# Modeling the effects of varying Ti concentration on the mechanical
properties of Cu-Ti alloys

Vasileios Fotopoulos<EMAIL_ADDRESS>[  Corey S. O’Hern  [  Mark
D. Shattuck  [  Alexander L. Shluger  [

###### Abstract

The mechanical properties of Cu-Ti alloys have been characterized extensively
through experimental studies. However, a detailed understanding of why the
strength of Cu increases after a small fraction of Ti atoms is added to the
alloy is still missing. In this work, we address this question using density
functional theory (DFT) and molecular dynamics (MD) simulations with modified
embedded atom method (MEAM) interatomic potentials. First, we performed
calculations of uniaxial tension deformations of small bicrystalline Cu cells
using DFT static simulations. We then carried out uniaxial tension
deformations on much larger bicrystalline and polycrystalline Cu cells using
MEAM MD simulations. In bicrystalline Cu, the inclusion of Ti increases the
grain boundary separation energy and maximum tensile stress. The DFT
calculations demonstrate that the increase in tensile stress can be attributed
to an increase in the local charge density arising from Ti. MEAM simulations
in larger bicrystalline systems have shown that increasing the Ti
concentration decreases the density of stacking faults. This observation is
enhanced in polycrystalline Cu, where the addition of Ti atoms, even at
concentrations as low as 1.5 at.%, increases the yield strength and elastic
modulus of the material compared to pure Cu. Under uniaxial tensile loading,
the addition of small amounts of Ti hinders the formation of partial Shockley
dislocations in the grain boundaries of Cu, leading to reduced local
deformation. These results shed light on the role of Ti in determining the
mechanical properties of polycrystalline Cu and will enable the engineering of
grain boundaries and the inclusion of Ti to improve degradation resistance.

###### keywords:

American Chemical Society, LaTeX

University College London] Department of Physics and Astronomy, University
College London, Gower Street, London WC1E 6BT, United Kingdom Yale
University1]Department of Mechanical Engineering $\&$ Materials Science, Yale
University, New Haven, 06520, CT, USA [Yale University]Department of Physics,
Yale University, New Haven, 06520, CT, USA [Yale University]Department of
Applied Physics, Yale University, New Haven, 06520, CT, USA The City College
of the City University of New York] Benjamin Levich Institute and Physics
Department, The City College of the City University of New York, New York, NY,
USA University College London] Department of Physics and Astronomy, University
College London, Gower Street, London WC1E 6BT, United Kingdom [Tohoku
University]WPI-Advanced Institute for Materials Research (WPI-AIMR), Tohoku
University, 2–1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan IR,NMR,UV

##  1 Introduction

Numerous experimental studies have shown that binary alloys made from Cu have
significantly improved mechanical properties compared to pure Cu 1, 2. For
instance, Cu-Be alloys have been widely used in numerous applications 3, 4.
However, Be is highly toxic even in small amounts, and thus there is
significant interest in identifying other Cu alloys with similar advantageous
properties. In particular, Cu-Ti alloys, used in a wide range of applications
5 are promising since they possess high strength and electrical conductivity,
as well as improved corrosion resistance 6, 7, 8. Furthermore, Cu-Ti alloys
have an increased mechanical strength over pure Cu even at concentrations less
than 5 atomic (at.) % Ti 9, 10.

A detailed understanding of the atomistic mechanisms that improve the strength
of Cu through the addition of small amounts of Ti can be achieved by using
theoretical and computational modeling. Kohn-Sham 11, 12 density functional
theory (DFT) and other quantum mechanics-based calculation methods can provide
highly accurate electronic structure measurements for alloys 13, 14. Several
theoretical studies highlight the strengthening effect of metallic solutes
when they are introduced into the grain boundaries (GBs) of metals, such as Cu
15, Ni 16, V 17 and Au 18. However, these calculations are limited to systems
with only $\sim 100$ atoms due to their computational cost 19. However,
understanding the structural and mechanical properties of polycrystalline
alloys requires much larger system sizes than $\sim 100$ atoms 20. Large-scale
molecular dynamics (MD) simulations using embedded atom method (EAM)
potentials can reproduce the structural and mechanical properties of many
alloys 21. The modified EAM (MEAM) proposed by Baskes, et al. 22 was developed
to extend EAM interatomic potentials to alloys with strong angular bonding. MD
simulations using MEAM potentials have been carried out to understand
important properties of alloys, such as ductile versus brittle mechanical
response 23, structure-property relationships 24, dislocation dynamics 25, and
fracture mechanics 26 in FCC metals and alloys. However, MEAM potentials have
been developed for only a small fraction of alloys 27, 28.

Several previous studies have investigated the effect of metallic dopants on
the mechanical properties of Cu and similar metals using either DFT 29 or
semi-emprical MD simulations. However, identifying the origins of the
experimentally-observed improvements in the mechanical properties of
polycrystalline Cu-Ti compared to those of pure Cu requires understanding the
role of topological defects, such as dislocations. In this study, we
investigate how the addition of Ti atoms affects the mechanical properties of
bicrystalline and polycrystalline Cu using both DFT calculations and MEAM MD
simulations. First, we use DFT calculations involving relatively small
periodic cells to assess the accuracy of a recent MEAM potential that was
fitted for Cu-Ti 30. We find that the MEAM potential accurately predicts the
most energetically favorable segregation sites of Ti at the Cu grain
boundaries. Using DFT calculations of CuTi alloys undergoing uniaxial tension
deformations, we show that the addition of Ti increases charge localization
and the separation energy of Cu GBs, which indicates the formation of covalent
bonding between Ti and its neighboring Cu atoms. We then carry out MEAM MD
simulations of bicrystalline Cu systems with Ti impurities undergoing uniaxial
tension deformation. We show that, similar to the DFT results, the presence of
Ti increases the yield strength of bicrystalline Cu by hindering the formation
of stacking faults. We then conduct MEAM MD simulations of polycrystalline Cu-
Ti undergoing uniaxial tension deformations. We find that the addition of even
a small amount of Ti increases the yield strength and Young’s modulus of the
Cu polycrystals. The increases in the yield strength and elastic modulus of
the polycrystalline Cu are caused by the fact that Ti hinders the emission of
Shockley dislocations from the GBs during the tensile deformation. This
strengthening increases with the concentration of Ti atoms at the GBs. In
light of recent experimental studies that developed advanced materials with
controllable interfaces 31, our findings highlight the potential of decorating
GBs in polycrystalline Cu with Ti atoms to improve its resistance to
degradation.

##  2 Methods

###  2.1 First-Principles Calculations

The DFT calculations were carried out using the Vienna Ab Initio Simulation
Package (VASP) 32, 33, 34 with the Perdew–Burke–Ernzerhof (PBE) GGA exchange-
correlation functional 35 in 76- and 108-atom periodic cells. (Additional
details concerning the DFT calculations can be found in Appendix A.) In line
with previous studies of Cu 36, a mixture of the Davidson 37 and RMM-DIIS 38,
39 algorithms is used to minimize the energy. Relevant details about the
functional form of the ground state energy ($\mathcal{E}$) obtained via
DFT/PBE can be found in the literature 35. Two $\Sigma 5$ twin boundary
symmetries were examined, (210)[100] and (012)[100], using different cell
sizes. These GBs are among the lowest energy reported in Cu 40. The GB
simulation cells are periodically translated in the $x$-, $y$-, and
$z$-directions. In the $z$-direction, $10$ Å of vacuum is added to the
simulation cell to avoid interactions between periodically translated images.
(See Figure 1(a).) For the 76-atom GB and 108-atom bulk cells, in line with
previous studies 41, 42, 43, converged 5$\times$4$\times$1 and
4$\times$4$\times$4 $k$-point grids were used, respectively, with a 450 eV
energy cutoff. To determine the number of $k$-points for various cell sizes,
the product of the length of the cell and number of $k$-points in the $x$-,
$y$-, and $z$-directions was chosen to be as close as possible to $35$
$k$-points $\times$ Å to ensure a converged $k$-point sampling for all unit
cells of different dimensions, given the Cu lattice constant of $3.62$ Å). In
the case of the GB cell, due to the added vacuum, just one $k$-point is used
in the $z$-direction. The atomic positions were relaxed using energy
minimization to an energy tolerance of $<10^{-5}$ eV.

###  2.2 Molecular Dynamics Simulations of Cu-Ti

All MD simulations were performed using the large-scale atomic molecular
massively parallel simulator (LAMMPS) 44. 120,000-atom bicrystalline
simulation cells of (210)[100] $\Sigma 5$ GBs were constructed using the
Atomsk code 45. For polycrystalline systems, we consider simulation cells with
volumes $8\times 10^{6}$ Å3 containing randomly oriented grains with 665,500
atoms. The polycrystals are then built using radical Voronoi tessellation of
randomly distributed points or nodes 45. Radical Voronoi tessellation allows
us to construct complex grain structures, such as curved grain boundaries
commonly found in experimental images of polycrystalline metals 46, 47.
Typically 12 nodes are introduced randomly within the periodic cells. The
grain sizes obtained from Voronoi tessellation follow a Gaussian distribution
with a mean grain volume of 6.6$\times$105 Å3 (with a standard deviation of
0.1$\times$106 Å3) and mean grain size (MGS) of approximately 89 Å. The MGS
values are computed using the mean linear intercept (MLI) method 48. This
value for the MGS is in line with the grain sizes of experimentally reported
nanocrystalline Cu 49, 50 and Cu-based alloys 2.

Table 1 summarizes the MGS, temperature (T), strain rate, and Young’s modulus
used in the MD simulations, together with previous theoretical and
experimental studies of polycrystalline Cu. Table 1 illustrates that the
elastic moduli obtained from the EAM and MEAM MD simulations are close to the
respective values reported in the experimental studies. The cell with an MGS
of 8.9 nm gives an elastic modulus range of 95.7-145.9 GPa and 92.5-125.3 GPa
using EAM and MEAM MD simulations, respectively. The latter ranges are in good
agreement with the reported experimental range of 108-116 GPa 51, where the
samples had considerably larger MGS of 54 nm.

Table 1:  Mean grain size (MGS) and Young’s moduli of polycrystalline pure Cu obtained from previous theoretical and experimental studies at various strain rates and temperatures (T). EMT indicates the many-body effective medium potential approach. For our results, both MEAM and EAM MD simulations are included. Values with ∗ are obtained using cells with dimensions of one unit cell (one translation, planar cells) along the $z$-axis. Method | Reference | MGS (nm) | T (K) |  Strain Rate (s-1) | Young’s modulus (GPa) |
---|---|---|---|---|---|---
EAM |  Chen 52 | 4.65-12.41 | 300 |  5$\times$108 | 54-92 |
EAM |  Zhou 53 | 2.6-53.1 | 300 |  108 |  25-75∗ |
EAM |  Rida 54 | 9-24 | 300 |  108 | 55-83 |
EAM |  Xiang 55 | 2.9-12.6 | 300 |  6.7$\times$107 | 60-112 |
EMT |  Schiotz 56 | 3.28-6.56 | 0 |  5$\times$108 | 90-120 |
Experimental |  Sanders 51 | 54 | 293 |  10-4 | 108-116 |
|  Cheng 57 | 54 | 297.3 |  10-4 | - |
|  Guduru 58 | 23-74 | 293 |  4$\times$10-4 | - |
EAM | (Our results) | 4.1 | 300 |  108 | 37.2-67.3 |
EAM | - | 6.1 | 300 |  108 | 79.9-104.7 |
EAM | - | 8.9 | 300 |  108 | 95.7-145.9 |
EAM | - | 10.2 | 300 |  108 | 88.8-104 |
EAM | - | 12.2 | 300 |  108 | 93.3-110-5 |
EAM | - | 16.3 | 300 |  108 | 97.7-136 |
MEAM | - | 8.9 | 300 |  108 | 92.5-125.3 |

The equilibration of the polycrystalline structures constructed via Voronoi
tessellation is crucial for obtaining multigrain simulation cells with
structural and mechanical properties comparable to those observed
experimentally 59. Initially, the total energy of all polycrystalline
structures is minimized using the conjugate gradient algorithm with a
$10^{-10}$ eV/ Å tolerance for the atomic forces. To further relax the grain
structures, the polycrystals are thermally annealed up to 750 K at a heating
rate of 3 K/ps for 250 ps using a time step of 1 fs at a constant pressure of
1 bar using the isothermal–isobaric NPT ensemble. The NPT equations of motion
are integrated using a leapfrog Verlet algorithm 60. The selected pressure is
in line with experimental conditions for Cu-Ti alloys 61. Previous MD studies
in polycrystalline metals, including Cu, have demonstrated that the selected
heating rate promotes structural relaxation of the grains without allowing
excessive grain growth 62. To regulate the temperature, a Nosé-Hoover
thermostat 63, 64 is used with a time constant of 1 ps, in line with previous
MD studies in polycrystalline Cu 65. The polycrystals are then cooled to 300 K
at a cooling rate of 3 K/ps. This cooling rate has been used in previous MD
studies for polycrystalline Cu 62 and Cu alloys 66. In the case of the
bicrystalline Cu undergoing uniaxial tension, following energy minimization,
the system is annealed to 300 K at a heating rate of 3 K/ps.

Experimentally, Cu samples containing various concentrations of Ti are
prepared in powder form after being mixed in plastic canisters with alumina
balls for three hours. Following the mixture process, the powders are sintered
at a temperature of 923 K, with a punch load of 50 MPa, dwelling time of 5
min, and a heating rate of 323 K/min. This process results in Cu-Ti samples
with a uniform distribution of Ti 9. Accordingly, Ti is introduced randomly in
polycrystalline simulation cells prior to equilibration. After equilibration,
Ti atoms are expected to occupy their most energetically favorable sites. As
an example, in the case where 1.5%at.Ti was randomly introduced into the
polycrystalline Cu cell prior to equilibration, 1.4% of the GB atoms were Ti
atoms (approximately 2000 atoms). After equilibration, 1.6% of the GB atoms
were Ti atoms. The 0.2% increase of Ti atoms in the GBs was due to the motion
of atoms near the GBs as the system was annealed. During annealing, no Ti
migration was observed from the bulk to the GBs. The latter indicates that the
temperature reached during equilibration is not sufficient for Ti to overcome
its diffusion barriers in the bulk. The corresponding effective Ti
concentrations per grain boundary volume for all simulation cells can be found
in Appendix A.

For the bicrystalline and polycrystalline Cu-Ti simulation cells, the uniaxial
tensile loading along the $y$-axis is conducted at a strain rate of 108 s-1,
in line with previous MD simulations of Cu 67. During the simulations, the
cells are kept at a constant temperature of 300 K, and the boundaries in the
$x$\- and $z$-directions are allowed to vary to maintain zero pressure.
However, 108 s-1 is a high strain rate compared to those used in most
experimental studies. Thus, it may be necessary to extrapolate the stress-
strain curves in the simulations to those obtained at much lower strain rates
in the experimental systems 68, 69.

###  2.3 Energetic Parameters

An important component of these studies is the comparison of the results
obtained from MEAM MD simulations and DFT calculations. To distinguish between
MEAM and DFT energies, $E$ and $\mathcal{E}$ will indicate the MEAM potential
energy and the ground state energy minimized by DFT, respectively. To
understand the main properties of Ti in Cu grain boundaries, three energetic
parameters need to be computed: the (i) segregation, (ii) strengthening, and
(iii) separation energies. The segregation energies provide information on
whether Ti would prefer to be located at the grain boundaries or in the bulk.
To calculate the segregation energies of Ti in Cu GB, five substitutional
sites are tested, as shown in Figure 1(a). Since the DFT calculations showed
that the octahedral and tetrahedral interstitial sites are unfavorable for Ti,
only substitutional sites are considered. The segregation energies of the
substitutional Ti sites in Cu grain boundaries are computed using

$E/\mathcal{E}_{seg}=(E/\mathcal{E}_{GB+Ti}-E/\mathcal{E}_{GB})-(E/\mathcal{E}_{Bulk+Ti}-E/\mathcal{E}_{Bulk}),$
(1)

where E$/\mathcal{E}_{GB+Ti}$ and E$/\mathcal{E}_{Bulk+Ti}$ are the MEAM/DFT
energies of the $76$-atom grain boundaries and $108$-atom bulk Cu cells,
respectively, each containing one substitutional Ti atom. E$/\mathcal{E}_{GB}$
and E$/\mathcal{E}_{Bulk}$ are the respective MEAM/DFT pristine Cu grain
boundary and bulk energies. Negative energies correspond to favorable
segregation of Ti at the GB, whereas positive energies correspond to mixing of
Ti in the bulk.

Predictions of the impact of impurities on the grain boundary strength can be
made using the strengthening energy, which provides information about whether
Ti prefers to be located at the grain boundaries rather than on the sample
surface. The MEAM/DFT strengthening energies (E$/\mathcal{E}_{str}$), based on
the Rice-Wang 70 model, are computed using

$E/\mathcal{E}_{str}=(E/\mathcal{E}_{GB+Ti}-E/\mathcal{E}_{GB})-(E/\mathcal{E}_{Sur+Ti}-E/\mathcal{E}_{Sur}),$
(2)

where E$/\mathcal{E}_{Sur}$ and E$/\mathcal{E}_{Sur+Ti}$ represent the
MEAM/DFT energies of the $108$-atom pure Cu (100) surface and Cu surface
simulation cells with one Ti atom, respectively. To obtain the Cu surface
simulation cells, a slab approach is employed. Starting from the conventional
FCC Cu unit cell, a 3$\times$3$\times$6 translation is performed to create a
cell with an extended $z$-dimension. Atoms along the $z$-direction are removed
to generate the slab model with a desired slab plus vacuum thickness. We set
the thickness to be 10.86 Å. A negative value of the strengthening energy
means that the impurity will enhance the grain boundary strength, while a
positive value implies that the grain boundary will be weakened by the
addition of a Ti atom.

Finally, the separation energy indicates whether Ti in Cu GBs promotes or
deters fracture initiation. DFT tensile tests use two stretching methods 71,
72. In this work, we selected a pre-crack fracture plane based on previous DFT
simulations in $\Sigma 5$ Cu GBs 40, 73. The separation energies were
calculated by subtracting the total energy of the GB cell at a given spacing
between the grains from the energy at the equilibrium separation.

##  3 Results

Before we perform MD simulations of uniaxial tensile strain in bicrystalline
and polycrystalline Cu, it is important to compare the results of DFT
calculations of GB energetics to those obtained from the MEAM interatomic
potential. In this section, we compare the MEAM and DFT energies for one Ti
atom introduced into Cu grain boundaries. In addition, we apply uniaxial
tension to small CuTi systems and employ DFT to calculate the separation
energy for Ti in Cu grain boundaries.

Figure 1:  (a) (i) Cu 296-atom (210)[100] $\Sigma 5$ GB simulation cell. Cu
atoms are shown in blue, and the numbered atoms correspond to the
substitutional Ti sites. Atoms labelled with a ‘P’ represent atoms that are
periodically translated. (ii) Comparison of Ti segregation (top) and
strengthening energies (bottom) obtained using DFT and MEAM for the five
substitutional sites. (b) (i) Cu 232-atom (012)[100] GB simulation cell for
the labelled segregation sites. (ii) Comparison of Ti segregation (top) and
strengthening energies (bottom) from DFT and MEAM.

###  3.1 Calculations of segregation and separation energies using DFT and
MEAM

In Figure 1(a)(i), we show the 296-atom (210)[100] $\Sigma 5$ GB simulation
cell for five labelled Ti substitutional sites. For the DFT calculations (see
Figure 1(a)(ii)), segregation sites 1 and 3 have the most favorable (lowest)
segregation energies, around -0.35 eV. Previous DFT studies on Al $\Sigma 5$
GB solutes showed similar negative segregation energies for Ti, around -0.2 eV
74. In agreement with DFT, MEAM identifies 1 and 3 as the two most favorable
Ti substitution sites. For all substitution sites we considered, the MEAM
interatomic potential gives segregation energies within 0.15 eV of those
obtained using DFT, which, based on previous comparisons between EAM and DFT
calculations 74, 75, is reasonable agreement between the two methods.

In Figure 1(a)(ii), we also display the strengthening energies of grain
boundaries in the presence of Ti. We used a formalism that is frequently used
in the literature 73 to determine the strengthening effect of non-metallic
impurities in Cu grain boundaries. We find that MEAM and DFT give negative
strengthening energies for Ti in all substitution sites. Thus, both methods
predict that Ti would have a strengthening effect when introduced into Cu GBs.
The two methods are also in good agreement when a different $\Sigma 5$ twin
boundary simulation cell is used (i.e. $\Sigma$5 (012)[100] in a simulation
cell with 232 atoms as shown in Figure 1(b)(i)). In both $\Sigma 5$ simulation
cells, the substitutional site closest to the center of symmetry of the grain
boundaries showed the lowest segregation energy (see Figure 1(b)(ii)),
indicating that our results are relevant to other $\Sigma 5$ symmetries.

Figure 2:  (a) Cu 76-atom (210)[100] $\Sigma 5$ GB simulation cell used for
the DFT calculations of applied unaxial tension. The yellow line shows the
fracture plane. (b) Schematic illustrating the Cu GB (i) at the equilibrium
separation and (ii) a separation of 2Å. The yellow and red lines represent the
equilibrium and separation distances, respectively. (c) Single point (rigid;
no relaxation) and full relaxation DFT calculations of the separation
energies, illustrating the energy difference as the separation distance of the
grain boundaries is increased for pure Cu and Cu with one substitutional Ti.
(d) Hydrostatic stress computed from DFT full-relaxation calculations. The
labels (i)-(iii) denote the three distinctive regions during the DFT uniaxial
tension calculations.

In Figure 2(a), we show the GB simulation cell used for the DFT uniaxial
tension tests. The yellow line illustrates the selected fracture plane. Due to
the computational cost of the DFT-based uniaxial tension calculations, we used
a smaller cell with 76 atoms. These smaller systems possessed similar
segregation and separation energies as those in Figure 1(a)(i) (see Appendix
B). The effect of Ti on the GB separation energy can be seen in Figure 2(c).
The separation distance refers to the displacement along the z-axis from the
equilibrium position, as shown in Figure 2(b). We show the separation energies
obtained from both single-point (rigid, no atom relaxation is allowed) and
full relaxation DFT calculations. The results follow the universal binding
energy relation 76, where the separation energy increases rapidly for small
separations and, at larger distances, reaches a plateau. The smallest
separations correspond to the pre-fracture regime 29 (region (i) in Figures
2(c) and (d)), while, after $\sim 1$Å, the system enters the plastic region
(region (ii)), where an intergranular fracture initiates. We define a fracture
as the point at which uninterrupted areas of zero electron density are formed
within the GB. The calculated separation energy in the case of Cu with a
single Ti atom is approximately 0.02 eV/Å2 higher than for pure Cu. Another
interesting point is that for Cu with a single Ti atom, the separation of the
GB into two free surfaces (intergranular fracture) occurs at a separation
distance of 2.0 Å. On the other hand, in pure Cu, the fracture initiates at a
separation distance of 1.8 Å. Finally, in the third deformation stage (region
(iii)), the separation energy slowly increases until it plateaus as the
remaining long-range interaction forces between the two fracture surfaces tend
to zero. The relaxed separation energy curves in the third stage follow the
universal binding energy relation.

The tensile strength of the Cu cells with and without Ti can be taken as the
maximum tensile stress, shown in Figure 2(d). The fully relaxed DFT
calculations show that the tensile strength of the grain boundary with one
substitutional Ti is higher compared to the pure Cu grain boundary. According
to these findings, doped Cu showed higher mechanical strength compared to that
of pure Cu. The observed effect of Ti is in good agreement with previous DFT
studies on dopants in Cu, where it was reported that the introduction of
transition metals, such as Nb, Mo, and Zr, into Cu, can significantly increase
the energy for the initiation of intergranular fracture 73. In addition, our
results are in good agreement with previous theoretical studies in grain
boundaries of Au and Fe, which showed that 3d block transition metals can have
a beneficial effect on grain boundary cohesion 18, 77.

What is the underlying physical mechanism that gives rise to the strengthening
of Cu grain boundaries with the addition of Ti? In Figure 3(a), we show the
total charge density distributions ($\rho$) for a pure Cu GB and a Cu GB with
one substitutional Ti atom at site 1 (cf. Figure 2(a)). The three columns
indicate the equilibrium GB separation (first column) and separations of $1.8$
Å (second column) and $2$ Å (third column). The three columns also represent
the three stages of grain boundary decohesion. At a separation of 1.8 Å, a
fracture initiates in the case of the GB cell without Ti, whereas no fracture
is observed in the presence of Ti. The latter effect is attributed to the
elongation of the Cu-Cu distance close to the Ti atom. This behavior agrees
with the lower separation energy in the case of a Cu GB with Ti as shown in
Figure 2(c).

In Figure 3(a), we plot the total electron charge density ($\rho$)
distribution contours for a pure Cu GB (top) and a Cu GB with a single Ti atom
(bottom). In Figure 3(b), we show the differential charge density
($\delta$$\rho$) contour maps at the equilibrium separation (first column) and
at separations of $1.8$ Å (second column) and $2$ Å (third column). Ti, which
has two extra valence electrons compared to Cu, significantly redistributes
the local charge density at a separation of $1.8$ Å. At this separation,
fracture is also initiated in pure Cu, but not in Cu with Ti (see Figure
3(a)). From the differential charge density distribution contour plots, we
find that the addition of Ti enhances the charge density near the GB. As noted
in previous studies 78, 74, this charge localization suggests covalent bonding
between Ti and the adjacent Cu atoms. Similar covalent bond formation was
reported in theoretical studies of Cu-Ti intermetallic compounds, which was
attributed to the polarized nature of Ti 78, 74.

Figure 3:  (a) Total electron charge density ($\rho$) distribution for a pure
Cu GB (top) and a Cu GB with a single Ti atom (bottom) at the equilibrium
separation (Equil., first column) and separations of $1.8$ Å (second column)
and $2$ Å (third column). (b) 2D contour plots of the differential electron
charge density ($\delta$$\rho$) for a Cu GB with a single Ti atom at the
equilibrium separation (Equil., first column) and at separations of $1.8$ Å
(second column) and $2$ Å (third column). Red (blue) shading corresponds to
areas of electron accumulation (depletion). We employ VESTA to visualize the
electron charge density distributions 79.

In accordance with previous DFT studies of segregants in Cu GBs 29, electron
redistribution with strong relaxation reduces the GB strength. However, in
cases where relaxation is minimal, such as Cu GBs with Ti substitutions,
charge redistribution increases the local strength. Furthermore, in agreement
with previous studies 29, dopants in Cu GB that lose electrons are expected to
increase the GB strength. Therefore, based on the DFT calculations of Ti in Cu
GBs, more energy compared to pure Cu is required to initiate a fracture. Ti
enhances the electron density between the two Cu grains, which leads to an
increase in the binding energy.

###  3.2 MD Simulations using the MEAM Interatomic Potential

####  3.2.1 Bicrystalline Cu-Ti under uniaxial tensile strain

Figure 4:  (a) Illustration of the simulation cell used for the MD simulations
(using the MEAM interatomic potential) of the bicrystalline (210)[100] $\Sigma
5$ Cu GB (containing 120,000 atoms) undergoing tensile strain. Strain is
applied along the $y$-axis. Atoms shaded green and white indicate FCC and
amorphous regions of the sample, respectively. (b) GB region of the simulation
cell. Yellow shading illustrates the $\Sigma 5$ kite-shaped structural units.
(c) Stress versus strain plots obtained from the MD simulations as a function
of Ti concentration. The vertical lines indicate the yield strength (in GPa)
for each concentration. $\sigma$ corresponds to the $yy$ component of the
stress tensor. (d) Number of atoms with HCP local order plotted as a function
of strain for several Ti concentrations. The insets provide snapshots of the
system at 8$\%$ strain for pure Cu (top) and Cu with 3.5$\%$Ti. The red
shading indicates atoms with local HCP order. Common neighbor analysis 80 was
used to classify the local crystalline structure surrounding each atom.
Structures are visualized using OVITO 81.

The results presented in the previous section demonstrate reasonable agreement
for the segregation and separation energies of Cu grain boundaries with Ti
between the MEAM and DFT methods. We will now carry out MD simulations using
the MEAM interatomic potential in larger and more complex Cu GB structures. We
will first perform MD simulations of bicrystalline Cu cells (similar to those
used in the DFT calculations) undergoing uniaxial tension. The DFT tensile
tests revealed that Ti significantly strengthens grain boundaries only for
separations $\geq 1.8$ Å. However, these DFT calculations only captured the
normal component of the stress, not the shear component, which can induce the
formation of topological defects, such as dislocations. By characterizing
larger bicrystalline GB cells with MD simulations of uniaxial tensile strain,
we can investigate effects of shear stress and dynamics on the yield strength
of Cu GBs with varying Ti concentrations.

In Figure 4(a), we illustrate the 120,000-atom simulation cell of a
bicrystalline Cu GB with the same symmetry as that used in the DFT
calculations of Cu GBs undergoing uniaxial tension (see Figure 4(b)). Constant
strain-rate MD MEAM simulations are carried out for pure Cu, as well as Cu
with 0.5$\%$, 1.5$\%$, and 3.5$\%$ concentrations of randomly distributed
substitutional Ti atoms. Uniaxial tensile tension is applied at constant
strain rate after the cell has been equilibrated at a temperature of 300 K.
Figure 4(c) depicts the resulting stress-strain curves, which show that the
presence of Ti improves the mechanical properties of the bicrystal, as
observed in DFT calculations (see Figure 2). Both the yield strength (vertical
lines in Figure 4(c)) and Young’s modulus increase as the Ti concentration
increases.

To better understand the underlying mechanism for the strengthening of Cu GBs,
we analyzed the local structural order of the bicrystalline GBs as a function
of strain and Ti concentration. As the system is strained beyond the yield
stress, dislocations are emitted by the GB to release large local
concentrations of stress. These dislocations correspond to the formation of
stacking faults and twins, which can be identified as planes of atoms with
local HCP order. As shown in Figure 4(d), the percentage of HCP atoms
decreases with increasing Ti concentration. This result is further illustrated
in the insets, where pure Cu possesses a higher density of stacking faults
(red-shaded atoms possess HCP order) compared to that for Cu-3.5$\%$ at 8$\%$
strain. Furthermore, Ti is found to hinder the nucleation of the HCP planes in
the case of Cu-3.5$\%$Ti, which results in the gradual increase of the number
of atoms with HCP symmetry between 5$\%$-6$\%$ strain (see (Figure 4(d))).

####  3.2.2 Polycrystalline Cu-Ti undergoing uniaxial tensile strain

Figure 5:  (a) 12-grain polycrystalline cells for Cu-Ti (cells 1-3), Cu2Ti,
and pure HCP Ti before (top row) and after (bottom row) equilibration at a
temperature of $300$ K. The unit cell of bulk Cu2Ti is also shown, where brown
and gray shading indicates the Cu and Ti atoms, respectively. (b) Mean-squared
error (MSE) of the stress from the average value for multiple MD MEAM
simulations of uniaxial tension with different random initial distributions of
Ti in Cu-1.5$\%$Ti and Cu-2.6$\%$Ti using cells 1-3. (c) Stress plotted versus
strain for polycrystalline pure Cu, Cu-1.5$\%$Ti, Cu-2.6$\%$Ti, Cu-3.5$\%$Ti,
Cu-20$\%$Ti, and Cu-33$\%$Ti (i.e. the Cu2Ti phase). The inset shows the yield
strength (top) and Young’s modulus (bottom) for several Ti concentrations. (d)
Stress versus strain for Cu-1.5$\%$Ti and different initial distributions of
Ti: random, only in the GBs, and only in the bulk. The inset shows the yield
strength (top) and Young’s modulus (bottom) for different initial
distributions of Ti. $\sigma$ corresponds to the $yy$ component of the stress
tensor.

MD simulations of bicrystalline Cu undergoing uniaxial tensile strain showed
that Ti hinders the formation of HCP stacking faults, leading to a significant
increase in the yield strength. We now examine whether similar strengthening
occurs for larger and more realistic polycrystalline models of Cu-Ti. We
randomly introduce Ti into the polycrystalline simulation cells with different
grain distributions. After adding the Ti atoms, the polycrystals are
equilibrated, as described in the Methods section. In Figure 5(a), we display
the polycrystalline Cu-Ti cells prior to (top) and after equilibration
(bottom). Cells 1-3 correspond to the cells used for the Cu-Ti polycrystals,
with Ti concentrations ranging from 1.5 to 20$\%$. Figure 5(a) also shows the
simulation cells for Cu2Ti and pure HCP Ti.

For each cell and Ti concentration, three MD simulations of uniaxial tension
are performed. Different initial random Ti distributions are considered for
each of the three runs. In Figure 5(b), we show the mean-squared error of the
stress (MSE; see Appendix A) relative to the average stress over all runs for
each of the three simulation cells (cells 1-3 in Figure 5(a)) for Cu-1.5$\%$Ti
and Cu-2.6$\%$Ti. At both concentrations and for cells 1 and 3, the MSE stress
is less than 0.02. The MSE in stress continue to decrease with increasing Ti
concentration. Interestingly, cell 2 shows the highest MSE in stress for both
Ti concentrations. This result can be attributed to the higher fraction of GBs
in this cell compared to cells 1 and 3. Even for cell 2, the MSE in stress is
$<0.05$ for Cu-1.5$\%$Ti and $<0.025$ for Cu-2.6$\%$Ti. Hence, the stress
self-averages for different initial random distributions of Ti.

In Figure 5(c), we show the stress versus strain averaged over multiple runs
for pure Cu and Cu-1.5$\%$Ti, Cu-2.6$\%$Ti, Cu-3.5$\%$Ti, and Cu-20$\%$Ti. The
20$\%$ concentration is selected as it corresponds to the point in the Cu-Ti
phase diagram where FCC Cu transitions to $\beta$-Cu4Ti 82. The stress-strain
curve for Cu-33$\%$Ti (Cu2Ti phase), using the cell in Figure 5(a), is also
included as a reference. The inset displays the yield strength and Young’s
modulus for pure Cu and Cu with several Ti concentrations. In all cases, the
inclusion of Ti increases the Young’s modulus. Furthermore, the yield strength
increases with increasing Ti concentration, in accordance with the DFT
calculations and MEAM MD simulations of bicrystalline Cu-Ti undergoing
uniaxial tension. Interestingly, for Ti concentrations higher than 1.5%, the
yield strength of Cu-Ti is higher than that of HCP Ti. The mechanical strength
of the crystal increases significantly when it transitions into the Cu2Ti
phase. This behavior is expected and supported by the solid solution hardening
theory for Cu 83, which shows that doping Cu with metallic substituents will
increase its mechanical strength due to the differing mass of the dopant and
host atoms 29.

Figure 6:  (a) Average nonaffine displacement D${}^{2}_{\rm min}$ plotted
versus strain for pure Cu, Cu-1.5$\%$Ti, Cu-2.6$\%$Ti, Cu-3.5$\%$Ti, and
Cu-20$\%$Ti undergoing unaxial tension. Only atoms within the GBs are
considered. The inset highlights the region from 2 to 4% strain. (b) Spatial
distributions of D${}^{2}_{\rm min}$ at a single $z$-slice ($z=0$) for pure Cu
(top) and Cu-3.5$\%$Ti (bottom). (c) D2min contour plots averaged for all $z$
for pure Cu (top) and Cu-3.5at.$\%$Ti. For pure Cu, triple junctions
surrounding smaller grains possess lower values of D${}^{2}_{\rm min}$ (red
and black circles in (b) and (c)).

We next seek to determine whether the increase in the yield strength of Cu-Ti
compared to pure Cu can be attributed to the presence of Ti in the GBs. In
Figure 5(d), we show the stress versus strain for Cu-1.5$\%$ when Ti was
initially randomly distributed, was located only in the bulk, and only in the
GBs. In the latter case, Ti is introduced at random sites, but only within the
grain boundary region, as defined by common neighbor analysis. As shown in the
inset of Figure 5(d), the inclusion of Ti, either only in the bulk or only in
the GBs does not affect the Young’s modulus for small strain. However, the
stress versus strain curve for randomly distributed Ti has a smaller Young’s
modulus for strains $>2\%$. Furthermore, when Ti is introduced only in the GB
region, we find a considerable increase in the yield strength compared to the
case where Ti is distributed randomly or only in the bulk (see inset in Figure
5(d)). These results illustrate that decorating Cu GBs with Ti atoms will
significantly improve the mechanical strength of the crystal.

Figure 7:  (a) Characterization of the local structural order for (top) pure
Cu and (bottom) Cu-3.5%Ti at three strains. Ti (blue) atoms prevent the
formation of HCP order (red shading) in smaller grains and reduce the number
of grain-boundary induced dislocations (black circles). (b) Percentages of
atoms with HCP order for Cu-3.5%Ti at several strains. The inset illustrates
the initiation of a stacking fault (HCP atoms in red) via the emission of
partial Shockley dislocations (green lines) from the GB (gray region). (c)
Total length of all dislocations and Shockley dislocations versus strain for
pure Cu, Cu-1.5$\%$Ti, Cu-3.5$\%$Ti, and Cu-20$\%$Ti.

To better understand the origin of the superior mechanical properties of Cu-
Ti, we examine the mechanical response between 2$\%$ and 5$\%$ strain,
approximately where pure Cu and all Cu-Ti alloys reach their yield strength
(see Figure 5(c)). (Because polycrystalline cell 1 (see Figure 5(a)(i)) has
the lowest MSE in stress among the three cells tested, this cell is used for
the following analysis.) In Figure 6(a), we show the average non-affine
displacement D2min as a function of strain for pure Cu and Cu with increasing
Ti concentration. (Only atoms within the GBs are included in D2min; see
Appendix A for more details.) For strains larger than 2$\%$, cells with a
higher concentration of Ti have lower D2min values. The spatial distribution
of D2min for pure Cu and Cu-3.5$\%$Ti at strains between 2$\%$-5$\%$ are shown
in Figures 6(b) and (c) for a single $z$-slice and averaged over all $z$,
respectively. In the case of pure Cu, the GBs of the smaller grains show
higher values of D2min (see the red and black circles in Figures 7(b) and (c),
respectively). The triple junctions and GBs surrounding smaller grains show
considerably smaller D2min in Cu-3.5$\%$Ti.

In Figure 7(a), we show that intrinsic stacking faults (ISFs), which are
characterized by atoms with local HCP order (red shading) begin to nucleate
from the GBs and propagate at 3$\%$ strain. Consistent with the results from
the tensile tests of bicrystalline Cu-Ti, the presence of Ti (blue atoms)
prevents the formation of HCP-ordered atoms in smaller grains. Thus, the GBs
in Cu-3.5$\%$Ti emit a lower density of partial dislocations compared to pure
Cu (see black circles in Figure 7(a)). Based on these results, the
dislocations emitted by the grain boundaries may be responsible for both the
higher values of D2min and the lower yield strength of pure Cu compared to the
Cu-Ti alloys.

Figure 6(b) shows the fraction of atoms with local HCP order for Cu-Ti alloys
up to 6$\%$ strain. Higher Ti concentrations reduce the number of atoms with
HCP order. This effect aligns with what was observed in the MD simulations of
bicrystalline Cu-Ti undergoing uniaxial tension. Pure Cu has more HCP atoms at
5$\%$ strain than Cu-20$\%$Ti has at 6$\%$ strain. The inset in Figure 6(b)
illustrates the initiation of a stacking fault (HCP atoms in red) via the
emission of Shockley dislocations by the grain boundaries. We find that Ti
inhibits the emission of Shockley dislocations and nucleation of stacking
faults. This dislocation inhibition phenomenon will become more prominent when
Ti is introduced in the GBs rather than randomly. The latter explains the
enhanced mechanical properties of Cu-Ti under strain when Ti was introduced
only in the GBs (see Figure 7(d)).

Since dislocations govern the mechanical properties of polycrystals 84, 74, we
now examine and quantify the effect of Ti on the dislocation emission process.
In Figure 7(c), we show the lengths of all dislocations in pure Cu,
Cu-1.5$\%$Ti, Cu-3.5$\%$Ti, and Cu-20$\%$Ti as a function of strain. Since
partial dislocations (stacking faults) are initially emitted as Shockley
dislocations by the GBs 85, the lengths of the Shockley dislocations are also
included for each concentration. Increasing the Ti concentration increases the
lengths of both the total and Shockley dislocations. In pure Cu, Shockley
dislocations have a larger total length than that for Cu-20$\%$Ti at 3$\%$
strain. Increasing the Ti concentration decreases both the total and Shockley
dislocation lengths. Thus, Ti is expected to reduce twinning, a process during
which a part of the metal undergoes shear deformation along a preferred slip
plane. The results in Figures 7(a) and (b) are consistent with twinning, where
the addition of Ti is found to alter the crystal’s ability to generate HCP
planes. A similar finding was reported in bicrystalline Al through MD
simulations, where the inclusion of Mg in the GB hindered the nucleation of
dislocations, leading to an increased yield strength 86.

Based on our analysis and taking into account the already established role of
dislocation emission in nanocrystalline FCC metals 84, we can identify a
mechanism where the inclusion of Ti atoms improves the mechanical properties
of the samples under tensile loading. Initial plastic deformation occurs under
tensile loading throught the nucleation of partial dislocations emitted by the
GBs at a strain of $\sim 1.5\%$. The continual process of partial dislocation
nucleation from GBs and annihilation of GBs at opposite ends of the grains
causes pronounced local distortion close to the GBs. With the addition of Ti,
the distortion becomes less pronounced. Ti, when introduced at the GBs,
increases the energy needed for the GBs to slide and emit partial
dislocations. Thus, the inclusion of Ti substantially decreases the density of
the emitted partial dislocations by the GBs, leading to an increase in the
yield stress. The latter observation is in agreement with the Hall–Petch
equation 87, 88, which predicts that as the stress needed for the activation
of the dislocations increases, the yield strength also increases.

##  4 Summary and Conclusions

Previous studies have mainly used DFT calculations to investigate the role of
metallic solutes in the grain boundaries of Cu in determining its mechanical
properties. To better understand the effect of Ti on the mechanical properties
of Cu, we employ both DFT calculations and MEAM MD simulations, which enables
studies of tensile deformation over a range of length scales. We first
compared the results from the MEAM potential for the most energetically
favorable Ti substitution sites in Cu GBs to the results from the DFT
calculations. For both the DFT calculations and MEAM MD simulations,
substitutional Ti prefers to segregate at the GB rather than in the bulk of
Cu. DFT calculations of tensile deformation show that Ti at the GBs induces
local charge localization, which increases the maximum stress, and the
separation and strengthening energies of the GB. An interesting future
direction would be to examine whether other elements with properties similar
to Ti, like the valency of electrons or radius, when added to Cu grain
boundaries would cause similar charge localization and increases in the
maximum stress and segregation and separation energies.

The changes in the mechanical properties of large bicrystalline Cu cells in
response to the addition of Ti was studied using MEAM MD simulations of
uniaxial tension. Similar to the results obtained from DFT calculations of
uniaxial tension, the addition of Ti increased the yield strength of the
crystal, due to the inhibition of stacking faults emitted by the GBs. This
phenomenon was also present when polycrystalline systems were considered. MEAM
MD simulations using nanocrystalline multigrain simulation cells showed that
the presence of Ti prevents the emission of partial Shockley dislocations from
GBs, which reduces local distortions. This effect becomes more pronounced as
the Ti concentration increases and when all Ti atoms are introduced at the GBs
of the polycrystal. Thus, the inclusion of Ti results in a significant
increase in the yield strength and Young’s modulus of Cu polycrystals compared
to that for pure Cu polycrystals. These mechanical properties were further
improved at higher Ti concentrations.

Our findings suggest that the addition of Ti to Cu has a significant effect on
the yield strength and Young’s modulus of the material. Thus, grain boundary
segregation engineering of polycrystalline Cu with Ti solutes can be used to
enhance the material’s durability and broaden its industrial applications.
However, due to the intrinsic limitations of the time and length scales of MD
simulations, the unrealistically high deformation rates used in MD simulations
are many orders of magnitude higher than those used in experiments. Alongside
the current data, future work should focus on extrapolating the mechanical
properties of Cu-Ti to significantly slower strain rates. In addition, further
investigations are needed for a broader range of Cu-Ti alloys. Finally, to
simulate polycrystalline Cu-Ti cells with mean grain sizes closer to those
studied experimentally, reliable and less computationally demanding MEAM
potentials for modeling polycrystalline Cu-Ti will be required.

A.L.S. acknowledges funding by EPSRC (grant EP/P013503/1). C.S.O. acknowledges
funding from NSF Grant No. 2244310. V.F. would like to acknowledge funding by
EPSRC (grant EP/L015862/1) as part of the CDT in molecular modeling and
materials science. V.F. gratefully acknowledges the UCL Doctoral School for
supporting the placement in the UCL-Yale Collaborative Exchange Program.
Computational resources on ARCHER2 (http://www.archer2.ac.uk) were provided
via our membership of the UK’s HPC Materials Chemistry Consortium, which is
funded by EPSRC (EP/L000202, EP/R029431).

## References

  * An et al. 2011 An, X.; Qu, S.; Wu, S.; Zhang, Z. Effects of stacking fault energy on the thermal stability and mechanical properties of nanostructured Cu–Al alloys during thermal annealing. _Journal of Materials Research_ 2011, _26_ , 407–415
  * Chad Hornbuckle et al. 2020 Chad Hornbuckle, B.; Williams, C. L.; Dean, S. W.; Zhou, X.; Kale, C.; Turnage, S. A.; Clayton, J. D.; Thompson, G. B.; Giri, A. K.; Solanki, K. N.; Darling, K. A. Stable microstructure in a nanocrystalline copper–tantalum alloy during shock loading. _Communications Materials_ 2020, _1_ , 1–6
  * Wang et al. 2022 Wang, Z.; Li, J.; Zhang, Y.; Lv, C.; Li, T.; Zhang, J.; Hui, S.; Peng, L.; Huang, G.; Xie, H.; Mi, X. Comparison of the Mechanical Properties and Microstructures of QB2.0 and C17200 Alloys. _Materials_ 2022, _15_ , 2570
  * Mazar Atabaki 2012 Mazar Atabaki, M. Characterization of transient liquid-phase bonded joints in a copper-beryllium alloy with silver-base interlayer. _Journal of Materials Engineering and Performance_ 2012, _21_ , 1040–1045
  * Semboshi et al. 2014 Semboshi, S.; Hinamoto, E.; Iwase, A. Age-hardening behavior of a single-crystal Cu–Ti alloy. _Materials Letters_ 2014, _131_ , 90–93
  * Datta and Soffa 1976 Datta, A.; Soffa, W. The structure and properties of age hardened Cu-Ti alloys. _Acta Metallurgica_ 1976, _24_ , 987–1001
  * Laughlin and Cahn 1975 Laughlin, D. E.; Cahn, J. W. Spinodal decomposition in age hardening copper-titanium alloys. _Acta Metallurgica_ 1975, _23_ , 329–339
  * Nagarjuna et al. 1995 Nagarjuna, S.; Balasubramanian, K.; DS, S. Effects of cold work on precipitation hardening of Cu-4.5 mass% Ti alloy. _Materials Transactions, JIM_ 1995, _36_ , 1058–1066
  * Eze et al. 2018 Eze, A. A.; Jamiru, T.; Sadiku, E. R.; Durowoju, M. O.; Kupolati, W. K.; Ibrahim, I. D.; Obadele, B. A.; Olubambi, P. A.; Diouf, S. Effect of titanium addition on the microstructure, electrical conductivity and mechanical properties of copper by using SPS for the preparation of Cu-Ti alloys. _Journal of Alloys and Compounds_ 2018, _736_ , 163–171
  * Ramesh and Shivananda Nayaka 2020 Ramesh, S.; Shivananda Nayaka, H. Investigation of tribological and corrosion behavior of Cu-Ti alloy processed by multiaxial cryoforging. _Journal of Materials Engineering and Performance_ 2020, _29_ , 3287–3296
  * Hohenberg and Kohn 1964 Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. _Physical Review_ 1964, _136_ , B864
  * Kohn and Sham 1965 Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. _Physical review_ 1965, _140_ , A1133
  * Yang et al. 2022 Yang, Y.; Guo, Z.; Gellman, A. J.; Kitchin, J. R. Simulating Segregation in a Ternary Cu–Pd–Au Alloy with Density Functional Theory, Machine Learning, and Monte Carlo Simulations. _The Journal of Physical Chemistry C_ 2022, _126_ , 1800–1808
  * Alidoust et al. 2020 Alidoust, M.; Kleiven, D.; Akola, J. Density functional simulations of pressurized Mg-Zn and Al-Zn alloys. _Physical Review Materials_ 2020, _4_ , 045002
  * Razumovskiy et al. 2018 Razumovskiy, V.; Divinski, S.; Romaner, L. Solute segregation in Cu: DFT vs. experiment. _Acta Materialia_ 2018, _147_ , 122–132
  * Bentria et al. 2019 Bentria, E. T.; Lefkaier, I. K.; Benghia, A.; Bentria, B.; Kanoun, M. B.; Goumri-Said, S. Toward a better understanding of the enhancing/embrittling effects of impurities in Nickel grain boundaries. _Scientific Reports_ 2019, _9_ , 14024
  * Wu et al. 2016 Wu, X.; Kong, X.-S.; You, Y.-W.; Liu, W.; Liu, C.; Chen, J.-L.; Luo, G.-N. Effect of transition metal impurities on the strength of grain boundaries in vanadium. _Journal of Applied Physics_ 2016, _120_ , 095901
  * Scheiber 2021 Scheiber, D. Segregation and embrittlement of gold grain boundaries. _Computational Materials Science_ 2021, _187_ , 110110
  * Fabian et al. 2022 Fabian, M. D.; Shpiro, B.; Baer, R. Linear Weak Scalability of Density Functional Theory Calculations without Imposing Electron Localization. _Journal of Chemical Theory and Computation_ 2022, _18_ , 2162–2170
  * Derlet and Dudarev 2007 Derlet, P. M.; Dudarev, S. Million-atom molecular dynamics simulations of magnetic iron. _Progress in Materials Science_ 2007, _52_ , 299–318
  * Daw et al. 1993 Daw, M. S.; Foiles, S. M.; Baskes, M. I. The embedded-atom method: A review of theory and applications. _Materials Science Reports_ 1993, _9_ , 251–310
  * Baskes 1997 Baskes, M. I. Determination of modified embedded atom method parameters for nickel. _Materials Chemistry and Physics_ 1997, _50_ , 152–158
  * Kang and Cai 2007 Kang, K.; Cai, W. Brittle and ductile fracture of semiconductor nanowires–molecular dynamics simulations. _Philosophical Magazine_ 2007, _87_ , 2169–2189
  * Gates et al. 2005 Gates, T.; Odegard, G.; Frankland, S.; Clancy, T. Computational materials: multi-scale modeling and simulation of nanostructured materials. _Composites Science and Technology_ 2005, _65_ , 2416–2434
  * Martinez et al. 2008 Martinez, E.; Marian, J.; Arsenlis, A.; Victoria, M.; Perlado, J. M. Atomistically informed dislocation dynamics in fcc crystals. _Journal of the Mechanics and Physics of Solids_ 2008, _56_ , 869–895
  * Potirniche et al. 2006 Potirniche, G.; Horstemeyer, M.; Wagner, G.; Gullett, P. A molecular dynamics study of void growth and coalescence in single crystal nickel. _International Journal of Plasticity_ 2006, _22_ , 257–278
  * Jelinek et al. 2012 Jelinek, B.; Groh, S.; Horstemeyer, M. F.; Houze, J.; Kim, S.-G.; Wagner, G. J.; Moitra, A.; Baskes, M. I. Modified embedded atom method potential for Al, Si, Mg, Cu, and Fe alloys. _Physical Review B_ 2012, _85_ , 245102
  * Kim and Lee 2008 Kim, Y.-M.; Lee, B.-J. Modified embedded-atom method interatomic potentials for the Ti–C and Ti–N binary systems. _Acta Materialia_ 2008, _56_ , 3481–3489
  * Huang et al. 2018 Huang, Z.; Chen, F.; Shen, Q.; Zhang, L.; Rupert, T. J. Uncovering the influence of common nonmetallic impurities on the stability and strength of a $\Sigma$5 (310) grain boundary in Cu. _Acta Materialia_ 2018, _148_ , 110–122
  * Miraz et al. 2020 Miraz, A. S. M.; Dhariwal, N.; Meng, W.; Ramachandran, B. R.; Wick, C. D. Development and application of interatomic potentials to study the stability and shear strength of Ti/TiN and Cu/TiN interfaces. _Materials & Design_ 2020, _196_ , 109123
  * Zhou et al. 2023 Zhou, X.; Ahmadian, A.; Gault, B.; Ophus, C.; Liebscher, C. H.; Dehm, G.; Raabe, D. Atomic motifs govern the decoration of grain boundaries by interstitial solutes. _Nature Communications_ 2023, _14_ , 3535
  * Kresse and Hafner 1993 Kresse, G.; Hafner, J. Ab initio molecular dynamics for open-shell transition metals. _Physical Review B_ 1993, _48_ , 13115
  * Kresse and Furthmüller 1996 Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. _Physical Review B_ 1996, _54_ , 11169
  * Kresse and Furthmüller 1996 Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. _Computational Materials Science_ 1996, _6_ , 15–50
  * Perdew et al. 1996 Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. _Physical Review Letters_ 1996, _77_ , 3865
  * Ganchenkova et al. 2014 Ganchenkova, M.; Yagodzinskyy, Y.; Borodin, V.; Hänninen, H. Effects of hydrogen and impurities on void nucleation in copper: Simulation point of view. _Philosophical Magazine_ 2014, _94_ , 3522–3548
  * Davidson 1975 Davidson, E. R. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. _Journal of Computational Physics_ 1975, _17_ , 87–94
  * Pulay 1980 Pulay, P. Convergence acceleration of iterative sequences. The case of SCF iteration. _Chemical Physics Letters_ 1980, _73_ , 393–398
  * Wood and Zunger 1985 Wood, D.; Zunger, A. A new method for diagonalising large matrices. _Journal of Physics A: Mathematical and General_ 1985, _18_ , 1343
  * Wu et al. 2016 Wu, X.; You, Y.-W.; Kong, X.-S.; Chen, J.-L.; Luo, G.-N.; Lu, G.-H.; Liu, C.; Wang, Z. First-principles determination of grain boundary strengthening in tungsten: Dependence on grain boundary structure and metallic radius of solute. _Acta Materialia_ 2016, _120_ , 315–326
  * Nazarov et al. 2012 Nazarov, R.; Hickel, T.; Neugebauer, J. Vacancy formation energies in fcc metals: influence of exchange-correlation functionals and correction schemes. _Physical Review B_ 2012, _85_ , 144118
  * Nazarov et al. 2014 Nazarov, R.; Hickel, T.; Neugebauer, J. Ab initio study of H-vacancy interactions in fcc metals: Implications for the formation of superabundant vacancies. _Physical Review B_ 2014, _89_ , 144108
  * Fotopoulos et al. 2023 Fotopoulos, V.; Grau-Crespo, R.; Shluger, A. L. Thermodynamic analysis of the interaction between metal vacancies and hydrogen in bulk Cu. _Physical Chemistry Chemical Physics_ 2023, _25_ , 9168–9175
  * Thompson et al. 2022 Thompson, A. P.; Aktulga, H. M.; Berger, R.; Bolintineanu, D. S.; Brown, W. M.; Crozier, P. S.; in’t Veld, P. J.; Kohlmeyer, A.; Moore, S. G.; Nguyen, T. D.; Shan, R.; Stevens, M. J.; Tranchida, J.; Trott, C.; Plimpton, S. J. LAMMPS-A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. _Computer Physics Communications_ 2022, _271_ , 108171
  * Hirel 2015 Hirel, P. Atomsk: A tool for manipulating and converting atomic data files. _Computer Physics Communications_ 2015, _197_ , 212–219
  * Bourne et al. 2020 Bourne, D. P.; Kok, P. J.; Roper, S. M.; Spanjer, W. D. Laguerre tessellations and polycrystalline microstructures: A fast algorithm for generating grains of given volumes. _Philosophical Magazine_ 2020, _100_ , 2677–2707
  * Mantisi 2016 Mantisi, B. Generation of polycrystalline material at the atomic scale. _Computational Materials Science_ 2016, _118_ , 245–250
  * Abrams 1971 Abrams, H. Grain size measurement by the intercept method. _Metallography_ 1971, _4_ , 59–78
  * Cao et al. 2019 Cao, C.; Yao, G.; Jiang, L.; Sokoluk, M.; Wang, X.; Ciston, J.; Javadi, A.; Guan, Z.; De Rosa, I.; Xie, W.; Lavernia, E. J. Bulk ultrafine grained/nanocrystalline metals via slow cooling. _Science Advances_ 2019, _5_ , 2398
  * Bober et al. 2019 Bober, D. B.; LaGrange, T.; Kumar, M.; Rupert, T. J. Pronounced grain boundary network evolution in nanocrystalline Cu subjected to large cyclic strains. _Journal of Materials Research_ 2019, _34_ , 35–47
  * Sanders et al. 1997 Sanders, P. G.; Eastman, J.; Weertman, J. Elastic and tensile behavior of nanocrystalline copper and palladium. _Acta Materialia_ 1997, _45_ , 4019–4025
  * Chen et al. 2019 Chen, P.; Zhang, Z.; Liu, C.; An, T.; Yu, H.; Qin, F. Temperature and grain size dependences of mechanical properties of nanocrystalline copper by molecular dynamics simulation. _Modelling and Simulation in Materials Science and Engineering_ 2019, _27_ , 065012
  * Zhou et al. 2014 Zhou, K.; Liu, B.; Yao, Y.; Zhong, K. Effects of grain size and shape on mechanical properties of nanocrystalline copper investigated by molecular dynamics. _Materials Science and Engineering: A_ 2014, _615_ , 92–97
  * Rida et al. 2017 Rida, A.; Rouhaud, E.; Makke, A.; Micoulaut, M.; Mantisi, B. Study of the effects of grain size on the mechanical properties of nanocrystalline copper using molecular dynamics simulation with initial realistic samples. _Philosophical Magazine_ 2017, _97_ , 2387–2405
  * Xiang et al. 2013 Xiang, M.; Cui, J.; Tian, X.; Chen, J. Molecular dynamics study of grain size and strain rate dependent tensile properties of nanocrystalline copper. _Journal of Computational and Theoretical Nanoscience_ 2013, _10_ , 1215–1221
  * Schiøtz et al. 1999 Schiøtz, J.; Vegge, T.; Di Tolla, F.; Jacobsen, K. W. Atomic-scale simulations of the mechanical deformation of nanocrystalline metals. _Physical Review B_ 1999, _60_ , 11971
  * Cheng et al. 2005 Cheng, S.; Ma, E.; Wang, Y.; Kecskes, L.; Youssef, K.; Koch, C.; Trociewitz, U.; Han, K. Tensile properties of in situ consolidated nanocrystalline Cu. _Acta Materialia_ 2005, _53_ , 1521–1533
  * Guduru et al. 2007 Guduru, R. K.; Murty, K. L.; Youssef, K. M.; Scattergood, R. O.; Koch, C. C. Mechanical behavior of nanocrystalline copper. _Materials Science and Engineering: A_ 2007, _463_ , 14–21
  * Li et al. 2019 Li, J.; Lu, B.; Zhou, H.; Tian, C.; Xian, Y.; Hu, G.; Xia, R. Molecular dynamics simulation of mechanical properties of nanocrystalline platinum: Grain-size and temperature effects. _Physics Letters A_ 2019, _383_ , 1922–1928
  * Kapoor and Nemat-Nasser 1998 Kapoor, R.; Nemat-Nasser, S. Determination of temperature rise during high strain rate deformation. _Mechanics of Materials_ 1998, _27_ , 1–12
  * Chen et al. 2015 Chen, S.; Duan, Y.-H.; Huang, B.; Hu, W.-C. Structural properties, phase stability, elastic properties and electronic structures of Cu–Ti intermetallics. _Philosophical Magazine_ 2015, _95_ , 3535–3553
  * Wagih et al. 2020 Wagih, M.; Larsen, P. M.; Schuh, C. A. Learning grain boundary segregation energy spectra in polycrystals. _Nature Communications_ 2020, _11_ , 1–9
  * Nosé 1984 Nosé, S. A molecular dynamics method for simulations in the canonical ensemble. _Molecular Physics_ 1984, _52_ , 255–268
  * Hoover 1985 Hoover, W. G. Canonical dynamics: Equilibrium phase-space distributions. _Physical Review A_ 1985, _31_ , 1695
  * Kuppart et al. 2021 Kuppart, K.; Vigonski, S.; Aabloo, A.; Wang, Y.; Djurabekova, F.; Kyritsakis, A.; Zadin, V. Mechanism of spontaneous surface modifications on polycrystalline Cu due to electric fields. _Micromachines_ 2021, _12_ , 1178
  * Sargent et al. 2021 Sargent, N.; Jones, M.; Otis, R.; Shapiro, A. A.; Delplanque, J.-P.; Xiong, W. Integration of processing and microstructure models for non-equilibrium solidification in additive manufacturing. _Metals_ 2021, _11_ , 570
  * Zhou et al. 2017 Zhou, K.; Liu, B.; Shao, S.; Yao, Y. Molecular dynamics simulations of tension–compression asymmetry in nanocrystalline copper. _Physics Letters A_ 2017, _381_ , 1163–1168
  * Brandl et al. 2009 Brandl, C.; Derlet, P. M.; Van Swygenhoven, H. Strain rates in molecular dynamics simulations of nanocrystalline metals. _Philosophical Magazine_ 2009, _89_ , 3465–3475
  * Özeren et al. 2021 Özeren, H. D.; Nilsson, F.; Olsson, R. T.; Hedenqvist, M. S. Prediction of real tensile properties using extrapolations from atomistic simulations; An assessment on thermoplastic starch. _Polymer_ 2021, _228_ , 123919
  * Rice and Wang 1989 Rice, J. R.; Wang, J.-S. Embrittlement of interfaces by solute segregation. _Materials Science and Engineering: A_ 1989, _107_ , 23–40
  * Lu et al. 2006 Lu, G.-H.; Zhang, Y.; Deng, S.; Wang, T.; Kohyama, M.; Yamamoto, R.; Liu, F.; Horikawa, K.; Kanno, M. Origin of intergranular embrittlement of Al alloys induced by Na and Ca segregation: Grain boundary weakening. _Physical Review B_ 2006, _73_ , 224115
  * Tian et al. 2009 Tian, Z.; Yan, J.; Xiao, W.; Geng, W. Effect of lateral contraction and magnetism on the energy release upon fracture in metals: First-principles computational tensile tests. _Physical Review B_ 2009, _79_ , 144114
  * Huang et al. 2019 Huang, Z.; Chen, F.; Shen, Q.; Zhang, L.; Rupert, T. J. Combined effects of nonmetallic impurities and planned metallic dopants on grain boundary energy and strength. _Acta Materialia_ 2019, _166_ , 113–125
  * Karkina et al. 2016 Karkina, L.; Karkin, I.; Kuznetsov, A.; Razumov, I.; Korzhavyi, P. A.; Gornostyrev, Y. N. Solute–grain boundary interaction and segregation formation in Al: First principles calculations and molecular dynamics modeling. _Computational Materials Science_ 2016, _112_ , 18–26
  * Mason et al. 2017 Mason, D. R.; Nguyen-Manh, D.; Becquart, C. S. An empirical potential for simulating vacancy clusters in tungsten. _Journal of Physics: Condensed Matter_ 2017, _29_ , 505501
  * Ferrante et al. 1981 Ferrante, J.; Smith, J.; Rose, J. _Tribology Series_ ; Elsevier, 1981; Vol. 7; pp 19–30
  * Yamaguchi et al. 2019 Yamaguchi, M.; Ebihara, K.-I.; Itakura, M.; Tsuru, T.; Matsuda, K.; Toda, H. First-principles calculation of multiple hydrogen segregation along aluminum grain boundaries. _Computational Materials Science_ 2019, _156_ , 368–375
  * Zhu et al. 2016 Zhu, Y.; Yan, M.; Zhang, Y.; Zhang, C. First-principles investigation of structural, mechanical and electronic properties for Cu–Ti intermetallics. _Computational Materials Science_ 2016, _123_ , 70–78
  * Momma and Izumi 2008 Momma, K.; Izumi, F. VESTA: A three-dimensional visualization system for electronic and structural analysis. _Journal of Applied Crystallography_ 2008, _41_ , 653–658
  * Faken and Jónsson 1994 Faken, D.; Jónsson, H. Systematic analysis of local atomic structure combined with 3D computer graphics. _Computational Materials Science_ 1994, _2_ , 279–286
  * Stukowski 2009 Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool. _Modelling and Simulation in Materials Science and Engineering_ 2009, _18_ , 015012
  * Bateni et al. 2003 Bateni, M. R.; Szpunar, J.; Ashrafizadeh, F.; Zandrahimi, M. The effect of novel Ti-Cu intermetallic compound coatings on tribological properties of copper. _Surfaces_ 2003, _10_
  * Čižek et al. 1974 Čižek, L.; Kratochvíl, P.; Smola, B. Solid solution hardening of copper crystals. _Journal of Materials Science_ 1974, _9_ , 1517–1520
  * Li et al. 2020 Li, J. C.; Feng, C.; Rath, B. B. Emission of dislocations from grain boundaries and its role in nanomaterials. _Crystals_ 2020, _11_ , 41
  * Hu et al. 2022 Hu, S.; Fu, T.; Liang, Q.; Weng, S.; Chen, X.; Zhao, Y.; Peng, X. Formation and anisotropic mechanical behavior of stacking fault tetrahedron in Ni and CoCrFeNiMn high-entropy alloy. _Frontiers in Materials_ 2022, _8_ , 608
  * Ma et al. 2022 Ma, N.; Zhao, D.; Shi, C.; He, C.; Liu, E.; Sha, J.; Li, Y.; Zhao, N. Unraveling the enhanced stability and strength of Al $\Sigma$9 (221)[1 1¯ 0] symmetric tilt grain boundary with Mg segregation. _Journal of Materials Science_ 2022, _57_ , 21591–21606
  * Hall 1951 Hall, E. The deformation and ageing of mild steel: II characteristics of the Lüders deformation. _Proceedings of the Physical Society. Section B_ 1951, _64_ , 742
  * Petch 1953 Petch, N. The cleavage strength of polycrystals. _Journal of the Iron and Steel Institute_ 1953, _174_ , 25–28