Melting line of polymeric nitrogen
We made an attempt to predict location of the melting line of polymeric nitrogen using two equations for Helmholtz free energy: proposed earlier for cubic gauche-structure and developed recently for liquid polymer-ized nitrogen. The P–T relation, orthobaric densities and latent heat of melting were...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1184512017-05-31T03:07:06Z Melting line of polymeric nitrogen Yakub, L.N. 9th International Conference on Cryocrystals and Quantum Crystals We made an attempt to predict location of the melting line of polymeric nitrogen using two equations for Helmholtz free energy: proposed earlier for cubic gauche-structure and developed recently for liquid polymer-ized nitrogen. The P–T relation, orthobaric densities and latent heat of melting were determined using a standard double tangent construction. The estimated melting temperature decreases with increasing pressure, alike the temperature of molecular–nonmolecular transition in solid. We discuss the possibility of a triple point (solid–molecular fluid–polymeric fluid) at ~ 80 GPa and observed maximum of melting temperature of nitrogen. 2013 Article Melting line of polymeric nitrogen / L.N. Yakub // Физика низких температур. — 2013. — Т. 39, № 5. — С. 552–555. — Бібліогр.: 18 назв. — англ. 0132-6414 PACS: 05.70.Ce, 61.66.Bi, 61.50.Ah, 64.70.dj http://dspace.nbuv.gov.ua/handle/123456789/118451 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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9th International Conference on Cryocrystals and Quantum Crystals 9th International Conference on Cryocrystals and Quantum Crystals Yakub, L.N. Melting line of polymeric nitrogen Физика низких температур |
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We made an attempt to predict location of the melting line of polymeric nitrogen using two equations for Helmholtz free energy: proposed earlier for cubic gauche-structure and developed recently for liquid polymer-ized nitrogen. The P–T relation, orthobaric densities and latent heat of melting were determined using a standard double tangent construction. The estimated melting temperature decreases with increasing pressure, alike the temperature of molecular–nonmolecular transition in solid. We discuss the possibility of a triple point (solid–molecular fluid–polymeric fluid) at ~ 80 GPa and observed maximum of melting temperature of nitrogen. |
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Yakub, L.N. |
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Yakub, L.N. |
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Yakub, L.N. |
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Melting line of polymeric nitrogen |
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Melting line of polymeric nitrogen |
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Melting line of polymeric nitrogen |
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Melting line of polymeric nitrogen |
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Melting line of polymeric nitrogen |
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melting line of polymeric nitrogen |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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9th International Conference on Cryocrystals and Quantum Crystals |
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Melting line of polymeric nitrogen / L.N. Yakub // Физика низких температур. — 2013. — Т. 39, № 5. — С. 552–555. — Бібліогр.: 18 назв. — англ. |
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Физика низких температур |
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AT yakubln meltinglineofpolymericnitrogen |
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2025-07-08T14:00:34Z |
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2025-07-08T14:00:34Z |
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1837087771927248896 |
| fulltext |
© L.N. Yakub, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5, pp. 552–555
Melting line of polymeric nitrogen
L.N. Yakub
Thermophysics Department, Odessa State Academy of Refrigeration, 1/3 Dvoryanskaya Str., Odessa 65082, Ukraine
E-mail: unive@icn.od.ua
Received December 5, 2012
We made an attempt to predict location of the melting line of polymeric nitrogen using two equations for
Helmholtz free energy: proposed earlier for cubic gauche-structure and developed recently for liquid polymer-
ized nitrogen. The P–T relation, orthobaric densities and latent heat of melting were determined using a standard
double tangent construction. The estimated melting temperature decreases with increasing pressure, alike the
temperature of molecular–nonmolecular transition in solid. We discuss the possibility of a triple point (solid–
molecular fluid–polymeric fluid) at ~ 80 GPa and observed maximum of melting temperature of nitrogen.
PACS: 05.70.Ce Thermodynamic functions and equations of state;
61.66.Bi Elemental solids;
61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling;
64.70.dj Melting of specific substances.
Keywords: solid nitrogen, polymerization, phase transition, equation of stare.
1. Introduction
Recent studies of solid nitrogen at high pressures [1]
revealed existence of its new crystalline phases. Polymor-
phism is typical for molecular cryocrystals but the specific
feature of nitrogen is that some high-pressure phases of
solid nitrogen are nonmolecular [2].
Experimental confirmation of the polymerization in so-
lid [3] and liquid nitrogen [4] allow theorists and experi-
mentalists to discuss the issue of a new configuration of
the phase diagram of solid nitrogen at high pressures.
The calculated P–T line of the molecular–to–polymeric
transition in solid nitrogen [5,6] reveals essential depend-
ence on the structure of polymeric phase. Phase transitions
with rearranging of chemical bonds are typical not only for
nitrogen, but for many simple molecular condensed sys-
tems build from molecules with multiple chemical bonds.
The possibility of molecular–to–polymer transition in
liquid nitrogen was discussed in relation to the discovery
of the temperature drop and increased conductivity of the
nitrogen fluid behind the reflected shock wave (shock
cooling) discovered by Nellis et al. [7].
Ab initio simulations of Boates and Bonev [8] reveal
that dense liquid nitrogen may also have complex struc-
ture, similar to that found in the solid nitrogen. The transi-
tion from the molecular to the atomic structure can be in-
terpreted as a break triple chemical bond in N2 and
formation of a network of ordinary chemical bonds con-
necting each N atom with three its nearest neighbors in the
polymeric structure.
Thus, the general idea, which may explain the phenom-
enon of polymerization in the liquid and solid phases, is
the same. This allows using the equation of state (EOS) of
the solid phase and liquid phase polymer nitrogen line to
predict the melting crystalline polymeric nitrogen into po-
lymeric liquid.
In this work we use a new EOS for highly compressed
polymer nitrogen liquid which was developed recently and
calibrated on results of ab initio simulations and applied to
the prediction of the liquid–liquid transition in highly com-
pressed nitrogen [9]. Using two equations of state: for po-
lymeric solid and polymeric liquid nitrogen one can calcu-
late the location of the melting line on P–T diagram, and
densities of coexisting phases.
2. EOS for polymeric nitrogen solid
We applied the modified Mie–Grüneisen model and
EOS for anharmonic polymeric solid proposed in our work
[10]. This EOS describes the thermodynamic properties of
solid nitrogen in a wide range of parameters of state in
cubic gauche (cg)-polymeric phase, and predicts the nega-
tive thermal expansion and significant deviations of heat
capacity from the Dulong–Petit law. This EOS was used
earlier in prediction of the molecular–to–polymer phase
transition in solid nitrogen [6].
mailto:unive@icn.od.ua
Melting line of polymeric nitrogen
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 553
The Helmholtz free energy of an anharmonic solid was
written [10] as a sum:
( ) (anh)
poly poly poly
h
F F F , (1)
where
( )
poly
h
F is the quasi-harmonic contribution, represent-
ed by the modified Mie–Grüneisen model, and
(anh)
polyF is
an anharmonic correction:
( ) (0)
poly poly( , ) ( ) 3 ( )
h
F DF T U RTD x . (2)
Here
(0)
polyU is the energy of static lattice, and ( )F DD x is
the Debye function:
2
3
0
3
ln (1 e ) ;
Dx
x D
F D D
D
D x x dx x
Tx
. (3)
Both thermal and caloric EOS are easily obtainable
from Eq. (1) by differentiation with respect to density and
temperature:
2, ,P T F T
, (4)
2( , ) ( , )BE T n k T F T n
T
. (5)
Here 1( )Bk T .
As it was shown in Ref. 10, the anharmonic contri-
butions to the heat capacity as well as to the thermal ex-
pansion and isothermal compressibility are important.
The anharmonic contribution includes anharmonic cor-
rections 1( )A and 2 ( )A :
anh
poly 21 2( ) ( )
2 6
F A A
T T
RT
, (6)
and 1( )A and 2 ( )A were found in Ref. 10 using devia-
tions of the heat capacity from the Dulong–Petit law, dedu-
ced from Monte Carlo data [11] for cubic gauche (cg)-solid
nitrogen:
* * *
1( ) 0.004918 ( 1.0468)( 0.8481)A , (7)
4 * * *
2( ) 4.03 10 ( 0.9666)( 0.8763)A , (8)
* 3
0 0, 7cm /molV V .
The quasi-harmonic thermal Grüneisen parameter,
ln / lnD , was determined by extracting the anhar-
monic corrections, calculated according to Eqs. (4) and (5)
from the Monte Carlo data [11] on pressure and energy.
Surprisingly, it was found to be almost independent of
temperature and decreasing nearly linear with the increas-
ing density. Equations for D and density-dependent Grü-
neisen parameter:
0
0
1
(9)
includes three constants: 0 , 0 and
(0)
0( )DD ,
where 0 is the density corresponding within quasi-
harmonic approximation to 0 . All the constants were
determined from Monte Carlo data [11]: 0 = 30.5, 0
= 1/7 cm
3
/mol, and 200 KD .
The linear decrease of the thermal quasi-harmonic Grü-
neisen parameter with density [10] gives the possibility for
a simple modification of the Mie–Grüneisen model. Ex-
pression for the quasi-harmonic contribution to the Helm-
holtz free energy as a function of temperature T and vo-
lume V remains the same as in the original Mie–Grüneisen
model.
Applying the standard thermodynamic relations one can
obtain expressions for quasi-harmonic contributions to all
thermodynamic functions. Equations for the energy and
heat capacity remain the same as in the original Mie–Grü-
neisen model, except for the new density dependence of
the Debye temperature.
All the EOS parameters used in this work were adopted
from Ref. 10 except the static lattice energy
(0)
polyU , which
was shifted by 0E — the difference in energies between
static molecular and static cg-lattices. The value of 0E
parameter is important in calculation of the phase equilib-
rium. Zhang et al. [12] refer to 0E value of 1 eV/atom. In
our calculations we adopted the value 0 0.97E eV/atom
obtained by Mailhiot et al. [13].
3. EOS for liquid polymeric phase
EOS of polymerizing fluid nitrogen was written [9] as
an expression for Helmholtz free energy F of a mixture of
2N molecules, dimers 4N and all other possible polymers
2N k is as follows:
(id) ( )
1 polypoly ( ) ( )HD
LF F F F . (10)
Here 1n is numerical density (concentration) of molecular
component 2(N ) , 3
1 /3nd is “molecular” and L
3 /3Lnd is “polymeric” packing fraction, n is the molar
density of nitrogen, is the degree of polymerization:
11 /n n .
The general expression for ideal-gas part for Helmholtz
free energy can be written as:
(id)
poly ln
1
k
k
k
n
F kTV n
Q T
k
, (11)
where kQ T are partition functions of kth component:
k = 1 ( 2N ), k = 2 4(N ), etc. The explicit expression of the
ideal-gas Helmholtz free energy
(id)
polyF , was developed in
Ref. 9 in terms of two dimensionless quantities and :
(id)
poly 2
2 1
1
1 2 1 ln (1 ) (1 ) ln .
F
Q T
(12)
Auxiliary variables and are defined [9] as follows:
L.N. Yakub
554 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5
2
;
1 2 1 4 1
(13)
2 ( ) 6
1 /2 1
exp L
L
nK T
. (14)
The polymerization degree can be expressed in terms of
and as:
1 / . (15)
Effect of nonideality
1( )
HD
F was included by the
hard-dumbbell contribution [14].
The contribution of the strong attraction between atoms
in polymerized fluid leads to formation of a network of
interatomic single bonds and is accounted by the last term
in Eq. (10). It is a function of the “polymeric” packing
fraction 3 /3L Lnd proposed in Ref. 15:
poly
6
1
L
L
L
F
. (16)
The above Eqs. (10)–(16) completely define the EOS
for the polymeric liquid.
EOS for polymeric liquid nitrogen was calibrated on
ab initio simulation data [8] and applied to prediction of
fluid–fluid transition in strongly compressed nitrogen [9].
The 0E value was fitted to reproduce the melting pres-
sure near the triple point at 1500 K [1].
4. Results and conclusions
The pressure–temperature relation, orthobaric volumes
and latent heat of melting were determined using a stand-
ard double tangent construction. Orthobaric volumes of the
coexisting liquid and solid phases of polymeric nitrogen
were obtained as abscissa of the point of contact of the
common tangent to the curves of the free energy Eq. (2)
and Eq. (10) and the equilibrium pressure of melting was
defined as the slope of this tangent. The results are present-
ed and compared with experimental data in Figs. 1 and 2.
It should be noted that EOS for polymeric liquid [9]
was calibrated on ab initio data [8] ranging from 2000 to
5000 K and hence the solution appear to be possible only
within the limited range of temperatures and pressures. At
temperatures above ~ 1750 K the predicted melting line
crosses the (extrapolated) liquid–liquid phase separation
line and therefore no above common tangent was found. At
temperatures below 1500 K the extrapolation of EOS for
liquid phase become, in our opinion, too far.
In Fig. 1 we present the predicted temperature–pressure
relation on the melting curve of polymeric nitrogen. Pre-
dicted melting temperature is compared here with experi-
mental data and the molecular–to–plastic transition line
predicted earlier [6]. Note that the estimated melting tem-
perature decreases with increasing pressure, alike the tem-
perature of molecular–nonmolecular transition in solid.
The predicted pressure dependence of the orthobaric
volumes of the solid and liquid polymeric nitrogen on the
melting curve are shown in Fig. 2 together with the only
available experimental data Eremets et al. [16] for the mo-
lecular (circles and squares) and polymeric nitrogen (dia-
monds) at room temperature. This comparison gives an idea
of the volume change during melting of polymeric cg-phase
in that limited range of pressures where the calculations
have been carried out. There is a reasonable quantitative
agreement with the room-temperature data of Eremets
Fig. 1. Temperature–pressure dependence of the nitrogen melting
line. Predicted melting temperature (dashed line) in comparison
with experiments [1,17,18], ab initio [8] data, and molecular–to–
polymeric solid–solid transition (dash-dotted line), predicted
earlier [6]. Light gray area: phase boundary of cg-N estimated in
Ref. 1.
Fig. 2. Predicted pressure–volume dependence of the solid (open
diamonds) and liquid (open squares) polymeric nitrogen on the
melting line. Solid symbols represent experimental data of
Eremets et al. [16] for molecular (solid circles and squares) and
polymeric nitrogen (solid diamonds) at room temperature.
Melting line of polymeric nitrogen
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 5 555
et al. [17]. As one can see in Fig. 2, volumes of polymeric
solid at high temperature are less than measured at room
temperature. And the unusual increase of both volumes
with pressure is not surprising. It due to negative slope of
P(T) relation and negative thermal expansion of polymeric
nitrogen [11].
The estimated latent heat L of cg-N melting is also neg-
ative and its absolute value increases with temperature. At
T = 1700 K the value of latent heat reaches –2.0 eV/atom.
Recent ab initio simulations [8] predict existence of at
least a second triple point (solid–molecular fluid–polyme-
ric fluid) at Ptr ~ 80 GPa on the melting line of nitrogen.
Our calculations support this prediction.
The predicted melting temperature decreases with the
increasing pressure. It drops from 1750 K at 80 GPa up to
1500 K at 95 GPa (see Fig. 1). Such behavior is in line
with the recent measurements of Goncharov et al. [1] who
observed the maximum of the melting temperature.
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http://dx.doi.org/10.1016/j.fluid.2012.09.011
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