LORENZ NUMBER AND THERMAL
CONDUCTIVITY OF LIQUID GALLIUM, MERCURY
AND MERCURY-INDIUM ALLOYS
G. Busch, H.-J. Güntherodt, W. Haller, P. Wyssmann
To cite this version:
G. Busch, H.-J. Güntherodt, W. Haller, P. Wyssmann.
LORENZ NUMBER AND
THERMAL CONDUCTIVITY OF LIQUID GALLIUM, MERCURY AND MERCURYINDIUM ALLOYS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-313-C4-316.
<10.1051/jphyscol:1974460>. <jpa-00215651>
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Submitted on 1 Jan 1974
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JOURNAL DE PHYSIQUE
Colloque C4, supplkment au no 5, Tome 35, Mai 1974, page C4-313
LORENZ NUMBER AND THERMAL CONDUCTIVITY OF LIQUID GALLIUM,
MERCURY AND MERCURY-INDIUM ALLOYS
G. BUSCH, H.-J. GUNTHERODT,
W. HALLER and P. WYSSMANN
Laboratorium fiir Festkorperphysik, ETH, Ziirich, Schweiz
RbumB. - Le nombre de Lorenz des metaux liquides de haute puret6 Ga et Hg, et des alliages
liquides Hg-In a Bte mesure de maniere directe en appliquant la methode de Kohlrausch sur un
Bchantillon dont la gBometrie prksente une constriction. Des valeurs independantes de la tempBrature et differant de moins d'l % de la valeur theorique, calculke d'apres le modele des electrons
libres, ont ainsi Bt6 obtenues pour le nombre de Lorenz. La conductivite thermique a ete calculee
a partir du nombre de Lorenz et la conductivitB Blectrique a kt6 deduite de la loi de WiedemannFranz.
Abstract. -The Lorenz number of liquid high purity Ga, Hg and Hg-In alloys has been carefully
measured directly by applying the Kohlrausch method to a constriction-shaped sample. We obtained
temperature independent values with deviations less than 1 % from the theoretical free electron
value for the Lorenz number. The thermal conductivity has been calculated from the Lorenz
number and the electrical conductivity by means of the Wiedemann-Franz law.
1. Introduction. - Recent experimental work of
the thermal conductivity gives the rather surprising
result that the indirectly determined Lorenz numbers
(derived from separate measurements of the thermal
and electrical conductivity) of the liquid metals exhibit
substantial deviations from the free-electron value.
There are not only significant discrepancies among
the many different sets of data but even both signs of
the temperature coefficient appear [I].
All the other properties of liquid normal metals.
are well explained by the free-electron model [2].
There is clearly need for more work to be undertaken
on both the thermal conductivity and the Lorenz
number in the hope of solving the controversy about
the Lorenz number of liquid metals.
2. Method. - A direct measurement of the Lorenz
number of liquid and solid metals can be realized
with the help of Kohlrausch method [3].
The Kohlrausch theory is valid for an arbitrary
shaped electrical conductor with phenomenological
thermal and electrical conductivity (Fig. 1).
A current I passes through the electrodes S, and
S, with the potentials cp, and cp,, held on the constant
temperature To. The voltage between the electrodes is
If there are no lateral losses of heat or electricity
through the surface S, i. e.
FIG. 1. - Kohlrausch theory.
the famous U - Tm relation is given, by solving the
heat flow equation and Ohm's law, to
Tm is the maximum temperature in the conductor.
Supposing the Wiedemann-Franz law
holds, eq. (3) is reduced to
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974460
G . BUSCH, H.-J. GUNTHERODT, W. HALLER AND P. WYSSMANN
C4-3 14
then the Lorenz number is given by
172
and with
e = T, - T,
(7)
~ 7 2
Figure 2 shows the modification used in our experiment for the constriction-shaped sample in a schematic manner. The electrodes S, and S, are half
spheres and the surface S gives due to symmetric
reasons a nearly ideal experimental realization for
eq. (2). Furthermore, S forms the constriction with a
small junction having a diameter of 1.5 mm. T, is
the temperature in the center of the constriction. Tm
and U can be varied by changing the current I through
the constriction.
Figure 3 shows the cross section of the sample
holder.
The thickness of the Polysulfon foil S is 0.4 mm.
The temperatures To and 8 are measured by thermocouples. The ideal conditions of the Kohlrausch
theory are fulfilled in the limit I 4 0 i. e. T, -, To.
This extrapolation also prevents the influence of heat
convection in the liquid state and other disturbing
effects.
3. Experimental results. - Tables 1-3 show the
measured Lorenz numbers of Gay Hg and Hg-In
alloys [4]. For comparison the values deduced from
previous experiments are also presented.
Lorenz number of solid and liquid Ga near the melting
point TM= 303 K
Temp.
(K)
Temp.
(K)
293.3
294.7
302.4
307.3
304.2
310.3
313.3
348.3
303
350
-
FIG. 2.
- Schematic representation of the constriction-shaped
sample.
-
L
(v2deg-2)
-
2.457
2.441
2.442
2.63
2.057
x lo-'
x lo-'
x lo-'
x lo-' (*)
x lo-' (**)
(*) Yurchack and Smirnov [5] ;L decreases with temperature.
(**) Duggin [6] ; L increases with temperature.
Lorenz number of solid and liquid Hg
(meltingpoint TM= 234.8 K )
Temp.
(K)
Temp.
(K)
227.7
231.5
234.9
242.4
273.0
282.2
291.2
304.8
321.6
339.1
373.4
303.0
-
-
The directly determined Lorenz numbers for Ga,
Hg and Hg-In alloys are in excellent agreement
with the free-electron value
---
@
Chromel - Alumel
Water
-
Insulation
Polysulfon
FIG. 3. - Cross section of the sample holder.
where k, is the Boltzmann constant and e the electronic
charge. The deviations are less than 1 %.
LOREN2 NUMBER AND THERMAL CONDUCTIVITY O F LIQUID GALLIUM, MERCURY
TABLEI11
Lorem number of liquid Hg-In alloys at T = 313 K
At.
% In
-
0
5
12.8
20.7
L
(10-8
V2 deg-2)
At.
% In
L
(10-8
V2 deg-2)
-
-
-
2.440
2.441
2.445
2.439
29.3
49.2
67
100
2.454
2.448
2.415
2.76 (*)
C4-315
Rubenstein [lo] (curve 4, these values are out of
the scale of this figure) are also presented. Fig. 5
shows our calculated values (curve 1) of the thermal
conductivity of Hg and the directly measured values
12
'E
(*) At the melting point [I].
This result gives support to the current theories of
the electronic transport properties of liquid normal
metals. The conduction electrons are considered to be
scattered elastically by the ions. For such elastic
scattering the electronic part of the thermal conductivity is related to the electrical conductivity by the
Wiedemann-Franz law [9].
4. Thermal conductivity. - New, careful direct
measurements on the thermal conductivity of liquid
metals are needed. In the meantime, one may deduce
the thermal conductivity for such liquid metals
which show free-electron values for the Lorenz
number by means of the Wiedemann-Franz law,
using the data for the Lorenz number and the electrical
conductivity.
This method gives more accurate values for the
thermal conductivity than the direct measurement,
because it is much easier to make precise measurements
of the electrical resistivity and the Lorenz number
than of the thermal conductivity. Figure 4 shows
FIG. 4. - Thermal conductivity of liquid Ga.
our calculated values (curve 1) of the thermal conductivity of liquid Ga in comparison with directly measured values [4]. The full dots indicate the range over
which the Lorenz number has been measured. The
empty dots are obtained by using the theoretical
value for the Lorenz number. For comparison, the
values measured by Yurchak and Smirnov [5]
(curve 3), Duggin [6] (curve 2) and Seidensticker and
X
6
200
300
500
400
T
(OK1
FIG. 5. - Thermal conductivity of liquid Hg.
by Ewing et a/. 171 (curve 2) and Powell and Tye [8]
(curve 3). Again the full dots indicate the range over
which the Lorenz number has been measured, whereas
the empty dots are obtained by using the theoretical
value of the Lorenz number. Figure 6 presents our
calculated values of the thermal conductivity and
the electrical resistivity of liquid Hg-In alloys as a
function of concentration. The thermal conductivity
of liquid Hg increases in alloying with In. Hg-In
alloys with 70 at. %. In are liquid at room temperature, but have a thermal conductivity nearly a
factor of 3 larger than pure liquid Hg.
FIG. 6. - Thermal conductivity and electrical resistivity of
liquid Hg-In alloys.
If the Wiedemann-Franz law is confirmed this
suggests that the theoretical treatment of the thermal
conductivity can be formulated in terms of the Ziman
theory [Ill of the electrical resistivity, i. e. using the
pseudopotential and the structure factor. The numerical calculations for liquid Ga, Hg and Hg-In alloys
are in excellent agreement with the experiments. The
increase of the thermal conductivity of liquid Hg-In
on alloying can be explained by the same effect
suggested for the decrease of the electrical resistivity.
This decrease of the electrical resistivity of Hg-rich
alloys on alloying with In is well explained by the
large negative pseudopotential of Hg [12]. This large
pseudopotential is due to the d-states at the bottom
of the conduction band of Hg.
It should be pointed out that the total thermal
conductivity enters into the Kohlrausch relation
whereas only the electronic part occurs in the Wiedemann-Franz law. Thus, the agreement with this law
shows that the tc lattice )) contribution is negligible
within the experimental error and therefore the
calculated thermal conductivity is the total thermal
conductivity.
5. Conclusion. - This new powerful method gives
for the first time reproducible experimental results
of the Lorenz number in liquid metals with a much
smaller experimental error than known so far. Therefore, a more careful development of this method
could answer all the interesting questions of the
contribution from inelastic scattering and from the
ct lattice )) to the thermal conductivity in liquid
metals.
Acknowledgments. - The authors are indebted
to the cc Eidgenossische Stiftung zur Forderung
Schweizerischer Volkswirtschaft durch wissenschaftliche Forschung)) and to the Research Center of
Alusuisse, for financial support.
References
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