity [13].Currently, understanding of size effects of microscale single crystal focuses on
the operative mechanisms for crystal plasticity under nominally uniform compression.
The dislocation starvation model [6] was proposed to
reflect the discrete stain bursts separated by periods of nearly elastic loading.
This model elucidates that the rate of dislocation escape from the free surface is greater than that of
dislocation multiplication in the small scale crystal. It is suggested that the
sample sizes physically limit the free distance that dislocations can travel
for breeding and replicating [6].
According to a theoretical analysis concerning the dislocation kinetics, the
microscale size effects are attributed to dislocation
nucleation, dislocation escape, and dislocation trapping
[7]. These stochastic effects alter the distribution
of dislocation sources and restrict the multiplication processes, leading to
reduction of the dislocation density and finally exhaustion hardening
[7]. Based on molecular dynamics (MD) simulations for homogeneous
shear of Ni3Al single crystal, a scaling model involving the statistical
distribution of atomic relative displacements was present to predict the size
dependence of incipient dislocation plasticity in microscale samples by extrapolation
[17]. Furthermore, a survival probability theory
[18] of homogeneous dislocation generation was
proposed in the stochastic nature of yield strength of microscale crystals. For the
prediction of incipient plasticity behavior, this methodology bridges both the
length and time scales between MD and experimental regimes
[18]. Combined this statistical methodology,
the dislocation nucleation from the free surfaces has been revealed by the
analysis of elastic instability [19].
The recent work rationalized the size effects by considering the statistical
variation of dislocation source lengths in the finite size single crystals
[20]. This work also suggests that the discussions
about size effects should be divided into two regimes: one is dislocation
nucleation and propagation in very small crystals and the other is operation of
exiting dislocation sources for larger microscale crystals. Excitingly, state-of-the-art
in situ nanocompression experiments give a clear evidence of dislocation
starvation model and further demonstrate that the strength increase in the small
pillar structure is controlled by the activation of new dislocation sources in
a source-limited regime [21].In this paper, we carry out molecular dynamics (MD) simulations to investigate the
compressive plastic yield of Ni single crystal nanopillars.
The compressions of nanopillars with two different orientations provide the observation
of distinct deformation mechanisms for single and octal slip behaviors. Furthermore,
the influences of finite temperature on yield strength are explored. To account
for the dependence of yield strength on sample size and temperature, we present
a dislocation mechanics-based model by considering incipient dislocation
plasticity.


## 2. Simulation Method

The simulated configuration is a pillar with a
fixed length-to-diameter ratio of 2:1. The diameters of the pillars range from
5 nm to 40 nm. Consequently, the simulated systems contain
approximately 18000 to 9000 000 Ni atoms. Referring to a perfect fcc lattice
arrangement, we orient the pillars in the directions of [134] (single slip) and [001]
(octal slip), respectively, with respect to the compressing axis. Throughout MD
simulations, interatomic interactions are described by an embedded atom method
(EAM) potential [22]. To improve the efficiency of simulations, we adopt
a multiple time step algorithm [23] as the numerical
integrator. The shorter and the longer time steps are set to be 0.5 femtosecond (fs)
and 1.5 femtoseconds, respectively. By a Gaussian thermostat
[24], the temperature of simulation is controlled at
the prescribed values from 100 K to 900 K.At the beginning of all simulations, single crystal
pillars were relaxed free for 50 picoseconds (ps) to minimize the potential
energy of system. Subsequently, uniaxial compressions were performed on the
nanopillars. In each loading step, an incremental compressive displacement of
0.04% in the length direction was imposed on the top face, followed by a
dynamic relaxation of 4 picoseconds for the overall pillar. During such
process, the side surface was kept free and the bottom face was adhered to the
rigid foundation, resembling a clamped end. Atoms in the loading region
retained the same plane all along. This loading procedure led to a relatively
high-strain rate of 108 s-1 (inevitable for MD simulations)
relative to the experimental strain rates (approximately 10-4 s-1 or
4×10−3 s-1). The overall
simulations last 2 nanoseconds, and, therefore, the maximal strain of simulated
system reaches 20%. The local crystalline order analysis
[25–27] assisted by coordination number technique was
employed to visualize defects appearing in the deformation. Moreover, the
stresses of nanopillar were calculated through averaging atomic stresses
described by the Virial theorem.Due to an inherent limitation of MD simulations
[28], the high deformation rate applied in simulations
may induce rather high stresses which are usually much larger than results of
experiments. However, this artificially high rate cannot change the underlying
deformation physics in the single crystals. To avoid probing unphysical
behavior, the applied stress is verified to lie well below the theoretical
shear strength of single crystal in the particular direction of straining
[28]. It is also essential that the time scale, on
which the simulated behavior occurs (e.g., the velocity of gliding dislocation),
is verified to lie well below any critical inherent time scale of single
crystal (such as the speed of sound) [28].
Although MD simulations are limited to relatively small systems consisting of
typically millions of atoms, such massive simulations represent a promising step toward elucidating the intrinsic mechanisms
of size effects for microscale single crystals.


## 3. Results and Discussion

Figure1 displays the stress-strain curves of 〈134〉- and 〈001〉-oriented nanopillars at 300 K. If the
maximal stress on the curve is taken as the yield strength, dramatic size effects
are observed for Ni single crystals under nominally single slip or octal slip
conditions. As illustrated in Figure 1, the flow
stress drops to a low level after yielding. Such a phenomenon differs from the
experimental observations in which the flow stress maintains at the level of yield
strength [5–7]. One interpretation comes from the initially
defect-free status of the nanopillars before compression. The similar stress drop
also occurred in the atomistic simulations for nanowires
[12, 29] and single crystal blocks
[30]. The high-frequency stress oscillations are
attenuated as the size increases, as evidenced in
Figure 1. This indicates that the local atomic
vibrations play a critical role at the scale below 20 nm. These local
effects are weakened by size increasing due to a global stress averaging procedure
[30]. Figure 2
shows the variation of activation volume with the diameter of 〈134〉-oriented
nanopillars. For the small size nanopillars, the present activation volume is
consistent with the result from a recent atomistic modelling about surface
dislocation nucleation [31]. Herein,
the activation volume is obtained by totalizing the Virial volumes
of atoms in the nucleated dislocation at the first stress drop. The present
activation volumes are atomic rather than the volume expected from an actual
dislocation [32].
The rise of activation volume with an increase of sample size implies that the
dislocation generation corresponding to initial yielding appears to follow
homogeneous nucleation kinetics [17].Stress-strain curves of〈134〉- and 〈001〉-oriented nanopillars at 300 K.
(a)(b)Figure 2
Variation of activation volume with the diameter of〈134〉-oriented at 300 K. The sign “b” represents the magnitude
of Burgers vector of extended dislocation.For the pillars less than 20 nm in diameter, their stress-strain
curves are featured by several sharp fluctuations that correspond to the
discrete nucleation of dislocations. To reveal dislocation nucleation events,
Figure3 depicts an image
of octahedral shear-stress distribution at initial yielding of 10 nm
〈134〉-oriented pillar. This stress distribution exhibits
dynamical propagation of stress waves, distinctly unlike the quasistatic stress
states reported in the literature [7].
In all of simulations, an   instantaneous stress drop-off concurred with dislocation
emission from the corners or the free surfaces, implying that the stress drop-off
after yielding is indicative to the onset of incipient dislocation plasticity
[17]. In addition, Figures
4(a)–4(c)
display the stress-strain relations of nanopillars at different temperatures. As
shown in Figure 4, the yield strength of nanopillars
decreases with an increase of temperature. It suggests that temperature is a
significant external factor for the occurrence of initial yielding of nanopillars
because the incipient dislocation nucleation is a thermally activated event
[17]. Meanwhile, the stress oscillations become more and
more prominent in the stress-strain curves with an increase of temperature since
the atomic vibrations are gradually quickened. Moreover, the curves shown in
Figure 4 also clearly indicate that the high temperature induces a thermal softening
of elastic properties.Figure 3
Octahedral shear stress distribution at initial yielding of
10 nm〈134〉-oriented nanopillar. The stress value of each
point is obtained by averaging the Virial stresses of atoms in a sphere
(centered at this point and with a radius of the cut-off distance used in
the EAM potentials). The length unit is
Angstrom, and the stress unit is MPa.Stress-strain curves of〈134〉-oriented nanopillars at
different temperatures.
(a)(b)(c)Figure5 depicts the microstructure evolution
of 20 nm 〈134〉-oriented
nanopillars under the compression. The extended dislocation initially nucleates
from a corner and then travels through the perfect crystal along a primary
glide plane. When reaching the free surface at the   opposite end, it is reflected after
leaving a tiny surface step. Such process substantiates that the overall sample
sizes limit the length scales available for the plasticity process
[5]. The reflected dislocation meets
with the emitted dislocations located at the same slip plane. Then, they either
annihilate or react with each other, leading to formation of a few defects,
such as vacancy chains and new dislocations. As the compression proceeds, the activation of secondary slip plane results
in further dislocation motion, reaction, and even multiplication (even involving
double-cross-slip). The successive emissions of dislocations on primary slip system
induce a global sliding of the top-left part of nanopillar, as shown in Figure
5(e). This inhomogeneous deformation mode can
explain glide offset and lateral displacement of the top portion of single crystal
relative to the fixed end [7]. Further developments of
single-slip offsets may cause plastic flow involving strain bursts governed by
accumulative growth of dense dislocation structures
[33].Snapshots for microstructure evolution in 20 nm〈134〉-oriented
nano-pillar. (a)–(c) The extended dislocations nucleate, glide and reflect at
the free surfaces. (d)–(f) The top-left part of the pillar undergoes
global sliding, leading to strain bursts. The symbol “◯” marks the surface step.
Slip lines and surface steps are clearly observed in the deformed states.
(a)(b)(c)(d)(e)(f)Figure6 illustrates the microstructure
evolution of 20 nm 〈001〉-oriented
nanopillars under compression. Several extended dislocations, namely, stack fault
(SF) areas towed by partial dislocation, firstly, nucleate from the corners,
then, glide on the slip planes of {111}-type, and finally, merge into the free
surface with remnant surface steps. Thus, perfect single crystal can be
transected by several SFs remaining in the crystal interior, which
circumscribes further dislocation motion. The local stress may induce nanoscale
twins from the coupled partial dislocations on the neighboring glide planes. Such
small scale twins are determined by dimensionally constrained structures and serve
as barriers to further dislocation activities. However, the intersection of
twins becomes new defect sources under high-applied stress. The emission and
propagation of partial dislocations from the active sources will broaden the twin spacing, which significantly reduces the ductility
of nanopillars [34]. When
two parallel twin boundaries encounter, the hcp atom layers recover the perfect
fcc structures. As a result, two nanoscale twins are incorporated into a new
twin.Snapshots for microstructure evolution in 20 nm〈001〉-oriented
nanopillar. (a) Poly-slip behavior concurs with yielding. (b)-(d) Nanoscale
twins are formed and then broaden by the successive emission of Shockley partial
dislocations. (e)-(f) Two parallel twin boundaries annihilate as a result of a
new nanoscale twin. The symbol “◯” marks the surface step.
Slip lines and surface steps are clearly observed in the deformed states.
(a)(b)(c)(d)(e)(f)Figures5(f) and
6(f) illustrate the final deformation
microstructures of 〈134〉- and 〈001〉-oriented pillars with a diameter of
20 nm, respectively. The appearances of
samples after compression are analogous to those in experiments
[5–7]. For nanopillars orientated to favor single slip, plastic
behaviors are governed by intense dislocation activities including nucleation,
propagation, multiplication, and interaction. For nanopillars with the
octal-slip orientation, nanoscale twinning is a dominant deformation mechanism
controlling plastic flow. The above mechanisms indicate that crystal internal
orientation is responsible to the physical nature of irreversible deformation.
Nevertheless, two dissimilar mechanisms are consistent in the aspect of
underlying physics, since they are strongly governed by dislocation plasticity.
All details of microstructure evolution predicted by different deformation
mechanisms are expectantly supported by the transmission electron microscopy
(TEM) analysis of experimental samples before and after compression.To better understand finite size and temperature effects, we develop a dislocation
dynamics-based model which is applicable to single-slip behaviors. Based on the
thermal activation analysis
[35, 36] and intrinsic stress scaling
[37], the shear strength
τS consists of the thermal stress
τT and the size-related
stress τD,
(1)τS=τT+τD=A1exp[−β0T+β1Tln(γ˙γ˙0)]+A2(DD0)n,
where τT represents a threshold   value of dislocation
nucleation; and τD depends on the sample diameter
D (normalized by a reference
diameter D0=1μm) by a power law of exponent n. In (1),
γ˙ and γ˙0 are the actual and the reference shear strain
rates; T is the absolute temperature; and parameters
A1, A2, β0, and β1 are constants for a given material under a specific condition.
Based on the experimental results [5, 7] and the present MD simulations, we use a nonlinear
least-squares fit to determine the related parameters as follows:
(2)n=−0.69,A1=2224.17±297.56MPa,A2=124.68±11.47MPa,β0=(12.01±2.07)×10−3K−1,β1=(4.40±0.35)×10−4K−1.These data are optimized values by repeated fit procedure.
The exponent n denotes a fractal dimension between the strength
and the internal structure [37].
Its value locates in the suggested range of −0.6~−0.7
[7].Yield behaviors in the nearly dislocation-free materials are not dominated by
the morphological distribution or assembling density of dislocations, but by the
nucleation and propagation of dislocations [30, 31]. During the
present simulations for single-slip oriented nanopillars, yielding occurs when the
first extended dislocation nucleates from the surface and sweeps through the
pillar. Figure 6(a) describes a glide process of the
first extended dislocation travelling through the crystals with different diameters. The observed
geometries suggest that the curvature radii of the leading and trailing partial
dislocations are proportional to the sample diameters, that is,
R1=η1D and R2=η2D. This implies
that the sample sizes inhibit the size and shape of incipient dislocations, which
further affects the deformation  mechanism and
yield strength of single crystals. In the light of incipient dislocation plasticity,
we focus on the nucleation of the first dislocation in the following discussion.Figure7(b) illustrates a schematic image of the
dislocation dynamics-based model related to incipient plasticity. If an extended
dislocation moves a small distance of Δs, the work done by the dimensional stress τD can be
written as
(3)Wτ=(R1+R2)ΔsϕτDbpcosπ6=24(η1+η2)ΔsDϕτDa, where a is the
lattice length and
ϕ is the enclosed angle of curved dislocation.
Additionally, the increased SF energy is
(4)ΔESF=(R1−R2)ΔsϕγSF=(η1−η2)ΔsDϕGa224πd,
where γSF is the SF
energy per unit area [38],
G is the shear
modulus, and d is the separating distance of two partials in SF.
The incremental dislocation line energy (ignoring the core energy of dislocation)
is (5)ΔEd=(2−ν)ΔsϕGbp28π(1−ν)(ln4R1ρ+ln4R2ρ−4)+(2+ν)ΔsϕGbp28π(1−ν)lndρ=(2−ν)ΔsϕGa248π(1−ν)(ln4η1Dρ+ln4η2Dρ−4)+(2+ν)ΔsϕGa248π(1−ν)lndρ,
where the first term denotes the increased
self-energy of two partials and the second one describes the increased
interaction energy. In (5),
ν is Possion’s ratio, and ρ is the cutoff radius of a dislocation core and generally,
ρ≅bp/8 [38]. The energy conversation law
gives (6)Wτ=ΔESF+ΔEd.By combining
(3)–(6) and
simplifying, an analytical expression between the dimensional stress
τD and the
sample diameter D is obtained,
(7)τD=2Ga24πD(1−ν)(η1+η2)×[(2−ν)(ln4η1Dρ+ln4η2Dρ−4)+(2+ν)lndρ]+2Ga(η1−η2)12πd(η1+η2). If the thermal activation stress adds to the analytical
dimensional stress, a semi-analytical relation among strength, sample size,
temperature, and strain rate is obtained. Under the actual deformation conditions (ε˙<10−4s−1), the dimensional stress is much greater than the
thermal stress. Therefore, this stress-scaling expression can be directly used
to estimate the yield strength of microscale single crystals. From the relevant numerical calculations, the
following parameters are obtainable directly or indirectly from the present MD
simulations: (8)G=70.97GPa,ν=0.312,a=0.352nm,d=3.10nm,η1=0.40,η2=0.25. Figure 8 shows
the comparison between the dislocation dynamics-based model, experimental, and
present MD results. As shown in Figure 8, the
dislocation dynamics-based model correlates well with the results from MD
simulations, but slightly differs from the experimental datum in the submicron and
micron range. One possible reason for this deviation is the neglect of surface
effects, such as image forces on dislocations by free surface and influences of
surface steps. Although the dislocation dynamics-based model is established on the
basis of MD simulations at the nanoscale, this model can be extended to the
submicron and micron range  because the size effects of single crystals in
the nanometer to micron range may be traced to the same physical origin, namely,
the dimensional constraint  [8]. Indeed, the dislocation density tends to decrease
owing to the escape of dislocations from the free surface or dislocation
reaction [6, 9], when the micron-scale single crystals are
subjected to a nominally uniform compression. Once the dislocation-starved
conditions are reached, very high stresses would be required to nucleate new dislocations,
either at the surface or in the bulk of the crystal
[6]. Thus, the stresses of dislocation
nucleation and sudden startup can be estimated by the proposed dislocation dynamic-based
model.(a) Glide process of the first extended dislocation traveling through the crystal. For nanopillars with different diameters, the configurations of the initial dislocation at different time are placed in a drawing.
The drawing plane is the primary slip plane of dislocations. (b) A schematic illustration of
dislocation-dynamics-based model.
(a)(b)Figure 8
Variations of shear yield strengths with sample diameters.
Purple-painted star-like data points come from experiments in
[5, 7], and other data points come from the present MD simulations. Solid and dashed lines represent the results of the fitting and
semi-analytical models, respectively.The fitting model in Figure8 shows a stress scaling
τ∝D−0.69 which is different from the usual Hall-Petch
relationship τ∝D−0.5. Power-law
exponents close to –0.5 are commonly understood to result from the kinematics
of glide obstacles within a linear-glide model, such as grain boundaries
[7]. The size effects following a power law with an
exponent in the range of −0.6~−0.7 arise from influence of both external
geometry (such as sample sizes, surface features) and internal structure (such
as distribution of dislocation source, obstacle limit) on crystal plasticity. For
nanoindentation experiments, plastic deformation is confined within a very
small volume, leading to nonuniform stresses and strains which further setup
strong gradients of strain [6].
The so-called indentation size effects have been interpreted via strain gradient
plasticity models which incorporate the concept of geometrically necessary
dislocations. The size effects occurring in the homogenous compression of microscale
crystals are associated with dislocation starvation in the absence of stain
gradients [6].The proposed model involves a   geometrical assumption
that the shape of initially nucleated dislocation is physically limited by the
sample sizes. Though it is an observation based on the present MD simulation, it
is necessary to verify if this assumption can be experimentally observed via
TEM. Notably, the present model predicts only the initial stress for crystal
plasticity, that is, critical resolved shear stress (CRSS) required for the
first percolation of a dislocation across the sample
[20]. Since the contributions of image stress to yield
strength are neglected, the analytic model is approximate for explaining size
effects in the microscale crystals. Nevertheless, this model is available for predicting
the yield strength of the single slip-oriented crystal from submicron to micron
meters because the image effect gradually diminishes for nucleation and motion
of dislocation as the sample size increases.


## 4. Conclusions

In summary, we show dramatic size dependence of
yield strength by performing MD simulations for the compression of Ni single
crystals. For single- and octal-slip behaviors, dislocation-mediated plasticity
is manifested by dislocation   glide, strain bursts, and nanoscale twinning. Both
impediment to dislocation motion and formation of nanoscale twins illuminate
that sample sizes limit the length scales available for crystal plasticity. Considering
sudden nucleation and propagation of the incipient dislocations, we develop a dislocation
dynamics-based model whose estimations are in agreement with the results from MD
simulations and experiments.



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