Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions
The thermal conductivity of (CH4)1–c(CD4)c solid solutions with c = 0, 0.03, 0.065, 0.13, 0.22, 0.4, 0.78, and 1.0 has been measured in the region of existence of three orientational phases: disordered (phase I), partially ordered (phase II) and completely ordered (phase III). The temperature range...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/7774 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions / A.I. Krivchikov, P. Stachowiak, E. Pisarska, A. Jezowski // Физика низких температур. — 2007. — Т. 33, № 12. — С. 1393-1400. — Бібліогр.: 37 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859608516465000448 |
|---|---|
| author | Krivchikov, A.I. Stachowiak, P. Pisarska, E. Jezowski, A. |
| author_facet | Krivchikov, A.I. Stachowiak, P. Pisarska, E. Jezowski, A. |
| citation_txt | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions / A.I. Krivchikov, P. Stachowiak, E. Pisarska, A. Jezowski // Физика низких температур. — 2007. — Т. 33, № 12. — С. 1393-1400. — Бібліогр.: 37 назв. — англ. |
| collection | DSpace DC |
| description | The thermal conductivity of (CH4)1–c(CD4)c solid solutions with c = 0, 0.03, 0.065, 0.13, 0.22, 0.4, 0.78, and 1.0 has been measured in the region of existence of three orientational phases: disordered (phase I), partially ordered (phase II) and completely ordered (phase III). The temperature range is 1.3–30 K. It is shown that the thermal conductivity has different temperature dependences k(T) in these phases. Its value increases with the degree of the orientational order in the phase. In phase I the thermal conductivity is independent of c and weakly dependent on T. The impurity effect in k(T) is much stronger in the low-temperature part of phase II than in phase III. As the concentration c grows, the k(T) curve of phase II approaches the dependence k(T) typical of phase I. There is a hysteresis in the vicinity of the II↔III phase transition. In phase III the impurity effect in k(T) can be considered as phonon scattering at rotational defects developing due to the difference between the moments of inertia of the CH4 and CD4 molecules. The obtained dependences of thermal conductivity on temperature and concentration can be explained qualitatively assuming that the dominant mechanism of phonon scattering is connected with the interaction of phonons with the rotational motion of the molecules in all of the three orientational phases of the CH4–CD4 system.
|
| first_indexed | 2025-11-28T08:59:24Z |
| format | Article |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 12, p. 1393–1400
Orientational isotopic effects in the thermal conductivity
of CH4/CD4 solid solutions
A.I. Krivchikov
Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: krivchikov@ilt.kharkov.ua
P. Stachowiak, E. Pisarska, and A. Jezowski
Institute for Low Temperatures and Structure Research,
Polish Academy of Sciences, PN 1410, 50-950 Wroclaw, Poland
Received June 1, 2007, revised July 13, 2007
The thermal conductivity of (CH4)1–c(CD4)c solid solutions with c � 0, 0.03, 0.065, 0.13, 0.22, 0.4, 0.78,
and 1.0 has been measured in the region of existence of three orientational phases: disordered (phase I), par-
tially ordered (phase II) and completely ordered (phase III). The temperature range is 1.3–30 K. It is shown
that the thermal conductivity has different temperature dependences �( )T in these phases. Its value increases
with the degree of the orientational order in the phase. In phase I the thermal conductivity is independent of c
and weakly dependent on T. The impurity effect in �( )T is much stronger in the low-temperature part of phase
II than in phase III. As the concentration c grows, the �( )T curve of phase II approaches the dependence �( )T
typical of phase I. There is a hysteresis in the vicinity of the II�III phase transition. In phase III the impurity
effect in �( )T can be considered as phonon scattering at rotational defects developing due to the difference be-
tween the moments of inertia of the CH4 and CD4 molecules. The obtained dependences of thermal conductiv-
ity on temperature and concentration can be explained qualitatively assuming that the dominant mechanism of
phonon scattering is connected with the interaction of phonons with the rotational motion of the molecules in
all of the three orientational phases of the CH4–CD4 system.
PACS: 63.20.–e Phonons in crystal lattices;
66.70.+f Nonelectronic thermal conduction and heat-pulse propagation in solids; thermal waves.
Keywords: thermal conductivity, molecular crystal, orientational disorder, phonon scattering, isotopic ef-
fects, phase transition.
Introduction
Crystalline methane, deuteromethane and their solid so-
lutions are very interesting physical quantum objects: they
undergo structural solid-state orientational transformations
[1,2] and, besides, the rotational motion of their molecules
can proceed as librations and as weakly hindered or free ro-
tation. The strong isotopic effects observed in the properties
of these substances [3] are first of all connected with the ro-
tational degrees of freedom, the quantum statistics of nu-
clear spins and the weak anisotropic molecular interaction.
The phase transformations from the orientational disorder
(phase I) to the partial orientational order (phase II) and then
from phase II to the completely orientationally ordered
phase (phase III) occurs under equilibrium vapor pressure in
pure CD4 and its concentrated solutions with CH4, Kr, Xe
and at P � 200 bar in pure CH4. Deuteration considerably
affects the dynamics of the molecules in the orientationally
ordered phases of solid methane, mainly because the rota-
tional constant B / Ik B� � 2 (I is the moment of inertia) of
CD4 (B � 3.75 K) is half as high as that of CH4 ( .B � 7 5 K).
Under equilibrium vapor pressure solid CH4 experiences
one phase transition from orientationally disordered phase I
(plastic phase with cubic symmetry) to phase II with partial
orientational order, at TI II� = 20.4 K. Apart from phases I
and II with TI II� = 27.4 K, CD4 can have a state with a
complete orientational order (phase III). Its structure
(orthorombic symmetry group Cmca) is not cubic and is
very close to the structure of phase II [4]. The orientational
© A.I. Krivchikov, P. Stachowiak, E. Pisarska, and A. Jezowski, 2007
phase transformation at TII III� = 22.1 K is a first-order phase
transition with a volume jump of 0.63% [5,4], which is
much larger than that during the I–II phase transition. The
considerable difference between the rotational constants,
different total nuclear spins of the CD4 and CH4 molecules
and the weak molecular field are the factors that generate
strong orientational fluctuations in the vicinity of the phase
transition temperature [6,7]. In solid CH4–CD4 solutions
phase III can exist under the equilibrium vapor pressure in a
wide range of temperatures at CH4 concentrations c above
0.15 [2].
In phase I the molecules occupy the sites of the fcc lat-
tice. They possess high orientational mobility: the ther-
mal orientational fluctuations whose frequency increases
with temperature provoke random jump-like reorienta-
tions of the molecules (rotational diffusion). The rota-
tional diffusion of CH4 molecules occurs when the tem-
perature decreases down to the point of the orientational
phase transition (TI II� = 20.4 K). The orientational fluctu-
ations can be viewed as random rotational moment
through which the neighboring atoms and molecules in-
fluence the molecule. The fluctuations are usually of the
order of k T/BB [3].
The structure of phase II in CD4 was predicted by James
in 1958 [8] and detected experimentally by Press in 1973
[9]. In contrast to CD4, there is no direct unambiguous in-
formation about the structure of CH4. However, the experi-
mental evidence shows that in both substances phases II
have identical crystalline structures with the space group
Fm3c. The unit cell of phase II consists of 32 molecules
and 8 sublattices. In six sublattices the molecules are
orientationally ordered in opposite directions. In the other
two sublattices the anisotropic contributions of the nearest
molecules to the molecular field are counter balanced and
the molecules located at these sites behave as weakly hin-
dered rotators. The symmetry of the site is Oh for free rota-
tors and D d2 for orientationally ordered molecules. The
molecules at the Oh sites are weakly hindered rotators. The
structure of their low-lying energy levels (the splitting is
about 12 K) is quite similar to that of a free molecule. The
molecular octupole interaction in phase II of CH4 (CD4) is
not strong enough to align all the molecules in the pre-
ferred directions. The interactions cannot suppress the
large oscillations of the D d2 molecules about their equilib-
rium orientations in the potential wells. This property sets
methane off from other molecular crystals.
The structure of phase III has the spatial symmetry
Cmca [4]. It has a tetragonal primitive lattice with c �
= 11.708 � and a b� � 8.187 � (this corresponds effec-
tively to 1.0 % of tetragonal distortion). The unit cell con-
tains 16 molecules. The arrangement of the carbon atoms
is nearly the same as in the fcc lattice. The frustration ef-
fects generated by the ordering of tetrahedral molecules
prohibits a complete orientational order in the fcc lattice.
In phase III, unlike phase II, eight of 16 molecules in the
unit cell change their orientations drastically: four mole-
cules rotate by 90° and four molecules become orienta-
tionally ordered. The remaining eight molecules (ordered
in phase II) practically hold their initial positions and ori-
entations, the deviation being only 5.2°. In phase II (4m2)
the deviation of the most symmetrical orientation is
±4.5°. The libration amplitude is about 15° at all the sites
at T � 18 K [4].
On transformation from phase I to phase III, the CD4
(or CH4 under pressure) molecules change from complete
orientational disordering (phase I) to a partially ordered
state (phase II) and finally to complete ordering (phase
III). The molecular rotation changes drastically: from
random orientational wandering (rotational diffusion)
caused by the thermal orientational fluctuations (phase I)
to anharmonic librations about the ordered orientations
(phase III). Near the phase transitions temperature the
heat capacity exhibits anomalous features and consider-
able entropy effects [10] which account for intensive
orientational motion of the molecules. In phases II and III
the thermal orientational fluctuations provoke the mole-
cules to hop between the neighboring orientations. The
frequency of these reorientations decreases as the temper-
ature lowers and the fluctuations attenuate. The main fea-
tures of phases II and III are visualized in the rotational
spectrum. It is structured at low temperatures (with one or
more frequencies of coherent tunnel rotation), broadens
with increasing temperature and attains the shape typical
for no coherent thermoactivated orientational motion
(random reorientations [3]).
The thermal conductivity � is an useful physical pro-
perty in investigations of molecular substances [11]. By
measuring � as a function of temperature, it is possible to
characterize the orientational phases. For dynamically
disordered phases and orientational glasses � has been
found to be almost independent of temperature (weak in-
crease with temperature [11–13]). In the orientationally-
ordered phases � decreases exponentially as a function of
temperature and approaches T �1 near or above the Debye
temperature [14]. A deviation from T �1 is observed when
the mean free path of the thermal phonons is at its allow-
able minimum (about the size of the unit cell) [14].
Since the rotation of CH4 molecules and the trans-
lational vibrations of the lattice are interrelated, thermal
conductivity is a very sensitive tool of investigating the
orientational phase transitions in methane [15–17]. In our
experiments the thermal conductivity was investigated in
different orientational phases of (CH4)1–c(CD4)c solid so-
lutions in a wide interval of concentrations.
In deuteromethane the increase in temperature from
absolute zero can change the rotational motion of the ini-
tially (at T � 0) ordered molecules as follows:
1394 Fizika Nizkikh Temperatur, 2007, v. 33, No. 12
A.I. Krivchikov, P. Stachowiak, E. Pisarska, and A. Jezowski
– zero-amplitude librations enhance up to anharmonic
librations (phase III);
– the molecules undergo random jump-like reorienta-
tions (phase II);
– the frequency of reorientations increases up to rota-
tional diffusion (phase I).
The investigation of the intra- and interphase evolu-
tion of the thermal conductivity can provide new informa-
tion about the molecular dynamics in the cause of solid-
state transformations.
Experimental results
The experiments were carried out using the home-de-
signed setup described earlier [18]. The thermal conductiv-
ity was measured in the temperature range from 1.3 to 30 K
by the steady-state heat-flow method. The temperature and
its gradient along the sample were measured with two ger-
manium thermometers separated by 12 mm from each other.
The relative error of the thermal conductivity measurements
did not exceed 6%. The random error was no more than 2%.
The temperature gradient between the two thermometers
was about 0.03T. The procedure of growing and cooling the
(CH4)1–c(CD4)c crystals is described elsewhere [17]. Here
we report new experimental data on the thermal conducti-
vity of (CH4)1–c(CD4)c at c � 0 4. , 0.78 and 1.0 and analyze
the dependences of thermal conductivity on temperature and
CD4 concentration. Thermal conductivity data for the
CH4–CD4 system have been published in brief reports
[17,19,20]. The thermal conductivities of (CH4)1–c(CD4)c
are shown in Fig. 1 (weak solutions with c � 0, 0.03, 0.065,
and 0.13 CD4) and Fig. 2 (concentrated solutions with c �
= 0.22, 0.4, 0.78, and 1.0 CD4). The dependences of thermal
conductivity on CD4 concentration in phases I, II and III are
shown in Fig. 3.
The thermal conductivity of the CH4–CD4 system is de-
termined predominantly by the orientational order of the
phase. It increases as the dynamical disorder (phase I)
changes into partial (phase II) and then into complete
orientational order (phase III). In the general case this behav-
ior accounts for the attenuating anharmonic rotation at a
growing degree of orientational ordering. This regularity is
clearly illustrated by the impurity-caused isotopic effect in
�( )T and the jump in the concentration dependence of the
thermal conductivity (Fig. 3) during the II–III phase transi-
tion. The impurity effect in �( )T is much stronger in the
low-temperature part of phase II than in phase III. As the con-
centration c increases, the curve �( )T in the low-temperature
part of phase II approaches the �( )T of phase I. The impurity
effect in the thermal conductivity of phase III can be inter-
preted as scattering of phonons at the rotational defects that
develop due to the difference between the rotational constants
of the CH4 and CD4 molecules. The behavior of �( )T in the
high-temperature part of phase II (c � 0.22) is similar to that
Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions
Fizika Nizkikh Temperatur, 2007, v. 33, No. 12 1395
0
5 10 15 20 25 30 35
0.5
1.0
1.5
0.13
0.065
0.03
phase II
T, K
phase I
c = 0
W
/(
m
��
Fig. 1. Thermal conductivity of (CH4)1–c(CD4)c for c � 0, 0.03,
0.065, and 0.13. The dotted line is the temperature of orientational
I–II phase transition for c � 0.
0
5 10 15 20 25 30 35
1
2
3
T, K
W
/(
m
��
Fig. 2. Thermal conductivity of (CH4)1–c(CD4)c for c � 0.22
(�,�), 0.40 (�,�), 0.78 (�,�) and 1.00 (�,�) on increa-
sing (open symbols) and decreasing (solid symbols) tempera-
tures in the vicinity of the orientational phase transition II–III.
0 0,2 0,4 0,6 0,8 1,0
0
1
2
10 K
T = 5 K
23 K
W
/(
m
��
c
Fig. 3. Concentration dependence of the thermal conductivity of
(CH4)1–c(CD4)c for three different temperatures: T � 5 (�,�),
10 (�,�) and 23 (�) K. Open and solid symbols are data for
phases III and II, respectively, and the symbols� are for phase I.
of phase I: the thermal conductivity is independent of the CD4
concentration and decreases very slowly with decreasing
temperature (d /dT� � 0) down to the transition to the low-
temperature phase II or to phase III where d /dT� only
changes sign. In phase II of pure CH4 and phase III of pure
CD4 �( )T decreases exponentially with increasing tempera-
ture in the interval from 7 K to the temperature of the orien-
tational transition.
The thermal conductivity has a hysteresis in the vicin-
ity of the II–III phase transition [20]. Its width increases
with the CH4 concentration. The smoothed thermal con-
ductivities (c � 0.4) in the hysteresis region are shown in
Fig. 4. The marked points characterize the temperature
behavior of phase transformation. T2 is the onset of pha-
se III at lowering temperature, T3 specifies the moment
when phase II disappears. At rising temperature phase II
appears at T4 and phase III disappears at T1. A quasi-sta-
tionary two-phase state is observed in the interval T T3 1� ,
where the thermal conductivity measured at the same tem-
perature is independent of time (at least during 24 hours
of observation). On changing from cooling to heating or
vice versa, the curve �( )T reverses in the T T3 1� interval
and shows a weak dependence on temperature. The in-
verse �( )T (dashed line in Fig. 4) is reversible if the tem-
perature is varied within the hysteresis region, i.e., the
curve is reproducible no matter whether the temperature
is increasing or decreasing. At increasing concentration c
the hysteresis of �( )T shifts towards low temperatures and
its T and � ranges extend.
The new phase diagram of (CH4)1–c(CD4)c with the
specified parameters of the hysteresis ( , , , )T T T T1 2 3 4 is
shown in Fig. 5. It is in good agreement with published
data [1,2,10] and provides additional information about
the region of existence of the heterogeneous state. It is of
fundamental importance that the heterogeneous state is
quasistationary and stable on the reversal of temperature.
This state can be considered as a mixed two-phase state of
coexistence of phases II and III. In this region �( )T is ac-
tually an effective thermal conductivity �eff ( )T deter-
mined by the relation between the thermal conductivities
of the two phases. The curve �eff ( )T can be influenced to
some extent by the complex spatial structure of the mixed
state which can vary depending on the concentrations of
the phases.
The diagram of the orientational phases is shown in
Fig. 5. The II–III phase boundary was drawn on the basis
of the �( )T behavior. The shaded region shows possible
local formations of phase III in phase II. Such zones with
orientationally ordered molecules form along the line ex-
trapolated from the dependences of the II–III transition
temperatures on CD4 concentrations. The curve �( )T has
a kink at the transition point, which is particularly distinct
at c � 0.13 and decreases as the concentration c lowers. In
the region with phase III inclusions, the thermal conduc-
tivity �( )T is influenced by the temperature prehistory of
the sample. The nuclear spin conversion of CH4 mole-
cules causing relaxation of �( )T [21,22] has little effect
on �( )T .
In terms of quality, the change of the thermal conduc-
tivity versus sample composition follows general expec-
tations: the thermal conductivity of the solutions is lower
than that of pure crystals. An increase of the isotopic
admixture causes lowering of the thermal conductivity,
which is most prominent at the maximum in the tempera-
ture dependence. However, the dependence of the thermal
conductivity on temperature measured for samples con-
taining 0.065 and 0.13 CD4 diverges from that observed
for other isotopic solid solutions, see, e.g., [23,24]. For
these two concentrations of CD4 the maximum disappears
1396 Fizika Nizkikh Temperatur, 2007, v. 33, No. 12
A.I. Krivchikov, P. Stachowiak, E. Pisarska, and A. Jezowski
12 13 14 15 16 17
0.3
0.4
0.5
0.6
0.7
T, K
T3
T4
T2
T1
W
/(
m
��
Fig. 4. Hysteresis of the thermal conductivity of (CH4)1–c(CD4)c
for c � 0.4 in the vicinity of orientational phase transition II–III.
Points T2, T3 (decreasing temperature) and points T4, T1 (increasing
temperature) correspond to the beginning and the finish of the
phase transition. Arrows show the direction of temperature change.
0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
phase I
phase II
phase III
T
,
K
c
Fig. 5. Part of the phase diagram of CH4–CD4 solid solution. We
show here the temperature of the phase transition II–III in depend-
ence on concentration of CD4 in the mixed crystal (CH4)1–c(CD4)c
obtained from a fixed thermal conductivity (solid lines), from
x-ray scattering (�) [1], from NMR (�,�) [2] and from heat ca-
pacity (�) [10] measurements. The shaded area contains local
orientational ordered regions of phase III.
and the dependence resembles that known rather for
«glassy» solids than the crystalline ones: initially, at the
lowest investigated temperatures, �( )T increases approxi-
mately as T 2, then the thermal conductivity saturates and,
starting from 5–6 K up to the temperature of the phase
transition, does not change within the accuracy of the ex-
periment. The value of thermal conductivity within the
plateau area amounts to 0.3 W/(m·K), which is consistent
with values found for some glasses.
Discussion
The obtained dependences of thermal conductivity on
temperature and concentration in solid solutions can be
explained quantitatively assuming that phonon scattering
is governed predominantly by the interaction between the
phonons and the rotational motion of molecules in three
orientational phases of the CH4–CD4 system. The �( )T
behavior in pure CH4 and CD4 is determined by the
orientational dynamics at lowering temperature. The rota-
tional motion of molecules in phase I can be interpreted as
follows [25]. Each molecule by itself is a hindered rota-
tor whose rotation in a short interval of time is randomly
modulated by the neighboring molecules through a weak
anisotropic molecular interaction. The rotation of the mo-
lecule at the site can be considered as a stochastic process
disturbed randomly by the neighboring molecules which
rotate randomly and independently of one another. The
modulated hindered rotation is a source of additional
phonon scattering.
At the point of the transition to the partially ordered
phase there is a kink in the thermal conductivity curve. In
the ordered phase the thermal conductivity passes through
the maximum and then decreases with the lowering temper-
ature. The initial increase of the thermal conductivity of the
ordered phase of methane is due to the decrease of the num-
ber of phonons with the energy sufficient to interact with the
scattering centers at lower temperatures. Its further decrease
is due to the reducing density of thermal phonons which are
scattered by the crystal boundaries and other defects.
The curve �( )T measured on pure substances (CH4 in
phase II and CD4 in phase III) can be described [17] by
the expression based on the approximation of additivity
of thermal resistance
�
� �
� �
( )
( ) ( )
( ) ( )
T
T T
T T
�
1 2
1 2
, (1)
where �1 1
2( )T Ñ T� and �2 2( ) exp( )T Ñ E/T� . Deutera-
tion of CH4 has a very little effect on the fitting parame-
ters, which are C1 � 0.08 W/(m·K3) and 0.1 W/(m·K3) and
C 2 � 0.08 W/(m·K) and 0.14 W/(m·K) for CD4 and CH4,
respectively. The exponential growth of the thermal con-
ductivity with lowering temperature in the low-tempera-
ture phase (phase II for CH4 and phase III for CD4) corre-
sponds to thermally activated scattering of phonons by
the rotational states against the background of pho-
non–phonon scattering (Umklapp process). The activa-
tion energies are E � 35 K for CD4 and 12 K for CH4. The
transition from the nonactivated mechanism in phase I to
the activated regime in phases II and III accounts for the
change in the rotational motion. The librational motions
of the ordered molecules are thermally activated and their
amplitudes decrease with lowering temperature. It is nat-
ural that the activation energy is lower for CH4 molecules
than for the CD4 ones (because of the different rotational
constants of these molecules).
Traditionally, the temperature dependence of thermal
conductivity is described quite accurately by the De-
bye–Pierls model of an isotropic solid. The model disre-
gards the difference between the phonon modes of differ-
ent polarizations
K T
k
s
T x
x
dxB
x
x
/T
( ) ( )
( )
�
��
4
2 3
4
2
0
2
e
1 e�
�
3
�
, (2)
where k B is the Boltzmann constant, � is Planck's con-
stant, � is the Debye temperature, s is the mean sound ve-
locity, x /k TB� �� , and
( )x is the effective relaxation
time of the phonons participating in scattering. The nor-
mal phonon–phonon processes can be omitted from con-
sideration because of their low intensity in CH4 and CD4
[26–28]. The inverse relaxation time (relaxation rate)
��1( ) is a sum of the rates of all resistive processes of
phonon scattering. The temperature dependence of the
thermal conductivity of a molecular crystal is determined
by the mechanisms of phonon scattering typical for crys-
tals with no rotational degrees of freedom (Umklapp pro-
cesses
U
�1, boundary scattering
B
�1, scattering at disloca-
tions
dis
�1 , and point defects
R
�1) and by the additional
mechanism allowing a coupling of the phonon gas and the
rotational degrees of freedom (librations and/or rotation
of molecules) of the molecular crystal.
The mechanism of the rotation-translation coupling is
quite clear in the general case [29]. There is no theoretical
model permitting a quantitative description of the thermal
conductivity in different orientational phases of the crys-
tal. It is known that the thermal conductivity of the orien-
tationally disordered phase is little dependent on tempe-
rature. The experimental data on the low-temperature
thermal conductivity of simple molecular crystals: H2
[30], N2 [31], CO [32], CH4 [17,26], CD4 [17], CO2 [33],
N2O [33,34], and others [35,36] describes only the gen-
eral behavior of the thermal conductivity of the orienta-
tionally ordered crystal. The temperature dependence of
the thermal conductivity of molecular crystal is identical
to that of an atomic crystal with defects and can be de-
scribed formally by the mechanisms typical of crystals
Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions
Fizika Nizkikh Temperatur, 2007, v. 33, No. 12 1397
having no rotational degrees of freedom. The total relax-
ation rate
��1( ) can be written down as
�
�
�
�� � � ��
1 1 1 1( , ) ( , ) ( ) ( )T TU Rdis . (3)
In Eq. (3) the relaxation rate of the thermal activated
U-processes is
� �U T A T b/T� � �1 2( , ) exp( ) , (4)
where A and b are the intensity and the activation energy
of the U-process, respectively. In the case of scattering at
dislocations, the relaxation time is assigned as
� �dis
� �1 ( ) D , (5)
where D is parameter dependent on the density of disloca-
tions.
In an isotropic continuum the relaxation rate of the
phonons scattered on an isotropic point defects is [24]
� �R C� �1 4( ) , (6)
where C is the Rayleigh scattering parameter, C �
�V / s0
34� � � , and V0 is the volume per molecule. The co-
efficient � determines the intensity of scattering.
The experimental data on �( )T in phases II and III exi-
sting at helium temperature are well described by the cal-
culated curves that allow for the thermally activated, Ray-
leigh and dislocation mechanisms (see Fig. 6). The fitting
parameters for these mechanisms are given in the Table.
Note that below the temperature of the orientation II–III
phase transition in pure CD4 (and the I–II transition in pure
CH4) the thermal conductivity changes as �( ) exp( )T b/T� .
This behavior can be explained assuming that the phonon
scattering at librations is a thermally activated process simi-
lar to phonon–phonon scattering (U-process) which is of
essential importance in high-temperature of phase I. It is ob-
vious that the activation energy b is more than two times
Table. Parameters used to describe the thermal conductivity of
(CH4)1–c(CD4)c for the thermoactivated processes ( ( , )
�U T� �1
� �A T b/T�2 exp( )), scattering at dislocations (
� �def
–1 ( ) � D ) and
Rayleigh scattering ( ( ) )
� �R C� �1 4
c A�10
16
, s/K b, K D�10
4
C �1012, s
3
Phase III
100 35 40 5 7.5
78 35 40 10 35
40 35 40 10 110
22 35 40 12 100
Phase II
13 9 9 33 30
6 9 9 23 10
3 9 9 6.7 10
0 15 16 4.2 5
higher in pure CD4. There are two main reasons for this
difference. First, the barrier hindering reorientations of the
molecules is stronger in phase III than in phase II. Second,
the moment of inertia of the CD4 molecule is twice as large
as that of the CH4 molecule. In the low-temperature region
the character of the dependence �( )T is determined by
phonon scattering at dislocations. Its intensity is almost in-
dependent of complete deuteration. The scattering in-
creases in the solution because the density of dislocations
becomes higher at the expense of dislocations generated in
the process of growing and subsequent cooling the sam-
ples. The curve �( )T measured at the background of dislo-
cation-induced scattering has no signs of resonance scat-
tering at the rotational tunnel states of the molecules. The
contribution of the Rayleigh scattering to the thermal con-
ductivity becomes evident in phase III of the concentrated
(CH4)1–c(CD4)c solutions. The intensity of the Rayleigh
scattering is shown in Fig. 7 as a function of CD4 concen-
tration. The parameter C is nearly an order of magnitude
higher than for scattering at isotopic point defects. The dif-
ference is due to local changes in the mass of molecules.
The parameter � describing local mass variations in a mo-
lecular crystalline two-component system consisting of
two kinds of molecules (CH4 and CD4) with different
masses M1 and M 2 is
�
�
� � �
��
�
��
c c
M
M
( )1
2
, (7)
1398 Fizika Nizkikh Temperatur, 2007, v. 33, No. 12
A.I. Krivchikov, P. Stachowiak, E. Pisarska, and A. Jezowski
0
5 10 15 20
1
2
3
T, K
W
/(
m
��
Fig. 6. Thermal conductivity of (CH4)1–c(CD4)c for c � 0 (�),
0.03 (�), 0.06 (�), 0.22 (�), 0.40 (�), 0.78 (�), and 1.00
(�). Solid lines are calculated curves using fitted parameters
from the Table.
where M ñM c M�
�1 21( ) is the average mass, and
�M M M� �1 2 is the mass difference.
The above fact suggests an additional mechanism of
strong scattering in the orientationally ordered phase of
the two-component solution. This is the interaction be-
tween the phonons and the librations of molecules. The
authors propose a phenomenological description of this
mechanism by analogy with the mechanism based on
local mass variations.
Diluting CH4 with CD4 does not produce a significant
change in the picture of tunneling and librational (rota-
tional) states of phase II of solid regular methane. As it
has been shown in high-resolution neutron spectroscopy
experiments [37], the shift of the lines is less than 10% in
the concentration range 0–0.15 of CD4 molecules in the
CH4–CD4 solid solution. The lines are broadened by a
factor of ~2–3 in this concentration range. These varia-
tions of the energy of the states and the line shapes are not
sufficient to explain the dramatic change in the depen-
dence of the thermal conductivity on temperature.
The CD4 molecule features a stronger effective octu-
pole moment than that of the molecule of CH4. There-
fore, partial replacement of CH4 molecules with their
heavier isotopic counterpart in the structure of phase II of
protonated methane changes (increases) the electrostatic
field at the position of the neighbors of the replaced mole-
cules. In particular, it affects the sites of almost freely ro-
tating molecules by lowering the symmetry of the field
experienced by the rotating molecule, which in turn re-
sults in stronger hindering of the molecule. It means a
stronger interaction of the rotating molecule with the
surroundings. Therefore, the glassy-like behavior of the
thermal conductivity coefficient of the CH4–CD4 crystal
can be explained by the enhancement of interaction of the
rotations with crystal phonons.
On the other hand, the Rayleigh scattering parameter C
of the molecular crystalline two-component system of
two-sorts of molecules (with moments I1 and I 2) is well
described by Eq. (7) if the mass of the molecule is re-
placed by its moment of inertia
�
�
� � �
��
�
��
c c
I
I
( )1
2
, (8)
where I cI c I�
�1 21( ) is the average moment of inertia,
and is the difference between the moments of inertia of
two molecules.
Equation (8) describes quantitatively with a good ac-
curacy the behavior of the parameter C in phase III at
varying concentration c (see Fig. 7). Note that a straight
forward calculation of thermal conductivity resulting
from the phonon-rotational coupling is a complicated and
unsolved yet theoretical problem of a three-dimensional
lattice. This is a challenge even for a structurally simple
orientationally ordered molecular crystal such as
(CH4)1–c(CD4)c. The influence of local changes in the
moments of inertia upon the thermal conductivity points
to the importance of kinetic processes for rotational dy-
namics of orientationally ordered molecules interacting
with thermal phonons.
Conclusions
The thermal conductivity of the solid (CH4)1–c(CD4)c
with c � 0, 0.03, 0.065, 0.13, 0.22, 0.4, 0.78 and 1.0 has
been measured in the region of existence of three orien-
tational phases: disordered (phase I), partially ordered
(phase II) and completely ordered (phase III). The tem-
perature interval of measurement was 1.3–30 K. The tem-
perature dependence of the thermal conductivity �( )T be-
haves differently in these phases. As the degree of the
orientational order increases, the thermal conductivity
grows too. In phase I the thermal conductivity is inde-
pendent of c and little dependent on T . The impurity effect
in �( )T is much stronger in the low-temperature region of
phase II than of phase III. At increasing c the curve �( )T
of phase II approaches the dependence typical for phase I.
There is a hysteresis in the vicinity of the II�III phase
transition. In phase III the impurity effect in �( )T can be
interpreted as phonon scattering on the rotational defects
appearing due to the difference between the moments of
inertia of the CH4 and CD4 molecules. The hysteresis of
the thermal conductivity observed in our experiments in-
dicates that the orientational phase transition from the
partially ordered phase to a complete order is a continu-
ous process with an intermediate mixed two-phase stage.
The temperature interval of the mixed state increases lin-
early with the CH4 concentration in the CD4 solution.
Such continuous transition is possible if the interphase
surface energy is low and there are strong temperature
Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions
Fizika Nizkikh Temperatur, 2007, v. 33, No. 12 1399
0 0.2 0.4 0.6 0.8 1.0
0
50
100
c
C
Fig. 7. The Rayleigh scattering parameter C dependence on CD4
concentration c get from: the fitting procedure (�), the calcu-
lations using the mass difference of molecules (doted line) and
the difference between the moments of inertia of two molecules
(solid line).
and pressure fluctuations in the presence of defects. The
transition is accompanied by the formation of a complex
inhomogeneous bulk structure.
The obtained dependences of thermal conductivity on
temperature and concentration can be explained qualita-
tively assuming that the dominant mechanism of phonon
scattering is connected with the interaction between the
phonons and the rotational motion of the molecules in all
of the three orientational phases of the CH4–CD4 system.
In the general case this behavior accounts for the lower-
ing degree of the anharmonic rotational motion at increas-
ing orientational order.
Authors are grateful to Prof. V.G. Manzhelii, Prof. A.I.
Prokhvatilov, Prof. Yu.A. Freiman, Prof. W. Press, Dr.
V.A. Konstantinov and Dr. M. Prager for fruitful discus-
sions.
This work was supported by the Polish State Commit-
tee for Scientific Research.
1. A.I. Prokhvatilov and A.P. Isakina, Fiz. Nizk. Temp. 10,
1206 (1984) [Sov. J. Low Temp. Phys. 10, 631 (1984)].
2. F. Lostak, K.O. Prins, and N.J. Trappeniers, Physica B162,
254 (1990); Physica B162, 21 (1990); Physica B+C 139–140,
272 (1986).
3. W. Press, Single-Particle Rotations in Molecular Crystals,
Springer Tracts of Modern Physics 92, Springer, Berlin,
Heidelberg, New York (1981).
4. M.A. Neumann, W. Press, C. Noldeke, B. Asmussen, M.
Prager, and R.M. Ibberson, J. Chem. Phys. 119, 1586 (2003).
5. A.I. Prokhvatilov and A.P. Isakina, Phys. State Solidi (a)
78, 147 (1983).
6. D. Marx and M. Muser, J. Phys.: Condens. Matter 11, R117
(1999).
7. W. Press, A, H�ller, H. Stiller, W. Stirling, and R. Currat,
Phys. Rev. Lett. 32, 1354 (1974).
8. H.M. James and T.A. Keenan, J. Chem. Phys. 31, 12 (1959).
9. W. Press and A. H�ller, Phys. Rev. Lett. 30, 1207 (1973).
10. E. Bartholome, G. Drikos, and A. Eucken, Z. Phys. Chem.
39, 371 (1938).
11. R.G. Ross, P. Andersson, B. Sundqvist, and G. Backstrom,
Rep. Prog. Phys. 47, 1347 (1984).
12. L.A. Koloskova, I.N. Krupskii, and V.G. Manzhelii, J.
Low Temp. Phys. 14, 403 (1974); V.A. Konstantinov, V.G.
Manzhelii, V.P. Revyakin, and S.A. Smirnov, Physica
B262, 421 (1999); G. Manzhelii and I.N. Krupskii, Fiz.
Tverd. Tela 10, 284 (1968).
13. A.I. Krivchikov, A.N. Yushchenko, V.G. Manzhelii, O.A.
Korolyuk, F.J. Bermejo, R. Fern�ndez-Perea, C. Cabrillo,
and M.A. González, Phys. Rev. B74, 060201(R) (2006).
14. V.A. Konstantinov, Fiz. Nizk. Temp. 29, 567 (2003) [Low
Temp. Phys. 29, 422 (2003)].
15. A. Jezowski, H. Misiorek, V.V. Sumarokov, and B.Ya.
Gorodilov, Phys. Rev. B55, 5578 (1997).
16. A.I. Krivchikov, B.Ya. Gorodilov, O.A. Korolyuk, V.G.
Manzhelii, and V.V. Dudkin, Phys. State Solidi (c) 1, 2959
(2004).
17. P. Stachowiak, E. Pisarska, A. Jezowski, and A.I. Kriv-
chikov, Phys. Rev. B73, 134301 (2006).
18. A. Jezowski and P. Stachowiak, Cryogenics 32, 601 (1992).
19. P. Stachowiak, E. Pisarska, A. Jezowski, and A.I. Kriv-
chikov, Europhys. Lett. 74, 96 (2006).
20. A.I. Krivchikov, P. Stachowiak, E. Pisarska, and A. Jezowski,
Phys. Rev. B75, 012303 (2007).
21. B.Ya. Gorodilov, A.I. Krivchikov, and O.A. Korolyuk, Fiz.
Nizk. Temp. 31, 1158 (2005) [J. Low Temp. Phys. 31, 884
(2005)].
22. E. Pisarska, P. Stachowiak, and A. Jezowski, Fiz. Nizk.
Temp. 33, 768 (2007) [Low Temp. Phys. 33, 587 (2007)].
23. R.M. Kimber and S.J. Rogers, J. Phys.C: Solid State Phys.
6, 2279 (1973).
24. R. Berman, Thermal Conduction in Solids, Clarendon Press,
Oxford (1976).
25. H. Yasuda, J. Low Temp. Phys. 31, 223 (1978).
26. A. Jezowski, H. Misiorek, V.V. Sumarokov, and B.Ya.
Gorodilov, Phys. Rev. B55, 5578 (1997).
27. O.A. Korolyuk, B.Ya. Gorodilov, A.I. Krivchikov, and
V.V. Dudkin, Fiz. Nizk. Temp. 26, 323 (2000) [Low Temp.
Phys. 26, 235 (2000)].
28. V.A. Konstantinov, V.G. Manzhelii, R.O. Pohl, and V.P.
Revyakin, Fiz. Nizk. Temp. 27, 1159 (2001) [Low Temp.
Phys. 27, 858 (2001)].
29. R.M. Lynden-Bell and K.H. Michel, Rev. Mod. Phys. 66,
721 (1994).
30. O.A. Korolyuk, B.Ya. Gorodilov, A.I. Krivchikov, A.S.
Pirogov, and V.V. Dudkin, J. Low Temp. Phys. 111, 515
(1998).
31. P. Stachowiak, V.V. Sumarokov, J. Mucha, and A. Jezowski,
Phys. Rev. B50, 543 (1994).
32. P. Stachowiak, V.V. Sumarokov, J. Mucha, and A. Jezowski,
J. Low Temp. Phys. 111, 379 (1998).
33. V.V. Sumarokov, P. Stachowiak, J. Mucha, and A. Jezowski,
Phys. Rev. B74, 224302 (2006).
34. P. Stachowiak, V.V. Sumarokov, J. Mucha, and A. Jezowski,
Phys. Rev. B67, 172102 (2003).
35. Structure and Thermodynamic Properties of Cryocrystals:
Handbook, V.G. Manzhelii (ed.), Begell House Hardcover,
Begell House (1998).
36. R.G. Ross, P. Andersson, B. Sundqvist, and G. Backstrom,
Rep. Prog. Phys. 47, 1347 (1984).
37. M. Prager and W.J. Press, Chem. Phys. 92, 5517 (1990).
1400 Fizika Nizkikh Temperatur, 2007, v. 33, No. 12
A.I. Krivchikov, P. Stachowiak, E. Pisarska, and A. Jezowski
|
| id | nasplib_isofts_kiev_ua-123456789-7774 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-11-28T08:59:24Z |
| publishDate | 2007 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Krivchikov, A.I. Stachowiak, P. Pisarska, E. Jezowski, A. 2010-04-12T13:15:22Z 2010-04-12T13:15:22Z 2007 Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions / A.I. Krivchikov, P. Stachowiak, E. Pisarska, A. Jezowski // Физика низких температур. — 2007. — Т. 33, № 12. — С. 1393-1400. — Бібліогр.: 37 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/7774 The thermal conductivity of (CH4)1–c(CD4)c solid solutions with c = 0, 0.03, 0.065, 0.13, 0.22, 0.4, 0.78, and 1.0 has been measured in the region of existence of three orientational phases: disordered (phase I), partially ordered (phase II) and completely ordered (phase III). The temperature range is 1.3–30 K. It is shown that the thermal conductivity has different temperature dependences k(T) in these phases. Its value increases with the degree of the orientational order in the phase. In phase I the thermal conductivity is independent of c and weakly dependent on T. The impurity effect in k(T) is much stronger in the low-temperature part of phase II than in phase III. As the concentration c grows, the k(T) curve of phase II approaches the dependence k(T) typical of phase I. There is a hysteresis in the vicinity of the II↔III phase transition. In phase III the impurity effect in k(T) can be considered as phonon scattering at rotational defects developing due to the difference between the moments of inertia of the CH4 and CD4 molecules. The obtained dependences of thermal conductivity on temperature and concentration can be explained qualitatively assuming that the dominant mechanism of phonon scattering is connected with the interaction of phonons with the rotational motion of the molecules in all of the three orientational phases of the CH4–CD4 system. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Динамика кристаллической решетки Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions Article published earlier |
| spellingShingle | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions Krivchikov, A.I. Stachowiak, P. Pisarska, E. Jezowski, A. Динамика кристаллической решетки |
| title | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions |
| title_full | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions |
| title_fullStr | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions |
| title_full_unstemmed | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions |
| title_short | Orientational isotopic effects in the thermal conductivity of CH4/CD4 solid solutions |
| title_sort | orientational isotopic effects in the thermal conductivity of ch4/cd4 solid solutions |
| topic | Динамика кристаллической решетки |
| topic_facet | Динамика кристаллической решетки |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/7774 |
| work_keys_str_mv | AT krivchikovai orientationalisotopiceffectsinthethermalconductivityofch4cd4solidsolutions AT stachowiakp orientationalisotopiceffectsinthethermalconductivityofch4cd4solidsolutions AT pisarskae orientationalisotopiceffectsinthethermalconductivityofch4cd4solidsolutions AT jezowskia orientationalisotopiceffectsinthethermalconductivityofch4cd4solidsolutions |