Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals
An attempt is made to explain the appearance of certain phases having different orientational and spatial structures in the phase diagram of crystals formed by tetrahedral molecules. The classical Monte Carlo method is applied to solid heavy methane CD₄ and the role of various contributions to the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1759532021-02-04T01:31:10Z Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals Yakub, E.S. Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) An attempt is made to explain the appearance of certain phases having different orientational and spatial structures in the phase diagram of crystals formed by tetrahedral molecules. The classical Monte Carlo method is applied to solid heavy methane CD₄ and the role of various contributions to the non-central intermolecular interactions in formation of the orientational structure in simple molecular crystals is assessed. Зроблено спробу пояснити причину появи деяких фаз, що мають різні орієнтаційні та просторові структури, на фазовій діаграмі кристалів, утворених тетраедричними молекулами. Класичний метод Монте-Карло застосовано до твердого важкого метану CD₄ та оцінено роль різних внесків у нецентральні міжмолекулярні взаємодії в формування орієнтаційної структури простих молекулярних кристалів. Сделана попытка объяснить причину появления некоторых фаз, имеющих разные ориентационные и пространственные структуры, на фазовой диаграмме кристаллов, образованных тетраэдрическими молекулами. Классический метод МонтеКарло применен к твердому тяжелому метану CD₄ и оценена роль различных вкладов в нецентральные межмолекулярные взаимодействия в формирование ориентационной структуры простых молекулярных кристаллов. 2019 Article Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals / E.S. Yakub // Физика низких температур. — 2019. — Т. 45, № 3. — С. 310-317. — Бібліогр.: 17 назв. — англ. 0132-6414 http://dspace.nbuv.gov.ua/handle/123456789/175953 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) |
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Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) Yakub, E.S. Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals Физика низких температур |
| description |
An attempt is made to explain the appearance of certain phases having different orientational and spatial
structures in the phase diagram of crystals formed by tetrahedral molecules. The classical Monte Carlo method is
applied to solid heavy methane CD₄ and the role of various contributions to the non-central intermolecular interactions in formation of the orientational structure in simple molecular crystals is assessed. |
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Article |
| author |
Yakub, E.S. |
| author_facet |
Yakub, E.S. |
| author_sort |
Yakub, E.S. |
| title |
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals |
| title_short |
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals |
| title_full |
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals |
| title_fullStr |
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals |
| title_full_unstemmed |
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals |
| title_sort |
role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2019 |
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Спеціальний випуск. “Proceedings of 12th International Conference on Cryocrystals and Quantum Crystals (CC-2018)” (Wrocław, Poland, August 26–31, 2018) |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/175953 |
| citation_txt |
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals / E.S. Yakub // Физика низких температур. — 2019. — Т. 45, № 3. — С. 310-317. — Бібліогр.: 17 назв. — англ. |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3, pp. 310–317
Role of short-range atom-atom forces in formation
of the orientational structure of simple molecular crystals
E.S. Yakub
Odessa National Economic University, 8 Preobrazhenskaya, Odessa 65082, Ukraine
E-mail: eugene.yakub@gmail.com
Received October 24, 2018
An attempt is made to explain the appearance of certain phases having different orientational and spatial
structures in the phase diagram of crystals formed by tetrahedral molecules. The classical Monte Carlo method is
applied to solid heavy methane CD4 and the role of various contributions to the non-central intermolecular inter-
actions in formation of the orientational structure in simple molecular crystals is assessed.
Keywords: phase diagram, Monte Carlo simulation, heavy methane, molecular rotation, atom-atom interactions.
1. Introduction
An extensive literature is devoted to crystals formed by
simple molecules and to explaining the structural complex-
ity of their solid-state phase diagram [1].
Despite the essential progress of modern quantum-
mechanical ab initio methods which can now provide ra-
ther accurate potential energy surfaces [2] and allow relia-
ble predictions of phase diagrams in the region of extreme
temperatures and pressures, they still cannot explain ap-
pearance of low-temperature phases having different spa-
tial and orientational structures and are unable answer the
question of what types of intermolecular interactions are
responsible for the appearance and location of a particular
phase on it, what is the role of quantum effects etc.
Even methane CH4 having simplest molecular structure
has very complex phase diagram, where phases differ in
both spatial and orientational structure. Special attention of
researchers was paid to the phase II, which has unusual
orientational structure. It has been established for a long
time that in phase II existing both in CH4 and CD4 [3] sol-
ids in a narrow low-temperature range and only at relative-
ly low-pressures part of molecules are orientationally dis-
ordered when the rest is spatially oriented.
According to Press [4] six of eight molecules in phase II
order with a certain local (cubic) symmetry, while the re-
maining two are orientationally disordered. Later, at pres-
sures above 5 kbar, a metastability region and phase IV were
discovered [5]. James and Keenan [6] developed the first
theoretical model of phase II in CD4 which accounts only
for one type of intermolecular interactions, namely the
octupole-octupole forces acting between nearest neighbors
fixed in sites of a static face-centered cubic (fcc) lattice.
The purpose of this paper is to elucidate the role of an-
other type of forces, namely atom-atom interactions, in the
formation of low-temperature phases in molecular crystals.
We restrict ourselves to considering the simplest type of
non-spherical molecules, namely, tetrahedral, and use the
classical Monte Carlo method. Additional methods needed
to describe the rotational degrees of freedom and orienta-
tional ordering of tetrahedral molecules have been devel-
oped and applied in the computer simulation. Heavy me-
thane CD4 was chosen as an object of study to minimize
the influence of quantum effects. Particular attention is paid
to the formation of phase II, which has the above-mentioned
unusual orientational structure.
Molecular interaction models used in simulations are de-
scribed in the next section. The basic simulation results are
presented in Sec. 3. In Sec. 4 these results are discussed and
compared with existing theoretical approaches and experi-
mental data. A sensitivity study is also presented here. In the
last section, we provide general conclusions and outline
ways to solve the problems that remain unresolved within
the current approach.
2. Molecular interaction model
The simplest model represents interaction energy of two
molecules as a sum of central and non-central octupole-
octupole (Ω–Ω) interactions
( ) ( )CC , ,ij ij ij i jr u rΩ−ΩΦ = Φ + ω ω (1)
and is based on the hypothesis of pair additivity of inter-
molecular interactions
0
N ij
i j N
U
< < ≤
= Φ∑ .
© E.S. Yakub, 2019
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals
Here { }, ,i = θ ϕ ψω determines the orientation of the tet-
rahedral molecule defined by three Eulerian angles θ, ϕ, and
ψ. The Ω–Ω interactions decay with 7th degree of the recip-
rocal intermolecular distance rij. Their explicit orientational
dependence is well known [7]. The central CC ( )ijrΦ interac-
tions are as usual described by the Lennard-Jones (12-6)
potential. This model was successfully applied recently to
the fcc phase of methane at elevated temperatures [8].
In this study we use a more advanced model including
two additional types of short-range atom-atom interactions
of heavy methane molecules:
( ) ( ) ( )( )CC CD , CD ,
1 4
ij ij i jk j ik
k
r r r
≤ ≤
Φ = Φ + Φ +Φ +∑
( ) ( )DD ,
1 , 4
, , .ik jl ij i j
k l
r u rΩ−Ω
≤ ≤
+ Φ +∑ ω ω (2)
Here ri, jk are distances between carbon atom of ith mole-
cule and kth (k = 1, …, 4) deuterium atom of jth molecule.
C–D bonds are considered to be rigid, and their length
fixed at L = 1.095 Å.
Such model much better reflects the nature of the short-
range interactions in CD4 solid, but its calibration is more
complicated. For the practical application of this model in
addition to the octupole moment at least two potential pa-
rameters are required for each type (C–C, C–D and D–D)
of atom-atom interaction.
To reduce the number of model parameters the usual
Lorentz–Berthelot combination rules were adopted:
( )CD CC DD CD CC DD1 , .2σ = σ + σ ε = ε ε (3)
Here σDD and εDD are parameters of the Lennard-Jones
(12-6) potential approximating atom-atom interaction en-
ergy of two hydrogen atoms. We accepted values σDD =
= 2.81 Å and εDD/kB = 8.6 K used in the recent simulations
of El-Sheikh et al. [9], because they provide an excellent
approximation for non-empirical potential of non-valent
interaction between two hydrogen atoms used earlier in our
computer simulations of hydrogen and deuterium fluids [10].
For the octupole moment it was also accepted the same
value (Ω = 2.3·10–34 esu) as in Ref. 9. This value is close to
that accepted by James and Keenan [6] but essentially dif-
fers from the value (Ω = 4.5·10–34 esu) recommended in [1]
and accepted in [8].
The last value provides reasonable results only within the
framework of the simplest interaction model (1) because it
effectively represents contributions of all other types of
short-range non-central forces acting between methane mol-
ecules which are ignored in this model.
Two remaining parameters (σCC and εCC) were fitted to
reproduce the low-temperature density of the CD4 solid.
The whole set of parameters adopted in this work is shown
in Table 1.
Table 1. Adopted parameters of the intermolecular interaction
potential (2)
Parameter Value
εDD/kB 8.6 K
σDD 2.8 Å
εCC/kB 50 K
σCC 3.63 Å
Ω 2.3·10–34 esu
The pairwise interaction energy (2) parameterized ac-
cordingly to Table 1 and pre-averaged over all orientations
is in reasonable agreement with the central Lennard-Jones
(12-6) potential recommended in the monograph of
Manzhelii et al. [1] as shown in Fig. 1.
3. Computer simulation
The adopted potential model (2) was applied in study of
structure and thermodynamic properties of crystalline CD4
using Monte Carlo simulation method in NVT ensemble.
Simulation processes was performed in cubic main cell
containing N molecules with usual periodic boundary con-
ditions and include equilibration and statistics collection
periods.
Each Monte Carlo step consisted of:
– random choice of a molecule to move;
– random displacement of its center of mass;
– random rotation of the molecule;
– application of the Metropolis rejection rule etc. [11].
A number of Metropolis Monte Carlo simulations of
crystalline CD4 were performed at temperatures from 10
up to 80 K and at densities, corresponding to pressures up
Fig. 1. Comparison of the pre-averaged pair interaction energy
Eq. (2) (dashed line) with the central Lennard-Jones (12-6) poten-
tial (ε/kB = 148 K, σ = 3.817 Å) [1] (solid line).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 311
E.S. Yakub
to 1 GPa. Cubic cells containing N molecules placed in the
sites of a fcc lattice were created at a start. The initial ori-
entations of all molecules were obtained by minimizing the
potential energy of the lattice.
Two different sizes of the main Monte Carlo cell were
used: a “small” cell N = 32 (160 atoms), and a “large” cell,
having twice the size of a “small” cell and containing N =
= 256 molecules (1280 atoms). Intermolecular potentials
were “cut” at ½ of the cell size (Rcut ∼ 5 Å in the “small”
cell and Rcut ∼ 10 Å in the “large” cell). Therefore the struc-
ture of the “small” cell was formed mainly by the short-
range forces acting between the nearest neighboring atoms.
Two types of the initial conditions in main cell were
used:
(a) “large” cell composed from eight relaxed “small”
cells;
(b) “large” cell with fcc spatial structure and molecular
orientations minimizing the potential energy of the lattice.
In the case (a) the “large” cell at the start of the new simu-
lation is filled by eight copies of a “small” 32-molecule cell
previously relaxed in a separate Monte Carlo run. In this case
the relaxation period of the “large” cell was essentially short-
er and this type of initial conditions was used throughout.
Sometimes the results of two runs with different initial
conditions were the same, but sometimes the structure inher-
ited from the “small” cell led to another stable spatial and
orientational structure of the “large” cell and the final states
have slightly lower energy and pressure. Below we discuss
such cases in connection with the problem of metastability.
The orientational structure was studied in two ways:
– via monitoring the numbers of the events of “rota-
tion”* for each molecule;
– by calculating orientational distributions of chemical
bonds.
The event of the molecular “rotation” within Monte Car-
lo procedure was treated as a change of the sign of any pro-
jection (x, y, or z) of the molecular orientation vector in a
single accepted Monte Carlo step**. When the frequency of
such “rotations” of the molecule reaches certain minimum
limit (i.e., such an event is not accidental), this molecule was
regarded as “rotating”.
Monte Carlo simulations started at relatively high temper-
atures and low pressures, were repeated with the step by step
increasing density and the changes appearing in the spatial
and orientational structures of the solid were monitored.
4. Results and discussion
4.1. Spatial structure
At T > 50 K and at relatively low densities the initial fcc
distribution of molecular centers survives in both “small”
and “large” cells, and a pronounced effect of molecular “ro-
tations” characteristic to phase I was observed. A typical
case is shown in Fig. 2.
When the density increases at a fixed temperature the
slope of the curves presented in Fig. 2 decreases and at a
certain density the molecular “rotation” stopped. We inter-
pret this effect as a transition to the phase III.
Special attention was paid to the formation of structures
which could be identified as corresponding to the phase II.
With a decrease in temperature, noticeable changes appear
not only in the orientational but also in the spatial structure
of solid. These changes appear first and were more pro-
nounced in “small” cells. Centers of molecules begin dis-
place from their initial fcc positions in such a way that in-
stead of four, eight different lattice positions are formed.
Only two of these eight lattice positions correspond to
molecules which continue “rotate” (Fig. 3). Such states were
interpreted as belonging to the phase II.
Displacements of molecular centers appearing in such
states are presented in Fig. 4 and in Table 2. In the first
lattice position occupied by molecules marked in Fig. 4 as
1, 13, 21, and 25 molecular centers are displaced along
X axis from their initial positions in the fcc lattice. Mole-
cules 2, 22, 14, 26 belonging to the second lattice position
are relocated along Y axis, and molecules 3, 15, 23, and 27
(third lattice position) moved along Z axis. Molecules
which belong to the lattice positions 5, 6 and 7 (see Ta-
ble 2) displace correspondingly along the same axes in the
opposite direction.
Centers of the remaining eight molecules occupying lat-
tice positions 4 and 8 are displaced along the main diagonal
* Here and below, we use quotation marks to emphasize the conventionality of this term within the Monte Carlo method used.
** The maximum possible angular displacement step was retained low enough to avoid the jump between the two regions, in fact
separated by high potential barriers insurmountable for molecules in their real dynamics.
Fig. 2. (Color online) Numbers of molecular “rotations” at differ-
ent lattice positions vs. number of successful steps per molecule
in phase I (T = 40 K, V = 31.75 cm3/mole).
312 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals
of the cubic cell. As a result, molecular centers form a new
cubic lattice which is alike to that discovered by Press [4] in
phase II.
Table 2. Directions of molecular centers displacements from
the initial fcc sites for different lattice positions in phase II
Lattice position Molecules Displaced along
1 1, 13, 21, 25 +X
2 2, 22, 14, 26 +Y
3 3, 15, 23, 27 +Z
4 4, 16, 24, 28 +XYZ
5 5, 9, 17, 29 –X
6 6, 10, 18, 30 –Y
7 7, 11, 19, 31 –Z
8 8, 12, 20, 32 –XYZ
Such structure of the “small” cell when transferred to the
“large” one according to the initial conditions of type (a)
mentioned above may survive there or may be destroyed by
the interaction of more distant molecules. In the first case
we obtain an orientational structure which resembles that
proposed for phase II: the “rotating” molecules in the
“large” cell are only those belonging to the lattice positions
4 and 8 (with molecular centers displaced along the diago-
nal of the cell). This case was interpreted as corresponding
to the formation of phase II and is illustrated in Fig. 4.
Results of the whole set of the Monte Carlo simulations
performed are summarized in Fig. 5. States where all mol-
ecules are “rotating” (phase I) are marked by circles. Tri-
angles denote such states, where only a quarter of all mole-
cules “rotate” as in Fig. 4 (phase II). And squares represent
such states where no systematic “rotations” were found
(phase III). As it can be seen, there is a qualitative agree-
ment with the experimental data of van der Putten [5]. The
deviations are of the same order as between data of van der
Putten [5] and of Stewart [3].
4.2. Orientational structure
Due to appearance of states with specific spatial struc-
tures attributed to phase II, special attention was paid to a
more detailed study of their orientational structure. Angu-
lar distributions of C–D chemical bonds directions of mol-
ecules occupying different lattice positions were evaluated.
Their analysis may help to understand the nature of molec-
ular “rotations” or “librations” and provide answers to such
questions as: does it really exist “free rotation” of CD4
molecules in phase I, does it really correspond to the uni-
form angular distribution of chemical bonds etc.?
The angular distribution the of the bonds directions in
phase III is the simplest one and is illustrated in Fig. 6. It
can be seen that there exist only four highly preferred di-
Fig. 3. (Color online) Numbers of molecular “rotations” at differ-
ent lattice positions vs. number of successful steps per molecule
at T = 30 K and V = 30 cm3/mole (phase II).
Fig. 4. Distortion of a fcc lattice induced by short-range atom-
atom forces in a “small” Monte Carlo cell.
Fig. 5. Comparison of the predicted low-temperature phase dia-
gram of CD4 with the experimental data of van der Putten [5]
(solid lines).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 313
E.S. Yakub
rections corresponding to the four chemical bonds which
can only slightly deviate from these preferred directions.
The same distributions corresponding to the states at-
tributed to phase II are shown in Fig. 7. As can be seen from
Fig. 7(a), in contrast to phase III the angular distribution of
chemical bonds of “non-rotating” molecules has eight pro-
nounced peaks. That means each bond has two preferred
directions and each “non-rotating” molecule librates be-
tween two available orientations.
The “rotating” molecules, belonging to lattice sites 4
and 8 (Table 2), have a very different angular distribution
presented in Fig. 7(b). It has one broad and several narrow
peaks which are interconnected by regions having smaller
but non-zero probability density. These regions allow tran-
sitions between different peaks and provide mechanism of
molecular “rotations”. At the same time it is clear that these
“rotations” are extremely hindered.
At low temperatures a similar picture is also observed
in phase I. All molecules are “rotating” but their rotations
remain highly hindered. As can be seen in Fig. 8, with in-
creasing temperature the orientational distribution is ap-
proaching to that which is characteristic for free rotation.
The only strong correlation remaining here between the
bonds directions is the intramolecular one.
4.3. Sensitivity study
It is clear that the results described above may be sensi-
tive both to the model parameters and to some details of the
simulation procedure adopted. Therefore a number of addi-
tional simulations aiming to reveal the sensitivity of the re-
sults to the model parameters and to the initial conditions
used in simulation was carried out.
First we studied the sensitivity of the obtained results to
the change in the parameters of the model. The purpose of
these tests was to assess the role of short-range atom-atom
forces in the formation of the orientational structure. All
the simulations were repeated using the simplest model of
interaction (1), including only the central forces represent-
ed by the Lennard-Jones (12-6) potential shown by solid
line in Fig. 1, and the octupole-octupole contribution.
The result was as follows: molecular “rotations” existed at
any value of the octupole moment (up to Ω = 4.5·10–34 esu
[1]) but their intensity (their number per successful Monte
Carlo step) decreases rapidly with decreasing temperature
and increasing density. No signs of the molecular centers
displacements and states where only part of molecules “ro-
tate”, characteristic for phase II (Table 2) were found.
Analysis of results obtained with the simple molecular
model (1) shows that when temperature decreases, no dis-
placements of molecular centers, similar to that shown in
Fig. 4 appear, and in this case no molecules stop “rotating”
Fig. 6. Low-temperature angular distributions of chemical bonds
directions in phase III (T = 10 K).
Fig. 7. Angular distributions of chemical bonds directions in phase II (T = 30 K): (a) “librating” molecules at lattice sites 2 and 6;
(b) “rotating” molecules at lattice sites 4 and 8 (see Table 2).
314 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals
at once and even the transition to states without “rotation”,
which could be attributed to phase III, appear gradually.
Within the simulation method applied the exact location of
the transition from phase I to phase III cannot be determined
but the absence of structures resembling phase II where only
part of molecules are “rotating”, was evident.
This contradicts the results of James and Keenan [6],
who found signs of phase II using actually the same model
(1) and assuming that the centers of all molecules are fixed
at the fcc lattice sites.
4.4. Metastability
The second set of tests was performed by changing the
initial conditions of the Monte Carlo simulations. Instead
of the mentioned above standard condition: (a) with the
“large” 256-molecule cell initially build from eight equili-
brated “small” cells, we tested another type of initial crys-
talline structure: (b) with initial orientations of molecules
minimizing the potential energy of the fcc lattice.
During these tests each Monte Carlo run with the
“large” cell was performed at least twice, using these two
different initial molecular configurations. Both at low and
high densities the results were the same; spatial structure
inherited in case (a) from “small” cells was slowly step-by-
step re-created in the “large” cell in the case (b).
But at intermediate densities in the case (b) the initial fcc
structure survives and led finally to another stable orienta-
tional structure in the “large” cell. This case usually corre-
sponds to slightly higher values of both internal energy and
pressure. A typical result of such double run is illustrated
in Fig. 9 where internal energy on the isotherm is presented
as a function of pressure for two different types of initial
conditions for “large” cell: (a) and (b).
Definitely one of these states could be interpreted as a
metastable one. Unfortunately the Metropolis Monte Carlo
method applied here does not allow calculating Helmholtz
free energy and we cannot state with confidence which of
the obtained states is metastable. Nevertheless, the very fact
of the appearance of a metastable state is beyond doubt. This
is in line with the results reported by van der Putten [5] who
discovered in the phase diagram of CD4 a broad area where
such metastability take place. The darkened area in Fig. 9
corresponds to that the metastability region found in [5]. Its
position on the phase diagram is not exactly the same as was
found in our simulations but is in outline close to it.
5. Conclusions
Despite the fact that the Metropolis Monte Carlo meth-
od applied does not allow calculating free energy and, ac-
cordingly, determining the exact location of the phase tran-
sition lines in the phase diagram, it allows us to identify
the role of the octupole-octupole and short-ranged atom-
atom forces in the formation of the spatial and orientational
structure of simple molecular crystals.
Both octupole-octupole and short-ranged atom-atom
forces are responsible for formation of low-temperature
crystalline phases in molecular solids. Nevertheless the role
Fig. 8. Angular distributions of C–D bonds directions in phase I at different temperatures on isochore V = 30 cm3/mole. Location of the
coordinate axes the same as in Fig. 7.
Fig. 9. Internal energy at T = 50 K as a function of pressure for two
different types of initial conditions: “large” cell build from 8 re-
laxed 32-molecule “small” cells (squares); start from 256-molecule
fcc lattice positions (circles). The darkened area corresponds to the
metastability region found in [5].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 315
E.S. Yakub
of the short-range atom-atom repulsive forces is crucial in
this respect.
As far as it possible to conclude on the basis of the pre-
sent Monte Carlo study of solid heavy methane, short-
range atom-atom forces are responsible for the transition of
the fcc phase I to the phase II. These forces determine for-
mation of another spatial cubic structure (Fm3c), which in
turn leads to formation of this specific orientational structure
of phase II where only a one fourth of all molecules “rotate”
and all other are librating around certain preferred orienta-
tions. It was found also that the same model allows repro-
ducing the transition from orientationally disordered phase I
to orientationally ordered phase III.
Angular distributions of the bonds directions predicted
on the basis of the adopted molecular model and presented
above in Figs. 6–9 allows us to conclude that molecular “ro-
tations” in both orientationally disordered phase I and par-
tially orientationally ordered phase II are highly hindered
and became more or less “free” only near the melting point.
This calls into question assessments carried out on the basis
of the idea that “one-fourth of the molecules in phase II are
free rotors” [12].
The results presented above show that the applicability
of simple models accounting only for central intermolecu-
lar and octupole-octupole interaction of tetrahedral me-
thane molecules is limited by the high-temperature region
near the melting line [8] and does not reproduce transitions
from orientationally disordered phase I to the phase II and
to orientationally ordered phase III.
Sure, not all of the issues arising in the field of this study
can be resolved on the basis of the classical Monte Carlo
simulation approach. Many problems remaining are related
to the limitations of the applied molecular model. Despite
the fact that the atom-atom model has found its application
for describing interactions in many molecular crystals [13],
chemical bonds within this model are rigid and short-range
atom-atom repulsion represented by Lennard-Jones (12-6)
potentials is too stiff, which limits applicability of the model
at high pressures [14]. Its application, for example, to me-
thane, where quantum effects are more important [15], is
limited, because it neglects these effects. Representation of
the short-ranged electrostatic interaction only by the
octupole-octupole forces is also limited.
Another problem worth to be noted in this context is the
manifestation of the metastable states discussed above. Ap-
pearance of such states during transitions into other high-
pressure phases in many solids build from deuterated me-
thane derivatives was discovered experimentally [12]. It is
probably responsible for some other effects observed in such
solids, e.g., for hysteresis of the thermal conductivity in a
mixed CD4−CH4 observed at low pressure [16], what make
this problem even more acute.
There remain also some much more general problems,
such as the absence of a consistent theoretical method for
estimating Helmholtz free energy of molecular solids and
determining the location of phase boundaries between dif-
ferent crystalline phases. This problem was solved only for
monatomic solids [17] by applying Mayer’s group expan-
sion technique for the Helmholtz free energy. The existence
of preferred molecular orientations in all phases revealed in
this work makes the extension of this approach to molecular
crystals a promising direction in future theoretical studies.
________
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___________________________
Роль короткодіючих атом-атомних сил
у формуванні орієнтаційної структури простих
молекулярних кристалів
Є.С. Якуб
Зроблено спробу пояснити причину появи деяких фаз, що
мають різні орієнтаційні та просторові структури, на фазовій
діаграмі кристалів, утворених тетраедричними молекулами.
Класичний метод Монте-Карло застосовано до твердого важ-
кого метану CD4 та оцінено роль різних внесків у нецентральні
міжмолекулярні взаємодії в формування орієнтаційної струк-
тури простих молекулярних кристалів.
Ключові слова: фазова діаграма, метод Монте-Карло, важкий
метан, молекулярне обертання, атом-атомні взаємодії.
316 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3
https://doi.org/10.1063/1.4961973
https://doi.org/10.1016/0022-3697(60)90029-9
https://doi.org/10.1063/1.1677586
https://doi.org/10.1016/0378-4363(84)90075-5
https://doi.org/10.1016/0378-4363(84)90075-5
https://doi.org/10.1063/1.1730276
https://doi.org/10.1080/00268977600101421
https://doi.org/10.1007/s10909-016-1721-7
https://doi.org/10.1063/1.2179422
https://doi.org/10.1023/A:1010731016500
https://doi.org/10.1007/BF00500571
https://doi.org/10.1063/1.481208
https://doi.org/10.1063/1.4905549
https://doi.org/10.1063/1.4905549
https://doi.org/10.1103/PhysRevLett.20.1
https://doi.org/10.1103/PhysRevB.75.012303
https://doi.org/10.1063/1.3702437
Role of short-range atom-atom forces in formation of the orientational structure of simple molecular crystals
Роль короткодействующих атом-атомных сил
в формировании ориентационной структуры
простых молекулярных кристаллов
Е.С. Якуб
Сделана попытка объяснить причину появления некоторых
фаз, имеющих разные ориентационные и пространственные
структуры, на фазовой диаграмме кристаллов, образованных
тетраэдрическими молекулами. Классический метод Монте-
Карло применен к твердому тяжелому метану CD4 и оценена
роль различных вкладов в нецентральные межмолекулярные
взаимодействия в формирование ориентационной структуры
простых молекулярных кристаллов.
Ключевые слова: фазовая диаграмма, метод Монте-Карло,
тяжелый метан, молекулярное вращение, атом-атомные
взаимодействия.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 3 317
1. Introduction
2. Molecular interaction model
3. Computer simulation
4. Results and discussion
4.1. Spatial structure
4.2. Orientational structure
4.3. Sensitivity study
4.4. Metastability
5. Conclusions
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