Heat transfer in Ar and N₂ doped solid CO
The measurements of thermal conductivity coefficient of a solid carbon monoxide crystal containing argon and nitrogen admixtures at different concentrations (1.5, 3, 6, 12.5, 25% for N2 and 0.5, 1, 1.25, 2, 4% for Ar) were performed in the temperature range from 1.5 to 40 K by steady-state heat fl...
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irk-123456789-1278282017-12-29T03:03:26Z Heat transfer in Ar and N₂ doped solid CO Romanova, T.V. Stachowiak, P. Jeżowski, A. 10th International Conference on Cryocrystals and Quantum Crystals The measurements of thermal conductivity coefficient of a solid carbon monoxide crystal containing argon and nitrogen admixtures at different concentrations (1.5, 3, 6, 12.5, 25% for N2 and 0.5, 1, 1.25, 2, 4% for Ar) were performed in the temperature range from 1.5 to 40 K by steady-state heat flow method. For analysis of the experimental data the Callaway method in the framework of the Debye model was used. The contribution of various mechanisms of phonon scattering, including scattering by disordered dipoles of the CO matrix, to the thermal conductivity of CO–N₂ and CO–Ar solid solutions were taken into account. 2015 Article Heat transfer in Ar and N₂ doped solid CO / T.V. Romanova, P. Stachowiak, and A. Jeżowski // Физика низких температур. — 2015. — Т. 41, № 6. — С. 559-563. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 44.10.+i, 63.20–e, 63.20.Mt, 65.40.–b http://dspace.nbuv.gov.ua/handle/123456789/127828 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals |
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10th International Conference on Cryocrystals and Quantum Crystals 10th International Conference on Cryocrystals and Quantum Crystals Romanova, T.V. Stachowiak, P. Jeżowski, A. Heat transfer in Ar and N₂ doped solid CO Физика низких температур |
| description |
The measurements of thermal conductivity coefficient of a solid carbon monoxide crystal containing argon
and nitrogen admixtures at different concentrations (1.5, 3, 6, 12.5, 25% for N2 and 0.5, 1, 1.25, 2, 4% for Ar)
were performed in the temperature range from 1.5 to 40 K by steady-state heat flow method. For analysis of the
experimental data the Callaway method in the framework of the Debye model was used. The contribution of various
mechanisms of phonon scattering, including scattering by disordered dipoles of the CO matrix, to the thermal
conductivity of CO–N₂ and CO–Ar solid solutions were taken into account. |
| format |
Article |
| author |
Romanova, T.V. Stachowiak, P. Jeżowski, A. |
| author_facet |
Romanova, T.V. Stachowiak, P. Jeżowski, A. |
| author_sort |
Romanova, T.V. |
| title |
Heat transfer in Ar and N₂ doped solid CO |
| title_short |
Heat transfer in Ar and N₂ doped solid CO |
| title_full |
Heat transfer in Ar and N₂ doped solid CO |
| title_fullStr |
Heat transfer in Ar and N₂ doped solid CO |
| title_full_unstemmed |
Heat transfer in Ar and N₂ doped solid CO |
| title_sort |
heat transfer in ar and n₂ doped solid co |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2015 |
| topic_facet |
10th International Conference on Cryocrystals and Quantum Crystals |
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http://dspace.nbuv.gov.ua/handle/123456789/127828 |
| citation_txt |
Heat transfer in Ar and N₂ doped solid CO / T.V. Romanova, P. Stachowiak, and A. Jeżowski // Физика низких температур. — 2015. — Т. 41, № 6. — С. 559-563. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT romanovatv heattransferinarandn2dopedsolidco AT stachowiakp heattransferinarandn2dopedsolidco AT jezowskia heattransferinarandn2dopedsolidco |
| first_indexed |
2025-07-09T07:49:11Z |
| last_indexed |
2025-07-09T07:49:11Z |
| _version_ |
1837154805366128640 |
| fulltext |
© T.V. Romanova, P. Stachowiak, and A. Jeżowski, 2015
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6, pp. 559–563
Heat transfer in Ar and N2 doped solid CO
T.V. Romanova, P. Stachowiak, and A. Jeżowski
Institute for Low Temperature and Structure Research, Polish Academy of Sciences
P.O. Box 1410, 50-950 Wrocław, Poland
E-mail: T.Romanova@int.pan.wroc.pl
Received February 10, 2015, published online April 23, 2015
The measurements of thermal conductivity coefficient of a solid carbon monoxide crystal containing argon
and nitrogen admixtures at different concentrations (1.5, 3, 6, 12.5, 25% for N2 and 0.5, 1, 1.25, 2, 4% for Ar)
were performed in the temperature range from 1.5 to 40 K by steady-state heat flow method. For analysis of the
experimental data the Callaway method in the framework of the Debye model was used. The contribution of var-
ious mechanisms of phonon scattering, including scattering by disordered dipoles of the CO matrix, to the ther-
mal conductivity of CO–N2 and CO–Ar solid solutions were taken into account.
PACS: 44.10.+i Heat conduction;
63.20–e Phonons in crystal lattice;
63.20.Mt Phonon-defect interactions;
65.40.–b Thermal properties of crystalline solids.
Keywords: thermal conductivity, molecular crystal, phonon scattering, carbon monoxide crystals, low tempera-
tures, relaxation time.
Introduction
Solid carbon monoxide, both in terms of its crystallo-
graphic structure and intermolecular interactions, belongs
to the group of the simplest molecular crystals [1,2]. This
van der Waals crystal at equilibrium vapor pressure can be,
depending on the temperature, in one of two crystallographic
phases. In the temperature range from 68.09 to 61.57 K crys-
talline CO exists in the high-temperature β-phase. This
phase features an hcp structure without long-range orien-
tational ordering of the molecules. The symmetry of the
phase belongs to P63/mmc space group. Below 61.57 K the
crystal undergoes a structural phase transition to the low-
temperature fcc α-phase. In this phase an orientational long-
range ordering of the axes of CO molecules is observed –
the axes of the molecules are oriented along space diagonals
of the elementary cubic cell [3,4]. At this point it should be
emphasized that the molecule of CO due to its asymmetry,
unlike symmetric linear molecules, has a non-zero perma-
nent electrostatic dipole moment which amounts to
3.7356·10
–31
C·m [5]. All experiments carried out so far
did not confirm existence of a new low-temperature
phase with dipolar ordered molecules as predicted by
theory [6–11]. On the contrary, all experiments show that
CO dipoles remain disordered down to the lowest inves-
tigated temperatures, forming a disordered structure in
the dipolar subsystem. This glassy state strongly influ-
ences the low-temperature physical properties of the car-
bon monoxide crystal and, among other things, its ther-
mal conductivity [12].
It is well known that the thermal conductivity of a crys-
tal can be modified by introducing foreign atoms or mole-
cules in the crystal structure. The impurities become scat-
tering centers for heat carriers and can modify the thermal
excitation spectrum of the structure influencing the thermal
conductivity of the crystal. The presence of an impurity in
the crystal usually leads to additional mechanism of pho-
non scattering due to difference of masses of the impurities
and the host atoms, resulting in the so called isotopic ef-
fect. But phonons can be scattered not only as a result of
the isotopic effect. Foreign atoms (molecules) in a crystal
have force constants which are different from those of the
host environment. This difference causes a deformation of
the structure in the vicinity of the impurity which entails an
extra phonon scattering due to the admixture.
To investigate the effect of impurities on the thermal
conductivity of solid carbon monoxide, we chose two types
of admixtures which we introduced into the structure of the
CO, namely, spherically symmetric atoms of argon and
linear symmetric molecules of nitrogen. While the Ar atom
differs strongly from CO molecule both in terms of its
symmetry and mass (40 atomic mass units for argon vs 28
mailto:T.Romanova@int.pan.wroc.pl
http://ufn.ru/en/pacs/63.20.Mt/
http://ufn.ru/en/pacs/65.40.-b/
T.V. Romanova, P. Stachowiak, and A. Jeżowski
560 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
for carbon monoxide), nitrogen and carbon monoxide are
considered alike molecules in many aspects. The molecular
masses are the same and the parameters of the Lennard-
Jones potential for both molecules are very close to each
other [2]. Due to the similarity of the molecules, solid mix-
tures of N2 and CO form homogeneous solid solutions for
any mutual concentrations [13]. The solubility of argon in
carbon monoxide is up to 5% in the whole temperature
range [14].
Experiment
Measurements of the thermal conductivity of solid car-
bon monoxide with argon and nitrogen admixtures were
carried out in the temperature range from 1.5 to 40 K by
the steady-state heat flow method. The concentrations of
nitrogen in the samples investigated were 1.5, 3, 6, 12.5,
25% and those of argon were 0.5, 1, 1.25, 2, 4%.
The measurements have been conducted with use of a
home-designed LHe cryostat [15]. The samples were grown
and studied in a thin-wall stainless steel ampoule with an
inner diameter of 5.5 mm and a length of 40.08 mm, the
thickness of the ampoule wall was 0.5 mm. The crystals
were grown directly from the gas phase, avoiding the liquid
phase. The temperature and pressure of condensation were
slightly below the triple-point values of the gas mixture and
were maintained constant during the whole crystal growth
procedure. The crystal gradually (at a constant rate of ap-
proximately 1 mm/h) filled the ampoule from bottom to top.
After growth the samples were cooled down to the starting
temperature of measurement at a rate of 0.2 K/h.
Results
The results of our measurements of the thermal conduc-
tivity as a function of temperature of pure carbon monoxi-
de and carbon monoxide containing nitrogen and argon
admixtures of different concentrations are shown in Figs. 1
and 2, respectively.
The temperature dependence of the thermal conductivi-
ty is typical of a dielectric crystal [16]. The experimental
curves feature a characteristic maximum. At low tempera-
tures the thermal conductivity increases with temperature
and then, after reaching the maximum value, starts to de-
crease. The crystals containing admixtures exhibit lower
thermal conductivity than the pure one. The height of their
maxima decreases with increasing concentration of the
admixture for both types of dopants. In addition, the max-
imum shifts towards higher temperatures for higher con-
centrations of both doping agents.
At the lowest investigated temperatures two curves of
argon-doped family (Fig. 2) show a slightly different be-
havior compared with the remaining ones. The thermal
conductivity for the samples containing 0.5 and 1.25% of
Ar as a function of T below the maximum is more sloping
than for other concentrations. At the same time, at the low-
est temperatures the thermal conductivity of these samples
is higher than one might expect and the maxima are shifted
to lower temperatures compared to pure CO.
Qualitatively, the admixture of nitrogen (Fig. 1) lowers
the thermal conductivity less than the argon dopant: 4% of
argon reduces the low-temperature thermal conductivity by
an order of magnitude while a similar admixture of nitro-
gen reduces the thermal conductivity by less than 40%.
For the purpose of a quantitative description of the con-
tributions due to various mechanisms of phonon scattering
to the thermal conductivity of CO–N2 and CO–Ar solid so-
lutions the Callaway method in the framework of the Debye
model was utilized [17]. In this method each of the scatter-
ing mechanisms is characterized by its own relaxation time
which is a measure of the average time between consecutive
phonon scatterings in a particular scattering process. The
relaxation time depends on phonon frequency.
The Callaway expression for the thermal conductivity κ
of a dielectric crystal can be written:
Fig. 1. (Color online) Thermal conductivity of pure and nitrogen
doped solid carbon monoxide.
Fig. 2. (Color online) Thermal conductivity of pure and argon
doped carbon monoxide.
Heat transfer in Ar and N2 doped solid CO
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 561
1 2 , (1)
where
/3 4
3
1 2 2
0
e
2 (e 1)
T x
cB B
x
xk k
T dx
v
(2)
and
2
/
4 2
3
03
2 2 /
4 2
0
( / )( e /(e 1) )
2
( / )( e /(e 1) )
T
x x
c N
B B
T
x x
c N r
x dx
k k
T
x dx
v
. (3)
Here the dimensionless variable / ,x w kT v=
3 3 1/32 )/3]l t[(= v v is the phonon velocity averaged over
longitudinal
l
v and transversal vt polarizations [18]. For
carbon monoxide v = 1225.5 m/s, and = 103.5 K is the
Debye temperature of solid carbon monoxide [2], kB is the
Boltzmann constant, ℏ is the Plank constant, τ is the relax-
ation time of phonon scattering, T is the temperature.
The combined inverse relaxation time (relaxation rate)
can be written as
1 1 1
c r n , (4)
where 1
r is the relaxation rate for resistive processes, 1
n
is the relaxation rate for normal processes.
The relaxation rate for resistive processes is a sum of
relaxation rates of all resistive processes taken into account
1 1 1 1 1 1
r b p d D U . (5)
This expression contains the relaxation rates for scatter-
ing by grain boundaries (b), by point defects (p), by dislo-
cation strain fields (d), by the subsystem of disordered di-
poles of CO molecules (D), and by Umklapp processes
(U). The three-phonon normal processes have not been
taken into account. For dielectric crystals such as discussed
here with their structure far from that of a perfect crystal.
these processes are by far less important than other phonon
scattering processes, which allowed us to omit them.
In our calculations the following expressions for the re-
laxation times were used:
1 1 4 4 1, ,b b p p d da a x T a xT ,
1 0.2 2.2 1 2 5
1 2, exp [ / ]D D U U Ua x T a x T a T . (6)
Here, ai are the parameters of the respective relaxation
rates. The values of these parameters are determined within
a fitting procedure to obtain the best match of Eq. (1) to the
thermal conductivities obtained in the experiment.
The values of ai found for the CO–N2 and CO–Ar solid
solutions studied at different concentrations of nitrogen
and argon admixtures are presented in Table 1 and Table 2,
respectively.
The relaxation time parameters obtained as a result of
fitting procedure carry a lot of information regarding the
structure of the investigated crystals. For example, from
the value of ab one can obtain an average grain size b of
the crystals, which in the case of pure carbon monoxide is
1.81 mm (from the dependence abound = d/v, where v is an
the average phonon velocity [19]).
The parameter ap yields information about the concen-
tration of point defects cp and about the strength Ap of the
phonon interaction with defects [20]:
p p pa c A .
For the samples containing 1.5 and 3% of N2, Ap are
1.19·10
5
s
–1
K
–4
and 1.186·10
5
s
–1
K
–4
, respectively. For
higher concentrations of nitrogen the parameter ap re-
mains constant. This can be ascribed to clusterization of
admixture molecules and, possibly, to some compensat-
ing mechanism which weakens the phonon-impurity in-
teraction at higher admixture concentrations. For the ar-
gon containing samples the parameter ap increases
linearly with the argon concentration. It is well under-
stood from Fig. 2. Here the maximum concentration of Ar
was 4%, so the probability of cluster formation is rela-
tively low and the mutual interaction of the atoms is neg-
Table 1. Phonon relaxation rate parameters obtained by Callaway method far sample of pure CO and the samples containing molar
admixture of 1.5, 3, 6, 12.5 and 25% of N2
CO CO–1.5%N2 CO–3%N2 CO–6%N2 CO–12.5%N2 CO–25%N2
ab 6.779·10
5
2.756·10
5
1.43·10
6
1.4847·10
6
9.275·10
5
1.665·10
6
ap 6.98·10
2
1.795·10
3
3.558·10
3
3.654·10
3
4.3·10
3
3.226·10
3
ad 9.618·10
4
1.991·10
5
1.586·10
4
1.087·10
4
2.899·10
5
9.566·10
5
1Ua 1.785·10
4
1.820·10
4
1.813·10
4
1.95·10
4
2.002·10
4
2.531·10
4
2Ua 8.6888 9.16708 9.1977 9.78 9.155 9.976
aD 6.841·10
5
4.413·10
5
3.231·10
5
3.615·10
5
3.83·10
5
1.348·10
6
T.V. Romanova, P. Stachowiak, and A. Jeżowski
562 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6
ligible. This is why we do not observe the effect of pho-
non-impurity interaction weakening.
For both admixtures (nitrogen and argon) the parameter
ap is smaller by two or three orders of magnitude com-
pared to other scattering mechanisms. Therefore, the scat-
tering of phonons by point defects affects little the thermal
conductivity of the samples investigated.
From the data presented in both Tables we can see that
the phonon-phonon scattering parameters are almost inde-
pendent of the impurity molecules in CO–N2 solid solutions.
Analyzing the data for carbon monoxide crystals con-
taining argon one can note that if we disregard the curves
which differ from other ones in shape (namely, as pointed
above, CO–0.5%Ar and CO–1.25%Ar), the results ob-
tained using Callaway’s method look smoother. The con-
tribution of dislocations increases gradually with increas-
ing argon content. The parameters of the phonon-phonon
scattering processes decrease monotonically with increas-
ing Ar concentration. It is particularly clearly seen with the
2U
a parameter, which scales the average energy of pho-
nons involved in U-processes.
As the nitrogen content in CO is varied, the parameter
aD, which quantitatively describes the significance of the
phonons scattering owing to the orientational disorder of
CO molecules. Firstly decreases with increasing N2 admix-
ture concentration and then begins to increase. For argon
doped crystals, the intensity of phonon scattering by the
subsystem of disordered carbon monoxide dipoles decreas-
es almost for all impurity concentrations.
Conclusions
The temperature dependence of the thermal conductivi-
ty of CO–N2 and CO–Ar solutions has been determined
within the temperature range 1.5–40 K for the samples
containing 1.5, 3.0, 6.0, 12.5 and 25% of N2 molecules and
0.5, 1.0, 1.25, 2.0 and 4.0% of Ar atoms. The thermal con-
ductivity as a function of temperature for all solutions
show a behavior typical of dielectric crystals. The height of
the thermal conductivity maxima decreases with increasing
concentration of both admixtures.
The experimental results were analyzed within the re-
laxation time approximation. The phonon scattering pro-
cesses due to grain boundaries, point defects, dislocation
strain fields, dipoles of orientationally disordered CO mol-
ecules, and Umklapp processes have been taken into ac-
count. Phonon scattering by point defects has the least sig-
nificant effect on thermal conductivity for all crystals
investigated. The parameters of phonon-phonon scattering
in CO–N2 solutions are almost independent of the nitrogen
content. The scattering of phonons by disordered CO di-
poles is significant for both solutions. A correlation be-
tween this type of phonon scattering and the concentration
of the admixtures was observed.
Acknowledgment
The work was supported by Wroclaw Research Centre
EIT+ within the project “The Application of Nanotechnol-
ogy in Advanced Materials” – NanoMat (POIG.01.01.02-
02-002/08) co-financed by the European Regional Devel-
opment Fund (operational Programme Innovative Econo-
my, 1.1.2)
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Table 2. Phonon relaxation rate parameters obtained by Callaway method far sample of pure CO and the samples containing molar
admixture of 0.5, 1, 1.25, 2 and 4% of Ar
CO CO–0.5%Ar CO–1% Ar CO–1.25%Ar CO–2%Ar CO–4%Ar
ab 6.779·10
5
5.166·10
5
3.817·10
5
1.262·10
5
3.44·10
6
1.35·10
7
ap 6.98·10
2
9.08·10
2
2.639·10
3
4.314·10
3
1.805·10
4
2.486·10
4
ad 9.618·10
4
5.623·10
3
6.5·10
5
6.18·10
5
5.618·10
6
8.87·10
6
1Ua 1.785·10
4
1.673·10
4
1.968·10
4
1.789·10
4
1.61·10
4
1.534·10
4
2Ua 8.6888 5.75 6.510 4.565 3.878 2.043
aD 6.841·10
5
8.056·10
5
6.772·10
5
4.512·10
5
6.608·10
4
4.106·10
4
Heat transfer in Ar and N2 doped solid CO
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 6 563
13. M.J. Angwin and J. Wassewman, J. Chem. Phys. 44, 417
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