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C: Physical Processes in Nanomaterials and Nanostructures
Mechanochemistry of Stable Diamane and Atomically Thin Diamond Films
Synthesis From Bi- and Multilayer Graphene: A Computational Study
Shiddartha Paul, and Kasra Momeni
J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b02149 • Publication Date (Web): 14 May 2019
Downloaded from http://pubs.acs.org on May 14, 2019
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1
Mechanochemistry of Stable Diamane and
Atomically Thin Diamond Films Synthesis From Bi-
and Multilayer Graphene: A Computational Study
Shiddartha Paul1, Kasra Momeni1,2,3,*
1 Department of Mechanical Engineering, Louisiana Tech University, Ruston, LA 71270, United
States.
2 Center for Two Dimensional and Layered Materials, Materials Research Institute, The
Pennsylvania State University, University Park, PA, 16802, USA
3 Center for Atomically Thin Multifunctional Coatings, Materials Research Institute, The
Pennsylvania State University, University Park, PA, 16802, USA
Keywords: Graphene, Diamond, Phase-Transformation, Molecular Dynamics, Passivation.
ABSTRACT — Mono- and few-layer graphene exhibit unique mechanical, thermal, and electrical
properties. However, their hardness and in-plane stiffness are still not comparable to the other
allotrope of carbon, i.e. diamond. This makes layered graphene structures to be less suitable for
* kzm5606@psu.edu; kmomeni@latech.edu
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application in harsh environments. Thus, there is an unmet need for the synthesis of atomically
thin diamond films for such applications that are also stable under ambient conditions. Here, we
demonstrate the possibility for the synthesis of such diamond films from multilayer graphene
using molecular dynamics approach with reactive force fields. We study the kinetics and
thermodynamics of the phase transformation as well as the stability of the formed diamond thin
films as a function of the layer thickness at different pressures and temperatures for pristine and
hydrogenated multilayer graphene. The results indicate that the transformation conditions
depend on the number of graphene layers and surface chemistry. We revealed a reduction in the
transformation strain by up to 50% while transformation stress has reduced by as much as five
times upon passivation with hydrogen atoms. While the multilayer pristine graphene to diamond
transformation is shown to be reversible, hydrogenated multilayer graphene structures had
formed a metastable diamond film. Our simulations have further revealed temperature-
independence of transformation strain, while transformation stresses are strong functions of
temperature.
Carbon can form different structures, such as graphite, Phagraphene1, diamond, graphene, and
nanotubes, which have diverse mechanical2 and electrical properties. For example, Graphene is
a semiconductor material without any band gap and a high mechanical strength. It can endure
high elongations where the strain can shift its band gap and this can result in a good mechanical
and electrical coupling for electromechanical devices. Graphene can also sustain a very high in-
plane tensile elastic strain, up to 25%3. Diamond has excellent mechanical hardness and thermal
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conductivity. Static compression applied on a few layers of pristine graphene at high
temperatures, transforms it to the diamond 4. Pressure applied on a multilayer graphene with a
certain radical group, e.g. hydogen (-H), hydroxyl (-OH), or flourine (-F), can transform the sp2
hybridization of carbon in the graphene structure into sp3 hybridization as in the, diamond
structure 5. The sp2 to sp3 hybridization opens the band gap and also change the electrical,
magnetic, and mechanical properties 6 . Density functional theory (DFT) 7 and ab initio 8 methods
have been used widely to study the phase transformation of multilayer graphene to diamond at
high compressive pressure 9,10,11,12. Moreover, existence of atomically thin nanodiamond has
been reported in several computational studies 13,14,15,16, where the interlayer gap between the
graphene layers reduced and formed a covalent bond. Effect of the pressurizing medium on the
transformation conditions of multilayer graphene to diamond has also been investigated using
DFT technique 17. Different methods can be adopted to apply the pressure and induce the
graphene-diamond phase transformation. For example, multilayer graphene can be placed
between two piston walls 18 or indentation process 19,20 can be adapted to pressurize the
structure in a certain direction. Conversion of multilayer graphene requires overcoming a certain
energy barrier for the transformation of the sp2 bond to sp3. Hydrogen and fluorine atoms have
been introduced 5,21,16 as chemical radicals to reduce the graphene to diamond transformation
pressure. This helps to get sp3 hybridized carbon bonds in graphene to create covalent bonds
between two adjacent graphene layers. Hydrogenation of graphene has a significant effect on
the mechanical characteristics 22 and its transformation to diamond. The concentration of
hydrogen on in graphene layers can decrease failure strength of the graphene structure and
facilitate the sp2 to sp3 hybridization. Recent experimental and computational investigations23
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have shown that shearing of multilayer graphene can reduce the compressive stress required for
graphene to diamond phase transformation.
Experimental studies are performed to prove formation of atomically thin diamond when two
or more layers of graphene were pressurized with water6,5. The multilayer graphene structure
has been pressurized at a high temperature in an aqua-ambient condition, which helped to
transform the sp2 hybridization to sp3 by hydrogenation. This transformation initiates the
formation of diamond structure. Diamond indenter has also been used to pressurize 20 the
multilayer graphene to form the diamond structure and existence of the newly formed diamond
has been confirmed by AFM microhardness and Raman spectroscopy.
In this study, we will use atomistic molecular dynamics technique with reactive force fields to
study the kinetics of bi- and few-layer graphene to diamond phase transformation at different
thermochemical conditions. This is a powerful technique which has been utilized to study phase
transformation in various materials. 1,24–27 We will calculate the transformation stress and strain
for the formation of diamond thin films at different thermochemical conditions. We then
analyzed the energy barrier for the diamond formation and the stability of the formed diamond
structures.
Computational Model
Graphene samples of infinite dimensions are considered by applying periodic boundary
conditions in the plane of the graphene layers. A simulation cell of 56.54×26.41Å in the plane of
graphene layers is considered. The dimension of simulation cell in the direction normal to the
graphene plane varies by change in the number of graphene layers. The initial interlayer distance
between graphene layers is assumed to be the length of -bond between graphene layers, i.e.
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3.4Å 28. We placed one graphene layer on top of the other layer while shifting it by 1.41Å, which
is the C-C bond length in the y-direction. As a result, graphene layers form the ABAB stacking
where carbon atoms in one layer are located in the middle of the hexagonal gap of its adjacent
graphene layer, see Figure 1(b) and Figure 1(c).
Figure 1 | Structure and configuration of the modeled system. (a) Bond length and configuration
of hydrogen and carbon for the hydrogenated graphene model; (b) Top view of ABA stacking,
where C atom of bottom layer positioned at the center of hexagonal adjacent top graphene
layers; (c) Different stackings of multilayer graphene structures, where the ABA stacking is
considered for our studies; The structure of (d) Cubic Diamond (CD) and (e) Hexagonal diamond
(HD).
Hydrogenated graphene is modeled by adding hydrogen atoms to the outer side of top and
bottom graphene layers. The C-H bond length is 1.10Å 10 and H atoms are situated perpendicular
to each carbon atom of their neighboring graphene layer. The motion of atoms in the system is
constrained in the plane of graphene layers, i.e. xy-plane 29. All the simulations are performed
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using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software
package28.
Interatomic Interactions Model — There are several potentials for modeling the C-C and C-H
bonds, e.g. ReaxFF or LCBOP. Here, we have chosen Adaptive Intermolecular Reactive Bond Order
(AIREBO) due to its capability for reproducing experimentally measured mechanical properties of
graphene30 which is the focus of this study. Orekhov et al.30 compared the MD simulation results
for AIREBO and LCBOP potentials with experimental results and found that AIREBO has very good
agreement with the experimental measurements in comparison with the LCBOP. The AIREBO
potential is used to model the interactions between the C-C and C-H atoms, which could
reproduce the thermophysical properties of the C-H system 31. The integration of pairwise
interactions in AIREBO potential can be represented by following equation
(1)
𝐸
=
1
2
∑
𝑖
∑
𝑗
≠
𝑖
[
𝐸
𝑅𝐸𝐵𝑂
𝑖𝑗
+
𝐸
𝑙𝑗
𝑖𝑗
+
∑
𝑘
≠
𝑖,𝑗
∑
𝑙
≠
𝑖,𝑗,𝑘
𝐸
𝑇𝑂𝑅𝑆𝐼𝑂𝑁
𝑖𝑗
]
.
The AIREBO potential is the modification over the reactive empirical bond-order (REBO)
potential, which also considers the torsional ( ) and Lennard-Jones ( ) interactions. The
𝐸
𝑇𝑂𝑅𝑆𝐼𝑂𝑁
𝑖𝑗
𝐸
𝑙𝑗
𝑖𝑗
REBO potential ( ) is a combination of attractive ( ) and repulsive ( ) interactions in
𝐸
𝑅𝐸𝐵𝑂
𝑖𝑗
𝑉
𝐴
𝑖𝑗
𝑉
𝑅
𝑖𝑗
certain ratio ( ) as
𝑏
𝑖𝑗
𝐸
𝑅𝐸𝐵𝑂
𝑖𝑗
=
𝑉
𝑅
𝑖𝑗
+
𝑏
𝑖𝑗
𝑉
𝐴
𝑖𝑗
.
(2)
The repulsive term is expressed by the Brenner equation 32,
𝑉
𝑅
𝑖𝑗
=
𝑤
𝑖𝑗
(
𝑟
𝑖𝑗
)
[
1
+
𝑄
𝑖𝑗
𝑟
𝑖𝑗
]
𝐴
𝑖𝑗
𝑒
―
𝛼
𝑖𝑗
𝑟
𝑖𝑗
.
(3)
Here, the , and parameters depend on i and j. The bond weighting parameter
𝑄
𝑖𝑗
𝑟
𝑖𝑗
𝛼
𝑖𝑗
𝑤
𝑖𝑗
(
𝑟
𝑖𝑗
)
depends on switching function S(t) as
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𝑤
𝑖𝑗
(
𝑟
𝑖𝑗
)
=
𝑆
(
𝑡
𝑐
(
𝑟
𝑖𝑗
)
)
,
(4)
𝑆
(
𝑡
)
=
Θ
(
―
𝑡
)
+
Θ
(
𝑡
)
Θ
(
1
―
𝑡
)
0.5
[
1
+
cos
(
𝜋𝑡
)
]
,
(5)
𝑡
𝑐
(
𝑟
𝑖𝑗
)
=
𝑟
𝑖𝑗
―
𝑟
𝑚𝑖𝑛
𝑖𝑗
𝑟
𝑚𝑎𝑥
𝑖𝑗
―
𝑟
𝑚𝑖𝑛
𝑖𝑗
.
(6)
The attractive interaction is expressed as
𝑉
𝐴
𝑖𝑗
=
―
𝑤
𝑖𝑗
(
𝑟
𝑖𝑗
)
3
∑
𝑛
=
1
𝐵
(𝑛)
𝑖𝑗
𝑒
𝐵
(𝑛)
𝑖𝑗
𝑟
𝑖𝑗
.
(7)
The attractive interaction ( ) is multiplied by the bond order interaction ratio between atoms
𝑉
𝐴
𝑖𝑗
i and j as
;
𝑏
𝑖𝑗
=
1
2
[
𝑝
𝜎𝜋
𝑖𝑗
+
𝑝
𝜎𝜋
𝑗𝑖
]
+
𝜋
𝑟𝑐
𝑖𝑗
+
𝜋
𝑑ℎ
𝑖𝑗
(8)
where is a many body potential term. Here, and are not necessarily equal as they
𝑏
𝑖𝑗
𝑝
𝜎𝜋
𝑖𝑗
𝑝
𝜎𝜋
𝑗𝑖
depend on the penalty function ( ) of bond angle between the vector vector , i.e.
𝑔
𝑖
𝜃
𝑗𝑖𝑘
𝑟
𝑖𝑗
𝑟
𝑘𝑖
𝑝
𝜎𝜋
𝑖𝑗
=
[
1
+
∑
𝑘
≠
𝑖,𝑗
𝑤
𝑖𝑘
(
𝑟
𝑖𝑘
)
𝑔
𝑖
(
cos
𝜃
𝑗𝑖𝑘
)
𝑒
𝜆
𝑗𝑖𝑘
+
𝑃
𝑖𝑗
]
―
1/2
.
(9)
The C-C bond length calculated by the AIREBO potential is 1.396Å which is in good agreement
with the experimentally measured value of 1.415Å 33 The AIREBO potential allows the smooth
formation and breaking of covalent bonds, as well as the associated change in hybridization. The
torsional and Lennard-Jones (LJ) interactions included in the AIREBO potential allow modeling of
hydrogenated graphene and its reactions during the phase transformations.
Compressing Walls Model — The compressive pressure is applied by sandwiching the
multilayer graphene structures between walls with a repulsive interaction force-field as
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𝐸
=
𝜀
[
2
15
(
𝜎
𝑟
)
9
+
(
𝜎
𝑟
)
3
]
,
(10)
which is the integration over a 3D half-lattice of LJ 12-6 potential. The σ and ɛ parameters are the
equilibrium distance and energy well of the LJ parameters, respectively. For the pristine
graphene, the repulsive parameters of the force-field of the wall were estimated based on the
C-C Lennard-Jones interaction parameters 34 where σC-C = 3.4Å and ɛC-C = 0.056 eV. In the case of
hydrogenated graphene, the repulsive force-field wall was modeled using C-H Lennard-Jones
interaction parameters 35 where σC-H = 2.82Å and ɛC-H = 0.00262216 eV. These piston walls are
initially placed at the corresponding LJ equilibrium distance from the structure for both cases of
pristine and hydrogenated graphene systems.
During loading, the bottom wall is fixed while the top one has moved toward the structure to
create the compressive stress. We investigated the formation of diamond from multilayer
pristine and hydrogenated graphene structures with two to eight layers at 0K, 500K, 800K, 1000K,
1200K and 1500K temperatures. Statistical analysis is performed by assigning different initial
seed velocities to the structures to investigate the sensitivity of the results. After each
compression step, all the multilayer graphene structures had been relaxed using the Nose-
Hoover thermostat 36 for 200ps to ensure that the structure reached equilibrium and C atoms
have enough time to form diamond to avoid reporting of the kinetically stabled diamond
structures. We applied compressive loading in the direction normal to the graphene planes at an
initial strain rate of 1 ps-1, but when the strain gets close to the critical transformation point, the
strain rate was changed to 0.1 ps-1. Unloading was performed at a strain rate 1 ps-1. These strain
rates are calculated with respect to the position of the walls. We also relaxed the structure using
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the same ensemble during each step of unloading. We calculated the stress for individual atoms
using the virial stress formulatoin 37,
𝜎
𝑖𝑗
=
1
𝛺
𝛼
(
1
2
𝑚
𝛼
𝑣
𝛼
𝑖
𝑣
𝛼
𝑗
+
𝑛
∑
𝛽
=
1,𝑛
𝑟
𝑗
𝛼𝛽
𝑓
𝑖
𝛼𝛽
)
,
(11)
where i and j denote indices in the Cartesian coordinate system; α and β are the atomic indices;
mα and νβ denote the mass and velocity of atom α; rαβ is the distance between atoms α and β;
and is the atomic volume of atom α. For virial stress calculations we have used the cell volume
𝛺
of the simulation.
Result and Discussion
In the following two subsections, we will elaborate our results for the transformation of pristine
and hydrogenated multilayer graphene structures to diamond. For each case, the simulations are
performed at various temperatures for different number of graphene layers. The phase
transformation stress is calculated by summing all the stress components normal to the graphene
plane,
zz. The transformation strain is also calculated as the change in the thickness of the
multilayer graphene film with respect to its thickness at the end of the relaxation step, i.e.
zz.
We used the engineering strain38, , in our calculations where L is the distance
𝜀
=
𝐿
𝐿
0
―
1
between the top and bottom graphene layers (thickness) at each time and is the initial value
𝐿
0
of L. Statistical analysis has been performed by assigning different seed velocity values for each
temperature. The reported experimental values are based on the earliest detection of diamond.
Thus, there is a variation in the reported experimental transformation stresses that depends not
only on the process but also on the sensitivity of the measurement devices. Generally, the
experimental measurements4,39 detect nucleation of diamond phase from a bulk graphite when
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at least 10% volume fraction of graphite has transformed to diamond and reported
transformation stresses between 1GPa to 15GPa. Thus, in this study we have calculated the
transformation stress as the stress that is needed to transform 10% of graphene layers to
diamond. We have also calculated the transformation stress for the formation of 50% diamond
(see Supporting Information). Our calculations indicate transformation stresses in the range of 1-
20GPa at different temperatures which is in agreement with the experimental studies. 4,39
Pristine Graphene — Our results for transformation of pristine graphene to diamond for
different number of graphene layers and temperatures are presented in this section. The general
mechanism can be depicted as under compressive loads the distance between atoms of adjacent
graphene layers reduces which results in the formation of new bonds. For phase transformation,
the carbon atoms in the graphite need to overcome the activation energy barrier. From the basic
definition of enthalpy, equation (12), we have the relation between enthalpy, internal
energy, pressure and volume as 40
, (12)
𝐻
=
𝐸
+
PV
where E is the internal energy, P is pressure, and V is volume. We know that the enthalpy
increases with pressure for compressible solids. Hence, at high pressures, the C atoms of
graphene structure can overcome the activation enthalpy for phase transformation and initialize
the diamond nucleation.
Our results indicate that for the pristine graphene, maximum volume fraction of formed
diamond increases by increasing temperature. Figure 2 shows the formation of diamond under
compressive loading in the three-layer (3L) and 8L pristine graphene. We also investigated the
reversibility of this phase transformation and stability of the formed diamond thin films by
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removing the applied compressive stress. We found that the formed diamond structure vanishes
after unloading even at 0K, see Figure 2 bottom row. Thus, the formed pristine diamond thin film
is not stable and transforms back to the multi-layer graphene through a barrier-less phase
transformation as the external load has removed.
Figure 2 | Three-layer and eight-layer pristine graphene phase transformation. The maximum
percentage of the diamond structure formed after compression (top row) and its reverse
transformation to graphene upon unloading (bottom row) are shown for three-layer graphene
at (a) 0K and (b) 1500K. The corresponding results for the eight-layer graphene for (c) 0K and (d)
1500K are shown.
The energy pathway for 8L pristine graphene at 0K has been shown in Figure 3. At point ‘A’, a
small portion of graphene transforms to diamond which results in a drop in the energy curve.
Beyond point ‘A’ the diamond fraction decreases up to point ‘B’ where there will be no diamond
remained. After that the energy increases rapidly as a function of strain and volume fraction of
diamond structure will also increase. At point ‘C’, the volume fraction of diamond maximizes and
reaches 91.9%, which will decrease upon further compression. We stopped applying any further
compression beyond point D, where the diamond fraction has reduced to 51.6%. Upon unloading
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the volume fraction of diamond structure decreases along with the total energy of the system.
Diamond structure completely vanishes at point ‘E’. The structure after unloading will have a
nonzero residual stress and its energy is lower than the initial energy of the structure due to
partial inter-layer bonding between adjacent graphene layers.
Figure 3 | Energy pathway during loading and unloading of 8L pristine graphene at 0K
temperature. The key transition points and associated structures are shown. Normalized energy
of the system is shown for loading (solid black line) and unloading (dashed red lines). A small
portion of graphene transforms to the diamond at = -18% (point A) during loading which will
disappear later at = -46% (point B). Upon further loading the energy and volume fraction of
diamond increase again and reaches a maximum at point ‘C’. At this point, increasing
compression although increase the energy of the system but results in the reduction of diamond
volume fraction. Unloading starts at point ‘D’ and at point ‘E’ all diamond will transform back to
graphene. The color map of the inset atomistic structures is the same as in Fig. 2.
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Transformation strain and stress are calculated for multilayer pristine graphene at different
temperatures (Figure 4). Our results indicate that the transformation strain is independent of
temperature and number of graphene layers (Figure 4a) when the numerical error (~2%) is
considered. Almost all the graphene layers will transform to diamond at ~46% compressive strain
normal to graphene plane,
zz. This is consistent with previous claims on temperature
independence of the transformation strain 41. Lagrangian formulation for strain components can
explain this temperature independence of graphite to diamond transformation strain as
formulated by Wentz-Covitch et al. 42,
𝐿
=
∑
𝑖
1
2
𝑚
𝑖
𝐪
𝑇
𝑖
𝑑
𝐪
𝑖
―
𝑬
(
{
𝐪
𝑖
}
, 𝜺
)
+
1
2
𝑊
Tr
(
𝜺
𝑇
𝜺
)
―
𝑃
𝑒𝑥𝑡
Ω
𝑐𝑒𝑙𝑙
.
(13)
Here is the mass of the i th atom, is the rescaled atomic coordinates, is the strain tensor,
𝑚
𝑖
𝐪
𝑖
𝜺
represent the fictious mass which is adjusted with the other dynamical variables, is the
𝑊
𝑃
𝑒𝑥𝑡
external pressure, is cell volume, superscript T represents transpose operation, and Tr() is
Ω
𝑐𝑒𝑙𝑙
the trace of a tensor. The relation between the coordinate of an atom, ri, and the rescaled
coordinate, , is . From equation (13) we can derive the following
𝐪
𝑖
𝑟 (
𝜺,
𝐪
𝑖
)
=
(1
+
𝜺
)
𝐪
𝑖
expression form and :
𝒒
𝑖
𝜺
𝒒
𝑖
=
―
1
𝑚
𝑖
(
1
+
𝜺
)
―
1
𝒇
𝑖
―
𝑑
―
1
𝑑
𝐪
𝑖
,
(14)
𝜺
=
Ω
𝑐𝑒𝑙𝑙
𝑊
(
Π
―
𝑃
𝑒𝑥𝑡
𝐈
)
(
1
+
𝜺
𝑇
)
―
1
.
(15)
Here , , and I is the identity tensor. Equation (15) indicates that
Π
=
∑
𝑁
𝑖
𝑚
𝑖
𝐯
𝑡
𝑖
𝐯
𝒊
Ω
𝑐𝑒𝑙𝑙
+
𝝈
𝐯
𝑖
=
(
1
+
𝜺
)
𝐫
𝑖
the strain is temperature-independent but depends on the and .
Ω
𝑐𝑒𝑙𝑙
𝑊
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Figure 4 | Transformation strain and stress for the pristine graphene to the diamond
transformation. Transformation strain (a) and stress (b) are calculated as a function of
temperature for a different number of graphene layers. It is shown that considering the 2% error
of the calculations, the transformation strain is independent of the temperature and number of
graphene layers. In contrast, the transformation stress increases with temperature.
The graphene to diamond transformation stress increases as temperature increases that is
consistent with previous reports 11, Figure 4(b). The transformation stress for 3L graphene is
slightly higher than its corresponding value for the 4L graphene which can be due to domination
of surface tension for 3L graphene. For 3L graphene at 0K suddenly transforms to diamond (first
order transformation), where almost the entire structure transforms to diamond. That explains
the sudden increase in the measured transformation strain. If we consider formation of 50%
graphene to diamond as the transformation criteria, then transformation stress increases to
80 GPa and remains constant as the number of layers increase (Figure S1 in the Supporting
Information). This can be interpreted by the lower contribution of thermal energy compared to
mechanical energy at higher volume fractions of diamond (higher stresses).
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Heterogenous structures form, where graphene partially transforms to diamond, during the
graphene to diamond phase transformation where the energy of carbon atoms in the graphene
phase increases and thus the activation barrier for graphene to diamond transformation will
reduce. By increasing the pressure, the height of this energy barrier reduces which facilitates
formation of the diamond phase. Thus, volume fraction of diamond increases as we increase the
pressure and all the graphene will transform to diamond when the energy barrier for graphene
to diamond transformation reduces to zero. This is the point where graphene becomes unstable
rather than being metastable. We found that the height of this energy barrier is a function of the
number of graphene layers.
Atoms at diamond phase can transform back to graphite. At high temperatures, the rate of
direct and reverse diamond to graphite transformation increases. This thermally activated
transformation can be modeled using the Arrhenius relation 43,
Г
𝐺
―
𝐷
𝑡ℎ
Г
𝐷
―
𝐺
𝑡ℎ
=
𝑒
―
∆𝐺
𝑘
𝐵
T
,
(16)
Here, is the rate of jumping from graphite to diamond interface and is vice versa;
Г
𝐺
―
𝐷
𝑡ℎ
Г
𝐷
―
𝐺
𝑡ℎ
is the difference of Gibbs free energy between graphite and diamond phase; kB is Boltzmann
∆𝐺
constant, and T is the temperature. Defining and are the direct and reverse diamond
𝑝
𝐷
―
𝐺
𝑝
𝐺
―
𝐷
to graphene and phase transformation probabilities, respectively, we have
Г
𝐺
―
𝐷
𝑡ℎ
=
𝑝
𝐷
―
𝐺
𝑣
𝑡ℎ
,
(17)
Г
𝐷
―
𝐺
𝑡ℎ
=
𝑝
𝐺
―
𝐷
𝑣
𝑡ℎ
.
(18)
Here is the total rate of thermally activated jumps. For these phase transformation
𝑣
𝑡ℎ
probabilities, we have
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,
𝑝
𝐷
―
𝐺
=
1
1
+
𝑒
―
∆𝐺
𝑘
𝐵
T
𝑝
𝐺
―
𝐷
=
𝑒
―
∆𝐺
𝑘
𝐵
T
1
+
𝑒
―
∆𝐺
𝑘
𝐵
T
.
(19)
Table 1: Percentage of Cubic Diamond (CD) and Hexagonal Diamond (HD) for different multilayer
pristine graphene structures (3L to 8L) at the different temperatures.
Temperature
0K
500K
800K
1000K
1200K
1500K
#Layers
CD
HD
CD
HD
CD
HD
CD
HD
CD
HD
CD
HD
3L
27.8*
0.0
0.1
0.0
0.1
0.0
0.0
0.1
0.0
0.1
0.1
0.0
4L
0.2
0.0
0.1
0.2
1.4
0.2
0.1
0.0
0.0
0.1
0.0
0.2
5L
0.3
0.0
1.0
0.0
1.1
0.0
0.1
0.0
0.1
0.0
0.1
0.0
6L
1.4
0.2
0.5
0.0
0.6
0.3
0.2
0.0
0.1
0.0
0.1
0.0
7L
5.3
0.0
0.3
0.0
0.1
0.0
0.1
0.0
0.1
0.0
0.1
0.0
8L
8.9
0.2
7.2
0.1
5.5
0.1
2.3
0.0
4.2
0.1
1.9
0.3
Note: Data is taken at 10% nondiamond volume fraction. The volume fractions of the 1st and
2nd nearest neighbors are not shown here. Thus, the sum of CD an HD volume fractions is
smaller than 10%.
* For 3L graphene at 0K the graphene to diamond transformation is of first order, where
almost the entire structure transforms to diamond. That is why the volume fraction of CD is
greater than 10% here.
From equation (19), we can see that the probability of thermally activated jumps from the
diamond to graphene phase is dominant at a higher temperature when is a weak function of
∆
𝐺
temperature. Thus, at high temperatures, a large deriving force is required to transform
graphene to diamond, which results in a larger transformation stress as temperature increases.
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During the diamond nucleation process, the compressive stress and thermal vibrations create
inter-layer distortions and stacking sequences which promote the formation of sp3 bonds. The
compressive transformation stress, , for multilayers of graphene as a function of temperature
𝜎
𝑐
𝑡
are shown in Figure 4(b). Static compression of the hexagonal graphite (HG) results in the
formation of cubic diamond (CD) and hexagonal diamond (HD) 12, see Figure 1. The HG HD
→
transformation requires higher pressure compared to HG CD due to its higher energy barrier
→
and the different stacking sequence of carbon atoms in the CD phase 12. Thus, the transformation
stress will be determined by the volume fraction mixture of the CD and HD phases, and is more
sensitive to the volume fraction of the HD phase. This is in agreement with the results presented
in Table 1 when the calculation errors are taken into account. For example, at 800K the 5L and
6L graphene systems have almost the same transformation. While 1.1% volume fraction of the
5L is diamond (1.1% CD and 0.0% HD), the 6L graphene system has an overall 0.9% volume
fraction of diamond (0.6% CD and 0.3% HD). This indicates that the weighting factor of the HD is
larger than the CD in the transformation stress. The volume fraction of CD in the 8L structures is
much higher than all other structures at all temperatures. Thus, the transformation stress of 8L
graphene structure is higher than its value for all other structures as shown in Figure 4(b). This
phenomenon become clearer when we see the transformation stress for 50% diamond (see
Supporting Information).
Hydrogenated graphene — We investigated the effect of surface passivation with hydrogen on
the formation of diamond from multilayer graphene. We followed the same procedure for
loading and unloading of the hydrogenated graphene structures as we explained in previous
section for the pristine graphene. Figure 5 shows that the hexagonal diamond fraction is
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predominant. The percentage of the converted diamond is lower than that for the pristine
graphene. However, the formed hydrogenated diamond films are metastable and remain even
after removing the external pressure. This is in contrast to pristine diamond films, where the
formed diamond structure becomes unstable after unloading and transforms back to multilayer
graphene. Figure 5 also shows that after unloading, the volume fraction of transformed diamond
increases, which is due to the absence of repulsive interactions of the wall after the unloading
stage. This implies that the transformation of hydrogenated graphene structures to diamond is
an irreversible process. Moreover, Figure 5 shows that the percentage of the diamond is higher
for hydrogenated graphene structures with smaller number of graphene layers.
In Figure 6, the energy pathway during the phase transformation has been represented for 8L
hydrogenated graphene at 0K. Transformation of the hydrogenated graphene starts at point ‘A’
that is detected by the slight drop of energy. The energy and diamond fraction both increase
upon further compression up to point ‘B’ where the diamond fraction decreases and the energy
drops. At point ‘C’ the diamond structure totally vanishes followed by a sudden drop in the energy
indicating formation of a new structure. During unloading (red dotted line in Figure 6) at point
‘D’ diamond structure reappears. At point ‘E’ the diamond percentage reaches the maximum and
remains even after bringing the walls back to their original location. This indicates stability of the
formed diamond thin films.
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Figure 5 | Two-layer and eight-layer hydrogenated graphene phase transformation. The
maximum percentage of the diamond structure formed after compression (above) and its reverse
transformation to graphene upon decompression (bottom) are shown for two-layers graphene
at (a) 0K and (b) 1500K. The corresponding results for the eight-layer graphene for (c) 0K and (d)
1500K are also shown.
Figure 7 shows the variation of transformation strain and stress for hydrogenated multilayer
graphene structures as a function of temperature using the same statistical approach adopted
for pristine graphene structure earlier. The results indicate temperature independence of the
transformation strain, Figure 7(a), as in the case of multilayer pristine graphene structures. While
the transformation strain monotonously increases by increasing the number of layers, the
hydrogenated graphene system with odd number of layers have a lower transformation strain
compared to the corresponding system of relatively same size but even number of graphene
layers. Although the transformation strain for both systems with odd and even number of
graphene layers increases with increasing the number of layers. It should be mentioned that the
systems with two and three graphene layers have a different behavior compared to the rest of
the system which is due to domination of surface atoms. Furthermore, the transformation stress
for hydrogenated graphene systems is a weak function of temperature that is in contrast to
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multilayer pristine graphene, i.e.
G in Eq. (19) is a linear function of temperature.
Transformation stress for the 2L and 3L systems is distinctly larger than the other systems which
is due to the domination of surface stresses. The transformation stress will also reduce by
increasing the system size although the rate of change is different for the systems with odd and
even number of layers.
Figure 6 | Energy pathway during compression of 8L hydrogenated graphene at 0K
temperature. A small portion of graphene transforms to the diamond at =-6.5% (point A) during
loading and the diamond structure retained up to =-51.1% (point B); beyond that point, the
diamond structure starts vanishing where at =-55.1% (point C ) diamond completely vanishes.
The diamond structure starts to form again at =-47.5% (point D) during unloading and gets to
the maximum volume fraction at =-37.14% (point E). Upon unloading, the hydrogenated
multilayer graphene won’t come back to its original shape and the formed diamond thin film
remains. The color map of the inset atomistic structures is the same as in Fig. 5.
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The 2L and 3L hydrogenated graphene have the largest transformation stress among all other
multilayer graphene systems, Figure 7 (b). This can be explained by the dominant surface tension
effect in the 2L and 3L graphene system. The surface stress, , in solid interfaces can be
𝜎
𝑠𝑟
𝑖𝑗
calculated using the Shuttleworth equation 25, i.e.
𝜎
𝑠𝑟
𝑖𝑗
=
𝛾
𝛿
𝑖𝑗
+
∂𝛾
∂
𝜀
𝑖𝑗
,
(20)
where is the surface energy and is the Kronecker delta function. The first term generates a
𝛾
𝛿
𝑖𝑗
hydrostatic pressure and the second term is the structural part of the stress tensor indicating the
work for stretching of the surface. The surface tension basically hinders the formation of the out-
of-plane sp3 bonds. Thus, higher pressure is needed to overcome this barrier.
Figure 7 | Transformation strain and stress of the hydrogenated graphene as a function of the
number of layers at different temperatures. Transformation strain (a) and stress (b) are
calculated as a function of temperature for a different number of hydrogenated graphene layers.
It is shown that considering the calculation error, the transformation strain is independent of the
temperature.
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Comparing the compressive transformation stresses, , in Figure 7(b) and volume fraction of
σ
c
t
CD and HD in Table 2, indicates that is a strong function of the phase compositions in the
σ
c
t
transformed diamond thin films. This is consistent with our results for the pristine multilayer
graphene. For example, 3L has a temperature-independent transformation stress which is
consistent with the temperature-independent composition reported in Table 2. It should be
noted that the volume fraction of different diamond structures in Table 2 are evaluated by
considering the hydrogen atoms in addition to the carbon atoms when coordination numbers are
calculated. This is key for correct determination of the crystal structure and volume fraction of
each diamond phase. Therefore, although comparing the composition of different diamond
phases in a specific system could give us an idea about the variation of the transformation stress
across different temperatures, same conclusions are not possible by comparing variation of
compositions for different systems. This is because some of the hydrogen atoms are located in
CD or HD coordination in a system, which increases the volume fraction of these phases.
Although, this artificial increase in the volume fraction of CD and HD remains consistent for a
system of specified graphene layers across different temperatures, this will not be the case for
systems with different number of layers. Excluding the hydrogen atoms when we are calculating
the volume fraction of CD and HD did not change this conclusion. This is because atoms beyond
the first nearest neighbors must be considered to determine whether a carbon atom belongs to
CD or HD crystal structure.
Table 2. Percentage of Cubic Diamond (CD) and Hexagonal Diamond (HD) for different multilayer
hydrogenated graphene structures (2L to 8L) at the different temperatures.
Temperature
#Layers
0K
500K
800K
1000K
1200K
1500K
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CD
HD
CD
HD
CD
HD
CD
HD
CD
HD
CD
HD
2L
16.6*
0.0
16.8
14.8
11.6
8.5
8.7
4.4
6.1
3.8
5.5
2.7
3L
0.1
0.0
0.6
0.0
0.7
0.0
1.4
0.3
1.0
0.2
0.9
0.1
4L
5.7
5.6
2.3
2.2
0.9
0.8
0.7
0.3
0.9
0.4
0.9
0.3
5L
1.7
1.9
0.6
0.5
1.5
1.2
0.5
0.5
0.3
0.6
0.3
0.8
6L
2.9
2.4
5.5
0.9
5.1
0.1
5.7
0.2
5.8
0.1
3.6
0.0
7L
5.2
3.5
1.3
0.1
1.3
1.9
1.9
1.9
1.4
0.8
1.4
0.7
8L
5.0
2.9
2.4
0.6
2.2
0.0
2.7
0.0
2.2
0.0
2.9
0.0
Note: Data is taken at 10% nondiamond volume fraction. The volume fractions of the 1st and 2nd
nearest neighbors are not shown here. Thus, the sum of CD an HD volume fractions is smaller
than 10%.
* For 2L graphene at 0K the graphene to diamond transformation is of first order, where almost
the entire structure transforms to diamond. That is why the volume fraction of CD is greater
than 10% here.
The transformation strain for hydrogenated graphene structures is also drastically reduced
compared to the pristine multilayer graphene, Figure 7(a). For example, for the 3L graphene
𝜀
𝑡
has reduced by a factor of five for the hydrogenated graphene and by a factor of three for the
thicker 8L graphene. However, the transformation stress for the hydrogenated 3L graphene has
increased compared to the pristine graphene. This can be due to distortion of the graphitic
structure44,16 and the change in surface stresses upon hydrogenation. The 3L system requires
more stress to straighten up the graphene structure, but less strain to nucleates the diamond
formation.
For the case which formation of 50% is considered as the transformation criteria, the
transformation stress reduces by 20% for the 3L graphene and cut by a half for the 8L graphene
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structure (see Figure S2 in Supporting Information). The sp3 hybridization due to hydrogen
absorption facilitates the formation of interlayer bonds and transformation to diamond. These
results emphasis the key role of surface effects and passivation on the phase transformation
characteristics.
Conclusions
We have presented an atomistic study of phase transformation of multilayer graphene to
diamond using the molecular dynamics approach with reactive force fields. We investigated the
effect of surface passivation with hydrogen atoms on transformation stress and strains at
different temperatures as a function of thickness of graphene layers. We further explored the
thermodynamics and kinetics of this phase transformation and analyzed the composition of the
formed diamonds. Compressive stress required for the synthesis of diamond from multilayer
graphene was calculated and its dependence of temperature was investigated. The mechanism
of this phase transformation was explained and the relation between thermodynamic conditions
and morphology of the converted diamond structure was reported.
Simulations were performed for both hydrogenated and pristine multilayer graphene. The
phase transformation mechanism and conditions significantly vary due to the dominance of
surface effects. For the pristine graphene no chemical radical like -H (hydrogen) or -OH (Hydroxyl
group) has been used to facilitate the sp3 hybridization process. The required transformation
stress is higher for pristine multilayer graphene compared to the corresponding structures which
are passivated with hydrogen atoms. We revealed a drop in the transformation stress of up to
five times and a drop in the transformation strain of up to 50%. The transformation strain found
to be independent of temperature for both pristine and hydrogenated graphene structures.
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However, the transformation strain for the hydrogenated graphene shows a strong dependence
on the number of layers in contrast to the pristine graphene structures that show thickness
independence. Furthermore, the transformation stress was shown to have a strong dependence
on the composition of the final diamond structure. Thus, we can conclude that transformation
of multilayer graphene structures to diamond is a strain-controlled process. We further revealed
that the diamond film with passivated surfaces is metastable and remains even after unloading
the structure, while the clean surface diamond structures formed from pristine graphene
transform back to graphene upon unloading. The effect of surface tension has shown to play a
significant role in ultrathin three-layer graphene structures which results in significant increase
in the transformation stress. In summary, the results presented in this study enlighten the kinetic
and thermodynamic conditions necessary for the formation of ultra-thin films of diamond. They
can guide the synthesis of diamond films with tailored characteristics for application in various
industries such as optoelectronics, defense, and coating industries.
Supporting Information:
The Supporting Information is available free of charge on the ACS Publications website.
The results for using transformation of 50% volume fraction from graphene to diamond has been
presented in the supplemental information. Transformation strain and stress for the to the
diamond transformation at different temperature for pristine and hydrogenated graphene.
Percentage of Cubic Diamond (CD) and Hexagonal Diamond (HD) for different multilayer pristine
graphene structures (3L to 8L) and hydrogenated graphene structure (2L to 8L) at the different
temperatures.
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Author Information:
Corresponding Author: kzm5606@psu.edu; kmomeni@latech.edu
Funding Sources
This project is supported by Louisiana Tech University, and the National Science Foundation 2D
Crystal Consortium – Material Innovation Platform (2DCC-MIP) under NSF cooperative
agreement DMR-1539916. This project is also partly supported by Louisiana EPSCoR-OIA-
1541079 (NSF(2018)-CIMMSeed-18 and NSF(2018)-CIMMSeed-19) and LEQSF(2015-18)-
LaSPACE. Calculations are performed using Louisiana Optical Network Initiative (LONI).
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Figure 1 | Structure and configuration of the modeled system. (a) Bond length and configuration of
hydrogen and carbon for the hydrogenated graphene model; (b) Top view of ABA stacking, where C atom of
bottom layer positioned at the center of hexagonal adjacent top graphene layers; (c) Different stackings of
multilayer graphene structures, where the ABA stacking is considered for our studies; The structure of (d)
Cubic Diamond (CD) and (e) Hexagonal diamond (HD).
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Figure 2 | Three-layer and eight-layer pristine graphene phase transformation. The maximum percentage of
the diamond structure formed after compression (top row) and its reverse transformation to graphene upon
unloading (bottom row) are shown for three-layer graphene at (a) 0K and (b) 1500K. The corresponding
results for the eight-layer graphene for (c) 0K and (d) 1500K are shown.
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Figure 3| Energy pathway during loading and unloading of 8L pristine graphene at 0K temperature. The key
transition points and associated structures are shown. Normalized energy of the system is shown for loading
(solid black line) and unloading (dashed red lines). A small portion of graphene transforms to the diamond
at e = -18% (point A) during loading which will disappear later at e = -46% (point B). Upon further loading
the energy and volume fraction of diamond increase again and reaches a maximum at point ‘C’. At this
point, increasing compression although increase the energy of the system but results in the reduction of
diamond volume fraction. Unloading starts at point ‘D’ and at point ‘E’ all diamond will transform back to
graphene. The color map of the inset atomistic structures is the same as in Fig. 2.
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Figure 4 | Transformation strain and stress for the pristine graphene to the diamond transformation.
Transformation strain (a) and stress (b) are calculated as a function of temperature for a different number
of graphene layers. It is shown that considering the 2% error of the calculations, the transformation strain is
independent of the temperature and number of graphene layers. In contrast, the transformation stress
increases with temperature.
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Figure 5 | Two-layer and eight-layer hydrogenated graphene phase transformation. The maximum
percentage of the diamond structure formed after compression (above) and its reverse transformation to
graphene upon decompression (bottom) are shown for two-layers graphene at (a) 0K and (b) 1500K. The
corresponding results for the eight-layer graphene for (c) 0K and (d) 1500K are also shown.
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Figure 6 | Energy pathway during compression of 8L hydrogenated graphene at 0K temperature. A small
portion of graphene transforms to the diamond at =-6.5% (point A) during loading and the diamond
structure retained up to = 51.1% (point B); beyond that point, the diamond structure starts vanishing
where at = 55.1% (point C ) diamond completely vanishes. The diamond structure starts to form again at
=-47.5% (point D) during unloading and gets to the maximum volume fraction at =-37.14% (point E).
Upon unloading, the hydrogenated multilayer graphene won’t come back to its original shape and the
formed diamond thin film remains. The color map of the inset atomistic structures is the same as in Fig. 5.
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Figure 7 | Transformation strain and stress of the hydrogenated graphene as a function of the number of
layers at different temperatures. Transformation strain (a) and stress (b) are calculated as a function of
temperature for a different number of hydrogenated graphene layers. It is shown that considering the
calculation error, the transformation strain is independent of the temperature.
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