Download Thermal transport properties of halide solid solutions: Experiments

THE JOURNAL OF CHEMICAL PHYSICS 142, 124109 (2015)
Thermal transport properties of halide solid solutions: Experiments
vs equilibrium molecular dynamics
Aïmen E. Gheribi,1,a) Mathieu Salanne,2 and Patrice Chartrand1
1
CRCT—Centre for Research in Computational Thermochemistry, Department of Chemical Engineering,
École Polytechnique, P.O. Box 6079, Station Downtown, Montréal, Québec H3C 3A7, Canada
2
Sorbonne Universits, UPMC Univ Paris 06, UMR 8234, PHENIX, F-75005 Paris, France
(Received 22 December 2014; accepted 9 March 2015; published online 31 March 2015)
The composition dependence of thermal transport properties of the (Na,K)Cl rocksalt solid solution
is investigated through equilibrium molecular dynamics (EMD) simulations in the entire range of
composition and the results are compared with experiments published in recent work [Gheribi et al.,
J. Chem. phys. 141, 104508 (2014)]. The thermal di↵usivity of the (Na,K)Cl solid solution has
been measured from 473 K to 823 K using the laser flash technique, and the thermal conductivity
was deduced from critically assessed data of heat capacity and density. The thermal conductivity
was also predicted at 900 K in the entire range of composition by a series of EMD simulations
in both NPT and NVT statistical ensembles using the Green-Kubo theory. The aim of the present
paper is to provide an objective analysis of the capability of EMD simulations in predicting the
composition dependence of the thermal transport properties of halide solid solutions. According to
the Klemens-Callaway [P. G. Klemens, Phys. Rev. 119, 507 (1960) and J. Callaway and H. C. von
Bayer, Phys. Rev. 120, 1149 (1960)] theory, the thermal conductivity degradation of the solid solution
is explained by mass and strain field fluctuations upon the phonon scattering cross section. A rigorous
analysis of the consistency between the theoretical approach and the EMD simulations is discussed
in detail. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4915524]
I. INTRODUCTION
In the last ten years, interest in renewable energy has
increased due to environmental concerns and the increasing
price of fossil fuel energy. The latent heat energy storage
technology using Phase Change Materials (PCMs) is one of
the most promising renewable energy sources for the near
future. Thermal energy storage in salt PCM represents a serious
option for solar energy applications. PCMs present many
possibilities for halide mixtures:1 from simple systems with
a single common anion, like NaCl-KCl, to complex reciprocal
systems with several common anions and cations. The heat
transfer mechanism is one of the most important issues in
designing a new PCM. It is usually preferable to maximize the
thermal conductivity of the substance in order to provide the
minimum temperature gradients and facilitate the charge and
discharge of heat. Consequently, a precise knowledge of the
thermal transport properties in the halide system is required. Of
course, other issues also have to be taken into consideration in
the design of PCM. For example, halides of transitions metals
are not suitable for a possible new PCM mixture because they
are usually highly corrosive and an inert and expensive coating
is necessary to protect the container.
The thermal conductivities of pure stoichiometric halide
compounds can be predicted in a rather large range of temperature with an appreciable precision either using a theoretical
approach2 or equilibrium molecular dynamics (EMD) simua)Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2015/142(12)/124109/8/$30.00
lations.3 These two methods have the same order of precision2–4 less than about ±15%. This is generally the typical
experimental range of error. Unlike pure compounds, the
thermal transport properties of halides’ solid solutions (SS)
are not well known. Although solid solutions are observed in
many halide systems,5 almost no experimental data on their
thermal conductivity are reported in the literature. This lack of
experimental data is a significant barrier in the design of new
PCM materials.
In other systems, it has been shown experimentally6–9 that
the thermal conductivity of solid solutions formed from oxides and semiconductors deviates considerably from a simple
linear behaviour. An analogous behaviour can reasonably be
expected for halide solid solutions. However, so far, EMD has
been applied to predict, with good accuracy, only the thermal
conductivity of halides’ liquid solutions.10–15 No study reporting EMD calculations of the thermal conductivity of halide
solid solutions can be found in the literature.
For solid stoichiometric compounds3 and liquid solutions,
Ohtori et al.12,13 have presented a predictive approach for the
determination of the thermal conductivity. Indeed, no experimental information has been used for the parametrization of
the interionic potentials, only density functional theory calculations. A similar approach is applied in this work.
The main purpose of this paper is to examine the predictive
capability of EMD simulations for the composition dependence of the thermal conductivity of halide solid solutions.
We perform a comparative study between EMD simulation
experiments in the NaCl-KCl system, which can be considered
a good prototype system because the interactions within the
142, 124109-1
© 2015 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12
124109-2
Gheribi, Salanne, and Chartrand
system are purely ionic. Both (i) the method of calculation of
the thermal conductivity from the phase trajectory generated
with EMD simulation and (ii) the interionic potential used in
the present work were already applied in predicting successfully the thermal conductivity of pure ionic compounds as a
function of temperature, with an error of about 15% or less.3,16
The original contribution of the present work is the study of the
predictive capability of EMD for the composition dependence
of the thermal transport of halide solid solutions.
II. EXPERIMENTS AND SIMULATION DETAILS
A. Experimental determination of thermal conductivity
of the (Na,K)Cl solid solution
The pair temperature-composition for which the thermal
di↵usivity has been measured17 with laser flash and the thermal
conductivity, directly predicted by the present EMD simulations, is shown in Figure 1 inside the NaCl-KCl coherent
phase diagram. As shown in Figure 1, there is no sample for
which the thermal conductivity has been both measured and
calculated. The reason is that during the experiments, it was
necessary to keep the sample at a temperature of at least 100 K
below the minimum liquidus temperature, i.e., below ⇠830 K.
This temperature is the safest maximum temperature at which
the flash apparatus can be used without damage by liquid
formation. On the other hand, the prediction of the thermal
conductivity by our EMD is limited to rather high temperatures
for practical reasons discussed below.
As also shown in Figure 1, all the studied samples are
out side the coherent miscibility gap. Thus only the thermal
conductivity of the single phase solid solution (with a rocksalt
structure Fm3̄m) is studied in this work.
The experimental thermal conductivity of the (Na,K)Cl
rocksalt solid solution as a function of both temperature and
composition has already been presented in a previous work.17
J. Chem. Phys. 142, 124109 (2015)
The thermal conductivity, , was deduced from measurement
of the thermal di↵usivity, a, using the laser flash method18 and
the critically evaluated heat capacity CP and density ⇢, using
the relation: = a· ⇢·CP .
B. Simulations details
The EMD simulations were carried out for (Na,K)Cl rocksalt solid solution at 900 K from pure KCl to pure NaCl by
varying the molar fraction x KCl by steps of 0.1 mole fraction
steps (the exact compositions are given in Table I together with
the calculated physical properties). The total number of ions is
512 (256 cations and 256 anions). A first series of simulations
were performed in the isobaric-isothermal statistical ensemble
(NPT) at T = 900 K and P = 105 Pa in order to determine the
equilibrium density, the heat capacity at constant pressure, CP ,
and the bulk modulus. It also generated thermally equilibrated
configurations. The two cationic positions (Na+ and K+) were
chosen randomly in the initial configuration of the NPT simulation. In a second step, starting with an initial configuration
perfectly thermally equilibrated (obtained from previous NPT
simulations), a new series of EMD simulations was performed
and was done in the canonical statistical ensemble (NVT), in
order to calculate the thermal conductivity of the (Na,K)Cl
solid solution. The temperature and the pressure for the NPT
simulations were controlled, respectively, with a weak NoséHoover thermostat19 and an extension of the Martyna barostat.20 The thermostat and barostat relaxation times were both
0.5 ps. The equations of motions were integrated using the
velocity Verlet algorithm21 with a time step of 1 fs for the NPT
and NVT simulations as well. The total simulation time for the
NPT simulation is 2.5 ns while the total simulation time for
the NVT simulation is 30 ns. All simulations were performed
using periodic boundary conditions and the minimum image
convention.
The so-called Polarizable Ion Model (DPIM) potential
was used to describe the ionic interaction; it has the form
TABLE I. Thermal conductivity ( ), heat capacity (C P ), density (⇢), and
thermal di↵usivity (a) of the (Na,K)Cl solid solution as a function of composition at 900 K calculated by the present EMD simulation. The rmse of each
calculated property is given in parenthesis next to each property. The units
of all the properties are in SI (i.e., in W m 1 K 1, C P in J kg 1 K 1, ⇢ in
kg m 3, and a in m2 m 1).
NVT
X KCl
FIG. 1. The coherent NaCl-KCl phase diagram along with the set of
composition-temperature points for which the thermal di↵usivity has been
measured by laser flash method17 (solid red square) and calculated by EMD
simulations (solid blue circle). Outside the miscibility gap, the SS is stable and inside the miscibility gap, phase separation into two distinct solid
solutions SS1 and SS2 occurs. All the studied compositions are below the
minimum liquidus temperature of 927 K and above the consolute temperature
of 453 K; thus they are in the stable solid solution region.
0.000
0.102
0.199
0.301
0.398
0.500
0.602
0.716
0.797
0.898
1.000
N PT
1017.62
991.46
971.52
960.83
944.31
929.99
903.69
880.17
863.54
835.10
807.65
107 ·a
⇢
CP
1.912 (0.009)
1.421 (0.007)
1.209 (0.008)
1.114 (0.0101)
0.949 (0.009)
0.934 (0.011)
0.982 (0.010)
0.968 (0.007)
1.031 (0.009)
1.107 (0.007)
1.185 (0.009)
C al c
(35.62)
(41.64)
(27.20)
(29.78)
(17.94)
(33.47)
(28.91)
(23.76)
(28.49)
(26.72)
(19.38)
1952.88
1943.47
1936.76
1934.45
1931.73
1930.29
1930.07
1933.02
1933.40
1935.84
1940.17
(13.67)
(13.55)
(17.41)
(7.73)
(11.22)
(9.81)
(17.91)
(8.02)
(11.07)
(10.12)
(15.67)
9.621
7.375
6.425
5.993
5.203
5.203
5.630
5.689
6.175
6.848
7.562
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12
124109-3
Gheribi, Salanne, and Chartrand
disp
pol
Vi j (r i j ) = Vicj (r i j ) + Virep
j (r i j ) + Vi j (r i j ) + Vi j (r i j ),
J. Chem. Phys. 142, 124109 (2015)
(1)
where Vi j is the total interionic pair potential describing the
total interaction between the i-th and j-th ions. In the DPIM,
Vi j is formulated as a sum of four independent contributions.
V c is the classical Coulomb potential describing the chargecharge interactions, V rep is the repulsive contribution to the
total potential, V disp is a dispersion term, and V pol takes into
account the charge-dipole, dipole-charge, and dipole-dipole
interactions. Additional details can be found in the literature.12–14,22 The parameters describing the total pairwise potentials between Na+-Na+, K+-K+, Cl -Cl , Cl -Na+, and Cl -K+
ionic pairs were taken from the previous work of Ohtori et al.13
The potentials were parametrized on the basis of atomistic
first principle electronic structure calculations. Specifically,
they are determined by matching the dipoles and forces on the
ions calculated from first-principles on condensed phase ionic
configurations.23 In other words, no experimental information
has been used in the parametrization of the potentials; thus, the
present EMD calculations of all physical properties presented
in this paper, in addition to the thermal conductivity, are purely
predictive. It may be pointed out that the predictive capability
of potentials has been already tested12 for many structural,
thermodynamic, and transport properties in both liquid and
solid phases.
Thermal conductivity is a non-equilibrium property. The
Green-Kubo formalism24 permits, via the fluctuation-dissipation theorem, its determination from the fluctuations of energy
and eventually of the charge currents obtained from EMD.
According to the Green-Kubo method, the thermal conductivity of stoichiometric ionic compounds and binary mixtures
is written, respectively, as3
(⌧,T, X) =
1
(L ee (⌧,T, X)
T2
and
(⌧,T, X) =
1
(L ee (⌧,T, X)
T2
L 2ez (⌧,T, X)
)
L z z (⌧,T, X)
(2)
N(⌧,T, X)
)
D(⌧,T, X)
(3)
comparison, the ionic conductivities in molten NaCl and KCl
are, respectively, 3570 S m 1 and 2030 S m 1.26 Consequently,
can be approximated as
(⌧,T, X) ⌘
1
· L ee (⌧, X,T)
T2
(6)
without any loss of precision.
To calculate the thermal conductivity, for each composition, the total NVT run has been divided into 30 blocks of
1 ns and the di↵erent transport coefficients were calculated
independently in each block. The correlation time, ⌧, was
chosen to be large enough to ensure the convergence of the
thermal conductivity. ⌧ was then set to 20 ps for all studied
compositions. Finally, the thermal conductivity of a perfectly
thermally equilibrated system is defined as an average value
over all the di↵erent blocks and expressed as
(T, X) = lim h (⌧,T, X)iall blocks.
⌧!1
(7)
Normally, should reach a plateau when ⌧ is large enough
(⌧ ! 1); nevertheless, we found that fluctuates slightly
around its average value in the plateau region due to numerical
uncertainties. Consequently, the value of thermal conductivity
reported in Table I is in fact the average value of in the plateau
region. The uncertainty associated in the thermal conductivity
value is defined as the root mean square error (rmse) in the
plateau region.
In Figure 2, we show the calculated thermal conductivity
of the (Na,K)Cl solid solution as a function of correlation time
at 900 K and at equimolar composition. In this case, the plateau
is reached at about 9 ps and the dispersion of the data in the
plateau region is about 1%; this value is weak compared to the
typical 10%–20 % of the experimental errors.
with the numerator N, and the denominator, D, respectively,
equal to
N = L 2ez 1 L z 2z 2 + L 2ez 2 L z 1z 1
D = L z 1z 1 L z 2z 2
L 2z 1z 2.
2L ez 1 L ez 2 L z 1z 2v,
(4a)
(4b)
L ↵ are the transport coefficients defined by the Green-Kubo
formula,24
⌅ ⌧
1
L ↵ (⌧,T, X) =
hJ↵ (t) · J (0)idt,
(5)
3k BV 0
where ↵ and can take the value of either e or z i . k B and V
are, respectively, the Boltzmann constant and the equilibrium
volume of the system. The two vectors Je and Jz i are the
energy and the charge flux of the i-th ion. Here, the ionic
conduction in the materials is negligible, so that we can safely
ignore the right end of Eqs. (2) and (3). Indeed, the electrical
conduction in chloride crystals near their melting temperatures
is small because it is due principally to the generation and
movement of negative ion vacancies. The magnitude of the
conductivity for NaCl and KCl is less than 10 3 S m 1.25 For
FIG. 2. Thermal conductivity (open dotted circle) of (Na,K)Cl solid solution
at 900 K at equimolar composition as a function of correlation time ⌧
averaged over 30 EMD simulation blocks of 1 ns each. The dashed line
represents the average value of the thermal conductivity in the plateau region.
The plateau is reached at about 9 ps.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12
124109-4
Gheribi, Salanne, and Chartrand
J. Chem. Phys. 142, 124109 (2015)
III. RESULTS AND DISCUSSION
The thermal transport properties within a material are
characterized by the heat capacity and by either the thermal
conductivity or the thermal di↵usivity. If CP is known, and a
are in principle equivalent and the value of one can be deduced
from the value of the other even though these two properties
are conceptually di↵erent. describes the ability of a material
to conduct heat and has physical significance in the context
of conduction processes in the steady state; a quantifies the
thermal inertia of the material and has physical significance
in the context of transient conduction processes. Normally, a
is the direct measured thermal transport property while can
be predicted through EMD simulations.
In Table I, we present the thermal properties and the density of the (Na,K)Cl solid solution as a function of composition
at 900 K obtained by the present EMD simulations. The equilibrium density of the system is obtained simply by averaging
the time of the instantaneous volume in the NPT ensemble
while the equilibrium heat capacity is determined from the
mean-squared enthalpy (H = E + PV ) variance, also in the
NPT ensemble through the well known statistical relationship,
CP =
hH 2i⌧ hHi⌧2
.
kB · T 2
(8)
For the calculation of both ⇢ and CP , the time average
was performed above 500 ps in order to let the system reach
thermal equilibrium. For all compositions, we have observed
that (⌧) reaches a plateau before 10 ps and the thermal conductivity values were calculated by averaging (⌧) for ⌧ 10 ps.
Figure 3 shows both the experimental and the calculated
thermal conductivities of the (Na,K)Cl rocksalt solid solu-
tion as functions of temperature and composition. The EMD
prediction of the thermal conductivity of pure NaCl at 900 K
is in very good agreement with the critically assessed value
reported by Gheribi et al.,17 with a di↵erence of about 5%.
Unlike the NaCl case, the accuracy of about 25% obtained for
KCl is relatively poor. This is surprising given that the thermal
conductivity of molten KCl was predicted by Ohtori et al.13
with an error of less than 1%, even though they used the same
interionic potentials and the Green-Kubo method. Both the
experiment and EMD simulation show that is a quadratic
function of composition, with a minimum in the vicinity of the
equimolar composition. In Figure 4, we show the temperature
dependence of both experimental and calculated in the vicinity of its minimum, i.e., for XKCl = {0.4, 0.5, 0.6}. Around
the composition of the minimum, is found to decrease as
1/T and, at a constant temperature, is almost constant in the
composition range 0.4 . XKCl . 0.6. As shown in Figure 4,
the agreement between experiment and EMD prediction is
excellent, as EMD data are very close (<5%) to the value
obtained by linearly extrapolation of experimental data.
For several types of properties (e.g., thermodynamic properties), the compositional dependence is often represented
through the deviation from a linear behaviour by the so-called
excess properties. For the thermal conductivity, the deviation
from linearity is of the same order of magnitude as the thermal
conductivity itself. The compositional dependence of cannot
properly be represented by an excess term. It is rather more
suitable to refer to the thermal conductivity degradation as
proposed in a prior work.17 The thermal conductivity degradation, D , is a measure of the relative deviation from a linear
dependence of with x and is defined as
D (X,T) = [1
FIG. 3. Experimental and calculated thermal conductivity of the (Na,K)Cl
rocksalt solid solution as a function of temperature and composition. The
experimental data (open symbols) were measured in the range 473 K  T
 773 K and were presented in a recent publication.17 The EMD simulations
(filled stars symbols) have been performed only at 900 K. The critically assessed values17 of the thermal conductivity of pure NaCl and KCl compounds
at 900 K are also reported (filled square). Point interpolation were splined
(dashed lines).
XNaCl ·
sol(X,T)
NaCl(T)
+ XKCl ·
KCl(T)
].
(9)
FIG. 4. Experimental (open symbol) and calculated (filled symbol) thermal
conductivities of the (Na,K)Cl rocksalt solid solution as a function of inverse
temperature in the vicinity of the equimolar composition. The solid line
represents the linear interpolation of the experimental data and the dashed line
is the extrapolation up to 1/900 K 1. The experimental value at X KCl = 0.5
was obtained by the spline interpolation shown in Figure 3.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12
124109-5
Gheribi, Salanne, and Chartrand
J. Chem. Phys. 142, 124109 (2015)
FIG. 5. Experimental and calculated thermal conductivity degradations of
(Na,K)Cl rocksalt solid solution as a function of temperature and composition. The experimental data (open symbols) were deduced from the experimental thermal conductivity,17 while the EMD data (filled stars symbols)
were deduced from the present calculations. The point were interpolated by
splines.
FIG. 6. Experimental and calculated maximum thermal conductivity degradations of the (Na,K)Cl rocksalt solid solution as a function of temperature.
The solid line represents the linear interpolation of the experimental points
(open squares), the dashed line is its extrapolation up to 900 K, and the
solid star is the maximum thermal conductivity degradation obtained from
the present EMD simulations.
The thermal conductivity degradation is a proper way
to define the composition-thermal conductivity relationship.
Such a definition clearly points out the strong deviation from
the linearity of the thermal transport properties relations.
Figure 5 shows the thermal conductivity degradation deduced,
respectively, from the experimental data of Gheribi et al.17 and
from the present EMD simulations. Both experiment and EMD
calculations show a quadratic shape of the thermal conductivity
degradation versus composition. The experimental maximum
D is observed around x KCl ⇠ 0.46 for all studied temperatures, which is very close to the value of x KCl ⇠ 0.43 for
the maximum D obtained by EMD simulations T = 900 K.
The temperature dependence of the maximum D is shown in
Figure 6; the value deduced from experimental data shows a
linear dependence with temperature. The linear extrapolation
up to 900 K leads to a value of 63% which is very close to
the value of 68% obtained by EMD simulation. However, the
predicted thermal conductivity degradation is clearly overestimated in the NaCl rich side while it fits almost perfectly
in the KCl rich side. The infinitesimal relative change of the
thermal conductivity due to the addition of one N a+ in the pure
K+ sublattice and vice versa can be defined by an expression
analogous to that of enthalpy at infinite dilution as follows:
K–Cl bond vibration at high temperature. The e↵ects on this
limitation must also be observed for all thermal properties of
the solid solution in the KCl rich side. Both the heat capacity
and the excess heat capacity, CPx s , are shown in Figure 8. The
di↵erence between the calculated and experimental heat capacity and the excess heat capacity is very similar to what was
obtained for the thermal conductivity and thermal conductivity
degradation. The prediction of CP is more accurate for NaCl
with only about 1.5% di↵erence, while for KCl, the di↵erence
is more than 5.5%. Like the thermal conductivity degradation
(Figure 5), the excess heat capacity is also predicted with a
poor accuracy on the NaCl rich side, in this case with the
opposite sign, and with very good accuracy in the KCl rich side.
This is also another hint of the limitation of the K+/Cl pair
potential to predict accurately the thermal excitation of K–Cl
bond vibration at high temperature.
i in j
sol
=
1
lim (
jCl X jC l !0
@ sol
).
@ X jCl
(10)
in K
Both Ksolin Na and Na
, derived from experimental
sol
data and EMD simulations, are shown in Figure 7. Ksolin Na
deduced from experiment is found to be almost independent
in K
of temperature with a value of about 0.8 whereas Na
sol
decreases with temperature. Its magnitude doubles from 473 K
to 773 K. Given both the experimental and the numerical
errors, it can be assumed that the accuracy of EMD prediction
in K
of Na
is satisfactory, unlike the prediction of Ksolin Na,
sol
which is clearly not acceptable. This reflects a limitation of the
K+/Cl potential to predict accurately the thermal excitation of
FIG. 7. Experimental (open symbols) and calculated (filled symbols)
K in Na and
K in Na.
sol
sol
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12
124109-6
Gheribi, Salanne, and Chartrand
J. Chem. Phys. 142, 124109 (2015)
FIG. 8. Calculated versus critically assessed heat capacity bottom and excess
heat capacity (top). The critical assessment of the heat capacity of NaCl-KCl
roksalt solid solution is due to Sangster and Pelton.27
At the microscopic scale, for insulating materials, the thermal conductivity degradation is explained by a modification
of the phonon-phonon relaxation time. In an isotropic lattice,
and assuming only the nearest neighbour interactions, it has
been shown by Klemens28,29 that the relaxation time due to the
atomic disorder is given by
⌧
1
=
!4a3
,
4⇡vg3
(11)
where !, a3, vg , and are, respectively, the phonon frequency,
the atomic volume, the phonon group velocity, and the disorder
parameter. The disorder parameter characterizes the phononphonon scattering cross section of the solute ion, i, in the
cationic sublattice. According to the Klemens-Abeles7,29,43
approximation, is expressed in terms of the fluctuation of
the mass and the elastic strain field induced by the alloying
e↵ect and, for a binary solid solution, it is expressed as
= X i X j {( M/M)2 + " anhar[( a/a)2]},
(12)
where M = Mi M j and a = ai a j . The two terms inside
the bracket are, respectively, the mass and the elastic strain
field fluctuation terms of the disorder parameter. " anhar is an
e↵ective parameter related to the average anharmonicity of the
bond. The mass fluctuation term depends only on the mass
of each cation constituting the solid solution. " is in general
considered as an adjustable parameter. The value of " anhar is
usually optimized to fit the experimental data by assuming
that the volume of the solid solution is perfectly known, or
by considering a mixing rule such as Vegard’s or Zen’s rule.30
Therefore, the two key parameters for the understanding of
thermal conductivity behaviour of the (Na,K)Cl rocksalt solid
solution with composition are the lattice constant and the anharmonicity of the interionic bond.
The calculated lattice parameter of the (Na,K)Cl rocksalt
solid solution, its absolute deviation from Vegard’s rule (i.e.,
linear behaviour of a with x), and its derivative versus the
composition are shown in Figure 9 in comparison with experimental values after the recent critical assessment by Walker
et al.31,32 As shown in Figure 9, the composition dependence
FIG. 9. EMD calculations versus critically assessed values for lattice parameter, a (bottom), the absolute deviation of a from Vegard’s rule a
(middle), and the first derivative of da/dx with composition (top). The
critical assessment of the lattice constant of NaCl-KCl roksalt solid solution
is from Walker et al.31,32
of the lattice parameter of the solid solution is reasonably well
predicted. Both available experimental data and the present
EMD simulations show that a deviates slightly from Vegard’s
rule (with respective maximum deviations at the equimolar
composition of 0.5% and 0.4%). Unlike CP , the compositional
dependence of a is well predicted on both NaCl and KCl rich
sides. The lattice parameters of pure NaCl are overestimated
by 1% while for pure KCl, it is underestimated by 1.6%.
The interaction potentials used in this work were constructed
from atomistic first-principles calculations using the generalized gradient approximation (GGA) according to the Perdew,
Burke, and Ernzerhof (PBE) scheme.33,34 Generally, for ionically bonded systems, the GGA-PBE method overestimates
the lattice parameter.35,36 At 0 K, the di↵erences between the
predicted lattice parameter with the GGA-PBE method36 and
the experimental value37 are, respectively, 1.05% and 1.5%
for NaCl and KCl. These di↵erences remain almost identical
at 900 K for NaCl but are significantly di↵erent for KCl, the
di↵erence decreases from 1.5% to 1.5% involving a sign
change. The thermal expansion of NaCl is accurately predicted
by EMD while it is underestimated for KCl by about 20%. One
may note that this instance is similar to what was also observed
for the thermal conductivity and heat capacity, except for the
fact that the excess property, a, in this case, is predicted
with satisfactory accuracy in the entire range of composition.
Moreover, the di↵erence observed for the CP of KCl is much
lower than what is observed for its thermal conductivity and
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12
124109-7
Gheribi, Salanne, and Chartrand
J. Chem. Phys. 142, 124109 (2015)
lattice parameter. The main contribution to the heat capacity is
the harmonic vibration of the lattice, contrary to the physical
origin of the thermal expansion, which is due purely to the
anharmonic vibration of the lattice.38 Consequently, it is clear
that the majority of the discrepancy observed between EMD
calculations and experimental values of the CP , , and a for
KCl is mainly due to the failure of the pair potential to describe
properly the anharmonic part of the K+–Cl bond vibration at
high temperature.
Finally, to end the discussion, we propose an objective
synthesis of the overall quality of the predictive capacity of the
EMD simulations. To be as rigorous as possible, the quality of
the present EMD predictions should be determined uniquely
by comparing the calculations to the experimental results.
Unfortunately, the experimental device, namely, the laser flash,
cannot be used at 900 K and EMD simulations failed to
converge for a long enough time at 773 K. All attempts at MD
simulations performed at 773 K failed before approximately
1 ns. This simulation time is too small to determine the thermal
conductivity properly. A same situation was also observed by
Salanne et al.3 in recent work. They managed to calculate the
thermal conductivity of pure NaCl only above 900 K.
In the range of temperature 473 K  T  773 K, the
experimental D can be fitted by a quadratic function of
composition given by
D
,exp(X,T)
XKCl · (1
XKCl)
= [2.4 · 10 3 ·T
0.25 · (2· XKCl
1)].
(13)
This equation and the value of the critically assessed thermal
conductivity of pure NaCl and KCl permit the extrapolation
of the experimental thermal conductivity for the solid solution
at 900 K. This equation is a linear relation that we assume
to be valid up to 900 K, i.e., 127 K above the experimental
temperature limit. This extrapolation is shown in Figure 10
along with the present EMD simulation results. The experi-
mental and numerical errors are also reported in this figure for
better visualization of the accuracy of the present EMD simulations. Except on the KCl rich side (x KCl & 0.9), EMD data
and experimental data extrapolations are in good agreement.
Therefore, the inability of the potential to describe properly
the anharmonic part of the K+–Cl high temperature bond
vibration has a negative impact on the thermal conductivity
only on the KCl rich side of the solid solution.
IV. CONCLUSION
This work was motivated primarily by the lack of experimental data on thermal conductivity and/or di↵usivity in the
literature for halide solid solutions, even though this information is critical for many industrial applications, such as thermal
energy storage involved in solar energy plants.
The aim of this work is to provide an objective analysis
of the accuracy of equilibrium molecular dynamics simulations in predicting the thermal transport properties of halide
solid solutions. The (Na,K)Cl rocksalt solid solution was presented as a prototype system. A series of equilibrium molecular
dynamics simulations was performed in order to predict the
thermal transport properties. The proposed approach is purely
predictive since the interionic potentials used in this study were
obtained from atomistic first principle calculation and thus no
experimental information has been used.
In conclusion, by using the theoretical method presented
in this work, the thermal conductivity of binary and higher
order halide solid solutions can be predicted with confidence.
Similar to what has been done for metal systems,39–42 a further
work will be proposed for a comprehensive study of the relations between equilibrium, structure, and thermal transport
properties of ionic solid solutions. Generally, as shown in
Figure 10, the accuracy of the predictions is within the margin
of experimental error, up to the critical composition of XKCl
. 0.9. The discrepancy between the calculated and experimental thermal conductivities is explained by the weakness of
the pair potential to describe the anharmonic part of the K+–Cl
high temperature bond vibration.
ACKNOWLEDGMENTS
This research was supported by funding from the Natural Sciences and Engineering Research Council of Canada
(NSERC) and Rio-Tinto-Alcan. Stimulating discussions with
Professor Laszlo Kiss are warmly acknowledged. We thank
Eve Bélisle and Dr. James Sangster for their help in writing
the manuscript. We are grateful to Calcul Québec (CQ) for
computer resources.
1A.
FIG. 10. Calculated thermal conductivity (filled stars) and extrapolation of
the thermal conductivity using experimental data (solid line) of Gheribi
et al.17 of the (Na,K)Cl rocksalt solid solution at 900 K. The two dashed
lines represent the experimental error of ±11% reported by Gheribi et al.17
for the experimental thermal conductivity at 773 K.
Sharma, V. Tyagi, C. Chen, and D. Buddhi, “Review on thermal energy
storage with phase change materials and applications,” Renewable Sustainable Energy Rev. 13, 318–345 (2009).
2A. E. Gheribi and P. Chartrand, “Application of the calphad method to predict
the thermal conductivity in dielectric and semiconductor crystals,” Calphad
39, 70–79 (2012).
3M. Salanne, D. Marrocchelli, C. Merlet, N. Ohtori, and P. A. Madden, “Thermal conductivity of ionic systems from equilibrium molecular dynamics,” J.
Phys.: Condens. Matter 23, 102101 (2011).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12
124109-8
4G.
Gheribi, Salanne, and Chartrand
V. Paolini, P. J. D. Lindan, and J. H. Harding, “The thermal conductivity
of defective crystals,” J. Chem. Phys. 106, 3681 (1997).
5See http://www.crct.polymtl.ca/fact/documentation/ftsalt/ for the description of halides system phases diagrams.
6M. Murabayashi, “Thermal conductivity of ceramic solid solutions,” J. Nucl.
Sci. Technol. 7, 559–563 (1970).
7B. Abeles, “Lattice thermal conductivity of disordered semiconductor alloys
at high temperatures,” Phys. Rev. 131, 1906–1911 (1963).
8J. L. Wang, H. Wang, G. J. Snyder, X. Zhang, Z. H. Ni, and Y. F. Chen,
“Characteristics of lattice thermal conductivity and carrier mobility of undoped pbse-pbs solid solutions,” J. Phys. D: Appl. Phys. 46, 405301 (2013).
9M. R. Winter and D. R. Clarke, “Thermal conductivity of yttria-stabilized
zirconiahafnia solid solutions,” Acta Mater. 54, 5051–5059 (2006).
10A. E. Gheribi, J. A. Torres, and P. Chartrand, “Recommended values for the
thermal conductivity of molten salts between the melting and boiling points,”
Sol. Energy Mater. Sol. Cells 126, 11–25 (2014).
11A. Gheribi, D. Corradini, L. Dewan, P. Chartrand, C. Simon, P. Madden, and
M. Salanne, “Prediction of the thermophysical properties of molten salt fast
reactor fuel from first-principles,” Mol. Phys. 112, 1305–1312 (2014).
12N. Ohtori, T. Oono, and K. Takase, “Thermal conductivity of molten alkali
halides: Temperature and density dependence,” J. Chem. Phys. 130, 044505
(2009).
13N. Ohtori, M. Salanne, and P. A. Madden, “Calculations of the thermal
conductivities of ionic materials by simulation with polarizable interaction
potentials,” J. Chem. Phys. 130, 104507 (2009).
14Y. Ishii, K. Sato, M. Salanne, P. A. Madden, and N. Ohtori, “Thermal
conductivity of molten alkali metal fluorides (lif, naf, kf) and their mixtures,”
J. Phys. Chem. B 118, 3385–3391 (2014).
15M. Salanne, C. Simon, P. Turq, and P. A. Madden, “Heat-transport properties
of molten fluorides: Determination from first-principles,” J. Fluorine Chem.
130, 38–44 (2009).
16V. Haigis, M. Salanne, S. Simon, M. Wilke, and S. Jahn, “Molecular dynamics simulations of y in silicate melts and implications for trace element
partitioning,” Chem. Geol. 346, 14–21 (2013).
17A. E. Gheribi, S. Poncsk, R. St-Pierre, L. I. Kiss, and P. Chartrand, “Thermal
conductivity of halide solid solutions: Measurement and prediction,” J.
Chem. Phys. 141, 104508 (2014).
18W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash method of
determining thermal di↵usivity, heat capacity, and thermal conductivity,” J.
Appl. Phys. 32, 1679 (1961).
19S. Nose, “A unified formulation of the constant temperature molecular
dynamics methods,” J. Chem. Phys. 81, 511–519 (1984).
20G. J. Martyna, D. J. Tobias, and M. L. Klein, “Constant pressure molecular
dynamics algorithms,” J. Chem. Phys. 101, 4177–4189 (1994).
21D. C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge
University Press, New York, NY, USA, 1996).
22M. Salanne, B. Rotenberg, S. Jahn, R. Vuilleumier, C. Simon, and P. Madden,
“Including many-body e↵ects in models for ionic liquids,” Theor. Chem.
Acc. 131, 1143 (2012).
23P. A. Madden, R. Heaton, A. Aguado, and S. Jahn, “From first-principles to
material properties,” J. Mol. Struct.: THEOCHEM 771, 9–18 (2006).
J. Chem. Phys. 142, 124109 (2015)
24P.
Sindzingre and M. J. Gillan, “A computer simulation study of transport
coefficients in alkali halides,” J. Phys.: Condens. Matter 2, 7033 (1990).
25C. E. Skov and E. A. Pearlstein, “Nonlinear ionic conductivity in alkalihalide crystals,” Phys. Rev. 137, A1483–A1495 (1965).
26S. Smedley, “Electrical conductivity in ionic liquids at high temperatures,”
in The Interpretation of Ionic Conductivity in Liquids (Springer, US, 1980),
pp. 101–132.
27J. Sangster and A. D. Pelton, “Phase diagrams and thermodynamic properties
of the 70 binary alkali halide systems having common ions,” J. Phys. Chem.
Ref. Data 16, 509 (1987).
28P. G. Klemens, “Thermal resistance due to point defects at high temperatures,” Phys. Rev. 119, 507–509 (1960).
29P. G. Klemens, “The scattering of low-frequency lattice waves by static
imperfections,” Proc. Phys. Soc. Sect. A 68, 1113 (1955).
30D. Sirdeshmukh and K. Srinivas, “Physical properties of mixed crystals of
alkali halides,” J. Mater. Sci. 21, 4117–4130 (1986).
31D. Walker, P. K. Verma, L. M. D. Cranswick, R. L. Jones, S. M. Clark, and S.
Buhre, “Halite-sylvite thermoelasticity,” Am. Mineral. 89, 204–210 (2004).
32D. Walker, P. K. Verma, L. M. D. Cranswick, S. M. Clark, R. L. Jones, and
S. Buhre, “Halite-sylvite thermoconsolution,” Am. Mineral. 90, 229–239
(2005).
33J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865–3868 (1996).
34J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)],” Phys. Rev. Lett. 78,
1396 (1997).
35P. Haas, F. Tran, and P. Blaha, “Calculation of the lattice constant of solids
with semilocal functionals,” Phys. Rev. B 79, 085104 (2009).
36L. Liu, X. Wu, R. Wang, H. Feng, and S. Wu, “On the generalized stacking
energy, core structure and peierls stress of the dislocations in alkali halide,”
Eur. Phys. J. B 85, 58 (2012).
37P. D. Pathak and N. G. Vasavada, “Thermal expansion of NaCl, KCl and CsBr
by x-ray di↵raction and the law of corresponding states,” Acta Crystallogr.,
Sect. A 26, 655–658 (1970).
38N. Ashcroft and N. Mermin, Solid State Physics, HRW international editions
(Holt, Rinehart and Winston, 1976).
39J.-P. Harvey, A. E. Gheribi, and P. Chartrand, “Accurate determination of the
Gibbs energy of cuzr melts using the thermodynamic integration method in
Monte Carlo simulations,” J. Chem. Phys. 135, 084502 (2011).
40J.-P. Harvey, A. E. Gheribi, and P. Chartrand, “On the determination of the
glass forming ability of alxzr1x alloys using molecular dynamics, Monte
Carlo simulations, and classical thermodynamics,” J. Appl. Phys. 112,
073508 (2012).
41J.-P. Harvey, A. E. Gheribi, and P. Chartrand, “Thermodynamic integration
based on classical atomistic simulations to determine the Gibbs energy of
condensed phases: Calculation of the aluminum-zirconium system,” Phys.
Rev. B 86, 224202 (2012).
42A. Gheribi, “Molecular dynamics study of stable and undercooled liquid
zirconium based on {MEAM} interatomic potential,” Mater. Chem. Phys.
116, 489–496 (2009).
43J. Callaway and H. C. von Bayer, Phys. Rev. 120, 1149 (1960).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
134.157.146.58 On: Tue, 31 Mar 2015 16:31:12