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Liquid Metals and Liquid Semiconductors.

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[13] M. Campagna. G. K . Werfhetm. E. Bucher. Struct. Bonding (Berlin) 30, 99
(1976).
I141 J. M. D. Coe.v, S. K . Ghatak, M. Avignon, F. Holtzberg. Phys. Rev. B 14,
3744 ( 1976).
[15] T. Penney. F. Holtzberg. Phys. Rev. Lett. 34, 322 (1975).
[ 161 B C. Sales. D. K . Wohlleben, Phys. Rev. Lett. 38, 1240 ( I 975).
[171 T. Penney. R. L. Melcher. F. Holtrberg, G. Guntherodt, AIP Conf. Proc. 29,
392 (1975).
[ l S ] S . uon Molnar, T. Penney, F Holtzberg. J. Phys. (Orsay, Fr.) 37. C4-241
(1976)
1191 P. D. Dernier, W. Weber, L. D. Longinorti, Phys. Rev. B 14. 3635 (1976).
[20] H. A. Maok, R. M . Nicklow. T. Penney, E Holtzberg. M . W. Shafer. Phys.
Rev. B 18. 2925 (1978).
1211 A. Jayaraman. R. G. Maines. E. Bucher. Solid State Commun. 27, 709
(1978).
[22] M. Crofi. A. Jayaraman, Solid State Commun. 29, 9 (1979).
Liquid Metals and Liquid Semiconductors[**]
By Friedrich Hensel“]
Dedicated to Professor Ernst Ulrich Franck on the occasion of his 60th birthday
Numerous liquid systems have electrical properties which resemble those of crystalline and
amorphous semiconductors. The existence of “semiconducting” behavior in these liquids is
mostly related to a continuous transition from a metallic to a “semiconducting” state when a
thermodynamic variable such as temperature, density or concentration is changed. Changes in
the nature of the chemical interaction and the associated changes in the structure of the liquid
are of fundamental importance for the transition to a “semiconducting” state. This will be demonstrated for the ionic liquid CsAu, for covalent liquid selenium, and for expanded liquid
metals.
1. Introduction
The discovery of noncrystalline semiconducting materials
and the application of such “amorphous semiconductors” for
technical purposes have stimulated an increasing interest in
the electrical and physicochemical properties of electronically conducting liquids during the past 15 years. According to
a proposal by Jofie and Regelill two classes of such liquids
can be distinguished: “liquid metals” and “liquid semiconductors”. At present the line of distinction between the two
classes is arbitrary since generally accepted theoretical or experimental criteria are still lacking. It has become established
practice to call those liquids which have electrical conductivities u larger than 2000 ohm - cm ’ metallic, since the mean
free paths of the electrons calculated from u within the
framework of the free electron
are mostly larger
than the mean interatomic distance. Liquids with conductivities u smaller than 300 ohm-’ cm-’ are called “semicond~cting”~
because
~’
they exhibit some properties which are
typical for solid crystalline and amorphous semiconductors,
e. g. a positive temperature dependence of u characterized by
the presence of an activation energy.
The difficulty in precisely defining the boundaries between the two types of conduction is essentially related to the
existence of a continuous transition from semiconducting to
metallic behavior which is observed in most semiconducting
liquids on changing a thermodynamic variable such as temperature, density, or composition. Conversely, it is always
possible, at least in principle, to transform any metallic liquid into a semiconducting or insulating state by expansion
to very low densities or by alloying with a nonmetal. Hence,
a comprehensive theoretical description of semiconducting
liquid systems must include not only the properties of the
semiconducting range, but must also account for the properties during the course of the metal-nonmetal transformation,
i. e. it must include a description of the electrical properties of
those liquids which lie at the borderline between typical metals and typical semiconductors. Accordingly, the most appropriate experiments for the investigation of electronically
conducting liquids are those which not only extend to the
semiconducting and the metallic state but also embrace the
transition range between the two. In Table 1 a list of groups
Table I . List of selected liquid systems in which a metal-nonmetal transition can
be induced.
Group
System
Inducing physical effect
1
Semiconducting alloys
(Mg,Bil, Li3Bi, CsAu, erc.)
Metal-salt mixtures
(Cs-CsCI. Bi-BiCli, etc.)
Alloys of metals with S, Se. Te
(TI2Te, In2Sez. Ga2Te3,etc.)
change of concentration
Se. Te and
(Se,Tec * )
change of chemical structure by
an increase of temperature and
pressure
expansion to low densities at high
temperature
2
3
4
I*]
[“I
their
mixtures
~
Prof. Dr. F. Hensel
Fachbereich Physikalische Chemie der Universitat
Hans-Meerwein-Strasse. D-3550 Marburg (Germany)
Lecture presented at the GDCh-meeting in Berlin, September 12, 1979
Angew. Chem. Inr. Ed. Engl. 19, 593-606 (1980)
5
Metals
(Hg, Cs, Rb, K, etc.)
0 Verlag Chemie, CmbH, 6940 Wernheim, 1980
0570-0833/80/0808-0893
change of concentration
change of concentration
$ 02.50/0
593
of liquids is assembled for which semiconducting properties
and a metal-nonmetal transition have been observed.
Changes in the chemical interaction and the associated
structural changes of the liquid are of fundamental importance for the transition from metallic to semiconducting behavior. The best evidence for this is provided by the semiconducting liquid alloys (Table 1, Group 1). Although e.g.
molten Li, Cs, Mg, Au, Bi, Pb and S b are very good metallic
conductors, their liquid mixtures have a relatively low conductivity at the stoichiometric compositions Li3Bi, Li4Pb,
Mg,Bi2, Cs3Sb and CsAu. At the same time the temperature
dependence of the conductivity is positive. In this article, interest is focussed on the role of charge transfer in these systems, i. e. whether the chemical interaction is essentially
ionic, as in salts. If this is the case the metal-nonmetal transition in this group would be similar to the transition observed
in metal-molten salt mixtures (Table 1, Group 2), whose
properties have been extensively studied by Bredigl'.'] and
about which there are recent detailed reviews by Nochtrieb['1
and WarredX1.
We shall therefore exclude from further consideration these metal-molten salt systems as also the alloys
of metals with the nonmetallic elements S, Se and T e which
are listed as Group 3 in Table 1. The properties of the latter
systems have also been reviewed recently[x 1' . In many respects they behave similarly to the alloys with two metallic
constituents (Table 1, Group 1); here also, the conductivity
has a minimum at the stoichiometric composition.
The chemical interactions in the systems mentioned above
vary widely from essentially ionic to predominantly covalent
depending o n the differences in electronegativity of the constituent atoms, whereas in the pure liquid elemental semiconductors s, Se and Te the chemical interaction is purely
covalent with a small contribution of van-der-waals-interaction. The electronic and structural properties of selenium are
especially interesting when considered over a larger temperature and pressure range. Near the melting point the properties of liquid selenium are consistent with the assumption
that it consists of long polymeric chains containing up to lo5
atoms[".'*]. However, the average chain length, the local liquid structure and the coordination number can change with
increasing temperature and pressure. This continuous
change in molecular structure is accompanied by a significant modification of the electronic structure: At high temperatures and pressures a continuous transformation into a liquid with conductivities approaching those in the metallic
range is observedl''l.
i n addition to the transitions between metaHic and predominantly ionic or purely covalent interaction, which will
be discussed in the present article in detail for liquid CsAu
and selenium, metal-nonmetal transitions in expanded fluid
metals like Hg, Cs or R b are of great theoretical and technical interest (Table 1, Group 5). Such a transition occurs in
any metal at very high temperatures due to thermal expansion of the liquid to low den~ities''~].
As in the other systems
this transition from metal to nonmetal implies that the nature of the chemical or interatomic interaction changes. A
knowledge of the physicochemical mechanism of the transition is of fundamental importance, since the thermodynamic
properties, as e.g. the free energy, the equation of state, or
the specific heat of the system are determined by the interatomic interaction. This is especially important in view of the
594
possible technical applications of liquid metals at high temperatures. In this case thermodynamic data are needed; a
theoretical prediction and treatment of such data is desirable.
2. Liquid Ionic Alloys
In some liquid binary alloys of metallic constituents the
conductivity passes through a minimum at well defined stoichiometric compositions. The most extensively studied system is the Cs-Au system, whose electrical conductivity G in
the concentration range between 0 and 60 atom-% Au a t
600 " C is plotted in Figure 1. m changes by almost four orders
1
,/I
'05r------7
600'C
I
10
30 50 70
Au [Atom-%l--r
,
90'
Fig. 1. Electrical conductivity u of liquid Cs-Au mixtures as a function of the
gold content at 600°C.
of magnitude and has a pronounced minimum of 3 o h m - '
cm- ' at the stoichiometric composition CsAu. This value is
of the same order of magnitude as the conductivities of molten salts; e. g. G of CsCl at the melting point is about 1 ohm - '
cm-I.
The transition to nonmetallic behavior is accompanied by
a change in the temperature dependence of d'5.'61.
In Figure
2 the logarithm of the conductivity is plotted versus the reciprocal temperature. At a concentration of 44 atom-% Au,
corresponding to a conductivity of about 100 o h m - ' cm-',
the temperature dependence changes from negative to positive. The corresponding apparent activation energy EA of the
conductivity calculated according to the relationship
u = a,,exp( - E A / R T )
(1)
approaches a maximum of about 20 KJ/mol at the stoichiometric composition. It is not possible, however, to decide
whether E , is the activation energy of the diffusive motion of
the ions Cs+ and A u - or that of thermally activated electrons, since eq. (1) is valid for both conduction mechanisms.
Measurements of the optical absorption are required. Analogously to crystalline semiconductors there also exists for nonAngew. Chem. Inr. Ed. Engl. 19, 593-606 (1980)
r
1
I
-
lo'
'E
o
103
2%
505%
48%
11
1.00
I
1.05
I
130
,
I
1.20
115
lo3/T I:]'-K
,
125
130
10
2.5
2.0
1.5
nwlevl
4
-
Flg. 2 Electrical conductivity u of liquid Cs-Au mixtures as a function of the re-
ciprocal temperature at different Au-concentrations in atom-%.
crystalline semiconducting systems a correlation between E,,
for thermally activated electrons and the distance between
valence- and conduction-band as derived from the optical
absorption edge. However, in disordered systems-including
liquids-the absorption edges are far from sharp, because
there are always a few allowed states for an electron in energy ranges which are forbidden in the corresponding crystalline arrangementi4).A schematic representation of this situation is given in Figure 3. The singularities in the electronic
'1
Semimnductdlnsulator
- ,E
14
-
2.5
1.5
2.0
EgleVl
Fig. 4. Top: Optical absorption of a 100 pm-thick sample of solid or liquid CsAu
as a function of the photon energy. Bottom: the temperature dependence of the
position of the absorption edge. The arrow indicates the abrupt change of the position of the edge at the melting point (590°C).
pendence of the position of the optical gap which corresponds roughly to the gap between valence- and conductionband. It is a plot of the photon energy corresponding to
K = lo4 cm- ' as a function of the temperature. There is a
striking quantitative similarity between the behavior of
CsAu and that of molten salts, as is demonstrated in Figure
5, where measurements of the absorption edges as a function
Density ot states
of a crystal
Density of states
of a disordered
system
0
3
Fig. 3 A schematic representation of the energy bands and the density of states
N ( E ) as a function of the energy E of a crystalline semiconductor or insulator
and a corresponding disordered semiconducting system. EL= bottom of the conduction band. E, = top of the valence band.
density of states N(E) at the band edges are smeared out. The
energy gap in the crystal [N(E)=O] is replaced by a minimum in N ( E ) . Figure 4 shows new results for the optical absorption edges of CsAu in the solid as well as in the liquid
state1l7I.The spectral dependence of the absorption coefficient K ( w ) for solid CsAu (from 25 "C to 560 "C; CsAu melts
at 590"CI'*1) and for liquid CsAu (626°C) is shown as a
function of the photon energy. The absorption edges of the
solid phase exhibit the expected red shift which is usually observed for crystalline semiconductors or ionic insulators with
increasing temperaturel"1. On melting, there is an abrupt
change in the position of the absorption edge of about 0.7 to
0.8 eV. The lower part of Figure 4 shows the temperature deAngew. Chem. I n ( . Ed. Engl 19, 593-606 (1980)
4
5
6
to lev]----lev]-----
Fig. 5. Position of the absorption edge of I mm-thick samples of salt in the solid
and liquid state. The abrupt change of the position takes place at the melting
point.
of temperature are reproduced for solid and molten KI and
KBr12'J. The experimental points correspond to the highest
energy of light transmitted through a 0.1 cm-thick layer of
the molten salt sample. The similarity between the data for
CsAu and those for the molten salts is consistent with the assumption of predominantly ionic bonds in solid and liquid
CsAu.
Comparison of the measured optical energy gap between
valence- and conduction-band in liquid CsAu of 1.3 eV with
the activation energy E , measured for the electrical conductivity CT leads directly to the conclusion that (T is determined
by ionic conduction. For a crystalline or noncrystalline intrinsic semiconductor the optical energy gap should be about
595
twice the activation energy for thermally activated electrons;
i e . one would expect an EA-value of about 63 KJ/mol. In
contrast to this the measured value of EA = 20 KJ/mol is of
the same order of magnitude as that observed for ion migration in salts12'I. Direct experimental evidence that liquid
CsAu is entirely ionic like fully ionized salts stems from a n
electromigration e ~ p e r i m e n t l ~ ' .When
~ ~ ] . electrolyzed, liquid
CsAu conforms to Faraday's law. Within experimental error,
one Cs and one Au are transported to the relevant electrodes per elementary charge. Measurements of the absolute
thermoelectric power also show that the contribution of electrons to the electrical transport in the liquid Cs-Au system
near the stoichiometric composition is negligibly small.
Our knowledge about the electrical properties of the solid
stoichiometric compound CsAu is quite poor compared to
what we know about molten CsAu. The phase diagram["]
shows the existence of a congruent melting phase with a narrow range of homogeneity. It melts at 590 " C and is known to
crystallize with the CsC1 structure with a lattice constant of
4.263
The corresponding calculated molar volume of
46.63 cm3 is 42%smaller than the volume obtained by simply
adding the atomic volumes of Cs and Au. The optical energy
gap at 25 "C (Fig. 4) is 2.6 eV117.251.
On the basis of electrical
conductivity u and thermoelectric power126.271
measurements
it can be deduced that solid CsAu is always an extrinsic ntype semiconductor. Since the measured conductivity (T is always relatively large (ranging from 5 to 1 0 0 ohm - l cm - ')
and not well reproducible, and since its temperature dependence is slightly negative (like in metals), it is suggested that a
broad impurity band is formed by excess cesium which may
eventually overlap with the conduction band. NMR measu r e m e n t ~ [show
~ ~ ] that the excess Cs is atomically dispersed
in the CsAu.
Several band structure calculations have been performed
for CsAu12' "1; the results compare well with photoemission
valence-band data127.301,
and they confirm the essential ionic
character of the bonds in CsAu. Further support for this ionicity stems from recent MODb a~ er [ ~
and
'] ESCA spectroscopHowever, with an energy gap of 2.6 eV
ic
(at 25 "C) between conduction- and valence-band, CsAu belongs to those compounds which lie on the borderline between electronic semiconductors and ionic insulators. Consequently, CsAu shows features of both (i.e. electronic and
ionic conductivity), as is clearly demonstrated by the behav+
~
ior of the electrical conductivity at the melting point (Fig. 6).
The marked change in o on melting indicates that the electrical transport mechanism in liquid CsAu is different from
that in solid CsAu. Other semiconducting substances show,
in contrast to the decrease in conductivity of CsAu, a more or
less large increase in u on melting. For these semiconductors
two groups must be distinguished. In the first group (Ge, Si,
InSb etc.) the coordination number and the nature of the
chemical interaction change drastically on melting. The melt
becomes a metallic conductor, i. e. a semiconductor-metal
transition accompanies the solid-liquid phase
In the second group (Se, In,Se3, Ga2Te3etc.) the coordination number changes only slightly in the melt and the shortrange order of the solid persists to a large extent, i e. at the
melting point a transition from a solid to a liquid semiconductor occurs. In these systems the short-range order is
changed only at very high temperatures and a continuous
transition to metallic behavior occurs (see Section 3).
It is evident from the results presented above that CsAu
does not belong to one of these two groups. It is possible for
semiconductors with essentially ionic bonds and energy gaps
of between 2.5 eV and 3 eV at room temperature that the solid-liquid phase transition coincides with a transition from a
solid electronic semiconductor to a liquid ionic conductor.
CsAu is the first example for which such a transition has
been observed.
The concentration dependence of the conductivity u in the
liquid Cs-Au system (Fig. 1 and 2) establishes the occurrence
of a transition from metallic to nonmetallic behavior. At uminimum for the stoichiometric composition CsAu the electrical transport is determined by the diffusive motion of Cs
and Au-. Therefore, it seems reasonable to assume that
bound states associated with the formation of Cs Au - ionic
assemblies remove electrons from the conduction band o f cesium if the concentration is changed between pure Cs and
+
+
I
c
I
I
G
e,1
I
1
I
I
L
MO
750
MO
U
750
-1003k 5
TLoCl--
XA""
Fig. 6. Comparison of the behavior of the electrical conductivity of semiconductors at the melting point: CsAu. InlTei and Ge.
Fig. 7 Molar volume of the mixture V M and excess volume V , of liquid Cs-Au
mixtures as a function of the mol fraction of gold.
596
Angew. Chem. Int. Ed. Engl 19, 593-606 (1980)
CsAu; i. e. the liquid alloys over the whole concentration
range-including the metallic range-are assumed to be binary mixtures of Cs' and Au- and the excess metal Cs or
Au. Direct evidence for the validity of this assumption comes
from very recent measurements of the concentration dependence of the molar volume of the mixture^^'^.^'] and of the radial distribution f ~ n c t i o n ~ ~ ~ . ~ ~ ~ .
Figure 7 shows the data compiled by Kempf and Schmutzfor the volume of the mixture V M ( x )as a function of
the mole fraction x together with those of the excess volumes
V E.
VE has a broad minimum in the vicinity of the stoichiometric
composition CsAu. Its unusually large negative value is comparable with the strong contraction observed on formation of
CsI or CsBr from their pure constituents. Assuming that the
packing fraction in the stoichiometric CsAu melt is about the
same as that in molten cesium halides, Kempf and Schmutzestimated from the V , and the well known ionic radius
of Cs + (1.65 A) the value rAu-= 1.8 A for the Au- ion. The
resulting value for the average nearest-neighbor distance of
3.45 A is in surprisingly good agreement with that of 3.55
obtained from the measured total pair distribution func8 shows the pair distribution function g ( R )
t i ~ n ' ~ Figure
'~.
Fig. 9 Partial molar volumes V'. and VA8<
as a function of the mol fraction of
gold.
A
Fig. 8. Pair distribution function g(R)of liquid Cs and liquid Cs-Au mixtures at
different concentrations. The numbers labeled at the curves are the concentrations of Au in atom-%.
for different liquid Cs-Au mixtures which were obtained
from neutron diffraction data and which are reproduced
from a paper by Martin et al.13']. In view of the hypothesis
that an Au- ion is formed even in very dilute solutions of Au
in Cs it is especially interesting that for a Au-concentration
of only 25 atom-% a first maximum is observed at 3.55 A.At
this concentration the conductivity is still higher than lo3
ohm - ' cm - ' and in the metallic range. This is a clear indication that Cs' and Au- are formed even in the diluted solutions. The attractive forces between the unlike charges and
the repulsive forces between the like charges lead to shortrange ordering like that which occurs in molten salts. Such
ordering in dilute solutions is convincingly demonstrated by
an analysis of the concentration dependence of the volume of
the mixtures[341.
Figure 9 shows the partial molar volumes of
Cs and Au, i. e.
Angew Chem Inl. Ed Engl 19, J93-606 (1980)
as a function of the mole fraction x of gold. For dilute solutions in the metallic range ( i e . x<0.3 and o>103 ohm-'
cm-') pcsis nearly independent of the concentration and
tends to
close to the value of pure liquid CsI3'1, whereas rAu
large negative values approaching - 107 cm3/mol for x+O.
The data available at present establish that liquid CsAu at
the stoichiometric composition has the properties of a liquid
salt. The metal-nonmetal transition in this system can be described as in the case of solutions of metals in their salts (e.g .
Cs in C S C ~ ~ ~i.e.
' ~ )as
, a solution of Cs or Au in CsAu. The
concentration dependence of the electrical conductivity (Fig.
1) and of the magnetic susceptibility (Fig. 10) can be satisfactorily explained within the framework of this
The
gram susceptibility shows a deep diamagnetic minimum at
the stoichiometric composition CsAu. The temperature dependence near the minimum is relatively small and negat i ~ e [ ~Since
~ I . this is once again consistent with the assumption of predominantly ionic bonding in liquid CsAu, Freyland and Steir~leitner'~'~,
using their experimental results, and
taking for the Cs' ion the value -36 x
cm3 mo1-i[4'1,
10
30 50 70 90
Au[Atom-%I-+
Fig. 10. Magnetic gram susceptibitity, xg, of liquid Cs-Au mixtures as a function
of the gold content.
597
calculated the molar diamagnetic susceptibility of the Au ion as -48 x lo-' cm3 mol-'. This value is in surprisingly
good agreement with the value of - 50 x lo-' cm3 mol-' obtained by extrapolation of the values for the isoelectronic sequence Bi+3, Pb", Tl', A u - [ ~ " .
There is relatively little direct experimental information
about the properties of the other alkali metal-gold systems.
Solid LiAu and RbAu compounds crystallize, like CsAu, in
the CsCl structure, whereas the structures of NaAu and KAu
are complex and have not been well
Recent experimental results for the conductivity of the liquid MAusystems (M = alkali metal) at the stoichiometric composition
are shown in Figure 11 as a function of temperature[421.The
Fig. 11. Electrical conductivity u of liquid MAu-compounds (M = alkali metal)
as a function of temperature. The arrows indicate the melting temperature.
arrows indicate the melting temperature of the different
compounds. The conductivities extend from values typical
for ionic melts (about 5 ohm - ' cm - I ) to values for metallic
melts (about lo3to lo4 ohm- ' cm-'). From the similarity of
the conductivity of CsAu and RbAu it can be concluded that
liquid RbAu is also a n ionic conductor. In contrast to RbAu,
liquid LiAu is a metallic conductor. On the other hand, the
conductivity of liquid NaAu is only 3000 o h m- ' c m - ' ,
which represents a lower limit for metallic conductors. Liquid KAu seems to be a borderline case with properties between metallic and semiconducting behavior.
The results indicate that the transition from an ionic semiconductor to a metal lies between RbAu and NaAu. This observation is consistent with band structure calculations for
the solid compounds CsAu, RbAu and L ~ A U ' ~The
' ~ . calculations show that CsAu and RbAu are semiconductors, while
LiAu is a metal. The most interesting aspect of the calculation, however, i s that the valence charge is centered essentially around the gold atom in all compounds. Thus LiAu is also
mainly ionic despite the fact that it is a metallic and surely
electronic conductor. The few data available at present demonstrate the importance of the ratio of the sizes of cations
and anions. The transition from an ionic semiconductor to a
metal occurs in parallel with a relatively large decrease in the
size of the alkali metal atoms.
The behavior of the alkali metal-gold systems (with the exception of CsAu and perhaps RbAu) is similar to that of the
598
well explored liquid semiconductors Li3Bi[43-4s1,Mg3Bi2[46491
and Li4Pb(s0-52j.Measurements of the thermodynamic propertiesI43.48-5 '1 and of the liquid structureis2]indicate that the
melt has a n essentially ionic structure, whereas the electrical
transport properties are dominated by the contribution of
electron^[^^.^^-^'^. The different behavior of CsAu, with respect to its electrical data, is essentially due to the larger energy gap between valence band and conduction band which
leads to a negligibly small contribution of the electrons to the
conductivity compared with the contribution of the ions,
which are highly mobile in the liquid state.
3. Liquid Elemental Semiconductors
The influence of chemical interaction on the electronic
structure of molten semiconductors becomes especially apparent when their behavior is studied over a wide range of
temperature (cf. Fig. 6). The predominantly ionic molten
CsAu retains its nonmetallic character over a wide range of
temperatures. In contrast, it is characteristic of covalent
semiconductors that they can be converted into a n essentially
metallic state by a sufficiently large increase in the temperature. It is demonstrated in Figure 6 that semiconductors such
as G e or In2Te3 become metallic immediately on melting
while a large number of substances such as e. g. In2Se, or Se
are continuously transformed into a metallic state when the
temperature is raised. The covalent structure is gradually
broken down with increasing temperature, so the bonding
electrons become delocalized and form a n almost free "electron gas". The following treatment of the interrelation between the change in chemical structure and the semiconductor-metal transition will be mainly concerned with selenium.
The tendency of the selenium atom to bond in two-fold
coordination favors formation of long helical chains (trigonal
crystalline Se) or of Se,-rings (monoclinic crystalline Se). It
was long assumed that liquid selenium near its melting point
also consists of a mixture of SeR-rings and long c h a i n ~ i ~ ~ !
However, recent investigations[541cast doubt on the previously determined concentration of Sex-rings[s51.In any case, the
physical properties of liquid selenium-e. g. the unusually
high viscosity of 25 poise at the melting p ~ i n t ' ' ~ . ~ ~ ~ - a r e
mainly influenced by the polymeric chains. Keezer and Baidetermined an average chain length of 105-10" atoms.
Structural investigations by X-ray and neutron diffraction
confirm the two-fold coordination in the melt[s7.5X1.
The distance between nearest-neighbor atoms (2.38
is the same
as in crystalline trigonal selenium. The structure of liquid selenium is, however, strongly dependent on temperature. The
average chain length must decrease very rapidly with increasing temperature. For example, there is a marked decrease in the viscosity, which is only 1 CP at the normal boiling point (684.9"C)-a value of the same order of magnitude
as that of other liquids. The strong influence of temperature
on the structure of liquid selenium must be accompanied by
modification of its electronic structure and related electrical
properties. However, a n estimation of the average degree of
polymerization in liquid selenium-using experimental data
of the viscosity, the magnetic susceptibility and electron spin
resonance[601-shows that even at 1000°C (the melting temperature is 217 "C) more than 90% of the atoms are still twoAngew. Chem. Ini. Ed. Engl. 19, 593-606 (1980)
fold coordinated. Since the semiconducting properties of the
liquid essentially result from this twofold coordination of
neighboring atoms, it is necessary to extend measurements
for investigating the influence of changes in the short range
order on the electronic properties to very high temperatures.
The electrical conducti~ity['~.",~~1,
the absolute thermoelectric p o ~ e r [ ' ~ . "and
~ I the nuclear magnetic resonance[6s1of liquid selenium have been measured up to supercritical temperatures and pressures. The critical pressure p' = 380 bar
and the critical temperature T,= 1590 "C have been determined["]. Figure 12 shows the electrical conductivity a of se-
This value would appear to increase rapidly with increasing
t e m p e ~ a t u r e [ ~ ~In. ~contrast,
~1.
the coordination number of
selenium remains two, even up to temperatures higher than
1000 0C[74,751.
Although these numbers cannot be definitely
identified with the number of bonds between an atom and its
neighbor they do, however, indicate that the disorder in liquid Te is greater than in liquid Se.
The strongest indication that the transition from semiconducting to nearly metallic behavior is accompanied by a
change in the local coordination number can be obtained
from measurements of the thermodynamic properties. Fig1
-31c, . ,
%Xl
, , , *
7M)
TY
, , , , ,
1
llM)
C
1
~
+%
Fig. 13. Isobaric thermal expansion coefficient apof liquid Se, Te and Te,,,Se5,,
as a function of temperature; a,,=
Flg. 12. Electrical conductivity u of Se, Te and a Se-Te mixture in the molten
state as a function of temperalure at constant pressures.
lenium at two pressures together with conductivity isobars of
liquid tellurium[661and of a selenium-tellurium mixture containing 50 atom-% of each component[671.At relatively low
temperatures u markedly increases with increasing temperature, as one would expect for a semiconductor. For temperatures higher than about 1300°C the selenium isobars flatten
off and, close to the critical temperature, reach maxima with
strongly pressure dependent conductivity values of almost
metallic order of magnitude. In the direct vicinity of the critical point, where relatively small changes in pressure and
temperature can lead to large changes in density, a drops
very rapidly with increasing temperature to values smaller
than 0.1 ohm-' cm-'. In this region selenium can be assumed to consist mainly of Sez and a small amount of atomic
Se. This assumption is supported by measurements of the
by extrapolation of the known
magnetic
composition of selenium vapor at low temperatures[701,and
by comparison with the optical data of supercritical sulfur
It is obvious from Figure 12 that the conductivity isobar
for the Se-Te system is very similar to the isobars for pure selenium; but the maximum is located at a lower temperature.
In contrast, pure tellurium has a flat maximum with a-values
in the metallic range. There exists a direct correlation between the levels and the positions of the maxima and the
bond strengths of the covalently bonded atoms in the chains
which decreases in the sequence Se-Se, Se-Tef7I1,Te-Te. This
tendency is also reflected in the structural properties of these
~ y s t e r n s751.
~ ~For
~ tellurium at the melting point an average
first coordination number of three is obtained from the area
under the first peak of the radial distribution function g ( R ) .
Angew. Chem. Inr. Ed. Engl. 19, 593-606 (1980)
I
Se
-I1
''1
200
Fig. 14. Adiabatic compressibility @. of liquid Se. Te and T e d e : , , as a functlon
of temperature.
ures 13 and 14 show the behavior of the isobaric thermal exand the
pansion coefficient aP derived from p V T data[77.7X-641
adiabatic compressibility & which was recently determined
by Yuo er U Z . [ ~ ~from
]
sonic measurements. A minimum in ap
with negative values and a maximum in
is observed for
the Se-Te mixture in the temperature range in which the
conductivity u approaches a maximum with value's nearly as
high as in the case of metals. Figure 13 demonstrates again
the very different behavior of tellurium, which becomes metallic just above the melting point, and selenium, which tends
to a metallic state at temperatures above 1300°C. The apcurves have different slopes, and negative values of ap are
only observed for Te in the supercooled melt (the melting
temperature is 449.5 "C).The temperature dependence of ap
of selenium fits into this picture. Figure 15[781shows a plot of
the volume of both sulfur and selenium at a fixed reduced
pressure p / p c = 3 as a function of the reduced temperature T /
T,. The sulfur isobar shows the behavior of normal molecu-
599
I
I
0
I
0.5
1.0
d
nb
.
d*
I
I
T I Tc-
Fig. 15. Volume V of sulfur and selenium at constant supercritical reduced pressure p/p< = 3 as a function of the reduced temperature T/T,.
E-
lar fluids with a continuous increase in volume as the critical
temperature is approached, whereas the curve of selenium
flattens out at a reduced temperature of about 0.8 and starts
rising again in the vicinity of the critical point T,. A plausible
explanation for the anomalous minimum or maximum in a p
or
seems to be that the packing becomes denser with increasing temperature leading to a higher density despite the
competing normal thermal expansion. The latter behavior
can be compared with the abrupt metal-nonmetal transition
of covalent semiconductors with diamond structure (Ge, Si
and InSb) on melting, which is accompanied by an increase
in the coordination number -and a corresponding decrease in
volume. For Se and Se-Te mixtures this metal-nonmetal
transition is smeared out over a wide range of temperatures.
Figure 16 gives a schematic illustration of the behavior of the
volume as a function of temperature.
temperature 4
Fig. 16. Schematic illustration of the temperature dependence of the volume of a
system for which the metal-nonmetal transition coincides with the liquid-solid
phase transition (---, e.g. Ge), and for a system with a continuous transition
spread over a wide temperature range (-, e.g. Se).
A model for the semiconductor-metal transition in liquid
selenium has recently been proposed by Warren and Du~ r e e [ ~It~is] mainly
.
based o n structural data, a critical analysis of the NMR data and on the assumption that the electronic structure of semiconducting polymeric selenium can
be derived from the band structure of crystalline trigonal selenium. Of the six valence electrons of the selenium atom
(4s24p4)the two 4s electrons are about 4-5 eV lower in energy than the lowest 4p-electrons and are unimportant for the
electrical transport properties. Two 4p electrons form bonds
to the two neighbors while the other two occupy a non-bonding state. In the condensed state (solid or liquid) the highest
occupied valence band is formed by the nonbonding states
(n-b.) while the empty antibonding o*(p) states form the
conduction band (cf. Fig. 17a). The number of polymeric selenium molecules, and hence the number of neutral chain
ends with unpaired electrons (Cy-centers), increases with increasing temperature. However, it cannot be ruled out that,
600
Fig. 17. Electronic structure of liquid selenium under different conditions (see
text).
in liquid Se, a chain end Cy reacts with the lone pair electrons of a Se-atom in a n appropriate configuration to give a
neutral threefold center (C$center). The existence of such
stable defects has been proposed for amorphous selenium by
Mott and StreetfRo'and by Kastner, Adler and Fritschel"].
Calculations[R21show that the C?- and Cp-centers lie close together in energy with a slightly lower value for Cy. The neutral centers introduce a narrow band of defect states in the
middle between valence and conduction band (indicated by
the arrows in Fig. 17) which increases with increasing temperature at the expense of the valence and conduction
bands["]. It can be expected that all bands become broader
with decreasing lengths of the polymeric chains. From the
behavior of the magnetic susceptibility it can be conc l ~ d e d ' that
~ ~ ' the width of the band of defect states is smaller than kT=0.16 eV up to 1550 " C and 400 bar. An additional increase in pressure results in a delocalization of the states
between valence and conduction band (Fig. 17c). Finally, at
still higher pressures the band becomes essentially metallic,
similar to that of liquid Te (Fig. 17d). The validity of this
model has been confirmed qualitatively by recent measurements of the optical reflectivity of liquid seleniumfx3]at temperatures up to 1400°C and pressures up to 1500 bar. The
characteristic features of the spectrum change only slightly
with increasing temperature as long as the pressure is smaller
than 500 bar, whereas application of higher pressures at temperatures above 1300°C leads to a rapid change in the shape
of the spectrum. Under high pressure it becomes very similar
to that of liquid tellurium[x41.
At high temperatures and slightly increased pressure the
electrical, optical and structural properties of liquid selenium
resemble those of liquid tellurium at its melting point. In
contrast, the electrical properties of sulfur are completely different from those of selenium and tellurium over the whole
liquid range. Near its melting point, sulfur is a n insulator
with a conductivity smaller than lo-'' ohm -' c m - '["l. This
is consistent with the large optical energy gap which can be
derived from the data shown in Figure 18. It shows the absorption
of liquid sulfur in the photon energy
range from 0.5-3.5 eV (1 eVg8065.46 c m -' ) at temperatures between 140°C and 1000°C and pressures slightly
higher than the vapor pressure. A continuous red shift of the
edge is observed with increasing temperature. A precise analysis of the data shows that the polymerization of liquid sulfur
Angew. Chem. In!. Ed. Engl. 19, 593-606 (1980)
145
1'0
-
1'5
20
h*C.VI
25
30
5
Fig. 18. Absorption spectrum o f liquid sulfur as a function of temperature (in
"C) at equilibrium pressure ( 1 eV s 8065.46 cm ').
~
at 160 0C[x71
does not affect the red shift. This result is qualitatively consistent with the fact that the number of first
neighbors and their separation are virtually unchanged by
the polymerization. This leads to nearly the same energy levels for both structures. However, new absorption shoulders
appear in the infrared region at energies of about 1.3 eV and
increase very rapidly with increasing temperatures above the
polymerization temperature. From a comparison of the intensities of these absorption bands with measurements of the
magnetic
and of the ESR-~ignal["~~~]
as a
function of temperature it has been concluded that the absorption at 1.3 eV derives from the chains which possess unpaired electron spin states at the chain endsfs6].This conclusion is also in agreement with theoretical results obtained on
the basis of the CNDO-S model["]. Except from the appearance of the low energy band, the form of the optical absorption edge of sulfur-especially at K-values higher than lo4
cm - '-is essentially determined by a series of overlapping
absorption bands which can be assigned to different species
I
LOO
mo
800
1
A lnml-
Fig. 19. Absorption spectrum of fluid sulfur at the constant supercritical temperature o f 1 1 0 0 "C as a function of pressure.
Angew. Chem. Int. Ed. Engl. 19, 593-606 (1980)
of sulfur, such as Sx, S2, S3, S, and probably S.5 or othe r ~ [ ~ ~Several
. ~ ~ , studies
~ ~ ] .on the composition of dilute sulfur vapor using different methods have established the existence of these species, and their concentration has been
measured as a function of pressure and t e m p e r a t ~ r e [ ~ An
~-~~].
efficient experimental approach for explaining the spectral
behavior of liquid sulfur is therefore provided by a comparison with the spectrum of the vapor, especially above the critical temperature where the density can be continuously
changed over a wide range by variation of pressure. Figure
19[931
shows the behavior of the absorption coefficient as a
function of the wavelength at a supercritical temperature of
1100°C. (The critical data of sulfur are: p,=205 bar,
T, = 1040 "C and pc = 0.6 g/cm3.) A strong increase in the intensity of the S3-band at 400 nm and of the s,-band at 520
nm is observed with increasing pressure or density (Fig. 19).
This is a clear indication that at high temperatures, liquid
sulfur contains different small highly absorbing molecules
besides the diradical chains detected by absorption bands at
1.3 eV[92.86,931.
Due to the logarithmic scale in Figure 19 the
bands are partly smeared out to shoulders.
Recent measurements of the electrical conductivity u of liquid sulfur over a wide range of temperature have shown
that the free-radical chain-ends of the polymeric molecules
give rise to carriers which determine d9'1.Figure 20 shows a
IMWll -]'K[
Fig 20 Electrical conductivity u of liquid sulfur as a function of the reciprocal
temperature
plot of the logarithm of the conductivity as a function of the
reciprocal temperature [cf. eq. (I)]. The dominant feature is
the relatively sharp change in the slope of the curve at about
550 "C, corresponding to activation energies of E, = 1.05 eV
for T>550"C and EA=1.9 eV for T<550"C. A comparison
of the activation energies of the conductivity u with the optical absorption band at 1.3 eV and the temperature dependence of the ESR signal of the free-radical chain-ends led Edeling et a1.[971to the conclusion that the chain-ends give rise
to charge carriers which determine the form of the temperature dependence of u in Figure 20. The inevitable presence
of a very small number N, of monovalent impurities X in liquid
influences the polymerization equilibrium[60].
There are two types of chain-ends. The chains are terminated
by either a S-X bond or an unpaired spin[601.The concentration of free-radical chain-ends N * is nearly independent
of X, but the average chain length decreases with increasing
N,. Under such circumstances, liquid sulfur behaves like a
doped semiconductor with (N* + N,) available defect levels
and N* acceptors[971.The activation energy of the conductivity of such a semiconductor changes by a factor of 2 at a well
defined temperature which is determined by the concentration N,[991.
601
4. Metals at High Temperatures
The possible application of liquid metals at high temperatures as working fluids for technical purposes has been extensively discussed in recent years. This discussion is stimulated by the fact that liquid metals have a favorable combination of physical properties such as, e. g., high latent heat of
vaporization, high thermal conductivity, and high-temperature liquid range. In spite of this technological interest many
fundamental physicochemical properties such as p VT-data,
critical data, or specific heats, are not known with sufficient
accuracy for most liquid metals at high temperatures. Reliable theoretical calculations of such data are still lacking. Experimental data relevant to properties like equation of state
and critical data have to be determined at very high temperatures and relatively high pressures; the liquid range of metals
extends to very high temperatures (Fig. 21). Only the critical
data for Hg['W. loll, Cs[~Oz.l031Rbrlo31 K [ w and Na[losl have
been measured with static experimental methods. Fucke and
SeydeZ['O6]have recently determined the values for Mo in a
transient experiment with exploding wires.
A
temperature 1
,
Fig. 21. Schematic illustration of the liquid-vapour equilibrium of a metal. Critical data of some metals (T,,p,): Hg 1490 "C, 1530 bar; Cs 1740 "C. I15 bar; Rb
182O"C,130 bar; K 195O"C, 155 bar; Na 2230°C. 250 bar. Mo 14ooO"C. 5700
bar.
The main difficulty in the theoretical calculation of the
properties of liquid metals, especially thermodynamic data,
is the absence of a n adequate interatomic-potential function
for the entire liquid range. Regardless of the way in which
the interatomic forces in a metal are described they must reflect the screening of the ionic charges by the electron gas
and must therefore be always implicit functions of the electron density, i. e. of the metal density. In addition it is well
known that a transition from metallic to nonmetallic, mostly
semiconducting, state occurs also in a pure liquid metal, if it
is continuously expanded to a low density fluid by increasing
the temperature at a pressure higher than the critical pressure (i e. avoiding evaporation)[14'. The transition to a nonmetallic state implies that the nature of the chemical interaction or of the interatomic forces changes; i. e. the description
of the forces of interaction in a liquid metal must take into
consideration the changes in density. This is in contrast to
non-conducting liquids like argon, for which to a first approximation the thermodynamic data are usually described
by reference to a single pair-potential over the whole density
range from the dilute gas to the liquid.
Since the electrical properties of expanded metals in the
transition range to nonmetallic behavior have already been
reviewed in detail["] we shall restrict ourselves here to only a
few basic problems, e.g.: where in the phase diagram (Fig.
21) does the metal-nonmetal transition occur? Does a "law of
corresponding states" exist for metals? How does the transi-
602
Fig. 22. Electrical conductivity -of Hg. Cs and Rb at the supercritical. constant
reduced temperature T / T , = 1.05 as a function of the reduced density p/pc.
tion affect the thermodynamic properties and the liquid
structure?
Most measurements over a wide range of temperaturemostly up to supercritical temperatures-have been performed on the metals Hg, Cs and Rb. This is simply because these
metals have the lowest critical temperatures. For Hg, extensive experimental data are available, including the electrical
conductivity[lW. 101.107.108], the thermoelectric power[logl, the
Hall
the optical absorption[' ' I and reflection" 12. I I31, the velocity of sound["41, the specific
heat["'], the nuclear magnetic resonance["61 and the p VTdata[ 100.101.108]. Less data are known for the more corrosive
alkali metals Cs and Rb. U p to now, the electrical conductivity" 17.103.11xl , the thermoelectric power['".' 'I, the magnetic
susceptibility["', 1201 and the p VT-data" IR.1031
have been measured. The radial distribution function and thus the structure
factor of R b have recently been measured by means of neutron scattering e ~ p e r i m e n t s [ ' ~ ~ ~ ' ~ ~ ~ ' ~ ~ ] .
Figure 22 gives a comparison of the behavior of the conductivity of Hg[loX1],
Cs and Rb[Io3]in the form of a plot of the
conductivity as a function of the reduced density p / p , at constant reduced temperature T/Tc. There is a striking difference between the two alkali metals and mercury. This is because Hg shows the characteristics of a liquid semiconductor
at densities p smaller than 9 g/cm3 (ie. in the liquid state,
pc = 5.3 g/cm3)[I41.The existence of a n activation energy derived from eq. (1) has recently been proven for densities
smaller than 9 g / ~ m ~ [ ' ~ ~ ] .
By way of contrast, it is apparent from Figure 22 that the
density dependence of the conductivity of Cs and R b is very
similar. The curves become identical at densities smaller
than 2p,. At the density 2 p c , the electron mean free path calculated with the free electron model[Zlbecomes equal to the
mean interatomic distance. At the same time, the temperature dependence of the conductivity changes in sign from negative to positive['03]. The value of the conductivity at the
critical point is about 300 ohm- I cm- for both metals. This
value is very close to the minimum metallic conductivity defined by Mott in numerous theoretical p a p e r ~ [ l ~ *On
. ~ the
~.
basis of this argument one can conclude that the metal-nonmetal transition in liquid alkali metals (a similar value of cr is
observed for potassium at the critical point) coincides with
the liquid-vapor phase transition.
Since the thermodynamic properties, like the electrical
conductivity, are mainly determined by the density of the
electron gas['] a similar behavior of the pVT-data of Cs and
Angew. Chem. In1 Ed. Engl. 19, 593-606 (1980)
I
1
'
I
060
t
-
-,-52-
4
5 0-
-
I
2
1
p/P,
80'
3
Fig. 23. Reduced diagram of the pVT data of Cs and Rb. Reduced density p / p ,
as a function of the reduced pressurep/p, at various reduced temperatures T/Tc.
The isotherms for Cs and Rb at T/T, =0.6 and 0.95 are indistinguishable.
Fig. 25. Average number N , and mean distance R , of next neighbors in liquid Rb
as a function of the density p .
Rb can be expected to occur in corresponding regions of the
phase diagram according to Figure 22. This suggestion is
supported by the plot in Figure 23['031.Both metals have the
same compressibility factor z, = p c V,/(RT,) = 0.22 which is
markedly different from the value zc = 0.385 for mercury,
which is already nonmetallic at the critical point. The theoretical reasoning of this observation should lead to a "law of
corresponding states" for groups of similar metals. The existence of such a law would be of considerable importance
with regard to possible technical applications of liquid metals at high temperatures. In this connection a knowledge of
the structure and of the isochoric thermal pressure coefficient
yv= (ap/a7'), of the metals over a wide range of temperatures
is of special interest. It is well known that at the melting
point both quantities can be reasonably modelledf'24.i251 In
'
a
first approximation by hard
However, it must be
emphasized that in a metal the diameters of the spheres will
vary not only with temperature but also with density, due to
the simultaneous variation in the number density of the free
electrons. Figure 24 shows new measurements of the static
structure factor S ( Q ) for liquid rubidium~'03.i2i~'20~'231
obtained by neutron scattering experiments. S ( Q ) is directly related to the pair distribution function, g ( R ) , which is the
probability of finding two particles of the liquid at a distance
of R from each other. g ( R ) depends only on R (see Fig. 24).
The density dependence of g ( R ) mainly reflects a linear decrease of the number of nearest neighbors, N , , (Fig. 25) as
calculated from the first peak of g ( R ) using the formula given in the legend of Figure 24. In contrast to N , , the nearest
neighbor distance R , , defined by the first maximum of g ( R ) ,
is almost constant (Fig. 25). A similar behavior is observed
for the inert gases Ar and Ne[l2'].From this similarity it can
be assumed that the application of the hard sphere model,
which provides an appropriate description of the liquid metal structure at the melting point, can be extended over a
wide range of densities. A comparison between the measured
structure factor S ( Q ) and that calculated with the hard
sphere model"261 is given in Figure 26. Allowing a decrease
0
1
z
a16'1
3
-
I
4
s
l
0
3
9
6
Rial
12
IS
S(P) using
!IS(Q)- 1) Q.sin (QW1 dQ
The average number of nearest neighbors is derived using RDF=4xR'g(R)
(right, bottom)
R!
N , = 2 x 4 n x n J g(R)R*dR
n
R ,=position of the first peak in G ( R ) .
Angew. Chem. Int. Ed. Engl. 19. 593-606 (1980)
I[K1
Fig. 26. Comparison between measured static structure factors Sip) (points) and
those calculated with the hard sphere model (curve).
~
Fig. 24. Left: Static structure factor S ( Q ) of liquid Rb as a function of temperature and density. The pair distribution function g(R) (right. top) is derived from
g(R) = 11 + 1/i2 n n R)1
I
A
of the hard sphere diameter from 4.4 A to 4 over the density range covered by the measurement yields a qualitatively
satisfactory fit for Q>O.I A-' over a surprisingly large density and temperature range. This observation has led to a
number of attempt^['^^.''*.'^^^ to describe the pVT-data of liquid metals by a van-der-Waals type of theoryl'2R1.The underlying idea is that the liquid consists of hard spheres immersed in a uniform potential which provides the cohesion
that the hard-sphere system otherwise lacks. This description
603
is especially plausible for liquid metals, the cohesion energy
of which can be attributed to the uniformly distributed gas of
conduction electrons rather than to short range forces between neighboring particles. With these assumptions the
pressure is given by:
(3)
where po is the hard-sphere pressure, 7"= ( a p / a 7)" is the isochoric thermal pressure coefficient and U is the internal energy of the liquid. Figure 27 shows the ?,-values of Rb de-
Fig. 27 Isochoric thermal pressure coefficient y v = ( $ / a r ) , of liquid Rb as a
function of the molar volume. ( x ) experimental points, ( + ) calculated with the
hard sphere model.
rived from experimental p VT-data. They are in good agreement, within experimental error, with those calculated with
the hard-sphere model. The experimentally determined internal pressure (a U/a V ) , is indeed to a first approximation
,150D". 2000bor
In [Atom-%l
-
Fig. 28. Comparison of the concentration dependence of the electrical conduct~vity o of dilute indium amalgams at constant pressure (left) and at constant number-density (right).
only volume dependentriw.'l']. Its density dependence between the melting point density and about twice the critical
density 2p, is essentially determined by the existence of the
electron gas[1o31.
For densities smaller than 2pc, also the electrical conductivity deviates from free-electron behavior. The
experimentally observed simple dependence of the terms in
eq. (3) on the volume V and temperature T is consistent with
the existence of a "law of corresponding states" for the group
of alkali metals.
The (divalent) metal mercury shows "semiconducting" behavior with respect to several properties at densities lower
than 9 g / ~ m (see
~ ~Fig.
' ~22).
~ In this range, for example, the
conductivity u has an activation energy, and addition of a
small concentration of excess electrons at constant pressure
leads to marked enhancement of ~ ' ' ' ~ 1 .However, it has recently been demonstrated that these increases in c have
nothing to do with the "semiconducting" character of the
,electronic structure of Hg but are caused by the liquid and
expanded state of m e r c ~ r y l ' ~ ~ . as
' ~ 'is
I , apparent from Figures 28 and 29. Figure 28 shows the effect of a few atomic
percent (trivalent) indium at different constant pressures
(left-hand side) or at constant number densities (right-hand
side). It is obvious that the enhancement of the conductivity
at constant pressure is caused by a corresponding volume
contraction on alloying with In. Such a large contraction is
generally observed in solvents having a very high compressibility on addition of small amounts of a solute, whose molecules exercise strong attractive forces on the molecules of the
~ o l v e n t ~ 'The
~ * ~relationship
.
between the compressibility xT
of pure mercury and the excess volume of mixing V , of In in
Hg is demonstrated in Figure 29. Both xT and V , rise quite
sharply at temperatures higher than 1400"C, i. e. as the density falls below the metal-nonmetal transition density of 9 g/
cm3. The transition implies a change in the nature of the interatomic interaction, and therefore anomalous behavior of
the thermal properties-e.g. of X-may
generally be connected with the transition range in the electronic structure
characteristic for metallic systems. It must be pointed out,
however, that the transition range from a well defined liquid
metal of high density to a gaseous dielectric of low density
extends over a wide density range. First measurements of the
dielectric behavior of gaseous Hg for densities smaller than
3.5 g/cm3 (i.e. for conductivities smaller than lo-* ohm-'
cm-') by Hefner[li3Ishow a drastic increase in the refractive
y [g/cm31-
Fig. 29. Isothermal compressibility ,yT of pure Hg and excess volume V, of In in
Hg (Hg with 2.8 atom-%In) as a function of temperature at constant pressure.
604
Fig. 30. Refractive index n of mercury vapour at A = 2 pm as a function of the
mercury density p at T = 1530 "C.The dotted line gives n calculated accordmg to
Clausius-Mosolli.
Angew Chem. Int. Ed Engl. 19, 593-606 (1980)
index n for densities larger than 2 g/cm3 (Fig. 30). For comparison, Figure 30 includes a graph of the behavior of n calculated according to the Clausius-Mosotti relationship using
the constant polarizability of the isolated Hg-atom. A striking upward deviation of about a factor of 2.5 from ClausiusMosotti behavior is observed within a relatively small density range. A rapid decrease in the ionization energy["'] is expected to occur in this range due to the far-reaching polarization-forces between the neutral Hg-atoms and the ions and
electrons of the slightly ionized (at 1530 "C) mercury. At the
relatively low density of 3 g/cm' (pc= 5.3 g/cm') and a temperature of 1530 "C the character of the optical properties of
Hg-vapor changes abruptly within a very small and experimentally undetectable density range. This change is indicated in Figure 30 by the onset of an inflection in the refractive index curve. According to its optical data, mercury vapor
above p = 3 g/cm3 has all the properties of a solution of small
metallic clusters in a solvent of a high dielectric constant.
The a. c. conductivity has a maximum in the infrared spectral
In
range which increases rapidly with increasing
contrast, the d. c. conductivity is smaller than l o - ' ohm-'
cm-' and nonmetallic. The abrupt formation of stable metal-clusters in the dense gaseous phase and particularly the
mechanism of their formation and the investigation of the
chemical interactions leading to their stabilization are of current interest in connection with metal-cluster c o m p l e ~ e s [ ~ ~ ~ 1 .
Received: March 27, 1980 [A 327 IE]
German version: Angew. Chem. 92. 598 (1980)
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The Influence of Density and Temperature on the Properties of
Pure Molten Salts
By Klaus Todheide'']
Dedicated to Prof. Dr. E. U. Franck at the occasion of his 60th birthday
Molten salts differ from most other types of liquids because of their high charge carrier density,
which is responsible for a number of their extraordinary properties such as high ionic conductivity. Experimental and theoretical investigations during the past few years have provided information on the static and dynamic microscopic structures of this class of liquids and their relationship with the macroscopic properties. A consistent, albeit incomplete picture of the behavior of molten salts over wide ranges of temperature and density is now available including
the transition from an insulator to an ionic conductor which a salt undergoes when it is compressed from vapor-like to liquid-like densities.
1. Introduction
number of
and review article^[^-'^]. Most of the
data collected is contained in a
Molten salts have been the subject of experimental and
theoretical investigation for many decades. A quantitative
knowledge of the equilibrium and transport properties of
pure salts and of salt mixtures was necessary for the development and improvement of technical processes based on molten salts and for searching for new applications. Since the statistical treatment of liquids was not sufficiently advanced to
correlate the experimental data and to interpret the results in
terms of the properties of the microscopic constituents, semiempirical models were developed for this purpose"]. The information accumulated up to about 1965 is presented in a
It is remarkable that up to that time the properties of molten salts were measured and discussed either at a fixed temperature or as a function of temperature at constant (normal)
pressure. Since the density changes with temperature at constant pressure the measured temperature dependencies always included a density dependence which was neglected in
the interpretation of the experimental results. The influence
of density on the properties of molten salts, however, can be
very important, as is demonstrated by an extreme example:
liquid potassium chloride is a good ionic conductor, whereas
the vapor in equilibrium with it contains nothing but uncharged particles such as molecules and associates of molecules and consequently behaves as an insulator. A transition
from an insulator to an ionic conductor must therefore occur
[*] Dr. K. Todheide
lnstitut fur Physikalische Chemie und Elektrochemie der Universitat
Postfach 6380, D-7500 Karlsruhe 1 (Germany)
606
0 Verlag Chemie. GmbH. 6940 Weinheim, 1980
0570-0833/80/0R08-006
$ 02.50/0
Angew. Chem. Int. Ed. Engl. 19, 606-619 (1980)
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