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Journal of Ceramic Processing Research. Vol. 7, No. 1, pp. 0~00 (2006)
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JOURNALOF
Ceramic
Processing Research
A novel delaunay simplex technique for the detection of crystalline nuclei in dense
packings of spheres
A.V. Anikeenkoa, M.L. Gavrilovab and N.N. Medvedeva
aInstitute of Chemical Kinetics and Combustion SB RAS, Novosibirsk, Russia
bDepartment of Computer Science, University of Calgary, Calgary, AB, Canada
The paper presents a new approach for revealing regions (nuclei) of crystalline structures in computer models of dense
packings of spherical atoms using the Voronoi-Delaunay method. A Delaunay simplex, comprised of four atoms, is the simplest
element of the structure. All atomic aggregates in an atomic structure consist of them. The shape of the Delaunay simplex and
the shape of its neighbors are used to determine whether the Delaunay simplex belongs to a given crystalline structure. The
characteristics of simplexes which define their relationship to FCC and HCP structures are studied. The possibility of using
this approach for investigation of other structures is demonstrated. In particular, polytetrahedral aggregates of atoms atypical
for crystals are discussed. The occurrence and growth of regions in FCC and HCP structures is studied on an example of
homogeneous nucleation of a Lennard-Jones liquid. The volume fraction of these structures in the model during the process
of crystallization is calculated.
Keywords: Voronoi diagram, Delaunay simplex, Crystallization, Homogeneous nucleation.
Introduction
Investigation of structural transformations in liquid,
amorphous and crystalline phases during crystallization,
ageing or relaxation processes is an important problem
of modern material science. A characteristic feature of
such processes is the structural heterogeneity, which
means that the sample may contain regions of different
structures, both crystalline and disordered. It is not an
easy task to investigate these structural features. While
the simulation of large computer models of atomic
systems is a rather routine problem, the analysis of
regions of different structures requires development of
special approaches. Recently, considerable progress in
this direction was achieved through the utilization of
the Voronoi-Delaunay method [1, 2]. An important
aspect of implementation of the method is based on the
Delaunay simplexes. The Delaunay simplex is describ-
ed by four atoms and represents the simplest three-
dimensional element (brick) of the structure. Any frag-
ment of the structure can be presented as a cluster of
Delaunay simplexes. Thus, one can determine regions
of the required structure by obtaining simplexes of a
given structural type [3-6]. The Delaunay simplexes
can be used for more precise identification of regions
of the given structure than methods based on Voronoi
polyhedra [7-9], spherical harmonics [10, 11], or the
distribution of angles between geometrical neighbours
[12, 13]. Voronoi polyhedra and spherical harmonics
characterize the nearest environment of the atom, i.e.
the structural unit that consists of large numbers of
atoms (15 on average). This does not present a problem
when relatively large heterogeneities are studied. How-
ever it is not suitable for studying small regions, such
as nuclei (embryos) of a new phase, since characteristic
features of structures are based on smaller groups of
atoms. For instance, a difference between face centered
cubic (FCC) and hexagonal close packing (HCP)
crystalline structures is observable on groups of 6
atoms, and the main parts of an icosahedron (fragments
with 5-fold symmetry) are detected on groups of 7
atoms [6].
The method based on using Delaunay simplexes
cannot be applied directly. First of all, we must point
out that an individual simplex, as a rule, does not
characterize the structure uniquely. For example, a
good tetrahedron can belong to both FCC and HCP
structures, and can also be found in an amorphous
phase. On the other hand, a crystalline structure is not
always represented by one particular type of simplex.
In practice, each crystalline structure consists of three
types of Delaunay simplexes of a different shape: tetra-
hedron, quartoctahedron (a quarter of an octahedron),
and, in a small proportion, simplexes close to a flat
square. All differences between the crystals are defined
by the mutual arrangement of the above types of
simplexes.
In our previous investigations related to the identi-
fication of simplexes of a given shape, we have developed
*Corresponding author:
Tel : 7-007-3832-33-28-54
Fax: 7-007-3832-34-23-50
E-mail: nikmed@ kinetics.nsc.ru
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A.V. Anikeenko, M.L. Gavrilova and N.N. Medvedev
a methodology based on measures of simplex forms T,
Q and K. These measures are defined as special vari-
ances (dispersions) of lengths of edges of the simplexes
[5, 6, 14]. To extract regions of a given structure, we
studied the arrangement of selected simplexes in the
Voronoi network of the model. Clusters of such
simplexes allow crystalline regions to be revealed as
well as specific non-crystalline aggregates of atoms.
However, the type of the structure can be established
only after a cluster of the simplexes is constructed.
Thus, cycles (rings) of tetrahedra and quartoctahedra
arranged in the form of a rhombus are typical for FCC
structure, and the trapezoid form is found in the HCP
structure [5, 6]. Such an analysis is rather qualitative,
as it demonstrates the presence of the above structures
in the model of a crystal. However, an open problem is
to determine the quantity of specific structures in a
given sample. To address this problem, one needs a
more specific, quantitative description of the simplexes
to a given structure type.
In this paper, we suggest a method to characterize the
relationship of the simplexes to a given structure con-
sidering the shape of both a given simplex and its
neighbors. We refer to this problem as a problem of
identification of a structural type of a Delaunay simplex.
In the case of neighboring simplexes we propose consi-
dering simplexes with adjacent faces. The structural
unit, which identifies a type of a given structure, is an
aggregate of eight atoms: four atoms of a given simplex
and four atoms of its faces. However, only the central
simplex is used for subsequent structural analysis.
The Voronoi-Delaunay Method
Methods for the using geometrical ideas of Voronoi
and Delaunay for the structural analysis of atomic
systems is discussed in detail in many articles (see, for
example, [3-7, 15]). A set of coordinates of all atoms
of the model {A} serves as the basic information for
structural analysis of atomic models. At the first step,
the Voronoi-Delaunay partitioning of the model studied
is calculated. Actually, for our analysis we deal only
with the Voronoi network, defined as a network of
edges and vertices of a set of Voronoi polyhedra. The
Voronoi network is represented by a set of coordinates
of vertexes {D} and a table of their connectivity {DD}.
Every vertex of the Voronoi network is incident on four
atoms of the system {A}, defining a Delaunay simplex.
This means that every vertex of the Voronoi network
determines the position of one of the Delaunay simplexes
of the system. Using coordinates of atoms of a given
simplex, one can calculate any geometrical characteri-
stics of it (in particular measures of its shape). Next,
using the connectivity of the Voronoi network, it is
convenient to study their mutual arrangement, and to
define clusters of simplexes with given structural
characteristics [5, 16].
Simplex Shape Measures
T, Q and K measures
A choice of simplex characteristics depends on the
problem being studied. In our case we study dense
packings of spherical atoms. The main configuration
for this study is the tetrahedral configuration of four
atoms. It is the densest local configuration and is a
preferable energy-wise configuration for spherical atoms.
In FCC and HCP crystals there are also octahedral
configurations, which together with tetrahedral configu-
rations ensure translation symmetry of the crystal. An
octahedral configuration is not a simplex, since it has
six vertexes. An ideal (perfect) octahedral configuration
provides an example of a degenerated configuration: all
six vertices lie on a sphere. However, in a computer
simulation of physical systems, atoms are typically
shifted from their ideal positions. Thus, every octa-
hedral configuration is represented through Delaunay
simplexes unambiguously. Usually, there appear four
similar simplexes (quartoctahedra [14]). A perfect
quartoctahedron has five equal edges, and the sixth
edge is 2 times longer than the five others.
However, at some specific displacements of atoms,
the perfect octahedron can be divided into five instead
of four simplexes. The fifth simplex springs up from
the flat configuration of four atoms of octahedron. This
simplex was found in the models of dense liquids and
was called a simplex Kize [17]. It has two opposite
edges (diagonals of a square) which are
longer then the other four edges. These simplexes are
rather rare, however they should also be taken into
account when studying crystalline structures.
To extract a good tetrahedral configurations we use
measure T, called tetrahedricity [6, 16]. It is the variance
of the lengths of edges of the simplex
times
.(1)
Here ei and ej are the lengths of the i-th and j-th edges,
and <e> is the mean edge length for a given simplex.
The number 15 used as normalization factor is the
number of possible pairs of six edges of the simplex.
For a perfect tetrahedron, the value of T is equal to
zero. A small value of T means unambiguously that the
simplex is close to a perfect tetrahedron.
For the unambiguous extraction of a good quartocta-
hedron, we use the special measure Q-quartoctahedri-
sity [6, 16]:
(2)
This measure is similar to the measure T, only now the
computation of the variance of edge lengths takes into
2
T=
i j ≠∑ eiej
–
()2/15 e 〈 〉2
Q=
i j <
i j ,
m
≠
∑ eiej
(
–
)2
i m
≠∑ eiem/ 2–
()
2
+
⎝⎠
⎜⎟
⎜⎟
⎛⎞
/15 e 〈 〉2
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A novel delaunay simplex technique for the detection of crystalline nuclei in dense packings of spheres
3
an account that one edge is
others. To compute Q, the longest edge m of a simplex
needs to be found first, and then the calculation is
carried out according to equation (2). It is obvious, that
for an almost perfect quartoctahedron, the value of the
measure Q approaches zero. The inverse is also true.
Note, the value of the measure T defined according to
equation (1) for simplexes of quartoctahedral shape is
equal to 0.050. However, the same value can corre-
spond to many simplexes of other shapes. Due to this
fact, one needs to introduce a special measure for every
different shape.
To extract Kizhe simplexes, a measure K was con-
structed following the same principle. Here we use the
fact that two edges are times longer than the others
[6, 17]:
times longer than the
(3)
The meaning of this expression is that when the value
K approaches zero, a simplex degenerates into a square.
To compute K, a pair of the longest opposite edges of
the simplex needs to be found first as the edges m and
n, and then the calculation is carried out according to
equation (3).
Note, the equations (2) and (3) differ from formulae
derived in [3, 6, 17]. In those references, every element
of the expression was normalized based on the number
of pairs of edges. In our work we introduce a common
normalization factor equal to 15, which is the total
number of different pairs. This method of normalization
is not crucial for selection of simplexes with a good
shape. However, below we introduce new characteristics
of simplexes, for which the unified equations for
measures of shape (1)-(3) are more convenient.
Calibration of the measures
In order to ascribe a Delaunay simplex to a given
shape, one should indicate boundary values of measures,
Tb, Qb u Kb, i.e. to make a calibration of the measures
(1)-(3). Following [16, 17], we calibrate our measures
with the help of a known structure, namely an FCC
crystal at a temperature below the melting point.
Calibration models were generated by the Monte Carlo
method, and consisted of 10000 Lennard-Jones atoms
in a cube with periodic boundary conditions. Initially
the atoms were settled on sites of a perfect FCC lattice,
and then the model was relaxed at a given low
temperature. Figure 1(a-c) demonstrate histograms for
T, Q and K, calculated for all Delaunay simplexes in
the model of a crystal at the temperatures of 0.32 and
0.48 (in reduced units). The peaks at small values of
these variables can definitely be related to the good
tetrahedra (a), quartoctahedra (b) and Kizhe simplexes
(c).
Boundary values which separate simplexes with an
appropriate shape from others can be chosen as locations
of minima on the histograms. We have assigned:
Tb, = 0.018, Qb = 0.013 Kb = 0.007. (4)
Thus, the simplexes having one of the measures of T, Q
or K less than in (4) are of interest to our analysis of
the structure. We will state that boundary values in (4)
determine “a full set” of simplexes typical for a crystal
structure. Indeed, all of them can be found in the model
2
2
K=
i j
<
i j ,
m n ,≠
∑
eiej
–
()2
i j ,
m n ,≠∑
eiem/ 2–
()
2
+
⎝
⎜
⎜
⎛
+
i m n ,
≠∑
eien/ 2–
()
2+ emen
(
–
)2
⎠
⎟
⎟
⎞
/14 e 〈 〉2
Fig. 1. Histograms of the distribution of the Delaunay
simplexes over different shapes for the models of an FCC
crystal: (a) tetrahedrisity T, (b) quartoctahedricity Q, (c)
measure for Kizhe simplex K, see formulae (1)-(3). Solid
lines for the model with a temperature T*=0.32, dashed
lines for T*=0.48. Arrows show boundary values for the
extraction of proper simplexes for crystalline phase.
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A.V. Anikeenko, M.L. Gavrilova and N.N. Medvedev
which is, from a physical point of view, a good crystal.
Decreasing the boundary values gives us, obviously,
higher-grade simplexes, but an excess of them intro-
duces simplexes whose shape could not agree with the
crystal structure.
Note, that variations of the boundary values (4)
within 10-20% does not influence the results obtained,
where we determine not only the shape, but also the
environment of the Delaunay simplexes (see below).
Structural types of the Delaunay simplexes
Crystal types
The essence of our approach to ascribe a structure to
a Delaunay simplex is a consideration of the environ-
ment of the simplexes together with their shape. In this
study, the simplest step is realized in this direction: as
an environment of the Delaunay simplex, only its
neighbors on adjacent faces are taken into account.
In the FCC configuration every tetrahedral configu-
ration is adjacent over faces only to octahedra, and
every octahedral configuration is adjacent only to the
tetrahedra. In terms of the Delaunay simplexes, this
represents a possibility of the following combinations
of neighboring simplexes for a simplex of a given
shape:
(I) T: Q Q Q Q
(II) Q: T T Q Q
(III) Q: T T Q K
(IV) K: Q Q Q Q.
(5)
Thus, a tetrahedral simplex may be adjacent to four
quartoctahedra (I). Quartoctahedron can be adjacent to
two tetrahedra and two quartoctahedra (II) or to two
tetrahedra, one quartoctahedron and one simplex Kizhe
(III). Simplex Kizhe can be adjacent only to four quart-
octahedra (IV).
Based on the above, a given Delaunay simplex can
identify the FCC structure if it satisfies one of the
above conditions (5). Situations (III) and (IV) arise in
the case when an octahedral configuration is divided
into Kizhe simplexes (see above).
The HCP structure has pairs of adjacent tetrahedra
(trigonal bipyramids), and the octahedra are organized
in chains in which they are also adjacent by faces.
Thus, we can formalize the neighborhoods of the
Delaunay simplexes:
(I) T: T Q Q Q
(II) Q: T Q Q Q
(III) Q: T Q Q K
(IV) Q: T T Q Q
(V) Q: T T Q K
(VI) K: Q Q Q Q
(6)
Note, combinations (I)-(III) are new ones, but (IV)-
(VI) are the same as for FCC (5). The similarity noted
of some combinations is not surprising due to the
inherent genetic proximity of the densest crystalline
structures. This also means that dissection of crystalline
simplexes between FCC and HCP is not an unambigu-
ous question in the general case. Further classification
of such “disputed” Delaunay simplexes requires addi-
tional considerations. The number of such questionable
Delaunay simplexes can be decreased by further ana-
lysis of the model. Indeed, if for instance the disputed
quartoctahedron (Q: T T Q Q) is adjacent to tetrahedra,
which belong to an FCC type, then it also can be
classified as of FCC type. Therefore, after determination
of the neighboring simplexes, we ascribe additional
simplexes to crystalline types: if a disputed simplex is
adjacent to an FCC type simplex (and does not have an
HCP type) then we assign it to the FCC type. Analog-
ously, a disputed simplex neighboring an HCP type and
not of an FCC type is classified as HCP type. If a
disputed simplex is adjacent both to FCC and HCP
types, we keep it as disputed. Such cases take place at
the regions bordering FCC and HCP structures. We
keep simplexes as disputed also is they do not have
neighbors of either FCC or HCP types. This case happens
for small aggregates of simplexes with good crystalline
shape in a disordered phase. Note, the residuary disputed
simplexes, nevertheless, represent regions of crystalline
structure.
Non-crystalline types
A proposed ideology to select Delaunay simplexes
related to FCC and HCP may be extended to other
structures, in particular, to non-crystalline ones. It is
known that a dense amorphous phase contains aggre-
gates of good tetrahedra adjacent by faces (polytetra-
hedral aggregates). Such an arrangement of more than
two tetrahedra is extraneous for crystals, since they are
incompatible with translational symmetry. For studying
such aggregates, one should extract Delaunay simplexes
with a good tetrahedral shape having also at least two
good tetrahedra in its neighborhood:
T: T T * *.(7)
The other pair of neighboring simplexes can have, in a
general case, an arbitrary shape. Polytetrahedral clusters,
and particularly, five-membered rings of tetrahera (pen-
tagonal bipyramids), are identified by these simplexes.
Recently [6, 9, 18], a significant number of penta-
gonal prisms was detected in models of dense packings
of hard spheres and in frozen Lennard-Jones liquids.
The existence of such configurations is not trivial.
They are not crystalline but also are unnatural for an
amorphous phase. In this paper, we suggest a method
to study them with the help of Delaunay simplexes of
the following structure type:
T: T T Q Q,(8)
i.e. a good tetrahedron, with two tetrahedra and two
quartoctahedra in their neighborhoods.
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A novel delaunay simplex technique for the detection of crystalline nuclei in dense packings of spheres
5
The Model
We studied a process of development of a crystalline
phase on a model of a rapidly cooled Lennard Jones
liquid. The model was generated by the Monte Carlo
method in an NPT (isobaric-isothermal) ensemble (It
contained 10000 atoms in a cube with periodic boun-
dary conditions. The initial configuration corresponded
to the liquid phase at a temperature T*=0.8 and density
ρ*=0.73. At every 500-th Monte Carlo step, the temper-
ature was decreased by ΔT*=0.00075. (500 steps are
enough for energy relaxation of the model). Cooling
was continued to zero temperature. The pressure was
kept constant and equal to zero. During the process, the
density increased to ρ*=1.037. The total number of
Monte Carlo steps was 533000. The result of such
gradual cooling, was a halfway crystallization happen-
ed. For full crystallization a slower cooling process
would be required.
Results
Figure 2 shows the volume fraction for different
structural components of our model as a function of the
temperature. Three upper curves belong to the crystal-
line phase. For extraction of the structural types, we
used simplexes with shapes corresponding to a full set
of crystalline shapes (4). First of all, we see that the
HCP structure is predominant. It arises before the FCC
structure appears and exists up to complete freezing of
the sample. This structure represents 30% of the model
volume. The total volume of the crystalline phase is
more then 60%, where 11% belongs to the disputed
simplexes representing cases bordering between the
FCC and HCP regions.
The two lowest curves belong to structural types not
typical for crystals (see formulas (7) and (8)). The
lowest curve shows the volume fraction occupied by
polytetrahedral simplexes (7), except for the volume of
simplexes that correspond to pentagonal prisms (8).
One can also note that the largest volume occupied by
non-crystalline simplexes is observed at the beginning
of crystallization. This fact corresponds to the sugges-
tion [6] that the existing polytetrahedral aggregates in
liquids (together with embryos of pentagonal prisms)
may initiate the appearance of crystalline nuclei.
Simplexes of pentagonal prism type represent 3% of
the volume when the model is completely frozen,
which is more then the other polytetrahedral simplexes.
They consist of central nuclei of five-fold twins of the
FCC structure. However, they can also be found in
other aggregates without 5-fold symmetry.
Approximately 30% of the model volume is not
related to the structural types mentioned above, and
represent a disordered structure in the model.
We calculate the volume of a phase as the sum of
volumes of the Delaunay simplexes of the correspond-
ing structural type. Note, that during the volume calcu-
lation we take into account all simplexes of a given
type, single as well as in clusters. Of course, from a
physical point of view, a separated simplex (four
atoms) does not present a region of a given structure.
Larger aggregates are usually treated as elements of an
individual phase. However we do not discuss the size
of aggregates in this paper. Our aim is to extract local
configurations of a specific structural type. Detailed
analysis of the structure, growth and stability of nuclei,
are important topics for further investigation.
Figure 3 demonstrates the fractions of FCC and HCP
structures in our model extracted with various criteria
for the shape of the Delaunay simplexes. A pair of
curves marked by the value 2 corresponds to boundary
measures T, Q and K which are twice as small as the
full set of crystalline simplexes (4). The pair of curves
marked by the value 3 corresponds to boundary
measures T, Q and K which are four times smaller. The
curves from Fig. 2 for the full set of forms are also
present (pair 1) for comparison. For more perfect
shapes of simplexes, the volumes of extracted phases
are obviously less. A nontrivial result that we obtained
here is the fact that the ratio between FCC and HCP
phases is changed. The pair of curves 1 demonstrate
that HCP is identified significantly earlier then FCC.
For the pair of curves 2 the HCP and FCC structures
practically coincide over the whole interval from the
moment when the crystalline phase originates to com-
plete freezing. For the pair of curves 3 the structure
FCC becomes predominant.
These results mean that the regions of the FCC
structure are comprised of the more perfect simplexes
than some aggregates of the HCP structure. This result
can shed light on the fact that HCP nuclei are always
present at the beginning stage of crystallization despite
the FCC structure being more predominant in systems
Fig. 2. Volume fraction occupied by Delaunay simplexes
that belong to different structural types as a function of
temperature of the model.
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A.V. Anikeenko, M.L. Gavrilova and N.N. Medvedev
of spherical atoms. An interpretation of this is usually
based on the suggestion that the appearance of FCC
and HCP is equally probable at the first stage of
crystallization. However, our analysis demonstrates that
HCP is even more likely than FCC. This happens
because the simplexes of HCP type are less affected by
the quality of their shape then FCC simplexes, Fig. 3.
Some polytetrahedral nuclei (aggregates of adjacent
tetrahedral) may also correspond to an HCP structure,
because HCP, in contrast to FCC, has pairs of terahedra
adjacent by face. The detailed analysis of the spatial
distribution of Delaunay simplexes of the HCP type
demonstrates that at the first stage they exist as small
nuclei uniformly distributed through the model.
Conclusions
A novel method for the extraction of crystalline
nuclei for FCC and HCP structures using the shape and
mutual arrangementsof the Delaunay simplexes is pro-
posed. Structural elements corresponding for each of
the defined structural types are identified and describ-
ed. The basis of such a structural element is a Delaunay
simplex of a shape similar to one of three characteristic
forms of dense crystalline structures: a tetrahedton, a
quarter of an octahedron (quartoctahedron), and a flat
square. The structural type of the simplex is determined
by the shape of neighboring simplexes, adjacent by
faces. Clusters of simplexes of a given structural type
represent the nuclei of a corresponding structure. This
approach can be also applied for extraction of other
specific structures, in particular, for polytertahedral
aggregates typical for an amorphous phase, and
pentagonal prisms.
The structure of a Lennard-Jones liquid in the
process of cooling has been studied as part of this
research. The non-trivial result is that nuclei of the
HCP structure appear earlier than FCC nuclei. This is
justified by the fact that elements of HCP can be
generated from simplexes whose shape is less perfect
than the one needed to form elements of FCC.
Acknowledgements
The research is supported by the Grant CRDF No-
008-X1, the RFFI 05-03-32647 Grant, the OTKA
Grant, the Canadian Foundation for Innovation Grant
and the NSERC Grant.
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Fig. 3. Volume fraction of FCC and HCP structures for
various criteria of shape quality of the Delaunay
simplexes. 1) T < 0.018, Q < 0.013, K < 0.007. 2) T <
0.009, Q < 0.0065, K < 0.0035. 3) T < 0.0045, Q <
0.00325, K < 0.00175.
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