# Comparative study on Mechanical characteristics of Functionally graded and
Coreshell nanospheres: An Atomistic approach

Prottay Malakar  Md Al Rifat Anan  Mahmudul Islam  Md Shajedul Hoque Thakur
Satyajit Mojumder<EMAIL_ADDRESS>
###### Abstract

Functionally Graded Material (FGM) is a type of advanced material consisting
of two (or more) distinct substances with a constantly changing composition
profile. FGM technologies have moved from their traditional use to advanced
micro and nanoscale electronics and energy conversion systems along with the
advent of nanotechnology. MD simulations are used in this analysis to examine
the effect of compressive load on Ag-Au FGM and Core-shell nanospheres. The
plasticity process is often started by the nucleation of partial dislocations
from the contact surfaces, and these dislocations spread towards the
nanosphere’s center. Also, we have found the formation of pyramidal-shaped
partial dislocations on the pseudo-plastic regime. For a given wt% range of Ag
in Au, Coreshell nanospheres have stronger mechanical strength than FGM
nanospheres, and we have also observed two distinct patterns in ultimate
stress variation for FGM and Coreshell nanospheres. The dislocation analysis
suggests a correlation between this stress variation and the Shockley & Hirth
partial dislocation density.

###### keywords:

Molecular Dynamics, Fracture Mechanics, Dislocation Density, Nanosphere,
Compression Test , Plane Indentation

††journal: arXiv

[label1] organization=Department of Mechanical Engineering, Bangladesh
University of Engineering and Technology, city=Dhaka, postcode=1000,
country=Bangladesh [label3] organization=Department of Materials Science and
NanoEngineering,Rice University, city=Houston, postcode=77005, state=TX,
country=USA [label2] organization= Theoretical and Applied Mechanics Program,
Northwestern University, city=Evanston, postcode=60208, state=IL, country=USA

##  1 Introduction

Functionally graded materials (FGMs) are relatively new and promising advanced
high-performance materials among the class of advanced composite materials,
where the material composition is continuously varied along a dimension
following a particular function to regulate the mechanical and physical
properties[1]. FGMs are still in the early stages of development, but they are
expected to have a significant impact on the design and development of novel
components and structures that perform better. Many studies have been
conducted to better understand the thermal and mechanical properties of FGMs.
Continuum mechanics and the finite element technique (FEM) are primarily used
in this process[2, 3, 4, 5, 6, 7, 8]. According to Udupa et al. [9] and
Howard[10], FGMS can interact as an interface layer between two incompatible
materials, improving bond strength, decreasing stress concentration, and
providing multifunctionality to control deformation, dynamic response,
corrosion, wear, and other properties.

Recently, there has been a surge of interest in learning more about the
plastic deformation of FGM nanoparticles. Despite this, the corresponding
literature is still inadequate due to the difficulties associated with
experimental mechanical testing. The mechanical properties of nanoparticles,
specifically the nanosphere, have been measured using the compression or
indentation approach. Gerberich et al. made one of the earliest observations
by compressing gold and silicon nanospheres and analyzing the plastic
deformation with transmission electron microscopy[11, 12, 13, 14, 15]. He
tested the silicon nanosphere’s hardness, which is up to four times that of
its bulk value [11]. The modulus of elasticity of gold nanoparticles reduced
with increasing indentation depth, according to Ramos et al[16]. Mordehai et
al. found that the nucleation process of dislocation in gold nanoparticles
begins at the surface and spreads to the adjacent surfaces of the faceted
particles, leading through defect-free particles during indentation, which is
consistent with Gerberich’s mechanism for silicon nanospheres [11, 17]. The
structure of the nanoparticle is thought to have an important impact in
dislocation propagation. The nucleation of first Shockley partial dislocations
is observed from the contact edge, followed by a pseudotwinning process[18].
All of these studies on the properties of nanoparticles are based on
theoretical models and finite element methods (FEM), both of which are related
to the classical mechanics approach.

Molecular dynamics (MD) simulations can now be an effective way to examine the
mechanical and thermal properties of nanoparticles and understand that too.
Several MD studies have already been carried out to understand the mechanical
and thermal properties of nanowires, nanoplates, nanospheres and nanobeams of
the same or different homogenous and non-homogeneic composites. By compressing
in the [001] crystallographic direction, Bian and Wang [19] conducted
molecular dynamics simulations on a copper nanosphere and thoroughly
characterized the nucleation of dislocations process. They demonstrated how
Shockley partial dislocations formed at the nanosphere-indenter contact edges
and propagated. However, the earlier research was limited to a single copper
nanosphere size (20nm). Further, Salah, Gerard and Pizzagalli performed
molecular dynamics simulations on aluminium nanosphere under compression to
show the influence of size on plasticity mechanism and yield stress[20]. The
elastic modulus and strength of gold nanoparticles are dimension-dependent
[21, 22]. Experiments confirmed the possession of higher modulus and stress in
gold nanoparticles compared with bulk gold [23]. Through in-situ high-
resolution transmission electron microscopy, it was discovered that silver
nanoparticles as small as 10 nm may deform like a liquid droplet at ambient
temperature[24]. In addition, some ultra-small silver nanoparticles showed
reversible plasticity dislocation[25]. The yield stress of gold nanoparticles
fluctuates according to surface atomic-scale morphologies, according to Yang
et al.[26] Under [0 0 1] compression, face-centered-cubic (fcc) nanoparticle
plasticity normally starts with partial dislocation nucleation from the
contact edge and the formation of distinctive pyramid hillocks [19, 20].

However, there has not been much research on the influence of dislocation
propagation on stress in FGM nanospheres. The goal of this research is to look
at the initiation and propagation of dislocations in Ag-Au FGMs and Coreshell
nanospheres. We will also look into how the stress-strain curve is affected by
dislocation propagation. In this study, we have put Ag-Au FGM and Coreshell
nanospheres through a compression test at a specific temperature and pressure
using MD simulations. The following section examines the deformation mechanism
of nanospheres and the development of dislocations in further detail. We have
also talked about how dislocation density affects nanosphere mechanical
characteristics. There has been relatively little investigation of the
mechanical properties of FGM and Coreshell nanospheres at any scale, according
to prior studies. Our findings will help scientists better understand the
alloying effect and deformation mechanism of FGM and Coreshell nanospheres.

##  2 Methodology

Figure 1: Compression Setup of a nanosphere by Plane Indenter.

In this study, the MD Simulation method has been considered to investigate the
contact stress of the Ag-Au alloy nanospheres and the effect of dislocations &
alloying structure on them. These simulations are conducted by using Large-
Scale Atomic/Molecular Massively Parallel Simulator, LAMMPS[27]. The selection
of interatomic potential has a significant influence on the accuracy of
simulation results. We has been used metal alloy embedded method potential
(EAM/alloy) developed by Zhou et al[28].This potential can successfully
simulate the dislocations and their nucleation and propagation accurately[28].
Previously, Embedded method potential (eam) was used for different atomistic
investigations of monatomic elements[29]. In the EAM, the total energy of the
crystal can be shown as below:

$E=\frac{1}{2}\sum_{i,j,j\neq i}\phi_{i,j}(r_{i,j})+\sum_{i}F_{i}(\rho_{i})$
(1)

Where the pair energy $\phi_{i,j}$ is considered between two atoms $i\&j$
which are separated by distance $r_{i,j}$ . $F_{i}$ is the embedded energy and
$\rho_{i}$ is the electron density associated with the atom $i$. The electron
density $\rho_{i}$ can be calculated by using the equation,

$\rho_{i}=\sum_{j,j\neq i}f_{j}(r_{i,j})$ (2)

This equation indicates the total sum of electron density of atom $i$ which is
$r_{i,j}$ distance away from the different atoms $j$. For an alloy, the metal
alloy EAM potential contains not only these three basic functions but also the
pair energy between different alloying elements. This indicates that the alloy
EAM potentials can be developed from elementary EAM alloys and such potentials
can be used in MD simulations to keep consistency with experimental
results[30].

In this investigation, Core-shell nanospheres are generated by replacing core
atoms with Ag atoms from pure Au nanospheres according to the required mass
percentage. On the other hand, FGM nanospheres are modeled by using a slight
modification of the source code of the nano-hub tool developed by Thakur el
al[31]. This tool’s validity has been demonstrated in previous research by
Islam et al[32]. The nanospheres used for this study are shown in Fig.1. For
modeling the radially graded Ag-Au FGM nanosphere, we have strictly maintained
three steps: (1) A pure Ag nanosphere is prepared with proper dimension (15 nm
dia) with the lattice constant of 0.40853 nm. (2) The nanosphere is then
subdivided into some annular spherical chunks of half lattice constant
thickness. (3) In each chunk, the Ag atoms are replaced by Au atoms according
to the grading function. Here, We have used the power-law function as our
grading function. The details of the function are described by using the
equation,

$g(r)=(\frac{r}{R})^{p}$ (3)

Here, the distribution of the mass function of Au in Ag, $g(r)$ is determined
by the parameter p, according to the percentage of Ag and Au along the radial
direction. Fig. 2 shows the Mass function distribution of Au in Ag of FGM and
Core-shell nanospheres for different percentages of Ag.

(a)

(b)

Figure 2: Mass function distribution of Au in Ag of (a) FGM and (b) Coreshell
nanospheres for different percentages of Ag.

10%

25%

50%

75%

90% (a)

10%

25%

50%

75%

90% (b)

Figure 3: Midplane view of (a) FGM and (b) Coreshell nanosphere for different
percentages of Ag.

The simulation is conducted by maintaining some steps. At first, In each
simulation, the system is minimized to a local minimum energy by using the
Polak-Ribiere version of the conjugate gradient (CG) algorithm[33]. After
minimization, the system iss equilibrated in canonical ensemble at a
temperature of 10 K for 150ps by using Nose-hoover thermostat[34]. After this
step, the nanosphere is placed between two planner repulsive force field in
order to perform uniaxial compression test. We have modeled this simulation
according to the experimental process. Experimentally, the compression is
performed by using indenters whose curvature is large enough compared to the
nanosphere’s radius. In this simulation, those indenters are modelled by using
this planner repulsive force field[35, 19]. The external forces acting on the
nanosphere by the planner repulsive force fields are calculated by using the
analytical function is described by:

$F(y)=-\alpha k(y-y_{i})^{2}$ (4)

Where, $y_{i}$ is the position of the indenter along [010] and the $y$ is the
coordinate of the atom along the same axis. Here, $k$ is the effective
stiffness constant of the indenters which value is considered $1000ev/$ Åin
this simulation. The value of $\alpha$ is the term which determined the
direction of force for the indenters. For upper indenter, $\alpha=1$ when,
$y\geq y_{i}$ and $\alpha=0$ when, $y<y_{i}$, while for the bottom indenter,
$\alpha=-1$ when, $y\leq y_{i}$ and $\alpha=0$ when, $y>y_{i}$. Then the
indenters were moving towards each other with a strain rate of $10^{9}$
$s^{-1}$ along [010] direction at 10K. No periodic boundary conditions were
used in all [100],[010], and [001] directions. We need to compute the contact
surface area between the nanosphere and the indenters at each deformation step
to calculate the contact stress.

The contact surface area is computed between the nanosphere and the plane
indenter according to the following procedures:

  * 1.

A nanosphere atom is considered to belong to the contact surface if $\mid
y-y_{i}\mid<0.1$Å.

  * 2.

For calculating the radius of contact area, molecules in contact with plane
indenter assumed between $y_{min}$ and $y_{min}+0.1$Åfor lower surface and
$y_{max}$ and $y_{max}-0.1$ Åfor upper surface.

  * 3.

The considered molecules are projected in $xz$ plane and effective radius, $a$
was calculated by following expression:

$\frac{a^{2}}{2}=\frac{1}{N}\sum_{i}[(x_{i}-x_{c})^{2}+(z_{i}-z_{c})^{2}]$ (5)

Here, $x_{c}$ and $z_{c}$ are the $x$ and $z$ coordinates of the center of
mass of the projected molecules.

  * 4.

So, the contact surface area is:

$A=\pi a^{2}$ (6)

All of the calculation steps has been done by using python Numpy and
MDanalysis[36] packages. Ovito[37] is used for visualized the Molecular
dynamics data. Dislocation density of the nanospheres has been calculated by
using Dislocation Extraction Algorithm (DXA), develpoed by Stukowski et
al.[38] The DXA converts an atomistic model of a displaced crystal into a
line-based dislocation network representation. It calculates each
dislocation’s Burgers vector and locates dislocation junctions. Partially
dislocations, as well as some secondary grain boundary dislocations, are
recognized by this algorithm.

##  3 Results and discussion

###  3.1 Effect on alloying percentage

(a)

(b)

Figure 4: Stress-Strain Curve of (a) FGM and (b) Core Shell nanosphere for
different percentages of Ag.

(a)

(b)

Figure 5: Total dislocation density-Strain Curve of (a) FGM and (b) Coreshell
nanosphere for different percentages of Ag.

(a)

(b)

Figure 6: Force-Displacement Curve of (a) FGM and (b) Coreshell nanosphere for
different percentages of Ag.

In this study, the influence of alloying percentages on mechanical properties
and dislocation density of Au-Ag nanospheres has been investigated using FGM
and Core-shell nanospheres. Figs.4(a) and 4(b) show stress-strain curves for
FGM and core-shell nanospheres with varing alloying percentages. Fig. 5 and 6
depict the dislocation density-strain and force-displacement curves,
respectively.

Fig. 4 shows the stress in nanospheres grow gradually at the beginning of
indentation. This propensity is due to the Ag-Au nanocrystal’s elastic
characteristics. Dislocation causes tiny flaws to emerge on the contact
surfaces. During the early stages of compression, certain dislocation loops
are formed inside the Ag-Au nanoparticle. The atomic coordination
representations clearly show these minor flaws [39]. Furthermore, dislocation-
driven plasticity is common in nanoscale Silver and Gold particles. As the
strain increases from roughly 1.6 to 2.2, the contact stress begins to
decrease significantly in the second stage. Contact stress is reduced at this
stage due to practically constant loading as seen in the inset of Fig.6 and
increased contact area. Following that, the contact tension rapidly builds,
reaches peak, and then quickly decreases. Similar to MD studies for Ag-Au
nanoparticles, this property might be due to the creation of a new structural
phase. The stress varies arbitrarily with increasing strain in the third
stage, indicating that the Ag-Au nanocrystal has amorphized under high
pressure.

The failure phenomena occur at 2.5 percent to 3.5 percent strain for Core-
shell and FGM Au-Ag nanospheres, as shown in Fig.4. We can see a yielding
point shortly before the failure in this stress-strain curve. For both type of
alloying, Fig. 4 shows a similar tendency for nanospheres. Following the first
drop in stress, we can see a fluctuating trend in stress, indicating that the
nanospheres are ductile. Because pure Ag and Au are both ductile in nature,
this ductile behavior is expected. The dislocation density-strain curve in
Figs. 5(a) and 5(b) explain the fluctuating pattern that follows the failure
point. As demonstrated in Fig. 5, the formation of dislocations results
permanent deformation in nanospheres. The dislocation density remains constant
until the strain reaches 4 to 4.5 percent. After this moment, the density of
dislocations begins to rise. The fluctuating pattern in Fig.4’s stress-strain
curves is due to this phenomenon.

The force-displacement curve for FGM and core-shell nanospheres are shown in
Fig. 6. It shows the relation between the displacement of surface atoms occurs
due to the forces exerted by the planar indenters and the forces. As the
percentage of silver in FGM and Core Shell nanospheres increases, the force
exerted by the planner indenter decreases.

(a)

(b)

Figure 7: Variation of Reduced Moduli of (a) FGM and (b) Coreshell nanosphere
obtained from molecular dynamics simulation with different percentages of Ag.

The nanospheres’ Reduced Modulus is determined using a method outlined in
earlier publications. The total elastic compressive response of nanoparticles
will approach the Hertzian contact model [40, 41], which yields the relation
between load F and compression depth $\delta$ as

$F=\frac{4}{3}E^{*}R^{\frac{1}{2}}\delta^{\frac{3}{2}}$ (7)

where $R$ is the radius of nanoparticle and $E^{*}$ is the reduced modulus, as
Bian et al. [42] pointed out. The Force-displacement curve of compression is
fitted using Hertzian contact theory. Fig. 7 depicts the fluctuation of
$E^{*}$ with alloying percentages for FGM and Core Shell nanospheres. As seen
in the graph, $E^{*}$ decreases as the amount of Ag in FGM and Core Shell
increases. The $E^{*}$ fluctuation pattern may be explained by the total
alloying percentages of Ag in Au, as well as the kind of alloying (FGM or Core
Shell). The values of $E^{*}$ for Core Shell nanospheres with a high Au
constituent will be larger than those for Core Shell nanospheres with a low Au
constituent. $E^{*}$ of FGM nanospheres, on the other hand, follows a similar
pattern to Core Shell nanospheres, with the exception of containing more
fluctuating behavior. Some fundamental characteristics of FGM architecture
will help explain this discrepancy. As previously stated, FGM nanospheres are
made by replacing Au atoms with Ag atoms; the error bar depicts the randomness
of the formations.

Fig. 8 depicts the ultimate stress variation for FGM and Core Shell
nanospheres. As the proportion of Ag in FGM nanospheres rises, the curve in
Fig. 8(a) displays a downward trend. Ultimate stress created in Core Shell
nanospheres, on the other hand, differed from that in FGM nanospheres in Fig.
8(b). The ultimate stress in core shell nanospheres grows as the proportion of
Ag increases until it reaches 75 percent. Ultimate stress begins to decrease
after this range. This may happen due to the Core Shell effect on the
nanospheres. We may conclude that until the proportion of Ag in nanospheres
reaches 75%, the ultimate stress of Core Shell and FGM nanospheres behaves in
opposite ways, and after that they follow a similar pattern.

(a)

(b)

Figure 8: Variation of Ultimate Stress of (a) FGM and (b) Coreshell nanosphere
obtained from molecular dynamics simulation with different percentages of Ag.

(a)

(b)

Figure 9: Dislocation Density vs. Strain Curves of FGM and Coreshell
nanosphere for different percentages of Ag. (a) Shockley and (b) Hirth
dislocation density are showed in different graphs.

###  3.2 Deformation Mechanism

We identify and describe the compression test, using the case of a 15 nm
nanoparticle as an example. As illustrated on the stress vs. deformation
curves in Fig.10, elastic behavior is first obtained. When two Shockley
partial dislocations are generated from the top and propagate in the four
possible 111 slip planes, the first indication of plastic deformation appears
in FGM nanospheres at a compression depth of $\delta|=1.19$Å(i.e., 1.6%
deformation), providing the slight stress drop[20]. In all nanoparticles
studied, we discovered that plasticity is initiated by the nucleation of these
partial dislocations from the edges of the surface in contact with indenters.
The edges form steps with the bottom layer and act as stress concentrators, a
well-known phenomena in uniaxial surface deformation[43]. Shockley partial
dislocations are a pair of dislocations that can cause stacking faults. By
enabling an alternate path for atomic mobility, this pair of partial
dislocations can permit dislocation motion. The Burgers vector of the Shockley
partial dislocation lies in the plane of the fault, making it glissile. The
term glissile refers to the mobile dislocations. A new Hirth dislocation is
formed from the contact surface when $\delta=1.33$Å(1.8%) and merges with
Shockley partial dislocation. Hirth dislocation is sessile known as Hirth
lock. When $\delta=1.48$Å(2%), five Hirth and two Shockley dislocations are
visible. All Shockley dislocations disappear at $\delta=1.56$Å(2.1%); only
eight Hirth dislocations are visible. Four Hirth dislocations unite in a point
at $\delta=1.63$Å(2.2%), forming a pyramid-like structure that vanishes
immediately. Again, four Hirth dislocations from each surface merge in a point
and form two pyramid-like structures in the upper and lower contact surfaces
during $\delta=2.82$Å(3.8 %). Two fresh Shockley partial dislocations are
nucleated from the pyramid’s pin-point as compression proceeds, and they
combine with Hirth’s dislocation at $\delta=3.33$Å(4.5% ). One of the newly
formed Shockley dislocations is fixed by a Hirth dislocation, while the other
propagates toward the contact surface. When $\delta=4.075$Å(5.5%), all Hirth
and Shockley dislocations converge in a point and start to propagate towards
the opposite surface. As the compression goes on, all dislocations merge and
create another new pin-point and begin to propagate sidewise towards the
opposing surface at $\delta=4.74$Å(6.4%). During $\delta=6.59$Å(8.9%), the
intersection of the Shockley partial dislocations forms two stair-rod
$\frac{1}{6}[110]$ dislocations according to the reaction:

$\frac{1}{6}[\bar{1}21]+\frac{1}{6}[2\bar{1}\bar{1}]=\frac{1}{6}[110]$ (8)

This stair-rod dislocation is sessile known as Lomer-Cottrell lock - it locks
slip in the two slip planes. All three types of dislocation merge in a point
and propagate at $\delta=6.89$Å(9.3%). When $\delta=7.41$Å(10%), one new
dislocation is introduced, known as Frank dislocation, it vanishes instantly.
As its Burgers vector is not included in one of the 111 planes, the Frank
partial is an edge dislocation that cannot glide and expand conservatively
under the influence of applied stress. Unlike the glissile Shockley partial,
this dislocation is stated to be sessile. But by climbing it can move [44]. A
perfect dislocation results from the interaction of a Shockley partial
dislocation with a Frank dislocation at $\delta=8.37$Å(11.3%), and it
disappears quickly. The reaction is:

$\frac{1}{6}[\bar{1}21]+\frac{1}{6}[2\bar{1}\bar{1}]=\frac{1}{6}[110]$ (9)

When $\delta=9.26$Å(12.5%), perfect dislocation is visible again and gains
stability. Two side Shockley dislocations meet with each other in the surface
area at $\delta=9.85$Å(13.3%) and start to propagate.

For core-shell of 15nm, Plastic deformation occurs when the compression depth
is $\delta=1.11$Å(1.5%). The top and bottom surfaces emit four shockley
partial dislocations, which propagate. All Shockley partial dislocations
vanish at $\delta=1.34$Å(1.8%), but two Hirth dislocations remain visible.
When $\delta=1.49$Å(2%), there are eight Hirth dislocations visible four on
each surface. At $\delta=2.3$Å(3.1%), four Hirth dislocations merge in a point
and create a pyramid-like structure. Two pyramid-like structure appears on
each surface at $\delta=2.45$Å(3.3%). As the compression goes on, two new
Shockley partial dislocations are visible from the pyramid’s pin-point at
$\delta=3.49$Å(4.7%). New Shockley partial dislocation starts to propagate
towards the contact surface. At $\delta=6.54$Å(8.8%), Stair-rod dislocation is
visible and merges with Hirth dislocation and multiplies. Perfect dislocation
is visual at $\delta=7.28$Å(9.8%) and disappears immediately. At
$\delta=7.5$Å(10.1%), Frank’s dislocation appears and disappears quickly. As
compression goes on, two surfaces’ Shockley dislocation merges with each other
at a compression depth of $\delta=10.10$Å(13.6%).

Here We are focused on how dislocations spread from the contact surface and
interact with one another. In our observation, plastic deformation begins with
dislocation propagation from the contact surface, which is related to the
nucleation of the initial dislocation in defect-free nanostructures. However,
when the load is released, the initial nucleated dislocations, which lead to
the formation of the pyramidal-like structure, spontaneously return to the
surface. After the formation of a pyramid-shaped structure, the materials
begin to act as amorphous materials with increasing compression depth ,
resulting in material failure.

###  3.3 Effect of Dislocation Density on Strength

Figure 10: Variation of stress and Dislocation Density of Core-shell
nanosphere with strain for 50% Ag. Here, the Yellow line indicates Hirth
dislocation, and the Green line means Shockley dislocation.

There is a link between dislocation density and stress. According to Stress
Vs. Strain and Dislocation density Vs. Strain curves shown in Fig.10, we
observed that stress is mostly regulated by dislocation density. Partial
Shockley dislocation is observed at 1.5-1.6% strain. This partial dislocation
may enable dislocation motion. The term glissile refers to this form of
partial dislocation. Shockley partials have the ability to glide on the glide
plane, causing stacking faults. A stacking fault ribbon connects two partial
dislocations as Shockley partial dislocation glides on the slip plane yielding
a small stress drop[20].

Shockley partial dislocation vanishes as compression proceeds, and Hirth
dislocation emerges at 2-2.1% strain. Hirth dislocation, also known as Hirth
lock, is a sessile dislocation (a sessile dislocation cannot glide or move)
having crystallographic orientations such as [100]. As these directions do not
allow slip, the Hirth dislocations become locked and function as obstacles to
any neighboring movable dislocations that share the same slip system,
resulting in strain hardening. Because of this strain hardening, the stress
curve reaches its maximum point as shown in Fig.10 Point A. The accumulation
of dislocation barriers, which function as impediments to uninhibited
dislocation movement, causes strain hardening. Immobile or sessile
dislocations are the most frequent barrier in FCC metals, and they play a
significant role in hardening behavior. Dislocation interactions between
dislocations on various glide planes can produce these. Some dislocation
interactions can cause dislocation annihilation. Dislocation annihilation can
occur as a result of some dislocation interactions. Some can generate
dislocation locks, which result in stable dislocation segments forming in
crystallographic directions (Hirth lock or lomer lock). These are not slip
directions, but rather crystallographic directions. These locks prevent
slippage and create barriers to other mobile dislocations, as well as causing
strain hardening[45].

(a)

(b)

Figure 11: Variation of dislocation density of (a) FGM and (b) Coreshell
nanosphere (Ag = 50%) with strain for different types of Partial dislocation.

The stress curve abruptly decreases after the pick value of stress, with the
growth of Hirth dislocation density, and falls to the lowest point at a
3.1-3.5 percent strain as shown in Fig. 10 Point B. With a reduction in Hirth
dislocation density from the lowest point, the stress curve begins to rise
again. Shockley partial dislocation resurfaces at 4.5-5 percent strain, unites
with Hirth dislocation, and propagates. Now stress largely depends on Shockley
partial dislocation. The stress curve rises as the Shockley partial
dislocation density increases and Hirth’s dislocation density falls as shown
in 10 Point C. The stress curve is now proportional to the density of Shockley
dislocations. Stair-rod dislocation is observable at 8.9% strain, which is
known as Lomer-Cottrell lock. As it locks slip in both slip planes, it is a
sessile dislocation. Strain hardening happens as it prevents plane slide, and
the stress curve grows as a result of this strain hardening.

We found a relationship between dislocation density and strain from the
Dislocation Density Vs Strain graph, as shown in Fig 10. In the case of FGM
and Core-shell, dislocation initiation begins at 1.5-1.7% strain as shown in
Fig. 11 Shockley Partial dislocation $(\frac{1}{6}<112>)$ is observed on both
surfaces at 1.5-1.7% strain. As Shockley is a glissile type dislocation, it
permits the plane to shift, lowering stress slightly. Shockley dislocations
vanish from both surfaces after 2% strain. On both surfaces, Hirth
$(\frac{1}{3}<001>)$ dislocation has taken over. After 1.8% strain, Hirth
dislocation is evident. From 2.5 percent strain, Hirth dislocation density
nearly stays the same. Only Hirth dislocation exists between 2% and 4.5%
strain. Because there is just one dislocation available in that range, total
dislocation density remains constant. After 4.5% strain, a new Shockley
partial dislocation forms. Hirth dislocation density remains constant from
4.5% strain onwards, although Shockley partial dislocation density increases.
Total dislocation density increases with strain starting at 4.5% strain, as it
now mostly depends on Shockley dislocation density. Other dislocations, such
as Stair-rod dislocation $(\frac{1}{6}<110>)$, Frank dislocation
$(\frac{1}{3}<111>)$, and Perfect dislocation $(\frac{1}{2}<110>)$, are
visible after 8.9% strain. However, with strain, their density remains low,
and the material begins to behave like amorphism. As a result of their low
density and lack of impact to material properties, certain forms of
dislocations are neglected.

We can deduce from our findings that the stress curve is first dominated by
Hirth dislocation density. After 2.7 percent strain, the relationship between
Stress and Hirth Dislocation density is inversely proportional. The stress
curve now depends on the density of Shockley partial dislocations as
compression progresses and new Shockley partial dislocations emerge. After 4.5
percent strain, the relationship between stress and Shockley partial
dislocation density is proportional.

###  3.4 Comparison of Different Percentage of Alloying

(a) $\epsilon=1.6\%$

(b) $\epsilon=2.6\%$

(c) $\epsilon=3.1\%$

(d) $\epsilon=6\%$

(e) Ag = 10%

(f) $\epsilon=1.6\%$

(g) $\epsilon=2.6\%$

(h) $\epsilon=3.2\%$

(i) $\epsilon=6\%$

(j) Ag = 25%

(k) $\epsilon=1.6\%$

(l) $\epsilon=2.7\%$

(m) $\epsilon=3.1\%$

(n) $\epsilon=6\%$

(o) Ag = 50%

(p) $\epsilon=1.7\%$

(q) $\epsilon=2.5\%$

(r) $\epsilon=3.3\%$

(s) $\epsilon=5.9\%$

(t) Ag = 75%

(u) $\epsilon=1.75\%$

(v) $\epsilon=2.6\%$

(w) $\epsilon=3.4\%$

(x) $\epsilon=6.1\%$

(y) Ag = 90%

Figure 13: Dislocation structures during uniaxial compression of FGM
nanosphere for different percentages of Ag. (Green line, Shockley partial
dislocation with Burgers vectors $\frac{1}{6}<112>$; yellow line, hirth
dislocation with Burgers vectors $\frac{1}{3}<100>$; Pink line, stair-rod
dislocation with Burgers vectors $\frac{1}{6}<110>$; Red line indicates all
other dislocation)

(a) $\epsilon=1.5\%$

(b) $\epsilon=2.6\%$

(c) $\epsilon=3.1\%$

(d) $\epsilon=6\%$

(e) Ag = 10%

(f) $\epsilon=1.5\%$

(g) $\epsilon=2.6\%$

(h) $\epsilon=3.1\%$

(i) $\epsilon=6\%$

(j) Ag = 25%

(k) $\epsilon=1.5\%$

(l) $\epsilon=2.6\%$

(m) $\epsilon=3.2\%$

(n) $\epsilon=6\%$

(o) Ag = 50%

(p) $\epsilon=1.5\%$

(q) $\epsilon=2.6\%$

(r) $\epsilon=3.4\%$

(s) $\epsilon=6.1\%$

(t) Ag = 75%

(u) $\epsilon=1.6\%$

(v) $\epsilon=2.7\%$

(w) $\epsilon=3.5\%$

(x) $\epsilon=6.1\%$

(y) Ag = 90%

Figure 15: Dislocation structures during uniaxial compression of Coreshell
nanosphere for different percentages of Ag. (Green line, Shockley partial
dislocation with Burgers vectors $\frac{1}{6}<112>$; yellow line, hirth
dislocation with Burgers vectors $\frac{1}{3}<100>$; Pink line, stair-rod
dislocation with Burgers vectors $\frac{1}{6}<110>$; Red line indicates all
other dislocation)

For FGMs, dislocation initiation starts from the contact surface under
compression at 1.6-1.7% strain. Two Shockley partial dislocations are visible
at 1.6% strain for 10%, 25% of Ag but for 75% & 90% of Ag dislocations are
visible at 1.7% strain as shown in Fig.13 as green line. So, we can conclude
that dislocation initiation starts earlier for a lower percentage of Silver.
However, at 1.5-1.6% strain range in the upper surface, there is no Shockley
Partial dislocation seen for 50% of Ag. At 1.6% strain, two Shockley Partial
dislocations are shown on the lower surface. At 1.8%strain, two Shockley
Partial dislocations and one Hirth dislocation are visible on the upper
surface. The inconsistent surface finish of FGM nanoparticles causes this
phenomenon. With the initiation of dislocation, the stress drops a little.
Subsequently, the stress increases rapidly and reaches a peak. The ultimate
compressive strength was obtained at 2.6% strain for 10%, 25% & 90% of Ag and
at 2.7% strain for 50% of Ag and at 2.5% strain for 75% of Ag. At the ultimate
compressive strength, four Hirth dislocations are visible which acts as a lock
as shown in Fig. 13 as yellow line. Hirth lock blocks the movement of the
plane which leads to stress hardening. As the compression goes on, the stress
abruptly falls to the lowest point at 3.1% strain for 10% & 50% of Ag, at 3.2%
strain for 25% of Ag, at 3.3% strain for 75% of Ag, at 3.45% strain for 90% of
Ag. In this stage, material failure occurs. Again, the stress starts to
increase and reaches a peak point at 6% strain for 10%, 25% & 50% of Ag, at
5.9% strain for 75%, at 6.1% strain for 90% of Ag. In this stage, Shockley
partial dislocation propagates towards the center of the nanosphere as shown
in Fig. 13. For Core-shell, dislocation initiation starts from the contact
surface under compression at 1.5-1.6%. Two Shockley p dislocations are visible
at 1.5% strain for 10%, 25%, 50%, 75% of Ag but at 1.6% strain for 90% of Ag
as shown in Fig. 15. So, we can say that dislocation initiation starts at the
same strain rate for all percentages of Silver except 90% of Ag. The ultimate
compressive strength was obtained at 2.6% strain for 10%, 25% & 75% of Ag and
at 2.5% strain for 50% of Ag and at 2.7% for 90% of Ag. At ultimate
compressive strength, four Hirth dislocations are visible which acts as a
lock. As the compression goes on, the stress falls to the lowest point at 3.1%
strain for 10% & 25% of Ag, at 3.2% strain for 50% of Ag, at 3.4% strain for
75% of Ag, at 3.5% strain for 90% of Ag. In this case, the strain rate is
increased with the percentage increase of Silver. Again, the stress reaches a
peak from the bottom at 6% strain for 10%, 25% & 50% of Ag, at 6.1% strain for
75% & 90% of Ag.

##  4 Conclusion

In this work, MD simulations were performed to study the compressive
deformation of Ag-Au functionally graded and core-shell nanospheres oriented
along [001]. We were especially interested in the effect of dislocations on
the plasticity mechanisms and yield stress of varying Ag-Au %wt. The
nucleation of Shockley partial dislocations from the contact edge top and
bottom surfaces appears to be the cause of first plastic deformation in many
circumstances. Hirth dislocations form a pyramid-like structure and Shockley
partial dislocation disappears. The pyramid-like structure may disappear
immediately. With increasing compression, the Shockley partial dislocation
becomes evident and propagates. With the visual of new partial dislocation,
materials trends to behave like amorphism. We observed that the stress is
mainly dependent on dislocation density. Initially, The stress level rises
gradually, but with partial dislocation, the stress level reduces slightly.
Then Hirth dislocation is visible and acts as a planar lock and helps to
concentrate stress. As a result, we got the maximum strength in stress strain
graph and observe the plasticity mechanisms. After that, stress drops abruptly
and again starts to increase and now creates some wave-like stress pattern,
indicating the amorphism of materials. We also found that the ultimate
strength is higher in core-shell for different wt% of Ag than FGMs. FGMs
maintain a decreasing order with the increasing wt% of Ag for the ultimate
strength. But Core-shell does not follow the same trend. Core-shell gives an
unusual trend, and ultimate strength increases upto a certain wt% of Ag and
then decreases. The phenomena may occur due to the core and shell effect on
the nanosphere. To better understand the mechanical properties of nanospheres,
we simulated the precisely [001] oriented compression of nanospheres with bare
surfaces at low temperatures. However, such optimal settings are uncommon in
real-life tests, which are typically conducted at room temperature with no
precise orientations. The [001] orientation is directly responsible for the
pyramidal-like formations in our simulation. A slight misorientation and the
effects of temperature are expected to delay but not prevent their creation.
In the end, a considerable number of additional calculations would be required
to narrow the gap between experiments and theory. We will try to construct
statistics for varied orientations and temperatures in a future investigation.

## Acknowledgment

The authors would like to acknowledge Multiscale Mechanical Modeling and
Research Networks (MMMRN),ME Dept, BUET for their technical assistance to
conduct the research.

## Data availability

The raw/processed data required to reproduce these findings cannot be shared
at this time as the data also forms part of an ongoing study.

## CRediT authorship contribution statement

Prottay Malakar: Methodology, Software, Data Curation, Visualization, Formal
analysis, Writing - Original Draft. Md Al Rifat Anan: Software, Data Curation,
Visualization, Formal analysis, Writing - Original Draft. Mahmudul Islam:
Conceptualization,Formal analysis, Writing-Review & Editing. Md Shajedul Hoque
Thakur: Software, Writing-Review & Editing. Satyajit Mojumder: Project
administration, Resources, Writing-Review & Editing.

## References

  * [1] P. Shanmugavel, G. Bhaskar, M. Chandrasekaran, P. Mani, S. Srinivasan, An overview of fracture analysis in functionally graded materials, European Journal of Scientific Research 68 (3) (2012) 412–439.
  * [2] J. S. Moita, A. L. Araújo, V. F. Correia, C. M. M. Soares, J. Herskovits, Material distribution and sizing optimization of functionally graded plate-shell structures, Composites Part B: Engineering 142 (2018) 263–272.
  * [3] F. Moleiro, V. F. Correia, A. Araújo, C. M. Soares, A. Ferreira, J. Reddy, Deformations and stresses of multilayered plates with embedded functionally graded material layers using a layerwise mixed model, Composites Part B: Engineering 156 (2019) 274–291.
  * [4] M. Song, J. Yang, S. Kitipornchai, Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets, Composites Part B: Engineering 134 (2018) 106–113.
  * [5] F. Z. Zaoui, D. Ouinas, A. Tounsi, New 2d and quasi-3d shear deformation theories for free vibration of functionally graded plates on elastic foundations, Composites Part B: Engineering 159 (2019) 231–247.
  * [6] J. Woo, S. Meguid, K. Liew, Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells, Acta Mechanica 165 (1) (2003) 99–115.
  * [7] Z. Belabed, M. S. A. Houari, A. Tounsi, S. Mahmoud, O. A. Bég, An efficient and simple higher order shear and normal deformation theory for functionally graded material (fgm) plates, Composites Part B: Engineering 60 (2014) 274–283.
  * [8] C. Sachdeva, S. S. Padhee, Functionally graded cylinders: Asymptotically exact analytical formulations, Applied Mathematical Modelling 54 (2018) 782–802.
  * [9] G. Udupa, S. S. Rao, K. Gangadharan, Functionally graded composite materials: an overview, Procedia Materials Science 5 (2014) 1291–1299.
  * [10] S. Howard, R. Stewart, W. Clegg, The delamination of ceramic laminates due to residual thermal stresses, in: Key Engineering Materials, Vol. 116, Trans Tech Publ, 1996, pp. 331–350.
  * [11] W. W. Gerberich, W. Mook, C. Perrey, C. Carter, M. Baskes, R. Mukherjee, A. Gidwani, J. Heberlein, P. H. McMurry, S. L. Girshick, Superhard silicon nanospheres, Journal of the Mechanics and Physics of Solids 51 (6) (2003) 979–992.
  * [12] S. Corcoran, R. Colton, E. Lilleodden, W. W. Gerberich, Anomalous plastic deformation at surfaces: Nanoindentation of gold single crystals, Physical Review B 55 (24) (1997) R16057.
  * [13] W. Mook, J. Nowak, C. Perrey, C. Carter, R. Mukherjee, S. Girshick, P. McMurry, W. Gerberich, Compressive stress effects on nanoparticle modulus and fracture, Physical Review B 75 (21) (2007) 214112.
  * [14] W. W. Gerberich, W. M. Mook, M. J. Cordill, C. B. Carter, C. R. Perrey, J. V. Heberlein, S. L. Girshick, Reverse plasticity in single crystal silicon nanospheres, International Journal of Plasticity 21 (12) (2005) 2391–2405.
  * [15] J. Deneen, W. M. Mook, A. Minor, W. W. Gerberich, C. B. Carter, In situ deformation of silicon nanospheres, Journal of materials science 41 (14) (2006) 4477–4483.
  * [16] M. Ramos, L. Ortiz-Jordan, A. Hurtado-Macias, S. Flores, J. T. Elizalde-Galindo, C. Rocha, B. Torres, M. Zarei-Chaleshtori, R. R. Chianelli, Hardness and elastic modulus on six-fold symmetry gold nanoparticles, Materials 6 (1) (2013) 198–205.
  * [17] D. Mordehai, M. Kazakevich, D. J. Srolovitz, E. Rabkin, Nanoindentation size effect in single-crystal nanoparticles and thin films: A comparative experimental and simulation study, Acta Materialia 59 (6) (2011) 2309–2321.
  * [18] A. Prakash, J. Guénolé, J. Wang, J. Müller, E. Spiecker, M. J. Mills, I. Povstugar, P. Choi, D. Raabe, E. Bitzek, Atom probe informed simulations of dislocation–precipitate interactions reveal the importance of local interface curvature, Acta Materialia 92 (2015) 33–45.
  * [19] J.-J. Bian, G.-F. Wang, Atomistic deformation mechanisms in copper nanoparticles, Journal of Computational and Theoretical Nanoscience 10 (9) (2013) 2299–2303.
  * [20] S. B. H. Salah, C. Gerard, L. Pizzagalli, Influence of surface atomic structure on the mechanical response of aluminum nanospheres under compression, Computational Materials Science 129 (2017) 273–278.
  * [21] P. Armstrong, W. Peukert, Size effects in the elastic deformation behavior of metallic nanoparticles, Journal of Nanoparticle Research 14 (12) (2012) 1–13.
  * [22] D. Mordehai, S.-W. Lee, B. Backes, D. J. Srolovitz, W. D. Nix, E. Rabkin, Size effect in compression of single-crystal gold microparticles, Acta Materialia 59 (13) (2011) 5202–5215.
  * [23] T. Kim, S. Myung, T. H. Kim, S. Hong, Robust single-nanoparticle probe for contact-mode analysis and dip-pen nanolithography, Small 4 (8) (2008) 1072–1075.
  * [24] J. Sun, L. He, Y.-C. Lo, T. Xu, H. Bi, L. Sun, Z. Zhang, S. X. Mao, J. Li, Liquid-like pseudoelasticity of sub-10-nm crystalline silver particles, Nature materials 13 (11) (2014) 1007–1012.
  * [25] C. Carlton, P. Ferreira, In situ tem nanoindentation of nanoparticles, Micron 43 (11) (2012) 1134–1139.
  * [26] L. Yang, J.-J. Bian, G.-F. Wang, Impact of atomic-scale surface morphology on the size-dependent yield stress of gold nanoparticles, Journal of Physics D: Applied Physics 50 (24) (2017) 245302.
  * [27] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of computational physics 117 (1) (1995) 1–19.
  * [28] X. Zhou, R. Johnson, H. Wadley, Misfit-energy-increasing dislocations in vapor-deposited cofe/nife multilayers, Physical Review B 69 (14) (2004) 144113\.
  * [29] Y. Gu, H. Zhan, Md investigations for mechanical properties of copper nanowires with and without surface defects, International Journal of Computational Methods 9 (01) (2012) 1240003.
  * [30] X. Zhou, H. Wadley, R. A. Johnson, D. Larson, N. Tabat, A. Cerezo, A. Petford-Long, G. Smith, P. Clifton, R. Martens, et al., Atomic scale structure of sputtered metal multilayers, Acta materialia 49 (19) (2001) 4005–4015.
  * [31] M. S. H. Thakur, M. Islam, A. Amin, S. Mojumder, M. M. Islam, Lammps input structure generator for functionally graded materials (fgm) (2019).
  * [32] M. Islam, M. S. H. Thakur, S. Mojumder, A. Al Amin, M. M. Islam, Mechanical and vibrational characteristics of functionally graded cu–ni nanowire: a molecular dynamics study, Composites Part B: Engineering 198 (2020) 108212.
  * [33] E. Polak, G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique 3 (R1) (1969) 35–43.
  * [34] W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Physical review A 31 (3) (1985) 1695.
  * [35] L. Hale, X. Zhou, J. Zimmerman, N. Moody, R. Ballarini, W. Gerberich, Phase transformations, dislocations and hardening behavior in uniaxially compressed silicon nanospheres, Computational materials science 50 (5) (2011) 1651–1660.
  * [36] N. Michaud-Agrawal, E. J. Denning, T. B. Woolf, O. Beckstein, Mdanalysis: a toolkit for the analysis of molecular dynamics simulations, Journal of computational chemistry 32 (10) (2011) 2319–2327.
  * [37] A. Stukowski, Visualization and analysis of atomistic simulation data with ovito–the open visualization tool, Modelling and Simulation in Materials Science and Engineering 18 (1) (2009) 015012.
  * [38] A. Stukowski, V. V. Bulatov, A. Arsenlis, Automated identification and indexing of dislocations in crystal interfaces, Modelling and Simulation in Materials Science and Engineering 20 (8) (2012) 085007.
  * [39] N. Zhang, Q. Deng, Y. Hong, L. Xiong, S. Li, M. Strasberg, W. Yin, Y. Zou, C. R. Taylor, G. Sawyer, et al., Deformation mechanisms in silicon nanoparticles, Journal of Applied Physics 109 (6) (2011) 063534.
  * [40] P. Valentini, W. Gerberich, T. Dumitrică, Phase-transition plasticity response in uniaxially compressed silicon nanospheres, Physical review letters 99 (17) (2007) 175701.
  * [41] J. Lian, J. Wang, Y.-Y. Kim, J. Greer, Sample boundary effect in nanoindentation of nano and microscale surface structures, Journal of the Mechanics and Physics of Solids 57 (5) (2009) 812–827.
  * [42] J. Bian, X. Niu, H. Zhang, G. Wang, Atomistic deformation mechanisms in twinned copper nanospheres, Nanoscale research letters 9 (1) (2014) 1–7.
  * [43] S. Brochard, P. Hirel, L. Pizzagalli, J. Godet, Elastic limit for surface step dislocation nucleation in face-centered cubic metals: Temperature and step height dependence, Acta Materialia 58 (12) (2010) 4182–4190.
  * [44] R. Smallman, A. Ngan, Introduction to dislocations, Modern Physical Metallurgy (Eighth Edition), edited by RE Smallman and AHW Ngan (Butterworth-Heinemann, City, 2014) (2014) 121.
  * [45] D. Hull, D. J. Bacon, Introduction to dislocations, Vol. 37, Elsevier, 2011.