Determination of the thermal conductivity in zirconia based inert matrix nuclear fuel by oscillating differential scanning calorimetry and laser flash

Article abstract of DOI:10.1016/s0040-6031(98)00504-8

The performances of oscillating differential scanning calorimetry (ODSC) and laser-flash technique in determining the thermal conductivity and the specific heat of zirconia-based materials, analogous to a potential nuclear fuel, were compared. The tested materials were (Zr_1-x-y-z,Y_x,Er_y,Me_z)O _2-(x+y)/2, with Me=Ce or Th. The measured specific heats were around 0.4 J K^-1 g^-1 and the thermal conductivities ranged from 2 to 3 W K^-1 m^-1. The ODSC measurements resulted in c_p values. The thermal conductivity was derived from two complementary measurements, one with a thin and the other with a thick sample. The laser-flash technique directly delivered the thermal diffusivity of the sample; consequently, the specific heat capacity c_p has to be known for determining the thermal conductivity. The ODSC measurements were affected by the position of the sample on the support. This, consequently, influenced the reproducibility of the measurements. The reproducibility of the scans by laser flash was excellent. Thermal conductivity decreased with increase in the stabilizer (Y, Er) concentration. This trend was justified on the basis of a model including concentration and size of the oxygen vacancies.

Full text of DOI:10.1016/s0040-6031(98)00504-8

Thermochimica Acta 323 (1998) 109±121
Determination of the thermal conductivity in zirconia based inert matrix nuclear
fuel by oscillating differential scanning calorimetry and laser ¯ash
M.A. Pouchona,b,*, C. Degueldrea,b, P. Tissotb
a Paul Scherrer Institute, Laboratory for Materials Behaviour, CH-5232 Villigen PSI, Switzerland
b
Â
University of Geneva, Groupe de Chimie Appliquee, CH-1211 Geneva 4, Switzerland
Received 20 March 1998; accepted 26 July 1998
Abstract
The performances of oscillating differential scanning calorimetry (ODSC) and laser-¯ash technique in determining the
thermal conductivity and the speci®c heat of zirconia-based materials, analogous to a potential nuclear fuel, were compared.
The tested materials were (Zr1
z,Yx,Ery,Mez)O2 (x‡y)/2, with MeˆCe or Th. The measured speci®c heats were around
x
y
0.4 J K 1 g 1 and the thermal conductivities ranged from 2 to 3 W K 1 m 1. The ODSC measurements resulted in cp values.
The thermal conductivity was derived from two complementary measurements, one with a thin and the other with a thick
sample. The laser-¯ash technique directly delivered the thermal diffusivity of the sample; consequently, the speci®c heat
capacity cp has to be known for determining the thermal conductivity. The ODSC measurements were affected by the position
of the sample on the support. This, consequently, in¯uenced the reproducibility of the measurements. The reproducibility of
the scans by laser ¯ash was excellent. Thermal conductivity decreased with increase in the stabilizer (Y, Er) concentration.
This trend was justi®ed on the basis of a model including concentration and size of the oxygen vacancies. # 1998 Elsevier
Science B.V. All rights reserved.
Keywords: Heat conductivity; Innovative nuclear fuel; Laser ¯ash; Oscillating differential scanning calorimetry; Zirconia-
based ceramics
1. Introduction
use of a U-free fuel is necessary. An inert-matrix
fuel material, based on yttria-stabilised zirconia
(Zrx,Yy)O2 y/2 (¯uorite structure) as inert component,
has been studied because of its high melting point,
its low neutron-capture cross section, and its low
chemical reactivity and because of the possibility
of recycling the cladding material of LWR's (i.e.
Zircalloy) from signi®cant waste stream.
A rare-earth oxide may be added as burnable poison
component. Erbium oxide was selected by Degueldre
et al. [1], while gadolinium oxide was chosen by
Nitani et al. [3]. The advantage of erbium in a sin-
gle-phase zirconia-based material is that it may con-
tribute to the stability of the solid solution [4]. These
inert-matrix components are selected to dissolve the
A zirconia-based inert matrix has been suggested
for burning in light-water reactor (LWR) plutonium
excesses which are due to the discontinuation of
nuclear military programs and to the worldwide
increase in use of nuclear energy [1]. The world
production of plutonium, estimated at the end of
1990, was ca. 650 tons [1,2] and Switzerland produces
annually (1993) ca. 750 kg of plutonium [1,2]. The use
of mixed oxide, (U,Pu)O2, is not suf®cient to balance
this production. An ef®cient strategy based on the
*Corresponding author. Fax: +41-56-310-2205, e-mail: pou-
chon@psi.ch
0040-6031/98/$ ± see front matter # 1998 Elsevier Science B.V. All rights reserved.
P I I : S 0 0 4 0 - 6 0 3 1 ( 9 8 ) 0 0 5 0 4 - 8

110
M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
actinide oxide, or in this work CeO2 or ThO2 taken as
analogous compounds of PuO2. The phases composed
of the inert-matrix material and ThO2 or CeO2 are
called simulated fuel.
called the sample. By applying the same heating rate
onto both samples, the heat capacity difference yields
a temperature gradient. The two samples are con-
nected and the temperature gradient yields a heat ¯ow
È (see Fig. 1(a)). In conventional DSC, the heating
rate is constant. The ODSC is derived from DSC by
applying a temperature modulation onto the DSC
signal. It allows the application of a suf®cient tem-
perature gradient with a small signal amplitude. The
analysis can, consequently, be made under almost
isothermal conditions.
For the laser-¯ash test, an energy impulse is applied
to one side of the sample. This photon energy shot
must be transformed to a thermal impulse. Because the
inert-matrix material is semi-transparent, a sample
coating is required (e.g. gold and carbon) in order
to absorb light and yield a heat impulse on the front
side. Without such a surface treatment, the energy
transfer would be driven by a combination of phonon
and photon transport. The back side of semi-transpar-
ent samples is also coated and the heat signal is
detected with the help of an infrared detector.
Nevertheless, zirconia-derived materials are
reported to have a rather low thermal conductivity.
This is a negative aspect about energy transfer from
the fuel pellet towards the coolant in the reactor. The
purpose of this paper is to present the measured
thermal conductivities of selected fuel analogue zir-
conia-based materials. The oscillating differential
scanning calorimetric (ODSC) and laser-¯ash meth-
ods were applied. Results are compared with emphasis
on the precision and accuracy of the results.
2. Methodological background
2.1. Principles of the measuring techniques
Analyses by differential scanning calorimetry
(DSC) are based on the heat capacity difference of
the system at two different positions. These positions
are called reference and sample positions. Both sides
are charged with a support device, this can be an
aluminium pot with cover or a simple aluminium
plate, but it is always the same device on both sides.
Both sides can be charged with a sample. In the
following descriptions the support±sample system is
2.2. Theoretical background and modelling
In a nuclear fuel, a ®ssion-product recoil yields a
thermal spike of the order of 3000 K over a 10 nm
range, for a period of the order of 10 11 s [5,6]. In a
Fig. 1. Schemes explaining the principle of the two measurement techniques: (a) `ODSC' method: A sinusoidal temperature signal is applied
to the heater block of the ODSC equipment and the temperature difference between sample and reference sides is measured; and (b) `laser-
flash' method: A laser flash induces an energy pulse onto the front side of the sample, the energy is then transferred to the backside of the
pellet from where it is detected by an IR detector.

M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
111
semi-transparent material, this energy can be trans-
ferred by phonons and by photons. In a conservative
way, this paper deals only with the phonon part of the
energy transfer.
The phonon conductivity ꢀ0 is currently modelled
on the basis of its inverse, whose variation with the
temperature (T) may be modelled as,
ity was observed as a function of the sample thickness
with variation for slim samples only. Since this effect
was observed for samples smaller than 0.8 mm, it is
not considered in this work because the samples here
are larger than 1 mm.
2.3. ODSC theoretical background
1
ꢀ0 ˆ A ‡ BT
which, according to Ambegaokar [7], yields:
(1)
In the ODSC, a sinusoidal temperature signal is
applied to the heater block (see Fig. 1(a)) of the
instrument. This temperature signal Tb(t) is given by:
4ꢁ2V0TD
A ˆ
Â
(2)
Tbꢀt† ˆ Tb ‡ qt ‡ AT sin ꢀ!t† (7)
where Tb (K) is the temperature of the block at time
v2h
and (e.g. Gibby [8]):
ꢂ2h3
1=3
3:81MꢀV0† ꢀkTD†
ˆÆ i may be estimated as follows (see Ref. [9] or
0
b
0
zero, q (K min 1) the underlying heating rate, t (s) the
time, AT (K) the amplitude of the modulated tem-
b
B ˆ
(3)
perature in the block, and ! (s 1) the angular fre-
quency of the modulation.
3
Here, the thermal properties of the materials are
investigated at a de®ned temperature. When the tem-
perature of interest is reached, the heating rate, q, is set
to zero and it remains a pure oscillating signal. This
state is studied for a suf®ciently long time period to
[8]):
"
#
ꢀ
ꢁ
ꢀ
ꢁ
2
2
xi
Mi
M
ri
r
ˆ
‡"
(4)
i
12
r
M
guarantee steady conditions. AT is the amplitude of
the induced temperature signal Tb(t) of the heater
b
for the materials of interest in this study, i.e. for binary,
ternary or quaternary mixtures ꢀM1
z; M0x;
with a matrix material M and the
x
y
M0y0; M0z00†O2
block. AT is small enough to assume isothermal
conditions of the test. The angular frequency ! is,
b
u
dopants M0, M00 and M000 (e.g. for M and M000 tetra-
together with AT , the factor which leads to a certain
temperature gradient; moreover, ! must be adapted to
AT and to the sample material.
valent and for M0 and M00 trivalent 2uˆx‡y). In
b
i
Eq. (4), xM the molar fraction of the component i,
b
i
i
MM its mass and rM its ionic radius, OV stands for
i
oxygen vacancy and xM ‡ ÆxM ‡ xO ‡ xOV ˆ 1. ꢃ is
given by 32(1‡1.6ꢂ)2, where ꢂ is the GruÈneisen
constant. In this equation, the average atom mass
and object radius are given by:
2.3.1. Determination of the heat capacity by ODSC
The temperature signal is induced symmetrically by
the heater block to the disc (see Fig. 1(a)). Two
thermocouples are used. One is placed at the refer-
ence, and the other at the sample position (see
Fig. 1(a)). The heat ¯ow between the heat block
and these two positions is given by Newton's law [15]:
X
i
i
M ˆ xMMM
‡
xM0 MM0 ‡ xOMO
(5)
and
X
i
i
r ˆ xMrM
‡
xM0 rM0 ‡ xOrO ‡ xOVrOV (6)
dQ
ˆ KꢀTr;s Tb†
(8)
dt
For phase heterogeneous mixtures, the values of the
phonon conductivity become much lower than that
dictated by this physical law. Its estimation, therefore,
remains subject to hypotheses; it is also linked to the
accuracy of the selected values of the parameters.
Recently, the thermal diffusivity of stabilised zir-
conia plasma-sprayed ®lms of 0.2 to 0.8 mm thickness
were investigated [10]. A dependence of the diffusiv-
where Q (J) is the heat ¯owing between two positions
(r or s and b), Tr,s (K) the temperature at reference and
sample positions, and K (J K 1 s 1) the Newton con-
stant.
Furthermore, the temperature of a body with a
thermal heat capacity Cp is in¯uenced by a quantity
of heat Q in the following manner:

112
M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
Q
AT
p
T ˆ
‡ T0
(9)
a ˆ
(14)
2L iÁ!=ꢄ
Cp
e
‡ 1
The heat ¯ow through the base of the rod is given by:
1
where T0 (K) is the initial temperature, and Cp (J K
the heat capacity.
Eqs. (5)±(7) lead to:
)
ꢀ
ꢁ
ꢀ
ꢁ
dQ
dt
dT
dx
ˆ
ꢀꢅS
(15)
ꢀ
ꢁ
xˆ0
xˆ0
dQ
dt
Q
where ꢀꢅ (W K 1 m 1) is the thermal conductivity of
the rod, S (m2) the cross-sectional area of the rod and ꢅ
the porosity of the sample.
ˆ K qt ‡ AT sin ꢀ!t†
(10)
b
Cp
Solving this differential equation under isothermal
conditions and then simplifying the solution leads
Inserting Eqs. (13) and (14) in Eq. (15) and multi-
plying the result by its complex conjugate yields the
amplitude of the heat ¯ow:
to the following simple equation:
r
AÈ K2
Cs Cr ˆ
‡ Cr2
(11)
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
2
2
dQ
dt
!
ꢄ
AT
!
2
ˆ ꢀꢀꢅT0S†
p
where Cr (J K 1) is the heat capacity of the reference
side, Cs (J K 1) the heat capacity of the sample side,
AÈ (K) the amplitude of the ODSC signal and AT (K)
the amplitude of the temperature signal. (Here, the
amplitude of the temperature signal is observed at the
reference position.)
ꢃ
ꢄ
p
p
L
L
2ꢄ!
2L 2ꢄ!
1
2e
cos L 2 ꢄ! ‡ e
cos L 2 ꢄ! ‡ e
ꢃ
ꢄ
p
p
Â
(16)
(17)
p
2ꢄ!
2L 2ꢄ!
1 ‡ 2e
furthermore:
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
ꢂ
dQ
dt
0
AÁ
ˆ
ˆ T0!CP
2.3.2. Determination of the thermal conductivity by
ODSC
To determine the thermal conductivity of a sample,
an additional measurement is necessary. For the deter-
mination of the speci®c heat, the sample was cut to a
thin disc to avoid any conductivity effects. Conse-
quently, the sample must be some kind of rod to affect
any conduction. When a periodic temperature varia-
tion is induced to one end of a rod, the temperature
distribution along the rod is given by [16]:
where ꢀꢅ (W K 1 m 1) is thermal conductivity of the
sample, AÁ (W s ) the heat ¯ow amplitude and CP0
1
(W K 1) the apparent heat capacity.
Let us assume that:
r
The thermal conductivity of the sample ꢀꢅ is then
given by:
!
L
2
ꢁ 0
(18)
ꢄ
Tꢀx; t† ˆ Tꢀx†ei!t
p
(12)
p
iÁ!
AÈ2
!ꢆꢅCPT02S2 ꢆCPA2
where ꢆꢅ (kg m 3) is the sample density.
!CP0
iÁ!
Tꢀx† ˆ a Á e
‡ b Á ex
(13)
x
ꢄ
ꢄ
ꢀꢅ ˆ
ꢂ
(19)
where ꢄ (m2 s 1) is the diffusivity constant of the rod,
x (m) the axis along the rod, a, b (K) constants given by
the boundary conditions
Heat losses at the surface of the sample lead to an
error. This error can be approximately determined by
measuring a reference material with similar dimen-
sions and heat conductivity values. This yields a
correction term, which is given by (the calculations
can again be found in Ref. [16]):
The heat ¯ux across the end of the rod is assumed to
be zero. This gives rise to the condition: (dT/
dx)xˆLˆ0, where L is the length of the rod. The other
end of the rod (base) just follows the temperature of
the induced signal. Consequently, the temperature
amplitude AT(0) is equal to the temperature amplitude
at the sample position AT : ATꢀ0† ˆ AT .
p
R ˆ ꢀꢅꢀM ꢀꢅ
(20)
s
s
where R (W m 1 K 1) is the correction term, ꢀM
(W m 1 K 1) the apparent thermal conductivity and
ꢀꢅ (W m 1 K 1)thethermalconductivityofthesample.
From the boundary conditions, we ®nd:
b ˆ AT
a

M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
113
The measurement of a sample with an unknown
thermal conductivity will then be corrected using the
following equation:
3. Experimental
3.1. Material preparation and analyses
p
ꢀM 2R ‡ ꢀ2M 4RꢀM
The preparation of all zirconia-based materials was
carried out by the wet route, because it was earlier
found that it yields the large material density [13]
required for the test. All solutions were prepared by
mixing nitrate solution of each element (Zr, Y, Er, Th
or Ce) composing the ®nal product. Co-precipitation
was carried out by adding ammonia under strong
stirring. The precipitate was ®ltered on a paper ®lter
and washed 10 times with ultrapure water. The ®ltered
phase was contacted with alcohol and milled, utilising
a Retch unit with zirconia container and bowls. The
wet powder was then dried 24 h at 420 K. The dried
powder was then manually desegregated prior to
calcination for 5 h at 900 K. The calcinated powder
was milled again, utilising the Retch unit with zirconia
vessel. The ®nal powder was then pressed, two dif-
ferent brands of pellets were produced, one 5 mm in
diameter and the other 11 mm in diameter. The 5 mm
pellets were pressed with 700 MPa for 1 min and the
11 mm pellets with 352 MPa for 10 s. Pelletised sam-
ples were then sintered in an oven with a temperature
programme including a ®rst step at 1475 K for 4 h
followed by a ®nal step of 48 h at 1875 K.
kꢅ ˆ
(21)
2
2.4. Laser flash
The phonon or thermal conductivity is derived from
the experimental thermal diffusivity. It may be
obtained by using the equation:
ꢀꢅ ˆ ꢄcpꢆꢅ
(22)
where ꢀꢅ (W m 1 K 1) is the thermal conductivity, ꢄ
(m2 s 1) the thermal diffusivity of the sample, cp
(J kg 1 K 1) the speci®c heat capacity and ꢆꢅ
(kg m 3) the sample density.
The thermal diffusivity is derived from the thermo-
gram recorded. (The scan can be studied in Fig. 1(b) in
the Section 2.1.) It is obtained from the temperature
variation measured on the back side of the sample. The
interpretation is derived from the method described in
Ref. [11]; it leads to the formula:
ꢀ
ꢁ
2TMÃ d
p
ꢁꢄ
L2
Ãp
lnꢀT t† ˆ ln
(23)
4ꢄt
The sintered pellets were ca. 10 or 5 mm in diameter.
The 10 mm pellets were cut to approximately 1 mm
thickness (L) and the 5 mm pellets were cut to two
differentdimensions,namely1 mmand5 mmthickness
(L). The cutting was performed by a diamond saw.
Table 2 indicates the masses of the pellets 5 mm in
diameterand1 mminheight.Thesewereusedtomeasure
the heat capacity of the different compositions. The
densityofthepelletsusedfortheconductivitymeasure-
ment was determined by measuring the sizes (diameter
and thickness) and weight utilising a Mettler balance
(AG204). Size (L), mass (m) and densities (ꢆꢅ and
where L (m) is the thickness of the sample, t (s) the
time after laser pulse, T* (K) the transferred tempera-
ture difference at time t and TMÃ (K) the maximal
temperature difference transfer.
2.5. Porosity correction of the conductivity
The investigated pellets had porosities from 6 to
20% (see Table 4 in Section 3.1). The thermal con-
ductivities of the porous samples are corrected to
obtain the conductivity values of the solid materials.
The correction equation used is:
1
1:7
1:7
ꢆꢅꢆ0 Â 100) are given in Table 4. ꢆ0 is the theoretical
ꢀ0 ˆ ꢀꢅꢀ1 ꢅ†
ꢂ ꢀꢅꢀꢆꢅ 1ꢆ0†
(24)
density and ꢆꢅ thegeometrical densityof the pellet. The
relative densities varied from 80 to 94%. For each com-
position, there were three different pellets which are
described in Appendix B.
where ꢀ0 (W m 1 K 1) is the thermal conductivity of
the solid material, ꢀꢅ (W m 1 K 1) the thermal con-
ductivity of the porous sample, ꢅ the porosity of the
sample, ꢆꢅꢆ0 1 the relative density of the sample and ꢆ0
(kg m 3) the theoretical density of the material.
This formula can be found in [12]. An overview of
the correction models is given in Appendix A.
The theoretical density was calculated from a sim-
ple approach based on the macroscopic density of each
component. X-ray diffraction (XRD) analysis was
carried out on a crushed pellet of the fabrication batch

114
M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
to identify the phases and to determine the lattice
parameter. The XRD unit was a Phillips diffractometer
PW1710, equipped with a Cu X-ray tube. The samples
were analysed after crushing and lattice parameters
already equipped with a programmer for sinusoidal
temperature variation. The output signal was recorded
as a raw data set and must be evaluated using an
separate computer unit with the calculation method
given by Wunderlich [15]. Sapphire was used for
calibration and glass with a well-known heat capacity
was measured to control the reliability of the
equipment. The parameters used for the ODSC mea-
surement were the following. The explanation of
the different parameters can be found in the Sec-
tion 2.3.
Ê
were measured with a precision better than Æ0.01 A.
Prior to measurement of thermal conductivity, the
samples were examined by X-ray diffraction to iden-
tify the phases and to con®rm the absence of any other
phases as the cubic phase in the inert matrix and
simulated fuel materials.
3.2. Laser flash
ttot
AT
!
total measuring time; around 30 min
0.5 K
The samples were measured by the laser-¯ash
method at JAERI, Japan [14]. Disc samples of
10 mm in diameter and ca. 1 mm in thickness were
subjected to a preliminary examination, and those
presenting any heterogeneity, small ®ssure or hole
were discarded. Since the samples were white or
slightly coloured, the surface of the disc samples
was coated with gold and carbon. Gold was vapour
deposited on the surface to allow good thermal contact
onto the semi-transparent material. Surface treatment
was completed by carbon powder spray on the surface
to ef®ciently absorb the laser beam. The coating was
very thin so that its in¯uence on the measured thermal
diffusivity was <1%. The measurements were per-
formed in vacuum below 3Â10 3 Pa.
b
1
2.0ꢁÂ10 2 s
At room temperature the natural cooling of the unit
block is slow. Therefore, a fast sinusoidal temperature
signal cannot be obtained without additional cooling.
The ODSC cell was consequently cooled by liquid
nitrogen.
Disc samples of 5 mm diameter and 1 mm thickness
were used for the cP measurements. The samples were
placed into an aluminium pot with cover. The sample-
containing pot±cover unit was tightened by a manual
press to ensure a good thermal contact of the unit. The
prepared samples were positioned manually on the
sample holder.
ATA-Instruments 2920 unit was also used, courtesy
Du Pont de Nemours. This unit was originally
equipped with the ODSC option and no additional
programming was required. The parameters have been
set to the following values:
The sample was heated in an electric furnace to the
required temperatures and the sample temperature was
measured with an IR detector. The method consists of
illuminating the front face of the disc with a short laser
pulse, creating at the surface a heat pulse when the
light is adsorbed. The energy of the laser pulse was 6 J,
which allows locally a temperature increase of the
sample surface of ca. 10 K. The thermal diffusivity, ꢄ,
is deduced from the thermal transient of the rear face,
called the thermogram. The temperature rise at the
rear surface was measured by an In±Sb infrared detec-
tor, after the front surface of the sample was heated by
a ruby laser pulse. The data of the temperature change
were analysed using a logarithmic method [14].
ttot
AT
total measuring time; around 30 min
1 K
b
1
!
2.5ꢁÂ10 2 s
The heat capacity measurements on the samples
already measured were repeated with the TA-Instru-
ments equipment. Further, heat conductivity measure-
ments were performed. Cylindrical pellets with 5 mm
diameter and 5 mm height were set on an aluminium
plate using a heat-conducting paste. This unit was then
placed on the sample holder again.
3.3. Oscillating differential scanning calorimetry
(ODSC)
For the thermal-loss correction, glass with a known
thermal conductivity was tested. This reference
sample had dimensions similar to the other pellets
and its thermal conductivity was also in a similar
range.
At the University of Geneva, a Seiko 220 DSC unit
was adapted for ODSC measurements. This unit was

M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
115
4. Results and discussion
4.1. Heat capacity
The heat capacity of the materials must ®rst be
obtained to calculate the thermal conductivity using
Eq. (19). Therefore, an oscillating temperature must
be applied to the equipment. An ODSC signal is,
consequently, measured as output. With the ampli-
tudes of these two signals, the heat capacity can be
calculated using Eq. (11). However, if the heat capa-
city is already known, the Newton constant, K, of the
instrument may be calculated.
Figs. 2 and 3 illustrate the ODSC measurement
with the simulated fuel material (Zr0.70,Y0.15,Er0.05,
Ce0.10)O1.9. Fig. 2(a) shows the temperature pro-
gramme induced by the Seiko instrument used for
the measurement of the simulated fuel mentioned
before. Fig. 2(b) shows a magni®cation of the average
temperature signal. The average is taken over one
period. A deviation from linearity means a difference
from a pure sinusoidal signal of the original data.
Fig. 2(c) shows the amplitude of the signal. The
resulting ODSC signal is presented in Fig. 3(a).
Again, a magni®cation of the average is displayed
in Fig. 3(b).
Fig. 3. ODSC signal measured between reference and sample
position. (a) ODSC signal; (b) average ODSC signal (average over
one period); and (c) signal amplitude.
Division of the amplitudes AÁ (Fig. 3(c)) and AT
(Fig. 2(c)) gives the required information about the
heat capacity (see Eq. (11)). Therefore, the average
value of a representative data area is taken (see Fig. 4).
Fig. 5(a and b) illustrate the heat capacity measured
with both equipments; with the Seiko as well as the
TA-Instruments machine. With both apparatuses the
speci®c heat capacity was detected by ODSC.
4.2. Thermal conductivity
The conductivities were measured using two tech-
niques, namely laser ¯ash and ODSC. Table 1 gives
the conductivity values for all studied compositions
obtained without porosity correction.
Fig. 6(a and b) shows the porosity corrected ther-
mal conductivities for the various material composi-
tions (for porosity correction see Section 2.5 and
Appendix A).
On increasing the concentration of dopants the
thermal conductivity shows a decreasingly trend.
The temperature independent term A in Eq. (1) is
coupled linearly to the scattering cross-section para-
meter (see Eq. (2)). i, the ith part of , is given in
Eq. (2). A large differences between the mass Mi and
Fig. 2. Temperature signal applied to the heater block. (a)
Temperature signal; (b) average temperature (average over one
period); and (c) signal amplitude.
the average mass of the system M yields an enlarge-
1
ment of the xiꢀMi M†M term (see Eq. (4)). Simi-

116
M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
Table 1
Uncorrected thermal conductivities of the studied samples at RT,
measured by laser flash and by ODSC
1
Sample
ꢀꢅ/W m 1 K
laser
ODSC
Zr
Zr10Y
1.99
1.93
1.30
1.46
1.55
1.73
1.59
1.59
1.10
1.46
2.30
1.80
2.01
1.78
2.02
1.69
1.53
1.74
1.42
1.07
Zr20Y
Zr30Y
Zr10Er
Zr10Y5Er
Zr20Y5Er
Zr15Y5Er10Ce
Zr15Y5Er10Th
Zr10Y7Er15Ce
1
Fig. 4. Statistics of AÈAT ratio used for the calculation of the cP
values. This statistical investigation only illustrates the stability
concerning one measurement. The statistics on the calculated cP
values can be found in Appendix C.
the structural changes from ZrO2 (monoclinic) to
(Zr1 z,Yx,Ery,Cez)O2 (¯uorite cubic).
larly, a large difference in the ionic radius ri and the
average ionic radius r yields an enlargement of the
term xiꢀri r†r 1 (see Eq. (4)). Consequently, impor-
tant differences exist in the different dopant ionic radii
x
y
(x‡y)/2
5. Concluding remarks
and molar mass increase
and the resistivity. The
Thermal conductivity measurements of zirconia-
based inert-matrix fuel have been achieved by the
oscillating differential scanning calorimetry and laser-
¯ash technique. Comparable results were obtained. A
conductivity as an inverse of the resistivity decreases.
This justi®es the experimental results shown in the
Fig. 6(a and b). Full modelling is, however, limited by
Fig. 5. Heat capacity of zirconia-based materials measured with two different ODSC instruments, with solid symbols: Seiko at 478C and open
symbols: TA-Instr. at 208C: (a): Ternary systems: (Zr1 y,Yx,Ery)O2 with an erbia part of (At%): (*) 0; (&) 5; and (~) 10. (b):
x
(x‡y)/2
Quaternary systems: (Zr1
z,Yx,Ery,Cez)O2
and (Zr1
x y
z,Yx,Ery,Thz)O2 (x‡y)/2. The different material compositions were: (!)
x
y
(x‡y)/2
(Zr0.70,Y0.10,Er0.05)O1.925; (^) (Zr0.70,Y0.20,Er0.05)O1.875; (~): (Zr0.70,Y0.15,Er0.05,Ce0.10)O1.900; (&) (Zr0.70,Y0.15,Er0.05,Th0.10)O1.900; and
(*) (Zr.68,Y.1,Er.07,Ce.15)O1.915
.

M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
117
Fig. 6. Thermal conductivity of zirconia based materials at 208C measured by the two studied techniques, solid symbols: laser flash, open
symbols: ODSC. (a): Ternary systems: (Zr1 y,Yx,Ery)O2 (x‡y)/2 with an erbia part of (At%): (*) 0, (&) 5; (~) 10. (b): Quaternary systems:
x
(Zr1
z,Yx,Ery,Cez)O2 (x‡y)/2. The different material compositions were: (!) (Zr0.70,Y0.10,Er0.05)O1.925; (^) (Zr0.70,Y0.20,Er0.05)O1.875;
x
y
(~) (Zr0.70,Y0.15,Er0.05,Ce0.10)O1.900; and (*) (Zr0.68,Y0.1,Er0.07,Ce0.15)O1.915
.
systematic
study
z,Yx,Ery,Mez)O2
was
carried
out
for
from theoretical models and three empirical formula
are presented here.
(Zr1
Th. The thermal conductivity decreases from 3 to
with MeˆCe or
x
y
(x‡y)/2
ꢃ The simple linear correction, which physically
supposes tube-shaped pores whose axes are parallel
to the temperature gradient:
2 W m 1 K when the dopant (Y, Er) concentration
1
increases up to 30 At%. This trend is justi®ed by the
theory based on the properties of the oxygen vacancies
in the cubic lattice generated by the addition of
dopant.
ꢀꢅ ˆ ꢀ0ꢀ1 ꢅ†
(A.1)
ꢃ The correction derived from a model supposing
closed spherical pores [17]:
1:5
Acknowledgements
ꢀꢅ ˆ ꢀ0ꢀ1 ꢅ†
(A.2)
ꢃ A model handling open porosity yields [17]:
Thanks are due to Mrs. H. Lartigue and Mr. A.
Desponds for their measurements by ODSC at the
University of Geneva and Du Pont de Nemours tech-
nical centre, Geneva. The technical assistance of Mr.
M. Takano is acknowledged for laser-¯ash tests at
JAERI. P. Heimgartner and T. Graber are thanked for
their technical support. G. Ledergerber, head of the
fuel section, is thanked for useful and constructive
discussions about this work.
ꢅ
ꢆ
(A.3)
1
cos2 cos2
ꢀ1 ꢅ†ꢀꢀ0 ꢀꢅ† 2
ˆ ꢅꢀ2 cos2 †
‡
ꢀ0 ꢀꢅ
ꢀꢅ
where is the orientation angle of the ellipsoidal
pores with respect to the heat gradient.
ꢃ Porosity correction for UO2 by determining the
structure of the pores and calculation by finite-
element method yields [12]:
1:7Æ0:2
Appendix A: Porosity corrections
ꢀꢅ ˆ ꢀ0ꢀ1 ꢅ†
(A.4)
Several equations have been proposed in order to
correct the thermal ꢀꢅ conductivity of solid solutions
with regard to porosity ꢅ. Four corrections derived
ꢃ A modified Loeb formula is suggested, based on
measurements on unirradiated UO2 fuel [18,19]:

118
M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
ꢀꢅ ˆ ꢀ0ꢀ1 ꢇꢅ†
where ꢅ is an empirical parameter.
(A.5)
tical models (Eqs. (A.2) and (A.3)) which are more
general than the empirical formulas (Eqs. (A.5)±
(A.7)). These empirical equations are derived from
data of UO2 samples.
ꢃ Another empirical correction proposed is the Max-
well±Eucken equation [20]:
1
ꢅ
ꢀꢅ ˆ ꢀ0
(A.6)
1 ‡ ꢈꢅ
where ꢈ is an empirical parameter.
Appendix B: Sample data
Tables 2±4
ꢃ The Loeb and the Maxwell±Euken equations can
be replaced with [21]:
2:5
Appendix C: Error calculation
ꢀꢅ ˆ ꢀ0ꢀ1 ꢅ†
(A.7)
C.1. Data from literature
The following figure shows a plot of the correction in
the porosity range from 0 to 30%. Parameters are
indicated in the legend.
The cP measurements by ODSD are calibrated
using a sapphire standard sample. The cP value of
sapphire is taken from Touloukian [22] and its error
is 0.2%.
C.2. Oscillating differential scanning calorimetry
(ODSC)
C.2.1. Heat capacity
The thermal conductivity was investigated with
both the equipments. For both of them a statistical
determination of the error was performed:
TA-Instruments: A series of nine calibration mea-
surements (determining of the Newton constant K, see
Section 2.3) with sapphire as standard was ful®lled.
Table 2
Characterisation of samples Ot produced for the heat capacity
measurements by ODSC: mass of the pellets and the measured heat
capacities, once at 208C investigated by the TA-Instr. equipment,
and once at 478C measured with the Seiko machine
Sample
m
cꢀPTA-Instr-20†
cꢀPSeiko-47†
1
1
(10 5 kg)
(J kg 1 K
)
(J kg 1 K
)
Zr_Ot
4.40
4.12
6.63
7.64
6.53
6.63
8.16
0.464
0.464
0.435
0.434
0.390
0.449
0.415
0.381
0.389
0.369
0.502
0.454
0.393
0.420
0.391
0.434
0.388
0.365
0.422
0.423
Zr10Y_Ot
Zr20Y_Ot
Zr30Y_Ot
Zr10Er_Ot
Zr10Y5Er_Ot
Zr20Y5Er_Ot
Zr15Y5Er10Ce_Ot 8.42
Zr15Y5Er10Th_Ot 6.74
Zr10Y7Er15Ce_Ot 8.49
In this work, the correction (Eq. (A.4)) by the ®nite-
element method was selected. It yielded corrected
conductivities between the values of the two theore-

M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
119
Table 3
Characterisation of samples OT produced for the thermal conductivity measurements by ODSC: thickness, weight, density, relative density and
the measured thermal conductivities of the porous and theoretically dense pellets. The equation applied for porosity correction can be found in
the Section 2.5 (Diameters can be calculated from thickness (L), mass (m) and the density (ꢆꢅ); the theoretical densities (ꢆ0) can be calculated
1
from ꢆꢅ and ꢆꢅꢆ0 Â 100
1
Sample
L
m
ꢆꢅ
ꢆꢅꢆ0
100%
ꢀꢅ
ꢀ0
3
1
1
(10 4 m)
(10 5 kg)
(10 3 kg m
)
(W m 1 K
)
(W m 1 K
)
Zr_OT
49.5
51.0
49.8
51.1
50.8
51.5
52.2
53.0
49.4
53.0
52.7
51.9
55.9
55.5
60.0
58.4
59.9
58.7
56.6
58.7
5.34
5.31
5.02
5.02
5.30
5.56
5.15
5.24
5.00
5.39
91
91
88
89
86
94
88
87
80
87
2.30
1.80
2.01
1.78
2.02
1.69
1.53
1.74
1.42
1.07
2.70
2.11
2.50
2.17
2.61
1.88
1.90
2.20
2.08
1.36
Zr10Y_OT
Zr20Y_OT
Zr30Y_OT
Zr10Er_OT
Zr10Y5Er_OT
Zr20Y5Er_OT
Zr15Y5Er10Ce_OT
Zr15Y5Er10Th_OT
Zr10Y7Er15Ce_OT
Table 4
Characterisation of samples OL produced for the thermal conductivity measurements by laser flash: thickness, weight, density, relative density
and the measured thermal conductivities of the porous and theoretically dense pellets. The equation applied for porosity correction can be
found in the Section 2.5 (Diameters can be calculated from thickness (L), mass (m) and the density (ꢆꢅ) the theoretical densities (ꢆ0) can be
1
calculated from ꢆꢅ and ꢆꢅꢆ0 Â 100)
1
Sample
L
m
ꢆꢅ
ꢆꢅꢆ0
100%
ꢀꢅ
ꢀ0
(10 4 m)
(10 5 kg)
(10 3 kg m
)
(W m 1 K
)
(W m 1 K
)
3
1
1
Zr_OL
Zr10Y_OL
8.5
8.4
7.8
8.3
8.5
8.3
8.5
8.2
8.3
8.6
28.6
30.2
27.2
29.7
32.5
32.3
32.6
31.2
32.2
30.1
5.89
5.80
5.70
5.61
6.18
5.94
5.85
6.03
6.23
6.20
83
88
81
85
83
92
86
87
80
84
1.99
1.93
1.30
1.46
1.55
1.73
1.59
1.59
1.10
1.46
2.72
2.39
1.87
1.91
2.10
1.97
2.05
1.93
1.60
1.97
Zr20Y_OL
Zr30Y_OL
Zr10Er_OL
Zr10Y5Er_OL
Zr20Y5Er_OL
Zr15Y5Er10Ce_OL
Zr15Y5Er10Th_OL
Zr10Y7Er15Ce_OL
The setup of the unit was maintained (for the setup, see
Section 3.3) and the standard was repositioned for
every measurement. A variation of 4.0% [23] was
found. The error from the literature value for the
sapphire heat capacity of 0.2% and a measurement
error for the mass of 0.1% were added to this statistical
error, resulting in a total error for the calibration of
4.3%. For determining the heat capacity, a second
similar measurement with the investigated material
was necessary. The statistical error was also assumed
to be 4.0% (same procedure) and again an error of
0.1% for the mass is added resulting in 4.1% error for a
sample measurement. In Eq. (11), these two errors
sum up to 8.4%.
Seiko: Again a series of calibration measurements
was performed. The setup was also maintained (see
Section 3.3) and for every four measurements a place-
ment and taking off procedure was ful®lled. This
resulted in a variation of 2.3%. Again the same
literature and the mass error, as before, were added
to this statistical error resulting in 2.6% total error for
the calibration. With the same calculation as above
this would result in an error of the heat capacity of
5.0% for a sample.
These errors were checked by selecting two materi-
als and making a statistical investigation about their
®nal cP value. For ZrO2, three measured cP values at
208C were taken into account. One from the TA-

120
M.A. Pouchon et al. / Thermochimica Acta 323 (1998) 109±121
Instruments equipment and two from the Seiko
machine. The mean value of these measurements
and const a known constantˆ0.13878 was used for the
calculation of error.
1
was 0.446 J g 1 K and the standard deviation was
The const is well known, and its error is negligible.
The thickness L of the sample is measured with a
precision of Æ10 5 m and L ranges between 0.78 and
0.86 mm. The maximal error is ÁL/Lˆ1.3%. The time
detection is precise, its error is estimated to 3%. It
results in a total error of 2.9% for the diffusivity. For
calculating the thermal conductivity, Eq. (22) is used.
Therefrom, Áꢆꢅ/ꢆꢅ and ÁcP/cP must also be consid-
ered. Áꢆꢅ/ꢆꢅ is 2% for the cross section (Æ10 5 m
precision for a 1-cm pellet) plus 1.3% for the height
plus 0.4% for the measured weight. ÁcP/cP is 8.4%.
Hence, the total error of the thermal conductivity
measured by laser ¯ash is 12.9%. (When using the
ODSC-measured heat conductivity, however, using
literature data precision should be better, but the
accuracy is doubtful. The heat conductivity of the
different compositions cannot be found in literature, a
calculation using the heat conductivity of every frac-
tion must be performed. Such a calculation does not
consider phase changes.)
0.018 J g 1 K 1, which is 4.1% of the mean value.
The literature value for this material at 208C is
1
0.450 J g 1 K
.
For the quaternary material
(Zr0.80Y0.15Er0.50Ce0.10)O1.9, again three different cP
measurements at 208C were calculated. One measured
by the TA-Instrument and two by the Seiko unit. The
1
mean value was 0.386 J g 1 K and the standard
deviation 0.006 J g 1 K 1, which is 1.49% of the
mean value.
C.2.2. Thermal conductivity
The thermal conductivity was investigated with the
TA-Instrument unit. To calculate the thermal conduc-
tivity, both CP and apparent CP (called CP0 ) values were
needed. The same Newton constant was taken for
determining both of them. The total error can be
derived from Eq. (19). It is the sum of the relative
errors for CP0 2=CP, the angular frequency !, twice the
cross section S and the porous sample density ꢆꢅ.
The error on CP0 2=CP is 16.2%. The statistical
error of three single ODSC measurements and one
calibration error are added. Presently, CP and CP0 are
calculated using the same Newton constant; therefore,
the calibration error (error of Newton constant) can-
cels in the ratio value CP0 =CP. Eq. (19) includes the
heat capacities, hence the error on the mass measure-
ment is ignored. The error of the angular frequency is
negligible. ÁS/S is 0.4% supposing a Æ10 5 m pre-
cision for a 5 mm pellet. Áꢆꢅ/ꢆꢅ is 0.7%, supposing
the same cross-sectional error plus an error of 0.2% for
the height (Æ10 5 m precision on a pellet height of
5 mm) and an error of 0.1% for the mass. So the total
error for the thermal conductivity is 17.7%.
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