Structure, elastic properties and strength of amorphous and nanocomposite carbon
ABSTRACT We study theoretically the equilibrium structure, as well as the response under external load, of characteristic carbon-based materials. The materials considered include diamond, amorphous carbon (a-C), ``amorphous diamond'' and nanocomposite amorphous carbon (na-C). A universal bulk-modulus versus density curve is obeyed by all structures we consider. We calculate the dependence of elastic constants on the density. The strength of a-C was found to increase in roughly a linear manner, with increasing concentration of four-fold atoms, with the maximum stress of the strongest a-C sample being about half that of diamond. The response of na-C to external load is essentially identical to the response of the embedding a-C matrix.
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ABSTRACT: We present atomistic simulations for the mechanical response of ultra nanocrystalline diamond, a polycrystalline form of diamond with grain diameters of the order of a few nm. We consider fully three-dimensional model structures, having several grains of random sizes and orientations, and employ state-of-the-art Monte Carlo simulations. We calculate structural properties, elastic constants and the hardness of the material; our results compare well with experimental observations for this material. Moreover, we verify that this material becomes softer at small grain sizes, in analogy to the observed reversal of the Hall-Petch effect in various nanocrystalline metals. The effect is attributed to the large concentration of grain boundary atoms at smaller grain sizes. Our analysis yields scaling relations for the elastic constants as a function of the average grain size. Comment: Proceedings of the IUTAM Symposium on Modelling Nanomaterials and Nanosystems, Aalborg, Denmark, May 19-22 2008; to be published in the IUTAM Bookseries by Springer07/2008;
arXiv:cond-mat/0611368v1 [cond-mat.mtrl-sci] 14 Nov 2006
Ioannis N. Remediakisa,∗Maria G. Fytaa,1Christos MathioudakisbGeorgios Kopidakisb
Pantelis C. Keliresa,c
aDepartment of Physics, University Of Crete, P.O. Box 2208, 71003 Heraklion, Crete, Greece
bDepartment of Materials Science and Technology, University of Crete, P.O. Box 2208, 71003 Heraklion, Crete, Greece
cIESL, Foundation for Research and Technology-Hellas (FORTH), 71003 Heraklion, Crete, Greece
We study theoretically the equilibrium structure, as well as the response under external load, of characteristic carbon-based mate-
rials. The materials considered include diamond, amorphous carbon (a-C), “amorphous diamond” and nanocomposite amorphous
carbon (na-C). A universal bulk-modulus versus density curve is obeyed by all structures we consider. We calculate the depen-
dence of elastic constants on the density. The strength of a-C was found to increase in roughly a linear manner, with increasing
concentration of four-fold atoms, with the maximum stress of the strongest a-C sample being about half that of diamond. The
response of na-C to external load is essentially identical to the response of the embedding a-C matrix.
Amorphous carbon, nanocomposite carbon, mechanical properties, fracture.
PACS: 81.07.Bc, 62.25.+g, 81.05.Gc.
Amorphous carbon (a-C) is a well-established carbon-
based material, with distinct mechanical properties such
as hardness and thermal stability. These properties render
a-C to be an ideal material for coating applications. Re-
cently, nanocomposite carbon (na-C) has been synthesized
in both hydrogenated and pure forms . It is considered
as a material with similar, and perhaps better properties
than those of a-C. Na-C consists of nanometer-scaleregions
of crystalline material embedded into an a-C matrix. This
material has mechanical properties between those of a-C
and diamond , that could in principle be fine-tuned by
adjusting either the size of the nanocrystalline region or
the density of the amorphous matrix. Such tailoring of ma-
∗Corresponding author. Present address: Department of Materials
Science and Technology, University of Crete, P.O. Box 2208, 71003
Heraklion, Crete, Greece.
Email address: firstname.lastname@example.org (Ioannis N. Remediakis).
1Present address: Department of Physics, Harvard University, Cam-
bridge, MA 02138, USA
terial properties to match those desired in the applications
could be invaluable.
The properties of a-C and na-C can be categorized into
two broad classes: The first class includes macroscopic,
mean-field properties, such as the density, that are implic-
itly only related to individual chemical bonds. Atomistic
properties,on the other hand, such as the strength, arisedi-
rectly from the individual chemical bonds between C atoms
of different electronic structure. The macroscopic proper-
ties are usually related to the minimum or the harmonic
regimeof the cohesiveenergyversusvolume curve,and can,
tion for the bonds. On the other hand, even a rough esti-
mate of atomistic properties requires employing some level
of quantum theory.
A quantum-mechanical treatment is able to accurately
describe the chemical bond, but can only be used to study
few hundreds of atoms for very short time scales. On the
other hand, a classical empirical potential method can han-
dle three orders of magnitude more atoms for realistic time
scales, at the price of a limited accuracy in the description
Preprint submitted to Diamond and Related Materials 6 February 2008
of the chemistry of carbon. This interplay between clas-
sical and quantum-mechanical simulations has been used
to verify or predict several experimental findings for na-C:
Diamond nanocrystallites were found to be stable, having
negative formation energy, when the amorphous matrix is
dense . The averageintrinsic stress of the material is zero
. Although the bulk modulus of na-C is higher than that
identical fracture mechanisms . In this work, we first es-
tablish the validity of our simulations by ensuring that dif-
ferent approximations yield identical qualitative and simi-
lar quantitative results. We then study in more detail the
elastic properties of these materials, including the depen-
dence of elastic moduli as a function of their density. Fi-
nally, we comment on their behavior under strain well be-
yond the elastic regime, and discuss briefly their fracture.
2. Computational Method
The quantum-mechanical calculations are based on two
different tight-binding hamiltonians. The so-called NRL-
TB was developed by Papaconstantopoulos, Mehl and
co-workers at the Naval Research Laboratory . The
assumptions to the previously published parametrization
for Si; see Ref.  for a review. The environment-dependent
tight-binding (EDTB) model of Wang, Ho and co-workers
 goes beyond the traditional two-center approximation
and allows the TB parameters to change according to the
bonding environment. In this respect, it is a considerable
improvement over the previous two-center model of Xu
et al. . Both NRL-TB and EDTB schemes have been
used successfully to simulate a-C systems [9,2]. The tight-
binding molecular-dynamics simulations simulations are
carried out in the canonical (N, V, T) ensemble, T being
controlled via a stochastic temperature control algorithm.
The supercells used in the tight-binding simulations con-
tain 512 C atoms each.
The empirical potential simulations are based on the
continuous-space Monte Carlo method. We employ the
many-body potential of Tersoff , which provides a very
good description of structure and energetics for a wide
range of carbon-based materials [11,12]. This method al-
lows for great statistical accuracy, as it is possible to have
samples at full thermodynamic equilibrium. Moreover,
through the use of relatively large supercells, it offers the
possibility to explore a larger portion of the configurational
space of the problem. The supercells used in the Monte
Carlo simulations contain 4096 C atoms each.
Na-C is modeled by a periodic repetition of cubic su-
percells that consist of a spherical crystalline region sur-
rounded by a-C. To construct such supercells, we first con-
sider a cubic supercell of diamond, and choose the radius
of the spherical crystalline phase. Keeping the atoms in-
side this sphere frozen to their diamond lattice positions,
we run the system at a veryhigh temperature (12000-15000
K) so that a liquid is created, and then quench it down to
50 K, at constant volume. After that, the system is fully
equilibrated by relaxing both the system volume and the
coordinates of all atoms. In the Monte Carlo simulations,
we perform an additional intermediate relaxation at room
temperature to ensure that the sample is fully relaxed.
By adjusting the volume or, equivalently, the pressure
of the liquid phase, we can create samples having low- or
high-density a-C. It has been shown that properties of a-
C can be described in terms of a single parameter, z .
This is the average coordination number, or number of
neighbors, for each atom in the sample. More precisely, z
it is the integral of the nearest-neighbor peak of the pair-
correlation function. By convention, individual atoms in
a-C are thought of having an electronic structure in one-
to-one correspondence to their coordination number: four-,
three- and two-fold atoms are usually labeled in the liter-
ature as sp3, sp2and sp1, respectively. Although for most
atoms such a relationship between coordination number
and hybridization holds to a good approximation, other
atomic electronic structures may be present as well. We
create samples with average coordination between 3.1 and
3.9, having concentrations of four-fold coordinated atoms
between 10 and 90%. Samples with z ? 3.8 are consid-
ered as tetrahedralamorphousCarbon (ta-C). We also con-
sider the fully tetrahedral Wooten-Winer-Weaire (WWW)
structure , as a model for “amorphous diamond”. The
radius of the nanocrystalline region is of the order of 2 nm
(1 nm for the quantum-mechanical simulations), while it
occupies about 30-40% of the total volume, in accordance
to experimental observations .
3. Equilibrium structural properties
and a-C are summarized in Table 1. For diamond, we calcu-
late the density (ρ) and the bulk modulus (B) using the two
tight-binding schemes and the Tersoff potential. All meth-
ods yield results in good agreement to experiment. The a-C
atoms, while two-fold atomsappear in low-density samples.
Using the EDTB model, we had found that the density of
a-C depends linearly on z, via ρ ≈ −3.3+1.7z . We find
that the linear relationship found using EDTB is also valid
within NRL-TB, at least for the denser samples.
Insight into the atomistic structure of a-C can be gained
by looking at the sizes and distributions of rings of atoms
in the material. Rings of atoms are defined in terms of the
shortest-path criterion of Franzblau . The number of
atoms that participate in a ring is the ring size, and can
have only specific values for a bulk crystalline material:
in diamond, for example, the smallest ring size is six. On
Structural properties of diamond (D), “amorphous diamond”
(WWW) and four characteristic a-C samples (A-D). All samples are
simulated using NRL-TB, except D (E), D (T) and D (e) that contain
results obtained from the EDTB, Tersoff potential and experiment,
respectively. N4, N3 and N2 is the percentage of four-, three- and
two-fold atoms, respectively. B is the bulk modulus (in GPa), ρ the
calculated density (in gr/cm3), and ρfitis the density (in gr/cm3)
according to the fit of ρ vs. z from Ref. .
D 4.010000 4803.7
D (E) 4.010000 4283.5
D (T)4.0 10000 422 3.5
D (e)4.0 10000 443 3.5
WWW 4.010000 434 3.4 3.5
A 3.878 220 3873.1 3.2
B 3.767 321 3612.93.0
C 3.547 521 325 2.52.7
D 3.2 18775 2111.6 2.1
the contrary, three-member rigs are common in a-C. As
the formation of rings is related to atomistic binding, a
quantum-mechanical treatment is necessary in order to get
reliable ring statistics.
Results for the relative ring numbers and concentrations,
found using the NRL-TB for three characteristic samples,
are shown in Fig. 1. Similar results have been obtained
with EDTB . For the low-density sample, we find signif-
icant numbers for three- and four- member rings, while the
most probable ring length is six. For ta-C, the most proba-
ble ring length is five, while there are still some three- and
four- member rings. Finally, for the “amorphous diamond”
sample, most rings are six-membered, with few five- and
seven-member ones. Although, by construction, only four-
fold atoms exist in this sample, its random topology allows
for rings that would not be present in the crystalline ma-
The bottom panel of Fig. 1 shows the composition of
rings as a function of their size, for the ta-C sample. Most
atoms (over 90%) of the smallest rings are four-fold. The
concentration of four-fold atoms decreases with increasing
ring size; for large rings, with more that eight members,
most atoms are three-fold. This behavior can be attributed
to the long-range correlations found for π-bonded atoms;
such π bonds are more likely to occur between three- than
between four-fold atoms. On the other hand, such double
bonds are more difficult to bend in order to form trian-
gles or quadrangles; this is why most atoms that partici-
pate in small rings are four-fold. 85 % of the three-member
rings and 75 % of the four-member rings consist of four-fold
atoms only, while there are no rings of sizes 8 and above
having only four-fold atoms. The ring statistics presented
here are in excellent agreement to experiments and state-
rings per atom
rings per atom
rings per atom
Fig. 1. Ring statistics for various a-C samples, as calculated using
the NRL-TB. The upper three panels show the number of rings
divided by the total number of atoms in the sample, as a function
of the number of atoms participating in each ring, for three cases:
low-density a-C (z=3.18), ta-C (z=3.78) and “amorphous diamond”
(z=4.00). Lowest panel shows the concentration of rings in four-fold
and three-fold atoms for the ta-C sample.
of-the-art ab initio simulations [16,17].
A typical sample of na-C, generated in the NRL-TB
scheme is shown in Fig. 2. The volume fraction of the
nanocrystalline region is 31%. The surrounding amorphous
material is ta-C (z=3.8 and ρ=3.0 gr/cm3). A sample cre-
ated using EDTB under identical conditions has the same
crystal volume fraction and a-C coordination, and only
slightly higher density of the amorphous phase, 3.1 gr/cm3.
In all cases, the surrounding a-C matrix obeys the same
density vs. coordination relationship as pure a-C samples
generated using the same recipe. The a-C atoms are cova-
lently bonded to the crystal, resulting thus in thin interface
As a first estimate of the hardness of nanocomposite
nd/a-C structures, we calculated their bulk modulus, B. In
Fig. 3, we plot B as a function of the total density of the
Fig. 2. Cross-section of a ball-and-stick representation for a typical
model structure of nanocomposite amorphous Carbon created using
EDTB: a spherical nanocrystal with a diameter of 1.24 nm is embed-
ded into amorphous-C with an average coordination of 3.8. Atoms
belonging to the crystal are represented by black spheres; four-and
three-fold atoms of the amorphous matrix are represented by white
and gray spheres, respectively.
na-C, d=1.05 nm
na-C, d=1.13 nm
na-C, d=1.24 nm
Fig. 3. Bulk modulus as a function of density, calculated within NR-
L-TB, for na-C with different radii of the crystalline region. Calcu-
lations for a-C of various densities, bulk diamond and “amorphous
diamond” are also shown.
samples, ρ. The bulk modulus of na-C is enhanced com-
pared to that of pure a-C . Replacement of some amor-
phous material by crystalline increases noticeably the bulk
modulus, rendering it for some samples to be higher even
than that of the “amorphous diamond”, and close to that
of diamond .
Interestingly, all samples, including pure a-C, na-C,
WWW, and even diamond, seem to follow the same univer-
salcurvein Fig.3.Such universalitieshavebeen observedin
the past: He and Thorpe  showed that B ∼ (z −z0)1.5,
where z0 is universal; Liu et al.  showed that B ∼ d−3.5,
where d is the average distance between atoms. Recently,
Mathioudakis et al.  showed that the two approaches are
equivalent, and that for a-C, B ∼ (d−const.)−3.5. The last
relationship would imply that B ∼ (ρ−1/3− const.)−3.5.
Indeed, this function fits very well the data of Fig. 3.
The universal dependence of B on ρ can be understood
by considering the microscopic response of the material to
the external pressure. We can think of two regimes: in low-
density materials, the pressure is undertaken by appropri-
ately adjusting the volume of the void regions of the mate-
rial. Such voids exist in every low-density material and are
a result of induced dipole (van der Waals) interactions. On
the other hand, for dense materials, strong covalent bonds
have to be deformed, resulting in higher bulk moduli. In
this case, the resistance of the electrons to compression fol-
lows from their quantum nature and the Pauli principle.
In the first case, the scaling of B with respect to ρ can
be found by considering a model solid bonded exclusively
through van der Waals interactions. Using the Lennard-
Jones potential we can find that B ∼ ρ . On the other
hand, for high densities, we can get the correct scaling by
using the free-electron approximation: in this case, B ∼
ρ5/3. This picture is demonstrated in Fig. 3: B ∼ ρ for
ρ ? 3.3 gr/cm3, while B ∼ ρ5/3for ρ ? 3.3 gr/cm3. These
relationships hold with surprisingly good accuracy.
Being such a fundamental average property, the B vs. ρ
curve should be reproduced well by both our tight-binding
models and the empirical potential, as the latter is known
to reproduce correctly the elastic response of the material.
On the other hand, the density dependence of elastic con-
stants associated with changes in shape, like the Young’s
modulus, can be different. Although the responseof materi-
als to volume changes can be addressed at almost any level
of theory, understanding the response to shape changes re-
quires a model that takes into account the directionality of
the chemical bonds. Fortunately, as the calculation of elas-
tic constants require only small deformations from the min-
imum energy structure, the empirical potential approach
should suffice for their calculation. As the Monte Carlosim-
ulations offer greater statistical accuracy and refer to more
realistic sizes of the crystalline regions, we prefer to employ
this method for the calculation of elastic constants.
To calculate the elastic constants, we apply the appro-
2.62.8 3.03.2 3.4
elastic constants (GPa)
Fig. 4. Elastic constants of na-C, calculated using the Tersoff po-
tential, as a function of the total density. The average diameter of
the crystalline region is 1.7 nm, and the surrounding a-C matrix has
z=3.8. The rightmost point for each data set, for a density ρ = 3.51
gr/cm3, corresponds to diamond and is taken from Ref. .
priate deformation to the system and compute its total en-
ergy as a function of the imposed strain. The curvature of
this function at its minimum yields the desired modulus.
The number of independent elastic constants depends on
the symmetry of the material: For a material with cubic
symmetry, there are three independent elastic constants,
while for an isotropic material, such as a-C or na-C, there
are only two . In Fig. 4 we plot c11, c12, c44 and the
Young’s modulus Y as a function of density for na-C and
diamond.All elasticconstantsincreasewith increasingden-
sity, similarto the previouslydescribed behaviorofthe bulk
modulus as a function of density. For an isotropic material,
2c44= c11−c12. This relationship holds within 4% or less,
for all data points presented in Fig. 4, demonstrating that
na-C is a highly isotropic material. The moduli c11and c44
are both associated with changes in shape, and this is why
their values for na-C are much lower than the correspond-
ing values for diamond, where strong directional bonds are
bent. On the other hand, c12describes simultaneous elon-
gation along two axes without shearing, and the value of
this modulus for diamond follows the trend observed for
4. Ideal Strength and fracture
The response of covalently-bonded materials, such as a-
C and na-C, under strain can be categorized into three
broad regimes: For small strains, the response of the mate-
rial is elastic, and Hooke’s law stands: the stress is propor-
tional to the applied strain. For example, if tensile strain
is applied to an isotropic material, the stress will equal the
strain times the Young’s modulus. The second regime cor-
0.000.05 0.10 0.15
Stress σ (GPa)
Fig. 5. Stress vs. strain curves for WWW model of “amorphous dia-
mond” (diamonds), ta-C (squares), low-density a-C (triangles) and
na-C (circles). The latter consists crystalline regions of 1.2 nm sur-
rounded by ta-C. Data shown are obtained by the NRL-TB method
for tensile load in the (111) direction of the crystal.
responds to strain beyond the elastic limit, and is usually
associated with plastic deformation of the material. The
stressexperienced by the materialincreaseswith increasing
strain until a maximum stress (strength) is reached. The
third regime is associated with strain beyond that giving
the maximum stress. For brittle materials like diamond,
the material breaks when the maximum stress is reached,
and further increase of the strain results therefore in zero
stress. Ductile materials, on the other hand, can be de-
formed beyond the strain corresponding to the maximum
of the stress.
To study the strength of a-C and na-C, we apply tensile
load on the  easy slip plane of the crystalline region.
Strain is simply the ratio of the volume change divided
by the initial volume of the sample; stress is the negative
three a-C samples: a typical a-C sample with average coor-
dination z=3.47, a ta-C sample with z=3.8 and an “amor-
phous diamond” sample. As graphite (z=3) is known to be
much softer than diamond (z=4), it is reasonable to expect
that the strength increases with increasing z, as was the
case for the elastic constants discussed in Section 3. This
is observed in Fig. 5. The maximum stresses are roughly
60, 40 and 30 GPa for z=4.00, 3.78 and 3.50, respectively,
so that the strength of a-C is roughly proportional to its
concentration of four-fold atoms. Interestingly, the stress
versus strain curve for the low-density a-C sample seems to
suggest a ductile behavior. We get similar results when ap-
plying shear strain. As a-C is highly isotropic, the energy
required to deform the material is a function of the change
in its volume and does not depend much on how this change
The strength of diamond does not fit into the simple
picture of the strength being proportional to the concen-
tration of four-fold atoms. If that was the case, then the
strength of the isotropic WWW sample under tensile load
would be higher than the strength of diamond under ten-
sile load perpendicular to its easy slip plane, (111). After
all, in the WWW model the number of bonds per unit area
for a given direction has to be higher than the number of
bonds per unit area on the  plane of diamond. This
justifies the name “easy slip plane” for the diamond (111).
One could naively expect that the strength of the WWW
sample, consisting of four-fold atoms only, could perhaps
be higher than that of diamond, due to the lack of such easy
slip planes. On the contrary, the calculation reveals that
the strength of diamond for tensile load along its easy slip
direction, (111), is about twice that of the isotropicWWW.
Apparently, the lack of easy slip planes in WWW is com-
pensated by its somehow distorted bonds and the lack of
order beyond the first-nearest-neighbor distance.
We applied the same methodology to a na-C sample
where crystalline regions having radii of about 1.2 nm are
embedded in ta-C with a volume fraction of about 30%.
The stress-strain curve for this sample follows exactly that
of the embedding ta-C. The crystalline phase remains un-
affected by the external load, which is almost completely
response of na-C to external load beyond the elastic regime
is identical to the response of the embedding matrix. As
atoms in the amorphous matrix form bonds that are always
weaker than the bonds in the crystal, the system prefers to
stretch or bend these bonds and keep the strong diamond
bonds untouched. By performing an atom-by-atom analy-
sis of the deformation, we can probe the four-fold atoms
of the amorphous atoms as the ones more extensively de-
formed when the material experiences large load .
We examined theoretically the structure, elastic and in-
elastic response to load of several carbon-based materials,
including diamond, amorphous carbon (a-C) “amorphous
diamond” (WWW) and nanocomposite amorphous carbon
(na-C). These materials are formed by covalently bonded
four-fold and three-fold atoms and are characterized by
their averagecoordination number (z). In a-C, three-, four-
and five-member rings are formed, their number increasing
atoms, while largerrings contain also three-fold atoms. The
bulk modulus of all these carbon-based materials seems to
follow a universal functional dependence of the density. All
elastic constants were also found to increase with increas-
The strength of a-C was found to increase in roughly a
linear manner, with increasing concentration of four-fold
atoms. High-density sample exhibited a brittle behavior,
analogousto that of diamond. The strongest a-C sample we
considered was the “amorphous diamond” WWW sample;
this has a maximum stress about half that of diamond. The
response of na-C to external load is essentially identical to
the response of the embedding a-C matrix.
This work is supported by the Ministry of National Ed-
ucation and Religious Affairs of Greece through the action
“EΠEAEK” (program “ΠΥΘAΓOPAΣ”.)
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