Applied Physics A
(2021) 127:358
https://doi.org/10.1007/s00339-021-04502-z
Effect of variation in inclination angle of Ʃ5 tilt grain boundary
on the shock response of Ni bicrystals
Tanmay Konnur1 · K. Vijay Reddy1 · Snehanshu Pal1,2
Received: 5 January 2021 / Accepted: 8 April 2021
© The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature 2021
Abstract
The intensity of damage production during shock wave propagation in polycrystalline metallic systems is mostly dependent
on the shock-defect interactions. Traditionally, different coincidence site lattice (CSL) grain boundaries (GBs) are introduced in the polycrystalline structure through various processing techniques to enhance the strength or plasticity. However,
their dynamic response to the impulsive loading condition has not been elaborately explored to date, which necessitates a
detailed study on the interaction between CSL GBs and shock wave. In this study, we have employed molecular dynamics
simulations to investigate the response of both symmetric and asymmetric ∑5[1 0 0] tilt GBs of nickel bicrystal under the
influence of shock-wave. We have also analysed the role of piston velocities and inclination angles of the GBs on the shock
response and the consequent deformation behaviour. This investigation gives insight into the mechanical response and the
underlying mechanisms which are inspected through atomic strain analysis, common neighbour analysis and pressure–time–
distance mapping. The results show that the stacking faults formation takes place when specimens are subjected to lower
shock velocities, whereas higher velocities facilitate phase transformation along with amorphization. There is a stark contrast
between the specimens with symmetric and asymmetric tilt GBs in the manner of plastic deformation behaviour in response
to shock-wave and GB failure at lower and higher piston velocities.
Keywords Shock deformation · Bicrystal Ni · Molecular dynamics simulation · CSL grain boundary · Inclination angle
1 Introduction
Grain boundary (GB) engineering involves the intentional
manipulation of the GB network in polycrystals with the
goal of creating a material, which has advanced interfacial
properties in terms of energy and structural features compared to the conventional metallic systems [1]. Such technique is essential because it suggests the construction of
polycrystalline materials by guiding the nature and distribution of individual GBs in order to augment the advantageous effects of the other neighbouring boundaries with
reference to the materials properties [2] such as strength
[3–5], creep [4], ductility [5] and corrosion resistance [3,
5]. The GB engineering, according to the coincidence site
* Snehanshu Pal
snehanshu.pal@gmail.com
1
Department of Metallurgical and Materials Engineering,
National Institute of Technology, Rourkela 769008, India
2
Centre for Nanomaterials, National Institute of Technology,
Rourkela 769008, India
lattice theory [6], states that hypothetically if two lattices
are allowed to interpenetrate, certain combinations of orientation relationship between the two lattices would result
in a periodic array of coinciding sites and the reciprocal
density of CSL points is denoted by ∑. In the view of contributing towards understanding the underlying deformation mechanism, investigations through bicrystals become
important as it is made by bonding together two crystals
with predetermined crystal orientations. This provides the
simplest form of a model incorporating a GB along with its
explicit structure and geometry. Along with the ability to
accurately manage degrees of freedom of the GB, bicrystal
also enables to scrutinize the GB structure and nucleation of
dislocations from GBs. The investigations performed using
polycrystalline aggregates show that the fundamental contribution of a GB cannot be discerned due to a large number
of grains in connection with a particular GB, which in turn
affects the mechanical properties of that GB under consideration. To counter this problem, bicrystal studies at smaller
length scales are employed to identify the influence of individual GB on the deformation behaviour and misorientation
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evolution [7] due to interaction with the lattice dislocations
as a result of high GB fraction [8].
The different modes of deformation reported in the literature are uniaxial, plane strain and equi-biaxial. Several
experimental studies are carried out reporting various deformations under different modes in bicrystal [9–18]. Li et al.
[16] studied the deformation behaviour of bicrystals with
inclined twin boundary under unidirectional and cyclic loadings. Similarly, Zaefferer et al. [10] analysed the deformation
of aluminium bicrystals having various misorientations in
a channel die experiment to study the effect of misorientation on the kinematics of deformation zones around GBs. Li
et al. [19] have reported that the strength of the crystalline
materials can be improved by the obstruction of the dislocation movement under monotonic loading. On the other
hand, Fensin et al. [20] investigated the influence of dynamic
loading conditions, which cause the metallic systems to fail
along the GB through void nucleation. Many experimental
studies related to deformation of GBs have been recorded
and analysed by in situ transmission electron microscopy
(TEM) experiments [21, 22], which has led to unique observations like GB motion and migration, dislocation pile-ups,
crack nucleation and propagation along the GB.
In this perspective, GBs also play a crucial role on the
defect evolution, deformation behaviour and damage of the
metallic specimens [23–27]. The shock-compression experiments in bicrystals have focused on reporting the evolution
of substructure as a function of the crystalline orientation
[28] in which the cases of [100] direction parallel to the
shock direction are studied extensively [29]. The investigation of the shock stress and the stress orientation arising
from the evolution of the substructure of copper has been
reported in the literature through experimental shock recovery studies on copper bicrystal, which focused on the analysis of twinning [30]. During such impact loading analysis,
an insight into the shock response and damage tolerance
will yield valuable apprehension regarding the design of the
microstructures adapted for implementations in the cases of
impact. The exploration of such relations in bicrystal materials with typical GBs using only experimental methods is
challenging due to the small length and time scales of the
underlying mechanism and processes [31]. It is also not economically viable to repeat such types of experiments as the
overall cost of a single set of analysis is expensive. These
are prone to errors because of factors like external atmosphere and sample purity. Here, the application of molecular
dynamics (MD) simulations in the study of GBs can be used
to explore the structure and its response under various loading attributing to plastic deformation. Moreover, the MD
simulations provide a means of maintaining the total purity
of the sample and firm control over the process parameters
[32]. A direct comparison can be made between the experiments and atomistic simulations owing to the increasing
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competence of MD simulations and advances in the features on a finer scale [33]. In this viewpoint, MD simulations
become increasingly important as it aids in getting valuable
insight into the deformation mechanisms operating for the
bicrystal of metals [34] and its associated GB. The strain
rates used generally in MD simulations, and sample dimensions are analogous to the experimental competence accomplished utilizing laser shock loading and hence can indicate nucleation and evolution [35]. Extensive studies using
experimental [36–39] and simulation approaches [40–42]
have been performed for decades on GBs to get insights
about the structure and its response to different loading processes. Our interest lies below the grain size of 10 nm, where
GB-mediated processes become the governing mechanisms
responsible for deformation. The apprehension of atomiclevel behaviour of bicrystals when subjected to deformation is limited; hence, MD simulations provide a detailed
analysis of the processes occurring at the atomic level in
the bicrystals. It helps uncover the real-time atomistic scale
phenomena and mechanisms that are difficult to gain insight
with experimental approaches [43–45]. The results obtained
from these simulations revealing the nucleation and propagation of dislocations in different crystal arrangements have
proved worthwhile [46]. MD simulations have varied GB
engineering applications like in tensile or bending tests in
nanowires where the mechanical properties have not been
evaluated yet because of difficulties in experimental testing
[47]. It also serves as a valuable tool to analyse phenomena
like the changeover of normal to inverse Hall–Petch behaviour in accordance to decreasing grain sizes during plastic
deformation and fracture mechanics [48]. There have also
been detailed studies of shock-wave interaction with different materials using MD simulations over the past years.
Xiang et al. [49] have enabled the analysis of melting and
spallation that nanocrystalline Pb undergoes when subjected
to strong shock-loading conditions and have indicated that
GBs make a substantial contribution to the processes leading to melting and spalling of nanocrystalline Pb. In this
line, Reddy et al. [50] studied the effect of shock loading on
crystalline Cu-amorphous ­Cu63Zr37 nanolaminates and concluded that the existence of crystalline -amorphous interface
is responsible for the transformation of the amorphous phase
formed as an intermediary to the BCC phase. Chen et al.
[51] employed MD simulations to analyse the spallation of
Ta bicrystals for gaining insight into the role of GB structure
leading to the dynamic failure of materials which led to findings that showed a correlation between spall strength and
GB misorientation angle. Similarly, Long et al. [52] used
MD simulations to study the spallation and deformation of
Cu bicrystals with (1 1 1) twist GB of different misorientation angles and inferred that twist GB furnish dislocation
sources for deformations and the single crystals are of higher
HEL than the bicrystals.
Effect of variation in inclination angle of Ʃ5 tilt grain boundary on the shock response of Ni…
A considerable number of studies have been reported in
the literature involving the use of MD simulations to analyse the shock-wave interaction of coincidence site lattice
(CSL) GB. Pham et al. [53] involved MD simulations for the
shock compression and spallation analysis of ∑5 GB in Pd
bicrystals and displayed the results that the GB is responsible for an increase in the amplitude of particle velocity and
serves as a site for a scattering of the wave. Similarly, Lin
et al. [54] used MD simulations to show that the presence
of GB has a considerable influence on the development of
pre-spall damage, spall strength and spall damage for a Cu
bicrystal sample under shock loading. Most of the studies
related to bicrystal behaviour under shock loading have been
performed by placing the GB plane normal to the shock
direction in order to study the effect of boundary on shock
attenuation [45, 53–55]. In this work, a comprehensive analysis of the ∑5 CSL GB with different inclination angles in
Ni bicrystal is presented. The orientation of GB is parallel
to the direction of shock propagation, which has not been
reported in the literature to date. Moreover, we have also
presented the effect of the shock wave on the deformation
behaviour of the Ni bicrystals having various ∑5 CSL GBs
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using MD simulations. We have also studied the influence
of shock intensity on the structural alteration and damage
in the specimen by varying the piston velocity. The plastic
deformation of specimens, phase transformation and GB
transitions are also analysed and reported comprehensively.
2 Simulation details
Six bicrystal Ni specimens have been constructed with a
cross section of (14 × 14) nm and a length of 55 nm having
various inclination angles, as shown in Fig. 1. The GBs are
modelled using the coincidence site lattice theory with the
reciprocal density of coincidence site (∑) equal to 5. For
each specimen, the GB normal and period vectors for the
upper and lower crystal are shown on the left–hand side
along with the inclination angles. In this case, the GBs with
inclination angle 0° (∑5 (− 310)/(310)) and 45°(∑5 (210)/
(120)) [56] are the two ∑5 symmetric tilt grain boundaries
(STGB), and the remaining GB structures are asymmetric
tilt grain boundaries (ATGB). Further, the four ∑5 ATGB
Fig. 1 Representation of six ∑5 grain boundary structures in Ni for various inclination angles along with the dimensions of the specimens
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with different inclination angles constitutes only two structural units corresponding to the two ∑5 STGB.
The arrangement of atoms that leads to the characteristic
structure of six ∑5 GB in Nickel at a temperature of 0 K is
magnified and marked in Fig. 1 in order of the increasing
inclination angle and are observed along [001] tilt axis. The
bicrystal specimens have been relaxed by energy minimization using the conjugate gradient method [57] prior to the
application of shock-loading. The NPT (N is the number of
particles, P is the pressure, and T is the temperature) ensemble is applied for the equilibration of the specimens at zero
pressure and temperature of 100 K. The equilibration time
step has been taken as 0.001 ps (~ 1 fs). After the specimens
are prepared, they are subjected to shock-loading along the
negative X-direction, which is performed by directing a rigid
piston (6 Å thickness) at one end of the length of the specimen with a constant inward velocity (Up). The direction of
propagation of the shock-wave in the specimens is parallel to
the orientation of the GB. The NVE (N is the number of particles, V is the volume, and E is the total energy) ensemble is
applied for performing the shock-loading process at a temperature of 100 K. The time step for simulation is considered
as 1 fs. In this work, the shock propagation has been probed
for different incremental piston velocities like 0.5 km/s,
0.8 km/s and 1.1 km/s for each specimen in order to explore
the effect of piston velocity on the deformation behaviour
of the specimens. In the process of shock-loading, periodic
boundary conditions are applied along the non-shock-loading direction (Y- and Z-directions), and free boundary conditions were applied to the ends of the bicrystal along the
shock-loading direction (negative X-direction). The current
MD simulations have been performed using the open-source
Large-scale Atomic/Molecular Massively Parallel simulator (LAMMPS) [57] software. The embedded atom method
(EAM) potential developed by Mendelev et al. [58] has been
used to describe the interatomic interaction between the Ni
atoms. The values of elastic constants obtained from the
interatomic potential, i.e. C11, C12 and C44 corresponding to
compression, are equivalent to the target value for pure Ni
[58]. Moreover, the change in energy (ΔE) required for the
phase transformation of nickel from FCC to BCC is quite
close to the desired values. Also, the value of lattice constant predicted through this interatomic potential is close to
the actual value [58]. These numerical inferences suggest
that the interatomic potential accurately predicts the atomic
interaction between the Ni atoms and can be implemented
for the shock deformation behaviour. Open Visualization
tool (OVITO) [59] software has been used to visualize and
distinguish various deformation behaviours during the shock
loading process. The common neighbour analysis (CNA)
[60] and atomic strain analysis [61] have been employed
to apprehend the deformation behaviour during the shockloading process in the bicrystal specimen.
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3 Results and discussion
3.1 Pressure contour during the shock propagation
The pattern of evolution of pressure during shock propagation for different piston velocities with different inclination angles of GB is portrayed as pressure contour, which
displays the pressure as a function of time by plotting the
average pressures along the direction of the shock at the
intermediate time to gain insight into the wave propagation behaviour. Figure 2 shows the representative compressive pressure variation for a sample with symmetric GB
(inclination angle = 0°) (refer Fig. 2a, b) and a sample with
asymmetric GB (inclination angle = 26.57°) (refer Fig. 2c,
d), each for lowest piston velocity (0.5 km/s) and highest
piston velocity (1.1 km/s), respectively, to display the time
evolution of shock pressure profile. The parts that are not
yet affected by the shock-wave are shown by the particular
colour corresponding to the near zero pressure depicted in
the colour legend and those sites which are under the influence of the final shock pressure are shown by red colour. It
is observed that the peak pressure increases significantly at
higher velocities (refer Fig. 2). Comparisons between the
specimens of the same GB inclination angles subjected to
different piston velocities reveal that both the peak pressure and pressure at a particular region are significantly
higher for higher piston velocities. Moreover, at higher piston velocity the pressure build-up is more pronounced and
discernible in regions that are in front of the progressing
shock. A close comparison between Fig. 2a and c reveals
that the pressure gradually decreases in the specimen with
GB inclination angle 0° as the shock progresses, which is
inferred by the subtle colour change from red to yellow.
A close comparison between Fig. 2b and d conveys that
the overall magnitude of the pressure experienced by the
region is greater for the specimen with GB having inclination angle 0°. Calculation of the pressure of entire system
of atoms corresponds to the system stress, which helps in
understanding the mechanical properties. The topologically constituent member unit forming the symmetric Ʃ5
(310) (inclination angle = 0°) and the symmetric Ʃ5 (210)
(inclination angle = 45°) are the same as shown in Fig. 1.
They differ only the direction and arrangement along the
boundary plane difference. It is stated that the mechanical behaviour of different Ʃ5 GBs can be associated with
their energy, and the ones with inclination angle 0° and
45° show comparatively lower energy [34]. Thus, these
structures are more stable. Hence, due to the similarly in
their structure and owing to their relatively higher stability, alikeness is seen in the pressure evolution behaviour between the specimen having two symmetric GBs
with inclination angle 0° and 45°. Also, similarity in the
Effect of variation in inclination angle of Ʃ5 tilt grain boundary on the shock response of Ni…
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Fig. 2 Evolution of pressure along the length of Ni bicrystal specimens having ∑5 GB inclination angle 0° and piston velocity of a 0.5 km/s, b
1.1 km/s and ∑5 GB inclination angle 26.57° and piston velocity of c 0.5 km/s, d 1.1 km/s, respectively
pressure evolution behaviour is seen between the specimen having asymmetric GBs with inclination angle 26.57°
and 11.31°,18.43°, 30.96° for which the compressive pressure variation figures are provided in the supplementary
material.
This is inferred by the persistent higher pressure almost
till the end of the length in Fig. 2b in contrast to the gradual
decrease in pressure in Fig. 2d. Also, in Fig. 2d the gradual
change of colour code yellow–green–blue signifies that the
region that comes in contact with the progressing shockwave preferentially experiences gentle elevation in pressure, which continues after the shock has passed through the
region till the pressure reaches a particular constant value.
This implies that the initial interaction with the shock-wave
induces the deformation processes, which leads to alteration
in pressures. On the same lines, it can be elucidated that
in cases where there is an abrupt pressure change (colour
alteration) the interaction of shock-wave, rise in pressure and
deformation process occur almost immediately.
3.2 Atomic shear strain analysis during the shock
at low piston velocity
The atomistic response of each bicrystal specimen under
shock compression can be demonstrated through the atomic
strain analysis. Figure 3 depicts the atomic shear strain snapshots at various intervals during the shock compression process of the bicrystal specimen for symmetric GB (inclination angle 0°) at a piston velocity of 0.5 km/s. As the shock
wave progresses in the bicrystal specimen, it leaves behind
the formation of high strain regions, as shown in Fig. 3a–c.
Structural alterations, along with shear leading to plastic
deformation are observed upon the interaction of the shock
wave with the specimen. Literature studies reveal that by
changing the shock loading direction from perpendicular to
parallel with respect to the GB, its ability to undergo plastic
deformation is altered during shock compression [20]. This
observation of the GB to experience plastic deformation
under shock compression can be elucidated by the strains
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T. Konnur et al.
Fig. 3 Atomic shear strain snapshots of the specimen with GB inclination angle 0° for a piston velocity of 0.5 km/s at different time steps. Shear
strain distribution in g upper and h lower crystal is plotted. The black coloured arrow shows the direction of propagation of shock-wave
developed at the GB. It is seen that the high strain planes are
along (111) plane as a virtue of the arrangement of atoms
for the GB with the particular inclination angle. The variation in the crystallographic orientation of the two grains
causes a discrepancy in the velocity of the piston wavefront
leading to the formation of shear stresses. This shear stress
is responsible for assisting the plasticity at the GB by the
modification of the resolved shear stress on the accessible
slip systems. The high strain is observed along the (111)
plane of the crystal as marked in Fig. 3. Figure 3g and h
portray graphs plotted between the atom fraction and atomic
shear strain for upper and lower crystals, respectively, in
the bicrystal with symmetric GB having inclination angle
0° for a lower piston velocity of 0.5 km/s after the shock
has completely passed. This particular analysis is performed
through the data obtained from OVITO by considering the
strain distribution in each crystal. The specimen is sliced
along the GB plane and each crystal is individually investigated further for shear strain. It can be seen from the figure
that the peaks for both crystals appear at the same value of
shear strain at 0.49. It is also worth noting that the shapes
of both curves are identical, which suggests that the strain
distribution in both crystals is similar. The dotted line in the
figure across the two subfigures shows that the maximum
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count for the particular strain is same in both crystals. The
reason for difference in atom fraction among the graphs is
the relatively greater number of shear bands formed in the
lower crystal. Hence, there is an almost equal amount of
atomic shear strain in the upper and lower crystal because of
the symmetric nature of the GB since the inclination angle
is 0°. Moreover, the pattern of strain generated along this
bicrystal specimen is symmetric concerning the GB for both
crystals. As the wave front progresses in the specimen, its
effect decreases which can be seen from Fig. 3d–f. But, the
accumulation of shear stress in the region close to the piston
increases with time. Dislocation emission is the governing
phenomena for the determination of activation of a particular
slip system, which can be evaluated by resolved shear stress
along a slip system [20]. Inferences about the mobility of
dislocations and the impelling force enabling the dislocation
glide away from the GB along with the mobile dislocations
can be drawn from the high resolved shear stress along a
slip plane [20]. The observation in this case of the parallel
loading state is that there is no promotion for void nucleation
since enhanced plasticity is observed near the GB acting as
the dissipative mechanism for the applied stress.
A similar effect of shock propagation in the bicrystal
specimen with asymmetric GB (inclination angle 26.57°)
Effect of variation in inclination angle of Ʃ5 tilt grain boundary on the shock response of Ni…
for a piston velocity of 0.5 km/s is observed and illustrated
in Fig. 4. The primary difference between the observations
of the two post-shocked specimens is the region under
stress and intensity of stress developed near the GB. It can
be observed from Fig. 4a–c that high shear strain planes
are formed only in the top grain of the specimen during the
initial time. Moreover, the region experiencing high atomic
strain is lesser than the specimen with GB inclination angle
0°. As discussed earlier, the weakening of the piston velocity
in the top grain compared to the bottom grain is due to the
interaction of the wave with the atomic planes having different lattice orientation and configuration. As a result, the
part of the wave which experiences the stress plane in preference (here, top-grain) leads to decrease in its velocity, thus
causing the discrepancy amongst the two grains pertaining
to the stressed region and the amount of plastic deformation
around the GB. This is the outcome of the slip mechanism
occurring along the closed packed orientation in the top
grain on the basis of the Ni crystal’s elastic anisotropy due
to dissimilarity in the lattice orientations of the specimen
having asymmetric GB. Similar observations are seen for
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the specimens with GB inclination angles 11.31°, 18.43°
and 30.96°. A comparison of the observations in Fig. 3 and
4 reveals that while the atomic shear strain distribution along
the bicrystal specimen is symmetric along the GB in Fig. 3,
it is asymmetric in the case discussed here.
3.3 Structural transformation in the specimen
during low piston velocity
Figure 5 illustrates the CNA snapshots of the bicrystal specimen with GB inclination angle 0° for a piston velocity of
0.5 km/s. The black coloured arrow indicates the direction of
shock propagation, and the advancement of the shock-wave
can be observed by the FCC–BCC phase transformation.
The aim of investigating the structural transformations during the shock propagation is to get an insight into the phase
transformation (or structural morphology at the atomic
level) and plasticity in each case. It can be inferred from the
initial snapshots that the lower piston velocity leads to the
generation of higher fraction of stacking faults. For a precise analysis of the stacking faults and their differentiation
Fig. 4 Atomic shear strain snapshots of specimen with GB inclination angle 26.57° for a piston velocity of 0.5 km/s at different time steps. The
black coloured arrow shows the direction of propagation of shock-wave
Fig. 5 Common neighbour analysis (CNA) snapshot during structural transformation of specimen with GB inclination angle 0° for a piston
velocity of 0.5 km/s at different time steps. The black coloured arrow shows the direction of propagation of shock-wave
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into intrinsic and extrinsic stacking fault, perfect atoms
are deleted from the bicrystal regions and are portrayed
separately. As the shock-wave propagates, different types
of stacking faults are observed during the deformation in
this case which is shown in the enlarged portion of the particular region near which it is seen. Emission of Shockley
partial dislocations results in the creation of a stacking fault
in between them. This is due to the reduction in the critical
shear stress for slip of partial dislocation compared to that
for perfect dislocation when the specimen size is around
the nanoscale [62]. It is also observed that the formation
of intrinsic and extrinsic stacking faults is along the {111}
plane. As per common neighbour analysis (CNA), a single
HCP coordinated layer represents a coherent twin boundary,
two HCP-coordinated-layers with a FCC coordinated layer
between them represent an extrinsic stacking fault, and the
two adjacent HCP-coordinated layers represent an intrinsic
stacking fault [63]. As the shock-wave progresses inside the
specimen, a greater number of such parallel stacking faults
are observed (extrinsic and intrinsic) as greater number of
slip systems are activated due to the increase in strain and
consequently, increase in the number of dislocations. Meanwhile, the FCC–BCC phase transformation does not occur
immediately after the interaction of a region with the shockwave. On the contrary, this transformation occurs after the
specimen accomplishes shock equilibrated state. From the
tabular data regarding the bulk properties of Nickel mentioned in [58], the values of ΔE for the phase transformation
of nickel from FCC to BCC are quite close to the target value
for the same, which suggests the validity of this particular
potential developed by Mendelev et al. Thus, the nucleation
of BCC phase may be through epitaxial Bain path leading
to the martensitic transformation [50, 64]. This martensitic
transformation is observed to be occurred majorly in the
lower part of the bicrystal around the GB because of the
orientation of the lower grain in which the GB period vector [130] interacts with the shock direction [100]. It is also
T. Konnur et al.
observed that as a virtue of the deformation process, the GB
region near the piston end expands leading to coarsening
while the GB in further half of the specimen is unchanged.
Figure 6 represents the stress profile along with shear
stress distribution map for specimen with GB inclination
angle 0° for a piston velocity of 0.5 km/s at different time
steps. The blue colour corresponds to compression while
the red colour corresponds to tension. Since shock-wave
propagation generates high compression, the wave moving
forward is represented by the specimen colouration turning
blue. In Fig. 6a, few regions with deeper blue colour are seen
representing higher stress. From the previously explained
CNA figures, it can be seen that the stacking faults are
nucleated and martensitic transformation is seen near these
regions. As the shock propagates, the region left behind
experiences minor alteration in the stress experienced due
to which the number of stacking faults increases along with
its dimensions. It can also be visualized through the stress
profile where the wave is exemplified through the disturbances and oscillations in the graph. From Fig. 6b and c it is
observed that the compressive stress levels relax a bit in the
region from where the shock has already passed.
In comparison with the CNA discussed above, the fraction of stacking faults is lesser and the martensitic transformation is almost negligible in the specimen with an inclination angle 26.57° for piston velocity 0.5 km/s as shown in
Fig. 7, which points towards the interpretation that this specimen hardly undergoes plastic deformation. It is observed
that the plastic deformation during the shock propagation at
the lower piston velocity (0.5 km/s) was mediated through
the formation of stacking faults. While the amount of stacking faults formation in the specimen with GB inclination
angle 0° is almost equal in the upper and lower crystal, an
unequal distribution of the same is found in the specimen
with an inclination angle 26.57°. The orientation of the
stacking faults is not symmetric with respect to the GB in the
specimen with an inclination angle 26.57° unlike in the one
Fig. 6 Shear stress snapshots and stress profiles of specimen with GB inclination angle 0° for a piston velocity of 0.5 km/s at different time steps
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Effect of variation in inclination angle of Ʃ5 tilt grain boundary on the shock response of Ni…
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Fig. 7 Common neighbour analysis (CNA) snapshot during structural transformation of specimen with GB inclination angle 26.57° for a piston
velocity of 0.5 km/s at different time steps. The black-coloured arrow shows the direction of propagation of shock-wave
with inclination angle 0°. In the upper part of the bicrystal,
only a small region near the piston undergoes a martensitic
transformation as shown in Fig. 7a–c. In Fig. 7d–f, only a
few stacking faults are formed near the piston with different
orientation in the upper and lower crystal of the specimen.
The intrinsic and extrinsic stacking faults observed are similar as mentioned above. This localized deformation results in
the distortion of the specific structure of the GB to a minor
extent leading to GB getting expanded and coarsened near
the piston end of the specimen while the latter half of the
GB structure is unchanged. A similar trend of stacking fault
formation and martensitic transformation is observed in the
case of specimens with GB inclination angles 11.31°, 18.43°
and 30.96°.
Figure 8 shows the graph plotted between the number of
dislocations and GB inclination angles for each type of dislocation after the shock-wave has passed completely through
each specimen. When metallic materials are subjected to
shock wave generating high pressure compression in the
specimen, the propagation of the shock wave leads to the
nucleation of defects consisting mainly dislocations [65]
of different types. The dislocations can be further distinguished based on their types, namely perfect, Shockley partial, stair rod, Hirth partials, Frank partials and other types.
It is observed that the number of dislocations of the type
Stair-rod, Hirth and Frank remains almost constant when the
GB inclination angle is varied, with the Stair rod and Frank
being almost zero. The Shockley partial show a relatively
greater number for symmetric GBs (inclination angle 0°
and 45°). In contrast, the other type of dislocations shows a
greater number for asymmetric GBs. The number of perfect
dislocations increases with the increase in GB inclination
angle and is maximum for the specimen with inclination
angle 45°. Since the deformation for specimens subjected
to high piston velocity (1.1 km/s) is governed majorly by
the martensitic transformation of Nickel, hardly any dislocations are found. So, the discussion regarding the number of
Fig. 8 Graph between the number of dislocations and GB inclination
angles for each type of dislocation when specimens are subjected to
shock-wave having piston velocity 0.5 km/s
dislocations upon variation of inclination angle is limited for
specimens with low piston velocity (0.5 km/s).
3.4 Structural transformation and atomic shear
strain analysis during the high piston velocity
shock
Figure 9 portrays the atomic shear strain snapshots at various intervals during shock compression process of the bicrystal specimen with GB inclination angle 0° for a higher
piston velocity of 1.1 km/s. Figure 9a–c shows the evolution
of the atomic shear strain through the specimen subjected
to shock loading for the first half of the time period. The
fundamental observation to be made is that the distribution of atomic shear strain is considerably symmetric with
respect to the upper and lower crystal. It is observed that
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T. Konnur et al.
Fig. 9 Atomic shear strain snapshots of the specimen with GB inclination angle 0° for a piston velocity of 1.1 km/s at different time steps. The
black coloured arrow shows the direction of propagation of shock-wave
there is a substantial amount of atomic shear strain experienced by the atoms and is almost uniform throughout the
specimen. An intriguing pattern of evolution of atomic shear
strain is observed with the advancement of a shock wave
into the specimen. A V-shaped curved outward from the
GB, which maintains its curvature throughout the process,
is distinguished. A closer inspection of the GB structure
with inclination angle 0° in Fig. 1 reveals that there is a
peculiar “convex kite-shaped” topological unit which organizes this GB. When the shock wave meets the vertex of the
kite-shaped unit, this particular crystallographic orientation of the two grains leads to the wave front getting curved
along the atomic arrangement. It is also worth mentioning
that after a certain period of time, few high strain planes
are formed which are almost symmetric in orientation with
respect to the GB.
Figure 10 shows the representative illustration of the
CNA snapshots of the bicrystal specimen with GB inclination angle 0° for a piston velocity of 1.1 km/s. In contrast
to the same specimen subjected to a lower shock velocity
of 0.5 km/s, here martensitic transformation takes place
throughout the specimen which is resulted by the nucleation of BCC phase through epitaxial Bain path [50, 64].
During the interaction of the specimen with shock wave
having high piston velocity (1.1 km/s), the deformation of
the specimen leads to the BCC phase of the Nickel being
stabilized. Figure 11 shows the spontaneous martensitic
transformation corresponding to the Bain model. The formation of this BCC phase may be a result of the swift
decrease in the temperature behind the shock front that
releases a significant amount of energy. Hence, for assimilation of this large energy, lattice reorientation occurs
which in turn leads to a structural phase transformation to
BCC from FCC. From the tabular data regarding the bulk
properties of Nickel mentioned in [58], the values of ΔE
for the phase transformation of nickel from FCC to BCC
is quite close to the target value for the same, which suggests the validity of this particular potential developed by
Mendelev et al. Nickel undergoes martensitic transformation when the equivalent strain in the sample during plastic
Fig. 10 CNA snapshot during structural transformation of specimen with GB inclination angle 0° for a piston velocity of 1.1 km/s at different
time steps. The black arrow shows the direction of the propagation of shock-wave
13
Effect of variation in inclination angle of Ʃ5 tilt grain boundary on the shock response of Ni…
Page 11 of 18
358
Fig. 11 Schematic representation of the Bain model along
with the orientation relationship
obtained during the FCC–BCC
phase transition in the specimen
deformation is considerably higher than what is observed
during a quintessential tensile test [66]. Zhang et al. [66]
put forward a compelling experimental demonstration
using XRD and HRTEM in which they showed the formation of a bcc structure in nanocrystalline nickel when subjected to large strains. They also stated that when nickel in
is the length scale of nanometers, the plastic strain can be
accommodating through a change in lattice structure into
an alternative form when subjected to mechanical loading.
In our study of shock-induced compression studies, large
strains are observed which aid in the plastic deformation in
the specimen through the mechanically induced martensitic transformation of nickel. It can be seen from Fig. 10a–c
that the martensitic transformation occurs almost entirely
across the bicrystal as the piston propagates and the manner of BCC transformation is analogous to the observation
recorded in the atomic shear strain evolution discussed
above. The important aspect of this discussion is the effect
of this martensitic transformation on the structural change
of GB. The original GB structure is destroyed and few
amorphous connections are intermittently formed in the
crystal. This structure gets disintegrated through BCC
phase transformation as the shock-wave propagates forward, leaving behind a loop of the amorphous Ni atoms.
Figure 12 portrays the atomic shear strain snapshots at
various intervals during shock compression process of the
bicrystal specimen with GB inclination angle 26.57° for a
higher piston velocity of 1.1 km/s. Figure 12a–c shows the
progression of the shock wave inside the specimen. It is seen
that from the very beginning, there is a large discrepancy in
the manner of interaction of the piston with the upper and
lower crystal, unlike in the above discussion for the specimen with GB inclination angle 0°. The region in the upper
crystal experiences almost uniform atomic shear strain while
the lower crystal has band formation of alternate low and
high strain regions. Also, the propagation of the wave in the
lower crystal is in the form of an elastic wave, which can be
inferred from the alternate bright and dark pattern trailing
the wave front. The specimen considered here embodies an
asymmetric GB giving rise to the difference in the lattice
orientations of the crystals. This leads to the mismatch in the
velocity of the shock wave, with the shock possessing higher
velocity in the lower crystal and thus leading with respect
to the upper crystal. While the nature of strain distribution
Fig. 12 Atomic shear strain snapshots of the specimen with GB inclination angle 26.57° for a piston velocity of 1.1 km/s at different time steps.
The black coloured arrow shows the direction of propagation of shock-wave
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358
Page 12 of 18
T. Konnur et al.
is uniform in the upper crystal as the shock propagates, the
lower crystal responds differently as observed in Fig. 12d–f.
There is a greater generation of regions with higher atomic
shear strain and the particular atomic planes (vertical) have
colouration which implies the same magnitude of atomic
shear strain. In addition, there are a few hotspots seen having extremely high values of atomic shear strain, particularly
near the GB. A closer look of Fig. 12a and b leads to the
observation that in the lower crystal, the atomic shear strain
is developed in the form of curvature. A similar trend of
atomic shear strain distribution is observed in the case of
specimens with GB inclination angles 11.31°, 18.43° and
30.96°.
Figure 13 shows the representative illustration of the
CNA snapshots of the bicrystal specimen with a GB inclination angle 26.57° for a piston velocity of 1.1 km/s. In this
case, a stark contrast is seen regarding the phase transformation between the upper and lower grains of the bicrystal. Almost all the region in the upper crystal undergoes a
complete martensitic transformation as the shock wave progresses in the specimen, implying that higher volume fraction of the BCC phase is formed with the increase in piston
velocity. As discussed above, this martensitic transformation
occurs after the specimen accomplishes shock equilibrated
state and the nucleation of BCC phase may be through epitaxial Bain path [50, 64]. It can be stated that at lower velocity, the gross plastic deformation during shock propagation
is dominated by the generation of stacking faults and slight
transformation from FCC to BCC, while in this case the
martensitic transformation plays a major role. The response
of the lower crystal of the above-mentioned specimen is discussed further. Figure 13a and b show the transformation of
the FCC phase to the amorphous phase in the initial stages
of shock loading. It is also the contributing factor for the GB
transformation into a crystalline–amorphous interface. This
amorphization is a result of the enhanced initial Gibbs free
energy before plastic deformation resulting from the GBs,
which also contributes to the intensification of defect density
[67, 68]. The numerical value for the difference between
the values of Gibbs free energy between the crystalline
and amorphous phase (ΔGv) is 2530.5 × ­106 J/m3 for lower
temperatures [68], similar to the temperature used in this
simulation study. The accumulation of defects is favoured at
low temperature leading to high energy status during plastic deformation, which leads to amorphization involved in
this shock loading process. The low temperature considered
during shock loading can subdue the dynamic recovery and
aid defect accumulation, thus contributing to the amorphous
transformation [67]. Meanwhile, it is worth mentioning that
some regions of the amorphous phase transforms to BCC
as the shock wave further penetrates the specimen. It is
observed that twin formation takes place in the lower grain
during the phase transformation of the specimen. This is a
result of deformation twinning in which a Shockley partial
dislocation is nucleated from the GB. This Shockley partial
which has been nucleated then becomes the reason for the
increase in dislocation activities. The occurrence of twinning in such cases depends on numerous factors like loading
condition, crystal orientation, etc. [30]. The most important
observation to be made here is that the formation of twins is
observed only in the lower grain [110] of the specimen and
not in the upper grain [170]. The twin density also varies
over the course of the shock loading of the specimen. The
formation of twins in the lower grain [110] of the specimen
and not in the upper grain [170] is due to the energetically
favourable process in which slip dislocations get dissociated
into Shockley partials and also the stress-orientation effect
on partial width [69]. On correlating the atomic shear strain
analysis of this specimen (Fig. 12) with its CNA (Fig. 13),
it is observed that the strain accumulation in the specimen
leads to the subsequent amorphization. A similar trend of
martensitic transformation and amorphization is observed
in the case of specimens with GB inclination angles 11.31°,
18.43° and 30.96°. The high strain rates employed here in
MD simulations are corresponding to those pertaining in
shock loading experiments [70, 71]. At higher strain rates
Fig. 13 CNA snapshot during structural transformation of specimen
with GB inclination angle 26.57° for a piston velocity of 1.1 km/s at
different time steps. The black coloured arrow shows the direction of
propagation of shock-wave. Twin in BCC is shown separately and the
corresponding region in the specimen is highlighted in yellow
13
Effect of variation in inclination angle of Ʃ5 tilt grain boundary on the shock response of Ni…
(high piston velocity), the dislocation segments are incapable of propagating quick enough to put up with the increasing strain and hence, the global stress escalates till ample
number of dislocation propagations lead to the reduction of
the global stress. This can be correlated with the suppression
of planar and cross-slip dislocation propagation leading to an
observation of an initial overshoot in the stress–strain curve
at higher strain rates [72].
3.5 Quantitative analysis of the structural
transformation during shock
Figure 14a presents a bar chart illustrating the after-shock
volume fraction of the BCC phase in all the specimens having varying GB inclination angle with respect to the increase
in the piston velocity. It is observed that the BCC phase
increases with an increase in the piston velocity indicating
that the effect of phase transformation is positively correlated to the increasing velocity. It means that martensitic
transformation occurs in the specimen leading to an increase
in BCC volume fraction. Also, another interesting trend is
that for particular piston velocity the BCC volume fraction is
highest for inclination angle 0° and progressively decreases
only to increase again till the inclination angle of 45°. This
observation is a result of the fact that Σ 5(310) and Σ 5(210)
boundaries are symmetric; hence, the shock wave passes
through the specimen uniformly causing deformations of
the same type and almost equal magnitude in both the grains
of the bicrystal across the GB. Figure 14b shows the piston
velocity-dependent variation in the volume fraction of HCP
Page 13 of 18
358
phase after the shock has traversed in the specimens having
varying inclination angles. A strong trend of decrease in
HCP phase is observed with the increase in piston velocity. This can be correlated to the stacking faults formation,
resulting from the dissociation of perfect dislocations into
partial dislocations. At lower piston velocities the deformation of the specimen is supported by defect generation
in the form of stacking faults. As discussed previously in
Sect. 3.1, the specimen with GB inclination angle 0° and 45°
is relatively more stable than the other GBs chosen in this
particular study. Hence, higher fraction of defects is generated in these specimens having symmetrical GBs. Also, in
the case of symmetric GBs (0° and 45°) the orientation of
the grains of the bicrystals is identical relative to the shock
loading direction and the maximum Schmid factor for each
grain is equal [34]. As a result, nucleation and emission of
dislocations take place simultaneously in both grains when
the shock wave interacts with the slip systems, thus producing higher fraction of dislocations in the bicrystal specimen.
In case of asymmetric GBs, the maximum Schmid factor for
each grain in the bicrystals is unequal due to the difference
in orientation of each crystal relative to the shock loading
direction, with the lower grains having relatively greater
Schmid factor. Hence, as the shock progresses through the
specimen the slip systems get activated easily in that region
where higher Schmid factor is observed (since such slip systems possess higher resolved stress). So, the fraction of dislocations generated is relatively lesser in case of asymmetric
GBs. This portrays the significance of variation in inclination angle in this study. As the piston velocity increases the
Fig. 14 Volume fraction analysis of the a BCC phase and b HCP phase concerning the piston velocity and grain boundary inclination angle after
the shock wave has propagated in the specimens
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358
Page 14 of 18
fraction of defects generated in the form of stacking faults
decreases since the deformation is now majorly supported
by martensitic transformation from FCC to BCC and amorphization of nickel due to low temperature and enhancement
of Gibbs free energy as discussed in Sect. 3.4 for explaining
Fig. 13.
3.6 Stacking faults formation and cross‑sectional
analysis
Figure 15 represents the CNA snapshots of the cross section
of different specimens showing the evolution of stacking
faults generation in the bulk of the bicrystal specimens at
different time instances for the piston velocity of 0.5 km/s.
Here, “ℓ” is the length at which the specimen is sliced, calculated from that end where the piston is considered (refer
Fig. 15a). Figure 15b shows pertaining to the specimen with
GB inclination angle 0° (here, ℓ = 100 Å) and illustrates that
the process of stacking faults generation is from the surface
of the specimen towards the GB, whereas Fig. 15d pertains
to the specimen with the GB inclination angle 45° (here,
ℓ = 114 Å) and shows that the way stacking faults are formed
is from the GB and propagate towards the surface of the
specimen. Corresponding planes along which the orientation
of stacking faults is observed are earmarked in the figure.
It can also be seen that the martensitic transformation also
follows the same pattern. The nucleation and emission of
Shockley partial dislocation from the GB during the deformation process indicate the onset of dislocation activity. The
initial single partial will advance through the cross section
of the entire grain only to be incorporated in the opposite
GB, provided there is no emission of another partial [69].
This leads to the formation of an extended stacking fault that
crosscuts the specimen in a transverse manner. Similarly,
a micro-twin is created if a trailing partial dislocation is
released upon the adjoining slip plane to the initially nucleated single partial dislocation. Deformation twinning is said
to commence after the formation of such micro-twin [69]. In
simulation studies involving Ni, extended stacking faults are
observed predominantly because there is a very negligible
difference in the high energy barriers that the full and twin
fault slip processes need to overcome [69]. This is the reason
why less twins and full dislocations are seen while more
extended stacking faults are observed in Ni. Moreover, after
the emission of a leading partial, stress relieving from the
GB can be observed as a consequence of the local atomic
shuffling. This relaxation warrants more time (in the order of
seconds) which can aid in building up the stress required to
overcome the barrier for a twin to be observed [69]. Hence,
twins are not seen frequently in MD simulations as the time
resolution up to seconds is not computationally viable. The
decrease in the Peierls barrier when the applied stress is
increased as the shock wave progresses leads to increased
13
T. Konnur et al.
dislocation actions [73]. The ratio of stacking fault energy
to unstable stacking fault energy (γsf/γusf) for Ni is 0.55 [73].
When this ratio is close to unity the energy barrier needed
to overcome in order to generate a trailing partial is quite
low; hence, full dislocations can be observed even though
there may be some presence of structural relaxations in the
GB. But, in case of Ni, γsf/γusf is lower, hence the energy
needed for the nucleation of trailing partial is considerably
higher. Hence, there are almost no full dislocations seen in
this study. Here, it is observed that for a specimen with a
GB inclination angle 0° the generation of extended stacking
faults is from the surface of the specimen towards the GB.
But, in case of specimen with GB inclination angle 18.43°
it is observed that extrinsic stacking faults are formed in one
of the grains of the bicrystal along with the extended and
intrinsic stacking faults in the other grain. Here, the intrinsic
and extrinsic stacking fault generates from the surface as
well as the GB. Moreover, their formation is seen in only
one of the grains in the beginning. Similar observations are
found in specimens with GB inclination angle 11.31°, 26.57°
and 30.96°. In this pictorial representation, some intrinsic
stacking faults might look wider. That is because of the overlap of numerous intrinsic stacking faults since the specimen
has been sliced along a particular plane and viewed along the
negative x-axis as shown in the figure. In these specimens of
asymmetric GB, the grains of the bicrystal differ in lattice
orientation with respect to the shock direction, which leads
to the dissimilar Schmid factor. Consequently, the slip system is activated preferentially in that grain which has greater
Schmid factor as it leads to greater resolved shear stress. The
stacking faults in specimen with GB inclination angle 45°
are generated from the GB towards the surface.
4 Conclusions
We have implemented molecular dynamics (MD) simulations to model and examine the shock response of a nickel
bicrystal specimen, which embodies Ʃ5 GB having different
inclination angles. The orientation of GB is considered parallel to the loading direction. The effect of the GB inclination angle on the shock response of the bicrystals has been
studied by considering different shock wave velocities. We
have carried out different analyses to get an insight into the
deformation behaviour, structural evolution and phase transformations in the specimen. Based on the results obtained
from MD simulations and various analyses, the following
conclusions can be made:
• Plastic deformation is assisted through shock compres-
sion wherein a mismatch between the shock velocities
across the GB leads to generation and consequent dispelling of shear stress. Additionally, asymmetric tilt bounda-
Effect of variation in inclination angle of Ʃ5 tilt grain boundary on the shock response of Ni…
Page 15 of 18
358
Fig. 15 a Common neighbour analysis (CNA) snapshot of the sliced
specimen for a piston velocity of 0.5 km/s, sequential CNA snapshots
of the cross section of the sliced specimen illustrating shock propaga-
tion at a piston velocity of 0.5 km/s with GB inclination angle b 0°, c
18.43° and d 45°
ries show a greater extent of mismatch in velocities of the
shock front, which is directly proportional to the shear
stresses generated.
• The cross-sectional analysis of stacking fault and twin
formation shows that in the case of Ni, stacking faults
formation is more prevalent. There are almost no full
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358
Page 16 of 18
dislocations since trailing partial is not emitted on the
same slip plane with respect to the leading partial.
• At lower shock velocities, significant stacking fault
generation and less martensitic phase transformation is
observed, but at higher shock velocities, twinning and
active deformation processes are less prominent, while
more significant martensitic transformation along with
amorphization is seen predominantly across the specimen. Also, in case of asymmetric tilt boundaries the GB
structure is greatly disintegrated at higher shock velocities.
• The response of bicrystals under shock compression suggests that alteration of the inclination angle influences the
effect of plasticity at the GB which can dictate the failure
at GB.
The above observations and conclusions in shock wave
simulations are made for the very high strain rate conditions of the order ­105/sec and ­106/sec which is generally
observed in case of ballistic impacts causing generation of
shock waves. It is anticipated that this work can help in discerning the atomistic deformation mechanisms in the course
of the shock loading process of Ni bicrystal specimen, which
can promote the accelerated design and development of bicrystals with superior capability to resist high shock loads.
Our study using MD simulations also aims to support the
advancement of grain boundary engineering in Nickel based
materials by providing a framework for modelling of related
polycrystalline solids at higher length scales.
Supplementary Information The online version contains supplementary material available at https://​doi.​org/​10.​1007/​s00339-​021-​04502-z.
Authors’ contribution TK has contributed towards Data curation, Formal analysis, Investigation, Software, Methodology, Visualization,
Validation, Writing – original draft. KVR has contributed towards Conceptualization, Software, Validation, Resources, Project administration,
Supervision, Writing – Review & Editing. SP has contributed towards
Conceptualization, Data curation, Funding acquisition, Investigation,
Methodology, Project administration, Resources, Software, Supervision, Validation, Writing – Review and editing.
Funding The authors did not receive support from any organization
for the submitted work.
Availability of data and material The raw/processed data required to
reproduce these findings can be shared upon request.
Code availability The code for the simulations can be provided upon
request.
Declarations
Conflict of interest All authors certify that they have no affiliations
with or involvement in any organization or entity with any financial
interest or non-financial interest in the subject matter or materials discussed in this manuscript.
13
T. Konnur et al.
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