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Shear deformation kinematics of bicrystalline grain boundaries in atomistic simulations
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2010 Modelling Simul. Mater. Sci. Eng. 18 015002
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IOP PUBLISHING
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002 (15pp)
doi:10.1088/0965-0393/18/1/015002
Shear deformation kinematics of bicrystalline grain
boundaries in atomistic simulations
G J Tucker1 , J A Zimmerman2 and D L McDowell1
1 School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive,
Atlanta, GA 30332, USA
2 Sandia National Laboratories, Livermore, CA 94550, USA
E-mail: [email protected]
Received 13 May 2009, in final form 27 August 2009
Published 10 December 2009
Online at stacks.iop.org/MSMSE/18/015002
Abstract
The shear deformation behavior of bicrystalline grain boundaries is analyzed
using continuum mechanical metrics extracted from atomistic simulations.
Calculating these quantities at this length-scale is premised on determining the
atomic deformation gradient tensor using interatomic distances. Employing
interatomic distance measurements in this manner permits extension of
the deformation gradient formulation to estimate important continuum-scale
quantities such as lattice curvature and vorticity. These continuum metrics
are calculated from atomic deformation fields produced in 2D and thin 3D
equilibrium bicrystalline grain boundary structures under shear at 10 K. Results
from these simulations show that interface structure strongly influences the
resulting accommodation mechanisms under shear and deformation fields
produced in the surrounding lattice. Calculating these continuum quantities
at the nanoscale lends insight into localized and collective atomic behavior
during shear deformation for various mechanisms, and it is shown that different
mechanisms lead to differing behavior. Additionally, the results of these
calculations can perhaps serve as an intermediary form to inform continuum
models seeking to explore larger-scaled grain boundary deformation behavior in
3D, and to evaluate the veracity of continuum models that overlap the nanoscale.
1. Introduction
Considerable progress has been made in understanding key structure/property relationships
governing the mechanical behavior in many engineered materials. However, questions
remain regarding behavior at the nanoscale and its relationship to larger-scaled behavior
commonly observed for these materials. For example, deviation from the Hall–Petch relation
[1, 2] in nanocrystalline metals has recently been measured experimentally and examined
computationally [3–6]. Researchers have found that as grain size is reduced to the nanometer
0965-0393/10/015002+15$30.00
© 2010 IOP Publishing Ltd
Printed in the UK
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
scale, inelastic deformation mechanisms accommodating plastic strain differ from those found
in larger-grained polycrystalline materials. It has been suggested that intercrystalline regions
such as grain boundaries and triple junctions possibly serve as nucleation sites and carriers of
the majority of inelastic deformation modes [7–10]. However, the influence of processes such
as dislocation/grain boundary interactions, interfacial dislocation nucleation and the onset of
grain boundary sliding and migration occurring within these structures on macroscopic material
behavior is not well understood.
To improve our knowledge of these types of processes, computational simulations have
emerged to provide valuable insight into these events that tend to evade traditional experimental
techniques. However, difficulty remains in that although dislocation mediation at a grain
boundary can be understand from an atomic perspective, the interactions of dislocation
populations are well beyond the reach of the nanometer scale modeling techniques. Molecular
dynamics simulations have the capability of probing atomic-scale behavior, such as dislocation
nucleation and slip transfer reactions at interfaces in nanocrystalline materials [8, 11–13],
but inherently lack the ability to connect to larger-scale computational methods founded
on continuum mechanics principles. There has been significant progress in the scientific
community to understand these material processes and the relationships therein through the
use of improved multiscale computational models [14–17], but severe limitations exist in
methods such as domain decomposition in exchanging dislocations between atomistic and
continuum domains.
In this paper, we outline a method for calculating continuum mechanical quantities based
on the concept of the atomic deformation gradient of Zimmerman et al [18] using atomic
position. The technique is premised on defining continuum metrics for each atom in the
current configuration that makes use of spatial and reference neighbor lists. Both 2D and thin
3D equilibrium bicrystalline grain boundary structures are used to analyze the low temperature
shear deformation behavior of the boundary and surrounding lattice during deformation. The
defined metrics are calculated for the entire domain, and estimates of lattice curvature and
vorticity are discussed.
2. Mathematical formulation
It is useful to briefly outline the methodology to estimate the deformation gradient tensor
F , rotation tensor R, velocity gradient tensor L and vorticity tensor W . Atomic strain
measurements are defined from the interatomic separation distance between an atom α and
its neighbor β. The deformation mapping F = ∂ x/∂ X will be outlined here, with more
details provided by Zimmerman et al in [18]. Reference configuration quantities will be noted
by upper case symbols while current configuration quantities will be lowercase, this includes
all subscripts which refer to coordinate components of each quantity. For example, in the
deformation gradient formulation below, X refers to reference configuration coordinates and
x refers to those in the current configuration. Using the interatomic separation distance, (x αβ )i ,
the deformation mapping of atom α and one of its neighbors β can be written as
(x αβ )i = FiI (X αβ )I .
(1)
Although equation (1) will be true for each atom–neighbor pair (i.e. α − β), it will not
completely determine F for all nearest neighbors. Therefore, this formulation requires
averaging over some finite domain incorporating multiple neighboring atoms. Thus, by
summing over all neighbors (β = 1 . . . n) and minimizing the squared errors with respect to
F , the atomic deformation gradient for each atom α is defined as a function of both reference
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
and current interatomic spacings based on the reference configuration neighbor list, i.e.
(ωα )iM = (F α )iI (ηα )I M ,
α
(2)
α
where ω and η are defined as
n
(x αβ )i (X αβ )M
(ωα )iM =
(3)
β=1
and
(ηα )I M =
n
(X αβ )I (X αβ )M .
(4)
β=1
It should be noted that this method for estimating F based on nearest neighbors can be
expanded to incorporate additional neighbor shells (e.g. 2nd, 3rd, etc) to provide various
averaging domains. A similar approach for determining a type of average atomic deformation
gradient has also been developed by Hartley and Mishin [19], although their method is more
particular with regard to neighbor acceptance and rejection. The method outlined here relies
on calculations based on retaining nearest neighbor identities throughout deformation, while
the Hartley and Mishin formulation of a lattice correspondent tensor from interatomic bond
angular deviations is not as neighbor specific.
After estimating F , the method outlined by Franca et al [20] is followed to determine the
right Cauchy–Green strain tensor (C ), the right stretch tensor (U ) and its inverse (U −1 ) in
order to calculate R. Once U −1 is calculated from this approach, R is determined from right
polar decomposition as
R = F U −1 .
(5)
Then the skew-symmetric part of R, Rskew , permits calculation of an associated axial vector
that defines the microrotation vector, φ, i.e.
Rskew = 21 (R − RT ),
(6)
φk = − 21 ij k (Rskew )ij .
(7)
Here, ij k is the permutation tensor.
We next extend consideration to current configuration kinematic quantities, which do not
rely on reference configuration neighbor lists, but rather on updated neighbor lists at each
timestep in the current configuration. In addition to estimating lattice curvature from φ and
its gradient, it is also instructive to calculate the vorticity or spin tensor during deformation
processes. At this point, we would like to recall an important conclusion made by Zimmerman
et al in [18] which states that currently it is not possible to separate the elastic and plastic
contributions to each deformation metric (such as F and L); therefore, it is not the case
that only information pertaining to inelastic deformation processes are captured with these
formulations.
To calculate the vorticity tensor, the velocity gradient tensor, L, is calculated as a function
of the instantaneous atomic velocities (v ) at each timestep, i.e.
∂v
L=
.
(8)
∂x
Then, W is found by utilizing the additive decomposition of L into the summation of the
symmetric part of L, the rate of deformation tensor (D ), and the skew-symmetric part of L,
the vorticity or spin tensor (W ), as follows:
L = D + W,
(9)
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
where W is calculated as
W = 21 (L − LT ).
(10)
As before, a dual vector representation of W exists in terms of the vorticity vector, ω , given by
ωk = − 21 ij k Wij .
(11)
For our analysis, we calculate L using three different approaches. Before we begin our
detailed formulations, a clear distinction between reference/current configuration coordinates
(denoted by X and x, respectively) and reference/current configuration neighbor lists needs to
be made. As previously mentioned in this section, upper case symbols refer to reference
configuration quantities while lower case symbols refer to current quantities. Therefore,
calculation of F always depends on reference coordinates (see equations (1)-(4)), but F can be
calculated for either reference or current neighbor lists while still utilizing reference coordinates
for those determined neighbors. This highlights one significant advantage of our approach for
determining kinematic quantities, we have the option of using either the reference/current
coordinates for atoms that are currently neighbors or the coordinates for atoms that were
neighbors in the reference configuration. Therefore, for current kinematic quantities such as
L, we wish to use updated neighbor lists but still retain reference coordinate information where
appropriate.
First, a method analogous to that used for F can be employed, where equation (8) is used
to define L in terms of the spatial atomic velocity and neighbor distances.
(v αβ )i = Lik (x αβ )k
(12)
can be rearranged.
(v αβ )i − Lik (x αβ )k = 0.
Then defining
Diα
(13)
as the summed squared errors over all neighbors
Diα =
n
(v αβ )i − (Lα )ik (x αβ )k
2
(14)
β=1
and minimizing by some choice of Lα and setting equal to zero, i.e.
∂ Dα
=0
∂ Lα
gives the following:
n
(v αβ )i (x αβ )m − (Lα )ik (x αβ )k (x αβ )m = 0.
(15)
(16)
β=1
This equation can be simplified to
(ρ α )im = (Lα )ik (τ α )km ,
α
(17)
α
where (ρ )im and (τ )km are defined as
n
(ρ α )im =
(v αβ )i (x αβ )m
(18)
β=1
and
(τ α )km =
n
(x αβ )k (x αβ )m .
β=1
Then ω can be calculated according to equations (10) and (11).
4
(19)
Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
The second approach is to utilize another formulation of L, using the material time rate
of change of F , Ḟ , and the inverse F −1 , i.e.
∂v
L = Ḟ F −1 =
,
(20)
∂x
where
∂v
Ḟ =
(21)
∂X
and once again minimizing the sum of the squared errors to estimate Ḟ .
(λα )iM = (Ḟ α )I K (ηα )KM ,
(22)
where
(λα )iM =
n
(v αβ )i (X αβ )M
(23)
β=1
and η has been previously defined in (4). Then ω can once again be calculated from equations
(10) and (11).
The final approach is to calculate Ḟ explicitly from the calculation of F in successive
timesteps, i.e.,
Fcurrent − Fpast
Ḟ =
,
(24)
δt
where δt is the time interval between the configurations used for calculating F . Then L is
calculated from equation (20) and ω follows from (10) and (11).
3. Computational method
Equilibrium bicrystalline grain boundary structures were produced of both 2D and ‘thin’ 3D
character. The interface structure in both setups is composed of a symmetric tilt grain boundary
located in the center of the simulation domain with the interface normal vector in the vertical (y)
direction and the grain boundary period vector in the shear (x) direction. In the 3D simulations,
the grain boundary tilt axis (z) direction is also considered. Periodic boundary conditions are
employed for directions parallel to the grain boundary (x and z), but not in the vertical (y)
direction. Free surfaces thus formed in the y direction are constrained such that all atoms
located within a specified distance from each free surface are forced to move as a rigid group
(free from interatomic interactions) in the x-direction during shear loading as displayed in
figure 1.
The interatomic potential used in the 2D simulations is a modified Lennard–Jones potential
that has been shifted and truncated so that the potential energy and its first derivative are
zero at the specified cutoff distance of 7.6364 Å. The important parameters are the atomic
mass (196.97 amu), finite distance at which the potential is zero (σ = 3.636 38 Å) and the
depth of potential ( = 1.5726 eV). These parameters lead to a lattice parameter of 4.08 Å
and a cohesive energy of −3.93 eV. Minimum energy configurations were calculated using a
conjugate gradient method in LAMMPS [21] using a relative energy convergence criterion
of 10−25 . For 3D simulations, an embedded atom method (EAM) potential for copper from
Mishin et al [22] is employed. After energy minimization, each simulation cell was then
allowed to equilibrate at 10 K for 10 ps. In both 2D and 3D, the time step used was 1 fs and
all atomic dynamics simulations were performed in the microcanonical ensemble (NVE).
To apply shear, the lower rigid atomic region containing all atoms within three times the
potential cutoff distance of the bottom free surface is held fixed, and the upper rigid atomic
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 1. A 2D schematic of the simulation cell and conditions for prescribing simple shear.
region is prescribed a constant velocity in the shear (x) direction. This velocity superimposes
on temperature induced fluctuations. Because of the inherent high strain rate condition of
atomistic simulations, a linearly ramped velocity field is also prescribed to each atom located
between the upper and lower rigid regions to alleviate possible shock wave generation [23, 24].
As shown in figure 1, atoms located near the lower rigid region are given an additional velocity
value close to zero and those near the upper region are given values near to the shear velocity.
The applied shear velocity corresponds to an approximate shear strain rate of 108 s−1 , where
shear strain is defined as γ = arctan(l/do ). In this equation, l is the shear displacement or
relative displacement of the upper boundary to the lower boundary in the x-direction and do is
the vertical distance between the lower and upper rigid atomic regions.
4. Simulation results
4.1. Two-dimensional simulations
Three different symmetric tilt bicrystalline structures were used in the 2D shear analysis.
Each structure is approximately 300 Å × 300 Å containing around 7000 atoms with a varying
disorientation (minimum misorientation) angle (). The resulting equilibrium structures are
shown in figures 2(a)–(c), colored according to potential energy (eV), and it is clear that
the atomic structure composing each bicrystal varies with . The three different values
are 9.4◦ , 15.2◦ and 27.8◦ . Each structure displays a different mechanical response under
shear. The deformation mechanisms are grain boundary migration, sliding and dissociation
respectively, and it is clear from figures 2(d)–(f ) that a unique deformation field accompanies
each mechanism, and that only the migration mechanism preserves the initial grain boundary
structure. These results show the relationship between structure and mechanical behavior,
and suggest the influence of atomic interface composition on deformation. These three
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 2. Initial grain boundary structures for (a) = 9.4◦ , (b) = 15.2◦ , (c) = 27.8◦ , and
after approximately 5% shear strain in (d), (e) and (f ), respectively. Atoms are colored according
to their potential energy (eV).
2D symmetric tilt bicrystal grain boundaries were chosen because each displayed a unique
deformation mechanism.
To gain additional insight into the shear deformation behavior of these boundaries and
obtain more useful information, the previously outlined continuum quantities were calculated
for each structure. Since F is a deformation mapping formulated using the reference neighbor
list calculated in the initial reference configuration (0% strain), a sense of path dependence or
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 3. (a) F12 and (b) F11 calculated for grain boundary migration ( = 9.4◦ ) at approximately
5% shear strain.
path history is found by calculating components of F for all atoms with the simulation domain.
For example, in the grain boundary migration mechanism, figures 2(a) and (d) have shown that
the initial defect atomic structure composing the grain boundary is preserved after boundary
migration at 5% shear strain. Accordingly, figure 3 shows F12 and F11 calculated from first
nearest neighbors only, and it is shown that these components of F provide detailed information
concerning the deformation path of atoms traversed by the migrating grain boundary. Although
atoms located within the red-colored region in figure 3(a) currently reside in their equilibrium
lattice positions, the deformation gradient captures some degree of their deformation history.
This region has undergone lattice rotation as a consequence of the migrating grain boundary,
so that the orientation vectors of this lattice region now correspond to those describing the
lower lattice region before migration. The relatively constant F12 value for these atoms results
from similar horizontal shifts with regard to the vertical position after boundary migration.
Figure 3(b) shows F11 for the grain boundary migration mechanism, this image varies
from 3(a) because it displays a different component of F . However, atoms traversed by the
interfacial defect structures show a different F11 value than those atoms located between the
defect structure migration path. Therefore, atoms directly involved in the defect structure
migration undergo a larger shift with regard to the horizontal direction. This difference is
seen in the highlighted migration paths of figure 3(b). The stress-driven mechanism of grain
boundary migration outlined here also suggests that coupled shear behavior detailed by Cahn
et al [25] exists in this boundary. The migration paths in figure 3(b) show that a small tangential
translation of the upper lattice with respect to the lower lattice occurs during shear deformation.
Figure 4 shows F12 and F11 for grain boundary dissociation, this deformation mechanism
differs from the grain boundary migration results shown in figure 3 leading to the conclusion
that as the mechanism changes, so does the resulting deformation field. In figure 4(a), lattice
regions where dissociation has occurred show higher F12 values. Atoms located within first
nearest neighbor distances of these dissociation regions experience a notable change in position
relative to their reference neighbors regarding vertical position, leading to a higher calculated
F12 . Figure 4(b) shows F11 for the dissociation mechanism, and it is clear that atomic movement
within the dissociation regions does not produce significant changes in the horizontal position
relative to the initial configuration. These examples show that calculating F for atoms during
deformation provides some insight into the path history of the atomic configurations.
Figures 5(a) and (b) display F12 and F11 calculated for grain boundary sliding, and it
is clear that this deformation mechanism differs from the others with regard to the extent of
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 4. (a) F12 and (b) F11 calculated for grain boundary dissociation ( = 27.8◦ ) at
approximately 5% shear strain.
Figure 5. (a) F12 and (b) F11 calculated for grain boundary sliding ( = 15.2◦ ) at approximately
5% shear strain.
lattice deformation. This mechanism produces limited deformation into the lattice apart from
that contained at the boundary; therefore, further analysis of this boundary and deformation
mechanism will be ignored in this work.
Calculating additional continuum quantities such as lattice curvature can also provide
useful insight into atomic behavior during grain boundary plasticity. According to equation (7),
components of the microrotation vector are calculated for each mechanism. Figures 6(a) and
(b) show φ3 for the migration and dissociation mechanisms, respectively.
In both cases, the calculation of φ3 shows atomic microrotation fields accompanying each
mechanism. In (a), there are two clear regions of microrotation, the migration path of the
boundary defect structures (colored red) and the regions between these paths (colored blue),
where atoms within each region have opposite microrotation values. In (b), regions where
dissociation has occurred coincide with high microrotation magnitude; however, the value
is of opposite sign on either side of the dissociation. In addition, it is noted that smaller
microrotation is experienced by atoms located within the boundary even where dissociation
has not occurred, and their microrotation value is approximately that of some atoms within the
dissociation regions.
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 6. φ3 for (a) grain boundary migration ( = 9.4◦ ) and (b) grain boundary dissociation
( = 27.8◦ ) at approximately 5% shear strain.
Figure 7. ω3 calculated with the (a) first, (b) second and (c) third approaches for the grain boundary
dissociation mechanism ( = 27.8◦ ).
Vorticity is an additional metric used to gain insight into the deformation behavior
of material substructures. Since vorticity is a measure of instantaneous atomic behavior,
the current atomic velocities along with an updated neighbor list is used for all vorticity
calculations. After calculating ω3 with each approach, we have come to the following
conclusions as the results of each approach give varying information.
The first approach contains much more calculated noise than the other two methods and no
apparent vorticity fields around the grain boundary dissociations. We speculate the source of
the noise is due to utilizing two correlated atomic properties that vary with time in the method,
v and x. The effect of this correlation could enhance thermal contributions to the velocity
gradient calculations using instantaneous atomic velocities. The second and third approaches
offer much smoother vorticity fields, but also differ from each other substantially. Calculated
atomic vorticity values in the second approach (figure 7(b)) are much greater than both methods
1 and 3, and both methods 2 and 3 capture vorticity fields near the leading and trailing edges
of the dissociations. Additionally, method 2 only uses one atomic property that varies in time,
v . Method 3 provides very clear vorticity fields in regions surrounding the dissociated planes
and their intersection with the grain boundary plane. The exact reason for varying effects in
the calculated vorticity fields is not clear at this time and future work is warranted to provide
explanations for this result.
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 8. The 3D 9 (2 2 1) symmetric tilt grain boundary structure at (a) 0% and (b) about 5%
shear strain colored according to centrosymmetry.
4.2. Three-dimensional simulations
A 9 (2 2 1) copper symmetric tilt grain boundary with a tilt axis of 1 1 0 and parallel to
the out-of-page direction is constructed to further analyze the shear deformation kinematics
using continuum quantities in 3D. The dimensions of the simulation cell are approximately
300 Å × 300 Å × 5 Å containing about 40 000 atoms. The initial and deformed structures are
shown in figure 8 colored according to centrosymmetry [26].
Partial dislocation nucleation along with grain boundary sliding are the observed
deformation accommodation mechanisms in this structure under shear. At approximately
5% shear strain, a partial dislocation is emitted into the lower lattice coinciding with extensive
deformation and structural reconfiguration at the interface. We note that the activation of partial
dislocation nucleation along with grain boundary sliding from this particular grain boundary
structure correlates with previous findings of a thin 3D bicrystalline 9 (2 2 1) grain boundary
performed by Sansoz and Molinari [27]. It is worth noting that due to the small dimension
size in the z-direction, the observed deformation mechanism of partial dislocation nucleation
from the grain boundary might change as the length of this periodic dimension increases.
Also, the emitted partial dislocation has a strong interaction with itself because of this thin
dimension length and periodic boundary conditions in the z-direction, which likely influences
its nucleation criteria from the grain boundary and motion through the lattice. In addition,
atomic movement now allowable in the z-direction during shear deformation is also thought
to lead to more complex interfacial structures than those observed in the true 2D structures
analyzed in section 4.1. Calculation of the presented continuum metrics again provides more
insight into nanoscale behavior during the shear deformation of this boundary.
Figure 9(a) shows F12 , and it is clear that significant sliding has occurred within the
boundary along with the nucleation of a partial dislocation as a result of the application of
shear. Atoms located on the slip plane trailing the leading partial dislocation have a lower
F12 value than most atoms located within the strained boundary region. This indicates that
atomic deformation resulting from partial dislocation slip can be up to a magnitude less than
that experienced by atoms located within interfacial regions experiencing sliding and more
complex deformation. Higher interfacial free volume due to the atomic composition of the
grain boundary (E structural unit) has been noted to lend higher potential mobility to atoms
within these particular structural units during grain boundary loading [27–30]. Therefore,
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 9. (a) F12 and (b) φ3 calculated during partial dislocation nucleation in the 9 boundary.
Figure 10. ω3 calculated with the (a) first, (b) second and (c) third approaches for the 9 grain
boundary. Note that the scale for each figure varies considerably.
atoms located near these higher free volume areas can undergo larger shifts than those atoms
within the surrounding lattice, which is observed in these calculations. Figure 9(b) shows the
calculation of φ3 once again resulting in larger microrotation fields observed within the grain
boundary region compared with the surrounding lattice regions. Slip plane atoms exhibit a
lower microrotation value than those in the grain boundary, once again pointing to larger atomic
motion and deformation due to grain boundary sliding than dislocation slip. The influence of
grain boundary structure on atomic-scale behavior during deformation is clearly seen in these
examples.
Figure 10 shows the resulting vorticity fields viewing along the tilt axis, 1 1 0, after using
the formulated vorticity measure by each of the three approaches outlined earlier. Different
ranges are used for each figure, unlike those shown in the 2D example, to show the resulting
vorticity fields from each method. Once again, the first method results in no distinct vorticity
field during deformation, and the presence of noise throughout the structure. The second
method does capture small vorticity fields both in the grain boundary region and on the slip
planes trailing the emitted leading partial dislocation. The non-uniform vorticity values and
rather complex fields observed for atoms located within the grain boundary agree with previous
results for this boundary using other kinematic quantities. Finally, the third method shows two
distinct vorticity fields around the slip plane near the leading partial dislocation. However,
the magnitude of the vorticity differs between the two slip planes located in the lower lattice.
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
Figure 11. φ3 calculated with (a) only nearest neighbors and (b) including second nearest neighbors
and the weight function.
Once again, the reasons for the different vorticity results are unclear at this time, and future
work is necessary.
4.3. Weight function
To obtain more nonlocal information to estimate each continuum quantity, additional neighbors
can be included in the calculation of each continuum quantity. However, as more neighbors are
considered, the influence of each must be duly weighted. A weight function is an appropriate
measure to implement in the calculations as neighbors further from the atom of interest are
included. As previously outlined by Gullett et al [31], the approximate form of the weight
function and cutoff radius can influence the calculated results. It is therefore vital to understand
the effect of including additional neighbors on each calculation. For example, Gullett et al have
found that in the case of slip, neighbors not directly involved in the slip process but included
in the calculation can have a distinct contribution to the calculated strain. In our calculations,
as a larger cutoff distance is used and a weight function is required, our method ensures that
nearest neighbors have the greatest influence in each calculation. Those neighboring atoms,
designated as nearest neighbors, are given the weighting value of unity, and all other neighbors’
values according to equation (25).
2
r − R1 2
W (r) = 1 −
.
(25)
Rc − R 1
In this equation, W (r) is the weight value dependent on the interatomic distance (r). R1 is the
first nearest neighbor distance and Rc is the cutoff distance.
Once W (r) is calculated for each atom, each neighbor atom’s contribution to the kinematic
quantity being calculated is weighted accordingly. For example, when F11 is being calculated
with two neighbor shells, a second neighbor shell atom’s influence on F11 will be multiplied by
its W (r) value, and a first neighbor shell atom’s weight will be one. Including more neighboring
atoms and their appropriate weight values leads to slightly different calculated values for each
of the continuum quantities, and less smooth fields. A representative comparison is shown
in figure 11, where (a) only nearest neighbors are considered and (b) the results including
second nearest neighbors with the appropriate weight values considered as well. As Gullett
et al pointed out, including more neighbors has a direct effect on the calculated values of
interesting phenomena. In addition, including more neighbors in the calculations must be
premised on the effective range of the process under consideration.
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Modelling Simul. Mater. Sci. Eng. 18 (2010) 015002
G J Tucker et al
5. Concluding remarks
This paper has outlined methods to estimate continuum mechanical quantities such as R, L,
φ and ω based on atomistic simulations, and has used these formulations to analyze the shear
deformation of 2D and thin 3D bicrystalline grain boundary structures. It has been shown that
insight into localized and collective atomic deformation behavior can be gained through the
calculation of each metric. Atomic values of each quantity have been calculated and visualized
to gain additional understanding into deformation phenomena at the nanoscale vital to each
mechanism. Deformation fields produced from each mechanism show vast differences in each
calculated quantity, and the influence of initial interface structure has been shown.
Three different mechanisms were seen in 2D and compared based on estimated kinematical
quantities. Each mechanism displayed a different behavior with different deformation fields.
Interfacial atomic behavior was analyzed and it was shown that enhanced understanding
is acquired through each metric. In the thin 3D structure, important information was
found concerning atomic deformation in varying regions of the deformation field. Atoms
traversed by the nucleated partial dislocation show different behavior with regard to slip and
microrotation than atoms within the boundary taking part in grain boundary sliding. Once
again, interfacial structure was found to be influential on deformation and the resulting atomicscale dynamics. Vorticity calculations were less significant for the presented structures and
strain values, but we note that more interesting vorticity fields are likely as defect concentrations
increase. Additionally, although more nonlocal influence resulted by the inclusion of additional
neighbors and the appropriate weight values, the analysis of important findings and insight did
not change.
We have shown that key continuum mechanical quantities can be formulated and
used in atomistic simulations based on interatomic distance calculations, and that they
provide meaningful and fundamental knowledge about interface phenomena integral to
material deformation. Future work includes investigating fully three-dimensional simulations
composed of more complex atomic structures (e.g. nanocrystals) containing multiple
intercrystalline regions, and outlining specific relationships to the measurements from
influential structural features.
Acknowledgments
GJT is grateful for the support of Sandia National Laboratories through the Enabling Predictive
Simulation Research Institute (EPSRI) intern program. GJT and DLM are grateful for the
support of the NSF grant (CMMI-0758265) on multiresolution, coarse-grained modeling of
3D dislocation nucleation and migration. Sandia is a multiprogram laboratory operated by
the Sandia Corporation, a Lockheed Martin Company, for the United States Department of
Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
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