# Simulations of Nanocrystalline Iron Formation under High Shear Strain

Ivan Tolkachev<EMAIL_ADDRESS>Department of Engineering Science,
University of Oxford, United Kingdom  Pui-Wai Ma<EMAIL_ADDRESS>United
Kingdom Atomic Energy Authority, Culham Science Centre, Abingdon, Oxfordshire,
OX14 3DB, United Kingdom  Felix Hofmann<EMAIL_ADDRESS>Department
of Engineering Science, University of Oxford, United Kingdom

###### Abstract

High-shear methods have long been used in experiments to refine grain
structures in metals, yet the underlying mechanisms remain elusive. We
demonstrate a refinement process using molecular dynamic simulations, wherein
nanocrystalline structures are generated from initially perfect lattices under
high-shear strain. The simulation cells undergo a highly disordered state,
followed by recrystallization and grain coarsening, resulting in nanograins.
We explore the dependence on parameters such as temperature, heat dissipation
rate, shear strain rate, and carbon impurity concentration. Higher
temperatures lead to the formation of larger and longer grains. The faster
heat dissipation sample initially yields more small grains, but their number
subsequently reduces, and is lower than the slower heat dissipation sample at
approximately 1.5 strain. Slower strain rates do not promote nanograin
formation. The presence of carbon impurities appears to have little effect on
grain formation. This detailed analysis affords insight into the mechanisms
that control the formation of nanograins under high-shear conditions.

##  I Introduction

Grain refinement can be achieved through various methods, including the use of
chemical refiners or spray-forming techniques [1]. An attractive approach to
accomplish grain refinement is through severe plastic deformation (SPD)
techniques, such as accumulative roll bonding (ARB) and equal channel angular
pressing (ECAP) [2]. High-shear methods, such as high-pressure torsion (HPT),
can significantly alter the microstructure of metals. Numerous materials,
including Al, Cu, and Mg alloys, have shown a reduction in grain size with
increasing shear strain [3, 4, 5, 6, 7]. Yusuf et al. [7] observed a reduction
in grain size with increasing shear strain for 316L stainless steel, which
also correlates with an increase in Vickers microhardness. Despite numerous
experimental studies attempting to elucidate the mechanisms of nanocrystal
formation with increased shear strain, the results are inconclusive [8].

The leading theory for grain refinement suggests that dislocations are
generated in the material due to shear strain. These dislocations accumulate
and form subgrain boundaries. As the shear strain increases, dislocations are
annihilated at these boundaries, leading to a rise in the misorientation angle
between grains. Some of the formed dislocations are not absorbed and persist,
forming low-angle grain boundaries (LAGBs), which perpetuates the grain
refinement process [4, 8, 9, 10].

Other factors can also influence grain refinement, such as the number and
distribution of precipitates within the material [11, 12, 13], as well as the
presence of twinning deformation [14, 15]. Isik et al. [16] observed a grain
subdivision and grain refinement through a deformation-induced martensitic
phase transformation, from $\gamma\rightarrow\epsilon$ phase in Co-28Cr-6Mo
alloy.

Numerous computational methods have been employed to explore the mechanism of
grain refinement through plastic shearing. Finite element analysis (FEA) has
been used to understand how the HPT process can modify the samples [17, 18,
19, 20]. FEA simulations greatly depend on the constitutive model used during
the simulation [21]. As such, the physics of a given FEA system are largely
defined by the constitutive model, which may not allow other physical
mechanisms of grain refinement to exist within the simulation, which in turn
affects the observable deformation modes. Nevertheless, numerous FEA studies
have been conducted that draw conclusions on possible grain refinement
mechanisms under shear strain. Wei et al. [22] investigated the mechanisms of
grain refinement in single crystal Ni subjected to HPT, employing crystal
plasticity finite element simulations. The results proposed the existence of
two rotation modes that directly contributed to the formation of grain
boundaries with high misorientations. Additionally, FEA simulations of high
angle pressing (HAP) were conducted by Frydrych et al. [23] to explore the
mechanism of grain refinement in an already nanocrystalline face-centered
cubic (FCC) material. This study confirmed that initial grain orientation was
a major indicator of their susceptibility to further refinement. The study
showed that grains which had larger Taylor factors were more prone to
refinement. A larger Taylor factor corresponds to greater plastic work done,
when compared over the same range of slip [24]. The work of Frydrych [23]
makes no further attempt to quantify the orientations, making it difficult to
obtain exact lattice orientations that are more prone to refinement.

Molecular dynamics (MD) simulations have been employed to study
microstructural evolution in metals subjected to SPD [25, 26, 27, 28, 29, 30,
31]. However, most of these studies start with simulation cells that are
already in a nanocrystalline state, and therefore, do not explore the
mechanisms by which a single crystal can transform into a nanocrystalline
state. Interestingly, the work of Nikonov [31] used MD simulations to shear a
perfect, single crystal simulation cell. However, the results were only used
for shear stress vs. strain comparisons with a polycrystalline box, and no
conclusions were reached about the single crystal shear. The work of Guan et
al. [32] attempted to replicate the HPT process using an MD simulation of
Aluminum. However, similar to the other MD investigations mentioned above, the
initial box was already in a nanocrystalline state before the shearing took
place, and thus, there was no investigation into the actual formation of
nanocrystals.

The precise underlying processes responsible for the development of a
nanocrystalline microstructure via HPT remains elusive. Despite the
significant disparity between shear rates in simulations and experimental
settings, we conducted molecular dynamics simulations on iron to gain insights
into potential mechanisms through which shear could induce the formation of
nanocrystalline structures. Additionally, we scrutinized the progression of
microstructures to facilitate meaningful comparisons with experimental
observations.

Iron is chosen as the material of the current study because of its diverse
uses in domestic and industrial applications. Especially, in advanced fission
and future fusion reactors [33, 34, 35], iron-based steels, such as
ferritic/martensitic (FM) steels, are chosen as the structural material. FM
steels have the same body-centred cubic (BCC) crystal structure as iron. It
was shown experimentally that FM steels exhibited less neutron irradiation-
induced swelling than austenitic steels with face-centred cubic (FCC)
structure [36].

In the following, we first discuss our simulation methods. Then, we present
our simulation results and discuss potential mechanisms underlying the
observed phenomena. We also draw comparisons with experimental HPT studies.

##  II Simulation setup

All MD simulations were carried out using LAMMPS [37]. Simulation cells were
created using Atomsk [38]. We constructed perfect crystal simulation cells
with $80\times 80\times 80$ unit cells, where each unit cell contains 2 atoms
in BCC structure with a lattice parameter of 2.8665 Å, corresponding to Fe.
Each simulation cell contains 1,024,000 atoms. The starting crystal
orientations are x = [100], y = [010], and z = [001]. Periodic boundary
conditions are applied in all three directions.

We also examined the effect of carbon impurities. We created simulation cells
by adding 102 carbon atoms into those perfect cells at random positions. This
corresponds to a carbon impurity content of 100 appm.

We adopted the interatomic potential for Fe developed by Ackland et al. [39],
which has been widely used to investigate the microstructure evolution of iron
[40, 41, 42, 43]. For Fe-C simulations, we used the Hepburn-Ackland FeC
interatomic potential [44], which was developed based on the aforementioned
Ackland Fe potential. It has been used in several studies concerned with the
effect of carbon on the microstructural evolution of iron [45, 46, 47, 48].

The trajectory of a system of interacting atoms is governed by the Langevin
equation of motion:

$\displaystyle\frac{d\mathbf{r}_{i}}{dt}$ $\displaystyle=$
$\displaystyle\mathbf{v}_{i}$ (1) $\displaystyle m\frac{d\mathbf{v}_{i}}{dt}$
$\displaystyle=$
$\displaystyle\mathbf{F}_{i}-\gamma\mathbf{v}_{i}+\mathbf{f}_{i}$ (2)

where $\mathbf{r}_{i}$, $\mathbf{v}_{i}$, and $\mathbf{F}_{i}=-\partial
U/\partial\mathbf{r}_{i}$ are position, velocity, and force associated with
atom $i$. $U$ is the interatomic potential energy. $m$ is the mass of an atom.
The temperature is controlled by the Langevin thermostat, where the damping
parameter $\gamma$ is related to the delta-correlated fluctuating force
$\mathbf{f}_{i}$ according to the fluctuation-dissipation theorem [49, 50],
namely,

$\langle\mathbf{f}_{i}(t)\rangle=0,$ (3) $\langle
f_{i\alpha}(t)f_{j\beta}(t^{\prime})\rangle=\mu\delta_{ij}\delta_{\alpha\beta}\delta(t-t^{\prime}),$
(4)

where $\alpha$ and $\beta$ are Cartesian coordinates, and

$\mu=2\gamma k_{B}T.$ (5)

The fluctuation and dissipation of atoms can be considered as a result of
electron-phonon interaction, so $\gamma$ describes the strength of electron-
phonon coupling. We used the electron-phonon coupling parameter for Fe
according to Mason et al. [51], such that $\gamma=6.875$ eV fs Å-2. We denote
this as $\gamma_{1}$. In some simulations, we used a damping parameter that is
ten times larger, which is denoted as $\gamma_{2}=10\gamma_{1}$. If we do not
explicitly mention the value of $\gamma$ below, we are using $\gamma_{1}$.

Before applying any shear, the simulation cells are thermalised to particular
temperatures. The cell volumes are also relaxed isotropically, so they attain
stress-free conditions. This is done under isobaric conditions with zero
hydrostatic stress.

Shear was imposed by continually deforming the simulation cell. The shear
strain is applied on the xy plane with displacement in the x direction. This
was done by imposing a cell tilt factor change, which is effectively an
engineering strain. A total simulation time of 335 ps was used, with a maximum
final shear strain of $\varepsilon=10$. This means the shear rate
$d\varepsilon/dt$ is approximately $2.985\times 10^{10}$ s-1. We also
performed the same simulation with a 10 times longer simulation time,
corresponding to a 10 times lower strain rate.

To avoid boundary self-interactions, we remap the simulation cell when the
tilting is more than 0.5 of the cell length. The cell vector and the positions
of atoms are remapped along the shearing direction, with the simulation cell
going from a 0.5 to a -0.5 tilt. It is important to note that, due to the
periodic boundary conditions, there is no change in the local atomic
environment.

We used OVITO [52] for analysis. Grains are identified using the grain
segmentation modifier. The minimum grain size was chosen as 50 atoms. We
discuss the effect of using a different number of atoms as the minimum grain
size in Appendix A3. Since this algorithm requires the crystal orientation of
each atom, the polyhedral template matching (PTM) modifier [53] is adopted
beforehand. PTM can determine the local crystal orientation of each atom in
terms of a quaternion, where each quaternion can be projected into Rodrigues
space [54]. Then, the orientation of each atom in this space can be visualised
by mapping orientation to an RGB colouring scheme [55]. Dislocation lines are
detected using the dislocation analysis (DXA) modifier [56].

##  III Results

We first present the results of our simulations exploring the influence of
shear strain on initially perfect, single-crystalline iron. Next, we
investigate the behaviour of samples under various conditions, including
differing temperatures, damping parameters, strain rates, and the presence of
carbon impurities.

###  III.1 Shear-induced nanocrystalline structure

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 1: Grain refinement process with atoms coloured by crystal
orientations. Temperature $T=$300 K. Strain rate $d\varepsilon/dt=1/33.5$
ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 and damping parameter
$\gamma=6.875$ eV fs Å2. (a) 0 Strain (b) Highly Disordered - 0.27 Strain (c)
Recrystallisation - 0.32 Strain (d) 1 Strain (e) 4 Strain (f) 7 Strain (g) 10
Strain.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Benchmark simulation data plotted as a function strain. Temperature
. Strain rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$
s-1 and damping parameter $\gamma=6.875$ eV fs Å2. (a) Kinetic and Potential
Energy (b) Kinetic and Potential Energy up to 2 strain (c) Temperature (d)
Number of Grains (e) Dislocation Density (f) von Mises Stress

Figure 1 shows a key result of this work. Figure 1(a) to 1(g) show the
simulation cell at different shear strain. The results reveal a process in
which nanograins are formed due to shear strain in a real-space
representation. This simulation was run for 335 ps, up to $\varepsilon=10$, at
a constant strain rate, at a temperature of 300 K, and with a constant damping
parameter $\gamma_{1}$. We will refer to this as the benchmark simulation. The
colouring of each atom in Figure 1 is based on its crystal orientation. Only
atoms detected as BCC structures are shown. Since a similar colour means a
similar crystal orientation, one can clearly observe the formation of
nanograins in real space. We are not aware of any similar simulations in the
literature showing the formation of nanograins under high shear strain.

Figure 1(a) shows the initially perfect crystal simulation cell. It is
subjected to a shear strain in the xy direction, which causes the crystal
lattice to lose its BCC structure. At around 0.27 strain, there is a high
level of atomic disorder, as shown in Figure 1(b). Most atoms deviate from the
BCC structure. We can consider that atoms are now in a disordered state.
Shortly after that, a recrystallization process begins, as shown in Figure
1(c), at around 0.3 strain. Many small grains are visible. Between Figure 1(c)
and 1(d), grain formation clearly occurs, with the atomic crystal orientations
still largely similar. By Figure 1(e), at $\varepsilon=4$, the atomic crystal
orientations are largely dissimilar, with grains exhibiting distinct
orientations. It appears that the grain refinement process continues between 4
and 7 strain, see Figure 1(f). Then, the grain number and sizes attain a
dynamical quasi-steady state.

Grain growth has been observed for different metals in simulations [32, 57]
and experiments [58]. These investigations typically focus on samples that
already possess a grain structure before applying any strain or stress. A
fundamental difference is that the current simulations start with a perfect
crystal structure. Our results show that when shear strain is applied, atoms
become highly disordered, followed by recrystallisation into a polycrystalline
structure. However, such a mechanism does not always occur under different
simulation conditions, as will be discussed below.

For a more detailed analysis, we can consider Figure 2 alongside Figure 1.
Figure 2(a) shows the average atomic potential and kinetic energy as a
function of shear strain. Figure 2(b) is an enlarged version that covers a
range up to $\varepsilon=1$. When the shear strain is first applied, there is
a gradual increase in the potential energy up to a value of around -3.73
eV/atom. The initial potential energy increment appears parabolic, suggesting
a linear elastic response, as the elastic potential energy is proportional to
$\varepsilon^{2}$. The maximum point of potential energy occurs at 0.27
strain, which is the point shown in Figure 1(b). The disordered state is a
state of high potential energy.

Then, the potential energy in the system gradually reduces to around -3.89
eV/atom and remains roughly at this value for the remaining shearing process.
At 0.27 strain, where the potential energy reaches its peak, we also observe a
rapid increase in the kinetic energy of the system, as shown in Figure 2(b),
from a value of 0.039 eV/atom to 0.072 eV/atom, peaking at 0.3 strain. This
increase is followed by a sharp decrease, reaching a value of around 0.042
eV/atom, at which the simulation plateaus for the continued shearing. Since
the estimated system temperature is directly proportional to the kinetic
energy, we observe a substantial increase in temperature at 0.27 strain, as
shown in Figure 2(c). The maximum temperature reaches 560 K, which is far
below 1,811 K, the melting temperature of iron.

Figure 2(d) illustrates the number of grains detected by OVITO. The number of
grains undergoes a sharp increase at $\varepsilon=0.25$ and peaks at
$\varepsilon=0.27$ with approximately 400 grains. This corresponds to the
highly disordered state, where OVITO identifies small pockets of BCC structure
as individual grains within a volume of highly disordered atoms. After that,
there is a rapid decrease in grain number, reaching a minimum of 25 grains at
around 0.3 strain, coinciding with the decrease in potential energy. Following
this, the number of grains once again rises to around 230 at $\varepsilon=2$.
Then, there is a decrease to about 160 at $\varepsilon=3$. Another increase in
grain count is observed after reaching $\varepsilon=6$, plateauing at a value
of around 240 until the end of the simulation. In Figure 2(e), we also observe
an increase in dislocation density starting at a strain value of 0.27. The
dislocation density is given as the total line length divided by the volume of
the simulation cell. Dislocation density rises to approximately $2.6\times
10^{-3}/$Å2 at 0.82 strain, followed by a decrease to $1.2\times 10^{-3}/$Å2
at around 2 strain. The dislocation density experiences a slight increase to
$1.6\times 10^{-3}/$Å2 at 3 strain and subsequently remains relatively
constant during further shearing.

The average atomic von Mises stress, as shown in Figure 2(f), and the
potential energy per atom are well correlated. There is an initial increase in
von Mises stress, followed by a rapid decrease when the highly disordered
state is formed at 0.27 strain. Nikonov [31] also found an increase in stress
during the single crystal shearing of BCC iron. The same rapid increase,
followed by a sharp decrease, was observed and attributed to lattice
reorientation. It is noteworthy that the potential energy never decreases to
its original value, as some excess energy, associated with dislocations and
grain boundaries generated during deformation, is stored within the material
[59].

As shown in Figure 1, the atoms become highly disordered, with peak disorder
occurring at 0.27 strain, which corresponds to a maximum in the potential
energy. This maximum is a threshold value after which the atoms within the
lattice move and reorganise themselves, as made evident by the increase in
kinetic energy in Figure 2(b), and the recrystallisation at 0.3 strain in
Figure 1(c). The increasing shear strain causes a breaking of the symmetry in
the crystal lattice that changes the potential energy landscape. Since atoms
already have a high potential energy, triggered by thermal excitations, the
unbalanced forces acting on atoms cause them to accelerate, triggering the
onset of the disordered phase.

Subsequently, there is a surge in system temperature, as shown in Figure 2(c),
which is due to the increase in atomic velocities. As our system is attached
to a Langevin thermostat, the system temperature reduces back to the target
temperature. Alongside the cool-down, the system undergoes a
recrystallization, causing grain nucleation. The grains are tiny at around 0.3
strain, see Figure 1(c). The microstructure remains dominated by highly
disoriented regions. Then, grain coarsening starts and continues, as observed
in Figure 2(d). This is driven by the excess free energy of the disordered
region manifesting as grain boundaries, which are known to possess excess free
energy [60]. This excess free energy provides the driving force for atomic
transport and subsequent grain growth [61].

In Fig 2(e), the dislocation density in the simulation cell increases and
peaks at around 0.82 strain. Dislocations also possess excess free energy
[60], which may contribute to grain growth. We note an important point
regarding the DXA: LAGBs can be recognised as arrays of dislocations.
Therefore, it is unclear if the detected dislocations are grain boundaries or
dislocations within grains. There is a new approach that may resolve this
situation. Ma et al. [62] suggested a new algorithm to calculate the shortest
distance of any atom from the determined grain boundaries. By eliminating
atoms close to the grain boundaries, one can estimate the dislocation line
density inside grains. However, we have not adopted this method in our current
work, because the definition of grains is somewhat ambiguous in such a highly
disordered structure. Additional analysis is provided in Appendix A2.

(a) Key; Total Energy Left; von Mises Stress Right

(b) 0 Strain - Benchmark

(c) Disordered Phase - 0.27 Strain - Benchmark

(d) Recrystallisation - 0.32 Strain - Benchmark

(e) 1 Strain - Benchmark

Figure 3: Benchmark simulation visualisations. Strain rate
$d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 and
damping parameter $\gamma=6.875$ eV fs Å2. Colour key found in (a). Left -
Atoms coloured by crystal orientations; Centre-left - Total energy; Centre-
right - von Mises stress; Right - Dislocation analysis; for different levels
of strain in benchmark simulation.

(a) 4 Strain - Benchmark

(b) 7 Strain - Benchmark

(c) 10 Strain - Benchmark

Figure 4: Benchmark simulation visualisations. Strain rate
$d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 and
damping parameter $\gamma=6.875$ eV fs Å2. Colour key found in (a). Left -
Atoms coloured by crystal orientations; Centre-left - Total energy; Centre-
right - von Mises stress; Right - Dislocation analysis; for different levels
of strain in benchmark simulation.

Experimental studies have exhibited similar trends for nanocrystal formation.
Studies on Fe-8% Al [63], TiAl [64], AZ91 Mg [65], and NiTi [66], all used
some form of hot deformation to induce strain into the material, followed by
quenching. These studies observed small, recrystallised grains compared to the
original structure.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Simulation data plotted as a function of strain at different
thermostat temperatures. Strain rate $d\varepsilon/dt=1/33.5$
ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 and damping parameter
$\gamma=6.875$ eV fs Å2. Red line - 300 K, Blue line - 500 K, Green line - 800
K, Black line - 1000 K. (a) Kinetic Energy (b) Kinetic Energy up to 2 strain
(c) Potential Energy (d) Potential Energy up to 2 strain (e) Temperature (f)
Number of Grains (g) Number of Grain up to 2 strain (h) Von Mises Stress up to
2 Strain (i) Dislocation Density.

(a)

(b)

(c)

Figure 6: Simulation data plotted as a function of strain at different
temperature simulations. Strain rate $d\varepsilon/dt=1/33.5$
ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 and damping parameter
$\gamma=6.875$ eV fs Å2. Red line - 300 K, Blue line - 500 K, Green line - 800
K, Black line - 1000 K. (a) Kinetic Energy (b) Kinetic Energy up to 2 strain
(c) Potential Energy (d) Potential Energy up to 2 strain (e) Temperature (f)
Number of Grains (g) Number of Grain up to 2 strain (h) Von Mises Stress up to
2 Strain (i) Dislocation Density.

Figure 3 shows the atomic crystal orientation, total energy, von Mises stress,
and dislocation lines at different shear strain values. The total energy is
the sum of atomic potential and kinetic energy. The figure keys for the total
energy and von Mises stress are shown in Figure 3(a). In the DXA analysis,
green lines represent dislocations with Burgers vector
$\mathbf{b}=\frac{1}{2}\langle 111\rangle$, and pink lines represent
$\mathbf{b}=\langle 100\rangle$. Figure 3 shows that most dislocation lines
have $\mathbf{b}=\frac{1}{2}\langle 111\rangle$. Only few dislocations have
$\mathbf{b}=\langle 100\rangle$. This agrees with the experimental findings in
the literature on iron, iron-chromium alloys and ferritic/martensitic steels
[67, 68, 69].

Next, we consider the total energy and von Mises stress, shown in Figure 3.
Initially, the atoms within the box have low energy and low stress. As the
atoms become disordered, they experience a high energy and stress state, as
shown by the colouring in Fig. 3(c). As recrystallisation occurs, it is
evident that some atoms return to the low energy and stress state, whilst
others remain in the high energy and stress state (Fig. 3(d)). As shear strain
is continually applied, we can observe areas of both high and low energy and
stress. The areas of low energy and low stress correspond to atoms within the
grains and are BCC in nature, whilst the areas of high energy and high stress
correspond to atoms which are not BCC in nature, and are not shown in the
crystal orientation images. As such, Figure 3 shows that the atoms between the
grains have high energy and high stress, which confirms the presence of grain
boundaries in the simulation box [60].

###  III.2 Temperature

To further probe the underlying mechanisms for nanocrystal formation from an
initially perfect crystal under high-shear strain, certain simulation
variables were altered one-by-one. We first changed the thermostat
temperature, whilst keeping all other conditions unchanged. Three additional
simulations were carried out with thermostat temperatures T = 500 K, 800 K,
and 1000 K. Figure 5 shows the kinetic energy, the potential energy, the
temperature, the number of grains, the dislocation density, and the von Mises
stress of different simulations.

Figure 5(a) and 5(b) show the change in kinetic energy. They are well
correlated with the system temperature in Figure 5(e) because of the linear
dependence between temperature and kinetic energy. We observe a notable
difference in the increase in kinetic energy associated with the formation of
the disordered phase at different thermostat temperatures. For example, the
kinetic energy increases from around 0.039 eV/atom to 0.072 eV/atom for the
300 K simulation, an increase of 0.033 eV/atom, whilst the 800 K simulation
experiences a spike from 0.1 eV/atom to around 0.117 eV/atom, an increase of
0.017 eV/atom, much less than the increment at 300 K. We can also observe from
Figure 5(b) that the initial increase in kinetic energy occurs earlier for the
higher temperature simulations, occurring at 0.27 strain for the 300 K, 0.25
strain for the 500 K, 0.2 strain for the 800 K, and 0.19 strain for the 1000 K
simulations.

The increase in kinetic energy corresponds to a decrease in potential energy.
Figure 5(c) and 5(d) show how the potential energy begins to decrease earlier
for the higher temperature simulations. Interestingly, the thermostat
temperature does not appear to significantly alter the maximum potential
energy. Whilst the 300 K simulation increases to a maximum of -3.73 eV/atom,
the higher temperature 500 K and 800 K simulations increase to -3.74 eV/atom,
and the 1000 K simulation increases to -3.72 eV/atom. The higher temperature
simulations generally experience higher potential energy after the initial
spike, with the 1000 K simulation hovering at around -3.82 eV/atom after
$\varepsilon=4$, whilst the 300 K simulation plateaus at around -3.89 eV/atom.

Inspection of Figure 5(f) and 6(a) provides interesting insight into the
effect of the thermostat temperature change. All simulations experience an
initial spike in grain number, with the spike occurring at a lower strain for
higher temperature simulations. The magnitude of the increase does not appear
to correlate with thermostat temperature. For example, whilst the 300 K
simulation increases to around 400 grains, the 1000 K simulation only
increases to around 195 grains, which is less than the 500 K simulation, but
more than the 800 K simulation.

All simulations show a rapid decrease in grain number between 0.19 and 0.25
strain, followed by an increase in grain number at $\sim$2 strain. The 300 K
simulation notably has the most grains at the local maximum point of 2 strain.
The other temperature simulations have fewer grains at this point, with the
500 K, 800 K, and 1000 K simulations having roughly the same number of grains
up to $\varepsilon=4$. The grain number in the 500 K and 800 K simulations
increases and saturates after $\varepsilon=7$. The 1000 K simulation
experiences a reduction in grain number after $\varepsilon=4$, and is as low
as 11 grains at $\varepsilon=10$.

(a) 0 strain

(b) Disordered Phase

(c) Recrystallisation

(d) 1 strain

Figure 7: Atoms coloured by orientation for different temperature simulations.
Strain rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$
s-1 and damping parameter $\gamma=6.875$ eV fs Å2. Left - T = 300 K
simulation; Centre-left - T = 500 K simulation; Centre-right - T = 800 K
simulation; Right - T = 1000 K simulation.

(a) 4 strain

(b) 7 strain

(c) 10 strain

Figure 8: Atoms coloured by orientation for different temperature simulations.
Strain rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$
s-1 and damping parameter $\gamma=6.875$ eV fs Å2. Left - T = 300 K
simulation; Centre-left - T = 500 K simulation; Centre-right - T = 800 K
simulation; Right - T = 1000 K simulation.

Figure 6(b) shows the von Mises stress. The stress directly correlates with
the potential energy. This is evident in the figure as the decrease in stress
after the initial increase occurs at exactly the same strain value as the
potential energy decrease (see Figure 5(c) and 5(d)). Nevertheless, whilst the
potential energy maximum is roughly the same value for all simulations, the
von Mises stress shows an unequal increase. As made evident by Figure 6(b),
the higher temperature simulations show a lower maximum von Mises stress and,
as the simulation temperature decreases, the maximum von Mises stress
increases. The subsequent decrease in von Mises stress is also larger in
magnitude for the lower temperature simulations, and the value at which the
stress plateaus with shearing is lower for the lower temperature simulations.

We observe in Figure 6(c) that the dislocation densities for the higher
temperature simulations also follow the same pattern as the benchmark
simulation. For each simulation, there is a marked increase in dislocation
density between 0.19 and 0.25 strain, which corresponds to the point when
kinetic energy begins to increase, and the potential energy decreases. This is
followed by a maximum point in dislocation density, occurring at 0.82 strain
for the 300 K simulation, 0.74 strain for the 500 K simulation, 0.7 strain for
the 800 K simulation, and 0.6 strain for the 1000 K simulation. We note that
the maximum value of dislocation density does not correlate with temperature.
Both the 500 K and 800 K simulations have larger dislocation densities, whilst
the 300 K and 1000 K simulations have lower maximum densities. As with the
benchmark simulation, the dislocation densities decrease up to
$\varepsilon=2$, after which there is another increase and saturation in
dislocation density for all simulations.

In Figure 5(d), there appears to be a nearly universal maximum potential
energy, around -3.73 eV/atom, for every simulation. We hypothesise that this
value corresponds to the maximum potential energy the material can sustain
before atoms are permanently displaced from their lattice positions, and it is
the value attained before the initiation of the disordered atomic state. After
this point, the potential energy decreases while the kinetic energy increases.
The higher temperature simulations exhibit higher potential energy at 0 strain
due to the principle of equipartition of energy. At higher temperatures, a
system under strain should reach the point of instability, in terms of
potential energy, sooner. This, in turn, causes the kinetic energy to increase
at a lower strain value, meaning that the higher temperature simulations
experience the initiation of the disordered state at a lower strain. This
observation could explain the discrepancy in the von Mises stress peak between
the simulations at different temperatures. As the temperature increases, the
stress applied to reach a certain potential energy is reduced because the
potential energy is already at a higher level. Similarly, this also explains
why the spike in grain number occurs earlier for the higher temperature
simulations.

Interestingly, upon analyzing Figure 5(f) and 6(c), we observe that the peak
in grain number after recrystallization (at 2 strain) directly corresponds to
a minimum value in dislocation density after the formation of the disordered
phase. This finding aligns with previous MD simulations of nanocrystalline Cu
[25]. The study found that at high dislocation density, a system is highly
unstable when nanocrystalline grains are present. With applied strain, the
dislocations were able to glide towards and annihilate at grain boundaries.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: Simulation data plotted as a function of strain using different
damping parameters. Strain rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx
2.985\times 10^{10}$ s-1. Damping parameter $\gamma=6.875$ eV fs Å2 for red
line and damping parameter $\gamma=68.75$ eV fs Å2 for blue line. (a) Kinetic
energy (b) Potential energy (c) Number of grains (d) von Mises stress (e)
Dislocation density (f) Percentage of BCC structure.

To further understand the effect of temperature on nanograin formation, we can
consider Figure 7. This figure shows atoms coloured according to their local
orientation, and only atoms in the BCC phase are visualized. The simulations,
from left to right are at 300 K, 500 K, 800 K, and 1000 K, respectively. All
simulations begin with a perfect box, thermalised to the desired temperature
(Figure 7(a)). With applied shear strain, the cells all enter a highly
disordered state (Figure 7(b)). The level of disorder appears to be larger for
the lower temperature simulations. This is visible in that the number of BCC
atoms is less for the 300 K simulation compared to, for example, the 1000 K
simulation.

Each simulation experiences a subsequent recrystallization (Figure 7(c)),
followed by a grain growth up to 1 strain (Figure 7(d)). At this point,
through inspecting the colouring of atoms, the majority of atoms have similar
crystal orientation to the initially perfect crystal lattice, meaning that the
misorientation between grains is small. As the shearing continues, it is
evident that grains develop different orientations, as shown in Figure 8(a) \-
8(c). It is also evident that the grains are larger and longer for the higher
temperature simulations compared to the lower temperature simulations. This is
observed by comparing the 300 K and 1000 K simulations in Figure 8(c), where
the grain number and size are very different. This is also consistent with the
lower grain numbers for the higher temperature simulations compared to the 300
K benchmark, seen in Figure 5(f).

As $\alpha$-iron is a BCC metal, it has a very high stacking fault energy
[70]. It is generally agreed that Dynamic Recovery (DRV) is the sole dynamic
restoration mechanism in $\alpha$-iron and other ferritic metals [71, 70].
However, some studies have also shown that Dynamic Recrystallisation (DRX)
occurs in BCC iron [72, 31]. Typically, DRV readily occurs in ferritic steels
and iron at temperatures $>0.4T_{m}$ [73]. On the other hand, the data
presented in Figure 6(c) suggests that this can also occur at room
temperature, as this figure shows a marked annihilation of dislocations. The
fundamental mechanisms of DRV are dislocation glide, climb, and cross-slip
[70]. We speculate that the accumulation of dislocations also contributes to
nanocrystal formation, as the dislocations accumulate and form LAGBs. In
general, the dislocation density stays fairly constant after around
$\varepsilon=4$ for all simulations, whilst the grains continue to elongate,
which is especially prominent for higher temperature simulations. This
behaviour is expected for a metallic material under constant strain
deformation [70, 74].

###  III.3 Heat dissipation

The heat dissipation from the lattice subsystem to the environment is
represented by the Langevin thermostat. In metals, the dominant heat transfer
mechanism is through electrons [75]. Therefore, we take the damping parameter
$\gamma$, which governs the speed of heat dissipation, according to the
phonon-electron coupling. The damping parameter has, so far, been kept at
$\gamma=6.875$ eV fs Å${}^{-2}=\gamma_{1}$. To investigate the effect of the
rate of heat dissipation on nanocrystal formation, a further simulation was
carried out with a damping parameter of $\gamma=68.75$ eV fs
Å${}^{-2}=\gamma_{2}$. All other conditions were kept the same. The Langevin
thermostat was set to 300 K.

Figure 9 shows various quantities plotted as a function of shear strain when
the two different damping parameters are used. We also compare the percentage
of BCC structure found in the simulation cell as a function of strain. This
will become important when considering Figure 10, which compares the evolution
of the atomic orientations in simulations using different damping parameters.

Figure 9(a) shows the change in kinetic energy between the lower (or
benchmark) and higher damping simulations. The average kinetic energy in the
lower damping simulation increases from around 0.038 eV/atom to 0.072 eV/atom,
a difference of 0.034 eV/atom, before reducing to a dynamic quasi-steady
state. Similarly, the higher damping simulation experiences an increase from
0.034 eV/atom to around 0.045 eV/atom, a difference of 0.011 eV/atom, before
it drops. The increase in kinetic energy is much lower for the higher damping
simulation because a higher damping parameter leads to a higher quenching
rate. The kinetic energy is also noticeably lower for the higher damping
simulation once both simulations have reached the dynamic quasi-steady state.
The lower damping simulation plateaus at 0.042 eV/atom whilst the higher
damping simulation sits at 0.039 eV/atom.

Figure 9(b) shows that both simulations experience an identical increase in
potential energy with increasing shear strain up to a value of -3.73 eV/atom
at 0.27 strain, corresponding to the highly disorder state. However, whilst
the benchmark simulation reduces to and subsequently stabilises at around
-3.89 eV/atom, the higher damping simulation only decreases to around -3.82
eV/atom, which suggests a higher state of disorder. Furthermore, the potential
energy shows a gradual decrease after around 4 strain, until it approximately
reaches the same value as the benchmark simulation at $\varepsilon=10$. As
before, the average atomic von Mises stress, as shown in Figure 9(d), and the
potential energy per atom, are well correlated for both simulations.

Next, we consider the grain counts in the simulation cells (Figure 9(c)). Both
the benchmark and higher damping simulations exhibit significant disordered
states at 0.27 strain. The benchmark cell contains about 400 grains, while the
higher damping simulation shows 390. Both simulations show a rapid decrease in
grain count. The benchmark simulation subsequently drops to around 30 grains,
whereas the higher damping simulation only drops to around 130 grains. The
benchmark simulation steadily increases in grain count until about 2 strain,
reaching $\sim$220 grains. In contrast, the higher damping simulation rapidly
rises to 320 grains around 1.1 strain, then sharply declines. By 3 strain, the
higher damping simulation maintains around 70 grains, plateauing until the end
of simulation, a lower count than the benchmark simulation’s plateau at 240
grains.

By observing Figure 9(e), which shows the dislocation density for the
simulations, we see that the initial spike in dislocation density is much
lower for the higher damping simulation. Whilst the benchmark simulation
increases to around $2.6\times 10^{-3}/$Å2 at 0.82 strain, the higher damping
simulation’s dislocation density only increases to $6.4\times 10^{-4}/$Å2 at
0.51 strain. With increased shearing, the dislocation density for the
benchmark simulation decreases and then saturates at around $1.6\times
10^{-3}/$Å2. This is in contrast to the higher damping simulation where the
dislocation density decreases to $2.5\times 10^{-5}/$Å2 at around 2 strain and
begins to increase at around 6.5 strain, with the dislocation density for the
higher damping simulation peaking at around $1.2\times 10^{-3}/$Å2 at
$\varepsilon=10$.

Figure 9(f) shows the percentage of the overall structure in the BCC phase as
a function of strain. Both simulations experience a highly disordered state
transition at 0.27 strain, where the percentage of BCC atoms within the cells
is below 10% for both simulations. However, the benchmark simulation rapidly
recovers, and around 50% of the atoms are considered to be BCC after
$\varepsilon=10$. Conversely, the higher damping simulation makes a slight
recovery to 17% at 0.44 strain, before reducing to 2% BCC. After 5 strain, the
higher damping cell begins to recover its BCC structure, increasing to
$\sim$32% BCC at 10 strain.

(a) 0 strain

(b) Disordered phase - 0.27 strain

(c) Recrystallisation - 0.32 strain

(d) 1 strain

(e) 4 strain

(f) 7 strain

(g) 10 strain

Figure 10: Grain refinement process for different damping parameter
simulations. Atoms are coloured according to atomic crystal orientations.
Strain rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$
s-1. Left - Benchmark simulation with damping parameter $\gamma=6.875$ eV fs
Å2; Right - Higher damping simulation with damping parameter $\gamma=68.75$ eV
fs Å2.

To further explore the data presented in Figure 9, a visual representation of
atomic crystal orientation is shown in Figure 10. Both simulations begin with
a perfect cell (Figure 10(a)) before experiencing a disordered state (Figure
10(b)) followed by a recrystallisation (Figure 10(c)). The number of atoms in
the disordered phase is much larger for the higher damping simulation. During
the recrystallisation, the grains appear to be much larger for the benchmark
simulation, with many small pockets of BCC structure being present in the
higher damping simulation.

In Figure 10(d), the higher damping simulation does not experience a grain
growth similar to the benchmark simulation, with many fragmented grains and
disordered atoms present. By Figure 10(e), most of the cell is in the
disordered phase. There are only a few pockets of BCC atoms still present
within the box. This is consistent with Figure 9(f) which shows that the
percentage of atoms in the BCC phase at 4 strain was 2%. Between 4 strain
(10(e)) and 7 strain (10(f)), there is a noticeable grain growth, which is
even more noticeable at 10 strain (10(g)), corresponding to a larger number of
atoms transitioning into the BCC phase as also seen in Figure 9(f).

Previous studies [70, 71] consider DRV to be the only dynamic recovery method
in ferritic steels. However, the data presented in Figures 9 and 10 appears to
show that DRX mechanisms are at play, which agrees with the observations made
by Tsuji et al. [72], which confirmed the occurrence of DRX in BCC iron.

Considering Figure 10(e) \- 10(g), the few remaining grains at 4 strain grow
much larger and recrystallisation occurs. This process continues to produce a
small number of large grains at $\varepsilon=10$. This coincides with an
increase in the percentage of BCC phase, as shown in 9(f). Interestingly,
previous analysis of Figure 9(c) showed that the grain number plateaued at
around $\varepsilon=3$ for the higher damping simulation. This provides
further evidence for DRX, as the grain number stays constant whilst the BCC
structure continues to recover.

A larger damping parameter fundamentally means that the quenching rate is
faster. This means that atoms lose their kinetic energy more quickly, and get
trapped in a disordered state. As shown in Figure 10(c) and 10(d), the higher
damping simulation experiences recrystallisation, similar to the benchmark
simulation, but not to the same extent. Increased shear strain causes the
atoms revert into a disordered state, as shown in Figure 10(e). This is
confirmed by considering Figure 9(f), which shows a recovery of the BCC phase
from the minimum at 0.27 strain, to 17% BCC at 0.44 strain. This is followed
by a return to the disordered phase. It is not immediately evident why the
atoms return to the disordered state post recrystallization. In the following
section, we will attempt to outline a possible explanation for this.

As previously mentioned, the system is quenched at a higher rate for the
higher damping simulation which causes many atoms to remain in the disordered
phase. This essentially means that after recrystallisation at 0.32 strain, the
grains are very small and the grain boundary volume is large. As such, the
dislocations formed as a result of shearing are heavily constrained within
small grains [76] and more readily meet and annihilate at the grain
boundaries. This is evident from Figure 9(e) which shows an increase in
dislocation density up to $6.0\times 10^{-4}/$Å2 post recrystallisation,
followed by a quick drop in dislocation density to $2.6\times 10^{-6}/$Å2 at
$\varepsilon=4$.

As dislocations are annihilated at grain boundaries and an increasing number
of atoms enter a disordered state, the percentage of BCC structure decreases.
The growth in grain size for the higher damped simulation provides the
necessary grain volume for the propagation of dislocations, and an increase in
dislocation density can be observed with the increase in BCC structure at
$\varepsilon=7$. It is also possible that, since there are many highly
disordered atoms that manifest as grain boundaries, the free energy in the
simulation is high, which drives the recrystallisation.

###  III.4 Strain Rate

The effect of strain rate was explored next. We performed a simulation at a
shear strain rate which is 10 times slower than the benchmark simulation,
whilst keeping other parameters unchanged. Figure 11 shows different
quantities plotted as a function of strain for the normal and slow rate
simulation.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Simulation data plotted as a function of strain for different
strain rate simulations. Damping parameter $\gamma=6.875$ eV fs Å2. Strain
rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 for
red line and strain rate $d\varepsilon/dt=1/335$ ps${}^{-1}\approx 2.985\times
10^{9}$ s-1 for blue line. (a) Kinetic energy (b) Kinetic energy up to 2
strain (c) Potential energy (d) Potential energy up to 2 strain (e) Number of
grains (f) Von Mises stress (g) Dislocation density (h) Dislocation density up
to 2 strain (i) percentage of BCC structure.

(a)

(b)

(c)

Figure 12: Simulation data plotted as a function of strain for different
strain rate simulations. Damping parameter $\gamma=6.875$ eV fs Å2. Strain
rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 for
red line and strain rate $d\varepsilon/dt=1/335$ ps${}^{-1}\approx 2.985\times
10^{9}$ s-1 for blue line. (a) Kinetic energy (b) Kinetic energy up to 2
strain (c) Potential energy (d) Potential energy up to 2 strain (e) Number of
grains (f) Von Mises stress (g) Dislocation density (h) Dislocation density up
to 2 strain (i) percentage of BCC structure.

(a) Recrystallisation

(b) 1 strain

(c) 4 strain

(d) 7 strain

(e) 10 strain

Figure 13: Dislocation network comparison for different strain rate
simulations. Damping parameter $\gamma=6.875$ eV fs Å2. Green dislocation
lines $\mathbf{b}=\frac{1}{2}\langle 111\rangle$ and pink dislocation lines
$\mathbf{b}=\langle 100\rangle$. Left - Benchmark simulation with strain rate
$d\varepsilon/dt=1/33.5$; Right - Slower rate simulation with strain rate
$d\varepsilon/dt=1/335$.

In Figure 11(a) and 11(b), the initial kinetic energy spike is not as
pronounced for the slow strain rate simulation. Whilst the benchmark
simulation shows an increase from 0.039 eV/atom to 0.072 eV/atom, the slow
rate simulation only shows an increase from 0.039 eV/atom to 0.056 eV/atom.
Furthermore, the spike for the benchmark simulation begins at 0.27 strain and
peaks at 0.3 strain, whilst the slow rate simulation experiences a rise
starting at 0.24 strain, and peaking at 0.25 strain. After the initial
increase and subsequent decrease in kinetic energy, the slow rate simulation
plateaus at a lower kinetic energy of around 0.039 eV/atom, compared with the
benchmark simulation at 0.042 eV/atom.

The difference in the potential energy is shown in Figure 11(c) and 11(d).
Both simulations have identical trajectories in the initial stage. However,
the slow rate simulation reaches the peak in potential energy at an earlier
stage than the benchmark. The benchmark simulation peaks at -3.73 eV/atom at
0.27 strain, while the slow rate simulation peaks at -3.78 eV/atom at a strain
of 0.24. This is followed by a rapid decrease for both simulations. The slow
rate simulation decreases and plateaus at a lower potential energy, i.e. -3.95
eV/atom, than the benchmark, i.e. -3.89 eV/atom. The von Mises stress again
correlates to the potential energy as shown in Figure 11(f).

Though we can observe grain refinement in the benchmark simulation, this does
not occur in the slow rate simulation, as shown in Figure 11(e). The
previously observed large spike in grains for the benchmark simulation does
not occur in the slow rate simulation. Whilst there is a smaller initial peak
of 80 grains at 0.27 strain for the slower rate simulation, this does not
result in the formation of a largely polycrystalline structure, and after 0.38
strain, a single grain is present. This is contrary to the data obtained for
the benchmark, and other previously discussed simulations that experienced the
generation of many nanocrystalline grains.

This difference is also inferred by Figure 12(c) which compares the percentage
of atoms in the BCC phase as a function of strain. A highly disordered state
is observed for the benchmark simulation at 0.27 strain, where only 7% of the
atoms are in the BCC phase, followed by a recovery where around 50% of the
atoms remain in the BCC phase. The slow rate simulation also experiences this
disordered state, but to a lesser extent. At 0.24 strain, 37% of the atoms in
the slow rate simulation are in the BCC phase, which is 30% more than the
lowest benchmark value. Furthermore, the vast majority of the atoms regain
their BCC phase, with the percentage of BCC atoms plateauing at around 93%.
This is logical as the structure in the slow rate simulation is largely a
single crystal, and there are no grain boundaries in the highly disordered
state.

Several differences between the simulations are observed when analysing the
dislocation densities in Figure 12(a) and 12(b). After a shear strain of 0.27,
the dislocation density for the benchmark simulation gradually increases up to
$\sim 2.6\times 10^{-3}/$Å2 at 0.82 strain, after which it decreases to
$1.2\times 10^{-3}/$Å2 at around 2 strain before plateauing at $1.6\times
10^{-3}/$Å2. However, in the slow rate simulation, dislocation densities show
a rapid increase to $3.6\times 10^{-3}/$Å2 at 0.3 strain followed by a
decrease to around $1.2\times 10^{-3}/$Å2 at 1.6 strain. After this point, the
dislocation densities for the simulations stay fairly similar, with the
benchmark simulation having a slightly higher dislocation density with
continued shearing.

A visual representation of dislocation density is shown in Figure 13 in pairs.
The left-hand side is the benchmark simulation, while the right-hand side
shows the slow rate simulation. Dislocations are colour-coded in the same way
as in Figure 3. During recrystallisation, in Figure 13(a), the benchmark
simulation shows few dislocation lines, and they are sparse and disconnected.
However, in the slow rate simulation, the dislocation density is much higher
with a dislocation network being formed. This is in agreement with Figure
12(b) which depicts a large dislocation density at recrystallisation for the
slow rate simulation compared with the benchmark.

At $\varepsilon=1$ (Figure 13(b)), a significant increase in dislocation
density is observed within the benchmark simulation. In contrast, the slow
rate simulation cell displays a reduction in dislocation density, consistent
with Figure 12(b). From 1 strain (Figure 13(b)) to 10 strain (Figure 13(e)),
the dislocation pattern in each simulation persists. The benchmark simulation
shows short dislocation lines, with dislocation pile-ups attributed to grain
boundary interactions. In contrast, the slow rate simulation displays long
dislocation lines, that form an extensive dislocation network. This behaviour
persists due to the absence of grain boundaries constraining dislocation
motion within the slow rate simulation cell.

These observations can be rationalised as follows: the slow rate simulation
does not show the formation of nanocrystals. During the initiation of plastic
deformation, many dislocation loops are formed, which causes a spike in
dislocation density, as seen in Figure 13(a). These loops do not form in the
benchmark simulation due to the presence of highly disordered atoms that
manifest as grain boundaries. Polycrystalline grains limit the volume in which
the dislocations can form, resulting in the short dislocation lines observed
in the benchmark simulation. With increased strain, the dislocations in the
slow rate simulation are allowed to evolve and are not annihilated, as there
are few grain boundaries. Instead, as shown in Figure 13, they combine and
form a large network of extended dislocations. The presence of grain
boundaries does not permit this phenomenon to occur in the benchmark
simulation, so short dislocations are observed.

We may speculate about the reason why the slower strain rate simulation does
not experience the same nanocrystal formation as the benchmark simulation. The
kinetic increases less in the slow rate simulation, which could point to fewer
atoms being able to accelerate within the lattice. As such, many atoms will
retain their original structure, which is confirmed by Figure 12(c). This
increase in kinetic energy comes at a lower strain for the slow rate
simulation, which results in a smaller increase in potential energy.
Ultimately, this means that fewer atoms reach the disordered state in the slow
rate simulation. It appears that, with increased strain, it is more
energetically favourable for the atoms to recrystallise back into a single
crystal structure rather than a polycrystalline one.

The DRV mechanism appears to be present in the slow rate simulation, as the
longer time scale gives dislocations more time to recover [77]. As a result
dislocations are less likely to pile up, and thus do not form grain
boundaries, limiting grain refinement. It is also possible that the
dislocations are simply unable to pile up, because the large dislocation
network, on different slip planes, impedes extended dislocation glide [78]. As
dislocation motion becomes constrained, the formation of LAGBs and thus,
further grain refinement, are inhibited.

Experimental works suggest that a lower strain rate correlates to a larger
mean grain size [66, 64] and it is possible that the present simulation cell
size is the limiting factor. In the future, it may be beneficial to carry out
such shearing simulations with a much larger number of atoms to ascertain
whether this is the case.

###  III.5 Carbon Impurities

(a)

(b)

(c)

(d)

(e)

(f)

Figure 14: Simulation data plotted as a function of strain for benchmark and
carbon-containing simulations. Strain rate $d\varepsilon/dt=1/33.5$
ps${}^{-1}\approx 2.985\times 10^{10}$ s-1 and damping parameter
$\gamma=6.875$ eV fs Å2. The red line is for the benchmark and the blue line
is for 100 appm carbon in iron simulation. (a) Kinetic Energy (b) Potential
Energy (c) Number of Grains (d) Von Mises Stress (e) Dislocation Density (f)
Temperature.

At which point, may one consider the carbon impurity content to be negligible?
Some sources point to a maximum carbon content of 0.006% [79] whilst others
argue that a carbon content as low as 0.001% can already immensely affect the
properties of iron [80]. Ferritic/martensitic steels that have been selected
for fusion applications typically contain less than 0.15% carbon [81]. The
atomic size of carbon is small enough that the atom can enter the iron lattice
as an interstitial solute atom [82]. Experiments showed that carbon can
greatly influence the microstructural behaviour of iron. For example, Stein
[83] showed that the dislocation velocity exponent increases in iron with an
increase in carbon content when held at room temperature. This alters the
yield point and rate of crack propagation within the material. Molecular
dynamics simulations have also been carried out which show the effect of
carbon interstitials. Carbon atoms can block dislocation motion in an iron
simulation cell [84].

In the current work, we inserted 102 carbon atoms into the simulation cell,
that is 100 appm or 0.01 atomic %. We attempt to understand the effect of
carbon interstitial atoms on the formation of nanograins under shear strain in
iron.

Figure 14 shows various simulation properties with and without carbon
impurities. Figure 14(a) shows the differences in kinetic energy between the
simulations. The benchmark simulation experiences a rapid increase in kinetic
energy, from 0.039 eV/atom at 0.27 strain, to 0.072 eV/atom at 0.3 strain,
before rapidly decreasing and plateauing at 0.042 eV/atom. Similarly, the
kinetic energy for the carbon-containing simulation increases from 0.039
eV/atom at 0.27 strain, to 0.067 eV/atom at 0.3 strain, before rapidly
decreasing. Hence, the difference in maximum kinetic energy between the
simulations is around 0.005 eV/atom.Furthermore, both simulations experience
kinetic energy reductions to 0.043 eV/atom at 0.32 strain. After this point,
both kinetic energies remain very similar, with only minor differences between
them.

In Figure 14(b), the behaviour of the potential energy is nearly identical for
both simulations up to 1 strain. Both cells experience a rapid increase from 0
strain, followed by a rapid decrease at 0.27 strain, to a value of around
-3.89 eV/atom at around 0.82 strain. However, the potential energy of the
carbon-containing simulation increases again from this point, reaching -3.85
eV/atom at $\varepsilon=2$, before decreasing slightly, and once again
increasing to -3.84 eV/atom at $\varepsilon=8$. This trend is also shown in
Figure 14(d), which shows the von Mises stress for both cells.

Figure 14(c) shows the number of grains found in each simulation cell. There
are some minor differences between the two simulations. The initial grain
number increases up to around 400 for the benchmark simulation, whilst the
maximum number of grains found in the cell with carbon atoms is 440, a
difference of 40 grains. Nevertheless, both simulations experience a nearly
identical trajectory from 0.27 strain to 3 strain. After this point, the
number of grains present in the benchmark simulation is noticeably larger than
the carbon-containing cell. For example, at $\varepsilon=8$, the number of
grains found in the benchmark simulation is around 260, whilst the carbon-
containing cell only has around 210.

A similar trend is observed when considering the dislocation densities of the
simulations in Figure 14(e). Again, the trajectories of the simulations are
nearly identical up to $\varepsilon=3$. With continued shearing, the
dislocation density of the carbon-containing cell is noticeably larger, until
the values become similar for both simulations at around $\varepsilon=8.5$.

In Figure 14(f), it is interesting to observe identical simulation
temperatures for the benchmark simulation and carbon-containing simulation,
whilst the kinetic energy between them is not identical. It is not immediately
apparent what could cause this.

Figure 15 shows the mean kinetic energy broken down by atom type for the
carbon-containing simulation. Evidently, the carbon atoms have higher kinetic
energy that the iron atoms, and the energy fluctuation is large. However, the
kinetic energy of the carbon atoms stays fairly consistent on average, at a
value of around 0.058 eV/atom.

Carbon atoms can move within the material by jumping between interstitial
sites through diffusion [85]. The work done by Wert [86] managed to
characterise the diffusion coefficient of carbon in iron as a function of the
temperature, and subsequent work has pointed to a diffusion barrier of around
0.87 eV for carbon in iron at room temperature [87]. This is further
reinforced by the work of Tapasa et al. [88], which also found that the
activation energy of C migration in Fe to be 0.82 - 0.86 eV, which is much
larger than the kinetic energy spikes in Figure 15. Fu et al. [89] performed
density functional theory (DFT) calculations to explore the migration energy
of C atoms in Fe. They found that C atoms will migrate from neighbouring
octahedral sites through a tetrahedral site, with an energy barrier of 0.87
eV. This value agrees with the work of Wert [86] and Tapasa [88]. Other DFT
calculations have also obtained similar values [90, 91].

Figure 15: Kinetic energy based on atom type for carbon-containing simulation.
Strain rate $d\varepsilon/dt=1/33.5$ ps${}^{-1}\approx 2.985\times 10^{10}$
s-1 and damping parameter $\gamma=6.875$ eV fs Å2.

Figure 15 suggests that these carbon atoms are not overcoming the energy
barrier as their average kinetic energy is never increases past the diffusion
energy barrier, therefore suggesting that the carbon atoms do not readily move
through diffusion. Instead, the movement of carbon atoms may be conducted
through a mobile Cottrell atmosphere. In Figure 14(e), the carbon-containing
cell initially shows a higher dislocation density compared to the benchmark
simulation up to around 8.5 strain, after which they become similar. Carbon,
behaving as a solute interstitial in the lattice, impedes dislocation motion
[92]. This prohibits dislocations from gliding effectively within the lattice,
hampering their movement toward grain boundaries and limiting the formation of
new LAGBs. This restraint likely contributes to the lower overall grain count
in the carbon-containing cell, as shown in Figure 14(c). Essentially, hindered
dislocation glide and limited LAGB formation slow grain refinement, yielding
higher dislocation density. Moreover, dislocations carry excess free energy
[60], likely contributing to the higher overall potential energy in the
carbon-containing cell (Figure 14(b)). Interestingly, the potential energy
notably decreases beyond $\varepsilon=8$, even surpassing the benchmark
simulation, aligning with the reduction in dislocation density in the carbon-
containing box.

##  IV Conclusion

Nanocrystalline formation in iron under high shear strain has been observed
through molecular dynamic simulations. The process of nucleation and growth of
nanograins during the shearing involves a disordered state, recrystallization,
and grain coarsening. The disordered state is caused by a sudden surge in
kinetic energy gained from the drop in potential energy, which was high and
unstable due to shearing. Following this, energy is dissipated into the
environment, mimicked by the thermostat. Atoms rearrange locally to achieve
energetically favourable configurations that lead to recrystallization and
grain coarsening.

We also examined the influence of various factors, such as thermostat
temperature, heat dissipation rate, shear strain rate, and carbon content.
Simulations at higher temperatures still experience nanocrystalline formation,
but with larger and longer grains forming. A faster rate of heat dissipation
altered the grain refinement process. This also involved a disordered state,
followed by recrystallization. However, dynamic restoration mechanisms were
observed to play a major role in nanocrystalline formation. Simulations with a
slower strain rate did not produce nanocrystalline material, with only a
single crystal structure being observed. The inclusion of carbon interstitial
atoms had a minor effect on nanocrystalline formation, with the process of
grain refinement being identical to that of the pristine material.
Nonetheless, a smaller number of grains were generally observed for the
carbon-containing simulations.

The current simulations demonstrate a possible mechanism of nanograin
formation using high-shear methods, which we have not found reported in the
literature.

##  V Data Availability

All input scripts and simulation data presented in the current work are
available at A link will be provided after the review process and before
publication.

###### Acknowledgements.

The authors gratefully acknowledge the Department of Engineering Science at
the University of Oxford for their contribution to the funding of the project.
This work has been carried out within the framework of the EUROfusion
Consortium, funded by the European Union via the Euratom Research and Training
Programme (Grant Agreement No 101052200 — EUROfusion) and from the EPSRC
[grant number EP/W006839/1]. To obtain further information on the data and
models underlying this paper please contact<EMAIL_ADDRESS>Views and opinions expressed are however those of the author(s) only and do
not necessarily reflect those of the European Union or the European
Commission. Neither the European Union nor the European Commission can be held
responsible for them. The authors acknowledge the use of the Cambridge Service
for Data Driven Discovery (CSD3) and associated support services provided by
the University of Cambridge Research Computing Services (www.csd3.cam.ac.uk)
in the completion of this work. This work used the ARCHER2 UK National
Supercomputing Service (https://www.archer2.ac.uk).

##  Appendix A Further analysis

###  A.1 Slip Systems

To analyze the slip systems within the current orientation, we calculated the
Schmid factor [93]. We apply a constant shear strain in the xy direction,
resolving to principal strain at 45°. Although the stress magnitude changes as
the simulation progresses, normalizing the applied stress direction to
$\sigma=[1\overline{1}0]$ is feasible. With shear strain confined to the xy
direction and periodic boundary conditions in place, a plane stress is
assumed. In BCC structures like $\alpha$-iron, slip direction predominantly
aligns with the $\langle 111\rangle$ family, while slip planes encompass the
{110}, {112}, and {123} families for Fe [94, 95]. Using these stress values
alongside slip planes and directions, we calculated the Schmid factor using:

$m=cos(\phi)cos(\lambda),$ (6)

where m is the Schmid factor, $\phi$ is the angle between the normal of the
slip plane and the direction of applied stress, and $\lambda$ is the angle
between the direction of applied stress and the slip direction.

The Schmid factor can be calculated using:

$\displaystyle
cos(\phi)=\frac{\vec{\sigma}\cdot\vec{n}}{|\vec{\sigma}||\vec{n}|},$ (7)
$\displaystyle
cos(\lambda)=\frac{\vec{\sigma}\cdot\vec{d}}{|\vec{\sigma}||\vec{d}|},$ (8)

where $\vec{n}$ is the vector normal to the slip plane, and $\vec{d}$ is the
vector in the direction of the slip.

There is a total of 48 slip systems in BCC metal. A full list of these can be
found in [96]. The Schmid factor was calculated for each system and this is
shown in Figure 16.

Figure 16: Schmid Factor for all slip systems in BCC system under shear
strain.

Analyzing Figure 16, we discover a multitude of potential slip systems that
can activate within the system when the crystal is oriented along $x=[100]$.
This observation may shed light on why the grain refinement process initiates
with numerous atoms entering a highly disordered state. During the shearing of
the cell, a diverse set of slip systems is simultaneously engaged. Upon
reaching the critical stress point, this leads to a mixing of atoms, inducing
high disorientation due to the involvement of multiple slip directions.
Subsequent quenching then triggers recrystallization.

###  A.2 Further Dislocation Analysis

Figure 17: Dislocation density comparison between standard DXA and DXA carried
out when $d<1$ atoms removed.

We performed additional DXA calculations after eliminating grain boundaries.
This is done by removing atoms that are sitting close to grain boundaries. We
adopt the code developed by Mason [97, 62]. It provides the distance $d$ of
each atom to the nearest grain boundary. Atoms with $d<1$ Å were excluded,
followed by a subsequent DXA calculation. Figure 17 displays the analysis
outcomes.

Figure 17 demonstrates that the standard DXA tends to overestimate dislocation
density, possibly due to grain boundaries. Dislocations can glide and
accumulate at grain boundaries, with some leading to the formation of LAGBs
[4, 8, 9, 10]. It’s likely that DXA interprets certain LAGBs as dislocation
lines, accounting for the discrepancy in dislocation count. Specifically,
conspicuous spikes in dislocation density around 4.8 strain are observed,
absent when $d<1$ atoms are excluded. Thus, we conclude that the standard DXA
encompasses dislocation pile-ups and LAGBs in its dislocation density
computation. Nevertheless, the current analysis remains a robust foundation
for comparing various simulations.

###  A.3 What Constitutes a Grain?

Figure 18: Total number of grains based on minimum number of atoms per grain.

In this work, we considered a minimum grain size of 50 atoms when determining
the number of grains present within the system. This number was selected as it
allowed us to observe the transition to the disordered state and the
subsequent recrystallisation clearly. Nevertheless, Figure 18 shows the
comparison of grain numbers for the benchmark simulation with different
numbers of atoms selected to form the minimum grain size. The Ovito default of
100 atoms was selected for comparison, as was 5,558 atoms, which corresponds
to a 5nm diameter spherical grain. A 5nm grain corresponds to a minimum grain
size which can accurately be resolved using transmission Kikuchi diffraction
(TKD, or t-EBSD) [98], allowing comparisons with future experimental HPT data.

Figure 18 shows that the number of grains between the 50-atom and 100-atom
analyses largely follow the same trajectory. At 0.27 strain, our-50 atom
analysis showed 400 grains whilst the 100-atom analysis showed 215. This means
that 185 grains were found to have less than 100 but more than 50 atoms at
this point. This further suggests that the spike is caused by a highly
disordered state of the atoms whereby only small pockets of BCC phase atoms
remain, which is flagged as a grain. At this strain value, the 5nm grain
analysis cannot pick up any grains due to the highly disordered state, and the
grain number is shown as 0. After 2 strain, the 50- and 100-atom trajectories
follow the same trajectory, however, there is always a difference of around 40
grains between the analyses. Therefore, we conclude that there are always
around 40 grains in the simulation cell which have less than 100 atoms but
more than 50. Interestingly, the 5nm grain analysis does not follow the same
trajectory, and instead hovers around 1 grain up to around 1.37 strain. After
this point, the grain number increases for the 5nm analysis, as shown in
Figure 18. A local maximum of grain number is reached at around 2.5 strain,
with 30 grains being present. This is followed by a minor decrease, after
which the grain number sits steadily at around 25 after 4 strain. As such, we
observe that the process of grain refinement still occurs if we define a grain
with a minimum of 5,558 atoms.

It further supports our hypothesis that the transition of atoms into the
disordered state, followed by a recrystallisation and grain growth, is a major
mechanism of grain refinement. It is clear that up to 1.37 strain, all of the
grains are small, with many containing less than 100 atoms. Through the
processes of grain growth described above, the grains expand, causing the 5nm
analysis value to increase. This analysis saturates at around 25-30 grains and
any subsequent changes in the simulation cell with shear occur through small
grain processing.

By increasing the minimum number of atoms that constitute a grain, we can
compare the effects of temperature and carbon on the formation of
experimentally observable grains. Figure 19 shows the number of 5nm spherical
grains found in different temperature simulations and carbon-containing
simulations.

(a)

(b)

Figure 19: Number of 5nm spherical grains found in the simulations; (a)
Dependence of temperature on 5nm grains, and (b) Dependence of carbon
inclusion on 5nm grains.

In Figure 19(a), there is no initial spike in 5nm grains for any simulation,
and the trajectories of the different simulations are largely similar until
around 2.5 strain. This is contrary to the data shown in Figure 5(f) which
shows that the 300 K simulation has the most grains at this point, suggesting
that at 300 K, the grain numbers are increased due to the presence of many
small grains. We also notice that the 1000 K simulation has around 2 grains
larger than 5nm at 10 strain, which is visible by considering Figure 8(c).
Ultimately, the 300 K simulation still shows the largest number of grains
overall. However, this is not to the same extent as in Figure 5(f), again
suggesting that the shearing process stimulates the production of many small
grains in the 300 K simulation. Nevertheless, the 500 K and 800 K simulations
also have many small grains, and this is visible by comparing the data in
Figure 19(a) to Figure 5(f). For example, at 6 strain, the total number of
grains present for the 500 K simulation is 120 and that number drops to 20
when considering only 5nm grains.

Figure 19(b) also shows the comparison of 5nm grains between the benchmark and
carbon-containing simulation. We show here that the deviation in grain number
observed in Figure 14(c) does not occur when considering 5nm grain, up to 8.5
strain. This suggests that the benchmark simulation has many smaller grains
than the carbon-containing simulation. After 8.5 strain, we notice that the
carbon-containing simulation begins to dip in grain number whilst the
benchmark simulation cell rises.

###  A.4 Thermal Stability of Grains

Figure 20: Number of grains present in different temperature simulations after
thermalising.

It was important to assess the stability of the newly formed grains to assess
their usefulness for future simulations. The benchmark simulation and the
variable temperature simulations from Section 3 were chosen for assessment.
These cells were thermalised for 1ns using the NPH ensemble, with the Langevin
thermostat keeping the temperature at 300 K, 500 K, 800 K, and 1000 K. Figure
20 shows the total number of grains present after the 1ns run time for each
simulation cell. For the purposes of this comparison, the minimum grain size
was taken as 50 atoms.

Inspection of Figure 20 shows that all cells follow similar trends. There is
first a reduction in the number of grains found in the cell, after which the
number of grains stabilises after around 700 ps of thermalisation. The 300 K,
500 K, and 800 K simulations all stabilise at around 150, 56, and 35 grains,
respectively. It is shown that the 1000 K simulation stabilises much earlier,
at a value of 2 grains. As such, it is inferred that the nanocrystalline
structure will remain stable at finite temperatures.

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